LMFDB - Embedded newform 2912.2.c.a.1457.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(1457,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 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("2912.1457"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [34] 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:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.5
Character	$\chi$	$=$	2912.1457
Dual form			2912.2.c.a.1457.30

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66332i q^{3} +2.77422i q^{5} +1.00000 q^{7} -4.09325 q^{9} -2.30522i q^{11} -1.00000i q^{13} +7.38863 q^{15} -6.69704 q^{17} +2.40379i q^{19} -2.66332i q^{21} -5.02802 q^{23} -2.69630 q^{25} +2.91168i q^{27} +4.22120i q^{29} +6.52105 q^{31} -6.13952 q^{33} +2.77422i q^{35} -7.64478i q^{37} -2.66332 q^{39} -1.66987 q^{41} -4.89973i q^{43} -11.3556i q^{45} -13.5343 q^{47} +1.00000 q^{49} +17.8363i q^{51} -0.579570i q^{53} +6.39518 q^{55} +6.40205 q^{57} -10.2153i q^{59} +7.92093i q^{61} -4.09325 q^{63} +2.77422 q^{65} +6.80101i q^{67} +13.3912i q^{69} -7.75298 q^{71} -5.71664 q^{73} +7.18109i q^{75} -2.30522i q^{77} -12.0537 q^{79} -4.52503 q^{81} +0.262559i q^{83} -18.5791i q^{85} +11.2424 q^{87} -14.0431 q^{89} -1.00000i q^{91} -17.3676i q^{93} -6.66864 q^{95} -15.9382 q^{97} +9.43584i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) $ 34 * q + 34 * q^7 - 26 * q^9 + 8 * q^15 - 20 * q^17 + 20 * q^23 - 22 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 34 * q^49 + 32 * q^55 + 8 * q^57 - 26 * q^63 - 20 * q^65 - 64 * q^71 - 20 * q^79 - 6 * q^81 + 56 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	− 2.66332i	− 1.53767i	−0.639449	−	0.768833i	$-0.720838\pi$
$3$	− 2.66332i	− 1.53767i	0.639449	−	0.768833i	$-0.279162\pi$
$4$	0	0
$5$	2.77422i	1.24067i	0.784337	+	0.620334i	$0.213003\pi$
$5$	2.77422i	1.24067i	−0.784337	+	0.620334i	$0.786997\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−4.09325	−1.36442
$10$	0	0
$11$	− 2.30522i	− 0.695049i	−0.937671	−	0.347524i	$-0.887022\pi$
$11$	− 2.30522i	− 0.695049i	0.937671	−	0.347524i	$-0.112978\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	7.38863	1.90773
$16$	0	0
$17$	−6.69704	−1.62427	−0.812135	−	0.583469i	$-0.801695\pi$
$17$	−6.69704	−1.62427	−0.812135	+	0.583469i	$0.801695\pi$
$18$	0	0
$19$	2.40379i	0.551467i	0.961234	+	0.275734i	$0.0889208\pi$
$19$	2.40379i	0.551467i	−0.961234	+	0.275734i	$0.911079\pi$
$20$	0	0
$21$	− 2.66332i	− 0.581183i
$22$	0	0
$23$	−5.02802	−1.04841	−0.524207	−	0.851591i	$-0.675638\pi$
$23$	−5.02802	−1.04841	−0.524207	+	0.851591i	$0.675638\pi$
$24$	0	0
$25$	−2.69630	−0.539259
$26$	0	0
$27$	2.91168i	0.560354i
$28$	0	0
$29$	4.22120i	0.783857i	0.919996	+	0.391929i	$0.128192\pi$
$29$	4.22120i	0.783857i	−0.919996	+	0.391929i	$0.871808\pi$
$30$	0	0
$31$	6.52105	1.17122	0.585608	−	0.810595i	$-0.300856\pi$
$31$	6.52105	1.17122	0.585608	+	0.810595i	$0.300856\pi$
$32$	0	0
$33$	−6.13952	−1.06875
$34$	0	0
$35$	2.77422i	0.468929i
$36$	0	0
$37$	− 7.64478i	− 1.25679i	−0.777893	−	0.628397i	$-0.783711\pi$
$37$	− 7.64478i	− 1.25679i	0.777893	−	0.628397i	$-0.216289\pi$
$38$	0	0
$39$	−2.66332	−0.426472
$40$	0	0
$41$	−1.66987	−0.260790	−0.130395	−	0.991462i	$-0.541625\pi$
$41$	−1.66987	−0.260790	−0.130395	+	0.991462i	$0.541625\pi$
$42$	0	0
$43$	− 4.89973i	− 0.747201i	−0.927590	−	0.373601i	$-0.878123\pi$
$43$	− 4.89973i	− 0.747201i	0.927590	−	0.373601i	$-0.121877\pi$
$44$	0	0
$45$	− 11.3556i	− 1.69279i
$46$	0	0
$47$	−13.5343	−1.97419	−0.987093	−	0.160148i	$-0.948803\pi$
$47$	−13.5343	−1.97419	−0.987093	+	0.160148i	$0.948803\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	17.8363i	2.49759i
$52$	0	0
$53$	− 0.579570i	− 0.0796101i	−0.999207	−	0.0398050i	$-0.987326\pi$
$53$	− 0.579570i	− 0.0796101i	0.999207	−	0.0398050i	$-0.0126737\pi$
$54$	0	0
$55$	6.39518	0.862325
$56$	0	0
$57$	6.40205	0.847973
$58$	0	0
$59$	− 10.2153i	− 1.32992i	−0.746880	−	0.664959i	$-0.768449\pi$
$59$	− 10.2153i	− 1.32992i	0.746880	−	0.664959i	$-0.231551\pi$
$60$	0	0
$61$	7.92093i	1.01417i	0.861896	+	0.507086i	$0.169277\pi$
$61$	7.92093i	1.01417i	−0.861896	+	0.507086i	$0.830723\pi$
$62$	0	0
$63$	−4.09325	−0.515702
$64$	0	0
$65$	2.77422	0.344100
$66$	0	0
$67$	6.80101i	0.830876i	0.909621	+	0.415438i	$0.136372\pi$
$67$	6.80101i	0.830876i	−0.909621	+	0.415438i	$0.863628\pi$
$68$	0	0
$69$	13.3912i	1.61211i
$70$	0	0
$71$	−7.75298	−0.920110	−0.460055	−	0.887890i	$-0.652170\pi$
$71$	−7.75298	−0.920110	−0.460055	+	0.887890i	$0.652170\pi$
$72$	0	0
$73$	−5.71664	−0.669082	−0.334541	−	0.942381i	$-0.608581\pi$
$73$	−5.71664	−0.669082	−0.334541	+	0.942381i	$0.608581\pi$
$74$	0	0
$75$	7.18109i	0.829201i
$76$	0	0
$77$	− 2.30522i	− 0.262704i
$78$	0	0
$79$	−12.0537	−1.35615	−0.678076	−	0.734992i	$-0.737186\pi$
$79$	−12.0537	−1.35615	−0.678076	+	0.734992i	$0.737186\pi$
$80$	0	0
$81$	−4.52503	−0.502781
$82$	0	0
$83$	0.262559i	0.0288196i	0.999896	+	0.0144098i	$0.00458695\pi$
$83$	0.262559i	0.0288196i	−0.999896	+	0.0144098i	$0.995413\pi$
$84$	0	0
$85$	− 18.5791i	− 2.01518i
$86$	0	0
$87$	11.2424	1.20531
$88$	0	0
$89$	−14.0431	−1.48857	−0.744283	−	0.667865i	$-0.767208\pi$
$89$	−14.0431	−1.48857	−0.744283	+	0.667865i	$0.767208\pi$
$90$	0	0
$91$	− 1.00000i	− 0.104828i
$92$	0	0
$93$	− 17.3676i	− 1.80094i
$94$	0	0
$95$	−6.66864	−0.684188
$96$	0	0
$97$	−15.9382	−1.61828	−0.809142	−	0.587613i	$-0.800068\pi$
$97$	−15.9382	−1.61828	−0.809142	+	0.587613i	$0.800068\pi$
$98$	0	0
$99$	9.43584i	0.948337i
$100$	0	0
$101$	− 11.8085i	− 1.17499i	−0.809229	−	0.587494i	$-0.800115\pi$
$101$	− 11.8085i	− 1.17499i	0.809229	−	0.587494i	$-0.199885\pi$
$102$	0	0
$103$	2.80041	0.275933	0.137966	−	0.990437i	$-0.455943\pi$
$103$	2.80041	0.275933	0.137966	+	0.990437i	$0.455943\pi$
$104$	0	0
$105$	7.38863	0.721056
$106$	0	0
$107$	− 2.80406i	− 0.271079i	−0.990772	−	0.135539i	$-0.956723\pi$
$107$	− 2.80406i	− 0.271079i	0.990772	−	0.135539i	$-0.0432767\pi$
$108$	0	0
$109$	9.23050i	0.884122i	0.896985	+	0.442061i	$0.145753\pi$
$109$	9.23050i	0.884122i	−0.896985	+	0.442061i	$0.854247\pi$
$110$	0	0
$111$	−20.3605	−1.93253
$112$	0	0
$113$	7.99544	0.752148	0.376074	−	0.926590i	$-0.377274\pi$
$113$	7.99544	0.752148	0.376074	+	0.926590i	$0.377274\pi$
$114$	0	0
$115$	− 13.9488i	− 1.30074i
$116$	0	0
$117$	4.09325i	0.378422i
$118$	0	0
$119$	−6.69704	−0.613917
$120$	0	0
$121$	5.68598	0.516907
$122$	0	0
$123$	4.44740i	0.401008i
$124$	0	0
$125$	6.39098i	0.571627i
$126$	0	0
$127$	9.95073	0.882985	0.441492	−	0.897265i	$-0.354449\pi$
$127$	9.95073	0.882985	0.441492	+	0.897265i	$0.354449\pi$
$128$	0	0
$129$	−13.0495	−1.14895
$130$	0	0
$131$	− 0.170370i	− 0.0148853i	−0.999972	−	0.00744265i	$-0.997631\pi$
$131$	− 0.170370i	− 0.0148853i	0.999972	−	0.00744265i	$-0.00236909\pi$
$132$	0	0
$133$	2.40379i	0.208435i
$134$	0	0
$135$	−8.07765	−0.695213
$136$	0	0
$137$	−21.3225	−1.82170	−0.910852	−	0.412734i	$-0.864574\pi$
$137$	−21.3225	−1.82170	−0.910852	+	0.412734i	$0.864574\pi$
$138$	0	0
$139$	− 2.75623i	− 0.233781i	−0.993145	−	0.116890i	$-0.962707\pi$
$139$	− 2.75623i	− 0.233781i	0.993145	−	0.116890i	$-0.0372926\pi$
$140$	0	0
$141$	36.0462i	3.03564i
$142$	0	0
$143$	−2.30522	−0.192772
$144$	0	0
$145$	−11.7105	−0.972507
$146$	0	0
$147$	− 2.66332i	− 0.219667i
$148$	0	0
$149$	− 4.75728i	− 0.389732i	−0.980830	−	0.194866i	$-0.937573\pi$
$149$	− 4.75728i	− 0.389732i	0.980830	−	0.194866i	$-0.0624271\pi$
$150$	0	0
$151$	6.72897	0.547596	0.273798	−	0.961787i	$-0.411720\pi$
$151$	6.72897	0.547596	0.273798	+	0.961787i	$0.411720\pi$
$152$	0	0
$153$	27.4127	2.21618
$154$	0	0
$155$	18.0908i	1.45309i
$156$	0	0
$157$	7.37175i	0.588330i	0.955755	+	0.294165i	$0.0950415\pi$
$157$	7.37175i	0.588330i	−0.955755	+	0.294165i	$0.904958\pi$
$158$	0	0
$159$	−1.54358	−0.122414
$160$	0	0
$161$	−5.02802	−0.396263
$162$	0	0
$163$	− 10.1735i	− 0.796853i	−0.917200	−	0.398426i	$-0.869556\pi$
$163$	− 10.1735i	− 0.796853i	0.917200	−	0.398426i	$-0.130444\pi$
$164$	0	0
$165$	− 17.0324i	− 1.32597i
$166$	0	0
$167$	20.1675	1.56061	0.780303	−	0.625401i	$-0.215065\pi$
$167$	20.1675	1.56061	0.780303	+	0.625401i	$0.215065\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 9.83932i	− 0.752432i
$172$	0	0
$173$	5.41669i	0.411823i	0.978571	+	0.205911i	$0.0660159\pi$
$173$	5.41669i	0.411823i	−0.978571	+	0.205911i	$0.933984\pi$
$174$	0	0
$175$	−2.69630	−0.203821
$176$	0	0
$177$	−27.2065	−2.04497
$178$	0	0
$179$	− 6.79698i	− 0.508030i	−0.967200	−	0.254015i	$-0.918249\pi$
$179$	− 6.79698i	− 0.508030i	0.967200	−	0.254015i	$-0.0817513\pi$
$180$	0	0
$181$	14.9174i	1.10880i	0.832249	+	0.554402i	$0.187053\pi$
$181$	14.9174i	1.10880i	−0.832249	+	0.554402i	$0.812947\pi$
$182$	0	0
$183$	21.0959	1.55946
$184$	0	0
$185$	21.2083	1.55927
$186$	0	0
$187$	15.4381i	1.12895i
$188$	0	0
$189$	2.91168i	0.211794i
$190$	0	0
$191$	13.5542	0.980744	0.490372	−	0.871513i	$-0.336861\pi$
$191$	13.5542	0.980744	0.490372	+	0.871513i	$0.336861\pi$
$192$	0	0
$193$	−4.61365	−0.332098	−0.166049	−	0.986118i	$-0.553101\pi$
$193$	−4.61365	−0.332098	−0.166049	+	0.986118i	$0.553101\pi$
$194$	0	0
$195$	− 7.38863i	− 0.529110i
$196$	0	0
$197$	− 5.21055i	− 0.371236i	−0.982622	−	0.185618i	$-0.940571\pi$
$197$	− 5.21055i	− 0.371236i	0.982622	−	0.185618i	$-0.0594287\pi$
$198$	0	0
$199$	−23.8017	−1.68726	−0.843629	−	0.536927i	$-0.819585\pi$
$199$	−23.8017	−1.68726	−0.843629	+	0.536927i	$0.819585\pi$
$200$	0	0
$201$	18.1133	1.27761
$202$	0	0
$203$	4.22120i	0.296270i
$204$	0	0
$205$	− 4.63259i	− 0.323554i
$206$	0	0
$207$	20.5810	1.43048
$208$	0	0
$209$	5.54125	0.383297
$210$	0	0
$211$	− 9.64867i	− 0.664242i	−0.943237	−	0.332121i	$-0.892236\pi$
$211$	− 9.64867i	− 0.664242i	0.943237	−	0.332121i	$-0.107764\pi$
$212$	0	0
$213$	20.6487i	1.41482i
$214$	0	0
$215$	13.5929	0.927029
$216$	0	0
$217$	6.52105	0.442678
$218$	0	0
$219$	15.2252i	1.02883i
$220$	0	0
$221$	6.69704i	0.450492i
$222$	0	0
$223$	1.12765	0.0755129	0.0377565	−	0.999287i	$-0.487979\pi$
$223$	1.12765	0.0755129	0.0377565	+	0.999287i	$0.487979\pi$
$224$	0	0
$225$	11.0366	0.735775
$226$	0	0
$227$	− 28.0410i	− 1.86115i	−0.366103	−	0.930574i	$-0.619308\pi$
$227$	− 28.0410i	− 1.86115i	0.366103	−	0.930574i	$-0.380692\pi$
$228$	0	0
$229$	19.7757i	1.30682i	0.757006	+	0.653408i	$0.226662\pi$
$229$	19.7757i	1.30682i	−0.757006	+	0.653408i	$0.773338\pi$
$230$	0	0
$231$	−6.13952	−0.403951
$232$	0	0
$233$	−26.7490	−1.75239	−0.876194	−	0.481958i	$-0.839926\pi$
$233$	−26.7490	−1.75239	−0.876194	+	0.481958i	$0.839926\pi$
$234$	0	0
$235$	− 37.5472i	− 2.44931i
$236$	0	0
$237$	32.1029i	2.08531i
$238$	0	0
$239$	−19.6982	−1.27417	−0.637086	−	0.770793i	$-0.719860\pi$
$239$	−19.6982	−1.27417	−0.637086	+	0.770793i	$0.719860\pi$
$240$	0	0
$241$	8.31281	0.535475	0.267738	−	0.963492i	$-0.413724\pi$
$241$	8.31281	0.535475	0.267738	+	0.963492i	$0.413724\pi$
$242$	0	0
$243$	20.7866i	1.33346i
$244$	0	0
$245$	2.77422i	0.177238i
$246$	0	0
$247$	2.40379	0.152949
$248$	0	0
$249$	0.699279	0.0443150
$250$	0	0
$251$	24.2860i	1.53292i	0.642292	+	0.766460i	$0.277984\pi$
$251$	24.2860i	1.53292i	−0.642292	+	0.766460i	$0.722016\pi$
$252$	0	0
$253$	11.5907i	0.728699i
$254$	0	0
$255$	−49.4819	−3.09868
$256$	0	0
$257$	11.7809	0.734871	0.367436	−	0.930049i	$-0.380236\pi$
$257$	11.7809	0.734871	0.367436	+	0.930049i	$0.380236\pi$
$258$	0	0
$259$	− 7.64478i	− 0.475024i
$260$	0	0
$261$	− 17.2784i	− 1.06951i
$262$	0	0
$263$	−16.2221	−1.00029	−0.500147	−	0.865940i	$-0.666721\pi$
$263$	−16.2221	−1.00029	−0.500147	+	0.865940i	$0.666721\pi$
$264$	0	0
$265$	1.60786	0.0987698
$266$	0	0
$267$	37.4012i	2.28892i
$268$	0	0
$269$	26.4552i	1.61300i	0.591233	+	0.806501i	$0.298641\pi$
$269$	26.4552i	1.61300i	−0.591233	+	0.806501i	$0.701359\pi$
$270$	0	0
$271$	21.9987	1.33633	0.668164	−	0.744014i	$-0.267080\pi$
$271$	21.9987	1.33633	0.668164	+	0.744014i	$0.267080\pi$
$272$	0	0
$273$	−2.66332	−0.161191
$274$	0	0
$275$	6.21554i	0.374811i
$276$	0	0
$277$	8.68422i	0.521784i	0.965368	+	0.260892i	$0.0840167\pi$
$277$	8.68422i	0.521784i	−0.965368	+	0.260892i	$0.915983\pi$
$278$	0	0
$279$	−26.6923	−1.59803
$280$	0	0
$281$	33.4194	1.99363	0.996817	−	0.0797253i	$-0.0254043\pi$
$281$	33.4194	1.99363	0.996817	+	0.0797253i	$0.0254043\pi$
$282$	0	0
$283$	23.1585i	1.37663i	0.725412	+	0.688314i	$0.241649\pi$
$283$	23.1585i	1.37663i	−0.725412	+	0.688314i	$0.758351\pi$
$284$	0	0
$285$	17.7607i	1.05205i
$286$	0	0
$287$	−1.66987	−0.0985695
$288$	0	0
$289$	27.8504	1.63826
$290$	0	0
$291$	42.4486i	2.48838i
$292$	0	0
$293$	− 20.2527i	− 1.18318i	−0.806241	−	0.591588i	$-0.798501\pi$
$293$	− 20.2527i	− 1.18318i	0.806241	−	0.591588i	$-0.201499\pi$
$294$	0	0
$295$	28.3395	1.64999
$296$	0	0
$297$	6.71206	0.389473
$298$	0	0
$299$	5.02802i	0.290778i
$300$	0	0
$301$	− 4.89973i	− 0.282416i
$302$	0	0
$303$	−31.4497	−1.80674
$304$	0	0
$305$	−21.9744	−1.25825
$306$	0	0
$307$	11.4584i	0.653968i	0.945030	+	0.326984i	$0.106032\pi$
$307$	11.4584i	0.653968i	−0.945030	+	0.326984i	$0.893968\pi$
$308$	0	0
$309$	− 7.45838i	− 0.424292i
$310$	0	0
$311$	−7.17869	−0.407066	−0.203533	−	0.979068i	$-0.565242\pi$
$311$	−7.17869	−0.407066	−0.203533	+	0.979068i	$0.565242\pi$
$312$	0	0
$313$	28.1120	1.58898	0.794492	−	0.607275i	$-0.207737\pi$
$313$	28.1120	1.58898	0.794492	+	0.607275i	$0.207737\pi$
$314$	0	0
$315$	− 11.3556i	− 0.639815i
$316$	0	0
$317$	− 15.6537i	− 0.879198i	−0.898194	−	0.439599i	$-0.855120\pi$
$317$	− 15.6537i	− 0.879198i	0.898194	−	0.439599i	$-0.144880\pi$
$318$	0	0
$319$	9.73078	0.544819
$320$	0	0
$321$	−7.46810	−0.416829
$322$	0	0
$323$	− 16.0983i	− 0.895732i
$324$	0	0
$325$	2.69630i	0.149564i
$326$	0	0
$327$	24.5838	1.35948
$328$	0	0
$329$	−13.5343	−0.746172
$330$	0	0
$331$	17.1765i	0.944106i	0.881570	+	0.472053i	$0.156487\pi$
$331$	17.1765i	0.944106i	−0.881570	+	0.472053i	$0.843513\pi$
$332$	0	0
$333$	31.2920i	1.71479i
$334$	0	0
$335$	−18.8675	−1.03084
$336$	0	0
$337$	−24.0567	−1.31045	−0.655227	−	0.755432i	$-0.727427\pi$
$337$	−24.0567	−1.31045	−0.655227	+	0.755432i	$0.727427\pi$
$338$	0	0
$339$	− 21.2944i	− 1.15655i
$340$	0	0
$341$	− 15.0324i	− 0.814052i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−37.1502	−2.00010
$346$	0	0
$347$	− 10.0923i	− 0.541785i	−0.962610	−	0.270893i	$-0.912681\pi$
$347$	− 10.0923i	− 0.541785i	0.962610	−	0.270893i	$-0.0873189\pi$
$348$	0	0
$349$	− 22.2058i	− 1.18865i	−0.804225	−	0.594325i	$-0.797419\pi$
$349$	− 22.2058i	− 1.18865i	0.804225	−	0.594325i	$-0.202581\pi$
$350$	0	0
$351$	2.91168	0.155414
$352$	0	0
$353$	2.84719	0.151540	0.0757702	−	0.997125i	$-0.475858\pi$
$353$	2.84719	0.151540	0.0757702	+	0.997125i	$0.475858\pi$
$354$	0	0
$355$	− 21.5085i	− 1.14155i
$356$	0	0
$357$	17.8363i	0.943999i
$358$	0	0
$359$	5.38588	0.284256	0.142128	−	0.989848i	$-0.454606\pi$
$359$	5.38588	0.284256	0.142128	+	0.989848i	$0.454606\pi$
$360$	0	0
$361$	13.2218	0.695884
$362$	0	0
$363$	− 15.1436i	− 0.794831i
$364$	0	0
$365$	− 15.8592i	− 0.830109i
$366$	0	0
$367$	−27.3949	−1.43000	−0.715001	−	0.699123i	$-0.753574\pi$
$367$	−27.3949	−1.43000	−0.715001	+	0.699123i	$0.753574\pi$
$368$	0	0
$369$	6.83521	0.355827
$370$	0	0
$371$	− 0.579570i	− 0.0300898i
$372$	0	0
$373$	1.81294i	0.0938703i	0.998898	+	0.0469351i	$0.0149454\pi$
$373$	1.81294i	0.0938703i	−0.998898	+	0.0469351i	$0.985055\pi$
$374$	0	0
$375$	17.0212	0.878971
$376$	0	0
$377$	4.22120	0.217403
$378$	0	0
$379$	− 14.4281i	− 0.741122i	−0.928808	−	0.370561i	$-0.879165\pi$
$379$	− 14.4281i	− 0.741122i	0.928808	−	0.370561i	$-0.120835\pi$
$380$	0	0
$381$	− 26.5019i	− 1.35774i
$382$	0	0
$383$	−20.0617	−1.02510	−0.512552	−	0.858656i	$-0.671300\pi$
$383$	−20.0617	−1.02510	−0.512552	+	0.858656i	$0.671300\pi$
$384$	0	0
$385$	6.39518	0.325928
$386$	0	0
$387$	20.0558i	1.01950i
$388$	0	0
$389$	− 20.5109i	− 1.03994i	−0.854183	−	0.519972i	$-0.825942\pi$
$389$	− 20.5109i	− 1.03994i	0.854183	−	0.519972i	$-0.174058\pi$
$390$	0	0
$391$	33.6729	1.70291
$392$	0	0
$393$	−0.453749	−0.0228886
$394$	0	0
$395$	− 33.4397i	− 1.68253i
$396$	0	0
$397$	− 0.791932i	− 0.0397459i	−0.999803	−	0.0198730i	$-0.993674\pi$
$397$	− 0.791932i	− 0.0397459i	0.999803	−	0.0198730i	$-0.00632618\pi$
$398$	0	0
$399$	6.40205	0.320504
$400$	0	0
$401$	22.0537	1.10131	0.550656	−	0.834732i	$-0.314378\pi$
$401$	22.0537	1.10131	0.550656	+	0.834732i	$0.314378\pi$
$402$	0	0
$403$	− 6.52105i	− 0.324837i
$404$	0	0
$405$	− 12.5534i	− 0.623785i
$406$	0	0
$407$	−17.6229	−0.873533
$408$	0	0
$409$	−22.6362	−1.11929	−0.559645	−	0.828732i	$-0.689063\pi$
$409$	−22.6362	−1.11929	−0.559645	+	0.828732i	$0.689063\pi$
$410$	0	0
$411$	56.7885i	2.80117i
$412$	0	0
$413$	− 10.2153i	− 0.502661i
$414$	0	0
$415$	−0.728397	−0.0357556
$416$	0	0
$417$	−7.34072	−0.359477
$418$	0	0
$419$	− 29.3054i	− 1.43166i	−0.698274	−	0.715831i	$-0.746048\pi$
$419$	− 29.3054i	− 1.43166i	0.698274	−	0.715831i	$-0.253952\pi$
$420$	0	0
$421$	− 12.9130i	− 0.629342i	−0.949201	−	0.314671i	$-0.898106\pi$
$421$	− 12.9130i	− 0.629342i	0.949201	−	0.314671i	$-0.101894\pi$
$422$	0	0
$423$	55.3995	2.69362
$424$	0	0
$425$	18.0572	0.875903
$426$	0	0
$427$	7.92093i	0.383321i
$428$	0	0
$429$	6.13952i	0.296419i
$430$	0	0
$431$	11.6196	0.559695	0.279847	−	0.960044i	$-0.409716\pi$
$431$	11.6196	0.559695	0.279847	+	0.960044i	$0.409716\pi$
$432$	0	0
$433$	11.7701	0.565634	0.282817	−	0.959174i	$-0.408731\pi$
$433$	11.7701	0.565634	0.282817	+	0.959174i	$0.408731\pi$
$434$	0	0
$435$	31.1889i	1.49539i
$436$	0	0
$437$	− 12.0863i	− 0.578166i
$438$	0	0
$439$	−0.365184	−0.0174293	−0.00871464	−	0.999962i	$-0.502774\pi$
$439$	−0.365184	−0.0174293	−0.00871464	+	0.999962i	$0.502774\pi$
$440$	0	0
$441$	−4.09325	−0.194917
$442$	0	0
$443$	− 36.1653i	− 1.71826i	−0.511755	−	0.859132i	$-0.671004\pi$
$443$	− 36.1653i	− 1.71826i	0.511755	−	0.859132i	$-0.328996\pi$
$444$	0	0
$445$	− 38.9586i	− 1.84682i
$446$	0	0
$447$	−12.6701	−0.599277
$448$	0	0
$449$	21.4503	1.01230	0.506152	−	0.862444i	$-0.331067\pi$
$449$	21.4503	1.01230	0.506152	+	0.862444i	$0.331067\pi$
$450$	0	0
$451$	3.84942i	0.181262i
$452$	0	0
$453$	− 17.9214i	− 0.842020i
$454$	0	0
$455$	2.77422	0.130057
$456$	0	0
$457$	2.80295	0.131116	0.0655582	−	0.997849i	$-0.479117\pi$
$457$	2.80295	0.131116	0.0655582	+	0.997849i	$0.479117\pi$
$458$	0	0
$459$	− 19.4997i	− 0.910166i
$460$	0	0
$461$	19.6840i	0.916774i	0.888753	+	0.458387i	$0.151573\pi$
$461$	19.6840i	0.916774i	−0.888753	+	0.458387i	$0.848427\pi$
$462$	0	0
$463$	−10.9945	−0.510959	−0.255479	−	0.966815i	$-0.582233\pi$
$463$	−10.9945	−0.510959	−0.255479	+	0.966815i	$0.582233\pi$
$464$	0	0
$465$	48.1816	2.23437
$466$	0	0
$467$	− 26.1563i	− 1.21037i	−0.796086	−	0.605184i	$-0.793099\pi$
$467$	− 26.1563i	− 1.21037i	0.796086	−	0.605184i	$-0.206901\pi$
$468$	0	0
$469$	6.80101i	0.314042i
$470$	0	0
$471$	19.6333	0.904655
$472$	0	0
$473$	−11.2949	−0.519341
$474$	0	0
$475$	− 6.48133i	− 0.297384i
$476$	0	0
$477$	2.37233i	0.108621i
$478$	0	0
$479$	−1.13276	−0.0517574	−0.0258787	−	0.999665i	$-0.508238\pi$
$479$	−1.13276	−0.0517574	−0.0258787	+	0.999665i	$0.508238\pi$
$480$	0	0
$481$	−7.64478	−0.348572
$482$	0	0
$483$	13.3912i	0.609321i
$484$	0	0
$485$	− 44.2162i	− 2.00775i
$486$	0	0
$487$	19.1216	0.866482	0.433241	−	0.901278i	$-0.357370\pi$
$487$	19.1216	0.866482	0.433241	+	0.901278i	$0.357370\pi$
$488$	0	0
$489$	−27.0953	−1.22529
$490$	0	0
$491$	− 41.9448i	− 1.89294i	−0.322788	−	0.946471i	$-0.604620\pi$
$491$	− 41.9448i	− 1.89294i	0.322788	−	0.946471i	$-0.395380\pi$
$492$	0	0
$493$	− 28.2695i	− 1.27320i
$494$	0	0
$495$	−26.1771	−1.17657
$496$	0	0
$497$	−7.75298	−0.347769
$498$	0	0
$499$	− 34.0302i	− 1.52340i	−0.647928	−	0.761702i	$-0.724364\pi$
$499$	− 34.0302i	− 1.52340i	0.647928	−	0.761702i	$-0.275636\pi$
$500$	0	0
$501$	− 53.7124i	− 2.39969i
$502$	0	0
$503$	−35.5484	−1.58502	−0.792512	−	0.609856i	$-0.791227\pi$
$503$	−35.5484	−1.58502	−0.792512	+	0.609856i	$0.791227\pi$
$504$	0	0
$505$	32.7593	1.45777
$506$	0	0
$507$	2.66332i	0.118282i
$508$	0	0
$509$	40.4806i	1.79427i	0.441755	+	0.897136i	$0.354356\pi$
$509$	40.4806i	1.79427i	−0.441755	+	0.897136i	$0.645644\pi$
$510$	0	0
$511$	−5.71664	−0.252889
$512$	0	0
$513$	−6.99907	−0.309017
$514$	0	0
$515$	7.76895i	0.342341i
$516$	0	0
$517$	31.1996i	1.37216i
$518$	0	0
$519$	14.4263	0.633246
$520$	0	0
$521$	23.0145	1.00828	0.504142	−	0.863621i	$-0.331809\pi$
$521$	23.0145	1.00828	0.504142	+	0.863621i	$0.331809\pi$
$522$	0	0
$523$	2.66001i	0.116314i	0.998307	+	0.0581570i	$0.0185224\pi$
$523$	2.66001i	0.116314i	−0.998307	+	0.0581570i	$0.981478\pi$
$524$	0	0
$525$	7.18109i	0.313408i
$526$	0	0
$527$	−43.6717	−1.90237
$528$	0	0
$529$	2.28098	0.0991731
$530$	0	0
$531$	41.8138i	1.81456i
$532$	0	0
$533$	1.66987i	0.0723302i
$534$	0	0
$535$	7.77908	0.336319
$536$	0	0
$537$	−18.1025	−0.781181
$538$	0	0
$539$	− 2.30522i	− 0.0992927i
$540$	0	0
$541$	− 8.51106i	− 0.365919i	−0.983120	−	0.182959i	$-0.941432\pi$
$541$	− 8.51106i	− 0.365919i	0.983120	−	0.182959i	$-0.0585677\pi$
$542$	0	0
$543$	39.7298	1.70497
$544$	0	0
$545$	−25.6074	−1.09690
$546$	0	0
$547$	− 8.16036i	− 0.348912i	−0.984665	−	0.174456i	$-0.944183\pi$
$547$	− 8.16036i	− 0.348912i	0.984665	−	0.174456i	$-0.0558166\pi$
$548$	0	0
$549$	− 32.4224i	− 1.38375i
$550$	0	0
$551$	−10.1469	−0.432271
$552$	0	0
$553$	−12.0537	−0.512577
$554$	0	0
$555$	− 56.4844i	− 2.39763i
$556$	0	0
$557$	− 30.1130i	− 1.27593i	−0.770066	−	0.637964i	$-0.779777\pi$
$557$	− 30.1130i	− 1.27593i	0.770066	−	0.637964i	$-0.220223\pi$
$558$	0	0
$559$	−4.89973	−0.207236
$560$	0	0
$561$	41.1166	1.73594
$562$	0	0
$563$	20.3344i	0.856993i	0.903543	+	0.428497i	$0.140957\pi$
$563$	20.3344i	0.856993i	−0.903543	+	0.428497i	$0.859043\pi$
$564$	0	0
$565$	22.1811i	0.933167i
$566$	0	0
$567$	−4.52503	−0.190033
$568$	0	0
$569$	41.3117	1.73188	0.865939	−	0.500150i	$-0.166722\pi$
$569$	41.3117	1.73188	0.865939	+	0.500150i	$0.166722\pi$
$570$	0	0
$571$	− 40.7491i	− 1.70530i	−0.522483	−	0.852649i	$-0.674994\pi$
$571$	− 40.7491i	− 1.70530i	0.522483	−	0.852649i	$-0.325006\pi$
$572$	0	0
$573$	− 36.0990i	− 1.50806i
$574$	0	0
$575$	13.5570	0.565367
$576$	0	0
$577$	18.2569	0.760043	0.380021	−	0.924978i	$-0.375917\pi$
$577$	18.2569	0.760043	0.380021	+	0.924978i	$0.375917\pi$
$578$	0	0
$579$	12.2876i	0.510656i
$580$	0	0
$581$	0.262559i	0.0108928i
$582$	0	0
$583$	−1.33603	−0.0553329
$584$	0	0
$585$	−11.3556	−0.469496
$586$	0	0
$587$	32.8881i	1.35744i	0.734399	+	0.678718i	$0.237464\pi$
$587$	32.8881i	1.35744i	−0.734399	+	0.678718i	$0.762536\pi$
$588$	0	0
$589$	15.6752i	0.645887i
$590$	0	0
$591$	−13.8773	−0.570837
$592$	0	0
$593$	−46.0425	−1.89074	−0.945369	−	0.326001i	$-0.894299\pi$
$593$	−46.0425	−1.89074	−0.945369	+	0.326001i	$0.894299\pi$
$594$	0	0
$595$	− 18.5791i	− 0.761667i
$596$	0	0
$597$	63.3915i	2.59444i
$598$	0	0
$599$	2.58321	0.105547	0.0527735	−	0.998607i	$-0.483194\pi$
$599$	2.58321	0.105547	0.0527735	+	0.998607i	$0.483194\pi$
$600$	0	0
$601$	5.32403	0.217172	0.108586	−	0.994087i	$-0.465368\pi$
$601$	5.32403	0.217172	0.108586	+	0.994087i	$0.465368\pi$
$602$	0	0
$603$	− 27.8383i	− 1.13366i
$604$	0	0
$605$	15.7742i	0.641311i
$606$	0	0
$607$	12.7131	0.516010	0.258005	−	0.966144i	$-0.416935\pi$
$607$	12.7131	0.516010	0.258005	+	0.966144i	$0.416935\pi$
$608$	0	0
$609$	11.2424	0.455565
$610$	0	0
$611$	13.5343i	0.547541i
$612$	0	0
$613$	− 25.9763i	− 1.04917i	−0.851357	−	0.524587i	$-0.824220\pi$
$613$	− 25.9763i	− 1.04917i	0.851357	−	0.524587i	$-0.175780\pi$
$614$	0	0
$615$	−12.3381	−0.497519
$616$	0	0
$617$	12.9002	0.519343	0.259671	−	0.965697i	$-0.416386\pi$
$617$	12.9002	0.519343	0.259671	+	0.965697i	$0.416386\pi$
$618$	0	0
$619$	8.88577i	0.357149i	0.983926	+	0.178575i	$0.0571486\pi$
$619$	8.88577i	0.357149i	−0.983926	+	0.178575i	$0.942851\pi$
$620$	0	0
$621$	− 14.6400i	− 0.587483i
$622$	0	0
$623$	−14.0431	−0.562625
$624$	0	0
$625$	−31.2115	−1.24846
$626$	0	0
$627$	− 14.7581i	− 0.589382i
$628$	0	0
$629$	51.1974i	2.04137i
$630$	0	0
$631$	−17.0995	−0.680719	−0.340359	−	0.940295i	$-0.610549\pi$
$631$	−17.0995	−0.680719	−0.340359	+	0.940295i	$0.610549\pi$
$632$	0	0
$633$	−25.6975	−1.02138
$634$	0	0
$635$	27.6055i	1.09549i
$636$	0	0
$637$	− 1.00000i	− 0.0396214i
$638$	0	0
$639$	31.7349	1.25541
$640$	0	0
$641$	−5.85739	−0.231353	−0.115677	−	0.993287i	$-0.536904\pi$
$641$	−5.85739	−0.231353	−0.115677	+	0.993287i	$0.536904\pi$
$642$	0	0
$643$	− 45.2251i	− 1.78350i	−0.452526	−	0.891751i	$-0.649477\pi$
$643$	− 45.2251i	− 1.78350i	0.452526	−	0.891751i	$-0.350523\pi$
$644$	0	0
$645$	− 36.2022i	− 1.42546i
$646$	0	0
$647$	26.6756	1.04873	0.524364	−	0.851494i	$-0.324303\pi$
$647$	26.6756	1.04873	0.524364	+	0.851494i	$0.324303\pi$
$648$	0	0
$649$	−23.5484	−0.924357
$650$	0	0
$651$	− 17.3676i	− 0.680691i
$652$	0	0
$653$	− 0.320551i	− 0.0125441i	−0.999980	−	0.00627206i	$-0.998004\pi$
$653$	− 0.320551i	− 0.0125441i	0.999980	−	0.00627206i	$-0.00199647\pi$
$654$	0	0
$655$	0.472644	0.0184677
$656$	0	0
$657$	23.3997	0.912908
$658$	0	0
$659$	− 18.9811i	− 0.739398i	−0.929151	−	0.369699i	$-0.879461\pi$
$659$	− 18.9811i	− 0.739398i	0.929151	−	0.369699i	$-0.120539\pi$
$660$	0	0
$661$	− 32.3812i	− 1.25948i	−0.776804	−	0.629742i	$-0.783161\pi$
$661$	− 32.3812i	− 1.25948i	0.776804	−	0.629742i	$-0.216839\pi$
$662$	0	0
$663$	17.8363	0.692706
$664$	0	0
$665$	−6.66864	−0.258599
$666$	0	0
$667$	− 21.2243i	− 0.821807i
$668$	0	0
$669$	− 3.00328i	− 0.116114i
$670$	0	0
$671$	18.2595	0.704898
$672$	0	0
$673$	−1.03242	−0.0397970	−0.0198985	−	0.999802i	$-0.506334\pi$
$673$	−1.03242	−0.0397970	−0.0198985	+	0.999802i	$0.506334\pi$
$674$	0	0
$675$	− 7.85076i	− 0.302176i
$676$	0	0
$677$	− 5.19777i	− 0.199767i	−0.994999	−	0.0998833i	$-0.968153\pi$
$677$	− 5.19777i	− 0.199767i	0.994999	−	0.0998833i	$-0.0318470\pi$
$678$	0	0
$679$	−15.9382	−0.611654
$680$	0	0
$681$	−74.6821	−2.86183
$682$	0	0
$683$	4.36371i	0.166973i	0.996509	+	0.0834865i	$0.0266055\pi$
$683$	4.36371i	0.166973i	−0.996509	+	0.0834865i	$0.973394\pi$
$684$	0	0
$685$	− 59.1533i	− 2.26013i
$686$	0	0
$687$	52.6690	2.00945
$688$	0	0
$689$	−0.579570	−0.0220799
$690$	0	0
$691$	− 5.66095i	− 0.215353i	−0.994186	−	0.107676i	$-0.965659\pi$
$691$	− 5.66095i	− 0.215353i	0.994186	−	0.107676i	$-0.0343410\pi$
$692$	0	0
$693$	9.43584i	0.358438i
$694$	0	0
$695$	7.64639	0.290044
$696$	0	0
$697$	11.1832	0.423594
$698$	0	0
$699$	71.2412i	2.69459i
$700$	0	0
$701$	13.3890i	0.505695i	0.967506	+	0.252848i	$0.0813671\pi$
$701$	13.3890i	0.505695i	−0.967506	+	0.252848i	$0.918633\pi$
$702$	0	0
$703$	18.3764	0.693081
$704$	0	0
$705$	−100.000	−3.76622
$706$	0	0
$707$	− 11.8085i	− 0.444103i
$708$	0	0
$709$	27.4963i	1.03265i	0.856394	+	0.516323i	$0.172699\pi$
$709$	27.4963i	1.03265i	−0.856394	+	0.516323i	$0.827301\pi$
$710$	0	0
$711$	49.3390	1.85036
$712$	0	0
$713$	−32.7880	−1.22792
$714$	0	0
$715$	− 6.39518i	− 0.239166i
$716$	0	0
$717$	52.4626i	1.95925i
$718$	0	0
$719$	5.61384	0.209361	0.104681	−	0.994506i	$-0.466618\pi$
$719$	5.61384	0.209361	0.104681	+	0.994506i	$0.466618\pi$
$720$	0	0
$721$	2.80041	0.104293
$722$	0	0
$723$	− 22.1396i	− 0.823382i
$724$	0	0
$725$	− 11.3816i	− 0.422702i
$726$	0	0
$727$	−41.5030	−1.53926	−0.769630	−	0.638490i	$-0.779559\pi$
$727$	−41.5030	−1.53926	−0.769630	+	0.638490i	$0.779559\pi$
$728$	0	0
$729$	41.7863	1.54764
$730$	0	0
$731$	32.8137i	1.21366i
$732$	0	0
$733$	− 15.7818i	− 0.582913i	−0.956584	−	0.291456i	$-0.905860\pi$
$733$	− 15.7818i	− 0.582913i	0.956584	−	0.291456i	$-0.0941398\pi$
$734$	0	0
$735$	7.38863	0.272534
$736$	0	0
$737$	15.6778	0.577499
$738$	0	0
$739$	35.4864i	1.30539i	0.757622	+	0.652694i	$0.226361\pi$
$739$	35.4864i	1.30539i	−0.757622	+	0.652694i	$0.773639\pi$
$740$	0	0
$741$	− 6.40205i	− 0.235185i
$742$	0	0
$743$	−43.4551	−1.59421	−0.797107	−	0.603838i	$-0.793637\pi$
$743$	−43.4551	−1.59421	−0.797107	+	0.603838i	$0.793637\pi$
$744$	0	0
$745$	13.1977	0.483528
$746$	0	0
$747$	− 1.07472i	− 0.0393220i
$748$	0	0
$749$	− 2.80406i	− 0.102458i
$750$	0	0
$751$	−0.732227	−0.0267193	−0.0133597	−	0.999911i	$-0.504253\pi$
$751$	−0.732227	−0.0267193	−0.0133597	+	0.999911i	$0.504253\pi$
$752$	0	0
$753$	64.6814	2.35712
$754$	0	0
$755$	18.6676i	0.679385i
$756$	0	0
$757$	− 14.3837i	− 0.522785i	−0.965233	−	0.261393i	$-0.915818\pi$
$757$	− 14.3837i	− 0.522785i	0.965233	−	0.261393i	$-0.0841817\pi$
$758$	0	0
$759$	30.8696	1.12050
$760$	0	0
$761$	1.97792	0.0716996	0.0358498	−	0.999357i	$-0.488586\pi$
$761$	1.97792	0.0716996	0.0358498	+	0.999357i	$0.488586\pi$
$762$	0	0
$763$	9.23050i	0.334167i
$764$	0	0
$765$	76.0488i	2.74955i
$766$	0	0
$767$	−10.2153	−0.368853
$768$	0	0
$769$	−14.1082	−0.508754	−0.254377	−	0.967105i	$-0.581870\pi$
$769$	−14.1082	−0.508754	−0.254377	+	0.967105i	$0.581870\pi$
$770$	0	0
$771$	− 31.3762i	− 1.12999i
$772$	0	0
$773$	39.4071i	1.41737i	0.705523	+	0.708687i	$0.250712\pi$
$773$	39.4071i	1.41737i	−0.705523	+	0.708687i	$0.749288\pi$
$774$	0	0
$775$	−17.5827	−0.631588
$776$	0	0
$777$	−20.3605	−0.730428
$778$	0	0
$779$	− 4.01402i	− 0.143817i
$780$	0	0
$781$	17.8723i	0.639521i
$782$	0	0
$783$	−12.2908	−0.439237
$784$	0	0
$785$	−20.4509	−0.729923
$786$	0	0
$787$	10.4800i	0.373571i	0.982401	+	0.186786i	$0.0598070\pi$
$787$	10.4800i	0.373571i	−0.982401	+	0.186786i	$0.940193\pi$
$788$	0	0
$789$	43.2045i	1.53812i
$790$	0	0
$791$	7.99544	0.284285
$792$	0	0
$793$	7.92093	0.281280
$794$	0	0
$795$	− 4.28223i	− 0.151875i
$796$	0	0
$797$	31.7295i	1.12392i	0.827166	+	0.561958i	$0.189952\pi$
$797$	31.7295i	1.12392i	−0.827166	+	0.561958i	$0.810048\pi$
$798$	0	0
$799$	90.6400	3.20661
$800$	0	0
$801$	57.4820	2.03103
$802$	0	0
$803$	13.1781i	0.465045i
$804$	0	0
$805$	− 13.9488i	− 0.491632i
$806$	0	0
$807$	70.4585	2.48026
$808$	0	0
$809$	17.7495	0.624039	0.312019	−	0.950076i	$-0.398995\pi$
$809$	17.7495	0.624039	0.312019	+	0.950076i	$0.398995\pi$
$810$	0	0
$811$	− 12.0825i	− 0.424273i	−0.977240	−	0.212136i	$-0.931958\pi$
$811$	− 12.0825i	− 0.424273i	0.977240	−	0.212136i	$-0.0680421\pi$
$812$	0	0
$813$	− 58.5896i	− 2.05483i
$814$	0	0
$815$	28.2236	0.988630
$816$	0	0
$817$	11.7779	0.412057
$818$	0	0
$819$	4.09325i	0.143030i
$820$	0	0
$821$	− 14.5237i	− 0.506882i	−0.967351	−	0.253441i	$-0.918438\pi$
$821$	− 14.5237i	− 0.506882i	0.967351	−	0.253441i	$-0.0815623\pi$
$822$	0	0
$823$	32.1613	1.12107	0.560536	−	0.828130i	$-0.310595\pi$
$823$	32.1613	1.12107	0.560536	+	0.828130i	$0.310595\pi$
$824$	0	0
$825$	16.5540	0.576335
$826$	0	0
$827$	27.0041i	0.939026i	0.882925	+	0.469513i	$0.155571\pi$
$827$	27.0041i	0.939026i	−0.882925	+	0.469513i	$0.844429\pi$
$828$	0	0
$829$	32.8357i	1.14043i	0.821496	+	0.570215i	$0.193140\pi$
$829$	32.8357i	1.14043i	−0.821496	+	0.570215i	$0.806860\pi$
$830$	0	0
$831$	23.1288	0.802330
$832$	0	0
$833$	−6.69704	−0.232039
$834$	0	0
$835$	55.9490i	1.93620i
$836$	0	0
$837$	18.9872i	0.656295i
$838$	0	0
$839$	−28.8041	−0.994429	−0.497215	−	0.867628i	$-0.665644\pi$
$839$	−28.8041	−0.994429	−0.497215	+	0.867628i	$0.665644\pi$
$840$	0	0
$841$	11.1815	0.385568
$842$	0	0
$843$	− 89.0064i	− 3.06554i
$844$	0	0
$845$	− 2.77422i	− 0.0954361i
$846$	0	0
$847$	5.68598	0.195373
$848$	0	0
$849$	61.6784	2.11680
$850$	0	0
$851$	38.4381i	1.31764i
$852$	0	0
$853$	− 34.6783i	− 1.18736i	−0.804700	−	0.593681i	$-0.797674\pi$
$853$	− 34.6783i	− 1.18736i	0.804700	−	0.593681i	$-0.202326\pi$
$854$	0	0
$855$	27.2964	0.933519
$856$	0	0
$857$	3.12688	0.106812	0.0534062	−	0.998573i	$-0.482992\pi$
$857$	3.12688	0.106812	0.0534062	+	0.998573i	$0.482992\pi$
$858$	0	0
$859$	− 36.7898i	− 1.25525i	−0.778515	−	0.627626i	$-0.784027\pi$
$859$	− 36.7898i	− 1.25525i	0.778515	−	0.627626i	$-0.215973\pi$
$860$	0	0
$861$	4.44740i	0.151567i
$862$	0	0
$863$	−16.4365	−0.559506	−0.279753	−	0.960072i	$-0.590253\pi$
$863$	−16.4365	−0.559506	−0.279753	+	0.960072i	$0.590253\pi$
$864$	0	0
$865$	−15.0271	−0.510936
$866$	0	0
$867$	− 74.1743i	− 2.51909i
$868$	0	0
$869$	27.7865i	0.942591i
$870$	0	0
$871$	6.80101	0.230444
$872$	0	0
$873$	65.2393	2.20802
$874$	0	0
$875$	6.39098i	0.216055i
$876$	0	0
$877$	− 10.0167i	− 0.338240i	−0.985595	−	0.169120i	$-0.945908\pi$
$877$	− 10.0167i	− 0.338240i	0.985595	−	0.169120i	$-0.0540925\pi$
$878$	0	0
$879$	−53.9394	−1.81933
$880$	0	0
$881$	20.3551	0.685779	0.342890	−	0.939376i	$-0.388594\pi$
$881$	20.3551	0.685779	0.342890	+	0.939376i	$0.388594\pi$
$882$	0	0
$883$	16.8609i	0.567415i	0.958911	+	0.283707i	$0.0915644\pi$
$883$	16.8609i	0.567415i	−0.958911	+	0.283707i	$0.908436\pi$
$884$	0	0
$885$	− 75.4769i	− 2.53713i
$886$	0	0
$887$	−3.98272	−0.133727	−0.0668633	−	0.997762i	$-0.521299\pi$
$887$	−3.98272	−0.133727	−0.0668633	+	0.997762i	$0.521299\pi$
$888$	0	0
$889$	9.95073	0.333737
$890$	0	0
$891$	10.4312i	0.349457i
$892$	0	0
$893$	− 32.5337i	− 1.08870i
$894$	0	0
$895$	18.8563	0.630297
$896$	0	0
$897$	13.3912	0.447119
$898$	0	0
$899$	27.5267i	0.918065i
$900$	0	0
$901$	3.88141i	0.129308i
$902$	0	0
$903$	−13.0495	−0.434261
$904$	0	0
$905$	−41.3842	−1.37566
$906$	0	0
$907$	16.0034i	0.531385i	0.964058	+	0.265692i	$0.0856005\pi$
$907$	16.0034i	0.531385i	−0.964058	+	0.265692i	$0.914399\pi$
$908$	0	0
$909$	48.3351i	1.60317i
$910$	0	0
$911$	−5.85697	−0.194050	−0.0970250	−	0.995282i	$-0.530933\pi$
$911$	−5.85697	−0.194050	−0.0970250	+	0.995282i	$0.530933\pi$
$912$	0	0
$913$	0.605256	0.0200310
$914$	0	0
$915$	58.5248i	1.93477i
$916$	0	0
$917$	− 0.170370i	− 0.00562611i
$918$	0	0
$919$	34.9195	1.15189	0.575943	−	0.817490i	$-0.304635\pi$
$919$	34.9195	1.15189	0.575943	+	0.817490i	$0.304635\pi$
$920$	0	0
$921$	30.5175	1.00559
$922$	0	0
$923$	7.75298i	0.255193i
$924$	0	0
$925$	20.6126i	0.677738i
$926$	0	0
$927$	−11.4628	−0.376487
$928$	0	0
$929$	−43.0245	−1.41159	−0.705794	−	0.708417i	$-0.749410\pi$
$929$	−43.0245	−1.41159	−0.705794	+	0.708417i	$0.749410\pi$
$930$	0	0
$931$	2.40379i	0.0787810i
$932$	0	0
$933$	19.1191i	0.625932i
$934$	0	0
$935$	−42.8287	−1.40065
$936$	0	0
$937$	−0.552460	−0.0180481	−0.00902404	−	0.999959i	$-0.502872\pi$
$937$	−0.552460	−0.0180481	−0.00902404	+	0.999959i	$0.502872\pi$
$938$	0	0
$939$	− 74.8711i	− 2.44333i
$940$	0	0
$941$	54.7997i	1.78642i	0.449641	+	0.893210i	$0.351552\pi$
$941$	54.7997i	1.78642i	−0.449641	+	0.893210i	$0.648448\pi$
$942$	0	0
$943$	8.39615	0.273416
$944$	0	0
$945$	−8.07765	−0.262766
$946$	0	0
$947$	33.7873i	1.09794i	0.835843	+	0.548969i	$0.184980\pi$
$947$	33.7873i	1.09794i	−0.835843	+	0.548969i	$0.815020\pi$
$948$	0	0
$949$	5.71664i	0.185570i
$950$	0	0
$951$	−41.6907	−1.35191
$952$	0	0
$953$	15.2367	0.493565	0.246782	−	0.969071i	$-0.420627\pi$
$953$	15.2367	0.493565	0.246782	+	0.969071i	$0.420627\pi$
$954$	0	0
$955$	37.6022i	1.21678i
$956$	0	0
$957$	− 25.9161i	− 0.837750i
$958$	0	0
$959$	−21.3225	−0.688539
$960$	0	0
$961$	11.5241	0.371745
$962$	0	0
$963$	11.4777i	0.369865i
$964$	0	0
$965$	− 12.7993i	− 0.412023i
$966$	0	0
$967$	14.5570	0.468121	0.234060	−	0.972222i	$-0.424799\pi$
$967$	14.5570	0.468121	0.234060	+	0.972222i	$0.424799\pi$
$968$	0	0
$969$	−42.8748	−1.37734
$970$	0	0
$971$	− 11.6461i	− 0.373742i	−0.982384	−	0.186871i	$-0.940165\pi$
$971$	− 11.6461i	− 0.373742i	0.982384	−	0.186871i	$-0.0598346\pi$
$972$	0	0
$973$	− 2.75623i	− 0.0883608i
$974$	0	0
$975$	7.18109	0.229979
$976$	0	0
$977$	58.9341	1.88547	0.942734	−	0.333545i	$-0.108245\pi$
$977$	58.9341	1.88547	0.942734	+	0.333545i	$0.108245\pi$
$978$	0	0
$979$	32.3724i	1.03463i
$980$	0	0
$981$	− 37.7828i	− 1.20631i
$982$	0	0
$983$	58.3762	1.86191	0.930957	−	0.365130i	$-0.118975\pi$
$983$	58.3762	1.86191	0.930957	+	0.365130i	$0.118975\pi$
$984$	0	0
$985$	14.4552	0.460581
$986$	0	0
$987$	36.0462i	1.14736i
$988$	0	0
$989$	24.6359i	0.783377i
$990$	0	0
$991$	4.43349	0.140834	0.0704172	−	0.997518i	$-0.477567\pi$
$991$	4.43349	0.140834	0.0704172	+	0.997518i	$0.477567\pi$
$992$	0	0
$993$	45.7465	1.45172
$994$	0	0
$995$	− 66.0312i	− 2.09333i
$996$	0	0
$997$	− 19.9632i	− 0.632241i	−0.948719	−	0.316121i	$-0.897620\pi$
$997$	− 19.9632i	− 0.632241i	0.948719	−	0.316121i	$-0.102380\pi$
$998$	0	0
$999$	22.2592	0.704249

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.c.a.1457.5		34
4.3	odd	2		728.2.c.a.365.10	yes	34
8.3	odd	2		728.2.c.a.365.9	✓	34
8.5	even	2	inner	2912.2.c.a.1457.30		34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.c.a.365.9	✓	34	8.3	odd	2
728.2.c.a.365.10	yes	34	4.3	odd	2
2912.2.c.a.1457.5		34	1.1	even	1	trivial
2912.2.c.a.1457.30		34	8.5	even	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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.66332i q^{3} +2.77422i q^{5} +1.00000 q^{7} -4.09325 q^{9} -2.30522i q^{11} -1.00000i q^{13} +7.38863 q^{15} -6.69704 q^{17} +2.40379i q^{19} -2.66332i q^{21} -5.02802 q^{23} -2.69630 q^{25} +2.91168i q^{27} +4.22120i q^{29} +6.52105 q^{31} -6.13952 q^{33} +2.77422i q^{35} -7.64478i q^{37} -2.66332 q^{39} -1.66987 q^{41} -4.89973i q^{43} -11.3556i q^{45} -13.5343 q^{47} +1.00000 q^{49} +17.8363i q^{51} -0.579570i q^{53} +6.39518 q^{55} +6.40205 q^{57} -10.2153i q^{59} +7.92093i q^{61} -4.09325 q^{63} +2.77422 q^{65} +6.80101i q^{67} +13.3912i q^{69} -7.75298 q^{71} -5.71664 q^{73} +7.18109i q^{75} -2.30522i q^{77} -12.0537 q^{79} -4.52503 q^{81} +0.262559i q^{83} -18.5791i q^{85} +11.2424 q^{87} -14.0431 q^{89} -1.00000i q^{91} -17.3676i q^{93} -6.66864 q^{95} -15.9382 q^{97} +9.43584i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) \) 34 * q + 34 * q^7 - 26 * q^9 + 8 * q^15 - 20 * q^17 + 20 * q^23 - 22 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 34 * q^49 + 32 * q^55 + 8 * q^57 - 26 * q^63 - 20 * q^65 - 64 * q^71 - 20 * q^79 - 6 * q^81 + 56 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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