LMFDB - Embedded newform 8020.2.a.c.1.13

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			1.13
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)

$f(q)$	$=$	$q-0.172341 q^{3} -1.00000 q^{5} +3.94901 q^{7} -2.97030 q^{9} +O(q^{10})$	$q-0.172341 q^{3} -1.00000 q^{5} +3.94901 q^{7} -2.97030 q^{9} +2.50533 q^{11} +5.08392 q^{13} +0.172341 q^{15} -5.79588 q^{17} -1.40336 q^{19} -0.680578 q^{21} -4.18550 q^{23} +1.00000 q^{25} +1.02893 q^{27} +3.24974 q^{29} -2.87221 q^{31} -0.431771 q^{33} -3.94901 q^{35} -2.92358 q^{37} -0.876169 q^{39} -11.1289 q^{41} -1.07418 q^{43} +2.97030 q^{45} +0.742520 q^{47} +8.59469 q^{49} +0.998869 q^{51} -7.39320 q^{53} -2.50533 q^{55} +0.241857 q^{57} -13.2989 q^{59} -13.5530 q^{61} -11.7297 q^{63} -5.08392 q^{65} -0.250028 q^{67} +0.721335 q^{69} +1.16282 q^{71} +9.19316 q^{73} -0.172341 q^{75} +9.89356 q^{77} +10.1231 q^{79} +8.73357 q^{81} -10.6880 q^{83} +5.79588 q^{85} -0.560065 q^{87} +14.6364 q^{89} +20.0765 q^{91} +0.495000 q^{93} +1.40336 q^{95} -3.29704 q^{97} -7.44157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.172341	−0.0995013	−0.0497506	−	0.998762i	$-0.515843\pi$
$3$	−0.172341	−0.0995013	−0.0497506	+	0.998762i	$0.515843\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.94901	1.49259	0.746293	−	0.665618i	$-0.231832\pi$
$7$	3.94901	1.49259	0.746293	+	0.665618i	$0.231832\pi$
$8$	0	0
$9$	−2.97030	−0.990099
$10$	0	0
$11$	2.50533	0.755384	0.377692	−	0.925931i	$-0.376718\pi$
$11$	2.50533	0.755384	0.377692	+	0.925931i	$0.376718\pi$
$12$	0	0
$13$	5.08392	1.41003	0.705013	−	0.709195i	$-0.250941\pi$
$13$	5.08392	1.41003	0.705013	+	0.709195i	$0.250941\pi$
$14$	0	0
$15$	0.172341	0.0444983
$16$	0	0
$17$	−5.79588	−1.40571	−0.702853	−	0.711335i	$-0.748091\pi$
$17$	−5.79588	−1.40571	−0.702853	+	0.711335i	$0.748091\pi$
$18$	0	0
$19$	−1.40336	−0.321953	−0.160976	−	0.986958i	$-0.551464\pi$
$19$	−1.40336	−0.321953	−0.160976	+	0.986958i	$0.551464\pi$
$20$	0	0
$21$	−0.680578	−0.148514
$22$	0	0
$23$	−4.18550	−0.872738	−0.436369	−	0.899768i	$-0.643736\pi$
$23$	−4.18550	−0.872738	−0.436369	+	0.899768i	$0.643736\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.02893	0.198017
$28$	0	0
$29$	3.24974	0.603462	0.301731	−	0.953393i	$-0.402436\pi$
$29$	3.24974	0.603462	0.301731	+	0.953393i	$0.402436\pi$
$30$	0	0
$31$	−2.87221	−0.515864	−0.257932	−	0.966163i	$-0.583041\pi$
$31$	−2.87221	−0.515864	−0.257932	+	0.966163i	$0.583041\pi$
$32$	0	0
$33$	−0.431771	−0.0751617
$34$	0	0
$35$	−3.94901	−0.667505
$36$	0	0
$37$	−2.92358	−0.480634	−0.240317	−	0.970695i	$-0.577251\pi$
$37$	−2.92358	−0.480634	−0.240317	+	0.970695i	$0.577251\pi$
$38$	0	0
$39$	−0.876169	−0.140299
$40$	0	0
$41$	−11.1289	−1.73805	−0.869023	−	0.494771i	$-0.835252\pi$
$41$	−11.1289	−1.73805	−0.869023	+	0.494771i	$0.835252\pi$
$42$	0	0
$43$	−1.07418	−0.163811	−0.0819053	−	0.996640i	$-0.526101\pi$
$43$	−1.07418	−0.163811	−0.0819053	+	0.996640i	$0.526101\pi$
$44$	0	0
$45$	2.97030	0.442786
$46$	0	0
$47$	0.742520	0.108308	0.0541539	−	0.998533i	$-0.482754\pi$
$47$	0.742520	0.108308	0.0541539	+	0.998533i	$0.482754\pi$
$48$	0	0
$49$	8.59469	1.22781
$50$	0	0
$51$	0.998869	0.139870
$52$	0	0
$53$	−7.39320	−1.01553	−0.507767	−	0.861494i	$-0.669529\pi$
$53$	−7.39320	−1.01553	−0.507767	+	0.861494i	$0.669529\pi$
$54$	0	0
$55$	−2.50533	−0.337818
$56$	0	0
$57$	0.241857	0.0320347
$58$	0	0
$59$	−13.2989	−1.73137	−0.865685	−	0.500589i	$-0.833117\pi$
$59$	−13.2989	−1.73137	−0.865685	+	0.500589i	$0.833117\pi$
$60$	0	0
$61$	−13.5530	−1.73528	−0.867642	−	0.497190i	$-0.834365\pi$
$61$	−13.5530	−1.73528	−0.867642	+	0.497190i	$0.834365\pi$
$62$	0	0
$63$	−11.7297	−1.47781
$64$	0	0
$65$	−5.08392	−0.630583
$66$	0	0
$67$	−0.250028	−0.0305458	−0.0152729	−	0.999883i	$-0.504862\pi$
$67$	−0.250028	−0.0305458	−0.0152729	+	0.999883i	$0.504862\pi$
$68$	0	0
$69$	0.721335	0.0868385
$70$	0	0
$71$	1.16282	0.138001	0.0690006	−	0.997617i	$-0.478019\pi$
$71$	1.16282	0.138001	0.0690006	+	0.997617i	$0.478019\pi$
$72$	0	0
$73$	9.19316	1.07598	0.537989	−	0.842952i	$-0.319184\pi$
$73$	9.19316	1.07598	0.537989	+	0.842952i	$0.319184\pi$
$74$	0	0
$75$	−0.172341	−0.0199003
$76$	0	0
$77$	9.89356	1.12748
$78$	0	0
$79$	10.1231	1.13894	0.569471	−	0.822012i	$-0.307148\pi$
$79$	10.1231	1.13894	0.569471	+	0.822012i	$0.307148\pi$
$80$	0	0
$81$	8.73357	0.970396
$82$	0	0
$83$	−10.6880	−1.17317	−0.586583	−	0.809889i	$-0.699527\pi$
$83$	−10.6880	−1.17317	−0.586583	+	0.809889i	$0.699527\pi$
$84$	0	0
$85$	5.79588	0.628651
$86$	0	0
$87$	−0.560065	−0.0600453
$88$	0	0
$89$	14.6364	1.55145	0.775726	−	0.631069i	$-0.217384\pi$
$89$	14.6364	1.55145	0.775726	+	0.631069i	$0.217384\pi$
$90$	0	0
$91$	20.0765	2.10458
$92$	0	0
$93$	0.495000	0.0513292
$94$	0	0
$95$	1.40336	0.143982
$96$	0	0
$97$	−3.29704	−0.334764	−0.167382	−	0.985892i	$-0.553531\pi$
$97$	−3.29704	−0.334764	−0.167382	+	0.985892i	$0.553531\pi$
$98$	0	0
$99$	−7.44157	−0.747906
$100$	0	0
$101$	8.66341	0.862042	0.431021	−	0.902342i	$-0.358153\pi$
$101$	8.66341	0.862042	0.431021	+	0.902342i	$0.358153\pi$
$102$	0	0
$103$	7.45638	0.734699	0.367350	−	0.930083i	$-0.380265\pi$
$103$	7.45638	0.734699	0.367350	+	0.930083i	$0.380265\pi$
$104$	0	0
$105$	0.680578	0.0664176
$106$	0	0
$107$	−15.2815	−1.47732	−0.738658	−	0.674080i	$-0.764540\pi$
$107$	−15.2815	−1.47732	−0.738658	+	0.674080i	$0.764540\pi$
$108$	0	0
$109$	−4.47754	−0.428871	−0.214435	−	0.976738i	$-0.568791\pi$
$109$	−4.47754	−0.428871	−0.214435	+	0.976738i	$0.568791\pi$
$110$	0	0
$111$	0.503854	0.0478237
$112$	0	0
$113$	−4.16788	−0.392081	−0.196041	−	0.980596i	$-0.562808\pi$
$113$	−4.16788	−0.392081	−0.196041	+	0.980596i	$0.562808\pi$
$114$	0	0
$115$	4.18550	0.390300
$116$	0	0
$117$	−15.1008	−1.39607
$118$	0	0
$119$	−22.8880	−2.09814
$120$	0	0
$121$	−4.72334	−0.429394
$122$	0	0
$123$	1.91797	0.172938
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.08740	−0.540170	−0.270085	−	0.962837i	$-0.587052\pi$
$127$	−6.08740	−0.540170	−0.270085	+	0.962837i	$0.587052\pi$
$128$	0	0
$129$	0.185125	0.0162994
$130$	0	0
$131$	−0.0374609	−0.00327298	−0.00163649	−	0.999999i	$-0.500521\pi$
$131$	−0.0374609	−0.00327298	−0.00163649	+	0.999999i	$0.500521\pi$
$132$	0	0
$133$	−5.54188	−0.480542
$134$	0	0
$135$	−1.02893	−0.0885561
$136$	0	0
$137$	−16.1557	−1.38027	−0.690136	−	0.723680i	$-0.742449\pi$
$137$	−16.1557	−1.38027	−0.690136	+	0.723680i	$0.742449\pi$
$138$	0	0
$139$	−5.26258	−0.446366	−0.223183	−	0.974777i	$-0.571645\pi$
$139$	−5.26258	−0.446366	−0.223183	+	0.974777i	$0.571645\pi$
$140$	0	0
$141$	−0.127967	−0.0107768
$142$	0	0
$143$	12.7369	1.06511
$144$	0	0
$145$	−3.24974	−0.269876
$146$	0	0
$147$	−1.48122	−0.122169
$148$	0	0
$149$	10.0907	0.826666	0.413333	−	0.910580i	$-0.364365\pi$
$149$	10.0907	0.826666	0.413333	+	0.910580i	$0.364365\pi$
$150$	0	0
$151$	−14.0956	−1.14708	−0.573541	−	0.819177i	$-0.694431\pi$
$151$	−14.0956	−1.14708	−0.573541	+	0.819177i	$0.694431\pi$
$152$	0	0
$153$	17.2155	1.39179
$154$	0	0
$155$	2.87221	0.230701
$156$	0	0
$157$	−4.50373	−0.359436	−0.179718	−	0.983718i	$-0.557519\pi$
$157$	−4.50373	−0.359436	−0.179718	+	0.983718i	$0.557519\pi$
$158$	0	0
$159$	1.27415	0.101047
$160$	0	0
$161$	−16.5286	−1.30264
$162$	0	0
$163$	−2.08183	−0.163062	−0.0815308	−	0.996671i	$-0.525981\pi$
$163$	−2.08183	−0.163062	−0.0815308	+	0.996671i	$0.525981\pi$
$164$	0	0
$165$	0.431771	0.0336133
$166$	0	0
$167$	−14.2807	−1.10508	−0.552538	−	0.833488i	$-0.686341\pi$
$167$	−14.2807	−1.10508	−0.552538	+	0.833488i	$0.686341\pi$
$168$	0	0
$169$	12.8462	0.988172
$170$	0	0
$171$	4.16839	0.318765
$172$	0	0
$173$	15.9731	1.21441	0.607206	−	0.794545i	$-0.292290\pi$
$173$	15.9731	1.21441	0.607206	+	0.794545i	$0.292290\pi$
$174$	0	0
$175$	3.94901	0.298517
$176$	0	0
$177$	2.29195	0.172274
$178$	0	0
$179$	−11.8862	−0.888414	−0.444207	−	0.895924i	$-0.646515\pi$
$179$	−11.8862	−0.888414	−0.444207	+	0.895924i	$0.646515\pi$
$180$	0	0
$181$	23.4450	1.74266	0.871328	−	0.490701i	$-0.163259\pi$
$181$	23.4450	1.74266	0.871328	+	0.490701i	$0.163259\pi$
$182$	0	0
$183$	2.33574	0.172663
$184$	0	0
$185$	2.92358	0.214946
$186$	0	0
$187$	−14.5206	−1.06185
$188$	0	0
$189$	4.06325	0.295558
$190$	0	0
$191$	11.0712	0.801084	0.400542	−	0.916278i	$-0.368822\pi$
$191$	11.0712	0.801084	0.400542	+	0.916278i	$0.368822\pi$
$192$	0	0
$193$	13.6691	0.983923	0.491961	−	0.870617i	$-0.336280\pi$
$193$	13.6691	0.983923	0.491961	+	0.870617i	$0.336280\pi$
$194$	0	0
$195$	0.876169	0.0627438
$196$	0	0
$197$	5.96193	0.424770	0.212385	−	0.977186i	$-0.431877\pi$
$197$	5.96193	0.424770	0.212385	+	0.977186i	$0.431877\pi$
$198$	0	0
$199$	12.2406	0.867712	0.433856	−	0.900982i	$-0.357153\pi$
$199$	12.2406	0.867712	0.433856	+	0.900982i	$0.357153\pi$
$200$	0	0
$201$	0.0430902	0.00303935
$202$	0	0
$203$	12.8333	0.900719
$204$	0	0
$205$	11.1289	0.777278
$206$	0	0
$207$	12.4322	0.864097
$208$	0	0
$209$	−3.51587	−0.243198
$210$	0	0
$211$	−16.8163	−1.15768	−0.578841	−	0.815441i	$-0.696495\pi$
$211$	−16.8163	−1.15768	−0.578841	+	0.815441i	$0.696495\pi$
$212$	0	0
$213$	−0.200402	−0.0137313
$214$	0	0
$215$	1.07418	0.0732583
$216$	0	0
$217$	−11.3424	−0.769972
$218$	0	0
$219$	−1.58436	−0.107061
$220$	0	0
$221$	−29.4658	−1.98208
$222$	0	0
$223$	12.1078	0.810798	0.405399	−	0.914140i	$-0.367133\pi$
$223$	12.1078	0.810798	0.405399	+	0.914140i	$0.367133\pi$
$224$	0	0
$225$	−2.97030	−0.198020
$226$	0	0
$227$	8.55660	0.567922	0.283961	−	0.958836i	$-0.408351\pi$
$227$	8.55660	0.567922	0.283961	+	0.958836i	$0.408351\pi$
$228$	0	0
$229$	19.5042	1.28887	0.644437	−	0.764657i	$-0.277092\pi$
$229$	19.5042	1.28887	0.644437	+	0.764657i	$0.277092\pi$
$230$	0	0
$231$	−1.70507	−0.112185
$232$	0	0
$233$	16.1430	1.05756	0.528782	−	0.848758i	$-0.322649\pi$
$233$	16.1430	1.05756	0.528782	+	0.848758i	$0.322649\pi$
$234$	0	0
$235$	−0.742520	−0.0484367
$236$	0	0
$237$	−1.74463	−0.113326
$238$	0	0
$239$	20.4042	1.31984	0.659919	−	0.751337i	$-0.270590\pi$
$239$	20.4042	1.31984	0.659919	+	0.751337i	$0.270590\pi$
$240$	0	0
$241$	−16.6841	−1.07472	−0.537358	−	0.843354i	$-0.680578\pi$
$241$	−16.6841	−1.07472	−0.537358	+	0.843354i	$0.680578\pi$
$242$	0	0
$243$	−4.59194	−0.294573
$244$	0	0
$245$	−8.59469	−0.549095
$246$	0	0
$247$	−7.13456	−0.453961
$248$	0	0
$249$	1.84199	0.116732
$250$	0	0
$251$	−9.38049	−0.592091	−0.296046	−	0.955174i	$-0.595668\pi$
$251$	−9.38049	−0.592091	−0.296046	+	0.955174i	$0.595668\pi$
$252$	0	0
$253$	−10.4861	−0.659253
$254$	0	0
$255$	−0.998869	−0.0625516
$256$	0	0
$257$	−13.7357	−0.856807	−0.428404	−	0.903587i	$-0.640924\pi$
$257$	−13.7357	−0.856807	−0.428404	+	0.903587i	$0.640924\pi$
$258$	0	0
$259$	−11.5453	−0.717387
$260$	0	0
$261$	−9.65271	−0.597488
$262$	0	0
$263$	−12.0127	−0.740735	−0.370368	−	0.928885i	$-0.620768\pi$
$263$	−12.0127	−0.740735	−0.370368	+	0.928885i	$0.620768\pi$
$264$	0	0
$265$	7.39320	0.454161
$266$	0	0
$267$	−2.52245	−0.154372
$268$	0	0
$269$	−16.4260	−1.00151	−0.500755	−	0.865589i	$-0.666944\pi$
$269$	−16.4260	−1.00151	−0.500755	+	0.865589i	$0.666944\pi$
$270$	0	0
$271$	−8.82884	−0.536314	−0.268157	−	0.963375i	$-0.586415\pi$
$271$	−8.82884	−0.536314	−0.268157	+	0.963375i	$0.586415\pi$
$272$	0	0
$273$	−3.46000	−0.209409
$274$	0	0
$275$	2.50533	0.151077
$276$	0	0
$277$	−9.90537	−0.595156	−0.297578	−	0.954698i	$-0.596179\pi$
$277$	−9.90537	−0.595156	−0.297578	+	0.954698i	$0.596179\pi$
$278$	0	0
$279$	8.53132	0.510757
$280$	0	0
$281$	−8.53454	−0.509128	−0.254564	−	0.967056i	$-0.581932\pi$
$281$	−8.53454	−0.509128	−0.254564	+	0.967056i	$0.581932\pi$
$282$	0	0
$283$	−25.4398	−1.51224	−0.756120	−	0.654433i	$-0.772907\pi$
$283$	−25.4398	−1.51224	−0.756120	+	0.654433i	$0.772907\pi$
$284$	0	0
$285$	−0.241857	−0.0143263
$286$	0	0
$287$	−43.9483	−2.59418
$288$	0	0
$289$	16.5922	0.976011
$290$	0	0
$291$	0.568216	0.0333094
$292$	0	0
$293$	−25.1395	−1.46867	−0.734334	−	0.678788i	$-0.762505\pi$
$293$	−25.1395	−1.46867	−0.734334	+	0.678788i	$0.762505\pi$
$294$	0	0
$295$	13.2989	0.774292
$296$	0	0
$297$	2.57780	0.149579
$298$	0	0
$299$	−21.2788	−1.23058
$300$	0	0
$301$	−4.24194	−0.244501
$302$	0	0
$303$	−1.49306	−0.0857743
$304$	0	0
$305$	13.5530	0.776042
$306$	0	0
$307$	11.7002	0.667767	0.333884	−	0.942614i	$-0.391641\pi$
$307$	11.7002	0.667767	0.333884	+	0.942614i	$0.391641\pi$
$308$	0	0
$309$	−1.28504	−0.0731035
$310$	0	0
$311$	3.84370	0.217956	0.108978	−	0.994044i	$-0.465242\pi$
$311$	3.84370	0.217956	0.108978	+	0.994044i	$0.465242\pi$
$312$	0	0
$313$	−26.2050	−1.48119	−0.740596	−	0.671951i	$-0.765457\pi$
$313$	−26.2050	−1.48119	−0.740596	+	0.671951i	$0.765457\pi$
$314$	0	0
$315$	11.7297	0.660896
$316$	0	0
$317$	27.3801	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$317$	27.3801	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$318$	0	0
$319$	8.14167	0.455846
$320$	0	0
$321$	2.63363	0.146995
$322$	0	0
$323$	8.13369	0.452571
$324$	0	0
$325$	5.08392	0.282005
$326$	0	0
$327$	0.771665	0.0426732
$328$	0	0
$329$	2.93222	0.161659
$330$	0	0
$331$	0.294394	0.0161814	0.00809068	−	0.999967i	$-0.497425\pi$
$331$	0.294394	0.0161814	0.00809068	+	0.999967i	$0.497425\pi$
$332$	0	0
$333$	8.68391	0.475875
$334$	0	0
$335$	0.250028	0.0136605
$336$	0	0
$337$	−18.2719	−0.995333	−0.497667	−	0.867368i	$-0.665810\pi$
$337$	−18.2719	−0.995333	−0.497667	+	0.867368i	$0.665810\pi$
$338$	0	0
$339$	0.718298	0.0390126
$340$	0	0
$341$	−7.19582	−0.389676
$342$	0	0
$343$	6.29746	0.340031
$344$	0	0
$345$	−0.721335	−0.0388354
$346$	0	0
$347$	−27.2663	−1.46373	−0.731866	−	0.681449i	$-0.761350\pi$
$347$	−27.2663	−1.46373	−0.731866	+	0.681449i	$0.761350\pi$
$348$	0	0
$349$	14.2024	0.760235	0.380117	−	0.924938i	$-0.375884\pi$
$349$	14.2024	0.760235	0.380117	+	0.924938i	$0.375884\pi$
$350$	0	0
$351$	5.23099	0.279210
$352$	0	0
$353$	−12.9886	−0.691316	−0.345658	−	0.938361i	$-0.612344\pi$
$353$	−12.9886	−0.691316	−0.345658	+	0.938361i	$0.612344\pi$
$354$	0	0
$355$	−1.16282	−0.0617160
$356$	0	0
$357$	3.94454	0.208767
$358$	0	0
$359$	−25.7593	−1.35952	−0.679762	−	0.733432i	$-0.737917\pi$
$359$	−25.7593	−1.35952	−0.679762	+	0.733432i	$0.737917\pi$
$360$	0	0
$361$	−17.0306	−0.896347
$362$	0	0
$363$	0.814026	0.0427253
$364$	0	0
$365$	−9.19316	−0.481192
$366$	0	0
$367$	2.93246	0.153073	0.0765366	−	0.997067i	$-0.475614\pi$
$367$	2.93246	0.153073	0.0765366	+	0.997067i	$0.475614\pi$
$368$	0	0
$369$	33.0562	1.72084
$370$	0	0
$371$	−29.1958	−1.51577
$372$	0	0
$373$	−3.42751	−0.177470	−0.0887350	−	0.996055i	$-0.528282\pi$
$373$	−3.42751	−0.177470	−0.0887350	+	0.996055i	$0.528282\pi$
$374$	0	0
$375$	0.172341	0.00889967
$376$	0	0
$377$	16.5214	0.850897
$378$	0	0
$379$	−9.04806	−0.464768	−0.232384	−	0.972624i	$-0.574653\pi$
$379$	−9.04806	−0.464768	−0.232384	+	0.972624i	$0.574653\pi$
$380$	0	0
$381$	1.04911	0.0537476
$382$	0	0
$383$	25.7643	1.31650	0.658248	−	0.752801i	$-0.271298\pi$
$383$	25.7643	1.31650	0.658248	+	0.752801i	$0.271298\pi$
$384$	0	0
$385$	−9.89356	−0.504223
$386$	0	0
$387$	3.19063	0.162189
$388$	0	0
$389$	27.9374	1.41648	0.708240	−	0.705971i	$-0.249489\pi$
$389$	27.9374	1.41648	0.708240	+	0.705971i	$0.249489\pi$
$390$	0	0
$391$	24.2587	1.22681
$392$	0	0
$393$	0.00645607	0.000325665 0
$394$	0	0
$395$	−10.1231	−0.509350
$396$	0	0
$397$	22.9558	1.15212	0.576060	−	0.817407i	$-0.304589\pi$
$397$	22.9558	1.15212	0.576060	+	0.817407i	$0.304589\pi$
$398$	0	0
$399$	0.955095	0.0478145
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−14.6021	−0.727382
$404$	0	0
$405$	−8.73357	−0.433975
$406$	0	0
$407$	−7.32452	−0.363063
$408$	0	0
$409$	12.4661	0.616410	0.308205	−	0.951320i	$-0.400272\pi$
$409$	12.4661	0.616410	0.308205	+	0.951320i	$0.400272\pi$
$410$	0	0
$411$	2.78429	0.137339
$412$	0	0
$413$	−52.5175	−2.58422
$414$	0	0
$415$	10.6880	0.524656
$416$	0	0
$417$	0.906960	0.0444140
$418$	0	0
$419$	−0.822135	−0.0401639	−0.0200820	−	0.999798i	$-0.506393\pi$
$419$	−0.822135	−0.0401639	−0.0200820	+	0.999798i	$0.506393\pi$
$420$	0	0
$421$	3.89826	0.189990	0.0949948	−	0.995478i	$-0.469717\pi$
$421$	3.89826	0.189990	0.0949948	+	0.995478i	$0.469717\pi$
$422$	0	0
$423$	−2.20551	−0.107235
$424$	0	0
$425$	−5.79588	−0.281141
$426$	0	0
$427$	−53.5209	−2.59006
$428$	0	0
$429$	−2.19509	−0.105980
$430$	0	0
$431$	−10.3758	−0.499787	−0.249893	−	0.968273i	$-0.580396\pi$
$431$	−10.3758	−0.499787	−0.249893	+	0.968273i	$0.580396\pi$
$432$	0	0
$433$	0.266162	0.0127909	0.00639547	−	0.999980i	$-0.497964\pi$
$433$	0.266162	0.0127909	0.00639547	+	0.999980i	$0.497964\pi$
$434$	0	0
$435$	0.560065	0.0268531
$436$	0	0
$437$	5.87376	0.280980
$438$	0	0
$439$	17.4599	0.833317	0.416658	−	0.909063i	$-0.363201\pi$
$439$	17.4599	0.833317	0.416658	+	0.909063i	$0.363201\pi$
$440$	0	0
$441$	−25.5288	−1.21566
$442$	0	0
$443$	−3.30611	−0.157078	−0.0785390	−	0.996911i	$-0.525026\pi$
$443$	−3.30611	−0.157078	−0.0785390	+	0.996911i	$0.525026\pi$
$444$	0	0
$445$	−14.6364	−0.693831
$446$	0	0
$447$	−1.73905	−0.0822544
$448$	0	0
$449$	−40.4136	−1.90724	−0.953619	−	0.301018i	$-0.902674\pi$
$449$	−40.4136	−1.90724	−0.953619	+	0.301018i	$0.902674\pi$
$450$	0	0
$451$	−27.8816	−1.31289
$452$	0	0
$453$	2.42925	0.114136
$454$	0	0
$455$	−20.0765	−0.941199
$456$	0	0
$457$	−10.7329	−0.502065	−0.251033	−	0.967979i	$-0.580770\pi$
$457$	−10.7329	−0.502065	−0.251033	+	0.967979i	$0.580770\pi$
$458$	0	0
$459$	−5.96355	−0.278354
$460$	0	0
$461$	0.208109	0.00969258	0.00484629	−	0.999988i	$-0.498457\pi$
$461$	0.208109	0.00969258	0.00484629	+	0.999988i	$0.498457\pi$
$462$	0	0
$463$	−16.3676	−0.760665	−0.380332	−	0.924850i	$-0.624190\pi$
$463$	−16.3676	−0.760665	−0.380332	+	0.924850i	$0.624190\pi$
$464$	0	0
$465$	−0.495000	−0.0229551
$466$	0	0
$467$	−21.3688	−0.988829	−0.494415	−	0.869226i	$-0.664618\pi$
$467$	−21.3688	−0.988829	−0.494415	+	0.869226i	$0.664618\pi$
$468$	0	0
$469$	−0.987364	−0.0455922
$470$	0	0
$471$	0.776178	0.0357644
$472$	0	0
$473$	−2.69117	−0.123740
$474$	0	0
$475$	−1.40336	−0.0643905
$476$	0	0
$477$	21.9600	1.00548
$478$	0	0
$479$	−29.5482	−1.35009	−0.675047	−	0.737775i	$-0.735877\pi$
$479$	−29.5482	−1.35009	−0.675047	+	0.737775i	$0.735877\pi$
$480$	0	0
$481$	−14.8632	−0.677706
$482$	0	0
$483$	2.84856	0.129614
$484$	0	0
$485$	3.29704	0.149711
$486$	0	0
$487$	20.3957	0.924218	0.462109	−	0.886823i	$-0.347093\pi$
$487$	20.3957	0.924218	0.462109	+	0.886823i	$0.347093\pi$
$488$	0	0
$489$	0.358786	0.0162249
$490$	0	0
$491$	15.0246	0.678053	0.339026	−	0.940777i	$-0.389902\pi$
$491$	15.0246	0.678053	0.339026	+	0.940777i	$0.389902\pi$
$492$	0	0
$493$	−18.8351	−0.848291
$494$	0	0
$495$	7.44157	0.334474
$496$	0	0
$497$	4.59198	0.205979
$498$	0	0
$499$	15.9864	0.715648	0.357824	−	0.933789i	$-0.383519\pi$
$499$	15.9864	0.715648	0.357824	+	0.933789i	$0.383519\pi$
$500$	0	0
$501$	2.46116	0.109957
$502$	0	0
$503$	11.8103	0.526594	0.263297	−	0.964715i	$-0.415190\pi$
$503$	11.8103	0.526594	0.263297	+	0.964715i	$0.415190\pi$
$504$	0	0
$505$	−8.66341	−0.385517
$506$	0	0
$507$	−2.21394	−0.0983244
$508$	0	0
$509$	−44.2108	−1.95961	−0.979805	−	0.199954i	$-0.935921\pi$
$509$	−44.2108	−1.95961	−0.979805	+	0.199954i	$0.935921\pi$
$510$	0	0
$511$	36.3039	1.60599
$512$	0	0
$513$	−1.44396	−0.0637522
$514$	0	0
$515$	−7.45638	−0.328568
$516$	0	0
$517$	1.86026	0.0818140
$518$	0	0
$519$	−2.75282	−0.120836
$520$	0	0
$521$	−14.7082	−0.644380	−0.322190	−	0.946675i	$-0.604419\pi$
$521$	−14.7082	−0.644380	−0.322190	+	0.946675i	$0.604419\pi$
$522$	0	0
$523$	−6.92390	−0.302761	−0.151380	−	0.988476i	$-0.548372\pi$
$523$	−6.92390	−0.302761	−0.151380	+	0.988476i	$0.548372\pi$
$524$	0	0
$525$	−0.680578	−0.0297028
$526$	0	0
$527$	16.6470	0.725154
$528$	0	0
$529$	−5.48156	−0.238329
$530$	0	0
$531$	39.5017	1.71423
$532$	0	0
$533$	−56.5786	−2.45069
$534$	0	0
$535$	15.2815	0.660676
$536$	0	0
$537$	2.04848	0.0883984
$538$	0	0
$539$	21.5325	0.927471
$540$	0	0
$541$	−33.3206	−1.43256	−0.716282	−	0.697811i	$-0.754157\pi$
$541$	−33.3206	−1.43256	−0.716282	+	0.697811i	$0.754157\pi$
$542$	0	0
$543$	−4.04055	−0.173397
$544$	0	0
$545$	4.47754	0.191797
$546$	0	0
$547$	−19.6932	−0.842021	−0.421011	−	0.907056i	$-0.638324\pi$
$547$	−19.6932	−0.842021	−0.421011	+	0.907056i	$0.638324\pi$
$548$	0	0
$549$	40.2564	1.71810
$550$	0	0
$551$	−4.56055	−0.194286
$552$	0	0
$553$	39.9764	1.69997
$554$	0	0
$555$	−0.503854	−0.0213874
$556$	0	0
$557$	−39.1600	−1.65926	−0.829632	−	0.558310i	$-0.811450\pi$
$557$	−39.1600	−1.65926	−0.829632	+	0.558310i	$0.811450\pi$
$558$	0	0
$559$	−5.46104	−0.230977
$560$	0	0
$561$	2.50249	0.105655
$562$	0	0
$563$	30.2992	1.27696	0.638480	−	0.769638i	$-0.279564\pi$
$563$	30.2992	1.27696	0.638480	+	0.769638i	$0.279564\pi$
$564$	0	0
$565$	4.16788	0.175344
$566$	0	0
$567$	34.4890	1.44840
$568$	0	0
$569$	−26.9939	−1.13164	−0.565822	−	0.824528i	$-0.691441\pi$
$569$	−26.9939	−1.13164	−0.565822	+	0.824528i	$0.691441\pi$
$570$	0	0
$571$	−42.1830	−1.76530	−0.882651	−	0.470029i	$-0.844244\pi$
$571$	−42.1830	−1.76530	−0.882651	+	0.470029i	$0.844244\pi$
$572$	0	0
$573$	−1.90802	−0.0797089
$574$	0	0
$575$	−4.18550	−0.174548
$576$	0	0
$577$	−13.0524	−0.543377	−0.271689	−	0.962385i	$-0.587582\pi$
$577$	−13.0524	−0.543377	−0.271689	+	0.962385i	$0.587582\pi$
$578$	0	0
$579$	−2.35575	−0.0979016
$580$	0	0
$581$	−42.2072	−1.75105
$582$	0	0
$583$	−18.5224	−0.767119
$584$	0	0
$585$	15.1008	0.624339
$586$	0	0
$587$	34.1703	1.41036	0.705179	−	0.709029i	$-0.250867\pi$
$587$	34.1703	1.41036	0.705179	+	0.709029i	$0.250867\pi$
$588$	0	0
$589$	4.03074	0.166084
$590$	0	0
$591$	−1.02749	−0.0422652
$592$	0	0
$593$	−12.6877	−0.521019	−0.260510	−	0.965471i	$-0.583891\pi$
$593$	−12.6877	−0.521019	−0.260510	+	0.965471i	$0.583891\pi$
$594$	0	0
$595$	22.8880	0.938316
$596$	0	0
$597$	−2.10956	−0.0863385
$598$	0	0
$599$	43.2204	1.76594	0.882969	−	0.469432i	$-0.155541\pi$
$599$	43.2204	1.76594	0.882969	+	0.469432i	$0.155541\pi$
$600$	0	0
$601$	5.16914	0.210854	0.105427	−	0.994427i	$-0.466379\pi$
$601$	5.16914	0.210854	0.105427	+	0.994427i	$0.466379\pi$
$602$	0	0
$603$	0.742658	0.0302434
$604$	0	0
$605$	4.72334	0.192031
$606$	0	0
$607$	−20.0138	−0.812333	−0.406167	−	0.913799i	$-0.633135\pi$
$607$	−20.0138	−0.812333	−0.406167	+	0.913799i	$0.633135\pi$
$608$	0	0
$609$	−2.21170	−0.0896227
$610$	0	0
$611$	3.77491	0.152717
$612$	0	0
$613$	44.9835	1.81687	0.908434	−	0.418029i	$-0.137279\pi$
$613$	44.9835	1.81687	0.908434	+	0.418029i	$0.137279\pi$
$614$	0	0
$615$	−1.91797	−0.0773402
$616$	0	0
$617$	5.50806	0.221746	0.110873	−	0.993835i	$-0.464635\pi$
$617$	5.50806	0.221746	0.110873	+	0.993835i	$0.464635\pi$
$618$	0	0
$619$	−2.84120	−0.114198	−0.0570988	−	0.998369i	$-0.518185\pi$
$619$	−2.84120	−0.114198	−0.0570988	+	0.998369i	$0.518185\pi$
$620$	0	0
$621$	−4.30659	−0.172817
$622$	0	0
$623$	57.7992	2.31568
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.605930	0.0241985
$628$	0	0
$629$	16.9447	0.675630
$630$	0	0
$631$	37.5017	1.49292	0.746459	−	0.665432i	$-0.231753\pi$
$631$	37.5017	1.49292	0.746459	+	0.665432i	$0.231753\pi$
$632$	0	0
$633$	2.89814	0.115191
$634$	0	0
$635$	6.08740	0.241571
$636$	0	0
$637$	43.6947	1.73125
$638$	0	0
$639$	−3.45392	−0.136635
$640$	0	0
$641$	0.515023	0.0203422	0.0101711	−	0.999948i	$-0.496762\pi$
$641$	0.515023	0.0203422	0.0101711	+	0.999948i	$0.496762\pi$
$642$	0	0
$643$	34.7070	1.36871	0.684356	−	0.729148i	$-0.260083\pi$
$643$	34.7070	1.36871	0.684356	+	0.729148i	$0.260083\pi$
$644$	0	0
$645$	−0.185125	−0.00728930
$646$	0	0
$647$	−2.06179	−0.0810574	−0.0405287	−	0.999178i	$-0.512904\pi$
$647$	−2.06179	−0.0810574	−0.0405287	+	0.999178i	$0.512904\pi$
$648$	0	0
$649$	−33.3181	−1.30785
$650$	0	0
$651$	1.95476	0.0766132
$652$	0	0
$653$	−4.21345	−0.164885	−0.0824426	−	0.996596i	$-0.526272\pi$
$653$	−4.21345	−0.164885	−0.0824426	+	0.996596i	$0.526272\pi$
$654$	0	0
$655$	0.0374609	0.00146372
$656$	0	0
$657$	−27.3064	−1.06533
$658$	0	0
$659$	40.3729	1.57271	0.786353	−	0.617777i	$-0.211967\pi$
$659$	40.3729	1.57271	0.786353	+	0.617777i	$0.211967\pi$
$660$	0	0
$661$	−2.51668	−0.0978876	−0.0489438	−	0.998802i	$-0.515586\pi$
$661$	−2.51668	−0.0978876	−0.0489438	+	0.998802i	$0.515586\pi$
$662$	0	0
$663$	5.07817	0.197220
$664$	0	0
$665$	5.54188	0.214905
$666$	0	0
$667$	−13.6018	−0.526664
$668$	0	0
$669$	−2.08667	−0.0806755
$670$	0	0
$671$	−33.9547	−1.31081
$672$	0	0
$673$	24.0961	0.928836	0.464418	−	0.885616i	$-0.346264\pi$
$673$	24.0961	0.928836	0.464418	+	0.885616i	$0.346264\pi$
$674$	0	0
$675$	1.02893	0.0396035
$676$	0	0
$677$	27.5605	1.05923	0.529617	−	0.848237i	$-0.322336\pi$
$677$	27.5605	1.05923	0.529617	+	0.848237i	$0.322336\pi$
$678$	0	0
$679$	−13.0201	−0.499664
$680$	0	0
$681$	−1.47466	−0.0565089
$682$	0	0
$683$	31.1261	1.19101	0.595504	−	0.803352i	$-0.296952\pi$
$683$	31.1261	1.19101	0.595504	+	0.803352i	$0.296952\pi$
$684$	0	0
$685$	16.1557	0.617276
$686$	0	0
$687$	−3.36138	−0.128245
$688$	0	0
$689$	−37.5864	−1.43193
$690$	0	0
$691$	−10.8852	−0.414094	−0.207047	−	0.978331i	$-0.566385\pi$
$691$	−10.8852	−0.414094	−0.207047	+	0.978331i	$0.566385\pi$
$692$	0	0
$693$	−29.3868	−1.11631
$694$	0	0
$695$	5.26258	0.199621
$696$	0	0
$697$	64.5019	2.44318
$698$	0	0
$699$	−2.78211	−0.105229
$700$	0	0
$701$	−35.7814	−1.35145	−0.675723	−	0.737156i	$-0.736168\pi$
$701$	−35.7814	−1.35145	−0.675723	+	0.737156i	$0.736168\pi$
$702$	0	0
$703$	4.10283	0.154741
$704$	0	0
$705$	0.127967	0.00481951
$706$	0	0
$707$	34.2119	1.28667
$708$	0	0
$709$	33.6583	1.26406	0.632031	−	0.774943i	$-0.282221\pi$
$709$	33.6583	1.26406	0.632031	+	0.774943i	$0.282221\pi$
$710$	0	0
$711$	−30.0687	−1.12767
$712$	0	0
$713$	12.0216	0.450214
$714$	0	0
$715$	−12.7369	−0.476332
$716$	0	0
$717$	−3.51649	−0.131326
$718$	0	0
$719$	10.6906	0.398690	0.199345	−	0.979929i	$-0.436119\pi$
$719$	10.6906	0.398690	0.199345	+	0.979929i	$0.436119\pi$
$720$	0	0
$721$	29.4453	1.09660
$722$	0	0
$723$	2.87536	0.106936
$724$	0	0
$725$	3.24974	0.120692
$726$	0	0
$727$	39.8500	1.47795	0.738977	−	0.673731i	$-0.235309\pi$
$727$	39.8500	1.47795	0.738977	+	0.673731i	$0.235309\pi$
$728$	0	0
$729$	−25.4093	−0.941086
$730$	0	0
$731$	6.22580	0.230270
$732$	0	0
$733$	−24.7794	−0.915248	−0.457624	−	0.889146i	$-0.651299\pi$
$733$	−24.7794	−0.915248	−0.457624	+	0.889146i	$0.651299\pi$
$734$	0	0
$735$	1.48122	0.0546356
$736$	0	0
$737$	−0.626402	−0.0230738
$738$	0	0
$739$	−25.5981	−0.941640	−0.470820	−	0.882229i	$-0.656042\pi$
$739$	−25.5981	−0.941640	−0.470820	+	0.882229i	$0.656042\pi$
$740$	0	0
$741$	1.22958	0.0451697
$742$	0	0
$743$	−33.9505	−1.24552	−0.622762	−	0.782411i	$-0.713989\pi$
$743$	−33.9505	−1.24552	−0.622762	+	0.782411i	$0.713989\pi$
$744$	0	0
$745$	−10.0907	−0.369696
$746$	0	0
$747$	31.7467	1.16155
$748$	0	0
$749$	−60.3468	−2.20502
$750$	0	0
$751$	−22.5974	−0.824589	−0.412295	−	0.911051i	$-0.635273\pi$
$751$	−22.5974	−0.824589	−0.412295	+	0.911051i	$0.635273\pi$
$752$	0	0
$753$	1.61665	0.0589139
$754$	0	0
$755$	14.0956	0.512991
$756$	0	0
$757$	−5.22886	−0.190046	−0.0950231	−	0.995475i	$-0.530292\pi$
$757$	−5.22886	−0.190046	−0.0950231	+	0.995475i	$0.530292\pi$
$758$	0	0
$759$	1.80718	0.0655965
$760$	0	0
$761$	−38.7689	−1.40537	−0.702686	−	0.711500i	$-0.748016\pi$
$761$	−38.7689	−1.40537	−0.702686	+	0.711500i	$0.748016\pi$
$762$	0	0
$763$	−17.6819	−0.640126
$764$	0	0
$765$	−17.2155	−0.622427
$766$	0	0
$767$	−67.6106	−2.44128
$768$	0	0
$769$	0.509032	0.0183562	0.00917808	−	0.999958i	$-0.497078\pi$
$769$	0.509032	0.0183562	0.00917808	+	0.999958i	$0.497078\pi$
$770$	0	0
$771$	2.36722	0.0852535
$772$	0	0
$773$	31.6275	1.13756	0.568781	−	0.822489i	$-0.307415\pi$
$773$	31.6275	1.13756	0.568781	+	0.822489i	$0.307415\pi$
$774$	0	0
$775$	−2.87221	−0.103173
$776$	0	0
$777$	1.98972	0.0713809
$778$	0	0
$779$	15.6179	0.559569
$780$	0	0
$781$	2.91324	0.104244
$782$	0	0
$783$	3.34375	0.119496
$784$	0	0
$785$	4.50373	0.160745
$786$	0	0
$787$	−18.1207	−0.645931	−0.322966	−	0.946411i	$-0.604680\pi$
$787$	−18.1207	−0.645931	−0.322966	+	0.946411i	$0.604680\pi$
$788$	0	0
$789$	2.07029	0.0737041
$790$	0	0
$791$	−16.4590	−0.585215
$792$	0	0
$793$	−68.9023	−2.44679
$794$	0	0
$795$	−1.27415	−0.0451896
$796$	0	0
$797$	6.00102	0.212567	0.106284	−	0.994336i	$-0.466105\pi$
$797$	6.00102	0.212567	0.106284	+	0.994336i	$0.466105\pi$
$798$	0	0
$799$	−4.30356	−0.152249
$800$	0	0
$801$	−43.4744	−1.53609
$802$	0	0
$803$	23.0319	0.812777
$804$	0	0
$805$	16.5286	0.582557
$806$	0	0
$807$	2.83087	0.0996515
$808$	0	0
$809$	−23.3365	−0.820468	−0.410234	−	0.911980i	$-0.634553\pi$
$809$	−23.3365	−0.820468	−0.410234	+	0.911980i	$0.634553\pi$
$810$	0	0
$811$	50.0159	1.75630	0.878149	−	0.478388i	$-0.158779\pi$
$811$	50.0159	1.75630	0.878149	+	0.478388i	$0.158779\pi$
$812$	0	0
$813$	1.52157	0.0533639
$814$	0	0
$815$	2.08183	0.0729234
$816$	0	0
$817$	1.50746	0.0527392
$818$	0	0
$819$	−59.6331	−2.08375
$820$	0	0
$821$	−41.1312	−1.43549	−0.717745	−	0.696306i	$-0.754826\pi$
$821$	−41.1312	−1.43549	−0.717745	+	0.696306i	$0.754826\pi$
$822$	0	0
$823$	−40.1532	−1.39965	−0.699827	−	0.714313i	$-0.746739\pi$
$823$	−40.1532	−1.39965	−0.699827	+	0.714313i	$0.746739\pi$
$824$	0	0
$825$	−0.431771	−0.0150323
$826$	0	0
$827$	20.0711	0.697941	0.348971	−	0.937134i	$-0.386531\pi$
$827$	20.0711	0.697941	0.348971	+	0.937134i	$0.386531\pi$
$828$	0	0
$829$	16.0369	0.556985	0.278492	−	0.960438i	$-0.410165\pi$
$829$	16.0369	0.556985	0.278492	+	0.960438i	$0.410165\pi$
$830$	0	0
$831$	1.70710	0.0592188
$832$	0	0
$833$	−49.8138	−1.72594
$834$	0	0
$835$	14.2807	0.494205
$836$	0	0
$837$	−2.95530	−0.102150
$838$	0	0
$839$	50.7575	1.75234	0.876172	−	0.481998i	$-0.160089\pi$
$839$	50.7575	1.75234	0.876172	+	0.481998i	$0.160089\pi$
$840$	0	0
$841$	−18.4392	−0.635833
$842$	0	0
$843$	1.47085	0.0506589
$844$	0	0
$845$	−12.8462	−0.441924
$846$	0	0
$847$	−18.6525	−0.640908
$848$	0	0
$849$	4.38433	0.150470
$850$	0	0
$851$	12.2367	0.419467
$852$	0	0
$853$	9.68631	0.331653	0.165826	−	0.986155i	$-0.446971\pi$
$853$	9.68631	0.331653	0.165826	+	0.986155i	$0.446971\pi$
$854$	0	0
$855$	−4.16839	−0.142556
$856$	0	0
$857$	3.41593	0.116686	0.0583430	−	0.998297i	$-0.481418\pi$
$857$	3.41593	0.116686	0.0583430	+	0.998297i	$0.481418\pi$
$858$	0	0
$859$	−46.1090	−1.57322	−0.786609	−	0.617451i	$-0.788165\pi$
$859$	−46.1090	−1.57322	−0.786609	+	0.617451i	$0.788165\pi$
$860$	0	0
$861$	7.57410	0.258125
$862$	0	0
$863$	0.419035	0.0142641	0.00713206	−	0.999975i	$-0.497730\pi$
$863$	0.419035	0.0142641	0.00713206	+	0.999975i	$0.497730\pi$
$864$	0	0
$865$	−15.9731	−0.543101
$866$	0	0
$867$	−2.85952	−0.0971143
$868$	0	0
$869$	25.3618	0.860338
$870$	0	0
$871$	−1.27112	−0.0430704
$872$	0	0
$873$	9.79320	0.331450
$874$	0	0
$875$	−3.94901	−0.133501
$876$	0	0
$877$	−27.8069	−0.938971	−0.469486	−	0.882940i	$-0.655561\pi$
$877$	−27.8069	−0.938971	−0.469486	+	0.882940i	$0.655561\pi$
$878$	0	0
$879$	4.33258	0.146134
$880$	0	0
$881$	−16.0861	−0.541955	−0.270978	−	0.962586i	$-0.587347\pi$
$881$	−16.0861	−0.541955	−0.270978	+	0.962586i	$0.587347\pi$
$882$	0	0
$883$	22.1778	0.746342	0.373171	−	0.927763i	$-0.378270\pi$
$883$	22.1778	0.746342	0.373171	+	0.927763i	$0.378270\pi$
$884$	0	0
$885$	−2.29195	−0.0770431
$886$	0	0
$887$	23.5203	0.789734	0.394867	−	0.918738i	$-0.370791\pi$
$887$	23.5203	0.789734	0.394867	+	0.918738i	$0.370791\pi$
$888$	0	0
$889$	−24.0392	−0.806250
$890$	0	0
$891$	21.8804	0.733022
$892$	0	0
$893$	−1.04202	−0.0348699
$894$	0	0
$895$	11.8862	0.397311
$896$	0	0
$897$	3.66721	0.122445
$898$	0	0
$899$	−9.33395	−0.311305
$900$	0	0
$901$	42.8501	1.42754
$902$	0	0
$903$	0.731062	0.0243282
$904$	0	0
$905$	−23.4450	−0.779340
$906$	0	0
$907$	−29.9251	−0.993646	−0.496823	−	0.867852i	$-0.665500\pi$
$907$	−29.9251	−0.993646	−0.496823	+	0.867852i	$0.665500\pi$
$908$	0	0
$909$	−25.7329	−0.853507
$910$	0	0
$911$	56.2978	1.86523	0.932614	−	0.360875i	$-0.117522\pi$
$911$	56.2978	1.86523	0.932614	+	0.360875i	$0.117522\pi$
$912$	0	0
$913$	−26.7771	−0.886191
$914$	0	0
$915$	−2.33574	−0.0772172
$916$	0	0
$917$	−0.147934	−0.00488520
$918$	0	0
$919$	12.1010	0.399175	0.199588	−	0.979880i	$-0.436040\pi$
$919$	12.1010	0.399175	0.199588	+	0.979880i	$0.436040\pi$
$920$	0	0
$921$	−2.01643	−0.0664437
$922$	0	0
$923$	5.91167	0.194585
$924$	0	0
$925$	−2.92358	−0.0961267
$926$	0	0
$927$	−22.1477	−0.727425
$928$	0	0
$929$	23.6374	0.775518	0.387759	−	0.921761i	$-0.373249\pi$
$929$	23.6374	0.775518	0.387759	+	0.921761i	$0.373249\pi$
$930$	0	0
$931$	−12.0614	−0.395298
$932$	0	0
$933$	−0.662428	−0.0216869
$934$	0	0
$935$	14.5206	0.474873
$936$	0	0
$937$	22.1206	0.722648	0.361324	−	0.932440i	$-0.382325\pi$
$937$	22.1206	0.722648	0.361324	+	0.932440i	$0.382325\pi$
$938$	0	0
$939$	4.51620	0.147381
$940$	0	0
$941$	−17.4606	−0.569200	−0.284600	−	0.958646i	$-0.591861\pi$
$941$	−17.4606	−0.569200	−0.284600	+	0.958646i	$0.591861\pi$
$942$	0	0
$943$	46.5802	1.51686
$944$	0	0
$945$	−4.06325	−0.132178
$946$	0	0
$947$	41.9493	1.36317	0.681585	−	0.731739i	$-0.261291\pi$
$947$	41.9493	1.36317	0.681585	+	0.731739i	$0.261291\pi$
$948$	0	0
$949$	46.7373	1.51716
$950$	0	0
$951$	−4.71873	−0.153015
$952$	0	0
$953$	−23.4034	−0.758112	−0.379056	−	0.925374i	$-0.623751\pi$
$953$	−23.4034	−0.758112	−0.379056	+	0.925374i	$0.623751\pi$
$954$	0	0
$955$	−11.0712	−0.358256
$956$	0	0
$957$	−1.40315	−0.0453573
$958$	0	0
$959$	−63.7989	−2.06017
$960$	0	0
$961$	−22.7504	−0.733884
$962$	0	0
$963$	45.3906	1.46269
$964$	0	0
$965$	−13.6691	−0.440024
$966$	0	0
$967$	26.4024	0.849045	0.424523	−	0.905417i	$-0.360442\pi$
$967$	26.4024	0.849045	0.424523	+	0.905417i	$0.360442\pi$
$968$	0	0
$969$	−1.40177	−0.0450314
$970$	0	0
$971$	−43.3895	−1.39243	−0.696217	−	0.717831i	$-0.745135\pi$
$971$	−43.3895	−1.39243	−0.696217	+	0.717831i	$0.745135\pi$
$972$	0	0
$973$	−20.7820	−0.666240
$974$	0	0
$975$	−0.876169	−0.0280599
$976$	0	0
$977$	−49.4544	−1.58219	−0.791093	−	0.611696i	$-0.790488\pi$
$977$	−49.4544	−1.58219	−0.791093	+	0.611696i	$0.790488\pi$
$978$	0	0
$979$	36.6689	1.17194
$980$	0	0
$981$	13.2996	0.424625
$982$	0	0
$983$	30.6175	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$983$	30.6175	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$984$	0	0
$985$	−5.96193	−0.189963
$986$	0	0
$987$	−0.505343	−0.0160852
$988$	0	0
$989$	4.49598	0.142964
$990$	0	0
$991$	−20.6310	−0.655365	−0.327682	−	0.944788i	$-0.606268\pi$
$991$	−20.6310	−0.655365	−0.327682	+	0.944788i	$0.606268\pi$
$992$	0	0
$993$	−0.0507362	−0.00161007
$994$	0	0
$995$	−12.2406	−0.388053
$996$	0	0
$997$	−10.6969	−0.338775	−0.169388	−	0.985550i	$-0.554179\pi$
$997$	−10.6969	−0.338775	−0.169388	+	0.985550i	$0.554179\pi$
$998$	0	0
$999$	−3.00816	−0.0951739

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.13	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-0.172341 q^{3} -1.00000 q^{5} +3.94901 q^{7} -2.97030 q^{9} +O(q^{10})\)	\(q-0.172341 q^{3} -1.00000 q^{5} +3.94901 q^{7} -2.97030 q^{9} +2.50533 q^{11} +5.08392 q^{13} +0.172341 q^{15} -5.79588 q^{17} -1.40336 q^{19} -0.680578 q^{21} -4.18550 q^{23} +1.00000 q^{25} +1.02893 q^{27} +3.24974 q^{29} -2.87221 q^{31} -0.431771 q^{33} -3.94901 q^{35} -2.92358 q^{37} -0.876169 q^{39} -11.1289 q^{41} -1.07418 q^{43} +2.97030 q^{45} +0.742520 q^{47} +8.59469 q^{49} +0.998869 q^{51} -7.39320 q^{53} -2.50533 q^{55} +0.241857 q^{57} -13.2989 q^{59} -13.5530 q^{61} -11.7297 q^{63} -5.08392 q^{65} -0.250028 q^{67} +0.721335 q^{69} +1.16282 q^{71} +9.19316 q^{73} -0.172341 q^{75} +9.89356 q^{77} +10.1231 q^{79} +8.73357 q^{81} -10.6880 q^{83} +5.79588 q^{85} -0.560065 q^{87} +14.6364 q^{89} +20.0765 q^{91} +0.495000 q^{93} +1.40336 q^{95} -3.29704 q^{97} -7.44157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 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

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