LMFDB - Embedded newform 531.5.c.d.235.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(235,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.c (of order $2$, degree $1$, 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:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.2
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.39

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.77798i q^{2} -44.4970 q^{4} -28.5836 q^{5} +72.7001 q^{7} +221.650i q^{8} +O(q^{10})$	\(q-7.77798i q^{2} -44.4970 q^{4} -28.5836 q^{5} +72.7001 q^{7} +221.650i q^{8} +222.323i q^{10} -9.33885i q^{11} +106.784i q^{13} -565.461i q^{14} +1012.03 q^{16} -337.515 q^{17} +546.806 q^{19} +1271.89 q^{20} -72.6374 q^{22} +39.3036i q^{23} +192.023 q^{25} +830.565 q^{26} -3234.94 q^{28} +444.446 q^{29} -1230.69i q^{31} -4325.19i q^{32} +2625.19i q^{34} -2078.03 q^{35} -1943.27i q^{37} -4253.05i q^{38} -6335.54i q^{40} +835.576 q^{41} +2242.91i q^{43} +415.551i q^{44} +305.703 q^{46} -3914.09i q^{47} +2884.31 q^{49} -1493.55i q^{50} -4751.57i q^{52} -3820.28 q^{53} +266.938i q^{55} +16113.9i q^{56} -3456.89i q^{58} +(3000.50 + 1764.76i) q^{59} +2344.07i q^{61} -9572.29 q^{62} -17448.7 q^{64} -3052.27i q^{65} +3650.67i q^{67} +15018.4 q^{68} +16162.9i q^{70} +2630.49 q^{71} +8924.21i q^{73} -15114.7 q^{74} -24331.3 q^{76} -678.935i q^{77} +1497.53 q^{79} -28927.6 q^{80} -6499.09i q^{82} -3985.30i q^{83} +9647.40 q^{85} +17445.3 q^{86} +2069.95 q^{88} -2531.49i q^{89} +7763.21i q^{91} -1748.89i q^{92} -30443.7 q^{94} -15629.7 q^{95} -9210.22i q^{97} -22434.1i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7	$ 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 3944 * q^16 + 528 * q^17 + 444 * q^19 - 444 * q^20 + 1304 * q^22 + 4880 * q^25 + 1452 * q^26 - 1160 * q^28 + 996 * q^29 - 10320 * q^35 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	7.77798i		−	1.94450i	−0.233953	−	0.972248i	$-0.575166\pi$
$2$		−	7.77798i		−	1.94450i	0.233953	−	0.972248i	$-0.424834\pi$
$3$		0			0
$3$		0			0
$4$	−44.4970			−2.78106
$5$	−28.5836			−1.14334			−0.571672	−	0.820482i	$-0.693705\pi$
$5$	−28.5836			−1.14334			−0.571672	+	0.820482i	$0.693705\pi$
$6$		0			0
$7$	72.7001			1.48368			0.741838	−	0.670579i	$-0.233954\pi$
$7$	72.7001			1.48368			0.741838	+	0.670579i	$0.233954\pi$
$8$			221.650i			3.46327i
$9$		0			0
$10$			222.323i			2.22323i
$11$		−	9.33885i		−	0.0771806i	−0.999255	−	0.0385903i	$-0.987713\pi$
$11$		−	9.33885i		−	0.0771806i	0.999255	−	0.0385903i	$-0.0122867\pi$
$12$		0			0
$13$			106.784i			0.631858i	0.948783	+	0.315929i	$0.102316\pi$
$13$			106.784i			0.631858i	−0.948783	+	0.315929i	$0.897684\pi$
$14$		−	565.461i		−	2.88500i
$15$		0			0
$16$	1012.03			3.95326
$17$	−337.515			−1.16787			−0.583936	−	0.811800i	$-0.698488\pi$
$17$	−337.515			−1.16787			−0.583936	+	0.811800i	$0.698488\pi$
$18$		0			0
$19$	546.806			1.51470			0.757349	−	0.653010i	$-0.226494\pi$
$19$	546.806			1.51470			0.757349	+	0.653010i	$0.226494\pi$
$20$	1271.89			3.17972
$21$		0			0
$22$	−72.6374			−0.150077
$23$			39.3036i			0.0742979i	0.999310	+	0.0371490i	$0.0118276\pi$
$23$			39.3036i			0.0742979i	−0.999310	+	0.0371490i	$0.988172\pi$
$24$		0			0
$25$	192.023			0.307237
$26$	830.565			1.22865
$27$		0			0
$28$	−3234.94			−4.12620
$29$	444.446			0.528473			0.264236	−	0.964458i	$-0.414880\pi$
$29$	444.446			0.528473			0.264236	+	0.964458i	$0.414880\pi$
$30$		0			0
$31$		−	1230.69i		−	1.28063i	−0.768110	−	0.640317i	$-0.778803\pi$
$31$		−	1230.69i		−	1.28063i	0.768110	−	0.640317i	$-0.221197\pi$
$32$		−	4325.19i		−	4.22382i
$33$		0			0
$34$			2625.19i			2.27092i
$35$	−2078.03			−1.69635
$36$		0			0
$37$		−	1943.27i		−	1.41948i	−0.704464	−	0.709740i	$-0.748812\pi$
$37$		−	1943.27i		−	1.41948i	0.704464	−	0.709740i	$-0.251188\pi$
$38$		−	4253.05i		−	2.94532i
$39$		0			0
$40$		−	6335.54i		−	3.95971i
$41$	835.576			0.497071			0.248535	−	0.968623i	$-0.420051\pi$
$41$	835.576			0.497071			0.248535	+	0.968623i	$0.420051\pi$
$42$		0			0
$43$			2242.91i			1.21304i	0.795069	+	0.606519i	$0.207435\pi$
$43$			2242.91i			1.21304i	−0.795069	+	0.606519i	$0.792565\pi$
$44$			415.551i			0.214644i
$45$		0			0
$46$	305.703			0.144472
$47$		−	3914.09i		−	1.77188i	−0.463797	−	0.885942i	$-0.653513\pi$
$47$		−	3914.09i		−	1.77188i	0.463797	−	0.885942i	$-0.346487\pi$
$48$		0			0
$49$	2884.31			1.20130
$50$		−	1493.55i		−	0.597421i
$51$		0			0
$52$		−	4751.57i		−	1.75724i
$53$	−3820.28			−1.36001			−0.680006	−	0.733206i	$-0.738023\pi$
$53$	−3820.28			−1.36001			−0.680006	+	0.733206i	$0.738023\pi$
$54$		0			0
$55$			266.938i			0.0882440i
$56$			16113.9i			5.13838i
$57$		0			0
$58$		−	3456.89i		−	1.02761i
$59$	3000.50	+	1764.76i	0.861964	+	0.506970i
$59$	3000.50	+	1764.76i	0.861964	+	0.506970i
$60$		0			0
$61$			2344.07i			0.629958i	0.949099	+	0.314979i	$0.101997\pi$
$61$			2344.07i			0.629958i	−0.949099	+	0.314979i	$0.898003\pi$
$62$	−9572.29			−2.49019
$63$		0			0
$64$	−17448.7			−4.25994
$65$		−	3052.27i		−	0.722432i
$66$		0			0
$67$			3650.67i			0.813249i	0.913595	+	0.406624i	$0.133294\pi$
$67$			3650.67i			0.813249i	−0.913595	+	0.406624i	$0.866706\pi$
$68$	15018.4			3.24793
$69$		0			0
$70$			16162.9i			3.29855i
$71$	2630.49			0.521819			0.260909	−	0.965363i	$-0.415978\pi$
$71$	2630.49			0.521819			0.260909	+	0.965363i	$0.415978\pi$
$72$		0			0
$73$			8924.21i			1.67465i	0.546705	+	0.837325i	$0.315882\pi$
$73$			8924.21i			1.67465i	−0.546705	+	0.837325i	$0.684118\pi$
$74$	−15114.7			−2.76017
$75$		0			0
$76$	−24331.3			−4.21247
$77$		−	678.935i		−	0.114511i
$78$		0			0
$79$	1497.53			0.239951			0.119976	−	0.992777i	$-0.461718\pi$
$79$	1497.53			0.239951			0.119976	+	0.992777i	$0.461718\pi$
$80$	−28927.6			−4.51993
$81$		0			0
$82$		−	6499.09i		−	0.966552i
$83$		−	3985.30i		−	0.578503i	−0.957253	−	0.289251i	$-0.906594\pi$
$83$		−	3985.30i		−	0.578503i	0.957253	−	0.289251i	$-0.0934063\pi$
$84$		0			0
$85$	9647.40			1.33528
$86$	17445.3			2.35875
$87$		0			0
$88$	2069.95			0.267297
$89$		−	2531.49i		−	0.319593i	−0.987150	−	0.159796i	$-0.948916\pi$
$89$		−	2531.49i		−	0.319593i	0.987150	−	0.159796i	$-0.0510838\pi$
$90$		0			0
$91$			7763.21i			0.937473i
$92$		−	1748.89i		−	0.206627i
$93$		0			0
$94$	−30443.7			−3.44542
$95$	−15629.7			−1.73182
$96$		0			0
$97$		−	9210.22i		−	0.978874i	−0.872039	−	0.489437i	$-0.837202\pi$
$97$		−	9210.22i		−	0.978874i	0.872039	−	0.489437i	$-0.162798\pi$
$98$		−	22434.1i		−	2.33591i
$99$		0			0
$100$	−8544.45			−0.854445
$101$			13290.3i			1.30284i	0.758717	+	0.651420i	$0.225826\pi$
$101$			13290.3i			1.30284i	−0.758717	+	0.651420i	$0.774174\pi$
$102$		0			0
$103$		−	14436.1i		−	1.36074i	−0.732867	−	0.680372i	$-0.761818\pi$
$103$		−	14436.1i		−	1.36074i	0.732867	−	0.680372i	$-0.238182\pi$
$104$	−23668.6			−2.18830
$105$		0			0
$106$			29714.0i			2.64454i
$107$	986.412			0.0861570			0.0430785	−	0.999072i	$-0.486283\pi$
$107$	986.412			0.0861570			0.0430785	+	0.999072i	$0.486283\pi$
$108$		0			0
$109$		−	14934.0i		−	1.25697i	−0.777824	−	0.628483i	$-0.783676\pi$
$109$		−	14934.0i		−	1.25697i	0.777824	−	0.628483i	$-0.216324\pi$
$110$	2076.24			0.171590
$111$		0			0
$112$	73575.0			5.86535
$113$		−	19908.8i		−	1.55915i	−0.626308	−	0.779576i	$-0.715435\pi$
$113$		−	19908.8i		−	1.55915i	0.626308	−	0.779576i	$-0.284565\pi$
$114$		0			0
$115$		−	1123.44i		−	0.0849481i
$116$	−19776.5			−1.46972
$117$		0			0
$118$	13726.3	−	23337.8i	0.985802	−	1.67608i
$119$	−24537.4			−1.73274
$120$		0			0
$121$	14553.8			0.994043
$122$	18232.2			1.22495
$123$		0			0
$124$			54762.1i			3.56153i
$125$	12376.0			0.792067
$126$		0			0
$127$	−15531.0			−0.962922			−0.481461	−	0.876468i	$-0.659894\pi$
$127$	−15531.0			−0.962922			−0.481461	+	0.876468i	$0.659894\pi$
$128$			66512.8i			4.05962i
$129$		0			0
$130$	−23740.5			−1.40477
$131$		−	1246.21i		−	0.0726188i	−0.999341	−	0.0363094i	$-0.988440\pi$
$131$		−	1246.21i		−	0.0726188i	0.999341	−	0.0363094i	$-0.0115602\pi$
$132$		0			0
$133$	39752.9			2.24732
$134$	28394.9			1.58136
$135$		0			0
$136$		−	74810.0i		−	4.04466i
$137$	−16792.4			−0.894689			−0.447344	−	0.894362i	$-0.647630\pi$
$137$	−16792.4			−0.894689			−0.447344	+	0.894362i	$0.647630\pi$
$138$		0			0
$139$	−35922.2			−1.85923			−0.929616	−	0.368530i	$-0.879861\pi$
$139$	−35922.2			−1.85923			−0.929616	+	0.368530i	$0.879861\pi$
$140$	92466.3			4.71767
$141$		0			0
$142$		−	20459.9i		−	1.01467i
$143$	997.240			0.0487672
$144$		0			0
$145$	−12703.9			−0.604227
$146$	69412.4			3.25635
$147$		0			0
$148$			86469.7i			3.94767i
$149$		−	25955.6i		−	1.16912i	−0.811351	−	0.584559i	$-0.801268\pi$
$149$		−	25955.6i		−	1.16912i	0.811351	−	0.584559i	$-0.198732\pi$
$150$		0			0
$151$		−	18565.1i		−	0.814222i	−0.913379	−	0.407111i	$-0.866536\pi$
$151$		−	18565.1i		−	0.814222i	0.913379	−	0.407111i	$-0.133464\pi$
$152$			121199.i			5.24581i
$153$		0			0
$154$	−5280.75			−0.222666
$155$			35177.6i			1.46421i
$156$		0			0
$157$		−	46527.1i		−	1.88759i	−0.330537	−	0.943793i	$-0.607230\pi$
$157$		−	46527.1i		−	1.88759i	0.330537	−	0.943793i	$-0.392770\pi$
$158$		−	11647.8i		−	0.466584i
$159$		0			0
$160$			123630.i			4.82928i
$161$			2857.38i			0.110234i
$162$		0			0
$163$	−437.670			−0.0164730			−0.00823648	−	0.999966i	$-0.502622\pi$
$163$	−437.670			−0.0164730			−0.00823648	+	0.999966i	$0.502622\pi$
$164$	−37180.6			−1.38239
$165$		0			0
$166$	−30997.6			−1.12490
$167$	−49721.0			−1.78282			−0.891408	−	0.453201i	$-0.850282\pi$
$167$	−49721.0			−1.78282			−0.891408	+	0.453201i	$0.850282\pi$
$168$		0			0
$169$	17158.2			0.600755
$170$		−	75037.3i		−	2.59645i
$171$		0			0
$172$		−	99802.8i		−	3.37354i
$173$		−	30766.7i		−	1.02799i	−0.857793	−	0.513995i	$-0.828165\pi$
$173$		−	30766.7i		−	1.02799i	0.857793	−	0.513995i	$-0.171835\pi$
$174$		0			0
$175$	13960.1			0.455840
$176$		−	9451.23i		−	0.305115i
$177$		0			0
$178$	−19689.9			−0.621447
$179$		−	20767.6i		−	0.648156i	−0.946030	−	0.324078i	$-0.894946\pi$
$179$		−	20767.6i		−	0.648156i	0.946030	−	0.324078i	$-0.105054\pi$
$180$		0			0
$181$	27789.6			0.848252			0.424126	−	0.905603i	$-0.360581\pi$
$181$	27789.6			0.848252			0.424126	+	0.905603i	$0.360581\pi$
$182$	60382.2			1.82291
$183$		0			0
$184$	−8711.63			−0.257314
$185$			55545.6i			1.62295i
$186$		0			0
$187$			3152.00i			0.0901370i
$188$			174165.i			4.92772i
$189$		0			0
$190$			121568.i			3.36752i
$191$		−	22222.5i		−	0.609153i	−0.952488	−	0.304576i	$-0.901485\pi$
$191$		−	22222.5i		−	0.609153i	0.952488	−	0.304576i	$-0.0985149\pi$
$192$		0			0
$193$	22533.4			0.604941			0.302470	−	0.953159i	$-0.402189\pi$
$193$	22533.4			0.604941			0.302470	+	0.953159i	$0.402189\pi$
$194$	−71637.0			−1.90342
$195$		0			0
$196$	−128343.			−3.34088
$197$	−11138.9			−0.287018			−0.143509	−	0.989649i	$-0.545839\pi$
$197$	−11138.9			−0.287018			−0.143509	+	0.989649i	$0.545839\pi$
$198$		0			0
$199$	−27923.7			−0.705126			−0.352563	−	0.935788i	$-0.614690\pi$
$199$	−27923.7			−0.705126			−0.352563	+	0.935788i	$0.614690\pi$
$200$			42561.8i			1.06404i
$201$		0			0
$202$	103372.			2.53337
$203$	32311.3			0.784083
$204$		0			0
$205$	−23883.8			−0.568323
$206$	−112284.			−2.64596
$207$		0			0
$208$			108069.i			2.49790i
$209$		−	5106.54i		−	0.116905i
$210$		0			0
$211$		−	4163.53i		−	0.0935182i	−0.998906	−	0.0467591i	$-0.985111\pi$
$211$		−	4163.53i		−	0.0935182i	0.998906	−	0.0467591i	$-0.0148893\pi$
$212$	169991.			3.78228
$213$		0			0
$214$		−	7672.29i		−	0.167532i
$215$		−	64110.4i		−	1.38692i
$216$		0			0
$217$		−	89471.3i		−	1.90005i
$218$	−116156.			−2.44416
$219$		0			0
$220$		−	11878.0i		−	0.245412i
$221$		−	36041.2i		−	0.737930i
$222$		0			0
$223$	−77429.7			−1.55703			−0.778517	−	0.627624i	$-0.784028\pi$
$223$	−77429.7			−1.55703			−0.778517	+	0.627624i	$0.784028\pi$
$224$		−	314442.i		−	6.26678i
$225$		0			0
$226$	−154850.			−3.03176
$227$		−	6994.10i		−	0.135731i	−0.997694	−	0.0678656i	$-0.978381\pi$
$227$		−	6994.10i		−	0.135731i	0.997694	−	0.0678656i	$-0.0216189\pi$
$228$		0			0
$229$		−	51473.3i		−	0.981547i	−0.871287	−	0.490773i	$-0.836714\pi$
$229$		−	51473.3i		−	0.981547i	0.871287	−	0.490773i	$-0.163286\pi$
$230$	−8738.09			−0.165181
$231$		0			0
$232$			98511.2i			1.83025i
$233$			79635.1i			1.46687i	0.679758	+	0.733437i	$0.262085\pi$
$233$			79635.1i			1.46687i	−0.679758	+	0.733437i	$0.737915\pi$
$234$		0			0
$235$			111879.i			2.02587i
$236$	−133513.	−	78526.7i	−2.39718	−	1.40992i
$237$		0			0
$238$			190851.i			3.36931i
$239$	27642.4			0.483927			0.241964	−	0.970285i	$-0.422208\pi$
$239$	27642.4			0.483927			0.241964	+	0.970285i	$0.422208\pi$
$240$		0			0
$241$	11767.3			0.202602			0.101301	−	0.994856i	$-0.467700\pi$
$241$	11767.3			0.202602			0.101301	+	0.994856i	$0.467700\pi$
$242$		−	113199.i		−	1.93291i
$243$		0			0
$244$		−	104304.i		−	1.75195i
$245$	−82444.0			−1.37349
$246$		0			0
$247$			58390.2i			0.957075i
$248$	272782.			4.43519
$249$		0			0
$250$		−	96260.7i		−	1.54017i
$251$	98763.5			1.56765			0.783825	−	0.620982i	$-0.213266\pi$
$251$	98763.5			1.56765			0.783825	+	0.620982i	$0.213266\pi$
$252$		0			0
$253$	367.050			0.00573436
$254$			120800.i			1.87240i
$255$		0			0
$256$	238156.			3.63398
$257$	−36087.2			−0.546370			−0.273185	−	0.961961i	$-0.588077\pi$
$257$	−36087.2			−0.546370			−0.273185	+	0.961961i	$0.588077\pi$
$258$		0			0
$259$		−	141276.i		−	2.10605i
$260$			135817.i			2.00913i
$261$		0			0
$262$	−9693.01			−0.141207
$263$	16587.3			0.239808			0.119904	−	0.992786i	$-0.461741\pi$
$263$	16587.3			0.239808			0.119904	+	0.992786i	$0.461741\pi$
$264$		0			0
$265$	109197.			1.55496
$266$		−	309197.i		−	4.36991i
$267$		0			0
$268$		−	162444.i		−	2.26170i
$269$			47175.2i			0.651943i	0.945380	+	0.325971i	$0.105691\pi$
$269$			47175.2i			0.651943i	−0.945380	+	0.325971i	$0.894309\pi$
$270$		0			0
$271$	37006.4			0.503893			0.251946	−	0.967741i	$-0.418929\pi$
$271$	37006.4			0.503893			0.251946	+	0.967741i	$0.418929\pi$
$272$	−341577.			−4.61690
$273$		0			0
$274$			130611.i			1.73972i
$275$		−	1793.27i		−	0.0237127i
$276$		0			0
$277$	−37499.2			−0.488722			−0.244361	−	0.969684i	$-0.578578\pi$
$277$	−37499.2			−0.488722			−0.244361	+	0.969684i	$0.578578\pi$
$278$			279402.i			3.61527i
$279$		0			0
$280$		−	460595.i		−	5.87494i
$281$	8246.03			0.104432			0.0522158	−	0.998636i	$-0.483372\pi$
$281$	8246.03			0.104432			0.0522158	+	0.998636i	$0.483372\pi$
$282$		0			0
$283$			3380.85i			0.0422137i	0.999777	+	0.0211069i	$0.00671902\pi$
$283$			3380.85i			0.0422137i	−0.999777	+	0.0211069i	$0.993281\pi$
$284$	−117049.			−1.45121
$285$		0			0
$286$		−	7756.52i		−	0.0948276i
$287$	60746.5			0.737492
$288$		0			0
$289$	30395.4			0.363925
$290$			98810.4i			1.17492i
$291$		0			0
$292$		−	397101.i		−	4.65731i
$293$	146875.			1.71085			0.855427	−	0.517923i	$-0.173295\pi$
$293$	146875.			1.71085			0.855427	+	0.517923i	$0.173295\pi$
$294$		0			0
$295$	−85765.0	−	50443.3i	−0.985521	−	0.579642i
$296$	430724.			4.91605
$297$		0			0
$298$	−201882.			−2.27334
$299$	−4197.00			−0.0469458
$300$		0			0
$301$			163060.i			1.79976i
$302$	−144399.			−1.58325
$303$		0			0
$304$	553386.			5.98799
$305$		−	67002.1i		−	0.720259i
$306$		0			0
$307$	92101.3			0.977213			0.488606	−	0.872504i	$-0.337505\pi$
$307$	92101.3			0.977213			0.488606	+	0.872504i	$0.337505\pi$
$308$			30210.6i			0.318462i
$309$		0			0
$310$	273611.			2.84714
$311$	32978.0			0.340960			0.170480	−	0.985361i	$-0.445468\pi$
$311$	32978.0			0.340960			0.170480	+	0.985361i	$0.445468\pi$
$312$		0			0
$313$		−	40714.0i		−	0.415581i	−0.978173	−	0.207790i	$-0.933373\pi$
$313$		−	40714.0i		−	0.415581i	0.978173	−	0.207790i	$-0.0666272\pi$
$314$	−361887.			−3.67040
$315$		0			0
$316$	−66635.9			−0.667320
$317$	−5097.47			−0.0507266			−0.0253633	−	0.999678i	$-0.508074\pi$
$317$	−5097.47			−0.0507266			−0.0253633	+	0.999678i	$0.508074\pi$
$318$		0			0
$319$		−	4150.61i		−	0.0407878i
$320$	498748.			4.87058
$321$		0			0
$322$	22224.6			0.214350
$323$	−184555.			−1.76897
$324$		0			0
$325$			20505.0i			0.194130i
$326$			3404.19i			0.0320316i
$327$		0			0
$328$			185205.i			1.72149i
$329$		−	284555.i		−	2.62890i
$330$		0			0
$331$	−18523.1			−0.169067			−0.0845334	−	0.996421i	$-0.526940\pi$
$331$	−18523.1			−0.169067			−0.0845334	+	0.996421i	$0.526940\pi$
$332$			177334.i			1.60885i
$333$		0			0
$334$			386729.i			3.46668i
$335$		−	104349.i		−	0.929823i
$336$		0			0
$337$		−	137124.i		−	1.20741i	−0.797209	−	0.603704i	$-0.793691\pi$
$337$		−	137124.i		−	1.20741i	0.797209	−	0.603704i	$-0.206309\pi$
$338$		−	133456.i		−	1.16817i
$339$		0			0
$340$	−429281.			−3.71350
$341$	−11493.2			−0.0988401
$342$		0			0
$343$	35136.7			0.298657
$344$	−497140.			−4.20108
$345$		0			0
$346$	−239303.			−1.99892
$347$		−	92388.3i		−	0.767287i	−0.923481	−	0.383643i	$-0.874669\pi$
$347$		−	92388.3i		−	0.767287i	0.923481	−	0.383643i	$-0.125331\pi$
$348$		0			0
$349$		−	192179.i		−	1.57781i	−0.614513	−	0.788907i	$-0.710648\pi$
$349$		−	192179.i		−	1.57781i	0.614513	−	0.788907i	$-0.289352\pi$
$350$		−	108581.i		−	0.886379i
$351$		0			0
$352$	−40392.3			−0.325997
$353$		−	133742.i		−	1.07329i	−0.843808	−	0.536645i	$-0.819692\pi$
$353$		−	133742.i		−	1.07329i	0.843808	−	0.536645i	$-0.180308\pi$
$354$		0			0
$355$	−75188.9			−0.596619
$356$			112644.i			0.888808i
$357$		0			0
$358$	−161530.			−1.26034
$359$	63441.2			0.492246			0.246123	−	0.969239i	$-0.420843\pi$
$359$	63441.2			0.492246			0.246123	+	0.969239i	$0.420843\pi$
$360$		0			0
$361$	168676.			1.29431
$362$		−	216147.i		−	1.64942i
$363$		0			0
$364$		−	345440.i		−	2.60717i
$365$		−	255086.i		−	1.91470i
$366$		0			0
$367$		−	62757.7i		−	0.465946i	−0.972483	−	0.232973i	$-0.925155\pi$
$367$		−	62757.7i		−	0.465946i	0.972483	−	0.232973i	$-0.0748453\pi$
$368$			39776.6i			0.293719i
$369$		0			0
$370$	432033.			3.15583
$371$	−277735.			−2.01782
$372$		0			0
$373$	−102419.			−0.736148			−0.368074	−	0.929797i	$-0.619983\pi$
$373$	−102419.			−0.736148			−0.368074	+	0.929797i	$0.619983\pi$
$374$	24516.2			0.175271
$375$		0			0
$376$	867556.			6.13652
$377$			47459.7i			0.333920i
$378$		0			0
$379$	−115222.			−0.802151			−0.401076	−	0.916045i	$-0.631364\pi$
$379$	−115222.			−0.802151			−0.401076	+	0.916045i	$0.631364\pi$
$380$	695475.			4.81631
$381$		0			0
$382$	−172846.			−1.18450
$383$	−68700.2			−0.468339			−0.234170	−	0.972196i	$-0.575237\pi$
$383$	−68700.2			−0.468339			−0.234170	+	0.972196i	$0.575237\pi$
$384$		0			0
$385$			19406.4i			0.130925i
$386$		−	175265.i		−	1.17631i
$387$		0			0
$388$			409828.i			2.72231i
$389$	−1971.29			−0.0130272			−0.00651360	−	0.999979i	$-0.502073\pi$
$389$	−1971.29			−0.0130272			−0.00651360	+	0.999979i	$0.502073\pi$
$390$		0			0
$391$		−	13265.6i		−	0.0867705i
$392$			639306.i			4.16041i
$393$		0			0
$394$			86638.0i			0.558105i
$395$	−42805.0			−0.274347
$396$		0			0
$397$			42440.9i			0.269280i	0.990895	+	0.134640i	$0.0429877\pi$
$397$			42440.9i			0.269280i	−0.990895	+	0.134640i	$0.957012\pi$
$398$			217190.i			1.37111i
$399$		0			0
$400$	194334.			1.21459
$401$			103710.i			0.644959i	0.946576	+	0.322480i	$0.104516\pi$
$401$			103710.i			0.644959i	−0.946576	+	0.322480i	$0.895484\pi$
$402$		0			0
$403$	131418.			0.809180
$404$		−	591378.i		−	3.62328i
$405$		0			0
$406$		−	251316.i		−	1.52465i
$407$	−18147.9			−0.109556
$408$		0			0
$409$			29250.4i			0.174858i	0.996171	+	0.0874289i	$0.0278651\pi$
$409$			29250.4i			0.174858i	−0.996171	+	0.0874289i	$0.972135\pi$
$410$			185768.i			1.10510i
$411$		0			0
$412$			642365.i			3.78432i
$413$	218136.	+	128299.i	1.27887	+	0.752180i
$414$		0			0
$415$			113914.i			0.661428i
$416$	461861.			2.66885
$417$		0			0
$418$	−39718.6			−0.227322
$419$		−	54737.0i		−	0.311784i	−0.987774	−	0.155892i	$-0.950175\pi$
$419$		−	54737.0i		−	0.311784i	0.987774	−	0.155892i	$-0.0498251\pi$
$420$		0			0
$421$			135111.i			0.762303i	0.924513	+	0.381151i	$0.124472\pi$
$421$			135111.i			0.762303i	−0.924513	+	0.381151i	$0.875528\pi$
$422$	−32383.8			−0.181846
$423$		0			0
$424$		−	846762.i		−	4.71010i
$425$	−64810.6			−0.358813
$426$		0			0
$427$			170414.i			0.934653i
$428$	−43892.4			−0.239608
$429$		0			0
$430$	−498650.			−2.69686
$431$			332324.i			1.78899i	0.447081	+	0.894493i	$0.352463\pi$
$431$			332324.i			1.78899i	−0.447081	+	0.894493i	$0.647537\pi$
$432$		0			0
$433$	245001.			1.30675			0.653375	−	0.757035i	$-0.273353\pi$
$433$	245001.			1.30675			0.653375	+	0.757035i	$0.273353\pi$
$434$	−695907.			−3.69463
$435$		0			0
$436$			664519.i			3.49570i
$437$			21491.5i			0.112539i
$438$		0			0
$439$	−181499.			−0.941771			−0.470885	−	0.882194i	$-0.656066\pi$
$439$	−181499.			−0.941771			−0.470885	+	0.882194i	$0.656066\pi$
$440$	−59166.7			−0.305613
$441$		0			0
$442$	−280328.			−1.43490
$443$		−	57457.9i		−	0.292781i	−0.989227	−	0.146390i	$-0.953234\pi$
$443$		−	57457.9i		−	0.292781i	0.989227	−	0.146390i	$-0.0467655\pi$
$444$		0			0
$445$			72359.3i			0.365405i
$446$			602247.i			3.02765i
$447$		0			0
$448$	−1.26852e6			−6.32038
$449$	−26596.2			−0.131925			−0.0659624	−	0.997822i	$-0.521012\pi$
$449$	−26596.2			−0.131925			−0.0659624	+	0.997822i	$0.521012\pi$
$450$		0			0
$451$		−	7803.31i		−	0.0383642i
$452$			885883.i			4.33610i
$453$		0			0
$454$	−54400.0			−0.263929
$455$		−	221901.i		−	1.07185i
$456$		0			0
$457$		−	300518.i		−	1.43892i	−0.694532	−	0.719462i	$-0.744388\pi$
$457$		−	300518.i		−	1.43892i	0.694532	−	0.719462i	$-0.255612\pi$
$458$	−400358.			−1.90861
$459$		0			0
$460$			49989.7i			0.236246i
$461$	140059.			0.659038			0.329519	−	0.944149i	$-0.393113\pi$
$461$	140059.			0.659038			0.329519	+	0.944149i	$0.393113\pi$
$462$		0			0
$463$			127338.i			0.594014i	0.954875	+	0.297007i	$0.0959885\pi$
$463$			127338.i			0.594014i	−0.954875	+	0.297007i	$0.904012\pi$
$464$	449794.			2.08919
$465$		0			0
$466$	619400.			2.85233
$467$			27447.8i			0.125856i	0.998018	+	0.0629280i	$0.0200439\pi$
$467$			27447.8i			0.125856i	−0.998018	+	0.0629280i	$0.979956\pi$
$468$		0			0
$469$			265404.i			1.20660i
$470$	870192.			3.93930
$471$		0			0
$472$	−391159.	+	665058.i	−1.75578	+	2.98522i
$473$	20946.2			0.0936230
$474$		0			0
$475$	104999.			0.465371
$476$	1.09184e6			4.81887
$477$		0			0
$478$		−	215002.i		−	0.940995i
$479$	403672.			1.75937			0.879687	−	0.475554i	$-0.157752\pi$
$479$	403672.			1.75937			0.879687	+	0.475554i	$0.157752\pi$
$480$		0			0
$481$	207510.			0.896910
$482$		−	91526.0i		−	0.393959i
$483$		0			0
$484$	−647600.			−2.76450
$485$			263261.i			1.11919i
$486$		0			0
$487$	160164.			0.675316			0.337658	−	0.941269i	$-0.390365\pi$
$487$	160164.			0.675316			0.337658	+	0.941269i	$0.390365\pi$
$488$	−519563.			−2.18172
$489$		0			0
$490$			641248.i			2.67075i
$491$	27802.8			0.115326			0.0576628	−	0.998336i	$-0.481635\pi$
$491$	27802.8			0.115326			0.0576628	+	0.998336i	$0.481635\pi$
$492$		0			0
$493$	−150007.			−0.617189
$494$	454158.			1.86103
$495$		0			0
$496$		−	1.24550e6i		−	5.06268i
$497$	191237.			0.774210
$498$		0			0
$499$	136913.			0.549848			0.274924	−	0.961466i	$-0.411347\pi$
$499$	136913.			0.549848			0.274924	+	0.961466i	$0.411347\pi$
$500$	−550697.			−2.20279
$501$		0			0
$502$		−	768181.i		−	3.04829i
$503$		−	45212.1i		−	0.178698i	−0.996000	−	0.0893489i	$-0.971521\pi$
$503$		−	45212.1i		−	0.178698i	0.996000	−	0.0893489i	$-0.0284786\pi$
$504$		0			0
$505$		−	379884.i		−	1.48960i
$506$		−	2854.91i		−	0.0111504i
$507$		0			0
$508$	691082.			2.67795
$509$		−	420229.i		−	1.62200i	−0.585048	−	0.810999i	$-0.698924\pi$
$509$		−	420229.i		−	1.62200i	0.585048	−	0.810999i	$-0.301076\pi$
$510$		0			0
$511$			648791.i			2.48464i
$512$		−	788170.i		−	3.00663i
$513$		0			0
$514$			280686.i			1.06241i
$515$			412637.i			1.55580i
$516$		0			0
$517$	−36553.1			−0.136755
$518$	−1.09884e6			−4.09520
$519$		0			0
$520$	676535.			2.50198
$521$	−176873.			−0.651608			−0.325804	−	0.945437i	$-0.605635\pi$
$521$	−176873.			−0.651608			−0.325804	+	0.945437i	$0.605635\pi$
$522$		0			0
$523$	−361972.			−1.32334			−0.661670	−	0.749795i	$-0.730152\pi$
$523$	−361972.			−1.32334			−0.661670	+	0.749795i	$0.730152\pi$
$524$			55452.7i			0.201958i
$525$		0			0
$526$		−	129015.i		−	0.466305i
$527$			415376.i			1.49562i
$528$		0			0
$529$	278296.			0.994480
$530$		−	849335.i		−	3.02362i
$531$		0			0
$532$	−1.76889e6			−6.24995
$533$			89226.1i			0.314078i
$534$		0			0
$535$	−28195.2			−0.0985071
$536$	−809170.			−2.81650
$537$		0			0
$538$	366928.			1.26770
$539$		−	26936.1i		−	0.0927166i
$540$		0			0
$541$		−	276519.i		−	0.944779i	−0.881390	−	0.472390i	$-0.843392\pi$
$541$		−	276519.i		−	0.944779i	0.881390	−	0.472390i	$-0.156608\pi$
$542$		−	287835.i		−	0.979818i
$543$		0			0
$544$			1.45982e6i			4.93288i
$545$			426868.i			1.43714i
$546$		0			0
$547$	−424332.			−1.41818			−0.709090	−	0.705118i	$-0.750894\pi$
$547$	−424332.			−1.41818			−0.709090	+	0.705118i	$0.750894\pi$
$548$	747213.			2.48819
$549$		0			0
$550$	−13948.1			−0.0461093
$551$	243026.			0.800477
$552$		0			0
$553$	108871.			0.356010
$554$			291668.i			0.950318i
$555$		0			0
$556$	1.59843e6			5.17064
$557$	408914.			1.31802			0.659010	−	0.752134i	$-0.270976\pi$
$557$	408914.			1.31802			0.659010	+	0.752134i	$0.270976\pi$
$558$		0			0
$559$	−239507.			−0.766468
$560$	−2.10304e6			−6.70612
$561$		0			0
$562$		−	64137.5i		−	0.203067i
$563$			165648.i			0.522600i	0.965258	+	0.261300i	$0.0841512\pi$
$563$			165648.i			0.522600i	−0.965258	+	0.261300i	$0.915849\pi$
$564$		0			0
$565$			569065.i			1.78265i
$566$	26296.2			0.0820844
$567$		0			0
$568$			583047.i			1.80720i
$569$			413247.i			1.27639i	0.769873	+	0.638197i	$0.220320\pi$
$569$			413247.i			1.27639i	−0.769873	+	0.638197i	$0.779680\pi$
$570$		0			0
$571$		−	531700.i		−	1.63078i	−0.578914	−	0.815388i	$-0.696523\pi$
$571$		−	531700.i		−	1.63078i	0.578914	−	0.815388i	$-0.303477\pi$
$572$	−44374.2			−0.135625
$573$		0			0
$574$		−	472485.i		−	1.43405i
$575$			7547.20i			0.0228271i
$576$		0			0
$577$	−94373.0			−0.283463			−0.141731	−	0.989905i	$-0.545267\pi$
$577$	−94373.0			−0.283463			−0.141731	+	0.989905i	$0.545267\pi$
$578$		−	236415.i		−	0.707651i
$579$		0			0
$580$	565284.			1.68039
$581$		−	289732.i		−	0.858310i
$582$		0			0
$583$			35677.0i			0.104967i
$584$	−1.97805e6			−5.79977
$585$		0			0
$586$		−	1.14239e6i		−	3.32675i
$587$			16716.5i			0.0485142i	0.999706	+	0.0242571i	$0.00772203\pi$
$587$			16716.5i			0.0485142i	−0.999706	+	0.0242571i	$0.992278\pi$
$588$		0			0
$589$		−	672949.i		−	1.93978i
$590$	−392347.	+	667079.i	−1.12711	+	1.91634i
$591$		0			0
$592$		−	1.96665e6i		−	5.61157i
$593$	406858.			1.15700			0.578500	−	0.815683i	$-0.303638\pi$
$593$	406858.			1.15700			0.578500	+	0.815683i	$0.303638\pi$
$594$		0			0
$595$	701367.			1.98112
$596$			1.15495e6i			3.25139i
$597$		0			0
$598$			32644.2i			0.0912858i
$599$	216469.			0.603313			0.301656	−	0.953417i	$-0.402460\pi$
$599$	216469.			0.603313			0.301656	+	0.953417i	$0.402460\pi$
$600$		0			0
$601$			114850.i			0.317966i	0.987281	+	0.158983i	$0.0508214\pi$
$601$			114850.i			0.317966i	−0.987281	+	0.158983i	$0.949179\pi$
$602$	1.26828e6			3.49962
$603$		0			0
$604$			826091.i			2.26440i
$605$	−416000.			−1.13653
$606$		0			0
$607$	460090.			1.24872			0.624360	−	0.781136i	$-0.285360\pi$
$607$	460090.			1.24872			0.624360	+	0.781136i	$0.285360\pi$
$608$		−	2.36504e6i		−	6.39781i
$609$		0			0
$610$	−521141.			−1.40054
$611$	417962.			1.11958
$612$		0			0
$613$			54204.8i			0.144250i	0.997396	+	0.0721252i	$0.0229781\pi$
$613$			54204.8i			0.144250i	−0.997396	+	0.0721252i	$0.977022\pi$
$614$		−	716363.i		−	1.90019i
$615$		0			0
$616$	150486.			0.396583
$617$	446671.			1.17332			0.586662	−	0.809832i	$-0.300442\pi$
$617$	446671.			1.17332			0.586662	+	0.809832i	$0.300442\pi$
$618$		0			0
$619$	608019.			1.58685			0.793425	−	0.608668i	$-0.208296\pi$
$619$	608019.			1.58685			0.793425	+	0.608668i	$0.208296\pi$
$620$		−	1.56530e6i		−	4.07205i
$621$		0			0
$622$		−	256503.i		−	0.662996i
$623$		−	184040.i		−	0.474172i
$624$		0			0
$625$	−473767.			−1.21284
$626$	−316673.			−0.808095
$627$		0			0
$628$			2.07032e6i			5.24950i
$629$			655882.i			1.65777i
$630$		0			0
$631$	−89180.6			−0.223981			−0.111991	−	0.993709i	$-0.535723\pi$
$631$	−89180.6			−0.223981			−0.111991	+	0.993709i	$0.535723\pi$
$632$			331928.i			0.831016i
$633$		0			0
$634$			39648.0i			0.0986377i
$635$	443931.			1.10095
$636$		0			0
$637$			307998.i			0.759048i
$638$	−32283.4			−0.0793118
$639$		0			0
$640$		−	1.90118e6i		−	4.64155i
$641$	−496660.			−1.20877			−0.604384	−	0.796693i	$-0.706581\pi$
$641$	−496660.			−1.20877			−0.604384	+	0.796693i	$0.706581\pi$
$642$		0			0
$643$	−477619.			−1.15521			−0.577603	−	0.816318i	$-0.696012\pi$
$643$	−477619.			−1.15521			−0.577603	+	0.816318i	$0.696012\pi$
$644$		−	127145.i		−	0.306568i
$645$		0			0
$646$			1.43547e6i			3.43976i
$647$	−255266.			−0.609796			−0.304898	−	0.952385i	$-0.598622\pi$
$647$	−255266.			−0.609796			−0.304898	+	0.952385i	$0.598622\pi$
$648$		0			0
$649$	16480.9	−	28021.2i	0.0391282	−	0.0665268i
$650$	159487.			0.377485
$651$		0			0
$652$	19475.0			0.0458124
$653$	−12720.0			−0.0298305			−0.0149152	−	0.999889i	$-0.504748\pi$
$653$	−12720.0			−0.0298305			−0.0149152	+	0.999889i	$0.504748\pi$
$654$		0			0
$655$			35621.2i			0.0830283i
$656$	845631.			1.96505
$657$		0			0
$658$	−2.21326e6			−5.11189
$659$		−	732103.i		−	1.68578i	−0.538084	−	0.842891i	$-0.680852\pi$
$659$		−	732103.i		−	1.68578i	0.538084	−	0.842891i	$-0.319148\pi$
$660$		0			0
$661$	601813.			1.37740			0.688698	−	0.725049i	$-0.258183\pi$
$661$	601813.			1.37740			0.688698	+	0.725049i	$0.258183\pi$
$662$			144073.i			0.328750i
$663$		0			0
$664$	883341.			2.00351
$665$	−1.13628e6			−2.56946
$666$		0			0
$667$			17468.3i			0.0392644i
$668$	2.21244e6			4.95813
$669$		0			0
$670$	−811628.			−1.80804
$671$	21890.9			0.0486205
$672$		0			0
$673$		−	177803.i		−	0.392563i	−0.980548	−	0.196282i	$-0.937113\pi$
$673$		−	177803.i		−	0.392563i	0.980548	−	0.196282i	$-0.0628867\pi$
$674$	−1.06655e6			−2.34780
$675$		0			0
$676$	−763488.			−1.67074
$677$	−238228.			−0.519774			−0.259887	−	0.965639i	$-0.583685\pi$
$677$	−238228.			−0.519774			−0.259887	+	0.965639i	$0.583685\pi$
$678$		0			0
$679$		−	669585.i		−	1.45233i
$680$			2.13834e6i			4.62444i
$681$		0			0
$682$			89394.1i			0.192194i
$683$		−	660787.i		−	1.41651i	−0.705956	−	0.708256i	$-0.749482\pi$
$683$		−	660787.i		−	1.41651i	0.705956	−	0.708256i	$-0.250518\pi$
$684$		0			0
$685$	479988.			1.02294
$686$		−	273293.i		−	0.580737i
$687$		0			0
$688$			2.26990e6i			4.79545i
$689$		−	407945.i		−	0.859335i
$690$		0			0
$691$			252072.i			0.527919i	0.964534	+	0.263960i	$0.0850286\pi$
$691$			252072.i			0.527919i	−0.964534	+	0.263960i	$0.914971\pi$
$692$			1.36903e6i			2.85891i
$693$		0			0
$694$	−718594.			−1.49199
$695$	1.02679e6			2.12574
$696$		0			0
$697$	−282019.			−0.580515
$698$	−1.49477e6			−3.06805
$699$		0			0
$700$	−621183.			−1.26772
$701$			601181.i			1.22340i	0.791089	+	0.611701i	$0.209514\pi$
$701$			601181.i			1.22340i	−0.791089	+	0.611701i	$0.790486\pi$
$702$		0			0
$703$		−	1.06259e6i		−	2.15008i
$704$			162951.i			0.328785i
$705$		0			0
$706$	−1.04024e6			−2.08701
$707$			966205.i			1.93299i
$708$		0			0
$709$	769466.			1.53072			0.765362	−	0.643600i	$-0.222560\pi$
$709$	769466.			1.53072			0.765362	+	0.643600i	$0.222560\pi$
$710$			584818.i			1.16012i
$711$		0			0
$712$	561105.			1.10684
$713$	48370.6			0.0951485
$714$		0			0
$715$	−28504.7			−0.0557577
$716$			924096.i			1.80256i
$717$		0			0
$718$		−	493444.i		−	0.957171i
$719$		−	412760.i		−	0.798435i	−0.916856	−	0.399218i	$-0.869282\pi$
$719$		−	412760.i		−	0.798435i	0.916856	−	0.399218i	$-0.130718\pi$
$720$		0			0
$721$		−	1.04951e6i		−	2.01890i
$722$		−	1.31196e6i		−	2.51678i
$723$		0			0
$724$	−1.23655e6			−2.35904
$725$	85343.8			0.162366
$726$		0			0
$727$	−352870.			−0.667645			−0.333822	−	0.942636i	$-0.608339\pi$
$727$	−352870.			−0.667645			−0.333822	+	0.942636i	$0.608339\pi$
$728$	−1.72071e6			−3.24673
$729$		0			0
$730$	−1.98406e6			−3.72313
$731$		−	757015.i		−	1.41667i
$732$		0			0
$733$	566886.			1.05509			0.527543	−	0.849528i	$-0.323113\pi$
$733$	566886.			1.05509			0.527543	+	0.849528i	$0.323113\pi$
$734$	−488129.			−0.906029
$735$		0			0
$736$	169996.			0.313821
$737$	34093.1			0.0627670
$738$		0			0
$739$		−	110960.i		−	0.203179i	−0.994826	−	0.101589i	$-0.967607\pi$
$739$		−	110960.i		−	0.203179i	0.994826	−	0.101589i	$-0.0323928\pi$
$740$		−	2.47162e6i		−	4.51354i
$741$		0			0
$742$			2.16022e6i			3.92364i
$743$	565178.			1.02378			0.511891	−	0.859050i	$-0.328945\pi$
$743$	565178.			1.02378			0.511891	+	0.859050i	$0.328945\pi$
$744$		0			0
$745$			741904.i			1.33670i
$746$			796617.i			1.43144i
$747$		0			0
$748$		−	140255.i		−	0.250677i
$749$	71712.3			0.127829
$750$		0			0
$751$		−	217676.i		−	0.385950i	−0.981204	−	0.192975i	$-0.938186\pi$
$751$		−	217676.i		−	0.385950i	0.981204	−	0.192975i	$-0.0618136\pi$
$752$		−	3.96119e6i		−	7.00471i
$753$		0			0
$754$	369141.			0.649306
$755$			530657.i			0.930936i
$756$		0			0
$757$	169980.			0.296623			0.148312	−	0.988941i	$-0.452616\pi$
$757$	169980.			0.296623			0.148312	+	0.988941i	$0.452616\pi$
$758$			896194.i			1.55978i
$759$		0			0
$760$		−	3.46431e6i		−	5.99777i
$761$	−880285.			−1.52004			−0.760019	−	0.649901i	$-0.774810\pi$
$761$	−880285.			−1.52004			−0.760019	+	0.649901i	$0.774810\pi$
$762$		0			0
$763$		−	1.08570e6i		−	1.86493i
$764$			988836.i			1.69409i
$765$		0			0
$766$			534349.i			0.910684i
$767$	−188449.	+	320405.i	−0.320333	+	0.544639i
$768$		0			0
$769$			619749.i			1.04801i	0.851717	+	0.524003i	$0.175562\pi$
$769$			619749.i			1.04801i	−0.851717	+	0.524003i	$0.824438\pi$
$770$	150943.			0.254584
$771$		0			0
$772$	−1.00267e6			−1.68238
$773$			1.11076e6i			1.85892i	0.368917	+	0.929462i	$0.379729\pi$
$773$			1.11076e6i			1.85892i	−0.368917	+	0.929462i	$0.620271\pi$
$774$		0			0
$775$		−	236321.i		−	0.393458i
$776$	2.04144e6			3.39011
$777$		0			0
$778$			15332.7i			0.0253313i
$779$	456898.			0.752912
$780$		0			0
$781$		−	24565.7i		−	0.0402743i
$782$	−103179.			−0.168725
$783$		0			0
$784$	2.91902e6			4.74903
$785$			1.32991e6i			2.15816i
$786$		0			0
$787$	243033.			0.392389			0.196194	−	0.980565i	$-0.437142\pi$
$787$	243033.			0.392389			0.196194	+	0.980565i	$0.437142\pi$
$788$	495647.			0.798216
$789$		0			0
$790$			332936.i			0.533466i
$791$		−	1.44737e6i		−	2.31328i
$792$		0			0
$793$	−250310.			−0.398044
$794$	330105.			0.523613
$795$		0			0
$796$	1.24252e6			1.96100
$797$			419997.i			0.661195i	0.943772	+	0.330597i	$0.107250\pi$
$797$			419997.i			0.661195i	−0.943772	+	0.330597i	$0.892750\pi$
$798$		0			0
$799$			1.32106e6i			2.06933i
$800$		−	830536.i		−	1.29771i
$801$		0			0
$802$	806655.			1.25412
$803$	83341.8			0.129250
$804$		0			0
$805$		−	81674.2i		−	0.126036i
$806$		−	1.02217e6i		−	1.57345i
$807$		0			0
$808$	−2.94578e6			−4.51209
$809$		−	573138.i		−	0.875713i	−0.899045	−	0.437857i	$-0.855738\pi$
$809$		−	573138.i		−	0.875713i	0.899045	−	0.437857i	$-0.144262\pi$
$810$		0			0
$811$			133816.i			0.203454i	0.994812	+	0.101727i	$0.0324368\pi$
$811$			133816.i			0.203454i	−0.994812	+	0.101727i	$0.967563\pi$
$812$	−1.43776e6			−2.18058
$813$		0			0
$814$			141154.i			0.213032i
$815$	12510.2			0.0188343
$816$		0			0
$817$			1.22644e6i			1.83739i
$818$	227509.			0.340010
$819$		0			0
$820$	1.06276e6			1.58054
$821$			197462.i			0.292952i	0.989214	+	0.146476i	$0.0467931\pi$
$821$			197462.i			0.292952i	−0.989214	+	0.146476i	$0.953207\pi$
$822$		0			0
$823$			562096.i			0.829871i	0.909851	+	0.414936i	$0.136196\pi$
$823$			562096.i			0.829871i	−0.909851	+	0.414936i	$0.863804\pi$
$824$	3.19976e6			4.71263
$825$		0			0
$826$	997904.	−	1.69666e6i	1.46261	−	2.48677i
$827$	567007.			0.829043			0.414522	−	0.910039i	$-0.363949\pi$
$827$	567007.			0.829043			0.414522	+	0.910039i	$0.363949\pi$
$828$		0			0
$829$	−430710.			−0.626723			−0.313362	−	0.949634i	$-0.601455\pi$
$829$	−430710.			−0.626723			−0.313362	+	0.949634i	$0.601455\pi$
$830$	886024.			1.28614
$831$		0			0
$832$		−	1.86325e6i		−	2.69168i
$833$	−973498.			−1.40296
$834$		0			0
$835$	1.42120e6			2.03837
$836$			227226.i			0.325121i
$837$		0			0
$838$	−425744.			−0.606262
$839$			53824.6i			0.0764640i	0.999269	+	0.0382320i	$0.0121726\pi$
$839$			53824.6i			0.0764640i	−0.999269	+	0.0382320i	$0.987827\pi$
$840$		0			0
$841$	−509749.			−0.720716
$842$	1.05089e6			1.48229
$843$		0			0
$844$			185265.i			0.260080i
$845$	−490442.			−0.686870
$846$		0			0
$847$	1.05806e6			1.47484
$848$	−3.86625e6			−5.37648
$849$		0			0
$850$			504096.i			0.697711i
$851$	76377.4			0.105464
$852$		0			0
$853$	−418548.			−0.575237			−0.287619	−	0.957745i	$-0.592864\pi$
$853$	−418548.			−0.575237			−0.287619	+	0.957745i	$0.592864\pi$
$854$	1.32548e6			1.81743
$855$		0			0
$856$			218638.i			0.298385i
$857$			1.10791e6i			1.50849i	0.656596	+	0.754243i	$0.271996\pi$
$857$			1.10791e6i			1.50849i	−0.656596	+	0.754243i	$0.728004\pi$
$858$		0			0
$859$			226154.i			0.306491i	0.988188	+	0.153245i	$0.0489725\pi$
$859$			226154.i			0.306491i	−0.988188	+	0.153245i	$0.951028\pi$
$860$			2.85272e6i			3.85712i
$861$		0			0
$862$	2.58481e6			3.47868
$863$			539905.i			0.724929i	0.931998	+	0.362464i	$0.118065\pi$
$863$			539905.i			0.724929i	−0.931998	+	0.362464i	$0.881935\pi$
$864$		0			0
$865$			879424.i			1.17535i
$866$		−	1.90561e6i		−	2.54097i
$867$		0			0
$868$			3.98121e6i			5.28416i
$869$		−	13985.2i		−	0.0185196i
$870$		0			0
$871$	−389834.			−0.513858
$872$	3.31011e6			4.35321
$873$		0			0
$874$	167160.			0.218832
$875$	899740.			1.17517
$876$		0			0
$877$	−288067.			−0.374537			−0.187269	−	0.982309i	$-0.559963\pi$
$877$	−288067.			−0.374537			−0.187269	+	0.982309i	$0.559963\pi$
$878$			1.41170e6i			1.83127i
$879$		0			0
$880$			270150.i			0.348851i
$881$			80450.5i			0.103652i	0.998656	+	0.0518259i	$0.0165041\pi$
$881$			80450.5i			0.103652i	−0.998656	+	0.0518259i	$0.983496\pi$
$882$		0			0
$883$	210993.			0.270611			0.135306	−	0.990804i	$-0.456798\pi$
$883$	210993.			0.270611			0.135306	+	0.990804i	$0.456798\pi$
$884$			1.60373e6i			2.05223i
$885$		0			0
$886$	−446907.			−0.569311
$887$		−	428946.i		−	0.545199i	−0.962128	−	0.272600i	$-0.912117\pi$
$887$		−	428946.i		−	0.545199i	0.962128	−	0.272600i	$-0.0878834\pi$
$888$		0			0
$889$	−1.12910e6			−1.42866
$890$	562809.			0.710528
$891$		0			0
$892$	3.44539e6			4.33021
$893$		−	2.14025e6i		−	2.68387i
$894$		0			0
$895$			593612.i			0.741066i
$896$			4.83549e6i			6.02316i
$897$		0			0
$898$			206865.i			0.256527i
$899$		−	546975.i		−	0.676781i
$900$		0			0
$901$	1.28940e6			1.58832
$902$	−60694.0			−0.0745990
$903$		0			0
$904$	4.41278e6			5.39977
$905$	−794327.			−0.969844
$906$		0			0
$907$	1.21610e6			1.47827			0.739134	−	0.673558i	$-0.235235\pi$
$907$	1.21610e6			1.47827			0.739134	+	0.673558i	$0.235235\pi$
$908$			311217.i			0.377477i
$909$		0			0
$910$	−1.72594e6			−2.08422
$911$	−789820.			−0.951681			−0.475841	−	0.879532i	$-0.657856\pi$
$911$	−789820.			−0.951681			−0.475841	+	0.879532i	$0.657856\pi$
$912$		0			0
$913$	−37218.1			−0.0446491
$914$	−2.33742e6			−2.79798
$915$		0			0
$916$			2.29041e6i			2.72975i
$917$		−	90599.7i		−	0.107743i
$918$		0			0
$919$			1.09607e6i			1.29780i	0.760872	+	0.648902i	$0.224772\pi$
$919$			1.09607e6i			1.29780i	−0.760872	+	0.648902i	$0.775228\pi$
$920$	249010.			0.294199
$921$		0			0
$922$		−	1.08938e6i		−	1.28150i
$923$			280894.i			0.329716i
$924$		0			0
$925$		−	373152.i		−	0.436116i
$926$	990435.			1.15506
$927$		0			0
$928$		−	1.92231e6i		−	2.23217i
$929$			346274.i			0.401225i	0.979671	+	0.200612i	$0.0642932\pi$
$929$			346274.i			0.401225i	−0.979671	+	0.200612i	$0.935707\pi$
$930$		0			0
$931$	1.57716e6			1.81960
$932$		−	3.54353e6i		−	4.07947i
$933$		0			0
$934$	213489.			0.244727
$935$		−	90095.6i		−	0.103058i
$936$		0			0
$937$		−	1.06972e6i		−	1.21840i	−0.793017	−	0.609200i	$-0.791491\pi$
$937$		−	1.06972e6i		−	1.21840i	0.793017	−	0.609200i	$-0.208509\pi$
$938$	2.06431e6			2.34622
$939$		0			0
$940$		−	4.97828e6i		−	5.63408i
$941$		−	1.10921e6i		−	1.25267i	−0.779555	−	0.626333i	$-0.784555\pi$
$941$		−	1.10921e6i		−	1.25267i	0.779555	−	0.626333i	$-0.215445\pi$
$942$		0			0
$943$			32841.1i			0.0369313i
$944$	3.03660e6	+	1.78600e6i	3.40756	+	2.00418i
$945$		0			0
$946$		−	162919.i		−	0.182050i
$947$	502348.			0.560151			0.280075	−	0.959978i	$-0.409641\pi$
$947$	502348.			0.560151			0.280075	+	0.959978i	$0.409641\pi$
$948$		0			0
$949$	−952963.			−1.05814
$950$		−	816683.i		−	0.904912i
$951$		0			0
$952$		−	5.43870e6i		−	6.00097i
$953$	−498865.			−0.549284			−0.274642	−	0.961547i	$-0.588559\pi$
$953$	−498865.			−0.549284			−0.274642	+	0.961547i	$0.588559\pi$
$954$		0			0
$955$			635200.i			0.696472i
$956$	−1.23001e6			−1.34583
$957$		0			0
$958$		−	3.13976e6i		−	3.42109i
$959$	−1.22081e6			−1.32743
$960$		0			0
$961$	−591077.			−0.640026
$962$		−	1.61401e6i		−	1.74404i
$963$		0			0
$964$	−523611.			−0.563449
$965$	−644087.			−0.691656
$966$		0			0
$967$		−	790210.i		−	0.845064i	−0.906348	−	0.422532i	$-0.861141\pi$
$967$		−	790210.i		−	0.845064i	0.906348	−	0.422532i	$-0.138859\pi$
$968$			3.22584e6i			3.44264i
$969$		0			0
$970$	2.04764e6			2.17626
$971$	−546860.			−0.580013			−0.290006	−	0.957025i	$-0.593657\pi$
$971$	−546860.			−0.580013			−0.290006	+	0.957025i	$0.593657\pi$
$972$		0			0
$973$	−2.61155e6			−2.75850
$974$		−	1.24575e6i		−	1.31315i
$975$		0			0
$976$			2.37228e6i			2.49038i
$977$		−	710431.i		−	0.744274i	−0.928178	−	0.372137i	$-0.878625\pi$
$977$		−	710431.i		−	0.744274i	0.928178	−	0.372137i	$-0.121375\pi$
$978$		0			0
$979$	−23641.2			−0.0246664
$980$	3.66851e6			3.81978
$981$		0			0
$982$		−	216250.i		−	0.224250i
$983$			1.52173e6i			1.57482i	0.616429	+	0.787410i	$0.288579\pi$
$983$			1.52173e6i			1.57482i	−0.616429	+	0.787410i	$0.711421\pi$
$984$		0			0
$985$	318389.			0.328160
$986$			1.16675e6i			1.20012i
$987$		0			0
$988$		−	2.59819e6i		−	2.66169i
$989$	−88154.4			−0.0901263
$990$		0			0
$991$		−	845241.i		−	0.860664i	−0.902671	−	0.430332i	$-0.858397\pi$
$991$		−	845241.i		−	0.860664i	0.902671	−	0.430332i	$-0.141603\pi$
$992$	−5.32297e6			−5.40917
$993$		0			0
$994$		−	1.48744e6i		−	1.50545i
$995$	798160.			0.806202
$996$		0			0
$997$	−168798.			−0.169816			−0.0849079	−	0.996389i	$-0.527060\pi$
$997$	−168798.			−0.169816			−0.0849079	+	0.996389i	$0.527060\pi$
$998$		−	1.06491e6i		−	1.06918i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.c.d.235.2		40
3.2	odd	2		177.5.c.a.58.39	yes	40
59.58	odd	2	inner	531.5.c.d.235.39		40
177.176	even	2		177.5.c.a.58.2	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.2	✓	40	177.176	even	2
177.5.c.a.58.39	yes	40	3.2	odd	2
531.5.c.d.235.2		40	1.1	even	1	trivial
531.5.c.d.235.39		40	59.58	odd	2	inner

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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.77798i q^{2} -44.4970 q^{4} -28.5836 q^{5} +72.7001 q^{7} +221.650i q^{8} +O(q^{10})\)	\(q-7.77798i q^{2} -44.4970 q^{4} -28.5836 q^{5} +72.7001 q^{7} +221.650i q^{8} +222.323i q^{10} -9.33885i q^{11} +106.784i q^{13} -565.461i q^{14} +1012.03 q^{16} -337.515 q^{17} +546.806 q^{19} +1271.89 q^{20} -72.6374 q^{22} +39.3036i q^{23} +192.023 q^{25} +830.565 q^{26} -3234.94 q^{28} +444.446 q^{29} -1230.69i q^{31} -4325.19i q^{32} +2625.19i q^{34} -2078.03 q^{35} -1943.27i q^{37} -4253.05i q^{38} -6335.54i q^{40} +835.576 q^{41} +2242.91i q^{43} +415.551i q^{44} +305.703 q^{46} -3914.09i q^{47} +2884.31 q^{49} -1493.55i q^{50} -4751.57i q^{52} -3820.28 q^{53} +266.938i q^{55} +16113.9i q^{56} -3456.89i q^{58} +(3000.50 + 1764.76i) q^{59} +2344.07i q^{61} -9572.29 q^{62} -17448.7 q^{64} -3052.27i q^{65} +3650.67i q^{67} +15018.4 q^{68} +16162.9i q^{70} +2630.49 q^{71} +8924.21i q^{73} -15114.7 q^{74} -24331.3 q^{76} -678.935i q^{77} +1497.53 q^{79} -28927.6 q^{80} -6499.09i q^{82} -3985.30i q^{83} +9647.40 q^{85} +17445.3 q^{86} +2069.95 q^{88} -2531.49i q^{89} +7763.21i q^{91} -1748.89i q^{92} -30443.7 q^{94} -15629.7 q^{95} -9210.22i q^{97} -22434.1i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 40 * q - 320 * q^4 + 80 * q^7	\( 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 + 80 * q^7 + 3944 * q^16 + 528 * q^17 + 444 * q^19 - 444 * q^20 + 1304 * q^22 + 4880 * q^25 + 1452 * q^26 - 1160 * q^28 + 996 * q^29 - 10320 * q^35 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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