LMFDB - Embedded newform 3168.2.k.a.1871.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3168,2,Mod(1871,3168)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3168.1871"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1871.12
Character	$\chi$	$=$	3168.1871
Dual form			3168.2.k.a.1871.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.89382 q^{5} -3.03135i q^{7} +1.00000i q^{11} +5.44282i q^{13} -6.40451i q^{17} +5.86923 q^{19} +2.92580 q^{23} -1.41343 q^{25} +0.600643 q^{29} +10.8348i q^{31} +5.74084i q^{35} +1.96238i q^{37} -8.98233i q^{41} -2.80132 q^{43} +0.286711 q^{47} -2.18907 q^{49} -10.9248 q^{53} -1.89382i q^{55} -6.10246i q^{59} -14.1808i q^{61} -10.3078i q^{65} +12.2822 q^{67} +9.74776 q^{71} +7.25976 q^{73} +3.03135 q^{77} -3.48834i q^{79} -9.17926i q^{83} +12.1290i q^{85} +9.80721i q^{89} +16.4991 q^{91} -11.1153 q^{95} +4.90282 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 32 q^{19} + 40 q^{25} - 8 q^{49} + 32 q^{73} - 32 q^{97}+O(q^{100}) $ 40 * q - 32 * q^19 + 40 * q^25 - 8 * q^49 + 32 * q^73 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$5$	−1.89382	−0.846944	−0.423472	−	0.905909i	$-0.639189\pi$
$5$	−1.89382	−0.846944	−0.423472	+	0.905909i	$0.639189\pi$
$6$	0	0
$7$	− 3.03135i	− 1.14574i	−0.819646	−	0.572871i	$-0.805830\pi$
$7$	− 3.03135i	− 1.14574i	0.819646	−	0.572871i	$-0.194170\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	5.44282i	1.50957i	0.655974	+	0.754784i	$0.272258\pi$
$13$	5.44282i	1.50957i	−0.655974	+	0.754784i	$0.727742\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.40451i	− 1.55332i	−0.629919	−	0.776661i	$-0.716912\pi$
$17$	− 6.40451i	− 1.55332i	0.629919	−	0.776661i	$-0.283088\pi$
$18$	0	0
$19$	5.86923	1.34649	0.673247	−	0.739418i	$-0.264899\pi$
$19$	5.86923	1.34649	0.673247	+	0.739418i	$0.264899\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.92580	0.610072	0.305036	−	0.952341i	$-0.401332\pi$
$23$	2.92580	0.610072	0.305036	+	0.952341i	$0.401332\pi$
$24$	0	0
$25$	−1.41343	−0.282685
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.600643	0.111537	0.0557683	−	0.998444i	$-0.482239\pi$
$29$	0.600643	0.111537	0.0557683	+	0.998444i	$0.482239\pi$
$30$	0	0
$31$	10.8348i	1.94599i	0.230823	+	0.972996i	$0.425858\pi$
$31$	10.8348i	1.94599i	−0.230823	+	0.972996i	$0.574142\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.74084i	0.970379i
$36$	0	0
$37$	1.96238i	0.322613i	0.986904	+	0.161307i	$0.0515708\pi$
$37$	1.96238i	0.322613i	−0.986904	+	0.161307i	$0.948429\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.98233i	− 1.40280i	−0.712766	−	0.701402i	$-0.752558\pi$
$41$	− 8.98233i	− 1.40280i	0.712766	−	0.701402i	$-0.247442\pi$
$42$	0	0
$43$	−2.80132	−0.427198	−0.213599	−	0.976921i	$-0.568519\pi$
$43$	−2.80132	−0.427198	−0.213599	+	0.976921i	$0.568519\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.286711	0.0418212	0.0209106	−	0.999781i	$-0.493343\pi$
$47$	0.286711	0.0418212	0.0209106	+	0.999781i	$0.493343\pi$
$48$	0	0
$49$	−2.18907	−0.312724
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.9248	−1.50063	−0.750316	−	0.661080i	$-0.770098\pi$
$53$	−10.9248	−1.50063	−0.750316	+	0.661080i	$0.770098\pi$
$54$	0	0
$55$	− 1.89382i	− 0.255363i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 6.10246i	− 0.794473i	−0.917716	−	0.397237i	$-0.869969\pi$
$59$	− 6.10246i	− 0.794473i	0.917716	−	0.397237i	$-0.130031\pi$
$60$	0	0
$61$	− 14.1808i	− 1.81567i	−0.419328	−	0.907835i	$-0.637734\pi$
$61$	− 14.1808i	− 1.81567i	0.419328	−	0.907835i	$-0.362266\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 10.3078i	− 1.27852i
$66$	0	0
$67$	12.2822	1.50051	0.750256	−	0.661147i	$-0.229930\pi$
$67$	12.2822	1.50051	0.750256	+	0.661147i	$0.229930\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.74776	1.15685	0.578423	−	0.815737i	$-0.303668\pi$
$71$	9.74776	1.15685	0.578423	+	0.815737i	$0.303668\pi$
$72$	0	0
$73$	7.25976	0.849690	0.424845	−	0.905266i	$-0.360329\pi$
$73$	7.25976	0.849690	0.424845	+	0.905266i	$0.360329\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.03135	0.345454
$78$	0	0
$79$	− 3.48834i	− 0.392469i	−0.980557	−	0.196234i	$-0.937129\pi$
$79$	− 3.48834i	− 0.392469i	0.980557	−	0.196234i	$-0.0628713\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.17926i	− 1.00755i	−0.863834	−	0.503777i	$-0.831943\pi$
$83$	− 9.17926i	− 1.00755i	0.863834	−	0.503777i	$-0.168057\pi$
$84$	0	0
$85$	12.1290i	1.31558i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.80721i	1.03956i	0.854299	+	0.519781i	$0.173986\pi$
$89$	9.80721i	1.03956i	−0.854299	+	0.519781i	$0.826014\pi$
$90$	0	0
$91$	16.4991	1.72957
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−11.1153	−1.14040
$96$	0	0
$97$	4.90282	0.497806	0.248903	−	0.968528i	$-0.419930\pi$
$97$	4.90282	0.497806	0.248903	+	0.968528i	$0.419930\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.89361	−0.884947	−0.442474	−	0.896782i	$-0.645899\pi$
$101$	−8.89361	−0.884947	−0.442474	+	0.896782i	$0.645899\pi$
$102$	0	0
$103$	− 18.9666i	− 1.86883i	−0.356181	−	0.934417i	$-0.615921\pi$
$103$	− 18.9666i	− 1.86883i	0.356181	−	0.934417i	$-0.384079\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.57243i	− 0.345360i	−0.984978	−	0.172680i	$-0.944757\pi$
$107$	− 3.57243i	− 0.345360i	0.984978	−	0.172680i	$-0.0552427\pi$
$108$	0	0
$109$	− 7.99636i	− 0.765912i	−0.923767	−	0.382956i	$-0.874906\pi$
$109$	− 7.99636i	− 0.765912i	0.923767	−	0.382956i	$-0.125094\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 16.1728i	− 1.52141i	−0.649100	−	0.760703i	$-0.724854\pi$
$113$	− 16.1728i	− 1.52141i	0.649100	−	0.760703i	$-0.275146\pi$
$114$	0	0
$115$	−5.54096	−0.516697
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−19.4143	−1.77971
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1459	1.08636
$126$	0	0
$127$	− 12.9725i	− 1.15113i	−0.817757	−	0.575563i	$-0.804783\pi$
$127$	− 12.9725i	− 1.15113i	0.817757	−	0.575563i	$-0.195217\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.79197i	0.243936i	0.992534	+	0.121968i	$0.0389205\pi$
$131$	2.79197i	0.243936i	−0.992534	+	0.121968i	$0.961080\pi$
$132$	0	0
$133$	− 17.7917i	− 1.54273i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.92047i	0.164077i	0.996629	+	0.0820386i	$0.0261431\pi$
$137$	1.92047i	0.164077i	−0.996629	+	0.0820386i	$0.973857\pi$
$138$	0	0
$139$	5.27856	0.447722	0.223861	−	0.974621i	$-0.428134\pi$
$139$	5.27856	0.447722	0.223861	+	0.974621i	$0.428134\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.44282	−0.455152
$144$	0	0
$145$	−1.13751	−0.0944654
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.86168	−0.562131	−0.281065	−	0.959689i	$-0.590688\pi$
$149$	−6.86168	−0.562131	−0.281065	+	0.959689i	$0.590688\pi$
$150$	0	0
$151$	3.42172i	0.278456i	0.990260	+	0.139228i	$0.0444621\pi$
$151$	3.42172i	0.278456i	−0.990260	+	0.139228i	$0.955538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 20.5193i	− 1.64815i
$156$	0	0
$157$	11.0872i	0.884852i	0.896805	+	0.442426i	$0.145882\pi$
$157$	11.0872i	0.884852i	−0.896805	+	0.442426i	$0.854118\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 8.86912i	− 0.698985i
$162$	0	0
$163$	−4.41416	−0.345744	−0.172872	−	0.984944i	$-0.555305\pi$
$163$	−4.41416	−0.345744	−0.172872	+	0.984944i	$0.555305\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.3233	0.798839	0.399419	−	0.916768i	$-0.369212\pi$
$167$	10.3233	0.798839	0.399419	+	0.916768i	$0.369212\pi$
$168$	0	0
$169$	−16.6243	−1.27879
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.3286	−0.861301	−0.430650	−	0.902519i	$-0.641716\pi$
$173$	−11.3286	−0.861301	−0.430650	+	0.902519i	$0.641716\pi$
$174$	0	0
$175$	4.28459i	0.323885i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 5.71663i	− 0.427281i	−0.976912	−	0.213640i	$-0.931468\pi$
$179$	− 5.71663i	− 0.427281i	0.976912	−	0.213640i	$-0.0685321\pi$
$180$	0	0
$181$	5.36910i	0.399082i	0.979889	+	0.199541i	$0.0639451\pi$
$181$	5.36910i	0.399082i	−0.979889	+	0.199541i	$0.936055\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 3.71640i	− 0.273235i
$186$	0	0
$187$	6.40451	0.468344
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.76090	0.706274	0.353137	−	0.935572i	$-0.385115\pi$
$191$	9.76090	0.706274	0.353137	+	0.935572i	$0.385115\pi$
$192$	0	0
$193$	−13.7187	−0.987492	−0.493746	−	0.869606i	$-0.664373\pi$
$193$	−13.7187	−0.987492	−0.493746	+	0.869606i	$0.664373\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.5373	1.17823	0.589116	−	0.808048i	$-0.299476\pi$
$197$	16.5373	1.17823	0.589116	+	0.808048i	$0.299476\pi$
$198$	0	0
$199$	− 16.6620i	− 1.18114i	−0.806986	−	0.590570i	$-0.798903\pi$
$199$	− 16.6620i	− 1.18114i	0.806986	−	0.590570i	$-0.201097\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.82076i	− 0.127792i
$204$	0	0
$205$	17.0110i	1.18810i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.86923i	0.405983i
$210$	0	0
$211$	13.5380	0.931997	0.465998	−	0.884786i	$-0.345695\pi$
$211$	13.5380	0.931997	0.465998	+	0.884786i	$0.345695\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.30521	0.361812
$216$	0	0
$217$	32.8441	2.22960
$218$	0	0
$219$	0	0
$220$	0	0
$221$	34.8586	2.34484
$222$	0	0
$223$	7.44081i	0.498274i	0.968468	+	0.249137i	$0.0801469\pi$
$223$	7.44081i	0.498274i	−0.968468	+	0.249137i	$0.919853\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 17.8423i	− 1.18424i	−0.805851	−	0.592118i	$-0.798292\pi$
$227$	− 17.8423i	− 1.18424i	0.805851	−	0.592118i	$-0.201708\pi$
$228$	0	0
$229$	− 1.69896i	− 0.112271i	−0.998423	−	0.0561353i	$-0.982122\pi$
$229$	− 1.69896i	− 0.112271i	0.998423	−	0.0561353i	$-0.0178778\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 20.6370i	− 1.35197i	−0.736914	−	0.675986i	$-0.763718\pi$
$233$	− 20.6370i	− 1.35197i	0.736914	−	0.675986i	$-0.236282\pi$
$234$	0	0
$235$	−0.542981	−0.0354202
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.1366	0.720369	0.360185	−	0.932881i	$-0.382714\pi$
$239$	11.1366	0.720369	0.360185	+	0.932881i	$0.382714\pi$
$240$	0	0
$241$	21.8983	1.41059	0.705297	−	0.708912i	$-0.250814\pi$
$241$	21.8983	1.41059	0.705297	+	0.708912i	$0.250814\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.14571	0.264860
$246$	0	0
$247$	31.9452i	2.03262i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4.82931i	− 0.304823i	−0.988317	−	0.152412i	$-0.951296\pi$
$251$	− 4.82931i	− 0.304823i	0.988317	−	0.152412i	$-0.0487040\pi$
$252$	0	0
$253$	2.92580i	0.183944i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.67491i	− 0.291613i	−0.989313	−	0.145806i	$-0.953422\pi$
$257$	− 4.67491i	− 0.291613i	0.989313	−	0.145806i	$-0.0465776\pi$
$258$	0	0
$259$	5.94865	0.369631
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.52091	−0.402096	−0.201048	−	0.979581i	$-0.564435\pi$
$263$	−6.52091	−0.402096	−0.201048	+	0.979581i	$0.564435\pi$
$264$	0	0
$265$	20.6896	1.27095
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.26314	−0.259928	−0.129964	−	0.991519i	$-0.541486\pi$
$269$	−4.26314	−0.259928	−0.129964	+	0.991519i	$0.541486\pi$
$270$	0	0
$271$	2.03841i	0.123824i	0.998082	+	0.0619122i	$0.0197199\pi$
$271$	2.03841i	0.123824i	−0.998082	+	0.0619122i	$0.980280\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.41343i	− 0.0852329i
$276$	0	0
$277$	− 13.8227i	− 0.830526i	−0.909701	−	0.415263i	$-0.863690\pi$
$277$	− 13.8227i	− 0.830526i	0.909701	−	0.415263i	$-0.136310\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.3473i	0.915543i	0.889070	+	0.457772i	$0.151352\pi$
$281$	15.3473i	0.915543i	−0.889070	+	0.457772i	$0.848648\pi$
$282$	0	0
$283$	6.72089	0.399516	0.199758	−	0.979845i	$-0.435984\pi$
$283$	6.72089	0.399516	0.199758	+	0.979845i	$0.435984\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−27.2286	−1.60725
$288$	0	0
$289$	−24.0178	−1.41281
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−28.4865	−1.66420	−0.832098	−	0.554628i	$-0.812861\pi$
$293$	−28.4865	−1.66420	−0.832098	+	0.554628i	$0.812861\pi$
$294$	0	0
$295$	11.5570i	0.672874i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	15.9246i	0.920944i
$300$	0	0
$301$	8.49178i	0.489458i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	26.8560i	1.53777i
$306$	0	0
$307$	−12.0385	−0.687071	−0.343536	−	0.939140i	$-0.611625\pi$
$307$	−12.0385	−0.687071	−0.343536	+	0.939140i	$0.611625\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.0967	1.02617	0.513084	−	0.858338i	$-0.328503\pi$
$311$	18.0967	1.02617	0.513084	+	0.858338i	$0.328503\pi$
$312$	0	0
$313$	29.1782	1.64925	0.824625	−	0.565680i	$-0.191386\pi$
$313$	29.1782	1.64925	0.824625	+	0.565680i	$0.191386\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.26052	−0.295460	−0.147730	−	0.989028i	$-0.547197\pi$
$317$	−5.26052	−0.295460	−0.147730	+	0.989028i	$0.547197\pi$
$318$	0	0
$319$	0.600643i	0.0336296i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 37.5895i	− 2.09154i
$324$	0	0
$325$	− 7.69303i	− 0.426733i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 0.869122i	− 0.0479162i
$330$	0	0
$331$	−24.3562	−1.33874	−0.669369	−	0.742930i	$-0.733435\pi$
$331$	−24.3562	−1.33874	−0.669369	+	0.742930i	$0.733435\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−23.2604	−1.27085
$336$	0	0
$337$	−11.1496	−0.607359	−0.303680	−	0.952774i	$-0.598215\pi$
$337$	−11.1496	−0.607359	−0.303680	+	0.952774i	$0.598215\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10.8348	−0.586739
$342$	0	0
$343$	− 14.5836i	− 0.787441i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.87537i	0.422772i	0.977403	+	0.211386i	$0.0677977\pi$
$347$	7.87537i	0.422772i	−0.977403	+	0.211386i	$0.932202\pi$
$348$	0	0
$349$	− 5.06335i	− 0.271035i	−0.990775	−	0.135517i	$-0.956730\pi$
$349$	− 5.06335i	− 0.271035i	0.990775	−	0.135517i	$-0.0432696\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.4608i	1.30192i	0.759114	+	0.650958i	$0.225633\pi$
$353$	24.4608i	1.30192i	−0.759114	+	0.650958i	$0.774367\pi$
$354$	0	0
$355$	−18.4605	−0.979784
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.36867	0.283348	0.141674	−	0.989913i	$-0.454752\pi$
$359$	5.36867	0.283348	0.141674	+	0.989913i	$0.454752\pi$
$360$	0	0
$361$	15.4478	0.813044
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−13.7487	−0.719640
$366$	0	0
$367$	28.9008i	1.50861i	0.656525	+	0.754304i	$0.272026\pi$
$367$	28.9008i	1.50861i	−0.656525	+	0.754304i	$0.727974\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	33.1167i	1.71934i
$372$	0	0
$373$	− 32.3825i	− 1.67670i	−0.545130	−	0.838352i	$-0.683520\pi$
$373$	− 32.3825i	− 1.67670i	0.545130	−	0.838352i	$-0.316480\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.26920i	0.168372i
$378$	0	0
$379$	16.7643	0.861123	0.430562	−	0.902561i	$-0.358315\pi$
$379$	16.7643	0.861123	0.430562	+	0.902561i	$0.358315\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.2458	−0.574634	−0.287317	−	0.957836i	$-0.592763\pi$
$383$	−11.2458	−0.574634	−0.287317	+	0.957836i	$0.592763\pi$
$384$	0	0
$385$	−5.74084	−0.292580
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.3472	0.778136	0.389068	−	0.921209i	$-0.372797\pi$
$389$	15.3472	0.778136	0.389068	+	0.921209i	$0.372797\pi$
$390$	0	0
$391$	− 18.7383i	− 0.947638i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.60630i	0.332399i
$396$	0	0
$397$	13.8736i	0.696297i	0.937439	+	0.348148i	$0.113189\pi$
$397$	13.8736i	0.696297i	−0.937439	+	0.348148i	$0.886811\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 8.36307i	− 0.417632i	−0.977955	−	0.208816i	$-0.933039\pi$
$401$	− 8.36307i	− 0.417632i	0.977955	−	0.208816i	$-0.0669609\pi$
$402$	0	0
$403$	−58.9720	−2.93761
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.96238	−0.0972715
$408$	0	0
$409$	20.6740	1.02226	0.511132	−	0.859502i	$-0.329226\pi$
$409$	20.6740	1.02226	0.511132	+	0.859502i	$0.329226\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−18.4987	−0.910261
$414$	0	0
$415$	17.3839i	0.853343i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.14867i	0.202676i	0.994852	+	0.101338i	$0.0323123\pi$
$419$	4.14867i	0.202676i	−0.994852	+	0.101338i	$0.967688\pi$
$420$	0	0
$421$	17.8077i	0.867896i	0.900938	+	0.433948i	$0.142880\pi$
$421$	17.8077i	0.867896i	−0.900938	+	0.433948i	$0.857120\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.05231i	0.439102i
$426$	0	0
$427$	−42.9870	−2.08029
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.5960	1.42559	0.712796	−	0.701372i	$-0.247429\pi$
$431$	29.5960	1.42559	0.712796	+	0.701372i	$0.247429\pi$
$432$	0	0
$433$	−13.0944	−0.629278	−0.314639	−	0.949211i	$-0.601883\pi$
$433$	−13.0944	−0.629278	−0.314639	+	0.949211i	$0.601883\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.1722	0.821457
$438$	0	0
$439$	2.26583i	0.108142i	0.998537	+	0.0540710i	$0.0172197\pi$
$439$	2.26583i	0.108142i	−0.998537	+	0.0540710i	$0.982780\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.704612i	0.0334771i	0.999860	+	0.0167386i	$0.00532830\pi$
$443$	0.704612i	0.0334771i	−0.999860	+	0.0167386i	$0.994672\pi$
$444$	0	0
$445$	− 18.5731i	− 0.880451i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 28.1301i	− 1.32754i	−0.747935	−	0.663771i	$-0.768955\pi$
$449$	− 28.1301i	− 1.32754i	0.747935	−	0.663771i	$-0.231045\pi$
$450$	0	0
$451$	8.98233	0.422961
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−31.2464	−1.46485
$456$	0	0
$457$	−1.92294	−0.0899512	−0.0449756	−	0.998988i	$-0.514321\pi$
$457$	−1.92294	−0.0899512	−0.0449756	+	0.998988i	$0.514321\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.1590	−0.985474	−0.492737	−	0.870178i	$-0.664003\pi$
$461$	−21.1590	−0.985474	−0.492737	+	0.870178i	$0.664003\pi$
$462$	0	0
$463$	9.13809i	0.424683i	0.977195	+	0.212342i	$0.0681090\pi$
$463$	9.13809i	0.424683i	−0.977195	+	0.212342i	$0.931891\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 28.4368i	− 1.31590i	−0.753063	−	0.657949i	$-0.771424\pi$
$467$	− 28.4368i	− 1.31590i	0.753063	−	0.657949i	$-0.228576\pi$
$468$	0	0
$469$	− 37.2317i	− 1.71920i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 2.80132i	− 0.128805i
$474$	0	0
$475$	−8.29573	−0.380634
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.6250	−1.39929	−0.699646	−	0.714490i	$-0.746659\pi$
$479$	−30.6250	−1.39929	−0.699646	+	0.714490i	$0.746659\pi$
$480$	0	0
$481$	−10.6809	−0.487006
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.28508	−0.421614
$486$	0	0
$487$	12.3155i	0.558067i	0.960281	+	0.279033i	$0.0900140\pi$
$487$	12.3155i	0.558067i	−0.960281	+	0.279033i	$0.909986\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	34.9081i	1.57538i	0.616072	+	0.787690i	$0.288723\pi$
$491$	34.9081i	1.57538i	−0.616072	+	0.787690i	$0.711277\pi$
$492$	0	0
$493$	− 3.84683i	− 0.173252i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 29.5488i	− 1.32545i
$498$	0	0
$499$	3.73964	0.167409	0.0837047	−	0.996491i	$-0.473325\pi$
$499$	3.73964	0.167409	0.0837047	+	0.996491i	$0.473325\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8.50418	−0.379182	−0.189591	−	0.981863i	$-0.560716\pi$
$503$	−8.50418	−0.379182	−0.189591	+	0.981863i	$0.560716\pi$
$504$	0	0
$505$	16.8429	0.749501
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.2510	0.587342	0.293671	−	0.955907i	$-0.405123\pi$
$509$	13.2510	0.587342	0.293671	+	0.955907i	$0.405123\pi$
$510$	0	0
$511$	− 22.0068i	− 0.973526i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	35.9194i	1.58280i
$516$	0	0
$517$	0.286711i	0.0126096i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	37.2904i	1.63372i	0.576836	+	0.816860i	$0.304287\pi$
$521$	37.2904i	1.63372i	−0.576836	+	0.816860i	$0.695713\pi$
$522$	0	0
$523$	20.1065	0.879196	0.439598	−	0.898195i	$-0.355121\pi$
$523$	20.1065	0.879196	0.439598	+	0.898195i	$0.355121\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	69.3917	3.02275
$528$	0	0
$529$	−14.4397	−0.627812
$530$	0	0
$531$	0	0
$532$	0	0
$533$	48.8892	2.11763
$534$	0	0
$535$	6.76557i	0.292501i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.18907i	− 0.0942899i
$540$	0	0
$541$	− 2.44647i	− 0.105182i	−0.998616	−	0.0525911i	$-0.983252\pi$
$541$	− 2.44647i	− 0.105182i	0.998616	−	0.0525911i	$-0.0167480\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	15.1437i	0.648685i
$546$	0	0
$547$	43.2279	1.84829	0.924147	−	0.382038i	$-0.124777\pi$
$547$	43.2279	1.84829	0.924147	+	0.382038i	$0.124777\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.52531	0.150183
$552$	0	0
$553$	−10.5744	−0.449668
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.3302	1.79359	0.896795	−	0.442447i	$-0.145890\pi$
$557$	42.3302	1.79359	0.896795	+	0.442447i	$0.145890\pi$
$558$	0	0
$559$	− 15.2471i	− 0.644883i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	46.1269i	1.94402i	0.234942	+	0.972009i	$0.424510\pi$
$563$	46.1269i	1.94402i	−0.234942	+	0.972009i	$0.575490\pi$
$564$	0	0
$565$	30.6284i	1.28855i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.6718i	1.20198i	0.799255	+	0.600992i	$0.205228\pi$
$569$	28.6718i	1.20198i	−0.799255	+	0.600992i	$0.794772\pi$
$570$	0	0
$571$	−8.59888	−0.359852	−0.179926	−	0.983680i	$-0.557586\pi$
$571$	−8.59888	−0.359852	−0.179926	+	0.983680i	$0.557586\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.13541	−0.172458
$576$	0	0
$577$	0.581402	0.0242041	0.0121020	−	0.999927i	$-0.496148\pi$
$577$	0.581402	0.0242041	0.0121020	+	0.999927i	$0.496148\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−27.8255	−1.15440
$582$	0	0
$583$	− 10.9248i	− 0.452457i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 44.5638i	− 1.83934i	−0.392686	−	0.919672i	$-0.628454\pi$
$587$	− 44.5638i	− 1.83934i	0.392686	−	0.919672i	$-0.371546\pi$
$588$	0	0
$589$	63.5920i	2.62026i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.3669i	1.16489i	0.812870	+	0.582445i	$0.197904\pi$
$593$	28.3669i	1.16489i	−0.812870	+	0.582445i	$0.802096\pi$
$594$	0	0
$595$	36.7673	1.50731
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.6490	0.843696	0.421848	−	0.906667i	$-0.361382\pi$
$599$	20.6490	0.843696	0.421848	+	0.906667i	$0.361382\pi$
$600$	0	0
$601$	−38.2236	−1.55917	−0.779587	−	0.626295i	$-0.784571\pi$
$601$	−38.2236	−1.55917	−0.779587	+	0.626295i	$0.784571\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.89382	0.0769949
$606$	0	0
$607$	− 40.4873i	− 1.64333i	−0.569971	−	0.821665i	$-0.693046\pi$
$607$	− 40.4873i	− 1.64333i	0.569971	−	0.821665i	$-0.306954\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.56052i	0.0631318i
$612$	0	0
$613$	3.43337i	0.138673i	0.997593	+	0.0693363i	$0.0220882\pi$
$613$	3.43337i	0.138673i	−0.997593	+	0.0693363i	$0.977912\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.0661i	1.04938i	0.851293	+	0.524690i	$0.175819\pi$
$617$	26.0661i	1.04938i	−0.851293	+	0.524690i	$0.824181\pi$
$618$	0	0
$619$	−37.0540	−1.48933	−0.744664	−	0.667440i	$-0.767390\pi$
$619$	−37.0540	−1.48933	−0.744664	+	0.667440i	$0.767390\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	29.7291	1.19107
$624$	0	0
$625$	−15.9351	−0.637403
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.5681	0.501122
$630$	0	0
$631$	19.3001i	0.768325i	0.923266	+	0.384162i	$0.125510\pi$
$631$	19.3001i	0.768325i	−0.923266	+	0.384162i	$0.874490\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	24.5677i	0.974940i
$636$	0	0
$637$	− 11.9147i	− 0.472078i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 26.9766i	− 1.06551i	−0.846269	−	0.532757i	$-0.821156\pi$
$641$	− 26.9766i	− 1.06551i	0.846269	−	0.532757i	$-0.178844\pi$
$642$	0	0
$643$	−29.2484	−1.15345	−0.576723	−	0.816940i	$-0.695669\pi$
$643$	−29.2484	−1.15345	−0.576723	+	0.816940i	$0.695669\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.7125	1.08949	0.544746	−	0.838601i	$-0.316626\pi$
$647$	27.7125	1.08949	0.544746	+	0.838601i	$0.316626\pi$
$648$	0	0
$649$	6.10246	0.239543
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−33.4035	−1.30718	−0.653590	−	0.756849i	$-0.726738\pi$
$653$	−33.4035	−1.30718	−0.653590	+	0.756849i	$0.726738\pi$
$654$	0	0
$655$	− 5.28750i	− 0.206600i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 7.79075i	− 0.303485i	−0.988420	−	0.151742i	$-0.951512\pi$
$659$	− 7.79075i	− 0.303485i	0.988420	−	0.151742i	$-0.0484884\pi$
$660$	0	0
$661$	− 9.59043i	− 0.373024i	−0.982453	−	0.186512i	$-0.940282\pi$
$661$	− 9.59043i	− 0.373024i	0.982453	−	0.186512i	$-0.0597184\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	33.6943i	1.30661i
$666$	0	0
$667$	1.75736	0.0680454
$668$	0	0
$669$	0	0
$670$	0	0
$671$	14.1808	0.547445
$672$	0	0
$673$	−17.3398	−0.668399	−0.334199	−	0.942502i	$-0.608466\pi$
$673$	−17.3398	−0.668399	−0.334199	+	0.942502i	$0.608466\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.5740	0.752290	0.376145	−	0.926561i	$-0.377249\pi$
$677$	19.5740	0.752290	0.376145	+	0.926561i	$0.377249\pi$
$678$	0	0
$679$	− 14.8621i	− 0.570357i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.6643i	1.13507i	0.823348	+	0.567537i	$0.192103\pi$
$683$	29.6643i	1.13507i	−0.823348	+	0.567537i	$0.807897\pi$
$684$	0	0
$685$	− 3.63704i	− 0.138964i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 59.4615i	− 2.26530i
$690$	0	0
$691$	−17.6568	−0.671695	−0.335848	−	0.941916i	$-0.609023\pi$
$691$	−17.6568	−0.671695	−0.335848	+	0.941916i	$0.609023\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.99668	−0.379196
$696$	0	0
$697$	−57.5274	−2.17901
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23.0321	0.869909	0.434955	−	0.900452i	$-0.356764\pi$
$701$	23.0321	0.869909	0.434955	+	0.900452i	$0.356764\pi$
$702$	0	0
$703$	11.5177i	0.434396i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	26.9596i	1.01392i
$708$	0	0
$709$	34.3471i	1.28993i	0.764211	+	0.644966i	$0.223128\pi$
$709$	34.3471i	1.28993i	−0.764211	+	0.644966i	$0.776872\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	31.7005i	1.18719i
$714$	0	0
$715$	10.3078	0.385488
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26.4769	−0.987422	−0.493711	−	0.869626i	$-0.664360\pi$
$719$	−26.4769	−0.987422	−0.493711	+	0.869626i	$0.664360\pi$
$720$	0	0
$721$	−57.4943	−2.14120
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.848966	−0.0315298
$726$	0	0
$727$	40.9300i	1.51801i	0.651085	+	0.759005i	$0.274314\pi$
$727$	40.9300i	1.51801i	−0.651085	+	0.759005i	$0.725686\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.9411i	0.663575i
$732$	0	0
$733$	− 22.4662i	− 0.829808i	−0.909865	−	0.414904i	$-0.863815\pi$
$733$	− 22.4662i	− 0.829808i	0.909865	−	0.414904i	$-0.136185\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.2822i	0.452422i
$738$	0	0
$739$	27.2642	1.00293	0.501465	−	0.865178i	$-0.332794\pi$
$739$	27.2642	1.00293	0.501465	+	0.865178i	$0.332794\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15.4473	−0.566706	−0.283353	−	0.959016i	$-0.591447\pi$
$743$	−15.4473	−0.566706	−0.283353	+	0.959016i	$0.591447\pi$
$744$	0	0
$745$	12.9948	0.476093
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.8293	−0.395694
$750$	0	0
$751$	− 44.7392i	− 1.63256i	−0.577658	−	0.816279i	$-0.696033\pi$
$751$	− 44.7392i	− 1.63256i	0.577658	−	0.816279i	$-0.303967\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 6.48014i	− 0.235837i
$756$	0	0
$757$	− 22.6158i	− 0.821985i	−0.911639	−	0.410992i	$-0.865182\pi$
$757$	− 22.6158i	− 0.821985i	0.911639	−	0.410992i	$-0.134818\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.47443i	0.162198i	0.996706	+	0.0810990i	$0.0258430\pi$
$761$	4.47443i	0.162198i	−0.996706	+	0.0810990i	$0.974157\pi$
$762$	0	0
$763$	−24.2397	−0.877538
$764$	0	0
$765$	0	0
$766$	0	0
$767$	33.2146	1.19931
$768$	0	0
$769$	−37.5456	−1.35393	−0.676965	−	0.736016i	$-0.736705\pi$
$769$	−37.5456	−1.35393	−0.676965	+	0.736016i	$0.736705\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.97430	−0.322783	−0.161392	−	0.986890i	$-0.551598\pi$
$773$	−8.97430	−0.322783	−0.161392	+	0.986890i	$0.551598\pi$
$774$	0	0
$775$	− 15.3142i	− 0.550104i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 52.7193i	− 1.88887i
$780$	0	0
$781$	9.74776i	0.348802i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 20.9971i	− 0.749420i
$786$	0	0
$787$	6.21327	0.221479	0.110740	−	0.993849i	$-0.464678\pi$
$787$	6.21327	0.221479	0.110740	+	0.993849i	$0.464678\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−49.0253	−1.74314
$792$	0	0
$793$	77.1838	2.74088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23.4519	−0.830708	−0.415354	−	0.909660i	$-0.636342\pi$
$797$	−23.4519	−0.830708	−0.415354	+	0.909660i	$0.636342\pi$
$798$	0	0
$799$	− 1.83625i	− 0.0649617i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.25976i	0.256191i
$804$	0	0
$805$	16.7966i	0.592001i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38.3475i	1.34823i	0.738628	+	0.674113i	$0.235474\pi$
$809$	38.3475i	1.34823i	−0.738628	+	0.674113i	$0.764526\pi$
$810$	0	0
$811$	20.3661	0.715152	0.357576	−	0.933884i	$-0.383603\pi$
$811$	20.3661	0.715152	0.357576	+	0.933884i	$0.383603\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.35965	0.292826
$816$	0	0
$817$	−16.4416	−0.575219
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.2233	−1.05480	−0.527401	−	0.849617i	$-0.676833\pi$
$821$	−30.2233	−1.05480	−0.527401	+	0.849617i	$0.676833\pi$
$822$	0	0
$823$	6.31287i	0.220053i	0.993929	+	0.110026i	$0.0350935\pi$
$823$	6.31287i	0.220053i	−0.993929	+	0.110026i	$0.964906\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.5499i	1.37528i	0.726050	+	0.687642i	$0.241354\pi$
$827$	39.5499i	1.37528i	−0.726050	+	0.687642i	$0.758646\pi$
$828$	0	0
$829$	11.5153i	0.399944i	0.979802	+	0.199972i	$0.0640851\pi$
$829$	11.5153i	0.399944i	−0.979802	+	0.199972i	$0.935915\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	14.0199i	0.485761i
$834$	0	0
$835$	−19.5505	−0.676572
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.8630	0.478603	0.239301	−	0.970945i	$-0.423082\pi$
$839$	13.8630	0.478603	0.239301	+	0.970945i	$0.423082\pi$
$840$	0	0
$841$	−28.6392	−0.987560
$842$	0	0
$843$	0	0
$844$	0	0
$845$	31.4835	1.08307
$846$	0	0
$847$	3.03135i	0.104158i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.74153i	0.196817i
$852$	0	0
$853$	13.8053i	0.472683i	0.971670	+	0.236341i	$0.0759484\pi$
$853$	13.8053i	0.472683i	−0.971670	+	0.236341i	$0.924052\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 14.3186i	− 0.489114i	−0.969635	−	0.244557i	$-0.921357\pi$
$857$	− 14.3186i	− 0.489114i	0.969635	−	0.244557i	$-0.0786425\pi$
$858$	0	0
$859$	50.8997	1.73667	0.868337	−	0.495974i	$-0.165189\pi$
$859$	50.8997	1.73667	0.868337	+	0.495974i	$0.165189\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.0310	−0.852066	−0.426033	−	0.904708i	$-0.640089\pi$
$863$	−25.0310	−0.852066	−0.426033	+	0.904708i	$0.640089\pi$
$864$	0	0
$865$	21.4545	0.729474
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.48834	0.118334
$870$	0	0
$871$	66.8500i	2.26512i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 36.8185i	− 1.24469i
$876$	0	0
$877$	− 51.9131i	− 1.75298i	−0.481420	−	0.876490i	$-0.659879\pi$
$877$	− 51.9131i	− 1.75298i	0.481420	−	0.876490i	$-0.340121\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	40.2674i	1.35664i	0.734765	+	0.678322i	$0.237292\pi$
$881$	40.2674i	1.35664i	−0.734765	+	0.678322i	$0.762708\pi$
$882$	0	0
$883$	14.7777	0.497309	0.248655	−	0.968592i	$-0.420012\pi$
$883$	14.7777	0.497309	0.248655	+	0.968592i	$0.420012\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.9704	−1.30850	−0.654249	−	0.756279i	$-0.727015\pi$
$887$	−38.9704	−1.30850	−0.654249	+	0.756279i	$0.727015\pi$
$888$	0	0
$889$	−39.3243	−1.31889
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.68277	0.0563119
$894$	0	0
$895$	10.8263i	0.361883i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.50787i	0.217049i
$900$	0	0
$901$	69.9677i	2.33096i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 10.1681i	− 0.338000i
$906$	0	0
$907$	13.4008	0.444967	0.222484	−	0.974936i	$-0.428584\pi$
$907$	13.4008	0.444967	0.222484	+	0.974936i	$0.428584\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	41.8465	1.38644	0.693218	−	0.720728i	$-0.256192\pi$
$911$	41.8465	1.38644	0.693218	+	0.720728i	$0.256192\pi$
$912$	0	0
$913$	9.17926	0.303789
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.46344	0.279487
$918$	0	0
$919$	9.95866i	0.328506i	0.986418	+	0.164253i	$0.0525213\pi$
$919$	9.95866i	0.328506i	−0.986418	+	0.164253i	$0.947479\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	53.0553i	1.74634i
$924$	0	0
$925$	− 2.77368i	− 0.0911981i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.24932i	0.0737976i	0.999319	+	0.0368988i	$0.0117479\pi$
$929$	2.24932i	0.0737976i	−0.999319	+	0.0368988i	$0.988252\pi$
$930$	0	0
$931$	−12.8481	−0.421081
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12.1290	−0.396661
$936$	0	0
$937$	53.8032	1.75767	0.878837	−	0.477123i	$-0.158320\pi$
$937$	53.8032	1.75767	0.878837	+	0.477123i	$0.158320\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	51.7764	1.68786	0.843931	−	0.536451i	$-0.180235\pi$
$941$	51.7764	1.68786	0.843931	+	0.536451i	$0.180235\pi$
$942$	0	0
$943$	− 26.2805i	− 0.855811i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.6828i	0.509622i	0.966991	+	0.254811i	$0.0820132\pi$
$947$	15.6828i	0.509622i	−0.966991	+	0.254811i	$0.917987\pi$
$948$	0	0
$949$	39.5136i	1.28266i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21.4309i	− 0.694215i	−0.937825	−	0.347107i	$-0.887164\pi$
$953$	− 21.4309i	− 0.694215i	0.937825	−	0.347107i	$-0.112836\pi$
$954$	0	0
$955$	−18.4854	−0.598175
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.82162	0.187990
$960$	0	0
$961$	−86.3934	−2.78688
$962$	0	0
$963$	0	0
$964$	0	0
$965$	25.9808	0.836351
$966$	0	0
$967$	− 20.8025i	− 0.668963i	−0.942402	−	0.334481i	$-0.891439\pi$
$967$	− 20.8025i	− 0.668963i	0.942402	−	0.334481i	$-0.108561\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 10.5120i	− 0.337346i	−0.985672	−	0.168673i	$-0.946052\pi$
$971$	− 10.5120i	− 0.337346i	0.985672	−	0.168673i	$-0.0539481\pi$
$972$	0	0
$973$	− 16.0012i	− 0.512974i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 57.0894i	− 1.82645i	−0.407453	−	0.913226i	$-0.633583\pi$
$977$	− 57.0894i	− 1.82645i	0.407453	−	0.913226i	$-0.366417\pi$
$978$	0	0
$979$	−9.80721	−0.313440
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.1034	0.545514	0.272757	−	0.962083i	$-0.412064\pi$
$983$	17.1034	0.545514	0.272757	+	0.962083i	$0.412064\pi$
$984$	0	0
$985$	−31.3187	−0.997898
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.19611	−0.260621
$990$	0	0
$991$	− 40.9703i	− 1.30147i	−0.759307	−	0.650733i	$-0.774462\pi$
$991$	− 40.9703i	− 1.30147i	0.759307	−	0.650733i	$-0.225538\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	31.5550i	1.00036i
$996$	0	0
$997$	21.1608i	0.670168i	0.942188	+	0.335084i	$0.108765\pi$
$997$	21.1608i	0.670168i	−0.942188	+	0.335084i	$0.891235\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.k.a.1871.12		40
3.2	odd	2	inner	3168.2.k.a.1871.30		40
4.3	odd	2		792.2.k.a.683.18	yes	40
8.3	odd	2	inner	3168.2.k.a.1871.29		40
8.5	even	2		792.2.k.a.683.24	yes	40
12.11	even	2		792.2.k.a.683.23	yes	40
24.5	odd	2		792.2.k.a.683.17	✓	40
24.11	even	2	inner	3168.2.k.a.1871.11		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.k.a.683.17	✓	40	24.5	odd	2
792.2.k.a.683.18	yes	40	4.3	odd	2
792.2.k.a.683.23	yes	40	12.11	even	2
792.2.k.a.683.24	yes	40	8.5	even	2
3168.2.k.a.1871.11		40	24.11	even	2	inner
3168.2.k.a.1871.12		40	1.1	even	1	trivial
3168.2.k.a.1871.29		40	8.3	odd	2	inner
3168.2.k.a.1871.30		40	3.2	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:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.89382 q^{5} -3.03135i q^{7} +1.00000i q^{11} +5.44282i q^{13} -6.40451i q^{17} +5.86923 q^{19} +2.92580 q^{23} -1.41343 q^{25} +0.600643 q^{29} +10.8348i q^{31} +5.74084i q^{35} +1.96238i q^{37} -8.98233i q^{41} -2.80132 q^{43} +0.286711 q^{47} -2.18907 q^{49} -10.9248 q^{53} -1.89382i q^{55} -6.10246i q^{59} -14.1808i q^{61} -10.3078i q^{65} +12.2822 q^{67} +9.74776 q^{71} +7.25976 q^{73} +3.03135 q^{77} -3.48834i q^{79} -9.17926i q^{83} +12.1290i q^{85} +9.80721i q^{89} +16.4991 q^{91} -11.1153 q^{95} +4.90282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 32 q^{19} + 40 q^{25} - 8 q^{49} + 32 q^{73} - 32 q^{97}+O(q^{100}) \) 40 * q - 32 * q^19 + 40 * q^25 - 8 * q^49 + 32 * q^73 - 32 * q^97

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