LMFDB - Embedded newform 7220.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7220,2,Mod(1,7220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7220, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7220 = 2^{2} \cdot 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7220.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:	$57.6519902594$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.133593.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 3x + 2 $ x^4 - x^3 - 8x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.709218$ of defining polynomial
Character	$\chi$	$=$	7220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.709218 q^{3} +1.00000 q^{5} +3.11079 q^{7} -2.49701 q^{9} +O(q^{10})$	$q-0.709218 q^{3} +1.00000 q^{5} +3.11079 q^{7} -2.49701 q^{9} -3.52922 q^{11} -0.401570 q^{13} -0.709218 q^{15} +3.49701 q^{17} -2.20623 q^{21} -7.31702 q^{23} +1.00000 q^{25} +3.89858 q^{27} +7.93079 q^{29} -5.73545 q^{31} +2.50299 q^{33} +3.11079 q^{35} +10.5292 q^{37} +0.284801 q^{39} +1.11079 q^{41} -8.60780 q^{43} -2.49701 q^{45} +7.52324 q^{47} +2.67700 q^{49} -2.48014 q^{51} -10.5555 q^{53} -3.52922 q^{55} +8.51236 q^{59} -9.22158 q^{61} -7.76767 q^{63} -0.401570 q^{65} -8.41246 q^{67} +5.18936 q^{69} +8.62466 q^{71} +1.74143 q^{73} -0.709218 q^{75} -10.9787 q^{77} +9.01089 q^{79} +4.72608 q^{81} +4.07857 q^{83} +3.49701 q^{85} -5.62466 q^{87} +13.2325 q^{89} -1.24920 q^{91} +4.06769 q^{93} +7.70324 q^{97} +8.81251 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} + 5 q^{9}+O(q^{10}) $ 4 * q - q^3 + 4 * q^5 + 5 * q^9	$ 4 q - q^{3} + 4 q^{5} + 5 q^{9} + 2 q^{11} + 9 q^{13} - q^{15} - q^{17} + 8 q^{21} + 4 q^{25} - 10 q^{27} + 5 q^{29} + 10 q^{31} + 25 q^{33} + 26 q^{37} - 27 q^{39} - 8 q^{41} - 7 q^{43} + 5 q^{45} - 16 q^{47} + 10 q^{49} + 12 q^{51} + 5 q^{53} + 2 q^{55} + 11 q^{59} - 12 q^{61} + 3 q^{63} + 9 q^{65} - 3 q^{69} + 14 q^{71} + 4 q^{73} - q^{75} - 22 q^{77} + 13 q^{79} + 24 q^{81} + 5 q^{83} - q^{85} - 2 q^{87} + 5 q^{89} - 46 q^{91} + 28 q^{93} - q^{97} - 24 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 + 5 * q^9 + 2 * q^11 + 9 * q^13 - q^15 - q^17 + 8 * q^21 + 4 * q^25 - 10 * q^27 + 5 * q^29 + 10 * q^31 + 25 * q^33 + 26 * q^37 - 27 * q^39 - 8 * q^41 - 7 * q^43 + 5 * q^45 - 16 * q^47 + 10 * q^49 + 12 * q^51 + 5 * q^53 + 2 * q^55 + 11 * q^59 - 12 * q^61 + 3 * q^63 + 9 * q^65 - 3 * q^69 + 14 * q^71 + 4 * q^73 - q^75 - 22 * q^77 + 13 * q^79 + 24 * q^81 + 5 * q^83 - q^85 - 2 * q^87 + 5 * q^89 - 46 * q^91 + 28 * q^93 - q^97 - 24 * 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
$3$	−0.709218	−0.409467	−0.204734	−	0.978818i	$-0.565633\pi$
$3$	−0.709218	−0.409467	−0.204734	+	0.978818i	$0.565633\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.11079	1.17577	0.587884	−	0.808945i	$-0.299961\pi$
$7$	3.11079	1.17577	0.587884	+	0.808945i	$0.299961\pi$
$8$	0	0
$9$	−2.49701	−0.832336
$10$	0	0
$11$	−3.52922	−1.06410	−0.532051	−	0.846713i	$-0.678578\pi$
$11$	−3.52922	−1.06410	−0.532051	+	0.846713i	$0.678578\pi$
$12$	0	0
$13$	−0.401570	−0.111375	−0.0556877	−	0.998448i	$-0.517735\pi$
$13$	−0.401570	−0.111375	−0.0556877	+	0.998448i	$0.517735\pi$
$14$	0	0
$15$	−0.709218	−0.183119
$16$	0	0
$17$	3.49701	0.848149	0.424075	−	0.905627i	$-0.360599\pi$
$17$	3.49701	0.848149	0.424075	+	0.905627i	$0.360599\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−2.20623	−0.481438
$22$	0	0
$23$	−7.31702	−1.52570	−0.762852	−	0.646574i	$-0.776201\pi$
$23$	−7.31702	−1.52570	−0.762852	+	0.646574i	$0.776201\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.89858	0.750282
$28$	0	0
$29$	7.93079	1.47271	0.736356	−	0.676595i	$-0.236545\pi$
$29$	7.93079	1.47271	0.736356	+	0.676595i	$0.236545\pi$
$30$	0	0
$31$	−5.73545	−1.03012	−0.515059	−	0.857155i	$-0.672230\pi$
$31$	−5.73545	−1.03012	−0.515059	+	0.857155i	$0.672230\pi$
$32$	0	0
$33$	2.50299	0.435715
$34$	0	0
$35$	3.11079	0.525819
$36$	0	0
$37$	10.5292	1.73099	0.865497	−	0.500914i	$-0.167003\pi$
$37$	10.5292	1.73099	0.865497	+	0.500914i	$0.167003\pi$
$38$	0	0
$39$	0.284801	0.0456046
$40$	0	0
$41$	1.11079	0.173476	0.0867380	−	0.996231i	$-0.472356\pi$
$41$	1.11079	0.173476	0.0867380	+	0.996231i	$0.472356\pi$
$42$	0	0
$43$	−8.60780	−1.31268	−0.656338	−	0.754467i	$-0.727896\pi$
$43$	−8.60780	−1.31268	−0.656338	+	0.754467i	$0.727896\pi$
$44$	0	0
$45$	−2.49701	−0.372232
$46$	0	0
$47$	7.52324	1.09738	0.548689	−	0.836027i	$-0.315127\pi$
$47$	7.52324	1.09738	0.548689	+	0.836027i	$0.315127\pi$
$48$	0	0
$49$	2.67700	0.382429
$50$	0	0
$51$	−2.48014	−0.347289
$52$	0	0
$53$	−10.5555	−1.44990	−0.724952	−	0.688800i	$-0.758138\pi$
$53$	−10.5555	−1.44990	−0.724952	+	0.688800i	$0.758138\pi$
$54$	0	0
$55$	−3.52922	−0.475881
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.51236	1.10821	0.554107	−	0.832445i	$-0.313060\pi$
$59$	8.51236	1.10821	0.554107	+	0.832445i	$0.313060\pi$
$60$	0	0
$61$	−9.22158	−1.18070	−0.590351	−	0.807147i	$-0.701011\pi$
$61$	−9.22158	−1.18070	−0.590351	+	0.807147i	$0.701011\pi$
$62$	0	0
$63$	−7.76767	−0.978634
$64$	0	0
$65$	−0.401570	−0.0498086
$66$	0	0
$67$	−8.41246	−1.02775	−0.513873	−	0.857867i	$-0.671790\pi$
$67$	−8.41246	−1.02775	−0.513873	+	0.857867i	$0.671790\pi$
$68$	0	0
$69$	5.18936	0.624726
$70$	0	0
$71$	8.62466	1.02356	0.511780	−	0.859117i	$-0.328986\pi$
$71$	8.62466	1.02356	0.511780	+	0.859117i	$0.328986\pi$
$72$	0	0
$73$	1.74143	0.203819	0.101910	−	0.994794i	$-0.467505\pi$
$73$	1.74143	0.203819	0.101910	+	0.994794i	$0.467505\pi$
$74$	0	0
$75$	−0.709218	−0.0818935
$76$	0	0
$77$	−10.9787	−1.25114
$78$	0	0
$79$	9.01089	1.01380	0.506902	−	0.862004i	$-0.330791\pi$
$79$	9.01089	1.01380	0.506902	+	0.862004i	$0.330791\pi$
$80$	0	0
$81$	4.72608	0.525121
$82$	0	0
$83$	4.07857	0.447682	0.223841	−	0.974626i	$-0.428140\pi$
$83$	4.07857	0.447682	0.223841	+	0.974626i	$0.428140\pi$
$84$	0	0
$85$	3.49701	0.379304
$86$	0	0
$87$	−5.62466	−0.603027
$88$	0	0
$89$	13.2325	1.40264	0.701319	−	0.712847i	$-0.252595\pi$
$89$	13.2325	1.40264	0.701319	+	0.712847i	$0.252595\pi$
$90$	0	0
$91$	−1.24920	−0.130952
$92$	0	0
$93$	4.06769	0.421800
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.70324	0.782145	0.391073	−	0.920360i	$-0.372104\pi$
$97$	7.70324	0.782145	0.391073	+	0.920360i	$0.372104\pi$
$98$	0	0
$99$	8.81251	0.885690
$100$	0	0
$101$	17.3755	1.72892	0.864462	−	0.502699i	$-0.167659\pi$
$101$	17.3755	1.72892	0.864462	+	0.502699i	$0.167659\pi$
$102$	0	0
$103$	7.12614	0.702159	0.351080	−	0.936346i	$-0.385815\pi$
$103$	7.12614	0.702159	0.351080	+	0.936346i	$0.385815\pi$
$104$	0	0
$105$	−2.20623	−0.215306
$106$	0	0
$107$	−0.735452	−0.0710989	−0.0355494	−	0.999368i	$-0.511318\pi$
$107$	−0.735452	−0.0710989	−0.0355494	+	0.999368i	$0.511318\pi$
$108$	0	0
$109$	16.0094	1.53342	0.766710	−	0.641994i	$-0.221893\pi$
$109$	16.0094	1.53342	0.766710	+	0.641994i	$0.221893\pi$
$110$	0	0
$111$	−7.46752	−0.708785
$112$	0	0
$113$	−4.34923	−0.409141	−0.204571	−	0.978852i	$-0.565580\pi$
$113$	−4.34923	−0.409141	−0.204571	+	0.978852i	$0.565580\pi$
$114$	0	0
$115$	−7.31702	−0.682315
$116$	0	0
$117$	1.00272	0.0927019
$118$	0	0
$119$	10.8785	0.997226
$120$	0	0
$121$	1.45543	0.132312
$122$	0	0
$123$	−0.787791	−0.0710327
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.64599	0.412265	0.206133	−	0.978524i	$-0.433912\pi$
$127$	4.64599	0.412265	0.206133	+	0.978524i	$0.433912\pi$
$128$	0	0
$129$	6.10481	0.537498
$130$	0	0
$131$	10.4955	0.916995	0.458498	−	0.888696i	$-0.348388\pi$
$131$	10.4955	0.916995	0.458498	+	0.888696i	$0.348388\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.89858	0.335536
$136$	0	0
$137$	−1.39220	−0.118944	−0.0594719	−	0.998230i	$-0.518942\pi$
$137$	−1.39220	−0.118944	−0.0594719	+	0.998230i	$0.518942\pi$
$138$	0	0
$139$	−6.07532	−0.515302	−0.257651	−	0.966238i	$-0.582948\pi$
$139$	−6.07532	−0.515302	−0.257651	+	0.966238i	$0.582948\pi$
$140$	0	0
$141$	−5.33562	−0.449340
$142$	0	0
$143$	1.41723	0.118515
$144$	0	0
$145$	7.93079	0.658617
$146$	0	0
$147$	−1.89858	−0.156592
$148$	0	0
$149$	−0.0845540	−0.00692694	−0.00346347	−	0.999994i	$-0.501102\pi$
$149$	−0.0845540	−0.00692694	−0.00346347	+	0.999994i	$0.501102\pi$
$150$	0	0
$151$	15.2216	1.23871	0.619357	−	0.785109i	$-0.287393\pi$
$151$	15.2216	1.23871	0.619357	+	0.785109i	$0.287393\pi$
$152$	0	0
$153$	−8.73207	−0.705946
$154$	0	0
$155$	−5.73545	−0.460683
$156$	0	0
$157$	16.2587	1.29759	0.648793	−	0.760965i	$-0.275274\pi$
$157$	16.2587	1.29759	0.648793	+	0.760965i	$0.275274\pi$
$158$	0	0
$159$	7.48612	0.593688
$160$	0	0
$161$	−22.7617	−1.79387
$162$	0	0
$163$	14.9461	1.17067	0.585336	−	0.810791i	$-0.300963\pi$
$163$	14.9461	1.17067	0.585336	+	0.810791i	$0.300963\pi$
$164$	0	0
$165$	2.50299	0.194858
$166$	0	0
$167$	−12.5292	−0.969541	−0.484770	−	0.874642i	$-0.661097\pi$
$167$	−12.5292	−0.969541	−0.484770	+	0.874642i	$0.661097\pi$
$168$	0	0
$169$	−12.8387	−0.987596
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.3848	−1.32174	−0.660872	−	0.750498i	$-0.729813\pi$
$173$	−17.3848	−1.32174	−0.660872	+	0.750498i	$0.729813\pi$
$174$	0	0
$175$	3.11079	0.235153
$176$	0	0
$177$	−6.03712	−0.453778
$178$	0	0
$179$	−20.7830	−1.55340	−0.776698	−	0.629873i	$-0.783107\pi$
$179$	−20.7830	−1.55340	−0.776698	+	0.629873i	$0.783107\pi$
$180$	0	0
$181$	−1.62913	−0.121092	−0.0605460	−	0.998165i	$-0.519284\pi$
$181$	−1.62913	−0.121092	−0.0605460	+	0.998165i	$0.519284\pi$
$182$	0	0
$183$	6.54011	0.483459
$184$	0	0
$185$	10.5292	0.774124
$186$	0	0
$187$	−12.3417	−0.902517
$188$	0	0
$189$	12.1277	0.882157
$190$	0	0
$191$	−13.4016	−0.969704	−0.484852	−	0.874596i	$-0.661126\pi$
$191$	−13.4016	−0.969704	−0.484852	+	0.874596i	$0.661126\pi$
$192$	0	0
$193$	10.6939	0.769762	0.384881	−	0.922966i	$-0.374243\pi$
$193$	10.6939	0.769762	0.384881	+	0.922966i	$0.374243\pi$
$194$	0	0
$195$	0.284801	0.0203950
$196$	0	0
$197$	−7.00611	−0.499165	−0.249582	−	0.968354i	$-0.580293\pi$
$197$	−7.00611	−0.499165	−0.249582	+	0.968354i	$0.580293\pi$
$198$	0	0
$199$	−20.7818	−1.47318	−0.736592	−	0.676338i	$-0.763566\pi$
$199$	−20.7818	−1.47318	−0.736592	+	0.676338i	$0.763566\pi$
$200$	0	0
$201$	5.96627	0.420828
$202$	0	0
$203$	24.6710	1.73157
$204$	0	0
$205$	1.11079	0.0775808
$206$	0	0
$207$	18.2707	1.26990
$208$	0	0
$209$	0	0
$210$	0	0
$211$	11.2325	0.773275	0.386637	−	0.922232i	$-0.373637\pi$
$211$	11.2325	0.773275	0.386637	+	0.922232i	$0.373637\pi$
$212$	0	0
$213$	−6.11677	−0.419114
$214$	0	0
$215$	−8.60780	−0.587047
$216$	0	0
$217$	−17.8418	−1.21118
$218$	0	0
$219$	−1.23506	−0.0834574
$220$	0	0
$221$	−1.40429	−0.0944630
$222$	0	0
$223$	10.9001	0.729925	0.364962	−	0.931022i	$-0.381082\pi$
$223$	10.9001	0.729925	0.364962	+	0.931022i	$0.381082\pi$
$224$	0	0
$225$	−2.49701	−0.166467
$226$	0	0
$227$	−3.96627	−0.263250	−0.131625	−	0.991300i	$-0.542020\pi$
$227$	−3.96627	−0.263250	−0.131625	+	0.991300i	$0.542020\pi$
$228$	0	0
$229$	−10.9139	−0.721213	−0.360606	−	0.932718i	$-0.617430\pi$
$229$	−10.9139	−0.721213	−0.360606	+	0.932718i	$0.617430\pi$
$230$	0	0
$231$	7.78627	0.512299
$232$	0	0
$233$	−15.5909	−1.02140	−0.510698	−	0.859760i	$-0.670613\pi$
$233$	−15.5909	−1.02140	−0.510698	+	0.859760i	$0.670613\pi$
$234$	0	0
$235$	7.52324	0.490762
$236$	0	0
$237$	−6.39068	−0.415120
$238$	0	0
$239$	−8.09544	−0.523650	−0.261825	−	0.965115i	$-0.584324\pi$
$239$	−8.09544	−0.523650	−0.261825	+	0.965115i	$0.584324\pi$
$240$	0	0
$241$	25.2849	1.62875	0.814373	−	0.580342i	$-0.197081\pi$
$241$	25.2849	1.62875	0.814373	+	0.580342i	$0.197081\pi$
$242$	0	0
$243$	−15.0476	−0.965302
$244$	0	0
$245$	2.67700	0.171027
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−2.89260	−0.183311
$250$	0	0
$251$	7.88921	0.497963	0.248981	−	0.968508i	$-0.419904\pi$
$251$	7.88921	0.497963	0.248981	+	0.968508i	$0.419904\pi$
$252$	0	0
$253$	25.8234	1.62350
$254$	0	0
$255$	−2.48014	−0.155313
$256$	0	0
$257$	9.18610	0.573013	0.286507	−	0.958078i	$-0.407506\pi$
$257$	9.18610	0.573013	0.286507	+	0.958078i	$0.407506\pi$
$258$	0	0
$259$	32.7542	2.03525
$260$	0	0
$261$	−19.8033	−1.22579
$262$	0	0
$263$	24.2433	1.49491	0.747454	−	0.664313i	$-0.231276\pi$
$263$	24.2433	1.49491	0.747454	+	0.664313i	$0.231276\pi$
$264$	0	0
$265$	−10.5555	−0.648417
$266$	0	0
$267$	−9.38470	−0.574335
$268$	0	0
$269$	−1.42442	−0.0868483	−0.0434241	−	0.999057i	$-0.513827\pi$
$269$	−1.42442	−0.0868483	−0.0434241	+	0.999057i	$0.513827\pi$
$270$	0	0
$271$	−10.3923	−0.631289	−0.315645	−	0.948877i	$-0.602221\pi$
$271$	−10.3923	−0.631289	−0.315645	+	0.948877i	$0.602221\pi$
$272$	0	0
$273$	0.885955	0.0536204
$274$	0	0
$275$	−3.52922	−0.212820
$276$	0	0
$277$	20.3078	1.22018	0.610088	−	0.792334i	$-0.291134\pi$
$277$	20.3078	1.22018	0.610088	+	0.792334i	$0.291134\pi$
$278$	0	0
$279$	14.3215	0.857405
$280$	0	0
$281$	−15.5909	−0.930077	−0.465038	−	0.885290i	$-0.653960\pi$
$281$	−15.5909	−0.930077	−0.465038	+	0.885290i	$0.653960\pi$
$282$	0	0
$283$	27.2647	1.62072	0.810358	−	0.585935i	$-0.199272\pi$
$283$	27.2647	1.62072	0.810358	+	0.585935i	$0.199272\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.45543	0.203967
$288$	0	0
$289$	−4.77092	−0.280643
$290$	0	0
$291$	−5.46328	−0.320263
$292$	0	0
$293$	12.6710	0.740249	0.370125	−	0.928982i	$-0.379315\pi$
$293$	12.6710	0.740249	0.370125	+	0.928982i	$0.379315\pi$
$294$	0	0
$295$	8.51236	0.495609
$296$	0	0
$297$	−13.7590	−0.798376
$298$	0	0
$299$	2.93829	0.169926
$300$	0	0
$301$	−26.7770	−1.54340
$302$	0	0
$303$	−12.3230	−0.707938
$304$	0	0
$305$	−9.22158	−0.528026
$306$	0	0
$307$	−2.57081	−0.146724	−0.0733619	−	0.997305i	$-0.523373\pi$
$307$	−2.57081	−0.146724	−0.0733619	+	0.997305i	$0.523373\pi$
$308$	0	0
$309$	−5.05399	−0.287511
$310$	0	0
$311$	19.7568	1.12030	0.560152	−	0.828390i	$-0.310743\pi$
$311$	19.7568	1.12030	0.560152	+	0.828390i	$0.310743\pi$
$312$	0	0
$313$	22.3755	1.26474	0.632368	−	0.774668i	$-0.282083\pi$
$313$	22.3755	1.26474	0.632368	+	0.774668i	$0.282083\pi$
$314$	0	0
$315$	−7.76767	−0.437658
$316$	0	0
$317$	−13.3063	−0.747354	−0.373677	−	0.927559i	$-0.621903\pi$
$317$	−13.3063	−0.747354	−0.373677	+	0.927559i	$0.621903\pi$
$318$	0	0
$319$	−27.9896	−1.56711
$320$	0	0
$321$	0.521596	0.0291127
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−0.401570	−0.0222751
$326$	0	0
$327$	−11.3541	−0.627885
$328$	0	0
$329$	23.4032	1.29026
$330$	0	0
$331$	19.6173	1.07826	0.539132	−	0.842221i	$-0.318752\pi$
$331$	19.6173	1.07826	0.539132	+	0.842221i	$0.318752\pi$
$332$	0	0
$333$	−26.2916	−1.44077
$334$	0	0
$335$	−8.41246	−0.459622
$336$	0	0
$337$	29.1819	1.58964	0.794819	−	0.606847i	$-0.207566\pi$
$337$	29.1819	1.58964	0.794819	+	0.606847i	$0.207566\pi$
$338$	0	0
$339$	3.08455	0.167530
$340$	0	0
$341$	20.2417	1.09615
$342$	0	0
$343$	−13.4479	−0.726120
$344$	0	0
$345$	5.18936	0.279386
$346$	0	0
$347$	−12.0787	−0.648419	−0.324209	−	0.945985i	$-0.605098\pi$
$347$	−12.0787	−0.648419	−0.324209	+	0.945985i	$0.605098\pi$
$348$	0	0
$349$	20.2894	1.08607	0.543033	−	0.839711i	$-0.317276\pi$
$349$	20.2894	1.08607	0.543033	+	0.839711i	$0.317276\pi$
$350$	0	0
$351$	−1.56555	−0.0835630
$352$	0	0
$353$	6.69387	0.356279	0.178139	−	0.984005i	$-0.442992\pi$
$353$	6.69387	0.356279	0.178139	+	0.984005i	$0.442992\pi$
$354$	0	0
$355$	8.62466	0.457750
$356$	0	0
$357$	−7.71520	−0.408332
$358$	0	0
$359$	9.89073	0.522013	0.261006	−	0.965337i	$-0.415946\pi$
$359$	9.89073	0.522013	0.261006	+	0.965337i	$0.415946\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−1.03222	−0.0541772
$364$	0	0
$365$	1.74143	0.0911508
$366$	0	0
$367$	16.1987	0.845567	0.422783	−	0.906231i	$-0.361053\pi$
$367$	16.1987	0.845567	0.422783	+	0.906231i	$0.361053\pi$
$368$	0	0
$369$	−2.77365	−0.144390
$370$	0	0
$371$	−32.8358	−1.70475
$372$	0	0
$373$	−10.3003	−0.533328	−0.266664	−	0.963790i	$-0.585921\pi$
$373$	−10.3003	−0.533328	−0.266664	+	0.963790i	$0.585921\pi$
$374$	0	0
$375$	−0.709218	−0.0366239
$376$	0	0
$377$	−3.18477	−0.164024
$378$	0	0
$379$	21.2587	1.09199	0.545993	−	0.837790i	$-0.316153\pi$
$379$	21.2587	1.09199	0.545993	+	0.837790i	$0.316153\pi$
$380$	0	0
$381$	−3.29502	−0.168809
$382$	0	0
$383$	13.5800	0.693908	0.346954	−	0.937882i	$-0.387216\pi$
$383$	13.5800	0.693908	0.346954	+	0.937882i	$0.387216\pi$
$384$	0	0
$385$	−10.9787	−0.559525
$386$	0	0
$387$	21.4938	1.09259
$388$	0	0
$389$	17.9754	0.911390	0.455695	−	0.890136i	$-0.349391\pi$
$389$	17.9754	0.911390	0.455695	+	0.890136i	$0.349391\pi$
$390$	0	0
$391$	−25.5877	−1.29402
$392$	0	0
$393$	−7.44359	−0.375480
$394$	0	0
$395$	9.01089	0.453387
$396$	0	0
$397$	24.0339	1.20622	0.603112	−	0.797656i	$-0.293927\pi$
$397$	24.0339	1.20622	0.603112	+	0.797656i	$0.293927\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.8215	0.590339	0.295169	−	0.955445i	$-0.404624\pi$
$401$	11.8215	0.590339	0.295169	+	0.955445i	$0.404624\pi$
$402$	0	0
$403$	2.30319	0.114730
$404$	0	0
$405$	4.72608	0.234841
$406$	0	0
$407$	−37.1600	−1.84195
$408$	0	0
$409$	31.2599	1.54570	0.772851	−	0.634587i	$-0.218830\pi$
$409$	31.2599	1.54570	0.772851	+	0.634587i	$0.218830\pi$
$410$	0	0
$411$	0.987375	0.0487036
$412$	0	0
$413$	26.4801	1.30300
$414$	0	0
$415$	4.07857	0.200209
$416$	0	0
$417$	4.30872	0.210999
$418$	0	0
$419$	−22.4217	−1.09537	−0.547686	−	0.836684i	$-0.684491\pi$
$419$	−22.4217	−1.09537	−0.547686	+	0.836684i	$0.684491\pi$
$420$	0	0
$421$	−27.2861	−1.32984	−0.664922	−	0.746912i	$-0.731535\pi$
$421$	−27.2861	−1.32984	−0.664922	+	0.746912i	$0.731535\pi$
$422$	0	0
$423$	−18.7856	−0.913388
$424$	0	0
$425$	3.49701	0.169630
$426$	0	0
$427$	−28.6864	−1.38823
$428$	0	0
$429$	−1.00513	−0.0485279
$430$	0	0
$431$	−27.3373	−1.31679	−0.658395	−	0.752673i	$-0.728764\pi$
$431$	−27.3373	−1.31679	−0.658395	+	0.752673i	$0.728764\pi$
$432$	0	0
$433$	7.35521	0.353469	0.176734	−	0.984259i	$-0.443447\pi$
$433$	7.35521	0.353469	0.176734	+	0.984259i	$0.443447\pi$
$434$	0	0
$435$	−5.62466	−0.269682
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−4.23724	−0.202232	−0.101116	−	0.994875i	$-0.532241\pi$
$439$	−4.23724	−0.202232	−0.101116	+	0.994875i	$0.532241\pi$
$440$	0	0
$441$	−6.68450	−0.318310
$442$	0	0
$443$	−22.2620	−1.05770	−0.528849	−	0.848716i	$-0.677376\pi$
$443$	−22.2620	−1.05770	−0.528849	+	0.848716i	$0.677376\pi$
$444$	0	0
$445$	13.2325	0.627279
$446$	0	0
$447$	0.0599673	0.00283635
$448$	0	0
$449$	−4.46328	−0.210635	−0.105318	−	0.994439i	$-0.533586\pi$
$449$	−4.46328	−0.210635	−0.105318	+	0.994439i	$0.533586\pi$
$450$	0	0
$451$	−3.92022	−0.184596
$452$	0	0
$453$	−10.7954	−0.507213
$454$	0	0
$455$	−1.24920	−0.0585634
$456$	0	0
$457$	10.7463	0.502692	0.251346	−	0.967897i	$-0.419127\pi$
$457$	10.7463	0.502692	0.251346	+	0.967897i	$0.419127\pi$
$458$	0	0
$459$	13.6334	0.636351
$460$	0	0
$461$	23.8326	1.11000	0.554998	−	0.831852i	$-0.312719\pi$
$461$	23.8326	1.11000	0.554998	+	0.831852i	$0.312719\pi$
$462$	0	0
$463$	−25.3695	−1.17902	−0.589510	−	0.807761i	$-0.700679\pi$
$463$	−25.3695	−1.17902	−0.589510	+	0.807761i	$0.700679\pi$
$464$	0	0
$465$	4.06769	0.188634
$466$	0	0
$467$	5.62020	0.260072	0.130036	−	0.991509i	$-0.458491\pi$
$467$	5.62020	0.260072	0.130036	+	0.991509i	$0.458491\pi$
$468$	0	0
$469$	−26.1694	−1.20839
$470$	0	0
$471$	−11.5310	−0.531319
$472$	0	0
$473$	30.3789	1.39682
$474$	0	0
$475$	0	0
$476$	0	0
$477$	26.3571	1.20681
$478$	0	0
$479$	13.6722	0.624700	0.312350	−	0.949967i	$-0.398884\pi$
$479$	13.6722	0.624700	0.312350	+	0.949967i	$0.398884\pi$
$480$	0	0
$481$	−4.22822	−0.192790
$482$	0	0
$483$	16.1430	0.734532
$484$	0	0
$485$	7.70324	0.349786
$486$	0	0
$487$	7.39233	0.334979	0.167489	−	0.985874i	$-0.446434\pi$
$487$	7.39233	0.334979	0.167489	+	0.985874i	$0.446434\pi$
$488$	0	0
$489$	−10.6001	−0.479352
$490$	0	0
$491$	1.95811	0.0883681	0.0441840	−	0.999023i	$-0.485931\pi$
$491$	1.95811	0.0883681	0.0441840	+	0.999023i	$0.485931\pi$
$492$	0	0
$493$	27.7341	1.24908
$494$	0	0
$495$	8.81251	0.396093
$496$	0	0
$497$	26.8295	1.20347
$498$	0	0
$499$	5.44467	0.243737	0.121868	−	0.992546i	$-0.461111\pi$
$499$	5.44467	0.243737	0.121868	+	0.992546i	$0.461111\pi$
$500$	0	0
$501$	8.88595	0.396995
$502$	0	0
$503$	−16.5926	−0.739827	−0.369913	−	0.929066i	$-0.620613\pi$
$503$	−16.5926	−0.739827	−0.369913	+	0.929066i	$0.620613\pi$
$504$	0	0
$505$	17.3755	0.773198
$506$	0	0
$507$	9.10547	0.404388
$508$	0	0
$509$	−12.1310	−0.537699	−0.268849	−	0.963182i	$-0.586643\pi$
$509$	−12.1310	−0.537699	−0.268849	+	0.963182i	$0.586643\pi$
$510$	0	0
$511$	5.41723	0.239644
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.12614	0.314015
$516$	0	0
$517$	−26.5512	−1.16772
$518$	0	0
$519$	12.3296	0.541211
$520$	0	0
$521$	−14.1249	−0.618824	−0.309412	−	0.950928i	$-0.600132\pi$
$521$	−14.1249	−0.618824	−0.309412	+	0.950928i	$0.600132\pi$
$522$	0	0
$523$	26.3680	1.15299	0.576495	−	0.817100i	$-0.304420\pi$
$523$	26.3680	1.15299	0.576495	+	0.817100i	$0.304420\pi$
$524$	0	0
$525$	−2.20623	−0.0962877
$526$	0	0
$527$	−20.0569	−0.873694
$528$	0	0
$529$	30.5387	1.32777
$530$	0	0
$531$	−21.2554	−0.922407
$532$	0	0
$533$	−0.446059	−0.0193210
$534$	0	0
$535$	−0.735452	−0.0317964
$536$	0	0
$537$	14.7397	0.636065
$538$	0	0
$539$	−9.44775	−0.406943
$540$	0	0
$541$	−29.3971	−1.26388	−0.631940	−	0.775017i	$-0.717741\pi$
$541$	−29.3971	−1.26388	−0.631940	+	0.775017i	$0.717741\pi$
$542$	0	0
$543$	1.15541	0.0495833
$544$	0	0
$545$	16.0094	0.685766
$546$	0	0
$547$	−21.7523	−0.930062	−0.465031	−	0.885294i	$-0.653957\pi$
$547$	−21.7523	−0.930062	−0.465031	+	0.885294i	$0.653957\pi$
$548$	0	0
$549$	23.0264	0.982741
$550$	0	0
$551$	0	0
$552$	0	0
$553$	28.0310	1.19200
$554$	0	0
$555$	−7.46752	−0.316978
$556$	0	0
$557$	−24.1431	−1.02298	−0.511489	−	0.859290i	$-0.670906\pi$
$557$	−24.1431	−1.02298	−0.511489	+	0.859290i	$0.670906\pi$
$558$	0	0
$559$	3.45663	0.146200
$560$	0	0
$561$	8.75298	0.369551
$562$	0	0
$563$	15.0693	0.635097	0.317548	−	0.948242i	$-0.397140\pi$
$563$	15.0693	0.635097	0.317548	+	0.948242i	$0.397140\pi$
$564$	0	0
$565$	−4.34923	−0.182974
$566$	0	0
$567$	14.7018	0.617420
$568$	0	0
$569$	−0.302873	−0.0126971	−0.00634855	−	0.999980i	$-0.502021\pi$
$569$	−0.302873	−0.0126971	−0.00634855	+	0.999980i	$0.502021\pi$
$570$	0	0
$571$	30.3107	1.26846	0.634232	−	0.773143i	$-0.281316\pi$
$571$	30.3107	1.26846	0.634232	+	0.773143i	$0.281316\pi$
$572$	0	0
$573$	9.50464	0.397062
$574$	0	0
$575$	−7.31702	−0.305141
$576$	0	0
$577$	18.6601	0.776832	0.388416	−	0.921484i	$-0.373022\pi$
$577$	18.6601	0.776832	0.388416	+	0.921484i	$0.373022\pi$
$578$	0	0
$579$	−7.58429	−0.315192
$580$	0	0
$581$	12.6876	0.526369
$582$	0	0
$583$	37.2526	1.54284
$584$	0	0
$585$	1.00272	0.0414575
$586$	0	0
$587$	−27.1972	−1.12255	−0.561275	−	0.827630i	$-0.689689\pi$
$587$	−27.1972	−1.12255	−0.561275	+	0.827630i	$0.689689\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	4.96886	0.204392
$592$	0	0
$593$	−29.5974	−1.21542	−0.607709	−	0.794160i	$-0.707911\pi$
$593$	−29.5974	−1.21542	−0.607709	+	0.794160i	$0.707911\pi$
$594$	0	0
$595$	10.8785	0.445973
$596$	0	0
$597$	14.7388	0.603221
$598$	0	0
$599$	−15.0881	−0.616482	−0.308241	−	0.951308i	$-0.599740\pi$
$599$	−15.0881	−0.616482	−0.308241	+	0.951308i	$0.599740\pi$
$600$	0	0
$601$	44.9249	1.83253	0.916263	−	0.400576i	$-0.131190\pi$
$601$	44.9249	1.83253	0.916263	+	0.400576i	$0.131190\pi$
$602$	0	0
$603$	21.0060	0.855430
$604$	0	0
$605$	1.45543	0.0591715
$606$	0	0
$607$	24.9570	1.01297	0.506487	−	0.862247i	$-0.330944\pi$
$607$	24.9570	1.01297	0.506487	+	0.862247i	$0.330944\pi$
$608$	0	0
$609$	−17.4971	−0.709020
$610$	0	0
$611$	−3.02111	−0.122221
$612$	0	0
$613$	−19.2863	−0.778967	−0.389484	−	0.921033i	$-0.627346\pi$
$613$	−19.2863	−0.778967	−0.389484	+	0.921033i	$0.627346\pi$
$614$	0	0
$615$	−0.787791	−0.0317668
$616$	0	0
$617$	19.4459	0.782862	0.391431	−	0.920208i	$-0.371980\pi$
$617$	19.4459	0.782862	0.391431	+	0.920208i	$0.371980\pi$
$618$	0	0
$619$	−1.70324	−0.0684589	−0.0342294	−	0.999414i	$-0.510898\pi$
$619$	−1.70324	−0.0684589	−0.0342294	+	0.999414i	$0.510898\pi$
$620$	0	0
$621$	−28.5260	−1.14471
$622$	0	0
$623$	41.1634	1.64918
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	36.8208	1.46814
$630$	0	0
$631$	−8.78627	−0.349776	−0.174888	−	0.984588i	$-0.555956\pi$
$631$	−8.78627	−0.349776	−0.174888	+	0.984588i	$0.555956\pi$
$632$	0	0
$633$	−7.96627	−0.316631
$634$	0	0
$635$	4.64599	0.184371
$636$	0	0
$637$	−1.07500	−0.0425932
$638$	0	0
$639$	−21.5359	−0.851946
$640$	0	0
$641$	35.8067	1.41428	0.707139	−	0.707075i	$-0.249986\pi$
$641$	35.8067	1.41428	0.707139	+	0.707075i	$0.249986\pi$
$642$	0	0
$643$	2.61517	0.103132	0.0515661	−	0.998670i	$-0.483579\pi$
$643$	2.61517	0.103132	0.0515661	+	0.998670i	$0.483579\pi$
$644$	0	0
$645$	6.10481	0.240377
$646$	0	0
$647$	30.8849	1.21421	0.607105	−	0.794622i	$-0.292331\pi$
$647$	30.8849	1.21421	0.607105	+	0.794622i	$0.292331\pi$
$648$	0	0
$649$	−30.0420	−1.17925
$650$	0	0
$651$	12.6537	0.495938
$652$	0	0
$653$	−17.2198	−0.673864	−0.336932	−	0.941529i	$-0.609389\pi$
$653$	−17.2198	−0.673864	−0.336932	+	0.941529i	$0.609389\pi$
$654$	0	0
$655$	10.4955	0.410093
$656$	0	0
$657$	−4.34838	−0.169646
$658$	0	0
$659$	0.153889	0.00599466	0.00299733	−	0.999996i	$-0.499046\pi$
$659$	0.153889	0.00599466	0.00299733	+	0.999996i	$0.499046\pi$
$660$	0	0
$661$	27.7190	1.07815	0.539073	−	0.842259i	$-0.318775\pi$
$661$	27.7190	1.07815	0.539073	+	0.842259i	$0.318775\pi$
$662$	0	0
$663$	0.995951	0.0386795
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−58.0298	−2.24692
$668$	0	0
$669$	−7.73055	−0.298880
$670$	0	0
$671$	32.5450	1.25639
$672$	0	0
$673$	48.0820	1.85342	0.926712	−	0.375773i	$-0.122623\pi$
$673$	48.0820	1.85342	0.926712	+	0.375773i	$0.122623\pi$
$674$	0	0
$675$	3.89858	0.150056
$676$	0	0
$677$	42.8392	1.64644	0.823222	−	0.567720i	$-0.192174\pi$
$677$	42.8392	1.64644	0.823222	+	0.567720i	$0.192174\pi$
$678$	0	0
$679$	23.9631	0.919621
$680$	0	0
$681$	2.81295	0.107792
$682$	0	0
$683$	−41.4022	−1.58421	−0.792106	−	0.610383i	$-0.791015\pi$
$683$	−41.4022	−1.58421	−0.792106	+	0.610383i	$0.791015\pi$
$684$	0	0
$685$	−1.39220	−0.0531933
$686$	0	0
$687$	7.74036	0.295313
$688$	0	0
$689$	4.23875	0.161484
$690$	0	0
$691$	16.9248	0.643850	0.321925	−	0.946765i	$-0.395670\pi$
$691$	16.9248	0.643850	0.321925	+	0.946765i	$0.395670\pi$
$692$	0	0
$693$	27.4138	1.04137
$694$	0	0
$695$	−6.07532	−0.230450
$696$	0	0
$697$	3.88444	0.147134
$698$	0	0
$699$	11.0574	0.418228
$700$	0	0
$701$	−37.5559	−1.41847	−0.709233	−	0.704974i	$-0.750959\pi$
$701$	−37.5559	−1.41847	−0.709233	+	0.704974i	$0.750959\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−5.33562	−0.200951
$706$	0	0
$707$	54.0514	2.03281
$708$	0	0
$709$	−30.7744	−1.15576	−0.577879	−	0.816122i	$-0.696120\pi$
$709$	−30.7744	−1.15576	−0.577879	+	0.816122i	$0.696120\pi$
$710$	0	0
$711$	−22.5003	−0.843826
$712$	0	0
$713$	41.9664	1.57165
$714$	0	0
$715$	1.41723	0.0530014
$716$	0	0
$717$	5.74143	0.214418
$718$	0	0
$719$	−46.3018	−1.72677	−0.863383	−	0.504549i	$-0.831659\pi$
$719$	−46.3018	−1.72677	−0.863383	+	0.504549i	$0.831659\pi$
$720$	0	0
$721$	22.1679	0.825576
$722$	0	0
$723$	−17.9325	−0.666918
$724$	0	0
$725$	7.93079	0.294542
$726$	0	0
$727$	−5.86909	−0.217672	−0.108836	−	0.994060i	$-0.534712\pi$
$727$	−5.86909	−0.217672	−0.108836	+	0.994060i	$0.534712\pi$
$728$	0	0
$729$	−3.50625	−0.129861
$730$	0	0
$731$	−30.1016	−1.11335
$732$	0	0
$733$	11.3955	0.420901	0.210450	−	0.977605i	$-0.432507\pi$
$733$	11.3955	0.420901	0.210450	+	0.977605i	$0.432507\pi$
$734$	0	0
$735$	−1.89858	−0.0700302
$736$	0	0
$737$	29.6894	1.09362
$738$	0	0
$739$	13.7324	0.505155	0.252578	−	0.967577i	$-0.418722\pi$
$739$	13.7324	0.505155	0.252578	+	0.967577i	$0.418722\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.67526	0.318265	0.159132	−	0.987257i	$-0.449130\pi$
$743$	8.67526	0.318265	0.159132	+	0.987257i	$0.449130\pi$
$744$	0	0
$745$	−0.0845540	−0.00309782
$746$	0	0
$747$	−10.1842	−0.372622
$748$	0	0
$749$	−2.28784	−0.0835957
$750$	0	0
$751$	−6.70627	−0.244715	−0.122358	−	0.992486i	$-0.539046\pi$
$751$	−6.70627	−0.244715	−0.122358	+	0.992486i	$0.539046\pi$
$752$	0	0
$753$	−5.59517	−0.203899
$754$	0	0
$755$	15.2216	0.553970
$756$	0	0
$757$	14.3896	0.522999	0.261500	−	0.965204i	$-0.415783\pi$
$757$	14.3896	0.522999	0.261500	+	0.965204i	$0.415783\pi$
$758$	0	0
$759$	−18.3144	−0.664771
$760$	0	0
$761$	−50.9650	−1.84748	−0.923740	−	0.383020i	$-0.874884\pi$
$761$	−50.9650	−1.84748	−0.923740	+	0.383020i	$0.874884\pi$
$762$	0	0
$763$	49.8018	1.80294
$764$	0	0
$765$	−8.73207	−0.315709
$766$	0	0
$767$	−3.41831	−0.123428
$768$	0	0
$769$	23.9975	0.865373	0.432687	−	0.901544i	$-0.357566\pi$
$769$	23.9975	0.865373	0.432687	+	0.901544i	$0.357566\pi$
$770$	0	0
$771$	−6.51495	−0.234630
$772$	0	0
$773$	43.9849	1.58203	0.791014	−	0.611799i	$-0.209554\pi$
$773$	43.9849	1.58203	0.791014	+	0.611799i	$0.209554\pi$
$774$	0	0
$775$	−5.73545	−0.206024
$776$	0	0
$777$	−23.2299	−0.833367
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−30.4384	−1.08917
$782$	0	0
$783$	30.9188	1.10495
$784$	0	0
$785$	16.2587	0.580298
$786$	0	0
$787$	−16.4815	−0.587501	−0.293751	−	0.955882i	$-0.594904\pi$
$787$	−16.4815	−0.587501	−0.293751	+	0.955882i	$0.594904\pi$
$788$	0	0
$789$	−17.1938	−0.612116
$790$	0	0
$791$	−13.5295	−0.481055
$792$	0	0
$793$	3.70311	0.131501
$794$	0	0
$795$	7.48612	0.265505
$796$	0	0
$797$	47.9346	1.69793	0.848966	−	0.528448i	$-0.177226\pi$
$797$	47.9346	1.69793	0.848966	+	0.528448i	$0.177226\pi$
$798$	0	0
$799$	26.3089	0.930740
$800$	0	0
$801$	−33.0416	−1.16747
$802$	0	0
$803$	−6.14591	−0.216884
$804$	0	0
$805$	−22.7617	−0.802244
$806$	0	0
$807$	1.01022	0.0355615
$808$	0	0
$809$	11.7879	0.414441	0.207221	−	0.978294i	$-0.433558\pi$
$809$	11.7879	0.414441	0.207221	+	0.978294i	$0.433558\pi$
$810$	0	0
$811$	−31.8863	−1.11968	−0.559840	−	0.828601i	$-0.689137\pi$
$811$	−31.8863	−1.11968	−0.559840	+	0.828601i	$0.689137\pi$
$812$	0	0
$813$	7.37043	0.258492
$814$	0	0
$815$	14.9461	0.523541
$816$	0	0
$817$	0	0
$818$	0	0
$819$	3.11926	0.108996
$820$	0	0
$821$	32.5159	1.13481	0.567406	−	0.823438i	$-0.307947\pi$
$821$	32.5159	1.13481	0.567406	+	0.823438i	$0.307947\pi$
$822$	0	0
$823$	49.3969	1.72187	0.860934	−	0.508716i	$-0.169880\pi$
$823$	49.3969	1.72187	0.860934	+	0.508716i	$0.169880\pi$
$824$	0	0
$825$	2.50299	0.0871429
$826$	0	0
$827$	−24.0295	−0.835587	−0.417794	−	0.908542i	$-0.637196\pi$
$827$	−24.0295	−0.835587	−0.417794	+	0.908542i	$0.637196\pi$
$828$	0	0
$829$	−8.42673	−0.292672	−0.146336	−	0.989235i	$-0.546748\pi$
$829$	−8.42673	−0.292672	−0.146336	+	0.989235i	$0.546748\pi$
$830$	0	0
$831$	−14.4026	−0.499622
$832$	0	0
$833$	9.36151	0.324357
$834$	0	0
$835$	−12.5292	−0.433592
$836$	0	0
$837$	−22.3601	−0.772879
$838$	0	0
$839$	−22.1617	−0.765108	−0.382554	−	0.923933i	$-0.624955\pi$
$839$	−22.1617	−0.765108	−0.382554	+	0.923933i	$0.624955\pi$
$840$	0	0
$841$	33.8975	1.16888
$842$	0	0
$843$	11.0574	0.380836
$844$	0	0
$845$	−12.8387	−0.441666
$846$	0	0
$847$	4.52752	0.155568
$848$	0	0
$849$	−19.3366	−0.663631
$850$	0	0
$851$	−77.0425	−2.64098
$852$	0	0
$853$	−38.8306	−1.32953	−0.664767	−	0.747051i	$-0.731469\pi$
$853$	−38.8306	−1.32953	−0.664767	+	0.747051i	$0.731469\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.8430	1.22437	0.612186	−	0.790713i	$-0.290290\pi$
$857$	35.8430	1.22437	0.612186	+	0.790713i	$0.290290\pi$
$858$	0	0
$859$	−18.1263	−0.618460	−0.309230	−	0.950987i	$-0.600071\pi$
$859$	−18.1263	−0.618460	−0.309230	+	0.950987i	$0.600071\pi$
$860$	0	0
$861$	−2.45065	−0.0835180
$862$	0	0
$863$	−48.5099	−1.65130	−0.825648	−	0.564186i	$-0.809190\pi$
$863$	−48.5099	−1.65130	−0.825648	+	0.564186i	$0.809190\pi$
$864$	0	0
$865$	−17.3848	−0.591102
$866$	0	0
$867$	3.38363	0.114914
$868$	0	0
$869$	−31.8014	−1.07879
$870$	0	0
$871$	3.37819	0.114466
$872$	0	0
$873$	−19.2351	−0.651008
$874$	0	0
$875$	3.11079	0.105164
$876$	0	0
$877$	−26.9417	−0.909756	−0.454878	−	0.890554i	$-0.650317\pi$
$877$	−26.9417	−0.909756	−0.454878	+	0.890554i	$0.650317\pi$
$878$	0	0
$879$	−8.98652	−0.303108
$880$	0	0
$881$	27.0603	0.911685	0.455843	−	0.890060i	$-0.349338\pi$
$881$	27.0603	0.911685	0.455843	+	0.890060i	$0.349338\pi$
$882$	0	0
$883$	36.6176	1.23228	0.616140	−	0.787636i	$-0.288695\pi$
$883$	36.6176	1.23228	0.616140	+	0.787636i	$0.288695\pi$
$884$	0	0
$885$	−6.03712	−0.202936
$886$	0	0
$887$	−42.4683	−1.42595	−0.712973	−	0.701191i	$-0.752652\pi$
$887$	−42.4683	−1.42595	−0.712973	+	0.701191i	$0.752652\pi$
$888$	0	0
$889$	14.4527	0.484728
$890$	0	0
$891$	−16.6794	−0.558781
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−20.7830	−0.694700
$896$	0	0
$897$	−2.08389	−0.0695791
$898$	0	0
$899$	−45.4867	−1.51707
$900$	0	0
$901$	−36.9125	−1.22973
$902$	0	0
$903$	18.9908	0.631973
$904$	0	0
$905$	−1.62913	−0.0541540
$906$	0	0
$907$	−53.6281	−1.78069	−0.890345	−	0.455286i	$-0.849537\pi$
$907$	−53.6281	−1.78069	−0.890345	+	0.455286i	$0.849537\pi$
$908$	0	0
$909$	−43.3867	−1.43905
$910$	0	0
$911$	10.1631	0.336719	0.168360	−	0.985726i	$-0.446153\pi$
$911$	10.1631	0.336719	0.168360	+	0.985726i	$0.446153\pi$
$912$	0	0
$913$	−14.3942	−0.476379
$914$	0	0
$915$	6.54011	0.216209
$916$	0	0
$917$	32.6493	1.07817
$918$	0	0
$919$	−16.6831	−0.550325	−0.275163	−	0.961398i	$-0.588732\pi$
$919$	−16.6831	−0.550325	−0.275163	+	0.961398i	$0.588732\pi$
$920$	0	0
$921$	1.82326	0.0600786
$922$	0	0
$923$	−3.46341	−0.113999
$924$	0	0
$925$	10.5292	0.346199
$926$	0	0
$927$	−17.7940	−0.584433
$928$	0	0
$929$	8.99687	0.295178	0.147589	−	0.989049i	$-0.452849\pi$
$929$	8.99687	0.295178	0.147589	+	0.989049i	$0.452849\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−14.0119	−0.458728
$934$	0	0
$935$	−12.3417	−0.403618
$936$	0	0
$937$	39.8373	1.30143	0.650714	−	0.759323i	$-0.274470\pi$
$937$	39.8373	1.30143	0.650714	+	0.759323i	$0.274470\pi$
$938$	0	0
$939$	−15.8691	−0.517868
$940$	0	0
$941$	−11.8447	−0.386127	−0.193063	−	0.981186i	$-0.561842\pi$
$941$	−11.8447	−0.386127	−0.193063	+	0.981186i	$0.561842\pi$
$942$	0	0
$943$	−8.12765	−0.264673
$944$	0	0
$945$	12.1277	0.394513
$946$	0	0
$947$	−13.3881	−0.435054	−0.217527	−	0.976054i	$-0.569799\pi$
$947$	−13.3881	−0.435054	−0.217527	+	0.976054i	$0.569799\pi$
$948$	0	0
$949$	−0.699307	−0.0227005
$950$	0	0
$951$	9.43704	0.306017
$952$	0	0
$953$	−46.1989	−1.49653	−0.748264	−	0.663401i	$-0.769112\pi$
$953$	−46.1989	−1.49653	−0.748264	+	0.663401i	$0.769112\pi$
$954$	0	0
$955$	−13.4016	−0.433665
$956$	0	0
$957$	19.8507	0.641682
$958$	0	0
$959$	−4.33085	−0.139850
$960$	0	0
$961$	1.89541	0.0611424
$962$	0	0
$963$	1.83643	0.0591782
$964$	0	0
$965$	10.6939	0.344248
$966$	0	0
$967$	23.7112	0.762501	0.381251	−	0.924472i	$-0.375493\pi$
$967$	23.7112	0.762501	0.381251	+	0.924472i	$0.375493\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	34.5447	1.10859	0.554296	−	0.832320i	$-0.312988\pi$
$971$	34.5447	1.10859	0.554296	+	0.832320i	$0.312988\pi$
$972$	0	0
$973$	−18.8990	−0.605875
$974$	0	0
$975$	0.284801	0.00912092
$976$	0	0
$977$	52.4467	1.67792	0.838959	−	0.544195i	$-0.183165\pi$
$977$	52.4467	1.67792	0.838959	+	0.544195i	$0.183165\pi$
$978$	0	0
$979$	−46.7003	−1.49255
$980$	0	0
$981$	−39.9755	−1.27632
$982$	0	0
$983$	−14.0293	−0.447464	−0.223732	−	0.974651i	$-0.571824\pi$
$983$	−14.0293	−0.447464	−0.223732	+	0.974651i	$0.571824\pi$
$984$	0	0
$985$	−7.00611	−0.223233
$986$	0	0
$987$	−16.5980	−0.528320
$988$	0	0
$989$	62.9834	2.00276
$990$	0	0
$991$	−9.88336	−0.313955	−0.156978	−	0.987602i	$-0.550175\pi$
$991$	−9.88336	−0.313955	−0.156978	+	0.987602i	$0.550175\pi$
$992$	0	0
$993$	−13.9129	−0.441514
$994$	0	0
$995$	−20.7818	−0.658828
$996$	0	0
$997$	−42.4421	−1.34415	−0.672077	−	0.740481i	$-0.734598\pi$
$997$	−42.4421	−1.34415	−0.672077	+	0.740481i	$0.734598\pi$
$998$	0	0
$999$	41.0490	1.29873

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7220.2.a.p.1.2		4
19.8	odd	6		380.2.i.c.121.2	✓	8
19.12	odd	6		380.2.i.c.201.2	yes	8
19.18	odd	2		7220.2.a.r.1.3		4
57.8	even	6		3420.2.t.w.1261.4		8
57.50	even	6		3420.2.t.w.3241.4		8
76.27	even	6		1520.2.q.m.881.3		8
76.31	even	6		1520.2.q.m.961.3		8
95.8	even	12		1900.2.s.d.349.6		16
95.12	even	12		1900.2.s.d.49.6		16
95.27	even	12		1900.2.s.d.349.3		16
95.69	odd	6		1900.2.i.d.201.3		8
95.84	odd	6		1900.2.i.d.501.3		8
95.88	even	12		1900.2.s.d.49.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.i.c.121.2	✓	8	19.8	odd	6
380.2.i.c.201.2	yes	8	19.12	odd	6
1520.2.q.m.881.3		8	76.27	even	6
1520.2.q.m.961.3		8	76.31	even	6
1900.2.i.d.201.3		8	95.69	odd	6
1900.2.i.d.501.3		8	95.84	odd	6
1900.2.s.d.49.3		16	95.88	even	12
1900.2.s.d.49.6		16	95.12	even	12
1900.2.s.d.349.3		16	95.27	even	12
1900.2.s.d.349.6		16	95.8	even	12
3420.2.t.w.1261.4		8	57.8	even	6
3420.2.t.w.3241.4		8	57.50	even	6
7220.2.a.p.1.2		4	1.1	even	1	trivial
7220.2.a.r.1.3		4	19.18	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:	\( N \)	\(=\)	\( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7220.a (trivial)

\(f(q)\)	\(=\)	\(q-0.709218 q^{3} +1.00000 q^{5} +3.11079 q^{7} -2.49701 q^{9} +O(q^{10})\)	\(q-0.709218 q^{3} +1.00000 q^{5} +3.11079 q^{7} -2.49701 q^{9} -3.52922 q^{11} -0.401570 q^{13} -0.709218 q^{15} +3.49701 q^{17} -2.20623 q^{21} -7.31702 q^{23} +1.00000 q^{25} +3.89858 q^{27} +7.93079 q^{29} -5.73545 q^{31} +2.50299 q^{33} +3.11079 q^{35} +10.5292 q^{37} +0.284801 q^{39} +1.11079 q^{41} -8.60780 q^{43} -2.49701 q^{45} +7.52324 q^{47} +2.67700 q^{49} -2.48014 q^{51} -10.5555 q^{53} -3.52922 q^{55} +8.51236 q^{59} -9.22158 q^{61} -7.76767 q^{63} -0.401570 q^{65} -8.41246 q^{67} +5.18936 q^{69} +8.62466 q^{71} +1.74143 q^{73} -0.709218 q^{75} -10.9787 q^{77} +9.01089 q^{79} +4.72608 q^{81} +4.07857 q^{83} +3.49701 q^{85} -5.62466 q^{87} +13.2325 q^{89} -1.24920 q^{91} +4.06769 q^{93} +7.70324 q^{97} +8.81251 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} + 5 q^{9}+O(q^{10}) \) 4 * q - q^3 + 4 * q^5 + 5 * q^9	\( 4 q - q^{3} + 4 q^{5} + 5 q^{9} + 2 q^{11} + 9 q^{13} - q^{15} - q^{17} + 8 q^{21} + 4 q^{25} - 10 q^{27} + 5 q^{29} + 10 q^{31} + 25 q^{33} + 26 q^{37} - 27 q^{39} - 8 q^{41} - 7 q^{43} + 5 q^{45} - 16 q^{47} + 10 q^{49} + 12 q^{51} + 5 q^{53} + 2 q^{55} + 11 q^{59} - 12 q^{61} + 3 q^{63} + 9 q^{65} - 3 q^{69} + 14 q^{71} + 4 q^{73} - q^{75} - 22 q^{77} + 13 q^{79} + 24 q^{81} + 5 q^{83} - q^{85} - 2 q^{87} + 5 q^{89} - 46 q^{91} + 28 q^{93} - q^{97} - 24 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 + 5 * q^9 + 2 * q^11 + 9 * q^13 - q^15 - q^17 + 8 * q^21 + 4 * q^25 - 10 * q^27 + 5 * q^29 + 10 * q^31 + 25 * q^33 + 26 * q^37 - 27 * q^39 - 8 * q^41 - 7 * q^43 + 5 * q^45 - 16 * q^47 + 10 * q^49 + 12 * q^51 + 5 * q^53 + 2 * q^55 + 11 * q^59 - 12 * q^61 + 3 * q^63 + 9 * q^65 - 3 * q^69 + 14 * q^71 + 4 * q^73 - q^75 - 22 * q^77 + 13 * q^79 + 24 * q^81 + 5 * q^83 - q^85 - 2 * q^87 + 5 * q^89 - 46 * q^91 + 28 * q^93 - q^97 - 24 * q^99

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