LMFDB - Embedded newform 4704.2.p.a.3919.5

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			3919.5
Character	$\chi$	$=$	4704.3919
Dual form			4704.2.p.a.3919.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -4.17553 q^{5} -1.00000 q^{9} -1.71097 q^{11} -1.54062 q^{13} +4.17553i q^{15} +2.33358i q^{17} +7.04013i q^{19} -0.468884i q^{23} +12.4350 q^{25} +1.00000i q^{27} +3.33885i q^{29} +3.16253 q^{31} +1.71097i q^{33} -8.94486i q^{37} +1.54062i q^{39} +5.31411i q^{41} +3.42772 q^{43} +4.17553 q^{45} -5.90095 q^{47} +2.33358 q^{51} -1.56389i q^{53} +7.14421 q^{55} +7.04013 q^{57} +6.08229i q^{59} -9.11917 q^{61} +6.43291 q^{65} +7.47317 q^{67} -0.468884 q^{69} +3.49263i q^{71} -14.5274i q^{73} -12.4350i q^{75} -1.68711i q^{79} +1.00000 q^{81} +2.72601i q^{83} -9.74391i q^{85} +3.33885 q^{87} -2.12267i q^{89} -3.16253i q^{93} -29.3963i q^{95} +1.95202i q^{97} +1.71097 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.17553	−1.86735	−0.933677	−	0.358117i	$-0.883419\pi$
$5$	−4.17553	−1.86735	−0.933677	+	0.358117i	$0.883419\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.71097	−0.515877	−0.257939	−	0.966161i	$-0.583043\pi$
$11$	−1.71097	−0.515877	−0.257939	+	0.966161i	$0.583043\pi$
$12$	0	0
$13$	−1.54062	−0.427292	−0.213646	−	0.976911i	$-0.568534\pi$
$13$	−1.54062	−0.427292	−0.213646	+	0.976911i	$0.568534\pi$
$14$	0	0
$15$	4.17553i	1.07812i
$16$	0	0
$17$	2.33358i	0.565975i	0.959124	+	0.282988i	$0.0913256\pi$
$17$	2.33358i	0.565975i	−0.959124	+	0.282988i	$0.908674\pi$
$18$	0	0
$19$	7.04013i	1.61512i	0.589788	+	0.807558i	$0.299211\pi$
$19$	7.04013i	1.61512i	−0.589788	+	0.807558i	$0.700789\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 0.468884i	− 0.0977691i	−0.998804	−	0.0488846i	$-0.984433\pi$
$23$	− 0.468884i	− 0.0977691i	0.998804	−	0.0488846i	$-0.0155666\pi$
$24$	0	0
$25$	12.4350	2.48701
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	3.33885i	0.620009i	0.950735	+	0.310005i	$0.100331\pi$
$29$	3.33885i	0.620009i	−0.950735	+	0.310005i	$0.899669\pi$
$30$	0	0
$31$	3.16253	0.568007	0.284003	−	0.958823i	$-0.408337\pi$
$31$	3.16253	0.568007	0.284003	+	0.958823i	$0.408337\pi$
$32$	0	0
$33$	1.71097i	0.297842i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.94486i	− 1.47053i	−0.677782	−	0.735263i	$-0.737059\pi$
$37$	− 8.94486i	− 1.47053i	0.677782	−	0.735263i	$-0.262941\pi$
$38$	0	0
$39$	1.54062i	0.246697i
$40$	0	0
$41$	5.31411i	0.829925i	0.909838	+	0.414963i	$0.136205\pi$
$41$	5.31411i	0.829925i	−0.909838	+	0.414963i	$0.863795\pi$
$42$	0	0
$43$	3.42772	0.522722	0.261361	−	0.965241i	$-0.415829\pi$
$43$	3.42772	0.522722	0.261361	+	0.965241i	$0.415829\pi$
$44$	0	0
$45$	4.17553	0.622451
$46$	0	0
$47$	−5.90095	−0.860742	−0.430371	−	0.902652i	$-0.641617\pi$
$47$	−5.90095	−0.860742	−0.430371	+	0.902652i	$0.641617\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.33358	0.326766
$52$	0	0
$53$	− 1.56389i	− 0.214817i	−0.994215	−	0.107409i	$-0.965745\pi$
$53$	− 1.56389i	− 0.214817i	0.994215	−	0.107409i	$-0.0342553\pi$
$54$	0	0
$55$	7.14421	0.963325
$56$	0	0
$57$	7.04013	0.932488
$58$	0	0
$59$	6.08229i	0.791846i	0.918284	+	0.395923i	$0.129575\pi$
$59$	6.08229i	0.791846i	−0.918284	+	0.395923i	$0.870425\pi$
$60$	0	0
$61$	−9.11917	−1.16759	−0.583795	−	0.811901i	$-0.698433\pi$
$61$	−9.11917	−1.16759	−0.583795	+	0.811901i	$0.698433\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.43291	0.797905
$66$	0	0
$67$	7.47317	0.912993	0.456496	−	0.889725i	$-0.349104\pi$
$67$	7.47317	0.912993	0.456496	+	0.889725i	$0.349104\pi$
$68$	0	0
$69$	−0.468884	−0.0564470
$70$	0	0
$71$	3.49263i	0.414499i	0.978288	+	0.207249i	$0.0664512\pi$
$71$	3.49263i	0.414499i	−0.978288	+	0.207249i	$0.933549\pi$
$72$	0	0
$73$	− 14.5274i	− 1.70031i	−0.526534	−	0.850154i	$-0.676509\pi$
$73$	− 14.5274i	− 1.70031i	0.526534	−	0.850154i	$-0.323491\pi$
$74$	0	0
$75$	− 12.4350i	− 1.43587i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 1.68711i	− 0.189815i	−0.995486	−	0.0949075i	$-0.969744\pi$
$79$	− 1.68711i	− 0.189815i	0.995486	−	0.0949075i	$-0.0302555\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.72601i	0.299219i	0.988745	+	0.149609i	$0.0478016\pi$
$83$	2.72601i	0.299219i	−0.988745	+	0.149609i	$0.952198\pi$
$84$	0	0
$85$	− 9.74391i	− 1.05688i
$86$	0	0
$87$	3.33885	0.357962
$88$	0	0
$89$	− 2.12267i	− 0.225003i	−0.993652	−	0.112501i	$-0.964114\pi$
$89$	− 2.12267i	− 0.225003i	0.993652	−	0.112501i	$-0.0358863\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 3.16253i	− 0.327939i
$94$	0	0
$95$	− 29.3963i	− 3.01599i
$96$	0	0
$97$	1.95202i	0.198198i	0.995078	+	0.0990990i	$0.0315960\pi$
$97$	1.95202i	0.198198i	−0.995078	+	0.0990990i	$0.968404\pi$
$98$	0	0
$99$	1.71097	0.171959
$100$	0	0
$101$	−13.7809	−1.37125	−0.685625	−	0.727955i	$-0.740471\pi$
$101$	−13.7809	−1.37125	−0.685625	+	0.727955i	$0.740471\pi$
$102$	0	0
$103$	−7.68258	−0.756987	−0.378494	−	0.925604i	$-0.623558\pi$
$103$	−7.68258	−0.756987	−0.378494	+	0.925604i	$0.623558\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.4083	1.39290	0.696450	−	0.717605i	$-0.254762\pi$
$107$	14.4083	1.39290	0.696450	+	0.717605i	$0.254762\pi$
$108$	0	0
$109$	− 3.37612i	− 0.323373i	−0.986842	−	0.161687i	$-0.948307\pi$
$109$	− 3.37612i	− 0.323373i	0.986842	−	0.161687i	$-0.0516934\pi$
$110$	0	0
$111$	−8.94486	−0.849009
$112$	0	0
$113$	−10.7090	−1.00742	−0.503710	−	0.863873i	$-0.668032\pi$
$113$	−10.7090	−1.00742	−0.503710	+	0.863873i	$0.668032\pi$
$114$	0	0
$115$	1.95784i	0.182569i
$116$	0	0
$117$	1.54062	0.142431
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.07258	−0.733871
$122$	0	0
$123$	5.31411	0.479157
$124$	0	0
$125$	−31.0452	−2.77677
$126$	0	0
$127$	9.49738i	0.842757i	0.906885	+	0.421378i	$0.138454\pi$
$127$	9.49738i	0.842757i	−0.906885	+	0.421378i	$0.861546\pi$
$128$	0	0
$129$	− 3.42772i	− 0.301794i
$130$	0	0
$131$	− 13.6675i	− 1.19414i	−0.802191	−	0.597068i	$-0.796332\pi$
$131$	− 13.6675i	− 1.19414i	0.802191	−	0.597068i	$-0.203668\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 4.17553i	− 0.359372i
$136$	0	0
$137$	−11.9892	−1.02431	−0.512153	−	0.858894i	$-0.671152\pi$
$137$	−11.9892	−1.02431	−0.512153	+	0.858894i	$0.671152\pi$
$138$	0	0
$139$	− 6.64909i	− 0.563968i	−0.959419	−	0.281984i	$-0.909007\pi$
$139$	− 6.64909i	− 0.563968i	0.959419	−	0.281984i	$-0.0909925\pi$
$140$	0	0
$141$	5.90095i	0.496950i
$142$	0	0
$143$	2.63596	0.220430
$144$	0	0
$145$	− 13.9415i	− 1.15778i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8.11897i	− 0.665132i	−0.943080	−	0.332566i	$-0.892086\pi$
$149$	− 8.11897i	− 0.665132i	0.943080	−	0.332566i	$-0.107914\pi$
$150$	0	0
$151$	1.23604i	0.100587i	0.998734	+	0.0502937i	$0.0160157\pi$
$151$	1.23604i	0.100587i	−0.998734	+	0.0502937i	$0.983984\pi$
$152$	0	0
$153$	− 2.33358i	− 0.188658i
$154$	0	0
$155$	−13.2052	−1.06067
$156$	0	0
$157$	9.60572	0.766620	0.383310	−	0.923620i	$-0.374784\pi$
$157$	9.60572	0.766620	0.383310	+	0.923620i	$0.374784\pi$
$158$	0	0
$159$	−1.56389	−0.124025
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.4092	1.52025	0.760123	−	0.649779i	$-0.225138\pi$
$163$	19.4092	1.52025	0.760123	+	0.649779i	$0.225138\pi$
$164$	0	0
$165$	− 7.14421i	− 0.556176i
$166$	0	0
$167$	17.7482	1.37340	0.686699	−	0.726942i	$-0.259059\pi$
$167$	17.7482	1.37340	0.686699	+	0.726942i	$0.259059\pi$
$168$	0	0
$169$	−10.6265	−0.817422
$170$	0	0
$171$	− 7.04013i	− 0.538372i
$172$	0	0
$173$	−7.14150	−0.542958	−0.271479	−	0.962444i	$-0.587513\pi$
$173$	−7.14150	−0.542958	−0.271479	+	0.962444i	$0.587513\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.08229	0.457173
$178$	0	0
$179$	−23.9162	−1.78758	−0.893791	−	0.448483i	$-0.851964\pi$
$179$	−23.9162	−1.78758	−0.893791	+	0.448483i	$0.851964\pi$
$180$	0	0
$181$	20.5572	1.52800	0.764002	−	0.645214i	$-0.223232\pi$
$181$	20.5572	1.52800	0.764002	+	0.645214i	$0.223232\pi$
$182$	0	0
$183$	9.11917i	0.674109i
$184$	0	0
$185$	37.3495i	2.74599i
$186$	0	0
$187$	− 3.99268i	− 0.291974i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 2.01521i	− 0.145816i	−0.997339	−	0.0729079i	$-0.976772\pi$
$191$	− 2.01521i	− 0.145816i	0.997339	−	0.0729079i	$-0.0232279\pi$
$192$	0	0
$193$	−3.57070	−0.257025	−0.128512	−	0.991708i	$-0.541020\pi$
$193$	−3.57070	−0.257025	−0.128512	+	0.991708i	$0.541020\pi$
$194$	0	0
$195$	− 6.43291i	− 0.460671i
$196$	0	0
$197$	− 17.5393i	− 1.24962i	−0.780775	−	0.624812i	$-0.785176\pi$
$197$	− 17.5393i	− 1.24962i	0.780775	−	0.624812i	$-0.214824\pi$
$198$	0	0
$199$	−17.7067	−1.25520	−0.627598	−	0.778537i	$-0.715962\pi$
$199$	−17.7067	−1.25520	−0.627598	+	0.778537i	$0.715962\pi$
$200$	0	0
$201$	− 7.47317i	− 0.527117i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 22.1892i	− 1.54976i
$206$	0	0
$207$	0.468884i	0.0325897i
$208$	0	0
$209$	− 12.0455i	− 0.833201i
$210$	0	0
$211$	4.23050	0.291240	0.145620	−	0.989341i	$-0.453482\pi$
$211$	4.23050	0.291240	0.145620	+	0.989341i	$0.453482\pi$
$212$	0	0
$213$	3.49263	0.239311
$214$	0	0
$215$	−14.3125	−0.976106
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−14.5274	−0.981673
$220$	0	0
$221$	− 3.59516i	− 0.241837i
$222$	0	0
$223$	1.43532	0.0961162	0.0480581	−	0.998845i	$-0.484697\pi$
$223$	1.43532	0.0961162	0.0480581	+	0.998845i	$0.484697\pi$
$224$	0	0
$225$	−12.4350	−0.829003
$226$	0	0
$227$	− 16.0144i	− 1.06291i	−0.847086	−	0.531455i	$-0.821645\pi$
$227$	− 16.0144i	− 1.06291i	0.847086	−	0.531455i	$-0.178355\pi$
$228$	0	0
$229$	16.2500	1.07383	0.536914	−	0.843637i	$-0.319590\pi$
$229$	16.2500	1.07383	0.536914	+	0.843637i	$0.319590\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.8611	0.777045	0.388522	−	0.921439i	$-0.372986\pi$
$233$	11.8611	0.777045	0.388522	+	0.921439i	$0.372986\pi$
$234$	0	0
$235$	24.6396	1.60731
$236$	0	0
$237$	−1.68711	−0.109590
$238$	0	0
$239$	0.846585i	0.0547610i	0.999625	+	0.0273805i	$0.00871657\pi$
$239$	0.846585i	0.0547610i	−0.999625	+	0.0273805i	$0.991283\pi$
$240$	0	0
$241$	− 0.879217i	− 0.0566354i	−0.999599	−	0.0283177i	$-0.990985\pi$
$241$	− 0.879217i	− 0.0566354i	0.999599	−	0.0283177i	$-0.00901501\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 10.8462i	− 0.690126i
$248$	0	0
$249$	2.72601	0.172754
$250$	0	0
$251$	− 18.1441i	− 1.14524i	−0.819820	−	0.572622i	$-0.805926\pi$
$251$	− 18.1441i	− 1.14524i	0.819820	−	0.572622i	$-0.194074\pi$
$252$	0	0
$253$	0.802247i	0.0504368i
$254$	0	0
$255$	−9.74391	−0.610188
$256$	0	0
$257$	− 18.3484i	− 1.14454i	−0.820064	−	0.572271i	$-0.806062\pi$
$257$	− 18.3484i	− 1.14454i	0.820064	−	0.572271i	$-0.193938\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 3.33885i	− 0.206670i
$262$	0	0
$263$	3.48770i	0.215061i	0.994202	+	0.107531i	$0.0342943\pi$
$263$	3.48770i	0.215061i	−0.994202	+	0.107531i	$0.965706\pi$
$264$	0	0
$265$	6.53008i	0.401140i
$266$	0	0
$267$	−2.12267	−0.129905
$268$	0	0
$269$	−14.7121	−0.897012	−0.448506	−	0.893780i	$-0.648044\pi$
$269$	−14.7121	−0.897012	−0.448506	+	0.893780i	$0.648044\pi$
$270$	0	0
$271$	19.9028	1.20901	0.604504	−	0.796602i	$-0.293371\pi$
$271$	19.9028	1.20901	0.604504	+	0.796602i	$0.293371\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−21.2760	−1.28299
$276$	0	0
$277$	− 26.4466i	− 1.58902i	−0.607251	−	0.794510i	$-0.707728\pi$
$277$	− 26.4466i	− 1.58902i	0.607251	−	0.794510i	$-0.292272\pi$
$278$	0	0
$279$	−3.16253	−0.189336
$280$	0	0
$281$	−20.2837	−1.21003	−0.605013	−	0.796216i	$-0.706832\pi$
$281$	−20.2837	−1.21003	−0.605013	+	0.796216i	$0.706832\pi$
$282$	0	0
$283$	− 6.58671i	− 0.391539i	−0.980650	−	0.195770i	$-0.937280\pi$
$283$	− 6.58671i	− 0.391539i	0.980650	−	0.195770i	$-0.0627205\pi$
$284$	0	0
$285$	−29.3963	−1.74128
$286$	0	0
$287$	0	0
$288$	0	0
$289$	11.5544	0.679672
$290$	0	0
$291$	1.95202	0.114430
$292$	0	0
$293$	12.9438	0.756187	0.378094	−	0.925767i	$-0.376580\pi$
$293$	12.9438	0.756187	0.378094	+	0.925767i	$0.376580\pi$
$294$	0	0
$295$	− 25.3968i	− 1.47866i
$296$	0	0
$297$	− 1.71097i	− 0.0992806i
$298$	0	0
$299$	0.722374i	0.0417759i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	13.7809i	0.791692i
$304$	0	0
$305$	38.0774	2.18030
$306$	0	0
$307$	− 11.8773i	− 0.677871i	−0.940810	−	0.338936i	$-0.889933\pi$
$307$	− 11.8773i	− 0.677871i	0.940810	−	0.338936i	$-0.110067\pi$
$308$	0	0
$309$	7.68258i	0.437047i
$310$	0	0
$311$	11.8370	0.671214	0.335607	−	0.942002i	$-0.391059\pi$
$311$	11.8370	0.671214	0.335607	+	0.942002i	$0.391059\pi$
$312$	0	0
$313$	14.8243i	0.837921i	0.908004	+	0.418961i	$0.137605\pi$
$313$	14.8243i	0.837921i	−0.908004	+	0.418961i	$0.862395\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.78361i	0.381006i	0.981687	+	0.190503i	$0.0610118\pi$
$317$	6.78361i	0.381006i	−0.981687	+	0.190503i	$0.938988\pi$
$318$	0	0
$319$	− 5.71268i	− 0.319848i
$320$	0	0
$321$	− 14.4083i	− 0.804191i
$322$	0	0
$323$	−16.4287	−0.914116
$324$	0	0
$325$	−19.1577	−1.06268
$326$	0	0
$327$	−3.37612	−0.186700
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.74570	0.260847	0.130424	−	0.991458i	$-0.458366\pi$
$331$	4.74570	0.260847	0.130424	+	0.991458i	$0.458366\pi$
$332$	0	0
$333$	8.94486i	0.490175i
$334$	0	0
$335$	−31.2044	−1.70488
$336$	0	0
$337$	16.5173	0.899754	0.449877	−	0.893090i	$-0.351468\pi$
$337$	16.5173	0.899754	0.449877	+	0.893090i	$0.351468\pi$
$338$	0	0
$339$	10.7090i	0.581634i
$340$	0	0
$341$	−5.41099	−0.293022
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.95784	0.105407
$346$	0	0
$347$	19.0997	1.02533	0.512663	−	0.858590i	$-0.328659\pi$
$347$	19.0997	1.02533	0.512663	+	0.858590i	$0.328659\pi$
$348$	0	0
$349$	−2.49767	−0.133697	−0.0668485	−	0.997763i	$-0.521294\pi$
$349$	−2.49767	−0.133697	−0.0668485	+	0.997763i	$0.521294\pi$
$350$	0	0
$351$	− 1.54062i	− 0.0822323i
$352$	0	0
$353$	− 25.7580i	− 1.37096i	−0.728091	−	0.685481i	$-0.759592\pi$
$353$	− 25.7580i	− 1.37096i	0.728091	−	0.685481i	$-0.240408\pi$
$354$	0	0
$355$	− 14.5836i	− 0.774016i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 33.2203i	− 1.75330i	−0.481126	−	0.876651i	$-0.659772\pi$
$359$	− 33.2203i	− 1.75330i	0.481126	−	0.876651i	$-0.340228\pi$
$360$	0	0
$361$	−30.5634	−1.60860
$362$	0	0
$363$	8.07258i	0.423701i
$364$	0	0
$365$	60.6597i	3.17507i
$366$	0	0
$367$	−4.35167	−0.227155	−0.113578	−	0.993529i	$-0.536231\pi$
$367$	−4.35167	−0.227155	−0.113578	+	0.993529i	$0.536231\pi$
$368$	0	0
$369$	− 5.31411i	− 0.276642i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.8107i	0.663310i	0.943401	+	0.331655i	$0.107607\pi$
$373$	12.8107i	0.663310i	−0.943401	+	0.331655i	$0.892393\pi$
$374$	0	0
$375$	31.0452i	1.60317i
$376$	0	0
$377$	− 5.14391i	− 0.264925i
$378$	0	0
$379$	24.0807	1.23694	0.618472	−	0.785807i	$-0.287752\pi$
$379$	24.0807	1.23694	0.618472	+	0.785807i	$0.287752\pi$
$380$	0	0
$381$	9.49738	0.486566
$382$	0	0
$383$	−17.5647	−0.897512	−0.448756	−	0.893654i	$-0.648133\pi$
$383$	−17.5647	−0.897512	−0.448756	+	0.893654i	$0.648133\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.42772	−0.174241
$388$	0	0
$389$	27.5999i	1.39937i	0.714451	+	0.699685i	$0.246676\pi$
$389$	27.5999i	1.39937i	−0.714451	+	0.699685i	$0.753324\pi$
$390$	0	0
$391$	1.09418	0.0553349
$392$	0	0
$393$	−13.6675	−0.689435
$394$	0	0
$395$	7.04459i	0.354452i
$396$	0	0
$397$	−17.3170	−0.869115	−0.434558	−	0.900644i	$-0.643095\pi$
$397$	−17.3170	−0.869115	−0.434558	+	0.900644i	$0.643095\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.75232	0.387132	0.193566	−	0.981087i	$-0.437995\pi$
$401$	7.75232	0.387132	0.193566	+	0.981087i	$0.437995\pi$
$402$	0	0
$403$	−4.87226	−0.242705
$404$	0	0
$405$	−4.17553	−0.207484
$406$	0	0
$407$	15.3044i	0.758611i
$408$	0	0
$409$	− 4.63836i	− 0.229352i	−0.993403	−	0.114676i	$-0.963417\pi$
$409$	− 4.63836i	− 0.229352i	0.993403	−	0.114676i	$-0.0365830\pi$
$410$	0	0
$411$	11.9892i	0.591383i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 11.3826i	− 0.558747i
$416$	0	0
$417$	−6.64909	−0.325607
$418$	0	0
$419$	− 14.2419i	− 0.695760i	−0.937539	−	0.347880i	$-0.886902\pi$
$419$	− 14.2419i	− 0.695760i	0.937539	−	0.347880i	$-0.113098\pi$
$420$	0	0
$421$	− 20.0126i	− 0.975356i	−0.873024	−	0.487678i	$-0.837844\pi$
$421$	− 20.0126i	− 0.975356i	0.873024	−	0.487678i	$-0.162156\pi$
$422$	0	0
$423$	5.90095	0.286914
$424$	0	0
$425$	29.0181i	1.40759i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 2.63596i	− 0.127265i
$430$	0	0
$431$	13.9833i	0.673552i	0.941585	+	0.336776i	$0.109336\pi$
$431$	13.9833i	0.673552i	−0.941585	+	0.336776i	$0.890664\pi$
$432$	0	0
$433$	0.984888i	0.0473307i	0.999720	+	0.0236653i	$0.00753362\pi$
$433$	0.984888i	0.0473307i	−0.999720	+	0.0236653i	$0.992466\pi$
$434$	0	0
$435$	−13.9415	−0.668442
$436$	0	0
$437$	3.30101	0.157908
$438$	0	0
$439$	−6.87387	−0.328072	−0.164036	−	0.986454i	$-0.552451\pi$
$439$	−6.87387	−0.328072	−0.164036	+	0.986454i	$0.552451\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.40966	−0.161998	−0.0809989	−	0.996714i	$-0.525811\pi$
$443$	−3.40966	−0.161998	−0.0809989	+	0.996714i	$0.525811\pi$
$444$	0	0
$445$	8.86327i	0.420159i
$446$	0	0
$447$	−8.11897	−0.384014
$448$	0	0
$449$	32.9924	1.55701	0.778503	−	0.627641i	$-0.215979\pi$
$449$	32.9924	1.55701	0.778503	+	0.627641i	$0.215979\pi$
$450$	0	0
$451$	− 9.09229i	− 0.428139i
$452$	0	0
$453$	1.23604	0.0580742
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.8449	−1.06864	−0.534319	−	0.845283i	$-0.679432\pi$
$457$	−22.8449	−1.06864	−0.534319	+	0.845283i	$0.679432\pi$
$458$	0	0
$459$	−2.33358	−0.108922
$460$	0	0
$461$	−15.8743	−0.739338	−0.369669	−	0.929164i	$-0.620529\pi$
$461$	−15.8743	−0.739338	−0.369669	+	0.929164i	$0.620529\pi$
$462$	0	0
$463$	− 2.72059i	− 0.126436i	−0.998000	−	0.0632182i	$-0.979864\pi$
$463$	− 2.72059i	− 0.126436i	0.998000	−	0.0632182i	$-0.0201364\pi$
$464$	0	0
$465$	13.2052i	0.612377i
$466$	0	0
$467$	23.9861i	1.10994i	0.831869	+	0.554972i	$0.187271\pi$
$467$	23.9861i	1.10994i	−0.831869	+	0.554972i	$0.812729\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 9.60572i	− 0.442608i
$472$	0	0
$473$	−5.86472	−0.269660
$474$	0	0
$475$	87.5443i	4.01681i
$476$	0	0
$477$	1.56389i	0.0716058i
$478$	0	0
$479$	19.7502	0.902411	0.451205	−	0.892420i	$-0.350994\pi$
$479$	19.7502	0.902411	0.451205	+	0.892420i	$0.350994\pi$
$480$	0	0
$481$	13.7807i	0.628344i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 8.15073i	− 0.370106i
$486$	0	0
$487$	− 37.1161i	− 1.68189i	−0.541119	−	0.840946i	$-0.681999\pi$
$487$	− 37.1161i	− 1.68189i	0.541119	−	0.840946i	$-0.318001\pi$
$488$	0	0
$489$	− 19.4092i	− 0.877715i
$490$	0	0
$491$	20.5746	0.928517	0.464259	−	0.885700i	$-0.346321\pi$
$491$	20.5746	0.928517	0.464259	+	0.885700i	$0.346321\pi$
$492$	0	0
$493$	−7.79146	−0.350910
$494$	0	0
$495$	−7.14421	−0.321108
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.03516	−0.0463400	−0.0231700	−	0.999732i	$-0.507376\pi$
$499$	−1.03516	−0.0463400	−0.0231700	+	0.999732i	$0.507376\pi$
$500$	0	0
$501$	− 17.7482i	− 0.792931i
$502$	0	0
$503$	−19.7898	−0.882382	−0.441191	−	0.897413i	$-0.645444\pi$
$503$	−19.7898	−0.882382	−0.441191	+	0.897413i	$0.645444\pi$
$504$	0	0
$505$	57.5425	2.56061
$506$	0	0
$507$	10.6265i	0.471939i
$508$	0	0
$509$	4.23698	0.187801	0.0939004	−	0.995582i	$-0.470066\pi$
$509$	4.23698	0.187801	0.0939004	+	0.995582i	$0.470066\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.04013	−0.310829
$514$	0	0
$515$	32.0788	1.41356
$516$	0	0
$517$	10.0963	0.444037
$518$	0	0
$519$	7.14150i	0.313477i
$520$	0	0
$521$	− 38.8638i	− 1.70265i	−0.524637	−	0.851326i	$-0.675799\pi$
$521$	− 38.8638i	− 1.70265i	0.524637	−	0.851326i	$-0.324201\pi$
$522$	0	0
$523$	17.1105i	0.748191i	0.927390	+	0.374095i	$0.122047\pi$
$523$	17.1105i	0.748191i	−0.927390	+	0.374095i	$0.877953\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.38000i	0.321478i
$528$	0	0
$529$	22.7801	0.990441
$530$	0	0
$531$	− 6.08229i	− 0.263949i
$532$	0	0
$533$	− 8.18704i	− 0.354620i
$534$	0	0
$535$	−60.1621	−2.60104
$536$	0	0
$537$	23.9162i	1.03206i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	43.4921i	1.86987i	0.354819	+	0.934935i	$0.384543\pi$
$541$	43.4921i	1.86987i	−0.354819	+	0.934935i	$0.615457\pi$
$542$	0	0
$543$	− 20.5572i	− 0.882193i
$544$	0	0
$545$	14.0971i	0.603852i
$546$	0	0
$547$	−10.8290	−0.463016	−0.231508	−	0.972833i	$-0.574366\pi$
$547$	−10.8290	−0.463016	−0.231508	+	0.972833i	$0.574366\pi$
$548$	0	0
$549$	9.11917	0.389197
$550$	0	0
$551$	−23.5059	−1.00139
$552$	0	0
$553$	0	0
$554$	0	0
$555$	37.3495	1.58540
$556$	0	0
$557$	8.13137i	0.344537i	0.985050	+	0.172269i	$0.0551097\pi$
$557$	8.13137i	0.344537i	−0.985050	+	0.172269i	$0.944890\pi$
$558$	0	0
$559$	−5.28082	−0.223355
$560$	0	0
$561$	−3.99268	−0.168571
$562$	0	0
$563$	− 22.6414i	− 0.954222i	−0.878843	−	0.477111i	$-0.841684\pi$
$563$	− 22.6414i	− 0.954222i	0.878843	−	0.477111i	$-0.158316\pi$
$564$	0	0
$565$	44.7158	1.88121
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.2971	−0.850898	−0.425449	−	0.904982i	$-0.639884\pi$
$569$	−20.2971	−0.850898	−0.425449	+	0.904982i	$0.639884\pi$
$570$	0	0
$571$	−36.7637	−1.53851	−0.769257	−	0.638940i	$-0.779373\pi$
$571$	−36.7637	−1.53851	−0.769257	+	0.638940i	$0.779373\pi$
$572$	0	0
$573$	−2.01521	−0.0841868
$574$	0	0
$575$	− 5.83059i	− 0.243153i
$576$	0	0
$577$	1.61560i	0.0672584i	0.999434	+	0.0336292i	$0.0107065\pi$
$577$	1.61560i	0.0672584i	−0.999434	+	0.0336292i	$0.989293\pi$
$578$	0	0
$579$	3.57070i	0.148393i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.67578i	0.110819i
$584$	0	0
$585$	−6.43291	−0.265968
$586$	0	0
$587$	3.68747i	0.152198i	0.997100	+	0.0760991i	$0.0242465\pi$
$587$	3.68747i	0.152198i	−0.997100	+	0.0760991i	$0.975753\pi$
$588$	0	0
$589$	22.2646i	0.917396i
$590$	0	0
$591$	−17.5393	−0.721471
$592$	0	0
$593$	− 37.3870i	− 1.53530i	−0.640869	−	0.767650i	$-0.721426\pi$
$593$	− 37.3870i	− 1.53530i	0.640869	−	0.767650i	$-0.278574\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.7067i	0.724688i
$598$	0	0
$599$	9.94700i	0.406423i	0.979135	+	0.203212i	$0.0651379\pi$
$599$	9.94700i	0.406423i	−0.979135	+	0.203212i	$0.934862\pi$
$600$	0	0
$601$	35.9296i	1.46560i	0.680445	+	0.732799i	$0.261786\pi$
$601$	35.9296i	1.46560i	−0.680445	+	0.732799i	$0.738214\pi$
$602$	0	0
$603$	−7.47317	−0.304331
$604$	0	0
$605$	33.7073	1.37040
$606$	0	0
$607$	29.8710	1.21242	0.606212	−	0.795303i	$-0.292688\pi$
$607$	29.8710	1.21242	0.606212	+	0.795303i	$0.292688\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.09113	0.367788
$612$	0	0
$613$	− 37.8926i	− 1.53047i	−0.643752	−	0.765234i	$-0.722623\pi$
$613$	− 37.8926i	− 1.53047i	0.643752	−	0.765234i	$-0.277377\pi$
$614$	0	0
$615$	−22.1892	−0.894756
$616$	0	0
$617$	39.0332	1.57142	0.785709	−	0.618597i	$-0.212298\pi$
$617$	39.0332	1.57142	0.785709	+	0.618597i	$0.212298\pi$
$618$	0	0
$619$	24.8651i	0.999413i	0.866195	+	0.499706i	$0.166559\pi$
$619$	24.8651i	0.999413i	−0.866195	+	0.499706i	$0.833441\pi$
$620$	0	0
$621$	0.468884	0.0188157
$622$	0	0
$623$	0	0
$624$	0	0
$625$	67.4550	2.69820
$626$	0	0
$627$	−12.0455	−0.481049
$628$	0	0
$629$	20.8735	0.832281
$630$	0	0
$631$	− 43.3823i	− 1.72702i	−0.504330	−	0.863511i	$-0.668260\pi$
$631$	− 43.3823i	− 1.72702i	0.504330	−	0.863511i	$-0.331740\pi$
$632$	0	0
$633$	− 4.23050i	− 0.168147i
$634$	0	0
$635$	− 39.6566i	− 1.57372i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 3.49263i	− 0.138166i
$640$	0	0
$641$	6.65118	0.262706	0.131353	−	0.991336i	$-0.458068\pi$
$641$	6.65118	0.262706	0.131353	+	0.991336i	$0.458068\pi$
$642$	0	0
$643$	18.1066i	0.714055i	0.934094	+	0.357027i	$0.116210\pi$
$643$	18.1066i	0.714055i	−0.934094	+	0.357027i	$0.883790\pi$
$644$	0	0
$645$	14.3125i	0.563555i
$646$	0	0
$647$	49.2843	1.93757	0.968783	−	0.247911i	$-0.0797440\pi$
$647$	49.2843	1.93757	0.968783	+	0.247911i	$0.0797440\pi$
$648$	0	0
$649$	− 10.4066i	− 0.408495i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25.5289i	0.999025i	0.866307	+	0.499512i	$0.166487\pi$
$653$	25.5289i	0.999025i	−0.866307	+	0.499512i	$0.833513\pi$
$654$	0	0
$655$	57.0691i	2.22987i
$656$	0	0
$657$	14.5274i	0.566769i
$658$	0	0
$659$	12.0942	0.471125	0.235562	−	0.971859i	$-0.424307\pi$
$659$	12.0942	0.471125	0.235562	+	0.971859i	$0.424307\pi$
$660$	0	0
$661$	10.8508	0.422048	0.211024	−	0.977481i	$-0.432320\pi$
$661$	10.8508	0.422048	0.211024	+	0.977481i	$0.432320\pi$
$662$	0	0
$663$	−3.59516	−0.139624
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.56553	0.0606177
$668$	0	0
$669$	− 1.43532i	− 0.0554927i
$670$	0	0
$671$	15.6026	0.602333
$672$	0	0
$673$	−48.1931	−1.85771	−0.928854	−	0.370446i	$-0.879205\pi$
$673$	−48.1931	−1.85771	−0.928854	+	0.370446i	$0.879205\pi$
$674$	0	0
$675$	12.4350i	0.478625i
$676$	0	0
$677$	−5.12187	−0.196849	−0.0984246	−	0.995145i	$-0.531380\pi$
$677$	−5.12187	−0.196849	−0.0984246	+	0.995145i	$0.531380\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−16.0144	−0.613672
$682$	0	0
$683$	−23.4381	−0.896835	−0.448418	−	0.893824i	$-0.648012\pi$
$683$	−23.4381	−0.896835	−0.448418	+	0.893824i	$0.648012\pi$
$684$	0	0
$685$	50.0612	1.91274
$686$	0	0
$687$	− 16.2500i	− 0.619975i
$688$	0	0
$689$	2.40937i	0.0917897i
$690$	0	0
$691$	13.8026i	0.525075i	0.964922	+	0.262537i	$0.0845593\pi$
$691$	13.8026i	0.525075i	−0.964922	+	0.262537i	$0.915441\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.7635i	1.05313i
$696$	0	0
$697$	−12.4009	−0.469717
$698$	0	0
$699$	− 11.8611i	− 0.448627i
$700$	0	0
$701$	11.8718i	0.448393i	0.974544	+	0.224196i	$0.0719757\pi$
$701$	11.8718i	0.448393i	−0.974544	+	0.224196i	$0.928024\pi$
$702$	0	0
$703$	62.9730	2.37507
$704$	0	0
$705$	− 24.6396i	− 0.927980i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 31.3202i	− 1.17625i	−0.808768	−	0.588127i	$-0.799865\pi$
$709$	− 31.3202i	− 1.17625i	0.808768	−	0.588127i	$-0.200135\pi$
$710$	0	0
$711$	1.68711i	0.0632717i
$712$	0	0
$713$	− 1.48286i	− 0.0555335i
$714$	0	0
$715$	−11.0065	−0.411621
$716$	0	0
$717$	0.846585	0.0316163
$718$	0	0
$719$	25.2547	0.941840	0.470920	−	0.882176i	$-0.343922\pi$
$719$	25.2547	0.941840	0.470920	+	0.882176i	$0.343922\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−0.879217	−0.0326984
$724$	0	0
$725$	41.5187i	1.54197i
$726$	0	0
$727$	−17.9342	−0.665144	−0.332572	−	0.943078i	$-0.607916\pi$
$727$	−17.9342	−0.665144	−0.332572	+	0.943078i	$0.607916\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	7.99884i	0.295848i
$732$	0	0
$733$	12.3356	0.455625	0.227813	−	0.973705i	$-0.426843\pi$
$733$	12.3356	0.455625	0.227813	+	0.973705i	$0.426843\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.7864	−0.470992
$738$	0	0
$739$	−43.4925	−1.59990	−0.799949	−	0.600068i	$-0.795140\pi$
$739$	−43.4925	−1.59990	−0.799949	+	0.600068i	$0.795140\pi$
$740$	0	0
$741$	−10.8462	−0.398444
$742$	0	0
$743$	32.8397i	1.20477i	0.798205	+	0.602386i	$0.205783\pi$
$743$	32.8397i	1.20477i	−0.798205	+	0.602386i	$0.794217\pi$
$744$	0	0
$745$	33.9010i	1.24204i
$746$	0	0
$747$	− 2.72601i	− 0.0997396i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	8.80656i	0.321356i	0.987007	+	0.160678i	$0.0513680\pi$
$751$	8.80656i	0.321356i	−0.987007	+	0.160678i	$0.948632\pi$
$752$	0	0
$753$	−18.1441	−0.661206
$754$	0	0
$755$	− 5.16112i	− 0.187832i
$756$	0	0
$757$	− 16.9328i	− 0.615433i	−0.951478	−	0.307717i	$-0.900435\pi$
$757$	− 16.9328i	− 0.615433i	0.951478	−	0.307717i	$-0.0995649\pi$
$758$	0	0
$759$	0.802247	0.0291197
$760$	0	0
$761$	− 12.5987i	− 0.456704i	−0.973579	−	0.228352i	$-0.926666\pi$
$761$	− 12.5987i	− 0.456704i	0.973579	−	0.228352i	$-0.0733336\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	9.74391i	0.352292i
$766$	0	0
$767$	− 9.37051i	− 0.338350i
$768$	0	0
$769$	− 16.0445i	− 0.578581i	−0.957241	−	0.289291i	$-0.906581\pi$
$769$	− 16.0445i	− 0.578581i	0.957241	−	0.289291i	$-0.0934194\pi$
$770$	0	0
$771$	−18.3484	−0.660802
$772$	0	0
$773$	18.3134	0.658688	0.329344	−	0.944210i	$-0.393172\pi$
$773$	18.3134	0.658688	0.329344	+	0.944210i	$0.393172\pi$
$774$	0	0
$775$	39.3261	1.41264
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−37.4120	−1.34043
$780$	0	0
$781$	− 5.97579i	− 0.213830i
$782$	0	0
$783$	−3.33885	−0.119321
$784$	0	0
$785$	−40.1090	−1.43155
$786$	0	0
$787$	− 4.49151i	− 0.160105i	−0.996791	−	0.0800526i	$-0.974491\pi$
$787$	− 4.49151i	− 0.160105i	0.996791	−	0.0800526i	$-0.0255088\pi$
$788$	0	0
$789$	3.48770	0.124166
$790$	0	0
$791$	0	0
$792$	0	0
$793$	14.0492	0.498902
$794$	0	0
$795$	6.53008	0.231598
$796$	0	0
$797$	−31.7698	−1.12534	−0.562672	−	0.826680i	$-0.690227\pi$
$797$	−31.7698	−1.12534	−0.562672	+	0.826680i	$0.690227\pi$
$798$	0	0
$799$	− 13.7703i	− 0.487159i
$800$	0	0
$801$	2.12267i	0.0750009i
$802$	0	0
$803$	24.8560i	0.877150i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.7121i	0.517890i
$808$	0	0
$809$	25.1568	0.884464	0.442232	−	0.896901i	$-0.354187\pi$
$809$	25.1568	0.884464	0.442232	+	0.896901i	$0.354187\pi$
$810$	0	0
$811$	− 48.0042i	− 1.68565i	−0.538184	−	0.842827i	$-0.680890\pi$
$811$	− 48.0042i	− 1.68565i	0.538184	−	0.842827i	$-0.319110\pi$
$812$	0	0
$813$	− 19.9028i	− 0.698021i
$814$	0	0
$815$	−81.0437	−2.83884
$816$	0	0
$817$	24.1316i	0.844257i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.8418i	0.587782i	0.955839	+	0.293891i	$0.0949503\pi$
$821$	16.8418i	0.587782i	−0.955839	+	0.293891i	$0.905050\pi$
$822$	0	0
$823$	− 19.7642i	− 0.688936i	−0.938798	−	0.344468i	$-0.888059\pi$
$823$	− 19.7642i	− 0.688936i	0.938798	−	0.344468i	$-0.111941\pi$
$824$	0	0
$825$	21.2760i	0.740735i
$826$	0	0
$827$	−49.1702	−1.70981	−0.854907	−	0.518781i	$-0.826386\pi$
$827$	−49.1702	−1.70981	−0.854907	+	0.518781i	$0.826386\pi$
$828$	0	0
$829$	15.9201	0.552929	0.276465	−	0.961024i	$-0.410837\pi$
$829$	15.9201	0.552929	0.276465	+	0.961024i	$0.410837\pi$
$830$	0	0
$831$	−26.4466	−0.917421
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−74.1081	−2.56462
$836$	0	0
$837$	3.16253i	0.109313i
$838$	0	0
$839$	38.3305	1.32332	0.661658	−	0.749806i	$-0.269853\pi$
$839$	38.3305	1.32332	0.661658	+	0.749806i	$0.269853\pi$
$840$	0	0
$841$	17.8521	0.615589
$842$	0	0
$843$	20.2837i	0.698609i
$844$	0	0
$845$	44.3712	1.52641
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−6.58671	−0.226055
$850$	0	0
$851$	−4.19410	−0.143772
$852$	0	0
$853$	22.5158	0.770927	0.385463	−	0.922723i	$-0.374042\pi$
$853$	22.5158	0.770927	0.385463	+	0.922723i	$0.374042\pi$
$854$	0	0
$855$	29.3963i	1.00533i
$856$	0	0
$857$	− 12.9226i	− 0.441427i	−0.975339	−	0.220713i	$-0.929161\pi$
$857$	− 12.9226i	− 0.441427i	0.975339	−	0.220713i	$-0.0708385\pi$
$858$	0	0
$859$	23.2577i	0.793544i	0.917917	+	0.396772i	$0.129870\pi$
$859$	23.2577i	0.793544i	−0.917917	+	0.396772i	$0.870130\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8.88440i	0.302428i	0.988501	+	0.151214i	$0.0483183\pi$
$863$	8.88440i	0.302428i	−0.988501	+	0.151214i	$0.951682\pi$
$864$	0	0
$865$	29.8195	1.01389
$866$	0	0
$867$	− 11.5544i	− 0.392409i
$868$	0	0
$869$	2.88660i	0.0979212i
$870$	0	0
$871$	−11.5133	−0.390114
$872$	0	0
$873$	− 1.95202i	− 0.0660660i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 46.4345i	− 1.56798i	−0.620773	−	0.783990i	$-0.713181\pi$
$877$	− 46.4345i	− 1.56798i	0.620773	−	0.783990i	$-0.286819\pi$
$878$	0	0
$879$	− 12.9438i	− 0.436585i
$880$	0	0
$881$	− 27.1901i	− 0.916058i	−0.888937	−	0.458029i	$-0.848556\pi$
$881$	− 27.1901i	− 0.916058i	0.888937	−	0.458029i	$-0.151444\pi$
$882$	0	0
$883$	21.7975	0.733543	0.366772	−	0.930311i	$-0.380463\pi$
$883$	21.7975	0.733543	0.366772	+	0.930311i	$0.380463\pi$
$884$	0	0
$885$	−25.3968	−0.853703
$886$	0	0
$887$	42.3322	1.42138	0.710688	−	0.703508i	$-0.248384\pi$
$887$	42.3322	1.42138	0.710688	+	0.703508i	$0.248384\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.71097	−0.0573197
$892$	0	0
$893$	− 41.5434i	− 1.39020i
$894$	0	0
$895$	99.8629	3.33805
$896$	0	0
$897$	0.722374	0.0241194
$898$	0	0
$899$	10.5592i	0.352169i
$900$	0	0
$901$	3.64946	0.121581
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−85.8371	−2.85332
$906$	0	0
$907$	45.1210	1.49822	0.749109	−	0.662446i	$-0.230482\pi$
$907$	45.1210	1.49822	0.749109	+	0.662446i	$0.230482\pi$
$908$	0	0
$909$	13.7809	0.457083
$910$	0	0
$911$	− 36.2714i	− 1.20173i	−0.799352	−	0.600863i	$-0.794824\pi$
$911$	− 36.2714i	− 1.20173i	0.799352	−	0.600863i	$-0.205176\pi$
$912$	0	0
$913$	− 4.66413i	− 0.154360i
$914$	0	0
$915$	− 38.0774i	− 1.25880i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	43.6367i	1.43944i	0.694264	+	0.719721i	$0.255730\pi$
$919$	43.6367i	1.43944i	−0.694264	+	0.719721i	$0.744270\pi$
$920$	0	0
$921$	−11.8773	−0.391369
$922$	0	0
$923$	− 5.38082i	− 0.177112i
$924$	0	0
$925$	− 111.230i	− 3.65721i
$926$	0	0
$927$	7.68258	0.252329
$928$	0	0
$929$	− 11.7866i	− 0.386707i	−0.981129	−	0.193354i	$-0.938064\pi$
$929$	− 11.7866i	− 0.386707i	0.981129	−	0.193354i	$-0.0619365\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 11.8370i	− 0.387525i
$934$	0	0
$935$	16.6716i	0.545218i
$936$	0	0
$937$	− 35.0529i	− 1.14513i	−0.819860	−	0.572565i	$-0.805949\pi$
$937$	− 35.0529i	− 1.14513i	0.819860	−	0.572565i	$-0.194051\pi$
$938$	0	0
$939$	14.8243	0.483774
$940$	0	0
$941$	−28.5971	−0.932238	−0.466119	−	0.884722i	$-0.654348\pi$
$941$	−28.5971	−0.932238	−0.466119	+	0.884722i	$0.654348\pi$
$942$	0	0
$943$	2.49170	0.0811410
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.4888	1.67316	0.836580	−	0.547845i	$-0.184552\pi$
$947$	51.4888	1.67316	0.836580	+	0.547845i	$0.184552\pi$
$948$	0	0
$949$	22.3813i	0.726528i
$950$	0	0
$951$	6.78361	0.219974
$952$	0	0
$953$	−11.8761	−0.384706	−0.192353	−	0.981326i	$-0.561612\pi$
$953$	−11.8761	−0.384706	−0.192353	+	0.981326i	$0.561612\pi$
$954$	0	0
$955$	8.41458i	0.272290i
$956$	0	0
$957$	−5.71268	−0.184665
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.9984	−0.677369
$962$	0	0
$963$	−14.4083	−0.464300
$964$	0	0
$965$	14.9096	0.479956
$966$	0	0
$967$	− 59.2193i	− 1.90437i	−0.305530	−	0.952183i	$-0.598833\pi$
$967$	− 59.2193i	− 1.90437i	0.305530	−	0.952183i	$-0.401167\pi$
$968$	0	0
$969$	16.4287i	0.527765i
$970$	0	0
$971$	− 5.23816i	− 0.168101i	−0.996462	−	0.0840503i	$-0.973214\pi$
$971$	− 5.23816i	− 0.168101i	0.996462	−	0.0840503i	$-0.0267857\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	19.1577i	0.613537i
$976$	0	0
$977$	−53.2097	−1.70233	−0.851165	−	0.524898i	$-0.824103\pi$
$977$	−53.2097	−1.70233	−0.851165	+	0.524898i	$0.824103\pi$
$978$	0	0
$979$	3.63183i	0.116074i
$980$	0	0
$981$	3.37612i	0.107791i
$982$	0	0
$983$	27.9475	0.891386	0.445693	−	0.895186i	$-0.352957\pi$
$983$	27.9475	0.891386	0.445693	+	0.895186i	$0.352957\pi$
$984$	0	0
$985$	73.2359i	2.33349i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.60720i	− 0.0511061i
$990$	0	0
$991$	21.7061i	0.689516i	0.938692	+	0.344758i	$0.112039\pi$
$991$	21.7061i	0.689516i	−0.938692	+	0.344758i	$0.887961\pi$
$992$	0	0
$993$	− 4.74570i	− 0.150600i
$994$	0	0
$995$	73.9349	2.34389
$996$	0	0
$997$	−33.6547	−1.06586	−0.532928	−	0.846161i	$-0.678908\pi$
$997$	−33.6547	−1.06586	−0.532928	+	0.846161i	$0.678908\pi$
$998$	0	0
$999$	8.94486	0.283003

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.p.a.3919.5		32
4.3	odd	2		1176.2.p.a.979.20		32
7.4	even	3		672.2.bb.a.271.8		32
7.5	odd	6		672.2.bb.a.367.1		32
7.6	odd	2	inner	4704.2.p.a.3919.24		32
8.3	odd	2	inner	4704.2.p.a.3919.23		32
8.5	even	2		1176.2.p.a.979.17		32
21.5	even	6		2016.2.bs.c.1711.16		32
21.11	odd	6		2016.2.bs.c.271.1		32
28.11	odd	6		168.2.t.a.19.1	✓	32
28.19	even	6		168.2.t.a.115.12	yes	32
28.27	even	2		1176.2.p.a.979.19		32
56.5	odd	6		168.2.t.a.115.1	yes	32
56.11	odd	6		672.2.bb.a.271.1		32
56.13	odd	2		1176.2.p.a.979.18		32
56.19	even	6		672.2.bb.a.367.8		32
56.27	even	2	inner	4704.2.p.a.3919.6		32
56.53	even	6		168.2.t.a.19.12	yes	32
84.11	even	6		504.2.bk.c.19.16		32
84.47	odd	6		504.2.bk.c.451.5		32
168.5	even	6		504.2.bk.c.451.16		32
168.11	even	6		2016.2.bs.c.271.16		32
168.53	odd	6		504.2.bk.c.19.5		32
168.131	odd	6		2016.2.bs.c.1711.1		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.t.a.19.1	✓	32	28.11	odd	6
168.2.t.a.19.12	yes	32	56.53	even	6
168.2.t.a.115.1	yes	32	56.5	odd	6
168.2.t.a.115.12	yes	32	28.19	even	6
504.2.bk.c.19.5		32	168.53	odd	6
504.2.bk.c.19.16		32	84.11	even	6
504.2.bk.c.451.5		32	84.47	odd	6
504.2.bk.c.451.16		32	168.5	even	6
672.2.bb.a.271.1		32	56.11	odd	6
672.2.bb.a.271.8		32	7.4	even	3
672.2.bb.a.367.1		32	7.5	odd	6
672.2.bb.a.367.8		32	56.19	even	6
1176.2.p.a.979.17		32	8.5	even	2
1176.2.p.a.979.18		32	56.13	odd	2
1176.2.p.a.979.19		32	28.27	even	2
1176.2.p.a.979.20		32	4.3	odd	2
2016.2.bs.c.271.1		32	21.11	odd	6
2016.2.bs.c.271.16		32	168.11	even	6
2016.2.bs.c.1711.1		32	168.131	odd	6
2016.2.bs.c.1711.16		32	21.5	even	6
4704.2.p.a.3919.5		32	1.1	even	1	trivial
4704.2.p.a.3919.6		32	56.27	even	2	inner
4704.2.p.a.3919.23		32	8.3	odd	2	inner
4704.2.p.a.3919.24		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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -4.17553 q^{5} -1.00000 q^{9} -1.71097 q^{11} -1.54062 q^{13} +4.17553i q^{15} +2.33358i q^{17} +7.04013i q^{19} -0.468884i q^{23} +12.4350 q^{25} +1.00000i q^{27} +3.33885i q^{29} +3.16253 q^{31} +1.71097i q^{33} -8.94486i q^{37} +1.54062i q^{39} +5.31411i q^{41} +3.42772 q^{43} +4.17553 q^{45} -5.90095 q^{47} +2.33358 q^{51} -1.56389i q^{53} +7.14421 q^{55} +7.04013 q^{57} +6.08229i q^{59} -9.11917 q^{61} +6.43291 q^{65} +7.47317 q^{67} -0.468884 q^{69} +3.49263i q^{71} -14.5274i q^{73} -12.4350i q^{75} -1.68711i q^{79} +1.00000 q^{81} +2.72601i q^{83} -9.74391i q^{85} +3.33885 q^{87} -2.12267i q^{89} -3.16253i q^{93} -29.3963i q^{95} +1.95202i q^{97} +1.71097 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{9} + 16 q^{11} + 32 q^{25} + 16 q^{43} + 16 q^{57} - 64 q^{67} + 32 q^{81} - 16 q^{99}+O(q^{100}) \) 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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