LMFDB - Embedded newform 4160.2.a.br.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4160,2,Mod(1,4160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4160, 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("4160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4160 = 2^{6} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4160.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:	$33.2177672409$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	4160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.32088 q^{3} -1.00000 q^{5} +5.02827 q^{7} +8.02827 q^{9} +O(q^{10})$	$q+3.32088 q^{3} -1.00000 q^{5} +5.02827 q^{7} +8.02827 q^{9} +1.70739 q^{11} -1.00000 q^{13} -3.32088 q^{15} -4.64177 q^{17} -4.34916 q^{19} +16.6983 q^{21} +0.679116 q^{23} +1.00000 q^{25} +16.6983 q^{27} +1.02827 q^{29} +2.29261 q^{31} +5.67004 q^{33} -5.02827 q^{35} +1.61350 q^{37} -3.32088 q^{39} +4.64177 q^{41} -3.32088 q^{43} -8.02827 q^{45} -1.02827 q^{47} +18.2835 q^{49} -15.4148 q^{51} +9.41478 q^{53} -1.70739 q^{55} -14.4431 q^{57} -8.93438 q^{59} -9.02827 q^{61} +40.3684 q^{63} +1.00000 q^{65} +5.61350 q^{67} +2.25526 q^{69} -1.70739 q^{71} +12.4431 q^{73} +3.32088 q^{75} +8.58522 q^{77} +2.64177 q^{79} +31.3684 q^{81} -8.25526 q^{83} +4.64177 q^{85} +3.41478 q^{87} +1.22699 q^{89} -5.02827 q^{91} +7.61350 q^{93} +4.34916 q^{95} -0.0565477 q^{97} +13.7074 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 11 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 11 * q^9	$ 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 11 q^{9} - 3 q^{13} - 2 q^{15} + 2 q^{17} + 8 q^{19} + 8 q^{21} + 10 q^{23} + 3 q^{25} + 8 q^{27} - 10 q^{29} + 12 q^{31} - 12 q^{33} - 2 q^{35} + 2 q^{37} - 2 q^{39} - 2 q^{41} - 2 q^{43} - 11 q^{45} + 10 q^{47} + 23 q^{49} - 36 q^{51} + 18 q^{53} - 20 q^{57} - 16 q^{59} - 14 q^{61} + 50 q^{63} + 3 q^{65} + 14 q^{67} - 12 q^{69} + 14 q^{73} + 2 q^{75} + 36 q^{77} - 8 q^{79} + 23 q^{81} - 6 q^{83} - 2 q^{85} - 2 q^{89} - 2 q^{91} + 20 q^{93} - 8 q^{95} + 26 q^{97} + 36 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 11 * q^9 - 3 * q^13 - 2 * q^15 + 2 * q^17 + 8 * q^19 + 8 * q^21 + 10 * q^23 + 3 * q^25 + 8 * q^27 - 10 * q^29 + 12 * q^31 - 12 * q^33 - 2 * q^35 + 2 * q^37 - 2 * q^39 - 2 * q^41 - 2 * q^43 - 11 * q^45 + 10 * q^47 + 23 * q^49 - 36 * q^51 + 18 * q^53 - 20 * q^57 - 16 * q^59 - 14 * q^61 + 50 * q^63 + 3 * q^65 + 14 * q^67 - 12 * q^69 + 14 * q^73 + 2 * q^75 + 36 * q^77 - 8 * q^79 + 23 * q^81 - 6 * q^83 - 2 * q^85 - 2 * q^89 - 2 * q^91 + 20 * q^93 - 8 * q^95 + 26 * q^97 + 36 * 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
$3$	3.32088	1.91731	0.958657	−	0.284565i	$-0.0918491\pi$
$3$	3.32088	1.91731	0.958657	+	0.284565i	$0.0918491\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	5.02827	1.90051	0.950254	−	0.311475i	$-0.100823\pi$
$7$	5.02827	1.90051	0.950254	+	0.311475i	$0.100823\pi$
$8$	0	0
$9$	8.02827	2.67609
$10$	0	0
$11$	1.70739	0.514797	0.257399	−	0.966305i	$-0.417135\pi$
$11$	1.70739	0.514797	0.257399	+	0.966305i	$0.417135\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.32088	−0.857449
$16$	0	0
$17$	−4.64177	−1.12579	−0.562897	−	0.826527i	$-0.690313\pi$
$17$	−4.64177	−1.12579	−0.562897	+	0.826527i	$0.690313\pi$
$18$	0	0
$19$	−4.34916	−0.997765	−0.498883	−	0.866670i	$-0.666256\pi$
$19$	−4.34916	−0.997765	−0.498883	+	0.866670i	$0.666256\pi$
$20$	0	0
$21$	16.6983	3.64387
$22$	0	0
$23$	0.679116	0.141605	0.0708027	−	0.997490i	$-0.477444\pi$
$23$	0.679116	0.141605	0.0708027	+	0.997490i	$0.477444\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	16.6983	3.21359
$28$	0	0
$29$	1.02827	0.190946	0.0954728	−	0.995432i	$-0.469564\pi$
$29$	1.02827	0.190946	0.0954728	+	0.995432i	$0.469564\pi$
$30$	0	0
$31$	2.29261	0.411765	0.205883	−	0.978577i	$-0.433994\pi$
$31$	2.29261	0.411765	0.205883	+	0.978577i	$0.433994\pi$
$32$	0	0
$33$	5.67004	0.987028
$34$	0	0
$35$	−5.02827	−0.849933
$36$	0	0
$37$	1.61350	0.265257	0.132628	−	0.991166i	$-0.457658\pi$
$37$	1.61350	0.265257	0.132628	+	0.991166i	$0.457658\pi$
$38$	0	0
$39$	−3.32088	−0.531767
$40$	0	0
$41$	4.64177	0.724923	0.362461	−	0.931999i	$-0.381937\pi$
$41$	4.64177	0.724923	0.362461	+	0.931999i	$0.381937\pi$
$42$	0	0
$43$	−3.32088	−0.506430	−0.253215	−	0.967410i	$-0.581488\pi$
$43$	−3.32088	−0.506430	−0.253215	+	0.967410i	$0.581488\pi$
$44$	0	0
$45$	−8.02827	−1.19678
$46$	0	0
$47$	−1.02827	−0.149989	−0.0749946	−	0.997184i	$-0.523894\pi$
$47$	−1.02827	−0.149989	−0.0749946	+	0.997184i	$0.523894\pi$
$48$	0	0
$49$	18.2835	2.61193
$50$	0	0
$51$	−15.4148	−2.15850
$52$	0	0
$53$	9.41478	1.29322	0.646610	−	0.762821i	$-0.276186\pi$
$53$	9.41478	1.29322	0.646610	+	0.762821i	$0.276186\pi$
$54$	0	0
$55$	−1.70739	−0.230224
$56$	0	0
$57$	−14.4431	−1.91303
$58$	0	0
$59$	−8.93438	−1.16316	−0.581579	−	0.813490i	$-0.697565\pi$
$59$	−8.93438	−1.16316	−0.581579	+	0.813490i	$0.697565\pi$
$60$	0	0
$61$	−9.02827	−1.15595	−0.577976	−	0.816054i	$-0.696157\pi$
$61$	−9.02827	−1.15595	−0.577976	+	0.816054i	$0.696157\pi$
$62$	0	0
$63$	40.3684	5.08594
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	5.61350	0.685798	0.342899	−	0.939372i	$-0.388591\pi$
$67$	5.61350	0.685798	0.342899	+	0.939372i	$0.388591\pi$
$68$	0	0
$69$	2.25526	0.271502
$70$	0	0
$71$	−1.70739	−0.202630	−0.101315	−	0.994854i	$-0.532305\pi$
$71$	−1.70739	−0.202630	−0.101315	+	0.994854i	$0.532305\pi$
$72$	0	0
$73$	12.4431	1.45635	0.728175	−	0.685392i	$-0.240369\pi$
$73$	12.4431	1.45635	0.728175	+	0.685392i	$0.240369\pi$
$74$	0	0
$75$	3.32088	0.383463
$76$	0	0
$77$	8.58522	0.978377
$78$	0	0
$79$	2.64177	0.297222	0.148611	−	0.988896i	$-0.452520\pi$
$79$	2.64177	0.297222	0.148611	+	0.988896i	$0.452520\pi$
$80$	0	0
$81$	31.3684	3.48537
$82$	0	0
$83$	−8.25526	−0.906133	−0.453066	−	0.891477i	$-0.649670\pi$
$83$	−8.25526	−0.906133	−0.453066	+	0.891477i	$0.649670\pi$
$84$	0	0
$85$	4.64177	0.503471
$86$	0	0
$87$	3.41478	0.366103
$88$	0	0
$89$	1.22699	0.130061	0.0650304	−	0.997883i	$-0.479286\pi$
$89$	1.22699	0.130061	0.0650304	+	0.997883i	$0.479286\pi$
$90$	0	0
$91$	−5.02827	−0.527106
$92$	0	0
$93$	7.61350	0.789483
$94$	0	0
$95$	4.34916	0.446214
$96$	0	0
$97$	−0.0565477	−0.00574155	−0.00287078	−	0.999996i	$-0.500914\pi$
$97$	−0.0565477	−0.00574155	−0.00287078	+	0.999996i	$0.500914\pi$
$98$	0	0
$99$	13.7074	1.37764
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−9.37743	−0.923986	−0.461993	−	0.886884i	$-0.652865\pi$
$103$	−9.37743	−0.923986	−0.461993	+	0.886884i	$0.652865\pi$
$104$	0	0
$105$	−16.6983	−1.62959
$106$	0	0
$107$	−13.3774	−1.29325	−0.646623	−	0.762810i	$-0.723819\pi$
$107$	−13.3774	−1.29325	−0.646623	+	0.762810i	$0.723819\pi$
$108$	0	0
$109$	−11.2835	−1.08077	−0.540383	−	0.841419i	$-0.681721\pi$
$109$	−11.2835	−1.08077	−0.540383	+	0.841419i	$0.681721\pi$
$110$	0	0
$111$	5.35823	0.508581
$112$	0	0
$113$	−18.6983	−1.75899	−0.879495	−	0.475908i	$-0.842119\pi$
$113$	−18.6983	−1.75899	−0.879495	+	0.475908i	$0.842119\pi$
$114$	0	0
$115$	−0.679116	−0.0633278
$116$	0	0
$117$	−8.02827	−0.742214
$118$	0	0
$119$	−23.3401	−2.13958
$120$	0	0
$121$	−8.08482	−0.734984
$122$	0	0
$123$	15.4148	1.38990
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.7357	−0.952636	−0.476318	−	0.879273i	$-0.658029\pi$
$127$	−10.7357	−0.952636	−0.476318	+	0.879273i	$0.658029\pi$
$128$	0	0
$129$	−11.0283	−0.970985
$130$	0	0
$131$	7.22699	0.631425	0.315713	−	0.948855i	$-0.397756\pi$
$131$	7.22699	0.631425	0.315713	+	0.948855i	$0.397756\pi$
$132$	0	0
$133$	−21.8688	−1.89626
$134$	0	0
$135$	−16.6983	−1.43716
$136$	0	0
$137$	17.3401	1.48146	0.740732	−	0.671801i	$-0.234479\pi$
$137$	17.3401	1.48146	0.740732	+	0.671801i	$0.234479\pi$
$138$	0	0
$139$	9.35823	0.793755	0.396877	−	0.917872i	$-0.370094\pi$
$139$	9.35823	0.793755	0.396877	+	0.917872i	$0.370094\pi$
$140$	0	0
$141$	−3.41478	−0.287576
$142$	0	0
$143$	−1.70739	−0.142779
$144$	0	0
$145$	−1.02827	−0.0853935
$146$	0	0
$147$	60.7175	5.00790
$148$	0	0
$149$	−17.3401	−1.42056	−0.710278	−	0.703922i	$-0.751431\pi$
$149$	−17.3401	−1.42056	−0.710278	+	0.703922i	$0.751431\pi$
$150$	0	0
$151$	−3.57615	−0.291023	−0.145511	−	0.989357i	$-0.546483\pi$
$151$	−3.57615	−0.291023	−0.145511	+	0.989357i	$0.546483\pi$
$152$	0	0
$153$	−37.2654	−3.01273
$154$	0	0
$155$	−2.29261	−0.184147
$156$	0	0
$157$	7.28354	0.581290	0.290645	−	0.956831i	$-0.406130\pi$
$157$	7.28354	0.581290	0.290645	+	0.956831i	$0.406130\pi$
$158$	0	0
$159$	31.2654	2.47951
$160$	0	0
$161$	3.41478	0.269122
$162$	0	0
$163$	0.840485	0.0658319	0.0329159	−	0.999458i	$-0.489521\pi$
$163$	0.840485	0.0658319	0.0329159	+	0.999458i	$0.489521\pi$
$164$	0	0
$165$	−5.67004	−0.441412
$166$	0	0
$167$	13.0283	1.00816	0.504079	−	0.863658i	$-0.331832\pi$
$167$	13.0283	1.00816	0.504079	+	0.863658i	$0.331832\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−34.9162	−2.67011
$172$	0	0
$173$	−25.9253	−1.97106	−0.985532	−	0.169488i	$-0.945789\pi$
$173$	−25.9253	−1.97106	−0.985532	+	0.169488i	$0.945789\pi$
$174$	0	0
$175$	5.02827	0.380102
$176$	0	0
$177$	−29.6700	−2.23014
$178$	0	0
$179$	−4.77301	−0.356751	−0.178376	−	0.983962i	$-0.557084\pi$
$179$	−4.77301	−0.356751	−0.178376	+	0.983962i	$0.557084\pi$
$180$	0	0
$181$	14.3118	1.06379	0.531894	−	0.846811i	$-0.321480\pi$
$181$	14.3118	1.06379	0.531894	+	0.846811i	$0.321480\pi$
$182$	0	0
$183$	−29.9819	−2.21632
$184$	0	0
$185$	−1.61350	−0.118627
$186$	0	0
$187$	−7.92531	−0.579556
$188$	0	0
$189$	83.9637	6.10746
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	4.05655	0.291997	0.145998	−	0.989285i	$-0.453361\pi$
$193$	4.05655	0.291997	0.145998	+	0.989285i	$0.453361\pi$
$194$	0	0
$195$	3.32088	0.237813
$196$	0	0
$197$	15.2835	1.08891	0.544453	−	0.838791i	$-0.316737\pi$
$197$	15.2835	1.08891	0.544453	+	0.838791i	$0.316737\pi$
$198$	0	0
$199$	8.77301	0.621902	0.310951	−	0.950426i	$-0.399352\pi$
$199$	8.77301	0.621902	0.310951	+	0.950426i	$0.399352\pi$
$200$	0	0
$201$	18.6418	1.31489
$202$	0	0
$203$	5.17044	0.362894
$204$	0	0
$205$	−4.64177	−0.324195
$206$	0	0
$207$	5.45213	0.378949
$208$	0	0
$209$	−7.42571	−0.513647
$210$	0	0
$211$	0.773010	0.0532162	0.0266081	−	0.999646i	$-0.491529\pi$
$211$	0.773010	0.0532162	0.0266081	+	0.999646i	$0.491529\pi$
$212$	0	0
$213$	−5.67004	−0.388505
$214$	0	0
$215$	3.32088	0.226482
$216$	0	0
$217$	11.5279	0.782563
$218$	0	0
$219$	41.3219	2.79228
$220$	0	0
$221$	4.64177	0.312239
$222$	0	0
$223$	14.3118	0.958390	0.479195	−	0.877709i	$-0.340929\pi$
$223$	14.3118	0.958390	0.479195	+	0.877709i	$0.340929\pi$
$224$	0	0
$225$	8.02827	0.535218
$226$	0	0
$227$	−13.7266	−0.911066	−0.455533	−	0.890219i	$-0.650551\pi$
$227$	−13.7266	−0.911066	−0.455533	+	0.890219i	$0.650551\pi$
$228$	0	0
$229$	−21.9253	−1.44887	−0.724433	−	0.689346i	$-0.757898\pi$
$229$	−21.9253	−1.44887	−0.724433	+	0.689346i	$0.757898\pi$
$230$	0	0
$231$	28.5105	1.87586
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	1.02827	0.0670772
$236$	0	0
$237$	8.77301	0.569868
$238$	0	0
$239$	15.7639	1.01968	0.509842	−	0.860268i	$-0.329704\pi$
$239$	15.7639	1.01968	0.509842	+	0.860268i	$0.329704\pi$
$240$	0	0
$241$	9.92531	0.639345	0.319673	−	0.947528i	$-0.396427\pi$
$241$	9.92531	0.639345	0.319673	+	0.947528i	$0.396427\pi$
$242$	0	0
$243$	54.0757	3.46896
$244$	0	0
$245$	−18.2835	−1.16809
$246$	0	0
$247$	4.34916	0.276730
$248$	0	0
$249$	−27.4148	−1.73734
$250$	0	0
$251$	24.6983	1.55894	0.779472	−	0.626437i	$-0.215487\pi$
$251$	24.6983	1.55894	0.779472	+	0.626437i	$0.215487\pi$
$252$	0	0
$253$	1.15951	0.0728981
$254$	0	0
$255$	15.4148	0.965311
$256$	0	0
$257$	−20.0565	−1.25109	−0.625547	−	0.780187i	$-0.715124\pi$
$257$	−20.0565	−1.25109	−0.625547	+	0.780187i	$0.715124\pi$
$258$	0	0
$259$	8.11310	0.504123
$260$	0	0
$261$	8.25526	0.510988
$262$	0	0
$263$	−26.0757	−1.60790	−0.803950	−	0.594697i	$-0.797272\pi$
$263$	−26.0757	−1.60790	−0.803950	+	0.594697i	$0.797272\pi$
$264$	0	0
$265$	−9.41478	−0.578345
$266$	0	0
$267$	4.07469	0.249367
$268$	0	0
$269$	12.8296	0.782232	0.391116	−	0.920341i	$-0.372089\pi$
$269$	12.8296	0.782232	0.391116	+	0.920341i	$0.372089\pi$
$270$	0	0
$271$	17.7074	1.07565	0.537824	−	0.843057i	$-0.319247\pi$
$271$	17.7074	1.07565	0.537824	+	0.843057i	$0.319247\pi$
$272$	0	0
$273$	−16.6983	−1.01063
$274$	0	0
$275$	1.70739	0.102959
$276$	0	0
$277$	15.4713	0.929582	0.464791	−	0.885420i	$-0.346129\pi$
$277$	15.4713	0.929582	0.464791	+	0.885420i	$0.346129\pi$
$278$	0	0
$279$	18.4057	1.10192
$280$	0	0
$281$	7.35823	0.438955	0.219478	−	0.975618i	$-0.429565\pi$
$281$	7.35823	0.438955	0.219478	+	0.975618i	$0.429565\pi$
$282$	0	0
$283$	−0.604422	−0.0359292	−0.0179646	−	0.999839i	$-0.505719\pi$
$283$	−0.604422	−0.0359292	−0.0179646	+	0.999839i	$0.505719\pi$
$284$	0	0
$285$	14.4431	0.855533
$286$	0	0
$287$	23.3401	1.37772
$288$	0	0
$289$	4.54602	0.267413
$290$	0	0
$291$	−0.187788	−0.0110084
$292$	0	0
$293$	−23.0101	−1.34427	−0.672133	−	0.740430i	$-0.734622\pi$
$293$	−23.0101	−1.34427	−0.672133	+	0.740430i	$0.734622\pi$
$294$	0	0
$295$	8.93438	0.520180
$296$	0	0
$297$	28.5105	1.65435
$298$	0	0
$299$	−0.679116	−0.0392743
$300$	0	0
$301$	−16.6983	−0.962475
$302$	0	0
$303$	−19.9253	−1.14468
$304$	0	0
$305$	9.02827	0.516957
$306$	0	0
$307$	20.3684	1.16248	0.581242	−	0.813731i	$-0.302567\pi$
$307$	20.3684	1.16248	0.581242	+	0.813731i	$0.302567\pi$
$308$	0	0
$309$	−31.1414	−1.77157
$310$	0	0
$311$	19.9253	1.12986	0.564930	−	0.825139i	$-0.308903\pi$
$311$	19.9253	1.12986	0.564930	+	0.825139i	$0.308903\pi$
$312$	0	0
$313$	31.4713	1.77886	0.889432	−	0.457067i	$-0.151100\pi$
$313$	31.4713	1.77886	0.889432	+	0.457067i	$0.151100\pi$
$314$	0	0
$315$	−40.3684	−2.27450
$316$	0	0
$317$	−1.72659	−0.0969750	−0.0484875	−	0.998824i	$-0.515440\pi$
$317$	−1.72659	−0.0969750	−0.0484875	+	0.998824i	$0.515440\pi$
$318$	0	0
$319$	1.75566	0.0982983
$320$	0	0
$321$	−44.4249	−2.47956
$322$	0	0
$323$	20.1878	1.12328
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−37.4713	−2.07217
$328$	0	0
$329$	−5.17044	−0.285056
$330$	0	0
$331$	10.4057	0.571949	0.285975	−	0.958237i	$-0.407683\pi$
$331$	10.4057	0.571949	0.285975	+	0.958237i	$0.407683\pi$
$332$	0	0
$333$	12.9536	0.709852
$334$	0	0
$335$	−5.61350	−0.306698
$336$	0	0
$337$	−10.6983	−0.582774	−0.291387	−	0.956605i	$-0.594117\pi$
$337$	−10.6983	−0.582774	−0.291387	+	0.956605i	$0.594117\pi$
$338$	0	0
$339$	−62.0950	−3.37253
$340$	0	0
$341$	3.91438	0.211976
$342$	0	0
$343$	56.7367	3.06349
$344$	0	0
$345$	−2.25526	−0.121419
$346$	0	0
$347$	−32.6044	−1.75030	−0.875149	−	0.483854i	$-0.839236\pi$
$347$	−32.6044	−1.75030	−0.875149	+	0.483854i	$0.839236\pi$
$348$	0	0
$349$	13.4148	0.718077	0.359038	−	0.933323i	$-0.383105\pi$
$349$	13.4148	0.718077	0.359038	+	0.933323i	$0.383105\pi$
$350$	0	0
$351$	−16.6983	−0.891290
$352$	0	0
$353$	−35.0101	−1.86340	−0.931701	−	0.363227i	$-0.881675\pi$
$353$	−35.0101	−1.86340	−0.931701	+	0.363227i	$0.881675\pi$
$354$	0	0
$355$	1.70739	0.0906188
$356$	0	0
$357$	−77.5097	−4.10225
$358$	0	0
$359$	−20.9344	−1.10487	−0.552437	−	0.833555i	$-0.686302\pi$
$359$	−20.9344	−1.10487	−0.552437	+	0.833555i	$0.686302\pi$
$360$	0	0
$361$	−0.0848216	−0.00446429
$362$	0	0
$363$	−26.8488	−1.40919
$364$	0	0
$365$	−12.4431	−0.651299
$366$	0	0
$367$	−11.4340	−0.596849	−0.298424	−	0.954433i	$-0.596461\pi$
$367$	−11.4340	−0.596849	−0.298424	+	0.954433i	$0.596461\pi$
$368$	0	0
$369$	37.2654	1.93996
$370$	0	0
$371$	47.3401	2.45777
$372$	0	0
$373$	14.1131	0.730748	0.365374	−	0.930861i	$-0.380941\pi$
$373$	14.1131	0.730748	0.365374	+	0.930861i	$0.380941\pi$
$374$	0	0
$375$	−3.32088	−0.171490
$376$	0	0
$377$	−1.02827	−0.0529588
$378$	0	0
$379$	−1.89518	−0.0973487	−0.0486744	−	0.998815i	$-0.515500\pi$
$379$	−1.89518	−0.0973487	−0.0486744	+	0.998815i	$0.515500\pi$
$380$	0	0
$381$	−35.6519	−1.82650
$382$	0	0
$383$	22.3118	1.14008	0.570040	−	0.821617i	$-0.306928\pi$
$383$	22.3118	1.14008	0.570040	+	0.821617i	$0.306928\pi$
$384$	0	0
$385$	−8.58522	−0.437543
$386$	0	0
$387$	−26.6610	−1.35525
$388$	0	0
$389$	28.6802	1.45414	0.727071	−	0.686562i	$-0.240881\pi$
$389$	28.6802	1.45414	0.727071	+	0.686562i	$0.240881\pi$
$390$	0	0
$391$	−3.15230	−0.159419
$392$	0	0
$393$	24.0000	1.21064
$394$	0	0
$395$	−2.64177	−0.132922
$396$	0	0
$397$	−15.5569	−0.780781	−0.390390	−	0.920649i	$-0.627660\pi$
$397$	−15.5569	−0.780781	−0.390390	+	0.920649i	$0.627660\pi$
$398$	0	0
$399$	−72.6236	−3.63573
$400$	0	0
$401$	−24.1696	−1.20697	−0.603487	−	0.797373i	$-0.706223\pi$
$401$	−24.1696	−1.20697	−0.603487	+	0.797373i	$0.706223\pi$
$402$	0	0
$403$	−2.29261	−0.114203
$404$	0	0
$405$	−31.3684	−1.55871
$406$	0	0
$407$	2.75486	0.136554
$408$	0	0
$409$	−29.9253	−1.47971	−0.739856	−	0.672766i	$-0.765106\pi$
$409$	−29.9253	−1.47971	−0.739856	+	0.672766i	$0.765106\pi$
$410$	0	0
$411$	57.5844	2.84043
$412$	0	0
$413$	−44.9245	−2.21059
$414$	0	0
$415$	8.25526	0.405235
$416$	0	0
$417$	31.0776	1.52188
$418$	0	0
$419$	−8.58522	−0.419416	−0.209708	−	0.977764i	$-0.567251\pi$
$419$	−8.58522	−0.419416	−0.209708	+	0.977764i	$0.567251\pi$
$420$	0	0
$421$	21.7375	1.05942	0.529711	−	0.848178i	$-0.322300\pi$
$421$	21.7375	1.05942	0.529711	+	0.848178i	$0.322300\pi$
$422$	0	0
$423$	−8.25526	−0.401385
$424$	0	0
$425$	−4.64177	−0.225159
$426$	0	0
$427$	−45.3966	−2.19690
$428$	0	0
$429$	−5.67004	−0.273752
$430$	0	0
$431$	−13.7074	−0.660262	−0.330131	−	0.943935i	$-0.607093\pi$
$431$	−13.7074	−0.660262	−0.330131	+	0.943935i	$0.607093\pi$
$432$	0	0
$433$	−40.5671	−1.94953	−0.974765	−	0.223235i	$-0.928338\pi$
$433$	−40.5671	−1.94953	−0.974765	+	0.223235i	$0.928338\pi$
$434$	0	0
$435$	−3.41478	−0.163726
$436$	0	0
$437$	−2.95358	−0.141289
$438$	0	0
$439$	−26.5671	−1.26798	−0.633989	−	0.773342i	$-0.718583\pi$
$439$	−26.5671	−1.26798	−0.633989	+	0.773342i	$0.718583\pi$
$440$	0	0
$441$	146.785	6.98977
$442$	0	0
$443$	−16.7922	−0.797822	−0.398911	−	0.916990i	$-0.630612\pi$
$443$	−16.7922	−0.797822	−0.398911	+	0.916990i	$0.630612\pi$
$444$	0	0
$445$	−1.22699	−0.0581649
$446$	0	0
$447$	−57.5844	−2.72365
$448$	0	0
$449$	−16.2443	−0.766618	−0.383309	−	0.923620i	$-0.625215\pi$
$449$	−16.2443	−0.766618	−0.383309	+	0.923620i	$0.625215\pi$
$450$	0	0
$451$	7.92531	0.373188
$452$	0	0
$453$	−11.8760	−0.557982
$454$	0	0
$455$	5.02827	0.235729
$456$	0	0
$457$	6.77301	0.316828	0.158414	−	0.987373i	$-0.449362\pi$
$457$	6.77301	0.316828	0.158414	+	0.987373i	$0.449362\pi$
$458$	0	0
$459$	−77.5097	−3.61784
$460$	0	0
$461$	22.8114	1.06243	0.531217	−	0.847236i	$-0.321735\pi$
$461$	22.8114	1.06243	0.531217	+	0.847236i	$0.321735\pi$
$462$	0	0
$463$	16.3300	0.758917	0.379459	−	0.925209i	$-0.376110\pi$
$463$	16.3300	0.758917	0.379459	+	0.925209i	$0.376110\pi$
$464$	0	0
$465$	−7.61350	−0.353067
$466$	0	0
$467$	10.6610	0.493331	0.246665	−	0.969101i	$-0.420665\pi$
$467$	10.6610	0.493331	0.246665	+	0.969101i	$0.420665\pi$
$468$	0	0
$469$	28.2262	1.30336
$470$	0	0
$471$	24.1878	1.11451
$472$	0	0
$473$	−5.67004	−0.260709
$474$	0	0
$475$	−4.34916	−0.199553
$476$	0	0
$477$	75.5844	3.46077
$478$	0	0
$479$	−6.48040	−0.296097	−0.148048	−	0.988980i	$-0.547299\pi$
$479$	−6.48040	−0.296097	−0.148048	+	0.988980i	$0.547299\pi$
$480$	0	0
$481$	−1.61350	−0.0735690
$482$	0	0
$483$	11.3401	0.515992
$484$	0	0
$485$	0.0565477	0.00256770
$486$	0	0
$487$	−23.7831	−1.07772	−0.538858	−	0.842396i	$-0.681144\pi$
$487$	−23.7831	−1.07772	−0.538858	+	0.842396i	$0.681144\pi$
$488$	0	0
$489$	2.79116	0.126220
$490$	0	0
$491$	−6.13124	−0.276699	−0.138350	−	0.990383i	$-0.544180\pi$
$491$	−6.13124	−0.276699	−0.138350	+	0.990383i	$0.544180\pi$
$492$	0	0
$493$	−4.77301	−0.214966
$494$	0	0
$495$	−13.7074	−0.616101
$496$	0	0
$497$	−8.58522	−0.385100
$498$	0	0
$499$	21.0475	0.942214	0.471107	−	0.882076i	$-0.343854\pi$
$499$	21.0475	0.942214	0.471107	+	0.882076i	$0.343854\pi$
$500$	0	0
$501$	43.2654	1.93296
$502$	0	0
$503$	−33.5652	−1.49660	−0.748300	−	0.663361i	$-0.769129\pi$
$503$	−33.5652	−1.49660	−0.748300	+	0.663361i	$0.769129\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	3.32088	0.147486
$508$	0	0
$509$	14.5852	0.646479	0.323239	−	0.946317i	$-0.395228\pi$
$509$	14.5852	0.646479	0.323239	+	0.946317i	$0.395228\pi$
$510$	0	0
$511$	62.5671	2.76780
$512$	0	0
$513$	−72.6236	−3.20641
$514$	0	0
$515$	9.37743	0.413219
$516$	0	0
$517$	−1.75566	−0.0772140
$518$	0	0
$519$	−86.0950	−3.77915
$520$	0	0
$521$	−3.48225	−0.152560	−0.0762802	−	0.997086i	$-0.524304\pi$
$521$	−3.48225	−0.152560	−0.0762802	+	0.997086i	$0.524304\pi$
$522$	0	0
$523$	−11.5087	−0.503239	−0.251620	−	0.967826i	$-0.580963\pi$
$523$	−11.5087	−0.503239	−0.251620	+	0.967826i	$0.580963\pi$
$524$	0	0
$525$	16.6983	0.728774
$526$	0	0
$527$	−10.6418	−0.463563
$528$	0	0
$529$	−22.5388	−0.979948
$530$	0	0
$531$	−71.7276	−3.11271
$532$	0	0
$533$	−4.64177	−0.201057
$534$	0	0
$535$	13.3774	0.578357
$536$	0	0
$537$	−15.8506	−0.684004
$538$	0	0
$539$	31.2171	1.34462
$540$	0	0
$541$	13.8122	0.593833	0.296917	−	0.954903i	$-0.404042\pi$
$541$	13.8122	0.593833	0.296917	+	0.954903i	$0.404042\pi$
$542$	0	0
$543$	47.5279	2.03962
$544$	0	0
$545$	11.2835	0.483334
$546$	0	0
$547$	−5.37743	−0.229922	−0.114961	−	0.993370i	$-0.536674\pi$
$547$	−5.37743	−0.229922	−0.114961	+	0.993370i	$0.536674\pi$
$548$	0	0
$549$	−72.4815	−3.09343
$550$	0	0
$551$	−4.47213	−0.190519
$552$	0	0
$553$	13.2835	0.564873
$554$	0	0
$555$	−5.35823	−0.227444
$556$	0	0
$557$	0.0674757	0.00285904	0.00142952	−	0.999999i	$-0.499545\pi$
$557$	0.0674757	0.00285904	0.00142952	+	0.999999i	$0.499545\pi$
$558$	0	0
$559$	3.32088	0.140458
$560$	0	0
$561$	−26.3190	−1.11119
$562$	0	0
$563$	−7.20779	−0.303772	−0.151886	−	0.988398i	$-0.548535\pi$
$563$	−7.20779	−0.303772	−0.151886	+	0.988398i	$0.548535\pi$
$564$	0	0
$565$	18.6983	0.786644
$566$	0	0
$567$	157.729	6.62398
$568$	0	0
$569$	−7.85783	−0.329417	−0.164709	−	0.986342i	$-0.552668\pi$
$569$	−7.85783	−0.329417	−0.164709	+	0.986342i	$0.552668\pi$
$570$	0	0
$571$	−23.9253	−1.00124	−0.500621	−	0.865666i	$-0.666895\pi$
$571$	−23.9253	−1.00124	−0.500621	+	0.865666i	$0.666895\pi$
$572$	0	0
$573$	−39.8506	−1.66478
$574$	0	0
$575$	0.679116	0.0283211
$576$	0	0
$577$	31.0101	1.29097	0.645484	−	0.763774i	$-0.276656\pi$
$577$	31.0101	1.29097	0.645484	+	0.763774i	$0.276656\pi$
$578$	0	0
$579$	13.4713	0.559849
$580$	0	0
$581$	−41.5097	−1.72211
$582$	0	0
$583$	16.0747	0.665746
$584$	0	0
$585$	8.02827	0.331928
$586$	0	0
$587$	47.4076	1.95672	0.978360	−	0.206911i	$-0.0663411\pi$
$587$	47.4076	1.95672	0.978360	+	0.206911i	$0.0663411\pi$
$588$	0	0
$589$	−9.97093	−0.410845
$590$	0	0
$591$	50.7549	2.08778
$592$	0	0
$593$	−14.8861	−0.611299	−0.305650	−	0.952144i	$-0.598874\pi$
$593$	−14.8861	−0.611299	−0.305650	+	0.952144i	$0.598874\pi$
$594$	0	0
$595$	23.3401	0.956850
$596$	0	0
$597$	29.1342	1.19238
$598$	0	0
$599$	−42.8680	−1.75154	−0.875769	−	0.482731i	$-0.839645\pi$
$599$	−42.8680	−1.75154	−0.875769	+	0.482731i	$0.839645\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	45.0667	1.83526
$604$	0	0
$605$	8.08482	0.328695
$606$	0	0
$607$	18.7357	0.760457	0.380229	−	0.924893i	$-0.375845\pi$
$607$	18.7357	0.760457	0.380229	+	0.924893i	$0.375845\pi$
$608$	0	0
$609$	17.1704	0.695781
$610$	0	0
$611$	1.02827	0.0415995
$612$	0	0
$613$	−36.6802	−1.48150	−0.740749	−	0.671782i	$-0.765529\pi$
$613$	−36.6802	−1.48150	−0.740749	+	0.671782i	$0.765529\pi$
$614$	0	0
$615$	−15.4148	−0.621584
$616$	0	0
$617$	−8.05655	−0.324344	−0.162172	−	0.986762i	$-0.551850\pi$
$617$	−8.05655	−0.324344	−0.162172	+	0.986762i	$0.551850\pi$
$618$	0	0
$619$	−33.1606	−1.33284	−0.666418	−	0.745578i	$-0.732173\pi$
$619$	−33.1606	−1.33284	−0.666418	+	0.745578i	$0.732173\pi$
$620$	0	0
$621$	11.3401	0.455062
$622$	0	0
$623$	6.16964	0.247182
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−24.6599	−0.984822
$628$	0	0
$629$	−7.48947	−0.298625
$630$	0	0
$631$	33.8205	1.34637	0.673186	−	0.739473i	$-0.264925\pi$
$631$	33.8205	1.34637	0.673186	+	0.739473i	$0.264925\pi$
$632$	0	0
$633$	2.56708	0.102032
$634$	0	0
$635$	10.7357	0.426032
$636$	0	0
$637$	−18.2835	−0.724420
$638$	0	0
$639$	−13.7074	−0.542256
$640$	0	0
$641$	12.8296	0.506737	0.253369	−	0.967370i	$-0.418461\pi$
$641$	12.8296	0.506737	0.253369	+	0.967370i	$0.418461\pi$
$642$	0	0
$643$	43.7084	1.72369	0.861846	−	0.507169i	$-0.169308\pi$
$643$	43.7084	1.72369	0.861846	+	0.507169i	$0.169308\pi$
$644$	0	0
$645$	11.0283	0.434238
$646$	0	0
$647$	−25.4158	−0.999200	−0.499600	−	0.866256i	$-0.666520\pi$
$647$	−25.4158	−0.999200	−0.499600	+	0.866256i	$0.666520\pi$
$648$	0	0
$649$	−15.2545	−0.598790
$650$	0	0
$651$	38.2827	1.50042
$652$	0	0
$653$	3.94345	0.154319	0.0771596	−	0.997019i	$-0.475415\pi$
$653$	3.94345	0.154319	0.0771596	+	0.997019i	$0.475415\pi$
$654$	0	0
$655$	−7.22699	−0.282382
$656$	0	0
$657$	99.8962	3.89732
$658$	0	0
$659$	−38.0950	−1.48397	−0.741984	−	0.670417i	$-0.766115\pi$
$659$	−38.0950	−1.48397	−0.741984	+	0.670417i	$0.766115\pi$
$660$	0	0
$661$	−18.1131	−0.704518	−0.352259	−	0.935903i	$-0.614586\pi$
$661$	−18.1131	−0.704518	−0.352259	+	0.935903i	$0.614586\pi$
$662$	0	0
$663$	15.4148	0.598660
$664$	0	0
$665$	21.8688	0.848034
$666$	0	0
$667$	0.698317	0.0270389
$668$	0	0
$669$	47.5279	1.83753
$670$	0	0
$671$	−15.4148	−0.595081
$672$	0	0
$673$	43.2088	1.66558	0.832789	−	0.553590i	$-0.186743\pi$
$673$	43.2088	1.66558	0.832789	+	0.553590i	$0.186743\pi$
$674$	0	0
$675$	16.6983	0.642719
$676$	0	0
$677$	30.6983	1.17983	0.589916	−	0.807465i	$-0.299161\pi$
$677$	30.6983	1.17983	0.589916	+	0.807465i	$0.299161\pi$
$678$	0	0
$679$	−0.284337	−0.0109119
$680$	0	0
$681$	−45.5844	−1.74680
$682$	0	0
$683$	−32.9536	−1.26093	−0.630467	−	0.776216i	$-0.717137\pi$
$683$	−32.9536	−1.26093	−0.630467	+	0.776216i	$0.717137\pi$
$684$	0	0
$685$	−17.3401	−0.662531
$686$	0	0
$687$	−72.8114	−2.77793
$688$	0	0
$689$	−9.41478	−0.358675
$690$	0	0
$691$	37.1222	1.41219	0.706097	−	0.708115i	$-0.250454\pi$
$691$	37.1222	1.41219	0.706097	+	0.708115i	$0.250454\pi$
$692$	0	0
$693$	68.9245	2.61823
$694$	0	0
$695$	−9.35823	−0.354978
$696$	0	0
$697$	−21.5460	−0.816114
$698$	0	0
$699$	−19.9253	−0.753644
$700$	0	0
$701$	7.39663	0.279367	0.139683	−	0.990196i	$-0.455391\pi$
$701$	7.39663	0.279367	0.139683	+	0.990196i	$0.455391\pi$
$702$	0	0
$703$	−7.01735	−0.264664
$704$	0	0
$705$	3.41478	0.128608
$706$	0	0
$707$	−30.1696	−1.13465
$708$	0	0
$709$	−28.7549	−1.07991	−0.539956	−	0.841693i	$-0.681559\pi$
$709$	−28.7549	−1.07991	−0.539956	+	0.841693i	$0.681559\pi$
$710$	0	0
$711$	21.2088	0.795394
$712$	0	0
$713$	1.55695	0.0583081
$714$	0	0
$715$	1.70739	0.0638527
$716$	0	0
$717$	52.3502	1.95505
$718$	0	0
$719$	39.4532	1.47136	0.735678	−	0.677332i	$-0.236864\pi$
$719$	39.4532	1.47136	0.735678	+	0.677332i	$0.236864\pi$
$720$	0	0
$721$	−47.1523	−1.75604
$722$	0	0
$723$	32.9608	1.22583
$724$	0	0
$725$	1.02827	0.0381891
$726$	0	0
$727$	0.866904	0.0321517	0.0160758	−	0.999871i	$-0.494883\pi$
$727$	0.866904	0.0321517	0.0160758	+	0.999871i	$0.494883\pi$
$728$	0	0
$729$	85.4742	3.16571
$730$	0	0
$731$	15.4148	0.570136
$732$	0	0
$733$	10.0000	0.369358	0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000	0.369358	0.184679	+	0.982799i	$0.440875\pi$
$734$	0	0
$735$	−60.7175	−2.23960
$736$	0	0
$737$	9.58442	0.353047
$738$	0	0
$739$	−31.5015	−1.15880	−0.579400	−	0.815043i	$-0.696713\pi$
$739$	−31.5015	−1.15880	−0.579400	+	0.815043i	$0.696713\pi$
$740$	0	0
$741$	14.4431	0.530579
$742$	0	0
$743$	31.8578	1.16875	0.584375	−	0.811484i	$-0.301340\pi$
$743$	31.8578	1.16875	0.584375	+	0.811484i	$0.301340\pi$
$744$	0	0
$745$	17.3401	0.635292
$746$	0	0
$747$	−66.2755	−2.42489
$748$	0	0
$749$	−67.2654	−2.45782
$750$	0	0
$751$	−46.0950	−1.68203	−0.841014	−	0.541013i	$-0.818041\pi$
$751$	−46.0950	−1.68203	−0.841014	+	0.541013i	$0.818041\pi$
$752$	0	0
$753$	82.0203	2.98898
$754$	0	0
$755$	3.57615	0.130149
$756$	0	0
$757$	−5.01735	−0.182359	−0.0911793	−	0.995834i	$-0.529064\pi$
$757$	−5.01735	−0.182359	−0.0911793	+	0.995834i	$0.529064\pi$
$758$	0	0
$759$	3.85061	0.139768
$760$	0	0
$761$	24.8296	0.900071	0.450035	−	0.893011i	$-0.351411\pi$
$761$	24.8296	0.900071	0.450035	+	0.893011i	$0.351411\pi$
$762$	0	0
$763$	−56.7367	−2.05401
$764$	0	0
$765$	37.2654	1.34733
$766$	0	0
$767$	8.93438	0.322602
$768$	0	0
$769$	36.9427	1.33219	0.666093	−	0.745869i	$-0.267965\pi$
$769$	36.9427	1.33219	0.666093	+	0.745869i	$0.267965\pi$
$770$	0	0
$771$	−66.6055	−2.39874
$772$	0	0
$773$	16.4431	0.591415	0.295708	−	0.955278i	$-0.404445\pi$
$773$	16.4431	0.591415	0.295708	+	0.955278i	$0.404445\pi$
$774$	0	0
$775$	2.29261	0.0823530
$776$	0	0
$777$	26.9427	0.966562
$778$	0	0
$779$	−20.1878	−0.723303
$780$	0	0
$781$	−2.91518	−0.104313
$782$	0	0
$783$	17.1704	0.613622
$784$	0	0
$785$	−7.28354	−0.259961
$786$	0	0
$787$	40.2553	1.43495	0.717473	−	0.696587i	$-0.245299\pi$
$787$	40.2553	1.43495	0.717473	+	0.696587i	$0.245299\pi$
$788$	0	0
$789$	−86.5946	−3.08285
$790$	0	0
$791$	−94.0203	−3.34298
$792$	0	0
$793$	9.02827	0.320603
$794$	0	0
$795$	−31.2654	−1.10887
$796$	0	0
$797$	−27.2835	−0.966432	−0.483216	−	0.875501i	$-0.660532\pi$
$797$	−27.2835	−0.966432	−0.483216	+	0.875501i	$0.660532\pi$
$798$	0	0
$799$	4.77301	0.168857
$800$	0	0
$801$	9.85061	0.348054
$802$	0	0
$803$	21.2451	0.749725
$804$	0	0
$805$	−3.41478	−0.120355
$806$	0	0
$807$	42.6055	1.49978
$808$	0	0
$809$	−28.4815	−1.00135	−0.500677	−	0.865634i	$-0.666916\pi$
$809$	−28.4815	−1.00135	−0.500677	+	0.865634i	$0.666916\pi$
$810$	0	0
$811$	31.6892	1.11276	0.556380	−	0.830928i	$-0.312190\pi$
$811$	31.6892	1.11276	0.556380	+	0.830928i	$0.312190\pi$
$812$	0	0
$813$	58.8042	2.06235
$814$	0	0
$815$	−0.840485	−0.0294409
$816$	0	0
$817$	14.4431	0.505298
$818$	0	0
$819$	−40.3684	−1.41058
$820$	0	0
$821$	6.65991	0.232433	0.116216	−	0.993224i	$-0.462923\pi$
$821$	6.65991	0.232433	0.116216	+	0.993224i	$0.462923\pi$
$822$	0	0
$823$	7.79301	0.271647	0.135824	−	0.990733i	$-0.456632\pi$
$823$	7.79301	0.271647	0.135824	+	0.990733i	$0.456632\pi$
$824$	0	0
$825$	5.67004	0.197406
$826$	0	0
$827$	26.4249	0.918884	0.459442	−	0.888208i	$-0.348049\pi$
$827$	26.4249	0.918884	0.459442	+	0.888208i	$0.348049\pi$
$828$	0	0
$829$	−11.7447	−0.407912	−0.203956	−	0.978980i	$-0.565380\pi$
$829$	−11.7447	−0.407912	−0.203956	+	0.978980i	$0.565380\pi$
$830$	0	0
$831$	51.3785	1.78230
$832$	0	0
$833$	−84.8680	−2.94050
$834$	0	0
$835$	−13.0283	−0.450862
$836$	0	0
$837$	38.2827	1.32325
$838$	0	0
$839$	44.5753	1.53891	0.769456	−	0.638700i	$-0.220527\pi$
$839$	44.5753	1.53891	0.769456	+	0.638700i	$0.220527\pi$
$840$	0	0
$841$	−27.9427	−0.963540
$842$	0	0
$843$	24.4358	0.841615
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−40.6527	−1.39684
$848$	0	0
$849$	−2.00722	−0.0688875
$850$	0	0
$851$	1.09575	0.0375618
$852$	0	0
$853$	−29.6135	−1.01395	−0.506973	−	0.861962i	$-0.669236\pi$
$853$	−29.6135	−1.01395	−0.506973	+	0.861962i	$0.669236\pi$
$854$	0	0
$855$	34.9162	1.19411
$856$	0	0
$857$	22.5105	0.768945	0.384472	−	0.923136i	$-0.374383\pi$
$857$	22.5105	0.768945	0.384472	+	0.923136i	$0.374383\pi$
$858$	0	0
$859$	−0.585221	−0.0199675	−0.00998375	−	0.999950i	$-0.503178\pi$
$859$	−0.585221	−0.0199675	−0.00998375	+	0.999950i	$0.503178\pi$
$860$	0	0
$861$	77.5097	2.64152
$862$	0	0
$863$	−45.6519	−1.55401	−0.777004	−	0.629495i	$-0.783262\pi$
$863$	−45.6519	−1.55401	−0.777004	+	0.629495i	$0.783262\pi$
$864$	0	0
$865$	25.9253	0.881487
$866$	0	0
$867$	15.0968	0.512714
$868$	0	0
$869$	4.51053	0.153009
$870$	0	0
$871$	−5.61350	−0.190206
$872$	0	0
$873$	−0.453981	−0.0153649
$874$	0	0
$875$	−5.02827	−0.169987
$876$	0	0
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	0	0
$879$	−76.4140	−2.57738
$880$	0	0
$881$	42.5380	1.43314	0.716571	−	0.697514i	$-0.245711\pi$
$881$	42.5380	1.43314	0.716571	+	0.697514i	$0.245711\pi$
$882$	0	0
$883$	6.22513	0.209492	0.104746	−	0.994499i	$-0.466597\pi$
$883$	6.22513	0.209492	0.104746	+	0.994499i	$0.466597\pi$
$884$	0	0
$885$	29.6700	0.997348
$886$	0	0
$887$	−22.0011	−0.738723	−0.369362	−	0.929286i	$-0.620424\pi$
$887$	−22.0011	−0.738723	−0.369362	+	0.929286i	$0.620424\pi$
$888$	0	0
$889$	−53.9819	−1.81049
$890$	0	0
$891$	53.5580	1.79426
$892$	0	0
$893$	4.47213	0.149654
$894$	0	0
$895$	4.77301	0.159544
$896$	0	0
$897$	−2.25526	−0.0753011
$898$	0	0
$899$	2.35743	0.0786247
$900$	0	0
$901$	−43.7012	−1.45590
$902$	0	0
$903$	−55.4532	−1.84537
$904$	0	0
$905$	−14.3118	−0.475741
$906$	0	0
$907$	28.2070	0.936598	0.468299	−	0.883570i	$-0.344867\pi$
$907$	28.2070	0.936598	0.468299	+	0.883570i	$0.344867\pi$
$908$	0	0
$909$	−48.1696	−1.59769
$910$	0	0
$911$	−40.5105	−1.34217	−0.671087	−	0.741379i	$-0.734172\pi$
$911$	−40.5105	−1.34217	−0.671087	+	0.741379i	$0.734172\pi$
$912$	0	0
$913$	−14.0950	−0.466475
$914$	0	0
$915$	29.9819	0.991170
$916$	0	0
$917$	36.3393	1.20003
$918$	0	0
$919$	−0.0746930	−0.00246389	−0.00123195	−	0.999999i	$-0.500392\pi$
$919$	−0.0746930	−0.00246389	−0.00123195	+	0.999999i	$0.500392\pi$
$920$	0	0
$921$	67.6410	2.22885
$922$	0	0
$923$	1.70739	0.0561994
$924$	0	0
$925$	1.61350	0.0530514
$926$	0	0
$927$	−75.2846	−2.47267
$928$	0	0
$929$	53.6410	1.75990	0.879952	−	0.475063i	$-0.157575\pi$
$929$	53.6410	1.75990	0.879952	+	0.475063i	$0.157575\pi$
$930$	0	0
$931$	−79.5180	−2.60610
$932$	0	0
$933$	66.1696	2.16630
$934$	0	0
$935$	7.92531	0.259185
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	104.513	3.41064
$940$	0	0
$941$	49.9253	1.62752	0.813759	−	0.581202i	$-0.197417\pi$
$941$	49.9253	1.62752	0.813759	+	0.581202i	$0.197417\pi$
$942$	0	0
$943$	3.15230	0.102653
$944$	0	0
$945$	−83.9637	−2.73134
$946$	0	0
$947$	−5.80128	−0.188516	−0.0942582	−	0.995548i	$-0.530048\pi$
$947$	−5.80128	−0.188516	−0.0942582	+	0.995548i	$0.530048\pi$
$948$	0	0
$949$	−12.4431	−0.403919
$950$	0	0
$951$	−5.73381	−0.185931
$952$	0	0
$953$	−13.2270	−0.428464	−0.214232	−	0.976783i	$-0.568725\pi$
$953$	−13.2270	−0.428464	−0.214232	+	0.976783i	$0.568725\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	5.83036	0.188469
$958$	0	0
$959$	87.1907	2.81553
$960$	0	0
$961$	−25.7439	−0.830450
$962$	0	0
$963$	−107.398	−3.46084
$964$	0	0
$965$	−4.05655	−0.130585
$966$	0	0
$967$	1.61350	0.0518865	0.0259433	−	0.999663i	$-0.491741\pi$
$967$	1.61350	0.0518865	0.0259433	+	0.999663i	$0.491741\pi$
$968$	0	0
$969$	67.0413	2.15368
$970$	0	0
$971$	16.7730	0.538271	0.269136	−	0.963102i	$-0.413262\pi$
$971$	16.7730	0.538271	0.269136	+	0.963102i	$0.413262\pi$
$972$	0	0
$973$	47.0557	1.50854
$974$	0	0
$975$	−3.32088	−0.106353
$976$	0	0
$977$	−47.6700	−1.52510	−0.762550	−	0.646929i	$-0.776053\pi$
$977$	−47.6700	−1.52510	−0.762550	+	0.646929i	$0.776053\pi$
$978$	0	0
$979$	2.09495	0.0669549
$980$	0	0
$981$	−90.5873	−2.89223
$982$	0	0
$983$	−30.4996	−0.972786	−0.486393	−	0.873740i	$-0.661688\pi$
$983$	−30.4996	−0.972786	−0.486393	+	0.873740i	$0.661688\pi$
$984$	0	0
$985$	−15.2835	−0.486974
$986$	0	0
$987$	−17.1704	−0.546541
$988$	0	0
$989$	−2.25526	−0.0717132
$990$	0	0
$991$	19.8506	0.630576	0.315288	−	0.948996i	$-0.397899\pi$
$991$	19.8506	0.630576	0.315288	+	0.948996i	$0.397899\pi$
$992$	0	0
$993$	34.5561	1.09661
$994$	0	0
$995$	−8.77301	−0.278123
$996$	0	0
$997$	33.6410	1.06542	0.532710	−	0.846298i	$-0.321174\pi$
$997$	33.6410	1.06542	0.532710	+	0.846298i	$0.321174\pi$
$998$	0	0
$999$	26.9427	0.852428

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4160.2.a.br.1.3		3
4.3	odd	2		4160.2.a.bo.1.1		3
8.3	odd	2		260.2.a.b.1.3	✓	3
8.5	even	2		1040.2.a.o.1.1		3
24.5	odd	2		9360.2.a.da.1.3		3
24.11	even	2		2340.2.a.n.1.1		3
40.3	even	4		1300.2.c.f.1249.6		6
40.19	odd	2		1300.2.a.i.1.1		3
40.27	even	4		1300.2.c.f.1249.1		6
40.29	even	2		5200.2.a.ci.1.3		3
104.51	odd	2		3380.2.a.o.1.3		3
104.83	even	4		3380.2.f.h.3041.5		6
104.99	even	4		3380.2.f.h.3041.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.a.b.1.3	✓	3	8.3	odd	2
1040.2.a.o.1.1		3	8.5	even	2
1300.2.a.i.1.1		3	40.19	odd	2
1300.2.c.f.1249.1		6	40.27	even	4
1300.2.c.f.1249.6		6	40.3	even	4
2340.2.a.n.1.1		3	24.11	even	2
3380.2.a.o.1.3		3	104.51	odd	2
3380.2.f.h.3041.5		6	104.83	even	4
3380.2.f.h.3041.6		6	104.99	even	4
4160.2.a.bo.1.1		3	4.3	odd	2
4160.2.a.br.1.3		3	1.1	even	1	trivial
5200.2.a.ci.1.3		3	40.29	even	2
9360.2.a.da.1.3		3	24.5	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:	\( N \)	\(=\)	\( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4160.a (trivial)

\(f(q)\)	\(=\)	\(q+3.32088 q^{3} -1.00000 q^{5} +5.02827 q^{7} +8.02827 q^{9} +O(q^{10})\)	\(q+3.32088 q^{3} -1.00000 q^{5} +5.02827 q^{7} +8.02827 q^{9} +1.70739 q^{11} -1.00000 q^{13} -3.32088 q^{15} -4.64177 q^{17} -4.34916 q^{19} +16.6983 q^{21} +0.679116 q^{23} +1.00000 q^{25} +16.6983 q^{27} +1.02827 q^{29} +2.29261 q^{31} +5.67004 q^{33} -5.02827 q^{35} +1.61350 q^{37} -3.32088 q^{39} +4.64177 q^{41} -3.32088 q^{43} -8.02827 q^{45} -1.02827 q^{47} +18.2835 q^{49} -15.4148 q^{51} +9.41478 q^{53} -1.70739 q^{55} -14.4431 q^{57} -8.93438 q^{59} -9.02827 q^{61} +40.3684 q^{63} +1.00000 q^{65} +5.61350 q^{67} +2.25526 q^{69} -1.70739 q^{71} +12.4431 q^{73} +3.32088 q^{75} +8.58522 q^{77} +2.64177 q^{79} +31.3684 q^{81} -8.25526 q^{83} +4.64177 q^{85} +3.41478 q^{87} +1.22699 q^{89} -5.02827 q^{91} +7.61350 q^{93} +4.34916 q^{95} -0.0565477 q^{97} +13.7074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 11 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 11 * q^9	\( 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 11 q^{9} - 3 q^{13} - 2 q^{15} + 2 q^{17} + 8 q^{19} + 8 q^{21} + 10 q^{23} + 3 q^{25} + 8 q^{27} - 10 q^{29} + 12 q^{31} - 12 q^{33} - 2 q^{35} + 2 q^{37} - 2 q^{39} - 2 q^{41} - 2 q^{43} - 11 q^{45} + 10 q^{47} + 23 q^{49} - 36 q^{51} + 18 q^{53} - 20 q^{57} - 16 q^{59} - 14 q^{61} + 50 q^{63} + 3 q^{65} + 14 q^{67} - 12 q^{69} + 14 q^{73} + 2 q^{75} + 36 q^{77} - 8 q^{79} + 23 q^{81} - 6 q^{83} - 2 q^{85} - 2 q^{89} - 2 q^{91} + 20 q^{93} - 8 q^{95} + 26 q^{97} + 36 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 11 * q^9 - 3 * q^13 - 2 * q^15 + 2 * q^17 + 8 * q^19 + 8 * q^21 + 10 * q^23 + 3 * q^25 + 8 * q^27 - 10 * q^29 + 12 * q^31 - 12 * q^33 - 2 * q^35 + 2 * q^37 - 2 * q^39 - 2 * q^41 - 2 * q^43 - 11 * q^45 + 10 * q^47 + 23 * q^49 - 36 * q^51 + 18 * q^53 - 20 * q^57 - 16 * q^59 - 14 * q^61 + 50 * q^63 + 3 * q^65 + 14 * q^67 - 12 * q^69 + 14 * q^73 + 2 * q^75 + 36 * q^77 - 8 * q^79 + 23 * q^81 - 6 * q^83 - 2 * q^85 - 2 * q^89 - 2 * q^91 + 20 * q^93 - 8 * q^95 + 26 * q^97 + 36 * q^99

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