LMFDB - Embedded newform 6004.2.a.g.1.23

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:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
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)

$f(q)$	$=$	$q+2.40855 q^{3} +0.936776 q^{5} -3.55545 q^{7} +2.80113 q^{9} +O(q^{10})$	$q+2.40855 q^{3} +0.936776 q^{5} -3.55545 q^{7} +2.80113 q^{9} -1.35047 q^{11} -2.20428 q^{13} +2.25627 q^{15} +1.19197 q^{17} +1.00000 q^{19} -8.56349 q^{21} -5.26841 q^{23} -4.12245 q^{25} -0.478999 q^{27} +9.37775 q^{29} +0.532765 q^{31} -3.25268 q^{33} -3.33066 q^{35} +4.51687 q^{37} -5.30913 q^{39} -8.05936 q^{41} +12.1234 q^{43} +2.62403 q^{45} -0.711383 q^{47} +5.64123 q^{49} +2.87092 q^{51} -12.0299 q^{53} -1.26509 q^{55} +2.40855 q^{57} -7.85491 q^{59} -11.6631 q^{61} -9.95926 q^{63} -2.06492 q^{65} +7.91918 q^{67} -12.6892 q^{69} -11.5274 q^{71} -7.79171 q^{73} -9.92914 q^{75} +4.80153 q^{77} -1.00000 q^{79} -9.55707 q^{81} -8.02598 q^{83} +1.11661 q^{85} +22.5868 q^{87} +14.4395 q^{89} +7.83722 q^{91} +1.28319 q^{93} +0.936776 q^{95} -15.0900 q^{97} -3.78284 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.40855	1.39058	0.695289	−	0.718730i	$-0.255276\pi$
$3$	2.40855	1.39058	0.695289	+	0.718730i	$0.255276\pi$
$4$	0	0
$5$	0.936776	0.418939	0.209469	−	0.977815i	$-0.432826\pi$
$5$	0.936776	0.418939	0.209469	+	0.977815i	$0.432826\pi$
$6$	0	0
$7$	−3.55545	−1.34383	−0.671917	−	0.740627i	$-0.734529\pi$
$7$	−3.55545	−1.34383	−0.671917	+	0.740627i	$0.734529\pi$
$8$	0	0
$9$	2.80113	0.933709
$10$	0	0
$11$	−1.35047	−0.407182	−0.203591	−	0.979056i	$-0.565261\pi$
$11$	−1.35047	−0.407182	−0.203591	+	0.979056i	$0.565261\pi$
$12$	0	0
$13$	−2.20428	−0.611358	−0.305679	−	0.952135i	$-0.598883\pi$
$13$	−2.20428	−0.611358	−0.305679	+	0.952135i	$0.598883\pi$
$14$	0	0
$15$	2.25627	0.582567
$16$	0	0
$17$	1.19197	0.289095	0.144547	−	0.989498i	$-0.453827\pi$
$17$	1.19197	0.289095	0.144547	+	0.989498i	$0.453827\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−8.56349	−1.86871
$22$	0	0
$23$	−5.26841	−1.09854	−0.549269	−	0.835645i	$-0.685094\pi$
$23$	−5.26841	−1.09854	−0.549269	+	0.835645i	$0.685094\pi$
$24$	0	0
$25$	−4.12245	−0.824490
$26$	0	0
$27$	−0.478999	−0.0921834
$28$	0	0
$29$	9.37775	1.74140	0.870702	−	0.491810i	$-0.163665\pi$
$29$	9.37775	1.74140	0.870702	+	0.491810i	$0.163665\pi$
$30$	0	0
$31$	0.532765	0.0956874	0.0478437	−	0.998855i	$-0.484765\pi$
$31$	0.532765	0.0956874	0.0478437	+	0.998855i	$0.484765\pi$
$32$	0	0
$33$	−3.25268	−0.566218
$34$	0	0
$35$	−3.33066	−0.562984
$36$	0	0
$37$	4.51687	0.742568	0.371284	−	0.928519i	$-0.378918\pi$
$37$	4.51687	0.742568	0.371284	+	0.928519i	$0.378918\pi$
$38$	0	0
$39$	−5.30913	−0.850142
$40$	0	0
$41$	−8.05936	−1.25866	−0.629330	−	0.777138i	$-0.716671\pi$
$41$	−8.05936	−1.25866	−0.629330	+	0.777138i	$0.716671\pi$
$42$	0	0
$43$	12.1234	1.84881	0.924403	−	0.381418i	$-0.124564\pi$
$43$	12.1234	1.84881	0.924403	+	0.381418i	$0.124564\pi$
$44$	0	0
$45$	2.62403	0.391167
$46$	0	0
$47$	−0.711383	−0.103766	−0.0518829	−	0.998653i	$-0.516522\pi$
$47$	−0.711383	−0.103766	−0.0518829	+	0.998653i	$0.516522\pi$
$48$	0	0
$49$	5.64123	0.805890
$50$	0	0
$51$	2.87092	0.402009
$52$	0	0
$53$	−12.0299	−1.65243	−0.826215	−	0.563356i	$-0.809510\pi$
$53$	−12.0299	−1.65243	−0.826215	+	0.563356i	$0.809510\pi$
$54$	0	0
$55$	−1.26509	−0.170584
$56$	0	0
$57$	2.40855	0.319021
$58$	0	0
$59$	−7.85491	−1.02262	−0.511311	−	0.859396i	$-0.670840\pi$
$59$	−7.85491	−1.02262	−0.511311	+	0.859396i	$0.670840\pi$
$60$	0	0
$61$	−11.6631	−1.49330	−0.746652	−	0.665215i	$-0.768340\pi$
$61$	−11.6631	−1.49330	−0.746652	+	0.665215i	$0.768340\pi$
$62$	0	0
$63$	−9.95926	−1.25475
$64$	0	0
$65$	−2.06492	−0.256122
$66$	0	0
$67$	7.91918	0.967481	0.483741	−	0.875211i	$-0.339278\pi$
$67$	7.91918	0.967481	0.483741	+	0.875211i	$0.339278\pi$
$68$	0	0
$69$	−12.6892	−1.52760
$70$	0	0
$71$	−11.5274	−1.36805	−0.684026	−	0.729458i	$-0.739773\pi$
$71$	−11.5274	−1.36805	−0.684026	+	0.729458i	$0.739773\pi$
$72$	0	0
$73$	−7.79171	−0.911951	−0.455975	−	0.889992i	$-0.650710\pi$
$73$	−7.79171	−0.911951	−0.455975	+	0.889992i	$0.650710\pi$
$74$	0	0
$75$	−9.92914	−1.14652
$76$	0	0
$77$	4.80153	0.547185
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−9.55707	−1.06190
$82$	0	0
$83$	−8.02598	−0.880965	−0.440483	−	0.897761i	$-0.645193\pi$
$83$	−8.02598	−0.880965	−0.440483	+	0.897761i	$0.645193\pi$
$84$	0	0
$85$	1.11661	0.121113
$86$	0	0
$87$	22.5868	2.42156
$88$	0	0
$89$	14.4395	1.53059	0.765294	−	0.643681i	$-0.222593\pi$
$89$	14.4395	1.53059	0.765294	+	0.643681i	$0.222593\pi$
$90$	0	0
$91$	7.83722	0.821564
$92$	0	0
$93$	1.28319	0.133061
$94$	0	0
$95$	0.936776	0.0961112
$96$	0	0
$97$	−15.0900	−1.53216	−0.766080	−	0.642745i	$-0.777796\pi$
$97$	−15.0900	−1.53216	−0.766080	+	0.642745i	$0.777796\pi$
$98$	0	0
$99$	−3.78284	−0.380189
$100$	0	0
$101$	−11.2921	−1.12361	−0.561805	−	0.827269i	$-0.689893\pi$
$101$	−11.2921	−1.12361	−0.561805	+	0.827269i	$0.689893\pi$
$102$	0	0
$103$	−10.5594	−1.04044	−0.520222	−	0.854031i	$-0.674151\pi$
$103$	−10.5594	−1.04044	−0.520222	+	0.854031i	$0.674151\pi$
$104$	0	0
$105$	−8.02207	−0.782874
$106$	0	0
$107$	−7.75718	−0.749915	−0.374957	−	0.927042i	$-0.622343\pi$
$107$	−7.75718	−0.749915	−0.374957	+	0.927042i	$0.622343\pi$
$108$	0	0
$109$	−17.2452	−1.65179	−0.825893	−	0.563827i	$-0.809329\pi$
$109$	−17.2452	−1.65179	−0.825893	+	0.563827i	$0.809329\pi$
$110$	0	0
$111$	10.8791	1.03260
$112$	0	0
$113$	12.0383	1.13247	0.566236	−	0.824243i	$-0.308399\pi$
$113$	12.0383	1.13247	0.566236	+	0.824243i	$0.308399\pi$
$114$	0	0
$115$	−4.93532	−0.460221
$116$	0	0
$117$	−6.17448	−0.570830
$118$	0	0
$119$	−4.23798	−0.388495
$120$	0	0
$121$	−9.17623	−0.834203
$122$	0	0
$123$	−19.4114	−1.75027
$124$	0	0
$125$	−8.54569	−0.764350
$126$	0	0
$127$	7.04160	0.624841	0.312420	−	0.949944i	$-0.398860\pi$
$127$	7.04160	0.624841	0.312420	+	0.949944i	$0.398860\pi$
$128$	0	0
$129$	29.1999	2.57091
$130$	0	0
$131$	11.8609	1.03629	0.518146	−	0.855292i	$-0.326622\pi$
$131$	11.8609	1.03629	0.518146	+	0.855292i	$0.326622\pi$
$132$	0	0
$133$	−3.55545	−0.308297
$134$	0	0
$135$	−0.448715	−0.0386192
$136$	0	0
$137$	2.92152	0.249602	0.124801	−	0.992182i	$-0.460171\pi$
$137$	2.92152	0.249602	0.124801	+	0.992182i	$0.460171\pi$
$138$	0	0
$139$	−1.79325	−0.152101	−0.0760507	−	0.997104i	$-0.524231\pi$
$139$	−1.79325	−0.152101	−0.0760507	+	0.997104i	$0.524231\pi$
$140$	0	0
$141$	−1.71340	−0.144295
$142$	0	0
$143$	2.97682	0.248934
$144$	0	0
$145$	8.78485	0.729542
$146$	0	0
$147$	13.5872	1.12065
$148$	0	0
$149$	10.6057	0.868851	0.434426	−	0.900708i	$-0.356951\pi$
$149$	10.6057	0.868851	0.434426	+	0.900708i	$0.356951\pi$
$150$	0	0
$151$	10.6241	0.864581	0.432290	−	0.901734i	$-0.357706\pi$
$151$	10.6241	0.864581	0.432290	+	0.901734i	$0.357706\pi$
$152$	0	0
$153$	3.33885	0.269930
$154$	0	0
$155$	0.499081	0.0400872
$156$	0	0
$157$	−0.709557	−0.0566288	−0.0283144	−	0.999599i	$-0.509014\pi$
$157$	−0.709557	−0.0566288	−0.0283144	+	0.999599i	$0.509014\pi$
$158$	0	0
$159$	−28.9746	−2.29783
$160$	0	0
$161$	18.7316	1.47625
$162$	0	0
$163$	−5.99174	−0.469310	−0.234655	−	0.972079i	$-0.575396\pi$
$163$	−5.99174	−0.469310	−0.234655	+	0.972079i	$0.575396\pi$
$164$	0	0
$165$	−3.04703	−0.237211
$166$	0	0
$167$	−1.48640	−0.115021	−0.0575107	−	0.998345i	$-0.518316\pi$
$167$	−1.48640	−0.115021	−0.0575107	+	0.998345i	$0.518316\pi$
$168$	0	0
$169$	−8.14113	−0.626241
$170$	0	0
$171$	2.80113	0.214207
$172$	0	0
$173$	−4.82189	−0.366601	−0.183301	−	0.983057i	$-0.558678\pi$
$173$	−4.82189	−0.366601	−0.183301	+	0.983057i	$0.558678\pi$
$174$	0	0
$175$	14.6572	1.10798
$176$	0	0
$177$	−18.9190	−1.42204
$178$	0	0
$179$	−3.48479	−0.260466	−0.130233	−	0.991483i	$-0.541572\pi$
$179$	−3.48479	−0.260466	−0.130233	+	0.991483i	$0.541572\pi$
$180$	0	0
$181$	−7.53606	−0.560151	−0.280075	−	0.959978i	$-0.590359\pi$
$181$	−7.53606	−0.560151	−0.280075	+	0.959978i	$0.590359\pi$
$182$	0	0
$183$	−28.0911	−2.07656
$184$	0	0
$185$	4.23129	0.311091
$186$	0	0
$187$	−1.60972	−0.117714
$188$	0	0
$189$	1.70306	0.123879
$190$	0	0
$191$	8.70692	0.630011	0.315005	−	0.949090i	$-0.397994\pi$
$191$	8.70692	0.630011	0.315005	+	0.949090i	$0.397994\pi$
$192$	0	0
$193$	−22.0259	−1.58546	−0.792730	−	0.609573i	$-0.791341\pi$
$193$	−22.0259	−1.58546	−0.792730	+	0.609573i	$0.791341\pi$
$194$	0	0
$195$	−4.97347	−0.356157
$196$	0	0
$197$	−24.7276	−1.76177	−0.880883	−	0.473335i	$-0.843050\pi$
$197$	−24.7276	−1.76177	−0.880883	+	0.473335i	$0.843050\pi$
$198$	0	0
$199$	10.6889	0.757716	0.378858	−	0.925455i	$-0.376317\pi$
$199$	10.6889	0.757716	0.378858	+	0.925455i	$0.376317\pi$
$200$	0	0
$201$	19.0738	1.34536
$202$	0	0
$203$	−33.3421	−2.34016
$204$	0	0
$205$	−7.54982	−0.527302
$206$	0	0
$207$	−14.7575	−1.02572
$208$	0	0
$209$	−1.35047	−0.0934139
$210$	0	0
$211$	16.3535	1.12582	0.562912	−	0.826517i	$-0.309681\pi$
$211$	16.3535	1.12582	0.562912	+	0.826517i	$0.309681\pi$
$212$	0	0
$213$	−27.7644	−1.90238
$214$	0	0
$215$	11.3569	0.774537
$216$	0	0
$217$	−1.89422	−0.128588
$218$	0	0
$219$	−18.7667	−1.26814
$220$	0	0
$221$	−2.62744	−0.176740
$222$	0	0
$223$	−5.94491	−0.398101	−0.199050	−	0.979989i	$-0.563786\pi$
$223$	−5.94491	−0.398101	−0.199050	+	0.979989i	$0.563786\pi$
$224$	0	0
$225$	−11.5475	−0.769834
$226$	0	0
$227$	17.8838	1.18699	0.593494	−	0.804838i	$-0.297748\pi$
$227$	17.8838	1.18699	0.593494	+	0.804838i	$0.297748\pi$
$228$	0	0
$229$	−11.2213	−0.741528	−0.370764	−	0.928727i	$-0.620904\pi$
$229$	−11.2213	−0.741528	−0.370764	+	0.928727i	$0.620904\pi$
$230$	0	0
$231$	11.5647	0.760903
$232$	0	0
$233$	−1.00557	−0.0658772	−0.0329386	−	0.999457i	$-0.510487\pi$
$233$	−1.00557	−0.0658772	−0.0329386	+	0.999457i	$0.510487\pi$
$234$	0	0
$235$	−0.666406	−0.0434716
$236$	0	0
$237$	−2.40855	−0.156452
$238$	0	0
$239$	−8.35601	−0.540506	−0.270253	−	0.962789i	$-0.587107\pi$
$239$	−8.35601	−0.540506	−0.270253	+	0.962789i	$0.587107\pi$
$240$	0	0
$241$	−19.6842	−1.26797	−0.633987	−	0.773344i	$-0.718583\pi$
$241$	−19.6842	−1.26797	−0.633987	+	0.773344i	$0.718583\pi$
$242$	0	0
$243$	−21.5817	−1.38447
$244$	0	0
$245$	5.28457	0.337619
$246$	0	0
$247$	−2.20428	−0.140255
$248$	0	0
$249$	−19.3310	−1.22505
$250$	0	0
$251$	28.7823	1.81672	0.908361	−	0.418187i	$-0.137334\pi$
$251$	28.7823	1.81672	0.908361	+	0.418187i	$0.137334\pi$
$252$	0	0
$253$	7.11482	0.447305
$254$	0	0
$255$	2.68941	0.168417
$256$	0	0
$257$	22.0606	1.37610	0.688050	−	0.725663i	$-0.258467\pi$
$257$	22.0606	1.37610	0.688050	+	0.725663i	$0.258467\pi$
$258$	0	0
$259$	−16.0595	−0.997888
$260$	0	0
$261$	26.2683	1.62596
$262$	0	0
$263$	29.8473	1.84046	0.920232	−	0.391374i	$-0.128000\pi$
$263$	29.8473	1.84046	0.920232	+	0.391374i	$0.128000\pi$
$264$	0	0
$265$	−11.2693	−0.692267
$266$	0	0
$267$	34.7784	2.12840
$268$	0	0
$269$	27.5517	1.67986	0.839929	−	0.542697i	$-0.182597\pi$
$269$	27.5517	1.67986	0.839929	+	0.542697i	$0.182597\pi$
$270$	0	0
$271$	−26.2579	−1.59506	−0.797528	−	0.603282i	$-0.793859\pi$
$271$	−26.2579	−1.59506	−0.797528	+	0.603282i	$0.793859\pi$
$272$	0	0
$273$	18.8764	1.14245
$274$	0	0
$275$	5.56724	0.335717
$276$	0	0
$277$	7.00196	0.420707	0.210353	−	0.977625i	$-0.432539\pi$
$277$	7.00196	0.420707	0.210353	+	0.977625i	$0.432539\pi$
$278$	0	0
$279$	1.49234	0.0893441
$280$	0	0
$281$	10.5668	0.630364	0.315182	−	0.949031i	$-0.397934\pi$
$281$	10.5668	0.630364	0.315182	+	0.949031i	$0.397934\pi$
$282$	0	0
$283$	12.4334	0.739089	0.369544	−	0.929213i	$-0.379514\pi$
$283$	12.4334	0.739089	0.369544	+	0.929213i	$0.379514\pi$
$284$	0	0
$285$	2.25627	0.133650
$286$	0	0
$287$	28.6547	1.69143
$288$	0	0
$289$	−15.5792	−0.916424
$290$	0	0
$291$	−36.3451	−2.13059
$292$	0	0
$293$	−4.15069	−0.242486	−0.121243	−	0.992623i	$-0.538688\pi$
$293$	−4.15069	−0.242486	−0.121243	+	0.992623i	$0.538688\pi$
$294$	0	0
$295$	−7.35829	−0.428416
$296$	0	0
$297$	0.646874	0.0375354
$298$	0	0
$299$	11.6131	0.671601
$300$	0	0
$301$	−43.1042	−2.48449
$302$	0	0
$303$	−27.1977	−1.56247
$304$	0	0
$305$	−10.9257	−0.625603
$306$	0	0
$307$	25.2948	1.44365	0.721825	−	0.692075i	$-0.243303\pi$
$307$	25.2948	1.44365	0.721825	+	0.692075i	$0.243303\pi$
$308$	0	0
$309$	−25.4328	−1.44682
$310$	0	0
$311$	11.6910	0.662935	0.331467	−	0.943467i	$-0.392456\pi$
$311$	11.6910	0.662935	0.331467	+	0.943467i	$0.392456\pi$
$312$	0	0
$313$	18.3489	1.03714	0.518570	−	0.855035i	$-0.326465\pi$
$313$	18.3489	1.03714	0.518570	+	0.855035i	$0.326465\pi$
$314$	0	0
$315$	−9.32960	−0.525663
$316$	0	0
$317$	−20.9077	−1.17429	−0.587147	−	0.809480i	$-0.699749\pi$
$317$	−20.9077	−1.17429	−0.587147	+	0.809480i	$0.699749\pi$
$318$	0	0
$319$	−12.6644	−0.709069
$320$	0	0
$321$	−18.6836	−1.04282
$322$	0	0
$323$	1.19197	0.0663229
$324$	0	0
$325$	9.08705	0.504059
$326$	0	0
$327$	−41.5359	−2.29694
$328$	0	0
$329$	2.52929	0.139444
$330$	0	0
$331$	−12.0338	−0.661439	−0.330719	−	0.943729i	$-0.607291\pi$
$331$	−12.0338	−0.661439	−0.330719	+	0.943729i	$0.607291\pi$
$332$	0	0
$333$	12.6523	0.693342
$334$	0	0
$335$	7.41849	0.405316
$336$	0	0
$337$	34.6249	1.88614	0.943069	−	0.332596i	$-0.107925\pi$
$337$	34.6249	1.88614	0.943069	+	0.332596i	$0.107925\pi$
$338$	0	0
$339$	28.9950	1.57479
$340$	0	0
$341$	−0.719482	−0.0389622
$342$	0	0
$343$	4.83105	0.260852
$344$	0	0
$345$	−11.8870	−0.639973
$346$	0	0
$347$	−22.9817	−1.23372	−0.616862	−	0.787071i	$-0.711596\pi$
$347$	−22.9817	−1.23372	−0.616862	+	0.787071i	$0.711596\pi$
$348$	0	0
$349$	26.7221	1.43040	0.715202	−	0.698918i	$-0.246335\pi$
$349$	26.7221	1.43040	0.715202	+	0.698918i	$0.246335\pi$
$350$	0	0
$351$	1.05585	0.0563571
$352$	0	0
$353$	28.4540	1.51445	0.757227	−	0.653151i	$-0.226554\pi$
$353$	28.4540	1.51445	0.757227	+	0.653151i	$0.226554\pi$
$354$	0	0
$355$	−10.7986	−0.573130
$356$	0	0
$357$	−10.2074	−0.540233
$358$	0	0
$359$	6.68284	0.352707	0.176354	−	0.984327i	$-0.443570\pi$
$359$	6.68284	0.352707	0.176354	+	0.984327i	$0.443570\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−22.1014	−1.16002
$364$	0	0
$365$	−7.29909	−0.382052
$366$	0	0
$367$	−3.58609	−0.187192	−0.0935961	−	0.995610i	$-0.529836\pi$
$367$	−3.58609	−0.187192	−0.0935961	+	0.995610i	$0.529836\pi$
$368$	0	0
$369$	−22.5753	−1.17522
$370$	0	0
$371$	42.7716	2.22059
$372$	0	0
$373$	17.1091	0.885876	0.442938	−	0.896552i	$-0.353936\pi$
$373$	17.1091	0.885876	0.442938	+	0.896552i	$0.353936\pi$
$374$	0	0
$375$	−20.5827	−1.06289
$376$	0	0
$377$	−20.6712	−1.06462
$378$	0	0
$379$	23.0695	1.18500	0.592500	−	0.805570i	$-0.298141\pi$
$379$	23.0695	1.18500	0.592500	+	0.805570i	$0.298141\pi$
$380$	0	0
$381$	16.9601	0.868890
$382$	0	0
$383$	10.6418	0.543768	0.271884	−	0.962330i	$-0.412353\pi$
$383$	10.6418	0.543768	0.271884	+	0.962330i	$0.412353\pi$
$384$	0	0
$385$	4.49796	0.229237
$386$	0	0
$387$	33.9592	1.72625
$388$	0	0
$389$	−35.9357	−1.82201	−0.911006	−	0.412392i	$-0.864693\pi$
$389$	−35.9357	−1.82201	−0.911006	+	0.412392i	$0.864693\pi$
$390$	0	0
$391$	−6.27977	−0.317582
$392$	0	0
$393$	28.5676	1.44105
$394$	0	0
$395$	−0.936776	−0.0471343
$396$	0	0
$397$	15.8086	0.793413	0.396707	−	0.917945i	$-0.370153\pi$
$397$	15.8086	0.793413	0.396707	+	0.917945i	$0.370153\pi$
$398$	0	0
$399$	−8.56349	−0.428711
$400$	0	0
$401$	−8.62286	−0.430605	−0.215302	−	0.976547i	$-0.569074\pi$
$401$	−8.62286	−0.430605	−0.215302	+	0.976547i	$0.569074\pi$
$402$	0	0
$403$	−1.17436	−0.0584993
$404$	0	0
$405$	−8.95283	−0.444870
$406$	0	0
$407$	−6.09989	−0.302360
$408$	0	0
$409$	6.61999	0.327338	0.163669	−	0.986515i	$-0.447667\pi$
$409$	6.61999	0.327338	0.163669	+	0.986515i	$0.447667\pi$
$410$	0	0
$411$	7.03663	0.347091
$412$	0	0
$413$	27.9277	1.37423
$414$	0	0
$415$	−7.51854	−0.369071
$416$	0	0
$417$	−4.31913	−0.211509
$418$	0	0
$419$	6.46135	0.315658	0.157829	−	0.987466i	$-0.449551\pi$
$419$	6.46135	0.315658	0.157829	+	0.987466i	$0.449551\pi$
$420$	0	0
$421$	12.3262	0.600740	0.300370	−	0.953823i	$-0.402890\pi$
$421$	12.3262	0.600740	0.300370	+	0.953823i	$0.402890\pi$
$422$	0	0
$423$	−1.99267	−0.0968871
$424$	0	0
$425$	−4.91383	−0.238356
$426$	0	0
$427$	41.4675	2.00675
$428$	0	0
$429$	7.16982	0.346162
$430$	0	0
$431$	−7.70060	−0.370925	−0.185463	−	0.982651i	$-0.559378\pi$
$431$	−7.70060	−0.370925	−0.185463	+	0.982651i	$0.559378\pi$
$432$	0	0
$433$	40.6940	1.95563	0.977815	−	0.209472i	$-0.0671745\pi$
$433$	40.6940	1.95563	0.977815	+	0.209472i	$0.0671745\pi$
$434$	0	0
$435$	21.1588	1.01449
$436$	0	0
$437$	−5.26841	−0.252022
$438$	0	0
$439$	−0.756586	−0.0361099	−0.0180549	−	0.999837i	$-0.505747\pi$
$439$	−0.756586	−0.0361099	−0.0180549	+	0.999837i	$0.505747\pi$
$440$	0	0
$441$	15.8018	0.752466
$442$	0	0
$443$	−8.48277	−0.403029	−0.201514	−	0.979486i	$-0.564586\pi$
$443$	−8.48277	−0.403029	−0.201514	+	0.979486i	$0.564586\pi$
$444$	0	0
$445$	13.5266	0.641223
$446$	0	0
$447$	25.5443	1.20821
$448$	0	0
$449$	14.6460	0.691188	0.345594	−	0.938384i	$-0.387677\pi$
$449$	14.6460	0.691188	0.345594	+	0.938384i	$0.387677\pi$
$450$	0	0
$451$	10.8839	0.512504
$452$	0	0
$453$	25.5888	1.20227
$454$	0	0
$455$	7.34172	0.344185
$456$	0	0
$457$	−24.2432	−1.13405	−0.567024	−	0.823701i	$-0.691905\pi$
$457$	−24.2432	−1.13405	−0.567024	+	0.823701i	$0.691905\pi$
$458$	0	0
$459$	−0.570952	−0.0266497
$460$	0	0
$461$	−7.26290	−0.338267	−0.169134	−	0.985593i	$-0.554097\pi$
$461$	−7.26290	−0.338267	−0.169134	+	0.985593i	$0.554097\pi$
$462$	0	0
$463$	8.47744	0.393980	0.196990	−	0.980405i	$-0.436883\pi$
$463$	8.47744	0.393980	0.196990	+	0.980405i	$0.436883\pi$
$464$	0	0
$465$	1.20206	0.0557443
$466$	0	0
$467$	−3.96076	−0.183282	−0.0916411	−	0.995792i	$-0.529211\pi$
$467$	−3.96076	−0.183282	−0.0916411	+	0.995792i	$0.529211\pi$
$468$	0	0
$469$	−28.1562	−1.30013
$470$	0	0
$471$	−1.70900	−0.0787468
$472$	0	0
$473$	−16.3723	−0.752800
$474$	0	0
$475$	−4.12245	−0.189151
$476$	0	0
$477$	−33.6972	−1.54289
$478$	0	0
$479$	−19.2114	−0.877792	−0.438896	−	0.898538i	$-0.644630\pi$
$479$	−19.2114	−0.877792	−0.438896	+	0.898538i	$0.644630\pi$
$480$	0	0
$481$	−9.95645	−0.453975
$482$	0	0
$483$	45.1159	2.05285
$484$	0	0
$485$	−14.1360	−0.641882
$486$	0	0
$487$	39.5738	1.79326	0.896629	−	0.442783i	$-0.146009\pi$
$487$	39.5738	1.79326	0.896629	+	0.442783i	$0.146009\pi$
$488$	0	0
$489$	−14.4314	−0.652612
$490$	0	0
$491$	−38.1386	−1.72117	−0.860585	−	0.509308i	$-0.829902\pi$
$491$	−38.1386	−1.72117	−0.860585	+	0.509308i	$0.829902\pi$
$492$	0	0
$493$	11.1780	0.503431
$494$	0	0
$495$	−3.54367	−0.159276
$496$	0	0
$497$	40.9851	1.83843
$498$	0	0
$499$	7.47262	0.334520	0.167260	−	0.985913i	$-0.446508\pi$
$499$	7.47262	0.334520	0.167260	+	0.985913i	$0.446508\pi$
$500$	0	0
$501$	−3.58008	−0.159946
$502$	0	0
$503$	−31.9768	−1.42578	−0.712888	−	0.701277i	$-0.752613\pi$
$503$	−31.9768	−1.42578	−0.712888	+	0.701277i	$0.752613\pi$
$504$	0	0
$505$	−10.5782	−0.470724
$506$	0	0
$507$	−19.6083	−0.870837
$508$	0	0
$509$	6.15479	0.272806	0.136403	−	0.990653i	$-0.456446\pi$
$509$	6.15479	0.272806	0.136403	+	0.990653i	$0.456446\pi$
$510$	0	0
$511$	27.7030	1.22551
$512$	0	0
$513$	−0.478999	−0.0211483
$514$	0	0
$515$	−9.89175	−0.435883
$516$	0	0
$517$	0.960701	0.0422516
$518$	0	0
$519$	−11.6138	−0.509788
$520$	0	0
$521$	−9.55471	−0.418599	−0.209300	−	0.977852i	$-0.567118\pi$
$521$	−9.55471	−0.418599	−0.209300	+	0.977852i	$0.567118\pi$
$522$	0	0
$523$	−4.66319	−0.203907	−0.101954	−	0.994789i	$-0.532509\pi$
$523$	−4.66319	−0.203907	−0.101954	+	0.994789i	$0.532509\pi$
$524$	0	0
$525$	35.3026	1.54073
$526$	0	0
$527$	0.635038	0.0276627
$528$	0	0
$529$	4.75611	0.206787
$530$	0	0
$531$	−22.0026	−0.954831
$532$	0	0
$533$	17.7651	0.769493
$534$	0	0
$535$	−7.26674	−0.314169
$536$	0	0
$537$	−8.39331	−0.362198
$538$	0	0
$539$	−7.61830	−0.328144
$540$	0	0
$541$	12.9631	0.557326	0.278663	−	0.960389i	$-0.410109\pi$
$541$	12.9631	0.557326	0.278663	+	0.960389i	$0.410109\pi$
$542$	0	0
$543$	−18.1510	−0.778934
$544$	0	0
$545$	−16.1548	−0.691998
$546$	0	0
$547$	−16.5915	−0.709403	−0.354701	−	0.934980i	$-0.615417\pi$
$547$	−16.5915	−0.709403	−0.354701	+	0.934980i	$0.615417\pi$
$548$	0	0
$549$	−32.6697	−1.39431
$550$	0	0
$551$	9.37775	0.399506
$552$	0	0
$553$	3.55545	0.151193
$554$	0	0
$555$	10.1913	0.432596
$556$	0	0
$557$	−4.94935	−0.209711	−0.104855	−	0.994487i	$-0.533438\pi$
$557$	−4.94935	−0.209711	−0.104855	+	0.994487i	$0.533438\pi$
$558$	0	0
$559$	−26.7235	−1.13028
$560$	0	0
$561$	−3.87709	−0.163691
$562$	0	0
$563$	35.4185	1.49271	0.746356	−	0.665547i	$-0.231802\pi$
$563$	35.4185	1.49271	0.746356	+	0.665547i	$0.231802\pi$
$564$	0	0
$565$	11.2772	0.474437
$566$	0	0
$567$	33.9797	1.42701
$568$	0	0
$569$	−16.5398	−0.693385	−0.346693	−	0.937979i	$-0.612695\pi$
$569$	−16.5398	−0.693385	−0.346693	+	0.937979i	$0.612695\pi$
$570$	0	0
$571$	8.56503	0.358435	0.179218	−	0.983809i	$-0.442643\pi$
$571$	8.56503	0.358435	0.179218	+	0.983809i	$0.442643\pi$
$572$	0	0
$573$	20.9711	0.876080
$574$	0	0
$575$	21.7187	0.905734
$576$	0	0
$577$	19.7332	0.821503	0.410751	−	0.911747i	$-0.365266\pi$
$577$	19.7332	0.821503	0.410751	+	0.911747i	$0.365266\pi$
$578$	0	0
$579$	−53.0506	−2.20471
$580$	0	0
$581$	28.5360	1.18387
$582$	0	0
$583$	16.2460	0.672839
$584$	0	0
$585$	−5.78410	−0.239143
$586$	0	0
$587$	21.0990	0.870847	0.435423	−	0.900226i	$-0.356599\pi$
$587$	21.0990	0.870847	0.435423	+	0.900226i	$0.356599\pi$
$588$	0	0
$589$	0.532765	0.0219522
$590$	0	0
$591$	−59.5576	−2.44987
$592$	0	0
$593$	−22.0395	−0.905055	−0.452527	−	0.891751i	$-0.649477\pi$
$593$	−22.0395	−0.905055	−0.452527	+	0.891751i	$0.649477\pi$
$594$	0	0
$595$	−3.97004	−0.162756
$596$	0	0
$597$	25.7448	1.05366
$598$	0	0
$599$	−27.3069	−1.11573	−0.557866	−	0.829931i	$-0.688380\pi$
$599$	−27.3069	−1.11573	−0.557866	+	0.829931i	$0.688380\pi$
$600$	0	0
$601$	21.4409	0.874592	0.437296	−	0.899318i	$-0.355936\pi$
$601$	21.4409	0.874592	0.437296	+	0.899318i	$0.355936\pi$
$602$	0	0
$603$	22.1826	0.903346
$604$	0	0
$605$	−8.59607	−0.349480
$606$	0	0
$607$	−39.6812	−1.61061	−0.805305	−	0.592861i	$-0.797999\pi$
$607$	−39.6812	−1.61061	−0.805305	+	0.592861i	$0.797999\pi$
$608$	0	0
$609$	−80.3063	−3.25417
$610$	0	0
$611$	1.56809	0.0634381
$612$	0	0
$613$	−45.2810	−1.82888	−0.914441	−	0.404719i	$-0.867369\pi$
$613$	−45.2810	−1.82888	−0.914441	+	0.404719i	$0.867369\pi$
$614$	0	0
$615$	−18.1841	−0.733255
$616$	0	0
$617$	29.4900	1.18722	0.593611	−	0.804752i	$-0.297702\pi$
$617$	29.4900	1.18722	0.593611	+	0.804752i	$0.297702\pi$
$618$	0	0
$619$	0.389269	0.0156460	0.00782302	−	0.999969i	$-0.497510\pi$
$619$	0.389269	0.0156460	0.00782302	+	0.999969i	$0.497510\pi$
$620$	0	0
$621$	2.52356	0.101267
$622$	0	0
$623$	−51.3391	−2.05686
$624$	0	0
$625$	12.6069	0.504274
$626$	0	0
$627$	−3.25268	−0.129899
$628$	0	0
$629$	5.38396	0.214673
$630$	0	0
$631$	39.8486	1.58635	0.793174	−	0.608995i	$-0.208427\pi$
$631$	39.8486	1.58635	0.793174	+	0.608995i	$0.208427\pi$
$632$	0	0
$633$	39.3884	1.56555
$634$	0	0
$635$	6.59640	0.261770
$636$	0	0
$637$	−12.4349	−0.492687
$638$	0	0
$639$	−32.2897	−1.27736
$640$	0	0
$641$	48.5283	1.91675	0.958377	−	0.285506i	$-0.0921617\pi$
$641$	48.5283	1.91675	0.958377	+	0.285506i	$0.0921617\pi$
$642$	0	0
$643$	12.6395	0.498452	0.249226	−	0.968445i	$-0.419824\pi$
$643$	12.6395	0.498452	0.249226	+	0.968445i	$0.419824\pi$
$644$	0	0
$645$	27.3538	1.07705
$646$	0	0
$647$	−1.24413	−0.0489117	−0.0244558	−	0.999701i	$-0.507785\pi$
$647$	−1.24413	−0.0489117	−0.0244558	+	0.999701i	$0.507785\pi$
$648$	0	0
$649$	10.6078	0.416393
$650$	0	0
$651$	−4.56232	−0.178812
$652$	0	0
$653$	9.93629	0.388837	0.194419	−	0.980919i	$-0.437718\pi$
$653$	9.93629	0.388837	0.194419	+	0.980919i	$0.437718\pi$
$654$	0	0
$655$	11.1110	0.434143
$656$	0	0
$657$	−21.8256	−0.851496
$658$	0	0
$659$	−7.53507	−0.293525	−0.146762	−	0.989172i	$-0.546885\pi$
$659$	−7.53507	−0.293525	−0.146762	+	0.989172i	$0.546885\pi$
$660$	0	0
$661$	−12.8718	−0.500654	−0.250327	−	0.968161i	$-0.580538\pi$
$661$	−12.8718	−0.500654	−0.250327	+	0.968161i	$0.580538\pi$
$662$	0	0
$663$	−6.32832	−0.245771
$664$	0	0
$665$	−3.33066	−0.129157
$666$	0	0
$667$	−49.4058	−1.91300
$668$	0	0
$669$	−14.3186	−0.553590
$670$	0	0
$671$	15.7506	0.608046
$672$	0	0
$673$	−30.4816	−1.17498	−0.587490	−	0.809231i	$-0.699884\pi$
$673$	−30.4816	−1.17498	−0.587490	+	0.809231i	$0.699884\pi$
$674$	0	0
$675$	1.97465	0.0760043
$676$	0	0
$677$	5.75782	0.221291	0.110646	−	0.993860i	$-0.464708\pi$
$677$	5.75782	0.221291	0.110646	+	0.993860i	$0.464708\pi$
$678$	0	0
$679$	53.6519	2.05897
$680$	0	0
$681$	43.0740	1.65060
$682$	0	0
$683$	−27.0527	−1.03514	−0.517572	−	0.855640i	$-0.673164\pi$
$683$	−27.0527	−1.03514	−0.517572	+	0.855640i	$0.673164\pi$
$684$	0	0
$685$	2.73681	0.104568
$686$	0	0
$687$	−27.0272	−1.03115
$688$	0	0
$689$	26.5172	1.01023
$690$	0	0
$691$	−20.6559	−0.785787	−0.392894	−	0.919584i	$-0.628526\pi$
$691$	−20.6559	−0.785787	−0.392894	+	0.919584i	$0.628526\pi$
$692$	0	0
$693$	13.4497	0.510911
$694$	0	0
$695$	−1.67987	−0.0637212
$696$	0	0
$697$	−9.60650	−0.363872
$698$	0	0
$699$	−2.42197	−0.0916074
$700$	0	0
$701$	−37.0156	−1.39806	−0.699031	−	0.715092i	$-0.746385\pi$
$701$	−37.0156	−1.39806	−0.699031	+	0.715092i	$0.746385\pi$
$702$	0	0
$703$	4.51687	0.170357
$704$	0	0
$705$	−1.60507	−0.0604506
$706$	0	0
$707$	40.1487	1.50995
$708$	0	0
$709$	−32.2592	−1.21152	−0.605761	−	0.795647i	$-0.707131\pi$
$709$	−32.2592	−1.21152	−0.605761	+	0.795647i	$0.707131\pi$
$710$	0	0
$711$	−2.80113	−0.105050
$712$	0	0
$713$	−2.80682	−0.105116
$714$	0	0
$715$	2.78861	0.104288
$716$	0	0
$717$	−20.1259	−0.751615
$718$	0	0
$719$	−41.5279	−1.54873	−0.774365	−	0.632739i	$-0.781931\pi$
$719$	−41.5279	−1.54873	−0.774365	+	0.632739i	$0.781931\pi$
$720$	0	0
$721$	37.5433	1.39818
$722$	0	0
$723$	−47.4105	−1.76322
$724$	0	0
$725$	−38.6593	−1.43577
$726$	0	0
$727$	−26.8936	−0.997428	−0.498714	−	0.866767i	$-0.666194\pi$
$727$	−26.8936	−0.997428	−0.498714	+	0.866767i	$0.666194\pi$
$728$	0	0
$729$	−23.3095	−0.863314
$730$	0	0
$731$	14.4507	0.534480
$732$	0	0
$733$	25.5539	0.943856	0.471928	−	0.881637i	$-0.343558\pi$
$733$	25.5539	0.943856	0.471928	+	0.881637i	$0.343558\pi$
$734$	0	0
$735$	12.7282	0.469485
$736$	0	0
$737$	−10.6946	−0.393941
$738$	0	0
$739$	50.0571	1.84138	0.920689	−	0.390297i	$-0.127628\pi$
$739$	50.0571	1.84138	0.920689	+	0.390297i	$0.127628\pi$
$740$	0	0
$741$	−5.30913	−0.195036
$742$	0	0
$743$	−37.9288	−1.39147	−0.695736	−	0.718297i	$-0.744922\pi$
$743$	−37.9288	−1.39147	−0.695736	+	0.718297i	$0.744922\pi$
$744$	0	0
$745$	9.93515	0.363996
$746$	0	0
$747$	−22.4818	−0.822565
$748$	0	0
$749$	27.5803	1.00776
$750$	0	0
$751$	−14.9353	−0.544998	−0.272499	−	0.962156i	$-0.587850\pi$
$751$	−14.9353	−0.544998	−0.272499	+	0.962156i	$0.587850\pi$
$752$	0	0
$753$	69.3237	2.52629
$754$	0	0
$755$	9.95244	0.362207
$756$	0	0
$757$	28.1795	1.02420	0.512100	−	0.858926i	$-0.328868\pi$
$757$	28.1795	1.02420	0.512100	+	0.858926i	$0.328868\pi$
$758$	0	0
$759$	17.1364	0.622013
$760$	0	0
$761$	14.9983	0.543687	0.271843	−	0.962342i	$-0.412367\pi$
$761$	14.9983	0.543687	0.271843	+	0.962342i	$0.412367\pi$
$762$	0	0
$763$	61.3143	2.21973
$764$	0	0
$765$	3.12776	0.113084
$766$	0	0
$767$	17.3144	0.625188
$768$	0	0
$769$	39.9234	1.43967	0.719837	−	0.694143i	$-0.244217\pi$
$769$	39.9234	1.43967	0.719837	+	0.694143i	$0.244217\pi$
$770$	0	0
$771$	53.1340	1.91358
$772$	0	0
$773$	−38.7050	−1.39212	−0.696060	−	0.717983i	$-0.745065\pi$
$773$	−38.7050	−1.39212	−0.696060	+	0.717983i	$0.745065\pi$
$774$	0	0
$775$	−2.19630	−0.0788933
$776$	0	0
$777$	−38.6801	−1.38764
$778$	0	0
$779$	−8.05936	−0.288757
$780$	0	0
$781$	15.5674	0.557046
$782$	0	0
$783$	−4.49193	−0.160529
$784$	0	0
$785$	−0.664696	−0.0237240
$786$	0	0
$787$	−31.6872	−1.12953	−0.564763	−	0.825253i	$-0.691032\pi$
$787$	−31.6872	−1.12953	−0.564763	+	0.825253i	$0.691032\pi$
$788$	0	0
$789$	71.8888	2.55931
$790$	0	0
$791$	−42.8017	−1.52185
$792$	0	0
$793$	25.7087	0.912944
$794$	0	0
$795$	−27.1427	−0.962651
$796$	0	0
$797$	31.2900	1.10835	0.554175	−	0.832400i	$-0.313034\pi$
$797$	31.2900	1.10835	0.554175	+	0.832400i	$0.313034\pi$
$798$	0	0
$799$	−0.847946	−0.0299982
$800$	0	0
$801$	40.4470	1.42912
$802$	0	0
$803$	10.5225	0.371330
$804$	0	0
$805$	17.5473	0.618460
$806$	0	0
$807$	66.3598	2.33597
$808$	0	0
$809$	−2.39222	−0.0841060	−0.0420530	−	0.999115i	$-0.513390\pi$
$809$	−2.39222	−0.0841060	−0.0420530	+	0.999115i	$0.513390\pi$
$810$	0	0
$811$	−19.8488	−0.696984	−0.348492	−	0.937312i	$-0.613306\pi$
$811$	−19.8488	−0.696984	−0.348492	+	0.937312i	$0.613306\pi$
$812$	0	0
$813$	−63.2436	−2.21805
$814$	0	0
$815$	−5.61292	−0.196612
$816$	0	0
$817$	12.1234	0.424145
$818$	0	0
$819$	21.9530	0.767101
$820$	0	0
$821$	−37.1381	−1.29613	−0.648064	−	0.761586i	$-0.724421\pi$
$821$	−37.1381	−1.29613	−0.648064	+	0.761586i	$0.724421\pi$
$822$	0	0
$823$	−13.6028	−0.474165	−0.237082	−	0.971490i	$-0.576191\pi$
$823$	−13.6028	−0.474165	−0.237082	+	0.971490i	$0.576191\pi$
$824$	0	0
$825$	13.4090	0.466841
$826$	0	0
$827$	31.8329	1.10694	0.553469	−	0.832870i	$-0.313304\pi$
$827$	31.8329	1.10694	0.553469	+	0.832870i	$0.313304\pi$
$828$	0	0
$829$	−36.8408	−1.27953	−0.639767	−	0.768569i	$-0.720969\pi$
$829$	−36.8408	−1.27953	−0.639767	+	0.768569i	$0.720969\pi$
$830$	0	0
$831$	16.8646	0.585026
$832$	0	0
$833$	6.72416	0.232978
$834$	0	0
$835$	−1.39243	−0.0481869
$836$	0	0
$837$	−0.255194	−0.00882079
$838$	0	0
$839$	18.1493	0.626584	0.313292	−	0.949657i	$-0.398568\pi$
$839$	18.1493	0.626584	0.313292	+	0.949657i	$0.398568\pi$
$840$	0	0
$841$	58.9422	2.03249
$842$	0	0
$843$	25.4508	0.876571
$844$	0	0
$845$	−7.62642	−0.262357
$846$	0	0
$847$	32.6256	1.12103
$848$	0	0
$849$	29.9465	1.02776
$850$	0	0
$851$	−23.7967	−0.815740
$852$	0	0
$853$	0.742831	0.0254341	0.0127170	−	0.999919i	$-0.495952\pi$
$853$	0.742831	0.0254341	0.0127170	+	0.999919i	$0.495952\pi$
$854$	0	0
$855$	2.62403	0.0897398
$856$	0	0
$857$	−6.22275	−0.212565	−0.106282	−	0.994336i	$-0.533895\pi$
$857$	−6.22275	−0.212565	−0.106282	+	0.994336i	$0.533895\pi$
$858$	0	0
$859$	14.9061	0.508589	0.254295	−	0.967127i	$-0.418157\pi$
$859$	14.9061	0.508589	0.254295	+	0.967127i	$0.418157\pi$
$860$	0	0
$861$	69.0163	2.35207
$862$	0	0
$863$	47.3403	1.61148	0.805742	−	0.592267i	$-0.201767\pi$
$863$	47.3403	1.61148	0.805742	+	0.592267i	$0.201767\pi$
$864$	0	0
$865$	−4.51703	−0.153584
$866$	0	0
$867$	−37.5234	−1.27436
$868$	0	0
$869$	1.35047	0.0458115
$870$	0	0
$871$	−17.4561	−0.591478
$872$	0	0
$873$	−42.2691	−1.43059
$874$	0	0
$875$	30.3838	1.02716
$876$	0	0
$877$	−55.6779	−1.88011	−0.940054	−	0.341026i	$-0.889226\pi$
$877$	−55.6779	−1.88011	−0.940054	+	0.341026i	$0.889226\pi$
$878$	0	0
$879$	−9.99715	−0.337196
$880$	0	0
$881$	−23.9339	−0.806352	−0.403176	−	0.915122i	$-0.632094\pi$
$881$	−23.9339	−0.806352	−0.403176	+	0.915122i	$0.632094\pi$
$882$	0	0
$883$	−41.9442	−1.41154	−0.705768	−	0.708444i	$-0.749398\pi$
$883$	−41.9442	−1.41154	−0.705768	+	0.708444i	$0.749398\pi$
$884$	0	0
$885$	−17.7228	−0.595746
$886$	0	0
$887$	13.6209	0.457344	0.228672	−	0.973503i	$-0.426562\pi$
$887$	13.6209	0.457344	0.228672	+	0.973503i	$0.426562\pi$
$888$	0	0
$889$	−25.0361	−0.839682
$890$	0	0
$891$	12.9065	0.432385
$892$	0	0
$893$	−0.711383	−0.0238055
$894$	0	0
$895$	−3.26447	−0.109119
$896$	0	0
$897$	27.9707	0.933914
$898$	0	0
$899$	4.99613	0.166630
$900$	0	0
$901$	−14.3392	−0.477709
$902$	0	0
$903$	−103.819	−3.45487
$904$	0	0
$905$	−7.05960	−0.234669
$906$	0	0
$907$	−17.6087	−0.584686	−0.292343	−	0.956313i	$-0.594435\pi$
$907$	−17.6087	−0.584686	−0.292343	+	0.956313i	$0.594435\pi$
$908$	0	0
$909$	−31.6307	−1.04912
$910$	0	0
$911$	−47.7812	−1.58306	−0.791531	−	0.611129i	$-0.790716\pi$
$911$	−47.7812	−1.58306	−0.791531	+	0.611129i	$0.790716\pi$
$912$	0	0
$913$	10.8388	0.358713
$914$	0	0
$915$	−26.3151	−0.869950
$916$	0	0
$917$	−42.1709	−1.39261
$918$	0	0
$919$	−39.4394	−1.30099	−0.650493	−	0.759512i	$-0.725438\pi$
$919$	−39.4394	−1.30099	−0.650493	+	0.759512i	$0.725438\pi$
$920$	0	0
$921$	60.9238	2.00751
$922$	0	0
$923$	25.4097	0.836370
$924$	0	0
$925$	−18.6206	−0.612240
$926$	0	0
$927$	−29.5781	−0.971472
$928$	0	0
$929$	27.5847	0.905025	0.452513	−	0.891758i	$-0.350528\pi$
$929$	27.5847	0.905025	0.452513	+	0.891758i	$0.350528\pi$
$930$	0	0
$931$	5.64123	0.184884
$932$	0	0
$933$	28.1583	0.921863
$934$	0	0
$935$	−1.50794	−0.0493150
$936$	0	0
$937$	−58.3070	−1.90481	−0.952403	−	0.304841i	$-0.901397\pi$
$937$	−58.3070	−1.90481	−0.952403	+	0.304841i	$0.901397\pi$
$938$	0	0
$939$	44.1942	1.44222
$940$	0	0
$941$	−32.4625	−1.05825	−0.529123	−	0.848545i	$-0.677479\pi$
$941$	−32.4625	−1.05825	−0.529123	+	0.848545i	$0.677479\pi$
$942$	0	0
$943$	42.4600	1.38269
$944$	0	0
$945$	1.59538	0.0518978
$946$	0	0
$947$	36.7606	1.19456	0.597279	−	0.802034i	$-0.296249\pi$
$947$	36.7606	1.19456	0.597279	+	0.802034i	$0.296249\pi$
$948$	0	0
$949$	17.1751	0.557529
$950$	0	0
$951$	−50.3573	−1.63295
$952$	0	0
$953$	−45.3134	−1.46784	−0.733922	−	0.679233i	$-0.762312\pi$
$953$	−45.3134	−1.46784	−0.733922	+	0.679233i	$0.762312\pi$
$954$	0	0
$955$	8.15644	0.263936
$956$	0	0
$957$	−30.5028	−0.986015
$958$	0	0
$959$	−10.3873	−0.335424
$960$	0	0
$961$	−30.7162	−0.990844
$962$	0	0
$963$	−21.7288	−0.700202
$964$	0	0
$965$	−20.6333	−0.664211
$966$	0	0
$967$	8.36218	0.268909	0.134455	−	0.990920i	$-0.457072\pi$
$967$	8.36218	0.268909	0.134455	+	0.990920i	$0.457072\pi$
$968$	0	0
$969$	2.87092	0.0922272
$970$	0	0
$971$	−41.6833	−1.33768	−0.668841	−	0.743405i	$-0.733209\pi$
$971$	−41.6833	−1.33768	−0.668841	+	0.743405i	$0.733209\pi$
$972$	0	0
$973$	6.37580	0.204399
$974$	0	0
$975$	21.8866	0.700933
$976$	0	0
$977$	10.4067	0.332939	0.166469	−	0.986047i	$-0.446763\pi$
$977$	10.4067	0.332939	0.166469	+	0.986047i	$0.446763\pi$
$978$	0	0
$979$	−19.5002	−0.623228
$980$	0	0
$981$	−48.3059	−1.54229
$982$	0	0
$983$	34.5652	1.10246	0.551229	−	0.834354i	$-0.314159\pi$
$983$	34.5652	1.10246	0.551229	+	0.834354i	$0.314159\pi$
$984$	0	0
$985$	−23.1642	−0.738072
$986$	0	0
$987$	6.09192	0.193908
$988$	0	0
$989$	−63.8711	−2.03098
$990$	0	0
$991$	45.4244	1.44295	0.721477	−	0.692438i	$-0.243464\pi$
$991$	45.4244	1.44295	0.721477	+	0.692438i	$0.243464\pi$
$992$	0	0
$993$	−28.9841	−0.919782
$994$	0	0
$995$	10.0131	0.317437
$996$	0	0
$997$	−46.4719	−1.47178	−0.735889	−	0.677102i	$-0.763236\pi$
$997$	−46.4719	−1.47178	−0.735889	+	0.677102i	$0.763236\pi$
$998$	0	0
$999$	−2.16357	−0.0684525

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.23	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.23	✓	27	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)

\(f(q)\)	\(=\)	\(q+2.40855 q^{3} +0.936776 q^{5} -3.55545 q^{7} +2.80113 q^{9} +O(q^{10})\)	\(q+2.40855 q^{3} +0.936776 q^{5} -3.55545 q^{7} +2.80113 q^{9} -1.35047 q^{11} -2.20428 q^{13} +2.25627 q^{15} +1.19197 q^{17} +1.00000 q^{19} -8.56349 q^{21} -5.26841 q^{23} -4.12245 q^{25} -0.478999 q^{27} +9.37775 q^{29} +0.532765 q^{31} -3.25268 q^{33} -3.33066 q^{35} +4.51687 q^{37} -5.30913 q^{39} -8.05936 q^{41} +12.1234 q^{43} +2.62403 q^{45} -0.711383 q^{47} +5.64123 q^{49} +2.87092 q^{51} -12.0299 q^{53} -1.26509 q^{55} +2.40855 q^{57} -7.85491 q^{59} -11.6631 q^{61} -9.95926 q^{63} -2.06492 q^{65} +7.91918 q^{67} -12.6892 q^{69} -11.5274 q^{71} -7.79171 q^{73} -9.92914 q^{75} +4.80153 q^{77} -1.00000 q^{79} -9.55707 q^{81} -8.02598 q^{83} +1.11661 q^{85} +22.5868 q^{87} +14.4395 q^{89} +7.83722 q^{91} +1.28319 q^{93} +0.936776 q^{95} -15.0900 q^{97} -3.78284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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