LMFDB - Embedded newform 648.4.i.t.217.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(217,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.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:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-67})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 16x^{2} - 17x + 289 $ x^4 - x^3 - 16x^2 - 17x + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$3.79436 + 1.61333i$ of defining polynomial
Character	$\chi$	$=$	648.217
Dual form			648.4.i.t.433.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.08872 - 8.81393i) q^{5} +(14.5887 - 25.2684i) q^{7} +O(q^{10})$	$q+(-5.08872 - 8.81393i) q^{5} +(14.5887 - 25.2684i) q^{7} +(9.76617 - 16.9155i) q^{11} +(-30.0887 - 52.1152i) q^{13} +102.420 q^{17} -113.887 q^{19} +(-7.41128 - 12.8367i) q^{23} +(10.7098 - 18.5499i) q^{25} +(34.0887 - 59.0434i) q^{29} +(79.4196 + 137.559i) q^{31} -296.952 q^{35} -75.8226 q^{37} +(182.177 + 315.541i) q^{41} +(-145.541 + 252.084i) q^{43} +(227.177 - 393.483i) q^{47} +(-254.162 - 440.221i) q^{49} -560.839 q^{53} -198.789 q^{55} +(179.532 + 310.959i) q^{59} +(-191.508 + 331.702i) q^{61} +(-306.226 + 530.400i) q^{65} +(-368.476 - 638.219i) q^{67} +360.468 q^{71} -1010.19 q^{73} +(-284.952 - 493.551i) q^{77} +(296.138 - 512.925i) q^{79} +(115.629 - 200.274i) q^{83} +(-521.185 - 902.719i) q^{85} -1323.00 q^{89} -1755.82 q^{91} +(579.541 + 1003.79i) q^{95} +(236.436 - 409.519i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 30 q^{7}+O(q^{10}) $ 4 * q + 8 * q^5 + 30 * q^7	$ 4 q + 8 q^{5} + 30 q^{7} - 46 q^{11} - 92 q^{13} - 44 q^{17} - 172 q^{19} - 58 q^{23} - 184 q^{25} + 108 q^{29} - 136 q^{31} - 564 q^{35} - 360 q^{37} + 672 q^{41} + 70 q^{43} + 852 q^{47} - 166 q^{49} - 1336 q^{53} - 2780 q^{55} + 548 q^{59} - 284 q^{61} - 34 q^{65} - 1162 q^{67} + 1612 q^{71} - 3020 q^{73} - 516 q^{77} + 22 q^{79} + 1540 q^{83} - 3304 q^{85} - 5292 q^{89} - 3564 q^{91} + 1666 q^{95} - 472 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 30 * q^7 - 46 * q^11 - 92 * q^13 - 44 * q^17 - 172 * q^19 - 58 * q^23 - 184 * q^25 + 108 * q^29 - 136 * q^31 - 564 * q^35 - 360 * q^37 + 672 * q^41 + 70 * q^43 + 852 * q^47 - 166 * q^49 - 1336 * q^53 - 2780 * q^55 + 548 * q^59 - 284 * q^61 - 34 * q^65 - 1162 * q^67 + 1612 * q^71 - 3020 * q^73 - 516 * q^77 + 22 * q^79 + 1540 * q^83 - 3304 * q^85 - 5292 * q^89 - 3564 * q^91 + 1666 * q^95 - 472 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−5.08872	−	8.81393i	−0.455149	−	0.788342i	0.543548	−	0.839378i	$-0.317081\pi$
$5$	−5.08872	−	8.81393i	−0.455149	−	0.788342i	−0.998697	+	0.0510368i	$0.983747\pi$
$6$		0			0
$7$	14.5887	−	25.2684i	0.787717	−	1.36437i	−0.139645	−	0.990202i	$-0.544596\pi$
$7$	14.5887	−	25.2684i	0.787717	−	1.36437i	0.927362	−	0.374164i	$-0.122070\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	9.76617	−	16.9155i	0.267692	−	0.463656i	−0.700573	−	0.713580i	$-0.747072\pi$
$11$	9.76617	−	16.9155i	0.267692	−	0.463656i	0.968265	+	0.249924i	$0.0804057\pi$
$12$		0			0
$13$	−30.0887	−	52.1152i	−0.641932	−	1.11186i	−0.985001	−	0.172548i	$-0.944800\pi$
$13$	−30.0887	−	52.1152i	−0.641932	−	1.11186i	0.343070	−	0.939310i	$-0.388533\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	102.420			1.46120			0.730600	−	0.682806i	$-0.239241\pi$
$17$	102.420			1.46120			0.730600	+	0.682806i	$0.239241\pi$
$18$		0			0
$19$	−113.887			−1.37513			−0.687566	−	0.726122i	$-0.741321\pi$
$19$	−113.887			−1.37513			−0.687566	+	0.726122i	$0.741321\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.41128	−	12.8367i	−0.0671895	−	0.116376i	0.830474	−	0.557058i	$-0.188070\pi$
$23$	−7.41128	−	12.8367i	−0.0671895	−	0.116376i	−0.897663	+	0.440682i	$0.854737\pi$
$24$		0			0
$25$	10.7098	−	18.5499i	0.0856783	−	0.148399i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	34.0887	−	59.0434i	0.218280	−	0.378072i	−0.736002	−	0.676979i	$-0.763289\pi$
$29$	34.0887	−	59.0434i	0.218280	−	0.378072i	0.954282	+	0.298907i	$0.0966221\pi$
$30$		0			0
$31$	79.4196	+	137.559i	0.460135	+	0.796977i	0.998967	−	0.0454362i	$-0.0144678\pi$
$31$	79.4196	+	137.559i	0.460135	+	0.796977i	−0.538833	+	0.842413i	$0.681134\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−296.952			−1.43412
$36$		0			0
$37$	−75.8226			−0.336896			−0.168448	−	0.985711i	$-0.553876\pi$
$37$	−75.8226			−0.336896			−0.168448	+	0.985711i	$0.553876\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	182.177	+	315.541i	0.693935	+	1.20193i	0.970538	+	0.240947i	$0.0774579\pi$
$41$	182.177	+	315.541i	0.693935	+	1.20193i	−0.276603	+	0.960984i	$0.589209\pi$
$42$		0			0
$43$	−145.541	+	252.084i	−0.516157	+	0.894010i	0.483667	+	0.875252i	$0.339304\pi$
$43$	−145.541	+	252.084i	−0.516157	+	0.894010i	−0.999824	+	0.0187578i	$0.994029\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	227.177	−	393.483i	0.705048	−	1.22118i	−0.261627	−	0.965169i	$-0.584259\pi$
$47$	227.177	−	393.483i	0.705048	−	1.22118i	0.966674	−	0.256009i	$-0.0824077\pi$
$48$		0			0
$49$	−254.162	−	440.221i	−0.740996	−	1.28344i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−560.839			−1.45353			−0.726766	−	0.686885i	$-0.758977\pi$
$53$	−560.839			−1.45353			−0.726766	+	0.686885i	$0.758977\pi$
$54$		0			0
$55$	−198.789			−0.487359
$56$		0			0
$57$		0			0
$58$		0			0
$59$	179.532	+	310.959i	0.396155	+	0.686160i	0.993248	−	0.116012i	$-0.0370111\pi$
$59$	179.532	+	310.959i	0.396155	+	0.686160i	−0.597093	+	0.802172i	$0.703678\pi$
$60$		0			0
$61$	−191.508	+	331.702i	−0.401969	+	0.696231i	−0.993964	−	0.109711i	$-0.965007\pi$
$61$	−191.508	+	331.702i	−0.401969	+	0.696231i	0.591994	+	0.805942i	$0.298341\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−306.226	+	530.400i	−0.584349	+	1.01212i
$66$		0			0
$67$	−368.476	−	638.219i	−0.671888	−	1.16374i	−0.977368	−	0.211546i	$-0.932150\pi$
$67$	−368.476	−	638.219i	−0.671888	−	1.16374i	0.305480	−	0.952199i	$-0.401183\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	360.468			0.602530			0.301265	−	0.953540i	$-0.402591\pi$
$71$	360.468			0.602530			0.301265	+	0.953540i	$0.402591\pi$
$72$		0			0
$73$	−1010.19			−1.61965			−0.809824	−	0.586673i	$-0.800437\pi$
$73$	−1010.19			−1.61965			−0.809824	+	0.586673i	$0.800437\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−284.952	−	493.551i	−0.421731	−	0.730459i
$78$		0			0
$79$	296.138	−	512.925i	0.421748	−	0.730489i	−0.574363	−	0.818601i	$-0.694750\pi$
$79$	296.138	−	512.925i	0.421748	−	0.730489i	0.996111	+	0.0881121i	$0.0280834\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	115.629	−	200.274i	0.152914	−	0.264855i	−0.779383	−	0.626547i	$-0.784468\pi$
$83$	115.629	−	200.274i	0.152914	−	0.264855i	0.932298	+	0.361692i	$0.117801\pi$
$84$		0			0
$85$	−521.185	−	902.719i	−0.665064	−	1.15192i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1323.00			−1.57570			−0.787852	−	0.615864i	$-0.788807\pi$
$89$	−1323.00			−1.57570			−0.787852	+	0.615864i	$0.788807\pi$
$90$		0			0
$91$	−1755.82			−2.02264
$92$		0			0
$93$		0			0
$94$		0			0
$95$	579.541	+	1003.79i	0.625891	+	1.08407i
$96$		0			0
$97$	236.436	−	409.519i	0.247489	−	0.428664i	−0.715339	−	0.698777i	$-0.753728\pi$
$97$	236.436	−	409.519i	0.247489	−	0.428664i	0.962829	+	0.270113i	$0.0870612\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	747.050	−	1293.93i	0.735982	−	1.27476i	−0.218308	−	0.975880i	$-0.570054\pi$
$101$	747.050	−	1293.93i	0.735982	−	1.27476i	0.954291	−	0.298879i	$-0.0966128\pi$
$102$		0			0
$103$	−682.275	−	1181.74i	−0.652685	−	1.13048i	−0.982469	−	0.186427i	$-0.940309\pi$
$103$	−682.275	−	1181.74i	−0.652685	−	1.13048i	0.329783	−	0.944057i	$-0.393024\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1239.87			1.12022			0.560108	−	0.828419i	$-0.310760\pi$
$107$	1239.87			1.12022			0.560108	+	0.828419i	$0.310760\pi$
$108$		0			0
$109$	197.405			0.173467			0.0867337	−	0.996232i	$-0.472357\pi$
$109$	197.405			0.173467			0.0867337	+	0.996232i	$0.472357\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	943.984	+	1635.03i	0.785863	+	1.36116i	0.928482	+	0.371378i	$0.121114\pi$
$113$	943.984	+	1635.03i	0.785863	+	1.36116i	−0.142618	+	0.989778i	$0.545552\pi$
$114$		0			0
$115$	−75.4279	+	130.645i	−0.0611625	+	0.105937i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1494.17	−	2587.98i	1.15101	−	1.99361i
$120$		0			0
$121$	474.744	+	822.280i	0.356682	+	0.617791i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1490.18			−1.06628
$126$		0			0
$127$	−1612.95			−1.12698			−0.563490	−	0.826123i	$-0.690541\pi$
$127$	−1612.95			−1.12698			−0.563490	+	0.826123i	$0.690541\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−991.833	−	1717.90i	−0.661502	−	1.14576i	−0.980221	−	0.197906i	$-0.936586\pi$
$131$	−991.833	−	1717.90i	−0.661502	−	1.14576i	0.318719	−	0.947849i	$-0.396748\pi$
$132$		0			0
$133$	−1661.47	+	2877.75i	−1.08322	+	1.87618i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−880.016	+	1524.23i	−0.548794	+	0.950540i	0.449563	+	0.893249i	$0.351580\pi$
$137$	−880.016	+	1524.23i	−0.548794	+	0.950540i	−0.998358	+	0.0572911i	$0.981754\pi$
$138$		0			0
$139$	1116.63	+	1934.06i	0.681377	+	1.18018i	0.974561	+	0.224123i	$0.0719517\pi$
$139$	1116.63	+	1934.06i	0.681377	+	1.18018i	−0.293184	+	0.956056i	$0.594715\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1175.41			−0.687360
$144$		0			0
$145$	−693.872			−0.397400
$146$		0			0
$147$		0			0
$148$		0			0
$149$	861.880	+	1492.82i	0.473879	+	0.820782i	0.999553	−	0.0299039i	$-0.00952014\pi$
$149$	861.880	+	1492.82i	0.473879	+	0.820782i	−0.525674	+	0.850686i	$0.676187\pi$
$150$		0			0
$151$	979.857	−	1697.16i	0.528077	−	0.914657i	−0.471387	−	0.881926i	$-0.656246\pi$
$151$	979.857	−	1697.16i	0.528077	−	0.914657i	0.999464	−	0.0327302i	$-0.0104202\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	808.289	−	1400.00i	0.418860	−	0.725487i
$156$		0			0
$157$	−1590.04	−	2754.03i	−0.808275	−	1.39997i	−0.914058	−	0.405584i	$-0.867068\pi$
$157$	−1590.04	−	2754.03i	−0.808275	−	1.39997i	0.105783	−	0.994389i	$-0.466265\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−432.484			−0.211705
$162$		0			0
$163$	−1654.55			−0.795056			−0.397528	−	0.917590i	$-0.630132\pi$
$163$	−1654.55			−0.795056			−0.397528	+	0.917590i	$0.630132\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	225.945	+	391.349i	0.104696	+	0.181338i	0.913614	−	0.406583i	$-0.133280\pi$
$167$	225.945	+	391.349i	0.104696	+	0.181338i	−0.808918	+	0.587921i	$0.799946\pi$
$168$		0			0
$169$	−712.163	+	1233.50i	−0.324152	+	0.561448i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1474.82	+	2554.46i	−0.648140	+	1.12261i	0.335427	+	0.942066i	$0.391120\pi$
$173$	−1474.82	+	2554.46i	−0.648140	+	1.12261i	−0.983567	+	0.180545i	$0.942214\pi$
$174$		0			0
$175$	−312.484	−	541.239i	−0.134981	−	0.233793i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	865.161			0.361258			0.180629	−	0.983551i	$-0.442187\pi$
$179$	865.161			0.361258			0.180629	+	0.983551i	$0.442187\pi$
$180$		0			0
$181$	2669.03			1.09606			0.548032	−	0.836457i	$-0.315377\pi$
$181$	2669.03			1.09606			0.548032	+	0.836457i	$0.315377\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	385.840	+	668.294i	0.153338	+	0.265589i
$186$		0			0
$187$	1000.25	−	1732.48i	0.391151	−	0.677494i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1146.66	+	1986.06i	−0.434393	+	0.752391i	−0.997246	−	0.0741664i	$-0.976370\pi$
$191$	−1146.66	+	1986.06i	−0.434393	+	0.752391i	0.562853	+	0.826557i	$0.309704\pi$
$192$		0			0
$193$	−1971.23	−	3414.27i	−0.735192	−	1.27339i	−0.954639	−	0.297766i	$-0.903759\pi$
$193$	−1971.23	−	3414.27i	−0.735192	−	1.27339i	0.219447	−	0.975624i	$-0.429575\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	27.6302			0.00999275			0.00499637	−	0.999988i	$-0.498410\pi$
$197$	27.6302			0.00999275			0.00499637	+	0.999988i	$0.498410\pi$
$198$		0			0
$199$	3076.20			1.09581			0.547904	−	0.836541i	$-0.315426\pi$
$199$	3076.20			1.09581			0.547904	+	0.836541i	$0.315426\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−994.622	−	1722.74i	−0.343886	−	0.595627i
$204$		0			0
$205$	1854.10	−	3211.40i	0.631688	−	1.09412i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1112.24	+	1926.46i	−0.368112	+	0.637589i
$210$		0			0
$211$	1493.52	+	2586.86i	0.487291	+	0.844013i	0.999893	−	0.0146133i	$-0.00465171\pi$
$211$	1493.52	+	2586.86i	0.487291	+	0.844013i	−0.512602	+	0.858626i	$0.671318\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2962.46			0.939713
$216$		0			0
$217$	4634.52			1.44982
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3081.67	−	5337.62i	−0.937990	−	1.62465i
$222$		0			0
$223$	−709.833	+	1229.47i	−0.213157	+	0.369198i	−0.952701	−	0.303910i	$-0.901708\pi$
$223$	−709.833	+	1229.47i	−0.213157	+	0.369198i	0.739544	+	0.673108i	$0.235041\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1192.33	−	2065.18i	0.348625	−	0.603837i	−0.637380	−	0.770549i	$-0.719982\pi$
$227$	1192.33	−	2065.18i	0.348625	−	0.603837i	0.986006	+	0.166713i	$0.0533152\pi$
$228$		0			0
$229$	973.168	+	1685.58i	0.280824	+	0.486402i	0.971588	−	0.236679i	$-0.0760588\pi$
$229$	973.168	+	1685.58i	0.280824	+	0.486402i	−0.690764	+	0.723081i	$0.742726\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2063.74			−0.580259			−0.290129	−	0.956987i	$-0.593698\pi$
$233$	−2063.74			−0.580259			−0.290129	+	0.956987i	$0.593698\pi$
$234$		0			0
$235$	−4624.17			−1.28361
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1058.50	+	1833.38i	0.286480	+	0.496198i	0.972967	−	0.230944i	$-0.0741815\pi$
$239$	1058.50	+	1833.38i	0.286480	+	0.496198i	−0.686487	+	0.727142i	$0.740848\pi$
$240$		0			0
$241$	−1417.98	+	2456.02i	−0.379006	+	0.656457i	−0.990918	−	0.134469i	$-0.957067\pi$
$241$	−1417.98	+	2456.02i	−0.379006	+	0.656457i	0.611912	+	0.790926i	$0.290401\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2586.72	+	4480.33i	−0.674528	+	1.16832i
$246$		0			0
$247$	3426.72	+	5935.26i	0.882741	+	1.52895i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4736.31			1.19105			0.595524	−	0.803337i	$-0.296944\pi$
$251$	4736.31			1.19105			0.595524	+	0.803337i	$0.296944\pi$
$252$		0			0
$253$	−289.519			−0.0719443
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1091.58	−	1890.67i	−0.264945	−	0.458899i	0.702604	−	0.711581i	$-0.252021\pi$
$257$	−1091.58	−	1890.67i	−0.264945	−	0.458899i	−0.967549	+	0.252682i	$0.918687\pi$
$258$		0			0
$259$	−1106.15	+	1915.92i	−0.265379	+	0.459649i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4090.67	−	7085.24i	0.959092	−	1.66120i	0.234381	−	0.972145i	$-0.424694\pi$
$263$	4090.67	−	7085.24i	0.959092	−	1.66120i	0.724712	−	0.689052i	$-0.241973\pi$
$264$		0			0
$265$	2853.96	+	4943.20i	0.661574	+	1.14588i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2165.62			−0.490856			−0.245428	−	0.969415i	$-0.578929\pi$
$269$	−2165.62			−0.490856			−0.245428	+	0.969415i	$0.578929\pi$
$270$		0			0
$271$	8529.79			1.91199			0.955993	−	0.293390i	$-0.0947835\pi$
$271$	8529.79			1.91199			0.955993	+	0.293390i	$0.0947835\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−209.187	−	362.323i	−0.0458708	−	0.0794505i
$276$		0			0
$277$	945.517	−	1637.68i	0.205093	−	0.355231i	−0.745070	−	0.666987i	$-0.767584\pi$
$277$	945.517	−	1637.68i	0.205093	−	0.355231i	0.950162	+	0.311756i	$0.100917\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−601.999	+	1042.69i	−0.127802	+	0.221359i	−0.922825	−	0.385220i	$-0.874125\pi$
$281$	−601.999	+	1042.69i	−0.127802	+	0.221359i	0.795023	+	0.606579i	$0.207459\pi$
$282$		0			0
$283$	1988.58	+	3444.32i	0.417699	+	0.723476i	0.995708	−	0.0925547i	$-0.0295033\pi$
$283$	1988.58	+	3444.32i	0.417699	+	0.723476i	−0.578009	+	0.816031i	$0.696170\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10630.9			2.18650
$288$		0			0
$289$	5576.77			1.13510
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3151.43	+	5458.44i	0.628357	+	1.08835i	0.987881	+	0.155211i	$0.0496056\pi$
$293$	3151.43	+	5458.44i	0.628357	+	1.08835i	−0.359524	+	0.933136i	$0.617061\pi$
$294$		0			0
$295$	1827.18	−	3164.77i	0.360619	−	0.624610i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−445.992	+	772.480i	−0.0862621	+	0.149410i
$300$		0			0
$301$	4246.50	+	7355.16i	0.813171	+	1.40845i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3898.13			0.731824
$306$		0			0
$307$	7852.27			1.45978			0.729890	−	0.683565i	$-0.239571\pi$
$307$	7852.27			1.45978			0.729890	+	0.683565i	$0.239571\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1303.94	−	2258.49i	−0.237748	−	0.411792i	0.722320	−	0.691559i	$-0.243076\pi$
$311$	−1303.94	−	2258.49i	−0.237748	−	0.411792i	−0.960068	+	0.279767i	$0.909743\pi$
$312$		0			0
$313$	828.241	−	1434.56i	0.149569	−	0.259060i	−0.781500	−	0.623906i	$-0.785545\pi$
$313$	828.241	−	1434.56i	0.149569	−	0.259060i	0.931068	+	0.364845i	$0.118878\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3101.40	+	5371.78i	−0.549501	+	0.951764i	0.448808	+	0.893628i	$0.351849\pi$
$317$	−3101.40	+	5371.78i	−0.549501	+	0.951764i	−0.998309	+	0.0581353i	$0.981485\pi$
$318$		0			0
$319$	−665.833	−	1153.26i	−0.116864	−	0.202414i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−11664.3			−2.00934
$324$		0			0
$325$	−1288.98			−0.219998
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6628.46	−	11480.8i	−1.11076	−	1.92389i
$330$		0			0
$331$	3811.21	−	6601.20i	0.632879	−	1.09618i	−0.354082	−	0.935214i	$-0.615207\pi$
$331$	3811.21	−	6601.20i	0.632879	−	1.09618i	0.986960	−	0.160963i	$-0.0514601\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3750.14	+	6495.44i	−0.611619	+	1.05935i
$336$		0			0
$337$	−2635.31	−	4564.49i	−0.425978	−	0.737816i	0.570533	−	0.821275i	$-0.306737\pi$
$337$	−2635.31	−	4564.49i	−0.425978	−	0.737816i	−0.996511	+	0.0834590i	$0.973403\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3102.50			0.492697
$342$		0			0
$343$	−4823.71			−0.759347
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4768.92	−	8260.01i	−0.737778	−	1.27787i	−0.953494	−	0.301412i	$-0.902542\pi$
$347$	−4768.92	−	8260.01i	−0.737778	−	1.27787i	0.215716	−	0.976456i	$-0.430791\pi$
$348$		0			0
$349$	−1044.34	+	1808.86i	−0.160179	+	0.277438i	−0.934933	−	0.354825i	$-0.884540\pi$
$349$	−1044.34	+	1808.86i	−0.160179	+	0.277438i	0.774754	+	0.632263i	$0.217874\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3236.56	−	5605.88i	0.488001	−	0.845243i	−0.511903	−	0.859043i	$-0.671059\pi$
$353$	3236.56	−	5605.88i	0.488001	−	0.845243i	0.999905	+	0.0137999i	$0.00439277\pi$
$354$		0			0
$355$	−1834.32	−	3177.14i	−0.274241	−	0.475000i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	11916.0			1.75182			0.875912	−	0.482470i	$-0.160260\pi$
$359$	11916.0			1.75182			0.875912	+	0.482470i	$0.160260\pi$
$360$		0			0
$361$	6111.30			0.890990
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5140.60	+	8903.78i	0.737181	+	1.27684i
$366$		0			0
$367$	−3255.07	+	5637.95i	−0.462979	+	0.801903i	−0.999108	−	0.0422329i	$-0.986553\pi$
$367$	−3255.07	+	5637.95i	−0.462979	+	0.801903i	0.536129	+	0.844136i	$0.319886\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8181.93	+	14171.5i	−1.14497	+	1.98315i
$372$		0			0
$373$	−2416.97	−	4186.32i	−0.335512	−	0.581124i	0.648071	−	0.761580i	$-0.275576\pi$
$373$	−2416.97	−	4186.32i	−0.335512	−	0.581124i	−0.983583	+	0.180456i	$0.942243\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4102.74			−0.560483
$378$		0			0
$379$	−1435.28			−0.194527			−0.0972633	−	0.995259i	$-0.531009\pi$
$379$	−1435.28			−0.194527			−0.0972633	+	0.995259i	$0.531009\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1482.46	−	2567.70i	−0.197781	−	0.342568i	0.750027	−	0.661407i	$-0.230040\pi$
$383$	−1482.46	−	2567.70i	−0.197781	−	0.342568i	−0.947809	+	0.318839i	$0.896707\pi$
$384$		0			0
$385$	−2900.08	+	5023.09i	−0.383901	+	0.664936i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−106.002	+	183.600i	−0.0138162	+	0.0239304i	−0.872851	−	0.487987i	$-0.837731\pi$
$389$	−106.002	+	183.600i	−0.0138162	+	0.0239304i	0.859035	+	0.511917i	$0.171065\pi$
$390$		0			0
$391$	−759.060	−	1314.73i	−0.0981772	−	0.170048i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6027.85			−0.767833
$396$		0			0
$397$	12508.1			1.58127			0.790635	−	0.612288i	$-0.209751\pi$
$397$	12508.1			1.58127			0.790635	+	0.612288i	$0.209751\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1003.26	+	1737.70i	0.124939	+	0.216400i	0.921709	−	0.387882i	$-0.126793\pi$
$401$	1003.26	+	1737.70i	0.124939	+	0.216400i	−0.796770	+	0.604282i	$0.793460\pi$
$402$		0			0
$403$	4779.27	−	8277.93i	0.590750	−	1.02321i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−740.496	+	1282.58i	−0.0901843	+	0.156204i
$408$		0			0
$409$	320.225	+	554.645i	0.0387142	+	0.0670549i	0.884733	−	0.466098i	$-0.154340\pi$
$409$	320.225	+	554.645i	0.0387142	+	0.0670549i	−0.846019	+	0.533153i	$0.821007\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	10476.6			1.24823
$414$		0			0
$415$	−2353.61			−0.278395
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1163.13	+	2014.60i	0.135615	+	0.234892i	0.925832	−	0.377935i	$-0.123366\pi$
$419$	1163.13	+	2014.60i	0.135615	+	0.234892i	−0.790217	+	0.612827i	$0.790032\pi$
$420$		0			0
$421$	−1713.31	+	2967.53i	−0.198341	+	0.343536i	−0.947991	−	0.318298i	$-0.896889\pi$
$421$	−1713.31	+	2967.53i	−0.198341	+	0.343536i	0.749650	+	0.661835i	$0.230222\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1096.89	−	1899.87i	0.125193	−	0.216841i
$426$		0			0
$427$	5587.72	+	9678.22i	0.633276	+	1.09687i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−13417.5			−1.49953			−0.749767	−	0.661701i	$-0.769835\pi$
$431$	−13417.5			−1.49953			−0.749767	+	0.661701i	$0.769835\pi$
$432$		0			0
$433$	−8862.95			−0.983663			−0.491832	−	0.870690i	$-0.663672\pi$
$433$	−8862.95			−0.983663			−0.491832	+	0.870690i	$0.663672\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	844.050	+	1461.94i	0.0923945	+	0.160032i
$438$		0			0
$439$	−3054.73	+	5290.94i	−0.332105	+	0.575223i	−0.982924	−	0.184010i	$-0.941092\pi$
$439$	−3054.73	+	5290.94i	−0.332105	+	0.575223i	0.650819	+	0.759233i	$0.274426\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7642.74	−	13237.6i	0.819678	−	1.41972i	−0.0862409	−	0.996274i	$-0.527485\pi$
$443$	7642.74	−	13237.6i	0.819678	−	1.41972i	0.905919	−	0.423450i	$-0.139181\pi$
$444$		0			0
$445$	6732.38	+	11660.8i	0.717181	+	1.24219i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2622.04			−0.275594			−0.137797	−	0.990460i	$-0.544002\pi$
$449$	−2622.04			−0.275594			−0.137797	+	0.990460i	$0.544002\pi$
$450$		0			0
$451$	7116.70			0.743043
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8934.90	+	15475.7i	0.920604	+	1.59453i
$456$		0			0
$457$	2563.62	−	4440.32i	0.262410	−	0.454507i	−0.704472	−	0.709732i	$-0.748816\pi$
$457$	2563.62	−	4440.32i	0.262410	−	0.454507i	0.966882	+	0.255225i	$0.0821495\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4903.04	−	8492.31i	0.495352	−	0.857975i	−0.504633	−	0.863334i	$-0.668372\pi$
$461$	4903.04	−	8492.31i	0.495352	−	0.857975i	0.999986	+	0.00535859i	$0.00170570\pi$
$462$		0			0
$463$	−9215.11	−	15961.0i	−0.924973	−	1.60210i	−0.791604	−	0.611035i	$-0.790754\pi$
$463$	−9215.11	−	15961.0i	−0.924973	−	1.60210i	−0.133369	−	0.991066i	$-0.542580\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17350.2			−1.71921			−0.859604	−	0.510960i	$-0.829290\pi$
$467$	−17350.2			−1.71921			−0.859604	+	0.510960i	$0.829290\pi$
$468$		0			0
$469$	−21502.4			−2.11703
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2842.75	+	4923.79i	0.276342	+	0.478638i
$474$		0			0
$475$	−1219.71	+	2112.60i	−0.117819	+	0.204069i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3460.76	−	5994.21i	0.330117	−	0.571780i	−0.652417	−	0.757860i	$-0.726245\pi$
$479$	3460.76	−	5994.21i	0.330117	−	0.571780i	0.982535	+	0.186080i	$0.0595784\pi$
$480$		0			0
$481$	2281.40	+	3951.51i	0.216264	+	0.374581i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4812.63			−0.450578
$486$		0			0
$487$	−6396.86			−0.595215			−0.297607	−	0.954688i	$-0.596189\pi$
$487$	−6396.86			−0.595215			−0.297607	+	0.954688i	$0.596189\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4869.89	−	8434.90i	−0.447607	−	0.775278i	0.550623	−	0.834754i	$-0.314390\pi$
$491$	−4869.89	−	8434.90i	−0.447607	−	0.775278i	−0.998230	+	0.0594759i	$0.981057\pi$
$492$		0			0
$493$	3491.35	−	6047.20i	0.318951	−	0.552439i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5258.76	−	9108.44i	0.474623	−	0.822072i
$498$		0			0
$499$	−1768.71	−	3063.50i	−0.158674	−	0.274832i	0.775716	−	0.631082i	$-0.217389\pi$
$499$	−1768.71	−	3063.50i	−0.158674	−	0.274832i	−0.934391	+	0.356249i	$0.884055\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17616.0			1.56155			0.780775	−	0.624812i	$-0.214824\pi$
$503$	17616.0			1.56155			0.780775	+	0.624812i	$0.214824\pi$
$504$		0			0
$505$	−15206.1			−1.33993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6467.63	−	11202.3i	−0.563207	−	0.975504i	−0.997214	−	0.0745941i	$-0.976234\pi$
$509$	−6467.63	−	11202.3i	−0.563207	−	0.975504i	0.434007	−	0.900910i	$-0.357099\pi$
$510$		0			0
$511$	−14737.4	+	25526.0i	−1.27582	+	2.20979i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6943.82	+	12027.1i	−0.594138	+	1.02908i
$516$		0			0
$517$	−4437.31	−	7685.64i	−0.377471	−	0.653799i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	22931.9			1.92834			0.964168	−	0.265291i	$-0.0854679\pi$
$521$	22931.9			1.92834			0.964168	+	0.265291i	$0.0854679\pi$
$522$		0			0
$523$	−7422.01			−0.620539			−0.310269	−	0.950649i	$-0.600419\pi$
$523$	−7422.01			−0.620539			−0.310269	+	0.950649i	$0.600419\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8134.12	+	14088.7i	0.672349	+	1.16454i
$528$		0			0
$529$	5973.65	−	10346.7i	0.490971	−	0.850387i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10963.0	−	18988.4i	0.890918	−	1.54311i
$534$		0			0
$535$	−6309.38	−	10928.2i	−0.509866	−	0.883113i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9928.75			−0.793435
$540$		0			0
$541$	−9659.86			−0.767670			−0.383835	−	0.923402i	$-0.625397\pi$
$541$	−9659.86			−0.767670			−0.383835	+	0.923402i	$0.625397\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1004.54	−	1739.91i	−0.0789535	−	0.136752i
$546$		0			0
$547$	2890.08	−	5005.77i	0.225907	−	0.391282i	−0.730684	−	0.682715i	$-0.760799\pi$
$547$	2890.08	−	5005.77i	0.225907	−	0.391282i	0.956591	+	0.291433i	$0.0941322\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3882.27	+	6724.29i	−0.300164	+	0.519899i
$552$		0			0
$553$	−8640.54	−	14965.9i	−0.664436	−	1.15084i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8160.01			−0.620737			−0.310369	−	0.950616i	$-0.600452\pi$
$557$	−8160.01			−0.620737			−0.310369	+	0.950616i	$0.600452\pi$
$558$		0			0
$559$	17516.5			1.32535
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2881.28	+	4990.52i	0.215686	+	0.373580i	0.953485	−	0.301441i	$-0.0974677\pi$
$563$	2881.28	+	4990.52i	0.215686	+	0.373580i	−0.737798	+	0.675021i	$0.764134\pi$
$564$		0			0
$565$	9607.35	−	16640.4i	0.715370	−	1.23906i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7012.59	+	12146.2i	−0.516666	+	0.894892i	0.483147	+	0.875540i	$0.339494\pi$
$569$	−7012.59	+	12146.2i	−0.516666	+	0.894892i	−0.999813	+	0.0193526i	$0.993839\pi$
$570$		0			0
$571$	−6983.86	−	12096.4i	−0.511848	−	0.886547i	−0.999906	−	0.0137356i	$-0.995628\pi$
$571$	−6983.86	−	12096.4i	−0.511848	−	0.886547i	0.488057	−	0.872812i	$-0.337706\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−317.493			−0.0230267
$576$		0			0
$577$	−13336.0			−0.962193			−0.481097	−	0.876668i	$-0.659761\pi$
$577$	−13336.0			−0.962193			−0.481097	+	0.876668i	$0.659761\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3373.74	−	5843.50i	−0.240906	−	0.417262i
$582$		0			0
$583$	−5477.25	+	9486.88i	−0.389099	+	0.673939i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9174.28	−	15890.3i	0.645082	−	1.11732i	−0.339200	−	0.940714i	$-0.610156\pi$
$587$	9174.28	−	15890.3i	0.645082	−	1.11732i	0.984283	−	0.176601i	$-0.0565102\pi$
$588$		0			0
$589$	−9044.88	−	15666.2i	−0.632746	−	1.09595i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−20361.5			−1.41003			−0.705016	−	0.709192i	$-0.749060\pi$
$593$	−20361.5			−1.41003			−0.705016	+	0.709192i	$0.749060\pi$
$594$		0			0
$595$	−30413.7			−2.09553
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3874.54	+	6710.91i	0.264290	+	0.457763i	0.967377	−	0.253340i	$-0.0815291\pi$
$599$	3874.54	+	6710.91i	0.264290	+	0.457763i	−0.703088	+	0.711103i	$0.748196\pi$
$600$		0			0
$601$	5265.91	−	9120.82i	0.357406	−	0.619045i	−0.630121	−	0.776497i	$-0.716995\pi$
$601$	5265.91	−	9120.82i	0.357406	−	0.619045i	0.987527	+	0.157452i	$0.0503280\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4831.68	−	8368.72i	0.324687	−	0.562375i
$606$		0			0
$607$	−3854.42	−	6676.06i	−0.257737	−	0.446413i	0.707899	−	0.706314i	$-0.249643\pi$
$607$	−3854.42	−	6676.06i	−0.257737	−	0.446413i	−0.965635	+	0.259901i	$0.916310\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−27341.9			−1.81037
$612$		0			0
$613$	−13630.8			−0.898112			−0.449056	−	0.893504i	$-0.648240\pi$
$613$	−13630.8			−0.898112			−0.449056	+	0.893504i	$0.648240\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9806.81	+	16985.9i	0.639882	+	1.10831i	0.985458	+	0.169918i	$0.0543502\pi$
$617$	9806.81	+	16985.9i	0.639882	+	1.10831i	−0.345576	+	0.938391i	$0.612316\pi$
$618$		0			0
$619$	8347.68	−	14458.6i	0.542038	−	0.938838i	−0.456749	−	0.889596i	$-0.650986\pi$
$619$	8347.68	−	14458.6i	0.542038	−	0.938838i	0.998787	−	0.0492419i	$-0.0156805\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−19300.9	+	33430.1i	−1.24121	+	2.14984i
$624$		0			0
$625$	6244.38	+	10815.6i	0.399640	+	0.692197i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7765.71			−0.492272
$630$		0			0
$631$	12302.0			0.776124			0.388062	−	0.921633i	$-0.373145\pi$
$631$	12302.0			0.776124			0.388062	+	0.921633i	$0.373145\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8207.87	+	14216.4i	0.512944	+	0.888445i
$636$		0			0
$637$	−15294.8	+	26491.4i	−0.951338	+	1.64777i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	13726.4	−	23774.8i	0.845802	−	1.46497i	−0.0391204	−	0.999235i	$-0.512456\pi$
$641$	13726.4	−	23774.8i	0.845802	−	1.46497i	0.884923	−	0.465738i	$-0.154211\pi$
$642$		0			0
$643$	2855.56	+	4945.98i	0.175136	+	0.303344i	0.940208	−	0.340600i	$-0.110630\pi$
$643$	2855.56	+	4945.98i	0.175136	+	0.303344i	−0.765072	+	0.643944i	$0.777297\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5973.11			−0.362948			−0.181474	−	0.983396i	$-0.558087\pi$
$647$	−5973.11			−0.362948			−0.181474	+	0.983396i	$0.558087\pi$
$648$		0			0
$649$	7013.37			0.424190
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7825.46	−	13554.1i	−0.468965	−	0.812270i	0.530406	−	0.847744i	$-0.322039\pi$
$653$	−7825.46	−	13554.1i	−0.468965	−	0.812270i	−0.999371	+	0.0354733i	$0.988706\pi$
$654$		0			0
$655$	−10094.3	+	17483.9i	−0.602165	+	1.04298i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1978.62	+	3427.07i	−0.116959	+	0.202579i	−0.918561	−	0.395279i	$-0.870648\pi$
$659$	−1978.62	+	3427.07i	−0.116959	+	0.202579i	0.801602	+	0.597858i	$0.203981\pi$
$660$		0			0
$661$	−6107.01	−	10577.7i	−0.359357	−	0.622425i	0.628496	−	0.777813i	$-0.283671\pi$
$661$	−6107.01	−	10577.7i	−0.359357	−	0.622425i	−0.987854	+	0.155387i	$0.950337\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	33819.0			1.97210
$666$		0			0
$667$	−1010.56			−0.0586644
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3740.61	+	6478.92i	0.215208	+	0.372751i
$672$		0			0
$673$	−11783.5	+	20409.6i	−0.674920	+	1.16899i	0.301573	+	0.953443i	$0.402488\pi$
$673$	−11783.5	+	20409.6i	−0.674920	+	1.16899i	−0.976492	+	0.215552i	$0.930845\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2051.64	+	3553.54i	−0.116471	+	0.201734i	−0.918367	−	0.395730i	$-0.870492\pi$
$677$	−2051.64	+	3553.54i	−0.116471	+	0.201734i	0.801896	+	0.597464i	$0.203825\pi$
$678$		0			0
$679$	−6898.60	−	11948.7i	−0.389903	−	0.675332i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−34200.5			−1.91602			−0.958012	−	0.286729i	$-0.907432\pi$
$683$	−34200.5			−1.91602			−0.958012	+	0.286729i	$0.907432\pi$
$684$		0			0
$685$	17912.6			0.999133
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16874.9	+	29228.2i	0.933068	+	1.61612i
$690$		0			0
$691$	−7560.79	+	13095.7i	−0.416246	+	0.720959i	−0.995558	−	0.0941463i	$-0.969988\pi$
$691$	−7560.79	+	13095.7i	−0.416246	+	0.720959i	0.579312	+	0.815106i	$0.303321\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	11364.4	−	19683.8i	0.620256	−	1.07432i
$696$		0			0
$697$	18658.5	+	32317.5i	1.01398	+	1.75626i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	24421.2			1.31580			0.657899	−	0.753106i	$-0.271445\pi$
$701$	24421.2			1.31580			0.657899	+	0.753106i	$0.271445\pi$
$702$		0			0
$703$	8635.22			0.463277
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−21797.0	−	37753.5i	−1.15949	−	2.00830i
$708$		0			0
$709$	1258.39	−	2179.59i	0.0666570	−	0.115453i	−0.830771	−	0.556615i	$-0.812100\pi$
$709$	1258.39	−	2179.59i	0.0666570	−	0.115453i	0.897428	+	0.441161i	$0.145433\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1177.20	−	2038.97i	0.0618324	−	0.107097i
$714$		0			0
$715$	5981.32	+	10359.9i	0.312851	+	0.541874i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4416.18			0.229062			0.114531	−	0.993420i	$-0.463463\pi$
$719$	4416.18			0.229062			0.114531	+	0.993420i	$0.463463\pi$
$720$		0			0
$721$	−39814.1			−2.05653
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−730.166	−	1264.68i	−0.0374037	−	0.0647851i
$726$		0			0
$727$	6731.65	−	11659.6i	0.343415	−	0.594813i	−0.641649	−	0.766998i	$-0.721749\pi$
$727$	6731.65	−	11659.6i	0.343415	−	0.594813i	0.985065	+	0.172185i	$0.0550828\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14906.2	+	25818.3i	−0.754208	+	1.30633i
$732$		0			0
$733$	10228.6	+	17716.5i	0.515420	+	0.892734i	0.999840	+	0.0178982i	$0.00569749\pi$
$733$	10228.6	+	17716.5i	0.515420	+	0.892734i	−0.484420	+	0.874836i	$0.660969\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14394.4			−0.719436
$738$		0			0
$739$	−7781.39			−0.387339			−0.193669	−	0.981067i	$-0.562039\pi$
$739$	−7781.39			−0.387339			−0.193669	+	0.981067i	$0.562039\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4369.41	+	7568.04i	0.215744	+	0.373680i	0.953503	−	0.301385i	$-0.0974489\pi$
$743$	4369.41	+	7568.04i	0.215744	+	0.373680i	−0.737758	+	0.675065i	$0.764116\pi$
$744$		0			0
$745$	8771.74	−	15193.1i	0.431371	−	0.747157i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18088.2	−	31329.6i	0.882413	−	1.52838i
$750$		0			0
$751$	−9690.18	−	16783.9i	−0.470838	−	0.815516i	0.528605	−	0.848868i	$-0.322715\pi$
$751$	−9690.18	−	16783.9i	−0.470838	−	0.815516i	−0.999444	+	0.0333518i	$0.989382\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−19944.9			−0.961416
$756$		0			0
$757$	20151.9			0.967548			0.483774	−	0.875193i	$-0.339266\pi$
$757$	20151.9			0.967548			0.483774	+	0.875193i	$0.339266\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	692.316	+	1199.13i	0.0329782	+	0.0571200i	0.882043	−	0.471168i	$-0.156167\pi$
$761$	692.316	+	1199.13i	0.0329782	+	0.0571200i	−0.849065	+	0.528288i	$0.822834\pi$
$762$		0			0
$763$	2879.88	−	4988.10i	0.136643	−	0.236673i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10803.8	−	18712.7i	0.508608	−	0.880935i
$768$		0			0
$769$	−9122.17	−	15800.1i	−0.427768	−	0.740916i	0.568906	−	0.822402i	$-0.307367\pi$
$769$	−9122.17	−	15800.1i	−0.427768	−	0.740916i	−0.996674	+	0.0814861i	$0.974033\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18353.9			0.854001			0.427001	−	0.904251i	$-0.359570\pi$
$773$	18353.9			0.854001			0.427001	+	0.904251i	$0.359570\pi$
$774$		0			0
$775$	3402.27			0.157694
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−20747.7	−	35936.0i	−0.954253	−	1.65281i
$780$		0			0
$781$	3520.39	−	6097.49i	0.161292	−	0.279367i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16182.6	+	28029.0i	−0.735772	+	1.27439i
$786$		0			0
$787$	−2413.44	−	4180.20i	−0.109314	−	0.189337i	0.806179	−	0.591672i	$-0.201532\pi$
$787$	−2413.44	−	4180.20i	−0.109314	−	0.189337i	−0.915492	+	0.402335i	$0.868199\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	55086.1			2.47615
$792$		0			0
$793$	23049.0			1.03215
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8211.82	+	14223.3i	0.364966	+	0.632139i	0.988771	−	0.149441i	$-0.0477474\pi$
$797$	8211.82	+	14223.3i	0.364966	+	0.632139i	−0.623805	+	0.781580i	$0.714414\pi$
$798$		0			0
$799$	23267.4	−	40300.3i	1.03022	−	1.78439i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9865.73	+	17087.9i	−0.433567	+	0.750959i
$804$		0			0
$805$	2200.79	+	3811.88i	0.0963575	+	0.166896i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16215.6			−0.704708			−0.352354	−	0.935867i	$-0.614619\pi$
$809$	−16215.6			−0.704708			−0.352354	+	0.935867i	$0.614619\pi$
$810$		0			0
$811$	−17229.9			−0.746023			−0.373012	−	0.927827i	$-0.621675\pi$
$811$	−17229.9			−0.746023			−0.373012	+	0.927827i	$0.621675\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8419.53	+	14583.1i	0.361869	+	0.626776i
$816$		0			0
$817$	16575.2	−	28709.1i	0.709784	−	1.22938i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	694.109	−	1202.23i	0.0295062	−	0.0511062i	−0.850895	−	0.525335i	$-0.823940\pi$
$821$	694.109	−	1202.23i	0.0295062	−	0.0511062i	0.880401	+	0.474229i	$0.157273\pi$
$822$		0			0
$823$	11627.5	+	20139.4i	0.492477	+	0.852996i	0.999962	−	0.00866465i	$-0.00275808\pi$
$823$	11627.5	+	20139.4i	0.492477	+	0.852996i	−0.507485	+	0.861661i	$0.669425\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	29307.4			1.23231			0.616154	−	0.787626i	$-0.288690\pi$
$827$	29307.4			1.23231			0.616154	+	0.787626i	$0.288690\pi$
$828$		0			0
$829$	18350.8			0.768818			0.384409	−	0.923163i	$-0.374405\pi$
$829$	18350.8			0.768818			0.384409	+	0.923163i	$0.374405\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−26031.1	−	45087.2i	−1.08274	−	1.87537i
$834$		0			0
$835$	2299.55	−	3982.93i	0.0953043	−	0.165072i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1531.63	−	2652.87i	0.0630249	−	0.109162i	−0.832791	−	0.553587i	$-0.813259\pi$
$839$	1531.63	−	2652.87i	0.0630249	−	0.109162i	0.895816	+	0.444425i	$0.146592\pi$
$840$		0			0
$841$	9870.42	+	17096.1i	0.404708	+	0.700974i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14496.0			0.590151
$846$		0			0
$847$	27703.6			1.12386
$848$		0			0
$849$		0			0
$850$		0			0
$851$	561.942	+	973.312i	0.0226359	+	0.0392065i
$852$		0			0
$853$	−1779.04	+	3081.39i	−0.0714105	+	0.123687i	−0.899520	−	0.436880i	$-0.856083\pi$
$853$	−1779.04	+	3081.39i	−0.0714105	+	0.123687i	0.828109	+	0.560567i	$0.189417\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3205.07	+	5551.34i	−0.127751	+	0.221272i	−0.922805	−	0.385267i	$-0.874109\pi$
$857$	−3205.07	+	5551.34i	−0.127751	+	0.221272i	0.795054	+	0.606539i	$0.207443\pi$
$858$		0			0
$859$	11806.0	+	20448.6i	0.468936	+	0.812222i	0.999369	−	0.0355052i	$-0.0113040\pi$
$859$	11806.0	+	20448.6i	0.468936	+	0.812222i	−0.530433	+	0.847727i	$0.677971\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	33640.6			1.32693			0.663463	−	0.748209i	$-0.269086\pi$
$863$	33640.6			1.32693			0.663463	+	0.748209i	$0.269086\pi$
$864$		0			0
$865$	30019.7			1.18000
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5784.26	−	10018.6i	−0.225797	−	0.391092i
$870$		0			0
$871$	−22173.9	+	38406.4i	−0.862612	+	1.49409i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−21739.8	+	37654.4i	−0.839930	+	1.45480i
$876$		0			0
$877$	−5293.72	−	9168.99i	−0.203827	−	0.353039i	0.745931	−	0.666023i	$-0.232005\pi$
$877$	−5293.72	−	9168.99i	−0.203827	−	0.353039i	−0.949758	+	0.312984i	$0.898671\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12213.9			0.467080			0.233540	−	0.972347i	$-0.424969\pi$
$881$	12213.9			0.467080			0.233540	+	0.972347i	$0.424969\pi$
$882$		0			0
$883$	34545.1			1.31657			0.658287	−	0.752767i	$-0.271281\pi$
$883$	34545.1			1.31657			0.658287	+	0.752767i	$0.271281\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8973.11	+	15541.9i	0.339670	+	0.588326i	0.984371	−	0.176109i	$-0.0563512\pi$
$887$	8973.11	+	15541.9i	0.339670	+	0.588326i	−0.644700	+	0.764435i	$0.723018\pi$
$888$		0			0
$889$	−23530.9	+	40756.7i	−0.887741	+	1.53761i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−25872.6	+	44812.7i	−0.969534	+	1.67928i
$894$		0			0
$895$	−4402.56	−	7625.47i	−0.164426	−	0.284795i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10829.2			0.401753
$900$		0			0
$901$	−57440.9			−2.12390
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−13582.0	−	23524.7i	−0.498873	−	0.864073i
$906$		0			0
$907$	7603.01	−	13168.8i	0.278339	−	0.482098i	−0.692633	−	0.721290i	$-0.743549\pi$
$907$	7603.01	−	13168.8i	0.278339	−	0.482098i	0.970972	+	0.239193i	$0.0768827\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13671.4	−	23679.6i	0.497205	−	0.861184i	−0.502790	−	0.864409i	$-0.667693\pi$
$911$	13671.4	−	23679.6i	0.497205	−	0.861184i	0.999995	+	0.00322453i	$0.00102640\pi$
$912$		0			0
$913$	−2258.50	−	3911.83i	−0.0818678	−	0.141799i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−57878.3			−2.08431
$918$		0			0
$919$	31391.3			1.12677			0.563386	−	0.826194i	$-0.309499\pi$
$919$	31391.3			1.12677			0.563386	+	0.826194i	$0.309499\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10846.0	−	18785.8i	−0.386783	−	0.669928i
$924$		0			0
$925$	−812.043	+	1406.50i	−0.0288647	+	0.0499951i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9115.15	−	15787.9i	0.321914	−	0.557572i	−0.658969	−	0.752170i	$-0.729007\pi$
$929$	9115.15	−	15787.9i	0.321914	−	0.557572i	0.980883	+	0.194598i	$0.0623404\pi$
$930$		0			0
$931$	28945.8	+	50135.6i	1.01897	+	1.76490i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−20359.9			−0.712129
$936$		0			0
$937$	19475.4			0.679012			0.339506	−	0.940604i	$-0.389740\pi$
$937$	19475.4			0.679012			0.339506	+	0.940604i	$0.389740\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3801.46	−	6584.32i	−0.131694	−	0.228101i	0.792636	−	0.609695i	$-0.208708\pi$
$941$	−3801.46	−	6584.32i	−0.131694	−	0.228101i	−0.924330	+	0.381595i	$0.875375\pi$
$942$		0			0
$943$	2700.33	−	4677.12i	0.0932503	−	0.161514i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12795.6	−	22162.6i	0.439071	−	0.760493i	−0.558547	−	0.829473i	$-0.688641\pi$
$947$	12795.6	−	22162.6i	0.439071	−	0.760493i	0.997618	+	0.0689796i	$0.0219743\pi$
$948$		0			0
$949$	30395.4	+	52646.5i	1.03970	+	1.80082i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−6621.51			−0.225070			−0.112535	−	0.993648i	$-0.535897\pi$
$953$	−6621.51			−0.225070			−0.112535	+	0.993648i	$0.535897\pi$
$954$		0			0
$955$	23340.0			0.790855
$956$		0			0
$957$		0			0
$958$		0			0
$959$	25676.6	+	44473.2i	0.864589	+	1.49751i
$960$		0			0
$961$	2280.56	−	3950.05i	0.0765521	−	0.132592i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−20062.1	+	34748.5i	−0.669244	+	1.15917i
$966$		0			0
$967$	−7626.03	−	13208.7i	−0.253606	−	0.439258i	0.710910	−	0.703283i	$-0.248283\pi$
$967$	−7626.03	−	13208.7i	−0.253606	−	0.439258i	−0.964516	+	0.264025i	$0.914950\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−55183.2			−1.82380			−0.911901	−	0.410410i	$-0.865386\pi$
$971$	−55183.2			−1.82380			−0.911901	+	0.410410i	$0.865386\pi$
$972$		0			0
$973$	65160.8			2.14693
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12146.5	−	21038.3i	−0.397749	−	0.688921i	0.595699	−	0.803208i	$-0.296875\pi$
$977$	−12146.5	−	21038.3i	−0.397749	−	0.688921i	−0.993448	+	0.114287i	$0.963542\pi$
$978$		0			0
$979$	−12920.6	+	22379.2i	−0.421803	+	0.730585i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	629.114	−	1089.66i	0.0204126	−	0.0353557i	−0.855639	−	0.517574i	$-0.826835\pi$
$983$	629.114	−	1089.66i	0.0204126	−	0.0353557i	0.876051	+	0.482218i	$0.160169\pi$
$984$		0			0
$985$	−140.603	−	243.531i	−0.00454819	−	0.00787770i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4314.57			0.138721
$990$		0			0
$991$	4198.63			0.134585			0.0672926	−	0.997733i	$-0.478564\pi$
$991$	4198.63			0.134585			0.0672926	+	0.997733i	$0.478564\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−15653.9	−	27113.4i	−0.498757	−	0.863872i
$996$		0			0
$997$	9029.08	−	15638.8i	0.286814	−	0.496777i	−0.686233	−	0.727382i	$-0.740737\pi$
$997$	9029.08	−	15638.8i	0.286814	−	0.496777i	0.973048	+	0.230605i	$0.0740704\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.t.217.1		4
3.2	odd	2		648.4.i.n.217.2		4
9.2	odd	6		648.4.a.f.1.1	yes	2
9.4	even	3	inner	648.4.i.t.433.1		4
9.5	odd	6		648.4.i.n.433.2		4
9.7	even	3		648.4.a.c.1.2	✓	2
36.7	odd	6		1296.4.a.m.1.2		2
36.11	even	6		1296.4.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.c.1.2	✓	2	9.7	even	3
648.4.a.f.1.1	yes	2	9.2	odd	6
648.4.i.n.217.2		4	3.2	odd	2
648.4.i.n.433.2		4	9.5	odd	6
648.4.i.t.217.1		4	1.1	even	1	trivial
648.4.i.t.433.1		4	9.4	even	3	inner
1296.4.a.m.1.2		2	36.7	odd	6
1296.4.a.q.1.1		2	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.08872 - 8.81393i) q^{5} +(14.5887 - 25.2684i) q^{7} +O(q^{10})\)	\(q+(-5.08872 - 8.81393i) q^{5} +(14.5887 - 25.2684i) q^{7} +(9.76617 - 16.9155i) q^{11} +(-30.0887 - 52.1152i) q^{13} +102.420 q^{17} -113.887 q^{19} +(-7.41128 - 12.8367i) q^{23} +(10.7098 - 18.5499i) q^{25} +(34.0887 - 59.0434i) q^{29} +(79.4196 + 137.559i) q^{31} -296.952 q^{35} -75.8226 q^{37} +(182.177 + 315.541i) q^{41} +(-145.541 + 252.084i) q^{43} +(227.177 - 393.483i) q^{47} +(-254.162 - 440.221i) q^{49} -560.839 q^{53} -198.789 q^{55} +(179.532 + 310.959i) q^{59} +(-191.508 + 331.702i) q^{61} +(-306.226 + 530.400i) q^{65} +(-368.476 - 638.219i) q^{67} +360.468 q^{71} -1010.19 q^{73} +(-284.952 - 493.551i) q^{77} +(296.138 - 512.925i) q^{79} +(115.629 - 200.274i) q^{83} +(-521.185 - 902.719i) q^{85} -1323.00 q^{89} -1755.82 q^{91} +(579.541 + 1003.79i) q^{95} +(236.436 - 409.519i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 30 q^{7}+O(q^{10}) \) 4 * q + 8 * q^5 + 30 * q^7	\( 4 q + 8 q^{5} + 30 q^{7} - 46 q^{11} - 92 q^{13} - 44 q^{17} - 172 q^{19} - 58 q^{23} - 184 q^{25} + 108 q^{29} - 136 q^{31} - 564 q^{35} - 360 q^{37} + 672 q^{41} + 70 q^{43} + 852 q^{47} - 166 q^{49} - 1336 q^{53} - 2780 q^{55} + 548 q^{59} - 284 q^{61} - 34 q^{65} - 1162 q^{67} + 1612 q^{71} - 3020 q^{73} - 516 q^{77} + 22 q^{79} + 1540 q^{83} - 3304 q^{85} - 5292 q^{89} - 3564 q^{91} + 1666 q^{95} - 472 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 30 * q^7 - 46 * q^11 - 92 * q^13 - 44 * q^17 - 172 * q^19 - 58 * q^23 - 184 * q^25 + 108 * q^29 - 136 * q^31 - 564 * q^35 - 360 * q^37 + 672 * q^41 + 70 * q^43 + 852 * q^47 - 166 * q^49 - 1336 * q^53 - 2780 * q^55 + 548 * q^59 - 284 * q^61 - 34 * q^65 - 1162 * q^67 + 1612 * q^71 - 3020 * q^73 - 516 * q^77 + 22 * q^79 + 1540 * q^83 - 3304 * q^85 - 5292 * q^89 - 3564 * q^91 + 1666 * q^95 - 472 * q^97

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