LMFDB - Embedded newform 1170.2.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1170,2,Mod(1,1170)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1170.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1170, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,-2,0,0,-2,0,2,0,0,-2,0,0,2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1170.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{10} +5.65685 q^{11} -1.00000 q^{13} -2.82843 q^{14} +1.00000 q^{16} +4.82843 q^{17} -2.82843 q^{19} -1.00000 q^{20} -5.65685 q^{22} -8.48528 q^{23} +1.00000 q^{25} +1.00000 q^{26} +2.82843 q^{28} +3.17157 q^{29} +4.00000 q^{31} -1.00000 q^{32} -4.82843 q^{34} -2.82843 q^{35} -0.343146 q^{37} +2.82843 q^{38} +1.00000 q^{40} -3.65685 q^{41} -1.65685 q^{43} +5.65685 q^{44} +8.48528 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} +9.31371 q^{53} -5.65685 q^{55} -2.82843 q^{56} -3.17157 q^{58} +13.6569 q^{59} +6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -5.65685 q^{67} +4.82843 q^{68} +2.82843 q^{70} -5.65685 q^{71} +2.48528 q^{73} +0.343146 q^{74} -2.82843 q^{76} +16.0000 q^{77} +13.6569 q^{79} -1.00000 q^{80} +3.65685 q^{82} -17.6569 q^{83} -4.82843 q^{85} +1.65685 q^{86} -5.65685 q^{88} +4.34315 q^{89} -2.82843 q^{91} -8.48528 q^{92} -8.00000 q^{94} +2.82843 q^{95} -8.82843 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 2 q^{13} + 2 q^{16} + 4 q^{17} - 2 q^{20} + 2 q^{25} + 2 q^{26} + 12 q^{29} + 8 q^{31} - 2 q^{32} - 4 q^{34} - 12 q^{37} + 2 q^{40} + 4 q^{41} + 8 q^{43}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 2 * q^13 + 2 * q^16 + 4 * q^17 - 2 * q^20 + 2 * q^25 + 2 * q^26 + 12 * q^29 + 8 * q^31 - 2 * q^32 - 4 * q^34 - 12 * q^37 + 2 * q^40 + 4 * q^41 + 8 * q^43 + 16 * q^47 + 2 * q^49 - 2 * q^50 - 2 * q^52 - 4 * q^53 - 12 * q^58 + 16 * q^59 + 12 * q^61 - 8 * q^62 + 2 * q^64 + 2 * q^65 + 4 * q^68 - 12 * q^73 + 12 * q^74 + 32 * q^77 + 16 * q^79 - 2 * q^80 - 4 * q^82 - 24 * q^83 - 4 * q^85 - 8 * q^86 + 20 * q^89 - 16 * q^94 - 12 * q^97 - 2 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.82843	−0.755929
$15$	0	0
$16$	1.00000	0.250000
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−5.65685	−1.20605
$23$	−8.48528	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	2.82843	0.534522
$29$	3.17157	0.588946	0.294473	−	0.955660i	$-0.404856\pi$
$29$	3.17157	0.588946	0.294473	+	0.955660i	$0.404856\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.82843	−0.828068
$35$	−2.82843	−0.478091
$36$	0	0
$37$	−0.343146	−0.0564128	−0.0282064	−	0.999602i	$-0.508980\pi$
$37$	−0.343146	−0.0564128	−0.0282064	+	0.999602i	$0.508980\pi$
$38$	2.82843	0.458831
$39$	0	0
$40$	1.00000	0.158114
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	0	0
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	5.65685	0.852803
$45$	0	0
$46$	8.48528	1.25109
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−1.00000	−0.138675
$53$	9.31371	1.27934	0.639668	−	0.768651i	$-0.279072\pi$
$53$	9.31371	1.27934	0.639668	+	0.768651i	$0.279072\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	−2.82843	−0.377964
$57$	0	0
$58$	−3.17157	−0.416448
$59$	13.6569	1.77797	0.888985	−	0.457935i	$-0.151411\pi$
$59$	13.6569	1.77797	0.888985	+	0.457935i	$0.151411\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	4.82843	0.585533
$69$	0	0
$70$	2.82843	0.338062
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	2.48528	0.290880	0.145440	−	0.989367i	$-0.453540\pi$
$73$	2.48528	0.290880	0.145440	+	0.989367i	$0.453540\pi$
$74$	0.343146	0.0398899
$75$	0	0
$76$	−2.82843	−0.324443
$77$	16.0000	1.82337
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	3.65685	0.403832
$83$	−17.6569	−1.93809	−0.969046	−	0.246881i	$-0.920594\pi$
$83$	−17.6569	−1.93809	−0.969046	+	0.246881i	$0.920594\pi$
$84$	0	0
$85$	−4.82843	−0.523716
$86$	1.65685	0.178663
$87$	0	0
$88$	−5.65685	−0.603023
$89$	4.34315	0.460373	0.230186	−	0.973147i	$-0.426066\pi$
$89$	4.34315	0.460373	0.230186	+	0.973147i	$0.426066\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	−8.48528	−0.884652
$93$	0	0
$94$	−8.00000	−0.825137
$95$	2.82843	0.290191
$96$	0	0
$97$	−8.82843	−0.896391	−0.448195	−	0.893936i	$-0.647933\pi$
$97$	−8.82843	−0.896391	−0.448195	+	0.893936i	$0.647933\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	12.1421	1.20819	0.604094	−	0.796913i	$-0.293535\pi$
$101$	12.1421	1.20819	0.604094	+	0.796913i	$0.293535\pi$
$102$	0	0
$103$	9.65685	0.951518	0.475759	−	0.879576i	$-0.342173\pi$
$103$	9.65685	0.951518	0.475759	+	0.879576i	$0.342173\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−9.31371	−0.904627
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	3.17157	0.303782	0.151891	−	0.988397i	$-0.451464\pi$
$109$	3.17157	0.303782	0.151891	+	0.988397i	$0.451464\pi$
$110$	5.65685	0.539360
$111$	0	0
$112$	2.82843	0.267261
$113$	10.4853	0.986372	0.493186	−	0.869924i	$-0.335832\pi$
$113$	10.4853	0.986372	0.493186	+	0.869924i	$0.335832\pi$
$114$	0	0
$115$	8.48528	0.791257
$116$	3.17157	0.294473
$117$	0	0
$118$	−13.6569	−1.25722
$119$	13.6569	1.25192
$120$	0	0
$121$	21.0000	1.90909
$122$	−6.00000	−0.543214
$123$	0	0
$124$	4.00000	0.359211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.65685	−0.147022	−0.0735110	−	0.997294i	$-0.523420\pi$
$127$	−1.65685	−0.147022	−0.0735110	+	0.997294i	$0.523420\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	−22.1421	−1.93457	−0.967284	−	0.253697i	$-0.918353\pi$
$131$	−22.1421	−1.93457	−0.967284	+	0.253697i	$0.918353\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	5.65685	0.488678
$135$	0	0
$136$	−4.82843	−0.414034
$137$	−5.31371	−0.453981	−0.226990	−	0.973897i	$-0.572889\pi$
$137$	−5.31371	−0.453981	−0.226990	+	0.973897i	$0.572889\pi$
$138$	0	0
$139$	17.6569	1.49763	0.748817	−	0.662776i	$-0.230622\pi$
$139$	17.6569	1.49763	0.748817	+	0.662776i	$0.230622\pi$
$140$	−2.82843	−0.239046
$141$	0	0
$142$	5.65685	0.474713
$143$	−5.65685	−0.473050
$144$	0	0
$145$	−3.17157	−0.263385
$146$	−2.48528	−0.205683
$147$	0	0
$148$	−0.343146	−0.0282064
$149$	7.65685	0.627274	0.313637	−	0.949543i	$-0.398453\pi$
$149$	7.65685	0.627274	0.313637	+	0.949543i	$0.398453\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	2.82843	0.229416
$153$	0	0
$154$	−16.0000	−1.28932
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−17.3137	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$157$	−17.3137	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$158$	−13.6569	−1.08648
$159$	0	0
$160$	1.00000	0.0790569
$161$	−24.0000	−1.89146
$162$	0	0
$163$	−11.3137	−0.886158	−0.443079	−	0.896483i	$-0.646114\pi$
$163$	−11.3137	−0.886158	−0.443079	+	0.896483i	$0.646114\pi$
$164$	−3.65685	−0.285552
$165$	0	0
$166$	17.6569	1.37044
$167$	24.9706	1.93228	0.966140	−	0.258018i	$-0.0830694\pi$
$167$	24.9706	1.93228	0.966140	+	0.258018i	$0.0830694\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	4.82843	0.370323
$171$	0	0
$172$	−1.65685	−0.126334
$173$	−13.3137	−1.01222	−0.506111	−	0.862468i	$-0.668917\pi$
$173$	−13.3137	−1.01222	−0.506111	+	0.862468i	$0.668917\pi$
$174$	0	0
$175$	2.82843	0.213809
$176$	5.65685	0.426401
$177$	0	0
$178$	−4.34315	−0.325533
$179$	−24.4853	−1.83012	−0.915058	−	0.403322i	$-0.867855\pi$
$179$	−24.4853	−1.83012	−0.915058	+	0.403322i	$0.867855\pi$
$180$	0	0
$181$	3.65685	0.271812	0.135906	−	0.990722i	$-0.456606\pi$
$181$	3.65685	0.271812	0.135906	+	0.990722i	$0.456606\pi$
$182$	2.82843	0.209657
$183$	0	0
$184$	8.48528	0.625543
$185$	0.343146	0.0252286
$186$	0	0
$187$	27.3137	1.99738
$188$	8.00000	0.583460
$189$	0	0
$190$	−2.82843	−0.205196
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	0	0
$193$	−14.4853	−1.04267	−0.521337	−	0.853351i	$-0.674566\pi$
$193$	−14.4853	−1.04267	−0.521337	+	0.853351i	$0.674566\pi$
$194$	8.82843	0.633844
$195$	0	0
$196$	1.00000	0.0714286
$197$	−9.31371	−0.663574	−0.331787	−	0.943354i	$-0.607652\pi$
$197$	−9.31371	−0.663574	−0.331787	+	0.943354i	$0.607652\pi$
$198$	0	0
$199$	21.6569	1.53521	0.767607	−	0.640921i	$-0.221447\pi$
$199$	21.6569	1.53521	0.767607	+	0.640921i	$0.221447\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−12.1421	−0.854318
$203$	8.97056	0.629610
$204$	0	0
$205$	3.65685	0.255406
$206$	−9.65685	−0.672825
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−16.0000	−1.10674
$210$	0	0
$211$	23.3137	1.60498	0.802491	−	0.596664i	$-0.203508\pi$
$211$	23.3137	1.60498	0.802491	+	0.596664i	$0.203508\pi$
$212$	9.31371	0.639668
$213$	0	0
$214$	−4.00000	−0.273434
$215$	1.65685	0.112997
$216$	0	0
$217$	11.3137	0.768025
$218$	−3.17157	−0.214806
$219$	0	0
$220$	−5.65685	−0.381385
$221$	−4.82843	−0.324795
$222$	0	0
$223$	5.17157	0.346314	0.173157	−	0.984894i	$-0.444603\pi$
$223$	5.17157	0.346314	0.173157	+	0.984894i	$0.444603\pi$
$224$	−2.82843	−0.188982
$225$	0	0
$226$	−10.4853	−0.697471
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−24.1421	−1.59536	−0.797679	−	0.603083i	$-0.793939\pi$
$229$	−24.1421	−1.59536	−0.797679	+	0.603083i	$0.793939\pi$
$230$	−8.48528	−0.559503
$231$	0	0
$232$	−3.17157	−0.208224
$233$	−22.4853	−1.47306	−0.736530	−	0.676405i	$-0.763537\pi$
$233$	−22.4853	−1.47306	−0.736530	+	0.676405i	$0.763537\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	13.6569	0.888985
$237$	0	0
$238$	−13.6569	−0.885242
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−17.3137	−1.11527	−0.557637	−	0.830085i	$-0.688292\pi$
$241$	−17.3137	−1.11527	−0.557637	+	0.830085i	$0.688292\pi$
$242$	−21.0000	−1.34993
$243$	0	0
$244$	6.00000	0.384111
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	2.82843	0.179969
$248$	−4.00000	−0.254000
$249$	0	0
$250$	1.00000	0.0632456
$251$	−5.17157	−0.326427	−0.163213	−	0.986591i	$-0.552186\pi$
$251$	−5.17157	−0.326427	−0.163213	+	0.986591i	$0.552186\pi$
$252$	0	0
$253$	−48.0000	−3.01773
$254$	1.65685	0.103960
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.828427	−0.0516759	−0.0258379	−	0.999666i	$-0.508225\pi$
$257$	−0.828427	−0.0516759	−0.0258379	+	0.999666i	$0.508225\pi$
$258$	0	0
$259$	−0.970563	−0.0603078
$260$	1.00000	0.0620174
$261$	0	0
$262$	22.1421	1.36795
$263$	0.485281	0.0299237	0.0149619	−	0.999888i	$-0.495237\pi$
$263$	0.485281	0.0299237	0.0149619	+	0.999888i	$0.495237\pi$
$264$	0	0
$265$	−9.31371	−0.572137
$266$	8.00000	0.490511
$267$	0	0
$268$	−5.65685	−0.345547
$269$	−2.48528	−0.151530	−0.0757651	−	0.997126i	$-0.524140\pi$
$269$	−2.48528	−0.151530	−0.0757651	+	0.997126i	$0.524140\pi$
$270$	0	0
$271$	−15.3137	−0.930242	−0.465121	−	0.885247i	$-0.653989\pi$
$271$	−15.3137	−0.930242	−0.465121	+	0.885247i	$0.653989\pi$
$272$	4.82843	0.292766
$273$	0	0
$274$	5.31371	0.321013
$275$	5.65685	0.341121
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−17.6569	−1.05899
$279$	0	0
$280$	2.82843	0.169031
$281$	−19.6569	−1.17263	−0.586315	−	0.810083i	$-0.699422\pi$
$281$	−19.6569	−1.17263	−0.586315	+	0.810083i	$0.699422\pi$
$282$	0	0
$283$	−6.34315	−0.377061	−0.188530	−	0.982067i	$-0.560372\pi$
$283$	−6.34315	−0.377061	−0.188530	+	0.982067i	$0.560372\pi$
$284$	−5.65685	−0.335673
$285$	0	0
$286$	5.65685	0.334497
$287$	−10.3431	−0.610537
$288$	0	0
$289$	6.31371	0.371395
$290$	3.17157	0.186241
$291$	0	0
$292$	2.48528	0.145440
$293$	−28.6274	−1.67243	−0.836216	−	0.548401i	$-0.815237\pi$
$293$	−28.6274	−1.67243	−0.836216	+	0.548401i	$0.815237\pi$
$294$	0	0
$295$	−13.6569	−0.795133
$296$	0.343146	0.0199449
$297$	0	0
$298$	−7.65685	−0.443550
$299$	8.48528	0.490716
$300$	0	0
$301$	−4.68629	−0.270113
$302$	−12.0000	−0.690522
$303$	0	0
$304$	−2.82843	−0.162221
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−10.3431	−0.590315	−0.295157	−	0.955449i	$-0.595372\pi$
$307$	−10.3431	−0.590315	−0.295157	+	0.955449i	$0.595372\pi$
$308$	16.0000	0.911685
$309$	0	0
$310$	4.00000	0.227185
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	2.97056	0.167906	0.0839531	−	0.996470i	$-0.473245\pi$
$313$	2.97056	0.167906	0.0839531	+	0.996470i	$0.473245\pi$
$314$	17.3137	0.977069
$315$	0	0
$316$	13.6569	0.768258
$317$	−2.68629	−0.150877	−0.0754386	−	0.997150i	$-0.524036\pi$
$317$	−2.68629	−0.150877	−0.0754386	+	0.997150i	$0.524036\pi$
$318$	0	0
$319$	17.9411	1.00451
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	24.0000	1.33747
$323$	−13.6569	−0.759888
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	11.3137	0.626608
$327$	0	0
$328$	3.65685	0.201916
$329$	22.6274	1.24749
$330$	0	0
$331$	8.48528	0.466393	0.233197	−	0.972430i	$-0.425081\pi$
$331$	8.48528	0.466393	0.233197	+	0.972430i	$0.425081\pi$
$332$	−17.6569	−0.969046
$333$	0	0
$334$	−24.9706	−1.36633
$335$	5.65685	0.309067
$336$	0	0
$337$	−22.9706	−1.25129	−0.625643	−	0.780109i	$-0.715163\pi$
$337$	−22.9706	−1.25129	−0.625643	+	0.780109i	$0.715163\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−4.82843	−0.261858
$341$	22.6274	1.22534
$342$	0	0
$343$	−16.9706	−0.916324
$344$	1.65685	0.0893316
$345$	0	0
$346$	13.3137	0.715749
$347$	−1.65685	−0.0889446	−0.0444723	−	0.999011i	$-0.514161\pi$
$347$	−1.65685	−0.0889446	−0.0444723	+	0.999011i	$0.514161\pi$
$348$	0	0
$349$	−16.1421	−0.864069	−0.432034	−	0.901857i	$-0.642204\pi$
$349$	−16.1421	−0.864069	−0.432034	+	0.901857i	$0.642204\pi$
$350$	−2.82843	−0.151186
$351$	0	0
$352$	−5.65685	−0.301511
$353$	17.3137	0.921516	0.460758	−	0.887526i	$-0.347578\pi$
$353$	17.3137	0.921516	0.460758	+	0.887526i	$0.347578\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	4.34315	0.230186
$357$	0	0
$358$	24.4853	1.29409
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	−3.65685	−0.192200
$363$	0	0
$364$	−2.82843	−0.148250
$365$	−2.48528	−0.130086
$366$	0	0
$367$	14.3431	0.748706	0.374353	−	0.927286i	$-0.377865\pi$
$367$	14.3431	0.748706	0.374353	+	0.927286i	$0.377865\pi$
$368$	−8.48528	−0.442326
$369$	0	0
$370$	−0.343146	−0.0178393
$371$	26.3431	1.36767
$372$	0	0
$373$	−25.3137	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$373$	−25.3137	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$374$	−27.3137	−1.41236
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−3.17157	−0.163344
$378$	0	0
$379$	−24.4853	−1.25772	−0.628862	−	0.777517i	$-0.716479\pi$
$379$	−24.4853	−1.25772	−0.628862	+	0.777517i	$0.716479\pi$
$380$	2.82843	0.145095
$381$	0	0
$382$	−11.3137	−0.578860
$383$	18.3431	0.937291	0.468645	−	0.883386i	$-0.344742\pi$
$383$	18.3431	0.937291	0.468645	+	0.883386i	$0.344742\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	14.4853	0.737281
$387$	0	0
$388$	−8.82843	−0.448195
$389$	−10.4853	−0.531625	−0.265812	−	0.964025i	$-0.585640\pi$
$389$	−10.4853	−0.531625	−0.265812	+	0.964025i	$0.585640\pi$
$390$	0	0
$391$	−40.9706	−2.07197
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	9.31371	0.469218
$395$	−13.6569	−0.687151
$396$	0	0
$397$	−26.2843	−1.31917	−0.659585	−	0.751630i	$-0.729268\pi$
$397$	−26.2843	−1.31917	−0.659585	+	0.751630i	$0.729268\pi$
$398$	−21.6569	−1.08556
$399$	0	0
$400$	1.00000	0.0500000
$401$	−6.97056	−0.348093	−0.174047	−	0.984737i	$-0.555684\pi$
$401$	−6.97056	−0.348093	−0.174047	+	0.984737i	$0.555684\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	12.1421	0.604094
$405$	0	0
$406$	−8.97056	−0.445202
$407$	−1.94113	−0.0962180
$408$	0	0
$409$	7.65685	0.378607	0.189304	−	0.981919i	$-0.439377\pi$
$409$	7.65685	0.378607	0.189304	+	0.981919i	$0.439377\pi$
$410$	−3.65685	−0.180599
$411$	0	0
$412$	9.65685	0.475759
$413$	38.6274	1.90073
$414$	0	0
$415$	17.6569	0.866741
$416$	1.00000	0.0490290
$417$	0	0
$418$	16.0000	0.782586
$419$	5.17157	0.252648	0.126324	−	0.991989i	$-0.459682\pi$
$419$	5.17157	0.252648	0.126324	+	0.991989i	$0.459682\pi$
$420$	0	0
$421$	4.14214	0.201875	0.100938	−	0.994893i	$-0.467816\pi$
$421$	4.14214	0.201875	0.100938	+	0.994893i	$0.467816\pi$
$422$	−23.3137	−1.13489
$423$	0	0
$424$	−9.31371	−0.452314
$425$	4.82843	0.234213
$426$	0	0
$427$	16.9706	0.821263
$428$	4.00000	0.193347
$429$	0	0
$430$	−1.65685	−0.0799006
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	10.9706	0.527212	0.263606	−	0.964630i	$-0.415088\pi$
$433$	10.9706	0.527212	0.263606	+	0.964630i	$0.415088\pi$
$434$	−11.3137	−0.543075
$435$	0	0
$436$	3.17157	0.151891
$437$	24.0000	1.14808
$438$	0	0
$439$	−22.6274	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$440$	5.65685	0.269680
$441$	0	0
$442$	4.82843	0.229665
$443$	−41.6569	−1.97918	−0.989588	−	0.143926i	$-0.954027\pi$
$443$	−41.6569	−1.97918	−0.989588	+	0.143926i	$0.954027\pi$
$444$	0	0
$445$	−4.34315	−0.205885
$446$	−5.17157	−0.244881
$447$	0	0
$448$	2.82843	0.133631
$449$	30.2843	1.42920	0.714602	−	0.699532i	$-0.246608\pi$
$449$	30.2843	1.42920	0.714602	+	0.699532i	$0.246608\pi$
$450$	0	0
$451$	−20.6863	−0.974079
$452$	10.4853	0.493186
$453$	0	0
$454$	4.00000	0.187729
$455$	2.82843	0.132599
$456$	0	0
$457$	15.1716	0.709696	0.354848	−	0.934924i	$-0.384533\pi$
$457$	15.1716	0.709696	0.354848	+	0.934924i	$0.384533\pi$
$458$	24.1421	1.12809
$459$	0	0
$460$	8.48528	0.395628
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	35.7990	1.66372	0.831860	−	0.554985i	$-0.187276\pi$
$463$	35.7990	1.66372	0.831860	+	0.554985i	$0.187276\pi$
$464$	3.17157	0.147237
$465$	0	0
$466$	22.4853	1.04161
$467$	15.3137	0.708634	0.354317	−	0.935125i	$-0.384713\pi$
$467$	15.3137	0.708634	0.354317	+	0.935125i	$0.384713\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	8.00000	0.369012
$471$	0	0
$472$	−13.6569	−0.628608
$473$	−9.37258	−0.430952
$474$	0	0
$475$	−2.82843	−0.129777
$476$	13.6569	0.625961
$477$	0	0
$478$	16.0000	0.731823
$479$	−11.3137	−0.516937	−0.258468	−	0.966020i	$-0.583218\pi$
$479$	−11.3137	−0.516937	−0.258468	+	0.966020i	$0.583218\pi$
$480$	0	0
$481$	0.343146	0.0156461
$482$	17.3137	0.788618
$483$	0	0
$484$	21.0000	0.954545
$485$	8.82843	0.400878
$486$	0	0
$487$	−0.485281	−0.0219902	−0.0109951	−	0.999940i	$-0.503500\pi$
$487$	−0.485281	−0.0219902	−0.0109951	+	0.999940i	$0.503500\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	1.00000	0.0451754
$491$	9.85786	0.444879	0.222440	−	0.974946i	$-0.428598\pi$
$491$	9.85786	0.444879	0.222440	+	0.974946i	$0.428598\pi$
$492$	0	0
$493$	15.3137	0.689695
$494$	−2.82843	−0.127257
$495$	0	0
$496$	4.00000	0.179605
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−16.4853	−0.737983	−0.368991	−	0.929433i	$-0.620297\pi$
$499$	−16.4853	−0.737983	−0.368991	+	0.929433i	$0.620297\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	5.17157	0.230819
$503$	40.4853	1.80515	0.902575	−	0.430533i	$-0.141674\pi$
$503$	40.4853	1.80515	0.902575	+	0.430533i	$0.141674\pi$
$504$	0	0
$505$	−12.1421	−0.540318
$506$	48.0000	2.13386
$507$	0	0
$508$	−1.65685	−0.0735110
$509$	14.6863	0.650958	0.325479	−	0.945549i	$-0.394474\pi$
$509$	14.6863	0.650958	0.325479	+	0.945549i	$0.394474\pi$
$510$	0	0
$511$	7.02944	0.310964
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	0.828427	0.0365404
$515$	−9.65685	−0.425532
$516$	0	0
$517$	45.2548	1.99031
$518$	0.970563	0.0426441
$519$	0	0
$520$	−1.00000	−0.0438529
$521$	6.97056	0.305386	0.152693	−	0.988274i	$-0.451205\pi$
$521$	6.97056	0.305386	0.152693	+	0.988274i	$0.451205\pi$
$522$	0	0
$523$	34.6274	1.51415	0.757076	−	0.653327i	$-0.226627\pi$
$523$	34.6274	1.51415	0.757076	+	0.653327i	$0.226627\pi$
$524$	−22.1421	−0.967284
$525$	0	0
$526$	−0.485281	−0.0211593
$527$	19.3137	0.841318
$528$	0	0
$529$	49.0000	2.13043
$530$	9.31371	0.404562
$531$	0	0
$532$	−8.00000	−0.346844
$533$	3.65685	0.158396
$534$	0	0
$535$	−4.00000	−0.172935
$536$	5.65685	0.244339
$537$	0	0
$538$	2.48528	0.107148
$539$	5.65685	0.243658
$540$	0	0
$541$	−2.48528	−0.106851	−0.0534253	−	0.998572i	$-0.517014\pi$
$541$	−2.48528	−0.106851	−0.0534253	+	0.998572i	$0.517014\pi$
$542$	15.3137	0.657780
$543$	0	0
$544$	−4.82843	−0.207017
$545$	−3.17157	−0.135855
$546$	0	0
$547$	23.3137	0.996822	0.498411	−	0.866941i	$-0.333917\pi$
$547$	23.3137	0.996822	0.498411	+	0.866941i	$0.333917\pi$
$548$	−5.31371	−0.226990
$549$	0	0
$550$	−5.65685	−0.241209
$551$	−8.97056	−0.382159
$552$	0	0
$553$	38.6274	1.64260
$554$	−26.0000	−1.10463
$555$	0	0
$556$	17.6569	0.748817
$557$	−33.3137	−1.41155	−0.705774	−	0.708437i	$-0.749400\pi$
$557$	−33.3137	−1.41155	−0.705774	+	0.708437i	$0.749400\pi$
$558$	0	0
$559$	1.65685	0.0700775
$560$	−2.82843	−0.119523
$561$	0	0
$562$	19.6569	0.829174
$563$	41.6569	1.75563	0.877814	−	0.479002i	$-0.159002\pi$
$563$	41.6569	1.75563	0.877814	+	0.479002i	$0.159002\pi$
$564$	0	0
$565$	−10.4853	−0.441119
$566$	6.34315	0.266622
$567$	0	0
$568$	5.65685	0.237356
$569$	−20.3431	−0.852829	−0.426415	−	0.904528i	$-0.640224\pi$
$569$	−20.3431	−0.852829	−0.426415	+	0.904528i	$0.640224\pi$
$570$	0	0
$571$	−12.9706	−0.542801	−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706	−0.542801	−0.271401	+	0.962466i	$0.587487\pi$
$572$	−5.65685	−0.236525
$573$	0	0
$574$	10.3431	0.431715
$575$	−8.48528	−0.353861
$576$	0	0
$577$	27.4558	1.14300	0.571501	−	0.820601i	$-0.306361\pi$
$577$	27.4558	1.14300	0.571501	+	0.820601i	$0.306361\pi$
$578$	−6.31371	−0.262616
$579$	0	0
$580$	−3.17157	−0.131692
$581$	−49.9411	−2.07191
$582$	0	0
$583$	52.6863	2.18204
$584$	−2.48528	−0.102842
$585$	0	0
$586$	28.6274	1.18259
$587$	42.6274	1.75942	0.879711	−	0.475509i	$-0.157736\pi$
$587$	42.6274	1.75942	0.879711	+	0.475509i	$0.157736\pi$
$588$	0	0
$589$	−11.3137	−0.466173
$590$	13.6569	0.562244
$591$	0	0
$592$	−0.343146	−0.0141032
$593$	11.6569	0.478690	0.239345	−	0.970935i	$-0.423067\pi$
$593$	11.6569	0.478690	0.239345	+	0.970935i	$0.423067\pi$
$594$	0	0
$595$	−13.6569	−0.559876
$596$	7.65685	0.313637
$597$	0	0
$598$	−8.48528	−0.346989
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	6.68629	0.272740	0.136370	−	0.990658i	$-0.456456\pi$
$601$	6.68629	0.272740	0.136370	+	0.990658i	$0.456456\pi$
$602$	4.68629	0.190999
$603$	0	0
$604$	12.0000	0.488273
$605$	−21.0000	−0.853771
$606$	0	0
$607$	−4.97056	−0.201749	−0.100874	−	0.994899i	$-0.532164\pi$
$607$	−4.97056	−0.201749	−0.100874	+	0.994899i	$0.532164\pi$
$608$	2.82843	0.114708
$609$	0	0
$610$	6.00000	0.242933
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−34.2843	−1.38473	−0.692364	−	0.721548i	$-0.743431\pi$
$613$	−34.2843	−1.38473	−0.692364	+	0.721548i	$0.743431\pi$
$614$	10.3431	0.417415
$615$	0	0
$616$	−16.0000	−0.644658
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	−29.1716	−1.17250	−0.586252	−	0.810129i	$-0.699397\pi$
$619$	−29.1716	−1.17250	−0.586252	+	0.810129i	$0.699397\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−24.0000	−0.962312
$623$	12.2843	0.492159
$624$	0	0
$625$	1.00000	0.0400000
$626$	−2.97056	−0.118728
$627$	0	0
$628$	−17.3137	−0.690892
$629$	−1.65685	−0.0660631
$630$	0	0
$631$	−22.3431	−0.889467	−0.444733	−	0.895663i	$-0.646702\pi$
$631$	−22.3431	−0.889467	−0.444733	+	0.895663i	$0.646702\pi$
$632$	−13.6569	−0.543240
$633$	0	0
$634$	2.68629	0.106686
$635$	1.65685	0.0657503
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	−17.9411	−0.710296
$639$	0	0
$640$	1.00000	0.0395285
$641$	−40.6274	−1.60469	−0.802343	−	0.596863i	$-0.796414\pi$
$641$	−40.6274	−1.60469	−0.802343	+	0.596863i	$0.796414\pi$
$642$	0	0
$643$	−39.5980	−1.56159	−0.780796	−	0.624786i	$-0.785186\pi$
$643$	−39.5980	−1.56159	−0.780796	+	0.624786i	$0.785186\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	13.6569	0.537322
$647$	8.48528	0.333591	0.166795	−	0.985992i	$-0.446658\pi$
$647$	8.48528	0.333591	0.166795	+	0.985992i	$0.446658\pi$
$648$	0	0
$649$	77.2548	3.03252
$650$	1.00000	0.0392232
$651$	0	0
$652$	−11.3137	−0.443079
$653$	−14.2843	−0.558987	−0.279493	−	0.960148i	$-0.590166\pi$
$653$	−14.2843	−0.558987	−0.279493	+	0.960148i	$0.590166\pi$
$654$	0	0
$655$	22.1421	0.865165
$656$	−3.65685	−0.142776
$657$	0	0
$658$	−22.6274	−0.882109
$659$	24.4853	0.953811	0.476906	−	0.878955i	$-0.341758\pi$
$659$	24.4853	0.953811	0.476906	+	0.878955i	$0.341758\pi$
$660$	0	0
$661$	20.1421	0.783438	0.391719	−	0.920085i	$-0.371880\pi$
$661$	20.1421	0.783438	0.391719	+	0.920085i	$0.371880\pi$
$662$	−8.48528	−0.329790
$663$	0	0
$664$	17.6569	0.685219
$665$	8.00000	0.310227
$666$	0	0
$667$	−26.9117	−1.04202
$668$	24.9706	0.966140
$669$	0	0
$670$	−5.65685	−0.218543
$671$	33.9411	1.31028
$672$	0	0
$673$	−12.6274	−0.486751	−0.243376	−	0.969932i	$-0.578255\pi$
$673$	−12.6274	−0.486751	−0.243376	+	0.969932i	$0.578255\pi$
$674$	22.9706	0.884793
$675$	0	0
$676$	1.00000	0.0384615
$677$	−23.6569	−0.909207	−0.454603	−	0.890694i	$-0.650219\pi$
$677$	−23.6569	−0.909207	−0.454603	+	0.890694i	$0.650219\pi$
$678$	0	0
$679$	−24.9706	−0.958282
$680$	4.82843	0.185162
$681$	0	0
$682$	−22.6274	−0.866449
$683$	−22.3431	−0.854937	−0.427468	−	0.904030i	$-0.640594\pi$
$683$	−22.3431	−0.854937	−0.427468	+	0.904030i	$0.640594\pi$
$684$	0	0
$685$	5.31371	0.203026
$686$	16.9706	0.647939
$687$	0	0
$688$	−1.65685	−0.0631670
$689$	−9.31371	−0.354824
$690$	0	0
$691$	11.7990	0.448855	0.224427	−	0.974491i	$-0.427949\pi$
$691$	11.7990	0.448855	0.224427	+	0.974491i	$0.427949\pi$
$692$	−13.3137	−0.506111
$693$	0	0
$694$	1.65685	0.0628933
$695$	−17.6569	−0.669763
$696$	0	0
$697$	−17.6569	−0.668801
$698$	16.1421	0.610989
$699$	0	0
$700$	2.82843	0.106904
$701$	28.1421	1.06291	0.531457	−	0.847085i	$-0.321645\pi$
$701$	28.1421	1.06291	0.531457	+	0.847085i	$0.321645\pi$
$702$	0	0
$703$	0.970563	0.0366055
$704$	5.65685	0.213201
$705$	0	0
$706$	−17.3137	−0.651610
$707$	34.3431	1.29161
$708$	0	0
$709$	−12.8284	−0.481782	−0.240891	−	0.970552i	$-0.577440\pi$
$709$	−12.8284	−0.481782	−0.240891	+	0.970552i	$0.577440\pi$
$710$	−5.65685	−0.212298
$711$	0	0
$712$	−4.34315	−0.162766
$713$	−33.9411	−1.27111
$714$	0	0
$715$	5.65685	0.211554
$716$	−24.4853	−0.915058
$717$	0	0
$718$	28.2843	1.05556
$719$	18.3431	0.684084	0.342042	−	0.939685i	$-0.388882\pi$
$719$	18.3431	0.684084	0.342042	+	0.939685i	$0.388882\pi$
$720$	0	0
$721$	27.3137	1.01722
$722$	11.0000	0.409378
$723$	0	0
$724$	3.65685	0.135906
$725$	3.17157	0.117789
$726$	0	0
$727$	21.9411	0.813751	0.406876	−	0.913484i	$-0.366618\pi$
$727$	21.9411	0.813751	0.406876	+	0.913484i	$0.366618\pi$
$728$	2.82843	0.104828
$729$	0	0
$730$	2.48528	0.0919844
$731$	−8.00000	−0.295891
$732$	0	0
$733$	−11.6569	−0.430556	−0.215278	−	0.976553i	$-0.569066\pi$
$733$	−11.6569	−0.430556	−0.215278	+	0.976553i	$0.569066\pi$
$734$	−14.3431	−0.529415
$735$	0	0
$736$	8.48528	0.312772
$737$	−32.0000	−1.17874
$738$	0	0
$739$	14.1421	0.520227	0.260113	−	0.965578i	$-0.416240\pi$
$739$	14.1421	0.520227	0.260113	+	0.965578i	$0.416240\pi$
$740$	0.343146	0.0126143
$741$	0	0
$742$	−26.3431	−0.967087
$743$	−20.2843	−0.744158	−0.372079	−	0.928201i	$-0.621355\pi$
$743$	−20.2843	−0.744158	−0.372079	+	0.928201i	$0.621355\pi$
$744$	0	0
$745$	−7.65685	−0.280525
$746$	25.3137	0.926801
$747$	0	0
$748$	27.3137	0.998688
$749$	11.3137	0.413394
$750$	0	0
$751$	−11.3137	−0.412843	−0.206422	−	0.978463i	$-0.566182\pi$
$751$	−11.3137	−0.412843	−0.206422	+	0.978463i	$0.566182\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	3.17157	0.115502
$755$	−12.0000	−0.436725
$756$	0	0
$757$	−47.9411	−1.74245	−0.871225	−	0.490884i	$-0.836674\pi$
$757$	−47.9411	−1.74245	−0.871225	+	0.490884i	$0.836674\pi$
$758$	24.4853	0.889345
$759$	0	0
$760$	−2.82843	−0.102598
$761$	−16.3431	−0.592439	−0.296219	−	0.955120i	$-0.595726\pi$
$761$	−16.3431	−0.592439	−0.296219	+	0.955120i	$0.595726\pi$
$762$	0	0
$763$	8.97056	0.324756
$764$	11.3137	0.409316
$765$	0	0
$766$	−18.3431	−0.662765
$767$	−13.6569	−0.493120
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	16.0000	0.576600
$771$	0	0
$772$	−14.4853	−0.521337
$773$	30.6863	1.10371	0.551855	−	0.833940i	$-0.313920\pi$
$773$	30.6863	1.10371	0.551855	+	0.833940i	$0.313920\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	8.82843	0.316922
$777$	0	0
$778$	10.4853	0.375916
$779$	10.3431	0.370582
$780$	0	0
$781$	−32.0000	−1.14505
$782$	40.9706	1.46510
$783$	0	0
$784$	1.00000	0.0357143
$785$	17.3137	0.617953
$786$	0	0
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	−9.31371	−0.331787
$789$	0	0
$790$	13.6569	0.485889
$791$	29.6569	1.05448
$792$	0	0
$793$	−6.00000	−0.213066
$794$	26.2843	0.932794
$795$	0	0
$796$	21.6569	0.767607
$797$	28.6274	1.01404	0.507018	−	0.861936i	$-0.330748\pi$
$797$	28.6274	1.01404	0.507018	+	0.861936i	$0.330748\pi$
$798$	0	0
$799$	38.6274	1.36654
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	6.97056	0.246139
$803$	14.0589	0.496127
$804$	0	0
$805$	24.0000	0.845889
$806$	4.00000	0.140894
$807$	0	0
$808$	−12.1421	−0.427159
$809$	9.31371	0.327453	0.163726	−	0.986506i	$-0.447649\pi$
$809$	9.31371	0.327453	0.163726	+	0.986506i	$0.447649\pi$
$810$	0	0
$811$	−30.1421	−1.05843	−0.529217	−	0.848487i	$-0.677514\pi$
$811$	−30.1421	−1.05843	−0.529217	+	0.848487i	$0.677514\pi$
$812$	8.97056	0.314805
$813$	0	0
$814$	1.94113	0.0680364
$815$	11.3137	0.396302
$816$	0	0
$817$	4.68629	0.163953
$818$	−7.65685	−0.267716
$819$	0	0
$820$	3.65685	0.127703
$821$	22.2843	0.777726	0.388863	−	0.921296i	$-0.372868\pi$
$821$	22.2843	0.777726	0.388863	+	0.921296i	$0.372868\pi$
$822$	0	0
$823$	−19.0294	−0.663324	−0.331662	−	0.943398i	$-0.607609\pi$
$823$	−19.0294	−0.663324	−0.331662	+	0.943398i	$0.607609\pi$
$824$	−9.65685	−0.336412
$825$	0	0
$826$	−38.6274	−1.34402
$827$	1.65685	0.0576145	0.0288072	−	0.999585i	$-0.490829\pi$
$827$	1.65685	0.0576145	0.0288072	+	0.999585i	$0.490829\pi$
$828$	0	0
$829$	−30.6863	−1.06578	−0.532889	−	0.846185i	$-0.678894\pi$
$829$	−30.6863	−1.06578	−0.532889	+	0.846185i	$0.678894\pi$
$830$	−17.6569	−0.612878
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	4.82843	0.167295
$834$	0	0
$835$	−24.9706	−0.864142
$836$	−16.0000	−0.553372
$837$	0	0
$838$	−5.17157	−0.178649
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−18.9411	−0.653142
$842$	−4.14214	−0.142747
$843$	0	0
$844$	23.3137	0.802491
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	59.3970	2.04090
$848$	9.31371	0.319834
$849$	0	0
$850$	−4.82843	−0.165614
$851$	2.91169	0.0998114
$852$	0	0
$853$	−18.2843	−0.626042	−0.313021	−	0.949746i	$-0.601341\pi$
$853$	−18.2843	−0.626042	−0.313021	+	0.949746i	$0.601341\pi$
$854$	−16.9706	−0.580721
$855$	0	0
$856$	−4.00000	−0.136717
$857$	15.1716	0.518251	0.259126	−	0.965844i	$-0.416566\pi$
$857$	15.1716	0.518251	0.259126	+	0.965844i	$0.416566\pi$
$858$	0	0
$859$	−29.9411	−1.02158	−0.510789	−	0.859706i	$-0.670647\pi$
$859$	−29.9411	−1.02158	−0.510789	+	0.859706i	$0.670647\pi$
$860$	1.65685	0.0564983
$861$	0	0
$862$	−16.0000	−0.544962
$863$	−28.2843	−0.962808	−0.481404	−	0.876499i	$-0.659873\pi$
$863$	−28.2843	−0.962808	−0.481404	+	0.876499i	$0.659873\pi$
$864$	0	0
$865$	13.3137	0.452680
$866$	−10.9706	−0.372795
$867$	0	0
$868$	11.3137	0.384012
$869$	77.2548	2.62069
$870$	0	0
$871$	5.65685	0.191675
$872$	−3.17157	−0.107403
$873$	0	0
$874$	−24.0000	−0.811812
$875$	−2.82843	−0.0956183
$876$	0	0
$877$	39.2548	1.32554	0.662771	−	0.748822i	$-0.269380\pi$
$877$	39.2548	1.32554	0.662771	+	0.748822i	$0.269380\pi$
$878$	22.6274	0.763638
$879$	0	0
$880$	−5.65685	−0.190693
$881$	−46.2843	−1.55936	−0.779678	−	0.626180i	$-0.784617\pi$
$881$	−46.2843	−1.55936	−0.779678	+	0.626180i	$0.784617\pi$
$882$	0	0
$883$	8.68629	0.292317	0.146158	−	0.989261i	$-0.453309\pi$
$883$	8.68629	0.292317	0.146158	+	0.989261i	$0.453309\pi$
$884$	−4.82843	−0.162398
$885$	0	0
$886$	41.6569	1.39949
$887$	23.5147	0.789547	0.394773	−	0.918778i	$-0.370823\pi$
$887$	23.5147	0.789547	0.394773	+	0.918778i	$0.370823\pi$
$888$	0	0
$889$	−4.68629	−0.157173
$890$	4.34315	0.145583
$891$	0	0
$892$	5.17157	0.173157
$893$	−22.6274	−0.757198
$894$	0	0
$895$	24.4853	0.818453
$896$	−2.82843	−0.0944911
$897$	0	0
$898$	−30.2843	−1.01060
$899$	12.6863	0.423112
$900$	0	0
$901$	44.9706	1.49819
$902$	20.6863	0.688778
$903$	0	0
$904$	−10.4853	−0.348735
$905$	−3.65685	−0.121558
$906$	0	0
$907$	−48.2843	−1.60325	−0.801626	−	0.597825i	$-0.796032\pi$
$907$	−48.2843	−1.60325	−0.801626	+	0.597825i	$0.796032\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	−2.82843	−0.0937614
$911$	8.97056	0.297208	0.148604	−	0.988897i	$-0.452522\pi$
$911$	8.97056	0.297208	0.148604	+	0.988897i	$0.452522\pi$
$912$	0	0
$913$	−99.8823	−3.30562
$914$	−15.1716	−0.501831
$915$	0	0
$916$	−24.1421	−0.797679
$917$	−62.6274	−2.06814
$918$	0	0
$919$	−25.9411	−0.855719	−0.427859	−	0.903845i	$-0.640732\pi$
$919$	−25.9411	−0.855719	−0.427859	+	0.903845i	$0.640732\pi$
$920$	−8.48528	−0.279751
$921$	0	0
$922$	14.0000	0.461065
$923$	5.65685	0.186198
$924$	0	0
$925$	−0.343146	−0.0112826
$926$	−35.7990	−1.17643
$927$	0	0
$928$	−3.17157	−0.104112
$929$	−45.5980	−1.49602	−0.748011	−	0.663687i	$-0.768991\pi$
$929$	−45.5980	−1.49602	−0.748011	+	0.663687i	$0.768991\pi$
$930$	0	0
$931$	−2.82843	−0.0926980
$932$	−22.4853	−0.736530
$933$	0	0
$934$	−15.3137	−0.501080
$935$	−27.3137	−0.893254
$936$	0	0
$937$	−28.6274	−0.935217	−0.467608	−	0.883936i	$-0.654884\pi$
$937$	−28.6274	−0.935217	−0.467608	+	0.883936i	$0.654884\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	−8.00000	−0.260931
$941$	−21.0294	−0.685540	−0.342770	−	0.939419i	$-0.611365\pi$
$941$	−21.0294	−0.685540	−0.342770	+	0.939419i	$0.611365\pi$
$942$	0	0
$943$	31.0294	1.01046
$944$	13.6569	0.444493
$945$	0	0
$946$	9.37258	0.304729
$947$	−41.6569	−1.35367	−0.676833	−	0.736137i	$-0.736648\pi$
$947$	−41.6569	−1.35367	−0.676833	+	0.736137i	$0.736648\pi$
$948$	0	0
$949$	−2.48528	−0.0806756
$950$	2.82843	0.0917663
$951$	0	0
$952$	−13.6569	−0.442621
$953$	56.1421	1.81862	0.909311	−	0.416117i	$-0.136609\pi$
$953$	56.1421	1.81862	0.909311	+	0.416117i	$0.136609\pi$
$954$	0	0
$955$	−11.3137	−0.366103
$956$	−16.0000	−0.517477
$957$	0	0
$958$	11.3137	0.365529
$959$	−15.0294	−0.485326
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−0.343146	−0.0110635
$963$	0	0
$964$	−17.3137	−0.557637
$965$	14.4853	0.466298
$966$	0	0
$967$	−24.4853	−0.787394	−0.393697	−	0.919240i	$-0.628804\pi$
$967$	−24.4853	−0.787394	−0.393697	+	0.919240i	$0.628804\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	−8.82843	−0.283464
$971$	−32.4853	−1.04250	−0.521251	−	0.853403i	$-0.674535\pi$
$971$	−32.4853	−1.04250	−0.521251	+	0.853403i	$0.674535\pi$
$972$	0	0
$973$	49.9411	1.60104
$974$	0.485281	0.0155494
$975$	0	0
$976$	6.00000	0.192055
$977$	19.6569	0.628878	0.314439	−	0.949278i	$-0.398184\pi$
$977$	19.6569	0.628878	0.314439	+	0.949278i	$0.398184\pi$
$978$	0	0
$979$	24.5685	0.785214
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−9.85786	−0.314577
$983$	13.6569	0.435586	0.217793	−	0.975995i	$-0.430114\pi$
$983$	13.6569	0.435586	0.217793	+	0.975995i	$0.430114\pi$
$984$	0	0
$985$	9.31371	0.296759
$986$	−15.3137	−0.487688
$987$	0	0
$988$	2.82843	0.0899843
$989$	14.0589	0.447046
$990$	0	0
$991$	58.9117	1.87139	0.935696	−	0.352808i	$-0.114773\pi$
$991$	58.9117	1.87139	0.935696	+	0.352808i	$0.114773\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	16.0000	0.507489
$995$	−21.6569	−0.686568
$996$	0	0
$997$	38.6863	1.22521	0.612604	−	0.790390i	$-0.290122\pi$
$997$	38.6863	1.22521	0.612604	+	0.790390i	$0.290122\pi$
$998$	16.4853	0.521832
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.a.o.1.2		2
3.2	odd	2		390.2.a.h.1.2	✓	2
4.3	odd	2		9360.2.a.ch.1.1		2
5.2	odd	4		5850.2.e.bk.5149.2		4
5.3	odd	4		5850.2.e.bk.5149.3		4
5.4	even	2		5850.2.a.cl.1.1		2
12.11	even	2		3120.2.a.bc.1.1		2
15.2	even	4		1950.2.e.o.1249.4		4
15.8	even	4		1950.2.e.o.1249.1		4
15.14	odd	2		1950.2.a.bd.1.1		2
39.5	even	4		5070.2.b.q.1351.2		4
39.8	even	4		5070.2.b.q.1351.3		4
39.38	odd	2		5070.2.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.h.1.2	✓	2	3.2	odd	2
1170.2.a.o.1.2		2	1.1	even	1	trivial
1950.2.a.bd.1.1		2	15.14	odd	2
1950.2.e.o.1249.1		4	15.8	even	4
1950.2.e.o.1249.4		4	15.2	even	4
3120.2.a.bc.1.1		2	12.11	even	2
5070.2.a.bc.1.1		2	39.38	odd	2
5070.2.b.q.1351.2		4	39.5	even	4
5070.2.b.q.1351.3		4	39.8	even	4
5850.2.a.cl.1.1		2	5.4	even	2
5850.2.e.bk.5149.2		4	5.2	odd	4
5850.2.e.bk.5149.3		4	5.3	odd	4
9360.2.a.ch.1.1		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{10} +5.65685 q^{11} -1.00000 q^{13} -2.82843 q^{14} +1.00000 q^{16} +4.82843 q^{17} -2.82843 q^{19} -1.00000 q^{20} -5.65685 q^{22} -8.48528 q^{23} +1.00000 q^{25} +1.00000 q^{26} +2.82843 q^{28} +3.17157 q^{29} +4.00000 q^{31} -1.00000 q^{32} -4.82843 q^{34} -2.82843 q^{35} -0.343146 q^{37} +2.82843 q^{38} +1.00000 q^{40} -3.65685 q^{41} -1.65685 q^{43} +5.65685 q^{44} +8.48528 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} +9.31371 q^{53} -5.65685 q^{55} -2.82843 q^{56} -3.17157 q^{58} +13.6569 q^{59} +6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -5.65685 q^{67} +4.82843 q^{68} +2.82843 q^{70} -5.65685 q^{71} +2.48528 q^{73} +0.343146 q^{74} -2.82843 q^{76} +16.0000 q^{77} +13.6569 q^{79} -1.00000 q^{80} +3.65685 q^{82} -17.6569 q^{83} -4.82843 q^{85} +1.65685 q^{86} -5.65685 q^{88} +4.34315 q^{89} -2.82843 q^{91} -8.48528 q^{92} -8.00000 q^{94} +2.82843 q^{95} -8.82843 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 2 q^{13} + 2 q^{16} + 4 q^{17} - 2 q^{20} + 2 q^{25} + 2 q^{26} + 12 q^{29} + 8 q^{31} - 2 q^{32} - 4 q^{34} - 12 q^{37} + 2 q^{40} + 4 q^{41} + 8 q^{43}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 2 * q^13 + 2 * q^16 + 4 * q^17 - 2 * q^20 + 2 * q^25 + 2 * q^26 + 12 * q^29 + 8 * q^31 - 2 * q^32 - 4 * q^34 - 12 * q^37 + 2 * q^40 + 4 * q^41 + 8 * q^43 + 16 * q^47 + 2 * q^49 - 2 * q^50 - 2 * q^52 - 4 * q^53 - 12 * q^58 + 16 * q^59 + 12 * q^61 - 8 * q^62 + 2 * q^64 + 2 * q^65 + 4 * q^68 - 12 * q^73 + 12 * q^74 + 32 * q^77 + 16 * q^79 - 2 * q^80 - 4 * q^82 - 24 * q^83 - 4 * q^85 - 8 * q^86 + 20 * q^89 - 16 * q^94 - 12 * q^97 - 2 * q^98

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