LMFDB - Embedded newform 405.4.e.g.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(136,405)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	405.271
Dual form			405.4.e.g.136.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(3.50000 - 6.06218i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(12.0000 + 20.7846i) q^{7} -15.0000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(3.50000 - 6.06218i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(12.0000 + 20.7846i) q^{7} -15.0000 q^{8} +5.00000 q^{10} +(-26.0000 - 45.0333i) q^{11} +(-11.0000 + 19.0526i) q^{13} +(12.0000 - 20.7846i) q^{14} +(-20.5000 - 35.5070i) q^{16} -14.0000 q^{17} -20.0000 q^{19} +(17.5000 + 30.3109i) q^{20} +(-26.0000 + 45.0333i) q^{22} +(84.0000 - 145.492i) q^{23} +(-12.5000 - 21.6506i) q^{25} +22.0000 q^{26} +168.000 q^{28} +(-115.000 - 199.186i) q^{29} +(144.000 - 249.415i) q^{31} +(-80.5000 + 139.430i) q^{32} +(7.00000 + 12.1244i) q^{34} -120.000 q^{35} -34.0000 q^{37} +(10.0000 + 17.3205i) q^{38} +(37.5000 - 64.9519i) q^{40} +(-61.0000 + 105.655i) q^{41} +(94.0000 + 162.813i) q^{43} -364.000 q^{44} -168.000 q^{46} +(-128.000 - 221.703i) q^{47} +(-116.500 + 201.784i) q^{49} +(-12.5000 + 21.6506i) q^{50} +(77.0000 + 133.368i) q^{52} -338.000 q^{53} +260.000 q^{55} +(-180.000 - 311.769i) q^{56} +(-115.000 + 199.186i) q^{58} +(-50.0000 + 86.6025i) q^{59} +(-371.000 - 642.591i) q^{61} -288.000 q^{62} -167.000 q^{64} +(-55.0000 - 95.2628i) q^{65} +(42.0000 - 72.7461i) q^{67} +(-49.0000 + 84.8705i) q^{68} +(60.0000 + 103.923i) q^{70} -328.000 q^{71} -38.0000 q^{73} +(17.0000 + 29.4449i) q^{74} +(-70.0000 + 121.244i) q^{76} +(624.000 - 1080.80i) q^{77} +(120.000 + 207.846i) q^{79} +205.000 q^{80} +122.000 q^{82} +(-606.000 - 1049.62i) q^{83} +(35.0000 - 60.6218i) q^{85} +(94.0000 - 162.813i) q^{86} +(390.000 + 675.500i) q^{88} +330.000 q^{89} -528.000 q^{91} +(-588.000 - 1018.45i) q^{92} +(-128.000 + 221.703i) q^{94} +(50.0000 - 86.6025i) q^{95} +(-433.000 - 749.978i) q^{97} +233.000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 7 q^{4} - 5 q^{5} + 24 q^{7} - 30 q^{8}+O(q^{10}) $ 2 * q - q^2 + 7 * q^4 - 5 * q^5 + 24 * q^7 - 30 * q^8	$ 2 q - q^{2} + 7 q^{4} - 5 q^{5} + 24 q^{7} - 30 q^{8} + 10 q^{10} - 52 q^{11} - 22 q^{13} + 24 q^{14} - 41 q^{16} - 28 q^{17} - 40 q^{19} + 35 q^{20} - 52 q^{22} + 168 q^{23} - 25 q^{25} + 44 q^{26} + 336 q^{28} - 230 q^{29} + 288 q^{31} - 161 q^{32} + 14 q^{34} - 240 q^{35} - 68 q^{37} + 20 q^{38} + 75 q^{40} - 122 q^{41} + 188 q^{43} - 728 q^{44} - 336 q^{46} - 256 q^{47} - 233 q^{49} - 25 q^{50} + 154 q^{52} - 676 q^{53} + 520 q^{55} - 360 q^{56} - 230 q^{58} - 100 q^{59} - 742 q^{61} - 576 q^{62} - 334 q^{64} - 110 q^{65} + 84 q^{67} - 98 q^{68} + 120 q^{70} - 656 q^{71} - 76 q^{73} + 34 q^{74} - 140 q^{76} + 1248 q^{77} + 240 q^{79} + 410 q^{80} + 244 q^{82} - 1212 q^{83} + 70 q^{85} + 188 q^{86} + 780 q^{88} + 660 q^{89} - 1056 q^{91} - 1176 q^{92} - 256 q^{94} + 100 q^{95} - 866 q^{97} + 466 q^{98}+O(q^{100}) $ 2 * q - q^2 + 7 * q^4 - 5 * q^5 + 24 * q^7 - 30 * q^8 + 10 * q^10 - 52 * q^11 - 22 * q^13 + 24 * q^14 - 41 * q^16 - 28 * q^17 - 40 * q^19 + 35 * q^20 - 52 * q^22 + 168 * q^23 - 25 * q^25 + 44 * q^26 + 336 * q^28 - 230 * q^29 + 288 * q^31 - 161 * q^32 + 14 * q^34 - 240 * q^35 - 68 * q^37 + 20 * q^38 + 75 * q^40 - 122 * q^41 + 188 * q^43 - 728 * q^44 - 336 * q^46 - 256 * q^47 - 233 * q^49 - 25 * q^50 + 154 * q^52 - 676 * q^53 + 520 * q^55 - 360 * q^56 - 230 * q^58 - 100 * q^59 - 742 * q^61 - 576 * q^62 - 334 * q^64 - 110 * q^65 + 84 * q^67 - 98 * q^68 + 120 * q^70 - 656 * q^71 - 76 * q^73 + 34 * q^74 - 140 * q^76 + 1248 * q^77 + 240 * q^79 + 410 * q^80 + 244 * q^82 - 1212 * q^83 + 70 * q^85 + 188 * q^86 + 780 * q^88 + 660 * q^89 - 1056 * q^91 - 1176 * q^92 - 256 * q^94 + 100 * q^95 - 866 * q^97 + 466 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.500000	−	0.866025i	−0.176777	−	0.306186i	0.763998	−	0.645219i	$-0.223234\pi$
$2$	−0.500000	−	0.866025i	−0.176777	−	0.306186i	−0.940775	+	0.339032i	$0.889900\pi$
$3$		0			0
$3$		0			0
$4$	3.50000	−	6.06218i	0.437500	−	0.757772i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	12.0000	+	20.7846i	0.647939	+	1.12226i	0.983614	+	0.180286i	$0.0577022\pi$
$7$	12.0000	+	20.7846i	0.647939	+	1.12226i	−0.335675	+	0.941978i	$0.608964\pi$
$8$	−15.0000			−0.662913
$9$		0			0
$10$	5.00000			0.158114
$11$	−26.0000	−	45.0333i	−0.712663	−	1.23437i	−0.963854	−	0.266431i	$-0.914155\pi$
$11$	−26.0000	−	45.0333i	−0.712663	−	1.23437i	0.251191	−	0.967938i	$-0.419178\pi$
$12$		0			0
$13$	−11.0000	+	19.0526i	−0.234681	+	0.406479i	−0.959180	−	0.282797i	$-0.908738\pi$
$13$	−11.0000	+	19.0526i	−0.234681	+	0.406479i	0.724499	+	0.689276i	$0.242071\pi$
$14$	12.0000	−	20.7846i	0.229081	−	0.396780i
$15$		0			0
$16$	−20.5000	−	35.5070i	−0.320312	−	0.554798i
$17$	−14.0000			−0.199735			−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000			−0.199735			−0.0998676	+	0.995001i	$0.531842\pi$
$18$		0			0
$19$	−20.0000			−0.241490			−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000			−0.241490			−0.120745	+	0.992684i	$0.538528\pi$
$20$	17.5000	+	30.3109i	0.195656	+	0.338886i
$21$		0			0
$22$	−26.0000	+	45.0333i	−0.251964	+	0.436415i
$23$	84.0000	−	145.492i	0.761531	−	1.31901i	−0.180530	−	0.983569i	$-0.557781\pi$
$23$	84.0000	−	145.492i	0.761531	−	1.31901i	0.942061	−	0.335441i	$-0.108885\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	22.0000			0.165944
$27$		0			0
$28$	168.000			1.13389
$29$	−115.000	−	199.186i	−0.736378	−	1.27544i	−0.954116	−	0.299437i	$-0.903201\pi$
$29$	−115.000	−	199.186i	−0.736378	−	1.27544i	0.217738	−	0.976007i	$-0.430132\pi$
$30$		0			0
$31$	144.000	−	249.415i	0.834296	−	1.44504i	−0.0603071	−	0.998180i	$-0.519208\pi$
$31$	144.000	−	249.415i	0.834296	−	1.44504i	0.894603	−	0.446862i	$-0.147459\pi$
$32$	−80.5000	+	139.430i	−0.444704	+	0.770250i
$33$		0			0
$34$	7.00000	+	12.1244i	0.0353085	+	0.0611562i
$35$	−120.000			−0.579534
$36$		0			0
$37$	−34.0000			−0.151069			−0.0755347	−	0.997143i	$-0.524066\pi$
$37$	−34.0000			−0.151069			−0.0755347	+	0.997143i	$0.524066\pi$
$38$	10.0000	+	17.3205i	0.0426898	+	0.0739410i
$39$		0			0
$40$	37.5000	−	64.9519i	0.148232	−	0.256745i
$41$	−61.0000	+	105.655i	−0.232356	+	0.402453i	−0.958501	−	0.285089i	$-0.907977\pi$
$41$	−61.0000	+	105.655i	−0.232356	+	0.402453i	0.726145	+	0.687542i	$0.241310\pi$
$42$		0			0
$43$	94.0000	+	162.813i	0.333369	+	0.577412i	0.983170	−	0.182692i	$-0.0584812\pi$
$43$	94.0000	+	162.813i	0.333369	+	0.577412i	−0.649801	+	0.760104i	$0.725148\pi$
$44$	−364.000			−1.24716
$45$		0			0
$46$	−168.000			−0.538484
$47$	−128.000	−	221.703i	−0.397249	−	0.688056i	0.596136	−	0.802883i	$-0.296702\pi$
$47$	−128.000	−	221.703i	−0.397249	−	0.688056i	−0.993385	+	0.114827i	$0.963369\pi$
$48$		0			0
$49$	−116.500	+	201.784i	−0.339650	+	0.588291i
$50$	−12.5000	+	21.6506i	−0.0353553	+	0.0612372i
$51$		0			0
$52$	77.0000	+	133.368i	0.205346	+	0.355669i
$53$	−338.000			−0.875998			−0.437999	−	0.898976i	$-0.644313\pi$
$53$	−338.000			−0.875998			−0.437999	+	0.898976i	$0.644313\pi$
$54$		0			0
$55$	260.000			0.637425
$56$	−180.000	−	311.769i	−0.429527	−	0.743963i
$57$		0			0
$58$	−115.000	+	199.186i	−0.260349	+	0.450938i
$59$	−50.0000	+	86.6025i	−0.110330	+	0.191096i	−0.915903	−	0.401399i	$-0.868524\pi$
$59$	−50.0000	+	86.6025i	−0.110330	+	0.191096i	0.805574	+	0.592496i	$0.201857\pi$
$60$		0			0
$61$	−371.000	−	642.591i	−0.778716	−	1.34878i	−0.932682	−	0.360700i	$-0.882538\pi$
$61$	−371.000	−	642.591i	−0.778716	−	1.34878i	0.153966	−	0.988076i	$-0.450795\pi$
$62$	−288.000			−0.589936
$63$		0			0
$64$	−167.000			−0.326172
$65$	−55.0000	−	95.2628i	−0.104952	−	0.181783i
$66$		0			0
$67$	42.0000	−	72.7461i	0.0765838	−	0.132647i	−0.825190	−	0.564855i	$-0.808932\pi$
$67$	42.0000	−	72.7461i	0.0765838	−	0.132647i	0.901774	+	0.432208i	$0.142265\pi$
$68$	−49.0000	+	84.8705i	−0.0873842	+	0.151354i
$69$		0			0
$70$	60.0000	+	103.923i	0.102448	+	0.177445i
$71$	−328.000			−0.548260			−0.274130	−	0.961693i	$-0.588390\pi$
$71$	−328.000			−0.548260			−0.274130	+	0.961693i	$0.588390\pi$
$72$		0			0
$73$	−38.0000			−0.0609255			−0.0304628	−	0.999536i	$-0.509698\pi$
$73$	−38.0000			−0.0609255			−0.0304628	+	0.999536i	$0.509698\pi$
$74$	17.0000	+	29.4449i	0.0267055	+	0.0462553i
$75$		0			0
$76$	−70.0000	+	121.244i	−0.105652	+	0.182995i
$77$	624.000	−	1080.80i	0.923525	−	1.59959i
$78$		0			0
$79$	120.000	+	207.846i	0.170899	+	0.296006i	0.938735	−	0.344641i	$-0.111999\pi$
$79$	120.000	+	207.846i	0.170899	+	0.296006i	−0.767835	+	0.640647i	$0.778666\pi$
$80$	205.000			0.286496
$81$		0			0
$82$	122.000			0.164301
$83$	−606.000	−	1049.62i	−0.801411	−	1.38809i	−0.918687	−	0.394986i	$-0.870750\pi$
$83$	−606.000	−	1049.62i	−0.801411	−	1.38809i	0.117276	−	0.993099i	$-0.462584\pi$
$84$		0			0
$85$	35.0000	−	60.6218i	0.0446622	−	0.0773571i
$86$	94.0000	−	162.813i	0.117864	−	0.204146i
$87$		0			0
$88$	390.000	+	675.500i	0.472433	+	0.818279i
$89$	330.000			0.393033			0.196516	−	0.980501i	$-0.437037\pi$
$89$	330.000			0.393033			0.196516	+	0.980501i	$0.437037\pi$
$90$		0			0
$91$	−528.000			−0.608236
$92$	−588.000	−	1018.45i	−0.666340	−	1.15413i
$93$		0			0
$94$	−128.000	+	221.703i	−0.140449	+	0.243265i
$95$	50.0000	−	86.6025i	0.0539989	−	0.0935288i
$96$		0			0
$97$	−433.000	−	749.978i	−0.453242	−	0.785038i	0.545343	−	0.838213i	$-0.316399\pi$
$97$	−433.000	−	749.978i	−0.453242	−	0.785038i	−0.998585	+	0.0531745i	$0.983066\pi$
$98$	233.000			0.240169
$99$		0			0
$100$	−175.000			−0.175000
$101$	609.000	+	1054.82i	0.599978	+	1.03919i	0.992824	+	0.119588i	$0.0381573\pi$
$101$	609.000	+	1054.82i	0.599978	+	1.03919i	−0.392846	+	0.919604i	$0.628509\pi$
$102$		0			0
$103$	44.0000	−	76.2102i	0.0420917	−	0.0729050i	−0.844212	−	0.536009i	$-0.819931\pi$
$103$	44.0000	−	76.2102i	0.0420917	−	0.0729050i	0.886304	+	0.463104i	$0.153264\pi$
$104$	165.000	−	285.788i	0.155573	−	0.269460i
$105$		0			0
$106$	169.000	+	292.717i	0.154856	+	0.268218i
$107$	36.0000			0.0325257			0.0162629	−	0.999868i	$-0.494823\pi$
$107$	36.0000			0.0325257			0.0162629	+	0.999868i	$0.494823\pi$
$108$		0			0
$109$	−970.000			−0.852378			−0.426189	−	0.904634i	$-0.640144\pi$
$109$	−970.000			−0.852378			−0.426189	+	0.904634i	$0.640144\pi$
$110$	−130.000	−	225.167i	−0.112682	−	0.195171i
$111$		0			0
$112$	492.000	−	852.169i	0.415086	−	0.718950i
$113$	−521.000	+	902.398i	−0.433731	+	0.751243i	−0.997191	−	0.0748996i	$-0.976136\pi$
$113$	−521.000	+	902.398i	−0.433731	+	0.751243i	0.563460	+	0.826143i	$0.309470\pi$
$114$		0			0
$115$	420.000	+	727.461i	0.340567	+	0.589879i
$116$	−1610.00			−1.28866
$117$		0			0
$118$	100.000			0.0780148
$119$	−168.000	−	290.985i	−0.129416	−	0.224156i
$120$		0			0
$121$	−686.500	+	1189.05i	−0.515778	+	0.893353i
$122$	−371.000	+	642.591i	−0.275318	+	0.476864i
$123$		0			0
$124$	−1008.00	−	1745.91i	−0.730009	−	1.26441i
$125$	125.000			0.0894427
$126$		0			0
$127$	1936.00			1.35269			0.676347	−	0.736583i	$-0.263562\pi$
$127$	1936.00			1.35269			0.676347	+	0.736583i	$0.263562\pi$
$128$	727.500	+	1260.07i	0.502363	+	0.870119i
$129$		0			0
$130$	−55.0000	+	95.2628i	−0.0371063	+	0.0642700i
$131$	−366.000	+	633.931i	−0.244104	+	0.422800i	−0.961879	−	0.273475i	$-0.911827\pi$
$131$	−366.000	+	633.931i	−0.244104	+	0.422800i	0.717776	+	0.696274i	$0.245160\pi$
$132$		0			0
$133$	−240.000	−	415.692i	−0.156471	−	0.271016i
$134$	−84.0000			−0.0541529
$135$		0			0
$136$	210.000			0.132407
$137$	1107.00	+	1917.38i	0.690346	+	1.19571i	0.971725	+	0.236117i	$0.0758750\pi$
$137$	1107.00	+	1917.38i	0.690346	+	1.19571i	−0.281379	+	0.959597i	$0.590792\pi$
$138$		0			0
$139$	−10.0000	+	17.3205i	−0.00610208	+	0.0105691i	−0.869060	−	0.494706i	$-0.835276\pi$
$139$	−10.0000	+	17.3205i	−0.00610208	+	0.0105691i	0.862958	+	0.505275i	$0.168609\pi$
$140$	−420.000	+	727.461i	−0.253546	+	0.439155i
$141$		0			0
$142$	164.000	+	284.056i	0.0969195	+	0.167870i
$143$	1144.00			0.668994
$144$		0			0
$145$	1150.00			0.658637
$146$	19.0000	+	32.9090i	0.0107702	+	0.0186546i
$147$		0			0
$148$	−119.000	+	206.114i	−0.0660928	+	0.114476i
$149$	665.000	−	1151.81i	0.365630	−	0.633290i	−0.623247	−	0.782025i	$-0.714187\pi$
$149$	665.000	−	1151.81i	0.365630	−	0.633290i	0.988877	+	0.148735i	$0.0475201\pi$
$150$		0			0
$151$	604.000	+	1046.16i	0.325515	+	0.563809i	0.981617	−	0.190864i	$-0.0611289\pi$
$151$	604.000	+	1046.16i	0.325515	+	0.563809i	−0.656101	+	0.754673i	$0.727796\pi$
$152$	300.000			0.160087
$153$		0			0
$154$	−1248.00			−0.653031
$155$	720.000	+	1247.08i	0.373108	+	0.646243i
$156$		0			0
$157$	1757.00	−	3043.21i	0.893146	−	1.54697i	0.0570627	−	0.998371i	$-0.481827\pi$
$157$	1757.00	−	3043.21i	0.893146	−	1.54697i	0.836083	−	0.548603i	$-0.184840\pi$
$158$	120.000	−	207.846i	0.0604221	−	0.104654i
$159$		0			0
$160$	−402.500	−	697.150i	−0.198878	−	0.344466i
$161$	4032.00			1.97370
$162$		0			0
$163$	−2068.00			−0.993732			−0.496866	−	0.867827i	$-0.665516\pi$
$163$	−2068.00			−0.993732			−0.496866	+	0.867827i	$0.665516\pi$
$164$	427.000	+	739.586i	0.203312	+	0.352146i
$165$		0			0
$166$	−606.000	+	1049.62i	−0.283342	+	0.490762i
$167$	12.0000	−	20.7846i	0.00556041	−	0.00963091i	−0.863232	−	0.504808i	$-0.831563\pi$
$167$	12.0000	−	20.7846i	0.00556041	−	0.00963091i	0.868792	+	0.495177i	$0.164897\pi$
$168$		0			0
$169$	856.500	+	1483.50i	0.389850	+	0.675240i
$170$	−70.0000			−0.0315809
$171$		0			0
$172$	1316.00			0.583396
$173$	309.000	+	535.204i	0.135797	+	0.235207i	0.925902	−	0.377765i	$-0.123307\pi$
$173$	309.000	+	535.204i	0.135797	+	0.235207i	−0.790105	+	0.612972i	$0.789974\pi$
$174$		0			0
$175$	300.000	−	519.615i	0.129588	−	0.224453i
$176$	−1066.00	+	1846.37i	−0.456550	+	0.790768i
$177$		0			0
$178$	−165.000	−	285.788i	−0.0694791	−	0.120341i
$179$	3340.00			1.39466			0.697328	−	0.716752i	$-0.254372\pi$
$179$	3340.00			1.39466			0.697328	+	0.716752i	$0.254372\pi$
$180$		0			0
$181$	−178.000			−0.0730974			−0.0365487	−	0.999332i	$-0.511636\pi$
$181$	−178.000			−0.0730974			−0.0365487	+	0.999332i	$0.511636\pi$
$182$	264.000	+	457.261i	0.107522	+	0.186233i
$183$		0			0
$184$	−1260.00	+	2182.38i	−0.504828	+	0.874389i
$185$	85.0000	−	147.224i	0.0337801	−	0.0585089i
$186$		0			0
$187$	364.000	+	630.466i	0.142344	+	0.246547i
$188$	−1792.00			−0.695186
$189$		0			0
$190$	−100.000			−0.0381830
$191$	944.000	+	1635.06i	0.357620	+	0.619416i	0.987563	−	0.157225i	$-0.0502549\pi$
$191$	944.000	+	1635.06i	0.357620	+	0.619416i	−0.629943	+	0.776642i	$0.716922\pi$
$192$		0			0
$193$	−961.000	+	1664.50i	−0.358416	+	0.620795i	−0.987696	−	0.156384i	$-0.950016\pi$
$193$	−961.000	+	1664.50i	−0.358416	+	0.620795i	0.629280	+	0.777178i	$0.283350\pi$
$194$	−433.000	+	749.978i	−0.160245	+	0.277553i
$195$		0			0
$196$	815.500	+	1412.49i	0.297194	+	0.514755i
$197$	2526.00			0.913554			0.456777	−	0.889581i	$-0.349004\pi$
$197$	2526.00			0.913554			0.456777	+	0.889581i	$0.349004\pi$
$198$		0			0
$199$	−1160.00			−0.413217			−0.206609	−	0.978424i	$-0.566243\pi$
$199$	−1160.00			−0.413217			−0.206609	+	0.978424i	$0.566243\pi$
$200$	187.500	+	324.760i	0.0662913	+	0.114820i
$201$		0			0
$202$	609.000	−	1054.82i	0.212124	−	0.367410i
$203$	2760.00	−	4780.46i	0.954256	−	1.65282i
$204$		0			0
$205$	−305.000	−	528.275i	−0.103913	−	0.179982i
$206$	−88.0000			−0.0297634
$207$		0			0
$208$	902.000			0.300685
$209$	520.000	+	900.666i	0.172101	+	0.298088i
$210$		0			0
$211$	2234.00	−	3869.40i	0.728886	−	1.26247i	−0.228469	−	0.973551i	$-0.573372\pi$
$211$	2234.00	−	3869.40i	0.728886	−	1.26247i	0.957354	−	0.288916i	$-0.0932948\pi$
$212$	−1183.00	+	2049.02i	−0.383249	+	0.663807i
$213$		0			0
$214$	−18.0000	−	31.1769i	−0.00574979	−	0.00995893i
$215$	−940.000			−0.298174
$216$		0			0
$217$	6912.00			2.16229
$218$	485.000	+	840.045i	0.150680	+	0.260986i
$219$		0			0
$220$	910.000	−	1576.17i	0.278874	−	0.483023i
$221$	154.000	−	266.736i	0.0468740	−	0.0811882i
$222$		0			0
$223$	−3016.00	−	5223.87i	−0.905678	−	1.56868i	−0.820004	−	0.572358i	$-0.806029\pi$
$223$	−3016.00	−	5223.87i	−0.905678	−	1.56868i	−0.0856746	−	0.996323i	$-0.527305\pi$
$224$	−3864.00			−1.15256
$225$		0			0
$226$	1042.00			0.306694
$227$	−1318.00	−	2282.84i	−0.385369	−	0.667478i	0.606451	−	0.795121i	$-0.292592\pi$
$227$	−1318.00	−	2282.84i	−0.385369	−	0.667478i	−0.991820	+	0.127642i	$0.959259\pi$
$228$		0			0
$229$	−2415.00	+	4182.90i	−0.696889	+	1.20705i	0.272650	+	0.962113i	$0.412100\pi$
$229$	−2415.00	+	4182.90i	−0.696889	+	1.20705i	−0.969540	+	0.244935i	$0.921233\pi$
$230$	420.000	−	727.461i	0.120409	−	0.208554i
$231$		0			0
$232$	1725.00	+	2987.79i	0.488154	+	0.845508i
$233$	2682.00			0.754093			0.377046	−	0.926194i	$-0.376940\pi$
$233$	2682.00			0.754093			0.377046	+	0.926194i	$0.376940\pi$
$234$		0			0
$235$	1280.00			0.355311
$236$	350.000	+	606.218i	0.0965384	+	0.167209i
$237$		0			0
$238$	−168.000	+	290.985i	−0.0457556	+	0.0792509i
$239$	−1160.00	+	2009.18i	−0.313950	+	0.543778i	−0.979214	−	0.202831i	$-0.934986\pi$
$239$	−1160.00	+	2009.18i	−0.313950	+	0.543778i	0.665263	+	0.746609i	$0.268319\pi$
$240$		0			0
$241$	−1001.00	−	1733.78i	−0.267552	−	0.463414i	0.700677	−	0.713479i	$-0.252881\pi$
$241$	−1001.00	−	1733.78i	−0.267552	−	0.463414i	−0.968229	+	0.250065i	$0.919548\pi$
$242$	1373.00			0.364710
$243$		0			0
$244$	−5194.00			−1.36275
$245$	−582.500	−	1008.92i	−0.151896	−	0.263092i
$246$		0			0
$247$	220.000	−	381.051i	0.0566731	−	0.0981608i
$248$	−2160.00	+	3741.23i	−0.553065	+	0.957937i
$249$		0			0
$250$	−62.5000	−	108.253i	−0.0158114	−	0.0273861i
$251$	132.000			0.0331943			0.0165971	−	0.999862i	$-0.494717\pi$
$251$	132.000			0.0331943			0.0165971	+	0.999862i	$0.494717\pi$
$252$		0			0
$253$	−8736.00			−2.17086
$254$	−968.000	−	1676.63i	−0.239125	−	0.414176i
$255$		0			0
$256$	59.5000	−	103.057i	0.0145264	−	0.0251604i
$257$	3807.00	−	6593.92i	0.924024	−	1.60046i	0.130900	−	0.991396i	$-0.458213\pi$
$257$	3807.00	−	6593.92i	0.924024	−	1.60046i	0.793124	−	0.609061i	$-0.208453\pi$
$258$		0			0
$259$	−408.000	−	706.677i	−0.0978837	−	0.169540i
$260$	−770.000			−0.183667
$261$		0			0
$262$	732.000			0.172607
$263$	2444.00	+	4233.13i	0.573017	+	0.992495i	0.996254	+	0.0864757i	$0.0275605\pi$
$263$	2444.00	+	4233.13i	0.573017	+	0.992495i	−0.423237	+	0.906019i	$0.639106\pi$
$264$		0			0
$265$	845.000	−	1463.58i	0.195879	−	0.339272i
$266$	−240.000	+	415.692i	−0.0553208	+	0.0958185i
$267$		0			0
$268$	−294.000	−	509.223i	−0.0670109	−	0.116066i
$269$	1270.00			0.287856			0.143928	−	0.989588i	$-0.454027\pi$
$269$	1270.00			0.287856			0.143928	+	0.989588i	$0.454027\pi$
$270$		0			0
$271$	1072.00			0.240293			0.120146	−	0.992756i	$-0.461664\pi$
$271$	1072.00			0.240293			0.120146	+	0.992756i	$0.461664\pi$
$272$	287.000	+	497.099i	0.0639777	+	0.110813i
$273$		0			0
$274$	1107.00	−	1917.38i	0.244074	−	0.422749i
$275$	−650.000	+	1125.83i	−0.142533	+	0.246874i
$276$		0			0
$277$	2697.00	+	4671.34i	0.585007	+	1.01326i	0.994875	+	0.101117i	$0.0322417\pi$
$277$	2697.00	+	4671.34i	0.585007	+	1.01326i	−0.409867	+	0.912145i	$0.634425\pi$
$278$	20.0000			0.00431482
$279$		0			0
$280$	1800.00			0.384181
$281$	−1221.00	−	2114.83i	−0.259213	−	0.448969i	0.706819	−	0.707395i	$-0.250130\pi$
$281$	−1221.00	−	2114.83i	−0.259213	−	0.448969i	−0.966031	+	0.258425i	$0.916796\pi$
$282$		0			0
$283$	−1386.00	+	2400.62i	−0.291128	+	0.504248i	−0.974077	−	0.226218i	$-0.927364\pi$
$283$	−1386.00	+	2400.62i	−0.291128	+	0.504248i	0.682949	+	0.730466i	$0.260697\pi$
$284$	−1148.00	+	1988.39i	−0.239864	+	0.415456i
$285$		0			0
$286$	−572.000	−	990.733i	−0.118262	−	0.204837i
$287$	−2928.00			−0.602210
$288$		0			0
$289$	−4717.00			−0.960106
$290$	−575.000	−	995.929i	−0.116432	−	0.201665i
$291$		0			0
$292$	−133.000	+	230.363i	−0.0266549	+	0.0461677i
$293$	−2271.00	+	3933.49i	−0.452810	+	0.784289i	−0.998559	−	0.0536589i	$-0.982912\pi$
$293$	−2271.00	+	3933.49i	−0.452810	+	0.784289i	0.545750	+	0.837948i	$0.316245\pi$
$294$		0			0
$295$	−250.000	−	433.013i	−0.0493409	−	0.0854609i
$296$	510.000			0.100146
$297$		0			0
$298$	−1330.00			−0.258540
$299$	1848.00	+	3200.83i	0.357433	+	0.619093i
$300$		0			0
$301$	−2256.00	+	3907.51i	−0.432006	+	0.748256i
$302$	604.000	−	1046.16i	0.115087	−	0.199337i
$303$		0			0
$304$	410.000	+	710.141i	0.0773523	+	0.133978i
$305$	3710.00			0.696505
$306$		0			0
$307$	5116.00			0.951093			0.475546	−	0.879691i	$-0.342250\pi$
$307$	5116.00			0.951093			0.475546	+	0.879691i	$0.342250\pi$
$308$	−4368.00	−	7565.60i	−0.808084	−	1.39964i
$309$		0			0
$310$	720.000	−	1247.08i	0.131914	−	0.228481i
$311$	1404.00	−	2431.80i	0.255992	−	0.443391i	−0.709172	−	0.705035i	$-0.750931\pi$
$311$	1404.00	−	2431.80i	0.255992	−	0.443391i	0.965165	+	0.261644i	$0.0842646\pi$
$312$		0			0
$313$	3659.00	+	6337.57i	0.660763	+	1.14448i	0.980415	+	0.196941i	$0.0631006\pi$
$313$	3659.00	+	6337.57i	0.660763	+	1.14448i	−0.319652	+	0.947535i	$0.603566\pi$
$314$	−3514.00			−0.631549
$315$		0			0
$316$	1680.00			0.299074
$317$	−1123.00	−	1945.09i	−0.198971	−	0.344629i	0.749224	−	0.662317i	$-0.230427\pi$
$317$	−1123.00	−	1945.09i	−0.198971	−	0.344629i	−0.948195	+	0.317688i	$0.897093\pi$
$318$		0			0
$319$	−5980.00	+	10357.7i	−1.04958	+	1.81792i
$320$	417.500	−	723.131i	0.0729342	−	0.126326i
$321$		0			0
$322$	−2016.00	−	3491.81i	−0.348905	−	0.604321i
$323$	280.000			0.0482341
$324$		0			0
$325$	550.000			0.0938723
$326$	1034.00	+	1790.94i	0.175669	+	0.304267i
$327$		0			0
$328$	915.000	−	1584.83i	0.154032	−	0.266791i
$329$	3072.00	−	5320.86i	0.514787	−	0.891637i
$330$		0			0
$331$	−666.000	−	1153.55i	−0.110594	−	0.191555i	0.805416	−	0.592710i	$-0.201942\pi$
$331$	−666.000	−	1153.55i	−0.110594	−	0.191555i	−0.916010	+	0.401155i	$0.868609\pi$
$332$	−8484.00			−1.40247
$333$		0			0
$334$	−24.0000			−0.00393180
$335$	210.000	+	363.731i	0.0342493	+	0.0593216i
$336$		0			0
$337$	5767.00	−	9988.74i	0.932191	−	1.61460i	0.152624	−	0.988284i	$-0.451228\pi$
$337$	5767.00	−	9988.74i	0.932191	−	1.61460i	0.779567	−	0.626319i	$-0.215439\pi$
$338$	856.500	−	1483.50i	0.137833	−	0.238733i
$339$		0			0
$340$	−245.000	−	424.352i	−0.0390794	−	0.0676875i
$341$	−14976.0			−2.37829
$342$		0			0
$343$	2640.00			0.415588
$344$	−1410.00	−	2442.19i	−0.220994	−	0.382774i
$345$		0			0
$346$	309.000	−	535.204i	0.0480114	−	0.0831582i
$347$	−5978.00	+	10354.2i	−0.924830	+	1.60185i	−0.132994	+	0.991117i	$0.542459\pi$
$347$	−5978.00	+	10354.2i	−0.924830	+	1.60185i	−0.791835	+	0.610735i	$0.790874\pi$
$348$		0			0
$349$	−2435.00	−	4217.54i	−0.373474	−	0.646877i	0.616623	−	0.787259i	$-0.288500\pi$
$349$	−2435.00	−	4217.54i	−0.373474	−	0.646877i	−0.990097	+	0.140382i	$0.955167\pi$
$350$	−600.000			−0.0916324
$351$		0			0
$352$	8372.00			1.26770
$353$	−5361.00	−	9285.52i	−0.808321	−	1.40005i	−0.914026	−	0.405655i	$-0.867043\pi$
$353$	−5361.00	−	9285.52i	−0.808321	−	1.40005i	0.105705	−	0.994397i	$-0.466290\pi$
$354$		0			0
$355$	820.000	−	1420.28i	0.122595	−	0.212340i
$356$	1155.00	−	2000.52i	0.171952	−	0.297829i
$357$		0			0
$358$	−1670.00	−	2892.52i	−0.246543	−	0.427024i
$359$	120.000			0.0176417			0.00882083	−	0.999961i	$-0.497192\pi$
$359$	120.000			0.0176417			0.00882083	+	0.999961i	$0.497192\pi$
$360$		0			0
$361$	−6459.00			−0.941682
$362$	89.0000	+	154.153i	0.0129219	+	0.0223814i
$363$		0			0
$364$	−1848.00	+	3200.83i	−0.266103	+	0.460904i
$365$	95.0000	−	164.545i	0.0136234	−	0.0235964i
$366$		0			0
$367$	−1968.00	−	3408.68i	−0.279915	−	0.484827i	0.691448	−	0.722426i	$-0.256973\pi$
$367$	−1968.00	−	3408.68i	−0.279915	−	0.484827i	−0.971363	+	0.237599i	$0.923640\pi$
$368$	−6888.00			−0.975711
$369$		0			0
$370$	−170.000			−0.0238862
$371$	−4056.00	−	7025.20i	−0.567593	−	0.983100i
$372$		0			0
$373$	−1511.00	+	2617.13i	−0.209750	+	0.363297i	−0.951636	−	0.307229i	$-0.900598\pi$
$373$	−1511.00	+	2617.13i	−0.209750	+	0.363297i	0.741886	+	0.670526i	$0.233932\pi$
$374$	364.000	−	630.466i	0.0503262	−	0.0871675i
$375$		0			0
$376$	1920.00	+	3325.54i	0.263342	+	0.456121i
$377$	5060.00			0.691255
$378$		0			0
$379$	−13340.0			−1.80799			−0.903997	−	0.427539i	$-0.859381\pi$
$379$	−13340.0			−1.80799			−0.903997	+	0.427539i	$0.859381\pi$
$380$	−350.000	−	606.218i	−0.0472490	−	0.0818377i
$381$		0			0
$382$	944.000	−	1635.06i	0.126438	−	0.218997i
$383$	504.000	−	872.954i	0.0672407	−	0.116464i	−0.830445	−	0.557101i	$-0.811914\pi$
$383$	504.000	−	872.954i	0.0672407	−	0.116464i	0.897686	+	0.440636i	$0.145247\pi$
$384$		0			0
$385$	3120.00	+	5404.00i	0.413013	+	0.715359i
$386$	1922.00			0.253438
$387$		0			0
$388$	−6062.00			−0.793174
$389$	−4815.00	−	8339.82i	−0.627584	−	1.08701i	−0.988035	−	0.154229i	$-0.950711\pi$
$389$	−4815.00	−	8339.82i	−0.627584	−	1.08701i	0.360451	−	0.932778i	$-0.382623\pi$
$390$		0			0
$391$	−1176.00	+	2036.89i	−0.152105	+	0.263453i
$392$	1747.50	−	3026.76i	0.225158	−	0.389986i
$393$		0			0
$394$	−1263.00	−	2187.58i	−0.161495	−	0.279718i
$395$	−1200.00			−0.152857
$396$		0			0
$397$	7126.00			0.900866			0.450433	−	0.892810i	$-0.351270\pi$
$397$	7126.00			0.900866			0.450433	+	0.892810i	$0.351270\pi$
$398$	580.000	+	1004.59i	0.0730472	+	0.126521i
$399$		0			0
$400$	−512.500	+	887.676i	−0.0640625	+	0.110960i
$401$	4359.00	−	7550.01i	0.542838	−	0.940223i	−0.455901	−	0.890030i	$-0.650683\pi$
$401$	4359.00	−	7550.01i	0.542838	−	0.940223i	0.998740	−	0.0501929i	$-0.0159836\pi$
$402$		0			0
$403$	3168.00	+	5487.14i	0.391586	+	0.678248i
$404$	8526.00			1.04996
$405$		0			0
$406$	−5520.00			−0.674761
$407$	884.000	+	1531.13i	0.107662	+	0.186475i
$408$		0			0
$409$	5435.00	−	9413.70i	0.657074	−	1.13809i	−0.324295	−	0.945956i	$-0.605127\pi$
$409$	5435.00	−	9413.70i	0.657074	−	1.13809i	0.981369	−	0.192130i	$-0.0615396\pi$
$410$	−305.000	+	528.275i	−0.0367387	+	0.0636333i
$411$		0			0
$412$	−308.000	−	533.472i	−0.0368303	−	0.0637919i
$413$	−2400.00			−0.285947
$414$		0			0
$415$	6060.00			0.716804
$416$	−1771.00	−	3067.46i	−0.208727	−	0.361526i
$417$		0			0
$418$	520.000	−	900.666i	0.0608470	−	0.105390i
$419$	4850.00	−	8400.45i	0.565484	−	0.979448i	−0.431520	−	0.902103i	$-0.642023\pi$
$419$	4850.00	−	8400.45i	0.565484	−	0.979448i	0.997004	−	0.0773445i	$-0.0246441\pi$
$420$		0			0
$421$	−431.000	−	746.514i	−0.0498947	−	0.0864201i	0.839999	−	0.542587i	$-0.182555\pi$
$421$	−431.000	−	746.514i	−0.0498947	−	0.0864201i	−0.889894	+	0.456167i	$0.849222\pi$
$422$	−4468.00			−0.515400
$423$		0			0
$424$	5070.00			0.580710
$425$	175.000	+	303.109i	0.0199735	+	0.0345952i
$426$		0			0
$427$	8904.00	−	15422.2i	1.00912	−	1.74785i
$428$	126.000	−	218.238i	0.0142300	−	0.0246471i
$429$		0			0
$430$	470.000	+	814.064i	0.0527103	+	0.0912969i
$431$	15792.0			1.76490			0.882452	−	0.470402i	$-0.155891\pi$
$431$	15792.0			1.76490			0.882452	+	0.470402i	$0.155891\pi$
$432$		0			0
$433$	11602.0			1.28766			0.643830	−	0.765169i	$-0.277345\pi$
$433$	11602.0			1.28766			0.643830	+	0.765169i	$0.277345\pi$
$434$	−3456.00	−	5985.97i	−0.382243	−	0.662064i
$435$		0			0
$436$	−3395.00	+	5880.31i	−0.372915	+	0.645908i
$437$	−1680.00	+	2909.85i	−0.183902	+	0.318528i
$438$		0			0
$439$	220.000	+	381.051i	0.0239181	+	0.0414273i	0.877737	−	0.479143i	$-0.159053\pi$
$439$	220.000	+	381.051i	0.0239181	+	0.0414273i	−0.853819	+	0.520571i	$0.825719\pi$
$440$	−3900.00			−0.422557
$441$		0			0
$442$	−308.000			−0.0331449
$443$	5094.00	+	8823.07i	0.546328	+	0.946268i	0.998522	+	0.0543481i	$0.0173081\pi$
$443$	5094.00	+	8823.07i	0.546328	+	0.946268i	−0.452194	+	0.891920i	$0.649359\pi$
$444$		0			0
$445$	−825.000	+	1428.94i	−0.0878848	+	0.152221i
$446$	−3016.00	+	5223.87i	−0.320206	+	0.554613i
$447$		0			0
$448$	−2004.00	−	3471.03i	−0.211340	−	0.366051i
$449$	−13310.0			−1.39897			−0.699485	−	0.714647i	$-0.746587\pi$
$449$	−13310.0			−1.39897			−0.699485	+	0.714647i	$0.746587\pi$
$450$		0			0
$451$	6344.00			0.662367
$452$	3647.00	+	6316.79i	0.379514	+	0.657338i
$453$		0			0
$454$	−1318.00	+	2282.84i	−0.136248	+	0.235989i
$455$	1320.00	−	2286.31i	0.136006	−	0.235569i
$456$		0			0
$457$	−1613.00	−	2793.80i	−0.165105	−	0.285970i	0.771588	−	0.636123i	$-0.219463\pi$
$457$	−1613.00	−	2793.80i	−0.165105	−	0.285970i	−0.936693	+	0.350153i	$0.886130\pi$
$458$	4830.00			0.492775
$459$		0			0
$460$	5880.00			0.595992
$461$	−3291.00	−	5700.18i	−0.332488	−	0.575887i	0.650511	−	0.759497i	$-0.274555\pi$
$461$	−3291.00	−	5700.18i	−0.332488	−	0.575887i	−0.982999	+	0.183610i	$0.941222\pi$
$462$		0			0
$463$	−7536.00	+	13052.7i	−0.756431	+	1.31018i	0.188229	+	0.982125i	$0.439725\pi$
$463$	−7536.00	+	13052.7i	−0.756431	+	1.31018i	−0.944660	+	0.328052i	$0.893608\pi$
$464$	−4715.00	+	8166.62i	−0.471742	+	0.817081i
$465$		0			0
$466$	−1341.00	−	2322.68i	−0.133306	−	0.230893i
$467$	476.000			0.0471663			0.0235831	−	0.999722i	$-0.492493\pi$
$467$	476.000			0.0471663			0.0235831	+	0.999722i	$0.492493\pi$
$468$		0			0
$469$	2016.00			0.198487
$470$	−640.000	−	1108.51i	−0.0628106	−	0.108791i
$471$		0			0
$472$	750.000	−	1299.04i	0.0731389	−	0.126680i
$473$	4888.00	−	8466.26i	0.475160	−	0.823001i
$474$		0			0
$475$	250.000	+	433.013i	0.0241490	+	0.0418273i
$476$	−2352.00			−0.226478
$477$		0			0
$478$	2320.00			0.221997
$479$	9840.00	+	17043.4i	0.938624	+	1.62575i	0.768040	+	0.640402i	$0.221232\pi$
$479$	9840.00	+	17043.4i	0.938624	+	1.62575i	0.170585	+	0.985343i	$0.445434\pi$
$480$		0			0
$481$	374.000	−	647.787i	0.0354531	−	0.0614065i
$482$	−1001.00	+	1733.78i	−0.0945940	+	0.163842i
$483$		0			0
$484$	4805.50	+	8323.37i	0.451305	+	0.781684i
$485$	4330.00			0.405392
$486$		0			0
$487$	−5944.00			−0.553077			−0.276538	−	0.961003i	$-0.589187\pi$
$487$	−5944.00			−0.553077			−0.276538	+	0.961003i	$0.589187\pi$
$488$	5565.00	+	9638.86i	0.516221	+	0.894121i
$489$		0			0
$490$	−582.500	+	1008.92i	−0.0537034	+	0.0930170i
$491$	−5386.00	+	9328.83i	−0.495044	+	0.857442i	−0.999984	−	0.00571287i	$-0.998182\pi$
$491$	−5386.00	+	9328.83i	−0.495044	+	0.857442i	0.504939	+	0.863155i	$0.331515\pi$
$492$		0			0
$493$	1610.00	+	2788.60i	0.147081	+	0.254751i
$494$	−440.000			−0.0400740
$495$		0			0
$496$	−11808.0			−1.06894
$497$	−3936.00	−	6817.35i	−0.355239	−	0.615292i
$498$		0			0
$499$	−4070.00	+	7049.45i	−0.365127	+	0.632418i	−0.988796	−	0.149270i	$-0.952308\pi$
$499$	−4070.00	+	7049.45i	−0.365127	+	0.632418i	0.623670	+	0.781688i	$0.285641\pi$
$500$	437.500	−	757.772i	0.0391312	−	0.0677772i
$501$		0			0
$502$	−66.0000	−	114.315i	−0.00586798	−	0.0101636i
$503$	−13768.0			−1.22045			−0.610223	−	0.792229i	$-0.708920\pi$
$503$	−13768.0			−1.22045			−0.610223	+	0.792229i	$0.708920\pi$
$504$		0			0
$505$	−6090.00			−0.536637
$506$	4368.00	+	7565.60i	0.383757	+	0.664687i
$507$		0			0
$508$	6776.00	−	11736.4i	0.591804	−	1.02503i
$509$	−11075.0	+	19182.5i	−0.964422	+	1.67043i	−0.253262	+	0.967398i	$0.581503\pi$
$509$	−11075.0	+	19182.5i	−0.964422	+	1.67043i	−0.711160	+	0.703030i	$0.751830\pi$
$510$		0			0
$511$	−456.000	−	789.815i	−0.0394760	−	0.0683745i
$512$	11521.0			0.994455
$513$		0			0
$514$	−7614.00			−0.653384
$515$	220.000	+	381.051i	0.0188240	+	0.0326041i
$516$		0			0
$517$	−6656.00	+	11528.5i	−0.566210	+	0.980704i
$518$	−408.000	+	706.677i	−0.0346071	+	0.0599413i
$519$		0			0
$520$	825.000	+	1428.94i	0.0695743	+	0.120506i
$521$	1562.00			0.131348			0.0656741	−	0.997841i	$-0.479080\pi$
$521$	1562.00			0.131348			0.0656741	+	0.997841i	$0.479080\pi$
$522$		0			0
$523$	−668.000			−0.0558501			−0.0279250	−	0.999610i	$-0.508890\pi$
$523$	−668.000			−0.0558501			−0.0279250	+	0.999610i	$0.508890\pi$
$524$	2562.00	+	4437.51i	0.213591	+	0.369950i
$525$		0			0
$526$	2444.00	−	4233.13i	0.202592	−	0.350900i
$527$	−2016.00	+	3491.81i	−0.166638	+	0.288626i
$528$		0			0
$529$	−8028.50	−	13905.8i	−0.659859	−	1.14291i
$530$	−1690.00			−0.138507
$531$		0			0
$532$	−3360.00			−0.273824
$533$	−1342.00	−	2324.41i	−0.109059	−	0.188896i
$534$		0			0
$535$	−90.0000	+	155.885i	−0.00727297	+	0.0125972i
$536$	−630.000	+	1091.19i	−0.0507684	+	0.0879334i
$537$		0			0
$538$	−635.000	−	1099.85i	−0.0508862	−	0.0881375i
$539$	12116.0			0.968225
$540$		0			0
$541$	−6138.00			−0.487788			−0.243894	−	0.969802i	$-0.578425\pi$
$541$	−6138.00			−0.487788			−0.243894	+	0.969802i	$0.578425\pi$
$542$	−536.000	−	928.379i	−0.0424782	−	0.0735744i
$543$		0			0
$544$	1127.00	−	1952.02i	0.0888230	−	0.153846i
$545$	2425.00	−	4200.22i	0.190597	−	0.330124i
$546$		0			0
$547$	5242.00	+	9079.41i	0.409747	+	0.709703i	0.994861	−	0.101248i	$-0.0322836\pi$
$547$	5242.00	+	9079.41i	0.409747	+	0.709703i	−0.585114	+	0.810951i	$0.698950\pi$
$548$	15498.0			1.20811
$549$		0			0
$550$	1300.00			0.100786
$551$	2300.00	+	3983.72i	0.177828	+	0.308007i
$552$		0			0
$553$	−2880.00	+	4988.31i	−0.221465	+	0.383588i
$554$	2697.00	−	4671.34i	0.206831	−	0.358242i
$555$		0			0
$556$	70.0000	+	121.244i	0.00533932	+	0.00924797i
$557$	3606.00			0.274311			0.137155	−	0.990550i	$-0.456204\pi$
$557$	3606.00			0.274311			0.137155	+	0.990550i	$0.456204\pi$
$558$		0			0
$559$	−4136.00			−0.312941
$560$	2460.00	+	4260.84i	0.185632	+	0.321524i
$561$		0			0
$562$	−1221.00	+	2114.83i	−0.0916455	+	0.158735i
$563$	−6126.00	+	10610.5i	−0.458579	+	0.794283i	−0.998886	−	0.0471855i	$-0.984975\pi$
$563$	−6126.00	+	10610.5i	−0.458579	+	0.794283i	0.540307	+	0.841468i	$0.318308\pi$
$564$		0			0
$565$	−2605.00	−	4511.99i	−0.193970	−	0.335966i
$566$	2772.00			0.205858
$567$		0			0
$568$	4920.00			0.363448
$569$	7275.00	+	12600.7i	0.536000	+	0.928379i	0.999114	+	0.0420803i	$0.0133985\pi$
$569$	7275.00	+	12600.7i	0.536000	+	0.928379i	−0.463114	+	0.886298i	$0.653268\pi$
$570$		0			0
$571$	12734.0	−	22055.9i	0.933277	−	1.61648i	0.155600	−	0.987820i	$-0.450269\pi$
$571$	12734.0	−	22055.9i	0.933277	−	1.61648i	0.777677	−	0.628663i	$-0.216398\pi$
$572$	4004.00	−	6935.13i	0.292685	−	0.506945i
$573$		0			0
$574$	1464.00	+	2535.72i	0.106457	+	0.184389i
$575$	−4200.00			−0.304612
$576$		0			0
$577$	12866.0			0.928282			0.464141	−	0.885761i	$-0.346363\pi$
$577$	12866.0			0.928282			0.464141	+	0.885761i	$0.346363\pi$
$578$	2358.50	+	4085.04i	0.169724	+	0.293971i
$579$		0			0
$580$	4025.00	−	6971.50i	0.288153	−	0.499096i
$581$	14544.0	−	25190.9i	1.03853	−	1.79879i
$582$		0			0
$583$	8788.00	+	15221.3i	0.624291	+	1.08130i
$584$	570.000			0.0403883
$585$		0			0
$586$	4542.00			0.320185
$587$	7422.00	+	12855.3i	0.521872	+	0.903908i	0.999676	+	0.0254422i	$0.00809939\pi$
$587$	7422.00	+	12855.3i	0.521872	+	0.903908i	−0.477805	+	0.878466i	$0.658567\pi$
$588$		0			0
$589$	−2880.00	+	4988.31i	−0.201474	+	0.348964i
$590$	−250.000	+	433.013i	−0.0174446	+	0.0302150i
$591$		0			0
$592$	697.000	+	1207.24i	0.0483894	+	0.0838129i
$593$	20402.0			1.41283			0.706416	−	0.707797i	$-0.250311\pi$
$593$	20402.0			1.41283			0.706416	+	0.707797i	$0.250311\pi$
$594$		0			0
$595$	1680.00			0.115753
$596$	−4655.00	−	8062.70i	−0.319927	−	0.554129i
$597$		0			0
$598$	1848.00	−	3200.83i	0.126372	−	0.218882i
$599$	−5380.00	+	9318.43i	−0.366980	+	0.635627i	−0.989092	−	0.147301i	$-0.952941\pi$
$599$	−5380.00	+	9318.43i	−0.366980	+	0.635627i	0.622112	+	0.782928i	$0.286275\pi$
$600$		0			0
$601$	−7141.00	−	12368.6i	−0.484671	−	0.839475i	0.515174	−	0.857086i	$-0.327727\pi$
$601$	−7141.00	−	12368.6i	−0.484671	−	0.839475i	−0.999845	+	0.0176105i	$0.994394\pi$
$602$	4512.00			0.305474
$603$		0			0
$604$	8456.00			0.569652
$605$	−3432.50	−	5945.26i	−0.230663	−	0.399520i
$606$		0			0
$607$	−5528.00	+	9574.78i	−0.369645	+	0.640244i	−0.989510	−	0.144464i	$-0.953854\pi$
$607$	−5528.00	+	9574.78i	−0.369645	+	0.640244i	0.619865	+	0.784709i	$0.287187\pi$
$608$	1610.00	−	2788.60i	0.107392	−	0.186008i
$609$		0			0
$610$	−1855.00	−	3212.95i	−0.123126	−	0.213260i
$611$	5632.00			0.372907
$612$		0			0
$613$	−16418.0			−1.08176			−0.540878	−	0.841101i	$-0.681908\pi$
$613$	−16418.0			−1.08176			−0.540878	+	0.841101i	$0.681908\pi$
$614$	−2558.00	−	4430.59i	−0.168131	−	0.291212i
$615$		0			0
$616$	−9360.00	+	16212.0i	−0.612216	+	1.06039i
$617$	5187.00	−	8984.15i	0.338445	−	0.586204i	−0.645695	−	0.763595i	$-0.723432\pi$
$617$	5187.00	−	8984.15i	0.338445	−	0.586204i	0.984140	+	0.177391i	$0.0567657\pi$
$618$		0			0
$619$	2630.00	+	4555.29i	0.170773	+	0.295788i	0.938690	−	0.344761i	$-0.112040\pi$
$619$	2630.00	+	4555.29i	0.170773	+	0.295788i	−0.767917	+	0.640549i	$0.778707\pi$
$620$	10080.0			0.652940
$621$		0			0
$622$	−2808.00			−0.181014
$623$	3960.00	+	6858.92i	0.254661	+	0.441086i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	3659.00	−	6337.57i	0.233615	−	0.404633i
$627$		0			0
$628$	−12299.0	−	21302.5i	−0.781502	−	1.35360i
$629$	476.000			0.0301739
$630$		0			0
$631$	21352.0			1.34708			0.673542	−	0.739149i	$-0.264772\pi$
$631$	21352.0			1.34708			0.673542	+	0.739149i	$0.264772\pi$
$632$	−1800.00	−	3117.69i	−0.113291	−	0.196226i
$633$		0			0
$634$	−1123.00	+	1945.09i	−0.0703470	+	0.121845i
$635$	−4840.00	+	8383.13i	−0.302472	+	0.523896i
$636$		0			0
$637$	−2563.00	−	4439.25i	−0.159419	−	0.276121i
$638$	11960.0			0.742164
$639$		0			0
$640$	−7275.00			−0.449328
$641$	14559.0	+	25216.9i	0.897108	+	1.55384i	0.831174	+	0.556012i	$0.187669\pi$
$641$	14559.0	+	25216.9i	0.897108	+	1.55384i	0.0659331	+	0.997824i	$0.478998\pi$
$642$		0			0
$643$	−2886.00	+	4998.70i	−0.177003	+	0.306578i	−0.940853	−	0.338816i	$-0.889973\pi$
$643$	−2886.00	+	4998.70i	−0.177003	+	0.306578i	0.763850	+	0.645394i	$0.223307\pi$
$644$	14112.0	−	24442.7i	0.863495	−	1.49562i
$645$		0			0
$646$	−140.000	−	242.487i	−0.00852667	−	0.0147686i
$647$	−14264.0			−0.866732			−0.433366	−	0.901218i	$-0.642674\pi$
$647$	−14264.0			−0.866732			−0.433366	+	0.901218i	$0.642674\pi$
$648$		0			0
$649$	5200.00			0.314511
$650$	−275.000	−	476.314i	−0.0165944	−	0.0287424i
$651$		0			0
$652$	−7238.00	+	12536.6i	−0.434758	+	0.753022i
$653$	−3451.00	+	5977.31i	−0.206812	+	0.358208i	−0.950708	−	0.310086i	$-0.899642\pi$
$653$	−3451.00	+	5977.31i	−0.206812	+	0.358208i	0.743897	+	0.668295i	$0.232975\pi$
$654$		0			0
$655$	−1830.00	−	3169.65i	−0.109166	−	0.189082i
$656$	5002.00			0.297706
$657$		0			0
$658$	−6144.00			−0.364009
$659$	−10070.0	−	17441.8i	−0.595253	−	1.03101i	−0.993511	−	0.113735i	$-0.963719\pi$
$659$	−10070.0	−	17441.8i	−0.595253	−	1.03101i	0.398258	−	0.917273i	$-0.369615\pi$
$660$		0			0
$661$	1609.00	−	2786.87i	0.0946790	−	0.163989i	−0.814796	−	0.579748i	$-0.803151\pi$
$661$	1609.00	−	2786.87i	0.0946790	−	0.163989i	0.909475	+	0.415759i	$0.136484\pi$
$662$	−666.000	+	1153.55i	−0.0391009	+	0.0677248i
$663$		0			0
$664$	9090.00	+	15744.3i	0.531266	+	0.920179i
$665$	2400.00			0.139952
$666$		0			0
$667$	−38640.0			−2.24310
$668$	−84.0000	−	145.492i	−0.00486536	−	0.00842704i
$669$		0			0
$670$	210.000	−	363.731i	0.0121090	−	0.0209733i
$671$	−19292.0	+	33414.7i	−1.10992	+	1.92245i
$672$		0			0
$673$	3759.00	+	6510.78i	0.215303	+	0.372915i	0.953366	−	0.301816i	$-0.0975928\pi$
$673$	3759.00	+	6510.78i	0.215303	+	0.372915i	−0.738063	+	0.674731i	$0.764259\pi$
$674$	−11534.0			−0.659159
$675$		0			0
$676$	11991.0			0.682237
$677$	9057.00	+	15687.2i	0.514164	+	0.890558i	0.999865	+	0.0164327i	$0.00523093\pi$
$677$	9057.00	+	15687.2i	0.514164	+	0.890558i	−0.485701	+	0.874125i	$0.661436\pi$
$678$		0			0
$679$	10392.0	−	17999.5i	0.587347	−	1.01731i
$680$	−525.000	+	909.327i	−0.0296071	+	0.0512810i
$681$		0			0
$682$	7488.00	+	12969.6i	0.420426	+	0.728199i
$683$	−23868.0			−1.33716			−0.668582	−	0.743638i	$-0.733099\pi$
$683$	−23868.0			−1.33716			−0.668582	+	0.743638i	$0.733099\pi$
$684$		0			0
$685$	−11070.0			−0.617464
$686$	−1320.00	−	2286.31i	−0.0734662	−	0.127247i
$687$		0			0
$688$	3854.00	−	6675.32i	0.213564	−	0.369905i
$689$	3718.00	−	6439.76i	0.205580	−	0.356075i
$690$		0			0
$691$	−86.0000	−	148.956i	−0.00473458	−	0.00820053i	0.863648	−	0.504095i	$-0.168174\pi$
$691$	−86.0000	−	148.956i	−0.00473458	−	0.00820053i	−0.868383	+	0.495894i	$0.834840\pi$
$692$	4326.00			0.237644
$693$		0			0
$694$	11956.0			0.653953
$695$	−50.0000	−	86.6025i	−0.00272893	−	0.00472665i
$696$		0			0
$697$	854.000	−	1479.17i	0.0464097	−	0.0803839i
$698$	−2435.00	+	4217.54i	−0.132043	+	0.228705i
$699$		0			0
$700$	−2100.00	−	3637.31i	−0.113389	−	0.196396i
$701$	−22138.0			−1.19278			−0.596391	−	0.802694i	$-0.703399\pi$
$701$	−22138.0			−1.19278			−0.596391	+	0.802694i	$0.703399\pi$
$702$		0			0
$703$	680.000			0.0364818
$704$	4342.00	+	7520.56i	0.232451	+	0.402616i
$705$		0			0
$706$	−5361.00	+	9285.52i	−0.285785	+	0.494993i
$707$	−14616.0	+	25315.7i	−0.777498	+	1.34667i
$708$		0			0
$709$	−1535.00	−	2658.70i	−0.0813091	−	0.140831i	0.822503	−	0.568760i	$-0.192577\pi$
$709$	−1535.00	−	2658.70i	−0.0813091	−	0.140831i	−0.903812	+	0.427929i	$0.859243\pi$
$710$	−1640.00			−0.0866875
$711$		0			0
$712$	−4950.00			−0.260546
$713$	−24192.0	−	41901.8i	−1.27068	−	2.20089i
$714$		0			0
$715$	−2860.00	+	4953.67i	−0.149592	+	0.259100i
$716$	11690.0	−	20247.7i	0.610162	−	1.05683i
$717$		0			0
$718$	−60.0000	−	103.923i	−0.00311864	−	0.00540163i
$719$	15600.0			0.809154			0.404577	−	0.914504i	$-0.367419\pi$
$719$	15600.0			0.809154			0.404577	+	0.914504i	$0.367419\pi$
$720$		0			0
$721$	2112.00			0.109092
$722$	3229.50	+	5593.66i	0.166468	+	0.288330i
$723$		0			0
$724$	−623.000	+	1079.07i	−0.0319801	+	0.0553912i
$725$	−2875.00	+	4979.65i	−0.147276	+	0.255089i
$726$		0			0
$727$	−10348.0	−	17923.3i	−0.527904	−	0.914356i	−0.999471	−	0.0325260i	$-0.989645\pi$
$727$	−10348.0	−	17923.3i	−0.527904	−	0.914356i	0.471567	−	0.881830i	$-0.343689\pi$
$728$	7920.00			0.403207
$729$		0			0
$730$	−190.000			−0.00963317
$731$	−1316.00	−	2279.38i	−0.0665855	−	0.115330i
$732$		0			0
$733$	15389.0	−	26654.5i	0.775451	−	1.34312i	−0.159089	−	0.987264i	$-0.550856\pi$
$733$	15389.0	−	26654.5i	0.775451	−	1.34312i	0.934541	−	0.355857i	$-0.115811\pi$
$734$	−1968.00	+	3408.68i	−0.0989649	+	0.171412i
$735$		0			0
$736$	13524.0	+	23424.3i	0.677311	+	1.17314i
$737$	−4368.00			−0.218314
$738$		0			0
$739$	11740.0			0.584388			0.292194	−	0.956359i	$-0.405615\pi$
$739$	11740.0			0.584388			0.292194	+	0.956359i	$0.405615\pi$
$740$	−595.000	−	1030.57i	−0.0295576	−	0.0511953i
$741$		0			0
$742$	−4056.00	+	7025.20i	−0.200674	+	0.347578i
$743$	−1316.00	+	2279.38i	−0.0649789	+	0.112547i	−0.896685	−	0.442670i	$-0.854031\pi$
$743$	−1316.00	+	2279.38i	−0.0649789	+	0.112547i	0.831706	+	0.555217i	$0.187365\pi$
$744$		0			0
$745$	3325.00	+	5759.07i	0.163515	+	0.283216i
$746$	3022.00			0.148315
$747$		0			0
$748$	5096.00			0.249102
$749$	432.000	+	748.246i	0.0210747	+	0.0365024i
$750$		0			0
$751$	10264.0	−	17777.8i	0.498720	−	0.863808i	−0.501279	−	0.865286i	$-0.667137\pi$
$751$	10264.0	−	17777.8i	0.498720	−	0.863808i	0.999999	+	0.00147745i	$0.000470287\pi$
$752$	−5248.00	+	9089.80i	−0.254488	+	0.440786i
$753$		0			0
$754$	−2530.00	−	4382.09i	−0.122198	−	0.211653i
$755$	−6040.00			−0.291150
$756$		0			0
$757$	21646.0			1.03928			0.519642	−	0.854384i	$-0.326066\pi$
$757$	21646.0			1.03928			0.519642	+	0.854384i	$0.326066\pi$
$758$	6670.00	+	11552.8i	0.319611	+	0.553583i
$759$		0			0
$760$	−750.000	+	1299.04i	−0.0357965	+	0.0620014i
$761$	−9141.00	+	15832.7i	−0.435428	+	0.754184i	−0.997331	−	0.0730197i	$-0.976736\pi$
$761$	−9141.00	+	15832.7i	−0.435428	+	0.754184i	0.561902	+	0.827204i	$0.310070\pi$
$762$		0			0
$763$	−11640.0	−	20161.1i	−0.552289	−	0.956592i
$764$	13216.0			0.625835
$765$		0			0
$766$	−1008.00			−0.0475464
$767$	−1100.00	−	1905.26i	−0.0517845	−	0.0896934i
$768$		0			0
$769$	12095.0	−	20949.2i	0.567174	−	0.982374i	−0.429670	−	0.902986i	$-0.641370\pi$
$769$	12095.0	−	20949.2i	0.567174	−	0.982374i	0.996844	−	0.0793882i	$-0.0252967\pi$
$770$	3120.00	−	5404.00i	0.146022	−	0.252918i
$771$		0			0
$772$	6727.00	+	11651.5i	0.313614	+	0.543195i
$773$	−25698.0			−1.19572			−0.597861	−	0.801600i	$-0.703982\pi$
$773$	−25698.0			−1.19572			−0.597861	+	0.801600i	$0.703982\pi$
$774$		0			0
$775$	−7200.00			−0.333718
$776$	6495.00	+	11249.7i	0.300460	+	0.520412i
$777$		0			0
$778$	−4815.00	+	8339.82i	−0.221884	+	0.384315i
$779$	1220.00	−	2113.10i	0.0561117	−	0.0971884i
$780$		0			0
$781$	8528.00	+	14770.9i	0.390724	+	0.676755i
$782$	2352.00			0.107554
$783$		0			0
$784$	9553.00			0.435177
$785$	8785.00	+	15216.1i	0.399427	+	0.691828i
$786$		0			0
$787$	−16718.0	+	28956.4i	−0.757220	+	1.31154i	0.187043	+	0.982352i	$0.440110\pi$
$787$	−16718.0	+	28956.4i	−0.757220	+	1.31154i	−0.944263	+	0.329192i	$0.893224\pi$
$788$	8841.00	−	15313.1i	0.399680	−	0.692266i
$789$		0			0
$790$	600.000	+	1039.23i	0.0270216	+	0.0468027i
$791$	−25008.0			−1.12412
$792$		0			0
$793$	16324.0			0.730999
$794$	−3563.00	−	6171.30i	−0.159252	−	0.275833i
$795$		0			0
$796$	−4060.00	+	7032.13i	−0.180783	+	0.313125i
$797$	18797.0	−	32557.4i	0.835413	−	1.44698i	−0.0582813	−	0.998300i	$-0.518562\pi$
$797$	18797.0	−	32557.4i	0.835413	−	1.44698i	0.893694	−	0.448677i	$-0.148105\pi$
$798$		0			0
$799$	1792.00	+	3103.84i	0.0793447	+	0.137429i
$800$	4025.00			0.177882
$801$		0			0
$802$	−8718.00			−0.383844
$803$	988.000	+	1711.27i	0.0434194	+	0.0752046i
$804$		0			0
$805$	−10080.0	+	17459.1i	−0.441333	+	0.764412i
$806$	3168.00	−	5487.14i	0.138447	−	0.239797i
$807$		0			0
$808$	−9135.00	−	15822.3i	−0.397733	−	0.688894i
$809$	4730.00			0.205560			0.102780	−	0.994704i	$-0.467226\pi$
$809$	4730.00			0.205560			0.102780	+	0.994704i	$0.467226\pi$
$810$		0			0
$811$	−8748.00			−0.378772			−0.189386	−	0.981903i	$-0.560650\pi$
$811$	−8748.00			−0.378772			−0.189386	+	0.981903i	$0.560650\pi$
$812$	−19320.0	−	33463.2i	−0.834974	−	1.44622i
$813$		0			0
$814$	884.000	−	1531.13i	0.0380641	−	0.0659290i
$815$	5170.00	−	8954.70i	0.222205	−	0.384871i
$816$		0			0
$817$	−1880.00	−	3256.26i	−0.0805054	−	0.139439i
$818$	−10870.0			−0.464622
$819$		0			0
$820$	−4270.00			−0.181847
$821$	−22071.0	−	38228.1i	−0.938226	−	1.62505i	−0.768779	−	0.639515i	$-0.779135\pi$
$821$	−22071.0	−	38228.1i	−0.938226	−	1.62505i	−0.169447	−	0.985539i	$-0.554198\pi$
$822$		0			0
$823$	−1996.00	+	3457.17i	−0.0845397	+	0.146427i	−0.905195	−	0.424997i	$-0.860275\pi$
$823$	−1996.00	+	3457.17i	−0.0845397	+	0.146427i	0.820655	+	0.571424i	$0.193609\pi$
$824$	−660.000	+	1143.15i	−0.0279031	+	0.0483297i
$825$		0			0
$826$	1200.00	+	2078.46i	0.0505488	+	0.0875532i
$827$	−14444.0			−0.607336			−0.303668	−	0.952778i	$-0.598211\pi$
$827$	−14444.0			−0.607336			−0.303668	+	0.952778i	$0.598211\pi$
$828$		0			0
$829$	42150.0			1.76590			0.882949	−	0.469468i	$-0.155554\pi$
$829$	42150.0			1.76590			0.882949	+	0.469468i	$0.155554\pi$
$830$	−3030.00	−	5248.11i	−0.126714	−	0.219476i
$831$		0			0
$832$	1837.00	−	3181.78i	0.0765463	−	0.132582i
$833$	1631.00	−	2824.97i	0.0678401	−	0.117502i
$834$		0			0
$835$	60.0000	+	103.923i	0.00248669	+	0.00430707i
$836$	7280.00			0.301177
$837$		0			0
$838$	−9700.00			−0.399858
$839$	−6700.00	−	11604.7i	−0.275697	−	0.477521i	0.694614	−	0.719383i	$-0.255575\pi$
$839$	−6700.00	−	11604.7i	−0.275697	−	0.477521i	−0.970311	+	0.241862i	$0.922242\pi$
$840$		0			0
$841$	−14255.5	+	24691.3i	−0.584505	+	1.01239i
$842$	−431.000	+	746.514i	−0.0176404	+	0.0305541i
$843$		0			0
$844$	−15638.0	−	27085.8i	−0.637775	−	1.10466i
$845$	−8565.00			−0.348692
$846$		0			0
$847$	−32952.0			−1.33677
$848$	6929.00	+	12001.4i	0.280593	+	0.486001i
$849$		0			0
$850$	175.000	−	303.109i	0.00706171	−	0.0122312i
$851$	−2856.00	+	4946.74i	−0.115044	+	0.199262i
$852$		0			0
$853$	4329.00	+	7498.05i	0.173766	+	0.300971i	0.939733	−	0.341908i	$-0.111073\pi$
$853$	4329.00	+	7498.05i	0.173766	+	0.300971i	−0.765968	+	0.642879i	$0.777740\pi$
$854$	−17808.0			−0.713556
$855$		0			0
$856$	−540.000			−0.0215617
$857$	−21413.0	−	37088.4i	−0.853505	−	1.47831i	−0.878025	−	0.478615i	$-0.841139\pi$
$857$	−21413.0	−	37088.4i	−0.853505	−	1.47831i	0.0245192	−	0.999699i	$-0.492195\pi$
$858$		0			0
$859$	17950.0	−	31090.3i	0.712976	−	1.23491i	−0.250759	−	0.968049i	$-0.580680\pi$
$859$	17950.0	−	31090.3i	0.712976	−	1.23491i	0.963735	−	0.266861i	$-0.0859863\pi$
$860$	−3290.00	+	5698.45i	−0.130451	+	0.225948i
$861$		0			0
$862$	−7896.00	−	13676.3i	−0.311994	−	0.540389i
$863$	−3088.00			−0.121804			−0.0609019	−	0.998144i	$-0.519398\pi$
$863$	−3088.00			−0.121804			−0.0609019	+	0.998144i	$0.519398\pi$
$864$		0			0
$865$	−3090.00			−0.121460
$866$	−5801.00	−	10047.6i	−0.227628	−	0.394264i
$867$		0			0
$868$	24192.0	−	41901.8i	0.946002	−	1.63852i
$869$	6240.00	−	10808.0i	0.243587	−	0.421906i
$870$		0			0
$871$	924.000	+	1600.41i	0.0359455	+	0.0622595i
$872$	14550.0			0.565052
$873$		0			0
$874$	3360.00			0.130039
$875$	1500.00	+	2598.08i	0.0579534	+	0.100378i
$876$		0			0
$877$	17637.0	−	30548.2i	0.679087	−	1.17621i	−0.296169	−	0.955135i	$-0.595709\pi$
$877$	17637.0	−	30548.2i	0.679087	−	1.17621i	0.975256	−	0.221078i	$-0.0709574\pi$
$878$	220.000	−	381.051i	0.00845631	−	0.0146468i
$879$		0			0
$880$	−5330.00	−	9231.83i	−0.204175	−	0.353642i
$881$	25042.0			0.957646			0.478823	−	0.877911i	$-0.341064\pi$
$881$	25042.0			0.957646			0.478823	+	0.877911i	$0.341064\pi$
$882$		0			0
$883$	12572.0			0.479141			0.239570	−	0.970879i	$-0.422993\pi$
$883$	12572.0			0.479141			0.239570	+	0.970879i	$0.422993\pi$
$884$	−1078.00	−	1867.15i	−0.0410148	−	0.0710397i
$885$		0			0
$886$	5094.00	−	8823.07i	0.193156	−	0.334556i
$887$	10932.0	−	18934.8i	0.413823	−	0.716762i	−0.581481	−	0.813560i	$-0.697527\pi$
$887$	10932.0	−	18934.8i	0.413823	−	0.716762i	0.995304	+	0.0967979i	$0.0308600\pi$
$888$		0			0
$889$	23232.0	+	40239.0i	0.876464	+	1.51808i
$890$	1650.00			0.0621440
$891$		0			0
$892$	−42224.0			−1.58494
$893$	2560.00	+	4434.05i	0.0959318	+	0.166159i
$894$		0			0
$895$	−8350.00	+	14462.6i	−0.311854	+	0.540148i
$896$	−17460.0	+	30241.6i	−0.651002	+	1.12757i
$897$		0			0
$898$	6655.00	+	11526.8i	0.247305	+	0.428345i
$899$	−66240.0			−2.45743
$900$		0			0
$901$	4732.00			0.174968
$902$	−3172.00	−	5494.07i	−0.117091	−	0.202807i
$903$		0			0
$904$	7815.00	−	13536.0i	0.287525	−	0.498009i
$905$	445.000	−	770.763i	0.0163451	−	0.0283105i
$906$		0			0
$907$	−15618.0	−	27051.2i	−0.571761	−	0.990319i	−0.996385	−	0.0849494i	$-0.972927\pi$
$907$	−15618.0	−	27051.2i	−0.571761	−	0.990319i	0.424624	−	0.905370i	$-0.360406\pi$
$908$	−18452.0			−0.674396
$909$		0			0
$910$	−2640.00			−0.0961705
$911$	−4136.00	−	7163.76i	−0.150419	−	0.260534i	0.780962	−	0.624578i	$-0.214729\pi$
$911$	−4136.00	−	7163.76i	−0.150419	−	0.260534i	−0.931382	+	0.364044i	$0.881396\pi$
$912$		0			0
$913$	−31512.0	+	54580.4i	−1.14227	+	1.97847i
$914$	−1613.00	+	2793.80i	−0.0583734	+	0.101106i
$915$		0			0
$916$	16905.0	+	29280.3i	0.609778	+	1.05617i
$917$	−17568.0			−0.632657
$918$		0			0
$919$	20200.0			0.725067			0.362533	−	0.931971i	$-0.381912\pi$
$919$	20200.0			0.725067			0.362533	+	0.931971i	$0.381912\pi$
$920$	−6300.00	−	10911.9i	−0.225766	−	0.391038i
$921$		0			0
$922$	−3291.00	+	5700.18i	−0.117552	+	0.203607i
$923$	3608.00	−	6249.24i	0.128666	−	0.222856i
$924$		0			0
$925$	425.000	+	736.122i	0.0151069	+	0.0261660i
$926$	15072.0			0.534878
$927$		0			0
$928$	37030.0			1.30988
$929$	−15505.0	−	26855.4i	−0.547581	−	0.948438i	−0.998440	−	0.0558425i	$-0.982216\pi$
$929$	−15505.0	−	26855.4i	−0.547581	−	0.948438i	0.450859	−	0.892595i	$-0.351118\pi$
$930$		0			0
$931$	2330.00	−	4035.68i	0.0820222	−	0.142067i
$932$	9387.00	−	16258.8i	0.329916	−	0.571431i
$933$		0			0
$934$	−238.000	−	412.228i	−0.00833790	−	0.0144417i
$935$	−3640.00			−0.127316
$936$		0			0
$937$	−39174.0			−1.36580			−0.682902	−	0.730510i	$-0.739283\pi$
$937$	−39174.0			−1.36580			−0.682902	+	0.730510i	$0.739283\pi$
$938$	−1008.00	−	1745.91i	−0.0350878	−	0.0607739i
$939$		0			0
$940$	4480.00	−	7759.59i	0.155448	−	0.269245i
$941$	2069.00	−	3583.61i	0.0716764	−	0.124147i	−0.827960	−	0.560788i	$-0.810498\pi$
$941$	2069.00	−	3583.61i	0.0716764	−	0.124147i	0.899636	+	0.436640i	$0.143832\pi$
$942$		0			0
$943$	10248.0	+	17750.1i	0.353893	+	0.612960i
$944$	4100.00			0.141360
$945$		0			0
$946$	−9776.00			−0.335989
$947$	−11838.0	−	20504.0i	−0.406213	−	0.703581i	0.588249	−	0.808680i	$-0.299817\pi$
$947$	−11838.0	−	20504.0i	−0.406213	−	0.703581i	−0.994462	+	0.105099i	$0.966484\pi$
$948$		0			0
$949$	418.000	−	723.997i	0.0142981	−	0.0247650i
$950$	250.000	−	433.013i	0.00853797	−	0.0147882i
$951$		0			0
$952$	2520.00	+	4364.77i	0.0857917	+	0.148596i
$953$	18922.0			0.643173			0.321586	−	0.946880i	$-0.395784\pi$
$953$	18922.0			0.643173			0.321586	+	0.946880i	$0.395784\pi$
$954$		0			0
$955$	−9440.00			−0.319865
$956$	8120.00	+	14064.3i	0.274707	+	0.475806i
$957$		0			0
$958$	9840.00	−	17043.4i	0.331854	−	0.574788i
$959$	−26568.0	+	46017.1i	−0.894604	+	1.54950i
$960$		0			0
$961$	−26576.5	−	46031.8i	−0.892098	−	1.54516i
$962$	−748.000			−0.0250691
$963$		0			0
$964$	−14014.0			−0.468216
$965$	−4805.00	−	8322.50i	−0.160289	−	0.277628i
$966$		0			0
$967$	−19828.0	+	34343.1i	−0.659385	+	1.14209i	0.321390	+	0.946947i	$0.395850\pi$
$967$	−19828.0	+	34343.1i	−0.659385	+	1.14209i	−0.980775	+	0.195142i	$0.937483\pi$
$968$	10297.5	−	17835.8i	0.341915	−	0.592215i
$969$		0			0
$970$	−2165.00	−	3749.89i	−0.0716639	−	0.124125i
$971$	−33228.0			−1.09818			−0.549092	−	0.835762i	$-0.685026\pi$
$971$	−33228.0			−1.09818			−0.549092	+	0.835762i	$0.685026\pi$
$972$		0			0
$973$	−480.000			−0.0158151
$974$	2972.00	+	5147.66i	0.0977711	+	0.169344i
$975$		0			0
$976$	−15211.0	+	26346.2i	−0.498865	+	0.864060i
$977$	487.000	−	843.509i	0.0159473	−	0.0276215i	−0.857942	−	0.513747i	$-0.828257\pi$
$977$	487.000	−	843.509i	0.0159473	−	0.0276215i	0.873889	+	0.486126i	$0.161590\pi$
$978$		0			0
$979$	−8580.00	−	14861.0i	−0.280100	−	0.485148i
$980$	−8155.00			−0.265818
$981$		0			0
$982$	10772.0			0.350049
$983$	6804.00	+	11784.9i	0.220767	+	0.382380i	0.955041	−	0.296474i	$-0.0958107\pi$
$983$	6804.00	+	11784.9i	0.220767	+	0.382380i	−0.734274	+	0.678853i	$0.762477\pi$
$984$		0			0
$985$	−6315.00	+	10937.9i	−0.204277	+	0.353818i
$986$	1610.00	−	2788.60i	0.0520009	−	0.0900681i
$987$		0			0
$988$	−1540.00	−	2667.36i	−0.0495890	−	0.0858907i
$989$	31584.0			1.01548
$990$		0			0
$991$	13472.0			0.431839			0.215919	−	0.976411i	$-0.430725\pi$
$991$	13472.0			0.431839			0.215919	+	0.976411i	$0.430725\pi$
$992$	23184.0	+	40155.9i	0.742029	+	1.28523i
$993$		0			0
$994$	−3936.00	+	6817.35i	−0.125596	+	0.217539i
$995$	2900.00	−	5022.95i	0.0923982	−	0.160038i
$996$		0			0
$997$	1617.00	+	2800.73i	0.0513650	+	0.0889668i	0.890565	−	0.454857i	$-0.150309\pi$
$997$	1617.00	+	2800.73i	0.0513650	+	0.0889668i	−0.839200	+	0.543823i	$0.816976\pi$
$998$	8140.00			0.258184
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.e.g.271.1		2
3.2	odd	2		405.4.e.i.271.1		2
9.2	odd	6		405.4.e.i.136.1		2
9.4	even	3		15.4.a.a.1.1	✓	1
9.5	odd	6		45.4.a.c.1.1		1
9.7	even	3	inner	405.4.e.g.136.1		2
36.23	even	6		720.4.a.n.1.1		1
36.31	odd	6		240.4.a.e.1.1		1
45.4	even	6		75.4.a.b.1.1		1
45.13	odd	12		75.4.b.b.49.1		2
45.14	odd	6		225.4.a.f.1.1		1
45.22	odd	12		75.4.b.b.49.2		2
45.23	even	12		225.4.b.e.199.2		2
45.32	even	12		225.4.b.e.199.1		2
63.13	odd	6		735.4.a.e.1.1		1
63.41	even	6		2205.4.a.l.1.1		1
72.13	even	6		960.4.a.b.1.1		1
72.67	odd	6		960.4.a.ba.1.1		1
99.76	odd	6		1815.4.a.e.1.1		1
180.67	even	12		1200.4.f.b.49.2		2
180.103	even	12		1200.4.f.b.49.1		2
180.139	odd	6		1200.4.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.a.1.1	✓	1	9.4	even	3
45.4.a.c.1.1		1	9.5	odd	6
75.4.a.b.1.1		1	45.4	even	6
75.4.b.b.49.1		2	45.13	odd	12
75.4.b.b.49.2		2	45.22	odd	12
225.4.a.f.1.1		1	45.14	odd	6
225.4.b.e.199.1		2	45.32	even	12
225.4.b.e.199.2		2	45.23	even	12
240.4.a.e.1.1		1	36.31	odd	6
405.4.e.g.136.1		2	9.7	even	3	inner
405.4.e.g.271.1		2	1.1	even	1	trivial
405.4.e.i.136.1		2	9.2	odd	6
405.4.e.i.271.1		2	3.2	odd	2
720.4.a.n.1.1		1	36.23	even	6
735.4.a.e.1.1		1	63.13	odd	6
960.4.a.b.1.1		1	72.13	even	6
960.4.a.ba.1.1		1	72.67	odd	6
1200.4.a.t.1.1		1	180.139	odd	6
1200.4.f.b.49.1		2	180.103	even	12
1200.4.f.b.49.2		2	180.67	even	12
1815.4.a.e.1.1		1	99.76	odd	6
2205.4.a.l.1.1		1	63.41	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(3.50000 - 6.06218i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(12.0000 + 20.7846i) q^{7} -15.0000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(3.50000 - 6.06218i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(12.0000 + 20.7846i) q^{7} -15.0000 q^{8} +5.00000 q^{10} +(-26.0000 - 45.0333i) q^{11} +(-11.0000 + 19.0526i) q^{13} +(12.0000 - 20.7846i) q^{14} +(-20.5000 - 35.5070i) q^{16} -14.0000 q^{17} -20.0000 q^{19} +(17.5000 + 30.3109i) q^{20} +(-26.0000 + 45.0333i) q^{22} +(84.0000 - 145.492i) q^{23} +(-12.5000 - 21.6506i) q^{25} +22.0000 q^{26} +168.000 q^{28} +(-115.000 - 199.186i) q^{29} +(144.000 - 249.415i) q^{31} +(-80.5000 + 139.430i) q^{32} +(7.00000 + 12.1244i) q^{34} -120.000 q^{35} -34.0000 q^{37} +(10.0000 + 17.3205i) q^{38} +(37.5000 - 64.9519i) q^{40} +(-61.0000 + 105.655i) q^{41} +(94.0000 + 162.813i) q^{43} -364.000 q^{44} -168.000 q^{46} +(-128.000 - 221.703i) q^{47} +(-116.500 + 201.784i) q^{49} +(-12.5000 + 21.6506i) q^{50} +(77.0000 + 133.368i) q^{52} -338.000 q^{53} +260.000 q^{55} +(-180.000 - 311.769i) q^{56} +(-115.000 + 199.186i) q^{58} +(-50.0000 + 86.6025i) q^{59} +(-371.000 - 642.591i) q^{61} -288.000 q^{62} -167.000 q^{64} +(-55.0000 - 95.2628i) q^{65} +(42.0000 - 72.7461i) q^{67} +(-49.0000 + 84.8705i) q^{68} +(60.0000 + 103.923i) q^{70} -328.000 q^{71} -38.0000 q^{73} +(17.0000 + 29.4449i) q^{74} +(-70.0000 + 121.244i) q^{76} +(624.000 - 1080.80i) q^{77} +(120.000 + 207.846i) q^{79} +205.000 q^{80} +122.000 q^{82} +(-606.000 - 1049.62i) q^{83} +(35.0000 - 60.6218i) q^{85} +(94.0000 - 162.813i) q^{86} +(390.000 + 675.500i) q^{88} +330.000 q^{89} -528.000 q^{91} +(-588.000 - 1018.45i) q^{92} +(-128.000 + 221.703i) q^{94} +(50.0000 - 86.6025i) q^{95} +(-433.000 - 749.978i) q^{97} +233.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 7 q^{4} - 5 q^{5} + 24 q^{7} - 30 q^{8}+O(q^{10}) \) 2 * q - q^2 + 7 * q^4 - 5 * q^5 + 24 * q^7 - 30 * q^8	\( 2 q - q^{2} + 7 q^{4} - 5 q^{5} + 24 q^{7} - 30 q^{8} + 10 q^{10} - 52 q^{11} - 22 q^{13} + 24 q^{14} - 41 q^{16} - 28 q^{17} - 40 q^{19} + 35 q^{20} - 52 q^{22} + 168 q^{23} - 25 q^{25} + 44 q^{26} + 336 q^{28} - 230 q^{29} + 288 q^{31} - 161 q^{32} + 14 q^{34} - 240 q^{35} - 68 q^{37} + 20 q^{38} + 75 q^{40} - 122 q^{41} + 188 q^{43} - 728 q^{44} - 336 q^{46} - 256 q^{47} - 233 q^{49} - 25 q^{50} + 154 q^{52} - 676 q^{53} + 520 q^{55} - 360 q^{56} - 230 q^{58} - 100 q^{59} - 742 q^{61} - 576 q^{62} - 334 q^{64} - 110 q^{65} + 84 q^{67} - 98 q^{68} + 120 q^{70} - 656 q^{71} - 76 q^{73} + 34 q^{74} - 140 q^{76} + 1248 q^{77} + 240 q^{79} + 410 q^{80} + 244 q^{82} - 1212 q^{83} + 70 q^{85} + 188 q^{86} + 780 q^{88} + 660 q^{89} - 1056 q^{91} - 1176 q^{92} - 256 q^{94} + 100 q^{95} - 866 q^{97} + 466 q^{98}+O(q^{100}) \) 2 * q - q^2 + 7 * q^4 - 5 * q^5 + 24 * q^7 - 30 * q^8 + 10 * q^10 - 52 * q^11 - 22 * q^13 + 24 * q^14 - 41 * q^16 - 28 * q^17 - 40 * q^19 + 35 * q^20 - 52 * q^22 + 168 * q^23 - 25 * q^25 + 44 * q^26 + 336 * q^28 - 230 * q^29 + 288 * q^31 - 161 * q^32 + 14 * q^34 - 240 * q^35 - 68 * q^37 + 20 * q^38 + 75 * q^40 - 122 * q^41 + 188 * q^43 - 728 * q^44 - 336 * q^46 - 256 * q^47 - 233 * q^49 - 25 * q^50 + 154 * q^52 - 676 * q^53 + 520 * q^55 - 360 * q^56 - 230 * q^58 - 100 * q^59 - 742 * q^61 - 576 * q^62 - 334 * q^64 - 110 * q^65 + 84 * q^67 - 98 * q^68 + 120 * q^70 - 656 * q^71 - 76 * q^73 + 34 * q^74 - 140 * q^76 + 1248 * q^77 + 240 * q^79 + 410 * q^80 + 244 * q^82 - 1212 * q^83 + 70 * q^85 + 188 * q^86 + 780 * q^88 + 660 * q^89 - 1056 * q^91 - 1176 * q^92 - 256 * q^94 + 100 * q^95 - 866 * q^97 + 466 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more