LMFDB - Embedded newform 5220.2.a.y.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.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:	$41.6819098551$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 22x^{5} + 38x^{4} + 81x^{3} - 75x^{2} - 42x + 18 $ x^7 - 3x^6 - 22x^5 + 38x^4 + 81x^3 - 75x^2 - 42x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.73648$ of defining polynomial
Character	$\chi$	$=$	5220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.23366 q^{7} -0.862773 q^{11} +6.19586 q^{13} +6.72420 q^{17} +3.79733 q^{19} -0.114019 q^{23} +1.00000 q^{25} -1.00000 q^{29} +1.48490 q^{31} -2.23366 q^{35} +5.84818 q^{37} -3.84818 q^{41} -2.66310 q^{43} -11.5650 q^{47} -2.01078 q^{49} +6.85596 q^{53} +0.862773 q^{55} +0.814916 q^{59} -0.312424 q^{61} -6.19586 q^{65} -6.15806 q^{67} +2.31242 q^{71} +10.9170 q^{73} -1.92714 q^{77} -6.57956 q^{79} -8.62530 q^{83} -6.72420 q^{85} -8.93763 q^{89} +13.8394 q^{91} -3.79733 q^{95} +6.87902 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{5} - 3 q^{11} + 2 q^{13} - 8 q^{17} + 6 q^{19} - q^{23} + 7 q^{25} - 7 q^{29} - 2 q^{31} + 15 q^{37} - q^{41} + 9 q^{43} - 4 q^{47} + 29 q^{49} - 17 q^{53} + 3 q^{55} + 4 q^{59} + 6 q^{61}+ \cdots + 15 q^{97}+O(q^{100}) $ 7 * q - 7 * q^5 - 3 * q^11 + 2 * q^13 - 8 * q^17 + 6 * q^19 - q^23 + 7 * q^25 - 7 * q^29 - 2 * q^31 + 15 * q^37 - q^41 + 9 * q^43 - 4 * q^47 + 29 * q^49 - 17 * q^53 + 3 * q^55 + 4 * q^59 + 6 * q^61 - 2 * q^65 + 24 * q^67 + 8 * q^71 + 29 * q^73 + 8 * q^79 - 7 * q^83 + 8 * q^85 + 4 * q^89 + 26 * q^91 - 6 * q^95 + 15 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.23366	0.844243	0.422122	−	0.906539i	$-0.361286\pi$
$7$	2.23366	0.844243	0.422122	+	0.906539i	$0.361286\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.862773	−0.260136	−0.130068	−	0.991505i	$-0.541520\pi$
$11$	−0.862773	−0.260136	−0.130068	+	0.991505i	$0.541520\pi$
$12$	0	0
$13$	6.19586	1.71842	0.859211	−	0.511621i	$-0.170955\pi$
$13$	6.19586	1.71842	0.859211	+	0.511621i	$0.170955\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.72420	1.63086	0.815430	−	0.578856i	$-0.196501\pi$
$17$	6.72420	1.63086	0.815430	+	0.578856i	$0.196501\pi$
$18$	0	0
$19$	3.79733	0.871167	0.435583	−	0.900148i	$-0.356542\pi$
$19$	3.79733	0.871167	0.435583	+	0.900148i	$0.356542\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.114019	−0.0237745	−0.0118873	−	0.999929i	$-0.503784\pi$
$23$	−0.114019	−0.0237745	−0.0118873	+	0.999929i	$0.503784\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.48490	0.266696	0.133348	−	0.991069i	$-0.457427\pi$
$31$	1.48490	0.266696	0.133348	+	0.991069i	$0.457427\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.23366	−0.377557
$36$	0	0
$37$	5.84818	0.961436	0.480718	−	0.876875i	$-0.340376\pi$
$37$	5.84818	0.961436	0.480718	+	0.876875i	$0.340376\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.84818	−0.600985	−0.300493	−	0.953784i	$-0.597151\pi$
$41$	−3.84818	−0.600985	−0.300493	+	0.953784i	$0.597151\pi$
$42$	0	0
$43$	−2.66310	−0.406119	−0.203059	−	0.979166i	$-0.565088\pi$
$43$	−2.66310	−0.406119	−0.203059	+	0.979166i	$0.565088\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.5650	−1.68692	−0.843462	−	0.537188i	$-0.819486\pi$
$47$	−11.5650	−1.68692	−0.843462	+	0.537188i	$0.819486\pi$
$48$	0	0
$49$	−2.01078	−0.287254
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.85596	0.941738	0.470869	−	0.882203i	$-0.343940\pi$
$53$	6.85596	0.941738	0.470869	+	0.882203i	$0.343940\pi$
$54$	0	0
$55$	0.862773	0.116336
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.814916	0.106093	0.0530465	−	0.998592i	$-0.483107\pi$
$59$	0.814916	0.106093	0.0530465	+	0.998592i	$0.483107\pi$
$60$	0	0
$61$	−0.312424	−0.0400017	−0.0200009	−	0.999800i	$-0.506367\pi$
$61$	−0.312424	−0.0400017	−0.0200009	+	0.999800i	$0.506367\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.19586	−0.768502
$66$	0	0
$67$	−6.15806	−0.752327	−0.376164	−	0.926553i	$-0.622757\pi$
$67$	−6.15806	−0.752327	−0.376164	+	0.926553i	$0.622757\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.31242	0.274434	0.137217	−	0.990541i	$-0.456184\pi$
$71$	2.31242	0.274434	0.137217	+	0.990541i	$0.456184\pi$
$72$	0	0
$73$	10.9170	1.27774	0.638868	−	0.769316i	$-0.279403\pi$
$73$	10.9170	1.27774	0.638868	+	0.769316i	$0.279403\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.92714	−0.219618
$78$	0	0
$79$	−6.57956	−0.740258	−0.370129	−	0.928980i	$-0.620687\pi$
$79$	−6.57956	−0.740258	−0.370129	+	0.928980i	$0.620687\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.62530	−0.946750	−0.473375	−	0.880861i	$-0.656964\pi$
$83$	−8.62530	−0.946750	−0.473375	+	0.880861i	$0.656964\pi$
$84$	0	0
$85$	−6.72420	−0.729342
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.93763	−0.947387	−0.473693	−	0.880690i	$-0.657080\pi$
$89$	−8.93763	−0.947387	−0.473693	+	0.880690i	$0.657080\pi$
$90$	0	0
$91$	13.8394	1.45077
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.79733	−0.389598
$96$	0	0
$97$	6.87902	0.698459	0.349229	−	0.937037i	$-0.386443\pi$
$97$	6.87902	0.698459	0.349229	+	0.937037i	$0.386443\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.54654	−0.651405	−0.325702	−	0.945472i	$-0.605601\pi$
$101$	−6.54654	−0.651405	−0.325702	+	0.945472i	$0.605601\pi$
$102$	0	0
$103$	16.1111	1.58747	0.793735	−	0.608264i	$-0.208134\pi$
$103$	16.1111	1.58747	0.793735	+	0.608264i	$0.208134\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.9837	1.64188	0.820939	−	0.571016i	$-0.193451\pi$
$107$	16.9837	1.64188	0.820939	+	0.571016i	$0.193451\pi$
$108$	0	0
$109$	−13.8805	−1.32951	−0.664754	−	0.747062i	$-0.731464\pi$
$109$	−13.8805	−1.32951	−0.664754	+	0.747062i	$0.731464\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.6770	1.09848	0.549238	−	0.835666i	$-0.314918\pi$
$113$	11.6770	1.09848	0.549238	+	0.835666i	$0.314918\pi$
$114$	0	0
$115$	0.114019	0.0106323
$116$	0	0
$117$	0	0
$118$	0	0
$119$	15.0196	1.37684
$120$	0	0
$121$	−10.2556	−0.932329
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.09773	−0.807293	−0.403647	−	0.914915i	$-0.632257\pi$
$127$	−9.09773	−0.807293	−0.403647	+	0.914915i	$0.632257\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.28904	0.112624	0.0563121	−	0.998413i	$-0.482066\pi$
$131$	1.28904	0.112624	0.0563121	+	0.998413i	$0.482066\pi$
$132$	0	0
$133$	8.48193	0.735476
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.3598	1.39771	0.698857	−	0.715261i	$-0.253692\pi$
$137$	16.3598	1.39771	0.698857	+	0.715261i	$0.253692\pi$
$138$	0	0
$139$	−3.73162	−0.316512	−0.158256	−	0.987398i	$-0.550587\pi$
$139$	−3.73162	−0.316512	−0.158256	+	0.987398i	$0.550587\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.34562	−0.447023
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.28223	−0.105044	−0.0525222	−	0.998620i	$-0.516726\pi$
$149$	−1.28223	−0.105044	−0.0525222	+	0.998620i	$0.516726\pi$
$150$	0	0
$151$	20.7941	1.69220	0.846100	−	0.533024i	$-0.178945\pi$
$151$	20.7941	1.69220	0.846100	+	0.533024i	$0.178945\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.48490	−0.119270
$156$	0	0
$157$	23.7487	1.89535	0.947677	−	0.319231i	$-0.103425\pi$
$157$	23.7487	1.89535	0.947677	+	0.319231i	$0.103425\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.254678	−0.0200715
$162$	0	0
$163$	−13.5073	−1.05797	−0.528987	−	0.848630i	$-0.677428\pi$
$163$	−13.5073	−1.05797	−0.528987	+	0.848630i	$0.677428\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.6135	1.44036	0.720180	−	0.693788i	$-0.244059\pi$
$167$	18.6135	1.44036	0.720180	+	0.693788i	$0.244059\pi$
$168$	0	0
$169$	25.3887	1.95298
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.88336	−0.143189	−0.0715947	−	0.997434i	$-0.522809\pi$
$173$	−1.88336	−0.143189	−0.0715947	+	0.997434i	$0.522809\pi$
$174$	0	0
$175$	2.23366	0.168849
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.11447	0.307530	0.153765	−	0.988107i	$-0.450860\pi$
$179$	4.11447	0.307530	0.153765	+	0.988107i	$0.450860\pi$
$180$	0	0
$181$	15.1334	1.12486	0.562429	−	0.826846i	$-0.309867\pi$
$181$	15.1334	1.12486	0.562429	+	0.826846i	$0.309867\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.84818	−0.429967
$186$	0	0
$187$	−5.80146	−0.424245
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.2790	−1.32262	−0.661311	−	0.750112i	$-0.730000\pi$
$191$	−18.2790	−1.32262	−0.661311	+	0.750112i	$0.730000\pi$
$192$	0	0
$193$	−8.34771	−0.600881	−0.300441	−	0.953801i	$-0.597134\pi$
$193$	−8.34771	−0.600881	−0.300441	+	0.953801i	$0.597134\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.8912	−1.48844	−0.744220	−	0.667935i	$-0.767179\pi$
$197$	−20.8912	−1.48844	−0.744220	+	0.667935i	$0.767179\pi$
$198$	0	0
$199$	1.29348	0.0916926	0.0458463	−	0.998949i	$-0.485402\pi$
$199$	1.29348	0.0916926	0.0458463	+	0.998949i	$0.485402\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.23366	−0.156772
$204$	0	0
$205$	3.84818	0.268769
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.27623	−0.226622
$210$	0	0
$211$	−17.2837	−1.18986	−0.594930	−	0.803778i	$-0.702820\pi$
$211$	−17.2837	−1.18986	−0.594930	+	0.803778i	$0.702820\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.66310	0.181622
$216$	0	0
$217$	3.31676	0.225157
$218$	0	0
$219$	0	0
$220$	0	0
$221$	41.6622	2.80250
$222$	0	0
$223$	19.9516	1.33606	0.668028	−	0.744136i	$-0.267139\pi$
$223$	19.9516	1.33606	0.668028	+	0.744136i	$0.267139\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.9351	1.12402	0.562011	−	0.827130i	$-0.310028\pi$
$227$	16.9351	1.12402	0.562011	+	0.827130i	$0.310028\pi$
$228$	0	0
$229$	26.4182	1.74576	0.872882	−	0.487931i	$-0.162248\pi$
$229$	26.4182	1.74576	0.872882	+	0.487931i	$0.162248\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.481568	−0.0315485	−0.0157743	−	0.999876i	$-0.505021\pi$
$233$	−0.481568	−0.0315485	−0.0157743	+	0.999876i	$0.505021\pi$
$234$	0	0
$235$	11.5650	0.754416
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.4889	−0.678468	−0.339234	−	0.940702i	$-0.610168\pi$
$239$	−10.4889	−0.678468	−0.339234	+	0.940702i	$0.610168\pi$
$240$	0	0
$241$	−6.18410	−0.398353	−0.199177	−	0.979964i	$-0.563827\pi$
$241$	−6.18410	−0.398353	−0.199177	+	0.979964i	$0.563827\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.01078	0.128464
$246$	0	0
$247$	23.5277	1.49703
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.1896	1.58995	0.794975	−	0.606642i	$-0.207484\pi$
$251$	25.1896	1.58995	0.794975	+	0.606642i	$0.207484\pi$
$252$	0	0
$253$	0.0983720	0.00618460
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.2072	−1.13574	−0.567868	−	0.823120i	$-0.692231\pi$
$257$	−18.2072	−1.13574	−0.567868	+	0.823120i	$0.692231\pi$
$258$	0	0
$259$	13.0628	0.811685
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.7119	−1.58546	−0.792732	−	0.609570i	$-0.791342\pi$
$263$	−25.7119	−1.58546	−0.792732	+	0.609570i	$0.791342\pi$
$264$	0	0
$265$	−6.85596	−0.421158
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.7347	−0.837419	−0.418710	−	0.908120i	$-0.637517\pi$
$269$	−13.7347	−0.837419	−0.418710	+	0.908120i	$0.637517\pi$
$270$	0	0
$271$	8.26714	0.502193	0.251096	−	0.967962i	$-0.419209\pi$
$271$	8.26714	0.502193	0.251096	+	0.967962i	$0.419209\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.862773	−0.0520271
$276$	0	0
$277$	−19.5738	−1.17607	−0.588037	−	0.808834i	$-0.700099\pi$
$277$	−19.5738	−1.17607	−0.588037	+	0.808834i	$0.700099\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8.15135	−0.486269	−0.243134	−	0.969993i	$-0.578176\pi$
$281$	−8.15135	−0.486269	−0.243134	+	0.969993i	$0.578176\pi$
$282$	0	0
$283$	−4.92175	−0.292567	−0.146284	−	0.989243i	$-0.546731\pi$
$283$	−4.92175	−0.292567	−0.146284	+	0.989243i	$0.546731\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.59552	−0.507378
$288$	0	0
$289$	28.2149	1.65970
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.04097	−0.352917	−0.176459	−	0.984308i	$-0.556464\pi$
$293$	−6.04097	−0.352917	−0.176459	+	0.984308i	$0.556464\pi$
$294$	0	0
$295$	−0.814916	−0.0474462
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.706443	−0.0408546
$300$	0	0
$301$	−5.94845	−0.342863
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.312424	0.0178893
$306$	0	0
$307$	−0.589954	−0.0336704	−0.0168352	−	0.999858i	$-0.505359\pi$
$307$	−0.589954	−0.0336704	−0.0168352	+	0.999858i	$0.505359\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.08054	0.344796	0.172398	−	0.985027i	$-0.444849\pi$
$311$	6.08054	0.344796	0.172398	+	0.985027i	$0.444849\pi$
$312$	0	0
$313$	−16.2276	−0.917241	−0.458620	−	0.888632i	$-0.651656\pi$
$313$	−16.2276	−0.917241	−0.458620	+	0.888632i	$0.651656\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.96348	0.391108	0.195554	−	0.980693i	$-0.437349\pi$
$317$	6.96348	0.391108	0.195554	+	0.980693i	$0.437349\pi$
$318$	0	0
$319$	0.862773	0.0483060
$320$	0	0
$321$	0	0
$322$	0	0
$323$	25.5340	1.42075
$324$	0	0
$325$	6.19586	0.343684
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−25.8322	−1.42417
$330$	0	0
$331$	−3.21645	−0.176792	−0.0883960	−	0.996085i	$-0.528174\pi$
$331$	−3.21645	−0.176792	−0.0883960	+	0.996085i	$0.528174\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.15806	0.336451
$336$	0	0
$337$	13.2124	0.719724	0.359862	−	0.933006i	$-0.382824\pi$
$337$	13.2124	0.719724	0.359862	+	0.933006i	$0.382824\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.28113	−0.0693773
$342$	0	0
$343$	−20.1270	−1.08675
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.9781	0.911434	0.455717	−	0.890125i	$-0.349383\pi$
$347$	16.9781	0.911434	0.455717	+	0.890125i	$0.349383\pi$
$348$	0	0
$349$	−0.578713	−0.0309778	−0.0154889	−	0.999880i	$-0.504930\pi$
$349$	−0.578713	−0.0309778	−0.0154889	+	0.999880i	$0.504930\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.09715	−0.430968	−0.215484	−	0.976507i	$-0.569133\pi$
$353$	−8.09715	−0.430968	−0.215484	+	0.976507i	$0.569133\pi$
$354$	0	0
$355$	−2.31242	−0.122731
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.9175	−0.734538	−0.367269	−	0.930115i	$-0.619707\pi$
$359$	−13.9175	−0.734538	−0.367269	+	0.930115i	$0.619707\pi$
$360$	0	0
$361$	−4.58031	−0.241069
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.9170	−0.571421
$366$	0	0
$367$	1.84218	0.0961612	0.0480806	−	0.998843i	$-0.484690\pi$
$367$	1.84218	0.0961612	0.0480806	+	0.998843i	$0.484690\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.3139	0.795056
$372$	0	0
$373$	30.9107	1.60050	0.800249	−	0.599668i	$-0.204701\pi$
$373$	30.9107	1.60050	0.800249	+	0.599668i	$0.204701\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.19586	−0.319103
$378$	0	0
$379$	14.1422	0.726438	0.363219	−	0.931704i	$-0.381678\pi$
$379$	14.1422	0.726438	0.363219	+	0.931704i	$0.381678\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−35.4290	−1.81034	−0.905170	−	0.425050i	$-0.860256\pi$
$383$	−35.4290	−1.81034	−0.905170	+	0.425050i	$0.860256\pi$
$384$	0	0
$385$	1.92714	0.0982160
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6.78581	−0.344054	−0.172027	−	0.985092i	$-0.555032\pi$
$389$	−6.78581	−0.344054	−0.172027	+	0.985092i	$0.555032\pi$
$390$	0	0
$391$	−0.766684	−0.0387729
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.57956	0.331054
$396$	0	0
$397$	6.07559	0.304925	0.152463	−	0.988309i	$-0.451280\pi$
$397$	6.07559	0.304925	0.152463	+	0.988309i	$0.451280\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	34.6086	1.72827	0.864135	−	0.503260i	$-0.167866\pi$
$401$	34.6086	1.72827	0.864135	+	0.503260i	$0.167866\pi$
$402$	0	0
$403$	9.20025	0.458297
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.04565	−0.250104
$408$	0	0
$409$	20.1817	0.997922	0.498961	−	0.866624i	$-0.333715\pi$
$409$	20.1817	0.997922	0.498961	+	0.866624i	$0.333715\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.82024	0.0895683
$414$	0	0
$415$	8.62530	0.423399
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.9108	0.972705	0.486353	−	0.873763i	$-0.338327\pi$
$419$	19.9108	0.972705	0.486353	+	0.873763i	$0.338327\pi$
$420$	0	0
$421$	−3.38394	−0.164923	−0.0824617	−	0.996594i	$-0.526278\pi$
$421$	−3.38394	−0.164923	−0.0824617	+	0.996594i	$0.526278\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.72420	0.326172
$426$	0	0
$427$	−0.697847	−0.0337712
$428$	0	0
$429$	0	0
$430$	0	0
$431$	40.3943	1.94573	0.972863	−	0.231382i	$-0.0743246\pi$
$431$	40.3943	1.94573	0.972863	+	0.231382i	$0.0743246\pi$
$432$	0	0
$433$	3.44005	0.165318	0.0826590	−	0.996578i	$-0.473659\pi$
$433$	3.44005	0.165318	0.0826590	+	0.996578i	$0.473659\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.432966	−0.0207115
$438$	0	0
$439$	−25.2149	−1.20344	−0.601721	−	0.798706i	$-0.705518\pi$
$439$	−25.2149	−1.20344	−0.601721	+	0.798706i	$0.705518\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−25.0925	−1.19218	−0.596091	−	0.802917i	$-0.703280\pi$
$443$	−25.0925	−1.19218	−0.596091	+	0.802917i	$0.703280\pi$
$444$	0	0
$445$	8.93763	0.423684
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−23.9624	−1.13086	−0.565429	−	0.824797i	$-0.691289\pi$
$449$	−23.9624	−1.13086	−0.565429	+	0.824797i	$0.691289\pi$
$450$	0	0
$451$	3.32011	0.156338
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−13.8394	−0.648802
$456$	0	0
$457$	11.0245	0.515705	0.257852	−	0.966184i	$-0.416985\pi$
$457$	11.0245	0.515705	0.257852	+	0.966184i	$0.416985\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	39.9276	1.85962	0.929808	−	0.368046i	$-0.119973\pi$
$461$	39.9276	1.85962	0.929808	+	0.368046i	$0.119973\pi$
$462$	0	0
$463$	−17.9300	−0.833279	−0.416639	−	0.909072i	$-0.636792\pi$
$463$	−17.9300	−0.833279	−0.416639	+	0.909072i	$0.636792\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.15485	−0.0997146	−0.0498573	−	0.998756i	$-0.515877\pi$
$467$	−2.15485	−0.0997146	−0.0498573	+	0.998756i	$0.515877\pi$
$468$	0	0
$469$	−13.7550	−0.635147
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.29765	0.105646
$474$	0	0
$475$	3.79733	0.174233
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.1212	1.23920	0.619600	−	0.784918i	$-0.287295\pi$
$479$	27.1212	1.23920	0.619600	+	0.784918i	$0.287295\pi$
$480$	0	0
$481$	36.2345	1.65215
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.87902	−0.312360
$486$	0	0
$487$	−29.9751	−1.35830	−0.679150	−	0.733999i	$-0.737651\pi$
$487$	−29.9751	−1.35830	−0.679150	+	0.733999i	$0.737651\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.62107	0.118287	0.0591437	−	0.998249i	$-0.481163\pi$
$491$	2.62107	0.118287	0.0591437	+	0.998249i	$0.481163\pi$
$492$	0	0
$493$	−6.72420	−0.302843
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.16516	0.231689
$498$	0	0
$499$	5.69586	0.254982	0.127491	−	0.991840i	$-0.459308\pi$
$499$	5.69586	0.254982	0.127491	+	0.991840i	$0.459308\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14.9136	−0.664966	−0.332483	−	0.943109i	$-0.607886\pi$
$503$	−14.9136	−0.664966	−0.332483	+	0.943109i	$0.607886\pi$
$504$	0	0
$505$	6.54654	0.291317
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.6905	1.27168	0.635842	−	0.771819i	$-0.280653\pi$
$509$	28.6905	1.27168	0.635842	+	0.771819i	$0.280653\pi$
$510$	0	0
$511$	24.3848	1.07872
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.1111	−0.709938
$516$	0	0
$517$	9.97794	0.438829
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.5920	0.683098	0.341549	−	0.939864i	$-0.389048\pi$
$521$	15.5920	0.683098	0.341549	+	0.939864i	$0.389048\pi$
$522$	0	0
$523$	8.48834	0.371169	0.185584	−	0.982628i	$-0.440582\pi$
$523$	8.48834	0.371169	0.185584	+	0.982628i	$0.440582\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.98479	0.434944
$528$	0	0
$529$	−22.9870	−0.999435
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−23.8428	−1.03275
$534$	0	0
$535$	−16.9837	−0.734270
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.73484	0.0747249
$540$	0	0
$541$	23.9357	1.02908	0.514538	−	0.857468i	$-0.327963\pi$
$541$	23.9357	1.02908	0.514538	+	0.857468i	$0.327963\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.8805	0.594574
$546$	0	0
$547$	24.2718	1.03779	0.518893	−	0.854839i	$-0.326344\pi$
$547$	24.2718	1.03779	0.518893	+	0.854839i	$0.326344\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.79733	−0.161772
$552$	0	0
$553$	−14.6965	−0.624958
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.0359	−1.01843	−0.509216	−	0.860639i	$-0.670065\pi$
$557$	−24.0359	−1.01843	−0.509216	+	0.860639i	$0.670065\pi$
$558$	0	0
$559$	−16.5002	−0.697884
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.7209	−0.536123	−0.268061	−	0.963402i	$-0.586383\pi$
$563$	−12.7209	−0.536123	−0.268061	+	0.963402i	$0.586383\pi$
$564$	0	0
$565$	−11.6770	−0.491253
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.11139	−0.381970	−0.190985	−	0.981593i	$-0.561168\pi$
$569$	−9.11139	−0.381970	−0.190985	+	0.981593i	$0.561168\pi$
$570$	0	0
$571$	10.7613	0.450347	0.225173	−	0.974319i	$-0.427705\pi$
$571$	10.7613	0.450347	0.225173	+	0.974319i	$0.427705\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.114019	−0.00475490
$576$	0	0
$577$	−1.34669	−0.0560636	−0.0280318	−	0.999607i	$-0.508924\pi$
$577$	−1.34669	−0.0560636	−0.0280318	+	0.999607i	$0.508924\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−19.2660	−0.799287
$582$	0	0
$583$	−5.91513	−0.244980
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.3469	−0.468335	−0.234168	−	0.972196i	$-0.575236\pi$
$587$	−11.3469	−0.468335	−0.234168	+	0.972196i	$0.575236\pi$
$588$	0	0
$589$	5.63866	0.232337
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22.5818	−0.927323	−0.463661	−	0.886012i	$-0.653465\pi$
$593$	−22.5818	−0.927323	−0.463661	+	0.886012i	$0.653465\pi$
$594$	0	0
$595$	−15.0196	−0.615742
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11.9871	−0.489782	−0.244891	−	0.969551i	$-0.578752\pi$
$599$	−11.9871	−0.489782	−0.244891	+	0.969551i	$0.578752\pi$
$600$	0	0
$601$	−21.3867	−0.872383	−0.436192	−	0.899854i	$-0.643673\pi$
$601$	−21.3867	−0.872383	−0.436192	+	0.899854i	$0.643673\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.2556	0.416950
$606$	0	0
$607$	3.91836	0.159042	0.0795208	−	0.996833i	$-0.474661\pi$
$607$	3.91836	0.159042	0.0795208	+	0.996833i	$0.474661\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−71.6549	−2.89885
$612$	0	0
$613$	31.5282	1.27341	0.636706	−	0.771106i	$-0.280296\pi$
$613$	31.5282	1.27341	0.636706	+	0.771106i	$0.280296\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.5381	−0.867092	−0.433546	−	0.901131i	$-0.642738\pi$
$617$	−21.5381	−0.867092	−0.433546	+	0.901131i	$0.642738\pi$
$618$	0	0
$619$	−1.44852	−0.0582209	−0.0291105	−	0.999576i	$-0.509267\pi$
$619$	−1.44852	−0.0582209	−0.0291105	+	0.999576i	$0.509267\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−19.9636	−0.799825
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	39.3244	1.56797
$630$	0	0
$631$	−39.9229	−1.58931	−0.794654	−	0.607063i	$-0.792348\pi$
$631$	−39.9229	−1.58931	−0.794654	+	0.607063i	$0.792348\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.09773	0.361032
$636$	0	0
$637$	−12.4585	−0.493623
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27.5532	1.08829	0.544143	−	0.838992i	$-0.316855\pi$
$641$	27.5532	1.08829	0.544143	+	0.838992i	$0.316855\pi$
$642$	0	0
$643$	−28.8043	−1.13593	−0.567965	−	0.823053i	$-0.692269\pi$
$643$	−28.8043	−1.13593	−0.567965	+	0.823053i	$0.692269\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.6477	0.851058	0.425529	−	0.904945i	$-0.360088\pi$
$647$	21.6477	0.851058	0.425529	+	0.904945i	$0.360088\pi$
$648$	0	0
$649$	−0.703087	−0.0275986
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.9102	−0.779147	−0.389574	−	0.920995i	$-0.627378\pi$
$653$	−19.9102	−0.779147	−0.389574	+	0.920995i	$0.627378\pi$
$654$	0	0
$655$	−1.28904	−0.0503671
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28.6588	1.11639	0.558195	−	0.829710i	$-0.311494\pi$
$659$	28.6588	1.11639	0.558195	+	0.829710i	$0.311494\pi$
$660$	0	0
$661$	11.3667	0.442113	0.221057	−	0.975261i	$-0.429049\pi$
$661$	11.3667	0.442113	0.221057	+	0.975261i	$0.429049\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.48193	−0.328915
$666$	0	0
$667$	0.114019	0.00441481
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.269550	0.0104059
$672$	0	0
$673$	4.10543	0.158253	0.0791264	−	0.996865i	$-0.474787\pi$
$673$	4.10543	0.158253	0.0791264	+	0.996865i	$0.474787\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.35675	−0.244310	−0.122155	−	0.992511i	$-0.538980\pi$
$677$	−6.35675	−0.244310	−0.122155	+	0.992511i	$0.538980\pi$
$678$	0	0
$679$	15.3654	0.589669
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−25.1306	−0.961595	−0.480798	−	0.876832i	$-0.659653\pi$
$683$	−25.1306	−0.961595	−0.480798	+	0.876832i	$0.659653\pi$
$684$	0	0
$685$	−16.3598	−0.625077
$686$	0	0
$687$	0	0
$688$	0	0
$689$	42.4786	1.61830
$690$	0	0
$691$	−39.9628	−1.52026	−0.760129	−	0.649772i	$-0.774864\pi$
$691$	−39.9628	−1.52026	−0.760129	+	0.649772i	$0.774864\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.73162	0.141548
$696$	0	0
$697$	−25.8760	−0.980122
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12.3476	0.466362	0.233181	−	0.972433i	$-0.425086\pi$
$701$	12.3476	0.466362	0.233181	+	0.972433i	$0.425086\pi$
$702$	0	0
$703$	22.2075	0.837570
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.6227	−0.549944
$708$	0	0
$709$	4.89993	0.184021	0.0920103	−	0.995758i	$-0.470671\pi$
$709$	4.89993	0.184021	0.0920103	+	0.995758i	$0.470671\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.169306	−0.00634058
$714$	0	0
$715$	5.34562	0.199915
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26.5380	−0.989699	−0.494850	−	0.868979i	$-0.664777\pi$
$719$	−26.5380	−0.989699	−0.494850	+	0.868979i	$0.664777\pi$
$720$	0	0
$721$	35.9866	1.34021
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−33.7317	−1.25104	−0.625520	−	0.780208i	$-0.715113\pi$
$727$	−33.7317	−1.25104	−0.625520	+	0.780208i	$0.715113\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−17.9072	−0.662323
$732$	0	0
$733$	52.0372	1.92204	0.961019	−	0.276482i	$-0.0891686\pi$
$733$	52.0372	1.92204	0.961019	+	0.276482i	$0.0891686\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.31301	0.195707
$738$	0	0
$739$	−23.4998	−0.864456	−0.432228	−	0.901764i	$-0.642272\pi$
$739$	−23.4998	−0.864456	−0.432228	+	0.901764i	$0.642272\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−42.6872	−1.56604	−0.783021	−	0.621995i	$-0.786322\pi$
$743$	−42.6872	−1.56604	−0.783021	+	0.621995i	$0.786322\pi$
$744$	0	0
$745$	1.28223	0.0469773
$746$	0	0
$747$	0	0
$748$	0	0
$749$	37.9358	1.38614
$750$	0	0
$751$	−39.1265	−1.42775	−0.713873	−	0.700276i	$-0.753060\pi$
$751$	−39.1265	−1.42775	−0.713873	+	0.700276i	$0.753060\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−20.7941	−0.756775
$756$	0	0
$757$	36.9259	1.34209	0.671047	−	0.741415i	$-0.265845\pi$
$757$	36.9259	1.34209	0.671047	+	0.741415i	$0.265845\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	50.8039	1.84164	0.920820	−	0.389988i	$-0.127521\pi$
$761$	50.8039	1.84164	0.920820	+	0.389988i	$0.127521\pi$
$762$	0	0
$763$	−31.0042	−1.12243
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.04910	0.182313
$768$	0	0
$769$	−5.85269	−0.211053	−0.105527	−	0.994416i	$-0.533653\pi$
$769$	−5.85269	−0.211053	−0.105527	+	0.994416i	$0.533653\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	11.3796	0.409297	0.204649	−	0.978835i	$-0.434395\pi$
$773$	11.3796	0.409297	0.204649	+	0.978835i	$0.434395\pi$
$774$	0	0
$775$	1.48490	0.0533393
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14.6128	−0.523558
$780$	0	0
$781$	−1.99510	−0.0713901
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−23.7487	−0.847628
$786$	0	0
$787$	24.5905	0.876557	0.438279	−	0.898839i	$-0.355588\pi$
$787$	24.5905	0.876557	0.438279	+	0.898839i	$0.355588\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	26.0823	0.927380
$792$	0	0
$793$	−1.93573	−0.0687399
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.93342	−0.139329	−0.0696644	−	0.997570i	$-0.522193\pi$
$797$	−3.93342	−0.139329	−0.0696644	+	0.997570i	$0.522193\pi$
$798$	0	0
$799$	−77.7652	−2.75114
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.41888	−0.332385
$804$	0	0
$805$	0.254678	0.00897623
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−48.9615	−1.72139	−0.860697	−	0.509117i	$-0.829972\pi$
$809$	−48.9615	−1.72139	−0.860697	+	0.509117i	$0.829972\pi$
$810$	0	0
$811$	−19.6362	−0.689522	−0.344761	−	0.938691i	$-0.612040\pi$
$811$	−19.6362	−0.689522	−0.344761	+	0.938691i	$0.612040\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.5073	0.473140
$816$	0	0
$817$	−10.1127	−0.353797
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.6248	0.580211	0.290106	−	0.956995i	$-0.406310\pi$
$821$	16.6248	0.580211	0.290106	+	0.956995i	$0.406310\pi$
$822$	0	0
$823$	−28.6690	−0.999337	−0.499669	−	0.866217i	$-0.666545\pi$
$823$	−28.6690	−0.999337	−0.499669	+	0.866217i	$0.666545\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.7188	1.62457	0.812286	−	0.583260i	$-0.198223\pi$
$827$	46.7188	1.62457	0.812286	+	0.583260i	$0.198223\pi$
$828$	0	0
$829$	9.89829	0.343782	0.171891	−	0.985116i	$-0.445012\pi$
$829$	9.89829	0.343782	0.171891	+	0.985116i	$0.445012\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−13.5209	−0.468470
$834$	0	0
$835$	−18.6135	−0.644148
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.0236	1.00201	0.501004	−	0.865445i	$-0.332964\pi$
$839$	29.0236	1.00201	0.501004	+	0.865445i	$0.332964\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−25.3887	−0.873397
$846$	0	0
$847$	−22.9075	−0.787113
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.666801	−0.0228577
$852$	0	0
$853$	−36.6959	−1.25644	−0.628222	−	0.778034i	$-0.716217\pi$
$853$	−36.6959	−1.25644	−0.628222	+	0.778034i	$0.716217\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.902417	0.0308260	0.0154130	−	0.999881i	$-0.495094\pi$
$857$	0.902417	0.0308260	0.0154130	+	0.999881i	$0.495094\pi$
$858$	0	0
$859$	−43.6590	−1.48963	−0.744813	−	0.667273i	$-0.767461\pi$
$859$	−43.6590	−1.48963	−0.744813	+	0.667273i	$0.767461\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.8040	0.640094	0.320047	−	0.947402i	$-0.396301\pi$
$863$	18.8040	0.640094	0.320047	+	0.947402i	$0.396301\pi$
$864$	0	0
$865$	1.88336	0.0640362
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.67666	0.192568
$870$	0	0
$871$	−38.1545	−1.29282
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.23366	−0.0755114
$876$	0	0
$877$	−8.49872	−0.286981	−0.143491	−	0.989652i	$-0.545833\pi$
$877$	−8.49872	−0.286981	−0.143491	+	0.989652i	$0.545833\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.0746	0.339421	0.169710	−	0.985494i	$-0.445717\pi$
$881$	10.0746	0.339421	0.169710	+	0.985494i	$0.445717\pi$
$882$	0	0
$883$	−17.4369	−0.586799	−0.293399	−	0.955990i	$-0.594787\pi$
$883$	−17.4369	−0.586799	−0.293399	+	0.955990i	$0.594787\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.96047	0.233710	0.116855	−	0.993149i	$-0.462719\pi$
$887$	6.96047	0.233710	0.116855	+	0.993149i	$0.462719\pi$
$888$	0	0
$889$	−20.3212	−0.681552
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−43.9160	−1.46959
$894$	0	0
$895$	−4.11447	−0.137532
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.48490	−0.0495243
$900$	0	0
$901$	46.1009	1.53584
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.1334	−0.503052
$906$	0	0
$907$	−23.0944	−0.766836	−0.383418	−	0.923575i	$-0.625253\pi$
$907$	−23.0944	−0.766836	−0.383418	+	0.923575i	$0.625253\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.39120	0.278013	0.139006	−	0.990291i	$-0.455609\pi$
$911$	8.39120	0.278013	0.139006	+	0.990291i	$0.455609\pi$
$912$	0	0
$913$	7.44167	0.246283
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.87928	0.0950822
$918$	0	0
$919$	−23.0180	−0.759293	−0.379647	−	0.925132i	$-0.623954\pi$
$919$	−23.0180	−0.759293	−0.379647	+	0.925132i	$0.623954\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.3275	0.471594
$924$	0	0
$925$	5.84818	0.192287
$926$	0	0
$927$	0	0
$928$	0	0
$929$	53.4134	1.75244	0.876219	−	0.481913i	$-0.160058\pi$
$929$	53.4134	1.75244	0.876219	+	0.481913i	$0.160058\pi$
$930$	0	0
$931$	−7.63557	−0.250246
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.80146	0.189728
$936$	0	0
$937$	−9.18564	−0.300082	−0.150041	−	0.988680i	$-0.547941\pi$
$937$	−9.18564	−0.300082	−0.150041	+	0.988680i	$0.547941\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.03797	−0.131634	−0.0658170	−	0.997832i	$-0.520965\pi$
$941$	−4.03797	−0.131634	−0.0658170	+	0.997832i	$0.520965\pi$
$942$	0	0
$943$	0.438764	0.0142881
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50.6471	1.64581	0.822905	−	0.568179i	$-0.192352\pi$
$947$	50.6471	1.64581	0.822905	+	0.568179i	$0.192352\pi$
$948$	0	0
$949$	67.6401	2.19569
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.24076	−0.104978	−0.0524892	−	0.998621i	$-0.516716\pi$
$953$	−3.24076	−0.104978	−0.0524892	+	0.998621i	$0.516716\pi$
$954$	0	0
$955$	18.2790	0.591495
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.5422	1.18001
$960$	0	0
$961$	−28.7951	−0.928873
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.34771	0.268722
$966$	0	0
$967$	−51.3561	−1.65150	−0.825750	−	0.564037i	$-0.809248\pi$
$967$	−51.3561	−1.65150	−0.825750	+	0.564037i	$0.809248\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−33.7059	−1.08167	−0.540836	−	0.841128i	$-0.681892\pi$
$971$	−33.7059	−1.08167	−0.540836	+	0.841128i	$0.681892\pi$
$972$	0	0
$973$	−8.33516	−0.267213
$974$	0	0
$975$	0	0
$976$	0	0
$977$	51.0353	1.63277	0.816383	−	0.577511i	$-0.195976\pi$
$977$	51.0353	1.63277	0.816383	+	0.577511i	$0.195976\pi$
$978$	0	0
$979$	7.71114	0.246449
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−47.8553	−1.52635	−0.763174	−	0.646193i	$-0.776360\pi$
$983$	−47.8553	−1.52635	−0.763174	+	0.646193i	$0.776360\pi$
$984$	0	0
$985$	20.8912	0.665650
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.303643	0.00965527
$990$	0	0
$991$	21.6652	0.688216	0.344108	−	0.938930i	$-0.388181\pi$
$991$	21.6652	0.688216	0.344108	+	0.938930i	$0.388181\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.29348	−0.0410062
$996$	0	0
$997$	11.7861	0.373270	0.186635	−	0.982429i	$-0.440242\pi$
$997$	11.7861	0.373270	0.186635	+	0.982429i	$0.440242\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.a.y.1.5	✓	7
3.2	odd	2		5220.2.a.z.1.5	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5220.2.a.y.1.5	✓	7	1.1	even	1	trivial
5220.2.a.z.1.5	yes	7	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:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.23366 q^{7} -0.862773 q^{11} +6.19586 q^{13} +6.72420 q^{17} +3.79733 q^{19} -0.114019 q^{23} +1.00000 q^{25} -1.00000 q^{29} +1.48490 q^{31} -2.23366 q^{35} +5.84818 q^{37} -3.84818 q^{41} -2.66310 q^{43} -11.5650 q^{47} -2.01078 q^{49} +6.85596 q^{53} +0.862773 q^{55} +0.814916 q^{59} -0.312424 q^{61} -6.19586 q^{65} -6.15806 q^{67} +2.31242 q^{71} +10.9170 q^{73} -1.92714 q^{77} -6.57956 q^{79} -8.62530 q^{83} -6.72420 q^{85} -8.93763 q^{89} +13.8394 q^{91} -3.79733 q^{95} +6.87902 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{5} - 3 q^{11} + 2 q^{13} - 8 q^{17} + 6 q^{19} - q^{23} + 7 q^{25} - 7 q^{29} - 2 q^{31} + 15 q^{37} - q^{41} + 9 q^{43} - 4 q^{47} + 29 q^{49} - 17 q^{53} + 3 q^{55} + 4 q^{59} + 6 q^{61}+ \cdots + 15 q^{97}+O(q^{100}) \) 7 * q - 7 * q^5 - 3 * q^11 + 2 * q^13 - 8 * q^17 + 6 * q^19 - q^23 + 7 * q^25 - 7 * q^29 - 2 * q^31 + 15 * q^37 - q^41 + 9 * q^43 - 4 * q^47 + 29 * q^49 - 17 * q^53 + 3 * q^55 + 4 * q^59 + 6 * q^61 - 2 * q^65 + 24 * q^67 + 8 * q^71 + 29 * q^73 + 8 * q^79 - 7 * q^83 + 8 * q^85 + 4 * q^89 + 26 * q^91 - 6 * q^95 + 15 * q^97

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