LMFDB - Embedded newform 1620.4.i.w.1081.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.i (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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904 $ x^12 + 306x^10 + 30777x^8 + 1381040x^6 + 28918584x^4 + 243888288*x^2 + 366645904
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{13} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.2
Root			$-5.63924i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.4.i.w.541.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 + 4.33013i) q^{5} +(-8.28565 - 14.3512i) q^{7} +O(q^{10})$	$q+(-2.50000 + 4.33013i) q^{5} +(-8.28565 - 14.3512i) q^{7} +(-36.2684 - 62.8188i) q^{11} +(29.9488 - 51.8728i) q^{13} -15.8529 q^{17} -136.646 q^{19} +(81.5503 - 141.249i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(5.79923 + 10.0446i) q^{29} +(20.5009 - 35.5086i) q^{31} +82.8565 q^{35} -242.545 q^{37} +(28.9332 - 50.1138i) q^{41} +(132.252 + 229.067i) q^{43} +(299.945 + 519.519i) q^{47} +(34.1961 - 59.2294i) q^{49} -592.285 q^{53} +362.684 q^{55} +(144.446 - 250.188i) q^{59} +(-412.690 - 714.800i) q^{61} +(149.744 + 259.364i) q^{65} +(-405.264 + 701.938i) q^{67} +966.126 q^{71} +802.840 q^{73} +(-601.015 + 1040.99i) q^{77} +(581.001 + 1006.32i) q^{79} +(251.187 + 435.069i) q^{83} +(39.6322 - 68.6450i) q^{85} -8.76580 q^{89} -992.580 q^{91} +(341.615 - 591.694i) q^{95} +(-69.1820 - 119.827i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 30 q^{5} - 12 q^{7}+O(q^{10}) $ 12 * q - 30 * q^5 - 12 * q^7	$ 12 q - 30 q^{5} - 12 q^{7} + 84 q^{13} - 24 q^{17} - 228 q^{19} + 30 q^{23} - 150 q^{25} + 168 q^{29} + 324 q^{31} + 120 q^{35} - 984 q^{37} + 312 q^{41} + 156 q^{43} + 462 q^{47} + 588 q^{49} - 2028 q^{53} + 1008 q^{59} - 36 q^{61} + 420 q^{65} - 144 q^{67} - 2424 q^{71} - 1800 q^{73} + 672 q^{77} + 936 q^{79} + 288 q^{83} + 60 q^{85} + 240 q^{89} + 4572 q^{91} + 570 q^{95} + 1188 q^{97}+O(q^{100}) $ 12 * q - 30 * q^5 - 12 * q^7 + 84 * q^13 - 24 * q^17 - 228 * q^19 + 30 * q^23 - 150 * q^25 + 168 * q^29 + 324 * q^31 + 120 * q^35 - 984 * q^37 + 312 * q^41 + 156 * q^43 + 462 * q^47 + 588 * q^49 - 2028 * q^53 + 1008 * q^59 - 36 * q^61 + 420 * q^65 - 144 * q^67 - 2424 * q^71 - 1800 * q^73 + 672 * q^77 + 936 * q^79 + 288 * q^83 + 60 * q^85 + 240 * q^89 + 4572 * q^91 + 570 * q^95 + 1188 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	−8.28565	−	14.3512i	−0.447383	−	0.774890i	0.550832	−	0.834616i	$-0.314311\pi$
$7$	−8.28565	−	14.3512i	−0.447383	−	0.774890i	−0.998215	+	0.0597263i	$0.980977\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−36.2684	−	62.8188i	−0.994122	−	1.72187i	−0.590822	−	0.806802i	$-0.701196\pi$
$11$	−36.2684	−	62.8188i	−0.994122	−	1.72187i	−0.403300	−	0.915068i	$-0.632137\pi$
$12$		0			0
$13$	29.9488	−	51.8728i	0.638946	−	1.10669i	−0.346719	−	0.937969i	$-0.612704\pi$
$13$	29.9488	−	51.8728i	0.638946	−	1.10669i	0.985664	−	0.168717i	$-0.0539625\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.8529			−0.226170			−0.113085	−	0.993585i	$-0.536073\pi$
$17$	−15.8529			−0.226170			−0.113085	+	0.993585i	$0.536073\pi$
$18$		0			0
$19$	−136.646			−1.64993			−0.824966	−	0.565182i	$-0.808806\pi$
$19$	−136.646			−1.64993			−0.824966	+	0.565182i	$0.808806\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	81.5503	−	141.249i	0.739322	−	1.28054i	−0.213479	−	0.976948i	$-0.568480\pi$
$23$	81.5503	−	141.249i	0.739322	−	1.28054i	0.952801	−	0.303595i	$-0.0981871\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.79923	+	10.0446i	0.0371342	+	0.0643183i	0.883995	−	0.467496i	$-0.154844\pi$
$29$	5.79923	+	10.0446i	0.0371342	+	0.0643183i	−0.846861	+	0.531814i	$0.821510\pi$
$30$		0			0
$31$	20.5009	−	35.5086i	0.118776	−	0.205727i	−0.800507	−	0.599324i	$-0.795436\pi$
$31$	20.5009	−	35.5086i	0.118776	−	0.205727i	0.919283	+	0.393597i	$0.128770\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	82.8565			0.400151
$36$		0			0
$37$	−242.545			−1.07768			−0.538841	−	0.842408i	$-0.681138\pi$
$37$	−242.545			−1.07768			−0.538841	+	0.842408i	$0.681138\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	28.9332	−	50.1138i	0.110210	−	0.190889i	−0.805645	−	0.592399i	$-0.798181\pi$
$41$	28.9332	−	50.1138i	0.110210	−	0.190889i	0.915855	+	0.401510i	$0.131514\pi$
$42$		0			0
$43$	132.252	+	229.067i	0.469029	+	0.812383i	0.999373	−	0.0354000i	$-0.0112705\pi$
$43$	132.252	+	229.067i	0.469029	+	0.812383i	−0.530344	+	0.847783i	$0.677937\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	299.945	+	519.519i	0.930882	+	1.61233i	0.781818	+	0.623506i	$0.214292\pi$
$47$	299.945	+	519.519i	0.930882	+	1.61233i	0.149063	+	0.988828i	$0.452374\pi$
$48$		0			0
$49$	34.1961	−	59.2294i	0.0996971	−	0.172680i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−592.285			−1.53503			−0.767516	−	0.641030i	$-0.778507\pi$
$53$	−592.285			−1.53503			−0.767516	+	0.641030i	$0.778507\pi$
$54$		0			0
$55$	362.684			0.889170
$56$		0			0
$57$		0			0
$58$		0			0
$59$	144.446	−	250.188i	0.318734	−	0.552063i	−0.661490	−	0.749954i	$-0.730076\pi$
$59$	144.446	−	250.188i	0.318734	−	0.552063i	0.980224	+	0.197890i	$0.0634090\pi$
$60$		0			0
$61$	−412.690	−	714.800i	−0.866222	−	1.50034i	−0.865829	−	0.500340i	$-0.833208\pi$
$61$	−412.690	−	714.800i	−0.866222	−	1.50034i	−0.000392970	−	1.00000i	$-0.500125\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	149.744	+	259.364i	0.285745	+	0.494925i
$66$		0			0
$67$	−405.264	+	701.938i	−0.738968	+	1.27993i	0.213992	+	0.976835i	$0.431353\pi$
$67$	−405.264	+	701.938i	−0.738968	+	1.27993i	−0.952960	+	0.303095i	$0.901980\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	966.126			1.61490			0.807451	−	0.589935i	$-0.200847\pi$
$71$	966.126			1.61490			0.807451	+	0.589935i	$0.200847\pi$
$72$		0			0
$73$	802.840			1.28720			0.643598	−	0.765364i	$-0.277441\pi$
$73$	802.840			1.28720			0.643598	+	0.765364i	$0.277441\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−601.015	+	1040.99i	−0.889506	+	1.54067i
$78$		0			0
$79$	581.001	+	1006.32i	0.827440	+	1.43317i	0.900040	+	0.435807i	$0.143537\pi$
$79$	581.001	+	1006.32i	0.827440	+	1.43317i	−0.0726002	+	0.997361i	$0.523130\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	251.187	+	435.069i	0.332186	+	0.575362i	0.982940	−	0.183926i	$-0.0588806\pi$
$83$	251.187	+	435.069i	0.332186	+	0.575362i	−0.650755	+	0.759288i	$0.725547\pi$
$84$		0			0
$85$	39.6322	−	68.6450i	0.0505731	−	0.0875952i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.76580			−0.0104401			−0.00522007	−	0.999986i	$-0.501662\pi$
$89$	−8.76580			−0.0104401			−0.00522007	+	0.999986i	$0.501662\pi$
$90$		0			0
$91$	−992.580			−1.14341
$92$		0			0
$93$		0			0
$94$		0			0
$95$	341.615	−	591.694i	0.368936	−	0.639016i
$96$		0			0
$97$	−69.1820	−	119.827i	−0.0724161	−	0.125428i	0.827544	−	0.561401i	$-0.189738\pi$
$97$	−69.1820	−	119.827i	−0.0724161	−	0.125428i	−0.899960	+	0.435973i	$0.856404\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−662.889	−	1148.16i	−0.653068	−	1.13115i	−0.982374	−	0.186924i	$-0.940148\pi$
$101$	−662.889	−	1148.16i	−0.653068	−	1.13115i	0.329306	−	0.944223i	$-0.393185\pi$
$102$		0			0
$103$	−166.772	+	288.857i	−0.159539	+	0.276330i	−0.934703	−	0.355431i	$-0.884334\pi$
$103$	−166.772	+	288.857i	−0.159539	+	0.276330i	0.775164	+	0.631761i	$0.217667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−584.004			−0.527643			−0.263821	−	0.964572i	$-0.584983\pi$
$107$	−584.004			−0.527643			−0.263821	+	0.964572i	$0.584983\pi$
$108$		0			0
$109$	−1915.42			−1.68315			−0.841577	−	0.540137i	$-0.818372\pi$
$109$	−1915.42			−1.68315			−0.841577	+	0.540137i	$0.818372\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	618.658	−	1071.55i	0.515030	−	0.892058i	−0.484818	−	0.874615i	$-0.661114\pi$
$113$	618.658	−	1071.55i	0.515030	−	0.892058i	0.999848	−	0.0174432i	$-0.00555262\pi$
$114$		0			0
$115$	407.751	+	706.246i	0.330635	+	0.572676i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	131.351	+	227.507i	0.101185	+	0.175257i
$120$		0			0
$121$	−1965.30	+	3403.99i	−1.47656	+	2.55747i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	−1785.42			−1.24749			−0.623743	−	0.781629i	$-0.714389\pi$
$127$	−1785.42			−1.24749			−0.623743	+	0.781629i	$0.714389\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	360.906	−	625.108i	0.240706	−	0.416915i	−0.720209	−	0.693757i	$-0.755954\pi$
$131$	360.906	−	625.108i	0.240706	−	0.416915i	0.960916	+	0.276841i	$0.0892876\pi$
$132$		0			0
$133$	1132.20	+	1961.03i	0.738152	+	1.27852i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−554.493	−	960.411i	−0.345792	−	0.598930i	0.639705	−	0.768621i	$-0.279057\pi$
$137$	−554.493	−	960.411i	−0.345792	−	0.598930i	−0.985497	+	0.169691i	$0.945723\pi$
$138$		0			0
$139$	−218.385	+	378.254i	−0.133260	+	0.230813i	−0.924932	−	0.380134i	$-0.875878\pi$
$139$	−218.385	+	378.254i	−0.133260	+	0.230813i	0.791671	+	0.610947i	$0.209211\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4344.78			−2.54076
$144$		0			0
$145$	−57.9923			−0.0332138
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−88.5585	+	153.388i	−0.0486912	+	0.0843357i	−0.889344	−	0.457239i	$-0.848838\pi$
$149$	−88.5585	+	153.388i	−0.0486912	+	0.0843357i	0.840653	+	0.541575i	$0.182172\pi$
$150$		0			0
$151$	−414.329	−	717.639i	−0.223296	−	0.386759i	0.732511	−	0.680755i	$-0.238348\pi$
$151$	−414.329	−	717.639i	−0.223296	−	0.386759i	−0.955807	+	0.293996i	$0.905015\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	102.504	+	177.543i	0.0531184	+	0.0920038i
$156$		0			0
$157$	433.262	−	750.432i	0.220242	−	0.381471i	−0.734639	−	0.678458i	$-0.762648\pi$
$157$	433.262	−	750.432i	0.220242	−	0.381471i	0.954882	+	0.296987i	$0.0959818\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2702.79			−1.32304
$162$		0			0
$163$	2984.70			1.43423			0.717116	−	0.696954i	$-0.245462\pi$
$163$	2984.70			1.43423			0.717116	+	0.696954i	$0.245462\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	979.720	−	1696.92i	0.453970	−	0.786299i	−0.544658	−	0.838658i	$-0.683341\pi$
$167$	979.720	−	1696.92i	0.453970	−	0.786299i	0.998628	+	0.0523588i	$0.0166739\pi$
$168$		0			0
$169$	−695.358	−	1204.40i	−0.316503	−	0.548200i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	253.472	+	439.027i	0.111394	+	0.192940i	0.916333	−	0.400418i	$-0.131135\pi$
$173$	253.472	+	439.027i	0.111394	+	0.192940i	−0.804939	+	0.593358i	$0.797802\pi$
$174$		0			0
$175$	−207.141	+	358.779i	−0.0894766	+	0.154978i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3122.26			1.30374			0.651869	−	0.758332i	$-0.273985\pi$
$179$	3122.26			1.30374			0.651869	+	0.758332i	$0.273985\pi$
$180$		0			0
$181$	−2828.38			−1.16150			−0.580751	−	0.814082i	$-0.697241\pi$
$181$	−2828.38			−1.16150			−0.580751	+	0.814082i	$0.697241\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	606.363	−	1050.25i	0.240977	−	0.417384i
$186$		0			0
$187$	574.959	+	995.858i	0.224840	+	0.389435i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2039.32	+	3532.20i	0.772565	+	1.33812i	0.936153	+	0.351593i	$0.114360\pi$
$191$	2039.32	+	3532.20i	0.772565	+	1.33812i	−0.163588	+	0.986529i	$0.552307\pi$
$192$		0			0
$193$	−1521.56	+	2635.42i	−0.567483	+	0.982909i	0.429331	+	0.903147i	$0.358749\pi$
$193$	−1521.56	+	2635.42i	−0.567483	+	0.982909i	−0.996814	+	0.0797619i	$0.974584\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1338.21			−0.483978			−0.241989	−	0.970279i	$-0.577800\pi$
$197$	−1338.21			−0.483978			−0.241989	+	0.970279i	$0.577800\pi$
$198$		0			0
$199$	453.645			0.161598			0.0807991	−	0.996730i	$-0.474253\pi$
$199$	453.645			0.161598			0.0807991	+	0.996730i	$0.474253\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	96.1008	−	166.451i	0.0332264	−	0.0575498i
$204$		0			0
$205$	144.666	+	250.569i	0.0492874	+	0.0853683i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4955.93	+	8583.92i	1.64023	+	2.84097i
$210$		0			0
$211$	−952.979	+	1650.61i	−0.310928	+	0.538543i	−0.978564	−	0.205945i	$-0.933973\pi$
$211$	−952.979	+	1650.61i	−0.310928	+	0.538543i	0.667636	+	0.744488i	$0.267306\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1322.52			−0.419513
$216$		0			0
$217$	−679.452			−0.212554
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−474.774	+	822.333i	−0.144510	+	0.250299i
$222$		0			0
$223$	−2227.76	−	3858.59i	−0.668976	−	1.15870i	−0.978191	−	0.207708i	$-0.933400\pi$
$223$	−2227.76	−	3858.59i	−0.668976	−	1.15870i	0.309215	−	0.950992i	$-0.399934\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	85.6173	+	148.294i	0.0250336	+	0.0433594i	0.878271	−	0.478164i	$-0.158697\pi$
$227$	85.6173	+	148.294i	0.0250336	+	0.0433594i	−0.853237	+	0.521523i	$0.825364\pi$
$228$		0			0
$229$	−2214.97	+	3836.44i	−0.639168	+	1.10707i	0.346448	+	0.938069i	$0.387388\pi$
$229$	−2214.97	+	3836.44i	−0.639168	+	1.10707i	−0.985616	+	0.169002i	$0.945946\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	762.810			0.214478			0.107239	−	0.994233i	$-0.465799\pi$
$233$	762.810			0.214478			0.107239	+	0.994233i	$0.465799\pi$
$234$		0			0
$235$	−2999.45			−0.832606
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1522.14	−	2636.43i	0.411963	−	0.713542i	−0.583141	−	0.812371i	$-0.698176\pi$
$239$	1522.14	−	2636.43i	0.411963	−	0.713542i	0.995104	+	0.0988294i	$0.0315098\pi$
$240$		0			0
$241$	1009.16	+	1747.91i	0.269733	+	0.467190i	0.968793	−	0.247872i	$-0.0797314\pi$
$241$	1009.16	+	1747.91i	0.269733	+	0.467190i	−0.699060	+	0.715063i	$0.746398\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	170.981	+	296.147i	0.0445859	+	0.0772250i
$246$		0			0
$247$	−4092.38	+	7088.20i	−1.05422	+	1.82596i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3031.55			−0.762351			−0.381175	−	0.924503i	$-0.624481\pi$
$251$	−3031.55			−0.762351			−0.381175	+	0.924503i	$0.624481\pi$
$252$		0			0
$253$	−11830.8			−2.93990
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2377.31	−	4117.62i	0.577013	−	0.999417i	−0.418806	−	0.908076i	$-0.637551\pi$
$257$	2377.31	−	4117.62i	0.577013	−	0.999417i	0.995820	−	0.0913409i	$-0.0291153\pi$
$258$		0			0
$259$	2009.65	+	3480.81i	0.482136	+	0.835084i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1982.43	−	3433.67i	−0.464798	−	0.805053i	0.534395	−	0.845235i	$-0.320540\pi$
$263$	−1982.43	−	3433.67i	−0.464798	−	0.805053i	−0.999192	+	0.0401818i	$0.987206\pi$
$264$		0			0
$265$	1480.71	−	2564.67i	0.343243	−	0.594515i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3989.09			−0.904161			−0.452080	−	0.891977i	$-0.649318\pi$
$269$	−3989.09			−0.904161			−0.452080	+	0.891977i	$0.649318\pi$
$270$		0			0
$271$	2684.00			0.601628			0.300814	−	0.953683i	$-0.402742\pi$
$271$	2684.00			0.601628			0.300814	+	0.953683i	$0.402742\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−906.711	+	1570.47i	−0.198824	+	0.344374i
$276$		0			0
$277$	−26.7606	−	46.3507i	−0.00580465	−	0.0100540i	0.863108	−	0.505019i	$-0.168514\pi$
$277$	−26.7606	−	46.3507i	−0.00580465	−	0.0100540i	−0.868913	+	0.494965i	$0.835181\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1819.03	+	3150.65i	0.386172	+	0.668869i	0.991931	−	0.126779i	$-0.0404638\pi$
$281$	1819.03	+	3150.65i	0.386172	+	0.668869i	−0.605759	+	0.795648i	$0.707130\pi$
$282$		0			0
$283$	−3663.02	+	6344.53i	−0.769412	+	1.33266i	0.168470	+	0.985707i	$0.446117\pi$
$283$	−3663.02	+	6344.53i	−0.769412	+	1.33266i	−0.937882	+	0.346954i	$0.887216\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−958.922			−0.197224
$288$		0			0
$289$	−4661.69			−0.948847
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1261.12	−	2184.32i	0.251452	−	0.435527i	−0.712474	−	0.701698i	$-0.752425\pi$
$293$	1261.12	−	2184.32i	0.251452	−	0.435527i	0.963926	+	0.266172i	$0.0857588\pi$
$294$		0			0
$295$	722.231	+	1250.94i	0.142542	+	0.246890i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4884.66	−	8460.48i	−0.944773	−	1.63640i
$300$		0			0
$301$	2191.59	−	3795.94i	0.419671	−	0.726892i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4126.90			0.774772
$306$		0			0
$307$	640.204			0.119017			0.0595087	−	0.998228i	$-0.481047\pi$
$307$	640.204			0.119017			0.0595087	+	0.998228i	$0.481047\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2437.17	+	4221.30i	−0.444371	+	0.769673i	−0.998008	−	0.0630852i	$-0.979906\pi$
$311$	−2437.17	+	4221.30i	−0.444371	+	0.769673i	0.553637	+	0.832758i	$0.313239\pi$
$312$		0			0
$313$	2581.99	+	4472.14i	0.466270	+	0.807604i	0.999258	−	0.0385190i	$-0.0122640\pi$
$313$	2581.99	+	4472.14i	0.466270	+	0.807604i	−0.532987	+	0.846123i	$0.678931\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3655.00	+	6330.64i	0.647587	+	1.12165i	0.983697	+	0.179831i	$0.0575552\pi$
$317$	3655.00	+	6330.64i	0.647587	+	1.12165i	−0.336110	+	0.941823i	$0.609111\pi$
$318$		0			0
$319$	420.658	−	728.601i	0.0738318	−	0.127880i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2166.23			0.373165
$324$		0			0
$325$	−1497.44			−0.255578
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4970.47	−	8609.11i	0.832921	−	1.44266i
$330$		0			0
$331$	5867.79	+	10163.3i	0.974390	+	1.68769i	0.681934	+	0.731414i	$0.261139\pi$
$331$	5867.79	+	10163.3i	0.974390	+	1.68769i	0.292456	+	0.956279i	$0.405527\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2026.32	−	3509.69i	−0.330477	−	0.572402i
$336$		0			0
$337$	−1125.51	+	1949.44i	−0.181930	+	0.315112i	−0.942538	−	0.334100i	$-0.891568\pi$
$337$	−1125.51	+	1949.44i	−0.181930	+	0.315112i	0.760608	+	0.649212i	$0.224901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2974.14			−0.472313
$342$		0			0
$343$	−6817.30			−1.07318
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3479.50	−	6026.66i	0.538297	−	0.932358i	−0.460699	−	0.887557i	$-0.652401\pi$
$347$	3479.50	−	6026.66i	0.538297	−	0.932358i	0.998996	−	0.0448014i	$-0.0142655\pi$
$348$		0			0
$349$	1462.00	+	2532.26i	0.224238	+	0.388392i	0.956091	−	0.293071i	$-0.0946772\pi$
$349$	1462.00	+	2532.26i	0.224238	+	0.388392i	−0.731852	+	0.681463i	$0.761344\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3276.24	−	5674.61i	−0.493984	−	0.855606i	0.505992	−	0.862538i	$-0.331127\pi$
$353$	−3276.24	−	5674.61i	−0.493984	−	0.855606i	−0.999976	+	0.00693255i	$0.997793\pi$
$354$		0			0
$355$	−2415.31	+	4183.45i	−0.361103	+	0.625449i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1324.66			0.194743			0.0973717	−	0.995248i	$-0.468956\pi$
$359$	1324.66			0.194743			0.0973717	+	0.995248i	$0.468956\pi$
$360$		0			0
$361$	11813.1			1.72228
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2007.10	+	3476.40i	−0.287826	+	0.498529i
$366$		0			0
$367$	1628.11	+	2819.96i	0.231571	+	0.401092i	0.958271	−	0.285863i	$-0.0922802\pi$
$367$	1628.11	+	2819.96i	0.231571	+	0.401092i	−0.726700	+	0.686955i	$0.758947\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4907.47	+	8499.98i	0.686747	+	1.18948i
$372$		0			0
$373$	2272.33	−	3935.79i	0.315433	−	0.546347i	−0.664096	−	0.747647i	$-0.731183\pi$
$373$	2272.33	−	3935.79i	0.315433	−	0.546347i	0.979530	+	0.201301i	$0.0645168\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	694.720			0.0949069
$378$		0			0
$379$	−2484.30			−0.336701			−0.168351	−	0.985727i	$-0.553844\pi$
$379$	−2484.30			−0.336701			−0.168351	+	0.985727i	$0.553844\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1414.43	+	2449.87i	−0.188705	+	0.326847i	−0.944819	−	0.327593i	$-0.893762\pi$
$383$	−1414.43	+	2449.87i	−0.188705	+	0.326847i	0.756114	+	0.654440i	$0.227096\pi$
$384$		0			0
$385$	−3005.07	−	5204.94i	−0.397799	−	0.689009i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1295.09	+	2243.16i	0.168801	+	0.292371i	0.937998	−	0.346639i	$-0.112677\pi$
$389$	1295.09	+	2243.16i	0.168801	+	0.292371i	−0.769198	+	0.639011i	$0.779344\pi$
$390$		0			0
$391$	−1292.81	+	2239.21i	−0.167212	+	0.289620i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5810.01			−0.740085
$396$		0			0
$397$	12207.4			1.54325			0.771625	−	0.636077i	$-0.219444\pi$
$397$	12207.4			1.54325			0.771625	+	0.636077i	$0.219444\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4760.20	−	8244.91i	0.592801	−	1.02676i	−0.401052	−	0.916055i	$-0.631355\pi$
$401$	4760.20	−	8244.91i	0.592801	−	1.02676i	0.993853	−	0.110706i	$-0.0353113\pi$
$402$		0			0
$403$	−1227.95	−	2126.88i	−0.151783	−	0.262896i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8796.74	+	15236.4i	1.07135	+	1.85563i
$408$		0			0
$409$	−5623.36	+	9739.94i	−0.679846	+	1.17753i	0.295181	+	0.955441i	$0.404620\pi$
$409$	−5623.36	+	9739.94i	−0.679846	+	1.17753i	−0.975027	+	0.222087i	$0.928713\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4787.32			−0.570384
$414$		0			0
$415$	−2511.87			−0.297116
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5340.99	+	9250.87i	−0.622731	+	1.07860i	0.366244	+	0.930519i	$0.380644\pi$
$419$	−5340.99	+	9250.87i	−0.622731	+	1.07860i	−0.988975	+	0.148083i	$0.952690\pi$
$420$		0			0
$421$	−5411.99	−	9373.84i	−0.626518	−	1.08516i	−0.988245	−	0.152877i	$-0.951146\pi$
$421$	−5411.99	−	9373.84i	−0.626518	−	1.08516i	0.361727	−	0.932284i	$-0.382187\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	198.161	+	343.225i	0.0226170	+	0.0391738i
$426$		0			0
$427$	−6838.81	+	11845.2i	−0.775066	+	1.34245i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	235.904			0.0263645			0.0131823	−	0.999913i	$-0.495804\pi$
$431$	235.904			0.0263645			0.0131823	+	0.999913i	$0.495804\pi$
$432$		0			0
$433$	−11895.5			−1.32024			−0.660118	−	0.751162i	$-0.729494\pi$
$433$	−11895.5			−1.32024			−0.660118	+	0.751162i	$0.729494\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11143.5	+	19301.1i	−1.21983	+	2.11281i
$438$		0			0
$439$	−1905.52	−	3300.46i	−0.207166	−	0.358821i	0.743655	−	0.668564i	$-0.233091\pi$
$439$	−1905.52	−	3300.46i	−0.207166	−	0.358821i	−0.950821	+	0.309742i	$0.899757\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4016.11	−	6956.12i	−0.430725	−	0.746038i	0.566210	−	0.824261i	$-0.308409\pi$
$443$	−4016.11	−	6956.12i	−0.430725	−	0.746038i	−0.996936	+	0.0782223i	$0.975076\pi$
$444$		0			0
$445$	21.9145	−	37.9570i	0.00233449	−	0.00404345i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12271.3			−1.28980			−0.644898	−	0.764269i	$-0.723100\pi$
$449$	−12271.3			−1.28980			−0.644898	+	0.764269i	$0.723100\pi$
$450$		0			0
$451$	−4197.45			−0.438249
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2481.45	−	4298.00i	0.255675	−	0.442842i
$456$		0			0
$457$	−4972.00	−	8611.76i	−0.508929	−	0.881490i	−0.999947	−	0.0103406i	$-0.996708\pi$
$457$	−4972.00	−	8611.76i	−0.508929	−	0.881490i	0.491018	−	0.871149i	$-0.336625\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3374.74	+	5845.22i	0.340949	+	0.590540i	0.984609	−	0.174771i	$-0.0559185\pi$
$461$	3374.74	+	5845.22i	0.340949	+	0.590540i	−0.643661	+	0.765311i	$0.722585\pi$
$462$		0			0
$463$	4780.08	−	8279.34i	0.479804	−	0.831044i	−0.519928	−	0.854210i	$-0.674041\pi$
$463$	4780.08	−	8279.34i	0.479804	−	0.831044i	0.999732	+	0.0231659i	$0.00737460\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6517.56			0.645818			0.322909	−	0.946430i	$-0.395339\pi$
$467$	6517.56			0.645818			0.322909	+	0.946430i	$0.395339\pi$
$468$		0			0
$469$	13431.5			1.32241
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9593.15	−	16615.8i	0.932545	−	1.61521i
$474$		0			0
$475$	1708.07	+	2958.47i	0.164993	+	0.285777i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2376.37	−	4115.99i	−0.226678	−	0.392619i	0.730143	−	0.683294i	$-0.239453\pi$
$479$	−2376.37	−	4115.99i	−0.226678	−	0.392619i	−0.956822	+	0.290676i	$0.906120\pi$
$480$		0			0
$481$	−7263.94	+	12581.5i	−0.688580	+	1.19266i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	691.820			0.0647710
$486$		0			0
$487$	−8412.56			−0.782771			−0.391386	−	0.920227i	$-0.628004\pi$
$487$	−8412.56			−0.782771			−0.391386	+	0.920227i	$0.628004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2785.22	+	4824.15i	−0.255999	+	0.443403i	−0.965166	−	0.261637i	$-0.915738\pi$
$491$	−2785.22	+	4824.15i	−0.255999	+	0.443403i	0.709167	+	0.705040i	$0.249071\pi$
$492$		0			0
$493$	−91.9345	−	159.235i	−0.00839863	−	0.0145469i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8004.98	−	13865.0i	−0.722479	−	1.25137i
$498$		0			0
$499$	7912.32	−	13704.5i	0.709828	−	1.22946i	−0.255093	−	0.966916i	$-0.582106\pi$
$499$	7912.32	−	13704.5i	0.709828	−	1.22946i	0.964921	−	0.262541i	$-0.0845605\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15946.4			1.41354			0.706772	−	0.707441i	$-0.250151\pi$
$503$	15946.4			1.41354			0.706772	+	0.707441i	$0.250151\pi$
$504$		0			0
$505$	6628.89			0.584122
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7319.34	+	12677.5i	−0.637375	+	1.10397i	0.348631	+	0.937260i	$0.386647\pi$
$509$	−7319.34	+	12677.5i	−0.637375	+	1.10397i	−0.986007	+	0.166707i	$0.946687\pi$
$510$		0			0
$511$	−6652.05	−	11521.7i	−0.575869	−	0.997435i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−833.859	−	1444.29i	−0.0713480	−	0.123578i
$516$		0			0
$517$	21757.0	−	37684.3i	1.85082	−	3.20571i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	13001.4			1.09329			0.546644	−	0.837365i	$-0.315905\pi$
$521$	13001.4			1.09329			0.546644	+	0.837365i	$0.315905\pi$
$522$		0			0
$523$	−1938.88			−0.162106			−0.0810531	−	0.996710i	$-0.525828\pi$
$523$	−1938.88			−0.162106			−0.0810531	+	0.996710i	$0.525828\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−324.998	+	562.913i	−0.0268636	+	0.0465292i
$528$		0			0
$529$	−7217.39	−	12500.9i	−0.593194	−	1.02744i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1733.03	−	3001.69i	−0.140836	−	0.243936i
$534$		0			0
$535$	1460.01	−	2528.81i	0.117984	−	0.204355i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4960.96			−0.396444
$540$		0			0
$541$	−7574.14			−0.601918			−0.300959	−	0.953637i	$-0.597307\pi$
$541$	−7574.14			−0.601918			−0.300959	+	0.953637i	$0.597307\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4788.55	−	8294.00i	0.376365	−	0.651883i
$546$		0			0
$547$	1570.94	+	2720.94i	0.122794	+	0.212686i	0.920869	−	0.389873i	$-0.127481\pi$
$547$	1570.94	+	2720.94i	0.122794	+	0.212686i	−0.798074	+	0.602559i	$0.794148\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−792.442	−	1372.55i	−0.0612689	−	0.106121i
$552$		0			0
$553$	9627.94	−	16676.1i	0.740365	−	1.28235i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12191.5			−0.927412			−0.463706	−	0.885989i	$-0.653481\pi$
$557$	−12191.5			−0.927412			−0.463706	+	0.885989i	$0.653481\pi$
$558$		0			0
$559$	15843.2			1.19874
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1091.42	+	1890.40i	−0.0817016	+	0.141511i	−0.903981	−	0.427573i	$-0.859369\pi$
$563$	−1091.42	+	1890.40i	−0.0817016	+	0.141511i	0.822279	+	0.569084i	$0.192702\pi$
$564$		0			0
$565$	3093.29	+	5357.73i	0.230328	+	0.398941i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−892.967	−	1546.66i	−0.0657911	−	0.113953i	0.831254	−	0.555893i	$-0.187624\pi$
$569$	−892.967	−	1546.66i	−0.0657911	−	0.113953i	−0.897045	+	0.441940i	$0.854290\pi$
$570$		0			0
$571$	−1702.15	+	2948.21i	−0.124751	+	0.216075i	−0.921636	−	0.388057i	$-0.873146\pi$
$571$	−1702.15	+	2948.21i	−0.124751	+	0.216075i	0.796885	+	0.604131i	$0.206480\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4077.51			−0.295729
$576$		0			0
$577$	−4174.78			−0.301210			−0.150605	−	0.988594i	$-0.548122\pi$
$577$	−4174.78			−0.301210			−0.150605	+	0.988594i	$0.548122\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4162.50	−	7209.66i	0.297228	−	0.514815i
$582$		0			0
$583$	21481.3	+	37206.6i	1.52601	+	2.64312i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9308.47	−	16122.8i	−0.654518	−	1.13366i	−0.982014	−	0.188806i	$-0.939538\pi$
$587$	−9308.47	−	16122.8i	−0.654518	−	1.13366i	0.327497	−	0.944852i	$-0.393795\pi$
$588$		0			0
$589$	−2801.36	+	4852.10i	−0.195973	+	0.339435i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24181.4			−1.67456			−0.837278	−	0.546777i	$-0.815855\pi$
$593$	−24181.4			−1.67456			−0.837278	+	0.546777i	$0.815855\pi$
$594$		0			0
$595$	−1313.51			−0.0905022
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2669.72	−	4624.08i	0.182106	−	0.315417i	−0.760491	−	0.649348i	$-0.775042\pi$
$599$	2669.72	−	4624.08i	0.182106	−	0.315417i	0.942598	+	0.333931i	$0.108375\pi$
$600$		0			0
$601$	−4582.86	−	7937.75i	−0.311046	−	0.538748i	0.667543	−	0.744571i	$-0.267346\pi$
$601$	−4582.86	−	7937.75i	−0.311046	−	0.538748i	−0.978589	+	0.205823i	$0.934013\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9826.49	−	17020.0i	−0.660336	−	1.14374i
$606$		0			0
$607$	−5715.85	+	9900.15i	−0.382206	+	0.662001i	−0.991377	−	0.131038i	$-0.958169\pi$
$607$	−5715.85	+	9900.15i	−0.382206	+	0.662001i	0.609171	+	0.793039i	$0.291502\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	35931.9			2.37913
$612$		0			0
$613$	29975.6			1.97505			0.987523	−	0.157477i	$-0.0503361\pi$
$613$	29975.6			1.97505			0.987523	+	0.157477i	$0.0503361\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3056.36	+	5293.77i	−0.199424	+	0.345412i	−0.948342	−	0.317251i	$-0.897240\pi$
$617$	−3056.36	+	5293.77i	−0.199424	+	0.345412i	0.748918	+	0.662663i	$0.230574\pi$
$618$		0			0
$619$	−693.987	−	1202.02i	−0.0450625	−	0.0780505i	0.842614	−	0.538517i	$-0.181015\pi$
$619$	−693.987	−	1202.02i	−0.0450625	−	0.0780505i	−0.887677	+	0.460467i	$0.847682\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	72.6303	+	125.799i	0.00467074	+	0.00808996i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3845.04			0.243739
$630$		0			0
$631$	18608.3			1.17399			0.586993	−	0.809592i	$-0.300312\pi$
$631$	18608.3			1.17399			0.586993	+	0.809592i	$0.300312\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4463.56	−	7731.12i	0.278947	−	0.483150i
$636$		0			0
$637$	−2048.26	−	3547.70i	−0.127402	−	0.220667i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4019.77	+	6962.45i	0.247693	+	0.429018i	0.962885	−	0.269910i	$-0.0869941\pi$
$641$	4019.77	+	6962.45i	0.247693	+	0.429018i	−0.715192	+	0.698928i	$0.753661\pi$
$642$		0			0
$643$	13330.7	−	23089.5i	0.817592	−	1.41611i	−0.0898593	−	0.995954i	$-0.528642\pi$
$643$	13330.7	−	23089.5i	0.817592	−	1.41611i	0.907452	−	0.420157i	$-0.138025\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6303.13			0.383001			0.191501	−	0.981492i	$-0.438665\pi$
$647$	6303.13			0.383001			0.191501	+	0.981492i	$0.438665\pi$
$648$		0			0
$649$	−20955.4			−1.26744
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11299.2	−	19570.7i	0.677136	−	1.17283i	−0.298703	−	0.954346i	$-0.596554\pi$
$653$	11299.2	−	19570.7i	0.677136	−	1.17283i	0.975839	−	0.218489i	$-0.0701127\pi$
$654$		0			0
$655$	1804.53	+	3125.54i	0.107647	+	0.186450i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1198.87	−	2076.50i	−0.0708667	−	0.122745i	0.828415	−	0.560115i	$-0.189243\pi$
$659$	−1198.87	−	2076.50i	−0.0708667	−	0.122745i	−0.899281	+	0.437370i	$0.855910\pi$
$660$		0			0
$661$	2531.81	−	4385.23i	0.148980	−	0.258042i	−0.781870	−	0.623441i	$-0.785734\pi$
$661$	2531.81	−	4385.23i	0.148980	−	0.258042i	0.930851	+	0.365399i	$0.119068\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−11322.0			−0.660223
$666$		0			0
$667$	1891.72			0.109816
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−29935.2	+	51849.3i	−1.72226	+	2.98304i
$672$		0			0
$673$	−3640.66	−	6305.81i	−0.208525	−	0.361176i	0.742725	−	0.669596i	$-0.233533\pi$
$673$	−3640.66	−	6305.81i	−0.208525	−	0.361176i	−0.951250	+	0.308421i	$0.900200\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14482.1	+	25083.8i	0.822146	+	1.42400i	0.904081	+	0.427361i	$0.140557\pi$
$677$	14482.1	+	25083.8i	0.822146	+	1.42400i	−0.0819349	+	0.996638i	$0.526110\pi$
$678$		0			0
$679$	−1146.43	+	1985.68i	−0.0647955	+	0.112229i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9158.42			0.513085			0.256543	−	0.966533i	$-0.417417\pi$
$683$	9158.42			0.513085			0.256543	+	0.966533i	$0.417417\pi$
$684$		0			0
$685$	5544.93			0.309286
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−17738.2	+	30723.5i	−0.980802	+	1.69880i
$690$		0			0
$691$	−5014.15	−	8684.76i	−0.276045	−	0.478124i	0.694353	−	0.719634i	$-0.255691\pi$
$691$	−5014.15	−	8684.76i	−0.276045	−	0.478124i	−0.970398	+	0.241510i	$0.922357\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1091.92	−	1891.27i	−0.0595958	−	0.103223i
$696$		0			0
$697$	−458.675	+	794.448i	−0.0249262	+	0.0431734i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27626.7			1.48851			0.744256	−	0.667894i	$-0.232804\pi$
$701$	27626.7			1.48851			0.744256	+	0.667894i	$0.232804\pi$
$702$		0			0
$703$	33142.8			1.77810
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10984.9	+	19026.4i	−0.584343	+	1.01211i
$708$		0			0
$709$	−4851.84	−	8403.64i	−0.257003	−	0.445142i	0.708435	−	0.705776i	$-0.249402\pi$
$709$	−4851.84	−	8403.64i	−0.257003	−	0.445142i	−0.965438	+	0.260635i	$0.916068\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3343.70	−	5791.47i	−0.175628	−	0.304197i
$714$		0			0
$715$	10861.9	−	18813.4i	0.568131	−	0.984032i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	32045.4			1.66216			0.831080	−	0.556153i	$-0.187723\pi$
$719$	32045.4			1.66216			0.831080	+	0.556153i	$0.187723\pi$
$720$		0			0
$721$	5527.25			0.285500
$722$		0			0
$723$		0			0
$724$		0			0
$725$	144.981	−	251.114i	0.00742683	−	0.0128637i
$726$		0			0
$727$	3723.63	+	6449.51i	0.189961	+	0.329022i	0.945237	−	0.326385i	$-0.105830\pi$
$727$	3723.63	+	6449.51i	0.189961	+	0.329022i	−0.755276	+	0.655407i	$0.772497\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2096.58	−	3631.38i	−0.106080	−	0.183736i
$732$		0			0
$733$	9072.37	−	15713.8i	0.457156	−	0.791818i	−0.541653	−	0.840602i	$-0.682201\pi$
$733$	9072.37	−	15713.8i	0.457156	−	0.791818i	0.998809	+	0.0487844i	$0.0155347\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	58793.1			2.93850
$738$		0			0
$739$	25920.2			1.29024			0.645121	−	0.764081i	$-0.276807\pi$
$739$	25920.2			1.29024			0.645121	+	0.764081i	$0.276807\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4656.55	−	8065.38i	0.229922	−	0.398237i	−0.727863	−	0.685723i	$-0.759486\pi$
$743$	4656.55	−	8065.38i	0.229922	−	0.398237i	0.957785	+	0.287486i	$0.0928195\pi$
$744$		0			0
$745$	−442.792	−	766.939i	−0.0217754	−	0.0377161i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4838.85	+	8381.13i	0.236058	+	0.408865i
$750$		0			0
$751$	2492.14	−	4316.52i	0.121091	−	0.209736i	−0.799107	−	0.601189i	$-0.794694\pi$
$751$	2492.14	−	4316.52i	0.121091	−	0.209736i	0.920198	+	0.391453i	$0.128027\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4143.29			0.199722
$756$		0			0
$757$	−10452.6			−0.501857			−0.250928	−	0.968006i	$-0.580736\pi$
$757$	−10452.6			−0.501857			−0.250928	+	0.968006i	$0.580736\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4702.97	+	8145.78i	−0.224024	+	0.388022i	−0.956026	−	0.293281i	$-0.905253\pi$
$761$	−4702.97	+	8145.78i	−0.224024	+	0.388022i	0.732002	+	0.681303i	$0.238586\pi$
$762$		0			0
$763$	15870.5	+	27488.5i	0.753014	+	1.30426i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8651.98	−	14985.7i	−0.407307	−	0.705477i
$768$		0			0
$769$	11784.1	−	20410.7i	0.552596	−	0.957125i	−0.445490	−	0.895287i	$-0.646970\pi$
$769$	11784.1	−	20410.7i	0.552596	−	0.957125i	0.998086	−	0.0618381i	$-0.0196962\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3172.70			−0.147625			−0.0738125	−	0.997272i	$-0.523517\pi$
$773$	−3172.70			−0.147625			−0.0738125	+	0.997272i	$0.523517\pi$
$774$		0			0
$775$	−1025.04			−0.0475106
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3953.60	+	6847.84i	−0.181839	+	0.314954i
$780$		0			0
$781$	−35039.9	−	60690.8i	−1.60541	−	2.78065i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2166.31	+	3752.16i	0.0984954	+	0.170599i
$786$		0			0
$787$	−7721.19	+	13373.5i	−0.349721	+	0.605735i	−0.986200	−	0.165559i	$-0.947057\pi$
$787$	−7721.19	+	13373.5i	−0.349721	+	0.605735i	0.636479	+	0.771294i	$0.280390\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−20503.9			−0.921663
$792$		0			0
$793$	−49438.2			−2.21388
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4984.27	−	8633.02i	0.221521	−	0.383685i	−0.733749	−	0.679420i	$-0.762231\pi$
$797$	4984.27	−	8633.02i	0.221521	−	0.383685i	0.955270	+	0.295735i	$0.0955646\pi$
$798$		0			0
$799$	−4754.99	−	8235.88i	−0.210537	−	0.364661i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−29117.7	−	50433.4i	−1.27963	−	2.21638i
$804$		0			0
$805$	6756.97	−	11703.4i	0.295841	−	0.512411i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10439.0			−0.453667			−0.226833	−	0.973934i	$-0.572837\pi$
$809$	−10439.0			−0.453667			−0.226833	+	0.973934i	$0.572837\pi$
$810$		0			0
$811$	17472.9			0.756544			0.378272	−	0.925694i	$-0.376518\pi$
$811$	17472.9			0.756544			0.378272	+	0.925694i	$0.376518\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7461.75	+	12924.1i	−0.320704	+	0.555476i
$816$		0			0
$817$	−18071.7	−	31301.1i	−0.773867	−	1.34038i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9750.89	+	16889.0i	0.414505	+	0.717943i	0.995376	−	0.0960521i	$-0.0306216\pi$
$821$	9750.89	+	16889.0i	0.414505	+	0.717943i	−0.580872	+	0.813995i	$0.697288\pi$
$822$		0			0
$823$	8296.21	−	14369.5i	0.351383	−	0.608612i	−0.635109	−	0.772422i	$-0.719045\pi$
$823$	8296.21	−	14369.5i	0.351383	−	0.608612i	0.986492	+	0.163810i	$0.0523783\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−32449.4			−1.36442			−0.682210	−	0.731156i	$-0.738981\pi$
$827$	−32449.4			−1.36442			−0.682210	+	0.731156i	$0.738981\pi$
$828$		0			0
$829$	7197.10			0.301527			0.150763	−	0.988570i	$-0.451827\pi$
$829$	7197.10			0.301527			0.150763	+	0.988570i	$0.451827\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−542.107	+	938.956i	−0.0225485	+	0.0390551i
$834$		0			0
$835$	4898.60	+	8484.62i	0.203022	+	0.351644i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7715.97	−	13364.4i	−0.317503	−	0.549931i	0.662464	−	0.749094i	$-0.269511\pi$
$839$	−7715.97	−	13364.4i	−0.317503	−	0.549931i	−0.979966	+	0.199163i	$0.936178\pi$
$840$		0			0
$841$	12127.2	−	21005.0i	0.497242	−	0.861249i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6953.58			0.283089
$846$		0			0
$847$	65135.0			2.64235
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−19779.6	+	34259.3i	−0.796753	+	1.38002i
$852$		0			0
$853$	−3957.37	−	6854.36i	−0.158848	−	0.275134i	0.775605	−	0.631218i	$-0.217445\pi$
$853$	−3957.37	−	6854.36i	−0.158848	−	0.275134i	−0.934454	+	0.356085i	$0.884111\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	13406.4	+	23220.6i	0.534369	+	0.925554i	0.999194	+	0.0401510i	$0.0127839\pi$
$857$	13406.4	+	23220.6i	0.534369	+	0.925554i	−0.464825	+	0.885403i	$0.653883\pi$
$858$		0			0
$859$	−809.265	+	1401.69i	−0.0321441	+	0.0556752i	−0.881650	−	0.471904i	$-0.843567\pi$
$859$	−809.265	+	1401.69i	−0.0321441	+	0.0556752i	0.849506	+	0.527579i	$0.176900\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−16678.5			−0.657869			−0.328935	−	0.944353i	$-0.606690\pi$
$863$	−16678.5			−0.657869			−0.328935	+	0.944353i	$0.606690\pi$
$864$		0			0
$865$	−2534.72			−0.0996338
$866$		0			0
$867$		0			0
$868$		0			0
$869$	42144.0	−	72995.6i	1.64515	−	2.84949i
$870$		0			0
$871$	24274.3	+	42044.3i	0.944321	+	1.63561i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1035.71	−	1793.90i	−0.0400151	−	0.0693083i
$876$		0			0
$877$	1064.16	−	1843.17i	0.0409738	−	0.0709686i	−0.844811	−	0.535064i	$-0.820287\pi$
$877$	1064.16	−	1843.17i	0.0409738	−	0.0709686i	0.885785	+	0.464096i	$0.153621\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2379.31			−0.0909886			−0.0454943	−	0.998965i	$-0.514486\pi$
$881$	−2379.31			−0.0909886			−0.0454943	+	0.998965i	$0.514486\pi$
$882$		0			0
$883$	−36926.5			−1.40733			−0.703666	−	0.710531i	$-0.748455\pi$
$883$	−36926.5			−1.40733			−0.703666	+	0.710531i	$0.748455\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2390.70	+	4140.81i	−0.0904981	+	0.156747i	−0.907721	−	0.419575i	$-0.862179\pi$
$887$	−2390.70	+	4140.81i	−0.0904981	+	0.156747i	0.817223	+	0.576322i	$0.195513\pi$
$888$		0			0
$889$	14793.4	+	25622.9i	0.558104	+	0.966665i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−40986.2	−	70990.2i	−1.53589	−	2.66024i
$894$		0			0
$895$	−7805.66	+	13519.8i	−0.291525	+	0.504935i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	475.558			0.0176426
$900$		0			0
$901$	9389.43			0.347178
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7070.95	−	12247.2i	0.259720	−	0.449847i
$906$		0			0
$907$	−24032.4	−	41625.3i	−0.879804	−	1.52387i	−0.851556	−	0.524264i	$-0.824340\pi$
$907$	−24032.4	−	41625.3i	−0.879804	−	1.52387i	−0.0282484	−	0.999601i	$-0.508993\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12817.2	+	22200.1i	0.466140	+	0.807378i	0.999252	−	0.0386666i	$-0.0123110\pi$
$911$	12817.2	+	22200.1i	0.466140	+	0.807378i	−0.533112	+	0.846044i	$0.678978\pi$
$912$		0			0
$913$	18220.3	−	31558.6i	0.660466	−	1.14396i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−11961.4			−0.430751
$918$		0			0
$919$	−9651.68			−0.346441			−0.173221	−	0.984883i	$-0.555417\pi$
$919$	−9651.68			−0.346441			−0.173221	+	0.984883i	$0.555417\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	28934.3	−	50115.6i	1.03183	−	1.78719i
$924$		0			0
$925$	3031.82	+	5251.26i	0.107768	+	0.186660i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	166.997	+	289.248i	0.00589775	+	0.0102152i	0.868959	−	0.494884i	$-0.164789\pi$
$929$	166.997	+	289.248i	0.00589775	+	0.0102152i	−0.863061	+	0.505099i	$0.831456\pi$
$930$		0			0
$931$	−4672.76	+	8093.45i	−0.164493	+	0.284911i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5749.59			−0.201103
$936$		0			0
$937$	11567.3			0.403295			0.201648	−	0.979458i	$-0.435370\pi$
$937$	11567.3			0.403295			0.201648	+	0.979458i	$0.435370\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10130.8	+	17547.1i	−0.350963	+	0.607886i	−0.986418	−	0.164252i	$-0.947479\pi$
$941$	−10130.8	+	17547.1i	−0.350963	+	0.607886i	0.635455	+	0.772138i	$0.280812\pi$
$942$		0			0
$943$	−4719.02	−	8173.59i	−0.162961	−	0.282257i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13676.7	+	23688.8i	0.469307	+	0.812863i	0.999384	−	0.0350859i	$-0.0111705\pi$
$947$	13676.7	+	23688.8i	0.469307	+	0.812863i	−0.530077	+	0.847949i	$0.677837\pi$
$948$		0			0
$949$	24044.1	−	41645.5i	0.822448	−	1.42452i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−45432.0			−1.54427			−0.772133	−	0.635460i	$-0.780810\pi$
$953$	−45432.0			−1.54427			−0.772133	+	0.635460i	$0.780810\pi$
$954$		0			0
$955$	−20393.2			−0.691003
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9188.67	+	15915.2i	−0.309403	+	0.535902i
$960$		0			0
$961$	14054.9	+	24343.8i	0.471784	+	0.817154i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7607.79	−	13177.1i	−0.253786	−	0.439570i
$966$		0			0
$967$	3567.09	−	6178.39i	0.118625	−	0.205464i	−0.800598	−	0.599202i	$-0.795485\pi$
$967$	3567.09	−	6178.39i	0.118625	−	0.205464i	0.919223	+	0.393738i	$0.128818\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−31719.4			−1.04833			−0.524163	−	0.851618i	$-0.675622\pi$
$971$	−31719.4			−1.04833			−0.524163	+	0.851618i	$0.675622\pi$
$972$		0			0
$973$	7237.84			0.238473
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8785.15	+	15216.3i	−0.287678	+	0.498274i	−0.973255	−	0.229727i	$-0.926217\pi$
$977$	−8785.15	+	15216.3i	−0.287678	+	0.498274i	0.685577	+	0.728000i	$0.259550\pi$
$978$		0			0
$979$	317.922	+	550.656i	0.0103788	+	0.0179766i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−28993.6	−	50218.5i	−0.940746	−	1.62942i	−0.764052	−	0.645154i	$-0.776793\pi$
$983$	−28993.6	−	50218.5i	−0.940746	−	1.62942i	−0.176694	−	0.984266i	$-0.556540\pi$
$984$		0			0
$985$	3345.53	−	5794.63i	0.108221	−	0.187444i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	43140.8			1.38705
$990$		0			0
$991$	20741.2			0.664851			0.332425	−	0.943130i	$-0.392133\pi$
$991$	20741.2			0.664851			0.332425	+	0.943130i	$0.392133\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1134.11	+	1964.34i	−0.0361344	+	0.0625867i
$996$		0			0
$997$	4960.70	+	8592.19i	0.157580	+	0.272936i	0.933995	−	0.357285i	$-0.116298\pi$
$997$	4960.70	+	8592.19i	0.157580	+	0.272936i	−0.776416	+	0.630221i	$0.782964\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.w.1081.2		12
3.2	odd	2		1620.4.i.x.1081.2		12
9.2	odd	6		1620.4.i.x.541.2		12
9.4	even	3		1620.4.a.j.1.5	yes	6
9.5	odd	6		1620.4.a.i.1.5	✓	6
9.7	even	3	inner	1620.4.i.w.541.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.i.1.5	✓	6	9.5	odd	6
1620.4.a.j.1.5	yes	6	9.4	even	3
1620.4.i.w.541.2		12	9.7	even	3	inner
1620.4.i.w.1081.2		12	1.1	even	1	trivial
1620.4.i.x.541.2		12	9.2	odd	6
1620.4.i.x.1081.2		12	3.2	odd	2

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 4.33013i) q^{5} +(-8.28565 - 14.3512i) q^{7} +O(q^{10})\)	\(q+(-2.50000 + 4.33013i) q^{5} +(-8.28565 - 14.3512i) q^{7} +(-36.2684 - 62.8188i) q^{11} +(29.9488 - 51.8728i) q^{13} -15.8529 q^{17} -136.646 q^{19} +(81.5503 - 141.249i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(5.79923 + 10.0446i) q^{29} +(20.5009 - 35.5086i) q^{31} +82.8565 q^{35} -242.545 q^{37} +(28.9332 - 50.1138i) q^{41} +(132.252 + 229.067i) q^{43} +(299.945 + 519.519i) q^{47} +(34.1961 - 59.2294i) q^{49} -592.285 q^{53} +362.684 q^{55} +(144.446 - 250.188i) q^{59} +(-412.690 - 714.800i) q^{61} +(149.744 + 259.364i) q^{65} +(-405.264 + 701.938i) q^{67} +966.126 q^{71} +802.840 q^{73} +(-601.015 + 1040.99i) q^{77} +(581.001 + 1006.32i) q^{79} +(251.187 + 435.069i) q^{83} +(39.6322 - 68.6450i) q^{85} -8.76580 q^{89} -992.580 q^{91} +(341.615 - 591.694i) q^{95} +(-69.1820 - 119.827i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 30 q^{5} - 12 q^{7}+O(q^{10}) \) 12 * q - 30 * q^5 - 12 * q^7	\( 12 q - 30 q^{5} - 12 q^{7} + 84 q^{13} - 24 q^{17} - 228 q^{19} + 30 q^{23} - 150 q^{25} + 168 q^{29} + 324 q^{31} + 120 q^{35} - 984 q^{37} + 312 q^{41} + 156 q^{43} + 462 q^{47} + 588 q^{49} - 2028 q^{53} + 1008 q^{59} - 36 q^{61} + 420 q^{65} - 144 q^{67} - 2424 q^{71} - 1800 q^{73} + 672 q^{77} + 936 q^{79} + 288 q^{83} + 60 q^{85} + 240 q^{89} + 4572 q^{91} + 570 q^{95} + 1188 q^{97}+O(q^{100}) \) 12 * q - 30 * q^5 - 12 * q^7 + 84 * q^13 - 24 * q^17 - 228 * q^19 + 30 * q^23 - 150 * q^25 + 168 * q^29 + 324 * q^31 + 120 * q^35 - 984 * q^37 + 312 * q^41 + 156 * q^43 + 462 * q^47 + 588 * q^49 - 2028 * q^53 + 1008 * q^59 - 36 * q^61 + 420 * q^65 - 144 * q^67 - 2424 * q^71 - 1800 * q^73 + 672 * q^77 + 936 * q^79 + 288 * q^83 + 60 * q^85 + 240 * q^89 + 4572 * q^91 + 570 * q^95 + 1188 * q^97

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