LMFDB - Embedded newform 8020.2.a.d.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, 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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.03867 q^{3} +1.00000 q^{5} -0.0217287 q^{7} -1.92117 q^{9} +O(q^{10})$	$q+1.03867 q^{3} +1.00000 q^{5} -0.0217287 q^{7} -1.92117 q^{9} -1.55147 q^{11} -0.436009 q^{13} +1.03867 q^{15} +7.24586 q^{17} -5.85124 q^{19} -0.0225689 q^{21} -8.05748 q^{23} +1.00000 q^{25} -5.11146 q^{27} -1.94015 q^{29} +10.0357 q^{31} -1.61146 q^{33} -0.0217287 q^{35} +5.01768 q^{37} -0.452868 q^{39} -7.09612 q^{41} -0.411986 q^{43} -1.92117 q^{45} +9.07623 q^{47} -6.99953 q^{49} +7.52603 q^{51} +6.43396 q^{53} -1.55147 q^{55} -6.07749 q^{57} +9.07361 q^{59} -11.0812 q^{61} +0.0417446 q^{63} -0.436009 q^{65} -13.3476 q^{67} -8.36903 q^{69} +10.5476 q^{71} -8.43416 q^{73} +1.03867 q^{75} +0.0337116 q^{77} -9.27918 q^{79} +0.454424 q^{81} -12.3781 q^{83} +7.24586 q^{85} -2.01517 q^{87} -14.0322 q^{89} +0.00947393 q^{91} +10.4237 q^{93} -5.85124 q^{95} -4.19522 q^{97} +2.98065 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * 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
$3$	1.03867	0.599674	0.299837	−	0.953990i	$-0.403068\pi$
$3$	1.03867	0.599674	0.299837	+	0.953990i	$0.403068\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.0217287	−0.00821269	−0.00410634	−	0.999992i	$-0.501307\pi$
$7$	−0.0217287	−0.00821269	−0.00410634	+	0.999992i	$0.501307\pi$
$8$	0	0
$9$	−1.92117	−0.640391
$10$	0	0
$11$	−1.55147	−0.467787	−0.233894	−	0.972262i	$-0.575147\pi$
$11$	−1.55147	−0.467787	−0.233894	+	0.972262i	$0.575147\pi$
$12$	0	0
$13$	−0.436009	−0.120927	−0.0604636	−	0.998170i	$-0.519258\pi$
$13$	−0.436009	−0.120927	−0.0604636	+	0.998170i	$0.519258\pi$
$14$	0	0
$15$	1.03867	0.268182
$16$	0	0
$17$	7.24586	1.75738	0.878690	−	0.477393i	$-0.158418\pi$
$17$	7.24586	1.75738	0.878690	+	0.477393i	$0.158418\pi$
$18$	0	0
$19$	−5.85124	−1.34237	−0.671184	−	0.741291i	$-0.734214\pi$
$19$	−5.85124	−1.34237	−0.671184	+	0.741291i	$0.734214\pi$
$20$	0	0
$21$	−0.0225689	−0.00492493
$22$	0	0
$23$	−8.05748	−1.68010	−0.840051	−	0.542508i	$-0.817475\pi$
$23$	−8.05748	−1.68010	−0.840051	+	0.542508i	$0.817475\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.11146	−0.983700
$28$	0	0
$29$	−1.94015	−0.360277	−0.180139	−	0.983641i	$-0.557655\pi$
$29$	−1.94015	−0.360277	−0.180139	+	0.983641i	$0.557655\pi$
$30$	0	0
$31$	10.0357	1.80246	0.901232	−	0.433338i	$-0.142664\pi$
$31$	10.0357	1.80246	0.901232	+	0.433338i	$0.142664\pi$
$32$	0	0
$33$	−1.61146	−0.280520
$34$	0	0
$35$	−0.0217287	−0.00367282
$36$	0	0
$37$	5.01768	0.824901	0.412451	−	0.910980i	$-0.364673\pi$
$37$	5.01768	0.824901	0.412451	+	0.910980i	$0.364673\pi$
$38$	0	0
$39$	−0.452868	−0.0725169
$40$	0	0
$41$	−7.09612	−1.10823	−0.554114	−	0.832441i	$-0.686943\pi$
$41$	−7.09612	−1.10823	−0.554114	+	0.832441i	$0.686943\pi$
$42$	0	0
$43$	−0.411986	−0.0628274	−0.0314137	−	0.999506i	$-0.510001\pi$
$43$	−0.411986	−0.0628274	−0.0314137	+	0.999506i	$0.510001\pi$
$44$	0	0
$45$	−1.92117	−0.286392
$46$	0	0
$47$	9.07623	1.32390	0.661952	−	0.749546i	$-0.269728\pi$
$47$	9.07623	1.32390	0.661952	+	0.749546i	$0.269728\pi$
$48$	0	0
$49$	−6.99953	−0.999933
$50$	0	0
$51$	7.52603	1.05385
$52$	0	0
$53$	6.43396	0.883772	0.441886	−	0.897071i	$-0.354310\pi$
$53$	6.43396	0.883772	0.441886	+	0.897071i	$0.354310\pi$
$54$	0	0
$55$	−1.55147	−0.209201
$56$	0	0
$57$	−6.07749	−0.804983
$58$	0	0
$59$	9.07361	1.18128	0.590642	−	0.806934i	$-0.298875\pi$
$59$	9.07361	1.18128	0.590642	+	0.806934i	$0.298875\pi$
$60$	0	0
$61$	−11.0812	−1.41880	−0.709401	−	0.704806i	$-0.751034\pi$
$61$	−11.0812	−1.41880	−0.709401	+	0.704806i	$0.751034\pi$
$62$	0	0
$63$	0.0417446	0.00525933
$64$	0	0
$65$	−0.436009	−0.0540803
$66$	0	0
$67$	−13.3476	−1.63067	−0.815335	−	0.578990i	$-0.803447\pi$
$67$	−13.3476	−1.63067	−0.815335	+	0.578990i	$0.803447\pi$
$68$	0	0
$69$	−8.36903	−1.00751
$70$	0	0
$71$	10.5476	1.25178	0.625888	−	0.779913i	$-0.284737\pi$
$71$	10.5476	1.25178	0.625888	+	0.779913i	$0.284737\pi$
$72$	0	0
$73$	−8.43416	−0.987143	−0.493572	−	0.869705i	$-0.664309\pi$
$73$	−8.43416	−0.987143	−0.493572	+	0.869705i	$0.664309\pi$
$74$	0	0
$75$	1.03867	0.119935
$76$	0	0
$77$	0.0337116	0.00384179
$78$	0	0
$79$	−9.27918	−1.04399	−0.521995	−	0.852949i	$-0.674812\pi$
$79$	−9.27918	−1.04399	−0.521995	+	0.852949i	$0.674812\pi$
$80$	0	0
$81$	0.454424	0.0504915
$82$	0	0
$83$	−12.3781	−1.35867	−0.679336	−	0.733827i	$-0.737732\pi$
$83$	−12.3781	−1.35867	−0.679336	+	0.733827i	$0.737732\pi$
$84$	0	0
$85$	7.24586	0.785924
$86$	0	0
$87$	−2.01517	−0.216049
$88$	0	0
$89$	−14.0322	−1.48741	−0.743707	−	0.668506i	$-0.766934\pi$
$89$	−14.0322	−1.48741	−0.743707	+	0.668506i	$0.766934\pi$
$90$	0	0
$91$	0.00947393	0.000993138 0
$92$	0	0
$93$	10.4237	1.08089
$94$	0	0
$95$	−5.85124	−0.600325
$96$	0	0
$97$	−4.19522	−0.425960	−0.212980	−	0.977057i	$-0.568317\pi$
$97$	−4.19522	−0.425960	−0.212980	+	0.977057i	$0.568317\pi$
$98$	0	0
$99$	2.98065	0.299567
$100$	0	0
$101$	−14.0559	−1.39862	−0.699309	−	0.714820i	$-0.746509\pi$
$101$	−14.0559	−1.39862	−0.699309	+	0.714820i	$0.746509\pi$
$102$	0	0
$103$	15.8917	1.56585	0.782927	−	0.622114i	$-0.213726\pi$
$103$	15.8917	1.56585	0.782927	+	0.622114i	$0.213726\pi$
$104$	0	0
$105$	−0.0225689	−0.00220250
$106$	0	0
$107$	1.54439	0.149301	0.0746507	−	0.997210i	$-0.476216\pi$
$107$	1.54439	0.149301	0.0746507	+	0.997210i	$0.476216\pi$
$108$	0	0
$109$	−18.3125	−1.75402	−0.877009	−	0.480474i	$-0.840465\pi$
$109$	−18.3125	−1.75402	−0.877009	+	0.480474i	$0.840465\pi$
$110$	0	0
$111$	5.21169	0.494672
$112$	0	0
$113$	−13.1118	−1.23345	−0.616727	−	0.787177i	$-0.711542\pi$
$113$	−13.1118	−1.23345	−0.616727	+	0.787177i	$0.711542\pi$
$114$	0	0
$115$	−8.05748	−0.751364
$116$	0	0
$117$	0.837649	0.0774407
$118$	0	0
$119$	−0.157443	−0.0144328
$120$	0	0
$121$	−8.59293	−0.781175
$122$	0	0
$123$	−7.37050	−0.664576
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.4053	0.923317	0.461658	−	0.887058i	$-0.347255\pi$
$127$	10.4053	0.923317	0.461658	+	0.887058i	$0.347255\pi$
$128$	0	0
$129$	−0.427916	−0.0376759
$130$	0	0
$131$	11.5697	1.01085	0.505427	−	0.862870i	$-0.331335\pi$
$131$	11.5697	1.01085	0.505427	+	0.862870i	$0.331335\pi$
$132$	0	0
$133$	0.127140	0.0110244
$134$	0	0
$135$	−5.11146	−0.439924
$136$	0	0
$137$	2.43826	0.208314	0.104157	−	0.994561i	$-0.466786\pi$
$137$	2.43826	0.208314	0.104157	+	0.994561i	$0.466786\pi$
$138$	0	0
$139$	4.39980	0.373186	0.186593	−	0.982437i	$-0.440255\pi$
$139$	4.39980	0.373186	0.186593	+	0.982437i	$0.440255\pi$
$140$	0	0
$141$	9.42717	0.793911
$142$	0	0
$143$	0.676458	0.0565682
$144$	0	0
$145$	−1.94015	−0.161121
$146$	0	0
$147$	−7.27017	−0.599634
$148$	0	0
$149$	−7.44958	−0.610293	−0.305147	−	0.952305i	$-0.598705\pi$
$149$	−7.44958	−0.610293	−0.305147	+	0.952305i	$0.598705\pi$
$150$	0	0
$151$	−17.0061	−1.38394	−0.691969	−	0.721927i	$-0.743256\pi$
$151$	−17.0061	−1.38394	−0.691969	+	0.721927i	$0.743256\pi$
$152$	0	0
$153$	−13.9206	−1.12541
$154$	0	0
$155$	10.0357	0.806086
$156$	0	0
$157$	12.1137	0.966775	0.483387	−	0.875407i	$-0.339406\pi$
$157$	12.1137	0.966775	0.483387	+	0.875407i	$0.339406\pi$
$158$	0	0
$159$	6.68274	0.529975
$160$	0	0
$161$	0.175079	0.0137981
$162$	0	0
$163$	22.3543	1.75092	0.875461	−	0.483288i	$-0.160558\pi$
$163$	22.3543	1.75092	0.875461	+	0.483288i	$0.160558\pi$
$164$	0	0
$165$	−1.61146	−0.125452
$166$	0	0
$167$	−20.4366	−1.58143	−0.790715	−	0.612185i	$-0.790291\pi$
$167$	−20.4366	−1.58143	−0.790715	+	0.612185i	$0.790291\pi$
$168$	0	0
$169$	−12.8099	−0.985377
$170$	0	0
$171$	11.2412	0.859640
$172$	0	0
$173$	−12.3307	−0.937482	−0.468741	−	0.883336i	$-0.655292\pi$
$173$	−12.3307	−0.937482	−0.468741	+	0.883336i	$0.655292\pi$
$174$	0	0
$175$	−0.0217287	−0.00164254
$176$	0	0
$177$	9.42445	0.708385
$178$	0	0
$179$	−6.71128	−0.501625	−0.250812	−	0.968036i	$-0.580698\pi$
$179$	−6.71128	−0.501625	−0.250812	+	0.968036i	$0.580698\pi$
$180$	0	0
$181$	22.3267	1.65953	0.829765	−	0.558114i	$-0.188475\pi$
$181$	22.3267	1.65953	0.829765	+	0.558114i	$0.188475\pi$
$182$	0	0
$183$	−11.5097	−0.850818
$184$	0	0
$185$	5.01768	0.368907
$186$	0	0
$187$	−11.2418	−0.822080
$188$	0	0
$189$	0.111065	0.00807882
$190$	0	0
$191$	−23.0064	−1.66468	−0.832341	−	0.554264i	$-0.813000\pi$
$191$	−23.0064	−1.66468	−0.832341	+	0.554264i	$0.813000\pi$
$192$	0	0
$193$	−1.08360	−0.0779994	−0.0389997	−	0.999239i	$-0.512417\pi$
$193$	−1.08360	−0.0779994	−0.0389997	+	0.999239i	$0.512417\pi$
$194$	0	0
$195$	−0.452868	−0.0324306
$196$	0	0
$197$	10.2813	0.732515	0.366258	−	0.930514i	$-0.380639\pi$
$197$	10.2813	0.732515	0.366258	+	0.930514i	$0.380639\pi$
$198$	0	0
$199$	3.97072	0.281477	0.140738	−	0.990047i	$-0.455052\pi$
$199$	3.97072	0.281477	0.140738	+	0.990047i	$0.455052\pi$
$200$	0	0
$201$	−13.8637	−0.977870
$202$	0	0
$203$	0.0421570	0.00295884
$204$	0	0
$205$	−7.09612	−0.495615
$206$	0	0
$207$	15.4798	1.07592
$208$	0	0
$209$	9.07805	0.627942
$210$	0	0
$211$	−6.16084	−0.424130	−0.212065	−	0.977256i	$-0.568019\pi$
$211$	−6.16084	−0.424130	−0.212065	+	0.977256i	$0.568019\pi$
$212$	0	0
$213$	10.9555	0.750657
$214$	0	0
$215$	−0.411986	−0.0280972
$216$	0	0
$217$	−0.218063	−0.0148031
$218$	0	0
$219$	−8.76027	−0.591964
$220$	0	0
$221$	−3.15926	−0.212515
$222$	0	0
$223$	−5.89231	−0.394578	−0.197289	−	0.980345i	$-0.563214\pi$
$223$	−5.89231	−0.394578	−0.197289	+	0.980345i	$0.563214\pi$
$224$	0	0
$225$	−1.92117	−0.128078
$226$	0	0
$227$	−8.08384	−0.536543	−0.268272	−	0.963343i	$-0.586452\pi$
$227$	−8.08384	−0.536543	−0.268272	+	0.963343i	$0.586452\pi$
$228$	0	0
$229$	−21.8256	−1.44227	−0.721137	−	0.692792i	$-0.756380\pi$
$229$	−21.8256	−1.44227	−0.721137	+	0.692792i	$0.756380\pi$
$230$	0	0
$231$	0.0350151	0.00230382
$232$	0	0
$233$	−13.0012	−0.851738	−0.425869	−	0.904785i	$-0.640032\pi$
$233$	−13.0012	−0.851738	−0.425869	+	0.904785i	$0.640032\pi$
$234$	0	0
$235$	9.07623	0.592068
$236$	0	0
$237$	−9.63797	−0.626053
$238$	0	0
$239$	22.9158	1.48230	0.741150	−	0.671340i	$-0.234281\pi$
$239$	22.9158	1.48230	0.741150	+	0.671340i	$0.234281\pi$
$240$	0	0
$241$	−12.3009	−0.792370	−0.396185	−	0.918171i	$-0.629666\pi$
$241$	−12.3009	−0.792370	−0.396185	+	0.918171i	$0.629666\pi$
$242$	0	0
$243$	15.8064	1.01398
$244$	0	0
$245$	−6.99953	−0.447183
$246$	0	0
$247$	2.55120	0.162329
$248$	0	0
$249$	−12.8567	−0.814760
$250$	0	0
$251$	−15.6332	−0.986758	−0.493379	−	0.869814i	$-0.664238\pi$
$251$	−15.6332	−0.986758	−0.493379	+	0.869814i	$0.664238\pi$
$252$	0	0
$253$	12.5010	0.785930
$254$	0	0
$255$	7.52603	0.471298
$256$	0	0
$257$	22.4259	1.39889	0.699443	−	0.714688i	$-0.253431\pi$
$257$	22.4259	1.39889	0.699443	+	0.714688i	$0.253431\pi$
$258$	0	0
$259$	−0.109028	−0.00677465
$260$	0	0
$261$	3.72737	0.230718
$262$	0	0
$263$	−29.8852	−1.84280	−0.921400	−	0.388616i	$-0.872953\pi$
$263$	−29.8852	−1.84280	−0.921400	+	0.388616i	$0.872953\pi$
$264$	0	0
$265$	6.43396	0.395235
$266$	0	0
$267$	−14.5748	−0.891963
$268$	0	0
$269$	−12.8851	−0.785618	−0.392809	−	0.919620i	$-0.628497\pi$
$269$	−12.8851	−0.785618	−0.392809	+	0.919620i	$0.628497\pi$
$270$	0	0
$271$	−30.9657	−1.88103	−0.940517	−	0.339747i	$-0.889659\pi$
$271$	−30.9657	−1.88103	−0.940517	+	0.339747i	$0.889659\pi$
$272$	0	0
$273$	0.00984025	0.000595559 0
$274$	0	0
$275$	−1.55147	−0.0935574
$276$	0	0
$277$	−11.6950	−0.702682	−0.351341	−	0.936248i	$-0.614274\pi$
$277$	−11.6950	−0.702682	−0.351341	+	0.936248i	$0.614274\pi$
$278$	0	0
$279$	−19.2803	−1.15428
$280$	0	0
$281$	−31.5687	−1.88323	−0.941616	−	0.336690i	$-0.890693\pi$
$281$	−31.5687	−1.88323	−0.941616	+	0.336690i	$0.890693\pi$
$282$	0	0
$283$	−7.88196	−0.468534	−0.234267	−	0.972172i	$-0.575269\pi$
$283$	−7.88196	−0.468534	−0.234267	+	0.972172i	$0.575269\pi$
$284$	0	0
$285$	−6.07749	−0.359999
$286$	0	0
$287$	0.154190	0.00910153
$288$	0	0
$289$	35.5025	2.08838
$290$	0	0
$291$	−4.35743	−0.255437
$292$	0	0
$293$	12.8849	0.752744	0.376372	−	0.926469i	$-0.377171\pi$
$293$	12.8849	0.752744	0.376372	+	0.926469i	$0.377171\pi$
$294$	0	0
$295$	9.07361	0.528286
$296$	0	0
$297$	7.93029	0.460162
$298$	0	0
$299$	3.51314	0.203170
$300$	0	0
$301$	0.00895194	0.000515981 0
$302$	0	0
$303$	−14.5994	−0.838715
$304$	0	0
$305$	−11.0812	−0.634507
$306$	0	0
$307$	11.4453	0.653216	0.326608	−	0.945160i	$-0.394094\pi$
$307$	11.4453	0.653216	0.326608	+	0.945160i	$0.394094\pi$
$308$	0	0
$309$	16.5061	0.939002
$310$	0	0
$311$	19.9729	1.13256	0.566279	−	0.824213i	$-0.308382\pi$
$311$	19.9729	1.13256	0.566279	+	0.824213i	$0.308382\pi$
$312$	0	0
$313$	5.22242	0.295189	0.147594	−	0.989048i	$-0.452847\pi$
$313$	5.22242	0.295189	0.147594	+	0.989048i	$0.452847\pi$
$314$	0	0
$315$	0.0417446	0.00235204
$316$	0	0
$317$	26.5317	1.49017	0.745084	−	0.666971i	$-0.232409\pi$
$317$	26.5317	1.49017	0.745084	+	0.666971i	$0.232409\pi$
$318$	0	0
$319$	3.01010	0.168533
$320$	0	0
$321$	1.60410	0.0895322
$322$	0	0
$323$	−42.3973	−2.35905
$324$	0	0
$325$	−0.436009	−0.0241855
$326$	0	0
$327$	−19.0206	−1.05184
$328$	0	0
$329$	−0.197215	−0.0108728
$330$	0	0
$331$	22.6215	1.24339	0.621696	−	0.783258i	$-0.286444\pi$
$331$	22.6215	1.24339	0.621696	+	0.783258i	$0.286444\pi$
$332$	0	0
$333$	−9.63983	−0.528259
$334$	0	0
$335$	−13.3476	−0.729257
$336$	0	0
$337$	−16.5660	−0.902408	−0.451204	−	0.892421i	$-0.649005\pi$
$337$	−16.5660	−0.902408	−0.451204	+	0.892421i	$0.649005\pi$
$338$	0	0
$339$	−13.6188	−0.739671
$340$	0	0
$341$	−15.5701	−0.843169
$342$	0	0
$343$	0.304192	0.0164248
$344$	0	0
$345$	−8.36903	−0.450574
$346$	0	0
$347$	−3.02136	−0.162195	−0.0810975	−	0.996706i	$-0.525843\pi$
$347$	−3.02136	−0.162195	−0.0810975	+	0.996706i	$0.525843\pi$
$348$	0	0
$349$	20.0829	1.07501	0.537507	−	0.843259i	$-0.319366\pi$
$349$	20.0829	1.07501	0.537507	+	0.843259i	$0.319366\pi$
$350$	0	0
$351$	2.22864	0.118956
$352$	0	0
$353$	21.9636	1.16901	0.584503	−	0.811391i	$-0.301289\pi$
$353$	21.9636	1.16901	0.584503	+	0.811391i	$0.301289\pi$
$354$	0	0
$355$	10.5476	0.559811
$356$	0	0
$357$	−0.163531	−0.00865498
$358$	0	0
$359$	16.8961	0.891740	0.445870	−	0.895098i	$-0.352894\pi$
$359$	16.8961	0.891740	0.445870	+	0.895098i	$0.352894\pi$
$360$	0	0
$361$	15.2370	0.801949
$362$	0	0
$363$	−8.92518	−0.468451
$364$	0	0
$365$	−8.43416	−0.441464
$366$	0	0
$367$	−16.8384	−0.878958	−0.439479	−	0.898253i	$-0.644837\pi$
$367$	−16.8384	−0.878958	−0.439479	+	0.898253i	$0.644837\pi$
$368$	0	0
$369$	13.6329	0.709699
$370$	0	0
$371$	−0.139802	−0.00725814
$372$	0	0
$373$	−14.5043	−0.751002	−0.375501	−	0.926822i	$-0.622529\pi$
$373$	−14.5043	−0.751002	−0.375501	+	0.926822i	$0.622529\pi$
$374$	0	0
$375$	1.03867	0.0536365
$376$	0	0
$377$	0.845924	0.0435673
$378$	0	0
$379$	3.19327	0.164027	0.0820136	−	0.996631i	$-0.473865\pi$
$379$	3.19327	0.164027	0.0820136	+	0.996631i	$0.473865\pi$
$380$	0	0
$381$	10.8076	0.553689
$382$	0	0
$383$	26.6694	1.36275	0.681373	−	0.731937i	$-0.261383\pi$
$383$	26.6694	1.36275	0.681373	+	0.731937i	$0.261383\pi$
$384$	0	0
$385$	0.0337116	0.00171810
$386$	0	0
$387$	0.791497	0.0402341
$388$	0	0
$389$	−6.73187	−0.341319	−0.170660	−	0.985330i	$-0.554590\pi$
$389$	−6.73187	−0.341319	−0.170660	+	0.985330i	$0.554590\pi$
$390$	0	0
$391$	−58.3834	−2.95258
$392$	0	0
$393$	12.0171	0.606183
$394$	0	0
$395$	−9.27918	−0.466886
$396$	0	0
$397$	8.65840	0.434553	0.217276	−	0.976110i	$-0.430283\pi$
$397$	8.65840	0.434553	0.217276	+	0.976110i	$0.430283\pi$
$398$	0	0
$399$	0.132056	0.00661107
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−4.37566	−0.217967
$404$	0	0
$405$	0.454424	0.0225805
$406$	0	0
$407$	−7.78480	−0.385878
$408$	0	0
$409$	37.3053	1.84463	0.922314	−	0.386440i	$-0.126295\pi$
$409$	37.3053	1.84463	0.922314	+	0.386440i	$0.126295\pi$
$410$	0	0
$411$	2.53253	0.124921
$412$	0	0
$413$	−0.197158	−0.00970151
$414$	0	0
$415$	−12.3781	−0.607617
$416$	0	0
$417$	4.56992	0.223790
$418$	0	0
$419$	−2.94394	−0.143821	−0.0719103	−	0.997411i	$-0.522910\pi$
$419$	−2.94394	−0.143821	−0.0719103	+	0.997411i	$0.522910\pi$
$420$	0	0
$421$	15.1158	0.736701	0.368350	−	0.929687i	$-0.379923\pi$
$421$	15.1158	0.736701	0.368350	+	0.929687i	$0.379923\pi$
$422$	0	0
$423$	−17.4370	−0.847816
$424$	0	0
$425$	7.24586	0.351476
$426$	0	0
$427$	0.240780	0.0116522
$428$	0	0
$429$	0.702614	0.0339225
$430$	0	0
$431$	−17.7899	−0.856907	−0.428454	−	0.903564i	$-0.640941\pi$
$431$	−17.7899	−0.856907	−0.428454	+	0.903564i	$0.640941\pi$
$432$	0	0
$433$	−12.4819	−0.599841	−0.299920	−	0.953964i	$-0.596960\pi$
$433$	−12.4819	−0.599841	−0.299920	+	0.953964i	$0.596960\pi$
$434$	0	0
$435$	−2.01517	−0.0966200
$436$	0	0
$437$	47.1463	2.25531
$438$	0	0
$439$	−3.27844	−0.156471	−0.0782357	−	0.996935i	$-0.524929\pi$
$439$	−3.27844	−0.156471	−0.0782357	+	0.996935i	$0.524929\pi$
$440$	0	0
$441$	13.4473	0.640348
$442$	0	0
$443$	22.0900	1.04953	0.524765	−	0.851247i	$-0.324153\pi$
$443$	22.0900	1.04953	0.524765	+	0.851247i	$0.324153\pi$
$444$	0	0
$445$	−14.0322	−0.665191
$446$	0	0
$447$	−7.73762	−0.365977
$448$	0	0
$449$	−3.91829	−0.184916	−0.0924578	−	0.995717i	$-0.529472\pi$
$449$	−3.91829	−0.184916	−0.0924578	+	0.995717i	$0.529472\pi$
$450$	0	0
$451$	11.0095	0.518415
$452$	0	0
$453$	−17.6637	−0.829912
$454$	0	0
$455$	0.00947393	0.000444145 0
$456$	0	0
$457$	−33.3676	−1.56087	−0.780436	−	0.625235i	$-0.785003\pi$
$457$	−33.3676	−1.56087	−0.780436	+	0.625235i	$0.785003\pi$
$458$	0	0
$459$	−37.0369	−1.72873
$460$	0	0
$461$	−6.40733	−0.298419	−0.149210	−	0.988806i	$-0.547673\pi$
$461$	−6.40733	−0.298419	−0.149210	+	0.988806i	$0.547673\pi$
$462$	0	0
$463$	21.7794	1.01218	0.506088	−	0.862482i	$-0.331091\pi$
$463$	21.7794	1.01218	0.506088	+	0.862482i	$0.331091\pi$
$464$	0	0
$465$	10.4237	0.483389
$466$	0	0
$467$	36.4959	1.68883	0.844414	−	0.535691i	$-0.179949\pi$
$467$	36.4959	1.68883	0.844414	+	0.535691i	$0.179949\pi$
$468$	0	0
$469$	0.290026	0.0133922
$470$	0	0
$471$	12.5820	0.579750
$472$	0	0
$473$	0.639187	0.0293898
$474$	0	0
$475$	−5.85124	−0.268473
$476$	0	0
$477$	−12.3607	−0.565960
$478$	0	0
$479$	−12.3319	−0.563460	−0.281730	−	0.959494i	$-0.590908\pi$
$479$	−12.3319	−0.563460	−0.281730	+	0.959494i	$0.590908\pi$
$480$	0	0
$481$	−2.18775	−0.0997530
$482$	0	0
$483$	0.181848	0.00827439
$484$	0	0
$485$	−4.19522	−0.190495
$486$	0	0
$487$	4.08880	0.185281	0.0926407	−	0.995700i	$-0.470469\pi$
$487$	4.08880	0.185281	0.0926407	+	0.995700i	$0.470469\pi$
$488$	0	0
$489$	23.2186	1.04998
$490$	0	0
$491$	26.2572	1.18497	0.592485	−	0.805581i	$-0.298147\pi$
$491$	26.2572	1.18497	0.592485	+	0.805581i	$0.298147\pi$
$492$	0	0
$493$	−14.0581	−0.633144
$494$	0	0
$495$	2.98065	0.133970
$496$	0	0
$497$	−0.229187	−0.0102804
$498$	0	0
$499$	0.877623	0.0392878	0.0196439	−	0.999807i	$-0.493747\pi$
$499$	0.877623	0.0392878	0.0196439	+	0.999807i	$0.493747\pi$
$500$	0	0
$501$	−21.2268	−0.948342
$502$	0	0
$503$	21.5067	0.958936	0.479468	−	0.877559i	$-0.340830\pi$
$503$	21.5067	0.958936	0.479468	+	0.877559i	$0.340830\pi$
$504$	0	0
$505$	−14.0559	−0.625481
$506$	0	0
$507$	−13.3052	−0.590905
$508$	0	0
$509$	40.3084	1.78664	0.893319	−	0.449422i	$-0.148370\pi$
$509$	40.3084	1.78664	0.893319	+	0.449422i	$0.148370\pi$
$510$	0	0
$511$	0.183263	0.00810710
$512$	0	0
$513$	29.9084	1.32049
$514$	0	0
$515$	15.8917	0.700271
$516$	0	0
$517$	−14.0815	−0.619305
$518$	0	0
$519$	−12.8074	−0.562184
$520$	0	0
$521$	−22.0448	−0.965800	−0.482900	−	0.875676i	$-0.660417\pi$
$521$	−22.0448	−0.965800	−0.482900	+	0.875676i	$0.660417\pi$
$522$	0	0
$523$	6.31245	0.276024	0.138012	−	0.990431i	$-0.455929\pi$
$523$	6.31245	0.276024	0.138012	+	0.990431i	$0.455929\pi$
$524$	0	0
$525$	−0.0225689	−0.000984987 0
$526$	0	0
$527$	72.7172	3.16761
$528$	0	0
$529$	41.9230	1.82274
$530$	0	0
$531$	−17.4320	−0.756483
$532$	0	0
$533$	3.09398	0.134015
$534$	0	0
$535$	1.54439	0.0667696
$536$	0	0
$537$	−6.97078	−0.300811
$538$	0	0
$539$	10.8596	0.467756
$540$	0	0
$541$	−7.04691	−0.302970	−0.151485	−	0.988460i	$-0.548406\pi$
$541$	−7.04691	−0.302970	−0.151485	+	0.988460i	$0.548406\pi$
$542$	0	0
$543$	23.1900	0.995177
$544$	0	0
$545$	−18.3125	−0.784421
$546$	0	0
$547$	38.3509	1.63976	0.819882	−	0.572532i	$-0.194039\pi$
$547$	38.3509	1.63976	0.819882	+	0.572532i	$0.194039\pi$
$548$	0	0
$549$	21.2889	0.908587
$550$	0	0
$551$	11.3523	0.483624
$552$	0	0
$553$	0.201625	0.00857395
$554$	0	0
$555$	5.21169	0.221224
$556$	0	0
$557$	8.21822	0.348217	0.174109	−	0.984726i	$-0.444296\pi$
$557$	8.21822	0.348217	0.174109	+	0.984726i	$0.444296\pi$
$558$	0	0
$559$	0.179630	0.00759754
$560$	0	0
$561$	−11.6764	−0.492980
$562$	0	0
$563$	−16.4499	−0.693282	−0.346641	−	0.937998i	$-0.612678\pi$
$563$	−16.4499	−0.693282	−0.346641	+	0.937998i	$0.612678\pi$
$564$	0	0
$565$	−13.1118	−0.551618
$566$	0	0
$567$	−0.00987405	−0.000414671 0
$568$	0	0
$569$	20.8612	0.874547	0.437273	−	0.899329i	$-0.355944\pi$
$569$	20.8612	0.874547	0.437273	+	0.899329i	$0.355944\pi$
$570$	0	0
$571$	44.5525	1.86446	0.932232	−	0.361862i	$-0.117859\pi$
$571$	44.5525	1.86446	0.932232	+	0.361862i	$0.117859\pi$
$572$	0	0
$573$	−23.8959	−0.998267
$574$	0	0
$575$	−8.05748	−0.336020
$576$	0	0
$577$	−41.9665	−1.74709	−0.873545	−	0.486744i	$-0.838184\pi$
$577$	−41.9665	−1.74709	−0.873545	+	0.486744i	$0.838184\pi$
$578$	0	0
$579$	−1.12550	−0.0467742
$580$	0	0
$581$	0.268960	0.0111583
$582$	0	0
$583$	−9.98213	−0.413417
$584$	0	0
$585$	0.837649	0.0346325
$586$	0	0
$587$	−8.21418	−0.339035	−0.169518	−	0.985527i	$-0.554221\pi$
$587$	−8.21418	−0.339035	−0.169518	+	0.985527i	$0.554221\pi$
$588$	0	0
$589$	−58.7213	−2.41957
$590$	0	0
$591$	10.6789	0.439270
$592$	0	0
$593$	−25.3554	−1.04122	−0.520612	−	0.853794i	$-0.674296\pi$
$593$	−25.3554	−1.04122	−0.520612	+	0.853794i	$0.674296\pi$
$594$	0	0
$595$	−0.157443	−0.00645455
$596$	0	0
$597$	4.12425	0.168794
$598$	0	0
$599$	−20.3510	−0.831521	−0.415761	−	0.909474i	$-0.636485\pi$
$599$	−20.3510	−0.831521	−0.415761	+	0.909474i	$0.636485\pi$
$600$	0	0
$601$	8.11961	0.331206	0.165603	−	0.986193i	$-0.447043\pi$
$601$	8.11961	0.331206	0.165603	+	0.986193i	$0.447043\pi$
$602$	0	0
$603$	25.6431	1.04427
$604$	0	0
$605$	−8.59293	−0.349352
$606$	0	0
$607$	21.5118	0.873136	0.436568	−	0.899671i	$-0.356194\pi$
$607$	21.5118	0.873136	0.436568	+	0.899671i	$0.356194\pi$
$608$	0	0
$609$	0.0437871	0.00177434
$610$	0	0
$611$	−3.95732	−0.160096
$612$	0	0
$613$	−45.7282	−1.84694	−0.923472	−	0.383667i	$-0.874661\pi$
$613$	−45.7282	−1.84694	−0.923472	+	0.383667i	$0.874661\pi$
$614$	0	0
$615$	−7.37050	−0.297207
$616$	0	0
$617$	−2.66481	−0.107281	−0.0536406	−	0.998560i	$-0.517083\pi$
$617$	−2.66481	−0.107281	−0.0536406	+	0.998560i	$0.517083\pi$
$618$	0	0
$619$	18.1803	0.730726	0.365363	−	0.930865i	$-0.380945\pi$
$619$	18.1803	0.730726	0.365363	+	0.930865i	$0.380945\pi$
$620$	0	0
$621$	41.1855	1.65272
$622$	0	0
$623$	0.304902	0.0122157
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	9.42907	0.376561
$628$	0	0
$629$	36.3574	1.44966
$630$	0	0
$631$	31.4159	1.25065	0.625323	−	0.780366i	$-0.284967\pi$
$631$	31.4159	1.25065	0.625323	+	0.780366i	$0.284967\pi$
$632$	0	0
$633$	−6.39906	−0.254340
$634$	0	0
$635$	10.4053	0.412920
$636$	0	0
$637$	3.05186	0.120919
$638$	0	0
$639$	−20.2639	−0.801626
$640$	0	0
$641$	−33.1709	−1.31017	−0.655087	−	0.755554i	$-0.727368\pi$
$641$	−33.1709	−1.31017	−0.655087	+	0.755554i	$0.727368\pi$
$642$	0	0
$643$	−41.8583	−1.65073	−0.825366	−	0.564599i	$-0.809031\pi$
$643$	−41.8583	−1.65073	−0.825366	+	0.564599i	$0.809031\pi$
$644$	0	0
$645$	−0.427916	−0.0168492
$646$	0	0
$647$	−13.1751	−0.517967	−0.258984	−	0.965882i	$-0.583388\pi$
$647$	−13.1751	−0.517967	−0.258984	+	0.965882i	$0.583388\pi$
$648$	0	0
$649$	−14.0775	−0.552589
$650$	0	0
$651$	−0.226494	−0.00887701
$652$	0	0
$653$	13.3848	0.523786	0.261893	−	0.965097i	$-0.415653\pi$
$653$	13.3848	0.523786	0.261893	+	0.965097i	$0.415653\pi$
$654$	0	0
$655$	11.5697	0.452067
$656$	0	0
$657$	16.2035	0.632158
$658$	0	0
$659$	9.27297	0.361224	0.180612	−	0.983554i	$-0.442192\pi$
$659$	9.27297	0.361224	0.180612	+	0.983554i	$0.442192\pi$
$660$	0	0
$661$	−44.4189	−1.72769	−0.863847	−	0.503755i	$-0.831952\pi$
$661$	−44.4189	−1.72769	−0.863847	+	0.503755i	$0.831952\pi$
$662$	0	0
$663$	−3.28142	−0.127440
$664$	0	0
$665$	0.127140	0.00493028
$666$	0	0
$667$	15.6327	0.605302
$668$	0	0
$669$	−6.12014	−0.236618
$670$	0	0
$671$	17.1922	0.663697
$672$	0	0
$673$	3.18921	0.122935	0.0614675	−	0.998109i	$-0.480422\pi$
$673$	3.18921	0.122935	0.0614675	+	0.998109i	$0.480422\pi$
$674$	0	0
$675$	−5.11146	−0.196740
$676$	0	0
$677$	3.49237	0.134223	0.0671114	−	0.997745i	$-0.478622\pi$
$677$	3.49237	0.134223	0.0671114	+	0.997745i	$0.478622\pi$
$678$	0	0
$679$	0.0911567	0.00349827
$680$	0	0
$681$	−8.39641	−0.321751
$682$	0	0
$683$	−31.3582	−1.19989	−0.599945	−	0.800041i	$-0.704811\pi$
$683$	−31.3582	−1.19989	−0.599945	+	0.800041i	$0.704811\pi$
$684$	0	0
$685$	2.43826	0.0931610
$686$	0	0
$687$	−22.6695	−0.864895
$688$	0	0
$689$	−2.80527	−0.106872
$690$	0	0
$691$	3.52124	0.133954	0.0669771	−	0.997755i	$-0.478665\pi$
$691$	3.52124	0.133954	0.0669771	+	0.997755i	$0.478665\pi$
$692$	0	0
$693$	−0.0647657	−0.00246025
$694$	0	0
$695$	4.39980	0.166894
$696$	0	0
$697$	−51.4175	−1.94758
$698$	0	0
$699$	−13.5039	−0.510765
$700$	0	0
$701$	27.4697	1.03752	0.518758	−	0.854921i	$-0.326395\pi$
$701$	27.4697	1.03752	0.518758	+	0.854921i	$0.326395\pi$
$702$	0	0
$703$	−29.3596	−1.10732
$704$	0	0
$705$	9.42717	0.355048
$706$	0	0
$707$	0.305418	0.0114864
$708$	0	0
$709$	−22.0853	−0.829430	−0.414715	−	0.909951i	$-0.636119\pi$
$709$	−22.0853	−0.829430	−0.414715	+	0.909951i	$0.636119\pi$
$710$	0	0
$711$	17.8269	0.668561
$712$	0	0
$713$	−80.8624	−3.02832
$714$	0	0
$715$	0.676458	0.0252981
$716$	0	0
$717$	23.8019	0.888897
$718$	0	0
$719$	−40.9389	−1.52676	−0.763382	−	0.645948i	$-0.776462\pi$
$719$	−40.9389	−1.52676	−0.763382	+	0.645948i	$0.776462\pi$
$720$	0	0
$721$	−0.345306	−0.0128599
$722$	0	0
$723$	−12.7765	−0.475164
$724$	0	0
$725$	−1.94015	−0.0720554
$726$	0	0
$727$	−27.8571	−1.03316	−0.516582	−	0.856238i	$-0.672796\pi$
$727$	−27.8571	−1.03316	−0.516582	+	0.856238i	$0.672796\pi$
$728$	0	0
$729$	15.0543	0.557565
$730$	0	0
$731$	−2.98520	−0.110411
$732$	0	0
$733$	19.2925	0.712585	0.356292	−	0.934375i	$-0.384041\pi$
$733$	19.2925	0.712585	0.356292	+	0.934375i	$0.384041\pi$
$734$	0	0
$735$	−7.27017	−0.268164
$736$	0	0
$737$	20.7085	0.762806
$738$	0	0
$739$	30.4989	1.12192	0.560959	−	0.827843i	$-0.310432\pi$
$739$	30.4989	1.12192	0.560959	+	0.827843i	$0.310432\pi$
$740$	0	0
$741$	2.64984	0.0973444
$742$	0	0
$743$	−42.1136	−1.54500	−0.772499	−	0.635017i	$-0.780993\pi$
$743$	−42.1136	−1.54500	−0.772499	+	0.635017i	$0.780993\pi$
$744$	0	0
$745$	−7.44958	−0.272931
$746$	0	0
$747$	23.7804	0.870081
$748$	0	0
$749$	−0.0335575	−0.00122617
$750$	0	0
$751$	−34.1795	−1.24723	−0.623614	−	0.781733i	$-0.714336\pi$
$751$	−34.1795	−1.24723	−0.623614	+	0.781733i	$0.714336\pi$
$752$	0	0
$753$	−16.2377	−0.591733
$754$	0	0
$755$	−17.0061	−0.618916
$756$	0	0
$757$	−22.7300	−0.826136	−0.413068	−	0.910700i	$-0.635543\pi$
$757$	−22.7300	−0.826136	−0.413068	+	0.910700i	$0.635543\pi$
$758$	0	0
$759$	12.9843	0.471302
$760$	0	0
$761$	−27.4202	−0.993981	−0.496991	−	0.867756i	$-0.665562\pi$
$761$	−27.4202	−0.993981	−0.496991	+	0.867756i	$0.665562\pi$
$762$	0	0
$763$	0.397907	0.0144052
$764$	0	0
$765$	−13.9206	−0.503299
$766$	0	0
$767$	−3.95618	−0.142849
$768$	0	0
$769$	−16.0889	−0.580179	−0.290090	−	0.956999i	$-0.593685\pi$
$769$	−16.0889	−0.580179	−0.290090	+	0.956999i	$0.593685\pi$
$770$	0	0
$771$	23.2930	0.838876
$772$	0	0
$773$	−6.74119	−0.242464	−0.121232	−	0.992624i	$-0.538684\pi$
$773$	−6.74119	−0.242464	−0.121232	+	0.992624i	$0.538684\pi$
$774$	0	0
$775$	10.0357	0.360493
$776$	0	0
$777$	−0.113243	−0.00406258
$778$	0	0
$779$	41.5211	1.48765
$780$	0	0
$781$	−16.3644	−0.585564
$782$	0	0
$783$	9.91700	0.354405
$784$	0	0
$785$	12.1137	0.432355
$786$	0	0
$787$	−31.4369	−1.12060	−0.560302	−	0.828288i	$-0.689315\pi$
$787$	−31.4369	−1.12060	−0.560302	+	0.828288i	$0.689315\pi$
$788$	0	0
$789$	−31.0407	−1.10508
$790$	0	0
$791$	0.284903	0.0101300
$792$	0	0
$793$	4.83150	0.171572
$794$	0	0
$795$	6.68274	0.237012
$796$	0	0
$797$	−19.6574	−0.696300	−0.348150	−	0.937439i	$-0.613190\pi$
$797$	−19.6574	−0.696300	−0.348150	+	0.937439i	$0.613190\pi$
$798$	0	0
$799$	65.7651	2.32660
$800$	0	0
$801$	26.9583	0.952526
$802$	0	0
$803$	13.0854	0.461773
$804$	0	0
$805$	0.175079	0.00617072
$806$	0	0
$807$	−13.3833	−0.471115
$808$	0	0
$809$	−28.5001	−1.00201	−0.501005	−	0.865444i	$-0.667036\pi$
$809$	−28.5001	−1.00201	−0.501005	+	0.865444i	$0.667036\pi$
$810$	0	0
$811$	10.9010	0.382785	0.191393	−	0.981514i	$-0.438700\pi$
$811$	10.9010	0.382785	0.191393	+	0.981514i	$0.438700\pi$
$812$	0	0
$813$	−32.1630	−1.12801
$814$	0	0
$815$	22.3543	0.783036
$816$	0	0
$817$	2.41063	0.0843374
$818$	0	0
$819$	−0.0182011	−0.000635996 0
$820$	0	0
$821$	51.0611	1.78204	0.891022	−	0.453960i	$-0.149989\pi$
$821$	51.0611	1.78204	0.891022	+	0.453960i	$0.149989\pi$
$822$	0	0
$823$	20.2828	0.707013	0.353507	−	0.935432i	$-0.384989\pi$
$823$	20.2828	0.707013	0.353507	+	0.935432i	$0.384989\pi$
$824$	0	0
$825$	−1.61146	−0.0561040
$826$	0	0
$827$	24.5423	0.853418	0.426709	−	0.904389i	$-0.359673\pi$
$827$	24.5423	0.853418	0.426709	+	0.904389i	$0.359673\pi$
$828$	0	0
$829$	7.45992	0.259094	0.129547	−	0.991573i	$-0.458648\pi$
$829$	7.45992	0.259094	0.129547	+	0.991573i	$0.458648\pi$
$830$	0	0
$831$	−12.1472	−0.421380
$832$	0	0
$833$	−50.7176	−1.75726
$834$	0	0
$835$	−20.4366	−0.707237
$836$	0	0
$837$	−51.2970	−1.77308
$838$	0	0
$839$	−30.9396	−1.06815	−0.534076	−	0.845436i	$-0.679340\pi$
$839$	−30.9396	−1.06815	−0.534076	+	0.845436i	$0.679340\pi$
$840$	0	0
$841$	−25.2358	−0.870200
$842$	0	0
$843$	−32.7893	−1.12933
$844$	0	0
$845$	−12.8099	−0.440674
$846$	0	0
$847$	0.186713	0.00641555
$848$	0	0
$849$	−8.18672	−0.280968
$850$	0	0
$851$	−40.4298	−1.38592
$852$	0	0
$853$	−29.7661	−1.01917	−0.509586	−	0.860420i	$-0.670202\pi$
$853$	−29.7661	−1.01917	−0.509586	+	0.860420i	$0.670202\pi$
$854$	0	0
$855$	11.2412	0.384443
$856$	0	0
$857$	−15.6025	−0.532971	−0.266485	−	0.963839i	$-0.585862\pi$
$857$	−15.6025	−0.532971	−0.266485	+	0.963839i	$0.585862\pi$
$858$	0	0
$859$	−15.9457	−0.544059	−0.272029	−	0.962289i	$-0.587695\pi$
$859$	−15.9457	−0.544059	−0.272029	+	0.962289i	$0.587695\pi$
$860$	0	0
$861$	0.160152	0.00545795
$862$	0	0
$863$	−23.2465	−0.791319	−0.395659	−	0.918397i	$-0.629484\pi$
$863$	−23.2465	−0.791319	−0.395659	+	0.918397i	$0.629484\pi$
$864$	0	0
$865$	−12.3307	−0.419255
$866$	0	0
$867$	36.8752	1.25235
$868$	0	0
$869$	14.3964	0.488365
$870$	0	0
$871$	5.81968	0.197192
$872$	0	0
$873$	8.05974	0.272781
$874$	0	0
$875$	−0.0217287	−0.000734565 0
$876$	0	0
$877$	−25.2722	−0.853381	−0.426690	−	0.904398i	$-0.640321\pi$
$877$	−25.2722	−0.853381	−0.426690	+	0.904398i	$0.640321\pi$
$878$	0	0
$879$	13.3831	0.451401
$880$	0	0
$881$	17.9260	0.603941	0.301970	−	0.953317i	$-0.402356\pi$
$881$	17.9260	0.603941	0.301970	+	0.953317i	$0.402356\pi$
$882$	0	0
$883$	−44.9169	−1.51157	−0.755787	−	0.654818i	$-0.772745\pi$
$883$	−44.9169	−1.51157	−0.755787	+	0.654818i	$0.772745\pi$
$884$	0	0
$885$	9.42445	0.316799
$886$	0	0
$887$	25.9239	0.870440	0.435220	−	0.900324i	$-0.356671\pi$
$887$	25.9239	0.870440	0.435220	+	0.900324i	$0.356671\pi$
$888$	0	0
$889$	−0.226093	−0.00758291
$890$	0	0
$891$	−0.705027	−0.0236193
$892$	0	0
$893$	−53.1072	−1.77716
$894$	0	0
$895$	−6.71128	−0.224333
$896$	0	0
$897$	3.64898	0.121836
$898$	0	0
$899$	−19.4708	−0.649386
$900$	0	0
$901$	46.6196	1.55312
$902$	0	0
$903$	0.00929808	0.000309421 0
$904$	0	0
$905$	22.3267	0.742164
$906$	0	0
$907$	11.1198	0.369228	0.184614	−	0.982811i	$-0.440897\pi$
$907$	11.1198	0.369228	0.184614	+	0.982811i	$0.440897\pi$
$908$	0	0
$909$	27.0039	0.895662
$910$	0	0
$911$	41.3975	1.37156	0.685780	−	0.727809i	$-0.259461\pi$
$911$	41.3975	1.37156	0.685780	+	0.727809i	$0.259461\pi$
$912$	0	0
$913$	19.2043	0.635569
$914$	0	0
$915$	−11.5097	−0.380498
$916$	0	0
$917$	−0.251396	−0.00830182
$918$	0	0
$919$	28.8774	0.952576	0.476288	−	0.879289i	$-0.341982\pi$
$919$	28.8774	0.952576	0.476288	+	0.879289i	$0.341982\pi$
$920$	0	0
$921$	11.8878	0.391717
$922$	0	0
$923$	−4.59887	−0.151374
$924$	0	0
$925$	5.01768	0.164980
$926$	0	0
$927$	−30.5307	−1.00276
$928$	0	0
$929$	56.8325	1.86461	0.932307	−	0.361668i	$-0.117793\pi$
$929$	56.8325	1.86461	0.932307	+	0.361668i	$0.117793\pi$
$930$	0	0
$931$	40.9559	1.34228
$932$	0	0
$933$	20.7452	0.679166
$934$	0	0
$935$	−11.2418	−0.367645
$936$	0	0
$937$	42.9182	1.40208	0.701038	−	0.713124i	$-0.252720\pi$
$937$	42.9182	1.40208	0.701038	+	0.713124i	$0.252720\pi$
$938$	0	0
$939$	5.42435	0.177017
$940$	0	0
$941$	2.65548	0.0865662	0.0432831	−	0.999063i	$-0.486218\pi$
$941$	2.65548	0.0865662	0.0432831	+	0.999063i	$0.486218\pi$
$942$	0	0
$943$	57.1769	1.86194
$944$	0	0
$945$	0.111065	0.00361296
$946$	0	0
$947$	34.8124	1.13125	0.565625	−	0.824662i	$-0.308635\pi$
$947$	34.8124	1.13125	0.565625	+	0.824662i	$0.308635\pi$
$948$	0	0
$949$	3.67737	0.119373
$950$	0	0
$951$	27.5576	0.893615
$952$	0	0
$953$	17.9886	0.582709	0.291355	−	0.956615i	$-0.405894\pi$
$953$	17.9886	0.582709	0.291355	+	0.956615i	$0.405894\pi$
$954$	0	0
$955$	−23.0064	−0.744468
$956$	0	0
$957$	3.12648	0.101065
$958$	0	0
$959$	−0.0529802	−0.00171082
$960$	0	0
$961$	69.7151	2.24887
$962$	0	0
$963$	−2.96703	−0.0956112
$964$	0	0
$965$	−1.08360	−0.0348824
$966$	0	0
$967$	39.2461	1.26207	0.631034	−	0.775755i	$-0.282631\pi$
$967$	39.2461	1.26207	0.631034	+	0.775755i	$0.282631\pi$
$968$	0	0
$969$	−44.0366	−1.41466
$970$	0	0
$971$	26.7996	0.860039	0.430020	−	0.902820i	$-0.358507\pi$
$971$	26.7996	0.860039	0.430020	+	0.902820i	$0.358507\pi$
$972$	0	0
$973$	−0.0956021	−0.00306486
$974$	0	0
$975$	−0.452868	−0.0145034
$976$	0	0
$977$	3.04735	0.0974933	0.0487466	−	0.998811i	$-0.484477\pi$
$977$	3.04735	0.0974933	0.0487466	+	0.998811i	$0.484477\pi$
$978$	0	0
$979$	21.7706	0.695793
$980$	0	0
$981$	35.1815	1.12326
$982$	0	0
$983$	−20.6907	−0.659931	−0.329966	−	0.943993i	$-0.607037\pi$
$983$	−20.6907	−0.659931	−0.329966	+	0.943993i	$0.607037\pi$
$984$	0	0
$985$	10.2813	0.327591
$986$	0	0
$987$	−0.204840	−0.00652014
$988$	0	0
$989$	3.31957	0.105556
$990$	0	0
$991$	16.8702	0.535899	0.267950	−	0.963433i	$-0.413654\pi$
$991$	16.8702	0.535899	0.267950	+	0.963433i	$0.413654\pi$
$992$	0	0
$993$	23.4962	0.745630
$994$	0	0
$995$	3.97072	0.125880
$996$	0	0
$997$	−27.6848	−0.876787	−0.438393	−	0.898783i	$-0.644452\pi$
$997$	−27.6848	−0.876787	−0.438393	+	0.898783i	$0.644452\pi$
$998$	0	0
$999$	−25.6476	−0.811455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.20	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.20	✓	29	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.03867 q^{3} +1.00000 q^{5} -0.0217287 q^{7} -1.92117 q^{9} +O(q^{10})\)	\(q+1.03867 q^{3} +1.00000 q^{5} -0.0217287 q^{7} -1.92117 q^{9} -1.55147 q^{11} -0.436009 q^{13} +1.03867 q^{15} +7.24586 q^{17} -5.85124 q^{19} -0.0225689 q^{21} -8.05748 q^{23} +1.00000 q^{25} -5.11146 q^{27} -1.94015 q^{29} +10.0357 q^{31} -1.61146 q^{33} -0.0217287 q^{35} +5.01768 q^{37} -0.452868 q^{39} -7.09612 q^{41} -0.411986 q^{43} -1.92117 q^{45} +9.07623 q^{47} -6.99953 q^{49} +7.52603 q^{51} +6.43396 q^{53} -1.55147 q^{55} -6.07749 q^{57} +9.07361 q^{59} -11.0812 q^{61} +0.0417446 q^{63} -0.436009 q^{65} -13.3476 q^{67} -8.36903 q^{69} +10.5476 q^{71} -8.43416 q^{73} +1.03867 q^{75} +0.0337116 q^{77} -9.27918 q^{79} +0.454424 q^{81} -12.3781 q^{83} +7.24586 q^{85} -2.01517 q^{87} -14.0322 q^{89} +0.00947393 q^{91} +10.4237 q^{93} -5.85124 q^{95} -4.19522 q^{97} +2.98065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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