LMFDB - Embedded newform 8020.2.a.c.1.27

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:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.27
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)

$f(q)$	$=$	$q+2.66187 q^{3} -1.00000 q^{5} +1.97303 q^{7} +4.08554 q^{9} +O(q^{10})$	$q+2.66187 q^{3} -1.00000 q^{5} +1.97303 q^{7} +4.08554 q^{9} -1.73654 q^{11} -1.28974 q^{13} -2.66187 q^{15} -5.75442 q^{17} +5.34938 q^{19} +5.25195 q^{21} -8.91698 q^{23} +1.00000 q^{25} +2.88957 q^{27} -9.63811 q^{29} -2.06666 q^{31} -4.62243 q^{33} -1.97303 q^{35} -1.44986 q^{37} -3.43311 q^{39} +1.37577 q^{41} +2.31413 q^{43} -4.08554 q^{45} -6.46605 q^{47} -3.10715 q^{49} -15.3175 q^{51} -2.84853 q^{53} +1.73654 q^{55} +14.2393 q^{57} -0.511825 q^{59} -0.496265 q^{61} +8.06090 q^{63} +1.28974 q^{65} +0.713348 q^{67} -23.7358 q^{69} -16.7638 q^{71} -5.39401 q^{73} +2.66187 q^{75} -3.42624 q^{77} +16.8023 q^{79} -4.56497 q^{81} -3.22875 q^{83} +5.75442 q^{85} -25.6554 q^{87} +3.45594 q^{89} -2.54469 q^{91} -5.50118 q^{93} -5.34938 q^{95} +10.7457 q^{97} -7.09469 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	2.66187	1.53683	0.768415	−	0.639952i	$-0.221046\pi$
$3$	2.66187	1.53683	0.768415	+	0.639952i	$0.221046\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.97303	0.745736	0.372868	−	0.927884i	$-0.378374\pi$
$7$	1.97303	0.745736	0.372868	+	0.927884i	$0.378374\pi$
$8$	0	0
$9$	4.08554	1.36185
$10$	0	0
$11$	−1.73654	−0.523585	−0.261793	−	0.965124i	$-0.584314\pi$
$11$	−1.73654	−0.523585	−0.261793	+	0.965124i	$0.584314\pi$
$12$	0	0
$13$	−1.28974	−0.357709	−0.178854	−	0.983876i	$-0.557239\pi$
$13$	−1.28974	−0.357709	−0.178854	+	0.983876i	$0.557239\pi$
$14$	0	0
$15$	−2.66187	−0.687291
$16$	0	0
$17$	−5.75442	−1.39565	−0.697825	−	0.716268i	$-0.745849\pi$
$17$	−5.75442	−1.39565	−0.697825	+	0.716268i	$0.745849\pi$
$18$	0	0
$19$	5.34938	1.22723	0.613615	−	0.789605i	$-0.289715\pi$
$19$	5.34938	1.22723	0.613615	+	0.789605i	$0.289715\pi$
$20$	0	0
$21$	5.25195	1.14607
$22$	0	0
$23$	−8.91698	−1.85932	−0.929660	−	0.368419i	$-0.879899\pi$
$23$	−8.91698	−1.85932	−0.929660	+	0.368419i	$0.879899\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.88957	0.556098
$28$	0	0
$29$	−9.63811	−1.78975	−0.894876	−	0.446314i	$-0.852736\pi$
$29$	−9.63811	−1.78975	−0.894876	+	0.446314i	$0.852736\pi$
$30$	0	0
$31$	−2.06666	−0.371184	−0.185592	−	0.982627i	$-0.559420\pi$
$31$	−2.06666	−0.371184	−0.185592	+	0.982627i	$0.559420\pi$
$32$	0	0
$33$	−4.62243	−0.804662
$34$	0	0
$35$	−1.97303	−0.333503
$36$	0	0
$37$	−1.44986	−0.238356	−0.119178	−	0.992873i	$-0.538026\pi$
$37$	−1.44986	−0.238356	−0.119178	+	0.992873i	$0.538026\pi$
$38$	0	0
$39$	−3.43311	−0.549738
$40$	0	0
$41$	1.37577	0.214859	0.107430	−	0.994213i	$-0.465738\pi$
$41$	1.37577	0.214859	0.107430	+	0.994213i	$0.465738\pi$
$42$	0	0
$43$	2.31413	0.352901	0.176450	−	0.984310i	$-0.443538\pi$
$43$	2.31413	0.352901	0.176450	+	0.984310i	$0.443538\pi$
$44$	0	0
$45$	−4.08554	−0.609037
$46$	0	0
$47$	−6.46605	−0.943171	−0.471586	−	0.881820i	$-0.656318\pi$
$47$	−6.46605	−0.943171	−0.471586	+	0.881820i	$0.656318\pi$
$48$	0	0
$49$	−3.10715	−0.443878
$50$	0	0
$51$	−15.3175	−2.14488
$52$	0	0
$53$	−2.84853	−0.391275	−0.195638	−	0.980676i	$-0.562678\pi$
$53$	−2.84853	−0.391275	−0.195638	+	0.980676i	$0.562678\pi$
$54$	0	0
$55$	1.73654	0.234154
$56$	0	0
$57$	14.2393	1.88605
$58$	0	0
$59$	−0.511825	−0.0666339	−0.0333170	−	0.999445i	$-0.510607\pi$
$59$	−0.511825	−0.0666339	−0.0333170	+	0.999445i	$0.510607\pi$
$60$	0	0
$61$	−0.496265	−0.0635402	−0.0317701	−	0.999495i	$-0.510114\pi$
$61$	−0.496265	−0.0635402	−0.0317701	+	0.999495i	$0.510114\pi$
$62$	0	0
$63$	8.06090	1.01558
$64$	0	0
$65$	1.28974	0.159972
$66$	0	0
$67$	0.713348	0.0871494	0.0435747	−	0.999050i	$-0.486125\pi$
$67$	0.713348	0.0871494	0.0435747	+	0.999050i	$0.486125\pi$
$68$	0	0
$69$	−23.7358	−2.85746
$70$	0	0
$71$	−16.7638	−1.98950	−0.994750	−	0.102337i	$-0.967368\pi$
$71$	−16.7638	−1.98950	−0.994750	+	0.102337i	$0.967368\pi$
$72$	0	0
$73$	−5.39401	−0.631321	−0.315660	−	0.948872i	$-0.602226\pi$
$73$	−5.39401	−0.631321	−0.315660	+	0.948872i	$0.602226\pi$
$74$	0	0
$75$	2.66187	0.307366
$76$	0	0
$77$	−3.42624	−0.390456
$78$	0	0
$79$	16.8023	1.89040	0.945201	−	0.326489i	$-0.105866\pi$
$79$	16.8023	1.89040	0.945201	+	0.326489i	$0.105866\pi$
$80$	0	0
$81$	−4.56497	−0.507219
$82$	0	0
$83$	−3.22875	−0.354402	−0.177201	−	0.984175i	$-0.556704\pi$
$83$	−3.22875	−0.354402	−0.177201	+	0.984175i	$0.556704\pi$
$84$	0	0
$85$	5.75442	0.624154
$86$	0	0
$87$	−25.6554	−2.75055
$88$	0	0
$89$	3.45594	0.366329	0.183165	−	0.983082i	$-0.441366\pi$
$89$	3.45594	0.366329	0.183165	+	0.983082i	$0.441366\pi$
$90$	0	0
$91$	−2.54469	−0.266756
$92$	0	0
$93$	−5.50118	−0.570446
$94$	0	0
$95$	−5.34938	−0.548834
$96$	0	0
$97$	10.7457	1.09107	0.545533	−	0.838090i	$-0.316327\pi$
$97$	10.7457	1.09107	0.545533	+	0.838090i	$0.316327\pi$
$98$	0	0
$99$	−7.09469	−0.713043
$100$	0	0
$101$	11.5503	1.14930	0.574649	−	0.818400i	$-0.305138\pi$
$101$	11.5503	1.14930	0.574649	+	0.818400i	$0.305138\pi$
$102$	0	0
$103$	−0.542742	−0.0534779	−0.0267390	−	0.999642i	$-0.508512\pi$
$103$	−0.542742	−0.0534779	−0.0267390	+	0.999642i	$0.508512\pi$
$104$	0	0
$105$	−5.25195	−0.512538
$106$	0	0
$107$	4.62705	0.447314	0.223657	−	0.974668i	$-0.428200\pi$
$107$	4.62705	0.447314	0.223657	+	0.974668i	$0.428200\pi$
$108$	0	0
$109$	−11.4937	−1.10090	−0.550450	−	0.834868i	$-0.685544\pi$
$109$	−11.4937	−1.10090	−0.550450	+	0.834868i	$0.685544\pi$
$110$	0	0
$111$	−3.85934	−0.366312
$112$	0	0
$113$	5.33432	0.501810	0.250905	−	0.968012i	$-0.419272\pi$
$113$	5.33432	0.501810	0.250905	+	0.968012i	$0.419272\pi$
$114$	0	0
$115$	8.91698	0.831513
$116$	0	0
$117$	−5.26928	−0.487145
$118$	0	0
$119$	−11.3536	−1.04079
$120$	0	0
$121$	−7.98444	−0.725858
$122$	0	0
$123$	3.66212	0.330202
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.50798	0.133812	0.0669058	−	0.997759i	$-0.478687\pi$
$127$	1.50798	0.133812	0.0669058	+	0.997759i	$0.478687\pi$
$128$	0	0
$129$	6.15990	0.542349
$130$	0	0
$131$	−13.8380	−1.20903	−0.604517	−	0.796593i	$-0.706634\pi$
$131$	−13.8380	−1.20903	−0.604517	+	0.796593i	$0.706634\pi$
$132$	0	0
$133$	10.5545	0.915190
$134$	0	0
$135$	−2.88957	−0.248695
$136$	0	0
$137$	16.9690	1.44976	0.724881	−	0.688874i	$-0.241894\pi$
$137$	16.9690	1.44976	0.724881	+	0.688874i	$0.241894\pi$
$138$	0	0
$139$	1.98272	0.168172	0.0840859	−	0.996459i	$-0.473203\pi$
$139$	1.98272	0.168172	0.0840859	+	0.996459i	$0.473203\pi$
$140$	0	0
$141$	−17.2118	−1.44949
$142$	0	0
$143$	2.23968	0.187291
$144$	0	0
$145$	9.63811	0.800402
$146$	0	0
$147$	−8.27082	−0.682165
$148$	0	0
$149$	−10.2818	−0.842317	−0.421158	−	0.906987i	$-0.638376\pi$
$149$	−10.2818	−0.842317	−0.421158	+	0.906987i	$0.638376\pi$
$150$	0	0
$151$	1.65717	0.134858	0.0674292	−	0.997724i	$-0.478520\pi$
$151$	1.65717	0.134858	0.0674292	+	0.997724i	$0.478520\pi$
$152$	0	0
$153$	−23.5099	−1.90066
$154$	0	0
$155$	2.06666	0.165998
$156$	0	0
$157$	8.97819	0.716538	0.358269	−	0.933618i	$-0.383367\pi$
$157$	8.97819	0.716538	0.358269	+	0.933618i	$0.383367\pi$
$158$	0	0
$159$	−7.58241	−0.601324
$160$	0	0
$161$	−17.5935	−1.38656
$162$	0	0
$163$	12.0071	0.940466	0.470233	−	0.882542i	$-0.344170\pi$
$163$	12.0071	0.940466	0.470233	+	0.882542i	$0.344170\pi$
$164$	0	0
$165$	4.62243	0.359856
$166$	0	0
$167$	−5.21043	−0.403196	−0.201598	−	0.979468i	$-0.564613\pi$
$167$	−5.21043	−0.403196	−0.201598	+	0.979468i	$0.564613\pi$
$168$	0	0
$169$	−11.3366	−0.872044
$170$	0	0
$171$	21.8551	1.67130
$172$	0	0
$173$	−2.76333	−0.210092	−0.105046	−	0.994467i	$-0.533499\pi$
$173$	−2.76333	−0.210092	−0.105046	+	0.994467i	$0.533499\pi$
$174$	0	0
$175$	1.97303	0.149147
$176$	0	0
$177$	−1.36241	−0.102405
$178$	0	0
$179$	−6.51685	−0.487092	−0.243546	−	0.969889i	$-0.578311\pi$
$179$	−6.51685	−0.487092	−0.243546	+	0.969889i	$0.578311\pi$
$180$	0	0
$181$	10.6708	0.793157	0.396578	−	0.918001i	$-0.370198\pi$
$181$	10.6708	0.793157	0.396578	+	0.918001i	$0.370198\pi$
$182$	0	0
$183$	−1.32099	−0.0976505
$184$	0	0
$185$	1.44986	0.106596
$186$	0	0
$187$	9.99275	0.730742
$188$	0	0
$189$	5.70121	0.414702
$190$	0	0
$191$	7.41805	0.536751	0.268376	−	0.963314i	$-0.413513\pi$
$191$	7.41805	0.536751	0.268376	+	0.963314i	$0.413513\pi$
$192$	0	0
$193$	−2.34074	−0.168490	−0.0842452	−	0.996445i	$-0.526848\pi$
$193$	−2.34074	−0.168490	−0.0842452	+	0.996445i	$0.526848\pi$
$194$	0	0
$195$	3.43311	0.245850
$196$	0	0
$197$	−3.02970	−0.215857	−0.107928	−	0.994159i	$-0.534422\pi$
$197$	−3.02970	−0.215857	−0.107928	+	0.994159i	$0.534422\pi$
$198$	0	0
$199$	−11.1750	−0.792173	−0.396087	−	0.918213i	$-0.629632\pi$
$199$	−11.1750	−0.792173	−0.396087	+	0.918213i	$0.629632\pi$
$200$	0	0
$201$	1.89884	0.133934
$202$	0	0
$203$	−19.0163	−1.33468
$204$	0	0
$205$	−1.37577	−0.0960880
$206$	0	0
$207$	−36.4307	−2.53211
$208$	0	0
$209$	−9.28938	−0.642560
$210$	0	0
$211$	18.7913	1.29365	0.646825	−	0.762639i	$-0.276096\pi$
$211$	18.7913	1.29365	0.646825	+	0.762639i	$0.276096\pi$
$212$	0	0
$213$	−44.6231	−3.05752
$214$	0	0
$215$	−2.31413	−0.157822
$216$	0	0
$217$	−4.07759	−0.276805
$218$	0	0
$219$	−14.3581	−0.970233
$220$	0	0
$221$	7.42168	0.499236
$222$	0	0
$223$	21.8092	1.46045	0.730225	−	0.683207i	$-0.239415\pi$
$223$	21.8092	1.46045	0.730225	+	0.683207i	$0.239415\pi$
$224$	0	0
$225$	4.08554	0.272369
$226$	0	0
$227$	−12.6754	−0.841295	−0.420648	−	0.907224i	$-0.638197\pi$
$227$	−12.6754	−0.841295	−0.420648	+	0.907224i	$0.638197\pi$
$228$	0	0
$229$	−10.0679	−0.665306	−0.332653	−	0.943049i	$-0.607944\pi$
$229$	−10.0679	−0.665306	−0.332653	+	0.943049i	$0.607944\pi$
$230$	0	0
$231$	−9.12020	−0.600065
$232$	0	0
$233$	13.1021	0.858350	0.429175	−	0.903221i	$-0.358804\pi$
$233$	13.1021	0.858350	0.429175	+	0.903221i	$0.358804\pi$
$234$	0	0
$235$	6.46605	0.421799
$236$	0	0
$237$	44.7254	2.90523
$238$	0	0
$239$	0.316974	0.0205033	0.0102517	−	0.999947i	$-0.496737\pi$
$239$	0.316974	0.0205033	0.0102517	+	0.999947i	$0.496737\pi$
$240$	0	0
$241$	23.1115	1.48874	0.744371	−	0.667766i	$-0.232749\pi$
$241$	23.1115	1.48874	0.744371	+	0.667766i	$0.232749\pi$
$242$	0	0
$243$	−20.8201	−1.33561
$244$	0	0
$245$	3.10715	0.198508
$246$	0	0
$247$	−6.89929	−0.438991
$248$	0	0
$249$	−8.59452	−0.544655
$250$	0	0
$251$	−16.8828	−1.06563	−0.532817	−	0.846230i	$-0.678867\pi$
$251$	−16.8828	−1.06563	−0.532817	+	0.846230i	$0.678867\pi$
$252$	0	0
$253$	15.4847	0.973512
$254$	0	0
$255$	15.3175	0.959219
$256$	0	0
$257$	1.38042	0.0861082	0.0430541	−	0.999073i	$-0.486291\pi$
$257$	1.38042	0.0861082	0.0430541	+	0.999073i	$0.486291\pi$
$258$	0	0
$259$	−2.86062	−0.177750
$260$	0	0
$261$	−39.3769	−2.43737
$262$	0	0
$263$	5.05477	0.311691	0.155845	−	0.987781i	$-0.450190\pi$
$263$	5.05477	0.311691	0.155845	+	0.987781i	$0.450190\pi$
$264$	0	0
$265$	2.84853	0.174984
$266$	0	0
$267$	9.19927	0.562986
$268$	0	0
$269$	−4.19688	−0.255888	−0.127944	−	0.991781i	$-0.540838\pi$
$269$	−4.19688	−0.255888	−0.127944	+	0.991781i	$0.540838\pi$
$270$	0	0
$271$	−10.0936	−0.613143	−0.306571	−	0.951848i	$-0.599182\pi$
$271$	−10.0936	−0.613143	−0.306571	+	0.951848i	$0.599182\pi$
$272$	0	0
$273$	−6.77363	−0.409959
$274$	0	0
$275$	−1.73654	−0.104717
$276$	0	0
$277$	−3.91290	−0.235103	−0.117552	−	0.993067i	$-0.537505\pi$
$277$	−3.91290	−0.235103	−0.117552	+	0.993067i	$0.537505\pi$
$278$	0	0
$279$	−8.44344	−0.505495
$280$	0	0
$281$	16.7534	0.999426	0.499713	−	0.866191i	$-0.333439\pi$
$281$	16.7534	0.999426	0.499713	+	0.866191i	$0.333439\pi$
$282$	0	0
$283$	6.22909	0.370281	0.185140	−	0.982712i	$-0.440726\pi$
$283$	6.22909	0.370281	0.185140	+	0.982712i	$0.440726\pi$
$284$	0	0
$285$	−14.2393	−0.843465
$286$	0	0
$287$	2.71444	0.160228
$288$	0	0
$289$	16.1133	0.947841
$290$	0	0
$291$	28.6038	1.67678
$292$	0	0
$293$	−1.84336	−0.107690	−0.0538451	−	0.998549i	$-0.517148\pi$
$293$	−1.84336	−0.107690	−0.0538451	+	0.998549i	$0.517148\pi$
$294$	0	0
$295$	0.511825	0.0297996
$296$	0	0
$297$	−5.01784	−0.291165
$298$	0	0
$299$	11.5006	0.665095
$300$	0	0
$301$	4.56584	0.263171
$302$	0	0
$303$	30.7454	1.76628
$304$	0	0
$305$	0.496265	0.0284160
$306$	0	0
$307$	−23.6153	−1.34779	−0.673897	−	0.738825i	$-0.735381\pi$
$307$	−23.6153	−1.34779	−0.673897	+	0.738825i	$0.735381\pi$
$308$	0	0
$309$	−1.44471	−0.0821865
$310$	0	0
$311$	18.9960	1.07716	0.538581	−	0.842573i	$-0.318960\pi$
$311$	18.9960	1.07716	0.538581	+	0.842573i	$0.318960\pi$
$312$	0	0
$313$	9.21347	0.520776	0.260388	−	0.965504i	$-0.416150\pi$
$313$	9.21347	0.520776	0.260388	+	0.965504i	$0.416150\pi$
$314$	0	0
$315$	−8.06090	−0.454180
$316$	0	0
$317$	22.7981	1.28047	0.640236	−	0.768178i	$-0.278837\pi$
$317$	22.7981	1.28047	0.640236	+	0.768178i	$0.278837\pi$
$318$	0	0
$319$	16.7369	0.937088
$320$	0	0
$321$	12.3166	0.687446
$322$	0	0
$323$	−30.7825	−1.71279
$324$	0	0
$325$	−1.28974	−0.0715417
$326$	0	0
$327$	−30.5948	−1.69189
$328$	0	0
$329$	−12.7577	−0.703356
$330$	0	0
$331$	−18.4874	−1.01616	−0.508080	−	0.861310i	$-0.669645\pi$
$331$	−18.4874	−1.01616	−0.508080	+	0.861310i	$0.669645\pi$
$332$	0	0
$333$	−5.92347	−0.324604
$334$	0	0
$335$	−0.713348	−0.0389744
$336$	0	0
$337$	−3.62704	−0.197578	−0.0987888	−	0.995108i	$-0.531497\pi$
$337$	−3.62704	−0.197578	−0.0987888	+	0.995108i	$0.531497\pi$
$338$	0	0
$339$	14.1992	0.771197
$340$	0	0
$341$	3.58884	0.194346
$342$	0	0
$343$	−19.9417	−1.07675
$344$	0	0
$345$	23.7358	1.27789
$346$	0	0
$347$	4.88595	0.262292	0.131146	−	0.991363i	$-0.458134\pi$
$347$	4.88595	0.262292	0.131146	+	0.991363i	$0.458134\pi$
$348$	0	0
$349$	−11.9219	−0.638163	−0.319082	−	0.947727i	$-0.603374\pi$
$349$	−11.9219	−0.638163	−0.319082	+	0.947727i	$0.603374\pi$
$350$	0	0
$351$	−3.72679	−0.198921
$352$	0	0
$353$	9.61133	0.511560	0.255780	−	0.966735i	$-0.417668\pi$
$353$	9.61133	0.511560	0.255780	+	0.966735i	$0.417668\pi$
$354$	0	0
$355$	16.7638	0.889731
$356$	0	0
$357$	−30.2219	−1.59951
$358$	0	0
$359$	−12.3091	−0.649648	−0.324824	−	0.945775i	$-0.605305\pi$
$359$	−12.3091	−0.649648	−0.324824	+	0.945775i	$0.605305\pi$
$360$	0	0
$361$	9.61581	0.506095
$362$	0	0
$363$	−21.2535	−1.11552
$364$	0	0
$365$	5.39401	0.282335
$366$	0	0
$367$	−18.6925	−0.975740	−0.487870	−	0.872916i	$-0.662226\pi$
$367$	−18.6925	−0.975740	−0.487870	+	0.872916i	$0.662226\pi$
$368$	0	0
$369$	5.62077	0.292606
$370$	0	0
$371$	−5.62024	−0.291788
$372$	0	0
$373$	11.4877	0.594813	0.297406	−	0.954751i	$-0.403878\pi$
$373$	11.4877	0.594813	0.297406	+	0.954751i	$0.403878\pi$
$374$	0	0
$375$	−2.66187	−0.137458
$376$	0	0
$377$	12.4306	0.640210
$378$	0	0
$379$	−28.4499	−1.46137	−0.730686	−	0.682714i	$-0.760800\pi$
$379$	−28.4499	−1.46137	−0.730686	+	0.682714i	$0.760800\pi$
$380$	0	0
$381$	4.01404	0.205646
$382$	0	0
$383$	4.01158	0.204982	0.102491	−	0.994734i	$-0.467319\pi$
$383$	4.01158	0.204982	0.102491	+	0.994734i	$0.467319\pi$
$384$	0	0
$385$	3.42624	0.174617
$386$	0	0
$387$	9.45446	0.480597
$388$	0	0
$389$	−8.17462	−0.414470	−0.207235	−	0.978291i	$-0.566446\pi$
$389$	−8.17462	−0.414470	−0.207235	+	0.978291i	$0.566446\pi$
$390$	0	0
$391$	51.3120	2.59496
$392$	0	0
$393$	−36.8350	−1.85808
$394$	0	0
$395$	−16.8023	−0.845414
$396$	0	0
$397$	−15.8586	−0.795922	−0.397961	−	0.917402i	$-0.630282\pi$
$397$	−15.8586	−0.795922	−0.397961	+	0.917402i	$0.630282\pi$
$398$	0	0
$399$	28.0946	1.40649
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	2.66545	0.132776
$404$	0	0
$405$	4.56497	0.226835
$406$	0	0
$407$	2.51774	0.124800
$408$	0	0
$409$	17.8477	0.882512	0.441256	−	0.897381i	$-0.354533\pi$
$409$	17.8477	0.882512	0.441256	+	0.897381i	$0.354533\pi$
$410$	0	0
$411$	45.1693	2.22804
$412$	0	0
$413$	−1.00985	−0.0496913
$414$	0	0
$415$	3.22875	0.158493
$416$	0	0
$417$	5.27773	0.258451
$418$	0	0
$419$	9.25199	0.451989	0.225995	−	0.974129i	$-0.427437\pi$
$419$	9.25199	0.451989	0.225995	+	0.974129i	$0.427437\pi$
$420$	0	0
$421$	−7.53037	−0.367008	−0.183504	−	0.983019i	$-0.558744\pi$
$421$	−7.53037	−0.367008	−0.183504	+	0.983019i	$0.558744\pi$
$422$	0	0
$423$	−26.4173	−1.28446
$424$	0	0
$425$	−5.75442	−0.279130
$426$	0	0
$427$	−0.979146	−0.0473842
$428$	0	0
$429$	5.96172	0.287835
$430$	0	0
$431$	13.0802	0.630053	0.315027	−	0.949083i	$-0.397987\pi$
$431$	13.0802	0.630053	0.315027	+	0.949083i	$0.397987\pi$
$432$	0	0
$433$	−27.7077	−1.33155	−0.665773	−	0.746154i	$-0.731898\pi$
$433$	−27.7077	−1.33155	−0.665773	+	0.746154i	$0.731898\pi$
$434$	0	0
$435$	25.6554	1.23008
$436$	0	0
$437$	−47.7003	−2.28181
$438$	0	0
$439$	−4.24857	−0.202773	−0.101387	−	0.994847i	$-0.532328\pi$
$439$	−4.24857	−0.202773	−0.101387	+	0.994847i	$0.532328\pi$
$440$	0	0
$441$	−12.6944	−0.604494
$442$	0	0
$443$	−26.3611	−1.25245	−0.626226	−	0.779642i	$-0.715401\pi$
$443$	−26.3611	−1.25245	−0.626226	+	0.779642i	$0.715401\pi$
$444$	0	0
$445$	−3.45594	−0.163827
$446$	0	0
$447$	−27.3688	−1.29450
$448$	0	0
$449$	−7.32803	−0.345831	−0.172916	−	0.984937i	$-0.555319\pi$
$449$	−7.32803	−0.345831	−0.172916	+	0.984937i	$0.555319\pi$
$450$	0	0
$451$	−2.38908	−0.112497
$452$	0	0
$453$	4.41116	0.207254
$454$	0	0
$455$	2.54469	0.119297
$456$	0	0
$457$	−6.66876	−0.311952	−0.155976	−	0.987761i	$-0.549852\pi$
$457$	−6.66876	−0.311952	−0.155976	+	0.987761i	$0.549852\pi$
$458$	0	0
$459$	−16.6278	−0.776119
$460$	0	0
$461$	−24.1286	−1.12378	−0.561890	−	0.827212i	$-0.689925\pi$
$461$	−24.1286	−1.12378	−0.561890	+	0.827212i	$0.689925\pi$
$462$	0	0
$463$	−29.1461	−1.35454	−0.677268	−	0.735737i	$-0.736836\pi$
$463$	−29.1461	−1.35454	−0.677268	+	0.735737i	$0.736836\pi$
$464$	0	0
$465$	5.50118	0.255111
$466$	0	0
$467$	−5.61590	−0.259873	−0.129936	−	0.991522i	$-0.541477\pi$
$467$	−5.61590	−0.259873	−0.129936	+	0.991522i	$0.541477\pi$
$468$	0	0
$469$	1.40746	0.0649904
$470$	0	0
$471$	23.8988	1.10120
$472$	0	0
$473$	−4.01856	−0.184774
$474$	0	0
$475$	5.34938	0.245446
$476$	0	0
$477$	−11.6378	−0.532858
$478$	0	0
$479$	19.4087	0.886807	0.443403	−	0.896322i	$-0.353771\pi$
$479$	19.4087	0.886807	0.443403	+	0.896322i	$0.353771\pi$
$480$	0	0
$481$	1.86994	0.0852620
$482$	0	0
$483$	−46.8315	−2.13091
$484$	0	0
$485$	−10.7457	−0.487939
$486$	0	0
$487$	−17.7626	−0.804901	−0.402451	−	0.915442i	$-0.631842\pi$
$487$	−17.7626	−0.804901	−0.402451	+	0.915442i	$0.631842\pi$
$488$	0	0
$489$	31.9612	1.44534
$490$	0	0
$491$	33.8764	1.52882	0.764411	−	0.644729i	$-0.223030\pi$
$491$	33.8764	1.52882	0.764411	+	0.644729i	$0.223030\pi$
$492$	0	0
$493$	55.4617	2.49787
$494$	0	0
$495$	7.09469	0.318883
$496$	0	0
$497$	−33.0755	−1.48364
$498$	0	0
$499$	−21.3145	−0.954168	−0.477084	−	0.878858i	$-0.658306\pi$
$499$	−21.3145	−0.954168	−0.477084	+	0.878858i	$0.658306\pi$
$500$	0	0
$501$	−13.8695	−0.619643
$502$	0	0
$503$	30.3950	1.35525	0.677624	−	0.735409i	$-0.263010\pi$
$503$	30.3950	1.35525	0.677624	+	0.735409i	$0.263010\pi$
$504$	0	0
$505$	−11.5503	−0.513982
$506$	0	0
$507$	−30.1765	−1.34018
$508$	0	0
$509$	10.0971	0.447544	0.223772	−	0.974641i	$-0.428163\pi$
$509$	10.0971	0.447544	0.223772	+	0.974641i	$0.428163\pi$
$510$	0	0
$511$	−10.6425	−0.470798
$512$	0	0
$513$	15.4574	0.682461
$514$	0	0
$515$	0.542742	0.0239161
$516$	0	0
$517$	11.2285	0.493831
$518$	0	0
$519$	−7.35561	−0.322876
$520$	0	0
$521$	24.9046	1.09109	0.545546	−	0.838081i	$-0.316322\pi$
$521$	24.9046	1.09109	0.545546	+	0.838081i	$0.316322\pi$
$522$	0	0
$523$	9.20182	0.402367	0.201184	−	0.979554i	$-0.435521\pi$
$523$	9.20182	0.402367	0.201184	+	0.979554i	$0.435521\pi$
$524$	0	0
$525$	5.25195	0.229214
$526$	0	0
$527$	11.8924	0.518043
$528$	0	0
$529$	56.5126	2.45707
$530$	0	0
$531$	−2.09108	−0.0907453
$532$	0	0
$533$	−1.77438	−0.0768571
$534$	0	0
$535$	−4.62705	−0.200045
$536$	0	0
$537$	−17.3470	−0.748578
$538$	0	0
$539$	5.39567	0.232408
$540$	0	0
$541$	−39.7255	−1.70793	−0.853967	−	0.520327i	$-0.825810\pi$
$541$	−39.7255	−1.70793	−0.853967	+	0.520327i	$0.825810\pi$
$542$	0	0
$543$	28.4044	1.21895
$544$	0	0
$545$	11.4937	0.492337
$546$	0	0
$547$	7.06670	0.302150	0.151075	−	0.988522i	$-0.451726\pi$
$547$	7.06670	0.302150	0.151075	+	0.988522i	$0.451726\pi$
$548$	0	0
$549$	−2.02751	−0.0865320
$550$	0	0
$551$	−51.5579	−2.19644
$552$	0	0
$553$	33.1514	1.40974
$554$	0	0
$555$	3.85934	0.163820
$556$	0	0
$557$	27.1028	1.14838	0.574191	−	0.818722i	$-0.305317\pi$
$557$	27.1028	1.14838	0.574191	+	0.818722i	$0.305317\pi$
$558$	0	0
$559$	−2.98461	−0.126236
$560$	0	0
$561$	26.5994	1.12303
$562$	0	0
$563$	−29.5228	−1.24424	−0.622118	−	0.782923i	$-0.713728\pi$
$563$	−29.5228	−1.24424	−0.622118	+	0.782923i	$0.713728\pi$
$564$	0	0
$565$	−5.33432	−0.224416
$566$	0	0
$567$	−9.00683	−0.378251
$568$	0	0
$569$	−10.3733	−0.434871	−0.217435	−	0.976075i	$-0.569769\pi$
$569$	−10.3733	−0.434871	−0.217435	+	0.976075i	$0.569769\pi$
$570$	0	0
$571$	6.48443	0.271365	0.135683	−	0.990752i	$-0.456677\pi$
$571$	6.48443	0.271365	0.135683	+	0.990752i	$0.456677\pi$
$572$	0	0
$573$	19.7459	0.824896
$574$	0	0
$575$	−8.91698	−0.371864
$576$	0	0
$577$	−14.6997	−0.611955	−0.305978	−	0.952039i	$-0.598983\pi$
$577$	−14.6997	−0.611955	−0.305978	+	0.952039i	$0.598983\pi$
$578$	0	0
$579$	−6.23075	−0.258941
$580$	0	0
$581$	−6.37043	−0.264290
$582$	0	0
$583$	4.94657	0.204866
$584$	0	0
$585$	5.26928	0.217858
$586$	0	0
$587$	−33.3101	−1.37486	−0.687428	−	0.726252i	$-0.741260\pi$
$587$	−33.3101	−1.37486	−0.687428	+	0.726252i	$0.741260\pi$
$588$	0	0
$589$	−11.0554	−0.455528
$590$	0	0
$591$	−8.06465	−0.331735
$592$	0	0
$593$	2.44030	0.100211	0.0501056	−	0.998744i	$-0.484044\pi$
$593$	2.44030	0.100211	0.0501056	+	0.998744i	$0.484044\pi$
$594$	0	0
$595$	11.3536	0.465454
$596$	0	0
$597$	−29.7463	−1.21744
$598$	0	0
$599$	−38.4577	−1.57134	−0.785669	−	0.618648i	$-0.787681\pi$
$599$	−38.4577	−1.57134	−0.785669	+	0.618648i	$0.787681\pi$
$600$	0	0
$601$	−1.96557	−0.0801773	−0.0400886	−	0.999196i	$-0.512764\pi$
$601$	−1.96557	−0.0801773	−0.0400886	+	0.999196i	$0.512764\pi$
$602$	0	0
$603$	2.91441	0.118684
$604$	0	0
$605$	7.98444	0.324614
$606$	0	0
$607$	−44.4099	−1.80254	−0.901271	−	0.433256i	$-0.857364\pi$
$607$	−44.4099	−1.80254	−0.901271	+	0.433256i	$0.857364\pi$
$608$	0	0
$609$	−50.6189	−2.05118
$610$	0	0
$611$	8.33951	0.337381
$612$	0	0
$613$	−7.45355	−0.301046	−0.150523	−	0.988607i	$-0.548096\pi$
$613$	−7.45355	−0.301046	−0.150523	+	0.988607i	$0.548096\pi$
$614$	0	0
$615$	−3.66212	−0.147671
$616$	0	0
$617$	23.5261	0.947125	0.473563	−	0.880760i	$-0.342968\pi$
$617$	23.5261	0.947125	0.473563	+	0.880760i	$0.342968\pi$
$618$	0	0
$619$	0.921010	0.0370185	0.0185093	−	0.999829i	$-0.494108\pi$
$619$	0.921010	0.0370185	0.0185093	+	0.999829i	$0.494108\pi$
$620$	0	0
$621$	−25.7663	−1.03396
$622$	0	0
$623$	6.81869	0.273185
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−24.7271	−0.987506
$628$	0	0
$629$	8.34311	0.332661
$630$	0	0
$631$	14.8621	0.591652	0.295826	−	0.955242i	$-0.404405\pi$
$631$	14.8621	0.591652	0.295826	+	0.955242i	$0.404405\pi$
$632$	0	0
$633$	50.0201	1.98812
$634$	0	0
$635$	−1.50798	−0.0598423
$636$	0	0
$637$	4.00740	0.158779
$638$	0	0
$639$	−68.4893	−2.70939
$640$	0	0
$641$	−36.3714	−1.43658	−0.718291	−	0.695742i	$-0.755076\pi$
$641$	−36.3714	−1.43658	−0.718291	+	0.695742i	$0.755076\pi$
$642$	0	0
$643$	−4.63830	−0.182917	−0.0914583	−	0.995809i	$-0.529153\pi$
$643$	−4.63830	−0.182917	−0.0914583	+	0.995809i	$0.529153\pi$
$644$	0	0
$645$	−6.15990	−0.242546
$646$	0	0
$647$	−33.0617	−1.29979	−0.649894	−	0.760025i	$-0.725187\pi$
$647$	−33.0617	−1.29979	−0.649894	+	0.760025i	$0.725187\pi$
$648$	0	0
$649$	0.888803	0.0348886
$650$	0	0
$651$	−10.8540	−0.425402
$652$	0	0
$653$	41.2781	1.61534	0.807668	−	0.589638i	$-0.200730\pi$
$653$	41.2781	1.61534	0.807668	+	0.589638i	$0.200730\pi$
$654$	0	0
$655$	13.8380	0.540696
$656$	0	0
$657$	−22.0374	−0.859762
$658$	0	0
$659$	6.56806	0.255855	0.127928	−	0.991783i	$-0.459167\pi$
$659$	6.56806	0.255855	0.127928	+	0.991783i	$0.459167\pi$
$660$	0	0
$661$	−46.4366	−1.80617	−0.903087	−	0.429458i	$-0.858704\pi$
$661$	−46.4366	−1.80617	−0.903087	+	0.429458i	$0.858704\pi$
$662$	0	0
$663$	19.7555	0.767242
$664$	0	0
$665$	−10.5545	−0.409285
$666$	0	0
$667$	85.9429	3.32772
$668$	0	0
$669$	58.0532	2.24446
$670$	0	0
$671$	0.861781	0.0332687
$672$	0	0
$673$	25.0349	0.965026	0.482513	−	0.875889i	$-0.339724\pi$
$673$	25.0349	0.965026	0.482513	+	0.875889i	$0.339724\pi$
$674$	0	0
$675$	2.88957	0.111220
$676$	0	0
$677$	19.3135	0.742280	0.371140	−	0.928577i	$-0.378967\pi$
$677$	19.3135	0.742280	0.371140	+	0.928577i	$0.378967\pi$
$678$	0	0
$679$	21.2017	0.813646
$680$	0	0
$681$	−33.7402	−1.29293
$682$	0	0
$683$	39.4966	1.51130	0.755649	−	0.654977i	$-0.227322\pi$
$683$	39.4966	1.51130	0.755649	+	0.654977i	$0.227322\pi$
$684$	0	0
$685$	−16.9690	−0.648354
$686$	0	0
$687$	−26.7994	−1.02246
$688$	0	0
$689$	3.67385	0.139963
$690$	0	0
$691$	−25.6669	−0.976415	−0.488207	−	0.872728i	$-0.662349\pi$
$691$	−25.6669	−0.976415	−0.488207	+	0.872728i	$0.662349\pi$
$692$	0	0
$693$	−13.9980	−0.531742
$694$	0	0
$695$	−1.98272	−0.0752087
$696$	0	0
$697$	−7.91676	−0.299869
$698$	0	0
$699$	34.8762	1.31914
$700$	0	0
$701$	36.8030	1.39003	0.695016	−	0.718994i	$-0.255397\pi$
$701$	36.8030	1.39003	0.695016	+	0.718994i	$0.255397\pi$
$702$	0	0
$703$	−7.75585	−0.292518
$704$	0	0
$705$	17.2118	0.648233
$706$	0	0
$707$	22.7891	0.857073
$708$	0	0
$709$	−21.7125	−0.815429	−0.407715	−	0.913109i	$-0.633674\pi$
$709$	−21.7125	−0.815429	−0.407715	+	0.913109i	$0.633674\pi$
$710$	0	0
$711$	68.6464	2.57444
$712$	0	0
$713$	18.4284	0.690149
$714$	0	0
$715$	−2.23968	−0.0837591
$716$	0	0
$717$	0.843742	0.0315101
$718$	0	0
$719$	19.5318	0.728413	0.364207	−	0.931318i	$-0.381340\pi$
$719$	19.5318	0.728413	0.364207	+	0.931318i	$0.381340\pi$
$720$	0	0
$721$	−1.07085	−0.0398804
$722$	0	0
$723$	61.5198	2.28794
$724$	0	0
$725$	−9.63811	−0.357951
$726$	0	0
$727$	−37.0169	−1.37288	−0.686440	−	0.727187i	$-0.740827\pi$
$727$	−37.0169	−1.37288	−0.686440	+	0.727187i	$0.740827\pi$
$728$	0	0
$729$	−41.7254	−1.54538
$730$	0	0
$731$	−13.3164	−0.492526
$732$	0	0
$733$	17.7039	0.653910	0.326955	−	0.945040i	$-0.393977\pi$
$733$	17.7039	0.653910	0.326955	+	0.945040i	$0.393977\pi$
$734$	0	0
$735$	8.27082	0.305074
$736$	0	0
$737$	−1.23875	−0.0456301
$738$	0	0
$739$	−24.1160	−0.887122	−0.443561	−	0.896244i	$-0.646285\pi$
$739$	−24.1160	−0.887122	−0.443561	+	0.896244i	$0.646285\pi$
$740$	0	0
$741$	−18.3650	−0.674655
$742$	0	0
$743$	−24.8881	−0.913056	−0.456528	−	0.889709i	$-0.650907\pi$
$743$	−24.8881	−0.913056	−0.456528	+	0.889709i	$0.650907\pi$
$744$	0	0
$745$	10.2818	0.376696
$746$	0	0
$747$	−13.1912	−0.482641
$748$	0	0
$749$	9.12932	0.333578
$750$	0	0
$751$	−32.7864	−1.19639	−0.598196	−	0.801349i	$-0.704116\pi$
$751$	−32.7864	−1.19639	−0.598196	+	0.801349i	$0.704116\pi$
$752$	0	0
$753$	−44.9398	−1.63770
$754$	0	0
$755$	−1.65717	−0.0603105
$756$	0	0
$757$	20.9084	0.759929	0.379965	−	0.925001i	$-0.375936\pi$
$757$	20.9084	0.759929	0.379965	+	0.925001i	$0.375936\pi$
$758$	0	0
$759$	41.2181	1.49612
$760$	0	0
$761$	24.8425	0.900538	0.450269	−	0.892893i	$-0.351328\pi$
$761$	24.8425	0.900538	0.450269	+	0.892893i	$0.351328\pi$
$762$	0	0
$763$	−22.6775	−0.820980
$764$	0	0
$765$	23.5099	0.850002
$766$	0	0
$767$	0.660120	0.0238355
$768$	0	0
$769$	25.5175	0.920184	0.460092	−	0.887871i	$-0.347816\pi$
$769$	25.5175	0.920184	0.460092	+	0.887871i	$0.347816\pi$
$770$	0	0
$771$	3.67449	0.132334
$772$	0	0
$773$	27.2242	0.979185	0.489593	−	0.871951i	$-0.337146\pi$
$773$	27.2242	0.979185	0.489593	+	0.871951i	$0.337146\pi$
$774$	0	0
$775$	−2.06666	−0.0742367
$776$	0	0
$777$	−7.61460	−0.273172
$778$	0	0
$779$	7.35952	0.263682
$780$	0	0
$781$	29.1110	1.04167
$782$	0	0
$783$	−27.8500	−0.995278
$784$	0	0
$785$	−8.97819	−0.320445
$786$	0	0
$787$	−11.4650	−0.408684	−0.204342	−	0.978900i	$-0.565505\pi$
$787$	−11.4650	−0.408684	−0.204342	+	0.978900i	$0.565505\pi$
$788$	0	0
$789$	13.4551	0.479016
$790$	0	0
$791$	10.5248	0.374218
$792$	0	0
$793$	0.640051	0.0227289
$794$	0	0
$795$	7.58241	0.268920
$796$	0	0
$797$	31.7969	1.12630	0.563152	−	0.826354i	$-0.309589\pi$
$797$	31.7969	1.12630	0.563152	+	0.826354i	$0.309589\pi$
$798$	0	0
$799$	37.2084	1.31634
$800$	0	0
$801$	14.1194	0.498885
$802$	0	0
$803$	9.36689	0.330550
$804$	0	0
$805$	17.5935	0.620089
$806$	0	0
$807$	−11.1715	−0.393257
$808$	0	0
$809$	12.4472	0.437620	0.218810	−	0.975767i	$-0.429782\pi$
$809$	12.4472	0.437620	0.218810	+	0.975767i	$0.429782\pi$
$810$	0	0
$811$	24.5391	0.861683	0.430842	−	0.902428i	$-0.358217\pi$
$811$	24.5391	0.861683	0.430842	+	0.902428i	$0.358217\pi$
$812$	0	0
$813$	−26.8679	−0.942297
$814$	0	0
$815$	−12.0071	−0.420589
$816$	0	0
$817$	12.3791	0.433091
$818$	0	0
$819$	−10.3964	−0.363281
$820$	0	0
$821$	30.8302	1.07598	0.537991	−	0.842951i	$-0.319184\pi$
$821$	30.8302	1.07598	0.537991	+	0.842951i	$0.319184\pi$
$822$	0	0
$823$	36.7269	1.28022	0.640110	−	0.768283i	$-0.278889\pi$
$823$	36.7269	1.28022	0.640110	+	0.768283i	$0.278889\pi$
$824$	0	0
$825$	−4.62243	−0.160932
$826$	0	0
$827$	45.1346	1.56948	0.784742	−	0.619823i	$-0.212796\pi$
$827$	45.1346	1.56948	0.784742	+	0.619823i	$0.212796\pi$
$828$	0	0
$829$	7.89661	0.274260	0.137130	−	0.990553i	$-0.456212\pi$
$829$	7.89661	0.274260	0.137130	+	0.990553i	$0.456212\pi$
$830$	0	0
$831$	−10.4156	−0.361314
$832$	0	0
$833$	17.8798	0.619499
$834$	0	0
$835$	5.21043	0.180315
$836$	0	0
$837$	−5.97177	−0.206415
$838$	0	0
$839$	36.7342	1.26820	0.634102	−	0.773249i	$-0.281370\pi$
$839$	36.7342	1.26820	0.634102	+	0.773249i	$0.281370\pi$
$840$	0	0
$841$	63.8932	2.20322
$842$	0	0
$843$	44.5954	1.53595
$844$	0	0
$845$	11.3366	0.389990
$846$	0	0
$847$	−15.7536	−0.541299
$848$	0	0
$849$	16.5810	0.569059
$850$	0	0
$851$	12.9284	0.443180
$852$	0	0
$853$	10.6265	0.363846	0.181923	−	0.983313i	$-0.441768\pi$
$853$	10.6265	0.363846	0.181923	+	0.983313i	$0.441768\pi$
$854$	0	0
$855$	−21.8551	−0.747429
$856$	0	0
$857$	3.13144	0.106968	0.0534841	−	0.998569i	$-0.482967\pi$
$857$	3.13144	0.106968	0.0534841	+	0.998569i	$0.482967\pi$
$858$	0	0
$859$	11.1292	0.379724	0.189862	−	0.981811i	$-0.439196\pi$
$859$	11.1292	0.379724	0.189862	+	0.981811i	$0.439196\pi$
$860$	0	0
$861$	7.22548	0.246244
$862$	0	0
$863$	2.79697	0.0952099	0.0476049	−	0.998866i	$-0.484841\pi$
$863$	2.79697	0.0952099	0.0476049	+	0.998866i	$0.484841\pi$
$864$	0	0
$865$	2.76333	0.0939559
$866$	0	0
$867$	42.8915	1.45667
$868$	0	0
$869$	−29.1777	−0.989787
$870$	0	0
$871$	−0.920032	−0.0311741
$872$	0	0
$873$	43.9022	1.48586
$874$	0	0
$875$	−1.97303	−0.0667006
$876$	0	0
$877$	−49.1702	−1.66036	−0.830179	−	0.557496i	$-0.811762\pi$
$877$	−49.1702	−1.66036	−0.830179	+	0.557496i	$0.811762\pi$
$878$	0	0
$879$	−4.90678	−0.165502
$880$	0	0
$881$	−26.9219	−0.907022	−0.453511	−	0.891251i	$-0.649829\pi$
$881$	−26.9219	−0.907022	−0.453511	+	0.891251i	$0.649829\pi$
$882$	0	0
$883$	−6.20825	−0.208924	−0.104462	−	0.994529i	$-0.533312\pi$
$883$	−6.20825	−0.208924	−0.104462	+	0.994529i	$0.533312\pi$
$884$	0	0
$885$	1.36241	0.0457969
$886$	0	0
$887$	−8.82398	−0.296280	−0.148140	−	0.988966i	$-0.547329\pi$
$887$	−8.82398	−0.296280	−0.148140	+	0.988966i	$0.547329\pi$
$888$	0	0
$889$	2.97529	0.0997880
$890$	0	0
$891$	7.92724	0.265572
$892$	0	0
$893$	−34.5894	−1.15749
$894$	0	0
$895$	6.51685	0.217834
$896$	0	0
$897$	30.6130	1.02214
$898$	0	0
$899$	19.9187	0.664327
$900$	0	0
$901$	16.3916	0.546084
$902$	0	0
$903$	12.1537	0.404449
$904$	0	0
$905$	−10.6708	−0.354710
$906$	0	0
$907$	−14.0418	−0.466251	−0.233125	−	0.972447i	$-0.574895\pi$
$907$	−14.0418	−0.466251	−0.233125	+	0.972447i	$0.574895\pi$
$908$	0	0
$909$	47.1893	1.56517
$910$	0	0
$911$	39.5330	1.30979	0.654893	−	0.755721i	$-0.272714\pi$
$911$	39.5330	1.30979	0.654893	+	0.755721i	$0.272714\pi$
$912$	0	0
$913$	5.60685	0.185560
$914$	0	0
$915$	1.32099	0.0436706
$916$	0	0
$917$	−27.3028	−0.901619
$918$	0	0
$919$	−2.12983	−0.0702565	−0.0351283	−	0.999383i	$-0.511184\pi$
$919$	−2.12983	−0.0702565	−0.0351283	+	0.999383i	$0.511184\pi$
$920$	0	0
$921$	−62.8607	−2.07133
$922$	0	0
$923$	21.6209	0.711661
$924$	0	0
$925$	−1.44986	−0.0476712
$926$	0	0
$927$	−2.21739	−0.0728288
$928$	0	0
$929$	−34.0872	−1.11836	−0.559182	−	0.829045i	$-0.688885\pi$
$929$	−34.0872	−1.11836	−0.559182	+	0.829045i	$0.688885\pi$
$930$	0	0
$931$	−16.6213	−0.544741
$932$	0	0
$933$	50.5648	1.65542
$934$	0	0
$935$	−9.99275	−0.326798
$936$	0	0
$937$	10.2010	0.333253	0.166627	−	0.986020i	$-0.446713\pi$
$937$	10.2010	0.333253	0.166627	+	0.986020i	$0.446713\pi$
$938$	0	0
$939$	24.5250	0.800345
$940$	0	0
$941$	−46.9720	−1.53124	−0.765621	−	0.643291i	$-0.777568\pi$
$941$	−46.9720	−1.53124	−0.765621	+	0.643291i	$0.777568\pi$
$942$	0	0
$943$	−12.2677	−0.399492
$944$	0	0
$945$	−5.70121	−0.185460
$946$	0	0
$947$	26.7792	0.870208	0.435104	−	0.900380i	$-0.356712\pi$
$947$	26.7792	0.870208	0.435104	+	0.900380i	$0.356712\pi$
$948$	0	0
$949$	6.95685	0.225829
$950$	0	0
$951$	60.6857	1.96787
$952$	0	0
$953$	31.1396	1.00871	0.504356	−	0.863496i	$-0.331730\pi$
$953$	31.1396	1.00871	0.504356	+	0.863496i	$0.331730\pi$
$954$	0	0
$955$	−7.41805	−0.240042
$956$	0	0
$957$	44.5515	1.44015
$958$	0	0
$959$	33.4804	1.08114
$960$	0	0
$961$	−26.7289	−0.862223
$962$	0	0
$963$	18.9040	0.609174
$964$	0	0
$965$	2.34074	0.0753512
$966$	0	0
$967$	−3.01844	−0.0970663	−0.0485332	−	0.998822i	$-0.515455\pi$
$967$	−3.01844	−0.0970663	−0.0485332	+	0.998822i	$0.515455\pi$
$968$	0	0
$969$	−81.9390	−2.63226
$970$	0	0
$971$	−25.6181	−0.822125	−0.411062	−	0.911607i	$-0.634842\pi$
$971$	−25.6181	−0.822125	−0.411062	+	0.911607i	$0.634842\pi$
$972$	0	0
$973$	3.91196	0.125412
$974$	0	0
$975$	−3.43311	−0.109948
$976$	0	0
$977$	−59.5149	−1.90405	−0.952025	−	0.306021i	$-0.901002\pi$
$977$	−59.5149	−1.90405	−0.952025	+	0.306021i	$0.901002\pi$
$978$	0	0
$979$	−6.00137	−0.191805
$980$	0	0
$981$	−46.9581	−1.49926
$982$	0	0
$983$	25.0208	0.798039	0.399019	−	0.916942i	$-0.369351\pi$
$983$	25.0208	0.798039	0.399019	+	0.916942i	$0.369351\pi$
$984$	0	0
$985$	3.02970	0.0965341
$986$	0	0
$987$	−33.9594	−1.08094
$988$	0	0
$989$	−20.6350	−0.656156
$990$	0	0
$991$	6.63951	0.210911	0.105456	−	0.994424i	$-0.466370\pi$
$991$	6.63951	0.210911	0.105456	+	0.994424i	$0.466370\pi$
$992$	0	0
$993$	−49.2111	−1.56167
$994$	0	0
$995$	11.1750	0.354271
$996$	0	0
$997$	−39.1152	−1.23879	−0.619396	−	0.785079i	$-0.712622\pi$
$997$	−39.1152	−1.23879	−0.619396	+	0.785079i	$0.712622\pi$
$998$	0	0
$999$	−4.18948	−0.132549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.27	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.27	✓	28	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)

\(f(q)\)	\(=\)	\(q+2.66187 q^{3} -1.00000 q^{5} +1.97303 q^{7} +4.08554 q^{9} +O(q^{10})\)	\(q+2.66187 q^{3} -1.00000 q^{5} +1.97303 q^{7} +4.08554 q^{9} -1.73654 q^{11} -1.28974 q^{13} -2.66187 q^{15} -5.75442 q^{17} +5.34938 q^{19} +5.25195 q^{21} -8.91698 q^{23} +1.00000 q^{25} +2.88957 q^{27} -9.63811 q^{29} -2.06666 q^{31} -4.62243 q^{33} -1.97303 q^{35} -1.44986 q^{37} -3.43311 q^{39} +1.37577 q^{41} +2.31413 q^{43} -4.08554 q^{45} -6.46605 q^{47} -3.10715 q^{49} -15.3175 q^{51} -2.84853 q^{53} +1.73654 q^{55} +14.2393 q^{57} -0.511825 q^{59} -0.496265 q^{61} +8.06090 q^{63} +1.28974 q^{65} +0.713348 q^{67} -23.7358 q^{69} -16.7638 q^{71} -5.39401 q^{73} +2.66187 q^{75} -3.42624 q^{77} +16.8023 q^{79} -4.56497 q^{81} -3.22875 q^{83} +5.75442 q^{85} -25.6554 q^{87} +3.45594 q^{89} -2.54469 q^{91} -5.50118 q^{93} -5.34938 q^{95} +10.7457 q^{97} -7.09469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more