LMFDB - Embedded newform 828.2.e.b.91.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [828,2,Mod(91,828)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(828, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("828.91");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 828 = 2^{2} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	828.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.61161328736$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8869743.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 92)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			91.6
Root			$1.33454 - 0.467979i$ of defining polynomial
Character	$\chi$	$=$	828.91
Dual form			828.2.e.b.91.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.33454 + 0.467979i) q^{2} +(1.56199 + 1.24907i) q^{4} +(1.50000 + 2.39792i) q^{8} +4.88325 q^{13} +(0.879635 + 3.90208i) q^{16} +4.79583i q^{23} +5.00000 q^{25} +(6.51690 + 2.28526i) q^{26} -6.70287 q^{29} +0.309728i q^{31} +(-0.652183 + 5.61913i) q^{32} +3.97345 q^{41} +(-2.24435 + 6.40023i) q^{46} -6.55848i q^{47} -7.00000 q^{49} +(6.67270 + 2.33989i) q^{50} +(7.62760 + 6.09954i) q^{52} +(-8.94525 - 3.13680i) q^{58} -9.59166i q^{59} +(-0.144946 + 0.413344i) q^{62} +(-3.50000 + 7.19375i) q^{64} -14.0461i q^{71} -7.61268 q^{73} +(5.30272 + 1.85949i) q^{82} +(-5.99034 + 7.49105i) q^{92} +(3.06923 - 8.75255i) q^{94} +(-9.34178 - 3.27585i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 9 q^{8} + 30 q^{25} + 27 q^{26} - 42 q^{49} + 3 q^{52} - 15 q^{58} - 45 q^{62} - 21 q^{64} + 33 q^{82} - 39 q^{94}+O(q^{100}) $ 6 * q + 9 * q^8 + 30 * q^25 + 27 * q^26 - 42 * q^49 + 3 * q^52 - 15 * q^58 - 45 * q^62 - 21 * q^64 + 33 * q^82 - 39 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/828\mathbb{Z}\right)^\times$.

$n$	$415$	$461$	$649$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	1.33454	+	0.467979i	0.943662	+	0.330911i
$2$	1.33454	+	0.467979i	0.943662	+	0.330911i
$3$		0			0
$3$		0			0
$4$	1.56199	+	1.24907i	0.780996	+	0.624536i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	1.50000	+	2.39792i	0.530330	+	0.847791i
$9$		0			0
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	4.88325			1.35437			0.677185	−	0.735812i	$-0.263199\pi$
$13$	4.88325			1.35437			0.677185	+	0.735812i	$0.263199\pi$
$14$		0			0
$15$		0			0
$16$	0.879635	+	3.90208i	0.219909	+	0.975520i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.79583i			1.00000i
$23$			4.79583i			1.00000i
$24$		0			0
$25$	5.00000			1.00000
$26$	6.51690	+	2.28526i	1.27807	+	0.448176i
$27$		0			0
$28$		0			0
$29$	−6.70287			−1.24469			−0.622346	−	0.782742i	$-0.713820\pi$
$29$	−6.70287			−1.24469			−0.622346	+	0.782742i	$0.713820\pi$
$30$		0			0
$31$			0.309728i			0.0556288i	0.999613	+	0.0278144i	$0.00885474\pi$
$31$			0.309728i			0.0556288i	−0.999613	+	0.0278144i	$0.991145\pi$
$32$	−0.652183	+	5.61913i	−0.115291	+	0.993332i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.97345			0.620548			0.310274	−	0.950647i	$-0.399579\pi$
$41$	3.97345			0.620548			0.310274	+	0.950647i	$0.399579\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$	−2.24435	+	6.40023i	−0.330911	+	0.943662i
$47$		−	6.55848i		−	0.956652i	−0.878182	−	0.478326i	$-0.841244\pi$
$47$		−	6.55848i		−	0.956652i	0.878182	−	0.478326i	$-0.158756\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$	6.67270	+	2.33989i	0.943662	+	0.330911i
$51$		0			0
$52$	7.62760	+	6.09954i	1.05776	+	0.845854i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−8.94525	−	3.13680i	−1.17457	−	0.411882i
$59$		−	9.59166i		−	1.24873i	−0.781133	−	0.624364i	$-0.785358\pi$
$59$		−	9.59166i		−	1.24873i	0.781133	−	0.624364i	$-0.214642\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	−0.144946	+	0.413344i	−0.0184082	+	0.0524948i
$63$		0			0
$64$	−3.50000	+	7.19375i	−0.437500	+	0.899218i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	14.0461i		−	1.66697i	−0.552542	−	0.833485i	$-0.686342\pi$
$71$		−	14.0461i		−	1.66697i	0.552542	−	0.833485i	$-0.313658\pi$
$72$		0			0
$73$	−7.61268			−0.890997			−0.445498	−	0.895283i	$-0.646973\pi$
$73$	−7.61268			−0.890997			−0.445498	+	0.895283i	$0.646973\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$	5.30272	+	1.85949i	0.585588	+	0.205346i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−5.99034	+	7.49105i	−0.624536	+	0.780996i
$93$		0			0
$94$	3.06923	−	8.75255i	0.316567	−	0.902756i
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−9.34178	−	3.27585i	−0.943662	−	0.330911i
$99$		0			0
$100$	7.80996	+	6.24536i	0.780996	+	0.624536i
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	7.32488	+	11.7096i	0.718264	+	1.14822i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$	−10.4698	−	8.37237i	−0.972099	−	0.777355i
$117$		0			0
$118$	4.48870	−	12.8005i	0.413218	−	1.17838i
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$	−0.386873	+	0.483792i	−0.0347422	+	0.0434458i
$125$		0			0
$126$		0			0
$127$		−	20.9143i		−	1.85585i	−0.372769	−	0.927924i	$-0.621592\pi$
$127$		−	20.9143i		−	1.85585i	0.372769	−	0.927924i	$-0.378408\pi$
$128$	−8.03741	+	7.96241i	−0.710413	+	0.703785i
$129$		0			0
$130$		0			0
$131$		−	2.81465i		−	0.245917i	−0.992412	−	0.122958i	$-0.960762\pi$
$131$		−	2.81465i		−	0.245917i	0.992412	−	0.122958i	$-0.0392382\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		−	10.9218i		−	0.926372i	−0.886261	−	0.463186i	$-0.846706\pi$
$139$		−	10.9218i		−	0.926372i	0.886261	−	0.463186i	$-0.153294\pi$
$140$		0			0
$141$		0			0
$142$	6.57330	−	18.7451i	0.551619	−	1.57306i
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−10.1594	−	3.56257i	−0.840800	−	0.294841i
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$			20.2949i			1.65157i	0.563982	+	0.825787i	$0.309269\pi$
$151$			20.2949i			1.65157i	−0.563982	+	0.825787i	$0.690731\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	9.68285i		−	0.758420i	−0.925311	−	0.379210i	$-0.876196\pi$
$163$		−	9.68285i		−	0.758420i	0.925311	−	0.379210i	$-0.123804\pi$
$164$	6.20649	+	4.96312i	0.484645	+	0.387555i
$165$		0			0
$166$		0			0
$167$			9.59166i			0.742225i	0.928588	+	0.371113i	$0.121024\pi$
$167$			9.59166i			0.742225i	−0.928588	+	0.371113i	$0.878976\pi$
$168$		0			0
$169$	10.8462			0.834321
$170$		0			0
$171$		0			0
$172$		0			0
$173$	18.0000			1.36851			0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000			1.36851			0.684257	+	0.729241i	$0.260127\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	17.7900i		−	1.32968i	−0.746984	−	0.664842i	$-0.768499\pi$
$179$		−	17.7900i		−	1.32968i	0.746984	−	0.664842i	$-0.231501\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−11.5000	+	7.19375i	−0.847791	+	0.530330i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	8.19201	−	10.2443i	0.597464	−	0.747141i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−27.1457			−1.95399			−0.976995	−	0.213262i	$-0.931591\pi$
$193$	−27.1457			−1.95399			−0.976995	+	0.213262i	$0.931591\pi$
$194$		0			0
$195$		0			0
$196$	−10.9339	−	8.74351i	−0.780996	−	0.624536i
$197$	−28.0555			−1.99887			−0.999436	−	0.0335834i	$-0.989308\pi$
$197$	−28.0555			−1.99887			−0.999436	+	0.0335834i	$0.989308\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	7.50000	+	11.9896i	0.530330	+	0.847791i
$201$		0			0
$202$	−8.00724	−	2.80787i	−0.563387	−	0.197561i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	4.29548	+	19.0549i	0.297838	+	1.32122i
$209$		0			0
$210$		0			0
$211$		−	28.7750i		−	1.98095i	−0.137686	−	0.990476i	$-0.543966\pi$
$211$		−	28.7750i		−	1.98095i	0.137686	−	0.990476i	$-0.456034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			28.7750i			1.92692i	0.267860	+	0.963458i	$0.413684\pi$
$223$			28.7750i			1.92692i	−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−10.0543	−	16.0729i	−0.660098	−	1.05524i
$233$	9.43229			0.617930			0.308965	−	0.951073i	$-0.400017\pi$
$233$	9.43229			0.617930			0.308965	+	0.951073i	$0.400017\pi$
$234$		0			0
$235$		0			0
$236$	11.9807	−	14.9821i	0.779876	−	0.975251i
$237$		0			0
$238$		0			0
$239$			27.1631i			1.75703i	0.477711	+	0.878517i	$0.341467\pi$
$239$			27.1631i			1.75703i	−0.477711	+	0.878517i	$0.658533\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−14.6799	−	5.14777i	−0.943662	−	0.330911i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−0.742702	+	0.464592i	−0.0471616	+	0.0295016i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	9.78747	−	27.9110i	0.614121	−	1.75129i
$255$		0			0
$256$	−14.4525	+	6.86482i	−0.903280	+	0.429051i
$257$	25.3261			1.57980			0.789899	−	0.613237i	$-0.210133\pi$
$257$	25.3261			1.57980			0.789899	+	0.613237i	$0.210133\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.31720	−	3.75626i	0.0813766	−	0.232062i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.24402			−0.0758493			−0.0379247	−	0.999281i	$-0.512075\pi$
$269$	−1.24402			−0.0758493			−0.0379247	+	0.999281i	$0.512075\pi$
$270$		0			0
$271$			28.7750i			1.74796i	0.485965	+	0.873978i	$0.338468\pi$
$271$			28.7750i			1.74796i	−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	29.8751			1.79502			0.897511	−	0.440992i	$-0.145373\pi$
$277$	29.8751			1.79502			0.897511	+	0.440992i	$0.145373\pi$
$278$	5.11115	−	14.5755i	0.306547	−	0.874182i
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	17.5446	−	21.9400i	1.04108	−	1.30190i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$	−11.8909	−	9.50879i	−0.695865	−	0.556460i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$			23.4193i			1.35437i
$300$		0			0
$301$		0			0
$302$	−9.49758	+	27.0843i	−0.546524	+	1.55853i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	28.7750i		−	1.64228i	−0.570730	−	0.821138i	$-0.693340\pi$
$307$		−	28.7750i		−	1.64228i	0.570730	−	0.821138i	$-0.306660\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			34.6508i			1.96486i	0.186621	+	0.982432i	$0.440246\pi$
$311$			34.6508i			1.96486i	−0.186621	+	0.982432i	$0.559754\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−30.0000			−1.68497			−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000			−1.68497			−0.842484	+	0.538721i	$0.818908\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	24.4163			1.35437
$326$	4.53137	−	12.9221i	0.250969	−	0.715692i
$327$		0			0
$328$	5.96017	+	9.52799i	0.329095	+	0.526095i
$329$		0			0
$330$		0			0
$331$			31.5264i			1.73285i	0.499310	+	0.866423i	$0.333587\pi$
$331$			31.5264i			1.73285i	−0.499310	+	0.866423i	$0.666413\pi$
$332$		0			0
$333$		0			0
$334$	−4.48870	+	12.8005i	−0.245610	+	0.700410i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	14.4746	+	5.07578i	0.787317	+	0.276086i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	24.0217	+	8.42362i	1.29142	+	0.452857i
$347$		−	9.59166i		−	0.514907i	−0.966291	−	0.257454i	$-0.917117\pi$
$347$		−	9.59166i		−	0.514907i	0.966291	−	0.257454i	$-0.0828835\pi$
$348$		0			0
$349$	10.3421			0.553600			0.276800	−	0.960928i	$-0.410726\pi$
$349$	10.3421			0.553600			0.276800	+	0.960928i	$0.410726\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	30.7849			1.63852			0.819258	−	0.573425i	$-0.194386\pi$
$353$	30.7849			1.63852			0.819258	+	0.573425i	$0.194386\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	8.32533	−	23.7414i	0.440007	−	1.25477i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−18.7137	+	4.21858i	−0.975520	+	0.219909i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	15.7267	−	9.83772i	0.811041	−	0.507341i
$377$	−32.7318			−1.68577
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	−36.2270	−	12.7036i	−1.84391	−	0.646597i
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−10.5000	−	16.7854i	−0.530330	−	0.847791i
$393$		0			0
$394$	−37.4412	−	13.1294i	−1.88626	−	0.661449i
$395$		0			0
$396$		0			0
$397$	−39.6416			−1.98956			−0.994778	−	0.102061i	$-0.967456\pi$
$397$	−39.6416			−1.98956			−0.994778	+	0.102061i	$0.967456\pi$
$398$		0			0
$399$		0			0
$400$	4.39818	+	19.5104i	0.219909	+	0.975520i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$			1.51248i			0.0753420i
$404$	−9.37195	−	7.49444i	−0.466272	−	0.372862i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	36.9122			1.82519			0.912595	−	0.408864i	$-0.134075\pi$
$409$	36.9122			1.82519			0.912595	+	0.408864i	$0.134075\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−3.18478	+	27.4397i	−0.156147	+	1.34534i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	13.4661	−	38.4014i	0.655519	−	1.86935i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			41.5190i			1.98159i	0.135364	+	0.990796i	$0.456780\pi$
$439$			41.5190i			1.98159i	−0.135364	+	0.990796i	$0.543220\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			38.3946i			1.82418i	0.409988	+	0.912091i	$0.365533\pi$
$443$			38.3946i			1.82418i	−0.409988	+	0.912091i	$0.634467\pi$
$444$		0			0
$445$		0			0
$446$	−13.4661	+	38.4014i	−0.637638	+	1.81836i
$447$		0			0
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	36.0024			1.67680			0.838399	−	0.545056i	$-0.183492\pi$
$461$	36.0024			1.67680			0.838399	+	0.545056i	$0.183492\pi$
$462$		0			0
$463$			28.7750i			1.33729i	0.743583	+	0.668644i	$0.233125\pi$
$463$			28.7750i			1.33729i	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−5.89608	−	26.1551i	−0.273719	−	1.21422i
$465$		0			0
$466$	12.5878	+	4.41411i	0.583117	+	0.204480i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	23.0000	−	14.3875i	1.05866	−	0.662238i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−12.7118	+	36.2502i	−0.581422	+	1.65805i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−17.1819	−	13.7398i	−0.780996	−	0.624536i
$485$		0			0
$486$		0			0
$487$		−	40.8995i		−	1.85333i	−0.375884	−	0.926667i	$-0.622661\pi$
$487$		−	40.8995i		−	1.85333i	0.375884	−	0.926667i	$-0.377339\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			4.67301i			0.210890i	0.994425	+	0.105445i	$0.0336267\pi$
$491$			4.67301i			0.210890i	−0.994425	+	0.105445i	$0.966373\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.20858	+	0.272448i	−0.0542670	+	0.0122333i
$497$		0			0
$498$		0			0
$499$			11.5412i			0.516656i	0.966057	+	0.258328i	$0.0831715\pi$
$499$			11.5412i			0.516656i	−0.966057	+	0.258328i	$0.916828\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	26.1235	−	32.6680i	1.15904	−	1.44941i
$509$	41.4612			1.83774			0.918869	−	0.394564i	$-0.129104\pi$
$509$	41.4612			1.83774			0.918869	+	0.394564i	$0.129104\pi$
$510$		0			0
$511$		0			0
$512$	−22.5000	+	2.39792i	−0.994369	+	0.105974i
$513$		0			0
$514$	33.7987	+	11.8521i	1.49079	+	0.522772i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	3.51570	−	4.39645i	0.153584	−	0.192060i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.4033			0.840452
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.66020	−	0.582176i	−0.0715761	−	0.0250994i
$539$		0			0
$540$		0			0
$541$	−0.575595			−0.0247468			−0.0123734	−	0.999923i	$-0.503939\pi$
$541$	−0.575595			−0.0247468			−0.0123734	+	0.999923i	$0.503939\pi$
$542$	−13.4661	+	38.4014i	−0.578418	+	1.64948i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			9.06340i			0.387523i	0.981049	+	0.193761i	$0.0620688\pi$
$547$			9.06340i			0.387523i	−0.981049	+	0.193761i	$0.937931\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	39.8695	+	13.9809i	1.69389	+	0.593993i
$555$		0			0
$556$	13.6421	−	17.0597i	0.578553	−	0.723493i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	33.6815	−	21.0692i	1.41324	−	0.884044i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			23.9792i			1.00000i
$576$		0			0
$577$	−32.6045			−1.35734			−0.678672	−	0.734441i	$-0.737444\pi$
$577$	−32.6045			−1.35734			−0.678672	+	0.734441i	$0.737444\pi$
$578$	22.6872	+	7.95564i	0.943662	+	0.330911i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−11.4190	−	18.2546i	−0.472522	−	0.755379i
$585$		0			0
$586$		0			0
$587$		−	25.2776i		−	1.04332i	−0.853154	−	0.521660i	$-0.825313\pi$
$587$		−	25.2776i		−	1.04332i	0.853154	−	0.521660i	$-0.174687\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−10.9597	+	31.2539i	−0.448176	+	1.27807i
$599$			9.59166i			0.391905i	0.980613	+	0.195952i	$0.0627798\pi$
$599$			9.59166i			0.391905i	−0.980613	+	0.195952i	$0.937220\pi$
$600$		0			0
$601$	42.3711			1.72835			0.864176	−	0.503190i	$-0.167841\pi$
$601$	42.3711			1.72835			0.864176	+	0.503190i	$0.167841\pi$
$602$		0			0
$603$		0			0
$604$	−25.3498	+	31.7004i	−1.03147	+	1.28987i
$605$		0			0
$606$		0			0
$607$			28.7750i			1.16794i	0.811775	+	0.583970i	$0.198502\pi$
$607$			28.7750i			1.16794i	−0.811775	+	0.583970i	$0.801498\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	32.0267i		−	1.29566i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	13.4661	−	38.4014i	0.543447	−	1.54975i
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	−16.2158	+	46.2428i	−0.650195	+	1.85417i
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$	−40.0362	−	14.0394i	−1.59004	−	0.557574i
$635$		0			0
$636$		0			0
$637$	−34.1828			−1.35437
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	47.7677i		−	1.87794i	−0.343996	−	0.938971i	$-0.611781\pi$
$647$		−	47.7677i		−	1.87794i	0.343996	−	0.938971i	$-0.388219\pi$
$648$		0			0
$649$		0			0
$650$	32.5845	+	11.4263i	1.27807	+	0.448176i
$651$		0			0
$652$	12.0946	−	15.1245i	0.473661	−	0.592322i
$653$	−12.1617			−0.475925			−0.237962	−	0.971274i	$-0.576480\pi$
$653$	−12.1617			−0.475925			−0.237962	+	0.971274i	$0.576480\pi$
$654$		0			0
$655$		0			0
$656$	3.49518	+	15.5047i	0.136464	+	0.605357i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−14.7537	+	42.0732i	−0.573418	+	1.63522i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	32.1458i		−	1.24469i
$668$	−11.9807	+	14.9821i	−0.463547	+	0.579675i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−21.6868			−0.835966			−0.417983	−	0.908455i	$-0.637263\pi$
$673$	−21.6868			−0.835966			−0.417983	+	0.908455i	$0.637263\pi$
$674$		0			0
$675$		0			0
$676$	16.9416	+	13.5477i	0.651601	+	0.521064i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	44.0239i		−	1.68453i	−0.539066	−	0.842263i	$-0.681223\pi$
$683$		−	44.0239i		−	1.68453i	0.539066	−	0.842263i	$-0.318777\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	28.7750i		−	1.09465i	−0.836919	−	0.547326i	$-0.815646\pi$
$691$		−	28.7750i		−	1.09465i	0.836919	−	0.547326i	$-0.184354\pi$
$692$	28.1158	+	22.4833i	1.06880	+	0.854687i
$693$		0			0
$694$	4.48870	−	12.8005i	0.170389	−	0.485899i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	13.8019	+	4.83989i	0.522411	+	0.183192i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	41.0837	+	14.4067i	1.54621	+	0.542203i
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.48540			−0.0556288
$714$		0			0
$715$		0			0
$716$	22.2210	−	27.7878i	0.830436	−	1.03848i
$717$		0			0
$718$		0			0
$719$			47.9583i			1.78854i	0.447524	+	0.894272i	$0.352306\pi$
$719$			47.9583i			1.78854i	−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$		0			0
$722$	−25.3563	−	8.89160i	−0.943662	−	0.330911i
$723$		0			0
$724$		0			0
$725$	−33.5144			−1.24469
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	−26.9484	−	3.12776i	−0.993332	−	0.115291i
$737$		0			0
$738$		0			0
$739$		−	52.1310i		−	1.91767i	−0.283964	−	0.958835i	$-0.591650\pi$
$739$		−	52.1310i		−	1.91767i	0.283964	−	0.958835i	$-0.408350\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	25.5917	−	5.76907i	0.933234	−	0.210376i
$753$		0			0
$754$	−43.6819	−	15.3178i	−1.59080	−	0.557841i
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−33.2730			−1.20614			−0.603072	−	0.797687i	$-0.706057\pi$
$761$	−33.2730			−1.20614			−0.603072	+	0.797687i	$0.706057\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	46.8385i		−	1.69124i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−42.4013	−	33.9069i	−1.52606	−	1.22034i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$			1.54864i			0.0556288i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−6.15745	−	27.3146i	−0.219909	−	0.975520i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−43.8225	−	35.0434i	−1.56111	−	1.24837i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−52.9033	−	18.5514i	−1.87747	−	0.658366i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−3.26092	+	28.0957i	−0.115291	+	0.993332i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−0.707809	+	2.01846i	−0.0249315	+	0.0710974i
$807$		0			0
$808$	−9.00000	−	14.3875i	−0.316619	−	0.506150i
$809$	−42.0000			−1.47664			−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000			−1.47664			−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$		−	29.6680i		−	1.04178i	−0.853622	−	0.520892i	$-0.825599\pi$
$811$		−	29.6680i		−	1.04178i	0.853622	−	0.520892i	$-0.174401\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	49.2608	+	17.2741i	1.72236	+	0.603976i
$819$		0			0
$820$		0			0
$821$	−54.0000			−1.88461			−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000			−1.88461			−0.942306	+	0.334751i	$0.891348\pi$
$822$		0			0
$823$		−	22.1533i		−	0.772214i	−0.922454	−	0.386107i	$-0.873820\pi$
$823$		−	22.1533i		−	0.772214i	0.922454	−	0.386107i	$-0.126180\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$		0			0
$832$	−17.0914	+	35.1289i	−0.592537	+	1.21788i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	15.9285			0.549258
$842$		0			0
$843$		0			0
$844$	35.9420	−	44.9463i	1.23718	−	1.54712i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−10.0000			−0.342393			−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000			−0.342393			−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−44.1907			−1.50952			−0.754762	−	0.655998i	$-0.772248\pi$
$857$	−44.1907			−1.50952			−0.754762	+	0.655998i	$0.772248\pi$
$858$		0			0
$859$		−	50.8921i		−	1.73642i	−0.496201	−	0.868208i	$-0.665272\pi$
$859$		−	50.8921i		−	1.73642i	0.496201	−	0.868208i	$-0.334728\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			12.1878i			0.414877i	0.978248	+	0.207438i	$0.0665126\pi$
$863$			12.1878i			0.414877i	−0.978248	+	0.207438i	$0.933487\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	14.0000			0.472746			0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000			0.472746			0.236373	+	0.971662i	$0.424041\pi$
$878$	−19.4300	+	55.4087i	−0.655731	+	1.86995i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		−	28.7750i		−	0.968355i	−0.874970	−	0.484178i	$-0.839119\pi$
$883$		−	28.7750i		−	0.968355i	0.874970	−	0.484178i	$-0.160881\pi$
$884$		0			0
$885$		0			0
$886$	−17.9679	+	51.2391i	−0.603642	+	1.72141i
$887$		−	29.0215i		−	0.974445i	−0.873278	−	0.487223i	$-0.838010\pi$
$887$		−	29.0215i		−	0.974445i	0.873278	−	0.487223i	$-0.161990\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−35.9420	+	44.9463i	−1.20343	+	1.50491i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−24.0217	−	8.42362i	−0.801615	−	0.281100i
$899$		−	2.07607i		−	0.0692407i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	48.0466	+	16.8484i	1.58233	+	0.554871i
$923$		−	68.5909i		−	2.25770i
$924$		0			0
$925$		0			0
$926$	−13.4661	+	38.4014i	−0.442523	+	1.26195i
$927$		0			0
$928$	4.37150	−	37.6643i	0.143502	−	1.23639i
$929$	−60.0845			−1.97131			−0.985653	−	0.168782i	$-0.946017\pi$
$929$	−60.0845			−1.97131			−0.985653	+	0.168782i	$0.946017\pi$
$930$		0			0
$931$		0			0
$932$	14.7332	+	11.7816i	0.482601	+	0.385920i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$			19.0560i			0.620548i
$944$	37.4275	−	8.43717i	1.21816	−	0.274606i
$945$		0			0
$946$		0			0
$947$		−	36.5362i		−	1.18727i	−0.804735	−	0.593634i	$-0.797693\pi$
$947$		−	36.5362i		−	1.18727i	0.804735	−	0.593634i	$-0.202307\pi$
$948$		0			0
$949$	−37.1746			−1.20674
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−33.9287	+	42.4285i	−1.09733	+	1.37224i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	30.9041			0.996905
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	62.1236i		−	1.99776i	−0.0473194	−	0.998880i	$-0.515068\pi$
$967$		−	62.1236i		−	1.99776i	0.0473194	−	0.998880i	$-0.484932\pi$
$968$	−16.5000	−	26.3771i	−0.530330	−	0.847791i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$	19.1401	−	54.5820i	0.613288	−	1.74892i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−2.18687	+	6.23632i	−0.0697859	+	0.199009i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			28.7750i			0.914068i	0.889449	+	0.457034i	$0.151088\pi$
$991$			28.7750i			0.914068i	−0.889449	+	0.457034i	$0.848912\pi$
$992$	−1.74040	−	0.201999i	−0.0552578	−	0.00641349i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−26.0000			−0.823428			−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000			−0.823428			−0.411714	+	0.911313i	$0.635070\pi$
$998$	−5.40105	+	15.4022i	−0.170967	+	0.487549i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	828.2.e.b.91.6		6
3.2	odd	2		92.2.b.b.91.1	✓	6
4.3	odd	2	inner	828.2.e.b.91.5		6
12.11	even	2		92.2.b.b.91.2	yes	6
23.22	odd	2	CM	828.2.e.b.91.6		6
24.5	odd	2		1472.2.c.c.1471.6		6
24.11	even	2		1472.2.c.c.1471.1		6
69.68	even	2		92.2.b.b.91.1	✓	6
92.91	even	2	inner	828.2.e.b.91.5		6
276.275	odd	2		92.2.b.b.91.2	yes	6
552.275	odd	2		1472.2.c.c.1471.1		6
552.413	even	2		1472.2.c.c.1471.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.2.b.b.91.1	✓	6	3.2	odd	2
92.2.b.b.91.1	✓	6	69.68	even	2
92.2.b.b.91.2	yes	6	12.11	even	2
92.2.b.b.91.2	yes	6	276.275	odd	2
828.2.e.b.91.5		6	4.3	odd	2	inner
828.2.e.b.91.5		6	92.91	even	2	inner
828.2.e.b.91.6		6	1.1	even	1	trivial
828.2.e.b.91.6		6	23.22	odd	2	CM
1472.2.c.c.1471.1		6	24.11	even	2
1472.2.c.c.1471.1		6	552.275	odd	2
1472.2.c.c.1471.6		6	24.5	odd	2
1472.2.c.c.1471.6		6	552.413	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	828.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.33454 + 0.467979i) q^{2} +(1.56199 + 1.24907i) q^{4} +(1.50000 + 2.39792i) q^{8} +4.88325 q^{13} +(0.879635 + 3.90208i) q^{16} +4.79583i q^{23} +5.00000 q^{25} +(6.51690 + 2.28526i) q^{26} -6.70287 q^{29} +0.309728i q^{31} +(-0.652183 + 5.61913i) q^{32} +3.97345 q^{41} +(-2.24435 + 6.40023i) q^{46} -6.55848i q^{47} -7.00000 q^{49} +(6.67270 + 2.33989i) q^{50} +(7.62760 + 6.09954i) q^{52} +(-8.94525 - 3.13680i) q^{58} -9.59166i q^{59} +(-0.144946 + 0.413344i) q^{62} +(-3.50000 + 7.19375i) q^{64} -14.0461i q^{71} -7.61268 q^{73} +(5.30272 + 1.85949i) q^{82} +(-5.99034 + 7.49105i) q^{92} +(3.06923 - 8.75255i) q^{94} +(-9.34178 - 3.27585i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 9 q^{8} + 30 q^{25} + 27 q^{26} - 42 q^{49} + 3 q^{52} - 15 q^{58} - 45 q^{62} - 21 q^{64} + 33 q^{82} - 39 q^{94}+O(q^{100}) \) 6 * q + 9 * q^8 + 30 * q^25 + 27 * q^26 - 42 * q^49 + 3 * q^52 - 15 * q^58 - 45 * q^62 - 21 * q^64 + 33 * q^82 - 39 * q^94

Introduction

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