LMFDB - Embedded newform 2268.2.t.b.1781.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(1781,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2268.1781");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.t (of order $6$, degree $2$, 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:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 $ x^16 - 2x^15 + 5x^14 - 17x^13 + 22x^12 - 31x^11 + 62x^10 - 52x^9 + 52x^8 - 156x^7 + 558x^6 - 837x^5 + 1782x^4 - 4131x^3 + 3645x^2 - 4374*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1781.5
Root			$1.68124 + 0.416458i$ of defining polynomial
Character	$\chi$	$=$	2268.1781
Dual form			2268.2.t.b.2105.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.349828 + 0.605920i) q^{5} +(-2.02556 - 1.70209i) q^{7} +O(q^{10})$	$q+(0.349828 + 0.605920i) q^{5} +(-2.02556 - 1.70209i) q^{7} +(-0.229685 - 0.132608i) q^{11} +1.31431i q^{13} +(-1.86392 + 3.22840i) q^{17} +(-0.382449 + 0.220807i) q^{19} +(4.29949 - 2.48231i) q^{23} +(2.25524 - 3.90619i) q^{25} +0.315564i q^{29} +(4.85521 + 2.80316i) q^{31} +(0.322736 - 1.82276i) q^{35} +(-0.351124 - 0.608164i) q^{37} +10.7871 q^{41} -7.46263 q^{43} +(3.50285 + 6.06712i) q^{47} +(1.20576 + 6.89537i) q^{49} +(8.51919 + 4.91856i) q^{53} -0.185561i q^{55} +(6.73182 - 11.6598i) q^{59} +(4.89484 - 2.82604i) q^{61} +(-0.796368 + 0.459783i) q^{65} +(2.97060 - 5.14523i) q^{67} -13.4323i q^{71} +(-6.66182 - 3.84620i) q^{73} +(0.239527 + 0.659550i) q^{77} +(-0.698360 - 1.20959i) q^{79} +7.44798 q^{83} -2.60820 q^{85} +(5.59261 + 9.68668i) q^{89} +(2.23708 - 2.66221i) q^{91} +(-0.267582 - 0.154489i) q^{95} +10.6028i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{7}+O(q^{10}) $ 16 * q + 2 * q^7	$ 16 q + 2 q^{7} + 6 q^{11} + 9 q^{17} - 21 q^{23} - 8 q^{25} - 6 q^{31} - 15 q^{35} + q^{37} + 12 q^{41} + 4 q^{43} + 18 q^{47} - 8 q^{49} + 15 q^{59} - 3 q^{61} - 39 q^{65} - 7 q^{67} - 48 q^{77} - q^{79} - 12 q^{85} + 21 q^{89} + 9 q^{91} + 6 q^{95}+O(q^{100}) $ 16 * q + 2 * q^7 + 6 * q^11 + 9 * q^17 - 21 * q^23 - 8 * q^25 - 6 * q^31 - 15 * q^35 + q^37 + 12 * q^41 + 4 * q^43 + 18 * q^47 - 8 * q^49 + 15 * q^59 - 3 * q^61 - 39 * q^65 - 7 * q^67 - 48 * q^77 - q^79 - 12 * q^85 + 21 * q^89 + 9 * q^91 + 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.349828	+	0.605920i	0.156448	+	0.270975i	0.933585	−	0.358355i	$-0.116662\pi$
$5$	0.349828	+	0.605920i	0.156448	+	0.270975i	−0.777137	+	0.629331i	$0.783329\pi$
$6$		0			0
$7$	−2.02556	−	1.70209i	−0.765588	−	0.643331i
$7$	−2.02556	−	1.70209i	−0.765588	−	0.643331i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.229685	−	0.132608i	−0.0692525	−	0.0399829i	0.464974	−	0.885324i	$-0.346064\pi$
$11$	−0.229685	−	0.132608i	−0.0692525	−	0.0399829i	−0.534226	+	0.845341i	$0.679397\pi$
$12$		0			0
$13$			1.31431i			0.364525i	0.983250	+	0.182262i	$0.0583420\pi$
$13$			1.31431i			0.364525i	−0.983250	+	0.182262i	$0.941658\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.86392	+	3.22840i	−0.452067	+	0.783003i	−0.998514	−	0.0544906i	$-0.982646\pi$
$17$	−1.86392	+	3.22840i	−0.452067	+	0.783003i	0.546447	+	0.837493i	$0.315980\pi$
$18$		0			0
$19$	−0.382449	+	0.220807i	−0.0877398	+	0.0506566i	−0.543228	−	0.839585i	$-0.682798\pi$
$19$	−0.382449	+	0.220807i	−0.0877398	+	0.0506566i	0.455488	+	0.890242i	$0.349465\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.29949	−	2.48231i	0.896507	−	0.517598i	0.0204414	−	0.999791i	$-0.493493\pi$
$23$	4.29949	−	2.48231i	0.896507	−	0.517598i	0.876065	+	0.482193i	$0.160160\pi$
$24$		0			0
$25$	2.25524	−	3.90619i	0.451048	−	0.781238i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			0.315564i			0.0585988i	0.999571	+	0.0292994i	$0.00932762\pi$
$29$			0.315564i			0.0585988i	−0.999571	+	0.0292994i	$0.990672\pi$
$30$		0			0
$31$	4.85521	+	2.80316i	0.872022	+	0.503462i	0.868020	−	0.496530i	$-0.165393\pi$
$31$	4.85521	+	2.80316i	0.872022	+	0.503462i	0.00400255	+	0.999992i	$0.498726\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.322736	−	1.82276i	0.0545523	−	0.308103i
$36$		0			0
$37$	−0.351124	−	0.608164i	−0.0577244	−	0.0999816i	0.835719	−	0.549157i	$-0.185051\pi$
$37$	−0.351124	−	0.608164i	−0.0577244	−	0.0999816i	−0.893444	+	0.449175i	$0.851718\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.7871			1.68466			0.842330	−	0.538962i	$-0.181183\pi$
$41$	10.7871			1.68466			0.842330	+	0.538962i	$0.181183\pi$
$42$		0			0
$43$	−7.46263			−1.13804			−0.569020	−	0.822324i	$-0.692677\pi$
$43$	−7.46263			−1.13804			−0.569020	+	0.822324i	$0.692677\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.50285	+	6.06712i	0.510943	+	0.884980i	0.999920	+	0.0126827i	$0.00403713\pi$
$47$	3.50285	+	6.06712i	0.510943	+	0.884980i	−0.488976	+	0.872297i	$0.662630\pi$
$48$		0			0
$49$	1.20576	+	6.89537i	0.172251	+	0.985053i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.51919	+	4.91856i	1.17020	+	0.675616i	0.953727	−	0.300672i	$-0.0972111\pi$
$53$	8.51919	+	4.91856i	1.17020	+	0.675616i	0.216474	+	0.976288i	$0.430544\pi$
$54$		0			0
$55$		−	0.185561i		−	0.0250210i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.73182	−	11.6598i	0.876408	−	1.51798i	0.0211522	−	0.999776i	$-0.493267\pi$
$59$	6.73182	−	11.6598i	0.876408	−	1.51798i	0.855256	−	0.518206i	$-0.173400\pi$
$60$		0			0
$61$	4.89484	−	2.82604i	0.626720	−	0.361837i	−0.152761	−	0.988263i	$-0.548816\pi$
$61$	4.89484	−	2.82604i	0.626720	−	0.361837i	0.779481	+	0.626426i	$0.215483\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.796368	+	0.459783i	−0.0987773	+	0.0570291i
$66$		0			0
$67$	2.97060	−	5.14523i	0.362916	−	0.628590i	−0.625523	−	0.780206i	$-0.715114\pi$
$67$	2.97060	−	5.14523i	0.362916	−	0.628590i	0.988439	+	0.151616i	$0.0484477\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	13.4323i		−	1.59412i	−0.603900	−	0.797060i	$-0.706387\pi$
$71$		−	13.4323i		−	1.59412i	0.603900	−	0.797060i	$-0.293613\pi$
$72$		0			0
$73$	−6.66182	−	3.84620i	−0.779707	−	0.450164i	0.0566194	−	0.998396i	$-0.481968\pi$
$73$	−6.66182	−	3.84620i	−0.779707	−	0.450164i	−0.836326	+	0.548232i	$0.815301\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.239527	+	0.659550i	0.0272966	+	0.0751627i
$78$		0			0
$79$	−0.698360	−	1.20959i	−0.0785716	−	0.136090i	0.824062	−	0.566499i	$-0.191703\pi$
$79$	−0.698360	−	1.20959i	−0.0785716	−	0.136090i	−0.902634	+	0.430409i	$0.858369\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.44798			0.817522			0.408761	−	0.912641i	$-0.365961\pi$
$83$	7.44798			0.817522			0.408761	+	0.912641i	$0.365961\pi$
$84$		0			0
$85$	−2.60820			−0.282899
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.59261	+	9.68668i	0.592815	+	1.02679i	0.993851	+	0.110724i	$0.0353168\pi$
$89$	5.59261	+	9.68668i	0.592815	+	1.02679i	−0.401036	+	0.916062i	$0.631350\pi$
$90$		0			0
$91$	2.23708	−	2.66221i	0.234510	−	0.279076i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.267582	−	0.154489i	−0.0274534	−	0.0158502i
$96$		0			0
$97$			10.6028i			1.07655i	0.842770	+	0.538273i	$0.180923\pi$
$97$			10.6028i			1.07655i	−0.842770	+	0.538273i	$0.819077\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.75357	−	15.1616i	0.871013	−	1.50864i	0.0100634	−	0.999949i	$-0.496797\pi$
$101$	8.75357	−	15.1616i	0.871013	−	1.50864i	0.860950	−	0.508690i	$-0.169870\pi$
$102$		0			0
$103$	−7.39775	+	4.27110i	−0.728922	+	0.420844i	−0.818028	−	0.575179i	$-0.804933\pi$
$103$	−7.39775	+	4.27110i	−0.728922	+	0.420844i	0.0891054	+	0.996022i	$0.471599\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.09489	−	5.25093i	0.879236	−	0.507627i	0.00882940	−	0.999961i	$-0.497189\pi$
$107$	9.09489	−	5.25093i	0.879236	−	0.507627i	0.870406	+	0.492334i	$0.163856\pi$
$108$		0			0
$109$	−7.12110	+	12.3341i	−0.682078	+	1.18139i	0.292268	+	0.956337i	$0.405590\pi$
$109$	−7.12110	+	12.3341i	−0.682078	+	1.18139i	−0.974346	+	0.225057i	$0.927743\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			15.4479i			1.45322i	0.687051	+	0.726609i	$0.258905\pi$
$113$			15.4479i			1.45322i	−0.687051	+	0.726609i	$0.741095\pi$
$114$		0			0
$115$	3.00817	+	1.73677i	0.280513	+	0.161954i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.27052	−	3.36675i	0.849827	−	0.308629i
$120$		0			0
$121$	−5.46483	−	9.46536i	−0.496803	−	0.860488i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.65406			0.595157
$126$		0			0
$127$	21.8304			1.93713			0.968566	−	0.248758i	$-0.0800225\pi$
$127$	21.8304			1.93713			0.968566	+	0.248758i	$0.0800225\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.60461	+	4.51132i	0.227566	+	0.394156i	0.957086	−	0.289803i	$-0.0935899\pi$
$131$	2.60461	+	4.51132i	0.227566	+	0.394156i	−0.729520	+	0.683959i	$0.760257\pi$
$132$		0			0
$133$	1.15051	+	0.203707i	0.0997615	+	0.0176636i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.33589	+	1.34863i	0.199568	+	0.115221i	0.596454	−	0.802647i	$-0.296576\pi$
$137$	2.33589	+	1.34863i	0.199568	+	0.115221i	−0.396886	+	0.917868i	$0.629909\pi$
$138$		0			0
$139$		−	11.7142i		−	0.993585i	−0.867869	−	0.496793i	$-0.834511\pi$
$139$		−	11.7142i		−	0.993585i	0.867869	−	0.496793i	$-0.165489\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.174289	−	0.301877i	0.0145748	−	0.0252442i
$144$		0			0
$145$	−0.191206	+	0.110393i	−0.0158788	+	0.00916765i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−16.3055	+	9.41399i	−1.33580	+	0.771224i	−0.986182	−	0.165668i	$-0.947022\pi$
$149$	−16.3055	+	9.41399i	−1.33580	+	0.771224i	−0.349618	+	0.936892i	$0.613689\pi$
$150$		0			0
$151$	−5.00143	+	8.66273i	−0.407010	+	0.704963i	−0.994553	−	0.104230i	$-0.966762\pi$
$151$	−5.00143	+	8.66273i	−0.407010	+	0.704963i	0.587543	+	0.809193i	$0.300095\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			3.92249i			0.315062i
$156$		0			0
$157$	0.218293	+	0.126032i	0.0174217	+	0.0100584i	0.508686	−	0.860952i	$-0.330132\pi$
$157$	0.218293	+	0.126032i	0.0174217	+	0.0100584i	−0.491264	+	0.871011i	$0.663465\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.9340	−	2.29007i	−1.01934	−	0.180483i
$162$		0			0
$163$	4.29780	+	7.44400i	0.336629	+	0.583059i	0.983796	−	0.179289i	$-0.0573797\pi$
$163$	4.29780	+	7.44400i	0.336629	+	0.583059i	−0.647167	+	0.762348i	$0.724046\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.48874			−0.347349			−0.173674	−	0.984803i	$-0.555564\pi$
$167$	−4.48874			−0.347349			−0.173674	+	0.984803i	$0.555564\pi$
$168$		0			0
$169$	11.2726			0.867122
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.56072	+	6.16736i	0.270717	+	0.468895i	0.969046	−	0.246882i	$-0.0794060\pi$
$173$	3.56072	+	6.16736i	0.270717	+	0.468895i	−0.698329	+	0.715777i	$0.746073\pi$
$174$		0			0
$175$	−11.2168	+	4.07358i	−0.847912	+	0.307934i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	22.1270	+	12.7750i	1.65385	+	0.954848i	0.975470	+	0.220134i	$0.0706494\pi$
$179$	22.1270	+	12.7750i	1.65385	+	0.954848i	0.678376	+	0.734715i	$0.262684\pi$
$180$		0			0
$181$		−	0.943175i		−	0.0701057i	−0.999385	−	0.0350528i	$-0.988840\pi$
$181$		−	0.943175i		−	0.0701057i	0.999385	−	0.0350528i	$-0.0111599\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.245666	−	0.425506i	0.0180617	−	0.0312838i
$186$		0			0
$187$	0.856227	−	0.494343i	0.0626135	−	0.0361499i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.57413	+	1.48617i	−0.186258	+	0.107536i	−0.590229	−	0.807236i	$-0.700963\pi$
$191$	−2.57413	+	1.48617i	−0.186258	+	0.107536i	0.403972	+	0.914771i	$0.367629\pi$
$192$		0			0
$193$	9.25721	−	16.0340i	0.666348	−	1.15415i	−0.312570	−	0.949895i	$-0.601190\pi$
$193$	9.25721	−	16.0340i	0.666348	−	1.15415i	0.978918	−	0.204254i	$-0.0654769\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.1774i		−	1.01010i	−0.863091	−	0.505048i	$-0.831475\pi$
$197$		−	14.1774i		−	1.01010i	0.863091	−	0.505048i	$-0.168525\pi$
$198$		0			0
$199$	−20.5293	−	11.8526i	−1.45529	−	0.840209i	−0.456512	−	0.889717i	$-0.650901\pi$
$199$	−20.5293	−	11.8526i	−1.45529	−	0.840209i	−0.998774	+	0.0495081i	$0.984235\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.537119	−	0.639193i	0.0376984	−	0.0448625i
$204$		0			0
$205$	3.77362	+	6.53611i	0.263561	+	0.456502i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.117123			0.00810160
$210$		0			0
$211$	−6.08007			−0.418569			−0.209285	−	0.977855i	$-0.567114\pi$
$211$	−6.08007			−0.418569			−0.209285	+	0.977855i	$0.567114\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.61063	−	4.52175i	−0.178044	−	0.308381i
$216$		0			0
$217$	−5.06327	−	13.9420i	−0.343717	−	0.946444i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.24313	−	2.44977i	−0.285424	−	0.164790i
$222$		0			0
$223$		−	0.919300i		−	0.0615609i	−0.999526	−	0.0307804i	$-0.990201\pi$
$223$		−	0.919300i		−	0.0615609i	0.999526	−	0.0307804i	$-0.00979926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.00297	−	8.66540i	0.332059	−	0.575143i	−0.650857	−	0.759201i	$-0.725590\pi$
$227$	5.00297	−	8.66540i	0.332059	−	0.575143i	0.982915	+	0.184058i	$0.0589234\pi$
$228$		0			0
$229$	2.38179	−	1.37513i	0.157393	−	0.0908710i	−0.419235	−	0.907878i	$-0.637702\pi$
$229$	2.38179	−	1.37513i	0.157393	−	0.0908710i	0.576628	+	0.817007i	$0.304368\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.55513	+	3.20725i	−0.363928	+	0.210114i	−0.670803	−	0.741636i	$-0.734050\pi$
$233$	−5.55513	+	3.20725i	−0.363928	+	0.210114i	0.306874	+	0.951750i	$0.400717\pi$
$234$		0			0
$235$	−2.45079	+	4.24489i	−0.159872	+	0.276906i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	13.1969i		−	0.853636i	−0.904338	−	0.426818i	$-0.859635\pi$
$239$		−	13.1969i		−	0.853636i	0.904338	−	0.426818i	$-0.140365\pi$
$240$		0			0
$241$	2.20722	+	1.27434i	0.142180	+	0.0820874i	0.569402	−	0.822059i	$-0.307175\pi$
$241$	2.20722	+	1.27434i	0.142180	+	0.0820874i	−0.427223	+	0.904146i	$0.640508\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.75623	+	3.14279i	−0.239977	+	0.200785i
$246$		0			0
$247$	−0.290209	−	0.502657i	−0.0184656	−	0.0319833i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	18.7893			1.18597			0.592986	−	0.805213i	$-0.297949\pi$
$251$	18.7893			1.18597			0.592986	+	0.805213i	$0.297949\pi$
$252$		0			0
$253$	−1.31670			−0.0827804
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.19727	+	12.4660i	0.448953	+	0.777610i	0.998318	−	0.0579725i	$-0.0184636\pi$
$257$	7.19727	+	12.4660i	0.448953	+	0.777610i	−0.549365	+	0.835583i	$0.685130\pi$
$258$		0			0
$259$	−0.323931	+	1.82952i	−0.0201281	+	0.113681i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.79810	−	3.92488i	−0.419189	−	0.242019i	0.275542	−	0.961289i	$-0.411143\pi$
$263$	−6.79810	−	3.92488i	−0.419189	−	0.242019i	−0.694730	+	0.719271i	$0.744476\pi$
$264$		0			0
$265$			6.88260i			0.422794i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.72267	+	13.3760i	−0.470859	+	0.815552i	−0.999444	−	0.0333281i	$-0.989389\pi$
$269$	−7.72267	+	13.3760i	−0.470859	+	0.815552i	0.528585	+	0.848880i	$0.322723\pi$
$270$		0			0
$271$	10.9476	−	6.32057i	0.665016	−	0.383947i	−0.129169	−	0.991623i	$-0.541231\pi$
$271$	10.9476	−	6.32057i	0.665016	−	0.383947i	0.794186	+	0.607675i	$0.207898\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.03599	+	0.598128i	−0.0624724	+	0.0360685i
$276$		0			0
$277$	5.94531	−	10.2976i	0.357219	−	0.618722i	−0.630276	−	0.776371i	$-0.717058\pi$
$277$	5.94531	−	10.2976i	0.357219	−	0.618722i	0.987495	+	0.157649i	$0.0503915\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.18018i			0.189714i	0.995491	+	0.0948568i	$0.0302393\pi$
$281$			3.18018i			0.189714i	−0.995491	+	0.0948568i	$0.969761\pi$
$282$		0			0
$283$	16.0195	+	9.24889i	0.952263	+	0.549789i	0.893783	−	0.448499i	$-0.148041\pi$
$283$	16.0195	+	9.24889i	0.952263	+	0.549789i	0.0584799	+	0.998289i	$0.481375\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−21.8499	−	18.3606i	−1.28976	−	1.08379i
$288$		0			0
$289$	1.55161	+	2.68746i	0.0912711	+	0.158086i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.85949			0.167053			0.0835266	−	0.996506i	$-0.473382\pi$
$293$	2.85949			0.167053			0.0835266	+	0.996506i	$0.473382\pi$
$294$		0			0
$295$	9.41991			0.548448
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.26254	+	5.65088i	0.188677	+	0.326799i
$300$		0			0
$301$	15.1160	+	12.7021i	0.871270	+	0.732136i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.42470	+	1.97725i	0.196098	+	0.113217i
$306$		0			0
$307$			21.6746i			1.23704i	0.785771	+	0.618518i	$0.212266\pi$
$307$			21.6746i			1.23704i	−0.785771	+	0.618518i	$0.787734\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.8462	−	20.5183i	0.671738	−	1.16348i	−0.305673	−	0.952136i	$-0.598881\pi$
$311$	11.8462	−	20.5183i	0.671738	−	1.16348i	0.977411	−	0.211348i	$-0.0677852\pi$
$312$		0			0
$313$	−23.6283	+	13.6418i	−1.33555	+	0.771081i	−0.986144	−	0.165890i	$-0.946950\pi$
$313$	−23.6283	+	13.6418i	−1.33555	+	0.771081i	−0.349407	+	0.936971i	$0.613617\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−21.2647	+	12.2772i	−1.19435	+	0.689556i	−0.959289	−	0.282426i	$-0.908861\pi$
$317$	−21.2647	+	12.2772i	−1.19435	+	0.689556i	−0.235057	+	0.971982i	$0.575528\pi$
$318$		0			0
$319$	0.0418465	−	0.0724802i	0.00234295	−	0.00405811i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.64626i		−	0.0916006i
$324$		0			0
$325$	5.13396	+	2.96409i	0.284781	+	0.164418i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.23157	−	18.2515i	0.178162	−	1.00624i
$330$		0			0
$331$	−8.15579	−	14.1262i	−0.448283	−	0.776449i	0.549991	−	0.835170i	$-0.314631\pi$
$331$	−8.15579	−	14.1262i	−0.448283	−	0.776449i	−0.998274	+	0.0587215i	$0.981298\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.15679			0.227110
$336$		0			0
$337$	−27.3160			−1.48800			−0.743998	−	0.668182i	$-0.767073\pi$
$337$	−27.3160			−1.48800			−0.743998	+	0.668182i	$0.767073\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.743445	−	1.28768i	−0.0402598	−	0.0697320i
$342$		0			0
$343$	9.29424	−	16.0193i	0.501842	−	0.864960i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.37986	−	3.10606i	−0.288806	−	0.166742i	0.348597	−	0.937273i	$-0.386658\pi$
$347$	−5.37986	−	3.10606i	−0.288806	−	0.166742i	−0.637403	+	0.770530i	$0.719991\pi$
$348$		0			0
$349$		−	28.4668i		−	1.52379i	−0.647700	−	0.761896i	$-0.724269\pi$
$349$		−	28.4668i		−	1.52379i	0.647700	−	0.761896i	$-0.275731\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.49346	−	2.58674i	0.0794887	−	0.137678i	−0.823541	−	0.567257i	$-0.808005\pi$
$353$	1.49346	−	2.58674i	0.0794887	−	0.137678i	0.903029	+	0.429579i	$0.141338\pi$
$354$		0			0
$355$	8.13889	−	4.69899i	0.431968	−	0.249397i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−26.5977	+	15.3562i	−1.40377	+	0.810468i	−0.994777	−	0.102070i	$-0.967454\pi$
$359$	−26.5977	+	15.3562i	−1.40377	+	0.810468i	−0.408994	+	0.912537i	$0.634120\pi$
$360$		0			0
$361$	−9.40249	+	16.2856i	−0.494868	+	0.857136i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	5.38204i		−	0.281709i
$366$		0			0
$367$	−16.4877	−	9.51918i	−0.860651	−	0.496897i	0.00357920	−	0.999994i	$-0.498861\pi$
$367$	−16.4877	−	9.51918i	−0.860651	−	0.496897i	−0.864230	+	0.503096i	$0.832194\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.88426	−	24.4633i	−0.461248	−	1.27007i
$372$		0			0
$373$	−2.05869	−	3.56576i	−0.106595	−	0.184628i	0.807794	−	0.589465i	$-0.200661\pi$
$373$	−2.05869	−	3.56576i	−0.106595	−	0.184628i	−0.914389	+	0.404837i	$0.867328\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.414750			−0.0213607
$378$		0			0
$379$	−11.2436			−0.577546			−0.288773	−	0.957398i	$-0.593247\pi$
$379$	−11.2436			−0.577546			−0.288773	+	0.957398i	$0.593247\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	15.8046	+	27.3745i	0.807580	+	1.39877i	0.914536	+	0.404505i	$0.132556\pi$
$383$	15.8046	+	27.3745i	0.807580	+	1.39877i	−0.106956	+	0.994264i	$0.534110\pi$
$384$		0			0
$385$	−0.315841	+	0.375863i	−0.0160968	+	0.0191558i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.4018	−	10.6243i	−0.933007	−	0.538672i	−0.0452458	−	0.998976i	$-0.514407\pi$
$389$	−18.4018	−	10.6243i	−0.933007	−	0.538672i	−0.887761	+	0.460304i	$0.847740\pi$
$390$		0			0
$391$			18.5073i			0.935956i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.488611	−	0.846300i	0.0245847	−	0.0425820i
$396$		0			0
$397$	20.6927	−	11.9469i	1.03854	−	0.599599i	0.119118	−	0.992880i	$-0.461993\pi$
$397$	20.6927	−	11.9469i	1.03854	−	0.599599i	0.919419	+	0.393281i	$0.128660\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−22.0121	+	12.7087i	−1.09923	+	0.634642i	−0.936019	−	0.351948i	$-0.885519\pi$
$401$	−22.0121	+	12.7087i	−1.09923	+	0.634642i	−0.163213	+	0.986591i	$0.552186\pi$
$402$		0			0
$403$	−3.68423	+	6.38127i	−0.183524	+	0.317874i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.186248i			0.00923197i
$408$		0			0
$409$	19.3831	+	11.1908i	0.958433	+	0.553351i	0.895690	−	0.444678i	$-0.146682\pi$
$409$	19.3831	+	11.1908i	0.958433	+	0.553351i	0.0627424	+	0.998030i	$0.480015\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−33.4818	+	12.1595i	−1.64753	+	0.598330i
$414$		0			0
$415$	2.60551	+	4.51288i	0.127900	+	0.221528i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.0836			−0.688030			−0.344015	−	0.938964i	$-0.611787\pi$
$419$	−14.0836			−0.688030			−0.344015	+	0.938964i	$0.611787\pi$
$420$		0			0
$421$	−16.1528			−0.787238			−0.393619	−	0.919274i	$-0.628777\pi$
$421$	−16.1528			−0.787238			−0.393619	+	0.919274i	$0.628777\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.40717	+	14.5617i	0.407808	+	0.706344i
$426$		0			0
$427$	−14.7250	−	2.60718i	−0.712590	−	0.126170i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.16179	−	4.13486i	−0.344971	−	0.199169i	0.317497	−	0.948259i	$-0.397158\pi$
$431$	−7.16179	−	4.13486i	−0.344971	−	0.199169i	−0.662468	+	0.749090i	$0.730491\pi$
$432$		0			0
$433$		−	4.35102i		−	0.209097i	−0.994520	−	0.104548i	$-0.966660\pi$
$433$		−	4.35102i		−	0.209097i	0.994520	−	0.104548i	$-0.0333397\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.09622	+	1.89872i	−0.0524395	+	0.0908279i
$438$		0			0
$439$	18.0200	−	10.4039i	0.860048	−	0.496549i	−0.00398054	−	0.999992i	$-0.501267\pi$
$439$	18.0200	−	10.4039i	0.860048	−	0.496549i	0.864028	+	0.503443i	$0.167934\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−26.7927	+	15.4688i	−1.27296	+	0.734945i	−0.975544	−	0.219803i	$-0.929459\pi$
$443$	−26.7927	+	15.4688i	−1.27296	+	0.734945i	−0.297417	+	0.954748i	$0.596125\pi$
$444$		0			0
$445$	−3.91290	+	6.77734i	−0.185489	+	0.321277i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			20.9215i			0.987346i	0.869648	+	0.493673i	$0.164346\pi$
$449$			20.9215i			0.987346i	−0.869648	+	0.493673i	$0.835654\pi$
$450$		0			0
$451$	−2.47763	−	1.43046i	−0.116667	−	0.0673577i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.39568	+	0.424175i	0.112311	+	0.0198857i
$456$		0			0
$457$	−1.15058	−	1.99286i	−0.0538217	−	0.0932218i	0.837859	−	0.545886i	$-0.183807\pi$
$457$	−1.15058	−	1.99286i	−0.0538217	−	0.0932218i	−0.891681	+	0.452664i	$0.850474\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.8499			−0.831355			−0.415677	−	0.909512i	$-0.636455\pi$
$461$	−17.8499			−0.831355			−0.415677	+	0.909512i	$0.636455\pi$
$462$		0			0
$463$	12.4807			0.580026			0.290013	−	0.957023i	$-0.406340\pi$
$463$	12.4807			0.580026			0.290013	+	0.957023i	$0.406340\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.42799	−	4.20541i	−0.112354	−	0.194603i	0.804365	−	0.594136i	$-0.202506\pi$
$467$	−2.42799	−	4.20541i	−0.112354	−	0.194603i	−0.916719	+	0.399533i	$0.869172\pi$
$468$		0			0
$469$	−14.7748	+	5.36571i	−0.682236	+	0.247766i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.71405	+	0.989607i	0.0788121	+	0.0455022i
$474$		0			0
$475$			1.99189i			0.0913942i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.40542	−	7.63041i	0.201289	−	0.348642i	−0.747655	−	0.664087i	$-0.768820\pi$
$479$	4.40542	−	7.63041i	0.201289	−	0.348642i	0.948944	+	0.315445i	$0.102154\pi$
$480$		0			0
$481$	0.799318	−	0.461486i	0.0364458	−	0.0210420i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.42441	+	3.70914i	−0.291718	+	0.168423i
$486$		0			0
$487$	4.66185	−	8.07456i	0.211249	−	0.365893i	−0.740857	−	0.671663i	$-0.765580\pi$
$487$	4.66185	−	8.07456i	0.211249	−	0.365893i	0.952106	+	0.305770i	$0.0989137\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	31.0836i		−	1.40278i	−0.712777	−	0.701391i	$-0.752563\pi$
$491$		−	31.0836i		−	1.40278i	0.712777	−	0.701391i	$-0.247437\pi$
$492$		0			0
$493$	−1.01877	−	0.588186i	−0.0458830	−	0.0264906i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−22.8630	+	27.2079i	−1.02555	+	1.22044i
$498$		0			0
$499$	11.1694	+	19.3459i	0.500010	+	0.866043i	1.00000		1.16519e-5i	$3.70891e-6\pi$
$499$	11.1694	+	19.3459i	0.500010	+	0.866043i	−0.499990	+	0.866031i	$0.666663\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.2396			−0.545738			−0.272869	−	0.962051i	$-0.587973\pi$
$503$	−12.2396			−0.545738			−0.272869	+	0.962051i	$0.587973\pi$
$504$		0			0
$505$	12.2490			0.545072
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.05496	−	12.2195i	−0.312706	−	0.541622i	0.666242	−	0.745736i	$-0.267902\pi$
$509$	−7.05496	−	12.2195i	−0.312706	−	0.541622i	−0.978947	+	0.204114i	$0.934569\pi$
$510$		0			0
$511$	6.94729	+	19.1297i	0.307330	+	0.846250i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.17588	−	2.98830i	−0.228077	−	0.131680i
$516$		0			0
$517$		−	1.85803i		−	0.0817161i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.81632	−	4.87800i	0.123385	−	0.213709i	−0.797715	−	0.603034i	$-0.793958\pi$
$521$	2.81632	−	4.87800i	0.123385	−	0.213709i	0.921101	+	0.389325i	$0.127292\pi$
$522$		0			0
$523$	−33.2293	+	19.1849i	−1.45302	+	0.838899i	−0.998651	−	0.0519176i	$-0.983467\pi$
$523$	−33.2293	+	19.1849i	−1.45302	+	0.838899i	−0.454364	+	0.890816i	$0.650133\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−18.0995	+	10.4497i	−0.788425	+	0.455197i
$528$		0			0
$529$	0.823769	−	1.42681i	0.0358161	−	0.0620352i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			14.1776i			0.614100i
$534$		0			0
$535$	6.36329	+	3.67385i	0.275109	+	0.158834i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.637441	−	1.74365i	0.0274565	−	0.0751045i
$540$		0			0
$541$	−3.21673	−	5.57154i	−0.138298	−	0.239539i	0.788555	−	0.614965i	$-0.210830\pi$
$541$	−3.21673	−	5.57154i	−0.138298	−	0.239539i	−0.926852	+	0.375426i	$0.877496\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.96464			−0.426838
$546$		0			0
$547$	13.0578			0.558310			0.279155	−	0.960246i	$-0.409946\pi$
$547$	13.0578			0.558310			0.279155	+	0.960246i	$0.409946\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.0696787	−	0.120687i	−0.00296841	−	0.00514144i
$552$		0			0
$553$	−0.644276	+	3.63878i	−0.0273974	+	0.154736i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.5409	−	14.7460i	−1.08220	−	0.624809i	−0.150712	−	0.988578i	$-0.548157\pi$
$557$	−25.5409	−	14.7460i	−1.08220	−	0.624809i	−0.931489	+	0.363769i	$0.881490\pi$
$558$		0			0
$559$		−	9.80822i		−	0.414844i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.25934	−	9.10944i	0.221655	−	0.383917i	−0.733656	−	0.679521i	$-0.762188\pi$
$563$	5.25934	−	9.10944i	0.221655	−	0.383917i	0.955311	+	0.295604i	$0.0955209\pi$
$564$		0			0
$565$	−9.36020	+	5.40411i	−0.393787	+	0.227353i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−22.8054	+	13.1667i	−0.956053	+	0.551977i	−0.894956	−	0.446155i	$-0.852793\pi$
$569$	−22.8054	+	13.1667i	−0.956053	+	0.551977i	−0.0610967	+	0.998132i	$0.519460\pi$
$570$		0			0
$571$	22.0295	−	38.1562i	0.921906	−	1.59679i	0.125444	−	0.992101i	$-0.459965\pi$
$571$	22.0295	−	38.1562i	0.921906	−	1.59679i	0.796463	−	0.604688i	$-0.206702\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	22.3929i		−	0.933847i
$576$		0			0
$577$	12.1535	+	7.01684i	0.505957	+	0.292115i	0.731170	−	0.682195i	$-0.238974\pi$
$577$	12.1535	+	7.01684i	0.505957	+	0.292115i	−0.225213	+	0.974310i	$0.572308\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−15.0863	−	12.6772i	−0.625885	−	0.525937i
$582$		0			0
$583$	−1.30448	−	2.25943i	−0.0540262	−	0.0935762i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.04939			−0.125862			−0.0629308	−	0.998018i	$-0.520045\pi$
$587$	−3.04939			−0.125862			−0.0629308	+	0.998018i	$0.520045\pi$
$588$		0			0
$589$	−2.47583			−0.102015
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.3041	+	23.0434i	0.546334	+	0.946278i	0.998522	+	0.0543552i	$0.0173103\pi$
$593$	13.3041	+	23.0434i	0.546334	+	0.946278i	−0.452188	+	0.891923i	$0.649356\pi$
$594$		0			0
$595$	5.28306	+	4.43941i	0.216585	+	0.181998i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.86333	−	2.23050i	−0.157852	−	0.0911356i	0.418993	−	0.907989i	$-0.362383\pi$
$599$	−3.86333	−	2.23050i	−0.157852	−	0.0911356i	−0.576845	+	0.816854i	$0.695716\pi$
$600$		0			0
$601$		−	6.06239i		−	0.247290i	−0.992327	−	0.123645i	$-0.960542\pi$
$601$		−	6.06239i		−	0.247290i	0.992327	−	0.123645i	$-0.0394584\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.82350	−	6.62250i	0.155447	−	0.269243i
$606$		0			0
$607$	−39.2581	+	22.6657i	−1.59344	+	0.919971i	−0.600725	+	0.799455i	$0.705121\pi$
$607$	−39.2581	+	22.6657i	−1.59344	+	0.919971i	−0.992711	+	0.120516i	$0.961545\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.97409	+	4.60384i	−0.322597	+	0.186251i
$612$		0			0
$613$	16.6294	−	28.8029i	0.671654	−	1.16334i	−0.305781	−	0.952102i	$-0.598917\pi$
$613$	16.6294	−	28.8029i	0.671654	−	1.16334i	0.977435	−	0.211237i	$-0.0677492\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			36.1423i			1.45503i	0.686090	+	0.727516i	$0.259325\pi$
$617$			36.1423i			1.45503i	−0.686090	+	0.727516i	$0.740675\pi$
$618$		0			0
$619$	22.9031	+	13.2231i	0.920554	+	0.531482i	0.883812	−	0.467843i	$-0.154969\pi$
$619$	22.9031	+	13.2231i	0.920554	+	0.531482i	0.0367423	+	0.999325i	$0.488302\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.15949	−	29.1401i	0.206711	−	1.16747i
$624$		0			0
$625$	−8.94843	−	15.4991i	−0.357937	−	0.619965i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.61787			0.104381
$630$		0			0
$631$	−32.0484			−1.27583			−0.637914	−	0.770107i	$-0.720203\pi$
$631$	−32.0484			−1.27583			−0.637914	+	0.770107i	$0.720203\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.63687	+	13.2274i	0.303060	+	0.524915i
$636$		0			0
$637$	−9.06267	+	1.58474i	−0.359076	+	0.0627898i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21.1444	+	12.2077i	0.835153	+	0.482176i	0.855614	−	0.517615i	$-0.173180\pi$
$641$	21.1444	+	12.2077i	0.835153	+	0.482176i	−0.0204610	+	0.999791i	$0.506513\pi$
$642$		0			0
$643$		−	36.8366i		−	1.45269i	−0.687329	−	0.726346i	$-0.741217\pi$
$643$		−	36.8366i		−	1.45269i	0.687329	−	0.726346i	$-0.258783\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	13.2847	−	23.0098i	0.522276	−	0.904608i	−0.477389	−	0.878692i	$-0.658417\pi$
$647$	13.2847	−	23.0098i	0.522276	−	0.904608i	0.999664	−	0.0259155i	$-0.00825009\pi$
$648$		0			0
$649$	−3.09239	+	1.78539i	−0.121387	+	0.0700827i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−26.2767	+	15.1709i	−1.02829	+	0.593683i	−0.916494	−	0.400049i	$-0.868993\pi$
$653$	−26.2767	+	15.1709i	−1.02829	+	0.593683i	−0.111794	+	0.993731i	$0.535660\pi$
$654$		0			0
$655$	−1.82233	+	3.15637i	−0.0712044	+	0.123330i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	47.3281i		−	1.84364i	−0.387618	−	0.921820i	$-0.626702\pi$
$659$		−	47.3281i		−	1.84364i	0.387618	−	0.921820i	$-0.373298\pi$
$660$		0			0
$661$	−30.4187	−	17.5623i	−1.18315	−	0.683092i	−0.226409	−	0.974032i	$-0.572699\pi$
$661$	−30.4187	−	17.5623i	−1.18315	−	0.683092i	−0.956741	+	0.290940i	$0.906032\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.279049	+	0.768376i	0.0108211	+	0.0297963i
$666$		0			0
$667$	0.783329	+	1.35677i	0.0303306	+	0.0525342i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.49903			−0.0578692
$672$		0			0
$673$	5.09516			0.196404			0.0982020	−	0.995167i	$-0.468691\pi$
$673$	5.09516			0.196404			0.0982020	+	0.995167i	$0.468691\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.42072	+	14.5851i	0.323635	+	0.560551i	0.981235	−	0.192815i	$-0.0617618\pi$
$677$	8.42072	+	14.5851i	0.323635	+	0.560551i	−0.657601	+	0.753367i	$0.728429\pi$
$678$		0			0
$679$	18.0469	−	21.4765i	0.692575	−	0.824191i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.7555	+	9.09645i	0.602868	+	0.348066i	0.770169	−	0.637840i	$-0.220172\pi$
$683$	15.7555	+	9.09645i	0.602868	+	0.348066i	−0.167301	+	0.985906i	$0.553505\pi$
$684$		0			0
$685$			1.88715i			0.0721042i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.46452	+	11.1969i	−0.246279	+	0.426567i
$690$		0			0
$691$	3.05405	−	1.76326i	0.116182	−	0.0670775i	−0.440783	−	0.897614i	$-0.645299\pi$
$691$	3.05405	−	1.76326i	0.116182	−	0.0670775i	0.556965	+	0.830536i	$0.311966\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.09786	−	4.09795i	0.269237	−	0.155444i
$696$		0			0
$697$	−20.1063	+	34.8251i	−0.761579	+	1.31909i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	13.3502i		−	0.504229i	−0.967697	−	0.252114i	$-0.918874\pi$
$701$		−	13.3502i		−	0.504229i	0.967697	−	0.252114i	$-0.0811259\pi$
$702$		0			0
$703$	0.268574	+	0.155061i	0.0101295	+	0.00584824i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−43.5374	+	15.8113i	−1.63739	+	0.594647i
$708$		0			0
$709$	21.1447	+	36.6237i	0.794107	+	1.37543i	0.923405	+	0.383827i	$0.125394\pi$
$709$	21.1447	+	36.6237i	0.794107	+	1.37543i	−0.129298	+	0.991606i	$0.541273\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	27.8333			1.04236
$714$		0			0
$715$	0.243884			0.00912076
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15.2035	+	26.3332i	0.566994	+	0.982062i	0.996861	+	0.0791697i	$0.0252269\pi$
$719$	15.2035	+	26.3332i	0.566994	+	0.982062i	−0.429868	+	0.902892i	$0.641440\pi$
$720$		0			0
$721$	22.2544	+	3.94032i	0.828796	+	0.146745i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.23265	+	0.711673i	0.0457796	+	0.0264309i
$726$		0			0
$727$			13.1256i			0.486802i	0.969926	+	0.243401i	$0.0782631\pi$
$727$			13.1256i			0.486802i	−0.969926	+	0.243401i	$0.921737\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	13.9097	−	24.0924i	0.514470	−	0.891089i
$732$		0			0
$733$	32.7001	−	18.8794i	1.20781	−	0.697327i	0.245527	−	0.969390i	$-0.421039\pi$
$733$	32.7001	−	18.8794i	1.20781	−	0.697327i	0.962280	+	0.272063i	$0.0877058\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.36460	+	0.787853i	−0.0502657	+	0.0290209i
$738$		0			0
$739$	−13.1215	+	22.7271i	−0.482683	+	0.836031i	−0.999802	−	0.0198820i	$-0.993671\pi$
$739$	−13.1215	+	22.7271i	−0.482683	+	0.836031i	0.517119	+	0.855913i	$0.327004\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	10.1426i		−	0.372098i	−0.982541	−	0.186049i	$-0.940432\pi$
$743$		−	10.1426i		−	0.372098i	0.982541	−	0.186049i	$-0.0595682\pi$
$744$		0			0
$745$	−11.4082	−	6.58655i	−0.417966	−	0.241313i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−27.3598	−	4.84428i	−0.999705	−	0.177006i
$750$		0			0
$751$	−3.95369	−	6.84798i	−0.144272	−	0.249886i	0.784829	−	0.619712i	$-0.212751\pi$
$751$	−3.95369	−	6.84798i	−0.144272	−	0.249886i	−0.929101	+	0.369826i	$0.879417\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.99855			−0.254703
$756$		0			0
$757$	29.8903			1.08638			0.543191	−	0.839609i	$-0.317216\pi$
$757$	29.8903			1.08638			0.543191	+	0.839609i	$0.317216\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.05687	−	5.29465i	−0.110811	−	0.191931i	0.805286	−	0.592886i	$-0.202012\pi$
$761$	−3.05687	−	5.29465i	−0.110811	−	0.191931i	−0.916098	+	0.400955i	$0.868678\pi$
$762$		0			0
$763$	35.4180	−	12.8627i	1.28222	−	0.465660i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	15.3247	+	8.84771i	0.553342	+	0.319472i
$768$		0			0
$769$		−	11.3145i		−	0.408011i	−0.978970	−	0.204005i	$-0.934604\pi$
$769$		−	11.3145i		−	0.408011i	0.978970	−	0.204005i	$-0.0653960\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.2106	−	33.2737i	0.690956	−	1.19677i	−0.280569	−	0.959834i	$-0.590523\pi$
$773$	19.2106	−	33.2737i	0.690956	−	1.19677i	0.971525	−	0.236937i	$-0.0761437\pi$
$774$		0			0
$775$	21.8994	−	12.6436i	0.786648	−	0.454172i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.12551	+	2.38186i	−0.147812	+	0.0853391i
$780$		0			0
$781$	−1.78124	+	3.08519i	−0.0637376	+	0.110397i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			0.176358i			0.00629447i
$786$		0			0
$787$	−41.2747	−	23.8300i	−1.47129	−	0.849447i	−0.471806	−	0.881703i	$-0.656398\pi$
$787$	−41.2747	−	23.8300i	−1.47129	−	0.849447i	−0.999480	+	0.0322557i	$0.989731\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	26.2938	−	31.2906i	0.934900	−	1.11257i
$792$		0			0
$793$	3.71430	+	6.43335i	0.131898	+	0.228455i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	30.2717			1.07228			0.536139	−	0.844129i	$-0.319882\pi$
$797$	30.2717			1.07228			0.536139	+	0.844129i	$0.319882\pi$
$798$		0			0
$799$	−26.1161			−0.923922
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.02008	+	1.76683i	0.0359978	+	0.0623500i
$804$		0			0
$805$	−3.13707	−	8.63810i	−0.110567	−	0.304453i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.219373	+	0.126655i	0.00771273	+	0.00445295i	0.503851	−	0.863790i	$-0.331916\pi$
$809$	0.219373	+	0.126655i	0.00771273	+	0.00445295i	−0.496139	+	0.868243i	$0.665249\pi$
$810$		0			0
$811$		−	22.0629i		−	0.774735i	−0.921925	−	0.387367i	$-0.873385\pi$
$811$		−	22.0629i		−	0.774735i	0.921925	−	0.387367i	$-0.126615\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.00698	+	5.20824i	−0.105330	+	0.182437i
$816$		0			0
$817$	2.85407	−	1.64780i	0.0998514	−	0.0576492i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	24.2467	−	13.9988i	0.846214	−	0.488562i	−0.0131576	−	0.999913i	$-0.504188\pi$
$821$	24.2467	−	13.9988i	0.846214	−	0.488562i	0.859372	+	0.511352i	$0.170855\pi$
$822$		0			0
$823$	−24.4771	+	42.3955i	−0.853217	+	1.47782i	0.0250719	+	0.999686i	$0.492019\pi$
$823$	−24.4771	+	42.3955i	−0.853217	+	1.47782i	−0.878289	+	0.478130i	$0.841315\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			0.641658i			0.0223126i	0.999938	+	0.0111563i	$0.00355124\pi$
$827$			0.641658i			0.0223126i	−0.999938	+	0.0111563i	$0.996449\pi$
$828$		0			0
$829$	9.57180	+	5.52628i	0.332442	+	0.191936i	0.656925	−	0.753956i	$-0.271857\pi$
$829$	9.57180	+	5.52628i	0.332442	+	0.191936i	−0.324483	+	0.945892i	$0.605190\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−24.5085	−	8.95975i	−0.849168	−	0.310437i
$834$		0			0
$835$	−1.57029	−	2.71981i	−0.0543420	−	0.0941230i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	9.24661			0.319228			0.159614	−	0.987179i	$-0.448975\pi$
$839$	9.24661			0.319228			0.159614	+	0.987179i	$0.448975\pi$
$840$		0			0
$841$	28.9004			0.996566
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.94346	+	6.83028i	0.135659	+	0.234969i
$846$		0			0
$847$	−5.04161	+	28.4743i	−0.173232	+	0.978388i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.01931	−	1.74320i	−0.103501	−	0.0597561i
$852$		0			0
$853$			39.6358i			1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$853$			39.6358i			1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11.9260	+	20.6565i	−0.407385	+	0.705612i	−0.994596	−	0.103822i	$-0.966893\pi$
$857$	−11.9260	+	20.6565i	−0.407385	+	0.705612i	0.587211	+	0.809434i	$0.300226\pi$
$858$		0			0
$859$	9.62480	−	5.55688i	0.328394	−	0.189598i	−0.326734	−	0.945116i	$-0.605948\pi$
$859$	9.62480	−	5.55688i	0.328394	−	0.189598i	0.655128	+	0.755518i	$0.272615\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−38.7780	+	22.3885i	−1.32002	+	0.762113i	−0.983731	−	0.179646i	$-0.942505\pi$
$863$	−38.7780	+	22.3885i	−1.32002	+	0.762113i	−0.336287	+	0.941759i	$0.609171\pi$
$864$		0			0
$865$	−2.49128	+	4.31503i	−0.0847061	+	0.146715i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			0.370434i			0.0125661i
$870$		0			0
$871$	6.76244	+	3.90429i	0.229136	+	0.132292i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−13.4782	−	11.3258i	−0.455646	−	0.382883i
$876$		0			0
$877$	−1.84096	−	3.18863i	−0.0621647	−	0.107672i	0.833268	−	0.552869i	$-0.186467\pi$
$877$	−1.84096	−	3.18863i	−0.0621647	−	0.107672i	−0.895433	+	0.445197i	$0.853134\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	17.3992			0.586194			0.293097	−	0.956083i	$-0.405314\pi$
$881$	17.3992			0.586194			0.293097	+	0.956083i	$0.405314\pi$
$882$		0			0
$883$	−2.02834			−0.0682592			−0.0341296	−	0.999417i	$-0.510866\pi$
$883$	−2.02834			−0.0682592			−0.0341296	+	0.999417i	$0.510866\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.1890	+	40.1645i	0.778610	+	1.34859i	0.932743	+	0.360542i	$0.117408\pi$
$887$	23.1890	+	40.1645i	0.778610	+	1.34859i	−0.154132	+	0.988050i	$0.549258\pi$
$888$		0			0
$889$	−44.2186	−	37.1573i	−1.48305	−	1.24622i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.67932	−	1.54691i	−0.0896601	−	0.0517653i
$894$		0			0
$895$			17.8762i			0.597536i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.884576	+	1.53213i	−0.0295023	+	0.0510994i
$900$		0			0
$901$	−31.7582	+	18.3356i	−1.05802	+	0.610847i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.571488	−	0.329949i	0.0189969	−	0.0109679i
$906$		0			0
$907$	−8.01957	+	13.8903i	−0.266285	+	0.461220i	−0.967900	−	0.251337i	$-0.919130\pi$
$907$	−8.01957	+	13.8903i	−0.266285	+	0.461220i	0.701614	+	0.712557i	$0.252463\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	20.5344i		−	0.680335i	−0.940365	−	0.340167i	$-0.889516\pi$
$911$		−	20.5344i		−	0.680335i	0.940365	−	0.340167i	$-0.110484\pi$
$912$		0			0
$913$	−1.71069	−	0.987665i	−0.0566154	−	0.0326869i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.40290	−	13.5712i	0.0793507	−	0.448161i
$918$		0			0
$919$	−17.7069	−	30.6693i	−0.584097	−	1.01169i	−0.994987	−	0.100001i	$-0.968115\pi$
$919$	−17.7069	−	30.6693i	−0.584097	−	1.01169i	0.410890	−	0.911685i	$-0.365218\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	17.6542			0.581096
$924$		0			0
$925$	−3.16748			−0.104146
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19.8626	+	34.4030i	0.651670	+	1.12873i	0.982718	+	0.185111i	$0.0592646\pi$
$929$	19.8626	+	34.4030i	0.651670	+	1.12873i	−0.331048	+	0.943614i	$0.607402\pi$
$930$		0			0
$931$	−1.98369	−	2.37089i	−0.0650127	−	0.0777027i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.599064	+	0.345870i	0.0195915	+	0.0113112i
$936$		0			0
$937$		−	23.2142i		−	0.758376i	−0.925320	−	0.379188i	$-0.876203\pi$
$937$		−	23.2142i		−	0.758376i	0.925320	−	0.379188i	$-0.123797\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	17.9616	−	31.1104i	0.585531	−	1.01417i	−0.409278	−	0.912410i	$-0.634219\pi$
$941$	17.9616	−	31.1104i	0.585531	−	1.01417i	0.994809	−	0.101760i	$-0.0324474\pi$
$942$		0			0
$943$	46.3790	−	26.7769i	1.51031	−	0.871977i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.7365	+	15.4363i	−0.868818	+	0.501612i	−0.866955	−	0.498386i	$-0.833926\pi$
$947$	−26.7365	+	15.4363i	−0.868818	+	0.501612i	−0.00186277	+	0.999998i	$0.500593\pi$
$948$		0			0
$949$	5.05511	−	8.75571i	0.164096	−	0.284222i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			30.4640i			0.986826i	0.869795	+	0.493413i	$0.164251\pi$
$953$			30.4640i			0.986826i	−0.869795	+	0.493413i	$0.835749\pi$
$954$		0			0
$955$	−1.80100	−	1.03981i	−0.0582791	−	0.0336475i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.43599	−	6.70762i	−0.0786621	−	0.216600i
$960$		0			0
$961$	0.215406	+	0.373095i	0.00694859	+	0.0120353i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12.9537			0.416995
$966$		0			0
$967$	13.5173			0.434687			0.217343	−	0.976095i	$-0.430261\pi$
$967$	13.5173			0.434687			0.217343	+	0.976095i	$0.430261\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	16.6428	+	28.8261i	0.534092	+	0.925074i	0.999207	+	0.0398238i	$0.0126797\pi$
$971$	16.6428	+	28.8261i	0.534092	+	0.925074i	−0.465115	+	0.885250i	$0.653987\pi$
$972$		0			0
$973$	−19.9387	+	23.7278i	−0.639204	+	0.760677i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	38.9127	+	22.4662i	1.24493	+	0.718758i	0.970093	−	0.242734i	$-0.0780441\pi$
$977$	38.9127	+	22.4662i	1.24493	+	0.718758i	0.274833	+	0.961492i	$0.411377\pi$
$978$		0			0
$979$		−	2.96651i		−	0.0948100i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13.4474	+	23.2916i	−0.428907	+	0.742888i	−0.996776	−	0.0802305i	$-0.974434\pi$
$983$	−13.4474	+	23.2916i	−0.428907	+	0.742888i	0.567870	+	0.823118i	$0.307768\pi$
$984$		0			0
$985$	8.59035	−	4.95964i	0.273711	−	0.158027i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.0855	+	18.5246i	−1.02026	+	0.589048i
$990$		0			0
$991$	17.7201	−	30.6920i	0.562896	−	0.974965i	−0.434346	−	0.900746i	$-0.643020\pi$
$991$	17.7201	−	30.6920i	0.562896	−	0.974965i	0.997242	−	0.0742186i	$-0.0236463\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	16.5855i		−	0.525796i
$996$		0			0
$997$	28.1418	+	16.2477i	0.891259	+	0.514568i	0.874354	−	0.485289i	$-0.161286\pi$
$997$	28.1418	+	16.2477i	0.891259	+	0.514568i	0.0169046	+	0.999857i	$0.494619\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.t.b.1781.5		16
3.2	odd	2		2268.2.t.a.1781.4		16
7.5	odd	6		2268.2.t.a.2105.4		16
9.2	odd	6		252.2.bm.a.185.8	yes	16
9.4	even	3		252.2.w.a.101.6	yes	16
9.5	odd	6		756.2.w.a.521.4		16
9.7	even	3		756.2.bm.a.17.4		16
21.5	even	6	inner	2268.2.t.b.2105.5		16
36.7	odd	6		3024.2.df.d.17.4		16
36.11	even	6		1008.2.df.d.689.1		16
36.23	even	6		3024.2.ca.d.2033.4		16
36.31	odd	6		1008.2.ca.d.353.3		16
63.2	odd	6		1764.2.w.b.509.3		16
63.4	even	3		1764.2.x.a.1469.6		16
63.5	even	6		756.2.bm.a.89.4		16
63.11	odd	6		1764.2.x.b.293.3		16
63.13	odd	6		1764.2.w.b.1109.3		16
63.16	even	3		5292.2.w.b.1097.5		16
63.20	even	6		1764.2.bm.a.1697.1		16
63.23	odd	6		5292.2.bm.a.4625.5		16
63.25	even	3		5292.2.x.b.881.5		16
63.31	odd	6		1764.2.x.b.1469.3		16
63.32	odd	6		5292.2.x.a.4409.4		16
63.34	odd	6		5292.2.bm.a.2285.5		16
63.38	even	6		1764.2.x.a.293.6		16
63.40	odd	6		252.2.bm.a.173.8	yes	16
63.41	even	6		5292.2.w.b.521.5		16
63.47	even	6		252.2.w.a.5.6	✓	16
63.52	odd	6		5292.2.x.a.881.4		16
63.58	even	3		1764.2.bm.a.1685.1		16
63.59	even	6		5292.2.x.b.4409.5		16
63.61	odd	6		756.2.w.a.341.4		16
252.47	odd	6		1008.2.ca.d.257.3		16
252.103	even	6		1008.2.df.d.929.1		16
252.131	odd	6		3024.2.df.d.1601.4		16
252.187	even	6		3024.2.ca.d.2609.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.w.a.5.6	✓	16	63.47	even	6
252.2.w.a.101.6	yes	16	9.4	even	3
252.2.bm.a.173.8	yes	16	63.40	odd	6
252.2.bm.a.185.8	yes	16	9.2	odd	6
756.2.w.a.341.4		16	63.61	odd	6
756.2.w.a.521.4		16	9.5	odd	6
756.2.bm.a.17.4		16	9.7	even	3
756.2.bm.a.89.4		16	63.5	even	6
1008.2.ca.d.257.3		16	252.47	odd	6
1008.2.ca.d.353.3		16	36.31	odd	6
1008.2.df.d.689.1		16	36.11	even	6
1008.2.df.d.929.1		16	252.103	even	6
1764.2.w.b.509.3		16	63.2	odd	6
1764.2.w.b.1109.3		16	63.13	odd	6
1764.2.x.a.293.6		16	63.38	even	6
1764.2.x.a.1469.6		16	63.4	even	3
1764.2.x.b.293.3		16	63.11	odd	6
1764.2.x.b.1469.3		16	63.31	odd	6
1764.2.bm.a.1685.1		16	63.58	even	3
1764.2.bm.a.1697.1		16	63.20	even	6
2268.2.t.a.1781.4		16	3.2	odd	2
2268.2.t.a.2105.4		16	7.5	odd	6
2268.2.t.b.1781.5		16	1.1	even	1	trivial
2268.2.t.b.2105.5		16	21.5	even	6	inner
3024.2.ca.d.2033.4		16	36.23	even	6
3024.2.ca.d.2609.4		16	252.187	even	6
3024.2.df.d.17.4		16	36.7	odd	6
3024.2.df.d.1601.4		16	252.131	odd	6
5292.2.w.b.521.5		16	63.41	even	6
5292.2.w.b.1097.5		16	63.16	even	3
5292.2.x.a.881.4		16	63.52	odd	6
5292.2.x.a.4409.4		16	63.32	odd	6
5292.2.x.b.881.5		16	63.25	even	3
5292.2.x.b.4409.5		16	63.59	even	6
5292.2.bm.a.2285.5		16	63.34	odd	6
5292.2.bm.a.4625.5		16	63.23	odd	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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.349828 + 0.605920i) q^{5} +(-2.02556 - 1.70209i) q^{7} +O(q^{10})\)	\(q+(0.349828 + 0.605920i) q^{5} +(-2.02556 - 1.70209i) q^{7} +(-0.229685 - 0.132608i) q^{11} +1.31431i q^{13} +(-1.86392 + 3.22840i) q^{17} +(-0.382449 + 0.220807i) q^{19} +(4.29949 - 2.48231i) q^{23} +(2.25524 - 3.90619i) q^{25} +0.315564i q^{29} +(4.85521 + 2.80316i) q^{31} +(0.322736 - 1.82276i) q^{35} +(-0.351124 - 0.608164i) q^{37} +10.7871 q^{41} -7.46263 q^{43} +(3.50285 + 6.06712i) q^{47} +(1.20576 + 6.89537i) q^{49} +(8.51919 + 4.91856i) q^{53} -0.185561i q^{55} +(6.73182 - 11.6598i) q^{59} +(4.89484 - 2.82604i) q^{61} +(-0.796368 + 0.459783i) q^{65} +(2.97060 - 5.14523i) q^{67} -13.4323i q^{71} +(-6.66182 - 3.84620i) q^{73} +(0.239527 + 0.659550i) q^{77} +(-0.698360 - 1.20959i) q^{79} +7.44798 q^{83} -2.60820 q^{85} +(5.59261 + 9.68668i) q^{89} +(2.23708 - 2.66221i) q^{91} +(-0.267582 - 0.154489i) q^{95} +10.6028i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{7}+O(q^{10}) \) 16 * q + 2 * q^7	\( 16 q + 2 q^{7} + 6 q^{11} + 9 q^{17} - 21 q^{23} - 8 q^{25} - 6 q^{31} - 15 q^{35} + q^{37} + 12 q^{41} + 4 q^{43} + 18 q^{47} - 8 q^{49} + 15 q^{59} - 3 q^{61} - 39 q^{65} - 7 q^{67} - 48 q^{77} - q^{79} - 12 q^{85} + 21 q^{89} + 9 q^{91} + 6 q^{95}+O(q^{100}) \) 16 * q + 2 * q^7 + 6 * q^11 + 9 * q^17 - 21 * q^23 - 8 * q^25 - 6 * q^31 - 15 * q^35 + q^37 + 12 * q^41 + 4 * q^43 + 18 * q^47 - 8 * q^49 + 15 * q^59 - 3 * q^61 - 39 * q^65 - 7 * q^67 - 48 * q^77 - q^79 - 12 * q^85 + 21 * q^89 + 9 * q^91 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more