LMFDB - Embedded newform 4024.2.a.e.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.137144 q^{3} +1.57958 q^{5} +4.44075 q^{7} -2.98119 q^{9} +O(q^{10})$	$q-0.137144 q^{3} +1.57958 q^{5} +4.44075 q^{7} -2.98119 q^{9} -3.56721 q^{11} -0.104051 q^{13} -0.216631 q^{15} -6.27710 q^{17} +3.22333 q^{19} -0.609023 q^{21} -0.996099 q^{23} -2.50491 q^{25} +0.820286 q^{27} -6.05098 q^{29} -0.827883 q^{31} +0.489222 q^{33} +7.01454 q^{35} +3.24249 q^{37} +0.0142700 q^{39} -10.9148 q^{41} -10.1934 q^{43} -4.70905 q^{45} -5.38152 q^{47} +12.7203 q^{49} +0.860868 q^{51} -2.80073 q^{53} -5.63471 q^{55} -0.442061 q^{57} -7.69410 q^{59} +0.864762 q^{61} -13.2387 q^{63} -0.164358 q^{65} +3.77761 q^{67} +0.136609 q^{69} -1.33473 q^{71} +10.1644 q^{73} +0.343534 q^{75} -15.8411 q^{77} -9.16131 q^{79} +8.83108 q^{81} -16.2039 q^{83} -9.91521 q^{85} +0.829857 q^{87} +17.9010 q^{89} -0.462066 q^{91} +0.113539 q^{93} +5.09152 q^{95} +6.97066 q^{97} +10.6345 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * 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
$3$	−0.137144	−0.0791803	−0.0395901	−	0.999216i	$-0.512605\pi$
$3$	−0.137144	−0.0791803	−0.0395901	+	0.999216i	$0.512605\pi$
$4$	0	0
$5$	1.57958	0.706412	0.353206	−	0.935546i	$-0.385091\pi$
$5$	1.57958	0.706412	0.353206	+	0.935546i	$0.385091\pi$
$6$	0	0
$7$	4.44075	1.67845	0.839223	−	0.543788i	$-0.183010\pi$
$7$	4.44075	1.67845	0.839223	+	0.543788i	$0.183010\pi$
$8$	0	0
$9$	−2.98119	−0.993730
$10$	0	0
$11$	−3.56721	−1.07555	−0.537777	−	0.843087i	$-0.680736\pi$
$11$	−3.56721	−1.07555	−0.537777	+	0.843087i	$0.680736\pi$
$12$	0	0
$13$	−0.104051	−0.0288586	−0.0144293	−	0.999896i	$-0.504593\pi$
$13$	−0.104051	−0.0288586	−0.0144293	+	0.999896i	$0.504593\pi$
$14$	0	0
$15$	−0.216631	−0.0559339
$16$	0	0
$17$	−6.27710	−1.52242	−0.761210	−	0.648505i	$-0.775394\pi$
$17$	−6.27710	−1.52242	−0.761210	+	0.648505i	$0.775394\pi$
$18$	0	0
$19$	3.22333	0.739482	0.369741	−	0.929135i	$-0.379446\pi$
$19$	3.22333	0.739482	0.369741	+	0.929135i	$0.379446\pi$
$20$	0	0
$21$	−0.609023	−0.132900
$22$	0	0
$23$	−0.996099	−0.207701	−0.103851	−	0.994593i	$-0.533116\pi$
$23$	−0.996099	−0.207701	−0.103851	+	0.994593i	$0.533116\pi$
$24$	0	0
$25$	−2.50491	−0.500982
$26$	0	0
$27$	0.820286	0.157864
$28$	0	0
$29$	−6.05098	−1.12364	−0.561820	−	0.827260i	$-0.689899\pi$
$29$	−6.05098	−1.12364	−0.561820	+	0.827260i	$0.689899\pi$
$30$	0	0
$31$	−0.827883	−0.148692	−0.0743461	−	0.997233i	$-0.523687\pi$
$31$	−0.827883	−0.148692	−0.0743461	+	0.997233i	$0.523687\pi$
$32$	0	0
$33$	0.489222	0.0851626
$34$	0	0
$35$	7.01454	1.18567
$36$	0	0
$37$	3.24249	0.533063	0.266531	−	0.963826i	$-0.414122\pi$
$37$	3.24249	0.533063	0.266531	+	0.963826i	$0.414122\pi$
$38$	0	0
$39$	0.0142700	0.00228503
$40$	0	0
$41$	−10.9148	−1.70461	−0.852303	−	0.523049i	$-0.824795\pi$
$41$	−10.9148	−1.70461	−0.852303	+	0.523049i	$0.824795\pi$
$42$	0	0
$43$	−10.1934	−1.55448	−0.777241	−	0.629203i	$-0.783381\pi$
$43$	−10.1934	−1.55448	−0.777241	+	0.629203i	$0.783381\pi$
$44$	0	0
$45$	−4.70905	−0.701983
$46$	0	0
$47$	−5.38152	−0.784976	−0.392488	−	0.919757i	$-0.628385\pi$
$47$	−5.38152	−0.784976	−0.392488	+	0.919757i	$0.628385\pi$
$48$	0	0
$49$	12.7203	1.81718
$50$	0	0
$51$	0.860868	0.120546
$52$	0	0
$53$	−2.80073	−0.384710	−0.192355	−	0.981325i	$-0.561612\pi$
$53$	−2.80073	−0.384710	−0.192355	+	0.981325i	$0.561612\pi$
$54$	0	0
$55$	−5.63471	−0.759784
$56$	0	0
$57$	−0.442061	−0.0585524
$58$	0	0
$59$	−7.69410	−1.00169	−0.500843	−	0.865538i	$-0.666977\pi$
$59$	−7.69410	−1.00169	−0.500843	+	0.865538i	$0.666977\pi$
$60$	0	0
$61$	0.864762	0.110721	0.0553607	−	0.998466i	$-0.482369\pi$
$61$	0.864762	0.110721	0.0553607	+	0.998466i	$0.482369\pi$
$62$	0	0
$63$	−13.2387	−1.66792
$64$	0	0
$65$	−0.164358	−0.0203861
$66$	0	0
$67$	3.77761	0.461508	0.230754	−	0.973012i	$-0.425881\pi$
$67$	3.77761	0.461508	0.230754	+	0.973012i	$0.425881\pi$
$68$	0	0
$69$	0.136609	0.0164458
$70$	0	0
$71$	−1.33473	−0.158403	−0.0792014	−	0.996859i	$-0.525237\pi$
$71$	−1.33473	−0.158403	−0.0792014	+	0.996859i	$0.525237\pi$
$72$	0	0
$73$	10.1644	1.18965	0.594824	−	0.803856i	$-0.297222\pi$
$73$	10.1644	1.18965	0.594824	+	0.803856i	$0.297222\pi$
$74$	0	0
$75$	0.343534	0.0396679
$76$	0	0
$77$	−15.8411	−1.80526
$78$	0	0
$79$	−9.16131	−1.03073	−0.515364	−	0.856972i	$-0.672343\pi$
$79$	−9.16131	−1.03073	−0.515364	+	0.856972i	$0.672343\pi$
$80$	0	0
$81$	8.83108	0.981231
$82$	0	0
$83$	−16.2039	−1.77861	−0.889307	−	0.457311i	$-0.848813\pi$
$83$	−16.2039	−1.77861	−0.889307	+	0.457311i	$0.848813\pi$
$84$	0	0
$85$	−9.91521	−1.07546
$86$	0	0
$87$	0.829857	0.0889701
$88$	0	0
$89$	17.9010	1.89750	0.948751	−	0.316024i	$-0.102348\pi$
$89$	17.9010	1.89750	0.948751	+	0.316024i	$0.102348\pi$
$90$	0	0
$91$	−0.462066	−0.0484376
$92$	0	0
$93$	0.113539	0.0117735
$94$	0	0
$95$	5.09152	0.522379
$96$	0	0
$97$	6.97066	0.707763	0.353882	−	0.935290i	$-0.384862\pi$
$97$	6.97066	0.707763	0.353882	+	0.935290i	$0.384862\pi$
$98$	0	0
$99$	10.6345	1.06881
$100$	0	0
$101$	−9.40013	−0.935348	−0.467674	−	0.883901i	$-0.654908\pi$
$101$	−9.40013	−0.935348	−0.467674	+	0.883901i	$0.654908\pi$
$102$	0	0
$103$	0.271379	0.0267397	0.0133699	−	0.999911i	$-0.495744\pi$
$103$	0.271379	0.0267397	0.0133699	+	0.999911i	$0.495744\pi$
$104$	0	0
$105$	−0.962004	−0.0938820
$106$	0	0
$107$	4.92068	0.475700	0.237850	−	0.971302i	$-0.423557\pi$
$107$	4.92068	0.475700	0.237850	+	0.971302i	$0.423557\pi$
$108$	0	0
$109$	−11.7504	−1.12548	−0.562742	−	0.826633i	$-0.690254\pi$
$109$	−11.7504	−1.12548	−0.562742	+	0.826633i	$0.690254\pi$
$110$	0	0
$111$	−0.444689	−0.0422080
$112$	0	0
$113$	13.5473	1.27442	0.637212	−	0.770688i	$-0.280087\pi$
$113$	13.5473	1.27442	0.637212	+	0.770688i	$0.280087\pi$
$114$	0	0
$115$	−1.57342	−0.146722
$116$	0	0
$117$	0.310197	0.0286777
$118$	0	0
$119$	−27.8750	−2.55530
$120$	0	0
$121$	1.72498	0.156816
$122$	0	0
$123$	1.49690	0.134971
$124$	0	0
$125$	−11.8546	−1.06031
$126$	0	0
$127$	18.8678	1.67425	0.837125	−	0.547012i	$-0.184235\pi$
$127$	18.8678	1.67425	0.837125	+	0.547012i	$0.184235\pi$
$128$	0	0
$129$	1.39797	0.123084
$130$	0	0
$131$	18.3612	1.60422	0.802111	−	0.597175i	$-0.203710\pi$
$131$	18.3612	1.60422	0.802111	+	0.597175i	$0.203710\pi$
$132$	0	0
$133$	14.3140	1.24118
$134$	0	0
$135$	1.29571	0.111517
$136$	0	0
$137$	−17.8295	−1.52328	−0.761640	−	0.648000i	$-0.775606\pi$
$137$	−17.8295	−1.52328	−0.761640	+	0.648000i	$0.775606\pi$
$138$	0	0
$139$	−1.65275	−0.140184	−0.0700922	−	0.997541i	$-0.522329\pi$
$139$	−1.65275	−0.140184	−0.0700922	+	0.997541i	$0.522329\pi$
$140$	0	0
$141$	0.738045	0.0621546
$142$	0	0
$143$	0.371173	0.0310390
$144$	0	0
$145$	−9.55804	−0.793752
$146$	0	0
$147$	−1.74451	−0.143885
$148$	0	0
$149$	23.1950	1.90020	0.950102	−	0.311938i	$-0.100978\pi$
$149$	23.1950	1.90020	0.950102	+	0.311938i	$0.100978\pi$
$150$	0	0
$151$	−22.7186	−1.84881	−0.924406	−	0.381410i	$-0.875439\pi$
$151$	−22.7186	−1.84881	−0.924406	+	0.381410i	$0.875439\pi$
$152$	0	0
$153$	18.7132	1.51288
$154$	0	0
$155$	−1.30771	−0.105038
$156$	0	0
$157$	20.3725	1.62590	0.812951	−	0.582331i	$-0.197859\pi$
$157$	20.3725	1.62590	0.812951	+	0.582331i	$0.197859\pi$
$158$	0	0
$159$	0.384104	0.0304614
$160$	0	0
$161$	−4.42343	−0.348615
$162$	0	0
$163$	6.93616	0.543282	0.271641	−	0.962399i	$-0.412434\pi$
$163$	6.93616	0.543282	0.271641	+	0.962399i	$0.412434\pi$
$164$	0	0
$165$	0.772768	0.0601599
$166$	0	0
$167$	0.925307	0.0716024	0.0358012	−	0.999359i	$-0.488602\pi$
$167$	0.925307	0.0716024	0.0358012	+	0.999359i	$0.488602\pi$
$168$	0	0
$169$	−12.9892	−0.999167
$170$	0	0
$171$	−9.60935	−0.734845
$172$	0	0
$173$	−14.7523	−1.12159	−0.560797	−	0.827953i	$-0.689505\pi$
$173$	−14.7523	−1.12159	−0.560797	+	0.827953i	$0.689505\pi$
$174$	0	0
$175$	−11.1237	−0.840872
$176$	0	0
$177$	1.05520	0.0793138
$178$	0	0
$179$	2.16479	0.161804	0.0809019	−	0.996722i	$-0.474220\pi$
$179$	2.16479	0.161804	0.0809019	+	0.996722i	$0.474220\pi$
$180$	0	0
$181$	18.8184	1.39876	0.699381	−	0.714749i	$-0.253459\pi$
$181$	18.8184	1.39876	0.699381	+	0.714749i	$0.253459\pi$
$182$	0	0
$183$	−0.118597	−0.00876695
$184$	0	0
$185$	5.12179	0.376562
$186$	0	0
$187$	22.3917	1.63745
$188$	0	0
$189$	3.64269	0.264966
$190$	0	0
$191$	−7.36926	−0.533221	−0.266610	−	0.963804i	$-0.585904\pi$
$191$	−7.36926	−0.533221	−0.266610	+	0.963804i	$0.585904\pi$
$192$	0	0
$193$	−11.7609	−0.846571	−0.423286	−	0.905996i	$-0.639123\pi$
$193$	−11.7609	−0.846571	−0.423286	+	0.905996i	$0.639123\pi$
$194$	0	0
$195$	0.0225407	0.00161417
$196$	0	0
$197$	14.4379	1.02866	0.514329	−	0.857593i	$-0.328041\pi$
$197$	14.4379	1.02866	0.514329	+	0.857593i	$0.328041\pi$
$198$	0	0
$199$	6.70039	0.474978	0.237489	−	0.971390i	$-0.423676\pi$
$199$	6.70039	0.474978	0.237489	+	0.971390i	$0.423676\pi$
$200$	0	0
$201$	−0.518077	−0.0365423
$202$	0	0
$203$	−26.8709	−1.88597
$204$	0	0
$205$	−17.2409	−1.20415
$206$	0	0
$207$	2.96956	0.206399
$208$	0	0
$209$	−11.4983	−0.795352
$210$	0	0
$211$	11.1352	0.766579	0.383289	−	0.923628i	$-0.374791\pi$
$211$	11.1352	0.766579	0.383289	+	0.923628i	$0.374791\pi$
$212$	0	0
$213$	0.183050	0.0125424
$214$	0	0
$215$	−16.1014	−1.09810
$216$	0	0
$217$	−3.67642	−0.249572
$218$	0	0
$219$	−1.39398	−0.0941967
$220$	0	0
$221$	0.653140	0.0439350
$222$	0	0
$223$	−18.7155	−1.25328	−0.626642	−	0.779308i	$-0.715571\pi$
$223$	−18.7155	−1.25328	−0.626642	+	0.779308i	$0.715571\pi$
$224$	0	0
$225$	7.46762	0.497841
$226$	0	0
$227$	−12.2064	−0.810166	−0.405083	−	0.914280i	$-0.632757\pi$
$227$	−12.2064	−0.810166	−0.405083	+	0.914280i	$0.632757\pi$
$228$	0	0
$229$	4.20381	0.277796	0.138898	−	0.990307i	$-0.455644\pi$
$229$	4.20381	0.277796	0.138898	+	0.990307i	$0.455644\pi$
$230$	0	0
$231$	2.17251	0.142941
$232$	0	0
$233$	14.5967	0.956263	0.478132	−	0.878288i	$-0.341314\pi$
$233$	14.5967	0.956263	0.478132	+	0.878288i	$0.341314\pi$
$234$	0	0
$235$	−8.50057	−0.554516
$236$	0	0
$237$	1.25642	0.0816133
$238$	0	0
$239$	−1.06640	−0.0689800	−0.0344900	−	0.999405i	$-0.510981\pi$
$239$	−1.06640	−0.0689800	−0.0344900	+	0.999405i	$0.510981\pi$
$240$	0	0
$241$	−0.501466	−0.0323023	−0.0161511	−	0.999870i	$-0.505141\pi$
$241$	−0.501466	−0.0323023	−0.0161511	+	0.999870i	$0.505141\pi$
$242$	0	0
$243$	−3.67199	−0.235558
$244$	0	0
$245$	20.0927	1.28368
$246$	0	0
$247$	−0.335391	−0.0213404
$248$	0	0
$249$	2.22228	0.140831
$250$	0	0
$251$	10.3338	0.652262	0.326131	−	0.945325i	$-0.394255\pi$
$251$	10.3338	0.652262	0.326131	+	0.945325i	$0.394255\pi$
$252$	0	0
$253$	3.55329	0.223394
$254$	0	0
$255$	1.35981	0.0851549
$256$	0	0
$257$	−7.84115	−0.489117	−0.244559	−	0.969634i	$-0.578643\pi$
$257$	−7.84115	−0.489117	−0.244559	+	0.969634i	$0.578643\pi$
$258$	0	0
$259$	14.3991	0.894717
$260$	0	0
$261$	18.0391	1.11659
$262$	0	0
$263$	−20.1755	−1.24407	−0.622036	−	0.782989i	$-0.713694\pi$
$263$	−20.1755	−1.24407	−0.622036	+	0.782989i	$0.713694\pi$
$264$	0	0
$265$	−4.42399	−0.271763
$266$	0	0
$267$	−2.45502	−0.150245
$268$	0	0
$269$	−1.42468	−0.0868645	−0.0434323	−	0.999056i	$-0.513829\pi$
$269$	−1.42468	−0.0868645	−0.0434323	+	0.999056i	$0.513829\pi$
$270$	0	0
$271$	−25.8290	−1.56900	−0.784500	−	0.620128i	$-0.787080\pi$
$271$	−25.8290	−1.56900	−0.784500	+	0.620128i	$0.787080\pi$
$272$	0	0
$273$	0.0633696	0.00383531
$274$	0	0
$275$	8.93554	0.538833
$276$	0	0
$277$	−11.0093	−0.661487	−0.330744	−	0.943721i	$-0.607300\pi$
$277$	−11.0093	−0.661487	−0.330744	+	0.943721i	$0.607300\pi$
$278$	0	0
$279$	2.46808	0.147760
$280$	0	0
$281$	7.50370	0.447633	0.223817	−	0.974631i	$-0.428148\pi$
$281$	7.50370	0.447633	0.223817	+	0.974631i	$0.428148\pi$
$282$	0	0
$283$	14.6706	0.872079	0.436039	−	0.899928i	$-0.356381\pi$
$283$	14.6706	0.872079	0.436039	+	0.899928i	$0.356381\pi$
$284$	0	0
$285$	−0.698272	−0.0413621
$286$	0	0
$287$	−48.4699	−2.86109
$288$	0	0
$289$	22.4020	1.31776
$290$	0	0
$291$	−0.955986	−0.0560409
$292$	0	0
$293$	−24.3818	−1.42440	−0.712200	−	0.701977i	$-0.752301\pi$
$293$	−24.3818	−1.42440	−0.712200	+	0.701977i	$0.752301\pi$
$294$	0	0
$295$	−12.1535	−0.707603
$296$	0	0
$297$	−2.92613	−0.169791
$298$	0	0
$299$	0.103645	0.00599397
$300$	0	0
$301$	−45.2664	−2.60911
$302$	0	0
$303$	1.28917	0.0740611
$304$	0	0
$305$	1.36597	0.0782149
$306$	0	0
$307$	−16.4237	−0.937349	−0.468675	−	0.883371i	$-0.655268\pi$
$307$	−16.4237	−0.937349	−0.468675	+	0.883371i	$0.655268\pi$
$308$	0	0
$309$	−0.0372180	−0.00211726
$310$	0	0
$311$	4.26482	0.241836	0.120918	−	0.992663i	$-0.461416\pi$
$311$	4.26482	0.241836	0.120918	+	0.992663i	$0.461416\pi$
$312$	0	0
$313$	14.1420	0.799354	0.399677	−	0.916656i	$-0.369122\pi$
$313$	14.1420	0.799354	0.399677	+	0.916656i	$0.369122\pi$
$314$	0	0
$315$	−20.9117	−1.17824
$316$	0	0
$317$	−0.151631	−0.00851646	−0.00425823	−	0.999991i	$-0.501355\pi$
$317$	−0.151631	−0.00851646	−0.00425823	+	0.999991i	$0.501355\pi$
$318$	0	0
$319$	21.5851	1.20853
$320$	0	0
$321$	−0.674843	−0.0376660
$322$	0	0
$323$	−20.2331	−1.12580
$324$	0	0
$325$	0.260639	0.0144577
$326$	0	0
$327$	1.61150	0.0891161
$328$	0	0
$329$	−23.8980	−1.31754
$330$	0	0
$331$	7.49245	0.411822	0.205911	−	0.978571i	$-0.433984\pi$
$331$	7.49245	0.411822	0.205911	+	0.978571i	$0.433984\pi$
$332$	0	0
$333$	−9.66649	−0.529721
$334$	0	0
$335$	5.96705	0.326015
$336$	0	0
$337$	−16.5644	−0.902322	−0.451161	−	0.892443i	$-0.648990\pi$
$337$	−16.5644	−0.902322	−0.451161	+	0.892443i	$0.648990\pi$
$338$	0	0
$339$	−1.85794	−0.100909
$340$	0	0
$341$	2.95323	0.159926
$342$	0	0
$343$	25.4022	1.37159
$344$	0	0
$345$	0.215786	0.0116175
$346$	0	0
$347$	−32.0620	−1.72118	−0.860589	−	0.509299i	$-0.829905\pi$
$347$	−32.0620	−1.72118	−0.860589	+	0.509299i	$0.829905\pi$
$348$	0	0
$349$	18.1181	0.969842	0.484921	−	0.874558i	$-0.338848\pi$
$349$	18.1181	0.969842	0.484921	+	0.874558i	$0.338848\pi$
$350$	0	0
$351$	−0.0853518	−0.00455574
$352$	0	0
$353$	−15.2174	−0.809940	−0.404970	−	0.914330i	$-0.632718\pi$
$353$	−15.2174	−0.809940	−0.404970	+	0.914330i	$0.632718\pi$
$354$	0	0
$355$	−2.10831	−0.111898
$356$	0	0
$357$	3.82290	0.202329
$358$	0	0
$359$	−26.1169	−1.37840	−0.689199	−	0.724572i	$-0.742038\pi$
$359$	−26.1169	−1.37840	−0.689199	+	0.724572i	$0.742038\pi$
$360$	0	0
$361$	−8.61017	−0.453167
$362$	0	0
$363$	−0.236571	−0.0124167
$364$	0	0
$365$	16.0555	0.840382
$366$	0	0
$367$	−12.8002	−0.668165	−0.334082	−	0.942544i	$-0.608426\pi$
$367$	−12.8002	−0.668165	−0.334082	+	0.942544i	$0.608426\pi$
$368$	0	0
$369$	32.5391	1.69392
$370$	0	0
$371$	−12.4373	−0.645714
$372$	0	0
$373$	−26.2157	−1.35740	−0.678700	−	0.734416i	$-0.737456\pi$
$373$	−26.2157	−1.35740	−0.678700	+	0.734416i	$0.737456\pi$
$374$	0	0
$375$	1.62580	0.0839558
$376$	0	0
$377$	0.629612	0.0324267
$378$	0	0
$379$	−16.0306	−0.823438	−0.411719	−	0.911311i	$-0.635071\pi$
$379$	−16.0306	−0.823438	−0.411719	+	0.911311i	$0.635071\pi$
$380$	0	0
$381$	−2.58762	−0.132568
$382$	0	0
$383$	−7.07663	−0.361599	−0.180799	−	0.983520i	$-0.557869\pi$
$383$	−7.07663	−0.361599	−0.180799	+	0.983520i	$0.557869\pi$
$384$	0	0
$385$	−25.0223	−1.27526
$386$	0	0
$387$	30.3885	1.54474
$388$	0	0
$389$	20.1616	1.02223	0.511116	−	0.859512i	$-0.329232\pi$
$389$	20.1616	1.02223	0.511116	+	0.859512i	$0.329232\pi$
$390$	0	0
$391$	6.25261	0.316208
$392$	0	0
$393$	−2.51813	−0.127023
$394$	0	0
$395$	−14.4711	−0.728118
$396$	0	0
$397$	−6.58287	−0.330385	−0.165192	−	0.986261i	$-0.552824\pi$
$397$	−6.58287	−0.330385	−0.165192	+	0.986261i	$0.552824\pi$
$398$	0	0
$399$	−1.96308	−0.0982769
$400$	0	0
$401$	27.0532	1.35097	0.675486	−	0.737373i	$-0.263934\pi$
$401$	27.0532	1.35097	0.675486	+	0.737373i	$0.263934\pi$
$402$	0	0
$403$	0.0861423	0.00429105
$404$	0	0
$405$	13.9494	0.693153
$406$	0	0
$407$	−11.5667	−0.573338
$408$	0	0
$409$	−5.81189	−0.287379	−0.143690	−	0.989623i	$-0.545897\pi$
$409$	−5.81189	−0.287379	−0.143690	+	0.989623i	$0.545897\pi$
$410$	0	0
$411$	2.44522	0.120614
$412$	0	0
$413$	−34.1676	−1.68128
$414$	0	0
$415$	−25.5955	−1.25643
$416$	0	0
$417$	0.226665	0.0110998
$418$	0	0
$419$	−18.6706	−0.912118	−0.456059	−	0.889949i	$-0.650739\pi$
$419$	−18.6706	−0.912118	−0.456059	+	0.889949i	$0.650739\pi$
$420$	0	0
$421$	6.85783	0.334230	0.167115	−	0.985937i	$-0.446555\pi$
$421$	6.85783	0.334230	0.167115	+	0.985937i	$0.446555\pi$
$422$	0	0
$423$	16.0433	0.780054
$424$	0	0
$425$	15.7236	0.762706
$426$	0	0
$427$	3.84019	0.185840
$428$	0	0
$429$	−0.0509042	−0.00245768
$430$	0	0
$431$	28.2264	1.35962	0.679809	−	0.733389i	$-0.262063\pi$
$431$	28.2264	1.35962	0.679809	+	0.733389i	$0.262063\pi$
$432$	0	0
$433$	−20.5692	−0.988491	−0.494246	−	0.869322i	$-0.664556\pi$
$433$	−20.5692	−0.988491	−0.494246	+	0.869322i	$0.664556\pi$
$434$	0	0
$435$	1.31083	0.0628495
$436$	0	0
$437$	−3.21075	−0.153591
$438$	0	0
$439$	−14.4953	−0.691822	−0.345911	−	0.938267i	$-0.612430\pi$
$439$	−14.4953	−0.691822	−0.345911	+	0.938267i	$0.612430\pi$
$440$	0	0
$441$	−37.9215	−1.80579
$442$	0	0
$443$	39.9584	1.89848	0.949240	−	0.314553i	$-0.101855\pi$
$443$	39.9584	1.89848	0.949240	+	0.314553i	$0.101855\pi$
$444$	0	0
$445$	28.2762	1.34042
$446$	0	0
$447$	−3.18105	−0.150459
$448$	0	0
$449$	35.6375	1.68184	0.840918	−	0.541162i	$-0.182016\pi$
$449$	35.6375	1.68184	0.840918	+	0.541162i	$0.182016\pi$
$450$	0	0
$451$	38.9354	1.83340
$452$	0	0
$453$	3.11572	0.146389
$454$	0	0
$455$	−0.729872	−0.0342169
$456$	0	0
$457$	−38.2192	−1.78782	−0.893910	−	0.448247i	$-0.852049\pi$
$457$	−38.2192	−1.78782	−0.893910	+	0.448247i	$0.852049\pi$
$458$	0	0
$459$	−5.14902	−0.240336
$460$	0	0
$461$	7.84710	0.365476	0.182738	−	0.983162i	$-0.441504\pi$
$461$	7.84710	0.365476	0.182738	+	0.983162i	$0.441504\pi$
$462$	0	0
$463$	−15.3588	−0.713784	−0.356892	−	0.934146i	$-0.616164\pi$
$463$	−15.3588	−0.713784	−0.356892	+	0.934146i	$0.616164\pi$
$464$	0	0
$465$	0.179345	0.00831693
$466$	0	0
$467$	−29.2438	−1.35324	−0.676620	−	0.736332i	$-0.736556\pi$
$467$	−29.2438	−1.35324	−0.676620	+	0.736332i	$0.736556\pi$
$468$	0	0
$469$	16.7754	0.774616
$470$	0	0
$471$	−2.79397	−0.128739
$472$	0	0
$473$	36.3621	1.67193
$474$	0	0
$475$	−8.07414	−0.370467
$476$	0	0
$477$	8.34951	0.382298
$478$	0	0
$479$	−30.2804	−1.38355	−0.691774	−	0.722114i	$-0.743170\pi$
$479$	−30.2804	−1.38355	−0.691774	+	0.722114i	$0.743170\pi$
$480$	0	0
$481$	−0.337386	−0.0153835
$482$	0	0
$483$	0.606648	0.0276034
$484$	0	0
$485$	11.0107	0.499972
$486$	0	0
$487$	18.1071	0.820510	0.410255	−	0.911971i	$-0.365440\pi$
$487$	18.1071	0.820510	0.410255	+	0.911971i	$0.365440\pi$
$488$	0	0
$489$	−0.951255	−0.0430172
$490$	0	0
$491$	26.5911	1.20004	0.600019	−	0.799986i	$-0.295160\pi$
$491$	26.5911	1.20004	0.600019	+	0.799986i	$0.295160\pi$
$492$	0	0
$493$	37.9826	1.71065
$494$	0	0
$495$	16.7981	0.755021
$496$	0	0
$497$	−5.92719	−0.265871
$498$	0	0
$499$	37.2818	1.66896	0.834480	−	0.551038i	$-0.185768\pi$
$499$	37.2818	1.66896	0.834480	+	0.551038i	$0.185768\pi$
$500$	0	0
$501$	−0.126901	−0.00566950
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−14.8483	−0.660741
$506$	0	0
$507$	1.78139	0.0791143
$508$	0	0
$509$	41.5372	1.84111	0.920553	−	0.390617i	$-0.127738\pi$
$509$	41.5372	1.84111	0.920553	+	0.390617i	$0.127738\pi$
$510$	0	0
$511$	45.1374	1.99676
$512$	0	0
$513$	2.64405	0.116738
$514$	0	0
$515$	0.428666	0.0188893
$516$	0	0
$517$	19.1970	0.844283
$518$	0	0
$519$	2.02319	0.0888081
$520$	0	0
$521$	27.4398	1.20216	0.601079	−	0.799189i	$-0.294738\pi$
$521$	27.4398	1.20216	0.601079	+	0.799189i	$0.294738\pi$
$522$	0	0
$523$	−5.86143	−0.256302	−0.128151	−	0.991755i	$-0.540904\pi$
$523$	−5.86143	−0.256302	−0.128151	+	0.991755i	$0.540904\pi$
$524$	0	0
$525$	1.52555	0.0665804
$526$	0	0
$527$	5.19670	0.226372
$528$	0	0
$529$	−22.0078	−0.956860
$530$	0	0
$531$	22.9376	0.995406
$532$	0	0
$533$	1.13570	0.0491926
$534$	0	0
$535$	7.77263	0.336040
$536$	0	0
$537$	−0.296888	−0.0128117
$538$	0	0
$539$	−45.3758	−1.95448
$540$	0	0
$541$	−1.80838	−0.0777482	−0.0388741	−	0.999244i	$-0.512377\pi$
$541$	−1.80838	−0.0777482	−0.0388741	+	0.999244i	$0.512377\pi$
$542$	0	0
$543$	−2.58084	−0.110754
$544$	0	0
$545$	−18.5607	−0.795055
$546$	0	0
$547$	−34.8669	−1.49080	−0.745401	−	0.666616i	$-0.767742\pi$
$547$	−34.8669	−1.49080	−0.745401	+	0.666616i	$0.767742\pi$
$548$	0	0
$549$	−2.57802	−0.110027
$550$	0	0
$551$	−19.5043	−0.830911
$552$	0	0
$553$	−40.6831	−1.73002
$554$	0	0
$555$	−0.702425	−0.0298163
$556$	0	0
$557$	12.6406	0.535601	0.267801	−	0.963474i	$-0.413703\pi$
$557$	12.6406	0.535601	0.267801	+	0.963474i	$0.413703\pi$
$558$	0	0
$559$	1.06064	0.0448602
$560$	0	0
$561$	−3.07090	−0.129653
$562$	0	0
$563$	11.8213	0.498209	0.249104	−	0.968477i	$-0.419864\pi$
$563$	11.8213	0.498209	0.249104	+	0.968477i	$0.419864\pi$
$564$	0	0
$565$	21.3991	0.900269
$566$	0	0
$567$	39.2166	1.64694
$568$	0	0
$569$	−7.65733	−0.321012	−0.160506	−	0.987035i	$-0.551313\pi$
$569$	−7.65733	−0.321012	−0.160506	+	0.987035i	$0.551313\pi$
$570$	0	0
$571$	1.37037	0.0573483	0.0286741	−	0.999589i	$-0.490871\pi$
$571$	1.37037	0.0573483	0.0286741	+	0.999589i	$0.490871\pi$
$572$	0	0
$573$	1.01065	0.0422206
$574$	0	0
$575$	2.49514	0.104055
$576$	0	0
$577$	−5.50469	−0.229163	−0.114582	−	0.993414i	$-0.536553\pi$
$577$	−5.50469	−0.229163	−0.114582	+	0.993414i	$0.536553\pi$
$578$	0	0
$579$	1.61295	0.0670318
$580$	0	0
$581$	−71.9577	−2.98531
$582$	0	0
$583$	9.99078	0.413776
$584$	0	0
$585$	0.489982	0.0202583
$586$	0	0
$587$	9.86409	0.407135	0.203567	−	0.979061i	$-0.434746\pi$
$587$	9.86409	0.407135	0.203567	+	0.979061i	$0.434746\pi$
$588$	0	0
$589$	−2.66854	−0.109955
$590$	0	0
$591$	−1.98008	−0.0814494
$592$	0	0
$593$	−29.6737	−1.21855	−0.609277	−	0.792958i	$-0.708540\pi$
$593$	−29.6737	−1.21855	−0.609277	+	0.792958i	$0.708540\pi$
$594$	0	0
$595$	−44.0310	−1.80509
$596$	0	0
$597$	−0.918920	−0.0376089
$598$	0	0
$599$	28.6620	1.17110	0.585548	−	0.810638i	$-0.300879\pi$
$599$	28.6620	1.17110	0.585548	+	0.810638i	$0.300879\pi$
$600$	0	0
$601$	−41.9195	−1.70993	−0.854966	−	0.518685i	$-0.826422\pi$
$601$	−41.9195	−1.70993	−0.854966	+	0.518685i	$0.826422\pi$
$602$	0	0
$603$	−11.2618	−0.458615
$604$	0	0
$605$	2.72475	0.110777
$606$	0	0
$607$	−19.7054	−0.799820	−0.399910	−	0.916555i	$-0.630959\pi$
$607$	−19.7054	−0.799820	−0.399910	+	0.916555i	$0.630959\pi$
$608$	0	0
$609$	3.68519	0.149331
$610$	0	0
$611$	0.559954	0.0226533
$612$	0	0
$613$	34.3426	1.38708	0.693542	−	0.720416i	$-0.256049\pi$
$613$	34.3426	1.38708	0.693542	+	0.720416i	$0.256049\pi$
$614$	0	0
$615$	2.36448	0.0953452
$616$	0	0
$617$	34.8171	1.40169	0.700843	−	0.713316i	$-0.252808\pi$
$617$	34.8171	1.40169	0.700843	+	0.713316i	$0.252808\pi$
$618$	0	0
$619$	21.3748	0.859126	0.429563	−	0.903037i	$-0.358668\pi$
$619$	21.3748	0.859126	0.429563	+	0.903037i	$0.358668\pi$
$620$	0	0
$621$	−0.817086	−0.0327885
$622$	0	0
$623$	79.4939	3.18486
$624$	0	0
$625$	−6.20086	−0.248035
$626$	0	0
$627$	1.57692	0.0629762
$628$	0	0
$629$	−20.3535	−0.811545
$630$	0	0
$631$	−13.7806	−0.548598	−0.274299	−	0.961644i	$-0.588446\pi$
$631$	−13.7806	−0.548598	−0.274299	+	0.961644i	$0.588446\pi$
$632$	0	0
$633$	−1.52713	−0.0606979
$634$	0	0
$635$	29.8034	1.18271
$636$	0	0
$637$	−1.32356	−0.0524413
$638$	0	0
$639$	3.97907	0.157410
$640$	0	0
$641$	27.3790	1.08141	0.540703	−	0.841214i	$-0.318158\pi$
$641$	27.3790	1.08141	0.540703	+	0.841214i	$0.318158\pi$
$642$	0	0
$643$	26.5203	1.04586	0.522929	−	0.852376i	$-0.324839\pi$
$643$	26.5203	1.04586	0.522929	+	0.852376i	$0.324839\pi$
$644$	0	0
$645$	2.20821	0.0869482
$646$	0	0
$647$	13.6398	0.536238	0.268119	−	0.963386i	$-0.413598\pi$
$647$	13.6398	0.536238	0.268119	+	0.963386i	$0.413598\pi$
$648$	0	0
$649$	27.4465	1.07737
$650$	0	0
$651$	0.504200	0.0197612
$652$	0	0
$653$	47.8602	1.87292	0.936458	−	0.350779i	$-0.114083\pi$
$653$	47.8602	1.87292	0.936458	+	0.350779i	$0.114083\pi$
$654$	0	0
$655$	29.0030	1.13324
$656$	0	0
$657$	−30.3019	−1.18219
$658$	0	0
$659$	43.7693	1.70501	0.852505	−	0.522718i	$-0.175082\pi$
$659$	43.7693	1.70501	0.852505	+	0.522718i	$0.175082\pi$
$660$	0	0
$661$	−39.6164	−1.54090	−0.770449	−	0.637502i	$-0.779968\pi$
$661$	−39.6164	−1.54090	−0.770449	+	0.637502i	$0.779968\pi$
$662$	0	0
$663$	−0.0895744	−0.00347878
$664$	0	0
$665$	22.6102	0.876784
$666$	0	0
$667$	6.02738	0.233381
$668$	0	0
$669$	2.56672	0.0992353
$670$	0	0
$671$	−3.08479	−0.119087
$672$	0	0
$673$	24.1058	0.929212	0.464606	−	0.885518i	$-0.346196\pi$
$673$	24.1058	0.929212	0.464606	+	0.885518i	$0.346196\pi$
$674$	0	0
$675$	−2.05474	−0.0790871
$676$	0	0
$677$	26.9028	1.03396	0.516978	−	0.855998i	$-0.327057\pi$
$677$	26.9028	1.03396	0.516978	+	0.855998i	$0.327057\pi$
$678$	0	0
$679$	30.9550	1.18794
$680$	0	0
$681$	1.67403	0.0641491
$682$	0	0
$683$	−4.13391	−0.158180	−0.0790898	−	0.996867i	$-0.525201\pi$
$683$	−4.13391	−0.158180	−0.0790898	+	0.996867i	$0.525201\pi$
$684$	0	0
$685$	−28.1633	−1.07606
$686$	0	0
$687$	−0.576529	−0.0219960
$688$	0	0
$689$	0.291419	0.0111022
$690$	0	0
$691$	−18.2722	−0.695107	−0.347554	−	0.937660i	$-0.612988\pi$
$691$	−18.2722	−0.695107	−0.347554	+	0.937660i	$0.612988\pi$
$692$	0	0
$693$	47.2253	1.79394
$694$	0	0
$695$	−2.61066	−0.0990280
$696$	0	0
$697$	68.5133	2.59513
$698$	0	0
$699$	−2.00186	−0.0757172
$700$	0	0
$701$	−32.9166	−1.24324	−0.621621	−	0.783318i	$-0.713525\pi$
$701$	−32.9166	−1.24324	−0.621621	+	0.783318i	$0.713525\pi$
$702$	0	0
$703$	10.4516	0.394190
$704$	0	0
$705$	1.16580	0.0439067
$706$	0	0
$707$	−41.7436	−1.56993
$708$	0	0
$709$	37.6661	1.41458	0.707289	−	0.706924i	$-0.249918\pi$
$709$	37.6661	1.41458	0.707289	+	0.706924i	$0.249918\pi$
$710$	0	0
$711$	27.3116	1.02427
$712$	0	0
$713$	0.824653	0.0308835
$714$	0	0
$715$	0.586299	0.0219263
$716$	0	0
$717$	0.146251	0.00546185
$718$	0	0
$719$	39.6610	1.47910	0.739552	−	0.673099i	$-0.235037\pi$
$719$	39.6610	1.47910	0.739552	+	0.673099i	$0.235037\pi$
$720$	0	0
$721$	1.20512	0.0448812
$722$	0	0
$723$	0.0687732	0.00255770
$724$	0	0
$725$	15.1572	0.562923
$726$	0	0
$727$	0.326534	0.0121105	0.00605524	−	0.999982i	$-0.498073\pi$
$727$	0.326534	0.0121105	0.00605524	+	0.999982i	$0.498073\pi$
$728$	0	0
$729$	−25.9896	−0.962579
$730$	0	0
$731$	63.9851	2.36658
$732$	0	0
$733$	15.2221	0.562239	0.281120	−	0.959673i	$-0.409294\pi$
$733$	15.2221	0.562239	0.281120	+	0.959673i	$0.409294\pi$
$734$	0	0
$735$	−2.75560	−0.101642
$736$	0	0
$737$	−13.4755	−0.496377
$738$	0	0
$739$	1.80611	0.0664389	0.0332195	−	0.999448i	$-0.489424\pi$
$739$	1.80611	0.0664389	0.0332195	+	0.999448i	$0.489424\pi$
$740$	0	0
$741$	0.0459970	0.00168974
$742$	0	0
$743$	−39.4995	−1.44910	−0.724548	−	0.689224i	$-0.757951\pi$
$743$	−39.4995	−1.44910	−0.724548	+	0.689224i	$0.757951\pi$
$744$	0	0
$745$	36.6384	1.34233
$746$	0	0
$747$	48.3071	1.76746
$748$	0	0
$749$	21.8515	0.798436
$750$	0	0
$751$	−36.2290	−1.32202	−0.661008	−	0.750379i	$-0.729871\pi$
$751$	−36.2290	−1.32202	−0.661008	+	0.750379i	$0.729871\pi$
$752$	0	0
$753$	−1.41722	−0.0516463
$754$	0	0
$755$	−35.8859	−1.30602
$756$	0	0
$757$	20.6460	0.750393	0.375197	−	0.926945i	$-0.377575\pi$
$757$	20.6460	0.750393	0.375197	+	0.926945i	$0.377575\pi$
$758$	0	0
$759$	−0.487314	−0.0176884
$760$	0	0
$761$	−14.9927	−0.543487	−0.271743	−	0.962370i	$-0.587600\pi$
$761$	−14.9927	−0.543487	−0.271743	+	0.962370i	$0.587600\pi$
$762$	0	0
$763$	−52.1806	−1.88906
$764$	0	0
$765$	29.5591	1.06871
$766$	0	0
$767$	0.800581	0.0289073
$768$	0	0
$769$	−8.34663	−0.300987	−0.150494	−	0.988611i	$-0.548086\pi$
$769$	−8.34663	−0.300987	−0.150494	+	0.988611i	$0.548086\pi$
$770$	0	0
$771$	1.07537	0.0387284
$772$	0	0
$773$	18.0952	0.650838	0.325419	−	0.945570i	$-0.394495\pi$
$773$	18.0952	0.650838	0.325419	+	0.945570i	$0.394495\pi$
$774$	0	0
$775$	2.07377	0.0744921
$776$	0	0
$777$	−1.97475	−0.0708439
$778$	0	0
$779$	−35.1820	−1.26052
$780$	0	0
$781$	4.76125	0.170371
$782$	0	0
$783$	−4.96354	−0.177382
$784$	0	0
$785$	32.1801	1.14856
$786$	0	0
$787$	7.53615	0.268635	0.134317	−	0.990938i	$-0.457116\pi$
$787$	7.53615	0.268635	0.134317	+	0.990938i	$0.457116\pi$
$788$	0	0
$789$	2.76695	0.0985059
$790$	0	0
$791$	60.1603	2.13905
$792$	0	0
$793$	−0.0899796	−0.00319527
$794$	0	0
$795$	0.606724	0.0215183
$796$	0	0
$797$	−35.8685	−1.27053	−0.635263	−	0.772296i	$-0.719108\pi$
$797$	−35.8685	−1.27053	−0.635263	+	0.772296i	$0.719108\pi$
$798$	0	0
$799$	33.7803	1.19506
$800$	0	0
$801$	−53.3663	−1.88561
$802$	0	0
$803$	−36.2584	−1.27953
$804$	0	0
$805$	−6.98718	−0.246266
$806$	0	0
$807$	0.195387	0.00687796
$808$	0	0
$809$	−35.8340	−1.25986	−0.629928	−	0.776654i	$-0.716916\pi$
$809$	−35.8340	−1.25986	−0.629928	+	0.776654i	$0.716916\pi$
$810$	0	0
$811$	−39.5546	−1.38895	−0.694475	−	0.719516i	$-0.744364\pi$
$811$	−39.5546	−1.38895	−0.694475	+	0.719516i	$0.744364\pi$
$812$	0	0
$813$	3.54230	0.124234
$814$	0	0
$815$	10.9563	0.383781
$816$	0	0
$817$	−32.8567	−1.14951
$818$	0	0
$819$	1.37751	0.0481340
$820$	0	0
$821$	−42.2012	−1.47283	−0.736416	−	0.676529i	$-0.763483\pi$
$821$	−42.2012	−1.47283	−0.736416	+	0.676529i	$0.763483\pi$
$822$	0	0
$823$	−29.8169	−1.03935	−0.519676	−	0.854363i	$-0.673947\pi$
$823$	−29.8169	−1.03935	−0.519676	+	0.854363i	$0.673947\pi$
$824$	0	0
$825$	−1.22546	−0.0426650
$826$	0	0
$827$	−5.40176	−0.187838	−0.0939188	−	0.995580i	$-0.529939\pi$
$827$	−5.40176	−0.187838	−0.0939188	+	0.995580i	$0.529939\pi$
$828$	0	0
$829$	−5.14687	−0.178758	−0.0893790	−	0.995998i	$-0.528488\pi$
$829$	−5.14687	−0.178758	−0.0893790	+	0.995998i	$0.528488\pi$
$830$	0	0
$831$	1.50987	0.0523767
$832$	0	0
$833$	−79.8463	−2.76651
$834$	0	0
$835$	1.46160	0.0505808
$836$	0	0
$837$	−0.679101	−0.0234732
$838$	0	0
$839$	−10.4855	−0.362000	−0.181000	−	0.983483i	$-0.557933\pi$
$839$	−10.4855	−0.362000	−0.181000	+	0.983483i	$0.557933\pi$
$840$	0	0
$841$	7.61439	0.262565
$842$	0	0
$843$	−1.02909	−0.0354437
$844$	0	0
$845$	−20.5175	−0.705824
$846$	0	0
$847$	7.66019	0.263207
$848$	0	0
$849$	−2.01199	−0.0690514
$850$	0	0
$851$	−3.22984	−0.110718
$852$	0	0
$853$	−18.4657	−0.632252	−0.316126	−	0.948717i	$-0.602382\pi$
$853$	−18.4657	−0.632252	−0.316126	+	0.948717i	$0.602382\pi$
$854$	0	0
$855$	−15.1788	−0.519104
$856$	0	0
$857$	−21.1416	−0.722184	−0.361092	−	0.932530i	$-0.617596\pi$
$857$	−21.1416	−0.722184	−0.361092	+	0.932530i	$0.617596\pi$
$858$	0	0
$859$	−2.62460	−0.0895503	−0.0447752	−	0.998997i	$-0.514257\pi$
$859$	−2.62460	−0.0895503	−0.0447752	+	0.998997i	$0.514257\pi$
$860$	0	0
$861$	6.64737	0.226542
$862$	0	0
$863$	26.5965	0.905356	0.452678	−	0.891674i	$-0.350469\pi$
$863$	26.5965	0.905356	0.452678	+	0.891674i	$0.350469\pi$
$864$	0	0
$865$	−23.3025	−0.792307
$866$	0	0
$867$	−3.07230	−0.104341
$868$	0	0
$869$	32.6803	1.10860
$870$	0	0
$871$	−0.393065	−0.0133185
$872$	0	0
$873$	−20.7809	−0.703326
$874$	0	0
$875$	−52.6435	−1.77968
$876$	0	0
$877$	51.7946	1.74898	0.874490	−	0.485043i	$-0.161196\pi$
$877$	51.7946	1.74898	0.874490	+	0.485043i	$0.161196\pi$
$878$	0	0
$879$	3.34382	0.112784
$880$	0	0
$881$	−33.0169	−1.11237	−0.556183	−	0.831060i	$-0.687735\pi$
$881$	−33.0169	−1.11237	−0.556183	+	0.831060i	$0.687735\pi$
$882$	0	0
$883$	−24.4735	−0.823597	−0.411799	−	0.911275i	$-0.635099\pi$
$883$	−24.4735	−0.823597	−0.411799	+	0.911275i	$0.635099\pi$
$884$	0	0
$885$	1.66678	0.0560282
$886$	0	0
$887$	18.1946	0.610916	0.305458	−	0.952206i	$-0.401191\pi$
$887$	18.1946	0.610916	0.305458	+	0.952206i	$0.401191\pi$
$888$	0	0
$889$	83.7874	2.81014
$890$	0	0
$891$	−31.5023	−1.05537
$892$	0	0
$893$	−17.3464	−0.580475
$894$	0	0
$895$	3.41946	0.114300
$896$	0	0
$897$	−0.0142144	−0.000474604 0
$898$	0	0
$899$	5.00951	0.167076
$900$	0	0
$901$	17.5804	0.585690
$902$	0	0
$903$	6.20803	0.206590
$904$	0	0
$905$	29.7253	0.988102
$906$	0	0
$907$	−23.9916	−0.796628	−0.398314	−	0.917249i	$-0.630405\pi$
$907$	−23.9916	−0.796628	−0.398314	+	0.917249i	$0.630405\pi$
$908$	0	0
$909$	28.0236	0.929483
$910$	0	0
$911$	−35.6670	−1.18170	−0.590850	−	0.806782i	$-0.701207\pi$
$911$	−35.6670	−1.18170	−0.590850	+	0.806782i	$0.701207\pi$
$912$	0	0
$913$	57.8028	1.91300
$914$	0	0
$915$	−0.187334	−0.00619308
$916$	0	0
$917$	81.5373	2.69260
$918$	0	0
$919$	42.5231	1.40271	0.701354	−	0.712813i	$-0.252579\pi$
$919$	42.5231	1.40271	0.701354	+	0.712813i	$0.252579\pi$
$920$	0	0
$921$	2.25241	0.0742196
$922$	0	0
$923$	0.138880	0.00457129
$924$	0	0
$925$	−8.12216	−0.267055
$926$	0	0
$927$	−0.809032	−0.0265721
$928$	0	0
$929$	3.09898	0.101674	0.0508371	−	0.998707i	$-0.483811\pi$
$929$	3.09898	0.101674	0.0508371	+	0.998707i	$0.483811\pi$
$930$	0	0
$931$	41.0015	1.34377
$932$	0	0
$933$	−0.584896	−0.0191486
$934$	0	0
$935$	35.3696	1.15671
$936$	0	0
$937$	22.4502	0.733414	0.366707	−	0.930336i	$-0.380485\pi$
$937$	22.4502	0.733414	0.366707	+	0.930336i	$0.380485\pi$
$938$	0	0
$939$	−1.93950	−0.0632931
$940$	0	0
$941$	−26.3639	−0.859440	−0.429720	−	0.902962i	$-0.641388\pi$
$941$	−26.3639	−0.859440	−0.429720	+	0.902962i	$0.641388\pi$
$942$	0	0
$943$	10.8722	0.354048
$944$	0	0
$945$	5.75393	0.187175
$946$	0	0
$947$	−35.5597	−1.15554	−0.577768	−	0.816201i	$-0.696076\pi$
$947$	−35.5597	−1.15554	−0.577768	+	0.816201i	$0.696076\pi$
$948$	0	0
$949$	−1.05761	−0.0343316
$950$	0	0
$951$	0.0207954	0.000674335 0
$952$	0	0
$953$	24.1635	0.782732	0.391366	−	0.920235i	$-0.372003\pi$
$953$	24.1635	0.782732	0.391366	+	0.920235i	$0.372003\pi$
$954$	0	0
$955$	−11.6404	−0.376674
$956$	0	0
$957$	−2.96027	−0.0956921
$958$	0	0
$959$	−79.1765	−2.55674
$960$	0	0
$961$	−30.3146	−0.977891
$962$	0	0
$963$	−14.6695	−0.472718
$964$	0	0
$965$	−18.5774	−0.598028
$966$	0	0
$967$	−25.2563	−0.812188	−0.406094	−	0.913831i	$-0.633109\pi$
$967$	−25.2563	−0.812188	−0.406094	+	0.913831i	$0.633109\pi$
$968$	0	0
$969$	2.77486	0.0891413
$970$	0	0
$971$	8.28223	0.265789	0.132895	−	0.991130i	$-0.457573\pi$
$971$	8.28223	0.265789	0.132895	+	0.991130i	$0.457573\pi$
$972$	0	0
$973$	−7.33945	−0.235292
$974$	0	0
$975$	−0.0357452	−0.00114476
$976$	0	0
$977$	32.2547	1.03192	0.515959	−	0.856613i	$-0.327436\pi$
$977$	32.2547	1.03192	0.515959	+	0.856613i	$0.327436\pi$
$978$	0	0
$979$	−63.8566	−2.04087
$980$	0	0
$981$	35.0302	1.11843
$982$	0	0
$983$	−40.3667	−1.28750	−0.643749	−	0.765237i	$-0.722622\pi$
$983$	−40.3667	−1.28750	−0.643749	+	0.765237i	$0.722622\pi$
$984$	0	0
$985$	22.8059	0.726656
$986$	0	0
$987$	3.27747	0.104323
$988$	0	0
$989$	10.1537	0.322867
$990$	0	0
$991$	−37.9851	−1.20664	−0.603319	−	0.797500i	$-0.706155\pi$
$991$	−37.9851	−1.20664	−0.603319	+	0.797500i	$0.706155\pi$
$992$	0	0
$993$	−1.02755	−0.0326082
$994$	0	0
$995$	10.5838	0.335530
$996$	0	0
$997$	19.5045	0.617712	0.308856	−	0.951109i	$-0.400054\pi$
$997$	19.5045	0.617712	0.308856	+	0.951109i	$0.400054\pi$
$998$	0	0
$999$	2.65977	0.0841515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.17	✓	29
4.3	odd	2		8048.2.a.w.1.13		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.17	✓	29	1.1	even	1	trivial
8048.2.a.w.1.13		29	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.137144 q^{3} +1.57958 q^{5} +4.44075 q^{7} -2.98119 q^{9} +O(q^{10})\)	\(q-0.137144 q^{3} +1.57958 q^{5} +4.44075 q^{7} -2.98119 q^{9} -3.56721 q^{11} -0.104051 q^{13} -0.216631 q^{15} -6.27710 q^{17} +3.22333 q^{19} -0.609023 q^{21} -0.996099 q^{23} -2.50491 q^{25} +0.820286 q^{27} -6.05098 q^{29} -0.827883 q^{31} +0.489222 q^{33} +7.01454 q^{35} +3.24249 q^{37} +0.0142700 q^{39} -10.9148 q^{41} -10.1934 q^{43} -4.70905 q^{45} -5.38152 q^{47} +12.7203 q^{49} +0.860868 q^{51} -2.80073 q^{53} -5.63471 q^{55} -0.442061 q^{57} -7.69410 q^{59} +0.864762 q^{61} -13.2387 q^{63} -0.164358 q^{65} +3.77761 q^{67} +0.136609 q^{69} -1.33473 q^{71} +10.1644 q^{73} +0.343534 q^{75} -15.8411 q^{77} -9.16131 q^{79} +8.83108 q^{81} -16.2039 q^{83} -9.91521 q^{85} +0.829857 q^{87} +17.9010 q^{89} -0.462066 q^{91} +0.113539 q^{93} +5.09152 q^{95} +6.97066 q^{97} +10.6345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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