LMFDB - Embedded newform 3120.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.a (trivial)

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:	yes
Analytic conductor:	$24.9133254306$
Analytic rank:	$1$
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			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +5.65685 q^{11} -1.00000 q^{13} -1.00000 q^{15} -4.82843 q^{17} +2.82843 q^{19} +2.82843 q^{21} -8.48528 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.17157 q^{29} -4.00000 q^{31} -5.65685 q^{33} -2.82843 q^{35} -0.343146 q^{37} +1.00000 q^{39} +3.65685 q^{41} +1.65685 q^{43} +1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} +4.82843 q^{51} -9.31371 q^{53} +5.65685 q^{55} -2.82843 q^{57} +13.6569 q^{59} +6.00000 q^{61} -2.82843 q^{63} -1.00000 q^{65} +5.65685 q^{67} +8.48528 q^{69} -5.65685 q^{71} +2.48528 q^{73} -1.00000 q^{75} -16.0000 q^{77} -13.6569 q^{79} +1.00000 q^{81} -17.6569 q^{83} -4.82843 q^{85} +3.17157 q^{87} -4.34315 q^{89} +2.82843 q^{91} +4.00000 q^{93} +2.82843 q^{95} -8.82843 q^{97} +5.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 2 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{25} - 2 q^{27} - 12 q^{29} - 8 q^{31} - 12 q^{37} + 2 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 16 q^{47} + 2 q^{49} + 4 q^{51} + 4 q^{53} + 16 q^{59} + 12 q^{61} - 2 q^{65} - 12 q^{73} - 2 q^{75} - 32 q^{77} - 16 q^{79} + 2 q^{81} - 24 q^{83} - 4 q^{85} + 12 q^{87} - 20 q^{89} + 8 q^{93} - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 - 2 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^25 - 2 * q^27 - 12 * q^29 - 8 * q^31 - 12 * q^37 + 2 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 + 16 * q^47 + 2 * q^49 + 4 * q^51 + 4 * q^53 + 16 * q^59 + 12 * q^61 - 2 * q^65 - 12 * q^73 - 2 * q^75 - 32 * q^77 - 16 * q^79 + 2 * q^81 - 24 * q^83 - 4 * q^85 + 12 * q^87 - 20 * q^89 + 8 * q^93 - 12 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	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
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	−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
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.17157	−0.588946	−0.294473	−	0.955660i	$-0.595144\pi$
$29$	−3.17157	−0.588946	−0.294473	+	0.955660i	$0.595144\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−5.65685	−0.984732
$34$	0	0
$35$	−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	$0.508980\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	3.65685	0.571105	0.285552	−	0.958363i	$-0.407823\pi$
$41$	3.65685	0.571105	0.285552	+	0.958363i	$0.407823\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	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
$50$	0	0
$51$	4.82843	0.676115
$52$	0	0
$53$	−9.31371	−1.27934	−0.639668	−	0.768651i	$-0.720928\pi$
$53$	−9.31371	−1.27934	−0.639668	+	0.768651i	$0.720928\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	−2.82843	−0.374634
$58$	0	0
$59$	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	$0.374508\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	0	0
$69$	8.48528	1.02151
$70$	0	0
$71$	−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	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−16.0000	−1.82337
$78$	0	0
$79$	−13.6569	−1.53652	−0.768258	−	0.640140i	$-0.778876\pi$
$79$	−13.6569	−1.53652	−0.768258	+	0.640140i	$0.778876\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−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
$86$	0	0
$87$	3.17157	0.340028
$88$	0	0
$89$	−4.34315	−0.460373	−0.230186	−	0.973147i	$-0.573934\pi$
$89$	−4.34315	−0.460373	−0.230186	+	0.973147i	$0.573934\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	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	$0.647933\pi$
$98$	0	0
$99$	5.65685	0.568535
$100$	0	0
$101$	−12.1421	−1.20819	−0.604094	−	0.796913i	$-0.706465\pi$
$101$	−12.1421	−1.20819	−0.604094	+	0.796913i	$0.706465\pi$
$102$	0	0
$103$	−9.65685	−0.951518	−0.475759	−	0.879576i	$-0.657827\pi$
$103$	−9.65685	−0.951518	−0.475759	+	0.879576i	$0.657827\pi$
$104$	0	0
$105$	2.82843	0.276026
$106$	0	0
$107$	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	$0.451464\pi$
$110$	0	0
$111$	0.343146	0.0325700
$112$	0	0
$113$	−10.4853	−0.986372	−0.493186	−	0.869924i	$-0.664168\pi$
$113$	−10.4853	−0.986372	−0.493186	+	0.869924i	$0.664168\pi$
$114$	0	0
$115$	−8.48528	−0.791257
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	−3.65685	−0.329727
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.65685	0.147022	0.0735110	−	0.997294i	$-0.476580\pi$
$127$	1.65685	0.147022	0.0735110	+	0.997294i	$0.476580\pi$
$128$	0	0
$129$	−1.65685	−0.145878
$130$	0	0
$131$	−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
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	5.31371	0.453981	0.226990	−	0.973897i	$-0.427111\pi$
$137$	5.31371	0.453981	0.226990	+	0.973897i	$0.427111\pi$
$138$	0	0
$139$	−17.6569	−1.49763	−0.748817	−	0.662776i	$-0.769378\pi$
$139$	−17.6569	−1.49763	−0.748817	+	0.662776i	$0.769378\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−5.65685	−0.473050
$144$	0	0
$145$	−3.17157	−0.263385
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−7.65685	−0.627274	−0.313637	−	0.949543i	$-0.601547\pi$
$149$	−7.65685	−0.627274	−0.313637	+	0.949543i	$0.601547\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	−4.82843	−0.390355
$154$	0	0
$155$	−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	$0.742782\pi$
$158$	0	0
$159$	9.31371	0.738625
$160$	0	0
$161$	24.0000	1.89146
$162$	0	0
$163$	11.3137	0.886158	0.443079	−	0.896483i	$-0.353886\pi$
$163$	11.3137	0.886158	0.443079	+	0.896483i	$0.353886\pi$
$164$	0	0
$165$	−5.65685	−0.440386
$166$	0	0
$167$	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
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	13.3137	1.01222	0.506111	−	0.862468i	$-0.331083\pi$
$173$	13.3137	1.01222	0.506111	+	0.862468i	$0.331083\pi$
$174$	0	0
$175$	−2.82843	−0.213809
$176$	0	0
$177$	−13.6569	−1.02651
$178$	0	0
$179$	−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	$0.456606\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−0.343146	−0.0252286
$186$	0	0
$187$	−27.3137	−1.99738
$188$	0	0
$189$	2.82843	0.205738
$190$	0	0
$191$	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	$0.674566\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	9.31371	0.663574	0.331787	−	0.943354i	$-0.392348\pi$
$197$	9.31371	0.663574	0.331787	+	0.943354i	$0.392348\pi$
$198$	0	0
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	0	0
$201$	−5.65685	−0.399004
$202$	0	0
$203$	8.97056	0.629610
$204$	0	0
$205$	3.65685	0.255406
$206$	0	0
$207$	−8.48528	−0.589768
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	−23.3137	−1.60498	−0.802491	−	0.596664i	$-0.796492\pi$
$211$	−23.3137	−1.60498	−0.802491	+	0.596664i	$0.796492\pi$
$212$	0	0
$213$	5.65685	0.387601
$214$	0	0
$215$	1.65685	0.112997
$216$	0	0
$217$	11.3137	0.768025
$218$	0	0
$219$	−2.48528	−0.167940
$220$	0	0
$221$	4.82843	0.324795
$222$	0	0
$223$	−5.17157	−0.346314	−0.173157	−	0.984894i	$-0.555397\pi$
$223$	−5.17157	−0.346314	−0.173157	+	0.984894i	$0.555397\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−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	$0.793939\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	0	0
$233$	22.4853	1.47306	0.736530	−	0.676405i	$-0.236463\pi$
$233$	22.4853	1.47306	0.736530	+	0.676405i	$0.236463\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	13.6569	0.887108
$238$	0	0
$239$	−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	$0.688292\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−2.82843	−0.179969
$248$	0	0
$249$	17.6569	1.11896
$250$	0	0
$251$	−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
$254$	0	0
$255$	4.82843	0.302368
$256$	0	0
$257$	0.828427	0.0516759	0.0258379	−	0.999666i	$-0.491775\pi$
$257$	0.828427	0.0516759	0.0258379	+	0.999666i	$0.491775\pi$
$258$	0	0
$259$	0.970563	0.0603078
$260$	0	0
$261$	−3.17157	−0.196315
$262$	0	0
$263$	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
$266$	0	0
$267$	4.34315	0.265796
$268$	0	0
$269$	2.48528	0.151530	0.0757651	−	0.997126i	$-0.475860\pi$
$269$	2.48528	0.151530	0.0757651	+	0.997126i	$0.475860\pi$
$270$	0	0
$271$	15.3137	0.930242	0.465121	−	0.885247i	$-0.346011\pi$
$271$	15.3137	0.930242	0.465121	+	0.885247i	$0.346011\pi$
$272$	0	0
$273$	−2.82843	−0.171184
$274$	0	0
$275$	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	$0.214662\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	19.6569	1.17263	0.586315	−	0.810083i	$-0.300578\pi$
$281$	19.6569	1.17263	0.586315	+	0.810083i	$0.300578\pi$
$282$	0	0
$283$	6.34315	0.377061	0.188530	−	0.982067i	$-0.439628\pi$
$283$	6.34315	0.377061	0.188530	+	0.982067i	$0.439628\pi$
$284$	0	0
$285$	−2.82843	−0.167542
$286$	0	0
$287$	−10.3431	−0.610537
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	8.82843	0.517532
$292$	0	0
$293$	28.6274	1.67243	0.836216	−	0.548401i	$-0.184763\pi$
$293$	28.6274	1.67243	0.836216	+	0.548401i	$0.184763\pi$
$294$	0	0
$295$	13.6569	0.795133
$296$	0	0
$297$	−5.65685	−0.328244
$298$	0	0
$299$	8.48528	0.490716
$300$	0	0
$301$	−4.68629	−0.270113
$302$	0	0
$303$	12.1421	0.697547
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	10.3431	0.590315	0.295157	−	0.955449i	$-0.404628\pi$
$307$	10.3431	0.590315	0.295157	+	0.955449i	$0.404628\pi$
$308$	0	0
$309$	9.65685	0.549359
$310$	0	0
$311$	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	$0.473245\pi$
$314$	0	0
$315$	−2.82843	−0.159364
$316$	0	0
$317$	2.68629	0.150877	0.0754386	−	0.997150i	$-0.475964\pi$
$317$	2.68629	0.150877	0.0754386	+	0.997150i	$0.475964\pi$
$318$	0	0
$319$	−17.9411	−1.00451
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−13.6569	−0.759888
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−3.17157	−0.175388
$328$	0	0
$329$	−22.6274	−1.24749
$330$	0	0
$331$	−8.48528	−0.466393	−0.233197	−	0.972430i	$-0.574919\pi$
$331$	−8.48528	−0.466393	−0.233197	+	0.972430i	$0.574919\pi$
$332$	0	0
$333$	−0.343146	−0.0188043
$334$	0	0
$335$	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	$0.715163\pi$
$338$	0	0
$339$	10.4853	0.569482
$340$	0	0
$341$	−22.6274	−1.22534
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	8.48528	0.456832
$346$	0	0
$347$	−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	$0.642204\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−17.3137	−0.921516	−0.460758	−	0.887526i	$-0.652422\pi$
$353$	−17.3137	−0.921516	−0.460758	+	0.887526i	$0.652422\pi$
$354$	0	0
$355$	−5.65685	−0.300235
$356$	0	0
$357$	−13.6569	−0.722797
$358$	0	0
$359$	−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
$362$	0	0
$363$	−21.0000	−1.10221
$364$	0	0
$365$	2.48528	0.130086
$366$	0	0
$367$	−14.3431	−0.748706	−0.374353	−	0.927286i	$-0.622135\pi$
$367$	−14.3431	−0.748706	−0.374353	+	0.927286i	$0.622135\pi$
$368$	0	0
$369$	3.65685	0.190368
$370$	0	0
$371$	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	$0.727478\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	3.17157	0.163344
$378$	0	0
$379$	24.4853	1.25772	0.628862	−	0.777517i	$-0.283521\pi$
$379$	24.4853	1.25772	0.628862	+	0.777517i	$0.283521\pi$
$380$	0	0
$381$	−1.65685	−0.0848832
$382$	0	0
$383$	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
$386$	0	0
$387$	1.65685	0.0842226
$388$	0	0
$389$	10.4853	0.531625	0.265812	−	0.964025i	$-0.414360\pi$
$389$	10.4853	0.531625	0.265812	+	0.964025i	$0.414360\pi$
$390$	0	0
$391$	40.9706	2.07197
$392$	0	0
$393$	22.1421	1.11692
$394$	0	0
$395$	−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	$0.729268\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	6.97056	0.348093	0.174047	−	0.984737i	$-0.444316\pi$
$401$	6.97056	0.348093	0.174047	+	0.984737i	$0.444316\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−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	$0.439377\pi$
$410$	0	0
$411$	−5.31371	−0.262106
$412$	0	0
$413$	−38.6274	−1.90073
$414$	0	0
$415$	−17.6569	−0.866741
$416$	0	0
$417$	17.6569	0.864660
$418$	0	0
$419$	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	$0.467816\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−4.82843	−0.234213
$426$	0	0
$427$	−16.9706	−0.821263
$428$	0	0
$429$	5.65685	0.273115
$430$	0	0
$431$	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	$0.415088\pi$
$434$	0	0
$435$	3.17157	0.152065
$436$	0	0
$437$	−24.0000	−1.14808
$438$	0	0
$439$	22.6274	1.07995	0.539974	−	0.841682i	$-0.318434\pi$
$439$	22.6274	1.07995	0.539974	+	0.841682i	$0.318434\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−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
$446$	0	0
$447$	7.65685	0.362157
$448$	0	0
$449$	−30.2843	−1.42920	−0.714602	−	0.699532i	$-0.753392\pi$
$449$	−30.2843	−1.42920	−0.714602	+	0.699532i	$0.753392\pi$
$450$	0	0
$451$	20.6863	0.974079
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	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	$0.384533\pi$
$458$	0	0
$459$	4.82843	0.225372
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	−35.7990	−1.66372	−0.831860	−	0.554985i	$-0.812724\pi$
$463$	−35.7990	−1.66372	−0.831860	+	0.554985i	$0.812724\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	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
$470$	0	0
$471$	17.3137	0.797774
$472$	0	0
$473$	9.37258	0.430952
$474$	0	0
$475$	2.82843	0.129777
$476$	0	0
$477$	−9.31371	−0.426445
$478$	0	0
$479$	−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
$482$	0	0
$483$	−24.0000	−1.09204
$484$	0	0
$485$	−8.82843	−0.400878
$486$	0	0
$487$	0.485281	0.0219902	0.0109951	−	0.999940i	$-0.496500\pi$
$487$	0.485281	0.0219902	0.0109951	+	0.999940i	$0.496500\pi$
$488$	0	0
$489$	−11.3137	−0.511624
$490$	0	0
$491$	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
$494$	0	0
$495$	5.65685	0.254257
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	16.4853	0.737983	0.368991	−	0.929433i	$-0.379703\pi$
$499$	16.4853	0.737983	0.368991	+	0.929433i	$0.379703\pi$
$500$	0	0
$501$	−24.9706	−1.11560
$502$	0	0
$503$	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
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−14.6863	−0.650958	−0.325479	−	0.945549i	$-0.605526\pi$
$509$	−14.6863	−0.650958	−0.325479	+	0.945549i	$0.605526\pi$
$510$	0	0
$511$	−7.02944	−0.310964
$512$	0	0
$513$	−2.82843	−0.124878
$514$	0	0
$515$	−9.65685	−0.425532
$516$	0	0
$517$	45.2548	1.99031
$518$	0	0
$519$	−13.3137	−0.584407
$520$	0	0
$521$	−6.97056	−0.305386	−0.152693	−	0.988274i	$-0.548795\pi$
$521$	−6.97056	−0.305386	−0.152693	+	0.988274i	$0.548795\pi$
$522$	0	0
$523$	−34.6274	−1.51415	−0.757076	−	0.653327i	$-0.773373\pi$
$523$	−34.6274	−1.51415	−0.757076	+	0.653327i	$0.773373\pi$
$524$	0	0
$525$	2.82843	0.123443
$526$	0	0
$527$	19.3137	0.841318
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	13.6569	0.592657
$532$	0	0
$533$	−3.65685	−0.158396
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	24.4853	1.05662
$538$	0	0
$539$	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	$0.517014\pi$
$542$	0	0
$543$	−3.65685	−0.156931
$544$	0	0
$545$	3.17157	0.135855
$546$	0	0
$547$	−23.3137	−0.996822	−0.498411	−	0.866941i	$-0.666083\pi$
$547$	−23.3137	−0.996822	−0.498411	+	0.866941i	$0.666083\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−8.97056	−0.382159
$552$	0	0
$553$	38.6274	1.64260
$554$	0	0
$555$	0.343146	0.0145657
$556$	0	0
$557$	33.3137	1.41155	0.705774	−	0.708437i	$-0.250600\pi$
$557$	33.3137	1.41155	0.705774	+	0.708437i	$0.250600\pi$
$558$	0	0
$559$	−1.65685	−0.0700775
$560$	0	0
$561$	27.3137	1.15319
$562$	0	0
$563$	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
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	20.3431	0.852829	0.426415	−	0.904528i	$-0.359776\pi$
$569$	20.3431	0.852829	0.426415	+	0.904528i	$0.359776\pi$
$570$	0	0
$571$	12.9706	0.542801	0.271401	−	0.962466i	$-0.412513\pi$
$571$	12.9706	0.542801	0.271401	+	0.962466i	$0.412513\pi$
$572$	0	0
$573$	−11.3137	−0.472637
$574$	0	0
$575$	−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	$0.306361\pi$
$578$	0	0
$579$	14.4853	0.601988
$580$	0	0
$581$	49.9411	2.07191
$582$	0	0
$583$	−52.6863	−2.18204
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	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
$590$	0	0
$591$	−9.31371	−0.383115
$592$	0	0
$593$	−11.6569	−0.478690	−0.239345	−	0.970935i	$-0.576933\pi$
$593$	−11.6569	−0.478690	−0.239345	+	0.970935i	$0.576933\pi$
$594$	0	0
$595$	13.6569	0.559876
$596$	0	0
$597$	21.6569	0.886356
$598$	0	0
$599$	−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	$0.456456\pi$
$602$	0	0
$603$	5.65685	0.230365
$604$	0	0
$605$	21.0000	0.853771
$606$	0	0
$607$	4.97056	0.201749	0.100874	−	0.994899i	$-0.467836\pi$
$607$	4.97056	0.201749	0.100874	+	0.994899i	$0.467836\pi$
$608$	0	0
$609$	−8.97056	−0.363506
$610$	0	0
$611$	−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	$0.743431\pi$
$614$	0	0
$615$	−3.65685	−0.147459
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	29.1716	1.17250	0.586252	−	0.810129i	$-0.300603\pi$
$619$	29.1716	1.17250	0.586252	+	0.810129i	$0.300603\pi$
$620$	0	0
$621$	8.48528	0.340503
$622$	0	0
$623$	12.2843	0.492159
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−16.0000	−0.638978
$628$	0	0
$629$	1.65685	0.0660631
$630$	0	0
$631$	22.3431	0.889467	0.444733	−	0.895663i	$-0.353298\pi$
$631$	22.3431	0.889467	0.444733	+	0.895663i	$0.353298\pi$
$632$	0	0
$633$	23.3137	0.926637
$634$	0	0
$635$	1.65685	0.0657503
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−5.65685	−0.223782
$640$	0	0
$641$	40.6274	1.60469	0.802343	−	0.596863i	$-0.203586\pi$
$641$	40.6274	1.60469	0.802343	+	0.596863i	$0.203586\pi$
$642$	0	0
$643$	39.5980	1.56159	0.780796	−	0.624786i	$-0.214814\pi$
$643$	39.5980	1.56159	0.780796	+	0.624786i	$0.214814\pi$
$644$	0	0
$645$	−1.65685	−0.0652386
$646$	0	0
$647$	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
$650$	0	0
$651$	−11.3137	−0.443419
$652$	0	0
$653$	14.2843	0.558987	0.279493	−	0.960148i	$-0.409834\pi$
$653$	14.2843	0.558987	0.279493	+	0.960148i	$0.409834\pi$
$654$	0	0
$655$	−22.1421	−0.865165
$656$	0	0
$657$	2.48528	0.0969601
$658$	0	0
$659$	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	$0.371880\pi$
$662$	0	0
$663$	−4.82843	−0.187521
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	26.9117	1.04202
$668$	0	0
$669$	5.17157	0.199945
$670$	0	0
$671$	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	$0.578255\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	23.6569	0.909207	0.454603	−	0.890694i	$-0.349781\pi$
$677$	23.6569	0.909207	0.454603	+	0.890694i	$0.349781\pi$
$678$	0	0
$679$	24.9706	0.958282
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	−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
$686$	0	0
$687$	24.1421	0.921080
$688$	0	0
$689$	9.31371	0.354824
$690$	0	0
$691$	−11.7990	−0.448855	−0.224427	−	0.974491i	$-0.572051\pi$
$691$	−11.7990	−0.448855	−0.224427	+	0.974491i	$0.572051\pi$
$692$	0	0
$693$	−16.0000	−0.607790
$694$	0	0
$695$	−17.6569	−0.669763
$696$	0	0
$697$	−17.6569	−0.668801
$698$	0	0
$699$	−22.4853	−0.850471
$700$	0	0
$701$	−28.1421	−1.06291	−0.531457	−	0.847085i	$-0.678355\pi$
$701$	−28.1421	−1.06291	−0.531457	+	0.847085i	$0.678355\pi$
$702$	0	0
$703$	−0.970563	−0.0366055
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	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	$0.577440\pi$
$710$	0	0
$711$	−13.6569	−0.512172
$712$	0	0
$713$	33.9411	1.27111
$714$	0	0
$715$	−5.65685	−0.211554
$716$	0	0
$717$	16.0000	0.597531
$718$	0	0
$719$	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
$722$	0	0
$723$	17.3137	0.643904
$724$	0	0
$725$	−3.17157	−0.117789
$726$	0	0
$727$	−21.9411	−0.813751	−0.406876	−	0.913484i	$-0.633382\pi$
$727$	−21.9411	−0.813751	−0.406876	+	0.913484i	$0.633382\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−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	$0.569066\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	32.0000	1.17874
$738$	0	0
$739$	−14.1421	−0.520227	−0.260113	−	0.965578i	$-0.583760\pi$
$739$	−14.1421	−0.520227	−0.260113	+	0.965578i	$0.583760\pi$
$740$	0	0
$741$	2.82843	0.103905
$742$	0	0
$743$	−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
$746$	0	0
$747$	−17.6569	−0.646031
$748$	0	0
$749$	−11.3137	−0.413394
$750$	0	0
$751$	11.3137	0.412843	0.206422	−	0.978463i	$-0.433818\pi$
$751$	11.3137	0.412843	0.206422	+	0.978463i	$0.433818\pi$
$752$	0	0
$753$	5.17157	0.188463
$754$	0	0
$755$	−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	$0.836674\pi$
$758$	0	0
$759$	48.0000	1.74229
$760$	0	0
$761$	16.3431	0.592439	0.296219	−	0.955120i	$-0.404274\pi$
$761$	16.3431	0.592439	0.296219	+	0.955120i	$0.404274\pi$
$762$	0	0
$763$	−8.97056	−0.324756
$764$	0	0
$765$	−4.82843	−0.174572
$766$	0	0
$767$	−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	$0.581229\pi$
$770$	0	0
$771$	−0.828427	−0.0298351
$772$	0	0
$773$	−30.6863	−1.10371	−0.551855	−	0.833940i	$-0.686080\pi$
$773$	−30.6863	−1.10371	−0.551855	+	0.833940i	$0.686080\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−0.970563	−0.0348187
$778$	0	0
$779$	10.3431	0.370582
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	3.17157	0.113343
$784$	0	0
$785$	−17.3137	−0.617953
$786$	0	0
$787$	24.0000	0.855508	0.427754	−	0.903895i	$-0.359305\pi$
$787$	24.0000	0.855508	0.427754	+	0.903895i	$0.359305\pi$
$788$	0	0
$789$	−0.485281	−0.0172765
$790$	0	0
$791$	29.6569	1.05448
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	9.31371	0.330323
$796$	0	0
$797$	−28.6274	−1.01404	−0.507018	−	0.861936i	$-0.669252\pi$
$797$	−28.6274	−1.01404	−0.507018	+	0.861936i	$0.669252\pi$
$798$	0	0
$799$	−38.6274	−1.36654
$800$	0	0
$801$	−4.34315	−0.153458
$802$	0	0
$803$	14.0589	0.496127
$804$	0	0
$805$	24.0000	0.845889
$806$	0	0
$807$	−2.48528	−0.0874860
$808$	0	0
$809$	−9.31371	−0.327453	−0.163726	−	0.986506i	$-0.552351\pi$
$809$	−9.31371	−0.327453	−0.163726	+	0.986506i	$0.552351\pi$
$810$	0	0
$811$	30.1421	1.05843	0.529217	−	0.848487i	$-0.322486\pi$
$811$	30.1421	1.05843	0.529217	+	0.848487i	$0.322486\pi$
$812$	0	0
$813$	−15.3137	−0.537075
$814$	0	0
$815$	11.3137	0.396302
$816$	0	0
$817$	4.68629	0.163953
$818$	0	0
$819$	2.82843	0.0988332
$820$	0	0
$821$	−22.2843	−0.777726	−0.388863	−	0.921296i	$-0.627132\pi$
$821$	−22.2843	−0.777726	−0.388863	+	0.921296i	$0.627132\pi$
$822$	0	0
$823$	19.0294	0.663324	0.331662	−	0.943398i	$-0.392391\pi$
$823$	19.0294	0.663324	0.331662	+	0.943398i	$0.392391\pi$
$824$	0	0
$825$	−5.65685	−0.196946
$826$	0	0
$827$	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	$0.678894\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	−4.82843	−0.167295
$834$	0	0
$835$	24.9706	0.864142
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−18.9411	−0.653142
$842$	0	0
$843$	−19.6569	−0.677018
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−59.3970	−2.04090
$848$	0	0
$849$	−6.34315	−0.217696
$850$	0	0
$851$	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	$0.601341\pi$
$854$	0	0
$855$	2.82843	0.0967302
$856$	0	0
$857$	−15.1716	−0.518251	−0.259126	−	0.965844i	$-0.583434\pi$
$857$	−15.1716	−0.518251	−0.259126	+	0.965844i	$0.583434\pi$
$858$	0	0
$859$	29.9411	1.02158	0.510789	−	0.859706i	$-0.329353\pi$
$859$	29.9411	1.02158	0.510789	+	0.859706i	$0.329353\pi$
$860$	0	0
$861$	10.3431	0.352493
$862$	0	0
$863$	−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
$866$	0	0
$867$	−6.31371	−0.214425
$868$	0	0
$869$	−77.2548	−2.62069
$870$	0	0
$871$	−5.65685	−0.191675
$872$	0	0
$873$	−8.82843	−0.298797
$874$	0	0
$875$	−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	$0.269380\pi$
$878$	0	0
$879$	−28.6274	−0.965579
$880$	0	0
$881$	46.2843	1.55936	0.779678	−	0.626180i	$-0.215383\pi$
$881$	46.2843	1.55936	0.779678	+	0.626180i	$0.215383\pi$
$882$	0	0
$883$	−8.68629	−0.292317	−0.146158	−	0.989261i	$-0.546691\pi$
$883$	−8.68629	−0.292317	−0.146158	+	0.989261i	$0.546691\pi$
$884$	0	0
$885$	−13.6569	−0.459070
$886$	0	0
$887$	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
$890$	0	0
$891$	5.65685	0.189512
$892$	0	0
$893$	22.6274	0.757198
$894$	0	0
$895$	−24.4853	−0.818453
$896$	0	0
$897$	−8.48528	−0.283315
$898$	0	0
$899$	12.6863	0.423112
$900$	0	0
$901$	44.9706	1.49819
$902$	0	0
$903$	4.68629	0.155950
$904$	0	0
$905$	3.65685	0.121558
$906$	0	0
$907$	48.2843	1.60325	0.801626	−	0.597825i	$-0.203968\pi$
$907$	48.2843	1.60325	0.801626	+	0.597825i	$0.203968\pi$
$908$	0	0
$909$	−12.1421	−0.402729
$910$	0	0
$911$	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
$914$	0	0
$915$	−6.00000	−0.198354
$916$	0	0
$917$	62.6274	2.06814
$918$	0	0
$919$	25.9411	0.855719	0.427859	−	0.903845i	$-0.359268\pi$
$919$	25.9411	0.855719	0.427859	+	0.903845i	$0.359268\pi$
$920$	0	0
$921$	−10.3431	−0.340818
$922$	0	0
$923$	5.65685	0.186198
$924$	0	0
$925$	−0.343146	−0.0112826
$926$	0	0
$927$	−9.65685	−0.317173
$928$	0	0
$929$	45.5980	1.49602	0.748011	−	0.663687i	$-0.231009\pi$
$929$	45.5980	1.49602	0.748011	+	0.663687i	$0.231009\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	−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	$0.654884\pi$
$938$	0	0
$939$	−2.97056	−0.0969407
$940$	0	0
$941$	21.0294	0.685540	0.342770	−	0.939419i	$-0.388635\pi$
$941$	21.0294	0.685540	0.342770	+	0.939419i	$0.388635\pi$
$942$	0	0
$943$	−31.0294	−1.01046
$944$	0	0
$945$	2.82843	0.0920087
$946$	0	0
$947$	−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
$950$	0	0
$951$	−2.68629	−0.0871090
$952$	0	0
$953$	−56.1421	−1.81862	−0.909311	−	0.416117i	$-0.863391\pi$
$953$	−56.1421	−1.81862	−0.909311	+	0.416117i	$0.863391\pi$
$954$	0	0
$955$	11.3137	0.366103
$956$	0	0
$957$	17.9411	0.579954
$958$	0	0
$959$	−15.0294	−0.485326
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	−14.4853	−0.466298
$966$	0	0
$967$	24.4853	0.787394	0.393697	−	0.919240i	$-0.371196\pi$
$967$	24.4853	0.787394	0.393697	+	0.919240i	$0.371196\pi$
$968$	0	0
$969$	13.6569	0.438721
$970$	0	0
$971$	−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	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−19.6569	−0.628878	−0.314439	−	0.949278i	$-0.601816\pi$
$977$	−19.6569	−0.628878	−0.314439	+	0.949278i	$0.601816\pi$
$978$	0	0
$979$	−24.5685	−0.785214
$980$	0	0
$981$	3.17157	0.101261
$982$	0	0
$983$	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
$986$	0	0
$987$	22.6274	0.720239
$988$	0	0
$989$	−14.0589	−0.447046
$990$	0	0
$991$	−58.9117	−1.87139	−0.935696	−	0.352808i	$-0.885227\pi$
$991$	−58.9117	−1.87139	−0.935696	+	0.352808i	$0.885227\pi$
$992$	0	0
$993$	8.48528	0.269272
$994$	0	0
$995$	−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	$0.290122\pi$
$998$	0	0
$999$	0.343146	0.0108567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.a.bc.1.1		2
3.2	odd	2		9360.2.a.ch.1.1		2
4.3	odd	2		390.2.a.h.1.2	✓	2
12.11	even	2		1170.2.a.o.1.2		2
20.3	even	4		1950.2.e.o.1249.1		4
20.7	even	4		1950.2.e.o.1249.4		4
20.19	odd	2		1950.2.a.bd.1.1		2
52.31	even	4		5070.2.b.q.1351.2		4
52.47	even	4		5070.2.b.q.1351.3		4
52.51	odd	2		5070.2.a.bc.1.1		2
60.23	odd	4		5850.2.e.bk.5149.3		4
60.47	odd	4		5850.2.e.bk.5149.2		4
60.59	even	2		5850.2.a.cl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.h.1.2	✓	2	4.3	odd	2
1170.2.a.o.1.2		2	12.11	even	2
1950.2.a.bd.1.1		2	20.19	odd	2
1950.2.e.o.1249.1		4	20.3	even	4
1950.2.e.o.1249.4		4	20.7	even	4
3120.2.a.bc.1.1		2	1.1	even	1	trivial
5070.2.a.bc.1.1		2	52.51	odd	2
5070.2.b.q.1351.2		4	52.31	even	4
5070.2.b.q.1351.3		4	52.47	even	4
5850.2.a.cl.1.1		2	60.59	even	2
5850.2.e.bk.5149.2		4	60.47	odd	4
5850.2.e.bk.5149.3		4	60.23	odd	4
9360.2.a.ch.1.1		2	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.82843 q^{7} +1.00000 q^{9} +5.65685 q^{11} -1.00000 q^{13} -1.00000 q^{15} -4.82843 q^{17} +2.82843 q^{19} +2.82843 q^{21} -8.48528 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.17157 q^{29} -4.00000 q^{31} -5.65685 q^{33} -2.82843 q^{35} -0.343146 q^{37} +1.00000 q^{39} +3.65685 q^{41} +1.65685 q^{43} +1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} +4.82843 q^{51} -9.31371 q^{53} +5.65685 q^{55} -2.82843 q^{57} +13.6569 q^{59} +6.00000 q^{61} -2.82843 q^{63} -1.00000 q^{65} +5.65685 q^{67} +8.48528 q^{69} -5.65685 q^{71} +2.48528 q^{73} -1.00000 q^{75} -16.0000 q^{77} -13.6569 q^{79} +1.00000 q^{81} -17.6569 q^{83} -4.82843 q^{85} +3.17157 q^{87} -4.34315 q^{89} +2.82843 q^{91} +4.00000 q^{93} +2.82843 q^{95} -8.82843 q^{97} +5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 2 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{25} - 2 q^{27} - 12 q^{29} - 8 q^{31} - 12 q^{37} + 2 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 16 q^{47} + 2 q^{49} + 4 q^{51} + 4 q^{53} + 16 q^{59} + 12 q^{61} - 2 q^{65} - 12 q^{73} - 2 q^{75} - 32 q^{77} - 16 q^{79} + 2 q^{81} - 24 q^{83} - 4 q^{85} + 12 q^{87} - 20 q^{89} + 8 q^{93} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 - 2 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^25 - 2 * q^27 - 12 * q^29 - 8 * q^31 - 12 * q^37 + 2 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 + 16 * q^47 + 2 * q^49 + 4 * q^51 + 4 * q^53 + 16 * q^59 + 12 * q^61 - 2 * q^65 - 12 * q^73 - 2 * q^75 - 32 * q^77 - 16 * q^79 + 2 * q^81 - 24 * q^83 - 4 * q^85 + 12 * q^87 - 20 * q^89 + 8 * q^93 - 12 * q^97

Introduction

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