LMFDB - Embedded newform 1216.2.c.i.609.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(609,1216)]

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, 0]))

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

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

chi := DirichletCharacter("1216.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.c (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:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2702336256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 $ x^8 + 9x^6 + 56x^4 + 225*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.5
Root			$1.52274 + 1.63746i$ of defining polynomial
Character	$\chi$	$=$	1216.609
Dual form			1216.2.c.i.609.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31342i q^{5} -4.77753 q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.31342i q^{5} -4.77753 q^{7} +3.00000 q^{9} +2.27492i q^{11} -6.09095i q^{13} -4.27492 q^{17} -1.00000i q^{19} -3.46410 q^{23} +3.27492 q^{25} -6.09095i q^{29} +2.62685 q^{31} -6.27492i q^{35} -0.837253i q^{37} -10.5498 q^{41} -10.2749i q^{43} +3.94027i q^{45} +4.77753 q^{47} +15.8248 q^{49} -10.3923i q^{53} -2.98793 q^{55} -8.54983i q^{59} +1.31342i q^{61} -14.3326 q^{63} +8.00000 q^{65} -8.54983i q^{67} -14.8087 q^{71} +4.27492 q^{73} -10.8685i q^{77} -4.30136 q^{79} +9.00000 q^{81} +13.0997i q^{83} -5.61478i q^{85} -10.0000 q^{89} +29.0997i q^{91} +1.31342 q^{95} +1.45017 q^{97} +6.82475i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} - 4 q^{17} - 4 q^{25} - 24 q^{41} + 36 q^{49} + 64 q^{65} + 4 q^{73} + 72 q^{81} - 80 q^{89} + 72 q^{97}+O(q^{100}) $ 8 * q + 24 * q^9 - 4 * q^17 - 4 * q^25 - 24 * q^41 + 36 * q^49 + 64 * q^65 + 4 * q^73 + 72 * q^81 - 80 * q^89 + 72 * 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
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	1.31342i	0.587381i	0.955901	+	0.293691i	$0.0948835\pi$
$5$	1.31342i	0.587381i	−0.955901	+	0.293691i	$0.905116\pi$
$6$	0	0
$7$	−4.77753	−1.80573	−0.902867	−	0.429919i	$-0.858542\pi$
$7$	−4.77753	−1.80573	−0.902867	+	0.429919i	$0.858542\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	2.27492i	0.685913i	0.939351	+	0.342957i	$0.111428\pi$
$11$	2.27492i	0.685913i	−0.939351	+	0.342957i	$0.888572\pi$
$12$	0	0
$13$	− 6.09095i	− 1.68933i	−0.535299	−	0.844663i	$-0.679801\pi$
$13$	− 6.09095i	− 1.68933i	0.535299	−	0.844663i	$-0.320199\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.27492	−1.03682	−0.518410	−	0.855132i	$-0.673476\pi$
$17$	−4.27492	−1.03682	−0.518410	+	0.855132i	$0.673476\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	3.27492	0.654983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.09095i	− 1.13106i	−0.824727	−	0.565530i	$-0.808671\pi$
$29$	− 6.09095i	− 1.13106i	0.824727	−	0.565530i	$-0.191329\pi$
$30$	0	0
$31$	2.62685	0.471796	0.235898	−	0.971778i	$-0.424197\pi$
$31$	2.62685	0.471796	0.235898	+	0.971778i	$0.424197\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 6.27492i	− 1.06065i
$36$	0	0
$37$	− 0.837253i	− 0.137644i	−0.997629	−	0.0688218i	$-0.978076\pi$
$37$	− 0.837253i	− 0.137644i	0.997629	−	0.0688218i	$-0.0219240\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.5498	−1.64761	−0.823804	−	0.566875i	$-0.808152\pi$
$41$	−10.5498	−1.64761	−0.823804	+	0.566875i	$0.808152\pi$
$42$	0	0
$43$	− 10.2749i	− 1.56691i	−0.621448	−	0.783455i	$-0.713455\pi$
$43$	− 10.2749i	− 1.56691i	0.621448	−	0.783455i	$-0.286545\pi$
$44$	0	0
$45$	3.94027i	0.587381i
$46$	0	0
$47$	4.77753	0.696874	0.348437	−	0.937332i	$-0.386713\pi$
$47$	4.77753	0.696874	0.348437	+	0.937332i	$0.386713\pi$
$48$	0	0
$49$	15.8248	2.26068
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.3923i	− 1.42749i	−0.700404	−	0.713746i	$-0.746997\pi$
$53$	− 10.3923i	− 1.42749i	0.700404	−	0.713746i	$-0.253003\pi$
$54$	0	0
$55$	−2.98793	−0.402893
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 8.54983i	− 1.11309i	−0.830816	−	0.556547i	$-0.812126\pi$
$59$	− 8.54983i	− 1.11309i	0.830816	−	0.556547i	$-0.187874\pi$
$60$	0	0
$61$	1.31342i	0.168167i	0.996459	+	0.0840834i	$0.0267962\pi$
$61$	1.31342i	0.168167i	−0.996459	+	0.0840834i	$0.973204\pi$
$62$	0	0
$63$	−14.3326	−1.80573
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	− 8.54983i	− 1.04453i	−0.852784	−	0.522264i	$-0.825087\pi$
$67$	− 8.54983i	− 1.04453i	0.852784	−	0.522264i	$-0.174913\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.8087	−1.75748	−0.878738	−	0.477305i	$-0.841614\pi$
$71$	−14.8087	−1.75748	−0.878738	+	0.477305i	$0.841614\pi$
$72$	0	0
$73$	4.27492	0.500341	0.250171	−	0.968202i	$-0.419513\pi$
$73$	4.27492	0.500341	0.250171	+	0.968202i	$0.419513\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.8685i	− 1.23858i
$78$	0	0
$79$	−4.30136	−0.483940	−0.241970	−	0.970284i	$-0.577794\pi$
$79$	−4.30136	−0.483940	−0.241970	+	0.970284i	$0.577794\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	13.0997i	1.43788i	0.695074	+	0.718938i	$0.255371\pi$
$83$	13.0997i	1.43788i	−0.695074	+	0.718938i	$0.744629\pi$
$84$	0	0
$85$	− 5.61478i	− 0.609008i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	29.0997i	3.05047i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.31342	0.134754
$96$	0	0
$97$	1.45017	0.147242	0.0736210	−	0.997286i	$-0.476544\pi$
$97$	1.45017	0.147242	0.0736210	+	0.997286i	$0.476544\pi$
$98$	0	0
$99$	6.82475i	0.685913i
$100$	0	0
$101$	13.8564i	1.37876i	0.724398	+	0.689382i	$0.242118\pi$
$101$	13.8564i	1.37876i	−0.724398	+	0.689382i	$0.757882\pi$
$102$	0	0
$103$	2.62685	0.258831	0.129416	−	0.991590i	$-0.458690\pi$
$103$	2.62685	0.258831	0.129416	+	0.991590i	$0.458690\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.5498i	− 1.59993i	−0.600045	−	0.799966i	$-0.704851\pi$
$107$	− 16.5498i	− 1.59993i	0.600045	−	0.799966i	$-0.295149\pi$
$108$	0	0
$109$	3.46410i	0.331801i	0.986143	+	0.165900i	$0.0530530\pi$
$109$	3.46410i	0.331801i	−0.986143	+	0.165900i	$0.946947\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.54983	−0.239868	−0.119934	−	0.992782i	$-0.538268\pi$
$113$	−2.54983	−0.239868	−0.119934	+	0.992782i	$0.538268\pi$
$114$	0	0
$115$	− 4.54983i	− 0.424274i
$116$	0	0
$117$	− 18.2728i	− 1.68933i
$118$	0	0
$119$	20.4235	1.87222
$120$	0	0
$121$	5.82475	0.529523
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.8685i	0.972106i
$126$	0	0
$127$	−17.4356	−1.54716	−0.773579	−	0.633699i	$-0.781536\pi$
$127$	−17.4356	−1.54716	−0.773579	+	0.633699i	$0.781536\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.27492i	0.198760i	0.995050	+	0.0993802i	$0.0316860\pi$
$131$	2.27492i	0.198760i	−0.995050	+	0.0993802i	$0.968314\pi$
$132$	0	0
$133$	4.77753i	0.414264i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.2749	−1.73220	−0.866102	−	0.499868i	$-0.833382\pi$
$137$	−20.2749	−1.73220	−0.866102	+	0.499868i	$0.833382\pi$
$138$	0	0
$139$	− 10.2749i	− 0.871507i	−0.900066	−	0.435754i	$-0.856482\pi$
$139$	− 10.2749i	− 0.871507i	0.900066	−	0.435754i	$-0.143518\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.8564	1.15873
$144$	0	0
$145$	8.00000	0.664364
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.98793i	0.244781i	0.992482	+	0.122390i	$0.0390560\pi$
$149$	2.98793i	0.244781i	−0.992482	+	0.122390i	$0.960944\pi$
$150$	0	0
$151$	9.55505	0.777579	0.388790	−	0.921327i	$-0.372893\pi$
$151$	9.55505	0.777579	0.388790	+	0.921327i	$0.372893\pi$
$152$	0	0
$153$	−12.8248	−1.03682
$154$	0	0
$155$	3.45017i	0.277124i
$156$	0	0
$157$	5.25370i	0.419291i	0.977778	+	0.209645i	$0.0672309\pi$
$157$	5.25370i	0.419291i	−0.977778	+	0.209645i	$0.932769\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	16.5498	1.30431
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.67451	0.129577	0.0647886	−	0.997899i	$-0.479363\pi$
$167$	1.67451	0.129577	0.0647886	+	0.997899i	$0.479363\pi$
$168$	0	0
$169$	−24.0997	−1.85382
$170$	0	0
$171$	− 3.00000i	− 0.229416i
$172$	0	0
$173$	− 8.71780i	− 0.662802i	−0.943490	−	0.331401i	$-0.892479\pi$
$173$	− 8.71780i	− 0.662802i	0.943490	−	0.331401i	$-0.107521\pi$
$174$	0	0
$175$	−15.6460	−1.18273
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 9.09967i	− 0.680141i	−0.940400	−	0.340071i	$-0.889549\pi$
$179$	− 9.09967i	− 0.680141i	0.940400	−	0.340071i	$-0.110451\pi$
$180$	0	0
$181$	− 3.46410i	− 0.257485i	−0.991678	−	0.128742i	$-0.958906\pi$
$181$	− 3.46410i	− 0.257485i	0.991678	−	0.128742i	$-0.0410940\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.09967	0.0808493
$186$	0	0
$187$	− 9.72508i	− 0.711168i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.0312	0.725834	0.362917	−	0.931822i	$-0.381781\pi$
$191$	10.0312	0.725834	0.362917	+	0.931822i	$0.381781\pi$
$192$	0	0
$193$	−19.0997	−1.37482	−0.687412	−	0.726268i	$-0.741253\pi$
$193$	−19.0997	−1.37482	−0.687412	+	0.726268i	$0.741253\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.1819i	0.867924i	0.900931	+	0.433962i	$0.142885\pi$
$197$	12.1819i	0.867924i	−0.900931	+	0.433962i	$0.857115\pi$
$198$	0	0
$199$	18.6339	1.32092	0.660462	−	0.750859i	$-0.270360\pi$
$199$	18.6339	1.32092	0.660462	+	0.750859i	$0.270360\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	29.0997i	2.04240i
$204$	0	0
$205$	− 13.8564i	− 0.967773i
$206$	0	0
$207$	−10.3923	−0.722315
$208$	0	0
$209$	2.27492	0.157359
$210$	0	0
$211$	11.4502i	0.788262i	0.919054	+	0.394131i	$0.128954\pi$
$211$	11.4502i	0.788262i	−0.919054	+	0.394131i	$0.871046\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	13.4953	0.920373
$216$	0	0
$217$	−12.5498	−0.851938
$218$	0	0
$219$	0	0
$220$	0	0
$221$	26.0383i	1.75153i
$222$	0	0
$223$	17.4356	1.16757	0.583787	−	0.811907i	$-0.301570\pi$
$223$	17.4356	1.16757	0.583787	+	0.811907i	$0.301570\pi$
$224$	0	0
$225$	9.82475	0.654983
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	22.0980i	1.46028i	0.683298	+	0.730140i	$0.260545\pi$
$229$	22.0980i	1.46028i	−0.683298	+	0.730140i	$0.739455\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.3746	1.13825	0.569123	−	0.822252i	$-0.307283\pi$
$233$	17.3746	1.13825	0.569123	+	0.822252i	$0.307283\pi$
$234$	0	0
$235$	6.27492i	0.409330i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.82518	0.247431	0.123715	−	0.992318i	$-0.460519\pi$
$239$	3.82518	0.247431	0.123715	+	0.992318i	$0.460519\pi$
$240$	0	0
$241$	2.54983	0.164249	0.0821246	−	0.996622i	$-0.473829\pi$
$241$	2.54983	0.164249	0.0821246	+	0.996622i	$0.473829\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.7846i	1.32788i
$246$	0	0
$247$	−6.09095	−0.387558
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.2749i	1.15350i	0.816920	+	0.576751i	$0.195680\pi$
$251$	18.2749i	1.15350i	−0.816920	+	0.576751i	$0.804320\pi$
$252$	0	0
$253$	− 7.88054i	− 0.495446i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.5498	1.40662	0.703310	−	0.710883i	$-0.251705\pi$
$257$	22.5498	1.40662	0.703310	+	0.710883i	$0.251705\pi$
$258$	0	0
$259$	4.00000i	0.248548i
$260$	0	0
$261$	− 18.2728i	− 1.13106i
$262$	0	0
$263$	−14.3326	−0.883785	−0.441892	−	0.897068i	$-0.645693\pi$
$263$	−14.3326	−0.883785	−0.441892	+	0.897068i	$0.645693\pi$
$264$	0	0
$265$	13.6495	0.838482
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.5742i	1.37637i	0.725534	+	0.688187i	$0.241593\pi$
$269$	22.5742i	1.37637i	−0.725534	+	0.688187i	$0.758407\pi$
$270$	0	0
$271$	−7.04329	−0.427849	−0.213925	−	0.976850i	$-0.568625\pi$
$271$	−7.04329	−0.427849	−0.213925	+	0.976850i	$0.568625\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.45017i	0.449262i
$276$	0	0
$277$	− 2.98793i	− 0.179527i	−0.995963	−	0.0897637i	$-0.971389\pi$
$277$	− 2.98793i	− 0.179527i	0.995963	−	0.0897637i	$-0.0286112\pi$
$278$	0	0
$279$	7.88054	0.471796
$280$	0	0
$281$	−19.0997	−1.13939	−0.569695	−	0.821856i	$-0.692939\pi$
$281$	−19.0997	−1.13939	−0.569695	+	0.821856i	$0.692939\pi$
$282$	0	0
$283$	− 11.3746i	− 0.676149i	−0.941119	−	0.338074i	$-0.890224\pi$
$283$	− 11.3746i	− 0.676149i	0.941119	−	0.338074i	$-0.109776\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	50.4021	2.97514
$288$	0	0
$289$	1.27492	0.0749951
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.6219i	1.26316i	0.775310	+	0.631581i	$0.217594\pi$
$293$	21.6219i	1.26316i	−0.775310	+	0.631581i	$0.782406\pi$
$294$	0	0
$295$	11.2296	0.653810
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.0997i	1.22023i
$300$	0	0
$301$	49.0887i	2.82942i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.72508	−0.0987780
$306$	0	0
$307$	16.5498i	0.944549i	0.881452	+	0.472274i	$0.156567\pi$
$307$	16.5498i	0.944549i	−0.881452	+	0.472274i	$0.843433\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.72987	−0.324911	−0.162455	−	0.986716i	$-0.551941\pi$
$311$	−5.72987	−0.324911	−0.162455	+	0.986716i	$0.551941\pi$
$312$	0	0
$313$	−15.0997	−0.853484	−0.426742	−	0.904373i	$-0.640339\pi$
$313$	−15.0997	−0.853484	−0.426742	+	0.904373i	$0.640339\pi$
$314$	0	0
$315$	− 18.8248i	− 1.06065i
$316$	0	0
$317$	8.71780i	0.489640i	0.969569	+	0.244820i	$0.0787289\pi$
$317$	8.71780i	0.489640i	−0.969569	+	0.244820i	$0.921271\pi$
$318$	0	0
$319$	13.8564	0.775810
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.27492i	0.237863i
$324$	0	0
$325$	− 19.9474i	− 1.10648i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−22.8248	−1.25837
$330$	0	0
$331$	− 24.5498i	− 1.34938i	−0.738101	−	0.674690i	$-0.764277\pi$
$331$	− 24.5498i	− 1.34938i	0.738101	−	0.674690i	$-0.235723\pi$
$332$	0	0
$333$	− 2.51176i	− 0.137644i
$334$	0	0
$335$	11.2296	0.613536
$336$	0	0
$337$	−11.6495	−0.634589	−0.317294	−	0.948327i	$-0.602774\pi$
$337$	−11.6495	−0.634589	−0.317294	+	0.948327i	$0.602774\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.97586i	0.323611i
$342$	0	0
$343$	−42.1605	−2.27645
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.8248i	0.795834i	0.917421	+	0.397917i	$0.130267\pi$
$347$	14.8248i	0.795834i	−0.917421	+	0.397917i	$0.869733\pi$
$348$	0	0
$349$	− 35.2323i	− 1.88594i	−0.332877	−	0.942970i	$-0.608019\pi$
$349$	− 35.2323i	− 1.88594i	0.332877	−	0.942970i	$-0.391981\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	− 19.4502i	− 1.03231i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.476171	−0.0251313	−0.0125657	−	0.999921i	$-0.504000\pi$
$359$	−0.476171	−0.0251313	−0.0125657	+	0.999921i	$0.504000\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.61478i	0.293891i
$366$	0	0
$367$	1.78959	0.0934161	0.0467080	−	0.998909i	$-0.485127\pi$
$367$	1.78959	0.0934161	0.0467080	+	0.998909i	$0.485127\pi$
$368$	0	0
$369$	−31.6495	−1.64761
$370$	0	0
$371$	49.6495i	2.57767i
$372$	0	0
$373$	− 25.2011i	− 1.30486i	−0.757849	−	0.652431i	$-0.773749\pi$
$373$	− 25.2011i	− 1.30486i	0.757849	−	0.652431i	$-0.226251\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−37.0997	−1.91073
$378$	0	0
$379$	21.6495i	1.11206i	0.831162	+	0.556030i	$0.187676\pi$
$379$	21.6495i	1.11206i	−0.831162	+	0.556030i	$0.812324\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.97586	0.305352	0.152676	−	0.988276i	$-0.451211\pi$
$383$	5.97586	0.305352	0.152676	+	0.988276i	$0.451211\pi$
$384$	0	0
$385$	14.2749	0.727517
$386$	0	0
$387$	− 30.8248i	− 1.56691i
$388$	0	0
$389$	− 17.7967i	− 0.902327i	−0.892441	−	0.451164i	$-0.851009\pi$
$389$	− 17.7967i	− 0.902327i	0.892441	−	0.451164i	$-0.148991\pi$
$390$	0	0
$391$	14.8087	0.748911
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 5.64950i	− 0.284257i
$396$	0	0
$397$	− 24.0027i	− 1.20466i	−0.798247	−	0.602331i	$-0.794239\pi$
$397$	− 24.0027i	− 1.20466i	0.798247	−	0.602331i	$-0.205761\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.90033	0.244711	0.122355	−	0.992486i	$-0.460955\pi$
$401$	4.90033	0.244711	0.122355	+	0.992486i	$0.460955\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	0	0
$405$	11.8208i	0.587381i
$406$	0	0
$407$	1.90468	0.0944116
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	40.8471i	2.00995i
$414$	0	0
$415$	−17.2054	−0.844581
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 12.0000i	− 0.586238i	−0.956076	−	0.293119i	$-0.905307\pi$
$419$	− 12.0000i	− 0.586238i	0.956076	−	0.293119i	$-0.0946933\pi$
$420$	0	0
$421$	− 24.2487i	− 1.18181i	−0.806741	−	0.590905i	$-0.798771\pi$
$421$	− 24.2487i	− 1.18181i	0.806741	−	0.590905i	$-0.201229\pi$
$422$	0	0
$423$	14.3326	0.696874
$424$	0	0
$425$	−14.0000	−0.679100
$426$	0	0
$427$	− 6.27492i	− 0.303665i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.1101	−0.920501	−0.460251	−	0.887789i	$-0.652240\pi$
$431$	−19.1101	−0.920501	−0.460251	+	0.887789i	$0.652240\pi$
$432$	0	0
$433$	27.6495	1.32875	0.664375	−	0.747399i	$-0.268698\pi$
$433$	27.6495	1.32875	0.664375	+	0.747399i	$0.268698\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.46410i	0.165710i
$438$	0	0
$439$	−3.57919	−0.170825	−0.0854127	−	0.996346i	$-0.527221\pi$
$439$	−3.57919	−0.170825	−0.0854127	+	0.996346i	$0.527221\pi$
$440$	0	0
$441$	47.4743	2.26068
$442$	0	0
$443$	− 27.3746i	− 1.30061i	−0.759675	−	0.650303i	$-0.774642\pi$
$443$	− 27.3746i	− 1.30061i	0.759675	−	0.650303i	$-0.225358\pi$
$444$	0	0
$445$	− 13.1342i	− 0.622623i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.45017	−0.257209	−0.128605	−	0.991696i	$-0.541050\pi$
$449$	−5.45017	−0.257209	−0.128605	+	0.991696i	$0.541050\pi$
$450$	0	0
$451$	− 24.0000i	− 1.13012i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−38.2202	−1.79179
$456$	0	0
$457$	−7.72508	−0.361364	−0.180682	−	0.983542i	$-0.557831\pi$
$457$	−7.72508	−0.361364	−0.180682	+	0.983542i	$0.557831\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 35.2323i	− 1.64093i	−0.571696	−	0.820465i	$-0.693714\pi$
$461$	− 35.2323i	− 1.64093i	0.571696	−	0.820465i	$-0.306286\pi$
$462$	0	0
$463$	33.4427	1.55421	0.777107	−	0.629369i	$-0.216687\pi$
$463$	33.4427	1.55421	0.777107	+	0.629369i	$0.216687\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.8248i	1.42640i	0.700961	+	0.713200i	$0.252755\pi$
$467$	30.8248i	1.42640i	−0.700961	+	0.713200i	$0.747245\pi$
$468$	0	0
$469$	40.8471i	1.88614i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	23.3746	1.07476
$474$	0	0
$475$	− 3.27492i	− 0.150264i
$476$	0	0
$477$	− 31.1769i	− 1.42749i
$478$	0	0
$479$	17.3205	0.791394	0.395697	−	0.918381i	$-0.370503\pi$
$479$	17.3205	0.791394	0.395697	+	0.918381i	$0.370503\pi$
$480$	0	0
$481$	−5.09967	−0.232525
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.90468i	0.0864872i
$486$	0	0
$487$	−12.9041	−0.584739	−0.292370	−	0.956305i	$-0.594444\pi$
$487$	−12.9041	−0.584739	−0.292370	+	0.956305i	$0.594444\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 12.0000i	− 0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$	− 12.0000i	− 0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$	0	0
$493$	26.0383i	1.17271i
$494$	0	0
$495$	−8.96379	−0.402893
$496$	0	0
$497$	70.7492	3.17353
$498$	0	0
$499$	34.2749i	1.53436i	0.641434	+	0.767178i	$0.278340\pi$
$499$	34.2749i	1.53436i	−0.641434	+	0.767178i	$0.721660\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.81312	0.303782	0.151891	−	0.988397i	$-0.451464\pi$
$503$	6.81312	0.303782	0.151891	+	0.988397i	$0.451464\pi$
$504$	0	0
$505$	−18.1993	−0.809860
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 13.0192i	− 0.577064i	−0.957470	−	0.288532i	$-0.906833\pi$
$509$	− 13.0192i	− 0.577064i	0.957470	−	0.288532i	$-0.0931671\pi$
$510$	0	0
$511$	−20.4235	−0.903484
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.45017i	0.152032i
$516$	0	0
$517$	10.8685i	0.477995i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.09967	0.135799	0.0678995	−	0.997692i	$-0.478370\pi$
$521$	3.09967	0.135799	0.0678995	+	0.997692i	$0.478370\pi$
$522$	0	0
$523$	20.5498i	0.898582i	0.893386	+	0.449291i	$0.148323\pi$
$523$	20.5498i	0.898582i	−0.893386	+	0.449291i	$0.851677\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.2296	−0.489167
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	− 25.6495i	− 1.11309i
$532$	0	0
$533$	64.2585i	2.78335i
$534$	0	0
$535$	21.7370	0.939770
$536$	0	0
$537$	0	0
$538$	0	0
$539$	36.0000i	1.55063i
$540$	0	0
$541$	− 7.28929i	− 0.313391i	−0.987647	−	0.156695i	$-0.949916\pi$
$541$	− 7.28929i	− 0.313391i	0.987647	−	0.156695i	$-0.0500841\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.54983	−0.194893
$546$	0	0
$547$	− 32.0000i	− 1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$	− 32.0000i	− 1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$	0	0
$549$	3.94027i	0.168167i
$550$	0	0
$551$	−6.09095	−0.259483
$552$	0	0
$553$	20.5498	0.873868
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 16.8443i	− 0.713717i	−0.934158	−	0.356859i	$-0.883848\pi$
$557$	− 16.8443i	− 0.713717i	0.934158	−	0.356859i	$-0.116152\pi$
$558$	0	0
$559$	−62.5840	−2.64702
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	− 3.34901i	− 0.140894i
$566$	0	0
$567$	−42.9977	−1.80573
$568$	0	0
$569$	−34.5498	−1.44840	−0.724202	−	0.689588i	$-0.757792\pi$
$569$	−34.5498	−1.44840	−0.724202	+	0.689588i	$0.757792\pi$
$570$	0	0
$571$	− 10.9003i	− 0.456165i	−0.973642	−	0.228082i	$-0.926754\pi$
$571$	− 10.9003i	− 0.456165i	0.973642	−	0.228082i	$-0.0732455\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11.3446	−0.473104
$576$	0	0
$577$	15.7251	0.654644	0.327322	−	0.944913i	$-0.393854\pi$
$577$	15.7251	0.654644	0.327322	+	0.944913i	$0.393854\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 62.5840i	− 2.59642i
$582$	0	0
$583$	23.6416	0.979136
$584$	0	0
$585$	24.0000	0.992278
$586$	0	0
$587$	− 29.7251i	− 1.22689i	−0.789739	−	0.613443i	$-0.789784\pi$
$587$	− 29.7251i	− 1.22689i	0.789739	−	0.613443i	$-0.210216\pi$
$588$	0	0
$589$	− 2.62685i	− 0.108237i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11.0997	−0.455809	−0.227904	−	0.973684i	$-0.573187\pi$
$593$	−11.0997	−0.455809	−0.227904	+	0.973684i	$0.573187\pi$
$594$	0	0
$595$	26.8248i	1.09971i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	35.5934	1.45431	0.727153	−	0.686476i	$-0.240843\pi$
$599$	35.5934	1.45431	0.727153	+	0.686476i	$0.240843\pi$
$600$	0	0
$601$	−19.0997	−0.779092	−0.389546	−	0.921007i	$-0.627368\pi$
$601$	−19.0997	−0.779092	−0.389546	+	0.921007i	$0.627368\pi$
$602$	0	0
$603$	− 25.6495i	− 1.04453i
$604$	0	0
$605$	7.65037i	0.311032i
$606$	0	0
$607$	−46.8229	−1.90048	−0.950242	−	0.311513i	$-0.899164\pi$
$607$	−46.8229	−1.90048	−0.950242	+	0.311513i	$0.899164\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 29.0997i	− 1.17725i
$612$	0	0
$613$	− 16.8443i	− 0.680336i	−0.940365	−	0.340168i	$-0.889516\pi$
$613$	− 16.8443i	− 0.680336i	0.940365	−	0.340168i	$-0.110484\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.2749	0.816237	0.408119	−	0.912929i	$-0.366185\pi$
$617$	20.2749	0.816237	0.408119	+	0.912929i	$0.366185\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	47.7753	1.91408
$624$	0	0
$625$	2.09967	0.0839868
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.57919i	0.142712i
$630$	0	0
$631$	−9.07888	−0.361425	−0.180712	−	0.983536i	$-0.557840\pi$
$631$	−9.07888	−0.361425	−0.180712	+	0.983536i	$0.557840\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 22.9003i	− 0.908772i
$636$	0	0
$637$	− 96.3878i	− 3.81902i
$638$	0	0
$639$	−44.4262	−1.75748
$640$	0	0
$641$	2.54983	0.100712	0.0503562	−	0.998731i	$-0.483964\pi$
$641$	2.54983	0.100712	0.0503562	+	0.998731i	$0.483964\pi$
$642$	0	0
$643$	7.92442i	0.312509i	0.987717	+	0.156254i	$0.0499419\pi$
$643$	7.92442i	0.312509i	−0.987717	+	0.156254i	$0.950058\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.82518	−0.150384	−0.0751918	−	0.997169i	$-0.523957\pi$
$647$	−3.82518	−0.150384	−0.0751918	+	0.997169i	$0.523957\pi$
$648$	0	0
$649$	19.4502	0.763486
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.98793i	− 0.116927i	−0.998290	−	0.0584634i	$-0.981380\pi$
$653$	− 2.98793i	− 0.116927i	0.998290	−	0.0584634i	$-0.0186201\pi$
$654$	0	0
$655$	−2.98793	−0.116748
$656$	0	0
$657$	12.8248	0.500341
$658$	0	0
$659$	− 28.5498i	− 1.11214i	−0.831134	−	0.556072i	$-0.812308\pi$
$659$	− 28.5498i	− 1.11214i	0.831134	−	0.556072i	$-0.187692\pi$
$660$	0	0
$661$	− 0.837253i	− 0.0325654i	−0.999867	−	0.0162827i	$-0.994817\pi$
$661$	− 0.837253i	− 0.0325654i	0.999867	−	0.0162827i	$-0.00518317\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.27492	−0.243331
$666$	0	0
$667$	21.0997i	0.816982i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.98793	−0.115348
$672$	0	0
$673$	−16.9003	−0.651460	−0.325730	−	0.945463i	$-0.605610\pi$
$673$	−16.9003	−0.651460	−0.325730	+	0.945463i	$0.605610\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 44.3112i	− 1.70302i	−0.524342	−	0.851508i	$-0.675688\pi$
$677$	− 44.3112i	− 1.70302i	0.524342	−	0.851508i	$-0.324312\pi$
$678$	0	0
$679$	−6.92820	−0.265880
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 38.1993i	− 1.46166i	−0.682561	−	0.730829i	$-0.739134\pi$
$683$	− 38.1993i	− 1.46166i	0.682561	−	0.730829i	$-0.260866\pi$
$684$	0	0
$685$	− 26.6296i	− 1.01746i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−63.2990	−2.41150
$690$	0	0
$691$	− 44.4743i	− 1.69188i	−0.533278	−	0.845940i	$-0.679040\pi$
$691$	− 44.4743i	− 1.69188i	0.533278	−	0.845940i	$-0.320960\pi$
$692$	0	0
$693$	− 32.6054i	− 1.23858i
$694$	0	0
$695$	13.4953	0.511907
$696$	0	0
$697$	45.0997	1.70827
$698$	0	0
$699$	0	0
$700$	0	0
$701$	48.7276i	1.84042i	0.391430	+	0.920208i	$0.371981\pi$
$701$	48.7276i	1.84042i	−0.391430	+	0.920208i	$0.628019\pi$
$702$	0	0
$703$	−0.837253	−0.0315776
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 66.1993i	− 2.48968i
$708$	0	0
$709$	3.34901i	0.125775i	0.998021	+	0.0628874i	$0.0200309\pi$
$709$	3.34901i	0.125775i	−0.998021	+	0.0628874i	$0.979969\pi$
$710$	0	0
$711$	−12.9041	−0.483940
$712$	0	0
$713$	−9.09967	−0.340785
$714$	0	0
$715$	18.1993i	0.680617i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−12.4279	−0.463482	−0.231741	−	0.972777i	$-0.574442\pi$
$719$	−12.4279	−0.463482	−0.231741	+	0.972777i	$0.574442\pi$
$720$	0	0
$721$	−12.5498	−0.467380
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 19.9474i	− 0.740826i
$726$	0	0
$727$	−19.5863	−0.726415	−0.363207	−	0.931708i	$-0.618318\pi$
$727$	−19.5863	−0.726415	−0.363207	+	0.931708i	$0.618318\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	43.9244i	1.62460i
$732$	0	0
$733$	− 1.67451i	− 0.0618493i	−0.999522	−	0.0309247i	$-0.990155\pi$
$733$	− 1.67451i	− 0.0618493i	0.999522	−	0.0309247i	$-0.00984520\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.4502	0.716456
$738$	0	0
$739$	− 14.8248i	− 0.545337i	−0.962108	−	0.272669i	$-0.912094\pi$
$739$	− 14.8248i	− 0.545337i	0.962108	−	0.272669i	$-0.0879063\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.7605	0.981746	0.490873	−	0.871231i	$-0.336678\pi$
$743$	26.7605	0.981746	0.490873	+	0.871231i	$0.336678\pi$
$744$	0	0
$745$	−3.92442	−0.143780
$746$	0	0
$747$	39.2990i	1.43788i
$748$	0	0
$749$	79.0673i	2.88905i
$750$	0	0
$751$	53.7511	1.96141	0.980703	−	0.195503i	$-0.0626340\pi$
$751$	53.7511	1.96141	0.980703	+	0.195503i	$0.0626340\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.5498i	0.456735i
$756$	0	0
$757$	− 1.31342i	− 0.0477372i	−0.999715	−	0.0238686i	$-0.992402\pi$
$757$	− 1.31342i	− 0.0477372i	0.999715	−	0.0238686i	$-0.00759833\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4.27492	−0.154966	−0.0774828	−	0.996994i	$-0.524688\pi$
$761$	−4.27492	−0.154966	−0.0774828	+	0.996994i	$0.524688\pi$
$762$	0	0
$763$	− 16.5498i	− 0.599144i
$764$	0	0
$765$	− 16.8443i	− 0.609008i
$766$	0	0
$767$	−52.0766	−1.88038
$768$	0	0
$769$	8.82475	0.318229	0.159114	−	0.987260i	$-0.449136\pi$
$769$	8.82475	0.318229	0.159114	+	0.987260i	$0.449136\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 0.115088i	− 0.00413942i	−0.999998	−	0.00206971i	$-0.999341\pi$
$773$	− 0.115088i	− 0.00413942i	0.999998	−	0.00206971i	$-0.000658809\pi$
$774$	0	0
$775$	8.60271	0.309018
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.5498i	0.377987i
$780$	0	0
$781$	− 33.6887i	− 1.20548i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.90033	−0.246283
$786$	0	0
$787$	46.1993i	1.64683i	0.567441	+	0.823414i	$0.307934\pi$
$787$	46.1993i	1.64683i	−0.567441	+	0.823414i	$0.692066\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.1819	0.433138
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.2487i	0.858933i	0.903083	+	0.429467i	$0.141298\pi$
$797$	24.2487i	0.858933i	−0.903083	+	0.429467i	$0.858702\pi$
$798$	0	0
$799$	−20.4235	−0.722532
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	9.72508i	0.343191i
$804$	0	0
$805$	21.7370i	0.766127i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.8248	0.591527	0.295763	−	0.955261i	$-0.404426\pi$
$809$	16.8248	0.591527	0.295763	+	0.955261i	$0.404426\pi$
$810$	0	0
$811$	− 43.2990i	− 1.52043i	−0.649669	−	0.760217i	$-0.725093\pi$
$811$	− 43.2990i	− 1.52043i	0.649669	−	0.760217i	$-0.274907\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.25370	0.184029
$816$	0	0
$817$	−10.2749	−0.359474
$818$	0	0
$819$	87.2990i	3.05047i
$820$	0	0
$821$	− 21.3759i	− 0.746023i	−0.927827	−	0.373011i	$-0.878325\pi$
$821$	− 21.3759i	− 0.746023i	0.927827	−	0.373011i	$-0.121675\pi$
$822$	0	0
$823$	−43.9501	−1.53200	−0.766002	−	0.642839i	$-0.777757\pi$
$823$	−43.9501	−1.53200	−0.766002	+	0.642839i	$0.777757\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.35050i	0.0817348i	0.999165	+	0.0408674i	$0.0130121\pi$
$827$	2.35050i	0.0817348i	−0.999165	+	0.0408674i	$0.986988\pi$
$828$	0	0
$829$	35.7084i	1.24021i	0.784521	+	0.620103i	$0.212909\pi$
$829$	35.7084i	1.24021i	−0.784521	+	0.620103i	$0.787091\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−67.6495	−2.34392
$834$	0	0
$835$	2.19934i	0.0761112i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.9424	1.34444	0.672220	−	0.740351i	$-0.265341\pi$
$839$	38.9424	1.34444	0.672220	+	0.740351i	$0.265341\pi$
$840$	0	0
$841$	−8.09967	−0.279299
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 31.6531i	− 1.08890i
$846$	0	0
$847$	−27.8279	−0.956178
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.90033i	0.0994221i
$852$	0	0
$853$	15.5309i	0.531768i	0.964005	+	0.265884i	$0.0856639\pi$
$853$	15.5309i	0.531768i	−0.964005	+	0.265884i	$0.914336\pi$
$854$	0	0
$855$	3.94027	0.134754
$856$	0	0
$857$	44.1993	1.50982	0.754910	−	0.655828i	$-0.227680\pi$
$857$	44.1993	1.50982	0.754910	+	0.655828i	$0.227680\pi$
$858$	0	0
$859$	− 12.4743i	− 0.425616i	−0.977094	−	0.212808i	$-0.931739\pi$
$859$	− 12.4743i	− 0.425616i	0.977094	−	0.212808i	$-0.0682609\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.9430	−0.951190	−0.475595	−	0.879664i	$-0.657767\pi$
$863$	−27.9430	−0.951190	−0.475595	+	0.879664i	$0.657767\pi$
$864$	0	0
$865$	11.4502	0.389317
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 9.78523i	− 0.331941i
$870$	0	0
$871$	−52.0766	−1.76455
$872$	0	0
$873$	4.35050	0.147242
$874$	0	0
$875$	− 51.9244i	− 1.75537i
$876$	0	0
$877$	38.3353i	1.29449i	0.762282	+	0.647245i	$0.224079\pi$
$877$	38.3353i	1.29449i	−0.762282	+	0.647245i	$0.775921\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20.8248	−0.701604	−0.350802	−	0.936450i	$-0.614091\pi$
$881$	−20.8248	−0.701604	−0.350802	+	0.936450i	$0.614091\pi$
$882$	0	0
$883$	51.3746i	1.72889i	0.502725	+	0.864446i	$0.332331\pi$
$883$	51.3746i	1.72889i	−0.502725	+	0.864446i	$0.667669\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.55505	−0.320827	−0.160414	−	0.987050i	$-0.551283\pi$
$887$	−9.55505	−0.320827	−0.160414	+	0.987050i	$0.551283\pi$
$888$	0	0
$889$	83.2990	2.79376
$890$	0	0
$891$	20.4743i	0.685913i
$892$	0	0
$893$	− 4.77753i	− 0.159874i
$894$	0	0
$895$	11.9517	0.399502
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 16.0000i	− 0.533630i
$900$	0	0
$901$	44.4262i	1.48005i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.54983	0.151242
$906$	0	0
$907$	− 21.6495i	− 0.718860i	−0.933172	−	0.359430i	$-0.882971\pi$
$907$	− 21.6495i	− 0.718860i	0.933172	−	0.359430i	$-0.117029\pi$
$908$	0	0
$909$	41.5692i	1.37876i
$910$	0	0
$911$	44.4262	1.47191	0.735954	−	0.677032i	$-0.236734\pi$
$911$	44.4262	1.47191	0.735954	+	0.677032i	$0.236734\pi$
$912$	0	0
$913$	−29.8007	−0.986258
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 10.8685i	− 0.358909i
$918$	0	0
$919$	34.7561	1.14650	0.573249	−	0.819381i	$-0.305683\pi$
$919$	34.7561	1.14650	0.573249	+	0.819381i	$0.305683\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	90.1993i	2.96895i
$924$	0	0
$925$	− 2.74194i	− 0.0901543i
$926$	0	0
$927$	7.88054	0.258831
$928$	0	0
$929$	23.0997	0.757876	0.378938	−	0.925422i	$-0.376289\pi$
$929$	23.0997	0.757876	0.378938	+	0.925422i	$0.376289\pi$
$930$	0	0
$931$	− 15.8248i	− 0.518635i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.7732	0.417727
$936$	0	0
$937$	37.9244	1.23894	0.619468	−	0.785022i	$-0.287348\pi$
$937$	37.9244	1.23894	0.619468	+	0.785022i	$0.287348\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 31.1769i	− 1.01634i	−0.861257	−	0.508169i	$-0.830322\pi$
$941$	− 31.1769i	− 1.01634i	0.861257	−	0.508169i	$-0.169678\pi$
$942$	0	0
$943$	36.5457	1.19009
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 30.1993i	− 0.981347i	−0.871344	−	0.490673i	$-0.836751\pi$
$947$	− 30.1993i	− 0.981347i	0.871344	−	0.490673i	$-0.163249\pi$
$948$	0	0
$949$	− 26.0383i	− 0.845239i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	45.2990	1.46738	0.733689	−	0.679485i	$-0.237797\pi$
$953$	45.2990	1.46738	0.733689	+	0.679485i	$0.237797\pi$
$954$	0	0
$955$	13.1752i	0.426341i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	96.8639	3.12790
$960$	0	0
$961$	−24.0997	−0.777409
$962$	0	0
$963$	− 49.6495i	− 1.59993i
$964$	0	0
$965$	− 25.0860i	− 0.807546i
$966$	0	0
$967$	−20.6695	−0.664687	−0.332344	−	0.943158i	$-0.607839\pi$
$967$	−20.6695	−0.664687	−0.332344	+	0.943158i	$0.607839\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 44.0000i	− 1.41203i	−0.708198	−	0.706014i	$-0.750492\pi$
$971$	− 44.0000i	− 1.41203i	0.708198	−	0.706014i	$-0.249508\pi$
$972$	0	0
$973$	49.0887i	1.57371i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.1993	1.03015	0.515074	−	0.857146i	$-0.327764\pi$
$977$	32.1993	1.03015	0.515074	+	0.857146i	$0.327764\pi$
$978$	0	0
$979$	− 22.7492i	− 0.727067i
$980$	0	0
$981$	10.3923i	0.331801i
$982$	0	0
$983$	−29.6175	−0.944651	−0.472326	−	0.881424i	$-0.656585\pi$
$983$	−29.6175	−0.944651	−0.472326	+	0.881424i	$0.656585\pi$
$984$	0	0
$985$	−16.0000	−0.509802
$986$	0	0
$987$	0	0
$988$	0	0
$989$	35.5934i	1.13180i
$990$	0	0
$991$	40.1249	1.27461	0.637305	−	0.770612i	$-0.280049\pi$
$991$	40.1249	1.27461	0.637305	+	0.770612i	$0.280049\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	24.4743i	0.775886i
$996$	0	0
$997$	− 11.5906i	− 0.367079i	−0.983012	−	0.183540i	$-0.941244\pi$
$997$	− 11.5906i	− 0.367079i	0.983012	−	0.183540i	$-0.0587556\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.c.i.609.5	yes	8
4.3	odd	2	inner	1216.2.c.i.609.6	yes	8
8.3	odd	2	inner	1216.2.c.i.609.4	yes	8
8.5	even	2	inner	1216.2.c.i.609.3	✓	8
16.3	odd	4		4864.2.a.bk.1.3		4
16.5	even	4		4864.2.a.bk.1.2		4
16.11	odd	4		4864.2.a.bj.1.2		4
16.13	even	4		4864.2.a.bj.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.i.609.3	✓	8	8.5	even	2	inner
1216.2.c.i.609.4	yes	8	8.3	odd	2	inner
1216.2.c.i.609.5	yes	8	1.1	even	1	trivial
1216.2.c.i.609.6	yes	8	4.3	odd	2	inner
4864.2.a.bj.1.2		4	16.11	odd	4
4864.2.a.bj.1.3		4	16.13	even	4
4864.2.a.bk.1.2		4	16.5	even	4
4864.2.a.bk.1.3		4	16.3	odd	4

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.31342i q^{5} -4.77753 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.31342i q^{5} -4.77753 q^{7} +3.00000 q^{9} +2.27492i q^{11} -6.09095i q^{13} -4.27492 q^{17} -1.00000i q^{19} -3.46410 q^{23} +3.27492 q^{25} -6.09095i q^{29} +2.62685 q^{31} -6.27492i q^{35} -0.837253i q^{37} -10.5498 q^{41} -10.2749i q^{43} +3.94027i q^{45} +4.77753 q^{47} +15.8248 q^{49} -10.3923i q^{53} -2.98793 q^{55} -8.54983i q^{59} +1.31342i q^{61} -14.3326 q^{63} +8.00000 q^{65} -8.54983i q^{67} -14.8087 q^{71} +4.27492 q^{73} -10.8685i q^{77} -4.30136 q^{79} +9.00000 q^{81} +13.0997i q^{83} -5.61478i q^{85} -10.0000 q^{89} +29.0997i q^{91} +1.31342 q^{95} +1.45017 q^{97} +6.82475i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} - 4 q^{17} - 4 q^{25} - 24 q^{41} + 36 q^{49} + 64 q^{65} + 4 q^{73} + 72 q^{81} - 80 q^{89} + 72 q^{97}+O(q^{100}) \) 8 * q + 24 * q^9 - 4 * q^17 - 4 * q^25 - 24 * q^41 + 36 * q^49 + 64 * q^65 + 4 * q^73 + 72 * q^81 - 80 * q^89 + 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more