LMFDB - Embedded newform 4704.2.p.a.3919.25

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4704,2,Mod(3919,4704)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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			3919.25
Character	$\chi$	$=$	4704.3919
Dual form			4704.2.p.a.3919.26

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.88284 q^{5} -1.00000 q^{9} +5.82680 q^{11} +1.04841 q^{13} +2.88284i q^{15} +6.82499i q^{17} +0.681229i q^{19} +2.14701i q^{23} +3.31079 q^{25} +1.00000i q^{27} -6.61515i q^{29} -3.83116 q^{31} -5.82680i q^{33} -2.38082i q^{37} -1.04841i q^{39} +1.19919i q^{41} -1.34319 q^{43} +2.88284 q^{45} -11.0534 q^{47} +6.82499 q^{51} -8.07845i q^{53} -16.7978 q^{55} +0.681229 q^{57} +7.87073i q^{59} -3.26942 q^{61} -3.02240 q^{65} -13.3126 q^{67} +2.14701 q^{69} +1.08533i q^{71} -5.64507i q^{73} -3.31079i q^{75} +12.6589i q^{79} +1.00000 q^{81} +0.482042i q^{83} -19.6754i q^{85} -6.61515 q^{87} +12.3909i q^{89} +3.83116i q^{93} -1.96388i q^{95} +3.63532i q^{97} -5.82680 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.88284	−1.28925	−0.644624	−	0.764500i	$-0.722986\pi$
$5$	−2.88284	−1.28925	−0.644624	+	0.764500i	$0.722986\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.82680	1.75685	0.878423	−	0.477883i	$-0.158596\pi$
$11$	5.82680	1.75685	0.878423	+	0.477883i	$0.158596\pi$
$12$	0	0
$13$	1.04841	0.290776	0.145388	−	0.989375i	$-0.453557\pi$
$13$	1.04841	0.290776	0.145388	+	0.989375i	$0.453557\pi$
$14$	0	0
$15$	2.88284i	0.744347i
$16$	0	0
$17$	6.82499i	1.65530i	0.561241	+	0.827652i	$0.310324\pi$
$17$	6.82499i	1.65530i	−0.561241	+	0.827652i	$0.689676\pi$
$18$	0	0
$19$	0.681229i	0.156285i	0.996942	+	0.0781423i	$0.0248989\pi$
$19$	0.681229i	0.156285i	−0.996942	+	0.0781423i	$0.975101\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.14701i	0.447683i	0.974626	+	0.223841i	$0.0718598\pi$
$23$	2.14701i	0.447683i	−0.974626	+	0.223841i	$0.928140\pi$
$24$	0	0
$25$	3.31079	0.662159
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 6.61515i	− 1.22840i	−0.789149	−	0.614201i	$-0.789478\pi$
$29$	− 6.61515i	− 1.22840i	0.789149	−	0.614201i	$-0.210522\pi$
$30$	0	0
$31$	−3.83116	−0.688096	−0.344048	−	0.938952i	$-0.611798\pi$
$31$	−3.83116	−0.688096	−0.344048	+	0.938952i	$0.611798\pi$
$32$	0	0
$33$	− 5.82680i	− 1.01432i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.38082i	− 0.391404i	−0.980663	−	0.195702i	$-0.937302\pi$
$37$	− 2.38082i	− 0.391404i	0.980663	−	0.195702i	$-0.0626985\pi$
$38$	0	0
$39$	− 1.04841i	− 0.167880i
$40$	0	0
$41$	1.19919i	0.187281i	0.995606	+	0.0936407i	$0.0298505\pi$
$41$	1.19919i	0.187281i	−0.995606	+	0.0936407i	$0.970149\pi$
$42$	0	0
$43$	−1.34319	−0.204835	−0.102418	−	0.994741i	$-0.532658\pi$
$43$	−1.34319	−0.204835	−0.102418	+	0.994741i	$0.532658\pi$
$44$	0	0
$45$	2.88284	0.429749
$46$	0	0
$47$	−11.0534	−1.61230	−0.806152	−	0.591708i	$-0.798454\pi$
$47$	−11.0534	−1.61230	−0.806152	+	0.591708i	$0.798454\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	6.82499	0.955690
$52$	0	0
$53$	− 8.07845i	− 1.10966i	−0.831963	−	0.554830i	$-0.812783\pi$
$53$	− 8.07845i	− 1.10966i	0.831963	−	0.554830i	$-0.187217\pi$
$54$	0	0
$55$	−16.7978	−2.26501
$56$	0	0
$57$	0.681229	0.0902310
$58$	0	0
$59$	7.87073i	1.02468i	0.858782	+	0.512341i	$0.171222\pi$
$59$	7.87073i	1.02468i	−0.858782	+	0.512341i	$0.828778\pi$
$60$	0	0
$61$	−3.26942	−0.418607	−0.209303	−	0.977851i	$-0.567120\pi$
$61$	−3.26942	−0.418607	−0.209303	+	0.977851i	$0.567120\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.02240	−0.374883
$66$	0	0
$67$	−13.3126	−1.62639	−0.813195	−	0.581991i	$-0.802274\pi$
$67$	−13.3126	−1.62639	−0.813195	+	0.581991i	$0.802274\pi$
$68$	0	0
$69$	2.14701	0.258470
$70$	0	0
$71$	1.08533i	0.128805i	0.997924	+	0.0644027i	$0.0205142\pi$
$71$	1.08533i	0.128805i	−0.997924	+	0.0644027i	$0.979486\pi$
$72$	0	0
$73$	− 5.64507i	− 0.660706i	−0.943858	−	0.330353i	$-0.892832\pi$
$73$	− 5.64507i	− 0.660706i	0.943858	−	0.330353i	$-0.107168\pi$
$74$	0	0
$75$	− 3.31079i	− 0.382298i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.6589i	1.42424i	0.702057	+	0.712120i	$0.252265\pi$
$79$	12.6589i	1.42424i	−0.702057	+	0.712120i	$0.747735\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.482042i	0.0529110i	0.999650	+	0.0264555i	$0.00842203\pi$
$83$	0.482042i	0.0529110i	−0.999650	+	0.0264555i	$0.991578\pi$
$84$	0	0
$85$	− 19.6754i	− 2.13410i
$86$	0	0
$87$	−6.61515	−0.709218
$88$	0	0
$89$	12.3909i	1.31343i	0.754140	+	0.656714i	$0.228054\pi$
$89$	12.3909i	1.31343i	−0.754140	+	0.656714i	$0.771946\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.83116i	0.397272i
$94$	0	0
$95$	− 1.96388i	− 0.201490i
$96$	0	0
$97$	3.63532i	0.369111i	0.982822	+	0.184556i	$0.0590846\pi$
$97$	3.63532i	0.369111i	−0.982822	+	0.184556i	$0.940915\pi$
$98$	0	0
$99$	−5.82680	−0.585616
$100$	0	0
$101$	2.50468	0.249225	0.124612	−	0.992206i	$-0.460231\pi$
$101$	2.50468	0.249225	0.124612	+	0.992206i	$0.460231\pi$
$102$	0	0
$103$	4.62644	0.455857	0.227928	−	0.973678i	$-0.426805\pi$
$103$	4.62644	0.455857	0.227928	+	0.973678i	$0.426805\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.06946	0.586757	0.293378	−	0.955996i	$-0.405220\pi$
$107$	6.06946	0.586757	0.293378	+	0.955996i	$0.405220\pi$
$108$	0	0
$109$	14.5645i	1.39503i	0.716573	+	0.697513i	$0.245710\pi$
$109$	14.5645i	1.39503i	−0.716573	+	0.697513i	$0.754290\pi$
$110$	0	0
$111$	−2.38082	−0.225977
$112$	0	0
$113$	−3.82875	−0.360179	−0.180089	−	0.983650i	$-0.557639\pi$
$113$	−3.82875	−0.360179	−0.180089	+	0.983650i	$0.557639\pi$
$114$	0	0
$115$	− 6.18950i	− 0.577174i
$116$	0	0
$117$	−1.04841	−0.0969254
$118$	0	0
$119$	0	0
$120$	0	0
$121$	22.9516	2.08651
$122$	0	0
$123$	1.19919	0.108127
$124$	0	0
$125$	4.86972	0.435561
$126$	0	0
$127$	0.550415i	0.0488415i	0.999702	+	0.0244207i	$0.00777413\pi$
$127$	0.550415i	0.0488415i	−0.999702	+	0.0244207i	$0.992226\pi$
$128$	0	0
$129$	1.34319i	0.118262i
$130$	0	0
$131$	2.80383i	0.244971i	0.992470	+	0.122486i	$0.0390866\pi$
$131$	2.80383i	0.244971i	−0.992470	+	0.122486i	$0.960913\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 2.88284i	− 0.248116i
$136$	0	0
$137$	12.7630	1.09041	0.545206	−	0.838302i	$-0.316451\pi$
$137$	12.7630	1.09041	0.545206	+	0.838302i	$0.316451\pi$
$138$	0	0
$139$	6.11761i	0.518889i	0.965758	+	0.259444i	$0.0835394\pi$
$139$	6.11761i	0.518889i	−0.965758	+	0.259444i	$0.916461\pi$
$140$	0	0
$141$	11.0534i	0.930864i
$142$	0	0
$143$	6.10887	0.510849
$144$	0	0
$145$	19.0704i	1.58371i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.02260i	0.165698i	0.996562	+	0.0828491i	$0.0264019\pi$
$149$	2.02260i	0.165698i	−0.996562	+	0.0828491i	$0.973598\pi$
$150$	0	0
$151$	14.0681i	1.14484i	0.819959	+	0.572422i	$0.193996\pi$
$151$	14.0681i	1.14484i	−0.819959	+	0.572422i	$0.806004\pi$
$152$	0	0
$153$	− 6.82499i	− 0.551768i
$154$	0	0
$155$	11.0446	0.887126
$156$	0	0
$157$	−5.23477	−0.417780	−0.208890	−	0.977939i	$-0.566985\pi$
$157$	−5.23477	−0.417780	−0.208890	+	0.977939i	$0.566985\pi$
$158$	0	0
$159$	−8.07845	−0.640663
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.976425	−0.0764795	−0.0382397	−	0.999269i	$-0.512175\pi$
$163$	−0.976425	−0.0764795	−0.0382397	+	0.999269i	$0.512175\pi$
$164$	0	0
$165$	16.7978i	1.30770i
$166$	0	0
$167$	2.08267	0.161162	0.0805808	−	0.996748i	$-0.474322\pi$
$167$	2.08267	0.161162	0.0805808	+	0.996748i	$0.474322\pi$
$168$	0	0
$169$	−11.9008	−0.915449
$170$	0	0
$171$	− 0.681229i	− 0.0520949i
$172$	0	0
$173$	21.3463	1.62293	0.811465	−	0.584402i	$-0.198671\pi$
$173$	21.3463	1.62293	0.811465	+	0.584402i	$0.198671\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.87073	0.591600
$178$	0	0
$179$	6.94974	0.519448	0.259724	−	0.965683i	$-0.416368\pi$
$179$	6.94974	0.519448	0.259724	+	0.965683i	$0.416368\pi$
$180$	0	0
$181$	−7.67619	−0.570566	−0.285283	−	0.958443i	$-0.592088\pi$
$181$	−7.67619	−0.570566	−0.285283	+	0.958443i	$0.592088\pi$
$182$	0	0
$183$	3.26942i	0.241683i
$184$	0	0
$185$	6.86352i	0.504616i
$186$	0	0
$187$	39.7679i	2.90812i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.8099i	1.28868i	0.764739	+	0.644340i	$0.222868\pi$
$191$	17.8099i	1.28868i	−0.764739	+	0.644340i	$0.777132\pi$
$192$	0	0
$193$	−13.7959	−0.993054	−0.496527	−	0.868021i	$-0.665392\pi$
$193$	−13.7959	−0.993054	−0.496527	+	0.868021i	$0.665392\pi$
$194$	0	0
$195$	3.02240i	0.216439i
$196$	0	0
$197$	− 1.13180i	− 0.0806374i	−0.999187	−	0.0403187i	$-0.987163\pi$
$197$	− 1.13180i	− 0.0806374i	0.999187	−	0.0403187i	$-0.0128373\pi$
$198$	0	0
$199$	−8.53186	−0.604808	−0.302404	−	0.953180i	$-0.597789\pi$
$199$	−8.53186	−0.604808	−0.302404	+	0.953180i	$0.597789\pi$
$200$	0	0
$201$	13.3126i	0.938997i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 3.45707i	− 0.241452i
$206$	0	0
$207$	− 2.14701i	− 0.149228i
$208$	0	0
$209$	3.96939i	0.274568i
$210$	0	0
$211$	−21.2436	−1.46247	−0.731237	−	0.682124i	$-0.761056\pi$
$211$	−21.2436	−1.46247	−0.731237	+	0.682124i	$0.761056\pi$
$212$	0	0
$213$	1.08533	0.0743658
$214$	0	0
$215$	3.87222	0.264083
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5.64507	−0.381459
$220$	0	0
$221$	7.15538i	0.481323i
$222$	0	0
$223$	−3.12207	−0.209069	−0.104535	−	0.994521i	$-0.533335\pi$
$223$	−3.12207	−0.209069	−0.104535	+	0.994521i	$0.533335\pi$
$224$	0	0
$225$	−3.31079	−0.220720
$226$	0	0
$227$	5.22675i	0.346911i	0.984842	+	0.173456i	$0.0554933\pi$
$227$	5.22675i	0.346911i	−0.984842	+	0.173456i	$0.944507\pi$
$228$	0	0
$229$	−28.4258	−1.87843	−0.939215	−	0.343329i	$-0.888446\pi$
$229$	−28.4258	−1.87843	−0.939215	+	0.343329i	$0.888446\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.11941	0.400896	0.200448	−	0.979704i	$-0.435760\pi$
$233$	6.11941	0.400896	0.200448	+	0.979704i	$0.435760\pi$
$234$	0	0
$235$	31.8652	2.07866
$236$	0	0
$237$	12.6589	0.822286
$238$	0	0
$239$	8.83528i	0.571507i	0.958303	+	0.285753i	$0.0922439\pi$
$239$	8.83528i	0.571507i	−0.958303	+	0.285753i	$0.907756\pi$
$240$	0	0
$241$	− 5.95545i	− 0.383624i	−0.981432	−	0.191812i	$-0.938564\pi$
$241$	− 5.95545i	− 0.383624i	0.981432	−	0.191812i	$-0.0614364\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.714206i	0.0454439i
$248$	0	0
$249$	0.482042	0.0305482
$250$	0	0
$251$	− 2.13955i	− 0.135047i	−0.997718	−	0.0675235i	$-0.978490\pi$
$251$	− 2.13955i	− 0.135047i	0.997718	−	0.0675235i	$-0.0215098\pi$
$252$	0	0
$253$	12.5102i	0.786510i
$254$	0	0
$255$	−19.6754	−1.23212
$256$	0	0
$257$	− 19.2869i	− 1.20309i	−0.798841	−	0.601543i	$-0.794553\pi$
$257$	− 19.2869i	− 1.20309i	0.798841	−	0.601543i	$-0.205447\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.61515i	0.409467i
$262$	0	0
$263$	24.3341i	1.50051i	0.661150	+	0.750254i	$0.270069\pi$
$263$	24.3341i	1.50051i	−0.661150	+	0.750254i	$0.729931\pi$
$264$	0	0
$265$	23.2889i	1.43063i
$266$	0	0
$267$	12.3909	0.758308
$268$	0	0
$269$	−20.9009	−1.27435	−0.637176	−	0.770718i	$-0.719898\pi$
$269$	−20.9009	−1.27435	−0.637176	+	0.770718i	$0.719898\pi$
$270$	0	0
$271$	−19.9939	−1.21455	−0.607273	−	0.794493i	$-0.707736\pi$
$271$	−19.9939	−1.21455	−0.607273	+	0.794493i	$0.707736\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	19.2913	1.16331
$276$	0	0
$277$	15.2920i	0.918807i	0.888228	+	0.459404i	$0.151937\pi$
$277$	15.2920i	0.918807i	−0.888228	+	0.459404i	$0.848063\pi$
$278$	0	0
$279$	3.83116	0.229365
$280$	0	0
$281$	5.59802	0.333950	0.166975	−	0.985961i	$-0.446600\pi$
$281$	5.59802	0.333950	0.166975	+	0.985961i	$0.446600\pi$
$282$	0	0
$283$	11.2342i	0.667804i	0.942608	+	0.333902i	$0.108365\pi$
$283$	11.2342i	0.667804i	−0.942608	+	0.333902i	$0.891635\pi$
$284$	0	0
$285$	−1.96388	−0.116330
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−29.5806	−1.74003
$290$	0	0
$291$	3.63532	0.213106
$292$	0	0
$293$	−17.7212	−1.03528	−0.517642	−	0.855598i	$-0.673190\pi$
$293$	−17.7212	−1.03528	−0.517642	+	0.855598i	$0.673190\pi$
$294$	0	0
$295$	− 22.6901i	− 1.32107i
$296$	0	0
$297$	5.82680i	0.338105i
$298$	0	0
$299$	2.25095i	0.130176i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 2.50468i	− 0.143890i
$304$	0	0
$305$	9.42524	0.539688
$306$	0	0
$307$	20.3724i	1.16271i	0.813649	+	0.581357i	$0.197478\pi$
$307$	20.3724i	1.16271i	−0.813649	+	0.581357i	$0.802522\pi$
$308$	0	0
$309$	− 4.62644i	− 0.263189i
$310$	0	0
$311$	28.8727	1.63722	0.818610	−	0.574350i	$-0.194745\pi$
$311$	28.8727	1.63722	0.818610	+	0.574350i	$0.194745\pi$
$312$	0	0
$313$	− 4.71947i	− 0.266760i	−0.991065	−	0.133380i	$-0.957417\pi$
$313$	− 4.71947i	− 0.266760i	0.991065	−	0.133380i	$-0.0425831\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.4437i	1.42906i	0.699606	+	0.714529i	$0.253359\pi$
$317$	25.4437i	1.42906i	−0.699606	+	0.714529i	$0.746641\pi$
$318$	0	0
$319$	− 38.5452i	− 2.15811i
$320$	0	0
$321$	− 6.06946i	− 0.338764i
$322$	0	0
$323$	−4.64938	−0.258699
$324$	0	0
$325$	3.47107	0.192540
$326$	0	0
$327$	14.5645	0.805418
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.7186	1.08383	0.541916	−	0.840433i	$-0.317699\pi$
$331$	19.7186	1.08383	0.541916	+	0.840433i	$0.317699\pi$
$332$	0	0
$333$	2.38082i	0.130468i
$334$	0	0
$335$	38.3781	2.09682
$336$	0	0
$337$	−24.7720	−1.34942	−0.674709	−	0.738084i	$-0.735731\pi$
$337$	−24.7720	−1.34942	−0.674709	+	0.738084i	$0.735731\pi$
$338$	0	0
$339$	3.82875i	0.207949i
$340$	0	0
$341$	−22.3234	−1.20888
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6.18950	−0.333232
$346$	0	0
$347$	14.9883	0.804612	0.402306	−	0.915505i	$-0.368209\pi$
$347$	14.9883	0.804612	0.402306	+	0.915505i	$0.368209\pi$
$348$	0	0
$349$	27.4546	1.46961	0.734806	−	0.678278i	$-0.237273\pi$
$349$	27.4546	1.46961	0.734806	+	0.678278i	$0.237273\pi$
$350$	0	0
$351$	1.04841i	0.0559599i
$352$	0	0
$353$	− 24.0043i	− 1.27762i	−0.769364	−	0.638811i	$-0.779427\pi$
$353$	− 24.0043i	− 1.27762i	0.769364	−	0.638811i	$-0.220573\pi$
$354$	0	0
$355$	− 3.12885i	− 0.166062i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.6543i	0.720645i	0.932828	+	0.360322i	$0.117333\pi$
$359$	13.6543i	0.720645i	−0.932828	+	0.360322i	$0.882667\pi$
$360$	0	0
$361$	18.5359	0.975575
$362$	0	0
$363$	− 22.9516i	− 1.20465i
$364$	0	0
$365$	16.2739i	0.851813i
$366$	0	0
$367$	34.6847	1.81053	0.905264	−	0.424849i	$-0.139673\pi$
$367$	34.6847	1.81053	0.905264	+	0.424849i	$0.139673\pi$
$368$	0	0
$369$	− 1.19919i	− 0.0624272i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6.08232i	0.314931i	0.987525	+	0.157465i	$0.0503322\pi$
$373$	6.08232i	0.314931i	−0.987525	+	0.157465i	$0.949668\pi$
$374$	0	0
$375$	− 4.86972i	− 0.251471i
$376$	0	0
$377$	− 6.93538i	− 0.357190i
$378$	0	0
$379$	−19.0628	−0.979189	−0.489594	−	0.871950i	$-0.662855\pi$
$379$	−19.0628	−0.979189	−0.489594	+	0.871950i	$0.662855\pi$
$380$	0	0
$381$	0.550415	0.0281986
$382$	0	0
$383$	−8.25003	−0.421557	−0.210778	−	0.977534i	$-0.567600\pi$
$383$	−8.25003	−0.421557	−0.210778	+	0.977534i	$0.567600\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.34319	0.0682784
$388$	0	0
$389$	− 19.6214i	− 0.994844i	−0.867509	−	0.497422i	$-0.834280\pi$
$389$	− 19.6214i	− 0.994844i	0.867509	−	0.497422i	$-0.165720\pi$
$390$	0	0
$391$	−14.6533	−0.741051
$392$	0	0
$393$	2.80383	0.141434
$394$	0	0
$395$	− 36.4937i	− 1.83620i
$396$	0	0
$397$	−31.3631	−1.57407	−0.787035	−	0.616908i	$-0.788385\pi$
$397$	−31.3631	−1.57407	−0.787035	+	0.616908i	$0.788385\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	34.9124	1.74344	0.871722	−	0.490002i	$-0.163004\pi$
$401$	34.9124	1.74344	0.871722	+	0.490002i	$0.163004\pi$
$402$	0	0
$403$	−4.01662	−0.200082
$404$	0	0
$405$	−2.88284	−0.143250
$406$	0	0
$407$	− 13.8725i	− 0.687636i
$408$	0	0
$409$	− 13.6465i	− 0.674776i	−0.941366	−	0.337388i	$-0.890457\pi$
$409$	− 13.6465i	− 0.674776i	0.941366	−	0.337388i	$-0.109543\pi$
$410$	0	0
$411$	− 12.7630i	− 0.629550i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 1.38965i	− 0.0682154i
$416$	0	0
$417$	6.11761	0.299581
$418$	0	0
$419$	− 4.22322i	− 0.206318i	−0.994665	−	0.103159i	$-0.967105\pi$
$419$	− 4.22322i	− 0.206318i	0.994665	−	0.103159i	$-0.0328950\pi$
$420$	0	0
$421$	− 11.6892i	− 0.569697i	−0.958573	−	0.284849i	$-0.908057\pi$
$421$	− 11.6892i	− 0.569697i	0.958573	−	0.284849i	$-0.0919433\pi$
$422$	0	0
$423$	11.0534	0.537435
$424$	0	0
$425$	22.5962i	1.09607i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 6.10887i	− 0.294939i
$430$	0	0
$431$	34.3020i	1.65227i	0.563471	+	0.826136i	$0.309465\pi$
$431$	34.3020i	1.65227i	−0.563471	+	0.826136i	$0.690535\pi$
$432$	0	0
$433$	− 14.1884i	− 0.681853i	−0.940090	−	0.340926i	$-0.889259\pi$
$433$	− 14.1884i	− 0.681853i	0.940090	−	0.340926i	$-0.110741\pi$
$434$	0	0
$435$	19.0704	0.914358
$436$	0	0
$437$	−1.46261	−0.0699659
$438$	0	0
$439$	−28.0504	−1.33877	−0.669386	−	0.742915i	$-0.733443\pi$
$439$	−28.0504	−1.33877	−0.669386	+	0.742915i	$0.733443\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11.0208	−0.523614	−0.261807	−	0.965120i	$-0.584318\pi$
$443$	−11.0208	−0.523614	−0.261807	+	0.965120i	$0.584318\pi$
$444$	0	0
$445$	− 35.7209i	− 1.69333i
$446$	0	0
$447$	2.02260	0.0956659
$448$	0	0
$449$	30.2270	1.42650	0.713250	−	0.700910i	$-0.247222\pi$
$449$	30.2270	1.42650	0.713250	+	0.700910i	$0.247222\pi$
$450$	0	0
$451$	6.98742i	0.329025i
$452$	0	0
$453$	14.0681	0.660976
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.4772	1.19177	0.595887	−	0.803068i	$-0.296801\pi$
$457$	25.4772	1.19177	0.595887	+	0.803068i	$0.296801\pi$
$458$	0	0
$459$	−6.82499	−0.318563
$460$	0	0
$461$	−27.9292	−1.30079	−0.650395	−	0.759596i	$-0.725397\pi$
$461$	−27.9292	−1.30079	−0.650395	+	0.759596i	$0.725397\pi$
$462$	0	0
$463$	− 19.4232i	− 0.902674i	−0.892354	−	0.451337i	$-0.850947\pi$
$463$	− 19.4232i	− 0.902674i	0.892354	−	0.451337i	$-0.149053\pi$
$464$	0	0
$465$	− 11.0446i	− 0.512182i
$466$	0	0
$467$	1.50989i	0.0698691i	0.999390	+	0.0349346i	$0.0111223\pi$
$467$	1.50989i	0.0698691i	−0.999390	+	0.0349346i	$0.988878\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.23477i	0.241206i
$472$	0	0
$473$	−7.82653	−0.359864
$474$	0	0
$475$	2.25541i	0.103485i
$476$	0	0
$477$	8.07845i	0.369887i
$478$	0	0
$479$	−14.2406	−0.650669	−0.325335	−	0.945599i	$-0.605477\pi$
$479$	−14.2406	−0.650669	−0.325335	+	0.945599i	$0.605477\pi$
$480$	0	0
$481$	− 2.49607i	− 0.113811i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 10.4801i	− 0.475876i
$486$	0	0
$487$	7.39097i	0.334917i	0.985879	+	0.167458i	$0.0535560\pi$
$487$	7.39097i	0.334917i	−0.985879	+	0.167458i	$0.946444\pi$
$488$	0	0
$489$	0.976425i	0.0441555i
$490$	0	0
$491$	−14.0578	−0.634420	−0.317210	−	0.948355i	$-0.602746\pi$
$491$	−14.0578	−0.634420	−0.317210	+	0.948355i	$0.602746\pi$
$492$	0	0
$493$	45.1484	2.03338
$494$	0	0
$495$	16.7978	0.755003
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−26.9919	−1.20832	−0.604162	−	0.796861i	$-0.706492\pi$
$499$	−26.9919	−1.20832	−0.604162	+	0.796861i	$0.706492\pi$
$500$	0	0
$501$	− 2.08267i	− 0.0930467i
$502$	0	0
$503$	3.15505	0.140677	0.0703384	−	0.997523i	$-0.477592\pi$
$503$	3.15505	0.140677	0.0703384	+	0.997523i	$0.477592\pi$
$504$	0	0
$505$	−7.22059	−0.321312
$506$	0	0
$507$	11.9008i	0.528535i
$508$	0	0
$509$	21.7630	0.964630	0.482315	−	0.875998i	$-0.339796\pi$
$509$	21.7630	0.964630	0.482315	+	0.875998i	$0.339796\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.681229	−0.0300770
$514$	0	0
$515$	−13.3373	−0.587712
$516$	0	0
$517$	−64.4060	−2.83257
$518$	0	0
$519$	− 21.3463i	− 0.936999i
$520$	0	0
$521$	40.3946i	1.76972i	0.465856	+	0.884860i	$0.345746\pi$
$521$	40.3946i	1.76972i	−0.465856	+	0.884860i	$0.654254\pi$
$522$	0	0
$523$	− 29.3834i	− 1.28485i	−0.766349	−	0.642424i	$-0.777929\pi$
$523$	− 29.3834i	− 1.28485i	0.766349	−	0.642424i	$-0.222071\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 26.1476i	− 1.13901i
$528$	0	0
$529$	18.3903	0.799580
$530$	0	0
$531$	− 7.87073i	− 0.341561i
$532$	0	0
$533$	1.25724i	0.0544570i
$534$	0	0
$535$	−17.4973	−0.756475
$536$	0	0
$537$	− 6.94974i	− 0.299903i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	30.6975i	1.31979i	0.751359	+	0.659893i	$0.229399\pi$
$541$	30.6975i	1.31979i	−0.751359	+	0.659893i	$0.770601\pi$
$542$	0	0
$543$	7.67619i	0.329417i
$544$	0	0
$545$	− 41.9872i	− 1.79853i
$546$	0	0
$547$	15.7691	0.674240	0.337120	−	0.941462i	$-0.390547\pi$
$547$	15.7691	0.674240	0.337120	+	0.941462i	$0.390547\pi$
$548$	0	0
$549$	3.26942	0.139536
$550$	0	0
$551$	4.50643	0.191980
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6.86352	0.291340
$556$	0	0
$557$	22.2196i	0.941474i	0.882274	+	0.470737i	$0.156012\pi$
$557$	22.2196i	0.941474i	−0.882274	+	0.470737i	$0.843988\pi$
$558$	0	0
$559$	−1.40822	−0.0595612
$560$	0	0
$561$	39.7679	1.67900
$562$	0	0
$563$	23.9957i	1.01130i	0.862740	+	0.505648i	$0.168747\pi$
$563$	23.9957i	1.01130i	−0.862740	+	0.505648i	$0.831253\pi$
$564$	0	0
$565$	11.0377	0.464359
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.26379	0.0529810	0.0264905	−	0.999649i	$-0.491567\pi$
$569$	1.26379	0.0529810	0.0264905	+	0.999649i	$0.491567\pi$
$570$	0	0
$571$	−20.7111	−0.866731	−0.433365	−	0.901218i	$-0.642674\pi$
$571$	−20.7111	−0.866731	−0.433365	+	0.901218i	$0.642674\pi$
$572$	0	0
$573$	17.8099	0.744019
$574$	0	0
$575$	7.10831i	0.296437i
$576$	0	0
$577$	31.4542i	1.30946i	0.755864	+	0.654728i	$0.227217\pi$
$577$	31.4542i	1.30946i	−0.755864	+	0.654728i	$0.772783\pi$
$578$	0	0
$579$	13.7959i	0.573340i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 47.0715i	− 1.94950i
$584$	0	0
$585$	3.02240	0.124961
$586$	0	0
$587$	− 5.93391i	− 0.244919i	−0.992474	−	0.122459i	$-0.960922\pi$
$587$	− 5.93391i	− 0.244919i	0.992474	−	0.122459i	$-0.0390781\pi$
$588$	0	0
$589$	− 2.60989i	− 0.107539i
$590$	0	0
$591$	−1.13180	−0.0465560
$592$	0	0
$593$	− 15.7040i	− 0.644885i	−0.946589	−	0.322443i	$-0.895496\pi$
$593$	− 15.7040i	− 0.644885i	0.946589	−	0.322443i	$-0.104504\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.53186i	0.349186i
$598$	0	0
$599$	17.3726i	0.709827i	0.934899	+	0.354914i	$0.115490\pi$
$599$	17.3726i	0.709827i	−0.934899	+	0.354914i	$0.884510\pi$
$600$	0	0
$601$	43.6846i	1.78193i	0.454069	+	0.890966i	$0.349972\pi$
$601$	43.6846i	1.78193i	−0.454069	+	0.890966i	$0.650028\pi$
$602$	0	0
$603$	13.3126	0.542130
$604$	0	0
$605$	−66.1659	−2.69003
$606$	0	0
$607$	19.2009	0.779341	0.389671	−	0.920954i	$-0.372589\pi$
$607$	19.2009	0.779341	0.389671	+	0.920954i	$0.372589\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.5885	−0.468820
$612$	0	0
$613$	− 17.1672i	− 0.693375i	−0.937981	−	0.346688i	$-0.887306\pi$
$613$	− 17.1672i	− 0.693375i	0.937981	−	0.346688i	$-0.112694\pi$
$614$	0	0
$615$	−3.45707	−0.139402
$616$	0	0
$617$	−16.8150	−0.676946	−0.338473	−	0.940976i	$-0.609910\pi$
$617$	−16.8150	−0.676946	−0.338473	+	0.940976i	$0.609910\pi$
$618$	0	0
$619$	− 34.8225i	− 1.39964i	−0.714322	−	0.699818i	$-0.753265\pi$
$619$	− 34.8225i	− 1.39964i	0.714322	−	0.699818i	$-0.246735\pi$
$620$	0	0
$621$	−2.14701	−0.0861566
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.5926	−1.22370
$626$	0	0
$627$	3.96939	0.158522
$628$	0	0
$629$	16.2491	0.647892
$630$	0	0
$631$	11.8869i	0.473210i	0.971606	+	0.236605i	$0.0760346\pi$
$631$	11.8869i	0.473210i	−0.971606	+	0.236605i	$0.923965\pi$
$632$	0	0
$633$	21.2436i	0.844359i
$634$	0	0
$635$	− 1.58676i	− 0.0629687i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 1.08533i	− 0.0429351i
$640$	0	0
$641$	−36.8895	−1.45705	−0.728523	−	0.685021i	$-0.759793\pi$
$641$	−36.8895	−1.45705	−0.728523	+	0.685021i	$0.759793\pi$
$642$	0	0
$643$	10.0475i	0.396235i	0.980178	+	0.198117i	$0.0634827\pi$
$643$	10.0475i	0.396235i	−0.980178	+	0.198117i	$0.936517\pi$
$644$	0	0
$645$	− 3.87222i	− 0.152469i
$646$	0	0
$647$	−39.1647	−1.53972	−0.769862	−	0.638210i	$-0.779675\pi$
$647$	−39.1647	−1.53972	−0.769862	+	0.638210i	$0.779675\pi$
$648$	0	0
$649$	45.8612i	1.80021i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 41.5024i	− 1.62412i	−0.583577	−	0.812058i	$-0.698347\pi$
$653$	− 41.5024i	− 1.62412i	0.583577	−	0.812058i	$-0.301653\pi$
$654$	0	0
$655$	− 8.08300i	− 0.315829i
$656$	0	0
$657$	5.64507i	0.220235i
$658$	0	0
$659$	35.4347	1.38034	0.690170	−	0.723647i	$-0.257536\pi$
$659$	35.4347	1.38034	0.690170	+	0.723647i	$0.257536\pi$
$660$	0	0
$661$	23.6331	0.919219	0.459610	−	0.888121i	$-0.347989\pi$
$661$	23.6331	0.919219	0.459610	+	0.888121i	$0.347989\pi$
$662$	0	0
$663$	7.15538	0.277892
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.2028	0.549935
$668$	0	0
$669$	3.12207i	0.120706i
$670$	0	0
$671$	−19.0503	−0.735428
$672$	0	0
$673$	8.69720	0.335253	0.167626	−	0.985851i	$-0.446390\pi$
$673$	8.69720	0.335253	0.167626	+	0.985851i	$0.446390\pi$
$674$	0	0
$675$	3.31079i	0.127433i
$676$	0	0
$677$	−10.9189	−0.419649	−0.209825	−	0.977739i	$-0.567289\pi$
$677$	−10.9189	−0.419649	−0.209825	+	0.977739i	$0.567289\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.22675	0.200289
$682$	0	0
$683$	−36.5517	−1.39861	−0.699305	−	0.714823i	$-0.746507\pi$
$683$	−36.5517	−1.39861	−0.699305	+	0.714823i	$0.746507\pi$
$684$	0	0
$685$	−36.7936	−1.40581
$686$	0	0
$687$	28.4258i	1.08451i
$688$	0	0
$689$	− 8.46952i	− 0.322663i
$690$	0	0
$691$	5.30956i	0.201985i	0.994887	+	0.100993i	$0.0322018\pi$
$691$	5.30956i	0.201985i	−0.994887	+	0.100993i	$0.967798\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 17.6361i	− 0.668976i
$696$	0	0
$697$	−8.18444	−0.310008
$698$	0	0
$699$	− 6.11941i	− 0.231457i
$700$	0	0
$701$	− 6.16681i	− 0.232917i	−0.993196	−	0.116459i	$-0.962846\pi$
$701$	− 6.16681i	− 0.232917i	0.993196	−	0.116459i	$-0.0371542\pi$
$702$	0	0
$703$	1.62188	0.0611704
$704$	0	0
$705$	− 31.8652i	− 1.20011i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.99834i	0.0750494i	0.999296	+	0.0375247i	$0.0119473\pi$
$709$	1.99834i	0.0750494i	−0.999296	+	0.0375247i	$0.988053\pi$
$710$	0	0
$711$	− 12.6589i	− 0.474747i
$712$	0	0
$713$	− 8.22554i	− 0.308049i
$714$	0	0
$715$	−17.6109	−0.658611
$716$	0	0
$717$	8.83528	0.329960
$718$	0	0
$719$	13.4489	0.501560	0.250780	−	0.968044i	$-0.419313\pi$
$719$	13.4489	0.501560	0.250780	+	0.968044i	$0.419313\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−5.95545	−0.221485
$724$	0	0
$725$	− 21.9014i	− 0.813398i
$726$	0	0
$727$	19.6532	0.728897	0.364448	−	0.931224i	$-0.381258\pi$
$727$	19.6532	0.728897	0.364448	+	0.931224i	$0.381258\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 9.16729i	− 0.339065i
$732$	0	0
$733$	25.2279	0.931814	0.465907	−	0.884834i	$-0.345728\pi$
$733$	25.2279	0.931814	0.465907	+	0.884834i	$0.345728\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−77.5698	−2.85732
$738$	0	0
$739$	−6.70955	−0.246815	−0.123407	−	0.992356i	$-0.539382\pi$
$739$	−6.70955	−0.246815	−0.123407	+	0.992356i	$0.539382\pi$
$740$	0	0
$741$	0.714206	0.0262370
$742$	0	0
$743$	26.0661i	0.956273i	0.878286	+	0.478137i	$0.158688\pi$
$743$	26.0661i	0.956273i	−0.878286	+	0.478137i	$0.841312\pi$
$744$	0	0
$745$	− 5.83085i	− 0.213626i
$746$	0	0
$747$	− 0.482042i	− 0.0176370i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 5.71302i	− 0.208471i	−0.994553	−	0.104236i	$-0.966760\pi$
$751$	− 5.71302i	− 0.208471i	0.994553	−	0.104236i	$-0.0332396\pi$
$752$	0	0
$753$	−2.13955	−0.0779694
$754$	0	0
$755$	− 40.5561i	− 1.47599i
$756$	0	0
$757$	− 4.75364i	− 0.172774i	−0.996262	−	0.0863869i	$-0.972468\pi$
$757$	− 4.75364i	− 0.172774i	0.996262	−	0.0863869i	$-0.0275321\pi$
$758$	0	0
$759$	12.5102	0.454092
$760$	0	0
$761$	− 27.3284i	− 0.990654i	−0.868707	−	0.495327i	$-0.835048\pi$
$761$	− 27.3284i	− 0.990654i	0.868707	−	0.495327i	$-0.164952\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	19.6754i	0.711366i
$766$	0	0
$767$	8.25174i	0.297953i
$768$	0	0
$769$	22.7151i	0.819127i	0.912282	+	0.409564i	$0.134319\pi$
$769$	22.7151i	0.819127i	−0.912282	+	0.409564i	$0.865681\pi$
$770$	0	0
$771$	−19.2869	−0.694602
$772$	0	0
$773$	18.1187	0.651685	0.325843	−	0.945424i	$-0.394352\pi$
$773$	18.1187	0.651685	0.325843	+	0.945424i	$0.394352\pi$
$774$	0	0
$775$	−12.6842	−0.455629
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.816920	−0.0292692
$780$	0	0
$781$	6.32402i	0.226291i
$782$	0	0
$783$	6.61515	0.236406
$784$	0	0
$785$	15.0910	0.538622
$786$	0	0
$787$	4.91161i	0.175080i	0.996161	+	0.0875400i	$0.0279006\pi$
$787$	4.91161i	0.175080i	−0.996161	+	0.0875400i	$0.972099\pi$
$788$	0	0
$789$	24.3341	0.866319
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−3.42769	−0.121721
$794$	0	0
$795$	23.2889	0.825973
$796$	0	0
$797$	23.8384	0.844398	0.422199	−	0.906503i	$-0.361258\pi$
$797$	23.8384	0.844398	0.422199	+	0.906503i	$0.361258\pi$
$798$	0	0
$799$	− 75.4394i	− 2.66885i
$800$	0	0
$801$	− 12.3909i	− 0.437809i
$802$	0	0
$803$	− 32.8927i	− 1.16076i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.9009i	0.735748i
$808$	0	0
$809$	−1.19583	−0.0420431	−0.0210216	−	0.999779i	$-0.506692\pi$
$809$	−1.19583	−0.0420431	−0.0210216	+	0.999779i	$0.506692\pi$
$810$	0	0
$811$	− 6.44559i	− 0.226335i	−0.993576	−	0.113168i	$-0.963900\pi$
$811$	− 6.44559i	− 0.226335i	0.993576	−	0.113168i	$-0.0360997\pi$
$812$	0	0
$813$	19.9939i	0.701218i
$814$	0	0
$815$	2.81488	0.0986010
$816$	0	0
$817$	− 0.915023i	− 0.0320126i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.16738i	− 0.0756420i	−0.999285	−	0.0378210i	$-0.987958\pi$
$821$	− 2.16738i	− 0.0756420i	0.999285	−	0.0378210i	$-0.0120417\pi$
$822$	0	0
$823$	5.99900i	0.209112i	0.994519	+	0.104556i	$0.0333422\pi$
$823$	5.99900i	0.209112i	−0.994519	+	0.104556i	$0.966658\pi$
$824$	0	0
$825$	− 19.2913i	− 0.671638i
$826$	0	0
$827$	−21.0143	−0.730738	−0.365369	−	0.930863i	$-0.619057\pi$
$827$	−21.0143	−0.730738	−0.365369	+	0.930863i	$0.619057\pi$
$828$	0	0
$829$	−13.1978	−0.458379	−0.229190	−	0.973382i	$-0.573608\pi$
$829$	−13.1978	−0.458379	−0.229190	+	0.973382i	$0.573608\pi$
$830$	0	0
$831$	15.2920	0.530474
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−6.00400	−0.207777
$836$	0	0
$837$	− 3.83116i	− 0.132424i
$838$	0	0
$839$	33.8661	1.16919	0.584594	−	0.811326i	$-0.301254\pi$
$839$	33.8661	1.16919	0.584594	+	0.811326i	$0.301254\pi$
$840$	0	0
$841$	−14.7602	−0.508973
$842$	0	0
$843$	− 5.59802i	− 0.192806i
$844$	0	0
$845$	34.3083	1.18024
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.2342	0.385557
$850$	0	0
$851$	5.11164	0.175225
$852$	0	0
$853$	50.0832	1.71482	0.857408	−	0.514638i	$-0.172074\pi$
$853$	50.0832	1.71482	0.857408	+	0.514638i	$0.172074\pi$
$854$	0	0
$855$	1.96388i	0.0671632i
$856$	0	0
$857$	− 49.2628i	− 1.68279i	−0.540424	−	0.841393i	$-0.681736\pi$
$857$	− 49.2628i	− 1.68279i	0.540424	−	0.841393i	$-0.318264\pi$
$858$	0	0
$859$	3.91059i	0.133428i	0.997772	+	0.0667139i	$0.0212515\pi$
$859$	3.91059i	0.133428i	−0.997772	+	0.0667139i	$0.978749\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 43.2124i	− 1.47097i	−0.677542	−	0.735484i	$-0.736955\pi$
$863$	− 43.2124i	− 1.47097i	0.677542	−	0.735484i	$-0.263045\pi$
$864$	0	0
$865$	−61.5381	−2.09236
$866$	0	0
$867$	29.5806i	1.00461i
$868$	0	0
$869$	73.7611i	2.50217i
$870$	0	0
$871$	−13.9570	−0.472916
$872$	0	0
$873$	− 3.63532i	− 0.123037i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 21.1390i	− 0.713815i	−0.934140	−	0.356907i	$-0.883831\pi$
$877$	− 21.1390i	− 0.713815i	0.934140	−	0.356907i	$-0.116169\pi$
$878$	0	0
$879$	17.7212i	0.597721i
$880$	0	0
$881$	49.1167i	1.65478i	0.561625	+	0.827392i	$0.310177\pi$
$881$	49.1167i	1.65478i	−0.561625	+	0.827392i	$0.689823\pi$
$882$	0	0
$883$	−37.8457	−1.27361	−0.636804	−	0.771025i	$-0.719744\pi$
$883$	−37.8457	−1.27361	−0.636804	+	0.771025i	$0.719744\pi$
$884$	0	0
$885$	−22.6901	−0.762719
$886$	0	0
$887$	17.9408	0.602393	0.301196	−	0.953562i	$-0.402614\pi$
$887$	17.9408	0.602393	0.301196	+	0.953562i	$0.402614\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.82680	0.195205
$892$	0	0
$893$	− 7.52989i	− 0.251978i
$894$	0	0
$895$	−20.0350	−0.669697
$896$	0	0
$897$	2.25095	0.0751569
$898$	0	0
$899$	25.3437i	0.845259i
$900$	0	0
$901$	55.1354	1.83683
$902$	0	0
$903$	0	0
$904$	0	0
$905$	22.1293	0.735601
$906$	0	0
$907$	−22.1774	−0.736388	−0.368194	−	0.929749i	$-0.620024\pi$
$907$	−22.1774	−0.736388	−0.368194	+	0.929749i	$0.620024\pi$
$908$	0	0
$909$	−2.50468	−0.0830748
$910$	0	0
$911$	35.2495i	1.16787i	0.811801	+	0.583934i	$0.198487\pi$
$911$	35.2495i	1.16787i	−0.811801	+	0.583934i	$0.801513\pi$
$912$	0	0
$913$	2.80877i	0.0929566i
$914$	0	0
$915$	− 9.42524i	− 0.311589i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	17.5326i	0.578348i	0.957277	+	0.289174i	$0.0933806\pi$
$919$	17.5326i	0.578348i	−0.957277	+	0.289174i	$0.906619\pi$
$920$	0	0
$921$	20.3724	0.671293
$922$	0	0
$923$	1.13787i	0.0374535i
$924$	0	0
$925$	− 7.88239i	− 0.259171i
$926$	0	0
$927$	−4.62644	−0.151952
$928$	0	0
$929$	− 51.8583i	− 1.70142i	−0.525638	−	0.850708i	$-0.676173\pi$
$929$	− 51.8583i	− 1.70142i	0.525638	−	0.850708i	$-0.323827\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 28.8727i	− 0.945249i
$934$	0	0
$935$	− 114.645i	− 3.74928i
$936$	0	0
$937$	− 36.9665i	− 1.20764i	−0.797120	−	0.603821i	$-0.793644\pi$
$937$	− 36.9665i	− 1.20764i	0.797120	−	0.603821i	$-0.206356\pi$
$938$	0	0
$939$	−4.71947	−0.154014
$940$	0	0
$941$	52.5502	1.71309	0.856544	−	0.516075i	$-0.172607\pi$
$941$	52.5502	1.71309	0.856544	+	0.516075i	$0.172607\pi$
$942$	0	0
$943$	−2.57467	−0.0838427
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.9341	−0.875242	−0.437621	−	0.899160i	$-0.644179\pi$
$947$	−26.9341	−0.875242	−0.437621	+	0.899160i	$0.644179\pi$
$948$	0	0
$949$	− 5.91834i	− 0.192118i
$950$	0	0
$951$	25.4437	0.825067
$952$	0	0
$953$	−31.5488	−1.02196	−0.510982	−	0.859591i	$-0.670718\pi$
$953$	−31.5488	−1.02196	−0.510982	+	0.859591i	$0.670718\pi$
$954$	0	0
$955$	− 51.3432i	− 1.66143i
$956$	0	0
$957$	−38.5452	−1.24599
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−16.3222	−0.526524
$962$	0	0
$963$	−6.06946	−0.195586
$964$	0	0
$965$	39.7716	1.28029
$966$	0	0
$967$	60.7635i	1.95402i	0.213187	+	0.977011i	$0.431616\pi$
$967$	60.7635i	1.95402i	−0.213187	+	0.977011i	$0.568384\pi$
$968$	0	0
$969$	4.64938i	0.149360i
$970$	0	0
$971$	16.0945i	0.516496i	0.966079	+	0.258248i	$0.0831452\pi$
$971$	16.0945i	0.516496i	−0.966079	+	0.258248i	$0.916855\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 3.47107i	− 0.111163i
$976$	0	0
$977$	−32.7166	−1.04670	−0.523349	−	0.852119i	$-0.675318\pi$
$977$	−32.7166	−1.04670	−0.523349	+	0.852119i	$0.675318\pi$
$978$	0	0
$979$	72.1991i	2.30749i
$980$	0	0
$981$	− 14.5645i	− 0.465008i
$982$	0	0
$983$	37.1569	1.18512	0.592561	−	0.805525i	$-0.298117\pi$
$983$	37.1569	1.18512	0.592561	+	0.805525i	$0.298117\pi$
$984$	0	0
$985$	3.26280i	0.103962i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2.88385i	− 0.0917012i
$990$	0	0
$991$	− 44.2149i	− 1.40453i	−0.711915	−	0.702266i	$-0.752172\pi$
$991$	− 44.2149i	− 1.40453i	0.711915	−	0.702266i	$-0.247828\pi$
$992$	0	0
$993$	− 19.7186i	− 0.625750i
$994$	0	0
$995$	24.5960	0.779747
$996$	0	0
$997$	−26.6863	−0.845165	−0.422582	−	0.906325i	$-0.638876\pi$
$997$	−26.6863	−0.845165	−0.422582	+	0.906325i	$0.638876\pi$
$998$	0	0
$999$	2.38082	0.0753257

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.p.a.3919.25		32
4.3	odd	2		1176.2.p.a.979.8		32
7.2	even	3		672.2.bb.a.367.15		32
7.3	odd	6		672.2.bb.a.271.10		32
7.6	odd	2	inner	4704.2.p.a.3919.30		32
8.3	odd	2	inner	4704.2.p.a.3919.29		32
8.5	even	2		1176.2.p.a.979.5		32
21.2	odd	6		2016.2.bs.c.1711.3		32
21.17	even	6		2016.2.bs.c.271.14		32
28.3	even	6		168.2.t.a.19.9	✓	32
28.23	odd	6		168.2.t.a.115.14	yes	32
28.27	even	2		1176.2.p.a.979.7		32
56.3	even	6		672.2.bb.a.271.15		32
56.13	odd	2		1176.2.p.a.979.6		32
56.27	even	2	inner	4704.2.p.a.3919.26		32
56.37	even	6		168.2.t.a.115.9	yes	32
56.45	odd	6		168.2.t.a.19.14	yes	32
56.51	odd	6		672.2.bb.a.367.10		32
84.23	even	6		504.2.bk.c.451.3		32
84.59	odd	6		504.2.bk.c.19.8		32
168.59	odd	6		2016.2.bs.c.271.3		32
168.101	even	6		504.2.bk.c.19.3		32
168.107	even	6		2016.2.bs.c.1711.14		32
168.149	odd	6		504.2.bk.c.451.8		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.t.a.19.9	✓	32	28.3	even	6
168.2.t.a.19.14	yes	32	56.45	odd	6
168.2.t.a.115.9	yes	32	56.37	even	6
168.2.t.a.115.14	yes	32	28.23	odd	6
504.2.bk.c.19.3		32	168.101	even	6
504.2.bk.c.19.8		32	84.59	odd	6
504.2.bk.c.451.3		32	84.23	even	6
504.2.bk.c.451.8		32	168.149	odd	6
672.2.bb.a.271.10		32	7.3	odd	6
672.2.bb.a.271.15		32	56.3	even	6
672.2.bb.a.367.10		32	56.51	odd	6
672.2.bb.a.367.15		32	7.2	even	3
1176.2.p.a.979.5		32	8.5	even	2
1176.2.p.a.979.6		32	56.13	odd	2
1176.2.p.a.979.7		32	28.27	even	2
1176.2.p.a.979.8		32	4.3	odd	2
2016.2.bs.c.271.3		32	168.59	odd	6
2016.2.bs.c.271.14		32	21.17	even	6
2016.2.bs.c.1711.3		32	21.2	odd	6
2016.2.bs.c.1711.14		32	168.107	even	6
4704.2.p.a.3919.25		32	1.1	even	1	trivial
4704.2.p.a.3919.26		32	56.27	even	2	inner
4704.2.p.a.3919.29		32	8.3	odd	2	inner
4704.2.p.a.3919.30		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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.88284 q^{5} -1.00000 q^{9} +5.82680 q^{11} +1.04841 q^{13} +2.88284i q^{15} +6.82499i q^{17} +0.681229i q^{19} +2.14701i q^{23} +3.31079 q^{25} +1.00000i q^{27} -6.61515i q^{29} -3.83116 q^{31} -5.82680i q^{33} -2.38082i q^{37} -1.04841i q^{39} +1.19919i q^{41} -1.34319 q^{43} +2.88284 q^{45} -11.0534 q^{47} +6.82499 q^{51} -8.07845i q^{53} -16.7978 q^{55} +0.681229 q^{57} +7.87073i q^{59} -3.26942 q^{61} -3.02240 q^{65} -13.3126 q^{67} +2.14701 q^{69} +1.08533i q^{71} -5.64507i q^{73} -3.31079i q^{75} +12.6589i q^{79} +1.00000 q^{81} +0.482042i q^{83} -19.6754i q^{85} -6.61515 q^{87} +12.3909i q^{89} +3.83116i q^{93} -1.96388i q^{95} +3.63532i q^{97} -5.82680 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{9} + 16 q^{11} + 32 q^{25} + 16 q^{43} + 16 q^{57} - 64 q^{67} + 32 q^{81} - 16 q^{99}+O(q^{100}) \) 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

Properties

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