LMFDB - Embedded newform 3024.2.b.r.1567.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1567,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3024.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.2
Root			$-1.32288 + 2.29129i$ of defining polynomial
Character	$\chi$	$=$	3024.1567
Dual form			3024.2.b.r.1567.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.46410i q^{5} +2.64575 q^{7} +O(q^{10})$	$q-3.46410i q^{5} +2.64575 q^{7} -4.58258i q^{13} -1.73205i q^{17} +5.29150 q^{19} -4.58258i q^{23} -7.00000 q^{25} -7.93725 q^{29} +2.64575 q^{31} -9.16515i q^{35} +4.00000 q^{37} +6.92820i q^{41} -1.73205i q^{43} -6.00000 q^{47} +7.00000 q^{49} +7.93725 q^{53} -3.00000 q^{59} -9.16515i q^{61} -15.8745 q^{65} +12.1244i q^{67} -4.58258i q^{71} +9.16515i q^{73} -6.92820i q^{79} +12.0000 q^{83} -6.00000 q^{85} +5.19615i q^{89} -12.1244i q^{91} -18.3303i q^{95} +9.16515i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 28 q^{25} + 16 q^{37} - 24 q^{47} + 28 q^{49} - 12 q^{59} + 48 q^{83} - 24 q^{85}+O(q^{100}) $ 4 * q - 28 * q^25 + 16 * q^37 - 24 * q^47 + 28 * q^49 - 12 * q^59 + 48 * q^83 - 24 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 3.46410i	− 1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	− 3.46410i	− 1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$	0	0
$7$	2.64575	1.00000
$7$	2.64575	1.00000
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 4.58258i	− 1.27098i	−0.772110	−	0.635489i	$-0.780799\pi$
$13$	− 4.58258i	− 1.27098i	0.772110	−	0.635489i	$-0.219201\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.73205i	− 0.420084i	−0.977692	−	0.210042i	$-0.932640\pi$
$17$	− 1.73205i	− 0.420084i	0.977692	−	0.210042i	$-0.0673601\pi$
$18$	0	0
$19$	5.29150	1.21395	0.606977	−	0.794719i	$-0.292382\pi$
$19$	5.29150	1.21395	0.606977	+	0.794719i	$0.292382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.58258i	− 0.955533i	−0.878487	−	0.477767i	$-0.841446\pi$
$23$	− 4.58258i	− 0.955533i	0.878487	−	0.477767i	$-0.158554\pi$
$24$	0	0
$25$	−7.00000	−1.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.93725	−1.47391	−0.736956	−	0.675941i	$-0.763737\pi$
$29$	−7.93725	−1.47391	−0.736956	+	0.675941i	$0.763737\pi$
$30$	0	0
$31$	2.64575	0.475191	0.237595	−	0.971364i	$-0.423641\pi$
$31$	2.64575	0.475191	0.237595	+	0.971364i	$0.423641\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 9.16515i	− 1.54919i
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.92820i	1.08200i	0.841021	+	0.541002i	$0.181955\pi$
$41$	6.92820i	1.08200i	−0.841021	+	0.541002i	$0.818045\pi$
$42$	0	0
$43$	− 1.73205i	− 0.264135i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	− 1.73205i	− 0.264135i	0.991241	−	0.132068i	$-0.0421616\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.93725	1.09027	0.545133	−	0.838350i	$-0.316479\pi$
$53$	7.93725	1.09027	0.545133	+	0.838350i	$0.316479\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	− 9.16515i	− 1.17348i	−0.809776	−	0.586739i	$-0.800412\pi$
$61$	− 9.16515i	− 1.17348i	0.809776	−	0.586739i	$-0.199588\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−15.8745	−1.96899
$66$	0	0
$67$	12.1244i	1.48123i	0.671932	+	0.740613i	$0.265465\pi$
$67$	12.1244i	1.48123i	−0.671932	+	0.740613i	$0.734535\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 4.58258i	− 0.543852i	−0.962318	−	0.271926i	$-0.912339\pi$
$71$	− 4.58258i	− 0.543852i	0.962318	−	0.271926i	$-0.0876605\pi$
$72$	0	0
$73$	9.16515i	1.07270i	0.843996	+	0.536350i	$0.180197\pi$
$73$	9.16515i	1.07270i	−0.843996	+	0.536350i	$0.819803\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 6.92820i	− 0.779484i	−0.920924	−	0.389742i	$-0.872564\pi$
$79$	− 6.92820i	− 0.779484i	0.920924	−	0.389742i	$-0.127436\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.19615i	0.550791i	0.961331	+	0.275396i	$0.0888088\pi$
$89$	5.19615i	0.550791i	−0.961331	+	0.275396i	$0.911191\pi$
$90$	0	0
$91$	− 12.1244i	− 1.27098i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 18.3303i	− 1.88065i
$96$	0	0
$97$	9.16515i	0.930580i	0.885158	+	0.465290i	$0.154050\pi$
$97$	9.16515i	0.930580i	−0.885158	+	0.465290i	$0.845950\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 3.46410i	− 0.344691i	−0.985037	−	0.172345i	$-0.944865\pi$
$101$	− 3.46410i	− 0.344691i	0.985037	−	0.172345i	$-0.0551346\pi$
$102$	0	0
$103$	−13.2288	−1.30347	−0.651734	−	0.758448i	$-0.725958\pi$
$103$	−13.2288	−1.30347	−0.651734	+	0.758448i	$0.725958\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 18.3303i	− 1.77206i	−0.463631	−	0.886029i	$-0.653453\pi$
$107$	− 18.3303i	− 1.77206i	0.463631	−	0.886029i	$-0.346547\pi$
$108$	0	0
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−15.8745	−1.49335	−0.746674	−	0.665190i	$-0.768350\pi$
$113$	−15.8745	−1.49335	−0.746674	+	0.665190i	$0.768350\pi$
$114$	0	0
$115$	−15.8745	−1.48031
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 4.58258i	− 0.420084i
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.92820i	0.619677i
$126$	0	0
$127$	13.8564i	1.22956i	0.788700	+	0.614779i	$0.210755\pi$
$127$	13.8564i	1.22956i	−0.788700	+	0.614779i	$0.789245\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	0	0
$133$	14.0000	1.21395
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.8745	−1.35625	−0.678125	−	0.734946i	$-0.737207\pi$
$137$	−15.8745	−1.35625	−0.678125	+	0.734946i	$0.737207\pi$
$138$	0	0
$139$	−5.29150	−0.448819	−0.224410	−	0.974495i	$-0.572045\pi$
$139$	−5.29150	−0.448819	−0.224410	+	0.974495i	$0.572045\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	27.4955i	2.28337i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.93725	0.650245	0.325123	−	0.945672i	$-0.394594\pi$
$149$	7.93725	0.650245	0.325123	+	0.945672i	$0.394594\pi$
$150$	0	0
$151$	3.46410i	0.281905i	0.990016	+	0.140952i	$0.0450164\pi$
$151$	3.46410i	0.281905i	−0.990016	+	0.140952i	$0.954984\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 9.16515i	− 0.736163i
$156$	0	0
$157$	− 22.9129i	− 1.82865i	−0.404985	−	0.914323i	$-0.632723\pi$
$157$	− 22.9129i	− 1.82865i	0.404985	−	0.914323i	$-0.367277\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 12.1244i	− 0.955533i
$162$	0	0
$163$	− 5.19615i	− 0.406994i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	− 5.19615i	− 0.406994i	0.979076	−	0.203497i	$-0.0652307\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.3205i	1.31685i	0.752645	+	0.658427i	$0.228778\pi$
$173$	17.3205i	1.31685i	−0.752645	+	0.658427i	$0.771222\pi$
$174$	0	0
$175$	−18.5203	−1.40000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 9.16515i	− 0.685036i	−0.939511	−	0.342518i	$-0.888720\pi$
$179$	− 9.16515i	− 0.685036i	0.939511	−	0.342518i	$-0.111280\pi$
$180$	0	0
$181$	13.7477i	1.02186i	0.859622	+	0.510930i	$0.170699\pi$
$181$	13.7477i	1.02186i	−0.859622	+	0.510930i	$0.829301\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 13.8564i	− 1.01874i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	13.2288	0.937762	0.468881	−	0.883261i	$-0.344657\pi$
$199$	13.2288	0.937762	0.468881	+	0.883261i	$0.344657\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−21.0000	−1.47391
$204$	0	0
$205$	24.0000	1.67623
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 12.1244i	− 0.834675i	−0.908752	−	0.417338i	$-0.862963\pi$
$211$	− 12.1244i	− 0.834675i	0.908752	−	0.417338i	$-0.137037\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	7.00000	0.475191
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.93725	−0.533917
$222$	0	0
$223$	5.29150	0.354345	0.177173	−	0.984180i	$-0.443305\pi$
$223$	5.29150	0.354345	0.177173	+	0.984180i	$0.443305\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.0000	−1.39382	−0.696909	−	0.717159i	$-0.745442\pi$
$227$	−21.0000	−1.39382	−0.696909	+	0.717159i	$0.745442\pi$
$228$	0	0
$229$	− 9.16515i	− 0.605650i	−0.953046	−	0.302825i	$-0.902070\pi$
$229$	− 9.16515i	− 0.605650i	0.953046	−	0.302825i	$-0.0979298\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.8745	−1.03997	−0.519987	−	0.854174i	$-0.674063\pi$
$233$	−15.8745	−1.03997	−0.519987	+	0.854174i	$0.674063\pi$
$234$	0	0
$235$	20.7846i	1.35584i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 18.3303i	− 1.18569i	−0.805317	−	0.592844i	$-0.798005\pi$
$239$	− 18.3303i	− 1.18569i	0.805317	−	0.592844i	$-0.201995\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 24.2487i	− 1.54919i
$246$	0	0
$247$	− 24.2487i	− 1.54291i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.92820i	− 0.432169i	−0.976375	−	0.216085i	$-0.930671\pi$
$257$	− 6.92820i	− 0.432169i	0.976375	−	0.216085i	$-0.0693287\pi$
$258$	0	0
$259$	10.5830	0.657596
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 22.9129i	− 1.41287i	−0.707779	−	0.706434i	$-0.750303\pi$
$263$	− 22.9129i	− 1.41287i	0.707779	−	0.706434i	$-0.249697\pi$
$264$	0	0
$265$	− 27.4955i	− 1.68903i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 13.8564i	− 0.844840i	−0.906400	−	0.422420i	$-0.861181\pi$
$269$	− 13.8564i	− 0.844840i	0.906400	−	0.422420i	$-0.138819\pi$
$270$	0	0
$271$	18.5203	1.12503	0.562513	−	0.826789i	$-0.309835\pi$
$271$	18.5203	1.12503	0.562513	+	0.826789i	$0.309835\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.8745	0.946994	0.473497	−	0.880795i	$-0.342992\pi$
$281$	15.8745	0.946994	0.473497	+	0.880795i	$0.342992\pi$
$282$	0	0
$283$	21.1660	1.25819	0.629094	−	0.777329i	$-0.283426\pi$
$283$	21.1660	1.25819	0.629094	+	0.777329i	$0.283426\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.3303i	1.08200i
$288$	0	0
$289$	14.0000	0.823529
$290$	0	0
$291$	0	0
$292$	0	0
$293$	31.1769i	1.82137i	0.413096	+	0.910687i	$0.364447\pi$
$293$	31.1769i	1.82137i	−0.413096	+	0.910687i	$0.635553\pi$
$294$	0	0
$295$	10.3923i	0.605063i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−21.0000	−1.21446
$300$	0	0
$301$	− 4.58258i	− 0.264135i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−31.7490	−1.81794
$306$	0	0
$307$	21.1660	1.20801	0.604004	−	0.796981i	$-0.293571\pi$
$307$	21.1660	1.20801	0.604004	+	0.796981i	$0.293571\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	− 27.4955i	− 1.55413i	−0.629417	−	0.777067i	$-0.716706\pi$
$313$	− 27.4955i	− 1.55413i	0.629417	−	0.777067i	$-0.283294\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.7490	1.78320	0.891601	−	0.452822i	$-0.149583\pi$
$317$	31.7490	1.78320	0.891601	+	0.452822i	$0.149583\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 9.16515i	− 0.509963i
$324$	0	0
$325$	32.0780i	1.77937i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−15.8745	−0.875190
$330$	0	0
$331$	29.4449i	1.61844i	0.587508	+	0.809218i	$0.300109\pi$
$331$	29.4449i	1.61844i	−0.587508	+	0.809218i	$0.699891\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	42.0000	2.29471
$336$	0	0
$337$	−1.00000	−0.0544735	−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000	−0.0544735	−0.0272367	+	0.999629i	$0.508671\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	18.5203	1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.4955i	1.47603i	0.674782	+	0.738017i	$0.264237\pi$
$347$	27.4955i	1.47603i	−0.674782	+	0.738017i	$0.735763\pi$
$348$	0	0
$349$	− 13.7477i	− 0.735899i	−0.929846	−	0.367949i	$-0.880060\pi$
$349$	− 13.7477i	− 0.735899i	0.929846	−	0.367949i	$-0.119940\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.73205i	− 0.0921878i	−0.998937	−	0.0460939i	$-0.985323\pi$
$353$	− 1.73205i	− 0.0921878i	0.998937	−	0.0460939i	$-0.0146773\pi$
$354$	0	0
$355$	−15.8745	−0.842531
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.0780i	1.69301i	0.532378	+	0.846507i	$0.321299\pi$
$359$	32.0780i	1.69301i	−0.532378	+	0.846507i	$0.678701\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	31.7490	1.66182
$366$	0	0
$367$	−2.64575	−0.138107	−0.0690535	−	0.997613i	$-0.521998\pi$
$367$	−2.64575	−0.138107	−0.0690535	+	0.997613i	$0.521998\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	21.0000	1.09027
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	36.3731i	1.87331i
$378$	0	0
$379$	− 10.3923i	− 0.533817i	−0.963722	−	0.266908i	$-0.913998\pi$
$379$	− 10.3923i	− 0.533817i	0.963722	−	0.266908i	$-0.0860021\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	31.7490	1.60974	0.804869	−	0.593452i	$-0.202235\pi$
$389$	31.7490	1.60974	0.804869	+	0.593452i	$0.202235\pi$
$390$	0	0
$391$	−7.93725	−0.401404
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−24.0000	−1.20757
$396$	0	0
$397$	9.16515i	0.459986i	0.973192	+	0.229993i	$0.0738703\pi$
$397$	9.16515i	0.459986i	−0.973192	+	0.229993i	$0.926130\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.8745	−0.792735	−0.396368	−	0.918092i	$-0.629729\pi$
$401$	−15.8745	−0.792735	−0.396368	+	0.918092i	$0.629729\pi$
$402$	0	0
$403$	− 12.1244i	− 0.603957i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 27.4955i	− 1.35956i	−0.733415	−	0.679781i	$-0.762075\pi$
$409$	− 27.4955i	− 1.35956i	0.733415	−	0.679781i	$-0.237925\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.93725	−0.390567
$414$	0	0
$415$	− 41.5692i	− 2.04055i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.1244i	0.588118i
$426$	0	0
$427$	− 24.2487i	− 1.17348i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.3303i	0.882940i	0.897276	+	0.441470i	$0.145543\pi$
$431$	18.3303i	0.882940i	−0.897276	+	0.441470i	$0.854457\pi$
$432$	0	0
$433$	18.3303i	0.880898i	0.897778	+	0.440449i	$0.145181\pi$
$433$	18.3303i	0.880898i	−0.897778	+	0.440449i	$0.854819\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 24.2487i	− 1.15997i
$438$	0	0
$439$	13.2288	0.631374	0.315687	−	0.948863i	$-0.397765\pi$
$439$	13.2288	0.631374	0.315687	+	0.948863i	$0.397765\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 18.3303i	− 0.870899i	−0.900213	−	0.435449i	$-0.856589\pi$
$443$	− 18.3303i	− 0.870899i	0.900213	−	0.435449i	$-0.143411\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.8745	−0.749164	−0.374582	−	0.927194i	$-0.622214\pi$
$449$	−15.8745	−0.749164	−0.374582	+	0.927194i	$0.622214\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−42.0000	−1.96899
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.1769i	1.45205i	0.687666	+	0.726027i	$0.258635\pi$
$461$	31.1769i	1.45205i	−0.687666	+	0.726027i	$0.741365\pi$
$462$	0	0
$463$	38.1051i	1.77090i	0.464739	+	0.885448i	$0.346148\pi$
$463$	38.1051i	1.77090i	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	32.0780i	1.48123i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−37.0405	−1.69954
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	− 18.3303i	− 0.835790i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	31.7490	1.44165
$486$	0	0
$487$	24.2487i	1.09881i	0.835555	+	0.549407i	$0.185146\pi$
$487$	24.2487i	1.09881i	−0.835555	+	0.549407i	$0.814854\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	36.6606i	1.65447i	0.561856	+	0.827235i	$0.310087\pi$
$491$	36.6606i	1.65447i	−0.561856	+	0.827235i	$0.689913\pi$
$492$	0	0
$493$	13.7477i	0.619166i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.1244i	− 0.543852i
$498$	0	0
$499$	31.1769i	1.39567i	0.716258	+	0.697835i	$0.245853\pi$
$499$	31.1769i	1.39567i	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 27.7128i	− 1.22835i	−0.789170	−	0.614174i	$-0.789489\pi$
$509$	− 27.7128i	− 1.22835i	0.789170	−	0.614174i	$-0.210511\pi$
$510$	0	0
$511$	24.2487i	1.07270i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	45.8258i	2.01932i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 25.9808i	− 1.13824i	−0.822255	−	0.569119i	$-0.807284\pi$
$521$	− 25.9808i	− 1.13824i	0.822255	−	0.569119i	$-0.192716\pi$
$522$	0	0
$523$	10.5830	0.462763	0.231381	−	0.972863i	$-0.425676\pi$
$523$	10.5830	0.462763	0.231381	+	0.972863i	$0.425676\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.58258i	− 0.199620i
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	0	0
$532$	0	0
$533$	31.7490	1.37520
$534$	0	0
$535$	−63.4980	−2.74526
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	55.4256i	2.37417i
$546$	0	0
$547$	− 38.1051i	− 1.62926i	−0.579983	−	0.814629i	$-0.696941\pi$
$547$	− 38.1051i	− 1.62926i	0.579983	−	0.814629i	$-0.303059\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−42.0000	−1.78926
$552$	0	0
$553$	− 18.3303i	− 0.779484i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.93725	0.336312	0.168156	−	0.985760i	$-0.446219\pi$
$557$	7.93725	0.336312	0.168156	+	0.985760i	$0.446219\pi$
$558$	0	0
$559$	−7.93725	−0.335710
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.0000	−0.885044	−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000	−0.885044	−0.442522	+	0.896758i	$0.645916\pi$
$564$	0	0
$565$	54.9909i	2.31348i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	− 15.5885i	− 0.652357i	−0.945308	−	0.326178i	$-0.894239\pi$
$571$	− 15.5885i	− 0.652357i	0.945308	−	0.326178i	$-0.105761\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	32.0780i	1.33775i
$576$	0	0
$577$	18.3303i	0.763100i	0.924348	+	0.381550i	$0.124610\pi$
$577$	18.3303i	0.763100i	−0.924348	+	0.381550i	$0.875390\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	31.7490	1.31717
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.0000	0.866763	0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000	0.866763	0.433381	+	0.901211i	$0.357320\pi$
$588$	0	0
$589$	14.0000	0.576860
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20.7846i	0.853522i	0.904365	+	0.426761i	$0.140345\pi$
$593$	20.7846i	0.853522i	−0.904365	+	0.426761i	$0.859655\pi$
$594$	0	0
$595$	−15.8745	−0.650791
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.0780i	1.31067i	0.755337	+	0.655336i	$0.227473\pi$
$599$	32.0780i	1.31067i	−0.755337	+	0.655336i	$0.772527\pi$
$600$	0	0
$601$	− 9.16515i	− 0.373854i	−0.982374	−	0.186927i	$-0.940147\pi$
$601$	− 9.16515i	− 0.373854i	0.982374	−	0.186927i	$-0.0598528\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 38.1051i	− 1.54919i
$606$	0	0
$607$	−18.5203	−0.751714	−0.375857	−	0.926678i	$-0.622652\pi$
$607$	−18.5203	−0.751714	−0.375857	+	0.926678i	$0.622652\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.4955i	1.11235i
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−5.29150	−0.212683	−0.106342	−	0.994330i	$-0.533914\pi$
$619$	−5.29150	−0.212683	−0.106342	+	0.994330i	$0.533914\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.7477i	0.550791i
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 6.92820i	− 0.276246i
$630$	0	0
$631$	− 10.3923i	− 0.413711i	−0.978371	−	0.206856i	$-0.933677\pi$
$631$	− 10.3923i	− 0.413711i	0.978371	−	0.206856i	$-0.0663230\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	48.0000	1.90482
$636$	0	0
$637$	− 32.0780i	− 1.27098i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.8745	0.627005	0.313503	−	0.949587i	$-0.398498\pi$
$641$	15.8745	0.627005	0.313503	+	0.949587i	$0.398498\pi$
$642$	0	0
$643$	5.29150	0.208676	0.104338	−	0.994542i	$-0.466728\pi$
$643$	5.29150	0.208676	0.104338	+	0.994542i	$0.466728\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23.8118	−0.931826	−0.465913	−	0.884830i	$-0.654274\pi$
$653$	−23.8118	−0.931826	−0.465913	+	0.884830i	$0.654274\pi$
$654$	0	0
$655$	− 72.7461i	− 2.84243i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.3303i	0.714047i	0.934095	+	0.357024i	$0.116208\pi$
$659$	18.3303i	0.714047i	−0.934095	+	0.357024i	$0.883792\pi$
$660$	0	0
$661$	9.16515i	0.356483i	0.983987	+	0.178242i	$0.0570408\pi$
$661$	9.16515i	0.356483i	−0.983987	+	0.178242i	$0.942959\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 48.4974i	− 1.88065i
$666$	0	0
$667$	36.3731i	1.40837i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	41.5692i	1.59763i	0.601574	+	0.798817i	$0.294541\pi$
$677$	41.5692i	1.59763i	−0.601574	+	0.798817i	$0.705459\pi$
$678$	0	0
$679$	24.2487i	0.930580i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 9.16515i	− 0.350695i	−0.984507	−	0.175347i	$-0.943895\pi$
$683$	− 9.16515i	− 0.350695i	0.984507	−	0.175347i	$-0.0561049\pi$
$684$	0	0
$685$	54.9909i	2.10109i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 36.3731i	− 1.38570i
$690$	0	0
$691$	−21.1660	−0.805193	−0.402596	−	0.915378i	$-0.631892\pi$
$691$	−21.1660	−0.805193	−0.402596	+	0.915378i	$0.631892\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.3303i	0.695308i
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31.7490	1.19914	0.599572	−	0.800321i	$-0.295338\pi$
$701$	31.7490	1.19914	0.599572	+	0.800321i	$0.295338\pi$
$702$	0	0
$703$	21.1660	0.798291
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.16515i	− 0.344691i
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 12.1244i	− 0.454061i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	−35.0000	−1.30347
$722$	0	0
$723$	0	0
$724$	0	0
$725$	55.5608	2.06348
$726$	0	0
$727$	34.3948	1.27563	0.637816	−	0.770189i	$-0.279838\pi$
$727$	34.3948	1.27563	0.637816	+	0.770189i	$0.279838\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	− 22.9129i	− 0.846306i	−0.906058	−	0.423153i	$-0.860923\pi$
$733$	− 22.9129i	− 0.846306i	0.906058	−	0.423153i	$-0.139077\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	24.2487i	0.892003i	0.895032	+	0.446002i	$0.147152\pi$
$739$	24.2487i	0.892003i	−0.895032	+	0.446002i	$0.852848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.7477i	0.504355i	0.967681	+	0.252178i	$0.0811467\pi$
$743$	13.7477i	0.504355i	−0.967681	+	0.252178i	$0.918853\pi$
$744$	0	0
$745$	− 27.4955i	− 1.00736i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 48.4974i	− 1.77206i
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 8.66025i	− 0.313934i	−0.987604	−	0.156967i	$-0.949828\pi$
$761$	− 8.66025i	− 0.313934i	0.987604	−	0.156967i	$-0.0501716\pi$
$762$	0	0
$763$	−42.3320	−1.53252
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.7477i	0.496402i
$768$	0	0
$769$	9.16515i	0.330504i	0.986251	+	0.165252i	$0.0528437\pi$
$769$	9.16515i	0.330504i	−0.986251	+	0.165252i	$0.947156\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.8564i	0.498380i	0.968455	+	0.249190i	$0.0801644\pi$
$773$	13.8564i	0.498380i	−0.968455	+	0.249190i	$0.919836\pi$
$774$	0	0
$775$	−18.5203	−0.665267
$776$	0	0
$777$	0	0
$778$	0	0
$779$	36.6606i	1.31350i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−79.3725	−2.83293
$786$	0	0
$787$	26.4575	0.943108	0.471554	−	0.881837i	$-0.343693\pi$
$787$	26.4575	0.943108	0.471554	+	0.881837i	$0.343693\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−42.0000	−1.49335
$792$	0	0
$793$	−42.0000	−1.49146
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.6410i	1.22705i	0.789676	+	0.613524i	$0.210249\pi$
$797$	34.6410i	1.22705i	−0.789676	+	0.613524i	$0.789751\pi$
$798$	0	0
$799$	10.3923i	0.367653i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−42.0000	−1.48031
$806$	0	0
$807$	0	0
$808$	0	0
$809$	47.6235	1.67435	0.837177	−	0.546932i	$-0.184204\pi$
$809$	47.6235	1.67435	0.837177	+	0.546932i	$0.184204\pi$
$810$	0	0
$811$	42.3320	1.48648	0.743239	−	0.669026i	$-0.233288\pi$
$811$	42.3320	1.48648	0.743239	+	0.669026i	$0.233288\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.0000	−0.630512
$816$	0	0
$817$	− 9.16515i	− 0.320648i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.93725	0.277012	0.138506	−	0.990362i	$-0.455770\pi$
$821$	7.93725	0.277012	0.138506	+	0.990362i	$0.455770\pi$
$822$	0	0
$823$	3.46410i	0.120751i	0.998176	+	0.0603755i	$0.0192298\pi$
$823$	3.46410i	0.120751i	−0.998176	+	0.0603755i	$0.980770\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.4955i	0.956111i	0.878330	+	0.478055i	$0.158658\pi$
$827$	27.4955i	0.956111i	−0.878330	+	0.478055i	$0.841342\pi$
$828$	0	0
$829$	− 45.8258i	− 1.59159i	−0.605563	−	0.795797i	$-0.707052\pi$
$829$	− 45.8258i	− 1.59159i	0.605563	−	0.795797i	$-0.292948\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 12.1244i	− 0.420084i
$834$	0	0
$835$	41.5692i	1.43856i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	34.0000	1.17241
$842$	0	0
$843$	0	0
$844$	0	0
$845$	27.7128i	0.953350i
$846$	0	0
$847$	29.1033	1.00000
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 18.3303i	− 0.628355i
$852$	0	0
$853$	− 32.0780i	− 1.09833i	−0.835714	−	0.549165i	$-0.814946\pi$
$853$	− 32.0780i	− 1.09833i	0.835714	−	0.549165i	$-0.185054\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.73205i	0.0591657i	0.999562	+	0.0295829i	$0.00941789\pi$
$857$	1.73205i	0.0591657i	−0.999562	+	0.0295829i	$0.990582\pi$
$858$	0	0
$859$	52.9150	1.80544	0.902719	−	0.430231i	$-0.141568\pi$
$859$	52.9150	1.80544	0.902719	+	0.430231i	$0.141568\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 41.2432i	− 1.40393i	−0.712209	−	0.701967i	$-0.752305\pi$
$863$	− 41.2432i	− 1.40393i	0.712209	−	0.701967i	$-0.247695\pi$
$864$	0	0
$865$	60.0000	2.04006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	55.5608	1.88261
$872$	0	0
$873$	0	0
$874$	0	0
$875$	18.3303i	0.619677i
$876$	0	0
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	50.2295i	1.69227i	0.532965	+	0.846137i	$0.321078\pi$
$881$	50.2295i	1.69227i	−0.532965	+	0.846137i	$0.678922\pi$
$882$	0	0
$883$	− 22.5167i	− 0.757746i	−0.925449	−	0.378873i	$-0.876312\pi$
$883$	− 22.5167i	− 0.757746i	0.925449	−	0.378873i	$-0.123688\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	0	0
$889$	36.6606i	1.22956i
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−31.7490	−1.06244
$894$	0	0
$895$	−31.7490	−1.06125
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−21.0000	−0.700389
$900$	0	0
$901$	− 13.7477i	− 0.458003i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	47.6235	1.58306
$906$	0	0
$907$	− 45.0333i	− 1.49531i	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	− 45.0333i	− 1.49531i	0.664089	−	0.747653i	$-0.268820\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	54.9909i	1.82193i	0.412484	+	0.910965i	$0.364661\pi$
$911$	54.9909i	1.82193i	−0.412484	+	0.910965i	$0.635339\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	55.5608	1.83478
$918$	0	0
$919$	− 41.5692i	− 1.37124i	−0.727959	−	0.685621i	$-0.759531\pi$
$919$	− 41.5692i	− 1.37124i	0.727959	−	0.685621i	$-0.240469\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−21.0000	−0.691223
$924$	0	0
$925$	−28.0000	−0.920634
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27.7128i	0.909228i	0.890689	+	0.454614i	$0.150223\pi$
$929$	27.7128i	0.909228i	−0.890689	+	0.454614i	$0.849777\pi$
$930$	0	0
$931$	37.0405	1.21395
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 27.4955i	− 0.898237i	−0.893472	−	0.449119i	$-0.851738\pi$
$937$	− 27.4955i	− 0.898237i	0.893472	−	0.449119i	$-0.148262\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.3923i	0.338779i	0.985549	+	0.169390i	$0.0541797\pi$
$941$	10.3923i	0.338779i	−0.985549	+	0.169390i	$0.945820\pi$
$942$	0	0
$943$	31.7490	1.03389
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.8258i	1.48914i	0.667546	+	0.744569i	$0.267345\pi$
$947$	45.8258i	1.48914i	−0.667546	+	0.744569i	$0.732655\pi$
$948$	0	0
$949$	42.0000	1.36338
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−42.0000	−1.35625
$960$	0	0
$961$	−24.0000	−0.774194
$962$	0	0
$963$	0	0
$964$	0	0
$965$	79.6743i	2.56481i
$966$	0	0
$967$	− 24.2487i	− 0.779786i	−0.920860	−	0.389893i	$-0.872512\pi$
$967$	− 24.2487i	− 0.779786i	0.920860	−	0.389893i	$-0.127488\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	0	0
$973$	−14.0000	−0.448819
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−47.6235	−1.52361	−0.761806	−	0.647806i	$-0.775687\pi$
$977$	−47.6235	−1.52361	−0.761806	+	0.647806i	$0.775687\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.93725	−0.252390
$990$	0	0
$991$	− 3.46410i	− 0.110041i	−0.998485	−	0.0550204i	$-0.982478\pi$
$991$	− 3.46410i	− 0.110041i	0.998485	−	0.0550204i	$-0.0175224\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 45.8258i	− 1.45277i
$996$	0	0
$997$	− 13.7477i	− 0.435395i	−0.976016	−	0.217697i	$-0.930145\pi$
$997$	− 13.7477i	− 0.435395i	0.976016	−	0.217697i	$-0.0698546\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.b.r.1567.2	yes	4
3.2	odd	2		3024.2.b.s.1567.4	yes	4
4.3	odd	2		3024.2.b.s.1567.1	yes	4
7.6	odd	2		3024.2.b.s.1567.3	yes	4
12.11	even	2	inner	3024.2.b.r.1567.3	yes	4
21.20	even	2	inner	3024.2.b.r.1567.1	✓	4
28.27	even	2	inner	3024.2.b.r.1567.4	yes	4
84.83	odd	2		3024.2.b.s.1567.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3024.2.b.r.1567.1	✓	4	21.20	even	2	inner
3024.2.b.r.1567.2	yes	4	1.1	even	1	trivial
3024.2.b.r.1567.3	yes	4	12.11	even	2	inner
3024.2.b.r.1567.4	yes	4	28.27	even	2	inner
3024.2.b.s.1567.1	yes	4	4.3	odd	2
3024.2.b.s.1567.2	yes	4	84.83	odd	2
3024.2.b.s.1567.3	yes	4	7.6	odd	2
3024.2.b.s.1567.4	yes	4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.46410i q^{5} +2.64575 q^{7} +O(q^{10})\)	\(q-3.46410i q^{5} +2.64575 q^{7} -4.58258i q^{13} -1.73205i q^{17} +5.29150 q^{19} -4.58258i q^{23} -7.00000 q^{25} -7.93725 q^{29} +2.64575 q^{31} -9.16515i q^{35} +4.00000 q^{37} +6.92820i q^{41} -1.73205i q^{43} -6.00000 q^{47} +7.00000 q^{49} +7.93725 q^{53} -3.00000 q^{59} -9.16515i q^{61} -15.8745 q^{65} +12.1244i q^{67} -4.58258i q^{71} +9.16515i q^{73} -6.92820i q^{79} +12.0000 q^{83} -6.00000 q^{85} +5.19615i q^{89} -12.1244i q^{91} -18.3303i q^{95} +9.16515i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 28 q^{25} + 16 q^{37} - 24 q^{47} + 28 q^{49} - 12 q^{59} + 48 q^{83} - 24 q^{85}+O(q^{100}) \) 4 * q - 28 * q^25 + 16 * q^37 - 24 * q^47 + 28 * q^49 - 12 * q^59 + 48 * q^83 - 24 * q^85

Introduction

L-functions

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