LMFDB - Embedded newform 6008.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.30310 q^{3} -2.23921 q^{5} +4.70410 q^{7} +7.91049 q^{9} +O(q^{10})$	$q-3.30310 q^{3} -2.23921 q^{5} +4.70410 q^{7} +7.91049 q^{9} +3.99705 q^{11} -1.48803 q^{13} +7.39635 q^{15} +7.43337 q^{17} -5.93362 q^{19} -15.5381 q^{21} -0.000615582 q^{23} +0.0140797 q^{25} -16.2198 q^{27} -9.83132 q^{29} -5.34712 q^{31} -13.2027 q^{33} -10.5335 q^{35} +8.90784 q^{37} +4.91513 q^{39} -4.14743 q^{41} +6.71528 q^{43} -17.7133 q^{45} -1.65117 q^{47} +15.1286 q^{49} -24.5532 q^{51} -6.85546 q^{53} -8.95025 q^{55} +19.5994 q^{57} +1.52391 q^{59} -10.3974 q^{61} +37.2118 q^{63} +3.33203 q^{65} -7.62953 q^{67} +0.00203333 q^{69} -11.4348 q^{71} -7.06623 q^{73} -0.0465066 q^{75} +18.8025 q^{77} +0.282220 q^{79} +29.8443 q^{81} +8.92638 q^{83} -16.6449 q^{85} +32.4739 q^{87} -2.42005 q^{89} -6.99987 q^{91} +17.6621 q^{93} +13.2867 q^{95} -14.2725 q^{97} +31.6186 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9	$ 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9} + 11 q^{11} - 36 q^{13} - 5 q^{15} - 10 q^{17} - 7 q^{19} - 42 q^{21} - 5 q^{23} + 29 q^{25} - 16 q^{27} - 57 q^{29} - 21 q^{31} - 32 q^{33} + 17 q^{35} - 52 q^{37} + 8 q^{39} - 16 q^{41} - 9 q^{43} - 84 q^{45} - q^{47} + 28 q^{49} - q^{51} - 52 q^{53} - 39 q^{55} - 15 q^{57} + 7 q^{59} - 85 q^{61} - 25 q^{63} - 9 q^{65} - 36 q^{67} - 72 q^{69} + 12 q^{71} - 60 q^{73} - 5 q^{75} - 81 q^{77} - 13 q^{79} + 20 q^{81} + 5 q^{83} - 72 q^{85} + 9 q^{87} - 37 q^{89} - 23 q^{91} - 60 q^{93} + 24 q^{95} - 79 q^{97} + 42 q^{99}+O(q^{100}) $ 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9 + 11 * q^11 - 36 * q^13 - 5 * q^15 - 10 * q^17 - 7 * q^19 - 42 * q^21 - 5 * q^23 + 29 * q^25 - 16 * q^27 - 57 * q^29 - 21 * q^31 - 32 * q^33 + 17 * q^35 - 52 * q^37 + 8 * q^39 - 16 * q^41 - 9 * q^43 - 84 * q^45 - q^47 + 28 * q^49 - q^51 - 52 * q^53 - 39 * q^55 - 15 * q^57 + 7 * q^59 - 85 * q^61 - 25 * q^63 - 9 * q^65 - 36 * q^67 - 72 * q^69 + 12 * q^71 - 60 * q^73 - 5 * q^75 - 81 * q^77 - 13 * q^79 + 20 * q^81 + 5 * q^83 - 72 * q^85 + 9 * q^87 - 37 * q^89 - 23 * q^91 - 60 * q^93 + 24 * q^95 - 79 * q^97 + 42 * 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
$3$	−3.30310	−1.90705	−0.953524	−	0.301318i	$-0.902573\pi$
$3$	−3.30310	−1.90705	−0.953524	+	0.301318i	$0.902573\pi$
$4$	0	0
$5$	−2.23921	−1.00141	−0.500703	−	0.865619i	$-0.666925\pi$
$5$	−2.23921	−1.00141	−0.500703	+	0.865619i	$0.666925\pi$
$6$	0	0
$7$	4.70410	1.77798	0.888992	−	0.457922i	$-0.151406\pi$
$7$	4.70410	1.77798	0.888992	+	0.457922i	$0.151406\pi$
$8$	0	0
$9$	7.91049	2.63683
$10$	0	0
$11$	3.99705	1.20516	0.602578	−	0.798060i	$-0.294140\pi$
$11$	3.99705	1.20516	0.602578	+	0.798060i	$0.294140\pi$
$12$	0	0
$13$	−1.48803	−0.412706	−0.206353	−	0.978478i	$-0.566160\pi$
$13$	−1.48803	−0.412706	−0.206353	+	0.978478i	$0.566160\pi$
$14$	0	0
$15$	7.39635	1.90973
$16$	0	0
$17$	7.43337	1.80286	0.901429	−	0.432928i	$-0.142519\pi$
$17$	7.43337	1.80286	0.901429	+	0.432928i	$0.142519\pi$
$18$	0	0
$19$	−5.93362	−1.36127	−0.680633	−	0.732624i	$-0.738295\pi$
$19$	−5.93362	−1.36127	−0.680633	+	0.732624i	$0.738295\pi$
$20$	0	0
$21$	−15.5381	−3.39070
$22$	0	0
$23$	−0.000615582 0	−0.000128358 0	−6.41789e−5	−	1.00000i	$-0.500020\pi$
$23$	−0.000615582 0	−0.000128358 0	−6.41789e−5		1.00000i	$0.500020\pi$
$24$	0	0
$25$	0.0140797	0.00281594
$26$	0	0
$27$	−16.2198	−3.12151
$28$	0	0
$29$	−9.83132	−1.82563	−0.912816	−	0.408372i	$-0.866097\pi$
$29$	−9.83132	−1.82563	−0.912816	+	0.408372i	$0.866097\pi$
$30$	0	0
$31$	−5.34712	−0.960372	−0.480186	−	0.877167i	$-0.659431\pi$
$31$	−5.34712	−0.960372	−0.480186	+	0.877167i	$0.659431\pi$
$32$	0	0
$33$	−13.2027	−2.29829
$34$	0	0
$35$	−10.5335	−1.78049
$36$	0	0
$37$	8.90784	1.46444	0.732220	−	0.681068i	$-0.238484\pi$
$37$	8.90784	1.46444	0.732220	+	0.681068i	$0.238484\pi$
$38$	0	0
$39$	4.91513	0.787051
$40$	0	0
$41$	−4.14743	−0.647719	−0.323860	−	0.946105i	$-0.604981\pi$
$41$	−4.14743	−0.647719	−0.323860	+	0.946105i	$0.604981\pi$
$42$	0	0
$43$	6.71528	1.02407	0.512035	−	0.858964i	$-0.328892\pi$
$43$	6.71528	1.02407	0.512035	+	0.858964i	$0.328892\pi$
$44$	0	0
$45$	−17.7133	−2.64054
$46$	0	0
$47$	−1.65117	−0.240848	−0.120424	−	0.992723i	$-0.538425\pi$
$47$	−1.65117	−0.240848	−0.120424	+	0.992723i	$0.538425\pi$
$48$	0	0
$49$	15.1286	2.16123
$50$	0	0
$51$	−24.5532	−3.43813
$52$	0	0
$53$	−6.85546	−0.941670	−0.470835	−	0.882221i	$-0.656047\pi$
$53$	−6.85546	−0.941670	−0.470835	+	0.882221i	$0.656047\pi$
$54$	0	0
$55$	−8.95025	−1.20685
$56$	0	0
$57$	19.5994	2.59600
$58$	0	0
$59$	1.52391	0.198396	0.0991979	−	0.995068i	$-0.468372\pi$
$59$	1.52391	0.198396	0.0991979	+	0.995068i	$0.468372\pi$
$60$	0	0
$61$	−10.3974	−1.33125	−0.665623	−	0.746289i	$-0.731834\pi$
$61$	−10.3974	−1.33125	−0.665623	+	0.746289i	$0.731834\pi$
$62$	0	0
$63$	37.2118	4.68824
$64$	0	0
$65$	3.33203	0.413287
$66$	0	0
$67$	−7.62953	−0.932095	−0.466048	−	0.884760i	$-0.654322\pi$
$67$	−7.62953	−0.932095	−0.466048	+	0.884760i	$0.654322\pi$
$68$	0	0
$69$	0.00203333	0.000244784 0
$70$	0	0
$71$	−11.4348	−1.35706	−0.678532	−	0.734571i	$-0.737384\pi$
$71$	−11.4348	−1.35706	−0.678532	+	0.734571i	$0.737384\pi$
$72$	0	0
$73$	−7.06623	−0.827040	−0.413520	−	0.910495i	$-0.635701\pi$
$73$	−7.06623	−0.827040	−0.413520	+	0.910495i	$0.635701\pi$
$74$	0	0
$75$	−0.0465066	−0.00537012
$76$	0	0
$77$	18.8025	2.14275
$78$	0	0
$79$	0.282220	0.0317523	0.0158761	−	0.999874i	$-0.494946\pi$
$79$	0.282220	0.0317523	0.0158761	+	0.999874i	$0.494946\pi$
$80$	0	0
$81$	29.8443	3.31604
$82$	0	0
$83$	8.92638	0.979798	0.489899	−	0.871779i	$-0.337034\pi$
$83$	8.92638	0.979798	0.489899	+	0.871779i	$0.337034\pi$
$84$	0	0
$85$	−16.6449	−1.80539
$86$	0	0
$87$	32.4739	3.48156
$88$	0	0
$89$	−2.42005	−0.256525	−0.128263	−	0.991740i	$-0.540940\pi$
$89$	−2.42005	−0.256525	−0.128263	+	0.991740i	$0.540940\pi$
$90$	0	0
$91$	−6.99987	−0.733786
$92$	0	0
$93$	17.6621	1.83147
$94$	0	0
$95$	13.2867	1.36318
$96$	0	0
$97$	−14.2725	−1.44915	−0.724575	−	0.689196i	$-0.757964\pi$
$97$	−14.2725	−1.44915	−0.724575	+	0.689196i	$0.757964\pi$
$98$	0	0
$99$	31.6186	3.17779
$100$	0	0
$101$	−14.1311	−1.40610	−0.703049	−	0.711141i	$-0.748179\pi$
$101$	−14.1311	−1.40610	−0.703049	+	0.711141i	$0.748179\pi$
$102$	0	0
$103$	−0.627884	−0.0618673	−0.0309336	−	0.999521i	$-0.509848\pi$
$103$	−0.627884	−0.0618673	−0.0309336	+	0.999521i	$0.509848\pi$
$104$	0	0
$105$	34.7932	3.39547
$106$	0	0
$107$	14.6007	1.41150	0.705751	−	0.708460i	$-0.250610\pi$
$107$	14.6007	1.41150	0.705751	+	0.708460i	$0.250610\pi$
$108$	0	0
$109$	7.25896	0.695282	0.347641	−	0.937628i	$-0.386983\pi$
$109$	7.25896	0.695282	0.347641	+	0.937628i	$0.386983\pi$
$110$	0	0
$111$	−29.4235	−2.79276
$112$	0	0
$113$	−9.06785	−0.853031	−0.426516	−	0.904480i	$-0.640259\pi$
$113$	−9.06785	−0.853031	−0.426516	+	0.904480i	$0.640259\pi$
$114$	0	0
$115$	0.00137842	0.000128538 0
$116$	0	0
$117$	−11.7711	−1.08824
$118$	0	0
$119$	34.9674	3.20545
$120$	0	0
$121$	4.97642	0.452401
$122$	0	0
$123$	13.6994	1.23523
$124$	0	0
$125$	11.1645	0.998587
$126$	0	0
$127$	14.1443	1.25510	0.627550	−	0.778576i	$-0.284058\pi$
$127$	14.1443	1.25510	0.627550	+	0.778576i	$0.284058\pi$
$128$	0	0
$129$	−22.1813	−1.95295
$130$	0	0
$131$	18.1197	1.58312	0.791562	−	0.611089i	$-0.209268\pi$
$131$	18.1197	1.58312	0.791562	+	0.611089i	$0.209268\pi$
$132$	0	0
$133$	−27.9124	−2.42031
$134$	0	0
$135$	36.3197	3.12590
$136$	0	0
$137$	6.76364	0.577856	0.288928	−	0.957351i	$-0.406701\pi$
$137$	6.76364	0.577856	0.288928	+	0.957351i	$0.406701\pi$
$138$	0	0
$139$	5.61603	0.476345	0.238173	−	0.971223i	$-0.423452\pi$
$139$	5.61603	0.476345	0.238173	+	0.971223i	$0.423452\pi$
$140$	0	0
$141$	5.45399	0.459309
$142$	0	0
$143$	−5.94775	−0.497376
$144$	0	0
$145$	22.0144	1.82820
$146$	0	0
$147$	−49.9713	−4.12157
$148$	0	0
$149$	−1.58463	−0.129818	−0.0649092	−	0.997891i	$-0.520676\pi$
$149$	−1.58463	−0.129818	−0.0649092	+	0.997891i	$0.520676\pi$
$150$	0	0
$151$	−13.8117	−1.12398	−0.561989	−	0.827144i	$-0.689964\pi$
$151$	−13.8117	−1.12398	−0.561989	+	0.827144i	$0.689964\pi$
$152$	0	0
$153$	58.8016	4.75383
$154$	0	0
$155$	11.9734	0.961723
$156$	0	0
$157$	−4.96025	−0.395871	−0.197935	−	0.980215i	$-0.563424\pi$
$157$	−4.96025	−0.395871	−0.197935	+	0.980215i	$0.563424\pi$
$158$	0	0
$159$	22.6443	1.79581
$160$	0	0
$161$	−0.00289576	−0.000228218 0
$162$	0	0
$163$	−10.6246	−0.832183	−0.416091	−	0.909323i	$-0.636600\pi$
$163$	−10.6246	−0.832183	−0.416091	+	0.909323i	$0.636600\pi$
$164$	0	0
$165$	29.5636	2.30152
$166$	0	0
$167$	−4.69671	−0.363443	−0.181721	−	0.983350i	$-0.558167\pi$
$167$	−4.69671	−0.363443	−0.181721	+	0.983350i	$0.558167\pi$
$168$	0	0
$169$	−10.7858	−0.829673
$170$	0	0
$171$	−46.9379	−3.58943
$172$	0	0
$173$	−8.58661	−0.652828	−0.326414	−	0.945227i	$-0.605840\pi$
$173$	−8.58661	−0.652828	−0.326414	+	0.945227i	$0.605840\pi$
$174$	0	0
$175$	0.0662323	0.00500669
$176$	0	0
$177$	−5.03362	−0.378350
$178$	0	0
$179$	6.56601	0.490767	0.245383	−	0.969426i	$-0.421086\pi$
$179$	6.56601	0.490767	0.245383	+	0.969426i	$0.421086\pi$
$180$	0	0
$181$	−0.526908	−0.0391648	−0.0195824	−	0.999808i	$-0.506234\pi$
$181$	−0.526908	−0.0391648	−0.0195824	+	0.999808i	$0.506234\pi$
$182$	0	0
$183$	34.3435	2.53875
$184$	0	0
$185$	−19.9466	−1.46650
$186$	0	0
$187$	29.7116	2.17272
$188$	0	0
$189$	−76.2998	−5.55000
$190$	0	0
$191$	−7.51236	−0.543575	−0.271788	−	0.962357i	$-0.587615\pi$
$191$	−7.51236	−0.543575	−0.271788	+	0.962357i	$0.587615\pi$
$192$	0	0
$193$	−24.3277	−1.75114	−0.875572	−	0.483087i	$-0.839516\pi$
$193$	−24.3277	−1.75114	−0.875572	+	0.483087i	$0.839516\pi$
$194$	0	0
$195$	−11.0060	−0.788158
$196$	0	0
$197$	−22.2405	−1.58457	−0.792285	−	0.610151i	$-0.791109\pi$
$197$	−22.2405	−1.58457	−0.792285	+	0.610151i	$0.791109\pi$
$198$	0	0
$199$	12.6507	0.896786	0.448393	−	0.893837i	$-0.351997\pi$
$199$	12.6507	0.896786	0.448393	+	0.893837i	$0.351997\pi$
$200$	0	0
$201$	25.2011	1.77755
$202$	0	0
$203$	−46.2476	−3.24594
$204$	0	0
$205$	9.28697	0.648630
$206$	0	0
$207$	−0.00486955	−0.000338457 0
$208$	0	0
$209$	−23.7170	−1.64054
$210$	0	0
$211$	19.2796	1.32727	0.663633	−	0.748059i	$-0.269014\pi$
$211$	19.2796	1.32727	0.663633	+	0.748059i	$0.269014\pi$
$212$	0	0
$213$	37.7704	2.58799
$214$	0	0
$215$	−15.0369	−1.02551
$216$	0	0
$217$	−25.1534	−1.70753
$218$	0	0
$219$	23.3405	1.57720
$220$	0	0
$221$	−11.0611	−0.744051
$222$	0	0
$223$	−23.5975	−1.58021	−0.790104	−	0.612973i	$-0.789973\pi$
$223$	−23.5975	−1.58021	−0.790104	+	0.612973i	$0.789973\pi$
$224$	0	0
$225$	0.111377	0.00742514
$226$	0	0
$227$	6.37329	0.423010	0.211505	−	0.977377i	$-0.432164\pi$
$227$	6.37329	0.423010	0.211505	+	0.977377i	$0.432164\pi$
$228$	0	0
$229$	−3.62314	−0.239424	−0.119712	−	0.992809i	$-0.538197\pi$
$229$	−3.62314	−0.239424	−0.119712	+	0.992809i	$0.538197\pi$
$230$	0	0
$231$	−62.1067	−4.08632
$232$	0	0
$233$	−8.72402	−0.571529	−0.285765	−	0.958300i	$-0.592248\pi$
$233$	−8.72402	−0.571529	−0.285765	+	0.958300i	$0.592248\pi$
$234$	0	0
$235$	3.69733	0.241187
$236$	0	0
$237$	−0.932203	−0.0605531
$238$	0	0
$239$	−2.88719	−0.186757	−0.0933784	−	0.995631i	$-0.529767\pi$
$239$	−2.88719	−0.186757	−0.0933784	+	0.995631i	$0.529767\pi$
$240$	0	0
$241$	−13.9447	−0.898256	−0.449128	−	0.893467i	$-0.648265\pi$
$241$	−13.9447	−0.898256	−0.449128	+	0.893467i	$0.648265\pi$
$242$	0	0
$243$	−49.9194	−3.20233
$244$	0	0
$245$	−33.8762	−2.16427
$246$	0	0
$247$	8.82944	0.561804
$248$	0	0
$249$	−29.4848	−1.86852
$250$	0	0
$251$	24.8644	1.56943	0.784714	−	0.619857i	$-0.212810\pi$
$251$	24.8644	1.56943	0.784714	+	0.619857i	$0.212810\pi$
$252$	0	0
$253$	−0.00246051	−0.000154691 0
$254$	0	0
$255$	54.9798	3.44297
$256$	0	0
$257$	20.6423	1.28763	0.643817	−	0.765180i	$-0.277350\pi$
$257$	20.6423	1.28763	0.643817	+	0.765180i	$0.277350\pi$
$258$	0	0
$259$	41.9034	2.60375
$260$	0	0
$261$	−77.7706	−4.81388
$262$	0	0
$263$	−8.25799	−0.509210	−0.254605	−	0.967045i	$-0.581945\pi$
$263$	−8.25799	−0.509210	−0.254605	+	0.967045i	$0.581945\pi$
$264$	0	0
$265$	15.3508	0.942995
$266$	0	0
$267$	7.99369	0.489206
$268$	0	0
$269$	−1.79340	−0.109346	−0.0546729	−	0.998504i	$-0.517412\pi$
$269$	−1.79340	−0.109346	−0.0546729	+	0.998504i	$0.517412\pi$
$270$	0	0
$271$	−13.4505	−0.817063	−0.408531	−	0.912744i	$-0.633959\pi$
$271$	−13.4505	−0.817063	−0.408531	+	0.912744i	$0.633959\pi$
$272$	0	0
$273$	23.1213	1.39936
$274$	0	0
$275$	0.0562772	0.00339364
$276$	0	0
$277$	−22.0971	−1.32768	−0.663842	−	0.747872i	$-0.731075\pi$
$277$	−22.0971	−1.32768	−0.663842	+	0.747872i	$0.731075\pi$
$278$	0	0
$279$	−42.2983	−2.53234
$280$	0	0
$281$	−1.72939	−0.103167	−0.0515834	−	0.998669i	$-0.516427\pi$
$281$	−1.72939	−0.103167	−0.0515834	+	0.998669i	$0.516427\pi$
$282$	0	0
$283$	−22.8173	−1.35635	−0.678174	−	0.734901i	$-0.737228\pi$
$283$	−22.8173	−1.35635	−0.678174	+	0.734901i	$0.737228\pi$
$284$	0	0
$285$	−43.8872	−2.59965
$286$	0	0
$287$	−19.5099	−1.15163
$288$	0	0
$289$	38.2550	2.25029
$290$	0	0
$291$	47.1434	2.76360
$292$	0	0
$293$	−5.72925	−0.334706	−0.167353	−	0.985897i	$-0.553522\pi$
$293$	−5.72925	−0.334706	−0.167353	+	0.985897i	$0.553522\pi$
$294$	0	0
$295$	−3.41236	−0.198675
$296$	0	0
$297$	−64.8315	−3.76191
$298$	0	0
$299$	0.000916007 0	5.29741e−5 0
$300$	0	0
$301$	31.5894	1.82078
$302$	0	0
$303$	46.6765	2.68150
$304$	0	0
$305$	23.2819	1.33312
$306$	0	0
$307$	−28.0111	−1.59868	−0.799340	−	0.600879i	$-0.794817\pi$
$307$	−28.0111	−1.59868	−0.799340	+	0.600879i	$0.794817\pi$
$308$	0	0
$309$	2.07397	0.117984
$310$	0	0
$311$	−10.4733	−0.593888	−0.296944	−	0.954895i	$-0.595968\pi$
$311$	−10.4733	−0.593888	−0.296944	+	0.954895i	$0.595968\pi$
$312$	0	0
$313$	−5.22444	−0.295303	−0.147652	−	0.989039i	$-0.547171\pi$
$313$	−5.22444	−0.295303	−0.147652	+	0.989039i	$0.547171\pi$
$314$	0	0
$315$	−83.3251	−4.69484
$316$	0	0
$317$	3.92911	0.220681	0.110340	−	0.993894i	$-0.464806\pi$
$317$	3.92911	0.220681	0.110340	+	0.993894i	$0.464806\pi$
$318$	0	0
$319$	−39.2963	−2.20017
$320$	0	0
$321$	−48.2276	−2.69180
$322$	0	0
$323$	−44.1068	−2.45417
$324$	0	0
$325$	−0.0209511	−0.00116216
$326$	0	0
$327$	−23.9771	−1.32594
$328$	0	0
$329$	−7.76729	−0.428225
$330$	0	0
$331$	14.6049	0.802757	0.401378	−	0.915912i	$-0.368531\pi$
$331$	14.6049	0.802757	0.401378	+	0.915912i	$0.368531\pi$
$332$	0	0
$333$	70.4654	3.86148
$334$	0	0
$335$	17.0842	0.933407
$336$	0	0
$337$	35.5246	1.93515	0.967573	−	0.252590i	$-0.0812824\pi$
$337$	35.5246	1.93515	0.967573	+	0.252590i	$0.0812824\pi$
$338$	0	0
$339$	29.9520	1.62677
$340$	0	0
$341$	−21.3727	−1.15740
$342$	0	0
$343$	38.2378	2.06465
$344$	0	0
$345$	−0.00455306	−0.000245129 0
$346$	0	0
$347$	24.2727	1.30302	0.651512	−	0.758638i	$-0.274135\pi$
$347$	24.2727	1.30302	0.651512	+	0.758638i	$0.274135\pi$
$348$	0	0
$349$	27.6548	1.48033	0.740164	−	0.672426i	$-0.234748\pi$
$349$	27.6548	1.48033	0.740164	+	0.672426i	$0.234748\pi$
$350$	0	0
$351$	24.1357	1.28827
$352$	0	0
$353$	7.39068	0.393366	0.196683	−	0.980467i	$-0.436983\pi$
$353$	7.39068	0.393366	0.196683	+	0.980467i	$0.436983\pi$
$354$	0	0
$355$	25.6050	1.35897
$356$	0	0
$357$	−115.501	−6.11295
$358$	0	0
$359$	24.5927	1.29795	0.648976	−	0.760809i	$-0.275198\pi$
$359$	24.5927	1.29795	0.648976	+	0.760809i	$0.275198\pi$
$360$	0	0
$361$	16.2079	0.853048
$362$	0	0
$363$	−16.4376	−0.862751
$364$	0	0
$365$	15.8228	0.828203
$366$	0	0
$367$	−1.41237	−0.0737253	−0.0368627	−	0.999320i	$-0.511736\pi$
$367$	−1.41237	−0.0737253	−0.0368627	+	0.999320i	$0.511736\pi$
$368$	0	0
$369$	−32.8082	−1.70792
$370$	0	0
$371$	−32.2488	−1.67427
$372$	0	0
$373$	6.30405	0.326411	0.163206	−	0.986592i	$-0.447817\pi$
$373$	6.30405	0.326411	0.163206	+	0.986592i	$0.447817\pi$
$374$	0	0
$375$	−36.8776	−1.90435
$376$	0	0
$377$	14.6293	0.753450
$378$	0	0
$379$	−8.06205	−0.414120	−0.207060	−	0.978328i	$-0.566389\pi$
$379$	−8.06205	−0.414120	−0.207060	+	0.978328i	$0.566389\pi$
$380$	0	0
$381$	−46.7200	−2.39354
$382$	0	0
$383$	−19.6579	−1.00447	−0.502236	−	0.864731i	$-0.667489\pi$
$383$	−19.6579	−1.00447	−0.502236	+	0.864731i	$0.667489\pi$
$384$	0	0
$385$	−42.1029	−2.14576
$386$	0	0
$387$	53.1211	2.70030
$388$	0	0
$389$	17.3641	0.880393	0.440197	−	0.897901i	$-0.354909\pi$
$389$	17.3641	0.880393	0.440197	+	0.897901i	$0.354909\pi$
$390$	0	0
$391$	−0.00457585	−0.000231411 0
$392$	0	0
$393$	−59.8512	−3.01909
$394$	0	0
$395$	−0.631952	−0.0317969
$396$	0	0
$397$	−36.2167	−1.81766	−0.908832	−	0.417163i	$-0.863024\pi$
$397$	−36.2167	−1.81766	−0.908832	+	0.417163i	$0.863024\pi$
$398$	0	0
$399$	92.1975	4.61565
$400$	0	0
$401$	32.0330	1.59965	0.799825	−	0.600233i	$-0.204926\pi$
$401$	32.0330	1.59965	0.799825	+	0.600233i	$0.204926\pi$
$402$	0	0
$403$	7.95670	0.396352
$404$	0	0
$405$	−66.8279	−3.32070
$406$	0	0
$407$	35.6051	1.76488
$408$	0	0
$409$	−33.6989	−1.66630	−0.833152	−	0.553044i	$-0.813466\pi$
$409$	−33.6989	−1.66630	−0.833152	+	0.553044i	$0.813466\pi$
$410$	0	0
$411$	−22.3410	−1.10200
$412$	0	0
$413$	7.16862	0.352745
$414$	0	0
$415$	−19.9881	−0.981177
$416$	0	0
$417$	−18.5503	−0.908413
$418$	0	0
$419$	5.69017	0.277983	0.138992	−	0.990294i	$-0.455614\pi$
$419$	5.69017	0.277983	0.138992	+	0.990294i	$0.455614\pi$
$420$	0	0
$421$	28.4916	1.38860	0.694298	−	0.719688i	$-0.255715\pi$
$421$	28.4916	1.38860	0.694298	+	0.719688i	$0.255715\pi$
$422$	0	0
$423$	−13.0616	−0.635076
$424$	0	0
$425$	0.104660	0.00507673
$426$	0	0
$427$	−48.9103	−2.36693
$428$	0	0
$429$	19.6460	0.948519
$430$	0	0
$431$	2.89536	0.139465	0.0697323	−	0.997566i	$-0.477785\pi$
$431$	2.89536	0.139465	0.0697323	+	0.997566i	$0.477785\pi$
$432$	0	0
$433$	16.1643	0.776808	0.388404	−	0.921489i	$-0.373027\pi$
$433$	16.1643	0.776808	0.388404	+	0.921489i	$0.373027\pi$
$434$	0	0
$435$	−72.7160	−3.48646
$436$	0	0
$437$	0.00365263	0.000174729 0
$438$	0	0
$439$	11.8859	0.567285	0.283643	−	0.958930i	$-0.408457\pi$
$439$	11.8859	0.567285	0.283643	+	0.958930i	$0.408457\pi$
$440$	0	0
$441$	119.675	5.69879
$442$	0	0
$443$	20.1084	0.955380	0.477690	−	0.878529i	$-0.341474\pi$
$443$	20.1084	0.955380	0.477690	+	0.878529i	$0.341474\pi$
$444$	0	0
$445$	5.41902	0.256886
$446$	0	0
$447$	5.23421	0.247570
$448$	0	0
$449$	−16.3822	−0.773122	−0.386561	−	0.922264i	$-0.626337\pi$
$449$	−16.3822	−0.773122	−0.386561	+	0.922264i	$0.626337\pi$
$450$	0	0
$451$	−16.5775	−0.780603
$452$	0	0
$453$	45.6214	2.14348
$454$	0	0
$455$	15.6742	0.734818
$456$	0	0
$457$	−24.9963	−1.16928	−0.584638	−	0.811294i	$-0.698763\pi$
$457$	−24.9963	−1.16928	−0.584638	+	0.811294i	$0.698763\pi$
$458$	0	0
$459$	−120.568	−5.62764
$460$	0	0
$461$	−8.64948	−0.402847	−0.201423	−	0.979504i	$-0.564557\pi$
$461$	−8.64948	−0.402847	−0.201423	+	0.979504i	$0.564557\pi$
$462$	0	0
$463$	−15.3884	−0.715158	−0.357579	−	0.933883i	$-0.616398\pi$
$463$	−15.3884	−0.715158	−0.357579	+	0.933883i	$0.616398\pi$
$464$	0	0
$465$	−39.5492	−1.83405
$466$	0	0
$467$	−8.93342	−0.413389	−0.206695	−	0.978406i	$-0.566271\pi$
$467$	−8.93342	−0.413389	−0.206695	+	0.978406i	$0.566271\pi$
$468$	0	0
$469$	−35.8901	−1.65725
$470$	0	0
$471$	16.3842	0.754945
$472$	0	0
$473$	26.8413	1.23416
$474$	0	0
$475$	−0.0835436	−0.00383324
$476$	0	0
$477$	−54.2300	−2.48302
$478$	0	0
$479$	−29.9294	−1.36751	−0.683754	−	0.729713i	$-0.739654\pi$
$479$	−29.9294	−1.36751	−0.683754	+	0.729713i	$0.739654\pi$
$480$	0	0
$481$	−13.2552	−0.604384
$482$	0	0
$483$	0.00956500	0.000435223 0
$484$	0	0
$485$	31.9591	1.45119
$486$	0	0
$487$	12.6093	0.571381	0.285690	−	0.958322i	$-0.407777\pi$
$487$	12.6093	0.571381	0.285690	+	0.958322i	$0.407777\pi$
$488$	0	0
$489$	35.0941	1.58701
$490$	0	0
$491$	4.44713	0.200696	0.100348	−	0.994952i	$-0.468004\pi$
$491$	4.44713	0.200696	0.100348	+	0.994952i	$0.468004\pi$
$492$	0	0
$493$	−73.0799	−3.29135
$494$	0	0
$495$	−70.8009	−3.18226
$496$	0	0
$497$	−53.7906	−2.41284
$498$	0	0
$499$	−9.12378	−0.408436	−0.204218	−	0.978925i	$-0.565465\pi$
$499$	−9.12378	−0.408436	−0.204218	+	0.978925i	$0.565465\pi$
$500$	0	0
$501$	15.5137	0.693102
$502$	0	0
$503$	9.84131	0.438802	0.219401	−	0.975635i	$-0.429590\pi$
$503$	9.84131	0.438802	0.219401	+	0.975635i	$0.429590\pi$
$504$	0	0
$505$	31.6426	1.40808
$506$	0	0
$507$	35.6265	1.58223
$508$	0	0
$509$	−24.8531	−1.10159	−0.550796	−	0.834640i	$-0.685676\pi$
$509$	−24.8531	−1.10159	−0.550796	+	0.834640i	$0.685676\pi$
$510$	0	0
$511$	−33.2403	−1.47046
$512$	0	0
$513$	96.2424	4.24921
$514$	0	0
$515$	1.40597	0.0619543
$516$	0	0
$517$	−6.59982	−0.290260
$518$	0	0
$519$	28.3625	1.24497
$520$	0	0
$521$	−14.6354	−0.641191	−0.320595	−	0.947216i	$-0.603883\pi$
$521$	−14.6354	−0.641191	−0.320595	+	0.947216i	$0.603883\pi$
$522$	0	0
$523$	−17.5319	−0.766617	−0.383308	−	0.923620i	$-0.625215\pi$
$523$	−17.5319	−0.766617	−0.383308	+	0.923620i	$0.625215\pi$
$524$	0	0
$525$	−0.218772	−0.00954800
$526$	0	0
$527$	−39.7471	−1.73141
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	12.0548	0.523136
$532$	0	0
$533$	6.17151	0.267318
$534$	0	0
$535$	−32.6941	−1.41349
$536$	0	0
$537$	−21.6882	−0.935915
$538$	0	0
$539$	60.4698	2.60462
$540$	0	0
$541$	−5.18605	−0.222966	−0.111483	−	0.993766i	$-0.535560\pi$
$541$	−5.18605	−0.222966	−0.111483	+	0.993766i	$0.535560\pi$
$542$	0	0
$543$	1.74043	0.0746891
$544$	0	0
$545$	−16.2544	−0.696260
$546$	0	0
$547$	−15.5872	−0.666459	−0.333229	−	0.942846i	$-0.608138\pi$
$547$	−15.5872	−0.666459	−0.333229	+	0.942846i	$0.608138\pi$
$548$	0	0
$549$	−82.2481	−3.51027
$550$	0	0
$551$	58.3354	2.48517
$552$	0	0
$553$	1.32759	0.0564550
$554$	0	0
$555$	65.8855	2.79669
$556$	0	0
$557$	27.9967	1.18626	0.593128	−	0.805108i	$-0.297893\pi$
$557$	27.9967	1.18626	0.593128	+	0.805108i	$0.297893\pi$
$558$	0	0
$559$	−9.99257	−0.422640
$560$	0	0
$561$	−98.1403	−4.14349
$562$	0	0
$563$	−7.27156	−0.306460	−0.153230	−	0.988191i	$-0.548967\pi$
$563$	−7.27156	−0.306460	−0.153230	+	0.988191i	$0.548967\pi$
$564$	0	0
$565$	20.3049	0.854232
$566$	0	0
$567$	140.391	5.89586
$568$	0	0
$569$	10.4973	0.440070	0.220035	−	0.975492i	$-0.429383\pi$
$569$	10.4973	0.440070	0.220035	+	0.975492i	$0.429383\pi$
$570$	0	0
$571$	22.0139	0.921252	0.460626	−	0.887594i	$-0.347625\pi$
$571$	22.0139	0.921252	0.460626	+	0.887594i	$0.347625\pi$
$572$	0	0
$573$	24.8141	1.03662
$574$	0	0
$575$	−8.66720e−6 0	−3.61447e−7 0
$576$	0	0
$577$	−38.4761	−1.60178	−0.800891	−	0.598810i	$-0.795640\pi$
$577$	−38.4761	−1.60178	−0.800891	+	0.598810i	$0.795640\pi$
$578$	0	0
$579$	80.3568	3.33952
$580$	0	0
$581$	41.9906	1.74207
$582$	0	0
$583$	−27.4016	−1.13486
$584$	0	0
$585$	26.3580	1.08977
$586$	0	0
$587$	4.51623	0.186405	0.0932024	−	0.995647i	$-0.470290\pi$
$587$	4.51623	0.186405	0.0932024	+	0.995647i	$0.470290\pi$
$588$	0	0
$589$	31.7278	1.30732
$590$	0	0
$591$	73.4627	3.02185
$592$	0	0
$593$	−31.7935	−1.30560	−0.652801	−	0.757529i	$-0.726406\pi$
$593$	−31.7935	−1.30560	−0.652801	+	0.757529i	$0.726406\pi$
$594$	0	0
$595$	−78.2994	−3.20996
$596$	0	0
$597$	−41.7866	−1.71021
$598$	0	0
$599$	11.9255	0.487261	0.243631	−	0.969868i	$-0.421662\pi$
$599$	11.9255	0.487261	0.243631	+	0.969868i	$0.421662\pi$
$600$	0	0
$601$	40.7812	1.66350	0.831751	−	0.555149i	$-0.187339\pi$
$601$	40.7812	1.66350	0.831751	+	0.555149i	$0.187339\pi$
$602$	0	0
$603$	−60.3533	−2.45778
$604$	0	0
$605$	−11.1433	−0.453038
$606$	0	0
$607$	10.9141	0.442990	0.221495	−	0.975161i	$-0.428906\pi$
$607$	10.9141	0.442990	0.221495	+	0.975161i	$0.428906\pi$
$608$	0	0
$609$	152.761	6.19017
$610$	0	0
$611$	2.45700	0.0993996
$612$	0	0
$613$	−9.21225	−0.372079	−0.186040	−	0.982542i	$-0.559565\pi$
$613$	−9.21225	−0.372079	−0.186040	+	0.982542i	$0.559565\pi$
$614$	0	0
$615$	−30.6758	−1.23697
$616$	0	0
$617$	3.72223	0.149851	0.0749257	−	0.997189i	$-0.476128\pi$
$617$	3.72223	0.149851	0.0749257	+	0.997189i	$0.476128\pi$
$618$	0	0
$619$	40.5539	1.63000	0.814999	−	0.579463i	$-0.196738\pi$
$619$	40.5539	1.63000	0.814999	+	0.579463i	$0.196738\pi$
$620$	0	0
$621$	0.00998464	0.000400670 0
$622$	0	0
$623$	−11.3842	−0.456098
$624$	0	0
$625$	−25.0702	−1.00281
$626$	0	0
$627$	78.3397	3.12859
$628$	0	0
$629$	66.2153	2.64018
$630$	0	0
$631$	37.1973	1.48080	0.740401	−	0.672165i	$-0.234636\pi$
$631$	37.1973	1.48080	0.740401	+	0.672165i	$0.234636\pi$
$632$	0	0
$633$	−63.6826	−2.53116
$634$	0	0
$635$	−31.6720	−1.25687
$636$	0	0
$637$	−22.5119	−0.891953
$638$	0	0
$639$	−90.4551	−3.57835
$640$	0	0
$641$	−28.4723	−1.12459	−0.562295	−	0.826937i	$-0.690081\pi$
$641$	−28.4723	−1.12459	−0.562295	+	0.826937i	$0.690081\pi$
$642$	0	0
$643$	14.3437	0.565661	0.282831	−	0.959170i	$-0.408727\pi$
$643$	14.3437	0.565661	0.282831	+	0.959170i	$0.408727\pi$
$644$	0	0
$645$	49.6686	1.95570
$646$	0	0
$647$	15.5766	0.612378	0.306189	−	0.951971i	$-0.400946\pi$
$647$	15.5766	0.612378	0.306189	+	0.951971i	$0.400946\pi$
$648$	0	0
$649$	6.09114	0.239098
$650$	0	0
$651$	83.0843	3.25633
$652$	0	0
$653$	−25.5460	−0.999691	−0.499845	−	0.866115i	$-0.666610\pi$
$653$	−25.5460	−0.999691	−0.499845	+	0.866115i	$0.666610\pi$
$654$	0	0
$655$	−40.5738	−1.58535
$656$	0	0
$657$	−55.8973	−2.18076
$658$	0	0
$659$	−23.2015	−0.903804	−0.451902	−	0.892068i	$-0.649254\pi$
$659$	−23.2015	−0.903804	−0.451902	+	0.892068i	$0.649254\pi$
$660$	0	0
$661$	−2.99953	−0.116668	−0.0583342	−	0.998297i	$-0.518579\pi$
$661$	−2.99953	−0.116668	−0.0583342	+	0.998297i	$0.518579\pi$
$662$	0	0
$663$	36.5360	1.41894
$664$	0	0
$665$	62.5018	2.42372
$666$	0	0
$667$	0.00605199	0.000234334 0
$668$	0	0
$669$	77.9451	3.01353
$670$	0	0
$671$	−41.5588	−1.60436
$672$	0	0
$673$	−46.3740	−1.78759	−0.893793	−	0.448480i	$-0.851966\pi$
$673$	−46.3740	−1.78759	−0.893793	+	0.448480i	$0.851966\pi$
$674$	0	0
$675$	−0.228370	−0.00878998
$676$	0	0
$677$	28.7598	1.10533	0.552664	−	0.833404i	$-0.313611\pi$
$677$	28.7598	1.10533	0.552664	+	0.833404i	$0.313611\pi$
$678$	0	0
$679$	−67.1392	−2.57656
$680$	0	0
$681$	−21.0516	−0.806700
$682$	0	0
$683$	−1.97587	−0.0756046	−0.0378023	−	0.999285i	$-0.512036\pi$
$683$	−1.97587	−0.0756046	−0.0378023	+	0.999285i	$0.512036\pi$
$684$	0	0
$685$	−15.1452	−0.578669
$686$	0	0
$687$	11.9676	0.456593
$688$	0	0
$689$	10.2012	0.388633
$690$	0	0
$691$	−50.0253	−1.90305	−0.951527	−	0.307566i	$-0.900486\pi$
$691$	−50.0253	−1.90305	−0.951527	+	0.307566i	$0.900486\pi$
$692$	0	0
$693$	148.737	5.65006
$694$	0	0
$695$	−12.5755	−0.477016
$696$	0	0
$697$	−30.8294	−1.16775
$698$	0	0
$699$	28.8163	1.08993
$700$	0	0
$701$	43.8744	1.65711	0.828556	−	0.559906i	$-0.189163\pi$
$701$	43.8744	1.65711	0.828556	+	0.559906i	$0.189163\pi$
$702$	0	0
$703$	−52.8558	−1.99349
$704$	0	0
$705$	−12.2127	−0.459955
$706$	0	0
$707$	−66.4743	−2.50002
$708$	0	0
$709$	−41.4964	−1.55843	−0.779214	−	0.626758i	$-0.784382\pi$
$709$	−41.4964	−1.55843	−0.779214	+	0.626758i	$0.784382\pi$
$710$	0	0
$711$	2.23250	0.0837253
$712$	0	0
$713$	0.00329159	0.000123271 0
$714$	0	0
$715$	13.3183	0.498076
$716$	0	0
$717$	9.53668	0.356154
$718$	0	0
$719$	26.1083	0.973677	0.486839	−	0.873492i	$-0.338150\pi$
$719$	26.1083	0.973677	0.486839	+	0.873492i	$0.338150\pi$
$720$	0	0
$721$	−2.95363	−0.109999
$722$	0	0
$723$	46.0607	1.71302
$724$	0	0
$725$	−0.138422	−0.00514086
$726$	0	0
$727$	−33.1694	−1.23018	−0.615092	−	0.788455i	$-0.710881\pi$
$727$	−33.1694	−1.23018	−0.615092	+	0.788455i	$0.710881\pi$
$728$	0	0
$729$	75.3558	2.79096
$730$	0	0
$731$	49.9172	1.84625
$732$	0	0
$733$	−24.8655	−0.918427	−0.459213	−	0.888326i	$-0.651869\pi$
$733$	−24.8655	−0.918427	−0.459213	+	0.888326i	$0.651869\pi$
$734$	0	0
$735$	111.897	4.12736
$736$	0	0
$737$	−30.4956	−1.12332
$738$	0	0
$739$	−17.0371	−0.626719	−0.313359	−	0.949635i	$-0.601454\pi$
$739$	−17.0371	−0.626719	−0.313359	+	0.949635i	$0.601454\pi$
$740$	0	0
$741$	−29.1645	−1.07139
$742$	0	0
$743$	1.94927	0.0715118	0.0357559	−	0.999361i	$-0.488616\pi$
$743$	1.94927	0.0715118	0.0357559	+	0.999361i	$0.488616\pi$
$744$	0	0
$745$	3.54834	0.130001
$746$	0	0
$747$	70.6120	2.58356
$748$	0	0
$749$	68.6832	2.50963
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	−82.1298	−2.99297
$754$	0	0
$755$	30.9273	1.12556
$756$	0	0
$757$	−33.2093	−1.20701	−0.603507	−	0.797358i	$-0.706230\pi$
$757$	−33.2093	−1.20701	−0.603507	+	0.797358i	$0.706230\pi$
$758$	0	0
$759$	0.00812733	0.000295003 0
$760$	0	0
$761$	5.15970	0.187039	0.0935195	−	0.995617i	$-0.470188\pi$
$761$	5.15970	0.187039	0.0935195	+	0.995617i	$0.470188\pi$
$762$	0	0
$763$	34.1469	1.23620
$764$	0	0
$765$	−131.669	−4.76051
$766$	0	0
$767$	−2.26763	−0.0818792
$768$	0	0
$769$	−20.6290	−0.743902	−0.371951	−	0.928252i	$-0.621311\pi$
$769$	−20.6290	−0.743902	−0.371951	+	0.928252i	$0.621311\pi$
$770$	0	0
$771$	−68.1837	−2.45558
$772$	0	0
$773$	29.1727	1.04927	0.524635	−	0.851327i	$-0.324202\pi$
$773$	29.1727	1.04927	0.524635	+	0.851327i	$0.324202\pi$
$774$	0	0
$775$	−0.0752858	−0.00270435
$776$	0	0
$777$	−138.411	−4.96548
$778$	0	0
$779$	24.6093	0.881719
$780$	0	0
$781$	−45.7056	−1.63547
$782$	0	0
$783$	159.463	5.69873
$784$	0	0
$785$	11.1071	0.396428
$786$	0	0
$787$	6.67011	0.237764	0.118882	−	0.992908i	$-0.462069\pi$
$787$	6.67011	0.237764	0.118882	+	0.992908i	$0.462069\pi$
$788$	0	0
$789$	27.2770	0.971087
$790$	0	0
$791$	−42.6561	−1.51668
$792$	0	0
$793$	15.4716	0.549413
$794$	0	0
$795$	−50.7054	−1.79834
$796$	0	0
$797$	40.4000	1.43104	0.715521	−	0.698591i	$-0.246189\pi$
$797$	40.4000	1.43104	0.715521	+	0.698591i	$0.246189\pi$
$798$	0	0
$799$	−12.2738	−0.434215
$800$	0	0
$801$	−19.1438	−0.676413
$802$	0	0
$803$	−28.2441	−0.996712
$804$	0	0
$805$	0.00648423	0.000228539 0
$806$	0	0
$807$	5.92380	0.208528
$808$	0	0
$809$	24.5291	0.862397	0.431198	−	0.902257i	$-0.358091\pi$
$809$	24.5291	0.862397	0.431198	+	0.902257i	$0.358091\pi$
$810$	0	0
$811$	−25.9682	−0.911868	−0.455934	−	0.890014i	$-0.650695\pi$
$811$	−25.9682	−0.911868	−0.455934	+	0.890014i	$0.650695\pi$
$812$	0	0
$813$	44.4285	1.55818
$814$	0	0
$815$	23.7908	0.833354
$816$	0	0
$817$	−39.8459	−1.39403
$818$	0	0
$819$	−55.3724	−1.93487
$820$	0	0
$821$	4.71834	0.164671	0.0823357	−	0.996605i	$-0.473762\pi$
$821$	4.71834	0.164671	0.0823357	+	0.996605i	$0.473762\pi$
$822$	0	0
$823$	30.8533	1.07548	0.537739	−	0.843112i	$-0.319279\pi$
$823$	30.8533	1.07548	0.537739	+	0.843112i	$0.319279\pi$
$824$	0	0
$825$	−0.185889	−0.00647184
$826$	0	0
$827$	−23.6140	−0.821139	−0.410569	−	0.911829i	$-0.634670\pi$
$827$	−23.6140	−0.821139	−0.410569	+	0.911829i	$0.634670\pi$
$828$	0	0
$829$	17.3307	0.601920	0.300960	−	0.953637i	$-0.402693\pi$
$829$	17.3307	0.601920	0.300960	+	0.953637i	$0.402693\pi$
$830$	0	0
$831$	72.9889	2.53196
$832$	0	0
$833$	112.457	3.89639
$834$	0	0
$835$	10.5169	0.363954
$836$	0	0
$837$	86.7295	2.99781
$838$	0	0
$839$	4.25150	0.146778	0.0733891	−	0.997303i	$-0.476619\pi$
$839$	4.25150	0.146778	0.0733891	+	0.997303i	$0.476619\pi$
$840$	0	0
$841$	67.6549	2.33293
$842$	0	0
$843$	5.71236	0.196744
$844$	0	0
$845$	24.1516	0.830841
$846$	0	0
$847$	23.4096	0.804363
$848$	0	0
$849$	75.3679	2.58662
$850$	0	0
$851$	−0.00548351	−0.000187972 0
$852$	0	0
$853$	3.38979	0.116064	0.0580320	−	0.998315i	$-0.481517\pi$
$853$	3.38979	0.116064	0.0580320	+	0.998315i	$0.481517\pi$
$854$	0	0
$855$	105.104	3.59448
$856$	0	0
$857$	20.2848	0.692914	0.346457	−	0.938066i	$-0.387385\pi$
$857$	20.2848	0.692914	0.346457	+	0.938066i	$0.387385\pi$
$858$	0	0
$859$	−13.2294	−0.451383	−0.225691	−	0.974199i	$-0.572464\pi$
$859$	−13.2294	−0.451383	−0.225691	+	0.974199i	$0.572464\pi$
$860$	0	0
$861$	64.4433	2.19622
$862$	0	0
$863$	48.0259	1.63482	0.817410	−	0.576056i	$-0.195409\pi$
$863$	48.0259	1.63482	0.817410	+	0.576056i	$0.195409\pi$
$864$	0	0
$865$	19.2273	0.653746
$866$	0	0
$867$	−126.360	−4.29142
$868$	0	0
$869$	1.12805	0.0382664
$870$	0	0
$871$	11.3530	0.384682
$872$	0	0
$873$	−112.902	−3.82116
$874$	0	0
$875$	52.5192	1.77547
$876$	0	0
$877$	−5.88274	−0.198646	−0.0993231	−	0.995055i	$-0.531668\pi$
$877$	−5.88274	−0.198646	−0.0993231	+	0.995055i	$0.531668\pi$
$878$	0	0
$879$	18.9243	0.638300
$880$	0	0
$881$	38.1457	1.28516	0.642581	−	0.766218i	$-0.277864\pi$
$881$	38.1457	1.28516	0.642581	+	0.766218i	$0.277864\pi$
$882$	0	0
$883$	−43.7871	−1.47355	−0.736777	−	0.676136i	$-0.763653\pi$
$883$	−43.7871	−1.47355	−0.736777	+	0.676136i	$0.763653\pi$
$884$	0	0
$885$	11.2714	0.378883
$886$	0	0
$887$	−45.2065	−1.51789	−0.758943	−	0.651157i	$-0.774284\pi$
$887$	−45.2065	−1.51789	−0.758943	+	0.651157i	$0.774284\pi$
$888$	0	0
$889$	66.5361	2.23155
$890$	0	0
$891$	119.289	3.99634
$892$	0	0
$893$	9.79744	0.327859
$894$	0	0
$895$	−14.7027	−0.491457
$896$	0	0
$897$	−0.00302567	−0.000101024 0
$898$	0	0
$899$	52.5693	1.75328
$900$	0	0
$901$	−50.9592	−1.69770
$902$	0	0
$903$	−104.343	−3.47232
$904$	0	0
$905$	1.17986	0.0392199
$906$	0	0
$907$	21.2172	0.704506	0.352253	−	0.935905i	$-0.385416\pi$
$907$	21.2172	0.704506	0.352253	+	0.935905i	$0.385416\pi$
$908$	0	0
$909$	−111.784	−3.70764
$910$	0	0
$911$	−11.2917	−0.374110	−0.187055	−	0.982349i	$-0.559894\pi$
$911$	−11.2917	−0.374110	−0.187055	+	0.982349i	$0.559894\pi$
$912$	0	0
$913$	35.6792	1.18081
$914$	0	0
$915$	−76.9025	−2.54232
$916$	0	0
$917$	85.2369	2.81477
$918$	0	0
$919$	−19.6587	−0.648480	−0.324240	−	0.945975i	$-0.605108\pi$
$919$	−19.6587	−0.648480	−0.324240	+	0.945975i	$0.605108\pi$
$920$	0	0
$921$	92.5236	3.04876
$922$	0	0
$923$	17.0154	0.560069
$924$	0	0
$925$	0.125420	0.00412377
$926$	0	0
$927$	−4.96687	−0.163133
$928$	0	0
$929$	−58.8052	−1.92934	−0.964668	−	0.263470i	$-0.915133\pi$
$929$	−58.8052	−1.92934	−0.964668	+	0.263470i	$0.915133\pi$
$930$	0	0
$931$	−89.7675	−2.94201
$932$	0	0
$933$	34.5945	1.13257
$934$	0	0
$935$	−66.5306	−2.17578
$936$	0	0
$937$	6.18168	0.201947	0.100973	−	0.994889i	$-0.467804\pi$
$937$	6.18168	0.201947	0.100973	+	0.994889i	$0.467804\pi$
$938$	0	0
$939$	17.2569	0.563157
$940$	0	0
$941$	−9.80793	−0.319729	−0.159865	−	0.987139i	$-0.551106\pi$
$941$	−9.80793	−0.319729	−0.159865	+	0.987139i	$0.551106\pi$
$942$	0	0
$943$	0.00255308	8.31398e−5 0
$944$	0	0
$945$	170.852	5.55780
$946$	0	0
$947$	−50.0948	−1.62786	−0.813931	−	0.580962i	$-0.802676\pi$
$947$	−50.0948	−1.62786	−0.813931	+	0.580962i	$0.802676\pi$
$948$	0	0
$949$	10.5148	0.341325
$950$	0	0
$951$	−12.9782	−0.420848
$952$	0	0
$953$	−27.6553	−0.895843	−0.447922	−	0.894073i	$-0.647836\pi$
$953$	−27.6553	−0.895843	−0.447922	+	0.894073i	$0.647836\pi$
$954$	0	0
$955$	16.8218	0.544340
$956$	0	0
$957$	129.800	4.19583
$958$	0	0
$959$	31.8169	1.02742
$960$	0	0
$961$	−2.40828	−0.0776865
$962$	0	0
$963$	115.499	3.72189
$964$	0	0
$965$	54.4749	1.75361
$966$	0	0
$967$	−30.4046	−0.977746	−0.488873	−	0.872355i	$-0.662592\pi$
$967$	−30.4046	−0.977746	−0.488873	+	0.872355i	$0.662592\pi$
$968$	0	0
$969$	145.689	4.68022
$970$	0	0
$971$	−53.2989	−1.71044	−0.855222	−	0.518262i	$-0.826579\pi$
$971$	−53.2989	−1.71044	−0.855222	+	0.518262i	$0.826579\pi$
$972$	0	0
$973$	26.4184	0.846935
$974$	0	0
$975$	0.0692035	0.00221628
$976$	0	0
$977$	−0.310992	−0.00994952	−0.00497476	−	0.999988i	$-0.501584\pi$
$977$	−0.310992	−0.00994952	−0.00497476	+	0.999988i	$0.501584\pi$
$978$	0	0
$979$	−9.67308	−0.309153
$980$	0	0
$981$	57.4219	1.83334
$982$	0	0
$983$	14.7721	0.471158	0.235579	−	0.971855i	$-0.424301\pi$
$983$	14.7721	0.471158	0.235579	+	0.971855i	$0.424301\pi$
$984$	0	0
$985$	49.8013	1.58680
$986$	0	0
$987$	25.6562	0.816644
$988$	0	0
$989$	−0.00413381	−0.000131447 0
$990$	0	0
$991$	1.52359	0.0483986	0.0241993	−	0.999707i	$-0.492296\pi$
$991$	1.52359	0.0483986	0.0241993	+	0.999707i	$0.492296\pi$
$992$	0	0
$993$	−48.2414	−1.53089
$994$	0	0
$995$	−28.3277	−0.898047
$996$	0	0
$997$	48.9610	1.55061	0.775305	−	0.631588i	$-0.217596\pi$
$997$	48.9610	1.55061	0.775305	+	0.631588i	$0.217596\pi$
$998$	0	0
$999$	−144.484	−4.57126

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.c.1.2	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.c.1.2	✓	44	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-3.30310 q^{3} -2.23921 q^{5} +4.70410 q^{7} +7.91049 q^{9} +O(q^{10})\)	\(q-3.30310 q^{3} -2.23921 q^{5} +4.70410 q^{7} +7.91049 q^{9} +3.99705 q^{11} -1.48803 q^{13} +7.39635 q^{15} +7.43337 q^{17} -5.93362 q^{19} -15.5381 q^{21} -0.000615582 q^{23} +0.0140797 q^{25} -16.2198 q^{27} -9.83132 q^{29} -5.34712 q^{31} -13.2027 q^{33} -10.5335 q^{35} +8.90784 q^{37} +4.91513 q^{39} -4.14743 q^{41} +6.71528 q^{43} -17.7133 q^{45} -1.65117 q^{47} +15.1286 q^{49} -24.5532 q^{51} -6.85546 q^{53} -8.95025 q^{55} +19.5994 q^{57} +1.52391 q^{59} -10.3974 q^{61} +37.2118 q^{63} +3.33203 q^{65} -7.62953 q^{67} +0.00203333 q^{69} -11.4348 q^{71} -7.06623 q^{73} -0.0465066 q^{75} +18.8025 q^{77} +0.282220 q^{79} +29.8443 q^{81} +8.92638 q^{83} -16.6449 q^{85} +32.4739 q^{87} -2.42005 q^{89} -6.99987 q^{91} +17.6621 q^{93} +13.2867 q^{95} -14.2725 q^{97} +31.6186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9	\( 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9} + 11 q^{11} - 36 q^{13} - 5 q^{15} - 10 q^{17} - 7 q^{19} - 42 q^{21} - 5 q^{23} + 29 q^{25} - 16 q^{27} - 57 q^{29} - 21 q^{31} - 32 q^{33} + 17 q^{35} - 52 q^{37} + 8 q^{39} - 16 q^{41} - 9 q^{43} - 84 q^{45} - q^{47} + 28 q^{49} - q^{51} - 52 q^{53} - 39 q^{55} - 15 q^{57} + 7 q^{59} - 85 q^{61} - 25 q^{63} - 9 q^{65} - 36 q^{67} - 72 q^{69} + 12 q^{71} - 60 q^{73} - 5 q^{75} - 81 q^{77} - 13 q^{79} + 20 q^{81} + 5 q^{83} - 72 q^{85} + 9 q^{87} - 37 q^{89} - 23 q^{91} - 60 q^{93} + 24 q^{95} - 79 q^{97} + 42 q^{99}+O(q^{100}) \) 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9 + 11 * q^11 - 36 * q^13 - 5 * q^15 - 10 * q^17 - 7 * q^19 - 42 * q^21 - 5 * q^23 + 29 * q^25 - 16 * q^27 - 57 * q^29 - 21 * q^31 - 32 * q^33 + 17 * q^35 - 52 * q^37 + 8 * q^39 - 16 * q^41 - 9 * q^43 - 84 * q^45 - q^47 + 28 * q^49 - q^51 - 52 * q^53 - 39 * q^55 - 15 * q^57 + 7 * q^59 - 85 * q^61 - 25 * q^63 - 9 * q^65 - 36 * q^67 - 72 * q^69 + 12 * q^71 - 60 * q^73 - 5 * q^75 - 81 * q^77 - 13 * q^79 + 20 * q^81 + 5 * q^83 - 72 * q^85 + 9 * q^87 - 37 * q^89 - 23 * q^91 - 60 * q^93 + 24 * q^95 - 79 * q^97 + 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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