LMFDB - Embedded newform 5265.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5265,2,Mod(1,5265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5265, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.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:	$42.0412366642$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1 $ x^8 - 3x^7 - 8x^6 + 31x^5 - x^4 - 70x^3 + 66x^2 - 19x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.63404$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.63404 q^{2} +0.670078 q^{4} +1.00000 q^{5} +2.13012 q^{7} +2.17314 q^{8} +O(q^{10})$	\(q-1.63404 q^{2} +0.670078 q^{4} +1.00000 q^{5} +2.13012 q^{7} +2.17314 q^{8} -1.63404 q^{10} +0.0526457 q^{11} -1.00000 q^{13} -3.48069 q^{14} -4.89115 q^{16} +2.48047 q^{17} +2.13105 q^{19} +0.670078 q^{20} -0.0860250 q^{22} -4.93083 q^{23} +1.00000 q^{25} +1.63404 q^{26} +1.42734 q^{28} +2.48694 q^{29} -8.16059 q^{31} +3.64604 q^{32} -4.05318 q^{34} +2.13012 q^{35} -1.11963 q^{37} -3.48221 q^{38} +2.17314 q^{40} +5.47417 q^{41} -9.47722 q^{43} +0.0352767 q^{44} +8.05716 q^{46} -9.76560 q^{47} -2.46260 q^{49} -1.63404 q^{50} -0.670078 q^{52} +3.64091 q^{53} +0.0526457 q^{55} +4.62905 q^{56} -4.06374 q^{58} -7.49087 q^{59} -5.78404 q^{61} +13.3347 q^{62} +3.82454 q^{64} -1.00000 q^{65} -6.23591 q^{67} +1.66211 q^{68} -3.48069 q^{70} -2.50050 q^{71} -1.10245 q^{73} +1.82952 q^{74} +1.42797 q^{76} +0.112141 q^{77} -15.6156 q^{79} -4.89115 q^{80} -8.94500 q^{82} -0.489152 q^{83} +2.48047 q^{85} +15.4861 q^{86} +0.114407 q^{88} +4.64901 q^{89} -2.13012 q^{91} -3.30404 q^{92} +15.9574 q^{94} +2.13105 q^{95} +7.34097 q^{97} +4.02399 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) $ 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8	$ 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} - 8 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} - 10 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 3 q^{26} - 34 q^{28} - 14 q^{29} - 31 q^{31} - q^{32} - 7 q^{34} - 11 q^{35} + q^{37} - 9 q^{38} + 6 q^{40} + 12 q^{41} + 15 q^{43} - 16 q^{44} - 32 q^{46} + 18 q^{47} + 17 q^{49} - 3 q^{50} - 9 q^{52} - 2 q^{53} - 6 q^{55} - 16 q^{56} - 42 q^{58} - 24 q^{59} - 9 q^{61} + 20 q^{62} - 30 q^{64} - 8 q^{65} - 18 q^{67} + 14 q^{68} - 10 q^{70} - 10 q^{71} + 6 q^{73} + 37 q^{74} - 53 q^{76} + 34 q^{77} - 3 q^{79} + 11 q^{80} - 34 q^{82} + 10 q^{83} + 2 q^{85} - 60 q^{86} - 14 q^{88} + 13 q^{89} + 11 q^{91} - 5 q^{92} + 17 q^{94} - 10 q^{95} - 34 q^{97} + 30 q^{98}+O(q^{100}) $ 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8 - 3 * q^10 - 6 * q^11 - 8 * q^13 - 10 * q^14 + 11 * q^16 + 2 * q^17 - 10 * q^19 + 9 * q^20 + 3 * q^22 - 6 * q^23 + 8 * q^25 + 3 * q^26 - 34 * q^28 - 14 * q^29 - 31 * q^31 - q^32 - 7 * q^34 - 11 * q^35 + q^37 - 9 * q^38 + 6 * q^40 + 12 * q^41 + 15 * q^43 - 16 * q^44 - 32 * q^46 + 18 * q^47 + 17 * q^49 - 3 * q^50 - 9 * q^52 - 2 * q^53 - 6 * q^55 - 16 * q^56 - 42 * q^58 - 24 * q^59 - 9 * q^61 + 20 * q^62 - 30 * q^64 - 8 * q^65 - 18 * q^67 + 14 * q^68 - 10 * q^70 - 10 * q^71 + 6 * q^73 + 37 * q^74 - 53 * q^76 + 34 * q^77 - 3 * q^79 + 11 * q^80 - 34 * q^82 + 10 * q^83 + 2 * q^85 - 60 * q^86 - 14 * q^88 + 13 * q^89 + 11 * q^91 - 5 * q^92 + 17 * q^94 - 10 * q^95 - 34 * q^97 + 30 * q^98

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 $

$2$	−1.63404	−1.15544	−0.577719	−	0.816235i	$-0.696057\pi$
$2$	−1.63404	−1.15544	−0.577719	+	0.816235i	$0.696057\pi$
$3$	0	0
$3$	0	0
$4$	0.670078	0.335039
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.13012	0.805108	0.402554	−	0.915396i	$-0.368123\pi$
$7$	2.13012	0.805108	0.402554	+	0.915396i	$0.368123\pi$
$8$	2.17314	0.768322
$9$	0	0
$10$	−1.63404	−0.516728
$11$	0.0526457	0.0158733	0.00793664	−	0.999969i	$-0.497474\pi$
$11$	0.0526457	0.0158733	0.00793664	+	0.999969i	$0.497474\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.48069	−0.930253
$15$	0	0
$16$	−4.89115	−1.22279
$17$	2.48047	0.601603	0.300801	−	0.953687i	$-0.402746\pi$
$17$	2.48047	0.601603	0.300801	+	0.953687i	$0.402746\pi$
$18$	0	0
$19$	2.13105	0.488895	0.244448	−	0.969662i	$-0.421393\pi$
$19$	2.13105	0.488895	0.244448	+	0.969662i	$0.421393\pi$
$20$	0.670078	0.149834
$21$	0	0
$22$	−0.0860250	−0.0183406
$23$	−4.93083	−1.02815	−0.514075	−	0.857745i	$-0.671865\pi$
$23$	−4.93083	−1.02815	−0.514075	+	0.857745i	$0.671865\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.63404	0.320461
$27$	0	0
$28$	1.42734	0.269743
$29$	2.48694	0.461812	0.230906	−	0.972976i	$-0.425831\pi$
$29$	2.48694	0.461812	0.230906	+	0.972976i	$0.425831\pi$
$30$	0	0
$31$	−8.16059	−1.46569	−0.732843	−	0.680398i	$-0.761807\pi$
$31$	−8.16059	−1.46569	−0.732843	+	0.680398i	$0.761807\pi$
$32$	3.64604	0.644535
$33$	0	0
$34$	−4.05318	−0.695115
$35$	2.13012	0.360055
$36$	0	0
$37$	−1.11963	−0.184066	−0.0920331	−	0.995756i	$-0.529337\pi$
$37$	−1.11963	−0.184066	−0.0920331	+	0.995756i	$0.529337\pi$
$38$	−3.48221	−0.564889
$39$	0	0
$40$	2.17314	0.343604
$41$	5.47417	0.854922	0.427461	−	0.904034i	$-0.359408\pi$
$41$	5.47417	0.854922	0.427461	+	0.904034i	$0.359408\pi$
$42$	0	0
$43$	−9.47722	−1.44526	−0.722632	−	0.691233i	$-0.757068\pi$
$43$	−9.47722	−1.44526	−0.722632	+	0.691233i	$0.757068\pi$
$44$	0.0352767	0.00531816
$45$	0	0
$46$	8.05716	1.18796
$47$	−9.76560	−1.42446	−0.712230	−	0.701946i	$-0.752315\pi$
$47$	−9.76560	−1.42446	−0.712230	+	0.701946i	$0.752315\pi$
$48$	0	0
$49$	−2.46260	−0.351801
$50$	−1.63404	−0.231088
$51$	0	0
$52$	−0.670078	−0.0929231
$53$	3.64091	0.500118	0.250059	−	0.968231i	$-0.419550\pi$
$53$	3.64091	0.500118	0.250059	+	0.968231i	$0.419550\pi$
$54$	0	0
$55$	0.0526457	0.00709875
$56$	4.62905	0.618582
$57$	0	0
$58$	−4.06374	−0.533596
$59$	−7.49087	−0.975228	−0.487614	−	0.873059i	$-0.662133\pi$
$59$	−7.49087	−0.975228	−0.487614	+	0.873059i	$0.662133\pi$
$60$	0	0
$61$	−5.78404	−0.740570	−0.370285	−	0.928918i	$-0.620740\pi$
$61$	−5.78404	−0.740570	−0.370285	+	0.928918i	$0.620740\pi$
$62$	13.3347	1.69351
$63$	0	0
$64$	3.82454	0.478067
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−6.23591	−0.761837	−0.380919	−	0.924609i	$-0.624392\pi$
$67$	−6.23591	−0.761837	−0.380919	+	0.924609i	$0.624392\pi$
$68$	1.66211	0.201560
$69$	0	0
$70$	−3.48069	−0.416022
$71$	−2.50050	−0.296755	−0.148377	−	0.988931i	$-0.547405\pi$
$71$	−2.50050	−0.296755	−0.148377	+	0.988931i	$0.547405\pi$
$72$	0	0
$73$	−1.10245	−0.129032	−0.0645159	−	0.997917i	$-0.520550\pi$
$73$	−1.10245	−0.129032	−0.0645159	+	0.997917i	$0.520550\pi$
$74$	1.82952	0.212677
$75$	0	0
$76$	1.42797	0.163799
$77$	0.112141	0.0127797
$78$	0	0
$79$	−15.6156	−1.75689	−0.878445	−	0.477844i	$-0.841418\pi$
$79$	−15.6156	−1.75689	−0.878445	+	0.477844i	$0.841418\pi$
$80$	−4.89115	−0.546847
$81$	0	0
$82$	−8.94500	−0.987810
$83$	−0.489152	−0.0536914	−0.0268457	−	0.999640i	$-0.508546\pi$
$83$	−0.489152	−0.0536914	−0.0268457	+	0.999640i	$0.508546\pi$
$84$	0	0
$85$	2.48047	0.269045
$86$	15.4861	1.66991
$87$	0	0
$88$	0.114407	0.0121958
$89$	4.64901	0.492794	0.246397	−	0.969169i	$-0.420753\pi$
$89$	4.64901	0.492794	0.246397	+	0.969169i	$0.420753\pi$
$90$	0	0
$91$	−2.13012	−0.223297
$92$	−3.30404	−0.344470
$93$	0	0
$94$	15.9574	1.64588
$95$	2.13105	0.218641
$96$	0	0
$97$	7.34097	0.745363	0.372682	−	0.927959i	$-0.378438\pi$
$97$	7.34097	0.745363	0.372682	+	0.927959i	$0.378438\pi$
$98$	4.02399	0.406484
$99$	0	0
$100$	0.670078	0.0670078
$101$	−0.552859	−0.0550115	−0.0275058	−	0.999622i	$-0.508756\pi$
$101$	−0.552859	−0.0550115	−0.0275058	+	0.999622i	$0.508756\pi$
$102$	0	0
$103$	−7.83795	−0.772296	−0.386148	−	0.922437i	$-0.626195\pi$
$103$	−7.83795	−0.772296	−0.386148	+	0.922437i	$0.626195\pi$
$104$	−2.17314	−0.213094
$105$	0	0
$106$	−5.94939	−0.577855
$107$	−19.4173	−1.87714	−0.938572	−	0.345084i	$-0.887851\pi$
$107$	−19.4173	−1.87714	−0.938572	+	0.345084i	$0.887851\pi$
$108$	0	0
$109$	−2.13502	−0.204498	−0.102249	−	0.994759i	$-0.532604\pi$
$109$	−2.13502	−0.204498	−0.102249	+	0.994759i	$0.532604\pi$
$110$	−0.0860250	−0.00820217
$111$	0	0
$112$	−10.4187	−0.984477
$113$	6.23529	0.586567	0.293283	−	0.956026i	$-0.405252\pi$
$113$	6.23529	0.586567	0.293283	+	0.956026i	$0.405252\pi$
$114$	0	0
$115$	−4.93083	−0.459802
$116$	1.66644	0.154725
$117$	0	0
$118$	12.2404	1.12682
$119$	5.28369	0.484355
$120$	0	0
$121$	−10.9972	−0.999748
$122$	9.45133	0.855683
$123$	0	0
$124$	−5.46823	−0.491062
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.6036	−0.940918	−0.470459	−	0.882422i	$-0.655912\pi$
$127$	−10.6036	−0.940918	−0.470459	+	0.882422i	$0.655912\pi$
$128$	−13.5415	−1.19691
$129$	0	0
$130$	1.63404	0.143315
$131$	16.1923	1.41473	0.707365	−	0.706849i	$-0.249884\pi$
$131$	16.1923	1.41473	0.707365	+	0.706849i	$0.249884\pi$
$132$	0	0
$133$	4.53938	0.393614
$134$	10.1897	0.880256
$135$	0	0
$136$	5.39042	0.462224
$137$	20.7686	1.77438	0.887189	−	0.461406i	$-0.152655\pi$
$137$	20.7686	1.77438	0.887189	+	0.461406i	$0.152655\pi$
$138$	0	0
$139$	−2.50075	−0.212111	−0.106055	−	0.994360i	$-0.533822\pi$
$139$	−2.50075	−0.212111	−0.106055	+	0.994360i	$0.533822\pi$
$140$	1.42734	0.120633
$141$	0	0
$142$	4.08591	0.342882
$143$	−0.0526457	−0.00440245
$144$	0	0
$145$	2.48694	0.206529
$146$	1.80144	0.149088
$147$	0	0
$148$	−0.750240	−0.0616693
$149$	1.89025	0.154856	0.0774279	−	0.996998i	$-0.475329\pi$
$149$	1.89025	0.154856	0.0774279	+	0.996998i	$0.475329\pi$
$150$	0	0
$151$	−4.88685	−0.397686	−0.198843	−	0.980031i	$-0.563718\pi$
$151$	−4.88685	−0.397686	−0.198843	+	0.980031i	$0.563718\pi$
$152$	4.63107	0.375629
$153$	0	0
$154$	−0.183243	−0.0147662
$155$	−8.16059	−0.655475
$156$	0	0
$157$	−17.9998	−1.43654	−0.718269	−	0.695765i	$-0.755065\pi$
$157$	−17.9998	−1.43654	−0.718269	+	0.695765i	$0.755065\pi$
$158$	25.5164	2.02998
$159$	0	0
$160$	3.64604	0.288245
$161$	−10.5032	−0.827772
$162$	0	0
$163$	9.41748	0.737634	0.368817	−	0.929502i	$-0.379763\pi$
$163$	9.41748	0.737634	0.368817	+	0.929502i	$0.379763\pi$
$164$	3.66812	0.286432
$165$	0	0
$166$	0.799292	0.0620371
$167$	15.4905	1.19869	0.599344	−	0.800492i	$-0.295428\pi$
$167$	15.4905	1.19869	0.599344	+	0.800492i	$0.295428\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−4.05318	−0.310865
$171$	0	0
$172$	−6.35048	−0.484219
$173$	9.67663	0.735700	0.367850	−	0.929885i	$-0.380094\pi$
$173$	9.67663	0.735700	0.367850	+	0.929885i	$0.380094\pi$
$174$	0	0
$175$	2.13012	0.161022
$176$	−0.257498	−0.0194096
$177$	0	0
$178$	−7.59665	−0.569393
$179$	3.31630	0.247872	0.123936	−	0.992290i	$-0.460448\pi$
$179$	3.31630	0.247872	0.123936	+	0.992290i	$0.460448\pi$
$180$	0	0
$181$	−22.6189	−1.68125	−0.840623	−	0.541621i	$-0.817811\pi$
$181$	−22.6189	−1.68125	−0.840623	+	0.541621i	$0.817811\pi$
$182$	3.48069	0.258006
$183$	0	0
$184$	−10.7154	−0.789950
$185$	−1.11963	−0.0823169
$186$	0	0
$187$	0.130586	0.00954940
$188$	−6.54371	−0.477249
$189$	0	0
$190$	−3.48221	−0.252626
$191$	−5.88211	−0.425614	−0.212807	−	0.977094i	$-0.568261\pi$
$191$	−5.88211	−0.425614	−0.212807	+	0.977094i	$0.568261\pi$
$192$	0	0
$193$	−0.459539	−0.0330783	−0.0165392	−	0.999863i	$-0.505265\pi$
$193$	−0.459539	−0.0330783	−0.0165392	+	0.999863i	$0.505265\pi$
$194$	−11.9954	−0.861221
$195$	0	0
$196$	−1.65014	−0.117867
$197$	2.31119	0.164665	0.0823326	−	0.996605i	$-0.473763\pi$
$197$	2.31119	0.164665	0.0823326	+	0.996605i	$0.473763\pi$
$198$	0	0
$199$	24.3734	1.72779	0.863893	−	0.503675i	$-0.168019\pi$
$199$	24.3734	1.72779	0.863893	+	0.503675i	$0.168019\pi$
$200$	2.17314	0.153664
$201$	0	0
$202$	0.903392	0.0635624
$203$	5.29746	0.371809
$204$	0	0
$205$	5.47417	0.382333
$206$	12.8075	0.892341
$207$	0	0
$208$	4.89115	0.339140
$209$	0.112190	0.00776037
$210$	0	0
$211$	27.3938	1.88587	0.942934	−	0.332980i	$-0.108054\pi$
$211$	27.3938	1.88587	0.942934	+	0.332980i	$0.108054\pi$
$212$	2.43969	0.167559
$213$	0	0
$214$	31.7286	2.16893
$215$	−9.47722	−0.646341
$216$	0	0
$217$	−17.3830	−1.18004
$218$	3.48870	0.236285
$219$	0	0
$220$	0.0352767	0.00237836
$221$	−2.48047	−0.166855
$222$	0	0
$223$	25.3359	1.69661	0.848307	−	0.529504i	$-0.177622\pi$
$223$	25.3359	1.69661	0.848307	+	0.529504i	$0.177622\pi$
$224$	7.76649	0.518920
$225$	0	0
$226$	−10.1887	−0.677742
$227$	21.2417	1.40986	0.704931	−	0.709276i	$-0.250978\pi$
$227$	21.2417	1.40986	0.704931	+	0.709276i	$0.250978\pi$
$228$	0	0
$229$	12.2337	0.808427	0.404214	−	0.914665i	$-0.367545\pi$
$229$	12.2337	0.808427	0.404214	+	0.914665i	$0.367545\pi$
$230$	8.05716	0.531274
$231$	0	0
$232$	5.40446	0.354820
$233$	−11.4048	−0.747154	−0.373577	−	0.927599i	$-0.621869\pi$
$233$	−11.4048	−0.747154	−0.373577	+	0.927599i	$0.621869\pi$
$234$	0	0
$235$	−9.76560	−0.637038
$236$	−5.01947	−0.326739
$237$	0	0
$238$	−8.63375	−0.559643
$239$	−14.8571	−0.961025	−0.480512	−	0.876988i	$-0.659549\pi$
$239$	−14.8571	−0.961025	−0.480512	+	0.876988i	$0.659549\pi$
$240$	0	0
$241$	−18.5908	−1.19754	−0.598768	−	0.800922i	$-0.704343\pi$
$241$	−18.5908	−1.19754	−0.598768	+	0.800922i	$0.704343\pi$
$242$	17.9699	1.15515
$243$	0	0
$244$	−3.87575	−0.248120
$245$	−2.46260	−0.157330
$246$	0	0
$247$	−2.13105	−0.135595
$248$	−17.7341	−1.12612
$249$	0	0
$250$	−1.63404	−0.103346
$251$	−30.0723	−1.89815	−0.949073	−	0.315058i	$-0.897976\pi$
$251$	−30.0723	−1.89815	−0.949073	+	0.315058i	$0.897976\pi$
$252$	0	0
$253$	−0.259587	−0.0163201
$254$	17.3267	1.08717
$255$	0	0
$256$	14.4783	0.904892
$257$	−3.04064	−0.189670	−0.0948348	−	0.995493i	$-0.530232\pi$
$257$	−3.04064	−0.189670	−0.0948348	+	0.995493i	$0.530232\pi$
$258$	0	0
$259$	−2.38494	−0.148193
$260$	−0.670078	−0.0415565
$261$	0	0
$262$	−26.4588	−1.63463
$263$	4.80102	0.296044	0.148022	−	0.988984i	$-0.452709\pi$
$263$	4.80102	0.296044	0.148022	+	0.988984i	$0.452709\pi$
$264$	0	0
$265$	3.64091	0.223659
$266$	−7.41751	−0.454797
$267$	0	0
$268$	−4.17854	−0.255245
$269$	16.8723	1.02873	0.514363	−	0.857573i	$-0.328029\pi$
$269$	16.8723	1.02873	0.514363	+	0.857573i	$0.328029\pi$
$270$	0	0
$271$	−2.34099	−0.142205	−0.0711026	−	0.997469i	$-0.522652\pi$
$271$	−2.34099	−0.142205	−0.0711026	+	0.997469i	$0.522652\pi$
$272$	−12.1324	−0.735632
$273$	0	0
$274$	−33.9366	−2.05019
$275$	0.0526457	0.00317466
$276$	0	0
$277$	−3.77233	−0.226657	−0.113329	−	0.993558i	$-0.536151\pi$
$277$	−3.77233	−0.226657	−0.113329	+	0.993558i	$0.536151\pi$
$278$	4.08632	0.245081
$279$	0	0
$280$	4.62905	0.276638
$281$	−10.9560	−0.653577	−0.326789	−	0.945097i	$-0.605967\pi$
$281$	−10.9560	−0.653577	−0.326789	+	0.945097i	$0.605967\pi$
$282$	0	0
$283$	17.5697	1.04441	0.522204	−	0.852821i	$-0.325110\pi$
$283$	17.5697	1.04441	0.522204	+	0.852821i	$0.325110\pi$
$284$	−1.67553	−0.0994243
$285$	0	0
$286$	0.0860250	0.00508677
$287$	11.6606	0.688305
$288$	0	0
$289$	−10.8473	−0.638074
$290$	−4.06374	−0.238631
$291$	0	0
$292$	−0.738726	−0.0432307
$293$	−24.5445	−1.43391	−0.716953	−	0.697122i	$-0.754464\pi$
$293$	−24.5445	−1.43391	−0.716953	+	0.697122i	$0.754464\pi$
$294$	0	0
$295$	−7.49087	−0.436135
$296$	−2.43312	−0.141422
$297$	0	0
$298$	−3.08875	−0.178926
$299$	4.93083	0.285157
$300$	0	0
$301$	−20.1876	−1.16359
$302$	7.98530	0.459502
$303$	0	0
$304$	−10.4233	−0.597815
$305$	−5.78404	−0.331193
$306$	0	0
$307$	−11.0689	−0.631735	−0.315868	−	0.948803i	$-0.602296\pi$
$307$	−11.0689	−0.631735	−0.315868	+	0.948803i	$0.602296\pi$
$308$	0.0751435	0.00428170
$309$	0	0
$310$	13.3347	0.757361
$311$	−14.7933	−0.838853	−0.419426	−	0.907789i	$-0.637769\pi$
$311$	−14.7933	−0.838853	−0.419426	+	0.907789i	$0.637769\pi$
$312$	0	0
$313$	27.0525	1.52910	0.764548	−	0.644567i	$-0.222962\pi$
$313$	27.0525	1.52910	0.764548	+	0.644567i	$0.222962\pi$
$314$	29.4123	1.65983
$315$	0	0
$316$	−10.4636	−0.588626
$317$	−0.577998	−0.0324636	−0.0162318	−	0.999868i	$-0.505167\pi$
$317$	−0.577998	−0.0324636	−0.0162318	+	0.999868i	$0.505167\pi$
$318$	0	0
$319$	0.130926	0.00733047
$320$	3.82454	0.213798
$321$	0	0
$322$	17.1627	0.956440
$323$	5.28600	0.294121
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−15.3885	−0.852291
$327$	0	0
$328$	11.8962	0.656855
$329$	−20.8019	−1.14684
$330$	0	0
$331$	−24.1140	−1.32542	−0.662712	−	0.748874i	$-0.730595\pi$
$331$	−24.1140	−1.32542	−0.662712	+	0.748874i	$0.730595\pi$
$332$	−0.327770	−0.0179887
$333$	0	0
$334$	−25.3120	−1.38501
$335$	−6.23591	−0.340704
$336$	0	0
$337$	6.13491	0.334190	0.167095	−	0.985941i	$-0.446561\pi$
$337$	6.13491	0.334190	0.167095	+	0.985941i	$0.446561\pi$
$338$	−1.63404	−0.0888799
$339$	0	0
$340$	1.66211	0.0901405
$341$	−0.429620	−0.0232652
$342$	0	0
$343$	−20.1564	−1.08835
$344$	−20.5954	−1.11043
$345$	0	0
$346$	−15.8120	−0.850057
$347$	3.72399	0.199914	0.0999572	−	0.994992i	$-0.468129\pi$
$347$	3.72399	0.199914	0.0999572	+	0.994992i	$0.468129\pi$
$348$	0	0
$349$	−3.34865	−0.179249	−0.0896246	−	0.995976i	$-0.528567\pi$
$349$	−3.34865	−0.179249	−0.0896246	+	0.995976i	$0.528567\pi$
$350$	−3.48069	−0.186051
$351$	0	0
$352$	0.191948	0.0102309
$353$	35.5626	1.89281	0.946403	−	0.322988i	$-0.104687\pi$
$353$	35.5626	1.89281	0.946403	+	0.322988i	$0.104687\pi$
$354$	0	0
$355$	−2.50050	−0.132713
$356$	3.11520	0.165105
$357$	0	0
$358$	−5.41896	−0.286401
$359$	−9.06902	−0.478645	−0.239322	−	0.970940i	$-0.576925\pi$
$359$	−9.06902	−0.478645	−0.239322	+	0.970940i	$0.576925\pi$
$360$	0	0
$361$	−14.4586	−0.760981
$362$	36.9600	1.94258
$363$	0	0
$364$	−1.42734	−0.0748131
$365$	−1.10245	−0.0577048
$366$	0	0
$367$	−24.6186	−1.28508	−0.642541	−	0.766252i	$-0.722120\pi$
$367$	−24.6186	−1.28508	−0.642541	+	0.766252i	$0.722120\pi$
$368$	24.1174	1.25721
$369$	0	0
$370$	1.82952	0.0951121
$371$	7.75557	0.402649
$372$	0	0
$373$	26.9758	1.39675	0.698376	−	0.715731i	$-0.253906\pi$
$373$	26.9758	1.39675	0.698376	+	0.715731i	$0.253906\pi$
$374$	−0.213383	−0.0110338
$375$	0	0
$376$	−21.2220	−1.09444
$377$	−2.48694	−0.128084
$378$	0	0
$379$	14.2429	0.731611	0.365805	−	0.930691i	$-0.380794\pi$
$379$	14.2429	0.731611	0.365805	+	0.930691i	$0.380794\pi$
$380$	1.42797	0.0732531
$381$	0	0
$382$	9.61158	0.491771
$383$	−25.9736	−1.32719	−0.663595	−	0.748092i	$-0.730970\pi$
$383$	−25.9736	−1.32719	−0.663595	+	0.748092i	$0.730970\pi$
$384$	0	0
$385$	0.112141	0.00571526
$386$	0.750903	0.0382200
$387$	0	0
$388$	4.91902	0.249726
$389$	−25.7906	−1.30764	−0.653819	−	0.756651i	$-0.726834\pi$
$389$	−25.7906	−1.30764	−0.653819	+	0.756651i	$0.726834\pi$
$390$	0	0
$391$	−12.2308	−0.618537
$392$	−5.35159	−0.270296
$393$	0	0
$394$	−3.77657	−0.190261
$395$	−15.6156	−0.785705
$396$	0	0
$397$	−19.2890	−0.968086	−0.484043	−	0.875044i	$-0.660832\pi$
$397$	−19.2890	−0.968086	−0.484043	+	0.875044i	$0.660832\pi$
$398$	−39.8271	−1.99635
$399$	0	0
$400$	−4.89115	−0.244558
$401$	16.4631	0.822128	0.411064	−	0.911607i	$-0.365157\pi$
$401$	16.4631	0.822128	0.411064	+	0.911607i	$0.365157\pi$
$402$	0	0
$403$	8.16059	0.406508
$404$	−0.370458	−0.0184310
$405$	0	0
$406$	−8.65625	−0.429602
$407$	−0.0589438	−0.00292173
$408$	0	0
$409$	−19.7526	−0.976704	−0.488352	−	0.872647i	$-0.662402\pi$
$409$	−19.7526	−0.976704	−0.488352	+	0.872647i	$0.662402\pi$
$410$	−8.94500	−0.441762
$411$	0	0
$412$	−5.25204	−0.258749
$413$	−15.9564	−0.785164
$414$	0	0
$415$	−0.489152	−0.0240115
$416$	−3.64604	−0.178762
$417$	0	0
$418$	−0.183323	−0.00896664
$419$	17.4583	0.852894	0.426447	−	0.904513i	$-0.359765\pi$
$419$	17.4583	0.852894	0.426447	+	0.904513i	$0.359765\pi$
$420$	0	0
$421$	−7.41596	−0.361432	−0.180716	−	0.983535i	$-0.557841\pi$
$421$	−7.41596	−0.361432	−0.180716	+	0.983535i	$0.557841\pi$
$422$	−44.7625	−2.17900
$423$	0	0
$424$	7.91222	0.384251
$425$	2.48047	0.120321
$426$	0	0
$427$	−12.3207	−0.596239
$428$	−13.0111	−0.628916
$429$	0	0
$430$	15.4861	0.746808
$431$	−33.6097	−1.61892	−0.809461	−	0.587174i	$-0.800240\pi$
$431$	−33.6097	−1.61892	−0.809461	+	0.587174i	$0.800240\pi$
$432$	0	0
$433$	30.2826	1.45529	0.727644	−	0.685955i	$-0.240615\pi$
$433$	30.2826	1.45529	0.727644	+	0.685955i	$0.240615\pi$
$434$	28.4045	1.36346
$435$	0	0
$436$	−1.43063	−0.0685147
$437$	−10.5078	−0.502658
$438$	0	0
$439$	−28.7370	−1.37154	−0.685772	−	0.727817i	$-0.740535\pi$
$439$	−28.7370	−1.37154	−0.685772	+	0.727817i	$0.740535\pi$
$440$	0.114407	0.00545412
$441$	0	0
$442$	4.05318	0.192790
$443$	22.4218	1.06529	0.532646	−	0.846338i	$-0.321198\pi$
$443$	22.4218	1.06529	0.532646	+	0.846338i	$0.321198\pi$
$444$	0	0
$445$	4.64901	0.220384
$446$	−41.3997	−1.96033
$447$	0	0
$448$	8.14671	0.384896
$449$	−16.0149	−0.755791	−0.377895	−	0.925848i	$-0.623352\pi$
$449$	−16.0149	−0.755791	−0.377895	+	0.925848i	$0.623352\pi$
$450$	0	0
$451$	0.288192	0.0135704
$452$	4.17813	0.196523
$453$	0	0
$454$	−34.7097	−1.62901
$455$	−2.13012	−0.0998614
$456$	0	0
$457$	1.06556	0.0498448	0.0249224	−	0.999689i	$-0.492066\pi$
$457$	1.06556	0.0498448	0.0249224	+	0.999689i	$0.492066\pi$
$458$	−19.9904	−0.934089
$459$	0	0
$460$	−3.30404	−0.154052
$461$	−24.6387	−1.14754	−0.573769	−	0.819017i	$-0.694519\pi$
$461$	−24.6387	−1.14754	−0.573769	+	0.819017i	$0.694519\pi$
$462$	0	0
$463$	7.12328	0.331047	0.165524	−	0.986206i	$-0.447069\pi$
$463$	7.12328	0.331047	0.165524	+	0.986206i	$0.447069\pi$
$464$	−12.1640	−0.564698
$465$	0	0
$466$	18.6359	0.863290
$467$	−20.4662	−0.947063	−0.473531	−	0.880777i	$-0.657021\pi$
$467$	−20.4662	−0.947063	−0.473531	+	0.880777i	$0.657021\pi$
$468$	0	0
$469$	−13.2832	−0.613361
$470$	15.9574	0.736058
$471$	0	0
$472$	−16.2787	−0.749289
$473$	−0.498935	−0.0229411
$474$	0	0
$475$	2.13105	0.0977791
$476$	3.54048	0.162278
$477$	0	0
$478$	24.2770	1.11041
$479$	14.4699	0.661145	0.330573	−	0.943781i	$-0.392758\pi$
$479$	14.4699	0.661145	0.330573	+	0.943781i	$0.392758\pi$
$480$	0	0
$481$	1.11963	0.0510508
$482$	30.3780	1.38368
$483$	0	0
$484$	−7.36900	−0.334954
$485$	7.34097	0.333336
$486$	0	0
$487$	29.8595	1.35306	0.676531	−	0.736414i	$-0.263483\pi$
$487$	29.8595	1.35306	0.676531	+	0.736414i	$0.263483\pi$
$488$	−12.5695	−0.568996
$489$	0	0
$490$	4.02399	0.181785
$491$	−22.0470	−0.994965	−0.497483	−	0.867474i	$-0.665742\pi$
$491$	−22.0470	−0.994965	−0.497483	+	0.867474i	$0.665742\pi$
$492$	0	0
$493$	6.16877	0.277827
$494$	3.48221	0.156672
$495$	0	0
$496$	39.9147	1.79222
$497$	−5.32635	−0.238920
$498$	0	0
$499$	21.1723	0.947802	0.473901	−	0.880578i	$-0.342845\pi$
$499$	21.1723	0.947802	0.473901	+	0.880578i	$0.342845\pi$
$500$	0.670078	0.0299668
$501$	0	0
$502$	49.1392	2.19319
$503$	40.3143	1.79752	0.898762	−	0.438436i	$-0.144467\pi$
$503$	40.3143	1.79752	0.898762	+	0.438436i	$0.144467\pi$
$504$	0	0
$505$	−0.552859	−0.0246019
$506$	0.424175	0.0188569
$507$	0	0
$508$	−7.10524	−0.315244
$509$	−19.0167	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$509$	−19.0167	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$510$	0	0
$511$	−2.34834	−0.103885
$512$	3.42501	0.151366
$513$	0	0
$514$	4.96851	0.219152
$515$	−7.83795	−0.345381
$516$	0	0
$517$	−0.514117	−0.0226108
$518$	3.89709	0.171228
$519$	0	0
$520$	−2.17314	−0.0952986
$521$	33.4407	1.46507	0.732533	−	0.680732i	$-0.238338\pi$
$521$	33.4407	1.46507	0.732533	+	0.680732i	$0.238338\pi$
$522$	0	0
$523$	−6.40250	−0.279962	−0.139981	−	0.990154i	$-0.544704\pi$
$523$	−6.40250	−0.279962	−0.139981	+	0.990154i	$0.544704\pi$
$524$	10.8501	0.473989
$525$	0	0
$526$	−7.84504	−0.342060
$527$	−20.2421	−0.881761
$528$	0	0
$529$	1.31310	0.0570915
$530$	−5.94939	−0.258425
$531$	0	0
$532$	3.04173	0.131876
$533$	−5.47417	−0.237113
$534$	0	0
$535$	−19.4173	−0.839484
$536$	−13.5515	−0.585336
$537$	0	0
$538$	−27.5700	−1.18863
$539$	−0.129646	−0.00558423
$540$	0	0
$541$	−17.4745	−0.751286	−0.375643	−	0.926764i	$-0.622578\pi$
$541$	−17.4745	−0.751286	−0.375643	+	0.926764i	$0.622578\pi$
$542$	3.82527	0.164309
$543$	0	0
$544$	9.04389	0.387754
$545$	−2.13502	−0.0914542
$546$	0	0
$547$	−10.0747	−0.430761	−0.215381	−	0.976530i	$-0.569099\pi$
$547$	−10.0747	−0.430761	−0.215381	+	0.976530i	$0.569099\pi$
$548$	13.9166	0.594486
$549$	0	0
$550$	−0.0860250	−0.00366812
$551$	5.29977	0.225778
$552$	0	0
$553$	−33.2630	−1.41449
$554$	6.16413	0.261889
$555$	0	0
$556$	−1.67570	−0.0710654
$557$	−21.2315	−0.899607	−0.449804	−	0.893127i	$-0.648506\pi$
$557$	−21.2315	−0.899607	−0.449804	+	0.893127i	$0.648506\pi$
$558$	0	0
$559$	9.47722	0.400844
$560$	−10.4187	−0.440271
$561$	0	0
$562$	17.9024	0.755169
$563$	−12.3449	−0.520276	−0.260138	−	0.965571i	$-0.583768\pi$
$563$	−12.3449	−0.520276	−0.260138	+	0.965571i	$0.583768\pi$
$564$	0	0
$565$	6.23529	0.262321
$566$	−28.7095	−1.20675
$567$	0	0
$568$	−5.43394	−0.228003
$569$	−7.75747	−0.325210	−0.162605	−	0.986691i	$-0.551990\pi$
$569$	−7.75747	−0.325210	−0.162605	+	0.986691i	$0.551990\pi$
$570$	0	0
$571$	−10.2139	−0.427440	−0.213720	−	0.976895i	$-0.568558\pi$
$571$	−10.2139	−0.427440	−0.213720	+	0.976895i	$0.568558\pi$
$572$	−0.0352767	−0.00147499
$573$	0	0
$574$	−19.0539	−0.795294
$575$	−4.93083	−0.205630
$576$	0	0
$577$	−44.5539	−1.85480	−0.927402	−	0.374065i	$-0.877964\pi$
$577$	−44.5539	−1.85480	−0.927402	+	0.374065i	$0.877964\pi$
$578$	17.7248	0.737256
$579$	0	0
$580$	1.66644	0.0691952
$581$	−1.04195	−0.0432274
$582$	0	0
$583$	0.191678	0.00793851
$584$	−2.39578	−0.0991380
$585$	0	0
$586$	40.1066	1.65679
$587$	−20.9069	−0.862921	−0.431461	−	0.902132i	$-0.642002\pi$
$587$	−20.9069	−0.862921	−0.431461	+	0.902132i	$0.642002\pi$
$588$	0	0
$589$	−17.3906	−0.716567
$590$	12.2404	0.503928
$591$	0	0
$592$	5.47628	0.225074
$593$	7.48843	0.307513	0.153756	−	0.988109i	$-0.450863\pi$
$593$	7.48843	0.307513	0.153756	+	0.988109i	$0.450863\pi$
$594$	0	0
$595$	5.28369	0.216610
$596$	1.26662	0.0518827
$597$	0	0
$598$	−8.05716	−0.329482
$599$	27.9314	1.14125	0.570623	−	0.821212i	$-0.306702\pi$
$599$	27.9314	1.14125	0.570623	+	0.821212i	$0.306702\pi$
$600$	0	0
$601$	32.7780	1.33704	0.668521	−	0.743693i	$-0.266927\pi$
$601$	32.7780	1.33704	0.668521	+	0.743693i	$0.266927\pi$
$602$	32.9873	1.34446
$603$	0	0
$604$	−3.27457	−0.133240
$605$	−10.9972	−0.447101
$606$	0	0
$607$	39.1704	1.58988	0.794940	−	0.606689i	$-0.207503\pi$
$607$	39.1704	1.58988	0.794940	+	0.606689i	$0.207503\pi$
$608$	7.76988	0.315110
$609$	0	0
$610$	9.45133	0.382673
$611$	9.76560	0.395074
$612$	0	0
$613$	−10.2019	−0.412051	−0.206026	−	0.978547i	$-0.566053\pi$
$613$	−10.2019	−0.412051	−0.206026	+	0.978547i	$0.566053\pi$
$614$	18.0870	0.729932
$615$	0	0
$616$	0.243699	0.00981893
$617$	9.68947	0.390084	0.195042	−	0.980795i	$-0.437516\pi$
$617$	9.68947	0.390084	0.195042	+	0.980795i	$0.437516\pi$
$618$	0	0
$619$	−19.8750	−0.798842	−0.399421	−	0.916768i	$-0.630789\pi$
$619$	−19.8750	−0.798842	−0.399421	+	0.916768i	$0.630789\pi$
$620$	−5.46823	−0.219610
$621$	0	0
$622$	24.1728	0.969243
$623$	9.90292	0.396752
$624$	0	0
$625$	1.00000	0.0400000
$626$	−44.2047	−1.76678
$627$	0	0
$628$	−12.0612	−0.481296
$629$	−2.77721	−0.110735
$630$	0	0
$631$	−38.9270	−1.54966	−0.774830	−	0.632170i	$-0.782165\pi$
$631$	−38.9270	−1.54966	−0.774830	+	0.632170i	$0.782165\pi$
$632$	−33.9349	−1.34986
$633$	0	0
$634$	0.944470	0.0375097
$635$	−10.6036	−0.420791
$636$	0	0
$637$	2.46260	0.0975719
$638$	−0.213939	−0.00846991
$639$	0	0
$640$	−13.5415	−0.535275
$641$	7.89412	0.311799	0.155899	−	0.987773i	$-0.450172\pi$
$641$	7.89412	0.311799	0.155899	+	0.987773i	$0.450172\pi$
$642$	0	0
$643$	35.7751	1.41083	0.705415	−	0.708794i	$-0.250760\pi$
$643$	35.7751	1.41083	0.705415	+	0.708794i	$0.250760\pi$
$644$	−7.03799	−0.277336
$645$	0	0
$646$	−8.63752	−0.339839
$647$	1.47889	0.0581412	0.0290706	−	0.999577i	$-0.490745\pi$
$647$	1.47889	0.0581412	0.0290706	+	0.999577i	$0.490745\pi$
$648$	0	0
$649$	−0.394362	−0.0154801
$650$	1.63404	0.0640922
$651$	0	0
$652$	6.31044	0.247136
$653$	6.95046	0.271992	0.135996	−	0.990709i	$-0.456577\pi$
$653$	6.95046	0.271992	0.135996	+	0.990709i	$0.456577\pi$
$654$	0	0
$655$	16.1923	0.632686
$656$	−26.7750	−1.04539
$657$	0	0
$658$	33.9910	1.32511
$659$	−30.9546	−1.20582	−0.602910	−	0.797810i	$-0.705992\pi$
$659$	−30.9546	−1.20582	−0.602910	+	0.797810i	$0.705992\pi$
$660$	0	0
$661$	3.57678	0.139121	0.0695603	−	0.997578i	$-0.477840\pi$
$661$	3.57678	0.139121	0.0695603	+	0.997578i	$0.477840\pi$
$662$	39.4032	1.53145
$663$	0	0
$664$	−1.06300	−0.0412522
$665$	4.53938	0.176029
$666$	0	0
$667$	−12.2627	−0.474812
$668$	10.3798	0.401607
$669$	0	0
$670$	10.1897	0.393663
$671$	−0.304505	−0.0117553
$672$	0	0
$673$	−28.0993	−1.08315	−0.541575	−	0.840653i	$-0.682172\pi$
$673$	−28.0993	−1.08315	−0.541575	+	0.840653i	$0.682172\pi$
$674$	−10.0247	−0.386136
$675$	0	0
$676$	0.670078	0.0257722
$677$	−44.2934	−1.70233	−0.851167	−	0.524895i	$-0.824105\pi$
$677$	−44.2934	−1.70233	−0.851167	+	0.524895i	$0.824105\pi$
$678$	0	0
$679$	15.6371	0.600098
$680$	5.39042	0.206713
$681$	0	0
$682$	0.702016	0.0268816
$683$	1.69041	0.0646816	0.0323408	−	0.999477i	$-0.489704\pi$
$683$	1.69041	0.0646816	0.0323408	+	0.999477i	$0.489704\pi$
$684$	0	0
$685$	20.7686	0.793526
$686$	32.9364	1.25752
$687$	0	0
$688$	46.3545	1.76725
$689$	−3.64091	−0.138708
$690$	0	0
$691$	−10.8144	−0.411398	−0.205699	−	0.978615i	$-0.565947\pi$
$691$	−10.8144	−0.411398	−0.205699	+	0.978615i	$0.565947\pi$
$692$	6.48409	0.246488
$693$	0	0
$694$	−6.08514	−0.230989
$695$	−2.50075	−0.0948589
$696$	0	0
$697$	13.5785	0.514323
$698$	5.47182	0.207111
$699$	0	0
$700$	1.42734	0.0539485
$701$	32.1755	1.21525	0.607626	−	0.794224i	$-0.292122\pi$
$701$	32.1755	1.21525	0.607626	+	0.794224i	$0.292122\pi$
$702$	0	0
$703$	−2.38598	−0.0899891
$704$	0.201346	0.00758850
$705$	0	0
$706$	−58.1106	−2.18702
$707$	−1.17765	−0.0442902
$708$	0	0
$709$	31.9120	1.19848	0.599240	−	0.800570i	$-0.295470\pi$
$709$	31.9120	1.19848	0.599240	+	0.800570i	$0.295470\pi$
$710$	4.08591	0.153341
$711$	0	0
$712$	10.1030	0.378624
$713$	40.2385	1.50694
$714$	0	0
$715$	−0.0526457	−0.00196884
$716$	2.22218	0.0830467
$717$	0	0
$718$	14.8191	0.553045
$719$	−20.6891	−0.771572	−0.385786	−	0.922588i	$-0.626070\pi$
$719$	−20.6891	−0.771572	−0.385786	+	0.922588i	$0.626070\pi$
$720$	0	0
$721$	−16.6958	−0.621782
$722$	23.6260	0.879267
$723$	0	0
$724$	−15.1564	−0.563283
$725$	2.48694	0.0923625
$726$	0	0
$727$	−24.6791	−0.915299	−0.457649	−	0.889133i	$-0.651308\pi$
$727$	−24.6791	−0.915299	−0.457649	+	0.889133i	$0.651308\pi$
$728$	−4.62905	−0.171564
$729$	0	0
$730$	1.80144	0.0666744
$731$	−23.5080	−0.869474
$732$	0	0
$733$	−21.7459	−0.803203	−0.401602	−	0.915814i	$-0.631546\pi$
$733$	−21.7459	−0.803203	−0.401602	+	0.915814i	$0.631546\pi$
$734$	40.2277	1.48483
$735$	0	0
$736$	−17.9780	−0.662678
$737$	−0.328294	−0.0120929
$738$	0	0
$739$	43.0460	1.58347	0.791736	−	0.610864i	$-0.209178\pi$
$739$	43.0460	1.58347	0.791736	+	0.610864i	$0.209178\pi$
$740$	−0.750240	−0.0275794
$741$	0	0
$742$	−12.6729	−0.465236
$743$	−7.89816	−0.289755	−0.144878	−	0.989450i	$-0.546279\pi$
$743$	−7.89816	−0.289755	−0.144878	+	0.989450i	$0.546279\pi$
$744$	0	0
$745$	1.89025	0.0692536
$746$	−44.0794	−1.61386
$747$	0	0
$748$	0.0875029	0.00319942
$749$	−41.3612	−1.51130
$750$	0	0
$751$	38.6450	1.41018	0.705089	−	0.709119i	$-0.250907\pi$
$751$	38.6450	1.41018	0.705089	+	0.709119i	$0.250907\pi$
$752$	47.7650	1.74181
$753$	0	0
$754$	4.06374	0.147993
$755$	−4.88685	−0.177851
$756$	0	0
$757$	−15.3437	−0.557678	−0.278839	−	0.960338i	$-0.589950\pi$
$757$	−15.3437	−0.557678	−0.278839	+	0.960338i	$0.589950\pi$
$758$	−23.2735	−0.845331
$759$	0	0
$760$	4.63107	0.167986
$761$	−48.7170	−1.76599	−0.882995	−	0.469382i	$-0.844477\pi$
$761$	−48.7170	−1.76599	−0.882995	+	0.469382i	$0.844477\pi$
$762$	0	0
$763$	−4.54784	−0.164643
$764$	−3.94147	−0.142597
$765$	0	0
$766$	42.4418	1.53349
$767$	7.49087	0.270480
$768$	0	0
$769$	6.37109	0.229747	0.114874	−	0.993380i	$-0.463354\pi$
$769$	6.37109	0.229747	0.114874	+	0.993380i	$0.463354\pi$
$770$	−0.183243	−0.00660363
$771$	0	0
$772$	−0.307927	−0.0110825
$773$	−29.8732	−1.07446	−0.537232	−	0.843435i	$-0.680530\pi$
$773$	−29.8732	−1.07446	−0.537232	+	0.843435i	$0.680530\pi$
$774$	0	0
$775$	−8.16059	−0.293137
$776$	15.9530	0.572679
$777$	0	0
$778$	42.1429	1.51090
$779$	11.6657	0.417967
$780$	0	0
$781$	−0.131641	−0.00471047
$782$	19.9856	0.714682
$783$	0	0
$784$	12.0450	0.430178
$785$	−17.9998	−0.642439
$786$	0	0
$787$	−23.6556	−0.843232	−0.421616	−	0.906774i	$-0.638537\pi$
$787$	−23.6556	−0.843232	−0.421616	+	0.906774i	$0.638537\pi$
$788$	1.54868	0.0551693
$789$	0	0
$790$	25.5164	0.907834
$791$	13.2819	0.472250
$792$	0	0
$793$	5.78404	0.205397
$794$	31.5189	1.11856
$795$	0	0
$796$	16.3321	0.578876
$797$	30.0006	1.06268	0.531339	−	0.847160i	$-0.321689\pi$
$797$	30.0006	1.06268	0.531339	+	0.847160i	$0.321689\pi$
$798$	0	0
$799$	−24.2233	−0.856959
$800$	3.64604	0.128907
$801$	0	0
$802$	−26.9013	−0.949918
$803$	−0.0580392	−0.00204816
$804$	0	0
$805$	−10.5032	−0.370191
$806$	−13.3347	−0.469695
$807$	0	0
$808$	−1.20144	−0.0422665
$809$	−37.0787	−1.30362	−0.651808	−	0.758384i	$-0.725989\pi$
$809$	−37.0787	−1.30362	−0.651808	+	0.758384i	$0.725989\pi$
$810$	0	0
$811$	−42.7582	−1.50144	−0.750722	−	0.660618i	$-0.770294\pi$
$811$	−42.7582	−1.50144	−0.750722	+	0.660618i	$0.770294\pi$
$812$	3.54971	0.124570
$813$	0	0
$814$	0.0963163	0.00337588
$815$	9.41748	0.329880
$816$	0	0
$817$	−20.1964	−0.706583
$818$	32.2765	1.12852
$819$	0	0
$820$	3.66812	0.128096
$821$	−28.5669	−0.996992	−0.498496	−	0.866892i	$-0.666114\pi$
$821$	−28.5669	−0.996992	−0.498496	+	0.866892i	$0.666114\pi$
$822$	0	0
$823$	32.0041	1.11559	0.557797	−	0.829977i	$-0.311647\pi$
$823$	32.0041	1.11559	0.557797	+	0.829977i	$0.311647\pi$
$824$	−17.0330	−0.593372
$825$	0	0
$826$	26.0734	0.907209
$827$	−31.4986	−1.09531	−0.547657	−	0.836703i	$-0.684480\pi$
$827$	−31.4986	−1.09531	−0.547657	+	0.836703i	$0.684480\pi$
$828$	0	0
$829$	32.3851	1.12478	0.562390	−	0.826872i	$-0.309882\pi$
$829$	32.3851	1.12478	0.562390	+	0.826872i	$0.309882\pi$
$830$	0.799292	0.0277438
$831$	0	0
$832$	−3.82454	−0.132592
$833$	−6.10842	−0.211644
$834$	0	0
$835$	15.4905	0.536070
$836$	0.0751763	0.00260003
$837$	0	0
$838$	−28.5275	−0.985467
$839$	−46.0763	−1.59073	−0.795365	−	0.606131i	$-0.792721\pi$
$839$	−46.0763	−1.59073	−0.795365	+	0.606131i	$0.792721\pi$
$840$	0	0
$841$	−22.8152	−0.786729
$842$	12.1180	0.417612
$843$	0	0
$844$	18.3560	0.631839
$845$	1.00000	0.0344010
$846$	0	0
$847$	−23.4254	−0.804905
$848$	−17.8083	−0.611538
$849$	0	0
$850$	−4.05318	−0.139023
$851$	5.52071	0.189248
$852$	0	0
$853$	48.7573	1.66942	0.834709	−	0.550692i	$-0.185636\pi$
$853$	48.7573	1.66942	0.834709	+	0.550692i	$0.185636\pi$
$854$	20.1324	0.688918
$855$	0	0
$856$	−42.1966	−1.44225
$857$	−31.9479	−1.09132	−0.545660	−	0.838007i	$-0.683721\pi$
$857$	−31.9479	−1.09132	−0.545660	+	0.838007i	$0.683721\pi$
$858$	0	0
$859$	−7.33499	−0.250267	−0.125133	−	0.992140i	$-0.539936\pi$
$859$	−7.33499	−0.250267	−0.125133	+	0.992140i	$0.539936\pi$
$860$	−6.35048	−0.216549
$861$	0	0
$862$	54.9195	1.87056
$863$	−28.4742	−0.969273	−0.484637	−	0.874716i	$-0.661048\pi$
$863$	−28.4742	−0.969273	−0.484637	+	0.874716i	$0.661048\pi$
$864$	0	0
$865$	9.67663	0.329015
$866$	−49.4829	−1.68150
$867$	0	0
$868$	−11.6480	−0.395358
$869$	−0.822093	−0.0278876
$870$	0	0
$871$	6.23591	0.211296
$872$	−4.63970	−0.157120
$873$	0	0
$874$	17.1702	0.580790
$875$	2.13012	0.0720111
$876$	0	0
$877$	45.8529	1.54834	0.774171	−	0.632977i	$-0.218167\pi$
$877$	45.8529	1.54834	0.774171	+	0.632977i	$0.218167\pi$
$878$	46.9574	1.58473
$879$	0	0
$880$	−0.257498	−0.00868026
$881$	37.4454	1.26157	0.630784	−	0.775959i	$-0.282733\pi$
$881$	37.4454	1.26157	0.630784	+	0.775959i	$0.282733\pi$
$882$	0	0
$883$	50.8068	1.70978	0.854892	−	0.518805i	$-0.173623\pi$
$883$	50.8068	1.70978	0.854892	+	0.518805i	$0.173623\pi$
$884$	−1.66211	−0.0559028
$885$	0	0
$886$	−36.6381	−1.23088
$887$	45.2968	1.52092	0.760459	−	0.649386i	$-0.224974\pi$
$887$	45.2968	1.52092	0.760459	+	0.649386i	$0.224974\pi$
$888$	0	0
$889$	−22.5869	−0.757541
$890$	−7.59665	−0.254640
$891$	0	0
$892$	16.9770	0.568432
$893$	−20.8109	−0.696412
$894$	0	0
$895$	3.31630	0.110852
$896$	−28.8450	−0.963644
$897$	0	0
$898$	26.1690	0.873270
$899$	−20.2949	−0.676872
$900$	0	0
$901$	9.03118	0.300872
$902$	−0.470916	−0.0156798
$903$	0	0
$904$	13.5502	0.450672
$905$	−22.6189	−0.751876
$906$	0	0
$907$	14.3690	0.477116	0.238558	−	0.971128i	$-0.423325\pi$
$907$	14.3690	0.477116	0.238558	+	0.971128i	$0.423325\pi$
$908$	14.2336	0.472359
$909$	0	0
$910$	3.48069	0.115384
$911$	−53.9063	−1.78600	−0.892998	−	0.450060i	$-0.851403\pi$
$911$	−53.9063	−1.78600	−0.892998	+	0.450060i	$0.851403\pi$
$912$	0	0
$913$	−0.0257517	−0.000852258 0
$914$	−1.74117	−0.0575926
$915$	0	0
$916$	8.19755	0.270855
$917$	34.4915	1.13901
$918$	0	0
$919$	−15.6573	−0.516487	−0.258244	−	0.966080i	$-0.583144\pi$
$919$	−15.6573	−0.516487	−0.258244	+	0.966080i	$0.583144\pi$
$920$	−10.7154	−0.353276
$921$	0	0
$922$	40.2605	1.32591
$923$	2.50050	0.0823049
$924$	0	0
$925$	−1.11963	−0.0368132
$926$	−11.6397	−0.382505
$927$	0	0
$928$	9.06746	0.297654
$929$	23.1439	0.759327	0.379664	−	0.925125i	$-0.376040\pi$
$929$	23.1439	0.759327	0.379664	+	0.925125i	$0.376040\pi$
$930$	0	0
$931$	−5.24792	−0.171994
$932$	−7.64211	−0.250326
$933$	0	0
$934$	33.4425	1.09427
$935$	0.130586	0.00427062
$936$	0	0
$937$	14.6354	0.478118	0.239059	−	0.971005i	$-0.423161\pi$
$937$	14.6354	0.478118	0.239059	+	0.971005i	$0.423161\pi$
$938$	21.7052	0.708702
$939$	0	0
$940$	−6.54371	−0.213432
$941$	22.0393	0.718462	0.359231	−	0.933249i	$-0.383039\pi$
$941$	22.0393	0.718462	0.359231	+	0.933249i	$0.383039\pi$
$942$	0	0
$943$	−26.9922	−0.878987
$944$	36.6390	1.19250
$945$	0	0
$946$	0.815279	0.0265070
$947$	50.3906	1.63747	0.818737	−	0.574168i	$-0.194674\pi$
$947$	50.3906	1.63747	0.818737	+	0.574168i	$0.194674\pi$
$948$	0	0
$949$	1.10245	0.0357870
$950$	−3.48221	−0.112978
$951$	0	0
$952$	11.4822	0.372141
$953$	7.28058	0.235841	0.117921	−	0.993023i	$-0.462377\pi$
$953$	7.28058	0.235841	0.117921	+	0.993023i	$0.462377\pi$
$954$	0	0
$955$	−5.88211	−0.190340
$956$	−9.95540	−0.321981
$957$	0	0
$958$	−23.6443	−0.763913
$959$	44.2395	1.42857
$960$	0	0
$961$	35.5953	1.14824
$962$	−1.82952	−0.0589861
$963$	0	0
$964$	−12.4573	−0.401221
$965$	−0.459539	−0.0147931
$966$	0	0
$967$	1.64327	0.0528441	0.0264220	−	0.999651i	$-0.491589\pi$
$967$	1.64327	0.0528441	0.0264220	+	0.999651i	$0.491589\pi$
$968$	−23.8985	−0.768128
$969$	0	0
$970$	−11.9954	−0.385150
$971$	−59.5135	−1.90988	−0.954939	−	0.296803i	$-0.904080\pi$
$971$	−59.5135	−1.90988	−0.954939	+	0.296803i	$0.904080\pi$
$972$	0	0
$973$	−5.32689	−0.170772
$974$	−48.7915	−1.56338
$975$	0	0
$976$	28.2906	0.905560
$977$	15.3273	0.490363	0.245181	−	0.969477i	$-0.421153\pi$
$977$	15.3273	0.490363	0.245181	+	0.969477i	$0.421153\pi$
$978$	0	0
$979$	0.244750	0.00782225
$980$	−1.65014	−0.0527117
$981$	0	0
$982$	36.0255	1.14962
$983$	3.26954	0.104282	0.0521411	−	0.998640i	$-0.483395\pi$
$983$	3.26954	0.104282	0.0521411	+	0.998640i	$0.483395\pi$
$984$	0	0
$985$	2.31119	0.0736405
$986$	−10.0800	−0.321013
$987$	0	0
$988$	−1.42797	−0.0454297
$989$	46.7306	1.48595
$990$	0	0
$991$	−39.2309	−1.24621	−0.623106	−	0.782138i	$-0.714129\pi$
$991$	−39.2309	−1.24621	−0.623106	+	0.782138i	$0.714129\pi$
$992$	−29.7538	−0.944685
$993$	0	0
$994$	8.70346	0.276057
$995$	24.3734	0.772690
$996$	0	0
$997$	−32.8254	−1.03959	−0.519795	−	0.854291i	$-0.673992\pi$
$997$	−32.8254	−1.03959	−0.519795	+	0.854291i	$0.673992\pi$
$998$	−34.5963	−1.09513
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.ba.1.3		8
3.2	odd	2		5265.2.a.bf.1.6		8
9.2	odd	6		585.2.i.e.391.3	yes	16
9.4	even	3		1755.2.i.f.586.6		16
9.5	odd	6		585.2.i.e.196.3	✓	16
9.7	even	3		1755.2.i.f.1171.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.e.196.3	✓	16	9.5	odd	6
585.2.i.e.391.3	yes	16	9.2	odd	6
1755.2.i.f.586.6		16	9.4	even	3
1755.2.i.f.1171.6		16	9.7	even	3
5265.2.a.ba.1.3		8	1.1	even	1	trivial
5265.2.a.bf.1.6		8	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:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q-1.63404 q^{2} +0.670078 q^{4} +1.00000 q^{5} +2.13012 q^{7} +2.17314 q^{8} +O(q^{10})\)	\(q-1.63404 q^{2} +0.670078 q^{4} +1.00000 q^{5} +2.13012 q^{7} +2.17314 q^{8} -1.63404 q^{10} +0.0526457 q^{11} -1.00000 q^{13} -3.48069 q^{14} -4.89115 q^{16} +2.48047 q^{17} +2.13105 q^{19} +0.670078 q^{20} -0.0860250 q^{22} -4.93083 q^{23} +1.00000 q^{25} +1.63404 q^{26} +1.42734 q^{28} +2.48694 q^{29} -8.16059 q^{31} +3.64604 q^{32} -4.05318 q^{34} +2.13012 q^{35} -1.11963 q^{37} -3.48221 q^{38} +2.17314 q^{40} +5.47417 q^{41} -9.47722 q^{43} +0.0352767 q^{44} +8.05716 q^{46} -9.76560 q^{47} -2.46260 q^{49} -1.63404 q^{50} -0.670078 q^{52} +3.64091 q^{53} +0.0526457 q^{55} +4.62905 q^{56} -4.06374 q^{58} -7.49087 q^{59} -5.78404 q^{61} +13.3347 q^{62} +3.82454 q^{64} -1.00000 q^{65} -6.23591 q^{67} +1.66211 q^{68} -3.48069 q^{70} -2.50050 q^{71} -1.10245 q^{73} +1.82952 q^{74} +1.42797 q^{76} +0.112141 q^{77} -15.6156 q^{79} -4.89115 q^{80} -8.94500 q^{82} -0.489152 q^{83} +2.48047 q^{85} +15.4861 q^{86} +0.114407 q^{88} +4.64901 q^{89} -2.13012 q^{91} -3.30404 q^{92} +15.9574 q^{94} +2.13105 q^{95} +7.34097 q^{97} +4.02399 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) \) 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8	\( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} - 8 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} - 10 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 3 q^{26} - 34 q^{28} - 14 q^{29} - 31 q^{31} - q^{32} - 7 q^{34} - 11 q^{35} + q^{37} - 9 q^{38} + 6 q^{40} + 12 q^{41} + 15 q^{43} - 16 q^{44} - 32 q^{46} + 18 q^{47} + 17 q^{49} - 3 q^{50} - 9 q^{52} - 2 q^{53} - 6 q^{55} - 16 q^{56} - 42 q^{58} - 24 q^{59} - 9 q^{61} + 20 q^{62} - 30 q^{64} - 8 q^{65} - 18 q^{67} + 14 q^{68} - 10 q^{70} - 10 q^{71} + 6 q^{73} + 37 q^{74} - 53 q^{76} + 34 q^{77} - 3 q^{79} + 11 q^{80} - 34 q^{82} + 10 q^{83} + 2 q^{85} - 60 q^{86} - 14 q^{88} + 13 q^{89} + 11 q^{91} - 5 q^{92} + 17 q^{94} - 10 q^{95} - 34 q^{97} + 30 q^{98}+O(q^{100}) \) 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8 - 3 * q^10 - 6 * q^11 - 8 * q^13 - 10 * q^14 + 11 * q^16 + 2 * q^17 - 10 * q^19 + 9 * q^20 + 3 * q^22 - 6 * q^23 + 8 * q^25 + 3 * q^26 - 34 * q^28 - 14 * q^29 - 31 * q^31 - q^32 - 7 * q^34 - 11 * q^35 + q^37 - 9 * q^38 + 6 * q^40 + 12 * q^41 + 15 * q^43 - 16 * q^44 - 32 * q^46 + 18 * q^47 + 17 * q^49 - 3 * q^50 - 9 * q^52 - 2 * q^53 - 6 * q^55 - 16 * q^56 - 42 * q^58 - 24 * q^59 - 9 * q^61 + 20 * q^62 - 30 * q^64 - 8 * q^65 - 18 * q^67 + 14 * q^68 - 10 * q^70 - 10 * q^71 + 6 * q^73 + 37 * q^74 - 53 * q^76 + 34 * q^77 - 3 * q^79 + 11 * q^80 - 34 * q^82 + 10 * q^83 + 2 * q^85 - 60 * q^86 - 14 * q^88 + 13 * q^89 + 11 * q^91 - 5 * q^92 + 17 * q^94 - 10 * q^95 - 34 * q^97 + 30 * q^98

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