LMFDB - Embedded newform 7644.2.e.b.4705.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7644,2,Mod(4705,7644)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7644, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7644.4705");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7644.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:	$61.0376473051$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4705.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	7644.4705
Dual form			7644.2.e.b.4705.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.00000i q^{5} +1.00000 q^{9} -6.00000i q^{11} +(-3.00000 + 2.00000i) q^{13} +2.00000i q^{15} +2.00000 q^{17} -8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +8.00000i q^{31} +6.00000i q^{33} +8.00000i q^{37} +(3.00000 - 2.00000i) q^{39} +2.00000i q^{41} -8.00000 q^{43} -2.00000i q^{45} +6.00000i q^{47} -2.00000 q^{51} +6.00000 q^{53} -12.0000 q^{55} +2.00000i q^{59} -2.00000 q^{61} +(4.00000 + 6.00000i) q^{65} +4.00000i q^{67} +8.00000 q^{69} +6.00000i q^{71} +4.00000i q^{73} -1.00000 q^{75} +1.00000 q^{81} -14.0000i q^{83} -4.00000i q^{85} -2.00000 q^{87} -6.00000i q^{89} -8.00000i q^{93} +12.0000i q^{97} -6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} + 4 q^{17} - 16 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} + 6 q^{39} - 16 q^{43} - 4 q^{51} + 12 q^{53} - 24 q^{55} - 4 q^{61} + 8 q^{65} + 16 q^{69} - 2 q^{75} + 2 q^{81}+ \cdots - 4 q^{87}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^13 + 4 * q^17 - 16 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 + 6 * q^39 - 16 * q^43 - 4 * q^51 + 12 * q^53 - 24 * q^55 - 4 * q^61 + 8 * q^65 + 16 * q^69 - 2 * q^75 + 2 * q^81 - 4 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7644\mathbb{Z}\right)^\times$.

$n$	$2549$	$3433$	$3823$	$5293$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	6.00000i		−	1.80907i	−0.426401	−	0.904534i	$-0.640219\pi$
$11$		−	6.00000i		−	1.80907i	0.426401	−	0.904534i	$-0.359781\pi$
$12$		0			0
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$14$		0			0
$15$			2.00000i			0.516398i
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.00000			−1.66812			−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000			−1.66812			−0.834058	+	0.551677i	$0.813988\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$			8.00000i			1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$			8.00000i			1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$		0			0
$33$			6.00000i			1.04447i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$	3.00000	−	2.00000i	0.480384	−	0.320256i
$40$		0			0
$41$			2.00000i			0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$			2.00000i			0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$		−	2.00000i		−	0.298142i
$46$		0			0
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	−12.0000			−1.61808
$56$		0			0
$57$		0			0
$58$		0			0
$59$			2.00000i			0.260378i	0.991489	+	0.130189i	$0.0415584\pi$
$59$			2.00000i			0.260378i	−0.991489	+	0.130189i	$0.958442\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.00000	+	6.00000i	0.496139	+	0.744208i
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		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
$81$	1.00000			0.111111
$82$		0			0
$83$		−	14.0000i		−	1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$		−	14.0000i		−	1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$		0			0
$85$		−	4.00000i		−	0.433861i
$86$		0			0
$87$	−2.00000			−0.214423
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	8.00000i		−	0.829561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		0			0
$99$		−	6.00000i		−	0.603023i
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	20.0000			1.93347			0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000			1.93347			0.966736	+	0.255774i	$0.0823304\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$		−	8.00000i		−	0.759326i
$112$		0			0
$113$	−10.0000			−0.940721			−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000			−0.940721			−0.470360	+	0.882474i	$0.655876\pi$
$114$		0			0
$115$			16.0000i			1.49201i
$116$		0			0
$117$	−3.00000	+	2.00000i	−0.277350	+	0.184900i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−25.0000			−2.27273
$122$		0			0
$123$		−	2.00000i		−	0.180334i
$124$		0			0
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$	8.00000			0.704361
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$			2.00000i			0.172133i
$136$		0			0
$137$			10.0000i			0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$			10.0000i			0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$		0			0
$141$		−	6.00000i		−	0.505291i
$142$		0			0
$143$	12.0000	+	18.0000i	1.00349	+	1.50524i
$144$		0			0
$145$		−	4.00000i		−	0.332182i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	10.0000i		−	0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$		−	10.0000i		−	0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$		0			0
$151$			16.0000i			1.30206i	0.759051	+	0.651031i	$0.225663\pi$
$151$			16.0000i			1.30206i	−0.759051	+	0.651031i	$0.774337\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$	16.0000			1.28515
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	8.00000i		−	0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$		−	8.00000i		−	0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$		0			0
$165$	12.0000			0.934199
$166$		0			0
$167$		−	18.0000i		−	1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$		−	18.0000i		−	1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	2.00000i		−	0.150329i
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		0			0
$185$	16.0000			1.17634
$186$		0			0
$187$		−	12.0000i		−	0.877527i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$		−	8.00000i		−	0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$		−	8.00000i		−	0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$		0			0
$195$	−4.00000	−	6.00000i	−0.286446	−	0.429669i
$196$		0			0
$197$			26.0000i			1.85242i	0.377004	+	0.926212i	$0.376954\pi$
$197$			26.0000i			1.85242i	−0.377004	+	0.926212i	$0.623046\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		−	4.00000i		−	0.282138i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.00000			0.279372
$206$		0			0
$207$	−8.00000			−0.556038
$208$		0			0
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$		−	6.00000i		−	0.411113i
$214$		0			0
$215$			16.0000i			1.09119i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	4.00000i		−	0.270295i
$220$		0			0
$221$	−6.00000	+	4.00000i	−0.403604	+	0.269069i
$222$		0			0
$223$			8.00000i			0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$			8.00000i			0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$		−	2.00000i		−	0.132745i	−0.997795	−	0.0663723i	$-0.978857\pi$
$227$		−	2.00000i		−	0.132745i	0.997795	−	0.0663723i	$-0.0211425\pi$
$228$		0			0
$229$			20.0000i			1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$			20.0000i			1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	10.0000i		−	0.646846i	−0.946254	−	0.323423i	$-0.895166\pi$
$239$		−	10.0000i		−	0.646846i	0.946254	−	0.323423i	$-0.104834\pi$
$240$		0			0
$241$		−	20.0000i		−	1.28831i	−0.764894	−	0.644157i	$-0.777208\pi$
$241$		−	20.0000i		−	1.28831i	0.764894	−	0.644157i	$-0.222792\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$			14.0000i			0.887214i
$250$		0			0
$251$	−20.0000			−1.26239			−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000			−1.26239			−0.631194	+	0.775625i	$0.717435\pi$
$252$		0			0
$253$			48.0000i			3.01773i
$254$		0			0
$255$			4.00000i			0.250490i
$256$		0			0
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		−	12.0000i		−	0.737154i
$266$		0			0
$267$			6.00000i			0.367194i
$268$		0			0
$269$	22.0000			1.34136			0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000			1.34136			0.670682	+	0.741745i	$0.266002\pi$
$270$		0			0
$271$		−	20.0000i		−	1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$		−	20.0000i		−	1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	6.00000i		−	0.361814i
$276$		0			0
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$		0			0
$279$			8.00000i			0.478947i
$280$		0			0
$281$		−	22.0000i		−	1.31241i	−0.754583	−	0.656205i	$-0.772161\pi$
$281$		−	22.0000i		−	1.31241i	0.754583	−	0.656205i	$-0.227839\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		−	12.0000i		−	0.703452i
$292$		0			0
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$			6.00000i			0.348155i
$298$		0			0
$299$	24.0000	−	16.0000i	1.38796	−	0.925304i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−10.0000			−0.574485
$304$		0			0
$305$			4.00000i			0.229039i
$306$		0			0
$307$			20.0000i			1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$			20.0000i			1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		0			0
$311$	16.0000			0.907277			0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000			0.907277			0.453638	+	0.891186i	$0.350126\pi$
$312$		0			0
$313$	14.0000			0.791327			0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000			0.791327			0.395663	+	0.918396i	$0.370515\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		0			0
$319$		−	12.0000i		−	0.671871i
$320$		0			0
$321$	−20.0000			−1.11629
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−3.00000	+	2.00000i	−0.166410	+	0.110940i
$326$		0			0
$327$		−	4.00000i		−	0.221201i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	28.0000i		−	1.53902i	−0.638635	−	0.769510i	$-0.720501\pi$
$331$		−	28.0000i		−	1.53902i	0.638635	−	0.769510i	$-0.279499\pi$
$332$		0			0
$333$			8.00000i			0.438397i
$334$		0			0
$335$	8.00000			0.437087
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		0			0
$339$	10.0000			0.543125
$340$		0			0
$341$	48.0000			2.59935
$342$		0			0
$343$		0			0
$344$		0			0
$345$		−	16.0000i		−	0.861411i
$346$		0			0
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$		−	8.00000i		−	0.428230i	−0.976808	−	0.214115i	$-0.931313\pi$
$349$		−	8.00000i		−	0.428230i	0.976808	−	0.214115i	$-0.0686868\pi$
$350$		0			0
$351$	3.00000	−	2.00000i	0.160128	−	0.106752i
$352$		0			0
$353$			10.0000i			0.532246i	0.963939	+	0.266123i	$0.0857428\pi$
$353$			10.0000i			0.532246i	−0.963939	+	0.266123i	$0.914257\pi$
$354$		0			0
$355$	12.0000			0.636894
$356$		0			0
$357$		0			0
$358$		0			0
$359$			10.0000i			0.527780i	0.964553	+	0.263890i	$0.0850056\pi$
$359$			10.0000i			0.527780i	−0.964553	+	0.263890i	$0.914994\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	25.0000			1.31216
$364$		0			0
$365$	8.00000			0.418739
$366$		0			0
$367$	28.0000			1.46159			0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000			1.46159			0.730794	+	0.682598i	$0.239150\pi$
$368$		0			0
$369$			2.00000i			0.104116i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$			12.0000i			0.619677i
$376$		0			0
$377$	−6.00000	+	4.00000i	−0.309016	+	0.206010i
$378$		0			0
$379$			16.0000i			0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$			16.0000i			0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$		0			0
$381$	−8.00000			−0.409852
$382$		0			0
$383$			22.0000i			1.12415i	0.827087	+	0.562074i	$0.189996\pi$
$383$			22.0000i			1.12415i	−0.827087	+	0.562074i	$0.810004\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8.00000			−0.406663
$388$		0			0
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$	4.00000			0.201773
$394$		0			0
$395$		0			0
$396$		0			0
$397$			16.0000i			0.803017i	0.915855	+	0.401508i	$0.131514\pi$
$397$			16.0000i			0.803017i	−0.915855	+	0.401508i	$0.868486\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			6.00000i			0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$			6.00000i			0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$		0			0
$403$	−16.0000	−	24.0000i	−0.797017	−	1.19553i
$404$		0			0
$405$		−	2.00000i		−	0.0993808i
$406$		0			0
$407$	48.0000			2.37927
$408$		0			0
$409$			20.0000i			0.988936i	0.869196	+	0.494468i	$0.164637\pi$
$409$			20.0000i			0.988936i	−0.869196	+	0.494468i	$0.835363\pi$
$410$		0			0
$411$		−	10.0000i		−	0.493264i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−28.0000			−1.37447
$416$		0			0
$417$	−8.00000			−0.391762
$418$		0			0
$419$	−28.0000			−1.36789			−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000			−1.36789			−0.683945	+	0.729534i	$0.739737\pi$
$420$		0			0
$421$		−	4.00000i		−	0.194948i	−0.995238	−	0.0974740i	$-0.968924\pi$
$421$		−	4.00000i		−	0.194948i	0.995238	−	0.0974740i	$-0.0310763\pi$
$422$		0			0
$423$			6.00000i			0.291730i
$424$		0			0
$425$	2.00000			0.0970143
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−12.0000	−	18.0000i	−0.579365	−	0.869048i
$430$		0			0
$431$			10.0000i			0.481683i	0.970564	+	0.240842i	$0.0774234\pi$
$431$			10.0000i			0.481683i	−0.970564	+	0.240842i	$0.922577\pi$
$432$		0			0
$433$	−22.0000			−1.05725			−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000			−1.05725			−0.528626	+	0.848855i	$0.677293\pi$
$434$		0			0
$435$			4.00000i			0.191785i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−36.0000			−1.71819			−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000			−1.71819			−0.859093	+	0.511819i	$0.828972\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.0000			0.950229			0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000			0.950229			0.475114	+	0.879924i	$0.342407\pi$
$444$		0			0
$445$	−12.0000			−0.568855
$446$		0			0
$447$			10.0000i			0.472984i
$448$		0			0
$449$			30.0000i			1.41579i	0.706319	+	0.707894i	$0.250354\pi$
$449$			30.0000i			1.41579i	−0.706319	+	0.707894i	$0.749646\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$		−	16.0000i		−	0.751746i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			4.00000i			0.187112i	0.995614	+	0.0935561i	$0.0298234\pi$
$457$			4.00000i			0.187112i	−0.995614	+	0.0935561i	$0.970177\pi$
$458$		0			0
$459$	−2.00000			−0.0933520
$460$		0			0
$461$			6.00000i			0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$			6.00000i			0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$		0			0
$463$			12.0000i			0.557687i	0.960337	+	0.278844i	$0.0899511\pi$
$463$			12.0000i			0.557687i	−0.960337	+	0.278844i	$0.910049\pi$
$464$		0			0
$465$	−16.0000			−0.741982
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−14.0000			−0.645086
$472$		0			0
$473$			48.0000i			2.20704i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$			18.0000i			0.822441i	0.911536	+	0.411220i	$0.134897\pi$
$479$			18.0000i			0.822441i	−0.911536	+	0.411220i	$0.865103\pi$
$480$		0			0
$481$	−16.0000	−	24.0000i	−0.729537	−	1.09431i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	24.0000			1.08978
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$			8.00000i			0.361773i
$490$		0			0
$491$	20.0000			0.902587			0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000			0.902587			0.451294	+	0.892375i	$0.350963\pi$
$492$		0			0
$493$	4.00000			0.180151
$494$		0			0
$495$	−12.0000			−0.539360
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	4.00000i		−	0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$		−	4.00000i		−	0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$		0			0
$501$			18.0000i			0.804181i
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$		−	20.0000i		−	0.889988i
$506$		0			0
$507$	−5.00000	+	12.0000i	−0.222058	+	0.532939i
$508$		0			0
$509$			30.0000i			1.32973i	0.746965	+	0.664863i	$0.231510\pi$
$509$			30.0000i			1.32973i	−0.746965	+	0.664863i	$0.768490\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	8.00000i		−	0.352522i
$516$		0			0
$517$	36.0000			1.58328
$518$		0			0
$519$	6.00000			0.263371
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	8.00000			0.349816			0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000			0.349816			0.174908	+	0.984585i	$0.444037\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	41.0000			1.78261
$530$		0			0
$531$			2.00000i			0.0867926i
$532$		0			0
$533$	−4.00000	−	6.00000i	−0.173259	−	0.259889i
$534$		0			0
$535$		−	40.0000i		−	1.72935i
$536$		0			0
$537$	12.0000			0.517838
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$		0			0
$543$	10.0000			0.429141
$544$		0			0
$545$	8.00000			0.342682
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$	−2.00000			−0.0853579
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−16.0000			−0.679162
$556$		0			0
$557$		−	30.0000i		−	1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$		−	30.0000i		−	1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$		0			0
$559$	24.0000	−	16.0000i	1.01509	−	0.676728i
$560$		0			0
$561$			12.0000i			0.506640i
$562$		0			0
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$		0			0
$565$			20.0000i			0.841406i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	34.0000			1.42535			0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000			1.42535			0.712677	+	0.701492i	$0.247483\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$	24.0000			1.00261
$574$		0			0
$575$	−8.00000			−0.333623
$576$		0			0
$577$			40.0000i			1.66522i	0.553858	+	0.832611i	$0.313155\pi$
$577$			40.0000i			1.66522i	−0.553858	+	0.832611i	$0.686845\pi$
$578$		0			0
$579$			8.00000i			0.332469i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	36.0000i		−	1.49097i
$584$		0			0
$585$	4.00000	+	6.00000i	0.165380	+	0.248069i
$586$		0			0
$587$			2.00000i			0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$			2.00000i			0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		−	26.0000i		−	1.06950i
$592$		0			0
$593$			34.0000i			1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$			34.0000i			1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$			4.00000i			0.162893i
$604$		0			0
$605$			50.0000i			2.03279i
$606$		0			0
$607$	−4.00000			−0.162355			−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000			−0.162355			−0.0811775	+	0.996700i	$0.525868\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000	−	18.0000i	−0.485468	−	0.728202i
$612$		0			0
$613$			16.0000i			0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$			16.0000i			0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$		0			0
$615$	−4.00000			−0.161296
$616$		0			0
$617$			14.0000i			0.563619i	0.959470	+	0.281809i	$0.0909346\pi$
$617$			14.0000i			0.563619i	−0.959470	+	0.281809i	$0.909065\pi$
$618$		0			0
$619$			20.0000i			0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$			20.0000i			0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$		0			0
$621$	8.00000			0.321029
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			16.0000i			0.637962i
$630$		0			0
$631$		−	8.00000i		−	0.318475i	−0.987240	−	0.159237i	$-0.949096\pi$
$631$		−	8.00000i		−	0.318475i	0.987240	−	0.159237i	$-0.0509036\pi$
$632$		0			0
$633$	−12.0000			−0.476957
$634$		0			0
$635$		−	16.0000i		−	0.634941i
$636$		0			0
$637$		0			0
$638$		0			0
$639$			6.00000i			0.237356i
$640$		0			0
$641$	−42.0000			−1.65890			−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000			−1.65890			−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$		−	36.0000i		−	1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$		−	36.0000i		−	1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$		0			0
$645$		−	16.0000i		−	0.629999i
$646$		0			0
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−14.0000			−0.547862			−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000			−0.547862			−0.273931	+	0.961749i	$0.588324\pi$
$654$		0			0
$655$			8.00000i			0.312586i
$656$		0			0
$657$			4.00000i			0.156055i
$658$		0			0
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$		−	32.0000i		−	1.24466i	−0.782757	−	0.622328i	$-0.786187\pi$
$661$		−	32.0000i		−	1.24466i	0.782757	−	0.622328i	$-0.213813\pi$
$662$		0			0
$663$	6.00000	−	4.00000i	0.233021	−	0.155347i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−16.0000			−0.619522
$668$		0			0
$669$		−	8.00000i		−	0.309298i
$670$		0			0
$671$			12.0000i			0.463255i
$672$		0			0
$673$	30.0000			1.15642			0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000			1.15642			0.578208	+	0.815890i	$0.303752\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			2.00000i			0.0766402i
$682$		0			0
$683$		−	6.00000i		−	0.229584i	−0.993390	−	0.114792i	$-0.963380\pi$
$683$		−	6.00000i		−	0.229584i	0.993390	−	0.114792i	$-0.0366201\pi$
$684$		0			0
$685$	20.0000			0.764161
$686$		0			0
$687$		−	20.0000i		−	0.763048i
$688$		0			0
$689$	−18.0000	+	12.0000i	−0.685745	+	0.457164i
$690$		0			0
$691$			4.00000i			0.152167i	0.997101	+	0.0760836i	$0.0242416\pi$
$691$			4.00000i			0.152167i	−0.997101	+	0.0760836i	$0.975758\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	16.0000i		−	0.606915i
$696$		0			0
$697$			4.00000i			0.151511i
$698$		0			0
$699$	−18.0000			−0.680823
$700$		0			0
$701$	10.0000			0.377695			0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000			0.377695			0.188847	+	0.982006i	$0.439525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−12.0000			−0.451946
$706$		0			0
$707$		0			0
$708$		0			0
$709$			36.0000i			1.35201i	0.736898	+	0.676004i	$0.236290\pi$
$709$			36.0000i			1.35201i	−0.736898	+	0.676004i	$0.763710\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	64.0000i		−	2.39682i
$714$		0			0
$715$	36.0000	−	24.0000i	1.34632	−	0.897549i
$716$		0			0
$717$			10.0000i			0.373457i
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			20.0000i			0.743808i
$724$		0			0
$725$	2.00000			0.0742781
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−16.0000			−0.591781
$732$		0			0
$733$			20.0000i			0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$			20.0000i			0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.0000			0.884051
$738$		0			0
$739$			28.0000i			1.03000i	0.857191	+	0.514998i	$0.172207\pi$
$739$			28.0000i			1.03000i	−0.857191	+	0.514998i	$0.827793\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			22.0000i			0.807102i	0.914957	+	0.403551i	$0.132224\pi$
$743$			22.0000i			0.807102i	−0.914957	+	0.403551i	$0.867776\pi$
$744$		0			0
$745$	−20.0000			−0.732743
$746$		0			0
$747$		−	14.0000i		−	0.512233i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	20.0000			0.729810			0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000			0.729810			0.364905	+	0.931045i	$0.381101\pi$
$752$		0			0
$753$	20.0000			0.728841
$754$		0			0
$755$	32.0000			1.16460
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		−	48.0000i		−	1.74229i
$760$		0			0
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		−	4.00000i		−	0.144620i
$766$		0			0
$767$	−4.00000	−	6.00000i	−0.144432	−	0.216647i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−6.00000			−0.216085
$772$		0			0
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		0			0
$775$			8.00000i			0.287368i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$	−2.00000			−0.0714742
$784$		0			0
$785$		−	28.0000i		−	0.999363i
$786$		0			0
$787$		−	44.0000i		−	1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$		−	44.0000i		−	1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.00000	−	4.00000i	0.213066	−	0.142044i
$794$		0			0
$795$			12.0000i			0.425596i
$796$		0			0
$797$	−10.0000			−0.354218			−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000			−0.354218			−0.177109	+	0.984191i	$0.556675\pi$
$798$		0			0
$799$			12.0000i			0.424529i
$800$		0			0
$801$		−	6.00000i		−	0.212000i
$802$		0			0
$803$	24.0000			0.846942
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−22.0000			−0.774437
$808$		0			0
$809$	22.0000			0.773479			0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000			0.773479			0.386739	+	0.922189i	$0.373601\pi$
$810$		0			0
$811$		−	40.0000i		−	1.40459i	−0.711886	−	0.702295i	$-0.752159\pi$
$811$		−	40.0000i		−	1.40459i	0.711886	−	0.702295i	$-0.247841\pi$
$812$		0			0
$813$			20.0000i			0.701431i
$814$		0			0
$815$	−16.0000			−0.560456
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			30.0000i			1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$			30.0000i			1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$		0			0
$823$	−36.0000			−1.25488			−0.627441	−	0.778664i	$-0.715897\pi$
$823$	−36.0000			−1.25488			−0.627441	+	0.778664i	$0.715897\pi$
$824$		0			0
$825$			6.00000i			0.208893i
$826$		0			0
$827$			18.0000i			0.625921i	0.949766	+	0.312961i	$0.101321\pi$
$827$			18.0000i			0.625921i	−0.949766	+	0.312961i	$0.898679\pi$
$828$		0			0
$829$	54.0000			1.87550			0.937749	−	0.347314i	$-0.112906\pi$
$829$	54.0000			1.87550			0.937749	+	0.347314i	$0.112906\pi$
$830$		0			0
$831$	−22.0000			−0.763172
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−36.0000			−1.24583
$836$		0			0
$837$		−	8.00000i		−	0.276520i
$838$		0			0
$839$			10.0000i			0.345238i	0.984989	+	0.172619i	$0.0552230\pi$
$839$			10.0000i			0.345238i	−0.984989	+	0.172619i	$0.944777\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			22.0000i			0.757720i
$844$		0			0
$845$	−24.0000	−	10.0000i	−0.825625	−	0.344010i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$		−	64.0000i		−	2.19389i
$852$		0			0
$853$			16.0000i			0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$			16.0000i			0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	22.0000			0.751506			0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000			0.751506			0.375753	+	0.926720i	$0.377384\pi$
$858$		0			0
$859$	−52.0000			−1.77422			−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000			−1.77422			−0.887109	+	0.461561i	$0.847290\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			34.0000i			1.15737i	0.815550	+	0.578687i	$0.196435\pi$
$863$			34.0000i			1.15737i	−0.815550	+	0.578687i	$0.803565\pi$
$864$		0			0
$865$			12.0000i			0.408012i
$866$		0			0
$867$	13.0000			0.441503
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−8.00000	−	12.0000i	−0.271070	−	0.406604i
$872$		0			0
$873$			12.0000i			0.406138i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	16.0000i		−	0.540282i	−0.962821	−	0.270141i	$-0.912930\pi$
$877$		−	16.0000i		−	0.540282i	0.962821	−	0.270141i	$-0.0870703\pi$
$878$		0			0
$879$		−	14.0000i		−	0.472208i
$880$		0			0
$881$	−42.0000			−1.41502			−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000			−1.41502			−0.707508	+	0.706705i	$0.750181\pi$
$882$		0			0
$883$	−12.0000			−0.403832			−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000			−0.403832			−0.201916	+	0.979403i	$0.564717\pi$
$884$		0			0
$885$	−4.00000			−0.134459
$886$		0			0
$887$	−24.0000			−0.805841			−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000			−0.805841			−0.402921	+	0.915235i	$0.632005\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	6.00000i		−	0.201008i
$892$		0			0
$893$		0			0
$894$		0			0
$895$			24.0000i			0.802232i
$896$		0			0
$897$	−24.0000	+	16.0000i	−0.801337	+	0.534224i
$898$		0			0
$899$			16.0000i			0.533630i
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$			20.0000i			0.664822i
$906$		0			0
$907$	−44.0000			−1.46100			−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000			−1.46100			−0.730498	+	0.682915i	$0.760712\pi$
$908$		0			0
$909$	10.0000			0.331679
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−84.0000			−2.77999
$914$		0			0
$915$		−	4.00000i		−	0.132236i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$		0			0
$921$		−	20.0000i		−	0.659022i
$922$		0			0
$923$	−12.0000	−	18.0000i	−0.394985	−	0.592477i
$924$		0			0
$925$			8.00000i			0.263038i
$926$		0			0
$927$	4.00000			0.131377
$928$		0			0
$929$		−	30.0000i		−	0.984268i	−0.870519	−	0.492134i	$-0.836217\pi$
$929$		−	30.0000i		−	0.984268i	0.870519	−	0.492134i	$-0.163783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−16.0000			−0.523816
$934$		0			0
$935$	−24.0000			−0.784884
$936$		0			0
$937$	50.0000			1.63343			0.816714	−	0.577042i	$-0.195793\pi$
$937$	50.0000			1.63343			0.816714	+	0.577042i	$0.195793\pi$
$938$		0			0
$939$	−14.0000			−0.456873
$940$		0			0
$941$		−	38.0000i		−	1.23876i	−0.785090	−	0.619382i	$-0.787383\pi$
$941$		−	38.0000i		−	1.23876i	0.785090	−	0.619382i	$-0.212617\pi$
$942$		0			0
$943$		−	16.0000i		−	0.521032i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			38.0000i			1.23483i	0.786636	+	0.617417i	$0.211821\pi$
$947$			38.0000i			1.23483i	−0.786636	+	0.617417i	$0.788179\pi$
$948$		0			0
$949$	−8.00000	−	12.0000i	−0.259691	−	0.389536i
$950$		0			0
$951$		−	18.0000i		−	0.583690i
$952$		0			0
$953$	−54.0000			−1.74923			−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000			−1.74923			−0.874616	+	0.484817i	$0.838886\pi$
$954$		0			0
$955$			48.0000i			1.55324i
$956$		0			0
$957$			12.0000i			0.387905i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−33.0000			−1.06452
$962$		0			0
$963$	20.0000			0.644491
$964$		0			0
$965$	−16.0000			−0.515058
$966$		0			0
$967$			8.00000i			0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$			8.00000i			0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	3.00000	−	2.00000i	0.0960769	−	0.0640513i
$976$		0			0
$977$		−	30.0000i		−	0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$		−	30.0000i		−	0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$		0			0
$979$	−36.0000			−1.15056
$980$		0			0
$981$			4.00000i			0.127710i
$982$		0			0
$983$			18.0000i			0.574111i	0.957914	+	0.287055i	$0.0926764\pi$
$983$			18.0000i			0.574111i	−0.957914	+	0.287055i	$0.907324\pi$
$984$		0			0
$985$	52.0000			1.65686
$986$		0			0
$987$		0			0
$988$		0			0
$989$	64.0000			2.03508
$990$		0			0
$991$	−44.0000			−1.39771			−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000			−1.39771			−0.698853	+	0.715265i	$0.746306\pi$
$992$		0			0
$993$			28.0000i			0.888553i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−6.00000			−0.190022			−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000			−0.190022			−0.0950110	+	0.995476i	$0.530289\pi$
$998$		0			0
$999$		−	8.00000i		−	0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7644.2.e.b.4705.1		2
7.6	odd	2		156.2.b.b.25.2	yes	2
13.12	even	2	inner	7644.2.e.b.4705.2		2
21.20	even	2		468.2.b.c.181.1		2
28.27	even	2		624.2.c.d.337.2		2
35.13	even	4		3900.2.j.e.649.2		2
35.27	even	4		3900.2.j.b.649.1		2
35.34	odd	2		3900.2.c.a.3301.1		2
56.13	odd	2		2496.2.c.b.961.1		2
56.27	even	2		2496.2.c.i.961.1		2
84.83	odd	2		1872.2.c.h.1585.1		2
91.6	even	12		2028.2.i.d.2005.1		2
91.20	even	12		2028.2.i.a.2005.1		2
91.34	even	4		2028.2.a.d.1.1		1
91.41	even	12		2028.2.i.d.529.1		2
91.48	odd	6		2028.2.q.e.361.2		4
91.55	odd	6		2028.2.q.e.1837.2		4
91.62	odd	6		2028.2.q.e.1837.1		4
91.69	odd	6		2028.2.q.e.361.1		4
91.76	even	12		2028.2.i.a.529.1		2
91.83	even	4		2028.2.a.f.1.1		1
91.90	odd	2		156.2.b.b.25.1	✓	2
273.83	odd	4		6084.2.a.d.1.1		1
273.125	odd	4		6084.2.a.n.1.1		1
273.272	even	2		468.2.b.c.181.2		2
364.83	odd	4		8112.2.a.l.1.1		1
364.307	odd	4		8112.2.a.d.1.1		1
364.363	even	2		624.2.c.d.337.1		2
455.272	even	4		3900.2.j.e.649.1		2
455.363	even	4		3900.2.j.b.649.2		2
455.454	odd	2		3900.2.c.a.3301.2		2
728.181	odd	2		2496.2.c.b.961.2		2
728.363	even	2		2496.2.c.i.961.2		2
1092.1091	odd	2		1872.2.c.h.1585.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.b.b.25.1	✓	2	91.90	odd	2
156.2.b.b.25.2	yes	2	7.6	odd	2
468.2.b.c.181.1		2	21.20	even	2
468.2.b.c.181.2		2	273.272	even	2
624.2.c.d.337.1		2	364.363	even	2
624.2.c.d.337.2		2	28.27	even	2
1872.2.c.h.1585.1		2	84.83	odd	2
1872.2.c.h.1585.2		2	1092.1091	odd	2
2028.2.a.d.1.1		1	91.34	even	4
2028.2.a.f.1.1		1	91.83	even	4
2028.2.i.a.529.1		2	91.76	even	12
2028.2.i.a.2005.1		2	91.20	even	12
2028.2.i.d.529.1		2	91.41	even	12
2028.2.i.d.2005.1		2	91.6	even	12
2028.2.q.e.361.1		4	91.69	odd	6
2028.2.q.e.361.2		4	91.48	odd	6
2028.2.q.e.1837.1		4	91.62	odd	6
2028.2.q.e.1837.2		4	91.55	odd	6
2496.2.c.b.961.1		2	56.13	odd	2
2496.2.c.b.961.2		2	728.181	odd	2
2496.2.c.i.961.1		2	56.27	even	2
2496.2.c.i.961.2		2	728.363	even	2
3900.2.c.a.3301.1		2	35.34	odd	2
3900.2.c.a.3301.2		2	455.454	odd	2
3900.2.j.b.649.1		2	35.27	even	4
3900.2.j.b.649.2		2	455.363	even	4
3900.2.j.e.649.1		2	455.272	even	4
3900.2.j.e.649.2		2	35.13	even	4
6084.2.a.d.1.1		1	273.83	odd	4
6084.2.a.n.1.1		1	273.125	odd	4
7644.2.e.b.4705.1		2	1.1	even	1	trivial
7644.2.e.b.4705.2		2	13.12	even	2	inner
8112.2.a.d.1.1		1	364.307	odd	4
8112.2.a.l.1.1		1	364.83	odd	4

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:	\( N \)	\(=\)	\( 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7644.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.00000i q^{5} +1.00000 q^{9} -6.00000i q^{11} +(-3.00000 + 2.00000i) q^{13} +2.00000i q^{15} +2.00000 q^{17} -8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +8.00000i q^{31} +6.00000i q^{33} +8.00000i q^{37} +(3.00000 - 2.00000i) q^{39} +2.00000i q^{41} -8.00000 q^{43} -2.00000i q^{45} +6.00000i q^{47} -2.00000 q^{51} +6.00000 q^{53} -12.0000 q^{55} +2.00000i q^{59} -2.00000 q^{61} +(4.00000 + 6.00000i) q^{65} +4.00000i q^{67} +8.00000 q^{69} +6.00000i q^{71} +4.00000i q^{73} -1.00000 q^{75} +1.00000 q^{81} -14.0000i q^{83} -4.00000i q^{85} -2.00000 q^{87} -6.00000i q^{89} -8.00000i q^{93} +12.0000i q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} + 4 q^{17} - 16 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{29} + 6 q^{39} - 16 q^{43} - 4 q^{51} + 12 q^{53} - 24 q^{55} - 4 q^{61} + 8 q^{65} + 16 q^{69} - 2 q^{75} + 2 q^{81}+ \cdots - 4 q^{87}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^13 + 4 * q^17 - 16 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^29 + 6 * q^39 - 16 * q^43 - 4 * q^51 + 12 * q^53 - 24 * q^55 - 4 * q^61 + 8 * q^65 + 16 * q^69 - 2 * q^75 + 2 * q^81 - 4 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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