LMFDB - Embedded newform 5520.2.a.bx.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, 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("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.71083 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.71083 q^{7} +1.00000 q^{9} -2.52444 q^{11} -6.72999 q^{13} +1.00000 q^{15} +4.44082 q^{17} +3.10278 q^{19} -1.71083 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.01916 q^{29} +1.13249 q^{31} +2.52444 q^{33} -1.71083 q^{35} +2.49472 q^{37} +6.72999 q^{39} -7.39194 q^{41} +6.78389 q^{43} -1.00000 q^{45} +13.3083 q^{47} -4.07306 q^{49} -4.44082 q^{51} -12.4408 q^{53} +2.52444 q^{55} -3.10278 q^{57} -11.0192 q^{59} -2.89722 q^{61} +1.71083 q^{63} +6.72999 q^{65} +0.0836184 q^{67} +1.00000 q^{69} +13.8030 q^{71} -5.57331 q^{73} -1.00000 q^{75} -4.31889 q^{77} +1.00000 q^{81} -1.97028 q^{83} -4.44082 q^{85} -5.01916 q^{87} -7.62721 q^{89} -11.5139 q^{91} -1.13249 q^{93} -3.10278 q^{95} +9.83276 q^{97} -2.52444 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{15} - 6 q^{17} + 2 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 6 q^{35} - 8 q^{37} - 14 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} + 5 q^{49} + 6 q^{51} - 18 q^{53} + 2 q^{55} - 2 q^{57} - 12 q^{59} - 16 q^{61} + 6 q^{63} + 14 q^{67} + 3 q^{69} + 4 q^{71} - 3 q^{75} - 22 q^{77} + 3 q^{81} + 4 q^{83} + 6 q^{85} + 6 q^{87} - 10 q^{89} + 2 q^{91} - 6 q^{93} - 2 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 - 2 * q^11 + 3 * q^15 - 6 * q^17 + 2 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 6 * q^29 + 6 * q^31 + 2 * q^33 - 6 * q^35 - 8 * q^37 - 14 * q^41 + 4 * q^43 - 3 * q^45 + 18 * q^47 + 5 * q^49 + 6 * q^51 - 18 * q^53 + 2 * q^55 - 2 * q^57 - 12 * q^59 - 16 * q^61 + 6 * q^63 + 14 * q^67 + 3 * q^69 + 4 * q^71 - 3 * q^75 - 22 * q^77 + 3 * q^81 + 4 * q^83 + 6 * q^85 + 6 * q^87 - 10 * q^89 + 2 * q^91 - 6 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

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
$7$	1.71083	0.646634	0.323317	−	0.946291i	$-0.395202\pi$
$7$	1.71083	0.646634	0.323317	+	0.946291i	$0.395202\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.52444	−0.761147	−0.380573	−	0.924751i	$-0.624273\pi$
$11$	−2.52444	−0.761147	−0.380573	+	0.924751i	$0.624273\pi$
$12$	0	0
$13$	−6.72999	−1.86656	−0.933281	−	0.359146i	$-0.883068\pi$
$13$	−6.72999	−1.86656	−0.933281	+	0.359146i	$0.883068\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.44082	1.07706	0.538528	−	0.842607i	$-0.318980\pi$
$17$	4.44082	1.07706	0.538528	+	0.842607i	$0.318980\pi$
$18$	0	0
$19$	3.10278	0.711825	0.355913	−	0.934519i	$-0.384170\pi$
$19$	3.10278	0.711825	0.355913	+	0.934519i	$0.384170\pi$
$20$	0	0
$21$	−1.71083	−0.373334
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.01916	0.932034	0.466017	−	0.884776i	$-0.345689\pi$
$29$	5.01916	0.932034	0.466017	+	0.884776i	$0.345689\pi$
$30$	0	0
$31$	1.13249	0.203402	0.101701	−	0.994815i	$-0.467572\pi$
$31$	1.13249	0.203402	0.101701	+	0.994815i	$0.467572\pi$
$32$	0	0
$33$	2.52444	0.439448
$34$	0	0
$35$	−1.71083	−0.289183
$36$	0	0
$37$	2.49472	0.410129	0.205065	−	0.978748i	$-0.434260\pi$
$37$	2.49472	0.410129	0.205065	+	0.978748i	$0.434260\pi$
$38$	0	0
$39$	6.72999	1.07766
$40$	0	0
$41$	−7.39194	−1.15443	−0.577214	−	0.816593i	$-0.695860\pi$
$41$	−7.39194	−1.15443	−0.577214	+	0.816593i	$0.695860\pi$
$42$	0	0
$43$	6.78389	1.03453	0.517267	−	0.855824i	$-0.326950\pi$
$43$	6.78389	1.03453	0.517267	+	0.855824i	$0.326950\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	13.3083	1.94122	0.970609	−	0.240660i	$-0.0773640\pi$
$47$	13.3083	1.94122	0.970609	+	0.240660i	$0.0773640\pi$
$48$	0	0
$49$	−4.07306	−0.581865
$50$	0	0
$51$	−4.44082	−0.621839
$52$	0	0
$53$	−12.4408	−1.70888	−0.854439	−	0.519552i	$-0.826099\pi$
$53$	−12.4408	−1.70888	−0.854439	+	0.519552i	$0.826099\pi$
$54$	0	0
$55$	2.52444	0.340395
$56$	0	0
$57$	−3.10278	−0.410973
$58$	0	0
$59$	−11.0192	−1.43457	−0.717286	−	0.696779i	$-0.754616\pi$
$59$	−11.0192	−1.43457	−0.717286	+	0.696779i	$0.754616\pi$
$60$	0	0
$61$	−2.89722	−0.370952	−0.185476	−	0.982649i	$-0.559383\pi$
$61$	−2.89722	−0.370952	−0.185476	+	0.982649i	$0.559383\pi$
$62$	0	0
$63$	1.71083	0.215545
$64$	0	0
$65$	6.72999	0.834752
$66$	0	0
$67$	0.0836184	0.0102156	0.00510781	−	0.999987i	$-0.498374\pi$
$67$	0.0836184	0.0102156	0.00510781	+	0.999987i	$0.498374\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	13.8030	1.63812	0.819060	−	0.573708i	$-0.194496\pi$
$71$	13.8030	1.63812	0.819060	+	0.573708i	$0.194496\pi$
$72$	0	0
$73$	−5.57331	−0.652307	−0.326154	−	0.945317i	$-0.605753\pi$
$73$	−5.57331	−0.652307	−0.326154	+	0.945317i	$0.605753\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−4.31889	−0.492183
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.97028	−0.216266	−0.108133	−	0.994136i	$-0.534487\pi$
$83$	−1.97028	−0.216266	−0.108133	+	0.994136i	$0.534487\pi$
$84$	0	0
$85$	−4.44082	−0.481675
$86$	0	0
$87$	−5.01916	−0.538110
$88$	0	0
$89$	−7.62721	−0.808483	−0.404241	−	0.914652i	$-0.632464\pi$
$89$	−7.62721	−0.808483	−0.404241	+	0.914652i	$0.632464\pi$
$90$	0	0
$91$	−11.5139	−1.20698
$92$	0	0
$93$	−1.13249	−0.117434
$94$	0	0
$95$	−3.10278	−0.318338
$96$	0	0
$97$	9.83276	0.998366	0.499183	−	0.866497i	$-0.333634\pi$
$97$	9.83276	0.998366	0.499183	+	0.866497i	$0.333634\pi$
$98$	0	0
$99$	−2.52444	−0.253716
$100$	0	0
$101$	−10.8136	−1.07599	−0.537997	−	0.842947i	$-0.680819\pi$
$101$	−10.8136	−1.07599	−0.537997	+	0.842947i	$0.680819\pi$
$102$	0	0
$103$	−15.0872	−1.48658	−0.743292	−	0.668967i	$-0.766737\pi$
$103$	−15.0872	−1.48658	−0.743292	+	0.668967i	$0.766737\pi$
$104$	0	0
$105$	1.71083	0.166960
$106$	0	0
$107$	13.8625	1.34014	0.670068	−	0.742299i	$-0.266265\pi$
$107$	13.8625	1.34014	0.670068	+	0.742299i	$0.266265\pi$
$108$	0	0
$109$	16.8277	1.61181	0.805903	−	0.592048i	$-0.201680\pi$
$109$	16.8277	1.61181	0.805903	+	0.592048i	$0.201680\pi$
$110$	0	0
$111$	−2.49472	−0.236788
$112$	0	0
$113$	−19.6952	−1.85277	−0.926386	−	0.376574i	$-0.877102\pi$
$113$	−19.6952	−1.85277	−0.926386	+	0.376574i	$0.877102\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−6.72999	−0.622188
$118$	0	0
$119$	7.59749	0.696461
$120$	0	0
$121$	−4.62721	−0.420656
$122$	0	0
$123$	7.39194	0.666509
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.9355	1.68026	0.840129	−	0.542387i	$-0.182479\pi$
$127$	18.9355	1.68026	0.840129	+	0.542387i	$0.182479\pi$
$128$	0	0
$129$	−6.78389	−0.597288
$130$	0	0
$131$	−12.4111	−1.08436	−0.542181	−	0.840261i	$-0.682401\pi$
$131$	−12.4111	−1.08436	−0.542181	+	0.840261i	$0.682401\pi$
$132$	0	0
$133$	5.30833	0.460290
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−13.4217	−1.14669	−0.573345	−	0.819314i	$-0.694355\pi$
$137$	−13.4217	−1.14669	−0.573345	+	0.819314i	$0.694355\pi$
$138$	0	0
$139$	−15.2786	−1.29591	−0.647957	−	0.761677i	$-0.724376\pi$
$139$	−15.2786	−1.29591	−0.647957	+	0.761677i	$0.724376\pi$
$140$	0	0
$141$	−13.3083	−1.12076
$142$	0	0
$143$	16.9894	1.42073
$144$	0	0
$145$	−5.01916	−0.416818
$146$	0	0
$147$	4.07306	0.335940
$148$	0	0
$149$	−1.68111	−0.137722	−0.0688610	−	0.997626i	$-0.521937\pi$
$149$	−1.68111	−0.137722	−0.0688610	+	0.997626i	$0.521937\pi$
$150$	0	0
$151$	2.78389	0.226550	0.113275	−	0.993564i	$-0.463866\pi$
$151$	2.78389	0.226550	0.113275	+	0.993564i	$0.463866\pi$
$152$	0	0
$153$	4.44082	0.359019
$154$	0	0
$155$	−1.13249	−0.0909641
$156$	0	0
$157$	−9.91638	−0.791413	−0.395707	−	0.918377i	$-0.629500\pi$
$157$	−9.91638	−0.791413	−0.395707	+	0.918377i	$0.629500\pi$
$158$	0	0
$159$	12.4408	0.986621
$160$	0	0
$161$	−1.71083	−0.134832
$162$	0	0
$163$	−7.42166	−0.581310	−0.290655	−	0.956828i	$-0.593873\pi$
$163$	−7.42166	−0.581310	−0.290655	+	0.956828i	$0.593873\pi$
$164$	0	0
$165$	−2.52444	−0.196527
$166$	0	0
$167$	5.94610	0.460123	0.230062	−	0.973176i	$-0.426107\pi$
$167$	5.94610	0.460123	0.230062	+	0.973176i	$0.426107\pi$
$168$	0	0
$169$	32.2927	2.48406
$170$	0	0
$171$	3.10278	0.237275
$172$	0	0
$173$	13.0872	0.995001	0.497500	−	0.867464i	$-0.334251\pi$
$173$	13.0872	0.995001	0.497500	+	0.867464i	$0.334251\pi$
$174$	0	0
$175$	1.71083	0.129327
$176$	0	0
$177$	11.0192	0.828251
$178$	0	0
$179$	−26.4494	−1.97692	−0.988461	−	0.151476i	$-0.951597\pi$
$179$	−26.4494	−1.97692	−0.988461	+	0.151476i	$0.951597\pi$
$180$	0	0
$181$	−24.0383	−1.78675	−0.893377	−	0.449308i	$-0.851671\pi$
$181$	−24.0383	−1.78675	−0.893377	+	0.449308i	$0.851671\pi$
$182$	0	0
$183$	2.89722	0.214169
$184$	0	0
$185$	−2.49472	−0.183415
$186$	0	0
$187$	−11.2106	−0.819798
$188$	0	0
$189$	−1.71083	−0.124445
$190$	0	0
$191$	−25.7194	−1.86099	−0.930496	−	0.366302i	$-0.880624\pi$
$191$	−25.7194	−1.86099	−0.930496	+	0.366302i	$0.880624\pi$
$192$	0	0
$193$	26.1361	1.88132	0.940658	−	0.339357i	$-0.110210\pi$
$193$	26.1361	1.88132	0.940658	+	0.339357i	$0.110210\pi$
$194$	0	0
$195$	−6.72999	−0.481944
$196$	0	0
$197$	−14.3033	−1.01907	−0.509534	−	0.860451i	$-0.670182\pi$
$197$	−14.3033	−1.01907	−0.509534	+	0.860451i	$0.670182\pi$
$198$	0	0
$199$	25.2927	1.79295	0.896477	−	0.443090i	$-0.146118\pi$
$199$	25.2927	1.79295	0.896477	+	0.443090i	$0.146118\pi$
$200$	0	0
$201$	−0.0836184	−0.00589799
$202$	0	0
$203$	8.58693	0.602684
$204$	0	0
$205$	7.39194	0.516276
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−7.83276	−0.541804
$210$	0	0
$211$	21.7875	1.49991	0.749955	−	0.661489i	$-0.230075\pi$
$211$	21.7875	1.49991	0.749955	+	0.661489i	$0.230075\pi$
$212$	0	0
$213$	−13.8030	−0.945769
$214$	0	0
$215$	−6.78389	−0.462657
$216$	0	0
$217$	1.93751	0.131527
$218$	0	0
$219$	5.57331	0.376610
$220$	0	0
$221$	−29.8867	−2.01039
$222$	0	0
$223$	−20.7738	−1.39112	−0.695560	−	0.718468i	$-0.744843\pi$
$223$	−20.7738	−1.39112	−0.695560	+	0.718468i	$0.744843\pi$
$224$	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
$229$	−9.19499	−0.607622	−0.303811	−	0.952732i	$-0.598259\pi$
$229$	−9.19499	−0.607622	−0.303811	+	0.952732i	$0.598259\pi$
$230$	0	0
$231$	4.31889	0.284162
$232$	0	0
$233$	6.98944	0.457893	0.228947	−	0.973439i	$-0.426472\pi$
$233$	6.98944	0.457893	0.228947	+	0.973439i	$0.426472\pi$
$234$	0	0
$235$	−13.3083	−0.868139
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.2353	−1.05017	−0.525086	−	0.851049i	$-0.675967\pi$
$239$	−16.2353	−1.05017	−0.525086	+	0.851049i	$0.675967\pi$
$240$	0	0
$241$	7.77886	0.501081	0.250540	−	0.968106i	$-0.419392\pi$
$241$	7.77886	0.501081	0.250540	+	0.968106i	$0.419392\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.07306	0.260218
$246$	0	0
$247$	−20.8816	−1.32867
$248$	0	0
$249$	1.97028	0.124861
$250$	0	0
$251$	19.0872	1.20477	0.602386	−	0.798205i	$-0.294217\pi$
$251$	19.0872	1.20477	0.602386	+	0.798205i	$0.294217\pi$
$252$	0	0
$253$	2.52444	0.158710
$254$	0	0
$255$	4.44082	0.278095
$256$	0	0
$257$	−27.2489	−1.69974	−0.849869	−	0.526993i	$-0.823319\pi$
$257$	−27.2489	−1.69974	−0.849869	+	0.526993i	$0.823319\pi$
$258$	0	0
$259$	4.26804	0.265203
$260$	0	0
$261$	5.01916	0.310678
$262$	0	0
$263$	−15.4303	−0.951470	−0.475735	−	0.879589i	$-0.657818\pi$
$263$	−15.4303	−0.951470	−0.475735	+	0.879589i	$0.657818\pi$
$264$	0	0
$265$	12.4408	0.764233
$266$	0	0
$267$	7.62721	0.466778
$268$	0	0
$269$	−12.9114	−0.787219	−0.393610	−	0.919278i	$-0.628774\pi$
$269$	−12.9114	−0.787219	−0.393610	+	0.919278i	$0.628774\pi$
$270$	0	0
$271$	3.74914	0.227744	0.113872	−	0.993495i	$-0.463675\pi$
$271$	3.74914	0.227744	0.113872	+	0.993495i	$0.463675\pi$
$272$	0	0
$273$	11.5139	0.696851
$274$	0	0
$275$	−2.52444	−0.152229
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	1.13249	0.0678007
$280$	0	0
$281$	−1.84835	−0.110263	−0.0551316	−	0.998479i	$-0.517558\pi$
$281$	−1.84835	−0.110263	−0.0551316	+	0.998479i	$0.517558\pi$
$282$	0	0
$283$	−1.40753	−0.0836689	−0.0418345	−	0.999125i	$-0.513320\pi$
$283$	−1.40753	−0.0836689	−0.0418345	+	0.999125i	$0.513320\pi$
$284$	0	0
$285$	3.10278	0.183793
$286$	0	0
$287$	−12.6464	−0.746492
$288$	0	0
$289$	2.72088	0.160052
$290$	0	0
$291$	−9.83276	−0.576407
$292$	0	0
$293$	27.5280	1.60820	0.804102	−	0.594492i	$-0.202647\pi$
$293$	27.5280	1.60820	0.804102	+	0.594492i	$0.202647\pi$
$294$	0	0
$295$	11.0192	0.641560
$296$	0	0
$297$	2.52444	0.146483
$298$	0	0
$299$	6.72999	0.389205
$300$	0	0
$301$	11.6061	0.668964
$302$	0	0
$303$	10.8136	0.621225
$304$	0	0
$305$	2.89722	0.165895
$306$	0	0
$307$	4.56275	0.260410	0.130205	−	0.991487i	$-0.458436\pi$
$307$	4.56275	0.260410	0.130205	+	0.991487i	$0.458436\pi$
$308$	0	0
$309$	15.0872	0.858280
$310$	0	0
$311$	−6.84333	−0.388049	−0.194025	−	0.980997i	$-0.562154\pi$
$311$	−6.84333	−0.388049	−0.194025	+	0.980997i	$0.562154\pi$
$312$	0	0
$313$	−25.7491	−1.45543	−0.727714	−	0.685881i	$-0.759417\pi$
$313$	−25.7491	−1.45543	−0.727714	+	0.685881i	$0.759417\pi$
$314$	0	0
$315$	−1.71083	−0.0963944
$316$	0	0
$317$	9.23884	0.518905	0.259452	−	0.965756i	$-0.416458\pi$
$317$	9.23884	0.518905	0.259452	+	0.965756i	$0.416458\pi$
$318$	0	0
$319$	−12.6705	−0.709415
$320$	0	0
$321$	−13.8625	−0.773728
$322$	0	0
$323$	13.7789	0.766676
$324$	0	0
$325$	−6.72999	−0.373313
$326$	0	0
$327$	−16.8277	−0.930576
$328$	0	0
$329$	22.7683	1.25526
$330$	0	0
$331$	−13.6030	−0.747690	−0.373845	−	0.927491i	$-0.621961\pi$
$331$	−13.6030	−0.747690	−0.373845	+	0.927491i	$0.621961\pi$
$332$	0	0
$333$	2.49472	0.136710
$334$	0	0
$335$	−0.0836184	−0.00456856
$336$	0	0
$337$	−9.25443	−0.504121	−0.252060	−	0.967712i	$-0.581108\pi$
$337$	−9.25443	−0.504121	−0.252060	+	0.967712i	$0.581108\pi$
$338$	0	0
$339$	19.6952	1.06970
$340$	0	0
$341$	−2.85891	−0.154819
$342$	0	0
$343$	−18.9441	−1.02289
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−22.9511	−1.23208	−0.616040	−	0.787715i	$-0.711264\pi$
$347$	−22.9511	−1.23208	−0.616040	+	0.787715i	$0.711264\pi$
$348$	0	0
$349$	−2.96526	−0.158727	−0.0793633	−	0.996846i	$-0.525289\pi$
$349$	−2.96526	−0.158727	−0.0793633	+	0.996846i	$0.525289\pi$
$350$	0	0
$351$	6.72999	0.359220
$352$	0	0
$353$	−19.2489	−1.02451	−0.512257	−	0.858832i	$-0.671191\pi$
$353$	−19.2489	−1.02451	−0.512257	+	0.858832i	$0.671191\pi$
$354$	0	0
$355$	−13.8030	−0.732589
$356$	0	0
$357$	−7.59749	−0.402102
$358$	0	0
$359$	−21.6116	−1.14062	−0.570309	−	0.821430i	$-0.693177\pi$
$359$	−21.6116	−1.14062	−0.570309	+	0.821430i	$0.693177\pi$
$360$	0	0
$361$	−9.37279	−0.493305
$362$	0	0
$363$	4.62721	0.242866
$364$	0	0
$365$	5.57331	0.291721
$366$	0	0
$367$	9.71083	0.506901	0.253451	−	0.967348i	$-0.418434\pi$
$367$	9.71083	0.506901	0.253451	+	0.967348i	$0.418434\pi$
$368$	0	0
$369$	−7.39194	−0.384809
$370$	0	0
$371$	−21.2841	−1.10502
$372$	0	0
$373$	−32.4494	−1.68017	−0.840083	−	0.542457i	$-0.817494\pi$
$373$	−32.4494	−1.68017	−0.840083	+	0.542457i	$0.817494\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−33.7789	−1.73970
$378$	0	0
$379$	−28.8222	−1.48050	−0.740248	−	0.672333i	$-0.765292\pi$
$379$	−28.8222	−1.48050	−0.740248	+	0.672333i	$0.765292\pi$
$380$	0	0
$381$	−18.9355	−0.970097
$382$	0	0
$383$	−2.54862	−0.130228	−0.0651141	−	0.997878i	$-0.520741\pi$
$383$	−2.54862	−0.130228	−0.0651141	+	0.997878i	$0.520741\pi$
$384$	0	0
$385$	4.31889	0.220111
$386$	0	0
$387$	6.78389	0.344844
$388$	0	0
$389$	−17.5577	−0.890212	−0.445106	−	0.895478i	$-0.646834\pi$
$389$	−17.5577	−0.890212	−0.445106	+	0.895478i	$0.646834\pi$
$390$	0	0
$391$	−4.44082	−0.224582
$392$	0	0
$393$	12.4111	0.626057
$394$	0	0
$395$	0	0
$396$	0	0
$397$	29.0872	1.45984	0.729922	−	0.683530i	$-0.239556\pi$
$397$	29.0872	1.45984	0.729922	+	0.683530i	$0.239556\pi$
$398$	0	0
$399$	−5.30833	−0.265749
$400$	0	0
$401$	−0.616650	−0.0307940	−0.0153970	−	0.999881i	$-0.504901\pi$
$401$	−0.616650	−0.0307940	−0.0153970	+	0.999881i	$0.504901\pi$
$402$	0	0
$403$	−7.62167	−0.379663
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−6.29776	−0.312168
$408$	0	0
$409$	−8.07357	−0.399212	−0.199606	−	0.979876i	$-0.563966\pi$
$409$	−8.07357	−0.399212	−0.199606	+	0.979876i	$0.563966\pi$
$410$	0	0
$411$	13.4217	0.662042
$412$	0	0
$413$	−18.8519	−0.927642
$414$	0	0
$415$	1.97028	0.0967173
$416$	0	0
$417$	15.2786	0.748197
$418$	0	0
$419$	−19.9844	−0.976303	−0.488151	−	0.872759i	$-0.662329\pi$
$419$	−19.9844	−0.976303	−0.488151	+	0.872759i	$0.662329\pi$
$420$	0	0
$421$	9.04334	0.440745	0.220373	−	0.975416i	$-0.429273\pi$
$421$	9.04334	0.440745	0.220373	+	0.975416i	$0.429273\pi$
$422$	0	0
$423$	13.3083	0.647073
$424$	0	0
$425$	4.44082	0.215411
$426$	0	0
$427$	−4.95666	−0.239870
$428$	0	0
$429$	−16.9894	−0.820258
$430$	0	0
$431$	−8.63778	−0.416067	−0.208034	−	0.978122i	$-0.566706\pi$
$431$	−8.63778	−0.416067	−0.208034	+	0.978122i	$0.566706\pi$
$432$	0	0
$433$	−34.6902	−1.66711	−0.833553	−	0.552440i	$-0.813697\pi$
$433$	−34.6902	−1.66711	−0.833553	+	0.552440i	$0.813697\pi$
$434$	0	0
$435$	5.01916	0.240650
$436$	0	0
$437$	−3.10278	−0.148426
$438$	0	0
$439$	−11.4217	−0.545126	−0.272563	−	0.962138i	$-0.587871\pi$
$439$	−11.4217	−0.545126	−0.272563	+	0.962138i	$0.587871\pi$
$440$	0	0
$441$	−4.07306	−0.193955
$442$	0	0
$443$	8.78943	0.417598	0.208799	−	0.977959i	$-0.433045\pi$
$443$	8.78943	0.417598	0.208799	+	0.977959i	$0.433045\pi$
$444$	0	0
$445$	7.62721	0.361565
$446$	0	0
$447$	1.68111	0.0795139
$448$	0	0
$449$	−11.5874	−0.546845	−0.273423	−	0.961894i	$-0.588156\pi$
$449$	−11.5874	−0.546845	−0.273423	+	0.961894i	$0.588156\pi$
$450$	0	0
$451$	18.6605	0.878689
$452$	0	0
$453$	−2.78389	−0.130798
$454$	0	0
$455$	11.5139	0.539779
$456$	0	0
$457$	10.6620	0.498745	0.249373	−	0.968408i	$-0.419776\pi$
$457$	10.6620	0.498745	0.249373	+	0.968408i	$0.419776\pi$
$458$	0	0
$459$	−4.44082	−0.207280
$460$	0	0
$461$	−22.3033	−1.03877	−0.519384	−	0.854541i	$-0.673839\pi$
$461$	−22.3033	−1.03877	−0.519384	+	0.854541i	$0.673839\pi$
$462$	0	0
$463$	27.9250	1.29778	0.648892	−	0.760881i	$-0.275233\pi$
$463$	27.9250	1.29778	0.648892	+	0.760881i	$0.275233\pi$
$464$	0	0
$465$	1.13249	0.0525182
$466$	0	0
$467$	−24.4308	−1.13052	−0.565261	−	0.824912i	$-0.691224\pi$
$467$	−24.4308	−1.13052	−0.565261	+	0.824912i	$0.691224\pi$
$468$	0	0
$469$	0.143057	0.00660576
$470$	0	0
$471$	9.91638	0.456923
$472$	0	0
$473$	−17.1255	−0.787431
$474$	0	0
$475$	3.10278	0.142365
$476$	0	0
$477$	−12.4408	−0.569626
$478$	0	0
$479$	−19.2106	−0.877753	−0.438877	−	0.898547i	$-0.644624\pi$
$479$	−19.2106	−0.877753	−0.438877	+	0.898547i	$0.644624\pi$
$480$	0	0
$481$	−16.7894	−0.765532
$482$	0	0
$483$	1.71083	0.0778455
$484$	0	0
$485$	−9.83276	−0.446483
$486$	0	0
$487$	−22.4933	−1.01927	−0.509634	−	0.860392i	$-0.670219\pi$
$487$	−22.4933	−1.01927	−0.509634	+	0.860392i	$0.670219\pi$
$488$	0	0
$489$	7.42166	0.335619
$490$	0	0
$491$	8.53857	0.385340	0.192670	−	0.981264i	$-0.438285\pi$
$491$	8.53857	0.385340	0.192670	+	0.981264i	$0.438285\pi$
$492$	0	0
$493$	22.2892	1.00385
$494$	0	0
$495$	2.52444	0.113465
$496$	0	0
$497$	23.6147	1.05926
$498$	0	0
$499$	−17.0247	−0.762130	−0.381065	−	0.924548i	$-0.624443\pi$
$499$	−17.0247	−0.762130	−0.381065	+	0.924548i	$0.624443\pi$
$500$	0	0
$501$	−5.94610	−0.265652
$502$	0	0
$503$	33.8625	1.50985	0.754927	−	0.655809i	$-0.227672\pi$
$503$	33.8625	1.50985	0.754927	+	0.655809i	$0.227672\pi$
$504$	0	0
$505$	10.8136	0.481199
$506$	0	0
$507$	−32.2927	−1.43417
$508$	0	0
$509$	34.7738	1.54132	0.770662	−	0.637244i	$-0.219926\pi$
$509$	34.7738	1.54132	0.770662	+	0.637244i	$0.219926\pi$
$510$	0	0
$511$	−9.53500	−0.421804
$512$	0	0
$513$	−3.10278	−0.136991
$514$	0	0
$515$	15.0872	0.664821
$516$	0	0
$517$	−33.5960	−1.47755
$518$	0	0
$519$	−13.0872	−0.574464
$520$	0	0
$521$	−12.7995	−0.560755	−0.280378	−	0.959890i	$-0.590460\pi$
$521$	−12.7995	−0.560755	−0.280378	+	0.959890i	$0.590460\pi$
$522$	0	0
$523$	24.4111	1.06742	0.533711	−	0.845667i	$-0.320797\pi$
$523$	24.4111	1.06742	0.533711	+	0.845667i	$0.320797\pi$
$524$	0	0
$525$	−1.71083	−0.0746668
$526$	0	0
$527$	5.02920	0.219076
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.0192	−0.478191
$532$	0	0
$533$	49.7477	2.15481
$534$	0	0
$535$	−13.8625	−0.599327
$536$	0	0
$537$	26.4494	1.14138
$538$	0	0
$539$	10.2822	0.442885
$540$	0	0
$541$	27.1849	1.16877	0.584386	−	0.811476i	$-0.301335\pi$
$541$	27.1849	1.16877	0.584386	+	0.811476i	$0.301335\pi$
$542$	0	0
$543$	24.0383	1.03158
$544$	0	0
$545$	−16.8277	−0.720821
$546$	0	0
$547$	−26.8433	−1.14774	−0.573869	−	0.818947i	$-0.694558\pi$
$547$	−26.8433	−1.14774	−0.573869	+	0.818947i	$0.694558\pi$
$548$	0	0
$549$	−2.89722	−0.123651
$550$	0	0
$551$	15.5733	0.663445
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.49472	0.105895
$556$	0	0
$557$	−32.8519	−1.39198	−0.695990	−	0.718051i	$-0.745034\pi$
$557$	−32.8519	−1.39198	−0.695990	+	0.718051i	$0.745034\pi$
$558$	0	0
$559$	−45.6555	−1.93102
$560$	0	0
$561$	11.2106	0.473311
$562$	0	0
$563$	26.7441	1.12713	0.563565	−	0.826072i	$-0.309429\pi$
$563$	26.7441	1.12713	0.563565	+	0.826072i	$0.309429\pi$
$564$	0	0
$565$	19.6952	0.828585
$566$	0	0
$567$	1.71083	0.0718482
$568$	0	0
$569$	−34.3799	−1.44128	−0.720641	−	0.693309i	$-0.756152\pi$
$569$	−34.3799	−1.44128	−0.720641	+	0.693309i	$0.756152\pi$
$570$	0	0
$571$	−39.9532	−1.67199	−0.835996	−	0.548736i	$-0.815109\pi$
$571$	−39.9532	−1.67199	−0.835996	+	0.548736i	$0.815109\pi$
$572$	0	0
$573$	25.7194	1.07444
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	23.5194	0.979126	0.489563	−	0.871968i	$-0.337156\pi$
$577$	23.5194	0.979126	0.489563	+	0.871968i	$0.337156\pi$
$578$	0	0
$579$	−26.1361	−1.08618
$580$	0	0
$581$	−3.37082	−0.139845
$582$	0	0
$583$	31.4061	1.30071
$584$	0	0
$585$	6.72999	0.278251
$586$	0	0
$587$	8.36274	0.345167	0.172584	−	0.984995i	$-0.444788\pi$
$587$	8.36274	0.345167	0.172584	+	0.984995i	$0.444788\pi$
$588$	0	0
$589$	3.51388	0.144787
$590$	0	0
$591$	14.3033	0.588359
$592$	0	0
$593$	5.87662	0.241324	0.120662	−	0.992694i	$-0.461498\pi$
$593$	5.87662	0.241324	0.120662	+	0.992694i	$0.461498\pi$
$594$	0	0
$595$	−7.59749	−0.311467
$596$	0	0
$597$	−25.2927	−1.03516
$598$	0	0
$599$	9.12550	0.372858	0.186429	−	0.982468i	$-0.440309\pi$
$599$	9.12550	0.372858	0.186429	+	0.982468i	$0.440309\pi$
$600$	0	0
$601$	15.3764	0.627215	0.313607	−	0.949553i	$-0.398462\pi$
$601$	15.3764	0.627215	0.313607	+	0.949553i	$0.398462\pi$
$602$	0	0
$603$	0.0836184	0.00340521
$604$	0	0
$605$	4.62721	0.188123
$606$	0	0
$607$	−11.1511	−0.452611	−0.226305	−	0.974056i	$-0.572665\pi$
$607$	−11.1511	−0.452611	−0.226305	+	0.974056i	$0.572665\pi$
$608$	0	0
$609$	−8.58693	−0.347960
$610$	0	0
$611$	−89.5649	−3.62341
$612$	0	0
$613$	5.47002	0.220932	0.110466	−	0.993880i	$-0.464766\pi$
$613$	5.47002	0.220932	0.110466	+	0.993880i	$0.464766\pi$
$614$	0	0
$615$	−7.39194	−0.298072
$616$	0	0
$617$	11.1169	0.447550	0.223775	−	0.974641i	$-0.428162\pi$
$617$	11.1169	0.447550	0.223775	+	0.974641i	$0.428162\pi$
$618$	0	0
$619$	18.8433	0.757377	0.378689	−	0.925524i	$-0.376375\pi$
$619$	18.8433	0.757377	0.378689	+	0.925524i	$0.376375\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−13.0489	−0.522792
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.83276	0.312810
$628$	0	0
$629$	11.0786	0.441733
$630$	0	0
$631$	4.10329	0.163349	0.0816747	−	0.996659i	$-0.473973\pi$
$631$	4.10329	0.163349	0.0816747	+	0.996659i	$0.473973\pi$
$632$	0	0
$633$	−21.7875	−0.865974
$634$	0	0
$635$	−18.9355	−0.751434
$636$	0	0
$637$	27.4116	1.08609
$638$	0	0
$639$	13.8030	0.546040
$640$	0	0
$641$	15.6116	0.616622	0.308311	−	0.951286i	$-0.400236\pi$
$641$	15.6116	0.616622	0.308311	+	0.951286i	$0.400236\pi$
$642$	0	0
$643$	0.602517	0.0237609	0.0118805	−	0.999929i	$-0.496218\pi$
$643$	0.602517	0.0237609	0.0118805	+	0.999929i	$0.496218\pi$
$644$	0	0
$645$	6.78389	0.267115
$646$	0	0
$647$	27.3466	1.07511	0.537554	−	0.843230i	$-0.319349\pi$
$647$	27.3466	1.07511	0.537554	+	0.843230i	$0.319349\pi$
$648$	0	0
$649$	27.8172	1.09192
$650$	0	0
$651$	−1.93751	−0.0759369
$652$	0	0
$653$	45.8172	1.79296	0.896482	−	0.443079i	$-0.146114\pi$
$653$	45.8172	1.79296	0.896482	+	0.443079i	$0.146114\pi$
$654$	0	0
$655$	12.4111	0.484942
$656$	0	0
$657$	−5.57331	−0.217436
$658$	0	0
$659$	−4.83779	−0.188453	−0.0942267	−	0.995551i	$-0.530038\pi$
$659$	−4.83779	−0.188453	−0.0942267	+	0.995551i	$0.530038\pi$
$660$	0	0
$661$	36.6933	1.42720	0.713602	−	0.700552i	$-0.247063\pi$
$661$	36.6933	1.42720	0.713602	+	0.700552i	$0.247063\pi$
$662$	0	0
$663$	29.8867	1.16070
$664$	0	0
$665$	−5.30833	−0.205848
$666$	0	0
$667$	−5.01916	−0.194343
$668$	0	0
$669$	20.7738	0.803163
$670$	0	0
$671$	7.31386	0.282349
$672$	0	0
$673$	37.2983	1.43774	0.718871	−	0.695143i	$-0.244659\pi$
$673$	37.2983	1.43774	0.718871	+	0.695143i	$0.244659\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	30.1275	1.15789	0.578946	−	0.815366i	$-0.303464\pi$
$677$	30.1275	1.15789	0.578946	+	0.815366i	$0.303464\pi$
$678$	0	0
$679$	16.8222	0.645577
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	20.8972	0.799610	0.399805	−	0.916600i	$-0.369078\pi$
$683$	20.8972	0.799610	0.399805	+	0.916600i	$0.369078\pi$
$684$	0	0
$685$	13.4217	0.512815
$686$	0	0
$687$	9.19499	0.350811
$688$	0	0
$689$	83.7266	3.18973
$690$	0	0
$691$	−1.21611	−0.0462631	−0.0231316	−	0.999732i	$-0.507364\pi$
$691$	−1.21611	−0.0462631	−0.0231316	+	0.999732i	$0.507364\pi$
$692$	0	0
$693$	−4.31889	−0.164061
$694$	0	0
$695$	15.2786	0.579551
$696$	0	0
$697$	−32.8263	−1.24338
$698$	0	0
$699$	−6.98944	−0.264365
$700$	0	0
$701$	2.42669	0.0916547	0.0458273	−	0.998949i	$-0.485408\pi$
$701$	2.42669	0.0916547	0.0458273	+	0.998949i	$0.485408\pi$
$702$	0	0
$703$	7.74055	0.291940
$704$	0	0
$705$	13.3083	0.501221
$706$	0	0
$707$	−18.5003	−0.695774
$708$	0	0
$709$	3.00502	0.112856	0.0564280	−	0.998407i	$-0.482029\pi$
$709$	3.00502	0.112856	0.0564280	+	0.998407i	$0.482029\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.13249	−0.0424122
$714$	0	0
$715$	−16.9894	−0.635369
$716$	0	0
$717$	16.2353	0.606317
$718$	0	0
$719$	−18.3814	−0.685510	−0.342755	−	0.939425i	$-0.611360\pi$
$719$	−18.3814	−0.685510	−0.342755	+	0.939425i	$0.611360\pi$
$720$	0	0
$721$	−25.8116	−0.961276
$722$	0	0
$723$	−7.77886	−0.289299
$724$	0	0
$725$	5.01916	0.186407
$726$	0	0
$727$	23.5335	0.872811	0.436405	−	0.899750i	$-0.356251\pi$
$727$	23.5335	0.872811	0.436405	+	0.899750i	$0.356251\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.1260	1.11425
$732$	0	0
$733$	−22.6308	−0.835887	−0.417944	−	0.908473i	$-0.637249\pi$
$733$	−22.6308	−0.835887	−0.417944	+	0.908473i	$0.637249\pi$
$734$	0	0
$735$	−4.07306	−0.150237
$736$	0	0
$737$	−0.211090	−0.00777558
$738$	0	0
$739$	2.18137	0.0802430	0.0401215	−	0.999195i	$-0.487226\pi$
$739$	2.18137	0.0802430	0.0401215	+	0.999195i	$0.487226\pi$
$740$	0	0
$741$	20.8816	0.767106
$742$	0	0
$743$	25.7633	0.945163	0.472582	−	0.881287i	$-0.343322\pi$
$743$	25.7633	0.945163	0.472582	+	0.881287i	$0.343322\pi$
$744$	0	0
$745$	1.68111	0.0615912
$746$	0	0
$747$	−1.97028	−0.0720888
$748$	0	0
$749$	23.7164	0.866577
$750$	0	0
$751$	−33.3850	−1.21823	−0.609117	−	0.793080i	$-0.708476\pi$
$751$	−33.3850	−1.21823	−0.609117	+	0.793080i	$0.708476\pi$
$752$	0	0
$753$	−19.0872	−0.695576
$754$	0	0
$755$	−2.78389	−0.101316
$756$	0	0
$757$	30.9935	1.12648	0.563239	−	0.826294i	$-0.309555\pi$
$757$	30.9935	1.12648	0.563239	+	0.826294i	$0.309555\pi$
$758$	0	0
$759$	−2.52444	−0.0916313
$760$	0	0
$761$	8.44082	0.305979	0.152990	−	0.988228i	$-0.451110\pi$
$761$	8.44082	0.305979	0.152990	+	0.988228i	$0.451110\pi$
$762$	0	0
$763$	28.7894	1.04225
$764$	0	0
$765$	−4.44082	−0.160558
$766$	0	0
$767$	74.1588	2.67772
$768$	0	0
$769$	25.4372	0.917291	0.458645	−	0.888619i	$-0.348335\pi$
$769$	25.4372	0.917291	0.458645	+	0.888619i	$0.348335\pi$
$770$	0	0
$771$	27.2489	0.981345
$772$	0	0
$773$	−35.8711	−1.29019	−0.645096	−	0.764101i	$-0.723183\pi$
$773$	−35.8711	−1.29019	−0.645096	+	0.764101i	$0.723183\pi$
$774$	0	0
$775$	1.13249	0.0406804
$776$	0	0
$777$	−4.26804	−0.153115
$778$	0	0
$779$	−22.9355	−0.821751
$780$	0	0
$781$	−34.8449	−1.24685
$782$	0	0
$783$	−5.01916	−0.179370
$784$	0	0
$785$	9.91638	0.353931
$786$	0	0
$787$	24.0524	0.857377	0.428689	−	0.903452i	$-0.358976\pi$
$787$	24.0524	0.857377	0.428689	+	0.903452i	$0.358976\pi$
$788$	0	0
$789$	15.4303	0.549332
$790$	0	0
$791$	−33.6952	−1.19807
$792$	0	0
$793$	19.4983	0.692405
$794$	0	0
$795$	−12.4408	−0.441230
$796$	0	0
$797$	−49.9291	−1.76858	−0.884289	−	0.466940i	$-0.845356\pi$
$797$	−49.9291	−1.76858	−0.884289	+	0.466940i	$0.845356\pi$
$798$	0	0
$799$	59.0999	2.09080
$800$	0	0
$801$	−7.62721	−0.269494
$802$	0	0
$803$	14.0695	0.496501
$804$	0	0
$805$	1.71083	0.0602989
$806$	0	0
$807$	12.9114	0.454501
$808$	0	0
$809$	20.8207	0.732019	0.366009	−	0.930611i	$-0.380724\pi$
$809$	20.8207	0.732019	0.366009	+	0.930611i	$0.380724\pi$
$810$	0	0
$811$	−11.0347	−0.387482	−0.193741	−	0.981053i	$-0.562062\pi$
$811$	−11.0347	−0.387482	−0.193741	+	0.981053i	$0.562062\pi$
$812$	0	0
$813$	−3.74914	−0.131488
$814$	0	0
$815$	7.42166	0.259970
$816$	0	0
$817$	21.0489	0.736407
$818$	0	0
$819$	−11.5139	−0.402327
$820$	0	0
$821$	−26.3627	−0.920066	−0.460033	−	0.887902i	$-0.652162\pi$
$821$	−26.3627	−0.920066	−0.460033	+	0.887902i	$0.652162\pi$
$822$	0	0
$823$	17.0177	0.593200	0.296600	−	0.955002i	$-0.404147\pi$
$823$	17.0177	0.593200	0.296600	+	0.955002i	$0.404147\pi$
$824$	0	0
$825$	2.52444	0.0878896
$826$	0	0
$827$	26.4408	0.919437	0.459719	−	0.888065i	$-0.347950\pi$
$827$	26.4408	0.919437	0.459719	+	0.888065i	$0.347950\pi$
$828$	0	0
$829$	14.1602	0.491806	0.245903	−	0.969294i	$-0.420916\pi$
$829$	14.1602	0.491806	0.245903	+	0.969294i	$0.420916\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−18.0877	−0.626702
$834$	0	0
$835$	−5.94610	−0.205773
$836$	0	0
$837$	−1.13249	−0.0391447
$838$	0	0
$839$	3.08719	0.106582	0.0532908	−	0.998579i	$-0.483029\pi$
$839$	3.08719	0.106582	0.0532908	+	0.998579i	$0.483029\pi$
$840$	0	0
$841$	−3.80807	−0.131313
$842$	0	0
$843$	1.84835	0.0636605
$844$	0	0
$845$	−32.2927	−1.11090
$846$	0	0
$847$	−7.91638	−0.272010
$848$	0	0
$849$	1.40753	0.0483063
$850$	0	0
$851$	−2.49472	−0.0855179
$852$	0	0
$853$	−16.2822	−0.557491	−0.278746	−	0.960365i	$-0.589919\pi$
$853$	−16.2822	−0.557491	−0.278746	+	0.960365i	$0.589919\pi$
$854$	0	0
$855$	−3.10278	−0.106113
$856$	0	0
$857$	−29.5889	−1.01074	−0.505369	−	0.862903i	$-0.668644\pi$
$857$	−29.5889	−1.01074	−0.505369	+	0.862903i	$0.668644\pi$
$858$	0	0
$859$	−30.8958	−1.05415	−0.527075	−	0.849819i	$-0.676711\pi$
$859$	−30.8958	−1.05415	−0.527075	+	0.849819i	$0.676711\pi$
$860$	0	0
$861$	12.6464	0.430987
$862$	0	0
$863$	−28.2127	−0.960371	−0.480186	−	0.877167i	$-0.659431\pi$
$863$	−28.2127	−0.960371	−0.480186	+	0.877167i	$0.659431\pi$
$864$	0	0
$865$	−13.0872	−0.444978
$866$	0	0
$867$	−2.72088	−0.0924059
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−0.562751	−0.0190681
$872$	0	0
$873$	9.83276	0.332789
$874$	0	0
$875$	−1.71083	−0.0578367
$876$	0	0
$877$	14.1078	0.476386	0.238193	−	0.971218i	$-0.423445\pi$
$877$	14.1078	0.476386	0.238193	+	0.971218i	$0.423445\pi$
$878$	0	0
$879$	−27.5280	−0.928497
$880$	0	0
$881$	−18.6222	−0.627398	−0.313699	−	0.949523i	$-0.601568\pi$
$881$	−18.6222	−0.627398	−0.313699	+	0.949523i	$0.601568\pi$
$882$	0	0
$883$	26.1133	0.878784	0.439392	−	0.898295i	$-0.355194\pi$
$883$	26.1133	0.878784	0.439392	+	0.898295i	$0.355194\pi$
$884$	0	0
$885$	−11.0192	−0.370405
$886$	0	0
$887$	−14.0383	−0.471360	−0.235680	−	0.971831i	$-0.575732\pi$
$887$	−14.0383	−0.471360	−0.235680	+	0.971831i	$0.575732\pi$
$888$	0	0
$889$	32.3955	1.08651
$890$	0	0
$891$	−2.52444	−0.0845719
$892$	0	0
$893$	41.2927	1.38181
$894$	0	0
$895$	26.4494	0.884106
$896$	0	0
$897$	−6.72999	−0.224708
$898$	0	0
$899$	5.68417	0.189578
$900$	0	0
$901$	−55.2474	−1.84056
$902$	0	0
$903$	−11.6061	−0.386226
$904$	0	0
$905$	24.0383	0.799061
$906$	0	0
$907$	−33.8953	−1.12547	−0.562737	−	0.826636i	$-0.690252\pi$
$907$	−33.8953	−1.12547	−0.562737	+	0.826636i	$0.690252\pi$
$908$	0	0
$909$	−10.8136	−0.358665
$910$	0	0
$911$	−14.6277	−0.484638	−0.242319	−	0.970197i	$-0.577908\pi$
$911$	−14.6277	−0.484638	−0.242319	+	0.970197i	$0.577908\pi$
$912$	0	0
$913$	4.97385	0.164610
$914$	0	0
$915$	−2.89722	−0.0957793
$916$	0	0
$917$	−21.2333	−0.701185
$918$	0	0
$919$	−5.68665	−0.187585	−0.0937927	−	0.995592i	$-0.529899\pi$
$919$	−5.68665	−0.187585	−0.0937927	+	0.995592i	$0.529899\pi$
$920$	0	0
$921$	−4.56275	−0.150348
$922$	0	0
$923$	−92.8943	−3.05765
$924$	0	0
$925$	2.49472	0.0820258
$926$	0	0
$927$	−15.0872	−0.495528
$928$	0	0
$929$	−27.7719	−0.911166	−0.455583	−	0.890193i	$-0.650569\pi$
$929$	−27.7719	−0.911166	−0.455583	+	0.890193i	$0.650569\pi$
$930$	0	0
$931$	−12.6378	−0.414186
$932$	0	0
$933$	6.84333	0.224040
$934$	0	0
$935$	11.2106	0.366625
$936$	0	0
$937$	−7.32391	−0.239262	−0.119631	−	0.992818i	$-0.538171\pi$
$937$	−7.32391	−0.239262	−0.119631	+	0.992818i	$0.538171\pi$
$938$	0	0
$939$	25.7491	0.840292
$940$	0	0
$941$	44.0711	1.43668	0.718338	−	0.695694i	$-0.244903\pi$
$941$	44.0711	1.43668	0.718338	+	0.695694i	$0.244903\pi$
$942$	0	0
$943$	7.39194	0.240715
$944$	0	0
$945$	1.71083	0.0556534
$946$	0	0
$947$	4.97225	0.161576	0.0807882	−	0.996731i	$-0.474256\pi$
$947$	4.97225	0.161576	0.0807882	+	0.996731i	$0.474256\pi$
$948$	0	0
$949$	37.5083	1.21757
$950$	0	0
$951$	−9.23884	−0.299590
$952$	0	0
$953$	−15.0177	−0.486471	−0.243236	−	0.969967i	$-0.578209\pi$
$953$	−15.0177	−0.486471	−0.243236	+	0.969967i	$0.578209\pi$
$954$	0	0
$955$	25.7194	0.832261
$956$	0	0
$957$	12.6705	0.409581
$958$	0	0
$959$	−22.9622	−0.741488
$960$	0	0
$961$	−29.7175	−0.958628
$962$	0	0
$963$	13.8625	0.446712
$964$	0	0
$965$	−26.1361	−0.841350
$966$	0	0
$967$	29.5834	0.951337	0.475668	−	0.879625i	$-0.342206\pi$
$967$	29.5834	0.951337	0.475668	+	0.879625i	$0.342206\pi$
$968$	0	0
$969$	−13.7789	−0.442641
$970$	0	0
$971$	33.2444	1.06686	0.533431	−	0.845843i	$-0.320902\pi$
$971$	33.2444	1.06686	0.533431	+	0.845843i	$0.320902\pi$
$972$	0	0
$973$	−26.1391	−0.837982
$974$	0	0
$975$	6.72999	0.215532
$976$	0	0
$977$	4.91136	0.157128	0.0785641	−	0.996909i	$-0.474966\pi$
$977$	4.91136	0.157128	0.0785641	+	0.996909i	$0.474966\pi$
$978$	0	0
$979$	19.2544	0.615374
$980$	0	0
$981$	16.8277	0.537268
$982$	0	0
$983$	5.83422	0.186083	0.0930413	−	0.995662i	$-0.470341\pi$
$983$	5.83422	0.186083	0.0930413	+	0.995662i	$0.470341\pi$
$984$	0	0
$985$	14.3033	0.455741
$986$	0	0
$987$	−22.7683	−0.724723
$988$	0	0
$989$	−6.78389	−0.215715
$990$	0	0
$991$	12.4353	0.395020	0.197510	−	0.980301i	$-0.436715\pi$
$991$	12.4353	0.395020	0.197510	+	0.980301i	$0.436715\pi$
$992$	0	0
$993$	13.6030	0.431679
$994$	0	0
$995$	−25.2927	−0.801834
$996$	0	0
$997$	14.8816	0.471306	0.235653	−	0.971837i	$-0.424277\pi$
$997$	14.8816	0.471306	0.235653	+	0.971837i	$0.424277\pi$
$998$	0	0
$999$	−2.49472	−0.0789294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bx.1.2		3
4.3	odd	2		2760.2.a.r.1.2	✓	3
12.11	even	2		8280.2.a.bm.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.r.1.2	✓	3	4.3	odd	2
5520.2.a.bx.1.2		3	1.1	even	1	trivial
8280.2.a.bm.1.2		3	12.11	even	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.71083 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.71083 q^{7} +1.00000 q^{9} -2.52444 q^{11} -6.72999 q^{13} +1.00000 q^{15} +4.44082 q^{17} +3.10278 q^{19} -1.71083 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.01916 q^{29} +1.13249 q^{31} +2.52444 q^{33} -1.71083 q^{35} +2.49472 q^{37} +6.72999 q^{39} -7.39194 q^{41} +6.78389 q^{43} -1.00000 q^{45} +13.3083 q^{47} -4.07306 q^{49} -4.44082 q^{51} -12.4408 q^{53} +2.52444 q^{55} -3.10278 q^{57} -11.0192 q^{59} -2.89722 q^{61} +1.71083 q^{63} +6.72999 q^{65} +0.0836184 q^{67} +1.00000 q^{69} +13.8030 q^{71} -5.57331 q^{73} -1.00000 q^{75} -4.31889 q^{77} +1.00000 q^{81} -1.97028 q^{83} -4.44082 q^{85} -5.01916 q^{87} -7.62721 q^{89} -11.5139 q^{91} -1.13249 q^{93} -3.10278 q^{95} +9.83276 q^{97} -2.52444 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{15} - 6 q^{17} + 2 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 6 q^{35} - 8 q^{37} - 14 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} + 5 q^{49} + 6 q^{51} - 18 q^{53} + 2 q^{55} - 2 q^{57} - 12 q^{59} - 16 q^{61} + 6 q^{63} + 14 q^{67} + 3 q^{69} + 4 q^{71} - 3 q^{75} - 22 q^{77} + 3 q^{81} + 4 q^{83} + 6 q^{85} + 6 q^{87} - 10 q^{89} + 2 q^{91} - 6 q^{93} - 2 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 - 2 * q^11 + 3 * q^15 - 6 * q^17 + 2 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 6 * q^29 + 6 * q^31 + 2 * q^33 - 6 * q^35 - 8 * q^37 - 14 * q^41 + 4 * q^43 - 3 * q^45 + 18 * q^47 + 5 * q^49 + 6 * q^51 - 18 * q^53 + 2 * q^55 - 2 * q^57 - 12 * q^59 - 16 * q^61 + 6 * q^63 + 14 * q^67 + 3 * q^69 + 4 * q^71 - 3 * q^75 - 22 * q^77 + 3 * q^81 + 4 * q^83 + 6 * q^85 + 6 * q^87 - 10 * q^89 + 2 * q^91 - 6 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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