LMFDB - Embedded newform 6004.2.a.h.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.20443 q^{3} -2.88512 q^{5} -4.09545 q^{7} -1.54935 q^{9} +O(q^{10})$	$q+1.20443 q^{3} -2.88512 q^{5} -4.09545 q^{7} -1.54935 q^{9} -1.48534 q^{11} -4.73551 q^{13} -3.47493 q^{15} -5.78107 q^{17} -1.00000 q^{19} -4.93268 q^{21} -5.93211 q^{23} +3.32394 q^{25} -5.47937 q^{27} +1.35179 q^{29} -3.88806 q^{31} -1.78899 q^{33} +11.8159 q^{35} +4.39821 q^{37} -5.70359 q^{39} -4.60652 q^{41} -4.40204 q^{43} +4.47006 q^{45} +5.66872 q^{47} +9.77267 q^{49} -6.96290 q^{51} +10.0207 q^{53} +4.28539 q^{55} -1.20443 q^{57} -10.2674 q^{59} -0.570401 q^{61} +6.34527 q^{63} +13.6625 q^{65} +7.78275 q^{67} -7.14481 q^{69} +4.68179 q^{71} -6.88061 q^{73} +4.00346 q^{75} +6.08313 q^{77} -1.00000 q^{79} -1.95147 q^{81} +3.98732 q^{83} +16.6791 q^{85} +1.62813 q^{87} +0.268808 q^{89} +19.3940 q^{91} -4.68290 q^{93} +2.88512 q^{95} -13.8388 q^{97} +2.30131 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.20443	0.695378	0.347689	−	0.937610i	$-0.386967\pi$
$3$	1.20443	0.695378	0.347689	+	0.937610i	$0.386967\pi$
$4$	0	0
$5$	−2.88512	−1.29027	−0.645133	−	0.764070i	$-0.723198\pi$
$5$	−2.88512	−1.29027	−0.645133	+	0.764070i	$0.723198\pi$
$6$	0	0
$7$	−4.09545	−1.54793	−0.773966	−	0.633227i	$-0.781730\pi$
$7$	−4.09545	−1.54793	−0.773966	+	0.633227i	$0.781730\pi$
$8$	0	0
$9$	−1.54935	−0.516449
$10$	0	0
$11$	−1.48534	−0.447847	−0.223923	−	0.974607i	$-0.571887\pi$
$11$	−1.48534	−0.447847	−0.223923	+	0.974607i	$0.571887\pi$
$12$	0	0
$13$	−4.73551	−1.31339	−0.656697	−	0.754154i	$-0.728047\pi$
$13$	−4.73551	−1.31339	−0.656697	+	0.754154i	$0.728047\pi$
$14$	0	0
$15$	−3.47493	−0.897223
$16$	0	0
$17$	−5.78107	−1.40212	−0.701058	−	0.713104i	$-0.747289\pi$
$17$	−5.78107	−1.40212	−0.701058	+	0.713104i	$0.747289\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.93268	−1.07640
$22$	0	0
$23$	−5.93211	−1.23693	−0.618465	−	0.785812i	$-0.712245\pi$
$23$	−5.93211	−1.23693	−0.618465	+	0.785812i	$0.712245\pi$
$24$	0	0
$25$	3.32394	0.664789
$26$	0	0
$27$	−5.47937	−1.05451
$28$	0	0
$29$	1.35179	0.251021	0.125510	−	0.992092i	$-0.459943\pi$
$29$	1.35179	0.251021	0.125510	+	0.992092i	$0.459943\pi$
$30$	0	0
$31$	−3.88806	−0.698317	−0.349158	−	0.937064i	$-0.613532\pi$
$31$	−3.88806	−0.698317	−0.349158	+	0.937064i	$0.613532\pi$
$32$	0	0
$33$	−1.78899	−0.311423
$34$	0	0
$35$	11.8159	1.99725
$36$	0	0
$37$	4.39821	0.723061	0.361530	−	0.932360i	$-0.382254\pi$
$37$	4.39821	0.723061	0.361530	+	0.932360i	$0.382254\pi$
$38$	0	0
$39$	−5.70359	−0.913305
$40$	0	0
$41$	−4.60652	−0.719418	−0.359709	−	0.933065i	$-0.617124\pi$
$41$	−4.60652	−0.719418	−0.359709	+	0.933065i	$0.617124\pi$
$42$	0	0
$43$	−4.40204	−0.671305	−0.335653	−	0.941986i	$-0.608957\pi$
$43$	−4.40204	−0.671305	−0.335653	+	0.941986i	$0.608957\pi$
$44$	0	0
$45$	4.47006	0.666358
$46$	0	0
$47$	5.66872	0.826868	0.413434	−	0.910534i	$-0.364329\pi$
$47$	5.66872	0.826868	0.413434	+	0.910534i	$0.364329\pi$
$48$	0	0
$49$	9.77267	1.39610
$50$	0	0
$51$	−6.96290	−0.975001
$52$	0	0
$53$	10.0207	1.37644	0.688222	−	0.725501i	$-0.258392\pi$
$53$	10.0207	1.37644	0.688222	+	0.725501i	$0.258392\pi$
$54$	0	0
$55$	4.28539	0.577842
$56$	0	0
$57$	−1.20443	−0.159531
$58$	0	0
$59$	−10.2674	−1.33671	−0.668353	−	0.743844i	$-0.733001\pi$
$59$	−10.2674	−1.33671	−0.668353	+	0.743844i	$0.733001\pi$
$60$	0	0
$61$	−0.570401	−0.0730324	−0.0365162	−	0.999333i	$-0.511626\pi$
$61$	−0.570401	−0.0730324	−0.0365162	+	0.999333i	$0.511626\pi$
$62$	0	0
$63$	6.34527	0.799429
$64$	0	0
$65$	13.6625	1.69463
$66$	0	0
$67$	7.78275	0.950814	0.475407	−	0.879766i	$-0.342301\pi$
$67$	7.78275	0.950814	0.475407	+	0.879766i	$0.342301\pi$
$68$	0	0
$69$	−7.14481	−0.860134
$70$	0	0
$71$	4.68179	0.555626	0.277813	−	0.960635i	$-0.410390\pi$
$71$	4.68179	0.555626	0.277813	+	0.960635i	$0.410390\pi$
$72$	0	0
$73$	−6.88061	−0.805314	−0.402657	−	0.915351i	$-0.631913\pi$
$73$	−6.88061	−0.805314	−0.402657	+	0.915351i	$0.631913\pi$
$74$	0	0
$75$	4.00346	0.462279
$76$	0	0
$77$	6.08313	0.693237
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−1.95147	−0.216830
$82$	0	0
$83$	3.98732	0.437665	0.218832	−	0.975762i	$-0.429775\pi$
$83$	3.98732	0.437665	0.218832	+	0.975762i	$0.429775\pi$
$84$	0	0
$85$	16.6791	1.80910
$86$	0	0
$87$	1.62813	0.174554
$88$	0	0
$89$	0.268808	0.0284936	0.0142468	−	0.999899i	$-0.495465\pi$
$89$	0.268808	0.0284936	0.0142468	+	0.999899i	$0.495465\pi$
$90$	0	0
$91$	19.3940	2.03305
$92$	0	0
$93$	−4.68290	−0.485594
$94$	0	0
$95$	2.88512	0.296008
$96$	0	0
$97$	−13.8388	−1.40511	−0.702557	−	0.711627i	$-0.747959\pi$
$97$	−13.8388	−1.40511	−0.702557	+	0.711627i	$0.747959\pi$
$98$	0	0
$99$	2.30131	0.231290
$100$	0	0
$101$	5.26675	0.524061	0.262031	−	0.965060i	$-0.415608\pi$
$101$	5.26675	0.524061	0.262031	+	0.965060i	$0.415608\pi$
$102$	0	0
$103$	11.6721	1.15008	0.575042	−	0.818124i	$-0.304986\pi$
$103$	11.6721	1.15008	0.575042	+	0.818124i	$0.304986\pi$
$104$	0	0
$105$	14.2314	1.38884
$106$	0	0
$107$	2.02358	0.195627	0.0978133	−	0.995205i	$-0.468815\pi$
$107$	2.02358	0.195627	0.0978133	+	0.995205i	$0.468815\pi$
$108$	0	0
$109$	−10.4703	−1.00287	−0.501436	−	0.865195i	$-0.667195\pi$
$109$	−10.4703	−1.00287	−0.501436	+	0.865195i	$0.667195\pi$
$110$	0	0
$111$	5.29733	0.502800
$112$	0	0
$113$	−18.1873	−1.71092	−0.855460	−	0.517868i	$-0.826726\pi$
$113$	−18.1873	−1.71092	−0.855460	+	0.517868i	$0.826726\pi$
$114$	0	0
$115$	17.1149	1.59597
$116$	0	0
$117$	7.33696	0.678302
$118$	0	0
$119$	23.6761	2.17038
$120$	0	0
$121$	−8.79377	−0.799433
$122$	0	0
$123$	−5.54823	−0.500267
$124$	0	0
$125$	4.83563	0.432512
$126$	0	0
$127$	−13.9537	−1.23819	−0.619096	−	0.785315i	$-0.712501\pi$
$127$	−13.9537	−1.23819	−0.619096	+	0.785315i	$0.712501\pi$
$128$	0	0
$129$	−5.30195	−0.466811
$130$	0	0
$131$	−15.6737	−1.36941	−0.684707	−	0.728818i	$-0.740070\pi$
$131$	−15.6737	−1.36941	−0.684707	+	0.728818i	$0.740070\pi$
$132$	0	0
$133$	4.09545	0.355120
$134$	0	0
$135$	15.8087	1.36059
$136$	0	0
$137$	−11.6170	−0.992505	−0.496253	−	0.868178i	$-0.665291\pi$
$137$	−11.6170	−0.992505	−0.496253	+	0.868178i	$0.665291\pi$
$138$	0	0
$139$	6.93527	0.588242	0.294121	−	0.955768i	$-0.404973\pi$
$139$	6.93527	0.588242	0.294121	+	0.955768i	$0.404973\pi$
$140$	0	0
$141$	6.82758	0.574986
$142$	0	0
$143$	7.03384	0.588199
$144$	0	0
$145$	−3.90007	−0.323884
$146$	0	0
$147$	11.7705	0.970815
$148$	0	0
$149$	−5.35330	−0.438559	−0.219280	−	0.975662i	$-0.570371\pi$
$149$	−5.35330	−0.438559	−0.219280	+	0.975662i	$0.570371\pi$
$150$	0	0
$151$	−3.90716	−0.317960	−0.158980	−	0.987282i	$-0.550821\pi$
$151$	−3.90716	−0.317960	−0.158980	+	0.987282i	$0.550821\pi$
$152$	0	0
$153$	8.95690	0.724122
$154$	0	0
$155$	11.2175	0.901015
$156$	0	0
$157$	−11.6369	−0.928727	−0.464363	−	0.885645i	$-0.653717\pi$
$157$	−11.6369	−0.928727	−0.464363	+	0.885645i	$0.653717\pi$
$158$	0	0
$159$	12.0692	0.957148
$160$	0	0
$161$	24.2946	1.91468
$162$	0	0
$163$	−10.3740	−0.812556	−0.406278	−	0.913749i	$-0.633174\pi$
$163$	−10.3740	−0.812556	−0.406278	+	0.913749i	$0.633174\pi$
$164$	0	0
$165$	5.16145	0.401819
$166$	0	0
$167$	2.22032	0.171813	0.0859066	−	0.996303i	$-0.472621\pi$
$167$	2.22032	0.171813	0.0859066	+	0.996303i	$0.472621\pi$
$168$	0	0
$169$	9.42506	0.725004
$170$	0	0
$171$	1.54935	0.118482
$172$	0	0
$173$	4.79455	0.364523	0.182262	−	0.983250i	$-0.441658\pi$
$173$	4.79455	0.364523	0.182262	+	0.983250i	$0.441658\pi$
$174$	0	0
$175$	−13.6130	−1.02905
$176$	0	0
$177$	−12.3664	−0.929516
$178$	0	0
$179$	−21.5061	−1.60744	−0.803719	−	0.595009i	$-0.797148\pi$
$179$	−21.5061	−1.60744	−0.803719	+	0.595009i	$0.797148\pi$
$180$	0	0
$181$	−10.9568	−0.814411	−0.407206	−	0.913337i	$-0.633497\pi$
$181$	−10.9568	−0.814411	−0.407206	+	0.913337i	$0.633497\pi$
$182$	0	0
$183$	−0.687008	−0.0507851
$184$	0	0
$185$	−12.6894	−0.932941
$186$	0	0
$187$	8.58686	0.627933
$188$	0	0
$189$	22.4405	1.63230
$190$	0	0
$191$	−17.7748	−1.28614	−0.643069	−	0.765808i	$-0.722339\pi$
$191$	−17.7748	−1.28614	−0.643069	+	0.765808i	$0.722339\pi$
$192$	0	0
$193$	16.5573	1.19182	0.595910	−	0.803051i	$-0.296791\pi$
$193$	16.5573	1.19182	0.595910	+	0.803051i	$0.296791\pi$
$194$	0	0
$195$	16.4556	1.17841
$196$	0	0
$197$	20.7745	1.48012	0.740059	−	0.672542i	$-0.234797\pi$
$197$	20.7745	1.48012	0.740059	+	0.672542i	$0.234797\pi$
$198$	0	0
$199$	1.89387	0.134253	0.0671263	−	0.997744i	$-0.478617\pi$
$199$	1.89387	0.134253	0.0671263	+	0.997744i	$0.478617\pi$
$200$	0	0
$201$	9.37377	0.661175
$202$	0	0
$203$	−5.53617	−0.388563
$204$	0	0
$205$	13.2904	0.928241
$206$	0	0
$207$	9.19090	0.638812
$208$	0	0
$209$	1.48534	0.102743
$210$	0	0
$211$	−3.37896	−0.232617	−0.116308	−	0.993213i	$-0.537106\pi$
$211$	−3.37896	−0.232617	−0.116308	+	0.993213i	$0.537106\pi$
$212$	0	0
$213$	5.63889	0.386370
$214$	0	0
$215$	12.7004	0.866163
$216$	0	0
$217$	15.9233	1.08095
$218$	0	0
$219$	−8.28721	−0.559998
$220$	0	0
$221$	27.3763	1.84153
$222$	0	0
$223$	10.3854	0.695456	0.347728	−	0.937596i	$-0.386953\pi$
$223$	10.3854	0.695456	0.347728	+	0.937596i	$0.386953\pi$
$224$	0	0
$225$	−5.14995	−0.343330
$226$	0	0
$227$	−23.5484	−1.56296	−0.781480	−	0.623931i	$-0.785535\pi$
$227$	−23.5484	−1.56296	−0.781480	+	0.623931i	$0.785535\pi$
$228$	0	0
$229$	−15.4585	−1.02152	−0.510762	−	0.859722i	$-0.670637\pi$
$229$	−15.4585	−1.02152	−0.510762	+	0.859722i	$0.670637\pi$
$230$	0	0
$231$	7.32670	0.482062
$232$	0	0
$233$	4.35258	0.285147	0.142573	−	0.989784i	$-0.454462\pi$
$233$	4.35258	0.285147	0.142573	+	0.989784i	$0.454462\pi$
$234$	0	0
$235$	−16.3550	−1.06688
$236$	0	0
$237$	−1.20443	−0.0782361
$238$	0	0
$239$	−0.339985	−0.0219918	−0.0109959	−	0.999940i	$-0.503500\pi$
$239$	−0.339985	−0.0219918	−0.0109959	+	0.999940i	$0.503500\pi$
$240$	0	0
$241$	−6.89088	−0.443880	−0.221940	−	0.975060i	$-0.571239\pi$
$241$	−6.89088	−0.443880	−0.221940	+	0.975060i	$0.571239\pi$
$242$	0	0
$243$	14.0877	0.903726
$244$	0	0
$245$	−28.1954	−1.80134
$246$	0	0
$247$	4.73551	0.301313
$248$	0	0
$249$	4.80244	0.304342
$250$	0	0
$251$	18.1135	1.14332	0.571658	−	0.820492i	$-0.306300\pi$
$251$	18.1135	1.14332	0.571658	+	0.820492i	$0.306300\pi$
$252$	0	0
$253$	8.81120	0.553955
$254$	0	0
$255$	20.0888	1.25801
$256$	0	0
$257$	−18.9151	−1.17989	−0.589947	−	0.807442i	$-0.700851\pi$
$257$	−18.9151	−1.17989	−0.589947	+	0.807442i	$0.700851\pi$
$258$	0	0
$259$	−18.0126	−1.11925
$260$	0	0
$261$	−2.09439	−0.129639
$262$	0	0
$263$	0.565823	0.0348902	0.0174451	−	0.999848i	$-0.494447\pi$
$263$	0.565823	0.0348902	0.0174451	+	0.999848i	$0.494447\pi$
$264$	0	0
$265$	−28.9108	−1.77598
$266$	0	0
$267$	0.323760	0.0198138
$268$	0	0
$269$	26.9433	1.64276	0.821380	−	0.570381i	$-0.193204\pi$
$269$	26.9433	1.64276	0.821380	+	0.570381i	$0.193204\pi$
$270$	0	0
$271$	−14.1136	−0.857340	−0.428670	−	0.903461i	$-0.641018\pi$
$271$	−14.1136	−0.857340	−0.428670	+	0.903461i	$0.641018\pi$
$272$	0	0
$273$	23.3587	1.41374
$274$	0	0
$275$	−4.93718	−0.297723
$276$	0	0
$277$	6.51479	0.391436	0.195718	−	0.980660i	$-0.437296\pi$
$277$	6.51479	0.391436	0.195718	+	0.980660i	$0.437296\pi$
$278$	0	0
$279$	6.02396	0.360645
$280$	0	0
$281$	−6.78055	−0.404494	−0.202247	−	0.979335i	$-0.564824\pi$
$281$	−6.78055	−0.404494	−0.202247	+	0.979335i	$0.564824\pi$
$282$	0	0
$283$	15.2465	0.906310	0.453155	−	0.891432i	$-0.350298\pi$
$283$	15.2465	0.906310	0.453155	+	0.891432i	$0.350298\pi$
$284$	0	0
$285$	3.47493	0.205837
$286$	0	0
$287$	18.8658	1.11361
$288$	0	0
$289$	16.4208	0.965929
$290$	0	0
$291$	−16.6678	−0.977086
$292$	0	0
$293$	15.0599	0.879807	0.439904	−	0.898045i	$-0.355013\pi$
$293$	15.0599	0.879807	0.439904	+	0.898045i	$0.355013\pi$
$294$	0	0
$295$	29.6228	1.72471
$296$	0	0
$297$	8.13873	0.472257
$298$	0	0
$299$	28.0916	1.62458
$300$	0	0
$301$	18.0283	1.03914
$302$	0	0
$303$	6.34343	0.364421
$304$	0	0
$305$	1.64568	0.0942313
$306$	0	0
$307$	−1.05243	−0.0600655	−0.0300328	−	0.999549i	$-0.509561\pi$
$307$	−1.05243	−0.0600655	−0.0300328	+	0.999549i	$0.509561\pi$
$308$	0	0
$309$	14.0582	0.799743
$310$	0	0
$311$	−0.977002	−0.0554007	−0.0277003	−	0.999616i	$-0.508818\pi$
$311$	−0.977002	−0.0554007	−0.0277003	+	0.999616i	$0.508818\pi$
$312$	0	0
$313$	−26.6321	−1.50533	−0.752667	−	0.658401i	$-0.771233\pi$
$313$	−26.6321	−1.50533	−0.752667	+	0.658401i	$0.771233\pi$
$314$	0	0
$315$	−18.3069	−1.03148
$316$	0	0
$317$	−7.34004	−0.412258	−0.206129	−	0.978525i	$-0.566087\pi$
$317$	−7.34004	−0.412258	−0.206129	+	0.978525i	$0.566087\pi$
$318$	0	0
$319$	−2.00786	−0.112419
$320$	0	0
$321$	2.43726	0.136034
$322$	0	0
$323$	5.78107	0.321667
$324$	0	0
$325$	−15.7406	−0.873129
$326$	0	0
$327$	−12.6107	−0.697375
$328$	0	0
$329$	−23.2159	−1.27994
$330$	0	0
$331$	2.37971	0.130801	0.0654004	−	0.997859i	$-0.479168\pi$
$331$	2.37971	0.130801	0.0654004	+	0.997859i	$0.479168\pi$
$332$	0	0
$333$	−6.81435	−0.373424
$334$	0	0
$335$	−22.4542	−1.22680
$336$	0	0
$337$	14.1300	0.769710	0.384855	−	0.922977i	$-0.374251\pi$
$337$	14.1300	0.769710	0.384855	+	0.922977i	$0.374251\pi$
$338$	0	0
$339$	−21.9054	−1.18974
$340$	0	0
$341$	5.77509	0.312739
$342$	0	0
$343$	−11.3553	−0.613130
$344$	0	0
$345$	20.6137	1.10980
$346$	0	0
$347$	21.7724	1.16880	0.584402	−	0.811464i	$-0.301329\pi$
$347$	21.7724	1.16880	0.584402	+	0.811464i	$0.301329\pi$
$348$	0	0
$349$	4.06784	0.217747	0.108873	−	0.994056i	$-0.465276\pi$
$349$	4.06784	0.217747	0.108873	+	0.994056i	$0.465276\pi$
$350$	0	0
$351$	25.9476	1.38498
$352$	0	0
$353$	−28.4939	−1.51658	−0.758290	−	0.651918i	$-0.773965\pi$
$353$	−28.4939	−1.51658	−0.758290	+	0.651918i	$0.773965\pi$
$354$	0	0
$355$	−13.5075	−0.716906
$356$	0	0
$357$	28.5162	1.50924
$358$	0	0
$359$	15.7596	0.831759	0.415880	−	0.909420i	$-0.363474\pi$
$359$	15.7596	0.831759	0.415880	+	0.909420i	$0.363474\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−10.5915	−0.555908
$364$	0	0
$365$	19.8514	1.03907
$366$	0	0
$367$	−29.3884	−1.53406	−0.767031	−	0.641610i	$-0.778267\pi$
$367$	−29.3884	−1.53406	−0.767031	+	0.641610i	$0.778267\pi$
$368$	0	0
$369$	7.13711	0.371543
$370$	0	0
$371$	−41.0391	−2.13064
$372$	0	0
$373$	−20.7389	−1.07382	−0.536911	−	0.843639i	$-0.680409\pi$
$373$	−20.7389	−1.07382	−0.536911	+	0.843639i	$0.680409\pi$
$374$	0	0
$375$	5.82418	0.300759
$376$	0	0
$377$	−6.40140	−0.329689
$378$	0	0
$379$	29.6041	1.52066	0.760330	−	0.649537i	$-0.225037\pi$
$379$	29.6041	1.52066	0.760330	+	0.649537i	$0.225037\pi$
$380$	0	0
$381$	−16.8063	−0.861011
$382$	0	0
$383$	−5.86367	−0.299620	−0.149810	−	0.988715i	$-0.547866\pi$
$383$	−5.86367	−0.299620	−0.149810	+	0.988715i	$0.547866\pi$
$384$	0	0
$385$	−17.5506	−0.894461
$386$	0	0
$387$	6.82030	0.346695
$388$	0	0
$389$	38.2820	1.94097	0.970487	−	0.241152i	$-0.0775251\pi$
$389$	38.2820	1.94097	0.970487	+	0.241152i	$0.0775251\pi$
$390$	0	0
$391$	34.2939	1.73432
$392$	0	0
$393$	−18.8778	−0.952260
$394$	0	0
$395$	2.88512	0.145166
$396$	0	0
$397$	−13.5865	−0.681887	−0.340944	−	0.940084i	$-0.610747\pi$
$397$	−13.5865	−0.681887	−0.340944	+	0.940084i	$0.610747\pi$
$398$	0	0
$399$	4.93268	0.246943
$400$	0	0
$401$	11.8205	0.590288	0.295144	−	0.955453i	$-0.404632\pi$
$401$	11.8205	0.590288	0.295144	+	0.955453i	$0.404632\pi$
$402$	0	0
$403$	18.4120	0.917165
$404$	0	0
$405$	5.63024	0.279769
$406$	0	0
$407$	−6.53283	−0.323820
$408$	0	0
$409$	−3.72994	−0.184434	−0.0922168	−	0.995739i	$-0.529395\pi$
$409$	−3.72994	−0.184434	−0.0922168	+	0.995739i	$0.529395\pi$
$410$	0	0
$411$	−13.9918	−0.690166
$412$	0	0
$413$	42.0497	2.06913
$414$	0	0
$415$	−11.5039	−0.564704
$416$	0	0
$417$	8.35305	0.409051
$418$	0	0
$419$	−26.4200	−1.29070	−0.645352	−	0.763885i	$-0.723289\pi$
$419$	−26.4200	−1.29070	−0.645352	+	0.763885i	$0.723289\pi$
$420$	0	0
$421$	−22.5095	−1.09704	−0.548522	−	0.836136i	$-0.684809\pi$
$421$	−22.5095	−1.09704	−0.548522	+	0.836136i	$0.684809\pi$
$422$	0	0
$423$	−8.78282	−0.427035
$424$	0	0
$425$	−19.2160	−0.932111
$426$	0	0
$427$	2.33605	0.113049
$428$	0	0
$429$	8.47177	0.409021
$430$	0	0
$431$	3.01283	0.145123	0.0725615	−	0.997364i	$-0.476883\pi$
$431$	3.01283	0.145123	0.0725615	+	0.997364i	$0.476883\pi$
$432$	0	0
$433$	−28.3292	−1.36142	−0.680708	−	0.732555i	$-0.738328\pi$
$433$	−28.3292	−1.36142	−0.680708	+	0.732555i	$0.738328\pi$
$434$	0	0
$435$	−4.69737	−0.225222
$436$	0	0
$437$	5.93211	0.283771
$438$	0	0
$439$	16.8910	0.806165	0.403082	−	0.915164i	$-0.367939\pi$
$439$	16.8910	0.806165	0.403082	+	0.915164i	$0.367939\pi$
$440$	0	0
$441$	−15.1413	−0.721013
$442$	0	0
$443$	−22.9063	−1.08831	−0.544154	−	0.838985i	$-0.683149\pi$
$443$	−22.9063	−1.08831	−0.544154	+	0.838985i	$0.683149\pi$
$444$	0	0
$445$	−0.775544	−0.0367643
$446$	0	0
$447$	−6.44768	−0.304965
$448$	0	0
$449$	33.7413	1.59235	0.796174	−	0.605067i	$-0.206854\pi$
$449$	33.7413	1.59235	0.796174	+	0.605067i	$0.206854\pi$
$450$	0	0
$451$	6.84225	0.322189
$452$	0	0
$453$	−4.70590	−0.221102
$454$	0	0
$455$	−55.9542	−2.62317
$456$	0	0
$457$	7.04927	0.329751	0.164875	−	0.986314i	$-0.447278\pi$
$457$	7.04927	0.329751	0.164875	+	0.986314i	$0.447278\pi$
$458$	0	0
$459$	31.6766	1.47854
$460$	0	0
$461$	−27.1110	−1.26268	−0.631342	−	0.775505i	$-0.717495\pi$
$461$	−27.1110	−1.26268	−0.631342	+	0.775505i	$0.717495\pi$
$462$	0	0
$463$	−4.85823	−0.225781	−0.112891	−	0.993607i	$-0.536011\pi$
$463$	−4.85823	−0.225781	−0.112891	+	0.993607i	$0.536011\pi$
$464$	0	0
$465$	13.5107	0.626546
$466$	0	0
$467$	−37.0427	−1.71413	−0.857066	−	0.515206i	$-0.827715\pi$
$467$	−37.0427	−1.71413	−0.857066	+	0.515206i	$0.827715\pi$
$468$	0	0
$469$	−31.8738	−1.47180
$470$	0	0
$471$	−14.0158	−0.645816
$472$	0	0
$473$	6.53853	0.300642
$474$	0	0
$475$	−3.32394	−0.152513
$476$	0	0
$477$	−15.5255	−0.710863
$478$	0	0
$479$	12.1954	0.557221	0.278610	−	0.960404i	$-0.410126\pi$
$479$	12.1954	0.557221	0.278610	+	0.960404i	$0.410126\pi$
$480$	0	0
$481$	−20.8277	−0.949664
$482$	0	0
$483$	29.2612	1.33143
$484$	0	0
$485$	39.9266	1.81297
$486$	0	0
$487$	14.5117	0.657587	0.328793	−	0.944402i	$-0.393358\pi$
$487$	14.5117	0.657587	0.328793	+	0.944402i	$0.393358\pi$
$488$	0	0
$489$	−12.4948	−0.565034
$490$	0	0
$491$	17.7295	0.800121	0.400061	−	0.916489i	$-0.368989\pi$
$491$	17.7295	0.800121	0.400061	+	0.916489i	$0.368989\pi$
$492$	0	0
$493$	−7.81478	−0.351960
$494$	0	0
$495$	−6.63956	−0.298426
$496$	0	0
$497$	−19.1740	−0.860072
$498$	0	0
$499$	−13.6072	−0.609143	−0.304571	−	0.952489i	$-0.598513\pi$
$499$	−13.6072	−0.609143	−0.304571	+	0.952489i	$0.598513\pi$
$500$	0	0
$501$	2.67422	0.119475
$502$	0	0
$503$	−16.7169	−0.745372	−0.372686	−	0.927958i	$-0.621563\pi$
$503$	−16.7169	−0.745372	−0.372686	+	0.927958i	$0.621563\pi$
$504$	0	0
$505$	−15.1952	−0.676179
$506$	0	0
$507$	11.3518	0.504152
$508$	0	0
$509$	−32.7371	−1.45105	−0.725523	−	0.688198i	$-0.758402\pi$
$509$	−32.7371	−1.45105	−0.725523	+	0.688198i	$0.758402\pi$
$510$	0	0
$511$	28.1791	1.24657
$512$	0	0
$513$	5.47937	0.241920
$514$	0	0
$515$	−33.6754	−1.48391
$516$	0	0
$517$	−8.41998	−0.370310
$518$	0	0
$519$	5.77470	0.253481
$520$	0	0
$521$	−1.93410	−0.0847347	−0.0423673	−	0.999102i	$-0.513490\pi$
$521$	−1.93410	−0.0847347	−0.0423673	+	0.999102i	$0.513490\pi$
$522$	0	0
$523$	23.4002	1.02322	0.511610	−	0.859218i	$-0.329049\pi$
$523$	23.4002	1.02322	0.511610	+	0.859218i	$0.329049\pi$
$524$	0	0
$525$	−16.3959	−0.715577
$526$	0	0
$527$	22.4772	0.979121
$528$	0	0
$529$	12.1899	0.529996
$530$	0	0
$531$	15.9078	0.690341
$532$	0	0
$533$	21.8142	0.944879
$534$	0	0
$535$	−5.83827	−0.252410
$536$	0	0
$537$	−25.9025	−1.11778
$538$	0	0
$539$	−14.5157	−0.625237
$540$	0	0
$541$	31.0596	1.33536	0.667679	−	0.744449i	$-0.267288\pi$
$541$	31.0596	1.33536	0.667679	+	0.744449i	$0.267288\pi$
$542$	0	0
$543$	−13.1967	−0.566324
$544$	0	0
$545$	30.2081	1.29397
$546$	0	0
$547$	−3.72479	−0.159260	−0.0796302	−	0.996824i	$-0.525374\pi$
$547$	−3.72479	−0.159260	−0.0796302	+	0.996824i	$0.525374\pi$
$548$	0	0
$549$	0.883750	0.0377175
$550$	0	0
$551$	−1.35179	−0.0575881
$552$	0	0
$553$	4.09545	0.174156
$554$	0	0
$555$	−15.2835	−0.648747
$556$	0	0
$557$	4.93471	0.209090	0.104545	−	0.994520i	$-0.466661\pi$
$557$	4.93471	0.209090	0.104545	+	0.994520i	$0.466661\pi$
$558$	0	0
$559$	20.8459	0.881688
$560$	0	0
$561$	10.3423	0.436651
$562$	0	0
$563$	−5.23272	−0.220533	−0.110266	−	0.993902i	$-0.535170\pi$
$563$	−5.23272	−0.220533	−0.110266	+	0.993902i	$0.535170\pi$
$564$	0	0
$565$	52.4727	2.20754
$566$	0	0
$567$	7.99215	0.335639
$568$	0	0
$569$	30.9476	1.29739	0.648696	−	0.761047i	$-0.275315\pi$
$569$	30.9476	1.29739	0.648696	+	0.761047i	$0.275315\pi$
$570$	0	0
$571$	−29.3128	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$571$	−29.3128	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$572$	0	0
$573$	−21.4085	−0.894352
$574$	0	0
$575$	−19.7180	−0.822297
$576$	0	0
$577$	−0.223971	−0.00932403	−0.00466201	−	0.999989i	$-0.501484\pi$
$577$	−0.223971	−0.00932403	−0.00466201	+	0.999989i	$0.501484\pi$
$578$	0	0
$579$	19.9421	0.828766
$580$	0	0
$581$	−16.3298	−0.677476
$582$	0	0
$583$	−14.8841	−0.616436
$584$	0	0
$585$	−21.1680	−0.875190
$586$	0	0
$587$	−39.3732	−1.62510	−0.812552	−	0.582888i	$-0.801922\pi$
$587$	−39.3732	−1.62510	−0.812552	+	0.582888i	$0.801922\pi$
$588$	0	0
$589$	3.88806	0.160205
$590$	0	0
$591$	25.0214	1.02924
$592$	0	0
$593$	−9.79825	−0.402366	−0.201183	−	0.979554i	$-0.564479\pi$
$593$	−9.79825	−0.402366	−0.201183	+	0.979554i	$0.564479\pi$
$594$	0	0
$595$	−68.3084	−2.80037
$596$	0	0
$597$	2.28103	0.0933563
$598$	0	0
$599$	−5.45134	−0.222736	−0.111368	−	0.993779i	$-0.535523\pi$
$599$	−5.45134	−0.222736	−0.111368	+	0.993779i	$0.535523\pi$
$600$	0	0
$601$	22.3135	0.910187	0.455094	−	0.890444i	$-0.349606\pi$
$601$	22.3135	0.910187	0.455094	+	0.890444i	$0.349606\pi$
$602$	0	0
$603$	−12.0582	−0.491047
$604$	0	0
$605$	25.3711	1.03148
$606$	0	0
$607$	−40.7620	−1.65448	−0.827240	−	0.561849i	$-0.810090\pi$
$607$	−40.7620	−1.65448	−0.827240	+	0.561849i	$0.810090\pi$
$608$	0	0
$609$	−6.66793	−0.270198
$610$	0	0
$611$	−26.8443	−1.08600
$612$	0	0
$613$	−33.8478	−1.36710	−0.683549	−	0.729905i	$-0.739564\pi$
$613$	−33.8478	−1.36710	−0.683549	+	0.729905i	$0.739564\pi$
$614$	0	0
$615$	16.0073	0.645478
$616$	0	0
$617$	−9.43136	−0.379692	−0.189846	−	0.981814i	$-0.560799\pi$
$617$	−9.43136	−0.379692	−0.189846	+	0.981814i	$0.560799\pi$
$618$	0	0
$619$	11.6549	0.468451	0.234226	−	0.972182i	$-0.424745\pi$
$619$	11.6549	0.468451	0.234226	+	0.972182i	$0.424745\pi$
$620$	0	0
$621$	32.5042	1.30435
$622$	0	0
$623$	−1.10089	−0.0441061
$624$	0	0
$625$	−30.5711	−1.22284
$626$	0	0
$627$	1.78899	0.0714453
$628$	0	0
$629$	−25.4263	−1.01381
$630$	0	0
$631$	−14.0107	−0.557756	−0.278878	−	0.960327i	$-0.589962\pi$
$631$	−14.0107	−0.557756	−0.278878	+	0.960327i	$0.589962\pi$
$632$	0	0
$633$	−4.06972	−0.161757
$634$	0	0
$635$	40.2582	1.59760
$636$	0	0
$637$	−46.2786	−1.83362
$638$	0	0
$639$	−7.25372	−0.286953
$640$	0	0
$641$	15.1055	0.596632	0.298316	−	0.954467i	$-0.403575\pi$
$641$	15.1055	0.596632	0.298316	+	0.954467i	$0.403575\pi$
$642$	0	0
$643$	−28.7278	−1.13291	−0.566457	−	0.824091i	$-0.691686\pi$
$643$	−28.7278	−1.13291	−0.566457	+	0.824091i	$0.691686\pi$
$644$	0	0
$645$	15.2968	0.602311
$646$	0	0
$647$	−4.26073	−0.167507	−0.0837534	−	0.996487i	$-0.526691\pi$
$647$	−4.26073	−0.167507	−0.0837534	+	0.996487i	$0.526691\pi$
$648$	0	0
$649$	15.2506	0.598640
$650$	0	0
$651$	19.1786	0.751667
$652$	0	0
$653$	−47.1169	−1.84383	−0.921913	−	0.387396i	$-0.873375\pi$
$653$	−47.1169	−1.84383	−0.921913	+	0.387396i	$0.873375\pi$
$654$	0	0
$655$	45.2205	1.76691
$656$	0	0
$657$	10.6605	0.415904
$658$	0	0
$659$	47.0657	1.83342	0.916709	−	0.399555i	$-0.130835\pi$
$659$	47.0657	1.83342	0.916709	+	0.399555i	$0.130835\pi$
$660$	0	0
$661$	23.0808	0.897738	0.448869	−	0.893597i	$-0.351827\pi$
$661$	23.0808	0.897738	0.448869	+	0.893597i	$0.351827\pi$
$662$	0	0
$663$	32.9729	1.28056
$664$	0	0
$665$	−11.8159	−0.458200
$666$	0	0
$667$	−8.01895	−0.310495
$668$	0	0
$669$	12.5084	0.483605
$670$	0	0
$671$	0.847240	0.0327073
$672$	0	0
$673$	−6.88022	−0.265213	−0.132606	−	0.991169i	$-0.542335\pi$
$673$	−6.88022	−0.265213	−0.132606	+	0.991169i	$0.542335\pi$
$674$	0	0
$675$	−18.2131	−0.701023
$676$	0	0
$677$	31.0825	1.19460	0.597298	−	0.802019i	$-0.296241\pi$
$677$	31.0825	1.19460	0.597298	+	0.802019i	$0.296241\pi$
$678$	0	0
$679$	56.6760	2.17502
$680$	0	0
$681$	−28.3623	−1.08685
$682$	0	0
$683$	−32.0161	−1.22506	−0.612531	−	0.790447i	$-0.709849\pi$
$683$	−32.0161	−1.22506	−0.612531	+	0.790447i	$0.709849\pi$
$684$	0	0
$685$	33.5164	1.28060
$686$	0	0
$687$	−18.6186	−0.710346
$688$	0	0
$689$	−47.4529	−1.80781
$690$	0	0
$691$	12.9347	0.492059	0.246030	−	0.969262i	$-0.420874\pi$
$691$	12.9347	0.492059	0.246030	+	0.969262i	$0.420874\pi$
$692$	0	0
$693$	−9.42489	−0.358022
$694$	0	0
$695$	−20.0091	−0.758989
$696$	0	0
$697$	26.6306	1.00871
$698$	0	0
$699$	5.24237	0.198285
$700$	0	0
$701$	−10.5226	−0.397434	−0.198717	−	0.980057i	$-0.563678\pi$
$701$	−10.5226	−0.397434	−0.198717	+	0.980057i	$0.563678\pi$
$702$	0	0
$703$	−4.39821	−0.165881
$704$	0	0
$705$	−19.6984	−0.741885
$706$	0	0
$707$	−21.5697	−0.811212
$708$	0	0
$709$	37.9437	1.42500	0.712502	−	0.701670i	$-0.247562\pi$
$709$	37.9437	1.42500	0.712502	+	0.701670i	$0.247562\pi$
$710$	0	0
$711$	1.54935	0.0581051
$712$	0	0
$713$	23.0644	0.863769
$714$	0	0
$715$	−20.2935	−0.758934
$716$	0	0
$717$	−0.409488	−0.0152926
$718$	0	0
$719$	16.1610	0.602704	0.301352	−	0.953513i	$-0.402562\pi$
$719$	16.1610	0.602704	0.301352	+	0.953513i	$0.402562\pi$
$720$	0	0
$721$	−47.8023	−1.78025
$722$	0	0
$723$	−8.29958	−0.308665
$724$	0	0
$725$	4.49326	0.166876
$726$	0	0
$727$	−42.3086	−1.56914	−0.784569	−	0.620041i	$-0.787116\pi$
$727$	−42.3086	−1.56914	−0.784569	+	0.620041i	$0.787116\pi$
$728$	0	0
$729$	22.8221	0.845262
$730$	0	0
$731$	25.4485	0.941248
$732$	0	0
$733$	−30.3710	−1.12178	−0.560890	−	0.827891i	$-0.689541\pi$
$733$	−30.3710	−1.12178	−0.560890	+	0.827891i	$0.689541\pi$
$734$	0	0
$735$	−33.9594	−1.25261
$736$	0	0
$737$	−11.5600	−0.425819
$738$	0	0
$739$	−23.2976	−0.857015	−0.428508	−	0.903538i	$-0.640960\pi$
$739$	−23.2976	−0.857015	−0.428508	+	0.903538i	$0.640960\pi$
$740$	0	0
$741$	5.70359	0.209527
$742$	0	0
$743$	−53.4097	−1.95941	−0.979705	−	0.200442i	$-0.935762\pi$
$743$	−53.4097	−1.95941	−0.979705	+	0.200442i	$0.935762\pi$
$744$	0	0
$745$	15.4449	0.565859
$746$	0	0
$747$	−6.17774	−0.226032
$748$	0	0
$749$	−8.28745	−0.302817
$750$	0	0
$751$	−17.8006	−0.649552	−0.324776	−	0.945791i	$-0.605289\pi$
$751$	−17.8006	−0.649552	−0.324776	+	0.945791i	$0.605289\pi$
$752$	0	0
$753$	21.8165	0.795037
$754$	0	0
$755$	11.2726	0.410253
$756$	0	0
$757$	26.8901	0.977337	0.488668	−	0.872470i	$-0.337483\pi$
$757$	26.8901	0.977337	0.488668	+	0.872470i	$0.337483\pi$
$758$	0	0
$759$	10.6125	0.385208
$760$	0	0
$761$	49.0269	1.77722	0.888612	−	0.458659i	$-0.151670\pi$
$761$	49.0269	1.77722	0.888612	+	0.458659i	$0.151670\pi$
$762$	0	0
$763$	42.8805	1.55238
$764$	0	0
$765$	−25.8418	−0.934311
$766$	0	0
$767$	48.6216	1.75562
$768$	0	0
$769$	1.28213	0.0462348	0.0231174	−	0.999733i	$-0.492641\pi$
$769$	1.28213	0.0462348	0.0231174	+	0.999733i	$0.492641\pi$
$770$	0	0
$771$	−22.7819	−0.820472
$772$	0	0
$773$	−9.80738	−0.352747	−0.176374	−	0.984323i	$-0.556437\pi$
$773$	−9.80738	−0.352747	−0.176374	+	0.984323i	$0.556437\pi$
$774$	0	0
$775$	−12.9237	−0.464233
$776$	0	0
$777$	−21.6949	−0.778301
$778$	0	0
$779$	4.60652	0.165046
$780$	0	0
$781$	−6.95405	−0.248835
$782$	0	0
$783$	−7.40695	−0.264703
$784$	0	0
$785$	33.5739	1.19831
$786$	0	0
$787$	−7.17375	−0.255717	−0.127858	−	0.991792i	$-0.540810\pi$
$787$	−7.17375	−0.255717	−0.127858	+	0.991792i	$0.540810\pi$
$788$	0	0
$789$	0.681495	0.0242618
$790$	0	0
$791$	74.4853	2.64839
$792$	0	0
$793$	2.70114	0.0959203
$794$	0	0
$795$	−34.8211	−1.23498
$796$	0	0
$797$	28.7626	1.01882	0.509412	−	0.860523i	$-0.329863\pi$
$797$	28.7626	1.01882	0.509412	+	0.860523i	$0.329863\pi$
$798$	0	0
$799$	−32.7713	−1.15936
$800$	0	0
$801$	−0.416477	−0.0147155
$802$	0	0
$803$	10.2200	0.360657
$804$	0	0
$805$	−70.0930	−2.47045
$806$	0	0
$807$	32.4513	1.14234
$808$	0	0
$809$	15.8104	0.555864	0.277932	−	0.960601i	$-0.410351\pi$
$809$	15.8104	0.555864	0.277932	+	0.960601i	$0.410351\pi$
$810$	0	0
$811$	6.54922	0.229974	0.114987	−	0.993367i	$-0.463317\pi$
$811$	6.54922	0.229974	0.114987	+	0.993367i	$0.463317\pi$
$812$	0	0
$813$	−16.9988	−0.596175
$814$	0	0
$815$	29.9304	1.04841
$816$	0	0
$817$	4.40204	0.154008
$818$	0	0
$819$	−30.0481	−1.04997
$820$	0	0
$821$	23.9957	0.837456	0.418728	−	0.908112i	$-0.362476\pi$
$821$	23.9957	0.837456	0.418728	+	0.908112i	$0.362476\pi$
$822$	0	0
$823$	20.7630	0.723751	0.361876	−	0.932226i	$-0.382136\pi$
$823$	20.7630	0.723751	0.361876	+	0.932226i	$0.382136\pi$
$824$	0	0
$825$	−5.94649	−0.207030
$826$	0	0
$827$	48.6589	1.69204	0.846018	−	0.533154i	$-0.178993\pi$
$827$	48.6589	1.69204	0.846018	+	0.533154i	$0.178993\pi$
$828$	0	0
$829$	−1.43031	−0.0496768	−0.0248384	−	0.999691i	$-0.507907\pi$
$829$	−1.43031	−0.0496768	−0.0248384	+	0.999691i	$0.507907\pi$
$830$	0	0
$831$	7.84661	0.272196
$832$	0	0
$833$	−56.4965	−1.95749
$834$	0	0
$835$	−6.40589	−0.221685
$836$	0	0
$837$	21.3041	0.736379
$838$	0	0
$839$	17.6197	0.608301	0.304151	−	0.952624i	$-0.401627\pi$
$839$	17.6197	0.608301	0.304151	+	0.952624i	$0.401627\pi$
$840$	0	0
$841$	−27.1727	−0.936989
$842$	0	0
$843$	−8.16670	−0.281276
$844$	0	0
$845$	−27.1925	−0.935449
$846$	0	0
$847$	36.0144	1.23747
$848$	0	0
$849$	18.3633	0.630228
$850$	0	0
$851$	−26.0906	−0.894375
$852$	0	0
$853$	−5.70417	−0.195307	−0.0976536	−	0.995220i	$-0.531134\pi$
$853$	−5.70417	−0.195307	−0.0976536	+	0.995220i	$0.531134\pi$
$854$	0	0
$855$	−4.47006	−0.152873
$856$	0	0
$857$	13.3639	0.456503	0.228251	−	0.973602i	$-0.426699\pi$
$857$	13.3639	0.456503	0.228251	+	0.973602i	$0.426699\pi$
$858$	0	0
$859$	−46.1394	−1.57426	−0.787128	−	0.616790i	$-0.788433\pi$
$859$	−46.1394	−1.57426	−0.787128	+	0.616790i	$0.788433\pi$
$860$	0	0
$861$	22.7225	0.774380
$862$	0	0
$863$	−22.5532	−0.767718	−0.383859	−	0.923392i	$-0.625405\pi$
$863$	−22.5532	−0.767718	−0.383859	+	0.923392i	$0.625405\pi$
$864$	0	0
$865$	−13.8329	−0.470332
$866$	0	0
$867$	19.7777	0.671686
$868$	0	0
$869$	1.48534	0.0503867
$870$	0	0
$871$	−36.8553	−1.24879
$872$	0	0
$873$	21.4411	0.725671
$874$	0	0
$875$	−19.8041	−0.669500
$876$	0	0
$877$	10.3700	0.350170	0.175085	−	0.984553i	$-0.443980\pi$
$877$	10.3700	0.350170	0.175085	+	0.984553i	$0.443980\pi$
$878$	0	0
$879$	18.1386	0.611799
$880$	0	0
$881$	−2.51707	−0.0848023	−0.0424012	−	0.999101i	$-0.513501\pi$
$881$	−2.51707	−0.0848023	−0.0424012	+	0.999101i	$0.513501\pi$
$882$	0	0
$883$	−14.9822	−0.504190	−0.252095	−	0.967702i	$-0.581120\pi$
$883$	−14.9822	−0.504190	−0.252095	+	0.967702i	$0.581120\pi$
$884$	0	0
$885$	35.6786	1.19932
$886$	0	0
$887$	−12.0170	−0.403490	−0.201745	−	0.979438i	$-0.564661\pi$
$887$	−12.0170	−0.403490	−0.201745	+	0.979438i	$0.564661\pi$
$888$	0	0
$889$	57.1467	1.91664
$890$	0	0
$891$	2.89860	0.0971068
$892$	0	0
$893$	−5.66872	−0.189696
$894$	0	0
$895$	62.0476	2.07402
$896$	0	0
$897$	33.8343	1.12969
$898$	0	0
$899$	−5.25583	−0.175292
$900$	0	0
$901$	−57.9301	−1.92993
$902$	0	0
$903$	21.7139	0.722592
$904$	0	0
$905$	31.6117	1.05081
$906$	0	0
$907$	49.1532	1.63211	0.816053	−	0.577977i	$-0.196158\pi$
$907$	49.1532	1.63211	0.816053	+	0.577977i	$0.196158\pi$
$908$	0	0
$909$	−8.16003	−0.270651
$910$	0	0
$911$	−4.11545	−0.136351	−0.0681755	−	0.997673i	$-0.521718\pi$
$911$	−4.11545	−0.136351	−0.0681755	+	0.997673i	$0.521718\pi$
$912$	0	0
$913$	−5.92252	−0.196007
$914$	0	0
$915$	1.98210	0.0655263
$916$	0	0
$917$	64.1906	2.11976
$918$	0	0
$919$	19.0523	0.628476	0.314238	−	0.949344i	$-0.398251\pi$
$919$	19.0523	0.628476	0.314238	+	0.949344i	$0.398251\pi$
$920$	0	0
$921$	−1.26758	−0.0417682
$922$	0	0
$923$	−22.1707	−0.729756
$924$	0	0
$925$	14.6194	0.480682
$926$	0	0
$927$	−18.0841	−0.593960
$928$	0	0
$929$	17.1059	0.561226	0.280613	−	0.959821i	$-0.409462\pi$
$929$	17.1059	0.561226	0.280613	+	0.959821i	$0.409462\pi$
$930$	0	0
$931$	−9.77267	−0.320286
$932$	0	0
$933$	−1.17673	−0.0385244
$934$	0	0
$935$	−24.7742	−0.810201
$936$	0	0
$937$	46.0301	1.50374	0.751869	−	0.659313i	$-0.229153\pi$
$937$	46.0301	1.50374	0.751869	+	0.659313i	$0.229153\pi$
$938$	0	0
$939$	−32.0765	−1.04678
$940$	0	0
$941$	53.5916	1.74704	0.873519	−	0.486790i	$-0.161832\pi$
$941$	53.5916	1.74704	0.873519	+	0.486790i	$0.161832\pi$
$942$	0	0
$943$	27.3264	0.889870
$944$	0	0
$945$	−64.7435	−2.10611
$946$	0	0
$947$	−21.9710	−0.713961	−0.356981	−	0.934112i	$-0.616194\pi$
$947$	−21.9710	−0.713961	−0.356981	+	0.934112i	$0.616194\pi$
$948$	0	0
$949$	32.5832	1.05769
$950$	0	0
$951$	−8.84056	−0.286675
$952$	0	0
$953$	1.43852	0.0465981	0.0232991	−	0.999729i	$-0.492583\pi$
$953$	1.43852	0.0465981	0.0232991	+	0.999729i	$0.492583\pi$
$954$	0	0
$955$	51.2824	1.65946
$956$	0	0
$957$	−2.41833	−0.0781736
$958$	0	0
$959$	47.5767	1.53633
$960$	0	0
$961$	−15.8830	−0.512354
$962$	0	0
$963$	−3.13523	−0.101031
$964$	0	0
$965$	−47.7699	−1.53777
$966$	0	0
$967$	−3.75592	−0.120782	−0.0603911	−	0.998175i	$-0.519235\pi$
$967$	−3.75592	−0.120782	−0.0603911	+	0.998175i	$0.519235\pi$
$968$	0	0
$969$	6.96290	0.223680
$970$	0	0
$971$	7.28950	0.233931	0.116966	−	0.993136i	$-0.462683\pi$
$971$	7.28950	0.233931	0.116966	+	0.993136i	$0.462683\pi$
$972$	0	0
$973$	−28.4030	−0.910559
$974$	0	0
$975$	−18.9584	−0.607155
$976$	0	0
$977$	0.768238	0.0245781	0.0122891	−	0.999924i	$-0.496088\pi$
$977$	0.768238	0.0245781	0.0122891	+	0.999924i	$0.496088\pi$
$978$	0	0
$979$	−0.399271	−0.0127608
$980$	0	0
$981$	16.2221	0.517932
$982$	0	0
$983$	18.8203	0.600276	0.300138	−	0.953896i	$-0.402967\pi$
$983$	18.8203	0.600276	0.300138	+	0.953896i	$0.402967\pi$
$984$	0	0
$985$	−59.9369	−1.90975
$986$	0	0
$987$	−27.9620	−0.890039
$988$	0	0
$989$	26.1134	0.830358
$990$	0	0
$991$	18.4535	0.586195	0.293098	−	0.956083i	$-0.405314\pi$
$991$	18.4535	0.586195	0.293098	+	0.956083i	$0.405314\pi$
$992$	0	0
$993$	2.86620	0.0909561
$994$	0	0
$995$	−5.46404	−0.173222
$996$	0	0
$997$	12.7836	0.404861	0.202430	−	0.979297i	$-0.435116\pi$
$997$	12.7836	0.404861	0.202430	+	0.979297i	$0.435116\pi$
$998$	0	0
$999$	−24.0994	−0.762471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.h.1.22	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.h.1.22	✓	31	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.20443 q^{3} -2.88512 q^{5} -4.09545 q^{7} -1.54935 q^{9} +O(q^{10})\)	\(q+1.20443 q^{3} -2.88512 q^{5} -4.09545 q^{7} -1.54935 q^{9} -1.48534 q^{11} -4.73551 q^{13} -3.47493 q^{15} -5.78107 q^{17} -1.00000 q^{19} -4.93268 q^{21} -5.93211 q^{23} +3.32394 q^{25} -5.47937 q^{27} +1.35179 q^{29} -3.88806 q^{31} -1.78899 q^{33} +11.8159 q^{35} +4.39821 q^{37} -5.70359 q^{39} -4.60652 q^{41} -4.40204 q^{43} +4.47006 q^{45} +5.66872 q^{47} +9.77267 q^{49} -6.96290 q^{51} +10.0207 q^{53} +4.28539 q^{55} -1.20443 q^{57} -10.2674 q^{59} -0.570401 q^{61} +6.34527 q^{63} +13.6625 q^{65} +7.78275 q^{67} -7.14481 q^{69} +4.68179 q^{71} -6.88061 q^{73} +4.00346 q^{75} +6.08313 q^{77} -1.00000 q^{79} -1.95147 q^{81} +3.98732 q^{83} +16.6791 q^{85} +1.62813 q^{87} +0.268808 q^{89} +19.3940 q^{91} -4.68290 q^{93} +2.88512 q^{95} -13.8388 q^{97} +2.30131 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more