LMFDB - Embedded newform 1216.3.g.e.417.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(417,1216)]

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([0, 1, 1]))

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

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

chi := DirichletCharacter("1216.417");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.g (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			417.5
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.e.417.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.81671 q^{3} +9.51528i q^{5} -8.13398 q^{7} +14.2007 q^{9} +O(q^{10})$	$q-4.81671 q^{3} +9.51528i q^{5} -8.13398 q^{7} +14.2007 q^{9} -2.56846i q^{11} +18.6905 q^{13} -45.8323i q^{15} -17.5853 q^{17} +(-18.8805 - 2.12779i) q^{19} +39.1791 q^{21} -23.2203 q^{23} -65.5405 q^{25} -25.0503 q^{27} -41.6989 q^{29} +15.2897i q^{31} +12.3715i q^{33} -77.3971i q^{35} +13.4331 q^{37} -90.0269 q^{39} -74.2281i q^{41} +45.3166i q^{43} +135.124i q^{45} +44.0123 q^{47} +17.1617 q^{49} +84.7031 q^{51} +42.9104 q^{53} +24.4396 q^{55} +(90.9418 + 10.2490i) q^{57} -23.3241 q^{59} -34.5816i q^{61} -115.508 q^{63} +177.846i q^{65} +36.9821 q^{67} +111.845 q^{69} +48.8452i q^{71} -67.0031 q^{73} +315.690 q^{75} +20.8918i q^{77} -91.4803i q^{79} -7.14632 q^{81} -37.4338i q^{83} -167.329i q^{85} +200.851 q^{87} +95.7801i q^{89} -152.029 q^{91} -73.6460i q^{93} +(20.2466 - 179.653i) q^{95} -111.631i q^{97} -36.4739i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 264 q^{9}+O(q^{10}) $ 48 * q + 264 * q^9	$ 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) $ 48 * q + 264 * q^9 + 24 * q^17 - 160 * q^25 + 328 * q^49 + 8 * q^57 - 472 * q^73 - 736 * q^81

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
$3$	−4.81671			−1.60557			−0.802785	−	0.596268i	$-0.796649\pi$
$3$	−4.81671			−1.60557			−0.802785	+	0.596268i	$0.796649\pi$
$4$		0			0
$5$			9.51528i			1.90306i	0.307563	+	0.951528i	$0.400486\pi$
$5$			9.51528i			1.90306i	−0.307563	+	0.951528i	$0.599514\pi$
$6$		0			0
$7$	−8.13398			−1.16200			−0.580999	−	0.813904i	$-0.697338\pi$
$7$	−8.13398			−1.16200			−0.580999	+	0.813904i	$0.697338\pi$
$8$		0			0
$9$	14.2007			1.57786
$10$		0			0
$11$		−	2.56846i		−	0.233496i	−0.993162	−	0.116748i	$-0.962753\pi$
$11$		−	2.56846i		−	0.233496i	0.993162	−	0.116748i	$-0.0372470\pi$
$12$		0			0
$13$	18.6905			1.43773			0.718867	−	0.695148i	$-0.244661\pi$
$13$	18.6905			1.43773			0.718867	+	0.695148i	$0.244661\pi$
$14$		0			0
$15$		−	45.8323i		−	3.05549i
$16$		0			0
$17$	−17.5853			−1.03443			−0.517213	−	0.855856i	$-0.673031\pi$
$17$	−17.5853			−1.03443			−0.517213	+	0.855856i	$0.673031\pi$
$18$		0			0
$19$	−18.8805	−	2.12779i	−0.993709	−	0.111989i
$19$	−18.8805	−	2.12779i	−0.993709	−	0.111989i
$20$		0			0
$21$	39.1791			1.86567
$22$		0			0
$23$	−23.2203			−1.00958			−0.504788	−	0.863243i	$-0.668429\pi$
$23$	−23.2203			−1.00958			−0.504788	+	0.863243i	$0.668429\pi$
$24$		0			0
$25$	−65.5405			−2.62162
$26$		0			0
$27$	−25.0503			−0.927789
$28$		0			0
$29$	−41.6989			−1.43789			−0.718946	−	0.695066i	$-0.755375\pi$
$29$	−41.6989			−1.43789			−0.718946	+	0.695066i	$0.755375\pi$
$30$		0			0
$31$			15.2897i			0.493216i	0.969115	+	0.246608i	$0.0793160\pi$
$31$			15.2897i			0.493216i	−0.969115	+	0.246608i	$0.920684\pi$
$32$		0			0
$33$			12.3715i			0.374895i
$34$		0			0
$35$		−	77.3971i		−	2.21135i
$36$		0			0
$37$	13.4331			0.363058			0.181529	−	0.983386i	$-0.441895\pi$
$37$	13.4331			0.363058			0.181529	+	0.983386i	$0.441895\pi$
$38$		0			0
$39$	−90.0269			−2.30838
$40$		0			0
$41$		−	74.2281i		−	1.81044i	−0.424942	−	0.905221i	$-0.639706\pi$
$41$		−	74.2281i		−	1.81044i	0.424942	−	0.905221i	$-0.360294\pi$
$42$		0			0
$43$			45.3166i			1.05387i	0.849905	+	0.526937i	$0.176660\pi$
$43$			45.3166i			1.05387i	−0.849905	+	0.526937i	$0.823340\pi$
$44$		0			0
$45$			135.124i			3.00275i
$46$		0			0
$47$	44.0123			0.936433			0.468216	−	0.883614i	$-0.344897\pi$
$47$	44.0123			0.936433			0.468216	+	0.883614i	$0.344897\pi$
$48$		0			0
$49$	17.1617			0.350239
$50$		0			0
$51$	84.7031			1.66085
$52$		0			0
$53$	42.9104			0.809631			0.404815	−	0.914398i	$-0.367336\pi$
$53$	42.9104			0.809631			0.404815	+	0.914398i	$0.367336\pi$
$54$		0			0
$55$	24.4396			0.444356
$56$		0			0
$57$	90.9418	+	10.2490i	1.59547	+	0.179806i
$58$		0			0
$59$	−23.3241			−0.395323			−0.197662	−	0.980270i	$-0.563335\pi$
$59$	−23.3241			−0.395323			−0.197662	+	0.980270i	$0.563335\pi$
$60$		0			0
$61$		−	34.5816i		−	0.566911i	−0.958985	−	0.283456i	$-0.908519\pi$
$61$		−	34.5816i		−	0.566911i	0.958985	−	0.283456i	$-0.0914809\pi$
$62$		0			0
$63$	−115.508			−1.83347
$64$		0			0
$65$			177.846i			2.73609i
$66$		0			0
$67$	36.9821			0.551972			0.275986	−	0.961162i	$-0.410996\pi$
$67$	36.9821			0.551972			0.275986	+	0.961162i	$0.410996\pi$
$68$		0			0
$69$	111.845			1.62095
$70$		0			0
$71$			48.8452i			0.687960i	0.938977	+	0.343980i	$0.111775\pi$
$71$			48.8452i			0.687960i	−0.938977	+	0.343980i	$0.888225\pi$
$72$		0			0
$73$	−67.0031			−0.917851			−0.458926	−	0.888475i	$-0.651766\pi$
$73$	−67.0031			−0.917851			−0.458926	+	0.888475i	$0.651766\pi$
$74$		0			0
$75$	315.690			4.20919
$76$		0			0
$77$			20.8918i			0.271322i
$78$		0			0
$79$		−	91.4803i		−	1.15798i	−0.815335	−	0.578989i	$-0.803447\pi$
$79$		−	91.4803i		−	1.15798i	0.815335	−	0.578989i	$-0.196553\pi$
$80$		0			0
$81$	−7.14632			−0.0882261
$82$		0			0
$83$		−	37.4338i		−	0.451010i	−0.974242	−	0.225505i	$-0.927597\pi$
$83$		−	37.4338i		−	0.451010i	0.974242	−	0.225505i	$-0.0724032\pi$
$84$		0			0
$85$		−	167.329i		−	1.96857i
$86$		0			0
$87$	200.851			2.30864
$88$		0			0
$89$			95.7801i			1.07618i	0.842887	+	0.538090i	$0.180854\pi$
$89$			95.7801i			1.07618i	−0.842887	+	0.538090i	$0.819146\pi$
$90$		0			0
$91$	−152.029			−1.67064
$92$		0			0
$93$		−	73.6460i		−	0.791893i
$94$		0			0
$95$	20.2466	−	179.653i	0.213122	−	1.89108i
$96$		0			0
$97$		−	111.631i		−	1.15083i	−0.817862	−	0.575415i	$-0.804841\pi$
$97$		−	111.631i		−	1.15083i	0.817862	−	0.575415i	$-0.195159\pi$
$98$		0			0
$99$		−	36.4739i		−	0.368423i
$100$		0			0
$101$			98.6276i			0.976511i	0.872701	+	0.488256i	$0.162367\pi$
$101$			98.6276i			0.976511i	−0.872701	+	0.488256i	$0.837633\pi$
$102$		0			0
$103$		−	136.985i		−	1.32995i	−0.746864	−	0.664977i	$-0.768441\pi$
$103$		−	136.985i		−	1.32995i	0.746864	−	0.664977i	$-0.231559\pi$
$104$		0			0
$105$			372.800i			3.55047i
$106$		0			0
$107$	111.644			1.04340			0.521701	−	0.853128i	$-0.325298\pi$
$107$	111.644			1.04340			0.521701	+	0.853128i	$0.325298\pi$
$108$		0			0
$109$	−47.3559			−0.434458			−0.217229	−	0.976121i	$-0.569702\pi$
$109$	−47.3559			−0.434458			−0.217229	+	0.976121i	$0.569702\pi$
$110$		0			0
$111$	−64.7036			−0.582915
$112$		0			0
$113$			165.852i			1.46772i	0.679301	+	0.733860i	$0.262283\pi$
$113$			165.852i			1.46772i	−0.679301	+	0.733860i	$0.737717\pi$
$114$		0			0
$115$		−	220.947i		−	1.92128i
$116$		0			0
$117$	265.419			2.26854
$118$		0			0
$119$	143.038			1.20200
$120$		0			0
$121$	114.403			0.945480
$122$		0			0
$123$			357.535i			2.90679i
$124$		0			0
$125$		−	385.754i		−	3.08603i
$126$		0			0
$127$			71.0170i			0.559189i	0.960118	+	0.279594i	$0.0902000\pi$
$127$			71.0170i			0.559189i	−0.960118	+	0.279594i	$0.909800\pi$
$128$		0			0
$129$		−	218.277i		−	1.69207i
$130$		0			0
$131$			165.526i			1.26356i	0.775149	+	0.631779i	$0.217675\pi$
$131$			165.526i			1.26356i	−0.775149	+	0.631779i	$0.782325\pi$
$132$		0			0
$133$	153.574	+	17.3074i	1.15469	+	0.130131i
$134$		0			0
$135$		−	238.360i		−	1.76563i
$136$		0			0
$137$	191.887			1.40063			0.700316	−	0.713833i	$-0.253042\pi$
$137$	191.887			1.40063			0.700316	+	0.713833i	$0.253042\pi$
$138$		0			0
$139$			140.783i			1.01282i	0.862292	+	0.506412i	$0.169028\pi$
$139$			140.783i			1.01282i	−0.862292	+	0.506412i	$0.830972\pi$
$140$		0			0
$141$	−211.995			−1.50351
$142$		0			0
$143$		−	48.0059i		−	0.335705i
$144$		0			0
$145$		−	396.776i		−	2.73639i
$146$		0			0
$147$	−82.6629			−0.562333
$148$		0			0
$149$			73.9940i			0.496604i	0.968683	+	0.248302i	$0.0798726\pi$
$149$			73.9940i			0.496604i	−0.968683	+	0.248302i	$0.920127\pi$
$150$		0			0
$151$			126.441i			0.837356i	0.908135	+	0.418678i	$0.137506\pi$
$151$			126.441i			0.837356i	−0.908135	+	0.418678i	$0.862494\pi$
$152$		0			0
$153$	−249.723			−1.63218
$154$		0			0
$155$	−145.486			−0.938617
$156$		0			0
$157$			41.1447i			0.262068i	0.991378	+	0.131034i	$0.0418297\pi$
$157$			41.1447i			0.262068i	−0.991378	+	0.131034i	$0.958170\pi$
$158$		0			0
$159$	−206.687			−1.29992
$160$		0			0
$161$	188.873			1.17313
$162$		0			0
$163$			122.474i			0.751373i	0.926747	+	0.375687i	$0.122593\pi$
$163$			122.474i			0.751373i	−0.926747	+	0.375687i	$0.877407\pi$
$164$		0			0
$165$	−117.718			−0.713445
$166$		0			0
$167$		−	276.340i		−	1.65473i	−0.561666	−	0.827364i	$-0.689839\pi$
$167$		−	276.340i		−	1.65473i	0.561666	−	0.827364i	$-0.310161\pi$
$168$		0			0
$169$	180.336			1.06708
$170$		0			0
$171$	−268.116	−	30.2162i	−1.56793	−	0.176703i
$172$		0			0
$173$	137.771			0.796367			0.398183	−	0.917306i	$-0.369641\pi$
$173$	137.771			0.796367			0.398183	+	0.917306i	$0.369641\pi$
$174$		0			0
$175$	533.105			3.04632
$176$		0			0
$177$	112.345			0.634719
$178$		0			0
$179$	36.7239			0.205161			0.102581	−	0.994725i	$-0.467290\pi$
$179$	36.7239			0.205161			0.102581	+	0.994725i	$0.467290\pi$
$180$		0			0
$181$	−16.0148			−0.0884796			−0.0442398	−	0.999021i	$-0.514087\pi$
$181$	−16.0148			−0.0884796			−0.0442398	+	0.999021i	$0.514087\pi$
$182$		0			0
$183$			166.569i			0.910216i
$184$		0			0
$185$			127.820i			0.690919i
$186$		0			0
$187$			45.1670i			0.241535i
$188$		0			0
$189$	203.759			1.07809
$190$		0			0
$191$	335.959			1.75895			0.879473	−	0.475949i	$-0.157895\pi$
$191$	335.959			1.75895			0.879473	+	0.475949i	$0.157895\pi$
$192$		0			0
$193$		−	272.697i		−	1.41294i	−0.707744	−	0.706469i	$-0.750287\pi$
$193$		−	272.697i		−	1.41294i	0.707744	−	0.706469i	$-0.249713\pi$
$194$		0			0
$195$		−	856.631i		−	4.39298i
$196$		0			0
$197$		−	161.948i		−	0.822073i	−0.911619	−	0.411036i	$-0.865167\pi$
$197$		−	161.948i		−	0.822073i	0.911619	−	0.411036i	$-0.134833\pi$
$198$		0			0
$199$	−56.7294			−0.285072			−0.142536	−	0.989790i	$-0.545526\pi$
$199$	−56.7294			−0.285072			−0.142536	+	0.989790i	$0.545526\pi$
$200$		0			0
$201$	−178.132			−0.886229
$202$		0			0
$203$	339.178			1.67083
$204$		0			0
$205$	706.301			3.44537
$206$		0			0
$207$	−329.744			−1.59297
$208$		0			0
$209$	−5.46515	+	48.4937i	−0.0261491	+	0.232027i
$210$		0			0
$211$	−276.139			−1.30872			−0.654359	−	0.756184i	$-0.727061\pi$
$211$	−276.139			−1.30872			−0.654359	+	0.756184i	$0.727061\pi$
$212$		0			0
$213$		−	235.273i		−	1.10457i
$214$		0			0
$215$	−431.200			−2.00558
$216$		0			0
$217$		−	124.366i		−	0.573116i
$218$		0			0
$219$	322.735			1.47367
$220$		0			0
$221$	−328.678			−1.48723
$222$		0			0
$223$		−	270.202i		−	1.21167i	−0.795592	−	0.605833i	$-0.792840\pi$
$223$		−	270.202i		−	1.21167i	0.795592	−	0.605833i	$-0.207160\pi$
$224$		0			0
$225$	−930.721			−4.13654
$226$		0			0
$227$	125.172			0.551419			0.275709	−	0.961241i	$-0.411087\pi$
$227$	125.172			0.551419			0.275709	+	0.961241i	$0.411087\pi$
$228$		0			0
$229$			164.839i			0.719820i	0.932987	+	0.359910i	$0.117193\pi$
$229$			164.839i			0.719820i	−0.932987	+	0.359910i	$0.882807\pi$
$230$		0			0
$231$		−	100.630i		−	0.435627i
$232$		0			0
$233$	21.3928			0.0918145			0.0459073	−	0.998946i	$-0.485382\pi$
$233$	21.3928			0.0918145			0.0459073	+	0.998946i	$0.485382\pi$
$234$		0			0
$235$			418.790i			1.78208i
$236$		0			0
$237$			440.634i			1.85922i
$238$		0			0
$239$	146.299			0.612132			0.306066	−	0.952010i	$-0.400987\pi$
$239$	146.299			0.612132			0.306066	+	0.952010i	$0.400987\pi$
$240$		0			0
$241$			60.6880i			0.251818i	0.992042	+	0.125909i	$0.0401847\pi$
$241$			60.6880i			0.251818i	−0.992042	+	0.125909i	$0.959815\pi$
$242$		0			0
$243$	259.874			1.06944
$244$		0			0
$245$			163.298i			0.666524i
$246$		0			0
$247$	−352.886	−	39.7696i	−1.42869	−	0.161011i
$248$		0			0
$249$			180.308i			0.724128i
$250$		0			0
$251$			304.384i			1.21269i	0.795203	+	0.606343i	$0.207364\pi$
$251$			304.384i			1.21269i	−0.795203	+	0.606343i	$0.792636\pi$
$252$		0			0
$253$			59.6403i			0.235732i
$254$		0			0
$255$			805.973i			3.16068i
$256$		0			0
$257$			110.188i			0.428749i	0.976752	+	0.214374i	$0.0687713\pi$
$257$			110.188i			0.428749i	−0.976752	+	0.214374i	$0.931229\pi$
$258$		0			0
$259$	−109.265			−0.421872
$260$		0			0
$261$	−592.154			−2.26879
$262$		0			0
$263$	−182.360			−0.693384			−0.346692	−	0.937979i	$-0.612695\pi$
$263$	−182.360			−0.693384			−0.346692	+	0.937979i	$0.612695\pi$
$264$		0			0
$265$			408.305i			1.54077i
$266$		0			0
$267$		−	461.345i		−	1.72788i
$268$		0			0
$269$	−255.904			−0.951316			−0.475658	−	0.879630i	$-0.657790\pi$
$269$	−255.904			−0.951316			−0.475658	+	0.879630i	$0.657790\pi$
$270$		0			0
$271$	−409.512			−1.51111			−0.755557	−	0.655083i	$-0.772634\pi$
$271$	−409.512			−1.51111			−0.755557	+	0.655083i	$0.772634\pi$
$272$		0			0
$273$	732.278			2.68234
$274$		0			0
$275$			168.338i			0.612138i
$276$		0			0
$277$		−	268.635i		−	0.969800i	−0.874570	−	0.484900i	$-0.838856\pi$
$277$		−	268.635i		−	0.969800i	0.874570	−	0.484900i	$-0.161144\pi$
$278$		0			0
$279$			217.124i			0.778223i
$280$		0			0
$281$			36.8664i			0.131197i	0.997846	+	0.0655985i	$0.0208957\pi$
$281$			36.8664i			0.131197i	−0.997846	+	0.0655985i	$0.979104\pi$
$282$		0			0
$283$		−	235.211i		−	0.831136i	−0.909562	−	0.415568i	$-0.863583\pi$
$283$		−	235.211i		−	0.831136i	0.909562	−	0.415568i	$-0.136417\pi$
$284$		0			0
$285$	−97.5218	+	865.337i	−0.342182	+	3.03627i
$286$		0			0
$287$			603.770i			2.10373i
$288$		0			0
$289$	20.2413			0.0700390
$290$		0			0
$291$			537.692i			1.84774i
$292$		0			0
$293$	94.9295			0.323991			0.161996	−	0.986791i	$-0.448207\pi$
$293$	94.9295			0.323991			0.161996	+	0.986791i	$0.448207\pi$
$294$		0			0
$295$		−	221.935i		−	0.752322i
$296$		0			0
$297$			64.3406i			0.216635i
$298$		0			0
$299$	−433.999			−1.45150
$300$		0			0
$301$		−	368.604i		−	1.22460i
$302$		0			0
$303$		−	475.061i		−	1.56786i
$304$		0			0
$305$	329.053			1.07886
$306$		0			0
$307$	54.2461			0.176698			0.0883488	−	0.996090i	$-0.471841\pi$
$307$	54.2461			0.176698			0.0883488	+	0.996090i	$0.471841\pi$
$308$		0			0
$309$			659.818i			2.13533i
$310$		0			0
$311$	161.297			0.518639			0.259319	−	0.965792i	$-0.416502\pi$
$311$	161.297			0.518639			0.259319	+	0.965792i	$0.416502\pi$
$312$		0			0
$313$	133.576			0.426759			0.213380	−	0.976969i	$-0.431553\pi$
$313$	133.576			0.426759			0.213380	+	0.976969i	$0.431553\pi$
$314$		0			0
$315$		−	1099.09i		−	3.48919i
$316$		0			0
$317$	178.496			0.563079			0.281539	−	0.959550i	$-0.409155\pi$
$317$	178.496			0.563079			0.281539	+	0.959550i	$0.409155\pi$
$318$		0			0
$319$			107.102i			0.335742i
$320$		0			0
$321$	−537.757			−1.67525
$322$		0			0
$323$	332.018	+	37.4178i	1.02792	+	0.115845i
$324$		0			0
$325$	−1224.99			−3.76919
$326$		0			0
$327$	228.100			0.697552
$328$		0			0
$329$	−357.996			−1.08813
$330$		0			0
$331$	34.4043			0.103941			0.0519703	−	0.998649i	$-0.483450\pi$
$331$	34.4043			0.103941			0.0519703	+	0.998649i	$0.483450\pi$
$332$		0			0
$333$	190.760			0.572853
$334$		0			0
$335$			351.895i			1.05043i
$336$		0			0
$337$		−	569.765i		−	1.69070i	−0.534216	−	0.845348i	$-0.679393\pi$
$337$		−	569.765i		−	1.69070i	0.534216	−	0.845348i	$-0.320607\pi$
$338$		0			0
$339$		−	798.862i		−	2.35653i
$340$		0			0
$341$	39.2709			0.115164
$342$		0			0
$343$	258.972			0.755021
$344$		0			0
$345$			1064.24i			3.08475i
$346$		0			0
$347$			490.606i			1.41385i	0.707288	+	0.706925i	$0.249918\pi$
$347$			490.606i			1.41385i	−0.707288	+	0.706925i	$0.750082\pi$
$348$		0			0
$349$			404.110i			1.15791i	0.815360	+	0.578954i	$0.196539\pi$
$349$			404.110i			1.15791i	−0.815360	+	0.578954i	$0.803461\pi$
$350$		0			0
$351$	−468.203			−1.33391
$352$		0			0
$353$	−370.269			−1.04892			−0.524460	−	0.851435i	$-0.675733\pi$
$353$	−370.269			−1.04892			−0.524460	+	0.851435i	$0.675733\pi$
$354$		0			0
$355$	−464.775			−1.30923
$356$		0			0
$357$	−688.974			−1.92990
$358$		0			0
$359$	67.6093			0.188327			0.0941634	−	0.995557i	$-0.469982\pi$
$359$	67.6093			0.188327			0.0941634	+	0.995557i	$0.469982\pi$
$360$		0			0
$361$	351.945	+	80.3475i	0.974917	+	0.222569i
$362$		0			0
$363$	−551.046			−1.51803
$364$		0			0
$365$		−	637.553i		−	1.74672i
$366$		0			0
$367$	148.148			0.403672			0.201836	−	0.979419i	$-0.435309\pi$
$367$	148.148			0.403672			0.201836	+	0.979419i	$0.435309\pi$
$368$		0			0
$369$		−	1054.09i		−	2.85662i
$370$		0			0
$371$	−349.033			−0.940789
$372$		0			0
$373$	−80.3745			−0.215481			−0.107741	−	0.994179i	$-0.534362\pi$
$373$	−80.3745			−0.215481			−0.107741	+	0.994179i	$0.534362\pi$
$374$		0			0
$375$			1858.07i			4.95484i
$376$		0			0
$377$	−779.375			−2.06731
$378$		0			0
$379$	−539.864			−1.42444			−0.712222	−	0.701954i	$-0.752311\pi$
$379$	−539.864			−1.42444			−0.712222	+	0.701954i	$0.752311\pi$
$380$		0			0
$381$		−	342.068i		−	0.897817i
$382$		0			0
$383$		−	148.152i		−	0.386819i	−0.981118	−	0.193409i	$-0.938045\pi$
$383$		−	148.152i		−	0.386819i	0.981118	−	0.193409i	$-0.0619545\pi$
$384$		0			0
$385$	−198.791			−0.516341
$386$		0			0
$387$			643.527i			1.66286i
$388$		0			0
$389$			221.407i			0.569169i	0.958651	+	0.284585i	$0.0918557\pi$
$389$			221.407i			0.569169i	−0.958651	+	0.284585i	$0.908144\pi$
$390$		0			0
$391$	408.334			1.04433
$392$		0			0
$393$		−	797.291i		−	2.02873i
$394$		0			0
$395$	870.461			2.20370
$396$		0			0
$397$		−	634.852i		−	1.59912i	−0.600585	−	0.799561i	$-0.705065\pi$
$397$		−	634.852i		−	1.59912i	0.600585	−	0.799561i	$-0.294935\pi$
$398$		0			0
$399$	−739.719	−	83.3650i	−1.85393	−	0.208935i
$400$		0			0
$401$		−	238.627i		−	0.595079i	−0.954709	−	0.297540i	$-0.903834\pi$
$401$		−	238.627i		−	0.595079i	0.954709	−	0.297540i	$-0.0961661\pi$
$402$		0			0
$403$			285.772i			0.709113i
$404$		0			0
$405$		−	67.9992i		−	0.167899i
$406$		0			0
$407$		−	34.5025i		−	0.0847726i
$408$		0			0
$409$		−	798.162i		−	1.95150i	−0.218898	−	0.975748i	$-0.570246\pi$
$409$		−	798.162i		−	1.95150i	0.218898	−	0.975748i	$-0.429754\pi$
$410$		0			0
$411$	−924.262			−2.24881
$412$		0			0
$413$	189.718			0.459365
$414$		0			0
$415$	356.193			0.858297
$416$		0			0
$417$		−	678.109i		−	1.62616i
$418$		0			0
$419$		−	520.299i		−	1.24176i	−0.783905	−	0.620881i	$-0.786775\pi$
$419$		−	520.299i		−	1.24176i	0.783905	−	0.620881i	$-0.213225\pi$
$420$		0			0
$421$	233.658			0.555007			0.277503	−	0.960725i	$-0.410493\pi$
$421$	233.658			0.555007			0.277503	+	0.960725i	$0.410493\pi$
$422$		0			0
$423$	625.006			1.47756
$424$		0			0
$425$	1152.55			2.71187
$426$		0			0
$427$			281.286i			0.658750i
$428$		0			0
$429$			231.230i			0.538999i
$430$		0			0
$431$		−	194.119i		−	0.450392i	−0.974314	−	0.225196i	$-0.927698\pi$
$431$		−	194.119i		−	0.450392i	0.974314	−	0.225196i	$-0.0723022\pi$
$432$		0			0
$433$			529.671i			1.22326i	0.791144	+	0.611630i	$0.209486\pi$
$433$			529.671i			1.22326i	−0.791144	+	0.611630i	$0.790514\pi$
$434$		0			0
$435$			1911.16i			4.39346i
$436$		0			0
$437$	438.410	+	49.4079i	1.00323	+	0.113062i
$438$		0			0
$439$			89.1328i			0.203036i	0.994834	+	0.101518i	$0.0323699\pi$
$439$			89.1328i			0.203036i	−0.994834	+	0.101518i	$0.967630\pi$
$440$		0			0
$441$	243.708			0.552626
$442$		0			0
$443$		−	57.6903i		−	0.130226i	−0.997878	−	0.0651132i	$-0.979259\pi$
$443$		−	57.6903i		−	0.130226i	0.997878	−	0.0651132i	$-0.0207409\pi$
$444$		0			0
$445$	−911.374			−2.04803
$446$		0			0
$447$		−	356.408i		−	0.797333i
$448$		0			0
$449$			240.506i			0.535648i	0.963468	+	0.267824i	$0.0863046\pi$
$449$			240.506i			0.535648i	−0.963468	+	0.267824i	$0.913695\pi$
$450$		0			0
$451$	−190.652			−0.422731
$452$		0			0
$453$		−	609.029i		−	1.34443i
$454$		0			0
$455$		−	1446.59i		−	3.17933i
$456$		0			0
$457$	110.164			0.241059			0.120530	−	0.992710i	$-0.461541\pi$
$457$	110.164			0.241059			0.120530	+	0.992710i	$0.461541\pi$
$458$		0			0
$459$	440.516			0.959730
$460$		0			0
$461$		−	755.483i		−	1.63879i	−0.573227	−	0.819396i	$-0.694309\pi$
$461$		−	755.483i		−	1.63879i	0.573227	−	0.819396i	$-0.305691\pi$
$462$		0			0
$463$	686.290			1.48227			0.741134	−	0.671358i	$-0.234289\pi$
$463$	686.290			1.48227			0.741134	+	0.671358i	$0.234289\pi$
$464$		0			0
$465$	700.762			1.50702
$466$		0			0
$467$			191.609i			0.410298i	0.978731	+	0.205149i	$0.0657678\pi$
$467$			191.609i			0.410298i	−0.978731	+	0.205149i	$0.934232\pi$
$468$		0			0
$469$	−300.812			−0.641390
$470$		0			0
$471$		−	198.182i		−	0.420769i
$472$		0			0
$473$	116.394			0.246075
$474$		0			0
$475$	1237.44	+	139.457i	2.60513	+	0.293593i
$476$		0			0
$477$	609.358			1.27748
$478$		0			0
$479$	−290.404			−0.606272			−0.303136	−	0.952947i	$-0.598034\pi$
$479$	−290.404			−0.606272			−0.303136	+	0.952947i	$0.598034\pi$
$480$		0			0
$481$	251.073			0.521981
$482$		0			0
$483$	−909.748			−1.88354
$484$		0			0
$485$	1062.20			2.19009
$486$		0			0
$487$			923.055i			1.89539i	0.319176	+	0.947695i	$0.396594\pi$
$487$			923.055i			1.89539i	−0.319176	+	0.947695i	$0.603406\pi$
$488$		0			0
$489$		−	589.921i		−	1.20638i
$490$		0			0
$491$		−	655.797i		−	1.33563i	−0.744325	−	0.667817i	$-0.767229\pi$
$491$		−	655.797i		−	1.33563i	0.744325	−	0.667817i	$-0.232771\pi$
$492$		0			0
$493$	733.286			1.48739
$494$		0			0
$495$	347.059			0.701130
$496$		0			0
$497$		−	397.306i		−	0.799408i
$498$		0			0
$499$		−	941.554i		−	1.88688i	−0.331541	−	0.943441i	$-0.607569\pi$
$499$		−	941.554i		−	1.88688i	0.331541	−	0.943441i	$-0.392431\pi$
$500$		0			0
$501$			1331.05i			2.65678i
$502$		0			0
$503$	192.197			0.382102			0.191051	−	0.981580i	$-0.438810\pi$
$503$	192.197			0.382102			0.191051	+	0.981580i	$0.438810\pi$
$504$		0			0
$505$	−938.469			−1.85835
$506$		0			0
$507$	−868.628			−1.71327
$508$		0			0
$509$	731.963			1.43804			0.719020	−	0.694989i	$-0.244591\pi$
$509$	731.963			1.43804			0.719020	+	0.694989i	$0.244591\pi$
$510$		0			0
$511$	545.003			1.06654
$512$		0			0
$513$	472.962	+	53.3019i	0.921952	+	0.103902i
$514$		0			0
$515$	1303.45			2.53097
$516$		0			0
$517$		−	113.044i		−	0.218654i
$518$		0			0
$519$	−663.605			−1.27862
$520$		0			0
$521$		−	27.3117i		−	0.0524216i	−0.999656	−	0.0262108i	$-0.991656\pi$
$521$		−	27.3117i		−	0.0524216i	0.999656	−	0.0262108i	$-0.00834412\pi$
$522$		0			0
$523$	−793.383			−1.51698			−0.758492	−	0.651682i	$-0.774064\pi$
$523$	−793.383			−1.51698			−0.758492	+	0.651682i	$0.774064\pi$
$524$		0			0
$525$	−2567.81			−4.89107
$526$		0			0
$527$		−	268.873i		−	0.510196i
$528$		0			0
$529$	10.1807			0.0192453
$530$		0			0
$531$	−331.218			−0.623763
$532$		0			0
$533$		−	1387.36i		−	2.60293i
$534$		0			0
$535$			1062.32i			1.98565i
$536$		0			0
$537$	−176.888			−0.329401
$538$		0			0
$539$		−	44.0791i		−	0.0817794i
$540$		0			0
$541$		−	465.412i		−	0.860282i	−0.902762	−	0.430141i	$-0.858464\pi$
$541$		−	465.412i		−	0.860282i	0.902762	−	0.430141i	$-0.141536\pi$
$542$		0			0
$543$	77.1387			0.142060
$544$		0			0
$545$		−	450.604i		−	0.826797i
$546$		0			0
$547$	151.934			0.277758			0.138879	−	0.990309i	$-0.455650\pi$
$547$	151.934			0.277758			0.138879	+	0.990309i	$0.455650\pi$
$548$		0			0
$549$		−	491.083i		−	0.894504i
$550$		0			0
$551$	787.295	+	88.7266i	1.42885	+	0.161028i
$552$		0			0
$553$			744.100i			1.34557i
$554$		0			0
$555$		−	615.672i		−	1.10932i
$556$		0			0
$557$			551.201i			0.989589i	0.869010	+	0.494795i	$0.164757\pi$
$557$			551.201i			0.989589i	−0.869010	+	0.494795i	$0.835243\pi$
$558$		0			0
$559$			846.991i			1.51519i
$560$		0			0
$561$		−	217.556i		−	0.387801i
$562$		0			0
$563$	−567.782			−1.00849			−0.504247	−	0.863559i	$-0.668230\pi$
$563$	−567.782			−1.00849			−0.504247	+	0.863559i	$0.668230\pi$
$564$		0			0
$565$	−1578.13			−2.79315
$566$		0			0
$567$	58.1280			0.102519
$568$		0			0
$569$			377.927i			0.664194i	0.943245	+	0.332097i	$0.107756\pi$
$569$			377.927i			0.664194i	−0.943245	+	0.332097i	$0.892244\pi$
$570$		0			0
$571$		−	233.894i		−	0.409621i	−0.978802	−	0.204811i	$-0.934342\pi$
$571$		−	233.894i		−	0.409621i	0.978802	−	0.204811i	$-0.0656578\pi$
$572$		0			0
$573$	−1618.22			−2.82411
$574$		0			0
$575$	1521.87			2.64673
$576$		0			0
$577$	−1055.47			−1.82924			−0.914621	−	0.404311i	$-0.867511\pi$
$577$	−1055.47			−1.82924			−0.914621	+	0.404311i	$0.867511\pi$
$578$		0			0
$579$			1313.50i			2.26857i
$580$		0			0
$581$			304.486i			0.524072i
$582$		0			0
$583$		−	110.214i		−	0.189046i
$584$		0			0
$585$			2525.53i			4.31715i
$586$		0			0
$587$		−	621.226i		−	1.05831i	−0.848526	−	0.529153i	$-0.822510\pi$
$587$		−	621.226i		−	1.05831i	0.848526	−	0.529153i	$-0.177490\pi$
$588$		0			0
$589$	32.5333	−	288.677i	0.0552348	−	0.490113i
$590$		0			0
$591$			780.058i			1.31990i
$592$		0			0
$593$	−167.997			−0.283301			−0.141650	−	0.989917i	$-0.545241\pi$
$593$	−167.997			−0.283301			−0.141650	+	0.989917i	$0.545241\pi$
$594$		0			0
$595$			1361.05i			2.28748i
$596$		0			0
$597$	273.249			0.457704
$598$		0			0
$599$		−	546.095i		−	0.911679i	−0.890062	−	0.455839i	$-0.849339\pi$
$599$		−	546.095i		−	0.911679i	0.890062	−	0.455839i	$-0.150661\pi$
$600$		0			0
$601$			188.849i			0.314225i	0.987581	+	0.157113i	$0.0502185\pi$
$601$			188.849i			0.314225i	−0.987581	+	0.157113i	$0.949781\pi$
$602$		0			0
$603$	525.172			0.870932
$604$		0			0
$605$			1088.58i			1.79930i
$606$		0			0
$607$		−	445.588i		−	0.734082i	−0.930205	−	0.367041i	$-0.880371\pi$
$607$		−	445.588i		−	0.734082i	0.930205	−	0.367041i	$-0.119629\pi$
$608$		0			0
$609$	−1633.72			−2.68263
$610$		0			0
$611$	822.614			1.34634
$612$		0			0
$613$		−	436.866i		−	0.712668i	−0.934359	−	0.356334i	$-0.884026\pi$
$613$		−	436.866i		−	0.712668i	0.934359	−	0.356334i	$-0.115974\pi$
$614$		0			0
$615$	−3402.05			−5.53178
$616$		0			0
$617$	619.544			1.00412			0.502062	−	0.864832i	$-0.332575\pi$
$617$	619.544			1.00412			0.502062	+	0.864832i	$0.332575\pi$
$618$		0			0
$619$		−	957.776i		−	1.54730i	−0.633616	−	0.773648i	$-0.718430\pi$
$619$		−	957.776i		−	1.54730i	0.633616	−	0.773648i	$-0.281570\pi$
$620$		0			0
$621$	581.674			0.936674
$622$		0			0
$623$		−	779.074i		−	1.25052i
$624$		0			0
$625$	2032.04			3.25127
$626$		0			0
$627$	26.3241	−	233.580i	0.0419841	−	0.372536i
$628$		0			0
$629$	−236.225			−0.375557
$630$		0			0
$631$	−168.597			−0.267191			−0.133595	−	0.991036i	$-0.542652\pi$
$631$	−168.597			−0.267191			−0.133595	+	0.991036i	$0.542652\pi$
$632$		0			0
$633$	1330.08			2.10124
$634$		0			0
$635$	−675.746			−1.06417
$636$		0			0
$637$	320.761			0.503550
$638$		0			0
$639$			693.636i			1.08550i
$640$		0			0
$641$		−	928.633i		−	1.44873i	−0.689419	−	0.724363i	$-0.742134\pi$
$641$		−	928.633i		−	1.44873i	0.689419	−	0.724363i	$-0.257866\pi$
$642$		0			0
$643$			688.613i			1.07094i	0.844555	+	0.535469i	$0.179865\pi$
$643$			688.613i			1.07094i	−0.844555	+	0.535469i	$0.820135\pi$
$644$		0			0
$645$	2076.96			3.22010
$646$		0			0
$647$	503.490			0.778191			0.389095	−	0.921197i	$-0.372788\pi$
$647$	503.490			0.778191			0.389095	+	0.921197i	$0.372788\pi$
$648$		0			0
$649$			59.9069i			0.0923065i
$650$		0			0
$651$			599.035i			0.920177i
$652$		0			0
$653$		−	320.595i		−	0.490958i	−0.969402	−	0.245479i	$-0.921055\pi$
$653$		−	320.595i		−	0.490958i	0.969402	−	0.245479i	$-0.0789452\pi$
$654$		0			0
$655$	−1575.03			−2.40462
$656$		0			0
$657$	−951.492			−1.44824
$658$		0			0
$659$	−104.356			−0.158356			−0.0791778	−	0.996861i	$-0.525229\pi$
$659$	−104.356			−0.158356			−0.0791778	+	0.996861i	$0.525229\pi$
$660$		0			0
$661$	−677.180			−1.02448			−0.512239	−	0.858843i	$-0.671184\pi$
$661$	−677.180			−1.02448			−0.512239	+	0.858843i	$0.671184\pi$
$662$		0			0
$663$	1583.15			2.38785
$664$		0			0
$665$	−164.685	+	1461.29i	−0.247647	+	2.19744i
$666$		0			0
$667$	968.259			1.45166
$668$		0			0
$669$			1301.48i			1.94542i
$670$		0			0
$671$	−88.8214			−0.132372
$672$		0			0
$673$			459.508i			0.682776i	0.939923	+	0.341388i	$0.110897\pi$
$673$			459.508i			0.682776i	−0.939923	+	0.341388i	$0.889103\pi$
$674$		0			0
$675$	1641.81			2.43231
$676$		0			0
$677$	401.257			0.592698			0.296349	−	0.955080i	$-0.404231\pi$
$677$	401.257			0.592698			0.296349	+	0.955080i	$0.404231\pi$
$678$		0			0
$679$			908.001i			1.33726i
$680$		0			0
$681$	−602.918			−0.885342
$682$		0			0
$683$	346.945			0.507972			0.253986	−	0.967208i	$-0.418258\pi$
$683$	346.945			0.507972			0.253986	+	0.967208i	$0.418258\pi$
$684$		0			0
$685$			1825.85i			2.66548i
$686$		0			0
$687$		−	793.980i		−	1.15572i
$688$		0			0
$689$	802.019			1.16403
$690$		0			0
$691$		−	647.035i		−	0.936374i	−0.883629	−	0.468187i	$-0.844907\pi$
$691$		−	647.035i		−	0.936374i	0.883629	−	0.468187i	$-0.155093\pi$
$692$		0			0
$693$			296.678i			0.428107i
$694$		0			0
$695$	−1339.58			−1.92746
$696$		0			0
$697$			1305.32i			1.87277i
$698$		0			0
$699$	−103.043			−0.147415
$700$		0			0
$701$			332.053i			0.473684i	0.971548	+	0.236842i	$0.0761124\pi$
$701$			332.053i			0.473684i	−0.971548	+	0.236842i	$0.923888\pi$
$702$		0			0
$703$	−253.624	−	28.5830i	−0.360774	−	0.0406585i
$704$		0			0
$705$		−	2017.19i		−	2.86126i
$706$		0			0
$707$		−	802.235i		−	1.13470i
$708$		0			0
$709$			313.146i			0.441673i	0.975311	+	0.220836i	$0.0708787\pi$
$709$			313.146i			0.441673i	−0.975311	+	0.220836i	$0.929121\pi$
$710$		0			0
$711$		−	1299.09i		−	1.82712i
$712$		0			0
$713$		−	355.031i		−	0.497939i
$714$		0			0
$715$	456.789			0.638866
$716$		0			0
$717$	−704.682			−0.982820
$718$		0			0
$719$	−786.113			−1.09334			−0.546671	−	0.837347i	$-0.684105\pi$
$719$	−786.113			−1.09334			−0.546671	+	0.837347i	$0.684105\pi$
$720$		0			0
$721$			1114.24i			1.54540i
$722$		0			0
$723$		−	292.317i		−	0.404311i
$724$		0			0
$725$	2732.97			3.76961
$726$		0			0
$727$	−101.332			−0.139384			−0.0696918	−	0.997569i	$-0.522202\pi$
$727$	−101.332			−0.139384			−0.0696918	+	0.997569i	$0.522202\pi$
$728$		0			0
$729$	−1187.42			−1.62884
$730$		0			0
$731$		−	796.903i		−	1.09015i
$732$		0			0
$733$			94.4717i			0.128884i	0.997921	+	0.0644418i	$0.0205267\pi$
$733$			94.4717i			0.128884i	−0.997921	+	0.0644418i	$0.979473\pi$
$734$		0			0
$735$		−	786.561i		−	1.07015i
$736$		0			0
$737$		−	94.9870i		−	0.128883i
$738$		0			0
$739$		−	552.138i		−	0.747142i	−0.927602	−	0.373571i	$-0.878133\pi$
$739$		−	552.138i		−	0.747142i	0.927602	−	0.373571i	$-0.121867\pi$
$740$		0			0
$741$	1699.75	+	191.559i	2.29386	+	0.258514i
$742$		0			0
$743$			436.458i			0.587426i	0.955894	+	0.293713i	$0.0948910\pi$
$743$			436.458i			0.587426i	−0.955894	+	0.293713i	$0.905109\pi$
$744$		0			0
$745$	−704.074			−0.945065
$746$		0			0
$747$		−	531.586i		−	0.711628i
$748$		0			0
$749$	−908.110			−1.21243
$750$		0			0
$751$		−	795.999i		−	1.05992i	−0.848023	−	0.529959i	$-0.822207\pi$
$751$		−	795.999i		−	1.05992i	0.848023	−	0.529959i	$-0.177793\pi$
$752$		0			0
$753$		−	1466.13i		−	1.94705i
$754$		0			0
$755$	−1203.12			−1.59354
$756$		0			0
$757$		−	627.876i		−	0.829427i	−0.909952	−	0.414714i	$-0.863882\pi$
$757$		−	627.876i		−	0.829427i	0.909952	−	0.414714i	$-0.136118\pi$
$758$		0			0
$759$		−	287.270i		−	0.378485i
$760$		0			0
$761$	−209.421			−0.275192			−0.137596	−	0.990488i	$-0.543938\pi$
$761$	−209.421			−0.275192			−0.137596	+	0.990488i	$0.543938\pi$
$762$		0			0
$763$	385.192			0.504839
$764$		0			0
$765$		−	2376.18i		−	3.10612i
$766$		0			0
$767$	−435.939			−0.568370
$768$		0			0
$769$	−1255.10			−1.63211			−0.816057	−	0.577972i	$-0.803844\pi$
$769$	−1255.10			−1.63211			−0.816057	+	0.577972i	$0.803844\pi$
$770$		0			0
$771$		−	530.746i		−	0.688387i
$772$		0			0
$773$	−594.561			−0.769161			−0.384580	−	0.923091i	$-0.625654\pi$
$773$	−594.561			−0.769161			−0.384580	+	0.923091i	$0.625654\pi$
$774$		0			0
$775$		−	1002.09i		−	1.29302i
$776$		0			0
$777$	526.298			0.677346
$778$		0			0
$779$	−157.942	+	1401.46i	−0.202750	+	1.79905i
$780$		0			0
$781$	125.457			0.160636
$782$		0			0
$783$	1044.57			1.33406
$784$		0			0
$785$	−391.503			−0.498730
$786$		0			0
$787$	518.213			0.658466			0.329233	−	0.944249i	$-0.393210\pi$
$787$	518.213			0.658466			0.329233	+	0.944249i	$0.393210\pi$
$788$		0			0
$789$	878.376			1.11328
$790$		0			0
$791$		−	1349.04i		−	1.70549i
$792$		0			0
$793$		−	646.348i		−	0.815067i
$794$		0			0
$795$		−	1966.69i		−	2.47382i
$796$		0			0
$797$	1179.23			1.47958			0.739790	−	0.672838i	$-0.234925\pi$
$797$	1179.23			1.47958			0.739790	+	0.672838i	$0.234925\pi$
$798$		0			0
$799$	−773.968			−0.968671
$800$		0			0
$801$			1360.14i			1.69806i
$802$		0			0
$803$			172.095i			0.214315i
$804$		0			0
$805$			1797.18i			2.23252i
$806$		0			0
$807$	1232.62			1.52740
$808$		0			0
$809$	−246.732			−0.304984			−0.152492	−	0.988305i	$-0.548730\pi$
$809$	−246.732			−0.304984			−0.152492	+	0.988305i	$0.548730\pi$
$810$		0			0
$811$	−751.394			−0.926503			−0.463251	−	0.886227i	$-0.653317\pi$
$811$	−751.394			−0.926503			−0.463251	+	0.886227i	$0.653317\pi$
$812$		0			0
$813$	1972.50			2.42620
$814$		0			0
$815$	−1165.37			−1.42991
$816$		0			0
$817$	96.4243	−	855.598i	0.118022	−	1.04724i
$818$		0			0
$819$	−2158.91			−2.63603
$820$		0			0
$821$			75.8243i			0.0923560i	0.998933	+	0.0461780i	$0.0147041\pi$
$821$			75.8243i			0.0923560i	−0.998933	+	0.0461780i	$0.985296\pi$
$822$		0			0
$823$	823.849			1.00103			0.500516	−	0.865727i	$-0.333144\pi$
$823$	823.849			1.00103			0.500516	+	0.865727i	$0.333144\pi$
$824$		0			0
$825$		−	810.836i		−	0.982831i
$826$		0			0
$827$	594.014			0.718276			0.359138	−	0.933284i	$-0.383071\pi$
$827$	594.014			0.718276			0.359138	+	0.933284i	$0.383071\pi$
$828$		0			0
$829$	785.953			0.948073			0.474037	−	0.880505i	$-0.342796\pi$
$829$	785.953			0.948073			0.474037	+	0.880505i	$0.342796\pi$
$830$		0			0
$831$			1293.93i			1.55708i
$832$		0			0
$833$	−301.793			−0.362296
$834$		0			0
$835$	2629.45			3.14904
$836$		0			0
$837$		−	383.011i		−	0.457600i
$838$		0			0
$839$		−	876.430i		−	1.04461i	−0.852758	−	0.522306i	$-0.825072\pi$
$839$		−	876.430i		−	1.04461i	0.852758	−	0.522306i	$-0.174928\pi$
$840$		0			0
$841$	897.797			1.06753
$842$		0			0
$843$		−	177.575i		−	0.210646i
$844$		0			0
$845$			1715.95i			2.03071i
$846$		0			0
$847$	−930.552			−1.09865
$848$		0			0
$849$			1132.95i			1.33445i
$850$		0			0
$851$	−311.921			−0.366535
$852$		0			0
$853$			16.0644i			0.0188329i	0.999956	+	0.00941644i	$0.00299739\pi$
$853$			16.0644i			0.0188329i	−0.999956	+	0.00941644i	$0.997003\pi$
$854$		0			0
$855$	287.515	−	2551.20i	0.336275	−	2.98386i
$856$		0			0
$857$		−	437.147i		−	0.510089i	−0.966929	−	0.255045i	$-0.917910\pi$
$857$		−	437.147i		−	0.510089i	0.966929	−	0.255045i	$-0.0820902\pi$
$858$		0			0
$859$			369.016i			0.429588i	0.976659	+	0.214794i	$0.0689081\pi$
$859$			369.016i			0.429588i	−0.976659	+	0.214794i	$0.931092\pi$
$860$		0			0
$861$		−	2908.19i		−	3.37768i
$862$		0			0
$863$			1051.04i			1.21789i	0.793211	+	0.608947i	$0.208408\pi$
$863$			1051.04i			1.21789i	−0.793211	+	0.608947i	$0.791592\pi$
$864$		0			0
$865$			1310.93i			1.51553i
$866$		0			0
$867$	−97.4963			−0.112452
$868$		0			0
$869$	−234.963			−0.270384
$870$		0			0
$871$	691.215			0.793588
$872$		0			0
$873$		−	1585.23i		−	1.81584i
$874$		0			0
$875$			3137.72i			3.58596i
$876$		0			0
$877$	676.417			0.771285			0.385642	−	0.922648i	$-0.373980\pi$
$877$	676.417			0.771285			0.385642	+	0.922648i	$0.373980\pi$
$878$		0			0
$879$	−457.248			−0.520191
$880$		0			0
$881$	1194.56			1.35592			0.677959	−	0.735100i	$-0.262865\pi$
$881$	1194.56			1.35592			0.677959	+	0.735100i	$0.262865\pi$
$882$		0			0
$883$		−	350.006i		−	0.396383i	−0.980163	−	0.198191i	$-0.936493\pi$
$883$		−	350.006i		−	0.396383i	0.980163	−	0.198191i	$-0.0635068\pi$
$884$		0			0
$885$			1069.00i			1.20791i
$886$		0			0
$887$		−	597.954i		−	0.674130i	−0.941481	−	0.337065i	$-0.890566\pi$
$887$		−	597.954i		−	0.674130i	0.941481	−	0.337065i	$-0.109434\pi$
$888$		0			0
$889$		−	577.651i		−	0.649776i
$890$		0			0
$891$			18.3550i			0.0206005i
$892$		0			0
$893$	−830.974	−	93.6492i	−0.930542	−	0.104870i
$894$		0			0
$895$			349.438i			0.390433i
$896$		0			0
$897$	2090.45			2.33049
$898$		0			0
$899$		−	637.563i		−	0.709191i
$900$		0			0
$901$	−754.591			−0.837504
$902$		0			0
$903$			1775.46i			1.96618i
$904$		0			0
$905$		−	152.385i		−	0.168382i
$906$		0			0
$907$	−1639.86			−1.80800			−0.904002	−	0.427529i	$-0.859384\pi$
$907$	−1639.86			−1.80800			−0.904002	+	0.427529i	$0.859384\pi$
$908$		0			0
$909$			1400.58i			1.54079i
$910$		0			0
$911$		−	767.932i		−	0.842955i	−0.906839	−	0.421477i	$-0.861512\pi$
$911$		−	767.932i		−	0.842955i	0.906839	−	0.421477i	$-0.138488\pi$
$912$		0			0
$913$	−96.1472			−0.105309
$914$		0			0
$915$	−1584.95			−1.73219
$916$		0			0
$917$		−	1346.39i		−	1.46825i
$918$		0			0
$919$	−892.788			−0.971478			−0.485739	−	0.874104i	$-0.661449\pi$
$919$	−892.788			−0.971478			−0.485739	+	0.874104i	$0.661449\pi$
$920$		0			0
$921$	−261.288			−0.283700
$922$		0			0
$923$			912.943i			0.989104i
$924$		0			0
$925$	−880.415			−0.951800
$926$		0			0
$927$		−	1945.29i		−	2.09847i
$928$		0			0
$929$	−1653.76			−1.78015			−0.890074	−	0.455817i	$-0.849347\pi$
$929$	−1653.76			−1.78015			−0.890074	+	0.455817i	$0.849347\pi$
$930$		0			0
$931$	−324.021	−	36.5166i	−0.348036	−	0.0392229i
$932$		0			0
$933$	−776.919			−0.832711
$934$		0			0
$935$	−429.777			−0.459654
$936$		0			0
$937$	958.381			1.02282			0.511409	−	0.859337i	$-0.329124\pi$
$937$	958.381			1.02282			0.511409	+	0.859337i	$0.329124\pi$
$938$		0			0
$939$	−643.395			−0.685192
$940$		0			0
$941$	447.751			0.475825			0.237912	−	0.971287i	$-0.423537\pi$
$941$	447.751			0.475825			0.237912	+	0.971287i	$0.423537\pi$
$942$		0			0
$943$			1723.60i			1.82778i
$944$		0			0
$945$			1938.82i			2.05166i
$946$		0			0
$947$		−	472.122i		−	0.498545i	−0.968433	−	0.249272i	$-0.919809\pi$
$947$		−	472.122i		−	0.498545i	0.968433	−	0.249272i	$-0.0801914\pi$
$948$		0			0
$949$	−1252.32			−1.31963
$950$		0			0
$951$	−859.763			−0.904062
$952$		0			0
$953$		−	1010.89i		−	1.06075i	−0.847763	−	0.530375i	$-0.822051\pi$
$953$		−	1010.89i		−	1.06075i	0.847763	−	0.530375i	$-0.177949\pi$
$954$		0			0
$955$			3196.74i			3.34737i
$956$		0			0
$957$		−	515.879i		−	0.539058i
$958$		0			0
$959$	−1560.80			−1.62753
$960$		0			0
$961$	727.225			0.756738
$962$		0			0
$963$	1585.42			1.64634
$964$		0			0
$965$	2594.79			2.68890
$966$		0			0
$967$	1006.28			1.04063			0.520313	−	0.853976i	$-0.325815\pi$
$967$	1006.28			1.04063			0.520313	+	0.853976i	$0.325815\pi$
$968$		0			0
$969$	−1599.24	−	180.231i	−1.65040	−	0.185997i
$970$		0			0
$971$	1357.39			1.39793			0.698966	−	0.715155i	$-0.253644\pi$
$971$	1357.39			1.39793			0.698966	+	0.715155i	$0.253644\pi$
$972$		0			0
$973$		−	1145.12i		−	1.17690i
$974$		0			0
$975$	5900.41			6.05170
$976$		0			0
$977$			1085.69i			1.11125i	0.831432	+	0.555627i	$0.187522\pi$
$977$			1085.69i			1.11125i	−0.831432	+	0.555627i	$0.812478\pi$
$978$		0			0
$979$	246.007			0.251284
$980$		0			0
$981$	−672.487			−0.685512
$982$		0			0
$983$		−	1209.78i		−	1.23070i	−0.788253	−	0.615352i	$-0.789014\pi$
$983$		−	1209.78i		−	1.23070i	0.788253	−	0.615352i	$-0.210986\pi$
$984$		0			0
$985$	1540.98			1.56445
$986$		0			0
$987$	1724.36			1.74707
$988$		0			0
$989$		−	1052.26i		−	1.06397i
$990$		0			0
$991$		−	1515.15i		−	1.52891i	−0.644675	−	0.764457i	$-0.723007\pi$
$991$		−	1515.15i		−	1.52891i	0.644675	−	0.764457i	$-0.276993\pi$
$992$		0			0
$993$	−165.716			−0.166884
$994$		0			0
$995$		−	539.796i		−	0.542508i
$996$		0			0
$997$			77.6455i			0.0778792i	0.999242	+	0.0389396i	$0.0123980\pi$
$997$			77.6455i			0.0778792i	−0.999242	+	0.0389396i	$0.987602\pi$
$998$		0			0
$999$	−336.504			−0.336841

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.g.e.417.5	✓	48
4.3	odd	2	inner	1216.3.g.e.417.41	yes	48
8.3	odd	2	inner	1216.3.g.e.417.6	yes	48
8.5	even	2	inner	1216.3.g.e.417.42	yes	48
19.18	odd	2	inner	1216.3.g.e.417.43	yes	48
76.75	even	2	inner	1216.3.g.e.417.7	yes	48
152.37	odd	2	inner	1216.3.g.e.417.8	yes	48
152.75	even	2	inner	1216.3.g.e.417.44	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.e.417.5	✓	48	1.1	even	1	trivial
1216.3.g.e.417.6	yes	48	8.3	odd	2	inner
1216.3.g.e.417.7	yes	48	76.75	even	2	inner
1216.3.g.e.417.8	yes	48	152.37	odd	2	inner
1216.3.g.e.417.41	yes	48	4.3	odd	2	inner
1216.3.g.e.417.42	yes	48	8.5	even	2	inner
1216.3.g.e.417.43	yes	48	19.18	odd	2	inner
1216.3.g.e.417.44	yes	48	152.75	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-4.81671 q^{3} +9.51528i q^{5} -8.13398 q^{7} +14.2007 q^{9} +O(q^{10})\)	\(q-4.81671 q^{3} +9.51528i q^{5} -8.13398 q^{7} +14.2007 q^{9} -2.56846i q^{11} +18.6905 q^{13} -45.8323i q^{15} -17.5853 q^{17} +(-18.8805 - 2.12779i) q^{19} +39.1791 q^{21} -23.2203 q^{23} -65.5405 q^{25} -25.0503 q^{27} -41.6989 q^{29} +15.2897i q^{31} +12.3715i q^{33} -77.3971i q^{35} +13.4331 q^{37} -90.0269 q^{39} -74.2281i q^{41} +45.3166i q^{43} +135.124i q^{45} +44.0123 q^{47} +17.1617 q^{49} +84.7031 q^{51} +42.9104 q^{53} +24.4396 q^{55} +(90.9418 + 10.2490i) q^{57} -23.3241 q^{59} -34.5816i q^{61} -115.508 q^{63} +177.846i q^{65} +36.9821 q^{67} +111.845 q^{69} +48.8452i q^{71} -67.0031 q^{73} +315.690 q^{75} +20.8918i q^{77} -91.4803i q^{79} -7.14632 q^{81} -37.4338i q^{83} -167.329i q^{85} +200.851 q^{87} +95.7801i q^{89} -152.029 q^{91} -73.6460i q^{93} +(20.2466 - 179.653i) q^{95} -111.631i q^{97} -36.4739i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 264 q^{9}+O(q^{10}) \) 48 * q + 264 * q^9	\( 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) \) 48 * q + 264 * q^9 + 24 * q^17 - 160 * q^25 + 328 * q^49 + 8 * q^57 - 472 * q^73 - 736 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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