LMFDB - Embedded newform 4704.2.p.a.3919.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(3919,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("4704.3919");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.p (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3919.1
Character	$\chi$	$=$	4704.3919
Dual form			4704.2.p.a.3919.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.50301 q^{5} -1.00000 q^{9} +5.67619 q^{11} +5.31228 q^{13} -2.50301i q^{15} -0.454858i q^{17} -3.69433i q^{19} -5.12043i q^{23} +1.26504 q^{25} +1.00000i q^{27} -2.57962i q^{29} -6.00666 q^{31} -5.67619i q^{33} -9.01565i q^{37} -5.31228i q^{39} -4.65692i q^{41} -3.66703 q^{43} -2.50301 q^{45} +0.957682 q^{47} -0.454858 q^{51} +6.24836i q^{53} +14.2075 q^{55} -3.69433 q^{57} +10.1222i q^{59} -5.00367 q^{61} +13.2967 q^{65} +9.30266 q^{67} -5.12043 q^{69} +7.35240i q^{71} +6.85362i q^{73} -1.26504i q^{75} +8.91293i q^{79} +1.00000 q^{81} +1.96259i q^{83} -1.13851i q^{85} -2.57962 q^{87} -6.82845i q^{89} +6.00666i q^{93} -9.24692i q^{95} -3.71270i q^{97} -5.67619 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 32 q^{9} + 16 q^{11} + 32 q^{25} + 16 q^{43} + 16 q^{57} - 64 q^{67} + 32 q^{81} - 16 q^{99}+O(q^{100}) $ 32 * q - 32 * q^9 + 16 * q^11 + 32 * q^25 + 16 * q^43 + 16 * q^57 - 64 * q^67 + 32 * q^81 - 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4704\mathbb{Z}\right)^\times$.

$n$	$1471$	$1765$	$3137$	$4609$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	2.50301	1.11938	0.559689	−	0.828703i	$-0.310920\pi$
$5$	2.50301	1.11938	0.559689	+	0.828703i	$0.310920\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.67619	1.71143	0.855717	−	0.517443i	$-0.173116\pi$
$11$	5.67619	1.71143	0.855717	+	0.517443i	$0.173116\pi$
$12$	0	0
$13$	5.31228	1.47336	0.736681	−	0.676241i	$-0.236392\pi$
$13$	5.31228	1.47336	0.736681	+	0.676241i	$0.236392\pi$
$14$	0	0
$15$	− 2.50301i	− 0.646273i
$16$	0	0
$17$	− 0.454858i	− 0.110319i	−0.998478	−	0.0551596i	$-0.982433\pi$
$17$	− 0.454858i	− 0.110319i	0.998478	−	0.0551596i	$-0.0175668\pi$
$18$	0	0
$19$	− 3.69433i	− 0.847537i	−0.905771	−	0.423768i	$-0.860707\pi$
$19$	− 3.69433i	− 0.847537i	0.905771	−	0.423768i	$-0.139293\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 5.12043i	− 1.06768i	−0.845584	−	0.533842i	$-0.820748\pi$
$23$	− 5.12043i	− 1.06768i	0.845584	−	0.533842i	$-0.179252\pi$
$24$	0	0
$25$	1.26504	0.253008
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 2.57962i	− 0.479024i	−0.970893	−	0.239512i	$-0.923013\pi$
$29$	− 2.57962i	− 0.479024i	0.970893	−	0.239512i	$-0.0769874\pi$
$30$	0	0
$31$	−6.00666	−1.07883	−0.539414	−	0.842041i	$-0.681354\pi$
$31$	−6.00666	−1.07883	−0.539414	+	0.842041i	$0.681354\pi$
$32$	0	0
$33$	− 5.67619i	− 0.988097i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.01565i	− 1.48216i	−0.671415	−	0.741082i	$-0.734313\pi$
$37$	− 9.01565i	− 1.48216i	0.671415	−	0.741082i	$-0.265687\pi$
$38$	0	0
$39$	− 5.31228i	− 0.850646i
$40$	0	0
$41$	− 4.65692i	− 0.727289i	−0.931538	−	0.363644i	$-0.881532\pi$
$41$	− 4.65692i	− 0.727289i	0.931538	−	0.363644i	$-0.118468\pi$
$42$	0	0
$43$	−3.66703	−0.559217	−0.279608	−	0.960114i	$-0.590205\pi$
$43$	−3.66703	−0.559217	−0.279608	+	0.960114i	$0.590205\pi$
$44$	0	0
$45$	−2.50301	−0.373126
$46$	0	0
$47$	0.957682	0.139692	0.0698461	−	0.997558i	$-0.477749\pi$
$47$	0.957682	0.139692	0.0698461	+	0.997558i	$0.477749\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.454858	−0.0636929
$52$	0	0
$53$	6.24836i	0.858278i	0.903239	+	0.429139i	$0.141183\pi$
$53$	6.24836i	0.858278i	−0.903239	+	0.429139i	$0.858817\pi$
$54$	0	0
$55$	14.2075	1.91574
$56$	0	0
$57$	−3.69433	−0.489326
$58$	0	0
$59$	10.1222i	1.31779i	0.752234	+	0.658896i	$0.228976\pi$
$59$	10.1222i	1.31779i	−0.752234	+	0.658896i	$0.771024\pi$
$60$	0	0
$61$	−5.00367	−0.640654	−0.320327	−	0.947307i	$-0.603793\pi$
$61$	−5.00367	−0.640654	−0.320327	+	0.947307i	$0.603793\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.2967	1.64925
$66$	0	0
$67$	9.30266	1.13650	0.568251	−	0.822856i	$-0.307620\pi$
$67$	9.30266	1.13650	0.568251	+	0.822856i	$0.307620\pi$
$68$	0	0
$69$	−5.12043	−0.616428
$70$	0	0
$71$	7.35240i	0.872569i	0.899809	+	0.436285i	$0.143706\pi$
$71$	7.35240i	0.872569i	−0.899809	+	0.436285i	$0.856294\pi$
$72$	0	0
$73$	6.85362i	0.802155i	0.916044	+	0.401078i	$0.131364\pi$
$73$	6.85362i	0.802155i	−0.916044	+	0.401078i	$0.868636\pi$
$74$	0	0
$75$	− 1.26504i	− 0.146074i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.91293i	1.00278i	0.865221	+	0.501391i	$0.167178\pi$
$79$	8.91293i	1.00278i	−0.865221	+	0.501391i	$0.832822\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.96259i	0.215423i	0.994182	+	0.107711i	$0.0343522\pi$
$83$	1.96259i	0.215423i	−0.994182	+	0.107711i	$0.965648\pi$
$84$	0	0
$85$	− 1.13851i	− 0.123489i
$86$	0	0
$87$	−2.57962	−0.276565
$88$	0	0
$89$	− 6.82845i	− 0.723814i	−0.932214	−	0.361907i	$-0.882126\pi$
$89$	− 6.82845i	− 0.723814i	0.932214	−	0.361907i	$-0.117874\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.00666i	0.622862i
$94$	0	0
$95$	− 9.24692i	− 0.948714i
$96$	0	0
$97$	− 3.71270i	− 0.376967i	−0.982076	−	0.188484i	$-0.939643\pi$
$97$	− 3.71270i	− 0.376967i	0.982076	−	0.188484i	$-0.0603573\pi$
$98$	0	0
$99$	−5.67619	−0.570478
$100$	0	0
$101$	5.68155	0.565335	0.282667	−	0.959218i	$-0.408781\pi$
$101$	5.68155	0.565335	0.282667	+	0.959218i	$0.408781\pi$
$102$	0	0
$103$	−5.90691	−0.582025	−0.291013	−	0.956719i	$-0.593992\pi$
$103$	−5.90691	−0.582025	−0.291013	+	0.956719i	$0.593992\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.6609	1.03063	0.515314	−	0.857002i	$-0.327675\pi$
$107$	10.6609	1.03063	0.515314	+	0.857002i	$0.327675\pi$
$108$	0	0
$109$	− 3.17911i	− 0.304504i	−0.988342	−	0.152252i	$-0.951348\pi$
$109$	− 3.17911i	− 0.304504i	0.988342	−	0.152252i	$-0.0486525\pi$
$110$	0	0
$111$	−9.01565	−0.855727
$112$	0	0
$113$	−0.302446	−0.0284518	−0.0142259	−	0.999899i	$-0.504528\pi$
$113$	−0.302446	−0.0284518	−0.0142259	+	0.999899i	$0.504528\pi$
$114$	0	0
$115$	− 12.8165i	− 1.19514i
$116$	0	0
$117$	−5.31228	−0.491121
$118$	0	0
$119$	0	0
$120$	0	0
$121$	21.2191	1.92901
$122$	0	0
$123$	−4.65692	−0.419900
$124$	0	0
$125$	−9.34863	−0.836167
$126$	0	0
$127$	− 6.39751i	− 0.567687i	−0.958871	−	0.283843i	$-0.908390\pi$
$127$	− 6.39751i	− 0.567687i	0.958871	−	0.283843i	$-0.0916096\pi$
$128$	0	0
$129$	3.66703i	0.322864i
$130$	0	0
$131$	− 14.5026i	− 1.26710i	−0.773703	−	0.633549i	$-0.781598\pi$
$131$	− 14.5026i	− 1.26710i	0.773703	−	0.633549i	$-0.218402\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.50301i	0.215424i
$136$	0	0
$137$	1.23112	0.105182	0.0525908	−	0.998616i	$-0.483252\pi$
$137$	1.23112	0.105182	0.0525908	+	0.998616i	$0.483252\pi$
$138$	0	0
$139$	15.7788i	1.33834i	0.743108	+	0.669172i	$0.233351\pi$
$139$	15.7788i	1.33834i	−0.743108	+	0.669172i	$0.766649\pi$
$140$	0	0
$141$	− 0.957682i	− 0.0806514i
$142$	0	0
$143$	30.1535	2.52156
$144$	0	0
$145$	− 6.45681i	− 0.536209i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 5.95556i	− 0.487898i	−0.969788	−	0.243949i	$-0.921557\pi$
$149$	− 5.95556i	− 0.487898i	0.969788	−	0.243949i	$-0.0784430\pi$
$150$	0	0
$151$	3.59257i	0.292359i	0.989258	+	0.146180i	$0.0466977\pi$
$151$	3.59257i	0.292359i	−0.989258	+	0.146180i	$0.953302\pi$
$152$	0	0
$153$	0.454858i	0.0367731i
$154$	0	0
$155$	−15.0347	−1.20762
$156$	0	0
$157$	−0.983524	−0.0784937	−0.0392469	−	0.999230i	$-0.512496\pi$
$157$	−0.983524	−0.0784937	−0.0392469	+	0.999230i	$0.512496\pi$
$158$	0	0
$159$	6.24836	0.495527
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.52167	−0.197512	−0.0987562	−	0.995112i	$-0.531486\pi$
$163$	−2.52167	−0.197512	−0.0987562	+	0.995112i	$0.531486\pi$
$164$	0	0
$165$	− 14.2075i	− 1.10606i
$166$	0	0
$167$	−5.26232	−0.407211	−0.203605	−	0.979053i	$-0.565266\pi$
$167$	−5.26232	−0.407211	−0.203605	+	0.979053i	$0.565266\pi$
$168$	0	0
$169$	15.2203	1.17079
$170$	0	0
$171$	3.69433i	0.282512i
$172$	0	0
$173$	−0.143623	−0.0109195	−0.00545974	−	0.999985i	$-0.501738\pi$
$173$	−0.143623	−0.0109195	−0.00545974	+	0.999985i	$0.501738\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.1222	0.760827
$178$	0	0
$179$	18.9248	1.41450	0.707252	−	0.706961i	$-0.249935\pi$
$179$	18.9248	1.41450	0.707252	+	0.706961i	$0.249935\pi$
$180$	0	0
$181$	−22.2260	−1.65204	−0.826022	−	0.563637i	$-0.809402\pi$
$181$	−22.2260	−1.65204	−0.826022	+	0.563637i	$0.809402\pi$
$182$	0	0
$183$	5.00367i	0.369882i
$184$	0	0
$185$	− 22.5662i	− 1.65910i
$186$	0	0
$187$	− 2.58186i	− 0.188804i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.7909i	0.853162i	0.904449	+	0.426581i	$0.140282\pi$
$191$	11.7909i	0.853162i	−0.904449	+	0.426581i	$0.859718\pi$
$192$	0	0
$193$	26.8224	1.93072	0.965361	−	0.260918i	$-0.0840252\pi$
$193$	26.8224	1.93072	0.965361	+	0.260918i	$0.0840252\pi$
$194$	0	0
$195$	− 13.2967i	− 0.952195i
$196$	0	0
$197$	16.5842i	1.18157i	0.806828	+	0.590786i	$0.201182\pi$
$197$	16.5842i	1.18157i	−0.806828	+	0.590786i	$0.798818\pi$
$198$	0	0
$199$	−11.4084	−0.808720	−0.404360	−	0.914600i	$-0.632506\pi$
$199$	−11.4084	−0.808720	−0.404360	+	0.914600i	$0.632506\pi$
$200$	0	0
$201$	− 9.30266i	− 0.656159i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 11.6563i	− 0.814112i
$206$	0	0
$207$	5.12043i	0.355895i
$208$	0	0
$209$	− 20.9697i	− 1.45050i
$210$	0	0
$211$	4.13300	0.284527	0.142264	−	0.989829i	$-0.454562\pi$
$211$	4.13300	0.284527	0.142264	+	0.989829i	$0.454562\pi$
$212$	0	0
$213$	7.35240	0.503778
$214$	0	0
$215$	−9.17860	−0.625975
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.85362	0.463125
$220$	0	0
$221$	− 2.41633i	− 0.162540i
$222$	0	0
$223$	−18.1063	−1.21249	−0.606245	−	0.795278i	$-0.707325\pi$
$223$	−18.1063	−1.21249	−0.606245	+	0.795278i	$0.707325\pi$
$224$	0	0
$225$	−1.26504	−0.0843361
$226$	0	0
$227$	− 7.92521i	− 0.526015i	−0.964794	−	0.263007i	$-0.915286\pi$
$227$	− 7.92521i	− 0.526015i	0.964794	−	0.263007i	$-0.0847143\pi$
$228$	0	0
$229$	2.15573	0.142455	0.0712274	−	0.997460i	$-0.477308\pi$
$229$	2.15573	0.142455	0.0712274	+	0.997460i	$0.477308\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.0144	−1.31119	−0.655594	−	0.755114i	$-0.727582\pi$
$233$	−20.0144	−1.31119	−0.655594	+	0.755114i	$0.727582\pi$
$234$	0	0
$235$	2.39708	0.156369
$236$	0	0
$237$	8.91293	0.578957
$238$	0	0
$239$	18.8923i	1.22204i	0.791614	+	0.611021i	$0.209241\pi$
$239$	18.8923i	1.22204i	−0.791614	+	0.611021i	$0.790759\pi$
$240$	0	0
$241$	− 18.3083i	− 1.17934i	−0.807643	−	0.589672i	$-0.799257\pi$
$241$	− 18.3083i	− 1.17934i	0.807643	−	0.589672i	$-0.200743\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 19.6253i	− 1.24873i
$248$	0	0
$249$	1.96259	0.124374
$250$	0	0
$251$	3.43251i	0.216658i	0.994115	+	0.108329i	$0.0345500\pi$
$251$	3.43251i	0.216658i	−0.994115	+	0.108329i	$0.965450\pi$
$252$	0	0
$253$	− 29.0645i	− 1.82727i
$254$	0	0
$255$	−1.13851	−0.0712964
$256$	0	0
$257$	− 25.0690i	− 1.56376i	−0.623426	−	0.781882i	$-0.714260\pi$
$257$	− 25.0690i	− 1.56376i	0.623426	−	0.781882i	$-0.285740\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.57962i	0.159675i
$262$	0	0
$263$	15.5092i	0.956339i	0.878268	+	0.478169i	$0.158699\pi$
$263$	15.5092i	0.956339i	−0.878268	+	0.478169i	$0.841301\pi$
$264$	0	0
$265$	15.6397i	0.960738i
$266$	0	0
$267$	−6.82845	−0.417894
$268$	0	0
$269$	−23.7333	−1.44705	−0.723523	−	0.690300i	$-0.757479\pi$
$269$	−23.7333	−1.44705	−0.723523	+	0.690300i	$0.757479\pi$
$270$	0	0
$271$	−14.7137	−0.893793	−0.446896	−	0.894586i	$-0.647471\pi$
$271$	−14.7137	−0.893793	−0.446896	+	0.894586i	$0.647471\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.18061	0.433007
$276$	0	0
$277$	− 7.12245i	− 0.427947i	−0.976840	−	0.213973i	$-0.931359\pi$
$277$	− 7.12245i	− 0.427947i	0.976840	−	0.213973i	$-0.0686406\pi$
$278$	0	0
$279$	6.00666	0.359609
$280$	0	0
$281$	−13.0561	−0.778861	−0.389430	−	0.921056i	$-0.627328\pi$
$281$	−13.0561	−0.778861	−0.389430	+	0.921056i	$0.627328\pi$
$282$	0	0
$283$	2.11369i	0.125646i	0.998025	+	0.0628230i	$0.0200104\pi$
$283$	2.11369i	0.125646i	−0.998025	+	0.0628230i	$0.979990\pi$
$284$	0	0
$285$	−9.24692	−0.547740
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.7931	0.987830
$290$	0	0
$291$	−3.71270	−0.217642
$292$	0	0
$293$	−0.334002	−0.0195126	−0.00975630	−	0.999952i	$-0.503106\pi$
$293$	−0.334002	−0.0195126	−0.00975630	+	0.999952i	$0.503106\pi$
$294$	0	0
$295$	25.3358i	1.47511i
$296$	0	0
$297$	5.67619i	0.329366i
$298$	0	0
$299$	− 27.2012i	− 1.57308i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 5.68155i	− 0.326396i
$304$	0	0
$305$	−12.5242	−0.717135
$306$	0	0
$307$	− 9.39141i	− 0.535996i	−0.963419	−	0.267998i	$-0.913638\pi$
$307$	− 9.39141i	− 0.535996i	0.963419	−	0.267998i	$-0.0863621\pi$
$308$	0	0
$309$	5.90691i	0.336032i
$310$	0	0
$311$	−15.8750	−0.900189	−0.450095	−	0.892981i	$-0.648610\pi$
$311$	−15.8750	−0.900189	−0.450095	+	0.892981i	$0.648610\pi$
$312$	0	0
$313$	32.1656i	1.81811i	0.416679	+	0.909054i	$0.363194\pi$
$313$	32.1656i	1.81811i	−0.416679	+	0.909054i	$0.636806\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.2712i	0.857714i	0.903373	+	0.428857i	$0.141083\pi$
$317$	15.2712i	0.857714i	−0.903373	+	0.428857i	$0.858917\pi$
$318$	0	0
$319$	− 14.6424i	− 0.819818i
$320$	0	0
$321$	− 10.6609i	− 0.595033i
$322$	0	0
$323$	−1.68039	−0.0934996
$324$	0	0
$325$	6.72025	0.372773
$326$	0	0
$327$	−3.17911	−0.175805
$328$	0	0
$329$	0	0
$330$	0	0
$331$	21.0379	1.15635	0.578174	−	0.815914i	$-0.303766\pi$
$331$	21.0379	1.15635	0.578174	+	0.815914i	$0.303766\pi$
$332$	0	0
$333$	9.01565i	0.494054i
$334$	0	0
$335$	23.2846	1.27217
$336$	0	0
$337$	1.18351	0.0644697	0.0322348	−	0.999480i	$-0.489738\pi$
$337$	1.18351	0.0644697	0.0322348	+	0.999480i	$0.489738\pi$
$338$	0	0
$339$	0.302446i	0.0164266i
$340$	0	0
$341$	−34.0949	−1.84634
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−12.8165	−0.690016
$346$	0	0
$347$	4.81341	0.258397	0.129199	−	0.991619i	$-0.458759\pi$
$347$	4.81341	0.258397	0.129199	+	0.991619i	$0.458759\pi$
$348$	0	0
$349$	35.4792	1.89916	0.949578	−	0.313530i	$-0.101512\pi$
$349$	35.4792	1.89916	0.949578	+	0.313530i	$0.101512\pi$
$350$	0	0
$351$	5.31228i	0.283549i
$352$	0	0
$353$	− 4.77713i	− 0.254261i	−0.991886	−	0.127131i	$-0.959423\pi$
$353$	− 4.77713i	− 0.254261i	0.991886	−	0.127131i	$-0.0405767\pi$
$354$	0	0
$355$	18.4031i	0.976735i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 36.9146i	− 1.94828i	−0.225947	−	0.974140i	$-0.572547\pi$
$359$	− 36.9146i	− 1.94828i	0.225947	−	0.974140i	$-0.427453\pi$
$360$	0	0
$361$	5.35195	0.281682
$362$	0	0
$363$	− 21.2191i	− 1.11371i
$364$	0	0
$365$	17.1547i	0.897916i
$366$	0	0
$367$	−0.568832	−0.0296928	−0.0148464	−	0.999890i	$-0.504726\pi$
$367$	−0.568832	−0.0296928	−0.0148464	+	0.999890i	$0.504726\pi$
$368$	0	0
$369$	4.65692i	0.242430i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	23.4322i	1.21328i	0.794979	+	0.606638i	$0.207482\pi$
$373$	23.4322i	1.21328i	−0.794979	+	0.606638i	$0.792518\pi$
$374$	0	0
$375$	9.34863i	0.482761i
$376$	0	0
$377$	− 13.7037i	− 0.705775i
$378$	0	0
$379$	14.7240	0.756320	0.378160	−	0.925740i	$-0.376557\pi$
$379$	14.7240	0.756320	0.378160	+	0.925740i	$0.376557\pi$
$380$	0	0
$381$	−6.39751	−0.327754
$382$	0	0
$383$	4.99508	0.255237	0.127618	−	0.991823i	$-0.459267\pi$
$383$	4.99508	0.255237	0.127618	+	0.991823i	$0.459267\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.66703	0.186406
$388$	0	0
$389$	31.8810i	1.61643i	0.588887	+	0.808216i	$0.299566\pi$
$389$	31.8810i	1.61643i	−0.588887	+	0.808216i	$0.700434\pi$
$390$	0	0
$391$	−2.32907	−0.117786
$392$	0	0
$393$	−14.5026	−0.731559
$394$	0	0
$395$	22.3091i	1.12249i
$396$	0	0
$397$	32.9060	1.65151	0.825753	−	0.564032i	$-0.190751\pi$
$397$	32.9060	1.65151	0.825753	+	0.564032i	$0.190751\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.212472	0.0106103	0.00530517	−	0.999986i	$-0.498311\pi$
$401$	0.212472	0.0106103	0.00530517	+	0.999986i	$0.498311\pi$
$402$	0	0
$403$	−31.9091	−1.58950
$404$	0	0
$405$	2.50301	0.124375
$406$	0	0
$407$	− 51.1745i	− 2.53663i
$408$	0	0
$409$	20.4715i	1.01225i	0.862461	+	0.506124i	$0.168922\pi$
$409$	20.4715i	1.01225i	−0.862461	+	0.506124i	$0.831078\pi$
$410$	0	0
$411$	− 1.23112i	− 0.0607266i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.91239i	0.241140i
$416$	0	0
$417$	15.7788	0.772693
$418$	0	0
$419$	− 11.8439i	− 0.578614i	−0.957236	−	0.289307i	$-0.906575\pi$
$419$	− 11.8439i	− 0.578614i	0.957236	−	0.289307i	$-0.0934248\pi$
$420$	0	0
$421$	21.7928i	1.06211i	0.847336	+	0.531057i	$0.178205\pi$
$421$	21.7928i	1.06211i	−0.847336	+	0.531057i	$0.821795\pi$
$422$	0	0
$423$	−0.957682	−0.0465641
$424$	0	0
$425$	− 0.575414i	− 0.0279117i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 30.1535i	− 1.45582i
$430$	0	0
$431$	− 16.5393i	− 0.796669i	−0.917240	−	0.398335i	$-0.869588\pi$
$431$	− 16.5393i	− 0.796669i	0.917240	−	0.398335i	$-0.130412\pi$
$432$	0	0
$433$	39.2724i	1.88731i	0.330929	+	0.943656i	$0.392638\pi$
$433$	39.2724i	1.88731i	−0.330929	+	0.943656i	$0.607362\pi$
$434$	0	0
$435$	−6.45681	−0.309580
$436$	0	0
$437$	−18.9166	−0.904901
$438$	0	0
$439$	31.0331	1.48113	0.740564	−	0.671986i	$-0.234558\pi$
$439$	31.0331	1.48113	0.740564	+	0.671986i	$0.234558\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.844928	0.0401438	0.0200719	−	0.999799i	$-0.493610\pi$
$443$	0.844928	0.0401438	0.0200719	+	0.999799i	$0.493610\pi$
$444$	0	0
$445$	− 17.0917i	− 0.810222i
$446$	0	0
$447$	−5.95556	−0.281688
$448$	0	0
$449$	−41.7433	−1.96999	−0.984995	−	0.172585i	$-0.944788\pi$
$449$	−41.7433	−1.96999	−0.984995	+	0.172585i	$0.944788\pi$
$450$	0	0
$451$	− 26.4336i	− 1.24471i
$452$	0	0
$453$	3.59257	0.168794
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.48572	−0.350167	−0.175083	−	0.984554i	$-0.556020\pi$
$457$	−7.48572	−0.350167	−0.175083	+	0.984554i	$0.556020\pi$
$458$	0	0
$459$	0.454858	0.0212310
$460$	0	0
$461$	−4.33499	−0.201901	−0.100950	−	0.994891i	$-0.532188\pi$
$461$	−4.33499	−0.201901	−0.100950	+	0.994891i	$0.532188\pi$
$462$	0	0
$463$	− 35.3200i	− 1.64146i	−0.571316	−	0.820730i	$-0.693567\pi$
$463$	− 35.3200i	− 1.64146i	0.571316	−	0.820730i	$-0.306433\pi$
$464$	0	0
$465$	15.0347i	0.697218i
$466$	0	0
$467$	10.4987i	0.485821i	0.970049	+	0.242911i	$0.0781022\pi$
$467$	10.4987i	0.485821i	−0.970049	+	0.242911i	$0.921898\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.983524i	0.0453184i
$472$	0	0
$473$	−20.8147	−0.957063
$474$	0	0
$475$	− 4.67348i	− 0.214434i
$476$	0	0
$477$	− 6.24836i	− 0.286093i
$478$	0	0
$479$	−16.9275	−0.773437	−0.386719	−	0.922198i	$-0.626392\pi$
$479$	−16.9275	−0.773437	−0.386719	+	0.922198i	$0.626392\pi$
$480$	0	0
$481$	− 47.8937i	− 2.18376i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9.29291i	− 0.421969i
$486$	0	0
$487$	23.7270i	1.07517i	0.843209	+	0.537586i	$0.180664\pi$
$487$	23.7270i	1.07517i	−0.843209	+	0.537586i	$0.819336\pi$
$488$	0	0
$489$	2.52167i	0.114034i
$490$	0	0
$491$	3.41198	0.153980	0.0769901	−	0.997032i	$-0.475469\pi$
$491$	3.41198	0.153980	0.0769901	+	0.997032i	$0.475469\pi$
$492$	0	0
$493$	−1.17336	−0.0528456
$494$	0	0
$495$	−14.2075	−0.638581
$496$	0	0
$497$	0	0
$498$	0	0
$499$	17.4600	0.781615	0.390808	−	0.920472i	$-0.372196\pi$
$499$	17.4600	0.781615	0.390808	+	0.920472i	$0.372196\pi$
$500$	0	0
$501$	5.26232i	0.235103i
$502$	0	0
$503$	7.08646	0.315970	0.157985	−	0.987442i	$-0.449500\pi$
$503$	7.08646	0.315970	0.157985	+	0.987442i	$0.449500\pi$
$504$	0	0
$505$	14.2209	0.632824
$506$	0	0
$507$	− 15.2203i	− 0.675959i
$508$	0	0
$509$	37.9306	1.68124	0.840622	−	0.541623i	$-0.182190\pi$
$509$	37.9306	1.68124	0.840622	+	0.541623i	$0.182190\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.69433	0.163109
$514$	0	0
$515$	−14.7850	−0.651506
$516$	0	0
$517$	5.43598	0.239074
$518$	0	0
$519$	0.143623i	0.00630436i
$520$	0	0
$521$	7.98089i	0.349649i	0.984600	+	0.174825i	$0.0559358\pi$
$521$	7.98089i	0.349649i	−0.984600	+	0.174825i	$0.944064\pi$
$522$	0	0
$523$	14.2336i	0.622393i	0.950346	+	0.311196i	$0.100730\pi$
$523$	14.2336i	0.622393i	−0.950346	+	0.311196i	$0.899270\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.73218i	0.119016i
$528$	0	0
$529$	−3.21883	−0.139949
$530$	0	0
$531$	− 10.1222i	− 0.439264i
$532$	0	0
$533$	− 24.7389i	− 1.07156i
$534$	0	0
$535$	26.6843	1.15366
$536$	0	0
$537$	− 18.9248i	− 0.816664i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 7.42798i	− 0.319354i	−0.987169	−	0.159677i	$-0.948955\pi$
$541$	− 7.42798i	− 0.319354i	0.987169	−	0.159677i	$-0.0510452\pi$
$542$	0	0
$543$	22.2260i	0.953809i
$544$	0	0
$545$	− 7.95734i	− 0.340855i
$546$	0	0
$547$	28.2287	1.20697	0.603485	−	0.797374i	$-0.293778\pi$
$547$	28.2287	1.20697	0.603485	+	0.797374i	$0.293778\pi$
$548$	0	0
$549$	5.00367	0.213551
$550$	0	0
$551$	−9.52997	−0.405990
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−22.5662	−0.957883
$556$	0	0
$557$	− 34.5692i	− 1.46474i	−0.680906	−	0.732371i	$-0.738414\pi$
$557$	− 34.5692i	− 1.46474i	0.680906	−	0.732371i	$-0.261586\pi$
$558$	0	0
$559$	−19.4803	−0.823929
$560$	0	0
$561$	−2.58186	−0.109006
$562$	0	0
$563$	11.7207i	0.493967i	0.969020	+	0.246983i	$0.0794393\pi$
$563$	11.7207i	0.493967i	−0.969020	+	0.246983i	$0.920561\pi$
$564$	0	0
$565$	−0.757026	−0.0318483
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.1065	−0.465608	−0.232804	−	0.972524i	$-0.574790\pi$
$569$	−11.1065	−0.465608	−0.232804	+	0.972524i	$0.574790\pi$
$570$	0	0
$571$	35.5114	1.48610	0.743052	−	0.669234i	$-0.233378\pi$
$571$	35.5114	1.48610	0.743052	+	0.669234i	$0.233378\pi$
$572$	0	0
$573$	11.7909	0.492573
$574$	0	0
$575$	− 6.47756i	− 0.270133i
$576$	0	0
$577$	− 27.0551i	− 1.12632i	−0.826348	−	0.563159i	$-0.809586\pi$
$577$	− 27.0551i	− 1.12632i	0.826348	−	0.563159i	$-0.190414\pi$
$578$	0	0
$579$	− 26.8224i	− 1.11470i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	35.4668i	1.46889i
$584$	0	0
$585$	−13.2967	−0.549750
$586$	0	0
$587$	− 35.3797i	− 1.46027i	−0.683300	−	0.730137i	$-0.739456\pi$
$587$	− 35.3797i	− 1.46027i	0.683300	−	0.730137i	$-0.260544\pi$
$588$	0	0
$589$	22.1906i	0.914347i
$590$	0	0
$591$	16.5842	0.682181
$592$	0	0
$593$	39.2204i	1.61059i	0.592875	+	0.805294i	$0.297993\pi$
$593$	39.2204i	1.61059i	−0.592875	+	0.805294i	$0.702007\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.4084i	0.466915i
$598$	0	0
$599$	− 47.6722i	− 1.94783i	−0.226908	−	0.973916i	$-0.572862\pi$
$599$	− 47.6722i	− 1.94783i	0.226908	−	0.973916i	$-0.427138\pi$
$600$	0	0
$601$	13.4207i	0.547442i	0.961809	+	0.273721i	$0.0882544\pi$
$601$	13.4207i	0.547442i	−0.961809	+	0.273721i	$0.911746\pi$
$602$	0	0
$603$	−9.30266	−0.378834
$604$	0	0
$605$	53.1116	2.15929
$606$	0	0
$607$	38.2781	1.55366	0.776831	−	0.629709i	$-0.216826\pi$
$607$	38.2781	1.55366	0.776831	+	0.629709i	$0.216826\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.08748	0.205817
$612$	0	0
$613$	− 19.5404i	− 0.789230i	−0.918847	−	0.394615i	$-0.870878\pi$
$613$	− 19.5404i	− 0.789230i	0.918847	−	0.394615i	$-0.129122\pi$
$614$	0	0
$615$	−11.6563	−0.470028
$616$	0	0
$617$	−1.16195	−0.0467785	−0.0233892	−	0.999726i	$-0.507446\pi$
$617$	−1.16195	−0.0467785	−0.0233892	+	0.999726i	$0.507446\pi$
$618$	0	0
$619$	− 35.8930i	− 1.44266i	−0.692592	−	0.721330i	$-0.743531\pi$
$619$	− 35.8930i	− 1.44266i	0.692592	−	0.721330i	$-0.256469\pi$
$620$	0	0
$621$	5.12043	0.205476
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.7249	−1.18900
$626$	0	0
$627$	−20.9697	−0.837449
$628$	0	0
$629$	−4.10084	−0.163511
$630$	0	0
$631$	− 8.26460i	− 0.329008i	−0.986376	−	0.164504i	$-0.947398\pi$
$631$	− 8.26460i	− 0.329008i	0.986376	−	0.164504i	$-0.0526024\pi$
$632$	0	0
$633$	− 4.13300i	− 0.164272i
$634$	0	0
$635$	− 16.0130i	− 0.635457i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 7.35240i	− 0.290856i
$640$	0	0
$641$	28.9910	1.14508	0.572539	−	0.819878i	$-0.305959\pi$
$641$	28.9910	1.14508	0.572539	+	0.819878i	$0.305959\pi$
$642$	0	0
$643$	39.1239i	1.54289i	0.636293	+	0.771447i	$0.280467\pi$
$643$	39.1239i	1.54289i	−0.636293	+	0.771447i	$0.719533\pi$
$644$	0	0
$645$	9.17860i	0.361407i
$646$	0	0
$647$	−12.1898	−0.479231	−0.239616	−	0.970868i	$-0.577021\pi$
$647$	−12.1898	−0.479231	−0.239616	+	0.970868i	$0.577021\pi$
$648$	0	0
$649$	57.4552i	2.25531i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.1636i	0.397732i	0.980027	+	0.198866i	$0.0637259\pi$
$653$	10.1636i	0.397732i	−0.980027	+	0.198866i	$0.936274\pi$
$654$	0	0
$655$	− 36.3001i	− 1.41836i
$656$	0	0
$657$	− 6.85362i	− 0.267385i
$658$	0	0
$659$	8.28558	0.322760	0.161380	−	0.986892i	$-0.448405\pi$
$659$	8.28558	0.322760	0.161380	+	0.986892i	$0.448405\pi$
$660$	0	0
$661$	16.6178	0.646358	0.323179	−	0.946338i	$-0.395248\pi$
$661$	16.6178	0.646358	0.323179	+	0.946338i	$0.395248\pi$
$662$	0	0
$663$	−2.41633	−0.0938426
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−13.2088	−0.511446
$668$	0	0
$669$	18.1063i	0.700031i
$670$	0	0
$671$	−28.4018	−1.09644
$672$	0	0
$673$	13.1599	0.507278	0.253639	−	0.967299i	$-0.418372\pi$
$673$	13.1599	0.507278	0.253639	+	0.967299i	$0.418372\pi$
$674$	0	0
$675$	1.26504i	0.0486915i
$676$	0	0
$677$	−20.4402	−0.785580	−0.392790	−	0.919628i	$-0.628490\pi$
$677$	−20.4402	−0.785580	−0.392790	+	0.919628i	$0.628490\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.92521	−0.303695
$682$	0	0
$683$	15.7455	0.602484	0.301242	−	0.953548i	$-0.402599\pi$
$683$	15.7455	0.602484	0.301242	+	0.953548i	$0.402599\pi$
$684$	0	0
$685$	3.08150	0.117738
$686$	0	0
$687$	− 2.15573i	− 0.0822464i
$688$	0	0
$689$	33.1930i	1.26455i
$690$	0	0
$691$	3.19097i	0.121390i	0.998156	+	0.0606952i	$0.0193318\pi$
$691$	3.19097i	0.121390i	−0.998156	+	0.0606952i	$0.980668\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	39.4945i	1.49811i
$696$	0	0
$697$	−2.11824	−0.0802340
$698$	0	0
$699$	20.0144i	0.757015i
$700$	0	0
$701$	− 39.2121i	− 1.48102i	−0.672045	−	0.740511i	$-0.734584\pi$
$701$	− 39.2121i	− 1.48102i	0.672045	−	0.740511i	$-0.265416\pi$
$702$	0	0
$703$	−33.3067	−1.25619
$704$	0	0
$705$	− 2.39708i	− 0.0902794i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	38.5269i	1.44691i	0.690372	+	0.723454i	$0.257447\pi$
$709$	38.5269i	1.44691i	−0.690372	+	0.723454i	$0.742553\pi$
$710$	0	0
$711$	− 8.91293i	− 0.334261i
$712$	0	0
$713$	30.7567i	1.15185i
$714$	0	0
$715$	75.4744	2.82258
$716$	0	0
$717$	18.8923	0.705547
$718$	0	0
$719$	−23.0018	−0.857824	−0.428912	−	0.903346i	$-0.641103\pi$
$719$	−23.0018	−0.857824	−0.428912	+	0.903346i	$0.641103\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−18.3083	−0.680894
$724$	0	0
$725$	− 3.26333i	− 0.121197i
$726$	0	0
$727$	21.8433	0.810125	0.405062	−	0.914289i	$-0.367250\pi$
$727$	21.8433	0.810125	0.405062	+	0.914289i	$0.367250\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	1.66798i	0.0616924i
$732$	0	0
$733$	−48.1361	−1.77795	−0.888974	−	0.457957i	$-0.848581\pi$
$733$	−48.1361	−1.77795	−0.888974	+	0.457957i	$0.848581\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	52.8037	1.94505
$738$	0	0
$739$	6.46911	0.237970	0.118985	−	0.992896i	$-0.462036\pi$
$739$	6.46911	0.237970	0.118985	+	0.992896i	$0.462036\pi$
$740$	0	0
$741$	−19.6253	−0.720953
$742$	0	0
$743$	16.6709i	0.611597i	0.952096	+	0.305798i	$0.0989234\pi$
$743$	16.6709i	0.611597i	−0.952096	+	0.305798i	$0.901077\pi$
$744$	0	0
$745$	− 14.9068i	− 0.546143i
$746$	0	0
$747$	− 1.96259i	− 0.0718076i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 24.5602i	− 0.896215i	−0.893980	−	0.448108i	$-0.852098\pi$
$751$	− 24.5602i	− 0.896215i	0.893980	−	0.448108i	$-0.147902\pi$
$752$	0	0
$753$	3.43251	0.125088
$754$	0	0
$755$	8.99223i	0.327261i
$756$	0	0
$757$	2.80888i	0.102091i	0.998696	+	0.0510453i	$0.0162553\pi$
$757$	2.80888i	0.102091i	−0.998696	+	0.0510453i	$0.983745\pi$
$758$	0	0
$759$	−29.0645	−1.05498
$760$	0	0
$761$	20.1736i	0.731292i	0.930754	+	0.365646i	$0.119152\pi$
$761$	20.1736i	0.731292i	−0.930754	+	0.365646i	$0.880848\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.13851i	0.0411630i
$766$	0	0
$767$	53.7717i	1.94158i
$768$	0	0
$769$	32.1388i	1.15895i	0.814988	+	0.579477i	$0.196743\pi$
$769$	32.1388i	1.15895i	−0.814988	+	0.579477i	$0.803257\pi$
$770$	0	0
$771$	−25.0690	−0.902840
$772$	0	0
$773$	−22.5863	−0.812371	−0.406186	−	0.913791i	$-0.633141\pi$
$773$	−22.5863	−0.812371	−0.406186	+	0.913791i	$0.633141\pi$
$774$	0	0
$775$	−7.59867	−0.272952
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17.2042	−0.616404
$780$	0	0
$781$	41.7336i	1.49335i
$782$	0	0
$783$	2.57962	0.0921882
$784$	0	0
$785$	−2.46177	−0.0878642
$786$	0	0
$787$	33.7858i	1.20433i	0.798371	+	0.602166i	$0.205696\pi$
$787$	33.7858i	1.20433i	−0.798371	+	0.602166i	$0.794304\pi$
$788$	0	0
$789$	15.5092	0.552142
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−26.5809	−0.943916
$794$	0	0
$795$	15.6397	0.554682
$796$	0	0
$797$	−27.0472	−0.958061	−0.479030	−	0.877798i	$-0.659012\pi$
$797$	−27.0472	−0.958061	−0.479030	+	0.877798i	$0.659012\pi$
$798$	0	0
$799$	− 0.435609i	− 0.0154108i
$800$	0	0
$801$	6.82845i	0.241271i
$802$	0	0
$803$	38.9024i	1.37284i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	23.7333i	0.835453i
$808$	0	0
$809$	−49.8449	−1.75245	−0.876226	−	0.481900i	$-0.839947\pi$
$809$	−49.8449	−1.75245	−0.876226	+	0.481900i	$0.839947\pi$
$810$	0	0
$811$	36.5330i	1.28285i	0.767188	+	0.641423i	$0.221655\pi$
$811$	36.5330i	1.28285i	−0.767188	+	0.641423i	$0.778345\pi$
$812$	0	0
$813$	14.7137i	0.516031i
$814$	0	0
$815$	−6.31175	−0.221091
$816$	0	0
$817$	13.5472i	0.473957i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 3.00119i	− 0.104742i	−0.998628	−	0.0523711i	$-0.983322\pi$
$821$	− 3.00119i	− 0.104742i	0.998628	−	0.0523711i	$-0.0166779\pi$
$822$	0	0
$823$	15.9975i	0.557639i	0.960343	+	0.278820i	$0.0899432\pi$
$823$	15.9975i	0.557639i	−0.960343	+	0.278820i	$0.910057\pi$
$824$	0	0
$825$	− 7.18061i	− 0.249997i
$826$	0	0
$827$	−4.57856	−0.159212	−0.0796060	−	0.996826i	$-0.525366\pi$
$827$	−4.57856	−0.159212	−0.0796060	+	0.996826i	$0.525366\pi$
$828$	0	0
$829$	−3.06153	−0.106331	−0.0531657	−	0.998586i	$-0.516931\pi$
$829$	−3.06153	−0.106331	−0.0531657	+	0.998586i	$0.516931\pi$
$830$	0	0
$831$	−7.12245	−0.247075
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13.1716	−0.455823
$836$	0	0
$837$	− 6.00666i	− 0.207621i
$838$	0	0
$839$	−39.0864	−1.34941	−0.674706	−	0.738087i	$-0.735730\pi$
$839$	−39.0864	−1.34941	−0.674706	+	0.738087i	$0.735730\pi$
$840$	0	0
$841$	22.3456	0.770536
$842$	0	0
$843$	13.0561i	0.449676i
$844$	0	0
$845$	38.0966	1.31056
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.11369	0.0725418
$850$	0	0
$851$	−46.1640	−1.58248
$852$	0	0
$853$	26.4897	0.906989	0.453495	−	0.891259i	$-0.350177\pi$
$853$	26.4897	0.906989	0.453495	+	0.891259i	$0.350177\pi$
$854$	0	0
$855$	9.24692i	0.316238i
$856$	0	0
$857$	44.9255i	1.53462i	0.641273	+	0.767312i	$0.278406\pi$
$857$	44.9255i	1.53462i	−0.641273	+	0.767312i	$0.721594\pi$
$858$	0	0
$859$	35.4056i	1.20802i	0.796976	+	0.604011i	$0.206432\pi$
$859$	35.4056i	1.20802i	−0.796976	+	0.604011i	$0.793568\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 2.77632i	− 0.0945072i	−0.998883	−	0.0472536i	$-0.984953\pi$
$863$	− 2.77632i	− 0.0945072i	0.998883	−	0.0472536i	$-0.0150469\pi$
$864$	0	0
$865$	−0.359490	−0.0122230
$866$	0	0
$867$	− 16.7931i	− 0.570324i
$868$	0	0
$869$	50.5915i	1.71620i
$870$	0	0
$871$	49.4184	1.67448
$872$	0	0
$873$	3.71270i	0.125656i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 16.5427i	− 0.558608i	−0.960203	−	0.279304i	$-0.909896\pi$
$877$	− 16.5427i	− 0.558608i	0.960203	−	0.279304i	$-0.0901036\pi$
$878$	0	0
$879$	0.334002i	0.0112656i
$880$	0	0
$881$	− 24.6158i	− 0.829327i	−0.909975	−	0.414664i	$-0.863899\pi$
$881$	− 24.6158i	− 0.829327i	0.909975	−	0.414664i	$-0.136101\pi$
$882$	0	0
$883$	−40.4623	−1.36167	−0.680833	−	0.732439i	$-0.738382\pi$
$883$	−40.4623	−1.36167	−0.680833	+	0.732439i	$0.738382\pi$
$884$	0	0
$885$	25.3358	0.851654
$886$	0	0
$887$	−9.28462	−0.311747	−0.155873	−	0.987777i	$-0.549819\pi$
$887$	−9.28462	−0.311747	−0.155873	+	0.987777i	$0.549819\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.67619	0.190159
$892$	0	0
$893$	− 3.53799i	− 0.118394i
$894$	0	0
$895$	47.3688	1.58337
$896$	0	0
$897$	−27.2012	−0.908221
$898$	0	0
$899$	15.4949i	0.516784i
$900$	0	0
$901$	2.84212	0.0946846
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−55.6318	−1.84926
$906$	0	0
$907$	−5.10654	−0.169560	−0.0847800	−	0.996400i	$-0.527019\pi$
$907$	−5.10654	−0.169560	−0.0847800	+	0.996400i	$0.527019\pi$
$908$	0	0
$909$	−5.68155	−0.188445
$910$	0	0
$911$	36.3702i	1.20500i	0.798119	+	0.602500i	$0.205829\pi$
$911$	36.3702i	1.20500i	−0.798119	+	0.602500i	$0.794171\pi$
$912$	0	0
$913$	11.1401i	0.368682i
$914$	0	0
$915$	12.5242i	0.414038i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 46.6786i	− 1.53978i	−0.638174	−	0.769892i	$-0.720310\pi$
$919$	− 46.6786i	− 1.53978i	0.638174	−	0.769892i	$-0.279690\pi$
$920$	0	0
$921$	−9.39141	−0.309458
$922$	0	0
$923$	39.0580i	1.28561i
$924$	0	0
$925$	− 11.4052i	− 0.375000i
$926$	0	0
$927$	5.90691	0.194008
$928$	0	0
$929$	− 39.1063i	− 1.28304i	−0.767108	−	0.641518i	$-0.778305\pi$
$929$	− 39.1063i	− 1.28304i	0.767108	−	0.641518i	$-0.221695\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15.8750i	0.519724i
$934$	0	0
$935$	− 6.46241i	− 0.211343i
$936$	0	0
$937$	14.4038i	0.470551i	0.971929	+	0.235275i	$0.0755992\pi$
$937$	14.4038i	0.470551i	−0.971929	+	0.235275i	$0.924401\pi$
$938$	0	0
$939$	32.1656	1.04969
$940$	0	0
$941$	−26.2537	−0.855848	−0.427924	−	0.903815i	$-0.640755\pi$
$941$	−26.2537	−0.855848	−0.427924	+	0.903815i	$0.640755\pi$
$942$	0	0
$943$	−23.8455	−0.776515
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.7071	−0.510412	−0.255206	−	0.966887i	$-0.582143\pi$
$947$	−15.7071	−0.510412	−0.255206	+	0.966887i	$0.582143\pi$
$948$	0	0
$949$	36.4084i	1.18187i
$950$	0	0
$951$	15.2712	0.495201
$952$	0	0
$953$	42.8466	1.38794	0.693968	−	0.720006i	$-0.255861\pi$
$953$	42.8466	1.38794	0.693968	+	0.720006i	$0.255861\pi$
$954$	0	0
$955$	29.5128i	0.955011i
$956$	0	0
$957$	−14.6424	−0.473322
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.07998	0.163870
$962$	0	0
$963$	−10.6609	−0.343543
$964$	0	0
$965$	67.1368	2.16121
$966$	0	0
$967$	0.672082i	0.0216127i	0.999942	+	0.0108063i	$0.00343983\pi$
$967$	0.672082i	0.0216127i	−0.999942	+	0.0108063i	$0.996560\pi$
$968$	0	0
$969$	1.68039i	0.0539820i
$970$	0	0
$971$	− 7.45749i	− 0.239322i	−0.992815	−	0.119661i	$-0.961819\pi$
$971$	− 7.45749i	− 0.239322i	0.992815	−	0.119661i	$-0.0381808\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 6.72025i	− 0.215220i
$976$	0	0
$977$	17.5740	0.562243	0.281121	−	0.959672i	$-0.409294\pi$
$977$	17.5740	0.562243	0.281121	+	0.959672i	$0.409294\pi$
$978$	0	0
$979$	− 38.7596i	− 1.23876i
$980$	0	0
$981$	3.17911i	0.101501i
$982$	0	0
$983$	42.7322	1.36294	0.681472	−	0.731844i	$-0.261340\pi$
$983$	42.7322	1.36294	0.681472	+	0.731844i	$0.261340\pi$
$984$	0	0
$985$	41.5103i	1.32263i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.7768i	0.597067i
$990$	0	0
$991$	45.4228i	1.44290i	0.692465	+	0.721451i	$0.256525\pi$
$991$	45.4228i	1.44290i	−0.692465	+	0.721451i	$0.743475\pi$
$992$	0	0
$993$	− 21.0379i	− 0.667617i
$994$	0	0
$995$	−28.5553	−0.905264
$996$	0	0
$997$	−43.7436	−1.38537	−0.692687	−	0.721239i	$-0.743573\pi$
$997$	−43.7436	−1.38537	−0.692687	+	0.721239i	$0.743573\pi$
$998$	0	0
$999$	9.01565	0.285242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.p.a.3919.1		32
4.3	odd	2		1176.2.p.a.979.27		32
7.4	even	3		672.2.bb.a.271.2		32
7.5	odd	6		672.2.bb.a.367.7		32
7.6	odd	2	inner	4704.2.p.a.3919.16		32
8.3	odd	2	inner	4704.2.p.a.3919.15		32
8.5	even	2		1176.2.p.a.979.26		32
21.5	even	6		2016.2.bs.c.1711.4		32
21.11	odd	6		2016.2.bs.c.271.13		32
28.11	odd	6		168.2.t.a.19.4	✓	32
28.19	even	6		168.2.t.a.115.8	yes	32
28.27	even	2		1176.2.p.a.979.28		32
56.5	odd	6		168.2.t.a.115.4	yes	32
56.11	odd	6		672.2.bb.a.271.7		32
56.13	odd	2		1176.2.p.a.979.25		32
56.19	even	6		672.2.bb.a.367.2		32
56.27	even	2	inner	4704.2.p.a.3919.2		32
56.53	even	6		168.2.t.a.19.8	yes	32
84.11	even	6		504.2.bk.c.19.13		32
84.47	odd	6		504.2.bk.c.451.9		32
168.5	even	6		504.2.bk.c.451.13		32
168.11	even	6		2016.2.bs.c.271.4		32
168.53	odd	6		504.2.bk.c.19.9		32
168.131	odd	6		2016.2.bs.c.1711.13		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.t.a.19.4	✓	32	28.11	odd	6
168.2.t.a.19.8	yes	32	56.53	even	6
168.2.t.a.115.4	yes	32	56.5	odd	6
168.2.t.a.115.8	yes	32	28.19	even	6
504.2.bk.c.19.9		32	168.53	odd	6
504.2.bk.c.19.13		32	84.11	even	6
504.2.bk.c.451.9		32	84.47	odd	6
504.2.bk.c.451.13		32	168.5	even	6
672.2.bb.a.271.2		32	7.4	even	3
672.2.bb.a.271.7		32	56.11	odd	6
672.2.bb.a.367.2		32	56.19	even	6
672.2.bb.a.367.7		32	7.5	odd	6
1176.2.p.a.979.25		32	56.13	odd	2
1176.2.p.a.979.26		32	8.5	even	2
1176.2.p.a.979.27		32	4.3	odd	2
1176.2.p.a.979.28		32	28.27	even	2
2016.2.bs.c.271.4		32	168.11	even	6
2016.2.bs.c.271.13		32	21.11	odd	6
2016.2.bs.c.1711.4		32	21.5	even	6
2016.2.bs.c.1711.13		32	168.131	odd	6
4704.2.p.a.3919.1		32	1.1	even	1	trivial
4704.2.p.a.3919.2		32	56.27	even	2	inner
4704.2.p.a.3919.15		32	8.3	odd	2	inner
4704.2.p.a.3919.16		32	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.50301 q^{5} -1.00000 q^{9} +5.67619 q^{11} +5.31228 q^{13} -2.50301i q^{15} -0.454858i q^{17} -3.69433i q^{19} -5.12043i q^{23} +1.26504 q^{25} +1.00000i q^{27} -2.57962i q^{29} -6.00666 q^{31} -5.67619i q^{33} -9.01565i q^{37} -5.31228i q^{39} -4.65692i q^{41} -3.66703 q^{43} -2.50301 q^{45} +0.957682 q^{47} -0.454858 q^{51} +6.24836i q^{53} +14.2075 q^{55} -3.69433 q^{57} +10.1222i q^{59} -5.00367 q^{61} +13.2967 q^{65} +9.30266 q^{67} -5.12043 q^{69} +7.35240i q^{71} +6.85362i q^{73} -1.26504i q^{75} +8.91293i q^{79} +1.00000 q^{81} +1.96259i q^{83} -1.13851i q^{85} -2.57962 q^{87} -6.82845i q^{89} +6.00666i q^{93} -9.24692i q^{95} -3.71270i q^{97} -5.67619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{9} + 16 q^{11} + 32 q^{25} + 16 q^{43} + 16 q^{57} - 64 q^{67} + 32 q^{81} - 16 q^{99}+O(q^{100}) \) 32 * q - 32 * q^9 + 16 * q^11 + 32 * q^25 + 16 * q^43 + 16 * q^57 - 64 * q^67 + 32 * q^81 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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