LMFDB - Embedded newform 1216.3.d.f.191.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,3,Mod(191,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.191"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 124 x^{18} + 6294 x^{16} + 169580 x^{14} + 2633777 x^{12} + 23965840 x^{10} + 123396288 x^{8} + \cdots + 6553600 $ x^20 + 124x^18 + 6294x^16 + 169580x^14 + 2633777x^12 + 23965840x^10 + 123396288x^8 + 323597312x^6 + 340403200x^4 + 121339904*x^2 + 6553600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 608)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.12
Root			$-0.760762i$ of defining polynomial
Character	$\chi$	$=$	1216.191
Dual form			1216.3.d.f.191.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.760762i q^{3} -3.86554 q^{5} -8.82674i q^{7} +8.42124 q^{9} -11.2858i q^{11} -10.3144 q^{13} -2.94076i q^{15} +2.39130 q^{17} +4.35890i q^{19} +6.71505 q^{21} +0.748138i q^{23} -10.0576 q^{25} +13.2534i q^{27} -26.1770 q^{29} -19.5534i q^{31} +8.58580 q^{33} +34.1201i q^{35} +26.2742 q^{37} -7.84682i q^{39} +22.0622 q^{41} +58.0818i q^{43} -32.5527 q^{45} +26.2030i q^{47} -28.9113 q^{49} +1.81921i q^{51} -56.1426 q^{53} +43.6257i q^{55} -3.31609 q^{57} -5.58045i q^{59} -92.1800 q^{61} -74.3321i q^{63} +39.8708 q^{65} +3.00658i q^{67} -0.569155 q^{69} +24.8541i q^{71} -31.0555 q^{73} -7.65144i q^{75} -99.6167 q^{77} -80.2428i q^{79} +65.7085 q^{81} +36.4566i q^{83} -9.24369 q^{85} -19.9145i q^{87} -162.602 q^{89} +91.0426i q^{91} +14.8755 q^{93} -16.8495i q^{95} -158.112 q^{97} -95.0403i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 68 q^{9} + 16 q^{13} + 8 q^{17} - 64 q^{21} + 196 q^{25} + 88 q^{29} - 184 q^{33} - 16 q^{37} - 16 q^{41} - 16 q^{45} + 52 q^{49} - 88 q^{53} + 208 q^{61} - 192 q^{65} - 248 q^{69} - 152 q^{73} - 312 q^{77}+ \cdots - 72 q^{97}+O(q^{100}) $ 20 * q - 68 * q^9 + 16 * q^13 + 8 * q^17 - 64 * q^21 + 196 * q^25 + 88 * q^29 - 184 * q^33 - 16 * q^37 - 16 * q^41 - 16 * q^45 + 52 * q^49 - 88 * q^53 + 208 * q^61 - 192 * q^65 - 248 * q^69 - 152 * q^73 - 312 * q^77 + 500 * q^81 + 488 * q^85 + 304 * q^89 + 240 * q^93 - 72 * q^97

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
$3$	0.760762i	0.253587i	0.991929	+	0.126794i	$0.0404686\pi$
$3$	0.760762i	0.253587i	−0.991929	+	0.126794i	$0.959531\pi$
$4$	0	0
$5$	−3.86554	−0.773108	−0.386554	−	0.922267i	$-0.626335\pi$
$5$	−3.86554	−0.773108	−0.386554	+	0.922267i	$0.626335\pi$
$6$	0	0
$7$	− 8.82674i	− 1.26096i	−0.776204	−	0.630481i	$-0.782858\pi$
$7$	− 8.82674i	− 1.26096i	0.776204	−	0.630481i	$-0.217142\pi$
$8$	0	0
$9$	8.42124	0.935693
$10$	0	0
$11$	− 11.2858i	− 1.02598i	−0.858394	−	0.512990i	$-0.828538\pi$
$11$	− 11.2858i	− 1.02598i	0.858394	−	0.512990i	$-0.171462\pi$
$12$	0	0
$13$	−10.3144	−0.793416	−0.396708	−	0.917945i	$-0.629847\pi$
$13$	−10.3144	−0.793416	−0.396708	+	0.917945i	$0.629847\pi$
$14$	0	0
$15$	− 2.94076i	− 0.196051i
$16$	0	0
$17$	2.39130	0.140665	0.0703325	−	0.997524i	$-0.477594\pi$
$17$	2.39130	0.140665	0.0703325	+	0.997524i	$0.477594\pi$
$18$	0	0
$19$	4.35890i	0.229416i
$19$	4.35890i	0.229416i
$20$	0	0
$21$	6.71505	0.319764
$22$	0	0
$23$	0.748138i	0.0325277i	0.999868	+	0.0162639i	$0.00517718\pi$
$23$	0.748138i	0.0325277i	−0.999868	+	0.0162639i	$0.994823\pi$
$24$	0	0
$25$	−10.0576	−0.402304
$26$	0	0
$27$	13.2534i	0.490868i
$28$	0	0
$29$	−26.1770	−0.902655	−0.451327	−	0.892358i	$-0.649049\pi$
$29$	−26.1770	−0.902655	−0.451327	+	0.892358i	$0.649049\pi$
$30$	0	0
$31$	− 19.5534i	− 0.630754i	−0.948967	−	0.315377i	$-0.897869\pi$
$31$	− 19.5534i	− 0.630754i	0.948967	−	0.315377i	$-0.102131\pi$
$32$	0	0
$33$	8.58580	0.260176
$34$	0	0
$35$	34.1201i	0.974861i
$36$	0	0
$37$	26.2742	0.710113	0.355056	−	0.934845i	$-0.384462\pi$
$37$	26.2742	0.710113	0.355056	+	0.934845i	$0.384462\pi$
$38$	0	0
$39$	− 7.84682i	− 0.201200i
$40$	0	0
$41$	22.0622	0.538102	0.269051	−	0.963126i	$-0.413290\pi$
$41$	22.0622	0.538102	0.269051	+	0.963126i	$0.413290\pi$
$42$	0	0
$43$	58.0818i	1.35074i	0.737480	+	0.675369i	$0.236016\pi$
$43$	58.0818i	1.35074i	−0.737480	+	0.675369i	$0.763984\pi$
$44$	0	0
$45$	−32.5527	−0.723392
$46$	0	0
$47$	26.2030i	0.557511i	0.960362	+	0.278755i	$0.0899219\pi$
$47$	26.2030i	0.557511i	−0.960362	+	0.278755i	$0.910078\pi$
$48$	0	0
$49$	−28.9113	−0.590027
$50$	0	0
$51$	1.81921i	0.0356709i
$52$	0	0
$53$	−56.1426	−1.05929	−0.529647	−	0.848218i	$-0.677676\pi$
$53$	−56.1426	−1.05929	−0.529647	+	0.848218i	$0.677676\pi$
$54$	0	0
$55$	43.6257i	0.793194i
$56$	0	0
$57$	−3.31609	−0.0581770
$58$	0	0
$59$	− 5.58045i	− 0.0945840i	−0.998881	−	0.0472920i	$-0.984941\pi$
$59$	− 5.58045i	− 0.0945840i	0.998881	−	0.0472920i	$-0.0150591\pi$
$60$	0	0
$61$	−92.1800	−1.51115	−0.755574	−	0.655063i	$-0.772642\pi$
$61$	−92.1800	−1.51115	−0.755574	+	0.655063i	$0.772642\pi$
$62$	0	0
$63$	− 74.3321i	− 1.17987i
$64$	0	0
$65$	39.8708	0.613397
$66$	0	0
$67$	3.00658i	0.0448744i	0.999748	+	0.0224372i	$0.00714257\pi$
$67$	3.00658i	0.0448744i	−0.999748	+	0.0224372i	$0.992857\pi$
$68$	0	0
$69$	−0.569155	−0.00824863
$70$	0	0
$71$	24.8541i	0.350058i	0.984563	+	0.175029i	$0.0560020\pi$
$71$	24.8541i	0.350058i	−0.984563	+	0.175029i	$0.943998\pi$
$72$	0	0
$73$	−31.0555	−0.425418	−0.212709	−	0.977116i	$-0.568229\pi$
$73$	−31.0555	−0.425418	−0.212709	+	0.977116i	$0.568229\pi$
$74$	0	0
$75$	− 7.65144i	− 0.102019i
$76$	0	0
$77$	−99.6167	−1.29372
$78$	0	0
$79$	− 80.2428i	− 1.01573i	−0.861436	−	0.507866i	$-0.830435\pi$
$79$	− 80.2428i	− 1.01573i	0.861436	−	0.507866i	$-0.169565\pi$
$80$	0	0
$81$	65.7085	0.811215
$82$	0	0
$83$	36.4566i	0.439237i	0.975586	+	0.219618i	$0.0704812\pi$
$83$	36.4566i	0.439237i	−0.975586	+	0.219618i	$0.929519\pi$
$84$	0	0
$85$	−9.24369	−0.108749
$86$	0	0
$87$	− 19.9145i	− 0.228902i
$88$	0	0
$89$	−162.602	−1.82699	−0.913496	−	0.406847i	$-0.866628\pi$
$89$	−162.602	−1.82699	−0.913496	+	0.406847i	$0.866628\pi$
$90$	0	0
$91$	91.0426i	1.00047i
$92$	0	0
$93$	14.8755	0.159951
$94$	0	0
$95$	− 16.8495i	− 0.177363i
$96$	0	0
$97$	−158.112	−1.63002	−0.815010	−	0.579447i	$-0.803269\pi$
$97$	−158.112	−1.63002	−0.815010	+	0.579447i	$0.803269\pi$
$98$	0	0
$99$	− 95.0403i	− 0.960003i
$100$	0	0
$101$	−85.3780	−0.845327	−0.422664	−	0.906287i	$-0.638905\pi$
$101$	−85.3780	−0.845327	−0.422664	+	0.906287i	$0.638905\pi$
$102$	0	0
$103$	− 130.762i	− 1.26954i	−0.772703	−	0.634768i	$-0.781096\pi$
$103$	− 130.762i	− 1.26954i	0.772703	−	0.634768i	$-0.218904\pi$
$104$	0	0
$105$	−25.9573	−0.247212
$106$	0	0
$107$	59.6796i	0.557753i	0.960327	+	0.278877i	$0.0899621\pi$
$107$	59.6796i	0.557753i	−0.960327	+	0.278877i	$0.910038\pi$
$108$	0	0
$109$	−162.073	−1.48691	−0.743456	−	0.668785i	$-0.766815\pi$
$109$	−162.073	−1.48691	−0.743456	+	0.668785i	$0.766815\pi$
$110$	0	0
$111$	19.9884i	0.180076i
$112$	0	0
$113$	−81.7194	−0.723181	−0.361590	−	0.932337i	$-0.617766\pi$
$113$	−81.7194	−0.723181	−0.361590	+	0.932337i	$0.617766\pi$
$114$	0	0
$115$	− 2.89196i	− 0.0251475i
$116$	0	0
$117$	−86.8601	−0.742394
$118$	0	0
$119$	− 21.1074i	− 0.177373i
$120$	0	0
$121$	−6.36896	−0.0526361
$122$	0	0
$123$	16.7841i	0.136456i
$124$	0	0
$125$	135.517	1.08413
$126$	0	0
$127$	116.320i	0.915906i	0.888976	+	0.457953i	$0.151417\pi$
$127$	116.320i	0.915906i	−0.888976	+	0.457953i	$0.848583\pi$
$128$	0	0
$129$	−44.1864	−0.342531
$130$	0	0
$131$	− 251.950i	− 1.92328i	−0.274313	−	0.961641i	$-0.588450\pi$
$131$	− 251.950i	− 1.92328i	0.274313	−	0.961641i	$-0.411550\pi$
$132$	0	0
$133$	38.4749	0.289285
$134$	0	0
$135$	− 51.2317i	− 0.379494i
$136$	0	0
$137$	47.2361	0.344789	0.172395	−	0.985028i	$-0.444850\pi$
$137$	47.2361	0.344789	0.172395	+	0.985028i	$0.444850\pi$
$138$	0	0
$139$	− 96.6697i	− 0.695466i	−0.937594	−	0.347733i	$-0.886952\pi$
$139$	− 96.6697i	− 0.695466i	0.937594	−	0.347733i	$-0.113048\pi$
$140$	0	0
$141$	−19.9343	−0.141378
$142$	0	0
$143$	116.406i	0.814030i
$144$	0	0
$145$	101.188	0.697850
$146$	0	0
$147$	− 21.9946i	− 0.149623i
$148$	0	0
$149$	36.1156	0.242387	0.121193	−	0.992629i	$-0.461328\pi$
$149$	36.1156	0.242387	0.121193	+	0.992629i	$0.461328\pi$
$150$	0	0
$151$	237.443i	1.57247i	0.617927	+	0.786235i	$0.287973\pi$
$151$	237.443i	1.57247i	−0.617927	+	0.786235i	$0.712027\pi$
$152$	0	0
$153$	20.1377	0.131619
$154$	0	0
$155$	75.5843i	0.487641i
$156$	0	0
$157$	132.029	0.840946	0.420473	−	0.907305i	$-0.361864\pi$
$157$	132.029	0.840946	0.420473	+	0.907305i	$0.361864\pi$
$158$	0	0
$159$	− 42.7112i	− 0.268624i
$160$	0	0
$161$	6.60362	0.0410163
$162$	0	0
$163$	− 262.316i	− 1.60930i	−0.593749	−	0.804650i	$-0.702353\pi$
$163$	− 262.316i	− 1.60930i	0.593749	−	0.804650i	$-0.297647\pi$
$164$	0	0
$165$	−33.1888	−0.201144
$166$	0	0
$167$	− 23.1647i	− 0.138711i	−0.997592	−	0.0693553i	$-0.977906\pi$
$167$	− 23.1647i	− 0.138711i	0.997592	−	0.0693553i	$-0.0220942\pi$
$168$	0	0
$169$	−62.6129	−0.370491
$170$	0	0
$171$	36.7073i	0.214663i
$172$	0	0
$173$	152.260	0.880117	0.440059	−	0.897969i	$-0.354958\pi$
$173$	152.260	0.880117	0.440059	+	0.897969i	$0.354958\pi$
$174$	0	0
$175$	88.7757i	0.507290i
$176$	0	0
$177$	4.24540	0.0239853
$178$	0	0
$179$	− 10.8248i	− 0.0604738i	−0.999543	−	0.0302369i	$-0.990374\pi$
$179$	− 10.8248i	− 0.0604738i	0.999543	−	0.0302369i	$-0.00962616\pi$
$180$	0	0
$181$	−37.8087	−0.208888	−0.104444	−	0.994531i	$-0.533306\pi$
$181$	−37.8087	−0.208888	−0.104444	+	0.994531i	$0.533306\pi$
$182$	0	0
$183$	− 70.1271i	− 0.383208i
$184$	0	0
$185$	−101.564	−0.548994
$186$	0	0
$187$	− 26.9877i	− 0.144319i
$188$	0	0
$189$	116.985	0.618966
$190$	0	0
$191$	247.848i	1.29763i	0.760945	+	0.648817i	$0.224736\pi$
$191$	247.848i	1.29763i	−0.760945	+	0.648817i	$0.775264\pi$
$192$	0	0
$193$	−266.743	−1.38209	−0.691043	−	0.722814i	$-0.742848\pi$
$193$	−266.743	−1.38209	−0.691043	+	0.722814i	$0.742848\pi$
$194$	0	0
$195$	30.3322i	0.155550i
$196$	0	0
$197$	210.230	1.06716	0.533579	−	0.845750i	$-0.320847\pi$
$197$	210.230	1.06716	0.533579	+	0.845750i	$0.320847\pi$
$198$	0	0
$199$	− 13.9023i	− 0.0698610i	−0.999390	−	0.0349305i	$-0.988879\pi$
$199$	− 13.9023i	− 0.0698610i	0.999390	−	0.0349305i	$-0.0111210\pi$
$200$	0	0
$201$	−2.28729	−0.0113796
$202$	0	0
$203$	231.057i	1.13821i
$204$	0	0
$205$	−85.2823	−0.416011
$206$	0	0
$207$	6.30025i	0.0304360i
$208$	0	0
$209$	49.1936	0.235376
$210$	0	0
$211$	− 118.599i	− 0.562079i	−0.959696	−	0.281040i	$-0.909321\pi$
$211$	− 118.599i	− 0.562079i	0.959696	−	0.281040i	$-0.0906792\pi$
$212$	0	0
$213$	−18.9081	−0.0887704
$214$	0	0
$215$	− 224.518i	− 1.04427i
$216$	0	0
$217$	−172.592	−0.795357
$218$	0	0
$219$	− 23.6258i	− 0.107881i
$220$	0	0
$221$	−24.6649	−0.111606
$222$	0	0
$223$	− 15.0828i	− 0.0676360i	−0.999428	−	0.0338180i	$-0.989233\pi$
$223$	− 15.0828i	− 0.0676360i	0.999428	−	0.0338180i	$-0.0107667\pi$
$224$	0	0
$225$	−84.6974	−0.376433
$226$	0	0
$227$	174.226i	0.767515i	0.923434	+	0.383758i	$0.125370\pi$
$227$	174.226i	0.767515i	−0.923434	+	0.383758i	$0.874630\pi$
$228$	0	0
$229$	−152.404	−0.665520	−0.332760	−	0.943012i	$-0.607980\pi$
$229$	−152.404	−0.665520	−0.332760	+	0.943012i	$0.607980\pi$
$230$	0	0
$231$	− 75.7846i	− 0.328072i
$232$	0	0
$233$	384.340	1.64953	0.824764	−	0.565477i	$-0.191308\pi$
$233$	384.340	1.64953	0.824764	+	0.565477i	$0.191308\pi$
$234$	0	0
$235$	− 101.289i	− 0.431016i
$236$	0	0
$237$	61.0457	0.257577
$238$	0	0
$239$	390.143i	1.63240i	0.577772	+	0.816199i	$0.303922\pi$
$239$	390.143i	1.63240i	−0.577772	+	0.816199i	$0.696078\pi$
$240$	0	0
$241$	−453.554	−1.88196	−0.940982	−	0.338456i	$-0.890096\pi$
$241$	−453.554	−1.88196	−0.940982	+	0.338456i	$0.890096\pi$
$242$	0	0
$243$	169.269i	0.696582i
$244$	0	0
$245$	111.758	0.456155
$246$	0	0
$247$	− 44.9595i	− 0.182022i
$248$	0	0
$249$	−27.7348	−0.111385
$250$	0	0
$251$	193.468i	0.770788i	0.922752	+	0.385394i	$0.125934\pi$
$251$	193.468i	0.770788i	−0.922752	+	0.385394i	$0.874066\pi$
$252$	0	0
$253$	8.44332	0.0333728
$254$	0	0
$255$	− 7.03225i	− 0.0275774i
$256$	0	0
$257$	73.9117	0.287594	0.143797	−	0.989607i	$-0.454069\pi$
$257$	73.9117	0.287594	0.143797	+	0.989607i	$0.454069\pi$
$258$	0	0
$259$	− 231.915i	− 0.895426i
$260$	0	0
$261$	−220.443	−0.844608
$262$	0	0
$263$	− 91.3741i	− 0.347430i	−0.984796	−	0.173715i	$-0.944423\pi$
$263$	− 91.3741i	− 0.347430i	0.984796	−	0.173715i	$-0.0555771\pi$
$264$	0	0
$265$	217.022	0.818950
$266$	0	0
$267$	− 123.702i	− 0.463302i
$268$	0	0
$269$	−110.840	−0.412045	−0.206023	−	0.978547i	$-0.566052\pi$
$269$	−110.840	−0.412045	−0.206023	+	0.978547i	$0.566052\pi$
$270$	0	0
$271$	532.042i	1.96326i	0.190806	+	0.981628i	$0.438890\pi$
$271$	532.042i	1.96326i	−0.190806	+	0.981628i	$0.561110\pi$
$272$	0	0
$273$	−69.2618	−0.253706
$274$	0	0
$275$	113.508i	0.412756i
$276$	0	0
$277$	−118.545	−0.427961	−0.213981	−	0.976838i	$-0.568643\pi$
$277$	−118.545	−0.427961	−0.213981	+	0.976838i	$0.568643\pi$
$278$	0	0
$279$	− 164.664i	− 0.590192i
$280$	0	0
$281$	−121.730	−0.433204	−0.216602	−	0.976260i	$-0.569497\pi$
$281$	−121.730	−0.433204	−0.216602	+	0.976260i	$0.569497\pi$
$282$	0	0
$283$	− 201.427i	− 0.711758i	−0.934532	−	0.355879i	$-0.884182\pi$
$283$	− 201.427i	− 0.711758i	0.934532	−	0.355879i	$-0.115818\pi$
$284$	0	0
$285$	12.8185	0.0449771
$286$	0	0
$287$	− 194.737i	− 0.678527i
$288$	0	0
$289$	−283.282	−0.980213
$290$	0	0
$291$	− 120.286i	− 0.413353i
$292$	0	0
$293$	464.387	1.58494	0.792469	−	0.609912i	$-0.208795\pi$
$293$	464.387	1.58494	0.792469	+	0.609912i	$0.208795\pi$
$294$	0	0
$295$	21.5715i	0.0731237i
$296$	0	0
$297$	149.575	0.503621
$298$	0	0
$299$	− 7.71660i	− 0.0258080i
$300$	0	0
$301$	512.673	1.70323
$302$	0	0
$303$	− 64.9524i	− 0.214364i
$304$	0	0
$305$	356.326	1.16828
$306$	0	0
$307$	− 361.537i	− 1.17764i	−0.808263	−	0.588822i	$-0.799592\pi$
$307$	− 361.537i	− 1.17764i	0.808263	−	0.588822i	$-0.200408\pi$
$308$	0	0
$309$	99.4790	0.321938
$310$	0	0
$311$	54.8958i	0.176514i	0.996098	+	0.0882570i	$0.0281297\pi$
$311$	54.8958i	0.176514i	−0.996098	+	0.0882570i	$0.971870\pi$
$312$	0	0
$313$	146.624	0.468447	0.234223	−	0.972183i	$-0.424745\pi$
$313$	146.624	0.468447	0.234223	+	0.972183i	$0.424745\pi$
$314$	0	0
$315$	287.334i	0.912171i
$316$	0	0
$317$	−323.315	−1.01992	−0.509961	−	0.860197i	$-0.670340\pi$
$317$	−323.315	−1.01992	−0.509961	+	0.860197i	$0.670340\pi$
$318$	0	0
$319$	295.428i	0.926106i
$320$	0	0
$321$	−45.4020	−0.141439
$322$	0	0
$323$	10.4235i	0.0322708i
$324$	0	0
$325$	103.738	0.319194
$326$	0	0
$327$	− 123.299i	− 0.377062i
$328$	0	0
$329$	231.287	0.703000
$330$	0	0
$331$	46.0501i	0.139124i	0.997578	+	0.0695620i	$0.0221602\pi$
$331$	46.0501i	0.139124i	−0.997578	+	0.0695620i	$0.977840\pi$
$332$	0	0
$333$	221.261	0.664448
$334$	0	0
$335$	− 11.6221i	− 0.0346927i
$336$	0	0
$337$	221.171	0.656294	0.328147	−	0.944627i	$-0.393576\pi$
$337$	221.171	0.656294	0.328147	+	0.944627i	$0.393576\pi$
$338$	0	0
$339$	− 62.1691i	− 0.183390i
$340$	0	0
$341$	−220.675	−0.647141
$342$	0	0
$343$	− 177.318i	− 0.516961i
$344$	0	0
$345$	2.20009	0.00637708
$346$	0	0
$347$	− 269.239i	− 0.775904i	−0.921679	−	0.387952i	$-0.873183\pi$
$347$	− 269.239i	− 0.775904i	0.921679	−	0.387952i	$-0.126817\pi$
$348$	0	0
$349$	−22.0942	−0.0633070	−0.0316535	−	0.999499i	$-0.510077\pi$
$349$	−22.0942	−0.0633070	−0.0316535	+	0.999499i	$0.510077\pi$
$350$	0	0
$351$	− 136.701i	− 0.389462i
$352$	0	0
$353$	534.345	1.51372	0.756862	−	0.653575i	$-0.226731\pi$
$353$	534.345	1.51372	0.756862	+	0.653575i	$0.226731\pi$
$354$	0	0
$355$	− 96.0747i	− 0.270633i
$356$	0	0
$357$	16.0577	0.0449796
$358$	0	0
$359$	− 635.968i	− 1.77150i	−0.464164	−	0.885750i	$-0.653645\pi$
$359$	− 635.968i	− 1.77150i	0.464164	−	0.885750i	$-0.346355\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	− 4.84527i	− 0.0133478i
$364$	0	0
$365$	120.046	0.328894
$366$	0	0
$367$	− 155.366i	− 0.423339i	−0.977341	−	0.211670i	$-0.932110\pi$
$367$	− 155.366i	− 0.423339i	0.977341	−	0.211670i	$-0.0678901\pi$
$368$	0	0
$369$	185.791	0.503499
$370$	0	0
$371$	495.556i	1.33573i
$372$	0	0
$373$	367.645	0.985642	0.492821	−	0.870131i	$-0.335966\pi$
$373$	367.645	0.985642	0.492821	+	0.870131i	$0.335966\pi$
$374$	0	0
$375$	103.096i	0.274922i
$376$	0	0
$377$	270.000	0.716181
$378$	0	0
$379$	256.207i	0.676008i	0.941145	+	0.338004i	$0.109752\pi$
$379$	256.207i	0.676008i	−0.941145	+	0.338004i	$0.890248\pi$
$380$	0	0
$381$	−88.4919	−0.232262
$382$	0	0
$383$	− 129.804i	− 0.338913i	−0.985538	−	0.169457i	$-0.945799\pi$
$383$	− 129.804i	− 0.338913i	0.985538	−	0.169457i	$-0.0542012\pi$
$384$	0	0
$385$	385.072	1.00019
$386$	0	0
$387$	489.121i	1.26388i
$388$	0	0
$389$	26.2241	0.0674141	0.0337071	−	0.999432i	$-0.489269\pi$
$389$	26.2241	0.0674141	0.0337071	+	0.999432i	$0.489269\pi$
$390$	0	0
$391$	1.78903i	0.00457551i
$392$	0	0
$393$	191.674	0.487720
$394$	0	0
$395$	310.182i	0.785270i
$396$	0	0
$397$	−439.813	−1.10784	−0.553920	−	0.832570i	$-0.686869\pi$
$397$	−439.813	−1.10784	−0.553920	+	0.832570i	$0.686869\pi$
$398$	0	0
$399$	29.2702i	0.0733590i
$400$	0	0
$401$	695.947	1.73553	0.867764	−	0.496976i	$-0.165556\pi$
$401$	695.947	1.73553	0.867764	+	0.496976i	$0.165556\pi$
$402$	0	0
$403$	201.681i	0.500450i
$404$	0	0
$405$	−253.999	−0.627157
$406$	0	0
$407$	− 296.525i	− 0.728562i
$408$	0	0
$409$	−323.243	−0.790324	−0.395162	−	0.918611i	$-0.629312\pi$
$409$	−323.243	−0.790324	−0.395162	+	0.918611i	$0.629312\pi$
$410$	0	0
$411$	35.9355i	0.0874342i
$412$	0	0
$413$	−49.2572	−0.119267
$414$	0	0
$415$	− 140.925i	− 0.339578i
$416$	0	0
$417$	73.5427	0.176361
$418$	0	0
$419$	− 662.516i	− 1.58118i	−0.612343	−	0.790592i	$-0.709773\pi$
$419$	− 662.516i	− 1.58118i	0.612343	−	0.790592i	$-0.290227\pi$
$420$	0	0
$421$	287.728	0.683440	0.341720	−	0.939802i	$-0.388991\pi$
$421$	287.728	0.683440	0.341720	+	0.939802i	$0.388991\pi$
$422$	0	0
$423$	220.662i	0.521659i
$424$	0	0
$425$	−24.0508	−0.0565900
$426$	0	0
$427$	813.649i	1.90550i
$428$	0	0
$429$	−88.5575	−0.206428
$430$	0	0
$431$	96.3997i	0.223665i	0.993727	+	0.111833i	$0.0356720\pi$
$431$	96.3997i	0.223665i	−0.993727	+	0.111833i	$0.964328\pi$
$432$	0	0
$433$	−280.807	−0.648514	−0.324257	−	0.945969i	$-0.605114\pi$
$433$	−280.807	−0.648514	−0.324257	+	0.945969i	$0.605114\pi$
$434$	0	0
$435$	76.9802i	0.176966i
$436$	0	0
$437$	−3.26106	−0.00746237
$438$	0	0
$439$	− 252.456i	− 0.575070i	−0.957770	−	0.287535i	$-0.907164\pi$
$439$	− 252.456i	− 0.575070i	0.957770	−	0.287535i	$-0.0928358\pi$
$440$	0	0
$441$	−243.469	−0.552084
$442$	0	0
$443$	− 152.899i	− 0.345145i	−0.984997	−	0.172573i	$-0.944792\pi$
$443$	− 152.899i	− 0.345145i	0.984997	−	0.172573i	$-0.0552079\pi$
$444$	0	0
$445$	628.546	1.41246
$446$	0	0
$447$	27.4754i	0.0614662i
$448$	0	0
$449$	504.216	1.12297	0.561487	−	0.827485i	$-0.310229\pi$
$449$	504.216	1.12297	0.561487	+	0.827485i	$0.310229\pi$
$450$	0	0
$451$	− 248.989i	− 0.552083i
$452$	0	0
$453$	−180.638	−0.398759
$454$	0	0
$455$	− 351.929i	− 0.773470i
$456$	0	0
$457$	260.361	0.569719	0.284859	−	0.958569i	$-0.408053\pi$
$457$	260.361	0.569719	0.284859	+	0.958569i	$0.408053\pi$
$458$	0	0
$459$	31.6930i	0.0690479i
$460$	0	0
$461$	98.7594	0.214229	0.107114	−	0.994247i	$-0.465839\pi$
$461$	98.7594	0.214229	0.107114	+	0.994247i	$0.465839\pi$
$462$	0	0
$463$	456.959i	0.986952i	0.869759	+	0.493476i	$0.164274\pi$
$463$	456.959i	0.986952i	−0.869759	+	0.493476i	$0.835726\pi$
$464$	0	0
$465$	−57.5017	−0.123660
$466$	0	0
$467$	− 170.991i	− 0.366148i	−0.983099	−	0.183074i	$-0.941395\pi$
$467$	− 170.991i	− 0.366148i	0.983099	−	0.183074i	$-0.0586048\pi$
$468$	0	0
$469$	26.5383	0.0565849
$470$	0	0
$471$	100.442i	0.213253i
$472$	0	0
$473$	655.498	1.38583
$474$	0	0
$475$	− 43.8400i	− 0.0922948i
$476$	0	0
$477$	−472.791	−0.991175
$478$	0	0
$479$	− 181.600i	− 0.379124i	−0.981869	−	0.189562i	$-0.939293\pi$
$479$	− 181.600i	− 0.379124i	0.981869	−	0.189562i	$-0.0607068\pi$
$480$	0	0
$481$	−271.003	−0.563415
$482$	0	0
$483$	5.02378i	0.0104012i
$484$	0	0
$485$	611.188	1.26018
$486$	0	0
$487$	455.016i	0.934324i	0.884172	+	0.467162i	$0.154724\pi$
$487$	455.016i	0.934324i	−0.884172	+	0.467162i	$0.845276\pi$
$488$	0	0
$489$	199.560	0.408099
$490$	0	0
$491$	662.583i	1.34946i	0.738066	+	0.674729i	$0.235739\pi$
$491$	662.583i	1.34946i	−0.738066	+	0.674729i	$0.764261\pi$
$492$	0	0
$493$	−62.5971	−0.126972
$494$	0	0
$495$	367.382i	0.742186i
$496$	0	0
$497$	219.381	0.441411
$498$	0	0
$499$	− 321.749i	− 0.644787i	−0.946606	−	0.322393i	$-0.895513\pi$
$499$	− 321.749i	− 0.644787i	0.946606	−	0.322393i	$-0.104487\pi$
$500$	0	0
$501$	17.6228	0.0351753
$502$	0	0
$503$	− 283.832i	− 0.564279i	−0.959373	−	0.282139i	$-0.908956\pi$
$503$	− 283.832i	− 0.564279i	0.959373	−	0.282139i	$-0.0910440\pi$
$504$	0	0
$505$	330.032	0.653529
$506$	0	0
$507$	− 47.6336i	− 0.0939518i
$508$	0	0
$509$	−599.831	−1.17845	−0.589225	−	0.807969i	$-0.700567\pi$
$509$	−599.831	−1.17845	−0.589225	+	0.807969i	$0.700567\pi$
$510$	0	0
$511$	274.119i	0.536436i
$512$	0	0
$513$	−57.7703	−0.112613
$514$	0	0
$515$	505.467i	0.981489i
$516$	0	0
$517$	295.722	0.571995
$518$	0	0
$519$	115.834i	0.223187i
$520$	0	0
$521$	−698.443	−1.34058	−0.670291	−	0.742099i	$-0.733831\pi$
$521$	−698.443	−1.34058	−0.670291	+	0.742099i	$0.733831\pi$
$522$	0	0
$523$	451.840i	0.863939i	0.901888	+	0.431969i	$0.142181\pi$
$523$	451.840i	0.863939i	−0.901888	+	0.431969i	$0.857819\pi$
$524$	0	0
$525$	−67.5372	−0.128642
$526$	0	0
$527$	− 46.7580i	− 0.0887249i
$528$	0	0
$529$	528.440	0.998942
$530$	0	0
$531$	− 46.9943i	− 0.0885016i
$532$	0	0
$533$	−227.559	−0.426939
$534$	0	0
$535$	− 230.694i	− 0.431204i
$536$	0	0
$537$	8.23510	0.0153354
$538$	0	0
$539$	326.287i	0.605356i
$540$	0	0
$541$	709.992	1.31237	0.656185	−	0.754600i	$-0.272169\pi$
$541$	709.992	1.31237	0.656185	+	0.754600i	$0.272169\pi$
$542$	0	0
$543$	− 28.7635i	− 0.0529714i
$544$	0	0
$545$	626.501	1.14954
$546$	0	0
$547$	− 642.779i	− 1.17510i	−0.809189	−	0.587549i	$-0.800093\pi$
$547$	− 642.779i	− 1.17510i	0.809189	−	0.587549i	$-0.199907\pi$
$548$	0	0
$549$	−776.270	−1.41397
$550$	0	0
$551$	− 114.103i	− 0.207083i
$552$	0	0
$553$	−708.282	−1.28080
$554$	0	0
$555$	− 77.2660i	− 0.139218i
$556$	0	0
$557$	1004.45	1.80331	0.901656	−	0.432453i	$-0.142352\pi$
$557$	1004.45	1.80331	0.901656	+	0.432453i	$0.142352\pi$
$558$	0	0
$559$	− 599.079i	− 1.07170i
$560$	0	0
$561$	20.5313	0.0365976
$562$	0	0
$563$	− 925.064i	− 1.64310i	−0.570138	−	0.821549i	$-0.693110\pi$
$563$	− 925.064i	− 1.64310i	0.570138	−	0.821549i	$-0.306890\pi$
$564$	0	0
$565$	315.890	0.559097
$566$	0	0
$567$	− 579.991i	− 1.02291i
$568$	0	0
$569$	−470.845	−0.827495	−0.413748	−	0.910392i	$-0.635780\pi$
$569$	−470.845	−0.827495	−0.413748	+	0.910392i	$0.635780\pi$
$570$	0	0
$571$	− 501.167i	− 0.877701i	−0.898560	−	0.438851i	$-0.855386\pi$
$571$	− 501.167i	− 0.877701i	0.898560	−	0.438851i	$-0.144614\pi$
$572$	0	0
$573$	−188.553	−0.329064
$574$	0	0
$575$	− 7.52446i	− 0.0130860i
$576$	0	0
$577$	179.702	0.311443	0.155721	−	0.987801i	$-0.450230\pi$
$577$	179.702	0.311443	0.155721	+	0.987801i	$0.450230\pi$
$578$	0	0
$579$	− 202.928i	− 0.350480i
$580$	0	0
$581$	321.793	0.553861
$582$	0	0
$583$	633.614i	1.08682i
$584$	0	0
$585$	335.761	0.573951
$586$	0	0
$587$	− 361.326i	− 0.615547i	−0.951460	−	0.307773i	$-0.900416\pi$
$587$	− 361.326i	− 0.615547i	0.951460	−	0.307773i	$-0.0995839\pi$
$588$	0	0
$589$	85.2311	0.144705
$590$	0	0
$591$	159.935i	0.270618i
$592$	0	0
$593$	−751.538	−1.26735	−0.633675	−	0.773600i	$-0.718454\pi$
$593$	−751.538	−1.26735	−0.633675	+	0.773600i	$0.718454\pi$
$594$	0	0
$595$	81.5916i	0.137129i
$596$	0	0
$597$	10.5764	0.0177159
$598$	0	0
$599$	− 1061.20i	− 1.77161i	−0.464056	−	0.885806i	$-0.653606\pi$
$599$	− 1061.20i	− 1.77161i	0.464056	−	0.885806i	$-0.346394\pi$
$600$	0	0
$601$	−1012.81	−1.68521	−0.842607	−	0.538529i	$-0.818980\pi$
$601$	−1012.81	−1.68521	−0.842607	+	0.538529i	$0.818980\pi$
$602$	0	0
$603$	25.3191i	0.0419886i
$604$	0	0
$605$	24.6195	0.0406934
$606$	0	0
$607$	− 1046.98i	− 1.72484i	−0.506195	−	0.862419i	$-0.668948\pi$
$607$	− 1046.98i	− 1.72484i	0.506195	−	0.862419i	$-0.331052\pi$
$608$	0	0
$609$	−175.780	−0.288637
$610$	0	0
$611$	− 270.269i	− 0.442338i
$612$	0	0
$613$	−826.300	−1.34796	−0.673981	−	0.738749i	$-0.735417\pi$
$613$	−826.300	−1.34796	−0.673981	+	0.738749i	$0.735417\pi$
$614$	0	0
$615$	− 64.8796i	− 0.105495i
$616$	0	0
$617$	−285.852	−0.463293	−0.231647	−	0.972800i	$-0.574411\pi$
$617$	−285.852	−0.463293	−0.231647	+	0.972800i	$0.574411\pi$
$618$	0	0
$619$	1149.76i	1.85745i	0.370772	+	0.928724i	$0.379093\pi$
$619$	1149.76i	1.85745i	−0.370772	+	0.928724i	$0.620907\pi$
$620$	0	0
$621$	−9.91539	−0.0159668
$622$	0	0
$623$	1435.25i	2.30377i
$624$	0	0
$625$	−272.405	−0.435848
$626$	0	0
$627$	37.4246i	0.0596884i
$628$	0	0
$629$	62.8295	0.0998880
$630$	0	0
$631$	− 249.014i	− 0.394634i	−0.980340	−	0.197317i	$-0.936777\pi$
$631$	− 249.014i	− 0.394634i	0.980340	−	0.197317i	$-0.0632229\pi$
$632$	0	0
$633$	90.2255	0.142536
$634$	0	0
$635$	− 449.640i	− 0.708094i
$636$	0	0
$637$	298.203	0.468137
$638$	0	0
$639$	209.303i	0.327547i
$640$	0	0
$641$	130.308	0.203289	0.101645	−	0.994821i	$-0.467590\pi$
$641$	130.308	0.203289	0.101645	+	0.994821i	$0.467590\pi$
$642$	0	0
$643$	− 902.547i	− 1.40365i	−0.712349	−	0.701825i	$-0.752369\pi$
$643$	− 902.547i	− 1.40365i	0.712349	−	0.701825i	$-0.247631\pi$
$644$	0	0
$645$	170.804	0.264813
$646$	0	0
$647$	− 826.308i	− 1.27714i	−0.769565	−	0.638569i	$-0.779527\pi$
$647$	− 826.308i	− 1.27714i	0.769565	−	0.638569i	$-0.220473\pi$
$648$	0	0
$649$	−62.9798	−0.0970413
$650$	0	0
$651$	− 131.302i	− 0.201693i
$652$	0	0
$653$	−116.405	−0.178262	−0.0891312	−	0.996020i	$-0.528409\pi$
$653$	−116.405	−0.178262	−0.0891312	+	0.996020i	$0.528409\pi$
$654$	0	0
$655$	973.923i	1.48690i
$656$	0	0
$657$	−261.526	−0.398060
$658$	0	0
$659$	− 203.861i	− 0.309349i	−0.987965	−	0.154674i	$-0.950567\pi$
$659$	− 203.861i	− 0.309349i	0.987965	−	0.154674i	$-0.0494329\pi$
$660$	0	0
$661$	231.009	0.349484	0.174742	−	0.984614i	$-0.444091\pi$
$661$	231.009	0.349484	0.174742	+	0.984614i	$0.444091\pi$
$662$	0	0
$663$	− 18.7641i	− 0.0283018i
$664$	0	0
$665$	−148.726	−0.223648
$666$	0	0
$667$	− 19.5840i	− 0.0293613i
$668$	0	0
$669$	11.4744	0.0171516
$670$	0	0
$671$	1040.32i	1.55041i
$672$	0	0
$673$	−850.912	−1.26436	−0.632179	−	0.774823i	$-0.717839\pi$
$673$	−850.912	−1.26436	−0.632179	+	0.774823i	$0.717839\pi$
$674$	0	0
$675$	− 133.298i	− 0.197478i
$676$	0	0
$677$	−560.684	−0.828189	−0.414095	−	0.910234i	$-0.635902\pi$
$677$	−560.684	−0.828189	−0.414095	+	0.910234i	$0.635902\pi$
$678$	0	0
$679$	1395.61i	2.05539i
$680$	0	0
$681$	−132.545	−0.194632
$682$	0	0
$683$	− 116.683i	− 0.170838i	−0.996345	−	0.0854191i	$-0.972777\pi$
$683$	− 116.683i	− 0.170838i	0.996345	−	0.0854191i	$-0.0272229\pi$
$684$	0	0
$685$	−182.593	−0.266559
$686$	0	0
$687$	− 115.943i	− 0.168767i
$688$	0	0
$689$	579.078	0.840462
$690$	0	0
$691$	515.820i	0.746483i	0.927734	+	0.373242i	$0.121754\pi$
$691$	515.820i	0.746483i	−0.927734	+	0.373242i	$0.878246\pi$
$692$	0	0
$693$	−838.896	−1.21053
$694$	0	0
$695$	373.681i	0.537670i
$696$	0	0
$697$	52.7574	0.0756921
$698$	0	0
$699$	292.392i	0.418300i
$700$	0	0
$701$	−675.831	−0.964095	−0.482048	−	0.876145i	$-0.660107\pi$
$701$	−675.831	−0.964095	−0.482048	+	0.876145i	$0.660107\pi$
$702$	0	0
$703$	114.526i	0.162911i
$704$	0	0
$705$	77.0567	0.109300
$706$	0	0
$707$	753.610i	1.06593i
$708$	0	0
$709$	−109.968	−0.155103	−0.0775517	−	0.996988i	$-0.524710\pi$
$709$	−109.968	−0.155103	−0.0775517	+	0.996988i	$0.524710\pi$
$710$	0	0
$711$	− 675.744i	− 0.950413i
$712$	0	0
$713$	14.6286	0.0205170
$714$	0	0
$715$	− 449.973i	− 0.629333i
$716$	0	0
$717$	−296.806	−0.413955
$718$	0	0
$719$	− 1090.82i	− 1.51713i	−0.651596	−	0.758566i	$-0.725900\pi$
$719$	− 1090.82i	− 1.51713i	0.651596	−	0.758566i	$-0.274100\pi$
$720$	0	0
$721$	−1154.20	−1.60084
$722$	0	0
$723$	− 345.046i	− 0.477243i
$724$	0	0
$725$	263.277	0.363141
$726$	0	0
$727$	1163.76i	1.60077i	0.599488	+	0.800383i	$0.295371\pi$
$727$	1163.76i	1.60077i	−0.599488	+	0.800383i	$0.704629\pi$
$728$	0	0
$729$	462.602	0.634571
$730$	0	0
$731$	138.891i	0.190002i
$732$	0	0
$733$	−657.756	−0.897347	−0.448674	−	0.893696i	$-0.648103\pi$
$733$	−657.756	−0.897347	−0.448674	+	0.893696i	$0.648103\pi$
$734$	0	0
$735$	85.0212i	0.115675i
$736$	0	0
$737$	33.9316	0.0460402
$738$	0	0
$739$	− 1091.82i	− 1.47744i	−0.674015	−	0.738718i	$-0.735432\pi$
$739$	− 1091.82i	− 1.47744i	0.674015	−	0.738718i	$-0.264568\pi$
$740$	0	0
$741$	34.2035	0.0461585
$742$	0	0
$743$	955.090i	1.28545i	0.766096	+	0.642726i	$0.222196\pi$
$743$	955.090i	1.28545i	−0.766096	+	0.642726i	$0.777804\pi$
$744$	0	0
$745$	−139.606	−0.187391
$746$	0	0
$747$	307.010i	0.410991i
$748$	0	0
$749$	526.776	0.703306
$750$	0	0
$751$	940.578i	1.25243i	0.779649	+	0.626217i	$0.215398\pi$
$751$	940.578i	1.25243i	−0.779649	+	0.626217i	$0.784602\pi$
$752$	0	0
$753$	−147.183	−0.195462
$754$	0	0
$755$	− 917.846i	− 1.21569i
$756$	0	0
$757$	177.412	0.234362	0.117181	−	0.993111i	$-0.462614\pi$
$757$	177.412	0.234362	0.117181	+	0.993111i	$0.462614\pi$
$758$	0	0
$759$	6.42336i	0.00846293i
$760$	0	0
$761$	515.326	0.677169	0.338585	−	0.940936i	$-0.390052\pi$
$761$	515.326	0.677169	0.338585	+	0.940936i	$0.390052\pi$
$762$	0	0
$763$	1430.58i	1.87494i
$764$	0	0
$765$	−77.8433	−0.101756
$766$	0	0
$767$	57.5591i	0.0750445i
$768$	0	0
$769$	−972.091	−1.26410	−0.632049	−	0.774929i	$-0.717786\pi$
$769$	−972.091	−1.26410	−0.632049	+	0.774929i	$0.717786\pi$
$770$	0	0
$771$	56.2293i	0.0729303i
$772$	0	0
$773$	407.151	0.526715	0.263357	−	0.964698i	$-0.415170\pi$
$773$	407.151	0.526715	0.263357	+	0.964698i	$0.415170\pi$
$774$	0	0
$775$	196.660i	0.253754i
$776$	0	0
$777$	176.432	0.227069
$778$	0	0
$779$	96.1669i	0.123449i
$780$	0	0
$781$	280.499	0.359153
$782$	0	0
$783$	− 346.935i	− 0.443084i
$784$	0	0
$785$	−510.362	−0.650142
$786$	0	0
$787$	333.101i	0.423254i	0.977350	+	0.211627i	$0.0678763\pi$
$787$	333.101i	0.423254i	−0.977350	+	0.211627i	$0.932124\pi$
$788$	0	0
$789$	69.5140	0.0881039
$790$	0	0
$791$	721.316i	0.911904i
$792$	0	0
$793$	950.783	1.19897
$794$	0	0
$795$	165.102i	0.207675i
$796$	0	0
$797$	652.883	0.819175	0.409588	−	0.912271i	$-0.365673\pi$
$797$	652.883	0.819175	0.409588	+	0.912271i	$0.365673\pi$
$798$	0	0
$799$	62.6594i	0.0784222i
$800$	0	0
$801$	−1369.31	−1.70950
$802$	0	0
$803$	350.486i	0.436470i
$804$	0	0
$805$	−25.5266	−0.0317100
$806$	0	0
$807$	− 84.3231i	− 0.104490i
$808$	0	0
$809$	−1517.67	−1.87598	−0.937990	−	0.346662i	$-0.887315\pi$
$809$	−1517.67	−1.87598	−0.937990	+	0.346662i	$0.887315\pi$
$810$	0	0
$811$	− 1542.04i	− 1.90140i	−0.310106	−	0.950702i	$-0.600365\pi$
$811$	− 1542.04i	− 1.90140i	0.310106	−	0.950702i	$-0.399635\pi$
$812$	0	0
$813$	−404.758	−0.497857
$814$	0	0
$815$	1013.99i	1.24416i
$816$	0	0
$817$	−253.173	−0.309881
$818$	0	0
$819$	766.692i	0.936131i
$820$	0	0
$821$	1456.93	1.77457	0.887287	−	0.461218i	$-0.152587\pi$
$821$	1456.93	1.77457	0.887287	+	0.461218i	$0.152587\pi$
$822$	0	0
$823$	− 1297.13i	− 1.57610i	−0.615609	−	0.788052i	$-0.711090\pi$
$823$	− 1297.13i	− 1.57610i	0.615609	−	0.788052i	$-0.288910\pi$
$824$	0	0
$825$	−86.3525	−0.104670
$826$	0	0
$827$	1222.01i	1.47764i	0.673902	+	0.738821i	$0.264617\pi$
$827$	1222.01i	1.47764i	−0.673902	+	0.738821i	$0.735383\pi$
$828$	0	0
$829$	−651.502	−0.785889	−0.392945	−	0.919562i	$-0.628544\pi$
$829$	−651.502	−0.785889	−0.392945	+	0.919562i	$0.628544\pi$
$830$	0	0
$831$	− 90.1848i	− 0.108526i
$832$	0	0
$833$	−69.1357	−0.0829961
$834$	0	0
$835$	89.5440i	0.107238i
$836$	0	0
$837$	259.149	0.309616
$838$	0	0
$839$	507.581i	0.604983i	0.953152	+	0.302491i	$0.0978184\pi$
$839$	507.581i	0.604983i	−0.953152	+	0.302491i	$0.902182\pi$
$840$	0	0
$841$	−155.765	−0.185214
$842$	0	0
$843$	− 92.6079i	− 0.109855i
$844$	0	0
$845$	242.033	0.286429
$846$	0	0
$847$	56.2172i	0.0663721i
$848$	0	0
$849$	153.238	0.180493
$850$	0	0
$851$	19.6567i	0.0230984i
$852$	0	0
$853$	−606.802	−0.711374	−0.355687	−	0.934605i	$-0.615753\pi$
$853$	−606.802	−0.711374	−0.355687	+	0.934605i	$0.615753\pi$
$854$	0	0
$855$	− 141.894i	− 0.165958i
$856$	0	0
$857$	111.500	0.130105	0.0650527	−	0.997882i	$-0.479278\pi$
$857$	111.500	0.130105	0.0650527	+	0.997882i	$0.479278\pi$
$858$	0	0
$859$	129.261i	0.150478i	0.997166	+	0.0752390i	$0.0239720\pi$
$859$	129.261i	0.150478i	−0.997166	+	0.0752390i	$0.976028\pi$
$860$	0	0
$861$	148.149	0.172066
$862$	0	0
$863$	− 1532.36i	− 1.77562i	−0.460214	−	0.887808i	$-0.652227\pi$
$863$	− 1532.36i	− 1.77562i	0.460214	−	0.887808i	$-0.347773\pi$
$864$	0	0
$865$	−588.568	−0.680426
$866$	0	0
$867$	− 215.510i	− 0.248570i
$868$	0	0
$869$	−905.603	−1.04212
$870$	0	0
$871$	− 31.0111i	− 0.0356040i
$872$	0	0
$873$	−1331.50	−1.52520
$874$	0	0
$875$	− 1196.17i	− 1.36705i
$876$	0	0
$877$	1151.28	1.31274	0.656372	−	0.754437i	$-0.272090\pi$
$877$	1151.28	1.31274	0.656372	+	0.754437i	$0.272090\pi$
$878$	0	0
$879$	353.288i	0.401921i
$880$	0	0
$881$	−782.531	−0.888230	−0.444115	−	0.895970i	$-0.646482\pi$
$881$	−782.531	−0.888230	−0.444115	+	0.895970i	$0.646482\pi$
$882$	0	0
$883$	− 364.109i	− 0.412355i	−0.978515	−	0.206177i	$-0.933898\pi$
$883$	− 364.109i	− 0.412355i	0.978515	−	0.206177i	$-0.0661024\pi$
$884$	0	0
$885$	−16.4108	−0.0185432
$886$	0	0
$887$	− 1209.99i	− 1.36413i	−0.731290	−	0.682066i	$-0.761082\pi$
$887$	− 1209.99i	− 1.36413i	0.731290	−	0.682066i	$-0.238918\pi$
$888$	0	0
$889$	1026.73	1.15492
$890$	0	0
$891$	− 741.572i	− 0.832291i
$892$	0	0
$893$	−114.216	−0.127902
$894$	0	0
$895$	41.8437i	0.0467528i
$896$	0	0
$897$	5.87050	0.00654459
$898$	0	0
$899$	511.848i	0.569353i
$900$	0	0
$901$	−134.254	−0.149006
$902$	0	0
$903$	390.022i	0.431918i
$904$	0	0
$905$	146.151	0.161493
$906$	0	0
$907$	− 634.641i	− 0.699715i	−0.936803	−	0.349857i	$-0.886230\pi$
$907$	− 634.641i	− 0.699715i	0.936803	−	0.349857i	$-0.113770\pi$
$908$	0	0
$909$	−718.989	−0.790967
$910$	0	0
$911$	− 444.113i	− 0.487500i	−0.969838	−	0.243750i	$-0.921622\pi$
$911$	− 444.113i	− 0.487500i	0.969838	−	0.243750i	$-0.0783776\pi$
$912$	0	0
$913$	411.442	0.450648
$914$	0	0
$915$	271.079i	0.296261i
$916$	0	0
$917$	−2223.90	−2.42519
$918$	0	0
$919$	− 156.620i	− 0.170425i	−0.996363	−	0.0852123i	$-0.972843\pi$
$919$	− 156.620i	− 0.170425i	0.996363	−	0.0852123i	$-0.0271568\pi$
$920$	0	0
$921$	275.044	0.298636
$922$	0	0
$923$	− 256.356i	− 0.277742i
$924$	0	0
$925$	−264.255	−0.285681
$926$	0	0
$927$	− 1101.18i	− 1.18790i
$928$	0	0
$929$	1369.54	1.47421	0.737106	−	0.675777i	$-0.236192\pi$
$929$	1369.54	1.47421	0.737106	+	0.675777i	$0.236192\pi$
$930$	0	0
$931$	− 126.022i	− 0.135361i
$932$	0	0
$933$	−41.7627	−0.0447617
$934$	0	0
$935$	104.322i	0.111575i
$936$	0	0
$937$	87.5411	0.0934270	0.0467135	−	0.998908i	$-0.485125\pi$
$937$	87.5411	0.0934270	0.0467135	+	0.998908i	$0.485125\pi$
$938$	0	0
$939$	111.546i	0.118792i
$940$	0	0
$941$	460.843	0.489738	0.244869	−	0.969556i	$-0.421255\pi$
$941$	460.843	0.489738	0.244869	+	0.969556i	$0.421255\pi$
$942$	0	0
$943$	16.5056i	0.0175033i
$944$	0	0
$945$	−452.209	−0.478528
$946$	0	0
$947$	− 1421.07i	− 1.50060i	−0.661096	−	0.750302i	$-0.729908\pi$
$947$	− 1421.07i	− 1.50060i	0.661096	−	0.750302i	$-0.270092\pi$
$948$	0	0
$949$	320.319	0.337533
$950$	0	0
$951$	− 245.966i	− 0.258640i
$952$	0	0
$953$	1079.05	1.13227	0.566134	−	0.824313i	$-0.308439\pi$
$953$	1079.05	1.13227	0.566134	+	0.824313i	$0.308439\pi$
$954$	0	0
$955$	− 958.067i	− 1.00321i
$956$	0	0
$957$	−224.750	−0.234849
$958$	0	0
$959$	− 416.941i	− 0.434766i
$960$	0	0
$961$	578.666	0.602150
$962$	0	0
$963$	502.576i	0.521886i
$964$	0	0
$965$	1031.10	1.06850
$966$	0	0
$967$	821.871i	0.849919i	0.905212	+	0.424959i	$0.139712\pi$
$967$	821.871i	0.849919i	−0.905212	+	0.424959i	$0.860288\pi$
$968$	0	0
$969$	−7.92977	−0.00818346
$970$	0	0
$971$	147.307i	0.151706i	0.997119	+	0.0758531i	$0.0241680\pi$
$971$	147.307i	0.151706i	−0.997119	+	0.0758531i	$0.975832\pi$
$972$	0	0
$973$	−853.279	−0.876956
$974$	0	0
$975$	78.9201i	0.0809436i
$976$	0	0
$977$	927.192	0.949020	0.474510	−	0.880250i	$-0.342625\pi$
$977$	927.192	0.949020	0.474510	+	0.880250i	$0.342625\pi$
$978$	0	0
$979$	1835.10i	1.87446i
$980$	0	0
$981$	−1364.86	−1.39129
$982$	0	0
$983$	− 81.6093i	− 0.0830206i	−0.999138	−	0.0415103i	$-0.986783\pi$
$983$	− 81.6093i	− 0.0830206i	0.999138	−	0.0415103i	$-0.0132169\pi$
$984$	0	0
$985$	−812.653	−0.825029
$986$	0	0
$987$	175.955i	0.178272i
$988$	0	0
$989$	−43.4532	−0.0439365
$990$	0	0
$991$	− 1540.57i	− 1.55456i	−0.629153	−	0.777281i	$-0.716598\pi$
$991$	− 1540.57i	− 1.55456i	0.629153	−	0.777281i	$-0.283402\pi$
$992$	0	0
$993$	−35.0332	−0.0352801
$994$	0	0
$995$	53.7400i	0.0540101i
$996$	0	0
$997$	−1222.11	−1.22579	−0.612895	−	0.790164i	$-0.709995\pi$
$997$	−1222.11	−1.22579	−0.612895	+	0.790164i	$0.709995\pi$
$998$	0	0
$999$	348.223i	0.348571i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.d.f.191.12		20
4.3	odd	2	inner	1216.3.d.f.191.9		20
8.3	odd	2		608.3.d.b.191.12	yes	20
8.5	even	2		608.3.d.b.191.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.d.b.191.9	✓	20	8.5	even	2
608.3.d.b.191.12	yes	20	8.3	odd	2
1216.3.d.f.191.9		20	4.3	odd	2	inner
1216.3.d.f.191.12		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.760762i q^{3} -3.86554 q^{5} -8.82674i q^{7} +8.42124 q^{9} -11.2858i q^{11} -10.3144 q^{13} -2.94076i q^{15} +2.39130 q^{17} +4.35890i q^{19} +6.71505 q^{21} +0.748138i q^{23} -10.0576 q^{25} +13.2534i q^{27} -26.1770 q^{29} -19.5534i q^{31} +8.58580 q^{33} +34.1201i q^{35} +26.2742 q^{37} -7.84682i q^{39} +22.0622 q^{41} +58.0818i q^{43} -32.5527 q^{45} +26.2030i q^{47} -28.9113 q^{49} +1.81921i q^{51} -56.1426 q^{53} +43.6257i q^{55} -3.31609 q^{57} -5.58045i q^{59} -92.1800 q^{61} -74.3321i q^{63} +39.8708 q^{65} +3.00658i q^{67} -0.569155 q^{69} +24.8541i q^{71} -31.0555 q^{73} -7.65144i q^{75} -99.6167 q^{77} -80.2428i q^{79} +65.7085 q^{81} +36.4566i q^{83} -9.24369 q^{85} -19.9145i q^{87} -162.602 q^{89} +91.0426i q^{91} +14.8755 q^{93} -16.8495i q^{95} -158.112 q^{97} -95.0403i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 68 q^{9} + 16 q^{13} + 8 q^{17} - 64 q^{21} + 196 q^{25} + 88 q^{29} - 184 q^{33} - 16 q^{37} - 16 q^{41} - 16 q^{45} + 52 q^{49} - 88 q^{53} + 208 q^{61} - 192 q^{65} - 248 q^{69} - 152 q^{73} - 312 q^{77}+ \cdots - 72 q^{97}+O(q^{100}) \) 20 * q - 68 * q^9 + 16 * q^13 + 8 * q^17 - 64 * q^21 + 196 * q^25 + 88 * q^29 - 184 * q^33 - 16 * q^37 - 16 * q^41 - 16 * q^45 + 52 * q^49 - 88 * q^53 + 208 * q^61 - 192 * q^65 - 248 * q^69 - 152 * q^73 - 312 * q^77 + 500 * q^81 + 488 * q^85 + 304 * q^89 + 240 * q^93 - 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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