LMFDB - Embedded newform 1216.3.f.b.799.37

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(799,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1216.799");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.f (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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.37
Character	$\chi$	$=$	1216.799
Dual form			1216.3.f.b.799.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.50746 q^{3} -0.252768i q^{5} +0.190530i q^{7} +3.30228 q^{9} +O(q^{10})$	$q+3.50746 q^{3} -0.252768i q^{5} +0.190530i q^{7} +3.30228 q^{9} -5.60842 q^{11} -14.5700i q^{13} -0.886574i q^{15} -18.9541 q^{17} -4.35890 q^{19} +0.668275i q^{21} -33.0016i q^{23} +24.9361 q^{25} -19.9845 q^{27} -45.4090i q^{29} -6.68424i q^{31} -19.6713 q^{33} +0.0481598 q^{35} +3.92687i q^{37} -51.1038i q^{39} -19.6255 q^{41} +2.22162 q^{43} -0.834712i q^{45} -5.41053i q^{47} +48.9637 q^{49} -66.4808 q^{51} -61.2112i q^{53} +1.41763i q^{55} -15.2887 q^{57} +25.8963 q^{59} +32.9728i q^{61} +0.629183i q^{63} -3.68284 q^{65} +29.2118 q^{67} -115.752i q^{69} -10.5166i q^{71} -91.2827 q^{73} +87.4624 q^{75} -1.06857i q^{77} -22.7410i q^{79} -99.8155 q^{81} +144.739 q^{83} +4.79099i q^{85} -159.270i q^{87} -37.1587 q^{89} +2.77602 q^{91} -23.4447i q^{93} +1.10179i q^{95} -117.690 q^{97} -18.5206 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 168 q^{9}+O(q^{10}) $ 48 * q + 168 * q^9	$ 48 q + 168 q^{9} + 100 q^{17} - 204 q^{25} - 8 q^{33} - 240 q^{41} - 460 q^{49} - 368 q^{65} - 132 q^{73} + 768 q^{81} + 696 q^{89} - 8 q^{97}+O(q^{100}) $ 48 * q + 168 * q^9 + 100 * q^17 - 204 * q^25 - 8 * q^33 - 240 * q^41 - 460 * q^49 - 368 * q^65 - 132 * q^73 + 768 * q^81 + 696 * q^89 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	3.50746	1.16915	0.584577	−	0.811338i	$-0.301261\pi$
$3$	3.50746	1.16915	0.584577	+	0.811338i	$0.301261\pi$
$4$	0	0
$5$	− 0.252768i	− 0.0505536i	−0.999680	−	0.0252768i	$-0.991953\pi$
$5$	− 0.252768i	− 0.0505536i	0.999680	−	0.0252768i	$-0.00804671\pi$
$6$	0	0
$7$	0.190530i	0.0272185i	0.999907	+	0.0136093i	$0.00433209\pi$
$7$	0.190530i	0.0272185i	−0.999907	+	0.0136093i	$0.995668\pi$
$8$	0	0
$9$	3.30228	0.366920
$10$	0	0
$11$	−5.60842	−0.509857	−0.254928	−	0.966960i	$-0.582052\pi$
$11$	−5.60842	−0.509857	−0.254928	+	0.966960i	$0.582052\pi$
$12$	0	0
$13$	− 14.5700i	− 1.12077i	−0.828232	−	0.560386i	$-0.810653\pi$
$13$	− 14.5700i	− 1.12077i	0.828232	−	0.560386i	$-0.189347\pi$
$14$	0	0
$15$	− 0.886574i	− 0.0591049i
$16$	0	0
$17$	−18.9541	−1.11495	−0.557474	−	0.830195i	$-0.688229\pi$
$17$	−18.9541	−1.11495	−0.557474	+	0.830195i	$0.688229\pi$
$18$	0	0
$19$	−4.35890	−0.229416
$19$	−4.35890	−0.229416
$20$	0	0
$21$	0.668275i	0.0318226i
$22$	0	0
$23$	− 33.0016i	− 1.43485i	−0.696636	−	0.717425i	$-0.745321\pi$
$23$	− 33.0016i	− 1.43485i	0.696636	−	0.717425i	$-0.254679\pi$
$24$	0	0
$25$	24.9361	0.997444
$26$	0	0
$27$	−19.9845	−0.740167
$28$	0	0
$29$	− 45.4090i	− 1.56583i	−0.622129	−	0.782914i	$-0.713732\pi$
$29$	− 45.4090i	− 1.56583i	0.622129	−	0.782914i	$-0.286268\pi$
$30$	0	0
$31$	− 6.68424i	− 0.215621i	−0.994171	−	0.107810i	$-0.965616\pi$
$31$	− 6.68424i	− 0.215621i	0.994171	−	0.107810i	$-0.0343839\pi$
$32$	0	0
$33$	−19.6713	−0.596101
$34$	0	0
$35$	0.0481598	0.00137599
$36$	0	0
$37$	3.92687i	0.106132i	0.998591	+	0.0530658i	$0.0168993\pi$
$37$	3.92687i	0.106132i	−0.998591	+	0.0530658i	$0.983101\pi$
$38$	0	0
$39$	− 51.1038i	− 1.31035i
$40$	0	0
$41$	−19.6255	−0.478671	−0.239335	−	0.970937i	$-0.576930\pi$
$41$	−19.6255	−0.478671	−0.239335	+	0.970937i	$0.576930\pi$
$42$	0	0
$43$	2.22162	0.0516657	0.0258328	−	0.999666i	$-0.491776\pi$
$43$	2.22162	0.0516657	0.0258328	+	0.999666i	$0.491776\pi$
$44$	0	0
$45$	− 0.834712i	− 0.0185491i
$46$	0	0
$47$	− 5.41053i	− 0.115118i	−0.998342	−	0.0575588i	$-0.981668\pi$
$47$	− 5.41053i	− 0.115118i	0.998342	−	0.0575588i	$-0.0183317\pi$
$48$	0	0
$49$	48.9637	0.999259
$50$	0	0
$51$	−66.4808	−1.30354
$52$	0	0
$53$	− 61.2112i	− 1.15493i	−0.816416	−	0.577464i	$-0.804042\pi$
$53$	− 61.2112i	− 1.15493i	0.816416	−	0.577464i	$-0.195958\pi$
$54$	0	0
$55$	1.41763i	0.0257751i
$56$	0	0
$57$	−15.2887	−0.268222
$58$	0	0
$59$	25.8963	0.438920	0.219460	−	0.975622i	$-0.429571\pi$
$59$	25.8963	0.438920	0.219460	+	0.975622i	$0.429571\pi$
$60$	0	0
$61$	32.9728i	0.540538i	0.962785	+	0.270269i	$0.0871127\pi$
$61$	32.9728i	0.540538i	−0.962785	+	0.270269i	$0.912887\pi$
$62$	0	0
$63$	0.629183i	0.00998702i
$64$	0	0
$65$	−3.68284	−0.0566590
$66$	0	0
$67$	29.2118	0.435997	0.217999	−	0.975949i	$-0.430047\pi$
$67$	29.2118	0.435997	0.217999	+	0.975949i	$0.430047\pi$
$68$	0	0
$69$	− 115.752i	− 1.67756i
$70$	0	0
$71$	− 10.5166i	− 0.148121i	−0.997254	−	0.0740604i	$-0.976404\pi$
$71$	− 10.5166i	− 0.148121i	0.997254	−	0.0740604i	$-0.0235958\pi$
$72$	0	0
$73$	−91.2827	−1.25045	−0.625224	−	0.780445i	$-0.714993\pi$
$73$	−91.2827	−1.25045	−0.625224	+	0.780445i	$0.714993\pi$
$74$	0	0
$75$	87.4624	1.16617
$76$	0	0
$77$	− 1.06857i	− 0.0138775i
$78$	0	0
$79$	− 22.7410i	− 0.287861i	−0.989588	−	0.143931i	$-0.954026\pi$
$79$	− 22.7410i	− 0.287861i	0.989588	−	0.143931i	$-0.0459742\pi$
$80$	0	0
$81$	−99.8155	−1.23229
$82$	0	0
$83$	144.739	1.74384	0.871920	−	0.489649i	$-0.162875\pi$
$83$	144.739	1.74384	0.871920	+	0.489649i	$0.162875\pi$
$84$	0	0
$85$	4.79099i	0.0563646i
$86$	0	0
$87$	− 159.270i	− 1.83069i
$88$	0	0
$89$	−37.1587	−0.417513	−0.208757	−	0.977968i	$-0.566942\pi$
$89$	−37.1587	−0.417513	−0.208757	+	0.977968i	$0.566942\pi$
$90$	0	0
$91$	2.77602	0.0305057
$92$	0	0
$93$	− 23.4447i	− 0.252094i
$94$	0	0
$95$	1.10179i	0.0115978i
$96$	0	0
$97$	−117.690	−1.21330	−0.606651	−	0.794968i	$-0.707487\pi$
$97$	−117.690	−1.21330	−0.606651	+	0.794968i	$0.707487\pi$
$98$	0	0
$99$	−18.5206	−0.187077
$100$	0	0
$101$	71.9079i	0.711959i	0.934494	+	0.355980i	$0.115853\pi$
$101$	71.9079i	0.711959i	−0.934494	+	0.355980i	$0.884147\pi$
$102$	0	0
$103$	− 147.791i	− 1.43486i	−0.696630	−	0.717431i	$-0.745318\pi$
$103$	− 147.791i	− 1.43486i	0.696630	−	0.717431i	$-0.254682\pi$
$104$	0	0
$105$	0.168919	0.00160875
$106$	0	0
$107$	19.4286	0.181576	0.0907880	−	0.995870i	$-0.471061\pi$
$107$	19.4286	0.181576	0.0907880	+	0.995870i	$0.471061\pi$
$108$	0	0
$109$	156.427i	1.43511i	0.696500	+	0.717557i	$0.254740\pi$
$109$	156.427i	1.43511i	−0.696500	+	0.717557i	$0.745260\pi$
$110$	0	0
$111$	13.7733i	0.124084i
$112$	0	0
$113$	−7.58861	−0.0671558	−0.0335779	−	0.999436i	$-0.510690\pi$
$113$	−7.58861	−0.0671558	−0.0335779	+	0.999436i	$0.510690\pi$
$114$	0	0
$115$	−8.34174	−0.0725368
$116$	0	0
$117$	− 48.1144i	− 0.411234i
$118$	0	0
$119$	− 3.61132i	− 0.0303472i
$120$	0	0
$121$	−89.5456	−0.740046
$122$	0	0
$123$	−68.8357	−0.559640
$124$	0	0
$125$	− 12.6223i	− 0.100978i
$126$	0	0
$127$	150.835i	1.18767i	0.804585	+	0.593837i	$0.202388\pi$
$127$	150.835i	1.18767i	−0.804585	+	0.593837i	$0.797612\pi$
$128$	0	0
$129$	7.79226	0.0604051
$130$	0	0
$131$	55.7865	0.425851	0.212926	−	0.977068i	$-0.431701\pi$
$131$	55.7865	0.425851	0.212926	+	0.977068i	$0.431701\pi$
$132$	0	0
$133$	− 0.830499i	− 0.00624435i
$134$	0	0
$135$	5.05145i	0.0374181i
$136$	0	0
$137$	−131.228	−0.957866	−0.478933	−	0.877851i	$-0.658976\pi$
$137$	−131.228	−0.957866	−0.478933	+	0.877851i	$0.658976\pi$
$138$	0	0
$139$	213.144	1.53341	0.766706	−	0.641998i	$-0.221894\pi$
$139$	213.144	1.53341	0.766706	+	0.641998i	$0.221894\pi$
$140$	0	0
$141$	− 18.9772i	− 0.134590i
$142$	0	0
$143$	81.7149i	0.571433i
$144$	0	0
$145$	−11.4780	−0.0791583
$146$	0	0
$147$	171.738	1.16829
$148$	0	0
$149$	− 108.251i	− 0.726514i	−0.931689	−	0.363257i	$-0.881665\pi$
$149$	− 108.251i	− 0.726514i	0.931689	−	0.363257i	$-0.118335\pi$
$150$	0	0
$151$	99.9886i	0.662176i	0.943600	+	0.331088i	$0.107416\pi$
$151$	99.9886i	0.662176i	−0.943600	+	0.331088i	$0.892584\pi$
$152$	0	0
$153$	−62.5918	−0.409097
$154$	0	0
$155$	−1.68956	−0.0109004
$156$	0	0
$157$	− 10.7332i	− 0.0683646i	−0.999416	−	0.0341823i	$-0.989117\pi$
$157$	− 10.7332i	− 0.0683646i	0.999416	−	0.0341823i	$-0.0108827\pi$
$158$	0	0
$159$	− 214.696i	− 1.35029i
$160$	0	0
$161$	6.28777	0.0390545
$162$	0	0
$163$	−28.7669	−0.176484	−0.0882419	−	0.996099i	$-0.528125\pi$
$163$	−28.7669	−0.176484	−0.0882419	+	0.996099i	$0.528125\pi$
$164$	0	0
$165$	4.97228i	0.0301350i
$166$	0	0
$167$	− 294.712i	− 1.76474i	−0.470554	−	0.882371i	$-0.655946\pi$
$167$	− 294.712i	− 1.76474i	0.470554	−	0.882371i	$-0.344054\pi$
$168$	0	0
$169$	−43.2857	−0.256128
$170$	0	0
$171$	−14.3943	−0.0841773
$172$	0	0
$173$	179.476i	1.03743i	0.854947	+	0.518716i	$0.173590\pi$
$173$	179.476i	1.03743i	−0.854947	+	0.518716i	$0.826410\pi$
$174$	0	0
$175$	4.75107i	0.0271489i
$176$	0	0
$177$	90.8301	0.513164
$178$	0	0
$179$	187.374	1.04678	0.523391	−	0.852093i	$-0.324667\pi$
$179$	187.374	1.04678	0.523391	+	0.852093i	$0.324667\pi$
$180$	0	0
$181$	− 76.7083i	− 0.423803i	−0.977291	−	0.211901i	$-0.932034\pi$
$181$	− 76.7083i	− 0.423803i	0.977291	−	0.211901i	$-0.0679655\pi$
$182$	0	0
$183$	115.651i	0.631972i
$184$	0	0
$185$	0.992587	0.00536534
$186$	0	0
$187$	106.303	0.568463
$188$	0	0
$189$	− 3.80764i	− 0.0201463i
$190$	0	0
$191$	109.452i	0.573046i	0.958073	+	0.286523i	$0.0924995\pi$
$191$	109.452i	0.573046i	−0.958073	+	0.286523i	$0.907500\pi$
$192$	0	0
$193$	−254.694	−1.31966	−0.659830	−	0.751415i	$-0.729372\pi$
$193$	−254.694	−1.31966	−0.659830	+	0.751415i	$0.729372\pi$
$194$	0	0
$195$	−12.9174	−0.0662431
$196$	0	0
$197$	− 96.8634i	− 0.491693i	−0.969309	−	0.245846i	$-0.920934\pi$
$197$	− 96.8634i	− 0.491693i	0.969309	−	0.245846i	$-0.0790659\pi$
$198$	0	0
$199$	− 123.397i	− 0.620087i	−0.950722	−	0.310044i	$-0.899656\pi$
$199$	− 123.397i	− 0.620087i	0.950722	−	0.310044i	$-0.100344\pi$
$200$	0	0
$201$	102.459	0.509748
$202$	0	0
$203$	8.65176	0.0426195
$204$	0	0
$205$	4.96070i	0.0241985i
$206$	0	0
$207$	− 108.980i	− 0.526476i
$208$	0	0
$209$	24.4466	0.116969
$210$	0	0
$211$	−47.1948	−0.223672	−0.111836	−	0.993727i	$-0.535673\pi$
$211$	−47.1948	−0.223672	−0.111836	+	0.993727i	$0.535673\pi$
$212$	0	0
$213$	− 36.8865i	− 0.173176i
$214$	0	0
$215$	− 0.561555i	− 0.00261189i
$216$	0	0
$217$	1.27355	0.00586887
$218$	0	0
$219$	−320.171	−1.46197
$220$	0	0
$221$	276.162i	1.24960i
$222$	0	0
$223$	208.654i	0.935666i	0.883817	+	0.467833i	$0.154965\pi$
$223$	208.654i	0.935666i	−0.883817	+	0.467833i	$0.845035\pi$
$224$	0	0
$225$	82.3461	0.365983
$226$	0	0
$227$	−252.772	−1.11353	−0.556766	−	0.830669i	$-0.687958\pi$
$227$	−252.772	−1.11353	−0.556766	+	0.830669i	$0.687958\pi$
$228$	0	0
$229$	− 272.172i	− 1.18853i	−0.804271	−	0.594263i	$-0.797444\pi$
$229$	− 272.172i	− 1.18853i	0.804271	−	0.594263i	$-0.202556\pi$
$230$	0	0
$231$	− 3.74797i	− 0.0162250i
$232$	0	0
$233$	−231.151	−0.992065	−0.496032	−	0.868304i	$-0.665210\pi$
$233$	−231.151	−0.992065	−0.496032	+	0.868304i	$0.665210\pi$
$234$	0	0
$235$	−1.36761	−0.00581961
$236$	0	0
$237$	− 79.7633i	− 0.336554i
$238$	0	0
$239$	129.434i	0.541564i	0.962641	+	0.270782i	$0.0872822\pi$
$239$	129.434i	0.541564i	−0.962641	+	0.270782i	$0.912718\pi$
$240$	0	0
$241$	255.763	1.06126	0.530629	−	0.847604i	$-0.321956\pi$
$241$	255.763	1.06126	0.530629	+	0.847604i	$0.321956\pi$
$242$	0	0
$243$	−170.238	−0.700569
$244$	0	0
$245$	− 12.3765i	− 0.0505161i
$246$	0	0
$247$	63.5093i	0.257123i
$248$	0	0
$249$	507.665	2.03882
$250$	0	0
$251$	176.045	0.701375	0.350688	−	0.936493i	$-0.385948\pi$
$251$	176.045	0.701375	0.350688	+	0.936493i	$0.385948\pi$
$252$	0	0
$253$	185.087i	0.731568i
$254$	0	0
$255$	16.8042i	0.0658989i
$256$	0	0
$257$	−309.494	−1.20426	−0.602128	−	0.798400i	$-0.705680\pi$
$257$	−309.494	−1.20426	−0.602128	+	0.798400i	$0.705680\pi$
$258$	0	0
$259$	−0.748185	−0.00288874
$260$	0	0
$261$	− 149.954i	− 0.574535i
$262$	0	0
$263$	− 296.602i	− 1.12777i	−0.825855	−	0.563883i	$-0.809307\pi$
$263$	− 296.602i	− 1.12777i	0.825855	−	0.563883i	$-0.190693\pi$
$264$	0	0
$265$	−15.4722	−0.0583858
$266$	0	0
$267$	−130.333	−0.488137
$268$	0	0
$269$	− 171.921i	− 0.639113i	−0.947567	−	0.319557i	$-0.896466\pi$
$269$	− 171.921i	− 0.639113i	0.947567	−	0.319557i	$-0.103534\pi$
$270$	0	0
$271$	− 295.403i	− 1.09005i	−0.838420	−	0.545024i	$-0.816521\pi$
$271$	− 295.403i	− 1.09005i	0.838420	−	0.545024i	$-0.183479\pi$
$272$	0	0
$273$	9.73678	0.0356659
$274$	0	0
$275$	−139.852	−0.508554
$276$	0	0
$277$	16.5342i	0.0596901i	0.999555	+	0.0298451i	$0.00950139\pi$
$277$	16.5342i	0.0596901i	−0.999555	+	0.0298451i	$0.990499\pi$
$278$	0	0
$279$	− 22.0733i	− 0.0791156i
$280$	0	0
$281$	238.073	0.847235	0.423618	−	0.905841i	$-0.360760\pi$
$281$	238.073	0.847235	0.423618	+	0.905841i	$0.360760\pi$
$282$	0	0
$283$	35.7247	0.126236	0.0631179	−	0.998006i	$-0.479896\pi$
$283$	35.7247	0.126236	0.0631179	+	0.998006i	$0.479896\pi$
$284$	0	0
$285$	3.86449i	0.0135596i
$286$	0	0
$287$	− 3.73924i	− 0.0130287i
$288$	0	0
$289$	70.2579	0.243107
$290$	0	0
$291$	−412.794	−1.41854
$292$	0	0
$293$	65.8283i	0.224670i	0.993670	+	0.112335i	$0.0358330\pi$
$293$	65.8283i	0.224670i	−0.993670	+	0.112335i	$0.964167\pi$
$294$	0	0
$295$	− 6.54574i	− 0.0221890i
$296$	0	0
$297$	112.082	0.377379
$298$	0	0
$299$	−480.834	−1.60814
$300$	0	0
$301$	0.423285i	0.00140626i
$302$	0	0
$303$	252.214i	0.832390i
$304$	0	0
$305$	8.33448	0.0273262
$306$	0	0
$307$	93.8652	0.305750	0.152875	−	0.988246i	$-0.451147\pi$
$307$	93.8652	0.305750	0.152875	+	0.988246i	$0.451147\pi$
$308$	0	0
$309$	− 518.370i	− 1.67757i
$310$	0	0
$311$	− 205.284i	− 0.660078i	−0.943967	−	0.330039i	$-0.892938\pi$
$311$	− 205.284i	− 0.660078i	0.943967	−	0.330039i	$-0.107062\pi$
$312$	0	0
$313$	605.663	1.93503	0.967513	−	0.252821i	$-0.0813584\pi$
$313$	605.663	1.93503	0.967513	+	0.252821i	$0.0813584\pi$
$314$	0	0
$315$	0.159037	0.000504880 0
$316$	0	0
$317$	341.870i	1.07845i	0.842160	+	0.539227i	$0.181284\pi$
$317$	341.870i	1.07845i	−0.842160	+	0.539227i	$0.818716\pi$
$318$	0	0
$319$	254.673i	0.798349i
$320$	0	0
$321$	68.1452	0.212290
$322$	0	0
$323$	82.6190	0.255786
$324$	0	0
$325$	− 363.320i	− 1.11791i
$326$	0	0
$327$	548.663i	1.67787i
$328$	0	0
$329$	1.03087	0.00313333
$330$	0	0
$331$	108.076	0.326513	0.163256	−	0.986584i	$-0.447800\pi$
$331$	108.076	0.326513	0.163256	+	0.986584i	$0.447800\pi$
$332$	0	0
$333$	12.9676i	0.0389419i
$334$	0	0
$335$	− 7.38381i	− 0.0220412i
$336$	0	0
$337$	507.918	1.50718	0.753588	−	0.657347i	$-0.228322\pi$
$337$	507.918	1.50718	0.753588	+	0.657347i	$0.228322\pi$
$338$	0	0
$339$	−26.6167	−0.0785155
$340$	0	0
$341$	37.4881i	0.109936i
$342$	0	0
$343$	18.6650i	0.0544168i
$344$	0	0
$345$	−29.2583	−0.0848067
$346$	0	0
$347$	461.716	1.33059	0.665297	−	0.746579i	$-0.268305\pi$
$347$	461.716	1.33059	0.665297	+	0.746579i	$0.268305\pi$
$348$	0	0
$349$	289.712i	0.830121i	0.909794	+	0.415060i	$0.136240\pi$
$349$	289.712i	0.830121i	−0.909794	+	0.415060i	$0.863760\pi$
$350$	0	0
$351$	291.175i	0.829558i
$352$	0	0
$353$	−17.1345	−0.0485398	−0.0242699	−	0.999705i	$-0.507726\pi$
$353$	−17.1345	−0.0485398	−0.0242699	+	0.999705i	$0.507726\pi$
$354$	0	0
$355$	−2.65825	−0.00748804
$356$	0	0
$357$	− 12.6666i	− 0.0354805i
$358$	0	0
$359$	455.304i	1.26826i	0.773228	+	0.634128i	$0.218641\pi$
$359$	455.304i	1.26826i	−0.773228	+	0.634128i	$0.781359\pi$
$360$	0	0
$361$	19.0000	0.0526316
$362$	0	0
$363$	−314.078	−0.865228
$364$	0	0
$365$	23.0734i	0.0632147i
$366$	0	0
$367$	186.157i	0.507240i	0.967304	+	0.253620i	$0.0816213\pi$
$367$	186.157i	0.507240i	−0.967304	+	0.253620i	$0.918379\pi$
$368$	0	0
$369$	−64.8089	−0.175634
$370$	0	0
$371$	11.6625	0.0314354
$372$	0	0
$373$	387.069i	1.03772i	0.854860	+	0.518859i	$0.173643\pi$
$373$	387.069i	1.03772i	−0.854860	+	0.518859i	$0.826357\pi$
$374$	0	0
$375$	− 44.2721i	− 0.118059i
$376$	0	0
$377$	−661.611	−1.75494
$378$	0	0
$379$	−561.796	−1.48231	−0.741156	−	0.671333i	$-0.765722\pi$
$379$	−561.796	−1.48231	−0.741156	+	0.671333i	$0.765722\pi$
$380$	0	0
$381$	529.047i	1.38857i
$382$	0	0
$383$	− 466.656i	− 1.21842i	−0.793008	−	0.609211i	$-0.791486\pi$
$383$	− 466.656i	− 1.21842i	0.793008	−	0.609211i	$-0.208514\pi$
$384$	0	0
$385$	−0.270100	−0.000701560 0
$386$	0	0
$387$	7.33643	0.0189572
$388$	0	0
$389$	− 213.259i	− 0.548224i	−0.961698	−	0.274112i	$-0.911616\pi$
$389$	− 213.259i	− 0.548224i	0.961698	−	0.274112i	$-0.0883839\pi$
$390$	0	0
$391$	625.515i	1.59978i
$392$	0	0
$393$	195.669	0.497885
$394$	0	0
$395$	−5.74820	−0.0145524
$396$	0	0
$397$	727.022i	1.83129i	0.401988	+	0.915645i	$0.368319\pi$
$397$	727.022i	1.83129i	−0.401988	+	0.915645i	$0.631681\pi$
$398$	0	0
$399$	− 2.91294i	− 0.00730061i
$400$	0	0
$401$	217.429	0.542216	0.271108	−	0.962549i	$-0.412610\pi$
$401$	217.429	0.542216	0.271108	+	0.962549i	$0.412610\pi$
$402$	0	0
$403$	−97.3895	−0.241661
$404$	0	0
$405$	25.2302i	0.0622967i
$406$	0	0
$407$	− 22.0236i	− 0.0541119i
$408$	0	0
$409$	504.230	1.23284	0.616418	−	0.787419i	$-0.288583\pi$
$409$	504.230	1.23284	0.616418	+	0.787419i	$0.288583\pi$
$410$	0	0
$411$	−460.276	−1.11989
$412$	0	0
$413$	4.93400i	0.0119467i
$414$	0	0
$415$	− 36.5853i	− 0.0881574i
$416$	0	0
$417$	747.595	1.79279
$418$	0	0
$419$	453.677	1.08276	0.541381	−	0.840778i	$-0.317902\pi$
$419$	453.677	1.08276	0.541381	+	0.840778i	$0.317902\pi$
$420$	0	0
$421$	− 7.49122i	− 0.0177939i	−0.999960	−	0.00889693i	$-0.997168\pi$
$421$	− 7.49122i	− 0.0177939i	0.999960	−	0.00889693i	$-0.00283202\pi$
$422$	0	0
$423$	− 17.8671i	− 0.0422390i
$424$	0	0
$425$	−472.641	−1.11210
$426$	0	0
$427$	−6.28230	−0.0147126
$428$	0	0
$429$	286.612i	0.668093i
$430$	0	0
$431$	576.583i	1.33778i	0.743362	+	0.668890i	$0.233230\pi$
$431$	576.583i	1.33778i	−0.743362	+	0.668890i	$0.766770\pi$
$432$	0	0
$433$	−272.573	−0.629498	−0.314749	−	0.949175i	$-0.601920\pi$
$433$	−272.573	−0.629498	−0.314749	+	0.949175i	$0.601920\pi$
$434$	0	0
$435$	−40.2585	−0.0925482
$436$	0	0
$437$	143.850i	0.329177i
$438$	0	0
$439$	447.790i	1.02002i	0.860168	+	0.510011i	$0.170359\pi$
$439$	447.790i	1.02002i	−0.860168	+	0.510011i	$0.829641\pi$
$440$	0	0
$441$	161.692	0.366649
$442$	0	0
$443$	134.192	0.302916	0.151458	−	0.988464i	$-0.451603\pi$
$443$	134.192	0.302916	0.151458	+	0.988464i	$0.451603\pi$
$444$	0	0
$445$	9.39253i	0.0211068i
$446$	0	0
$447$	− 379.685i	− 0.849407i
$448$	0	0
$449$	279.068	0.621533	0.310767	−	0.950486i	$-0.399414\pi$
$449$	279.068	0.621533	0.310767	+	0.950486i	$0.399414\pi$
$450$	0	0
$451$	110.068	0.244053
$452$	0	0
$453$	350.706i	0.774186i
$454$	0	0
$455$	− 0.701689i	− 0.00154217i
$456$	0	0
$457$	−158.492	−0.346809	−0.173405	−	0.984851i	$-0.555477\pi$
$457$	−158.492	−0.346809	−0.173405	+	0.984851i	$0.555477\pi$
$458$	0	0
$459$	378.789	0.825247
$460$	0	0
$461$	28.3312i	0.0614559i	0.999528	+	0.0307279i	$0.00978255\pi$
$461$	28.3312i	0.0614559i	−0.999528	+	0.0307279i	$0.990217\pi$
$462$	0	0
$463$	− 161.928i	− 0.349737i	−0.984592	−	0.174869i	$-0.944050\pi$
$463$	− 161.928i	− 0.349737i	0.984592	−	0.174869i	$-0.0559501\pi$
$464$	0	0
$465$	−5.92607	−0.0127442
$466$	0	0
$467$	329.554	0.705683	0.352842	−	0.935683i	$-0.385215\pi$
$467$	329.554	0.705683	0.352842	+	0.935683i	$0.385215\pi$
$468$	0	0
$469$	5.56571i	0.0118672i
$470$	0	0
$471$	− 37.6464i	− 0.0799287i
$472$	0	0
$473$	−12.4598	−0.0263421
$474$	0	0
$475$	−108.694	−0.228829
$476$	0	0
$477$	− 202.137i	− 0.423767i
$478$	0	0
$479$	− 387.409i	− 0.808786i	−0.914585	−	0.404393i	$-0.867483\pi$
$479$	− 387.409i	− 0.808786i	0.914585	−	0.404393i	$-0.132517\pi$
$480$	0	0
$481$	57.2146	0.118949
$482$	0	0
$483$	22.0541	0.0456607
$484$	0	0
$485$	29.7483i	0.0613368i
$486$	0	0
$487$	682.598i	1.40164i	0.713339	+	0.700819i	$0.247182\pi$
$487$	682.598i	1.40164i	−0.713339	+	0.700819i	$0.752818\pi$
$488$	0	0
$489$	−100.899	−0.206337
$490$	0	0
$491$	923.123	1.88009	0.940044	−	0.341054i	$-0.110784\pi$
$491$	923.123	1.88009	0.940044	+	0.341054i	$0.110784\pi$
$492$	0	0
$493$	860.687i	1.74582i
$494$	0	0
$495$	4.68142i	0.00945741i
$496$	0	0
$497$	2.00372	0.00403163
$498$	0	0
$499$	−711.486	−1.42582	−0.712912	−	0.701254i	$-0.752624\pi$
$499$	−711.486	−1.42582	−0.712912	+	0.701254i	$0.752624\pi$
$500$	0	0
$501$	− 1033.69i	− 2.06326i
$502$	0	0
$503$	762.691i	1.51628i	0.652090	+	0.758142i	$0.273892\pi$
$503$	762.691i	1.51628i	−0.652090	+	0.758142i	$0.726108\pi$
$504$	0	0
$505$	18.1760	0.0359921
$506$	0	0
$507$	−151.823	−0.299453
$508$	0	0
$509$	− 526.156i	− 1.03371i	−0.856074	−	0.516853i	$-0.827104\pi$
$509$	− 526.156i	− 1.03371i	0.856074	−	0.516853i	$-0.172896\pi$
$510$	0	0
$511$	− 17.3921i	− 0.0340353i
$512$	0	0
$513$	87.1105	0.169806
$514$	0	0
$515$	−37.3568	−0.0725374
$516$	0	0
$517$	30.3445i	0.0586935i
$518$	0	0
$519$	629.504i	1.21292i
$520$	0	0
$521$	−902.077	−1.73143	−0.865717	−	0.500534i	$-0.833137\pi$
$521$	−902.077	−1.73143	−0.865717	+	0.500534i	$0.833137\pi$
$522$	0	0
$523$	493.138	0.942902	0.471451	−	0.881892i	$-0.343730\pi$
$523$	493.138	0.942902	0.471451	+	0.881892i	$0.343730\pi$
$524$	0	0
$525$	16.6642i	0.0317413i
$526$	0	0
$527$	126.694i	0.240406i
$528$	0	0
$529$	−560.103	−1.05880
$530$	0	0
$531$	85.5168	0.161049
$532$	0	0
$533$	285.944i	0.536480i
$534$	0	0
$535$	− 4.91094i	− 0.00917932i
$536$	0	0
$537$	657.206	1.22385
$538$	0	0
$539$	−274.609	−0.509479
$540$	0	0
$541$	689.780i	1.27501i	0.770447	+	0.637504i	$0.220033\pi$
$541$	689.780i	1.27501i	−0.770447	+	0.637504i	$0.779967\pi$
$542$	0	0
$543$	− 269.051i	− 0.495490i
$544$	0	0
$545$	39.5398	0.0725502
$546$	0	0
$547$	−8.71102	−0.0159251	−0.00796254	−	0.999968i	$-0.502535\pi$
$547$	−8.71102	−0.0159251	−0.00796254	+	0.999968i	$0.502535\pi$
$548$	0	0
$549$	108.886i	0.198335i
$550$	0	0
$551$	197.933i	0.359226i
$552$	0	0
$553$	4.33284	0.00783515
$554$	0	0
$555$	3.48146	0.00627290
$556$	0	0
$557$	271.055i	0.486633i	0.969947	+	0.243317i	$0.0782354\pi$
$557$	271.055i	0.486633i	−0.969947	+	0.243317i	$0.921765\pi$
$558$	0	0
$559$	− 32.3691i	− 0.0579054i
$560$	0	0
$561$	372.852	0.664621
$562$	0	0
$563$	194.488	0.345450	0.172725	−	0.984970i	$-0.444743\pi$
$563$	194.488	0.345450	0.172725	+	0.984970i	$0.444743\pi$
$564$	0	0
$565$	1.91816i	0.00339497i
$566$	0	0
$567$	− 19.0178i	− 0.0335411i
$568$	0	0
$569$	1046.92	1.83992	0.919961	−	0.392010i	$-0.128220\pi$
$569$	1046.92	1.83992	0.919961	+	0.392010i	$0.128220\pi$
$570$	0	0
$571$	374.483	0.655838	0.327919	−	0.944706i	$-0.393653\pi$
$571$	374.483	0.655838	0.327919	+	0.944706i	$0.393653\pi$
$572$	0	0
$573$	383.898i	0.669979i
$574$	0	0
$575$	− 822.930i	− 1.43118i
$576$	0	0
$577$	−972.710	−1.68581	−0.842903	−	0.538066i	$-0.819155\pi$
$577$	−972.710	−1.68581	−0.842903	+	0.538066i	$0.819155\pi$
$578$	0	0
$579$	−893.331	−1.54289
$580$	0	0
$581$	27.5770i	0.0474647i
$582$	0	0
$583$	343.298i	0.588848i
$584$	0	0
$585$	−12.1618	−0.0207894
$586$	0	0
$587$	154.819	0.263746	0.131873	−	0.991267i	$-0.457901\pi$
$587$	154.819	0.263746	0.131873	+	0.991267i	$0.457901\pi$
$588$	0	0
$589$	29.1359i	0.0494668i
$590$	0	0
$591$	− 339.745i	− 0.574864i
$592$	0	0
$593$	417.721	0.704420	0.352210	−	0.935921i	$-0.385430\pi$
$593$	417.721	0.704420	0.352210	+	0.935921i	$0.385430\pi$
$594$	0	0
$595$	−0.912825	−0.00153416
$596$	0	0
$597$	− 432.811i	− 0.724977i
$598$	0	0
$599$	− 759.367i	− 1.26772i	−0.773446	−	0.633862i	$-0.781469\pi$
$599$	− 759.367i	− 1.26772i	0.773446	−	0.633862i	$-0.218531\pi$
$600$	0	0
$601$	362.886	0.603803	0.301902	−	0.953339i	$-0.402379\pi$
$601$	362.886	0.603803	0.301902	+	0.953339i	$0.402379\pi$
$602$	0	0
$603$	96.4657	0.159976
$604$	0	0
$605$	22.6343i	0.0374120i
$606$	0	0
$607$	1076.91i	1.77415i	0.461624	+	0.887076i	$0.347267\pi$
$607$	1076.91i	1.77415i	−0.461624	+	0.887076i	$0.652733\pi$
$608$	0	0
$609$	30.3457	0.0498288
$610$	0	0
$611$	−78.8315	−0.129021
$612$	0	0
$613$	− 468.337i	− 0.764009i	−0.924161	−	0.382004i	$-0.875234\pi$
$613$	− 468.337i	− 0.764009i	0.924161	−	0.382004i	$-0.124766\pi$
$614$	0	0
$615$	17.3995i	0.0282918i
$616$	0	0
$617$	−99.9309	−0.161962	−0.0809812	−	0.996716i	$-0.525805\pi$
$617$	−99.9309	−0.161962	−0.0809812	+	0.996716i	$0.525805\pi$
$618$	0	0
$619$	−693.446	−1.12027	−0.560134	−	0.828402i	$-0.689250\pi$
$619$	−693.446	−1.12027	−0.560134	+	0.828402i	$0.689250\pi$
$620$	0	0
$621$	659.520i	1.06203i
$622$	0	0
$623$	− 7.07983i	− 0.0113641i
$624$	0	0
$625$	620.212	0.992340
$626$	0	0
$627$	85.7453	0.136755
$628$	0	0
$629$	− 74.4303i	− 0.118331i
$630$	0	0
$631$	− 735.284i	− 1.16527i	−0.812735	−	0.582634i	$-0.802022\pi$
$631$	− 735.284i	− 1.16527i	0.812735	−	0.582634i	$-0.197978\pi$
$632$	0	0
$633$	−165.534	−0.261507
$634$	0	0
$635$	38.1262	0.0600412
$636$	0	0
$637$	− 713.402i	− 1.11994i
$638$	0	0
$639$	− 34.7287i	− 0.0543486i
$640$	0	0
$641$	−927.322	−1.44668	−0.723340	−	0.690492i	$-0.757394\pi$
$641$	−927.322	−1.44668	−0.723340	+	0.690492i	$0.757394\pi$
$642$	0	0
$643$	1230.08	1.91304	0.956519	−	0.291671i	$-0.0942112\pi$
$643$	1230.08	1.91304	0.956519	+	0.291671i	$0.0942112\pi$
$644$	0	0
$645$	− 1.96963i	− 0.00305370i
$646$	0	0
$647$	− 760.883i	− 1.17602i	−0.808855	−	0.588009i	$-0.799912\pi$
$647$	− 760.883i	− 1.17602i	0.808855	−	0.588009i	$-0.200088\pi$
$648$	0	0
$649$	−145.237	−0.223786
$650$	0	0
$651$	4.46691	0.00686161
$652$	0	0
$653$	− 598.576i	− 0.916655i	−0.888783	−	0.458328i	$-0.848449\pi$
$653$	− 598.576i	− 0.916655i	0.888783	−	0.458328i	$-0.151551\pi$
$654$	0	0
$655$	− 14.1010i	− 0.0215283i
$656$	0	0
$657$	−301.442	−0.458815
$658$	0	0
$659$	−536.626	−0.814304	−0.407152	−	0.913360i	$-0.633478\pi$
$659$	−536.626	−0.814304	−0.407152	+	0.913360i	$0.633478\pi$
$660$	0	0
$661$	104.138i	0.157547i	0.996893	+	0.0787733i	$0.0251003\pi$
$661$	104.138i	0.157547i	−0.996893	+	0.0787733i	$0.974900\pi$
$662$	0	0
$663$	968.627i	1.46098i
$664$	0	0
$665$	−0.209924	−0.000315675 0
$666$	0	0
$667$	−1498.57	−2.24673
$668$	0	0
$669$	731.844i	1.09394i
$670$	0	0
$671$	− 184.926i	− 0.275597i
$672$	0	0
$673$	−115.118	−0.171052	−0.0855261	−	0.996336i	$-0.527257\pi$
$673$	−115.118	−0.171052	−0.0855261	+	0.996336i	$0.527257\pi$
$674$	0	0
$675$	−498.336	−0.738276
$676$	0	0
$677$	− 938.327i	− 1.38601i	−0.720934	−	0.693004i	$-0.756287\pi$
$677$	− 938.327i	− 1.38601i	0.720934	−	0.693004i	$-0.243713\pi$
$678$	0	0
$679$	− 22.4235i	− 0.0330243i
$680$	0	0
$681$	−886.588	−1.30189
$682$	0	0
$683$	732.196	1.07203	0.536015	−	0.844209i	$-0.319929\pi$
$683$	732.196	1.07203	0.536015	+	0.844209i	$0.319929\pi$
$684$	0	0
$685$	33.1702i	0.0484236i
$686$	0	0
$687$	− 954.634i	− 1.38957i
$688$	0	0
$689$	−891.849	−1.29441
$690$	0	0
$691$	565.948	0.819028	0.409514	−	0.912304i	$-0.365698\pi$
$691$	565.948	0.819028	0.409514	+	0.912304i	$0.365698\pi$
$692$	0	0
$693$	− 3.52872i	− 0.00509195i
$694$	0	0
$695$	− 53.8761i	− 0.0775195i
$696$	0	0
$697$	371.984	0.533692
$698$	0	0
$699$	−810.753	−1.15988
$700$	0	0
$701$	− 786.820i	− 1.12243i	−0.827672	−	0.561213i	$-0.810335\pi$
$701$	− 786.820i	− 1.12243i	0.827672	−	0.561213i	$-0.189665\pi$
$702$	0	0
$703$	− 17.1168i	− 0.0243483i
$704$	0	0
$705$	−4.79683	−0.00680402
$706$	0	0
$707$	−13.7006	−0.0193785
$708$	0	0
$709$	− 529.321i	− 0.746574i	−0.927716	−	0.373287i	$-0.878231\pi$
$709$	− 529.321i	− 0.746574i	0.927716	−	0.373287i	$-0.121769\pi$
$710$	0	0
$711$	− 75.0973i	− 0.105622i
$712$	0	0
$713$	−220.590	−0.309383
$714$	0	0
$715$	20.6549	0.0288880
$716$	0	0
$717$	453.984i	0.633171i
$718$	0	0
$719$	− 1027.92i	− 1.42965i	−0.699301	−	0.714827i	$-0.746505\pi$
$719$	− 1027.92i	− 1.42965i	0.699301	−	0.714827i	$-0.253495\pi$
$720$	0	0
$721$	28.1585	0.0390548
$722$	0	0
$723$	897.080	1.24077
$724$	0	0
$725$	− 1132.32i	− 1.56183i
$726$	0	0
$727$	− 1195.70i	− 1.64471i	−0.568975	−	0.822355i	$-0.692660\pi$
$727$	− 1195.70i	− 1.64471i	0.568975	−	0.822355i	$-0.307340\pi$
$728$	0	0
$729$	301.235	0.413217
$730$	0	0
$731$	−42.1089	−0.0576045
$732$	0	0
$733$	− 1054.08i	− 1.43804i	−0.694988	−	0.719021i	$-0.744590\pi$
$733$	− 1054.08i	− 1.43804i	0.694988	−	0.719021i	$-0.255410\pi$
$734$	0	0
$735$	− 43.4099i	− 0.0590611i
$736$	0	0
$737$	−163.832	−0.222296
$738$	0	0
$739$	−127.708	−0.172812	−0.0864059	−	0.996260i	$-0.527538\pi$
$739$	−127.708	−0.172812	−0.0864059	+	0.996260i	$0.527538\pi$
$740$	0	0
$741$	222.756i	0.300616i
$742$	0	0
$743$	− 761.866i	− 1.02539i	−0.858570	−	0.512696i	$-0.828647\pi$
$743$	− 761.866i	− 1.02539i	0.858570	−	0.512696i	$-0.171353\pi$
$744$	0	0
$745$	−27.3623	−0.0367279
$746$	0	0
$747$	477.968	0.639850
$748$	0	0
$749$	3.70173i	0.00494223i
$750$	0	0
$751$	− 1067.78i	− 1.42181i	−0.703289	−	0.710904i	$-0.748286\pi$
$751$	− 1067.78i	− 1.42181i	0.703289	−	0.710904i	$-0.251714\pi$
$752$	0	0
$753$	617.472	0.820015
$754$	0	0
$755$	25.2739	0.0334754
$756$	0	0
$757$	− 728.624i	− 0.962515i	−0.876579	−	0.481257i	$-0.840180\pi$
$757$	− 728.624i	− 0.962515i	0.876579	−	0.481257i	$-0.159820\pi$
$758$	0	0
$759$	649.185i	0.855316i
$760$	0	0
$761$	−854.584	−1.12297	−0.561487	−	0.827485i	$-0.689771\pi$
$761$	−854.584	−1.12297	−0.561487	+	0.827485i	$0.689771\pi$
$762$	0	0
$763$	−29.8040	−0.0390617
$764$	0	0
$765$	15.8212i	0.0206813i
$766$	0	0
$767$	− 377.309i	− 0.491928i
$768$	0	0
$769$	1220.06	1.58656	0.793279	−	0.608859i	$-0.208372\pi$
$769$	1220.06	1.58656	0.793279	+	0.608859i	$0.208372\pi$
$770$	0	0
$771$	−1085.54	−1.40796
$772$	0	0
$773$	− 400.055i	− 0.517536i	−0.965940	−	0.258768i	$-0.916684\pi$
$773$	− 400.055i	− 0.517536i	0.965940	−	0.258768i	$-0.0833165\pi$
$774$	0	0
$775$	− 166.679i	− 0.215070i
$776$	0	0
$777$	−2.62423	−0.00337739
$778$	0	0
$779$	85.5455	0.109815
$780$	0	0
$781$	58.9814i	0.0755204i
$782$	0	0
$783$	907.478i	1.15898i
$784$	0	0
$785$	−2.71302	−0.00345608
$786$	0	0
$787$	470.101	0.597333	0.298666	−	0.954358i	$-0.403458\pi$
$787$	470.101	0.597333	0.298666	+	0.954358i	$0.403458\pi$
$788$	0	0
$789$	− 1040.32i	− 1.31853i
$790$	0	0
$791$	− 1.44585i	− 0.00182788i
$792$	0	0
$793$	480.415	0.605820
$794$	0	0
$795$	−54.2682	−0.0682619
$796$	0	0
$797$	349.833i	0.438938i	0.975619	+	0.219469i	$0.0704325\pi$
$797$	349.833i	0.438938i	−0.975619	+	0.219469i	$0.929568\pi$
$798$	0	0
$799$	102.552i	0.128350i
$800$	0	0
$801$	−122.709	−0.153194
$802$	0	0
$803$	511.952	0.637550
$804$	0	0
$805$	− 1.58935i	− 0.00197434i
$806$	0	0
$807$	− 603.008i	− 0.747221i
$808$	0	0
$809$	202.721	0.250582	0.125291	−	0.992120i	$-0.460014\pi$
$809$	202.721	0.250582	0.125291	+	0.992120i	$0.460014\pi$
$810$	0	0
$811$	1087.10	1.34045	0.670225	−	0.742158i	$-0.266198\pi$
$811$	1087.10	1.34045	0.670225	+	0.742158i	$0.266198\pi$
$812$	0	0
$813$	− 1036.11i	− 1.27443i
$814$	0	0
$815$	7.27134i	0.00892190i
$816$	0	0
$817$	−9.68383	−0.0118529
$818$	0	0
$819$	9.16721	0.0111932
$820$	0	0
$821$	− 296.394i	− 0.361016i	−0.983574	−	0.180508i	$-0.942226\pi$
$821$	− 296.394i	− 0.361016i	0.983574	−	0.180508i	$-0.0577742\pi$
$822$	0	0
$823$	1308.89i	1.59038i	0.606358	+	0.795192i	$0.292630\pi$
$823$	1308.89i	1.59038i	−0.606358	+	0.795192i	$0.707370\pi$
$824$	0	0
$825$	−490.526	−0.594578
$826$	0	0
$827$	−746.411	−0.902553	−0.451276	−	0.892384i	$-0.649031\pi$
$827$	−746.411	−0.902553	−0.451276	+	0.892384i	$0.649031\pi$
$828$	0	0
$829$	875.724i	1.05636i	0.849132	+	0.528181i	$0.177126\pi$
$829$	875.724i	1.05636i	−0.849132	+	0.528181i	$0.822874\pi$
$830$	0	0
$831$	57.9930i	0.0697870i
$832$	0	0
$833$	−928.063	−1.11412
$834$	0	0
$835$	−74.4938	−0.0892141
$836$	0	0
$837$	133.581i	0.159595i
$838$	0	0
$839$	− 1366.87i	− 1.62917i	−0.580047	−	0.814583i	$-0.696966\pi$
$839$	− 1366.87i	− 1.62917i	0.580047	−	0.814583i	$-0.303034\pi$
$840$	0	0
$841$	−1220.98	−1.45182
$842$	0	0
$843$	835.032	0.990548
$844$	0	0
$845$	10.9412i	0.0129482i
$846$	0	0
$847$	− 17.0611i	− 0.0201429i
$848$	0	0
$849$	125.303	0.147589
$850$	0	0
$851$	129.593	0.152283
$852$	0	0
$853$	1507.40i	1.76718i	0.468265	+	0.883588i	$0.344879\pi$
$853$	1507.40i	1.76718i	−0.468265	+	0.883588i	$0.655121\pi$
$854$	0	0
$855$	3.63842i	0.00425547i
$856$	0	0
$857$	236.639	0.276124	0.138062	−	0.990424i	$-0.455913\pi$
$857$	236.639	0.276124	0.138062	+	0.990424i	$0.455913\pi$
$858$	0	0
$859$	−993.051	−1.15605	−0.578027	−	0.816018i	$-0.696177\pi$
$859$	−993.051	−1.15605	−0.578027	+	0.816018i	$0.696177\pi$
$860$	0	0
$861$	− 13.1152i	− 0.0152326i
$862$	0	0
$863$	− 39.9720i	− 0.0463175i	−0.999732	−	0.0231588i	$-0.992628\pi$
$863$	− 39.9720i	− 0.0463175i	0.999732	−	0.0231588i	$-0.00737232\pi$
$864$	0	0
$865$	45.3657	0.0524459
$866$	0	0
$867$	246.427	0.284229
$868$	0	0
$869$	127.541i	0.146768i
$870$	0	0
$871$	− 425.617i	− 0.488653i
$872$	0	0
$873$	−388.647	−0.445185
$874$	0	0
$875$	2.40491	0.00274847
$876$	0	0
$877$	− 738.081i	− 0.841597i	−0.907154	−	0.420799i	$-0.861750\pi$
$877$	− 738.081i	− 0.841597i	0.907154	−	0.420799i	$-0.138250\pi$
$878$	0	0
$879$	230.890i	0.262674i
$880$	0	0
$881$	−403.793	−0.458335	−0.229168	−	0.973387i	$-0.573600\pi$
$881$	−403.793	−0.458335	−0.229168	+	0.973387i	$0.573600\pi$
$882$	0	0
$883$	−1279.75	−1.44932	−0.724661	−	0.689106i	$-0.758003\pi$
$883$	−1279.75	−1.44932	−0.724661	+	0.689106i	$0.758003\pi$
$884$	0	0
$885$	− 22.9589i	− 0.0259423i
$886$	0	0
$887$	− 1343.82i	− 1.51502i	−0.652826	−	0.757508i	$-0.726417\pi$
$887$	− 1343.82i	− 1.51502i	0.652826	−	0.757508i	$-0.273583\pi$
$888$	0	0
$889$	−28.7384	−0.0323267
$890$	0	0
$891$	559.808	0.628291
$892$	0	0
$893$	23.5839i	0.0264098i
$894$	0	0
$895$	− 47.3621i	− 0.0529186i
$896$	0	0
$897$	−1686.51	−1.88016
$898$	0	0
$899$	−303.525	−0.337625
$900$	0	0
$901$	1160.20i	1.28768i
$902$	0	0
$903$	1.48466i	0.00164414i
$904$	0	0
$905$	−19.3894	−0.0214247
$906$	0	0
$907$	939.932	1.03631	0.518154	−	0.855287i	$-0.326619\pi$
$907$	939.932	1.03631	0.518154	+	0.855287i	$0.326619\pi$
$908$	0	0
$909$	237.460i	0.261232i
$910$	0	0
$911$	809.909i	0.889033i	0.895771	+	0.444516i	$0.146624\pi$
$911$	809.909i	0.889033i	−0.895771	+	0.444516i	$0.853376\pi$
$912$	0	0
$913$	−811.756	−0.889109
$914$	0	0
$915$	29.2329	0.0319485
$916$	0	0
$917$	10.6290i	0.0115910i
$918$	0	0
$919$	− 1333.20i	− 1.45070i	−0.688379	−	0.725351i	$-0.741677\pi$
$919$	− 1333.20i	− 1.45070i	0.688379	−	0.725351i	$-0.258323\pi$
$920$	0	0
$921$	329.229	0.357469
$922$	0	0
$923$	−153.227	−0.166010
$924$	0	0
$925$	97.9209i	0.105860i
$926$	0	0
$927$	− 488.047i	− 0.526480i
$928$	0	0
$929$	−394.413	−0.424556	−0.212278	−	0.977209i	$-0.568088\pi$
$929$	−394.413	−0.424556	−0.212278	+	0.977209i	$0.568088\pi$
$930$	0	0
$931$	−213.428	−0.229246
$932$	0	0
$933$	− 720.027i	− 0.771733i
$934$	0	0
$935$	− 26.8699i	− 0.0287379i
$936$	0	0
$937$	−334.853	−0.357368	−0.178684	−	0.983907i	$-0.557184\pi$
$937$	−334.853	−0.357368	−0.178684	+	0.983907i	$0.557184\pi$
$938$	0	0
$939$	2124.34	2.26234
$940$	0	0
$941$	− 1801.39i	− 1.91434i	−0.289530	−	0.957169i	$-0.593499\pi$
$941$	− 1801.39i	− 1.91434i	0.289530	−	0.957169i	$-0.406501\pi$
$942$	0	0
$943$	647.672i	0.686821i
$944$	0	0
$945$	−0.962450	−0.00101847
$946$	0	0
$947$	1602.98	1.69269	0.846345	−	0.532636i	$-0.178798\pi$
$947$	1602.98	1.69269	0.846345	+	0.532636i	$0.178798\pi$
$948$	0	0
$949$	1329.99i	1.40147i
$950$	0	0
$951$	1199.10i	1.26088i
$952$	0	0
$953$	−1518.38	−1.59327	−0.796634	−	0.604462i	$-0.793388\pi$
$953$	−1518.38	−1.59327	−0.796634	+	0.604462i	$0.793388\pi$
$954$	0	0
$955$	27.6659	0.0289696
$956$	0	0
$957$	893.256i	0.933392i
$958$	0	0
$959$	− 25.0028i	− 0.0260717i
$960$	0	0
$961$	916.321	0.953508
$962$	0	0
$963$	64.1589	0.0666239
$964$	0	0
$965$	64.3786i	0.0667136i
$966$	0	0
$967$	− 1080.13i	− 1.11699i	−0.829508	−	0.558495i	$-0.811379\pi$
$967$	− 1080.13i	− 1.11699i	0.829508	−	0.558495i	$-0.188621\pi$
$968$	0	0
$969$	289.783	0.299054
$970$	0	0
$971$	110.309	0.113604	0.0568018	−	0.998385i	$-0.481910\pi$
$971$	110.309	0.113604	0.0568018	+	0.998385i	$0.481910\pi$
$972$	0	0
$973$	40.6103i	0.0417372i
$974$	0	0
$975$	− 1274.33i	− 1.30701i
$976$	0	0
$977$	−1429.36	−1.46301	−0.731503	−	0.681839i	$-0.761181\pi$
$977$	−1429.36	−1.46301	−0.731503	+	0.681839i	$0.761181\pi$
$978$	0	0
$979$	208.402	0.212872
$980$	0	0
$981$	516.568i	0.526573i
$982$	0	0
$983$	661.986i	0.673434i	0.941606	+	0.336717i	$0.109317\pi$
$983$	661.986i	0.673434i	−0.941606	+	0.336717i	$0.890683\pi$
$984$	0	0
$985$	−24.4840	−0.0248568
$986$	0	0
$987$	3.61572	0.00366334
$988$	0	0
$989$	− 73.3170i	− 0.0741325i
$990$	0	0
$991$	668.232i	0.674301i	0.941451	+	0.337150i	$0.109463\pi$
$991$	668.232i	0.674301i	−0.941451	+	0.337150i	$0.890537\pi$
$992$	0	0
$993$	379.072	0.381744
$994$	0	0
$995$	−31.1909	−0.0313476
$996$	0	0
$997$	− 1799.71i	− 1.80512i	−0.430560	−	0.902562i	$-0.641684\pi$
$997$	− 1799.71i	− 1.80512i	0.430560	−	0.902562i	$-0.358316\pi$
$998$	0	0
$999$	− 78.4766i	− 0.0785552i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.f.b.799.37	yes	48
4.3	odd	2	inner	1216.3.f.b.799.11	✓	48
8.3	odd	2	inner	1216.3.f.b.799.38	yes	48
8.5	even	2	inner	1216.3.f.b.799.12	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.f.b.799.11	✓	48	4.3	odd	2	inner
1216.3.f.b.799.12	yes	48	8.5	even	2	inner
1216.3.f.b.799.37	yes	48	1.1	even	1	trivial
1216.3.f.b.799.38	yes	48	8.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.50746 q^{3} -0.252768i q^{5} +0.190530i q^{7} +3.30228 q^{9} +O(q^{10})\)	\(q+3.50746 q^{3} -0.252768i q^{5} +0.190530i q^{7} +3.30228 q^{9} -5.60842 q^{11} -14.5700i q^{13} -0.886574i q^{15} -18.9541 q^{17} -4.35890 q^{19} +0.668275i q^{21} -33.0016i q^{23} +24.9361 q^{25} -19.9845 q^{27} -45.4090i q^{29} -6.68424i q^{31} -19.6713 q^{33} +0.0481598 q^{35} +3.92687i q^{37} -51.1038i q^{39} -19.6255 q^{41} +2.22162 q^{43} -0.834712i q^{45} -5.41053i q^{47} +48.9637 q^{49} -66.4808 q^{51} -61.2112i q^{53} +1.41763i q^{55} -15.2887 q^{57} +25.8963 q^{59} +32.9728i q^{61} +0.629183i q^{63} -3.68284 q^{65} +29.2118 q^{67} -115.752i q^{69} -10.5166i q^{71} -91.2827 q^{73} +87.4624 q^{75} -1.06857i q^{77} -22.7410i q^{79} -99.8155 q^{81} +144.739 q^{83} +4.79099i q^{85} -159.270i q^{87} -37.1587 q^{89} +2.77602 q^{91} -23.4447i q^{93} +1.10179i q^{95} -117.690 q^{97} -18.5206 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 168 q^{9}+O(q^{10}) \) 48 * q + 168 * q^9	\( 48 q + 168 q^{9} + 100 q^{17} - 204 q^{25} - 8 q^{33} - 240 q^{41} - 460 q^{49} - 368 q^{65} - 132 q^{73} + 768 q^{81} + 696 q^{89} - 8 q^{97}+O(q^{100}) \) 48 * q + 168 * q^9 + 100 * q^17 - 204 * q^25 - 8 * q^33 - 240 * q^41 - 460 * q^49 - 368 * q^65 - 132 * q^73 + 768 * q^81 + 696 * q^89 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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