LMFDB - Embedded newform 3120.2.a.bi.1.1

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.940.1
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.602705$ of defining polynomial
Character	$\chi$	$=$	3120.1

$q$-expansion

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -3.43134 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -3.43134 q^{7} +1.00000 q^{9} +2.63675 q^{11} -1.00000 q^{13} -1.00000 q^{15} +7.84216 q^{17} -0.794590 q^{19} +3.43134 q^{21} +3.43134 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.06808 q^{29} -9.27349 q^{31} -2.63675 q^{33} -3.43134 q^{35} -0.636747 q^{37} +1.00000 q^{39} -4.63675 q^{41} -2.41082 q^{43} +1.00000 q^{45} -5.27349 q^{47} +4.77407 q^{49} -7.84216 q^{51} +11.4994 q^{53} +2.63675 q^{55} +0.794590 q^{57} +12.1362 q^{59} -11.4994 q^{61} -3.43134 q^{63} -1.00000 q^{65} +6.41082 q^{67} -3.43134 q^{69} +10.6367 q^{71} +16.0681 q^{73} -1.00000 q^{75} -9.04757 q^{77} -10.6367 q^{79} +1.00000 q^{81} +16.1362 q^{83} +7.84216 q^{85} +4.06808 q^{87} +8.63675 q^{89} +3.43134 q^{91} +9.27349 q^{93} -0.794590 q^{95} +12.2940 q^{97} +2.63675 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} - 5 q^{11} - 3 q^{13} - 3 q^{15} + 7 q^{17} - 6 q^{19} + q^{21} + q^{23} + 3 q^{25} - 3 q^{27} + 10 q^{29} - 2 q^{31} + 5 q^{33} - q^{35} + 11 q^{37} + 3 q^{39} - q^{41} + 3 q^{45} + 10 q^{47} + 20 q^{49} - 7 q^{51} + 3 q^{53} - 5 q^{55} + 6 q^{57} - 8 q^{59} - 3 q^{61} - q^{63} - 3 q^{65} + 12 q^{67} - q^{69} + 19 q^{71} + 26 q^{73} - 3 q^{75} - 7 q^{77} - 19 q^{79} + 3 q^{81} + 4 q^{83} + 7 q^{85} - 10 q^{87} + 13 q^{89} + q^{91} + 2 q^{93} - 6 q^{95} + 9 q^{97} - 5 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 - 5 * q^11 - 3 * q^13 - 3 * q^15 + 7 * q^17 - 6 * q^19 + q^21 + q^23 + 3 * q^25 - 3 * q^27 + 10 * q^29 - 2 * q^31 + 5 * q^33 - q^35 + 11 * q^37 + 3 * q^39 - q^41 + 3 * q^45 + 10 * q^47 + 20 * q^49 - 7 * q^51 + 3 * q^53 - 5 * q^55 + 6 * q^57 - 8 * q^59 - 3 * q^61 - q^63 - 3 * q^65 + 12 * q^67 - q^69 + 19 * q^71 + 26 * q^73 - 3 * q^75 - 7 * q^77 - 19 * q^79 + 3 * q^81 + 4 * q^83 + 7 * q^85 - 10 * q^87 + 13 * q^89 + q^91 + 2 * q^93 - 6 * q^95 + 9 * q^97 - 5 * 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
$7$	−3.43134	−1.29692	−0.648462	−	0.761247i	$-0.724587\pi$
$7$	−3.43134	−1.29692	−0.648462	+	0.761247i	$0.724587\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.63675	0.795009	0.397505	−	0.917600i	$-0.369876\pi$
$11$	2.63675	0.795009	0.397505	+	0.917600i	$0.369876\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	7.84216	1.90200	0.951001	−	0.309187i	$-0.100057\pi$
$17$	7.84216	1.90200	0.951001	+	0.309187i	$0.100057\pi$
$18$	0	0
$19$	−0.794590	−0.182291	−0.0911457	−	0.995838i	$-0.529053\pi$
$19$	−0.794590	−0.182291	−0.0911457	+	0.995838i	$0.529053\pi$
$20$	0	0
$21$	3.43134	0.748779
$22$	0	0
$23$	3.43134	0.715483	0.357742	−	0.933821i	$-0.383547\pi$
$23$	3.43134	0.715483	0.357742	+	0.933821i	$0.383547\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.06808	−0.755424	−0.377712	−	0.925923i	$-0.623289\pi$
$29$	−4.06808	−0.755424	−0.377712	+	0.925923i	$0.623289\pi$
$30$	0	0
$31$	−9.27349	−1.66557	−0.832784	−	0.553598i	$-0.813255\pi$
$31$	−9.27349	−1.66557	−0.832784	+	0.553598i	$0.813255\pi$
$32$	0	0
$33$	−2.63675	−0.458999
$34$	0	0
$35$	−3.43134	−0.580002
$36$	0	0
$37$	−0.636747	−0.104681	−0.0523403	−	0.998629i	$-0.516668\pi$
$37$	−0.636747	−0.104681	−0.0523403	+	0.998629i	$0.516668\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−4.63675	−0.724138	−0.362069	−	0.932151i	$-0.617930\pi$
$41$	−4.63675	−0.724138	−0.362069	+	0.932151i	$0.617930\pi$
$42$	0	0
$43$	−2.41082	−0.367647	−0.183823	−	0.982959i	$-0.558847\pi$
$43$	−2.41082	−0.367647	−0.183823	+	0.982959i	$0.558847\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−5.27349	−0.769218	−0.384609	−	0.923080i	$-0.625664\pi$
$47$	−5.27349	−0.769218	−0.384609	+	0.923080i	$0.625664\pi$
$48$	0	0
$49$	4.77407	0.682010
$50$	0	0
$51$	−7.84216	−1.09812
$52$	0	0
$53$	11.4994	1.57957	0.789783	−	0.613386i	$-0.210193\pi$
$53$	11.4994	1.57957	0.789783	+	0.613386i	$0.210193\pi$
$54$	0	0
$55$	2.63675	0.355539
$56$	0	0
$57$	0.794590	0.105246
$58$	0	0
$59$	12.1362	1.57999	0.789997	−	0.613110i	$-0.210082\pi$
$59$	12.1362	1.57999	0.789997	+	0.613110i	$0.210082\pi$
$60$	0	0
$61$	−11.4994	−1.47235	−0.736175	−	0.676791i	$-0.763370\pi$
$61$	−11.4994	−1.47235	−0.736175	+	0.676791i	$0.763370\pi$
$62$	0	0
$63$	−3.43134	−0.432308
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	6.41082	0.783206	0.391603	−	0.920134i	$-0.371921\pi$
$67$	6.41082	0.783206	0.391603	+	0.920134i	$0.371921\pi$
$68$	0	0
$69$	−3.43134	−0.413084
$70$	0	0
$71$	10.6367	1.26235	0.631175	−	0.775641i	$-0.282573\pi$
$71$	10.6367	1.26235	0.631175	+	0.775641i	$0.282573\pi$
$72$	0	0
$73$	16.0681	1.88063	0.940313	−	0.340310i	$-0.110532\pi$
$73$	16.0681	1.88063	0.940313	+	0.340310i	$0.110532\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−9.04757	−1.03107
$78$	0	0
$79$	−10.6367	−1.19673	−0.598364	−	0.801225i	$-0.704182\pi$
$79$	−10.6367	−1.19673	−0.598364	+	0.801225i	$0.704182\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.1362	1.77117	0.885587	−	0.464473i	$-0.153756\pi$
$83$	16.1362	1.77117	0.885587	+	0.464473i	$0.153756\pi$
$84$	0	0
$85$	7.84216	0.850601
$86$	0	0
$87$	4.06808	0.436144
$88$	0	0
$89$	8.63675	0.915493	0.457747	−	0.889083i	$-0.348657\pi$
$89$	8.63675	0.915493	0.457747	+	0.889083i	$0.348657\pi$
$90$	0	0
$91$	3.43134	0.359702
$92$	0	0
$93$	9.27349	0.961616
$94$	0	0
$95$	−0.794590	−0.0815232
$96$	0	0
$97$	12.2940	1.24827	0.624134	−	0.781317i	$-0.285452\pi$
$97$	12.2940	1.24827	0.624134	+	0.781317i	$0.285452\pi$
$98$	0	0
$99$	2.63675	0.265003
$100$	0	0
$101$	5.34158	0.531507	0.265753	−	0.964041i	$-0.414379\pi$
$101$	5.34158	0.531507	0.265753	+	0.964041i	$0.414379\pi$
$102$	0	0
$103$	10.4108	1.02581	0.512904	−	0.858446i	$-0.328570\pi$
$103$	10.4108	1.02581	0.512904	+	0.858446i	$0.328570\pi$
$104$	0	0
$105$	3.43134	0.334864
$106$	0	0
$107$	−5.04757	−0.487967	−0.243983	−	0.969779i	$-0.578454\pi$
$107$	−5.04757	−0.487967	−0.243983	+	0.969779i	$0.578454\pi$
$108$	0	0
$109$	6.79459	0.650804	0.325402	−	0.945576i	$-0.394500\pi$
$109$	6.79459	0.650804	0.325402	+	0.945576i	$0.394500\pi$
$110$	0	0
$111$	0.636747	0.0604373
$112$	0	0
$113$	−8.06808	−0.758981	−0.379491	−	0.925196i	$-0.623901\pi$
$113$	−8.06808	−0.758981	−0.379491	+	0.925196i	$0.623901\pi$
$114$	0	0
$115$	3.43134	0.319974
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−26.9091	−2.46675
$120$	0	0
$121$	−4.04757	−0.367961
$122$	0	0
$123$	4.63675	0.418081
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.6843	1.39176	0.695879	−	0.718159i	$-0.255015\pi$
$127$	15.6843	1.39176	0.695879	+	0.718159i	$0.255015\pi$
$128$	0	0
$129$	2.41082	0.212261
$130$	0	0
$131$	−12.9308	−1.12977	−0.564883	−	0.825171i	$-0.691079\pi$
$131$	−12.9308	−1.12977	−0.564883	+	0.825171i	$0.691079\pi$
$132$	0	0
$133$	2.72651	0.236418
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−19.9102	−1.68876	−0.844382	−	0.535741i	$-0.820032\pi$
$139$	−19.9102	−1.68876	−0.844382	+	0.535741i	$0.820032\pi$
$140$	0	0
$141$	5.27349	0.444108
$142$	0	0
$143$	−2.63675	−0.220496
$144$	0	0
$145$	−4.06808	−0.337836
$146$	0	0
$147$	−4.77407	−0.393759
$148$	0	0
$149$	8.63675	0.707550	0.353775	−	0.935331i	$-0.384898\pi$
$149$	8.63675	0.707550	0.353775	+	0.935331i	$0.384898\pi$
$150$	0	0
$151$	17.7253	1.44247	0.721234	−	0.692691i	$-0.243575\pi$
$151$	17.7253	1.44247	0.721234	+	0.692691i	$0.243575\pi$
$152$	0	0
$153$	7.84216	0.634001
$154$	0	0
$155$	−9.27349	−0.744865
$156$	0	0
$157$	20.5470	1.63983	0.819914	−	0.572487i	$-0.194021\pi$
$157$	20.5470	1.63983	0.819914	+	0.572487i	$0.194021\pi$
$158$	0	0
$159$	−11.4994	−0.911963
$160$	0	0
$161$	−11.7741	−0.927927
$162$	0	0
$163$	4.22593	0.331000	0.165500	−	0.986210i	$-0.447076\pi$
$163$	4.22593	0.331000	0.165500	+	0.986210i	$0.447076\pi$
$164$	0	0
$165$	−2.63675	−0.205270
$166$	0	0
$167$	14.4108	1.11514	0.557571	−	0.830129i	$-0.311733\pi$
$167$	14.4108	1.11514	0.557571	+	0.830129i	$0.311733\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.794590	−0.0607638
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−3.43134	−0.259385
$176$	0	0
$177$	−12.1362	−0.912210
$178$	0	0
$179$	20.9308	1.56444	0.782219	−	0.623003i	$-0.214088\pi$
$179$	20.9308	1.56444	0.782219	+	0.623003i	$0.214088\pi$
$180$	0	0
$181$	−5.08860	−0.378233	−0.189116	−	0.981955i	$-0.560562\pi$
$181$	−5.08860	−0.378233	−0.189116	+	0.981955i	$0.560562\pi$
$182$	0	0
$183$	11.4994	0.850062
$184$	0	0
$185$	−0.636747	−0.0468146
$186$	0	0
$187$	20.6778	1.51211
$188$	0	0
$189$	3.43134	0.249593
$190$	0	0
$191$	13.7253	0.993131	0.496566	−	0.867999i	$-0.334594\pi$
$191$	13.7253	0.993131	0.496566	+	0.867999i	$0.334594\pi$
$192$	0	0
$193$	−5.11565	−0.368233	−0.184116	−	0.982904i	$-0.558942\pi$
$193$	−5.11565	−0.368233	−0.184116	+	0.982904i	$0.558942\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−20.9988	−1.49611	−0.748053	−	0.663639i	$-0.769011\pi$
$197$	−20.9988	−1.49611	−0.748053	+	0.663639i	$0.769011\pi$
$198$	0	0
$199$	−9.58918	−0.679759	−0.339879	−	0.940469i	$-0.610386\pi$
$199$	−9.58918	−0.679759	−0.339879	+	0.940469i	$0.610386\pi$
$200$	0	0
$201$	−6.41082	−0.452184
$202$	0	0
$203$	13.9590	0.979727
$204$	0	0
$205$	−4.63675	−0.323844
$206$	0	0
$207$	3.43134	0.238494
$208$	0	0
$209$	−2.09513	−0.144923
$210$	0	0
$211$	8.82164	0.607307	0.303653	−	0.952783i	$-0.401794\pi$
$211$	8.82164	0.607307	0.303653	+	0.952783i	$0.401794\pi$
$212$	0	0
$213$	−10.6367	−0.728818
$214$	0	0
$215$	−2.41082	−0.164417
$216$	0	0
$217$	31.8205	2.16011
$218$	0	0
$219$	−16.0681	−1.08578
$220$	0	0
$221$	−7.84216	−0.527521
$222$	0	0
$223$	9.93192	0.665090	0.332545	−	0.943087i	$-0.392093\pi$
$223$	9.93192	0.665090	0.332545	+	0.943087i	$0.392093\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−9.27349	−0.615503	−0.307752	−	0.951467i	$-0.599577\pi$
$227$	−9.27349	−0.615503	−0.307752	+	0.951467i	$0.599577\pi$
$228$	0	0
$229$	12.0681	0.797481	0.398741	−	0.917064i	$-0.369447\pi$
$229$	12.0681	0.797481	0.398741	+	0.917064i	$0.369447\pi$
$230$	0	0
$231$	9.04757	0.595286
$232$	0	0
$233$	13.1157	0.859235	0.429617	−	0.903011i	$-0.358648\pi$
$233$	13.1157	0.859235	0.429617	+	0.903011i	$0.358648\pi$
$234$	0	0
$235$	−5.27349	−0.344005
$236$	0	0
$237$	10.6367	0.690931
$238$	0	0
$239$	22.7729	1.47306	0.736529	−	0.676406i	$-0.236464\pi$
$239$	22.7729	1.47306	0.736529	+	0.676406i	$0.236464\pi$
$240$	0	0
$241$	6.82164	0.439420	0.219710	−	0.975565i	$-0.429489\pi$
$241$	6.82164	0.439420	0.219710	+	0.975565i	$0.429489\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.77407	0.305004
$246$	0	0
$247$	0.794590	0.0505586
$248$	0	0
$249$	−16.1362	−1.02259
$250$	0	0
$251$	−9.24644	−0.583630	−0.291815	−	0.956475i	$-0.594259\pi$
$251$	−9.24644	−0.583630	−0.291815	+	0.956475i	$0.594259\pi$
$252$	0	0
$253$	9.04757	0.568816
$254$	0	0
$255$	−7.84216	−0.491095
$256$	0	0
$257$	−11.2464	−0.701534	−0.350767	−	0.936463i	$-0.614079\pi$
$257$	−11.2464	−0.701534	−0.350767	+	0.936463i	$0.614079\pi$
$258$	0	0
$259$	2.18489	0.135763
$260$	0	0
$261$	−4.06808	−0.251808
$262$	0	0
$263$	22.0681	1.36078	0.680388	−	0.732852i	$-0.261811\pi$
$263$	22.0681	1.36078	0.680388	+	0.732852i	$0.261811\pi$
$264$	0	0
$265$	11.4994	0.706404
$266$	0	0
$267$	−8.63675	−0.528560
$268$	0	0
$269$	−13.2054	−0.805148	−0.402574	−	0.915387i	$-0.631884\pi$
$269$	−13.2054	−0.805148	−0.402574	+	0.915387i	$0.631884\pi$
$270$	0	0
$271$	−14.0951	−0.856218	−0.428109	−	0.903727i	$-0.640820\pi$
$271$	−14.0951	−0.856218	−0.428109	+	0.903727i	$0.640820\pi$
$272$	0	0
$273$	−3.43134	−0.207674
$274$	0	0
$275$	2.63675	0.159002
$276$	0	0
$277$	−30.2723	−1.81889	−0.909444	−	0.415826i	$-0.863493\pi$
$277$	−30.2723	−1.81889	−0.909444	+	0.415826i	$0.863493\pi$
$278$	0	0
$279$	−9.27349	−0.555190
$280$	0	0
$281$	2.13617	0.127433	0.0637165	−	0.997968i	$-0.479705\pi$
$281$	2.13617	0.127433	0.0637165	+	0.997968i	$0.479705\pi$
$282$	0	0
$283$	−0.136167	−0.00809431	−0.00404715	−	0.999992i	$-0.501288\pi$
$283$	−0.136167	−0.00809431	−0.00404715	+	0.999992i	$0.501288\pi$
$284$	0	0
$285$	0.794590	0.0470675
$286$	0	0
$287$	15.9102	0.939152
$288$	0	0
$289$	44.4994	2.61761
$290$	0	0
$291$	−12.2940	−0.720688
$292$	0	0
$293$	−15.2735	−0.892287	−0.446144	−	0.894961i	$-0.647203\pi$
$293$	−15.2735	−0.892287	−0.446144	+	0.894961i	$0.647203\pi$
$294$	0	0
$295$	12.1362	0.706595
$296$	0	0
$297$	−2.63675	−0.153000
$298$	0	0
$299$	−3.43134	−0.198439
$300$	0	0
$301$	8.27233	0.476809
$302$	0	0
$303$	−5.34158	−0.306866
$304$	0	0
$305$	−11.4994	−0.658455
$306$	0	0
$307$	−7.91024	−0.451461	−0.225731	−	0.974190i	$-0.572477\pi$
$307$	−7.91024	−0.451461	−0.225731	+	0.974190i	$0.572477\pi$
$308$	0	0
$309$	−10.4108	−0.592251
$310$	0	0
$311$	−2.54699	−0.144426	−0.0722132	−	0.997389i	$-0.523006\pi$
$311$	−2.54699	−0.144426	−0.0722132	+	0.997389i	$0.523006\pi$
$312$	0	0
$313$	19.5892	1.10725	0.553623	−	0.832767i	$-0.313245\pi$
$313$	19.5892	1.10725	0.553623	+	0.832767i	$0.313245\pi$
$314$	0	0
$315$	−3.43134	−0.193334
$316$	0	0
$317$	−23.2735	−1.30717	−0.653585	−	0.756853i	$-0.726736\pi$
$317$	−23.2735	−1.30717	−0.653585	+	0.756853i	$0.726736\pi$
$318$	0	0
$319$	−10.7265	−0.600569
$320$	0	0
$321$	5.04757	0.281728
$322$	0	0
$323$	−6.23130	−0.346719
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−6.79459	−0.375742
$328$	0	0
$329$	18.0951	0.997617
$330$	0	0
$331$	20.4789	1.12562	0.562811	−	0.826586i	$-0.309720\pi$
$331$	20.4789	1.12562	0.562811	+	0.826586i	$0.309720\pi$
$332$	0	0
$333$	−0.636747	−0.0348935
$334$	0	0
$335$	6.41082	0.350260
$336$	0	0
$337$	−25.6843	−1.39911	−0.699557	−	0.714577i	$-0.746619\pi$
$337$	−25.6843	−1.39911	−0.699557	+	0.714577i	$0.746619\pi$
$338$	0	0
$339$	8.06808	0.438198
$340$	0	0
$341$	−24.4519	−1.32414
$342$	0	0
$343$	7.63791	0.412408
$344$	0	0
$345$	−3.43134	−0.184737
$346$	0	0
$347$	−9.81511	−0.526903	−0.263451	−	0.964673i	$-0.584861\pi$
$347$	−9.81511	−0.526903	−0.263451	+	0.964673i	$0.584861\pi$
$348$	0	0
$349$	12.5199	0.670177	0.335088	−	0.942187i	$-0.391234\pi$
$349$	12.5199	0.670177	0.335088	+	0.942187i	$0.391234\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	10.6367	0.564540
$356$	0	0
$357$	26.9091	1.42418
$358$	0	0
$359$	28.1362	1.48497	0.742485	−	0.669863i	$-0.233647\pi$
$359$	28.1362	1.48497	0.742485	+	0.669863i	$0.233647\pi$
$360$	0	0
$361$	−18.3686	−0.966770
$362$	0	0
$363$	4.04757	0.212442
$364$	0	0
$365$	16.0681	0.841042
$366$	0	0
$367$	−2.41082	−0.125844	−0.0629219	−	0.998018i	$-0.520042\pi$
$367$	−2.41082	−0.125844	−0.0629219	+	0.998018i	$0.520042\pi$
$368$	0	0
$369$	−4.63675	−0.241379
$370$	0	0
$371$	−39.4584	−2.04858
$372$	0	0
$373$	−5.54815	−0.287272	−0.143636	−	0.989631i	$-0.545879\pi$
$373$	−5.54815	−0.287272	−0.143636	+	0.989631i	$0.545879\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.06808	0.209517
$378$	0	0
$379$	14.5199	0.745839	0.372920	−	0.927864i	$-0.378357\pi$
$379$	14.5199	0.745839	0.372920	+	0.927864i	$0.378357\pi$
$380$	0	0
$381$	−15.6843	−0.803532
$382$	0	0
$383$	22.8627	1.16823	0.584114	−	0.811672i	$-0.301442\pi$
$383$	22.8627	1.16823	0.584114	+	0.811672i	$0.301442\pi$
$384$	0	0
$385$	−9.04757	−0.461107
$386$	0	0
$387$	−2.41082	−0.122549
$388$	0	0
$389$	5.34158	0.270829	0.135414	−	0.990789i	$-0.456763\pi$
$389$	5.34158	0.270829	0.135414	+	0.990789i	$0.456763\pi$
$390$	0	0
$391$	26.9091	1.36085
$392$	0	0
$393$	12.9308	0.652270
$394$	0	0
$395$	−10.6367	−0.535193
$396$	0	0
$397$	−21.4584	−1.07697	−0.538483	−	0.842637i	$-0.681002\pi$
$397$	−21.4584	−1.07697	−0.538483	+	0.842637i	$0.681002\pi$
$398$	0	0
$399$	−2.72651	−0.136496
$400$	0	0
$401$	−17.3145	−0.864646	−0.432323	−	0.901719i	$-0.642306\pi$
$401$	−17.3145	−0.864646	−0.432323	+	0.901719i	$0.642306\pi$
$402$	0	0
$403$	9.27349	0.461946
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−1.67894	−0.0832220
$408$	0	0
$409$	16.4108	0.811463	0.405731	−	0.913992i	$-0.367017\pi$
$409$	16.4108	0.811463	0.405731	+	0.913992i	$0.367017\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	−41.6433	−2.04913
$414$	0	0
$415$	16.1362	0.792093
$416$	0	0
$417$	19.9102	0.975009
$418$	0	0
$419$	−25.7524	−1.25809	−0.629043	−	0.777370i	$-0.716553\pi$
$419$	−25.7524	−1.25809	−0.629043	+	0.777370i	$0.716553\pi$
$420$	0	0
$421$	13.4777	0.656865	0.328433	−	0.944527i	$-0.393480\pi$
$421$	13.4777	0.656865	0.328433	+	0.944527i	$0.393480\pi$
$422$	0	0
$423$	−5.27349	−0.256406
$424$	0	0
$425$	7.84216	0.380400
$426$	0	0
$427$	39.4584	1.90953
$428$	0	0
$429$	2.63675	0.127303
$430$	0	0
$431$	24.2723	1.16916	0.584579	−	0.811337i	$-0.301260\pi$
$431$	24.2723	1.16916	0.584579	+	0.811337i	$0.301260\pi$
$432$	0	0
$433$	3.13733	0.150770	0.0753851	−	0.997154i	$-0.475981\pi$
$433$	3.13733	0.150770	0.0753851	+	0.997154i	$0.475981\pi$
$434$	0	0
$435$	4.06808	0.195050
$436$	0	0
$437$	−2.72651	−0.130426
$438$	0	0
$439$	−8.59571	−0.410251	−0.205125	−	0.978736i	$-0.565760\pi$
$439$	−8.59571	−0.410251	−0.205125	+	0.978736i	$0.565760\pi$
$440$	0	0
$441$	4.77407	0.227337
$442$	0	0
$443$	−0.911399	−0.0433019	−0.0216509	−	0.999766i	$-0.506892\pi$
$443$	−0.911399	−0.0433019	−0.0216509	+	0.999766i	$0.506892\pi$
$444$	0	0
$445$	8.63675	0.409421
$446$	0	0
$447$	−8.63675	−0.408504
$448$	0	0
$449$	38.8139	1.83174	0.915872	−	0.401471i	$-0.131501\pi$
$449$	38.8139	1.83174	0.915872	+	0.401471i	$0.131501\pi$
$450$	0	0
$451$	−12.2259	−0.575696
$452$	0	0
$453$	−17.7253	−0.832809
$454$	0	0
$455$	3.43134	0.160864
$456$	0	0
$457$	−6.25298	−0.292502	−0.146251	−	0.989248i	$-0.546721\pi$
$457$	−6.25298	−0.292502	−0.146251	+	0.989248i	$0.546721\pi$
$458$	0	0
$459$	−7.84216	−0.366041
$460$	0	0
$461$	−27.9513	−1.30182	−0.650910	−	0.759155i	$-0.725613\pi$
$461$	−27.9513	−1.30182	−0.650910	+	0.759155i	$0.725613\pi$
$462$	0	0
$463$	11.8832	0.552259	0.276129	−	0.961120i	$-0.410948\pi$
$463$	11.8832	0.552259	0.276129	+	0.961120i	$0.410948\pi$
$464$	0	0
$465$	9.27349	0.430048
$466$	0	0
$467$	28.8680	1.33585	0.667927	−	0.744227i	$-0.267182\pi$
$467$	28.8680	1.33585	0.667927	+	0.744227i	$0.267182\pi$
$468$	0	0
$469$	−21.9977	−1.01576
$470$	0	0
$471$	−20.5470	−0.946755
$472$	0	0
$473$	−6.35672	−0.292282
$474$	0	0
$475$	−0.794590	−0.0364583
$476$	0	0
$477$	11.4994	0.526522
$478$	0	0
$479$	12.0464	0.550414	0.275207	−	0.961385i	$-0.411254\pi$
$479$	12.0464	0.550414	0.275207	+	0.961385i	$0.411254\pi$
$480$	0	0
$481$	0.636747	0.0290332
$482$	0	0
$483$	11.7741	0.535739
$484$	0	0
$485$	12.2940	0.558242
$486$	0	0
$487$	−11.2518	−0.509869	−0.254934	−	0.966958i	$-0.582054\pi$
$487$	−11.2518	−0.509869	−0.254934	+	0.966958i	$0.582054\pi$
$488$	0	0
$489$	−4.22593	−0.191103
$490$	0	0
$491$	−31.0259	−1.40018	−0.700089	−	0.714055i	$-0.746857\pi$
$491$	−31.0259	−1.40018	−0.700089	+	0.714055i	$0.746857\pi$
$492$	0	0
$493$	−31.9025	−1.43682
$494$	0	0
$495$	2.63675	0.118513
$496$	0	0
$497$	−36.4983	−1.63717
$498$	0	0
$499$	5.61623	0.251417	0.125708	−	0.992067i	$-0.459880\pi$
$499$	5.61623	0.251417	0.125708	+	0.992067i	$0.459880\pi$
$500$	0	0
$501$	−14.4108	−0.643828
$502$	0	0
$503$	3.52110	0.156998	0.0784990	−	0.996914i	$-0.474987\pi$
$503$	3.52110	0.156998	0.0784990	+	0.996914i	$0.474987\pi$
$504$	0	0
$505$	5.34158	0.237697
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−13.7741	−0.610525	−0.305263	−	0.952268i	$-0.598744\pi$
$509$	−13.7741	−0.610525	−0.305263	+	0.952268i	$0.598744\pi$
$510$	0	0
$511$	−55.1350	−2.43903
$512$	0	0
$513$	0.794590	0.0350820
$514$	0	0
$515$	10.4108	0.458756
$516$	0	0
$517$	−13.9049	−0.611535
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	25.3145	1.10905	0.554525	−	0.832167i	$-0.312900\pi$
$521$	25.3145	1.10905	0.554525	+	0.832167i	$0.312900\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	3.43134	0.149756
$526$	0	0
$527$	−72.7242	−3.16792
$528$	0	0
$529$	−11.2259	−0.488084
$530$	0	0
$531$	12.1362	0.526665
$532$	0	0
$533$	4.63675	0.200840
$534$	0	0
$535$	−5.04757	−0.218225
$536$	0	0
$537$	−20.9308	−0.903229
$538$	0	0
$539$	12.5880	0.542204
$540$	0	0
$541$	−26.6151	−1.14427	−0.572136	−	0.820159i	$-0.693885\pi$
$541$	−26.6151	−1.14427	−0.572136	+	0.820159i	$0.693885\pi$
$542$	0	0
$543$	5.08860	0.218373
$544$	0	0
$545$	6.79459	0.291048
$546$	0	0
$547$	17.7253	0.757881	0.378941	−	0.925421i	$-0.376288\pi$
$547$	17.7253	0.757881	0.378941	+	0.925421i	$0.376288\pi$
$548$	0	0
$549$	−11.4994	−0.490783
$550$	0	0
$551$	3.23246	0.137707
$552$	0	0
$553$	36.4983	1.55206
$554$	0	0
$555$	0.636747	0.0270284
$556$	0	0
$557$	30.2723	1.28268	0.641340	−	0.767257i	$-0.278379\pi$
$557$	30.2723	1.28268	0.641340	+	0.767257i	$0.278379\pi$
$558$	0	0
$559$	2.41082	0.101967
$560$	0	0
$561$	−20.6778	−0.873017
$562$	0	0
$563$	−22.6367	−0.954025	−0.477013	−	0.878896i	$-0.658280\pi$
$563$	−22.6367	−0.954025	−0.477013	+	0.878896i	$0.658280\pi$
$564$	0	0
$565$	−8.06808	−0.339427
$566$	0	0
$567$	−3.43134	−0.144103
$568$	0	0
$569$	20.0410	0.840164	0.420082	−	0.907486i	$-0.362001\pi$
$569$	20.0410	0.840164	0.420082	+	0.907486i	$0.362001\pi$
$570$	0	0
$571$	−20.3621	−0.852127	−0.426064	−	0.904693i	$-0.640100\pi$
$571$	−20.3621	−0.852127	−0.426064	+	0.904693i	$0.640100\pi$
$572$	0	0
$573$	−13.7253	−0.573385
$574$	0	0
$575$	3.43134	0.143097
$576$	0	0
$577$	2.70483	0.112604	0.0563018	−	0.998414i	$-0.482069\pi$
$577$	2.70483	0.112604	0.0563018	+	0.998414i	$0.482069\pi$
$578$	0	0
$579$	5.11565	0.212599
$580$	0	0
$581$	−55.3686	−2.29708
$582$	0	0
$583$	30.3211	1.25577
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−33.2735	−1.37334	−0.686672	−	0.726967i	$-0.740929\pi$
$587$	−33.2735	−1.37334	−0.686672	+	0.726967i	$0.740929\pi$
$588$	0	0
$589$	7.36863	0.303619
$590$	0	0
$591$	20.9988	0.863777
$592$	0	0
$593$	−28.2313	−1.15932	−0.579660	−	0.814858i	$-0.696815\pi$
$593$	−28.2313	−1.15932	−0.579660	+	0.814858i	$0.696815\pi$
$594$	0	0
$595$	−26.9091	−1.10316
$596$	0	0
$597$	9.58918	0.392459
$598$	0	0
$599$	−29.0940	−1.18875	−0.594374	−	0.804189i	$-0.702600\pi$
$599$	−29.0940	−1.18875	−0.594374	+	0.804189i	$0.702600\pi$
$600$	0	0
$601$	46.0464	1.87827	0.939136	−	0.343545i	$-0.111628\pi$
$601$	46.0464	1.87827	0.939136	+	0.343545i	$0.111628\pi$
$602$	0	0
$603$	6.41082	0.261069
$604$	0	0
$605$	−4.04757	−0.164557
$606$	0	0
$607$	−24.1362	−0.979657	−0.489828	−	0.871819i	$-0.662941\pi$
$607$	−24.1362	−0.979657	−0.489828	+	0.871819i	$0.662941\pi$
$608$	0	0
$609$	−13.9590	−0.565646
$610$	0	0
$611$	5.27349	0.213343
$612$	0	0
$613$	3.73304	0.150776	0.0753880	−	0.997154i	$-0.475980\pi$
$613$	3.73304	0.150776	0.0753880	+	0.997154i	$0.475980\pi$
$614$	0	0
$615$	4.63675	0.186972
$616$	0	0
$617$	−13.3686	−0.538201	−0.269100	−	0.963112i	$-0.586726\pi$
$617$	−13.3686	−0.538201	−0.269100	+	0.963112i	$0.586726\pi$
$618$	0	0
$619$	21.8886	0.879776	0.439888	−	0.898053i	$-0.355018\pi$
$619$	21.8886	0.879776	0.439888	+	0.898053i	$0.355018\pi$
$620$	0	0
$621$	−3.43134	−0.137695
$622$	0	0
$623$	−29.6356	−1.18732
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.09513	0.0836716
$628$	0	0
$629$	−4.99347	−0.199103
$630$	0	0
$631$	−21.5892	−0.859452	−0.429726	−	0.902959i	$-0.641390\pi$
$631$	−21.5892	−0.859452	−0.429726	+	0.902959i	$0.641390\pi$
$632$	0	0
$633$	−8.82164	−0.350629
$634$	0	0
$635$	15.6843	0.622413
$636$	0	0
$637$	−4.77407	−0.189156
$638$	0	0
$639$	10.6367	0.420783
$640$	0	0
$641$	−5.54815	−0.219139	−0.109569	−	0.993979i	$-0.534947\pi$
$641$	−5.54815	−0.219139	−0.109569	+	0.993979i	$0.534947\pi$
$642$	0	0
$643$	34.6367	1.36594	0.682970	−	0.730446i	$-0.260688\pi$
$643$	34.6367	1.36594	0.682970	+	0.730446i	$0.260688\pi$
$644$	0	0
$645$	2.41082	0.0949259
$646$	0	0
$647$	−29.9783	−1.17857	−0.589285	−	0.807925i	$-0.700590\pi$
$647$	−29.9783	−1.17857	−0.589285	+	0.807925i	$0.700590\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	−31.8205	−1.24714
$652$	0	0
$653$	32.6832	1.27899	0.639495	−	0.768795i	$-0.279143\pi$
$653$	32.6832	1.27899	0.639495	+	0.768795i	$0.279143\pi$
$654$	0	0
$655$	−12.9308	−0.505247
$656$	0	0
$657$	16.0681	0.626876
$658$	0	0
$659$	−23.9296	−0.932165	−0.466082	−	0.884741i	$-0.654335\pi$
$659$	−23.9296	−0.932165	−0.466082	+	0.884741i	$0.654335\pi$
$660$	0	0
$661$	−34.1632	−1.32880	−0.664398	−	0.747379i	$-0.731312\pi$
$661$	−34.1632	−1.32880	−0.664398	+	0.747379i	$0.731312\pi$
$662$	0	0
$663$	7.84216	0.304564
$664$	0	0
$665$	2.72651	0.105729
$666$	0	0
$667$	−13.9590	−0.540493
$668$	0	0
$669$	−9.93192	−0.383990
$670$	0	0
$671$	−30.3211	−1.17053
$672$	0	0
$673$	−6.90371	−0.266118	−0.133059	−	0.991108i	$-0.542480\pi$
$673$	−6.90371	−0.266118	−0.133059	+	0.991108i	$0.542480\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	49.2789	1.89394	0.946970	−	0.321321i	$-0.104127\pi$
$677$	49.2789	1.89394	0.946970	+	0.321321i	$0.104127\pi$
$678$	0	0
$679$	−42.1849	−1.61891
$680$	0	0
$681$	9.27349	0.355361
$682$	0	0
$683$	−3.04219	−0.116406	−0.0582031	−	0.998305i	$-0.518537\pi$
$683$	−3.04219	−0.116406	−0.0582031	+	0.998305i	$0.518537\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−12.0681	−0.460426
$688$	0	0
$689$	−11.4994	−0.438093
$690$	0	0
$691$	−13.8886	−0.528346	−0.264173	−	0.964475i	$-0.585099\pi$
$691$	−13.8886	−0.528346	−0.264173	+	0.964475i	$0.585099\pi$
$692$	0	0
$693$	−9.04757	−0.343689
$694$	0	0
$695$	−19.9102	−0.755238
$696$	0	0
$697$	−36.3621	−1.37731
$698$	0	0
$699$	−13.1157	−0.496079
$700$	0	0
$701$	−5.20541	−0.196606	−0.0983028	−	0.995157i	$-0.531341\pi$
$701$	−5.20541	−0.196606	−0.0983028	+	0.995157i	$0.531341\pi$
$702$	0	0
$703$	0.505953	0.0190824
$704$	0	0
$705$	5.27349	0.198611
$706$	0	0
$707$	−18.3288	−0.689324
$708$	0	0
$709$	−19.7524	−0.741817	−0.370908	−	0.928669i	$-0.620954\pi$
$709$	−19.7524	−0.741817	−0.370908	+	0.928669i	$0.620954\pi$
$710$	0	0
$711$	−10.6367	−0.398909
$712$	0	0
$713$	−31.8205	−1.19169
$714$	0	0
$715$	−2.63675	−0.0986087
$716$	0	0
$717$	−22.7729	−0.850470
$718$	0	0
$719$	4.13617	0.154253	0.0771265	−	0.997021i	$-0.475425\pi$
$719$	4.13617	0.154253	0.0771265	+	0.997021i	$0.475425\pi$
$720$	0	0
$721$	−35.7230	−1.33040
$722$	0	0
$723$	−6.82164	−0.253700
$724$	0	0
$725$	−4.06808	−0.151085
$726$	0	0
$727$	−6.09513	−0.226056	−0.113028	−	0.993592i	$-0.536055\pi$
$727$	−6.09513	−0.226056	−0.113028	+	0.993592i	$0.536055\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.9060	−0.699265
$732$	0	0
$733$	−40.0054	−1.47763	−0.738816	−	0.673907i	$-0.764615\pi$
$733$	−40.0054	−1.47763	−0.738816	+	0.673907i	$0.764615\pi$
$734$	0	0
$735$	−4.77407	−0.176094
$736$	0	0
$737$	16.9037	0.622656
$738$	0	0
$739$	−12.0270	−0.442422	−0.221211	−	0.975226i	$-0.571001\pi$
$739$	−12.0270	−0.442422	−0.221211	+	0.975226i	$0.571001\pi$
$740$	0	0
$741$	−0.794590	−0.0291900
$742$	0	0
$743$	−10.0410	−0.368370	−0.184185	−	0.982892i	$-0.558965\pi$
$743$	−10.0410	−0.368370	−0.184185	+	0.982892i	$0.558965\pi$
$744$	0	0
$745$	8.63675	0.316426
$746$	0	0
$747$	16.1362	0.590391
$748$	0	0
$749$	17.3199	0.632855
$750$	0	0
$751$	−12.6778	−0.462619	−0.231309	−	0.972880i	$-0.574301\pi$
$751$	−12.6778	−0.462619	−0.231309	+	0.972880i	$0.574301\pi$
$752$	0	0
$753$	9.24644	0.336959
$754$	0	0
$755$	17.7253	0.645091
$756$	0	0
$757$	13.6302	0.495399	0.247699	−	0.968837i	$-0.420326\pi$
$757$	13.6302	0.495399	0.247699	+	0.968837i	$0.420326\pi$
$758$	0	0
$759$	−9.04757	−0.328406
$760$	0	0
$761$	−11.8615	−0.429980	−0.214990	−	0.976616i	$-0.568972\pi$
$761$	−11.8615	−0.429980	−0.214990	+	0.976616i	$0.568972\pi$
$762$	0	0
$763$	−23.3145	−0.844043
$764$	0	0
$765$	7.84216	0.283534
$766$	0	0
$767$	−12.1362	−0.438212
$768$	0	0
$769$	20.9988	0.757238	0.378619	−	0.925553i	$-0.376399\pi$
$769$	20.9988	0.757238	0.378619	+	0.925553i	$0.376399\pi$
$770$	0	0
$771$	11.2464	0.405031
$772$	0	0
$773$	14.4519	0.519797	0.259899	−	0.965636i	$-0.416311\pi$
$773$	14.4519	0.519797	0.259899	+	0.965636i	$0.416311\pi$
$774$	0	0
$775$	−9.27349	−0.333114
$776$	0	0
$777$	−2.18489	−0.0783826
$778$	0	0
$779$	3.68431	0.132004
$780$	0	0
$781$	28.0464	1.00358
$782$	0	0
$783$	4.06808	0.145381
$784$	0	0
$785$	20.5470	0.733353
$786$	0	0
$787$	10.5470	0.375959	0.187980	−	0.982173i	$-0.439806\pi$
$787$	10.5470	0.375959	0.187980	+	0.982173i	$0.439806\pi$
$788$	0	0
$789$	−22.0681	−0.785645
$790$	0	0
$791$	27.6843	0.984341
$792$	0	0
$793$	11.4994	0.408356
$794$	0	0
$795$	−11.4994	−0.407842
$796$	0	0
$797$	40.7729	1.44425	0.722125	−	0.691762i	$-0.243165\pi$
$797$	40.7729	1.44425	0.722125	+	0.691762i	$0.243165\pi$
$798$	0	0
$799$	−41.3556	−1.46305
$800$	0	0
$801$	8.63675	0.305164
$802$	0	0
$803$	42.3675	1.49512
$804$	0	0
$805$	−11.7741	−0.414982
$806$	0	0
$807$	13.2054	0.464852
$808$	0	0
$809$	20.2747	0.712819	0.356409	−	0.934330i	$-0.384001\pi$
$809$	20.2747	0.712819	0.356409	+	0.934330i	$0.384001\pi$
$810$	0	0
$811$	15.0259	0.527630	0.263815	−	0.964573i	$-0.415019\pi$
$811$	15.0259	0.527630	0.263815	+	0.964573i	$0.415019\pi$
$812$	0	0
$813$	14.0951	0.494338
$814$	0	0
$815$	4.22593	0.148028
$816$	0	0
$817$	1.91561	0.0670188
$818$	0	0
$819$	3.43134	0.119901
$820$	0	0
$821$	−2.36209	−0.0824377	−0.0412188	−	0.999150i	$-0.513124\pi$
$821$	−2.36209	−0.0824377	−0.0412188	+	0.999150i	$0.513124\pi$
$822$	0	0
$823$	16.1362	0.562471	0.281236	−	0.959639i	$-0.409256\pi$
$823$	16.1362	0.562471	0.281236	+	0.959639i	$0.409256\pi$
$824$	0	0
$825$	−2.63675	−0.0917997
$826$	0	0
$827$	−24.5880	−0.855009	−0.427505	−	0.904013i	$-0.640607\pi$
$827$	−24.5880	−0.855009	−0.427505	+	0.904013i	$0.640607\pi$
$828$	0	0
$829$	−22.8216	−0.792628	−0.396314	−	0.918115i	$-0.629711\pi$
$829$	−22.8216	−0.792628	−0.396314	+	0.918115i	$0.629711\pi$
$830$	0	0
$831$	30.2723	1.05014
$832$	0	0
$833$	37.4390	1.29719
$834$	0	0
$835$	14.4108	0.498707
$836$	0	0
$837$	9.27349	0.320539
$838$	0	0
$839$	−46.3211	−1.59918	−0.799590	−	0.600546i	$-0.794950\pi$
$839$	−46.3211	−1.59918	−0.799590	+	0.600546i	$0.794950\pi$
$840$	0	0
$841$	−12.4507	−0.429334
$842$	0	0
$843$	−2.13617	−0.0735735
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	13.8886	0.477217
$848$	0	0
$849$	0.136167	0.00467325
$850$	0	0
$851$	−2.18489	−0.0748972
$852$	0	0
$853$	47.1837	1.61554	0.807770	−	0.589498i	$-0.200674\pi$
$853$	47.1837	1.61554	0.807770	+	0.589498i	$0.200674\pi$
$854$	0	0
$855$	−0.794590	−0.0271744
$856$	0	0
$857$	−11.8422	−0.404520	−0.202260	−	0.979332i	$-0.564829\pi$
$857$	−11.8422	−0.404520	−0.202260	+	0.979332i	$0.564829\pi$
$858$	0	0
$859$	−10.3211	−0.352150	−0.176075	−	0.984377i	$-0.556340\pi$
$859$	−10.3211	−0.352150	−0.176075	+	0.984377i	$0.556340\pi$
$860$	0	0
$861$	−15.9102	−0.542220
$862$	0	0
$863$	−1.13733	−0.0387150	−0.0193575	−	0.999813i	$-0.506162\pi$
$863$	−1.13733	−0.0387150	−0.0193575	+	0.999813i	$0.506162\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	−44.4994	−1.51128
$868$	0	0
$869$	−28.0464	−0.951409
$870$	0	0
$871$	−6.41082	−0.217222
$872$	0	0
$873$	12.2940	0.416089
$874$	0	0
$875$	−3.43134	−0.116000
$876$	0	0
$877$	−54.5447	−1.84184	−0.920921	−	0.389749i	$-0.872562\pi$
$877$	−54.5447	−1.84184	−0.920921	+	0.389749i	$0.872562\pi$
$878$	0	0
$879$	15.2735	0.515162
$880$	0	0
$881$	48.9555	1.64935	0.824676	−	0.565605i	$-0.191357\pi$
$881$	48.9555	1.64935	0.824676	+	0.565605i	$0.191357\pi$
$882$	0	0
$883$	−27.3686	−0.921028	−0.460514	−	0.887653i	$-0.652335\pi$
$883$	−27.3686	−0.921028	−0.460514	+	0.887653i	$0.652335\pi$
$884$	0	0
$885$	−12.1362	−0.407953
$886$	0	0
$887$	21.5265	0.722788	0.361394	−	0.932413i	$-0.382301\pi$
$887$	21.5265	0.722788	0.361394	+	0.932413i	$0.382301\pi$
$888$	0	0
$889$	−53.8182	−1.80500
$890$	0	0
$891$	2.63675	0.0883343
$892$	0	0
$893$	4.19027	0.140222
$894$	0	0
$895$	20.9308	0.699638
$896$	0	0
$897$	3.43134	0.114569
$898$	0	0
$899$	37.7253	1.25821
$900$	0	0
$901$	90.1803	3.00434
$902$	0	0
$903$	−8.27233	−0.275286
$904$	0	0
$905$	−5.08860	−0.169151
$906$	0	0
$907$	39.6843	1.31770	0.658848	−	0.752276i	$-0.271044\pi$
$907$	39.6843	1.31770	0.658848	+	0.752276i	$0.271044\pi$
$908$	0	0
$909$	5.34158	0.177169
$910$	0	0
$911$	4.13617	0.137037	0.0685187	−	0.997650i	$-0.478173\pi$
$911$	4.13617	0.137037	0.0685187	+	0.997650i	$0.478173\pi$
$912$	0	0
$913$	42.5470	1.40810
$914$	0	0
$915$	11.4994	0.380159
$916$	0	0
$917$	44.3698	1.46522
$918$	0	0
$919$	−17.6789	−0.583174	−0.291587	−	0.956544i	$-0.594183\pi$
$919$	−17.6789	−0.583174	−0.291587	+	0.956544i	$0.594183\pi$
$920$	0	0
$921$	7.91024	0.260651
$922$	0	0
$923$	−10.6367	−0.350113
$924$	0	0
$925$	−0.636747	−0.0209361
$926$	0	0
$927$	10.4108	0.341936
$928$	0	0
$929$	27.8151	0.912584	0.456292	−	0.889830i	$-0.349177\pi$
$929$	27.8151	0.912584	0.456292	+	0.889830i	$0.349177\pi$
$930$	0	0
$931$	−3.79343	−0.124325
$932$	0	0
$933$	2.54699	0.0833846
$934$	0	0
$935$	20.6778	0.676236
$936$	0	0
$937$	−54.5447	−1.78190	−0.890948	−	0.454105i	$-0.849959\pi$
$937$	−54.5447	−1.78190	−0.890948	+	0.454105i	$0.849959\pi$
$938$	0	0
$939$	−19.5892	−0.639269
$940$	0	0
$941$	−12.1849	−0.397216	−0.198608	−	0.980079i	$-0.563642\pi$
$941$	−12.1849	−0.397216	−0.198608	+	0.980079i	$0.563642\pi$
$942$	0	0
$943$	−15.9102	−0.518109
$944$	0	0
$945$	3.43134	0.111621
$946$	0	0
$947$	−39.5048	−1.28373	−0.641867	−	0.766816i	$-0.721840\pi$
$947$	−39.5048	−1.28373	−0.641867	+	0.766816i	$0.721840\pi$
$948$	0	0
$949$	−16.0681	−0.521592
$950$	0	0
$951$	23.2735	0.754695
$952$	0	0
$953$	19.0205	0.616135	0.308067	−	0.951365i	$-0.400318\pi$
$953$	19.0205	0.616135	0.308067	+	0.951365i	$0.400318\pi$
$954$	0	0
$955$	13.7253	0.444142
$956$	0	0
$957$	10.7265	0.346739
$958$	0	0
$959$	−6.86267	−0.221607
$960$	0	0
$961$	54.9977	1.77412
$962$	0	0
$963$	−5.04757	−0.162656
$964$	0	0
$965$	−5.11565	−0.164679
$966$	0	0
$967$	38.3404	1.23294	0.616472	−	0.787377i	$-0.288561\pi$
$967$	38.3404	1.23294	0.616472	+	0.787377i	$0.288561\pi$
$968$	0	0
$969$	6.23130	0.200178
$970$	0	0
$971$	−15.7114	−0.504202	−0.252101	−	0.967701i	$-0.581121\pi$
$971$	−15.7114	−0.504202	−0.252101	+	0.967701i	$0.581121\pi$
$972$	0	0
$973$	68.3187	2.19020
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	53.9566	1.72623	0.863113	−	0.505011i	$-0.168511\pi$
$977$	53.9566	1.72623	0.863113	+	0.505011i	$0.168511\pi$
$978$	0	0
$979$	22.7729	0.727825
$980$	0	0
$981$	6.79459	0.216935
$982$	0	0
$983$	45.0399	1.43655	0.718274	−	0.695760i	$-0.244932\pi$
$983$	45.0399	1.43655	0.718274	+	0.695760i	$0.244932\pi$
$984$	0	0
$985$	−20.9988	−0.669079
$986$	0	0
$987$	−18.0951	−0.575974
$988$	0	0
$989$	−8.27233	−0.263045
$990$	0	0
$991$	−7.63791	−0.242626	−0.121313	−	0.992614i	$-0.538710\pi$
$991$	−7.63791	−0.242626	−0.121313	+	0.992614i	$0.538710\pi$
$992$	0	0
$993$	−20.4789	−0.649878
$994$	0	0
$995$	−9.58918	−0.303997
$996$	0	0
$997$	−16.0951	−0.509738	−0.254869	−	0.966976i	$-0.582032\pi$
$997$	−16.0951	−0.509738	−0.254869	+	0.966976i	$0.582032\pi$
$998$	0	0
$999$	0.636747	0.0201458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.a.bi.1.1		3
3.2	odd	2		9360.2.a.cy.1.1		3
4.3	odd	2		1560.2.a.q.1.3	✓	3
12.11	even	2		4680.2.a.bh.1.3		3
20.19	odd	2		7800.2.a.bi.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.q.1.3	✓	3	4.3	odd	2
3120.2.a.bi.1.1		3	1.1	even	1	trivial
4680.2.a.bh.1.3		3	12.11	even	2
7800.2.a.bi.1.1		3	20.19	odd	2
9360.2.a.cy.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.43134 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.43134 q^{7} +1.00000 q^{9} +2.63675 q^{11} -1.00000 q^{13} -1.00000 q^{15} +7.84216 q^{17} -0.794590 q^{19} +3.43134 q^{21} +3.43134 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.06808 q^{29} -9.27349 q^{31} -2.63675 q^{33} -3.43134 q^{35} -0.636747 q^{37} +1.00000 q^{39} -4.63675 q^{41} -2.41082 q^{43} +1.00000 q^{45} -5.27349 q^{47} +4.77407 q^{49} -7.84216 q^{51} +11.4994 q^{53} +2.63675 q^{55} +0.794590 q^{57} +12.1362 q^{59} -11.4994 q^{61} -3.43134 q^{63} -1.00000 q^{65} +6.41082 q^{67} -3.43134 q^{69} +10.6367 q^{71} +16.0681 q^{73} -1.00000 q^{75} -9.04757 q^{77} -10.6367 q^{79} +1.00000 q^{81} +16.1362 q^{83} +7.84216 q^{85} +4.06808 q^{87} +8.63675 q^{89} +3.43134 q^{91} +9.27349 q^{93} -0.794590 q^{95} +12.2940 q^{97} +2.63675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} - 5 q^{11} - 3 q^{13} - 3 q^{15} + 7 q^{17} - 6 q^{19} + q^{21} + q^{23} + 3 q^{25} - 3 q^{27} + 10 q^{29} - 2 q^{31} + 5 q^{33} - q^{35} + 11 q^{37} + 3 q^{39} - q^{41} + 3 q^{45} + 10 q^{47} + 20 q^{49} - 7 q^{51} + 3 q^{53} - 5 q^{55} + 6 q^{57} - 8 q^{59} - 3 q^{61} - q^{63} - 3 q^{65} + 12 q^{67} - q^{69} + 19 q^{71} + 26 q^{73} - 3 q^{75} - 7 q^{77} - 19 q^{79} + 3 q^{81} + 4 q^{83} + 7 q^{85} - 10 q^{87} + 13 q^{89} + q^{91} + 2 q^{93} - 6 q^{95} + 9 q^{97} - 5 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 - 5 * q^11 - 3 * q^13 - 3 * q^15 + 7 * q^17 - 6 * q^19 + q^21 + q^23 + 3 * q^25 - 3 * q^27 + 10 * q^29 - 2 * q^31 + 5 * q^33 - q^35 + 11 * q^37 + 3 * q^39 - q^41 + 3 * q^45 + 10 * q^47 + 20 * q^49 - 7 * q^51 + 3 * q^53 - 5 * q^55 + 6 * q^57 - 8 * q^59 - 3 * q^61 - q^63 - 3 * q^65 + 12 * q^67 - q^69 + 19 * q^71 + 26 * q^73 - 3 * q^75 - 7 * q^77 - 19 * q^79 + 3 * q^81 + 4 * q^83 + 7 * q^85 - 10 * q^87 + 13 * q^89 + q^91 + 2 * q^93 - 6 * q^95 + 9 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more