LMFDB - Embedded newform 6024.2.a.q.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 57 x^{16} + 51 x^{15} + 1328 x^{14} - 1116 x^{13} - 16275 x^{12} + 13699 x^{11} + \cdots + 44032 $ x^18 - x^17 - 57x^16 + 51x^15 + 1328x^14 - 1116x^13 - 16275x^12 + 13699x^11 + 112394x^10 - 101250x^9 - 432956x^8 + 439806x^7 + 844117x^6 - 1010785x^5 - 593552x^4 + 965980x^3 - 88040x^2 - 184992x + 44032
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.493430$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.493430 q^{5} -4.29258 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.493430 q^{5} -4.29258 q^{7} +1.00000 q^{9} -1.99729 q^{11} -4.66762 q^{13} -0.493430 q^{15} +3.80423 q^{17} -1.23769 q^{19} -4.29258 q^{21} +3.36111 q^{23} -4.75653 q^{25} +1.00000 q^{27} -4.48698 q^{29} -3.11454 q^{31} -1.99729 q^{33} +2.11809 q^{35} +3.94623 q^{37} -4.66762 q^{39} +2.87078 q^{41} +5.68421 q^{43} -0.493430 q^{45} +1.89602 q^{47} +11.4263 q^{49} +3.80423 q^{51} -0.733412 q^{53} +0.985524 q^{55} -1.23769 q^{57} +6.68629 q^{59} -13.0294 q^{61} -4.29258 q^{63} +2.30314 q^{65} +9.04753 q^{67} +3.36111 q^{69} +2.61945 q^{71} -2.19011 q^{73} -4.75653 q^{75} +8.57355 q^{77} -8.73852 q^{79} +1.00000 q^{81} +9.59151 q^{83} -1.87712 q^{85} -4.48698 q^{87} +17.9364 q^{89} +20.0361 q^{91} -3.11454 q^{93} +0.610713 q^{95} -13.0636 q^{97} -1.99729 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.493430	−0.220668	−0.110334	−	0.993895i	$-0.535192\pi$
$5$	−0.493430	−0.220668	−0.110334	+	0.993895i	$0.535192\pi$
$6$	0	0
$7$	−4.29258	−1.62244	−0.811222	−	0.584738i	$-0.801197\pi$
$7$	−4.29258	−1.62244	−0.811222	+	0.584738i	$0.801197\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.99729	−0.602206	−0.301103	−	0.953592i	$-0.597355\pi$
$11$	−1.99729	−0.602206	−0.301103	+	0.953592i	$0.597355\pi$
$12$	0	0
$13$	−4.66762	−1.29456	−0.647282	−	0.762251i	$-0.724094\pi$
$13$	−4.66762	−1.29456	−0.647282	+	0.762251i	$0.724094\pi$
$14$	0	0
$15$	−0.493430	−0.127403
$16$	0	0
$17$	3.80423	0.922662	0.461331	−	0.887228i	$-0.347372\pi$
$17$	3.80423	0.922662	0.461331	+	0.887228i	$0.347372\pi$
$18$	0	0
$19$	−1.23769	−0.283945	−0.141973	−	0.989871i	$-0.545345\pi$
$19$	−1.23769	−0.283945	−0.141973	+	0.989871i	$0.545345\pi$
$20$	0	0
$21$	−4.29258	−0.936718
$22$	0	0
$23$	3.36111	0.700839	0.350419	−	0.936593i	$-0.386039\pi$
$23$	3.36111	0.700839	0.350419	+	0.936593i	$0.386039\pi$
$24$	0	0
$25$	−4.75653	−0.951305
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.48698	−0.833211	−0.416605	−	0.909087i	$-0.636780\pi$
$29$	−4.48698	−0.833211	−0.416605	+	0.909087i	$0.636780\pi$
$30$	0	0
$31$	−3.11454	−0.559387	−0.279694	−	0.960089i	$-0.590233\pi$
$31$	−3.11454	−0.559387	−0.279694	+	0.960089i	$0.590233\pi$
$32$	0	0
$33$	−1.99729	−0.347684
$34$	0	0
$35$	2.11809	0.358022
$36$	0	0
$37$	3.94623	0.648756	0.324378	−	0.945928i	$-0.394845\pi$
$37$	3.94623	0.648756	0.324378	+	0.945928i	$0.394845\pi$
$38$	0	0
$39$	−4.66762	−0.747417
$40$	0	0
$41$	2.87078	0.448341	0.224170	−	0.974550i	$-0.428033\pi$
$41$	2.87078	0.448341	0.224170	+	0.974550i	$0.428033\pi$
$42$	0	0
$43$	5.68421	0.866834	0.433417	−	0.901193i	$-0.357308\pi$
$43$	5.68421	0.866834	0.433417	+	0.901193i	$0.357308\pi$
$44$	0	0
$45$	−0.493430	−0.0735562
$46$	0	0
$47$	1.89602	0.276563	0.138282	−	0.990393i	$-0.455842\pi$
$47$	1.89602	0.276563	0.138282	+	0.990393i	$0.455842\pi$
$48$	0	0
$49$	11.4263	1.63232
$50$	0	0
$51$	3.80423	0.532699
$52$	0	0
$53$	−0.733412	−0.100742	−0.0503709	−	0.998731i	$-0.516040\pi$
$53$	−0.733412	−0.100742	−0.0503709	+	0.998731i	$0.516040\pi$
$54$	0	0
$55$	0.985524	0.132888
$56$	0	0
$57$	−1.23769	−0.163936
$58$	0	0
$59$	6.68629	0.870481	0.435241	−	0.900314i	$-0.356663\pi$
$59$	6.68629	0.870481	0.435241	+	0.900314i	$0.356663\pi$
$60$	0	0
$61$	−13.0294	−1.66824	−0.834121	−	0.551581i	$-0.814025\pi$
$61$	−13.0294	−1.66824	−0.834121	+	0.551581i	$0.814025\pi$
$62$	0	0
$63$	−4.29258	−0.540815
$64$	0	0
$65$	2.30314	0.285669
$66$	0	0
$67$	9.04753	1.10533	0.552666	−	0.833403i	$-0.313611\pi$
$67$	9.04753	1.10533	0.552666	+	0.833403i	$0.313611\pi$
$68$	0	0
$69$	3.36111	0.404630
$70$	0	0
$71$	2.61945	0.310871	0.155436	−	0.987846i	$-0.450322\pi$
$71$	2.61945	0.310871	0.155436	+	0.987846i	$0.450322\pi$
$72$	0	0
$73$	−2.19011	−0.256333	−0.128167	−	0.991753i	$-0.540909\pi$
$73$	−2.19011	−0.256333	−0.128167	+	0.991753i	$0.540909\pi$
$74$	0	0
$75$	−4.75653	−0.549236
$76$	0	0
$77$	8.57355	0.977046
$78$	0	0
$79$	−8.73852	−0.983160	−0.491580	−	0.870832i	$-0.663581\pi$
$79$	−8.73852	−0.983160	−0.491580	+	0.870832i	$0.663581\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.59151	1.05281	0.526403	−	0.850235i	$-0.323540\pi$
$83$	9.59151	1.05281	0.526403	+	0.850235i	$0.323540\pi$
$84$	0	0
$85$	−1.87712	−0.203602
$86$	0	0
$87$	−4.48698	−0.481054
$88$	0	0
$89$	17.9364	1.90125	0.950625	−	0.310342i	$-0.100444\pi$
$89$	17.9364	1.90125	0.950625	+	0.310342i	$0.100444\pi$
$90$	0	0
$91$	20.0361	2.10036
$92$	0	0
$93$	−3.11454	−0.322962
$94$	0	0
$95$	0.610713	0.0626578
$96$	0	0
$97$	−13.0636	−1.32641	−0.663203	−	0.748440i	$-0.730803\pi$
$97$	−13.0636	−1.32641	−0.663203	+	0.748440i	$0.730803\pi$
$98$	0	0
$99$	−1.99729	−0.200735
$100$	0	0
$101$	−6.34553	−0.631403	−0.315702	−	0.948858i	$-0.602240\pi$
$101$	−6.34553	−0.631403	−0.315702	+	0.948858i	$0.602240\pi$
$102$	0	0
$103$	19.7514	1.94616	0.973081	−	0.230462i	$-0.0740236\pi$
$103$	19.7514	1.94616	0.973081	+	0.230462i	$0.0740236\pi$
$104$	0	0
$105$	2.11809	0.206704
$106$	0	0
$107$	15.9003	1.53714	0.768572	−	0.639764i	$-0.220968\pi$
$107$	15.9003	1.53714	0.768572	+	0.639764i	$0.220968\pi$
$108$	0	0
$109$	5.22996	0.500939	0.250470	−	0.968124i	$-0.419415\pi$
$109$	5.22996	0.500939	0.250470	+	0.968124i	$0.419415\pi$
$110$	0	0
$111$	3.94623	0.374559
$112$	0	0
$113$	−8.36338	−0.786761	−0.393380	−	0.919376i	$-0.628694\pi$
$113$	−8.36338	−0.786761	−0.393380	+	0.919376i	$0.628694\pi$
$114$	0	0
$115$	−1.65847	−0.154653
$116$	0	0
$117$	−4.66762	−0.431521
$118$	0	0
$119$	−16.3300	−1.49697
$120$	0	0
$121$	−7.01082	−0.637347
$122$	0	0
$123$	2.87078	0.258850
$124$	0	0
$125$	4.81416	0.430592
$126$	0	0
$127$	−1.13400	−0.100626	−0.0503132	−	0.998733i	$-0.516022\pi$
$127$	−1.13400	−0.100626	−0.0503132	+	0.998733i	$0.516022\pi$
$128$	0	0
$129$	5.68421	0.500467
$130$	0	0
$131$	14.0545	1.22795	0.613974	−	0.789326i	$-0.289570\pi$
$131$	14.0545	1.22795	0.613974	+	0.789326i	$0.289570\pi$
$132$	0	0
$133$	5.31288	0.460685
$134$	0	0
$135$	−0.493430	−0.0424677
$136$	0	0
$137$	−14.6779	−1.25402	−0.627010	−	0.779011i	$-0.715722\pi$
$137$	−14.6779	−1.25402	−0.627010	+	0.779011i	$0.715722\pi$
$138$	0	0
$139$	−21.5709	−1.82962	−0.914812	−	0.403880i	$-0.867661\pi$
$139$	−21.5709	−1.82962	−0.914812	+	0.403880i	$0.867661\pi$
$140$	0	0
$141$	1.89602	0.159674
$142$	0	0
$143$	9.32260	0.779595
$144$	0	0
$145$	2.21401	0.183863
$146$	0	0
$147$	11.4263	0.942423
$148$	0	0
$149$	16.8655	1.38168	0.690839	−	0.723008i	$-0.257241\pi$
$149$	16.8655	1.38168	0.690839	+	0.723008i	$0.257241\pi$
$150$	0	0
$151$	15.8121	1.28677	0.643384	−	0.765543i	$-0.277530\pi$
$151$	15.8121	1.28677	0.643384	+	0.765543i	$0.277530\pi$
$152$	0	0
$153$	3.80423	0.307554
$154$	0	0
$155$	1.53681	0.123439
$156$	0	0
$157$	4.64457	0.370677	0.185338	−	0.982675i	$-0.440662\pi$
$157$	4.64457	0.370677	0.185338	+	0.982675i	$0.440662\pi$
$158$	0	0
$159$	−0.733412	−0.0581634
$160$	0	0
$161$	−14.4278	−1.13707
$162$	0	0
$163$	5.48381	0.429525	0.214762	−	0.976666i	$-0.431102\pi$
$163$	5.48381	0.429525	0.214762	+	0.976666i	$0.431102\pi$
$164$	0	0
$165$	0.985524	0.0767229
$166$	0	0
$167$	−12.9215	−0.999898	−0.499949	−	0.866055i	$-0.666648\pi$
$167$	−12.9215	−0.999898	−0.499949	+	0.866055i	$0.666648\pi$
$168$	0	0
$169$	8.78664	0.675895
$170$	0	0
$171$	−1.23769	−0.0946485
$172$	0	0
$173$	19.5378	1.48543	0.742717	−	0.669605i	$-0.233537\pi$
$173$	19.5378	1.48543	0.742717	+	0.669605i	$0.233537\pi$
$174$	0	0
$175$	20.4178	1.54344
$176$	0	0
$177$	6.68629	0.502573
$178$	0	0
$179$	−1.34288	−0.100371	−0.0501857	−	0.998740i	$-0.515981\pi$
$179$	−1.34288	−0.100371	−0.0501857	+	0.998740i	$0.515981\pi$
$180$	0	0
$181$	10.7737	0.800803	0.400401	−	0.916340i	$-0.368871\pi$
$181$	10.7737	0.800803	0.400401	+	0.916340i	$0.368871\pi$
$182$	0	0
$183$	−13.0294	−0.963160
$184$	0	0
$185$	−1.94719	−0.143160
$186$	0	0
$187$	−7.59817	−0.555633
$188$	0	0
$189$	−4.29258	−0.312239
$190$	0	0
$191$	9.53124	0.689656	0.344828	−	0.938666i	$-0.387937\pi$
$191$	9.53124	0.689656	0.344828	+	0.938666i	$0.387937\pi$
$192$	0	0
$193$	20.9876	1.51072	0.755360	−	0.655310i	$-0.227462\pi$
$193$	20.9876	1.51072	0.755360	+	0.655310i	$0.227462\pi$
$194$	0	0
$195$	2.30314	0.164931
$196$	0	0
$197$	−1.06445	−0.0758390	−0.0379195	−	0.999281i	$-0.512073\pi$
$197$	−1.06445	−0.0758390	−0.0379195	+	0.999281i	$0.512073\pi$
$198$	0	0
$199$	25.2124	1.78726	0.893630	−	0.448804i	$-0.148150\pi$
$199$	25.2124	1.78726	0.893630	+	0.448804i	$0.148150\pi$
$200$	0	0
$201$	9.04753	0.638163
$202$	0	0
$203$	19.2607	1.35184
$204$	0	0
$205$	−1.41653	−0.0989347
$206$	0	0
$207$	3.36111	0.233613
$208$	0	0
$209$	2.47203	0.170994
$210$	0	0
$211$	8.53574	0.587625	0.293812	−	0.955863i	$-0.405076\pi$
$211$	8.53574	0.587625	0.293812	+	0.955863i	$0.405076\pi$
$212$	0	0
$213$	2.61945	0.179482
$214$	0	0
$215$	−2.80476	−0.191283
$216$	0	0
$217$	13.3694	0.907575
$218$	0	0
$219$	−2.19011	−0.147994
$220$	0	0
$221$	−17.7567	−1.19444
$222$	0	0
$223$	5.59870	0.374917	0.187458	−	0.982273i	$-0.439975\pi$
$223$	5.59870	0.374917	0.187458	+	0.982273i	$0.439975\pi$
$224$	0	0
$225$	−4.75653	−0.317102
$226$	0	0
$227$	−4.56455	−0.302960	−0.151480	−	0.988460i	$-0.548404\pi$
$227$	−4.56455	−0.302960	−0.151480	+	0.988460i	$0.548404\pi$
$228$	0	0
$229$	0.319634	0.0211220	0.0105610	−	0.999944i	$-0.496638\pi$
$229$	0.319634	0.0211220	0.0105610	+	0.999944i	$0.496638\pi$
$230$	0	0
$231$	8.57355	0.564098
$232$	0	0
$233$	6.51711	0.426950	0.213475	−	0.976949i	$-0.431522\pi$
$233$	6.51711	0.426950	0.213475	+	0.976949i	$0.431522\pi$
$234$	0	0
$235$	−0.935554	−0.0610288
$236$	0	0
$237$	−8.73852	−0.567628
$238$	0	0
$239$	−17.4297	−1.12744	−0.563718	−	0.825967i	$-0.690629\pi$
$239$	−17.4297	−1.12744	−0.563718	+	0.825967i	$0.690629\pi$
$240$	0	0
$241$	14.1158	0.909281	0.454640	−	0.890675i	$-0.349768\pi$
$241$	14.1158	0.909281	0.454640	+	0.890675i	$0.349768\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−5.63806	−0.360202
$246$	0	0
$247$	5.77706	0.367585
$248$	0	0
$249$	9.59151	0.607837
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−6.71311	−0.422050
$254$	0	0
$255$	−1.87712	−0.117550
$256$	0	0
$257$	−22.4856	−1.40261	−0.701305	−	0.712861i	$-0.747399\pi$
$257$	−22.4856	−1.40261	−0.701305	+	0.712861i	$0.747399\pi$
$258$	0	0
$259$	−16.9395	−1.05257
$260$	0	0
$261$	−4.48698	−0.277737
$262$	0	0
$263$	−28.2275	−1.74058	−0.870291	−	0.492537i	$-0.836069\pi$
$263$	−28.2275	−1.74058	−0.870291	+	0.492537i	$0.836069\pi$
$264$	0	0
$265$	0.361887	0.0222306
$266$	0	0
$267$	17.9364	1.09769
$268$	0	0
$269$	−6.66577	−0.406419	−0.203210	−	0.979135i	$-0.565137\pi$
$269$	−6.66577	−0.406419	−0.203210	+	0.979135i	$0.565137\pi$
$270$	0	0
$271$	−23.2529	−1.41251	−0.706256	−	0.707956i	$-0.749617\pi$
$271$	−23.2529	−1.41251	−0.706256	+	0.707956i	$0.749617\pi$
$272$	0	0
$273$	20.0361	1.21264
$274$	0	0
$275$	9.50018	0.572882
$276$	0	0
$277$	21.3705	1.28403	0.642015	−	0.766692i	$-0.278099\pi$
$277$	21.3705	1.28403	0.642015	+	0.766692i	$0.278099\pi$
$278$	0	0
$279$	−3.11454	−0.186462
$280$	0	0
$281$	13.2701	0.791626	0.395813	−	0.918331i	$-0.370463\pi$
$281$	13.2701	0.791626	0.395813	+	0.918331i	$0.370463\pi$
$282$	0	0
$283$	13.3285	0.792298	0.396149	−	0.918186i	$-0.370346\pi$
$283$	13.3285	0.792298	0.396149	+	0.918186i	$0.370346\pi$
$284$	0	0
$285$	0.610713	0.0361755
$286$	0	0
$287$	−12.3231	−0.727408
$288$	0	0
$289$	−2.52781	−0.148695
$290$	0	0
$291$	−13.0636	−0.765800
$292$	0	0
$293$	3.87130	0.226164	0.113082	−	0.993586i	$-0.463928\pi$
$293$	3.87130	0.226164	0.113082	+	0.993586i	$0.463928\pi$
$294$	0	0
$295$	−3.29922	−0.192088
$296$	0	0
$297$	−1.99729	−0.115895
$298$	0	0
$299$	−15.6883	−0.907281
$300$	0	0
$301$	−24.3999	−1.40639
$302$	0	0
$303$	−6.34553	−0.364541
$304$	0	0
$305$	6.42909	0.368129
$306$	0	0
$307$	23.1709	1.32243	0.661217	−	0.750195i	$-0.270040\pi$
$307$	23.1709	1.32243	0.661217	+	0.750195i	$0.270040\pi$
$308$	0	0
$309$	19.7514	1.12362
$310$	0	0
$311$	−10.5663	−0.599160	−0.299580	−	0.954071i	$-0.596847\pi$
$311$	−10.5663	−0.599160	−0.299580	+	0.954071i	$0.596847\pi$
$312$	0	0
$313$	−10.0366	−0.567302	−0.283651	−	0.958928i	$-0.591546\pi$
$313$	−10.0366	−0.567302	−0.283651	+	0.958928i	$0.591546\pi$
$314$	0	0
$315$	2.11809	0.119341
$316$	0	0
$317$	−16.5926	−0.931933	−0.465966	−	0.884802i	$-0.654293\pi$
$317$	−16.5926	−0.931933	−0.465966	+	0.884802i	$0.654293\pi$
$318$	0	0
$319$	8.96181	0.501765
$320$	0	0
$321$	15.9003	0.887470
$322$	0	0
$323$	−4.70846	−0.261986
$324$	0	0
$325$	22.2016	1.23153
$326$	0	0
$327$	5.22996	0.289217
$328$	0	0
$329$	−8.13884	−0.448709
$330$	0	0
$331$	12.6881	0.697403	0.348702	−	0.937234i	$-0.386623\pi$
$331$	12.6881	0.697403	0.348702	+	0.937234i	$0.386623\pi$
$332$	0	0
$333$	3.94623	0.216252
$334$	0	0
$335$	−4.46432	−0.243912
$336$	0	0
$337$	−6.74151	−0.367234	−0.183617	−	0.982998i	$-0.558781\pi$
$337$	−6.74151	−0.367234	−0.183617	+	0.982998i	$0.558781\pi$
$338$	0	0
$339$	−8.36338	−0.454237
$340$	0	0
$341$	6.22064	0.336867
$342$	0	0
$343$	−19.0001	−1.02591
$344$	0	0
$345$	−1.65847	−0.0892890
$346$	0	0
$347$	11.6214	0.623871	0.311936	−	0.950103i	$-0.399023\pi$
$347$	11.6214	0.623871	0.311936	+	0.950103i	$0.399023\pi$
$348$	0	0
$349$	−25.1191	−1.34460	−0.672298	−	0.740281i	$-0.734693\pi$
$349$	−25.1191	−1.34460	−0.672298	+	0.740281i	$0.734693\pi$
$350$	0	0
$351$	−4.66762	−0.249139
$352$	0	0
$353$	24.2922	1.29294	0.646472	−	0.762938i	$-0.276244\pi$
$353$	24.2922	1.29294	0.646472	+	0.762938i	$0.276244\pi$
$354$	0	0
$355$	−1.29251	−0.0685995
$356$	0	0
$357$	−16.3300	−0.864274
$358$	0	0
$359$	−2.19737	−0.115973	−0.0579864	−	0.998317i	$-0.518468\pi$
$359$	−2.19737	−0.115973	−0.0579864	+	0.998317i	$0.518468\pi$
$360$	0	0
$361$	−17.4681	−0.919375
$362$	0	0
$363$	−7.01082	−0.367973
$364$	0	0
$365$	1.08067	0.0565646
$366$	0	0
$367$	−3.90149	−0.203656	−0.101828	−	0.994802i	$-0.532469\pi$
$367$	−3.90149	−0.203656	−0.101828	+	0.994802i	$0.532469\pi$
$368$	0	0
$369$	2.87078	0.149447
$370$	0	0
$371$	3.14823	0.163448
$372$	0	0
$373$	−6.89020	−0.356761	−0.178381	−	0.983962i	$-0.557086\pi$
$373$	−6.89020	−0.356761	−0.178381	+	0.983962i	$0.557086\pi$
$374$	0	0
$375$	4.81416	0.248602
$376$	0	0
$377$	20.9435	1.07864
$378$	0	0
$379$	−3.65496	−0.187743	−0.0938714	−	0.995584i	$-0.529924\pi$
$379$	−3.65496	−0.187743	−0.0938714	+	0.995584i	$0.529924\pi$
$380$	0	0
$381$	−1.13400	−0.0580967
$382$	0	0
$383$	−8.24314	−0.421205	−0.210602	−	0.977572i	$-0.567543\pi$
$383$	−8.24314	−0.421205	−0.210602	+	0.977572i	$0.567543\pi$
$384$	0	0
$385$	−4.23044	−0.215603
$386$	0	0
$387$	5.68421	0.288945
$388$	0	0
$389$	9.82091	0.497940	0.248970	−	0.968511i	$-0.419908\pi$
$389$	9.82091	0.497940	0.248970	+	0.968511i	$0.419908\pi$
$390$	0	0
$391$	12.7864	0.646637
$392$	0	0
$393$	14.0545	0.708957
$394$	0	0
$395$	4.31185	0.216952
$396$	0	0
$397$	−10.0640	−0.505097	−0.252548	−	0.967584i	$-0.581269\pi$
$397$	−10.0640	−0.505097	−0.252548	+	0.967584i	$0.581269\pi$
$398$	0	0
$399$	5.31288	0.265977
$400$	0	0
$401$	24.5321	1.22508	0.612538	−	0.790441i	$-0.290149\pi$
$401$	24.5321	1.22508	0.612538	+	0.790441i	$0.290149\pi$
$402$	0	0
$403$	14.5375	0.724163
$404$	0	0
$405$	−0.493430	−0.0245187
$406$	0	0
$407$	−7.88177	−0.390685
$408$	0	0
$409$	8.05897	0.398490	0.199245	−	0.979950i	$-0.436151\pi$
$409$	8.05897	0.398490	0.199245	+	0.979950i	$0.436151\pi$
$410$	0	0
$411$	−14.6779	−0.724009
$412$	0	0
$413$	−28.7015	−1.41231
$414$	0	0
$415$	−4.73274	−0.232321
$416$	0	0
$417$	−21.5709	−1.05633
$418$	0	0
$419$	33.9827	1.66016	0.830081	−	0.557643i	$-0.188294\pi$
$419$	33.9827	1.66016	0.830081	+	0.557643i	$0.188294\pi$
$420$	0	0
$421$	−0.636441	−0.0310183	−0.0155091	−	0.999880i	$-0.504937\pi$
$421$	−0.636441	−0.0310183	−0.0155091	+	0.999880i	$0.504937\pi$
$422$	0	0
$423$	1.89602	0.0921878
$424$	0	0
$425$	−18.0949	−0.877733
$426$	0	0
$427$	55.9297	2.70663
$428$	0	0
$429$	9.32260	0.450099
$430$	0	0
$431$	−20.7641	−1.00017	−0.500085	−	0.865977i	$-0.666698\pi$
$431$	−20.7641	−1.00017	−0.500085	+	0.865977i	$0.666698\pi$
$432$	0	0
$433$	−17.9140	−0.860893	−0.430447	−	0.902616i	$-0.641644\pi$
$433$	−17.9140	−0.860893	−0.430447	+	0.902616i	$0.641644\pi$
$434$	0	0
$435$	2.21401	0.106154
$436$	0	0
$437$	−4.16000	−0.199000
$438$	0	0
$439$	29.2346	1.39529	0.697646	−	0.716443i	$-0.254231\pi$
$439$	29.2346	1.39529	0.697646	+	0.716443i	$0.254231\pi$
$440$	0	0
$441$	11.4263	0.544108
$442$	0	0
$443$	−8.41823	−0.399962	−0.199981	−	0.979800i	$-0.564088\pi$
$443$	−8.41823	−0.399962	−0.199981	+	0.979800i	$0.564088\pi$
$444$	0	0
$445$	−8.85033	−0.419546
$446$	0	0
$447$	16.8655	0.797713
$448$	0	0
$449$	3.50474	0.165399	0.0826995	−	0.996575i	$-0.473646\pi$
$449$	3.50474	0.165399	0.0826995	+	0.996575i	$0.473646\pi$
$450$	0	0
$451$	−5.73379	−0.269994
$452$	0	0
$453$	15.8121	0.742916
$454$	0	0
$455$	−9.88642	−0.463483
$456$	0	0
$457$	−17.0704	−0.798518	−0.399259	−	0.916838i	$-0.630733\pi$
$457$	−17.0704	−0.798518	−0.399259	+	0.916838i	$0.630733\pi$
$458$	0	0
$459$	3.80423	0.177566
$460$	0	0
$461$	−13.9846	−0.651328	−0.325664	−	0.945485i	$-0.605588\pi$
$461$	−13.9846	−0.651328	−0.325664	+	0.945485i	$0.605588\pi$
$462$	0	0
$463$	19.2724	0.895662	0.447831	−	0.894118i	$-0.352197\pi$
$463$	19.2724	0.895662	0.447831	+	0.894118i	$0.352197\pi$
$464$	0	0
$465$	1.53681	0.0712676
$466$	0	0
$467$	−24.9610	−1.15506	−0.577528	−	0.816371i	$-0.695983\pi$
$467$	−24.9610	−1.15506	−0.577528	+	0.816371i	$0.695983\pi$
$468$	0	0
$469$	−38.8373	−1.79334
$470$	0	0
$471$	4.64457	0.214010
$472$	0	0
$473$	−11.3530	−0.522013
$474$	0	0
$475$	5.88710	0.270119
$476$	0	0
$477$	−0.733412	−0.0335806
$478$	0	0
$479$	36.4313	1.66459	0.832295	−	0.554333i	$-0.187027\pi$
$479$	36.4313	1.66459	0.832295	+	0.554333i	$0.187027\pi$
$480$	0	0
$481$	−18.4195	−0.839856
$482$	0	0
$483$	−14.4278	−0.656489
$484$	0	0
$485$	6.44596	0.292696
$486$	0	0
$487$	29.1787	1.32221	0.661107	−	0.750292i	$-0.270087\pi$
$487$	29.1787	1.32221	0.661107	+	0.750292i	$0.270087\pi$
$488$	0	0
$489$	5.48381	0.247986
$490$	0	0
$491$	−33.9828	−1.53362	−0.766811	−	0.641874i	$-0.778157\pi$
$491$	−33.9828	−1.53362	−0.766811	+	0.641874i	$0.778157\pi$
$492$	0	0
$493$	−17.0695	−0.768772
$494$	0	0
$495$	0.985524	0.0442960
$496$	0	0
$497$	−11.2442	−0.504371
$498$	0	0
$499$	16.8808	0.755688	0.377844	−	0.925869i	$-0.376666\pi$
$499$	16.8808	0.755688	0.377844	+	0.925869i	$0.376666\pi$
$500$	0	0
$501$	−12.9215	−0.577291
$502$	0	0
$503$	−3.84479	−0.171431	−0.0857154	−	0.996320i	$-0.527318\pi$
$503$	−3.84479	−0.171431	−0.0857154	+	0.996320i	$0.527318\pi$
$504$	0	0
$505$	3.13107	0.139331
$506$	0	0
$507$	8.78664	0.390228
$508$	0	0
$509$	−8.54376	−0.378695	−0.189348	−	0.981910i	$-0.560637\pi$
$509$	−8.54376	−0.378695	−0.189348	+	0.981910i	$0.560637\pi$
$510$	0	0
$511$	9.40123	0.415886
$512$	0	0
$513$	−1.23769	−0.0546453
$514$	0	0
$515$	−9.74593	−0.429457
$516$	0	0
$517$	−3.78691	−0.166548
$518$	0	0
$519$	19.5378	0.857616
$520$	0	0
$521$	18.4464	0.808151	0.404075	−	0.914726i	$-0.367593\pi$
$521$	18.4464	0.808151	0.404075	+	0.914726i	$0.367593\pi$
$522$	0	0
$523$	−11.8777	−0.519378	−0.259689	−	0.965692i	$-0.583620\pi$
$523$	−11.8777	−0.519378	−0.259689	+	0.965692i	$0.583620\pi$
$524$	0	0
$525$	20.4178	0.891105
$526$	0	0
$527$	−11.8484	−0.516126
$528$	0	0
$529$	−11.7030	−0.508825
$530$	0	0
$531$	6.68629	0.290160
$532$	0	0
$533$	−13.3997	−0.580406
$534$	0	0
$535$	−7.84570	−0.339199
$536$	0	0
$537$	−1.34288	−0.0579495
$538$	0	0
$539$	−22.8216	−0.982996
$540$	0	0
$541$	25.9687	1.11648	0.558242	−	0.829678i	$-0.311476\pi$
$541$	25.9687	1.11648	0.558242	+	0.829678i	$0.311476\pi$
$542$	0	0
$543$	10.7737	0.462344
$544$	0	0
$545$	−2.58062	−0.110542
$546$	0	0
$547$	2.28631	0.0977554	0.0488777	−	0.998805i	$-0.484436\pi$
$547$	2.28631	0.0977554	0.0488777	+	0.998805i	$0.484436\pi$
$548$	0	0
$549$	−13.0294	−0.556081
$550$	0	0
$551$	5.55348	0.236586
$552$	0	0
$553$	37.5108	1.59512
$554$	0	0
$555$	−1.94719	−0.0826535
$556$	0	0
$557$	−27.1349	−1.14974	−0.574870	−	0.818245i	$-0.694947\pi$
$557$	−27.1349	−1.14974	−0.574870	+	0.818245i	$0.694947\pi$
$558$	0	0
$559$	−26.5317	−1.12217
$560$	0	0
$561$	−7.59817	−0.320795
$562$	0	0
$563$	−17.5460	−0.739474	−0.369737	−	0.929136i	$-0.620552\pi$
$563$	−17.5460	−0.739474	−0.369737	+	0.929136i	$0.620552\pi$
$564$	0	0
$565$	4.12674	0.173613
$566$	0	0
$567$	−4.29258	−0.180272
$568$	0	0
$569$	−17.8736	−0.749301	−0.374650	−	0.927166i	$-0.622237\pi$
$569$	−17.8736	−0.749301	−0.374650	+	0.927166i	$0.622237\pi$
$570$	0	0
$571$	−5.25321	−0.219840	−0.109920	−	0.993940i	$-0.535059\pi$
$571$	−5.25321	−0.219840	−0.109920	+	0.993940i	$0.535059\pi$
$572$	0	0
$573$	9.53124	0.398173
$574$	0	0
$575$	−15.9872	−0.666712
$576$	0	0
$577$	29.4045	1.22413	0.612063	−	0.790809i	$-0.290340\pi$
$577$	29.4045	1.22413	0.612063	+	0.790809i	$0.290340\pi$
$578$	0	0
$579$	20.9876	0.872215
$580$	0	0
$581$	−41.1724	−1.70812
$582$	0	0
$583$	1.46484	0.0606674
$584$	0	0
$585$	2.30314	0.0952231
$586$	0	0
$587$	35.6607	1.47187	0.735937	−	0.677051i	$-0.236742\pi$
$587$	35.6607	1.47187	0.735937	+	0.677051i	$0.236742\pi$
$588$	0	0
$589$	3.85483	0.158835
$590$	0	0
$591$	−1.06445	−0.0437857
$592$	0	0
$593$	21.3565	0.877006	0.438503	−	0.898730i	$-0.355509\pi$
$593$	21.3565	0.877006	0.438503	+	0.898730i	$0.355509\pi$
$594$	0	0
$595$	8.05770	0.330333
$596$	0	0
$597$	25.2124	1.03188
$598$	0	0
$599$	−4.90053	−0.200230	−0.100115	−	0.994976i	$-0.531921\pi$
$599$	−4.90053	−0.200230	−0.100115	+	0.994976i	$0.531921\pi$
$600$	0	0
$601$	23.9343	0.976301	0.488150	−	0.872759i	$-0.337672\pi$
$601$	23.9343	0.976301	0.488150	+	0.872759i	$0.337672\pi$
$602$	0	0
$603$	9.04753	0.368444
$604$	0	0
$605$	3.45935	0.140642
$606$	0	0
$607$	−43.5564	−1.76790	−0.883950	−	0.467581i	$-0.845126\pi$
$607$	−43.5564	−1.76790	−0.883950	+	0.467581i	$0.845126\pi$
$608$	0	0
$609$	19.2607	0.780484
$610$	0	0
$611$	−8.84991	−0.358029
$612$	0	0
$613$	−35.1442	−1.41946	−0.709731	−	0.704473i	$-0.751183\pi$
$613$	−35.1442	−1.41946	−0.709731	+	0.704473i	$0.751183\pi$
$614$	0	0
$615$	−1.41653	−0.0571200
$616$	0	0
$617$	8.38519	0.337575	0.168788	−	0.985652i	$-0.446015\pi$
$617$	8.38519	0.337575	0.168788	+	0.985652i	$0.446015\pi$
$618$	0	0
$619$	27.7476	1.11527	0.557635	−	0.830087i	$-0.311709\pi$
$619$	27.7476	1.11527	0.557635	+	0.830087i	$0.311709\pi$
$620$	0	0
$621$	3.36111	0.134877
$622$	0	0
$623$	−76.9933	−3.08467
$624$	0	0
$625$	21.4072	0.856287
$626$	0	0
$627$	2.47203	0.0987233
$628$	0	0
$629$	15.0124	0.598582
$630$	0	0
$631$	25.3150	1.00777	0.503887	−	0.863770i	$-0.331903\pi$
$631$	25.3150	1.00777	0.503887	+	0.863770i	$0.331903\pi$
$632$	0	0
$633$	8.53574	0.339265
$634$	0	0
$635$	0.559550	0.0222051
$636$	0	0
$637$	−53.3334	−2.11315
$638$	0	0
$639$	2.61945	0.103624
$640$	0	0
$641$	−25.2658	−0.997938	−0.498969	−	0.866620i	$-0.666288\pi$
$641$	−25.2658	−0.997938	−0.498969	+	0.866620i	$0.666288\pi$
$642$	0	0
$643$	−22.6105	−0.891670	−0.445835	−	0.895115i	$-0.647093\pi$
$643$	−22.6105	−0.891670	−0.445835	+	0.895115i	$0.647093\pi$
$644$	0	0
$645$	−2.80476	−0.110437
$646$	0	0
$647$	45.8168	1.80124	0.900622	−	0.434603i	$-0.143111\pi$
$647$	45.8168	1.80124	0.900622	+	0.434603i	$0.143111\pi$
$648$	0	0
$649$	−13.3545	−0.524209
$650$	0	0
$651$	13.3694	0.523988
$652$	0	0
$653$	21.9502	0.858979	0.429489	−	0.903072i	$-0.358694\pi$
$653$	21.9502	0.858979	0.429489	+	0.903072i	$0.358694\pi$
$654$	0	0
$655$	−6.93492	−0.270970
$656$	0	0
$657$	−2.19011	−0.0854443
$658$	0	0
$659$	−34.5995	−1.34780	−0.673902	−	0.738820i	$-0.735383\pi$
$659$	−34.5995	−1.34780	−0.673902	+	0.738820i	$0.735383\pi$
$660$	0	0
$661$	−6.18566	−0.240594	−0.120297	−	0.992738i	$-0.538385\pi$
$661$	−6.18566	−0.240594	−0.120297	+	0.992738i	$0.538385\pi$
$662$	0	0
$663$	−17.7567	−0.689613
$664$	0	0
$665$	−2.62153	−0.101659
$666$	0	0
$667$	−15.0812	−0.583946
$668$	0	0
$669$	5.59870	0.216458
$670$	0	0
$671$	26.0235	1.00463
$672$	0	0
$673$	−36.0414	−1.38930	−0.694648	−	0.719350i	$-0.744440\pi$
$673$	−36.0414	−1.38930	−0.694648	+	0.719350i	$0.744440\pi$
$674$	0	0
$675$	−4.75653	−0.183079
$676$	0	0
$677$	−31.1490	−1.19715	−0.598577	−	0.801065i	$-0.704267\pi$
$677$	−31.1490	−1.19715	−0.598577	+	0.801065i	$0.704267\pi$
$678$	0	0
$679$	56.0765	2.15202
$680$	0	0
$681$	−4.56455	−0.174914
$682$	0	0
$683$	−0.260366	−0.00996262	−0.00498131	−	0.999988i	$-0.501586\pi$
$683$	−0.260366	−0.00996262	−0.00498131	+	0.999988i	$0.501586\pi$
$684$	0	0
$685$	7.24253	0.276723
$686$	0	0
$687$	0.319634	0.0121948
$688$	0	0
$689$	3.42329	0.130417
$690$	0	0
$691$	5.35251	0.203619	0.101809	−	0.994804i	$-0.467537\pi$
$691$	5.35251	0.203619	0.101809	+	0.994804i	$0.467537\pi$
$692$	0	0
$693$	8.57355	0.325682
$694$	0	0
$695$	10.6437	0.403740
$696$	0	0
$697$	10.9211	0.413667
$698$	0	0
$699$	6.51711	0.246500
$700$	0	0
$701$	29.2452	1.10458	0.552288	−	0.833653i	$-0.313755\pi$
$701$	29.2452	1.10458	0.552288	+	0.833653i	$0.313755\pi$
$702$	0	0
$703$	−4.88420	−0.184211
$704$	0	0
$705$	−0.935554	−0.0352350
$706$	0	0
$707$	27.2387	1.02442
$708$	0	0
$709$	1.10287	0.0414192	0.0207096	−	0.999786i	$-0.493407\pi$
$709$	1.10287	0.0414192	0.0207096	+	0.999786i	$0.493407\pi$
$710$	0	0
$711$	−8.73852	−0.327720
$712$	0	0
$713$	−10.4683	−0.392041
$714$	0	0
$715$	−4.60005	−0.172032
$716$	0	0
$717$	−17.4297	−0.650925
$718$	0	0
$719$	−15.1000	−0.563133	−0.281567	−	0.959542i	$-0.590854\pi$
$719$	−15.1000	−0.563133	−0.281567	+	0.959542i	$0.590854\pi$
$720$	0	0
$721$	−84.7845	−3.15754
$722$	0	0
$723$	14.1158	0.524973
$724$	0	0
$725$	21.3424	0.792638
$726$	0	0
$727$	28.2226	1.04672	0.523359	−	0.852112i	$-0.324679\pi$
$727$	28.2226	1.04672	0.523359	+	0.852112i	$0.324679\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	21.6241	0.799795
$732$	0	0
$733$	−11.4717	−0.423716	−0.211858	−	0.977301i	$-0.567951\pi$
$733$	−11.4717	−0.423716	−0.211858	+	0.977301i	$0.567951\pi$
$734$	0	0
$735$	−5.63806	−0.207963
$736$	0	0
$737$	−18.0706	−0.665638
$738$	0	0
$739$	25.6878	0.944939	0.472470	−	0.881347i	$-0.343363\pi$
$739$	25.6878	0.944939	0.472470	+	0.881347i	$0.343363\pi$
$740$	0	0
$741$	5.77706	0.212226
$742$	0	0
$743$	−13.5063	−0.495498	−0.247749	−	0.968824i	$-0.579691\pi$
$743$	−13.5063	−0.495498	−0.247749	+	0.968824i	$0.579691\pi$
$744$	0	0
$745$	−8.32196	−0.304893
$746$	0	0
$747$	9.59151	0.350935
$748$	0	0
$749$	−68.2535	−2.49393
$750$	0	0
$751$	43.8221	1.59909	0.799546	−	0.600605i	$-0.205074\pi$
$751$	43.8221	1.59909	0.799546	+	0.600605i	$0.205074\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	−7.80215	−0.283949
$756$	0	0
$757$	24.9747	0.907720	0.453860	−	0.891073i	$-0.350047\pi$
$757$	24.9747	0.907720	0.453860	+	0.891073i	$0.350047\pi$
$758$	0	0
$759$	−6.71311	−0.243671
$760$	0	0
$761$	−34.4559	−1.24903	−0.624513	−	0.781015i	$-0.714702\pi$
$761$	−34.4559	−1.24903	−0.624513	+	0.781015i	$0.714702\pi$
$762$	0	0
$763$	−22.4500	−0.812746
$764$	0	0
$765$	−1.87712	−0.0678675
$766$	0	0
$767$	−31.2090	−1.12689
$768$	0	0
$769$	30.8996	1.11427	0.557135	−	0.830422i	$-0.311901\pi$
$769$	30.8996	1.11427	0.557135	+	0.830422i	$0.311901\pi$
$770$	0	0
$771$	−22.4856	−0.809797
$772$	0	0
$773$	−37.5561	−1.35080	−0.675400	−	0.737451i	$-0.736029\pi$
$773$	−37.5561	−1.35080	−0.675400	+	0.737451i	$0.736029\pi$
$774$	0	0
$775$	14.8144	0.532148
$776$	0	0
$777$	−16.9395	−0.607702
$778$	0	0
$779$	−3.55314	−0.127304
$780$	0	0
$781$	−5.23181	−0.187209
$782$	0	0
$783$	−4.48698	−0.160351
$784$	0	0
$785$	−2.29177	−0.0817967
$786$	0	0
$787$	9.11299	0.324843	0.162421	−	0.986721i	$-0.448070\pi$
$787$	9.11299	0.324843	0.162421	+	0.986721i	$0.448070\pi$
$788$	0	0
$789$	−28.2275	−1.00493
$790$	0	0
$791$	35.9005	1.27648
$792$	0	0
$793$	60.8162	2.15965
$794$	0	0
$795$	0.361887	0.0128348
$796$	0	0
$797$	13.0375	0.461813	0.230906	−	0.972976i	$-0.425831\pi$
$797$	13.0375	0.461813	0.230906	+	0.972976i	$0.425831\pi$
$798$	0	0
$799$	7.21292	0.255175
$800$	0	0
$801$	17.9364	0.633750
$802$	0	0
$803$	4.37429	0.154365
$804$	0	0
$805$	7.11912	0.250916
$806$	0	0
$807$	−6.66577	−0.234646
$808$	0	0
$809$	−23.6861	−0.832759	−0.416380	−	0.909191i	$-0.636701\pi$
$809$	−23.6861	−0.832759	−0.416380	+	0.909191i	$0.636701\pi$
$810$	0	0
$811$	−38.6963	−1.35881	−0.679406	−	0.733763i	$-0.737762\pi$
$811$	−38.6963	−1.35881	−0.679406	+	0.733763i	$0.737762\pi$
$812$	0	0
$813$	−23.2529	−0.815514
$814$	0	0
$815$	−2.70587	−0.0947826
$816$	0	0
$817$	−7.03529	−0.246134
$818$	0	0
$819$	20.0361	0.700119
$820$	0	0
$821$	29.4503	1.02782	0.513911	−	0.857843i	$-0.328196\pi$
$821$	29.4503	1.02782	0.513911	+	0.857843i	$0.328196\pi$
$822$	0	0
$823$	48.9685	1.70694	0.853468	−	0.521145i	$-0.174495\pi$
$823$	48.9685	1.70694	0.853468	+	0.521145i	$0.174495\pi$
$824$	0	0
$825$	9.50018	0.330754
$826$	0	0
$827$	−23.7773	−0.826816	−0.413408	−	0.910546i	$-0.635662\pi$
$827$	−23.7773	−0.826816	−0.413408	+	0.910546i	$0.635662\pi$
$828$	0	0
$829$	−25.9555	−0.901471	−0.450735	−	0.892658i	$-0.648838\pi$
$829$	−25.9555	−0.901471	−0.450735	+	0.892658i	$0.648838\pi$
$830$	0	0
$831$	21.3705	0.741335
$832$	0	0
$833$	43.4682	1.50608
$834$	0	0
$835$	6.37587	0.220646
$836$	0	0
$837$	−3.11454	−0.107654
$838$	0	0
$839$	−27.4901	−0.949065	−0.474533	−	0.880238i	$-0.657383\pi$
$839$	−27.4901	−0.949065	−0.474533	+	0.880238i	$0.657383\pi$
$840$	0	0
$841$	−8.86704	−0.305760
$842$	0	0
$843$	13.2701	0.457046
$844$	0	0
$845$	−4.33559	−0.149149
$846$	0	0
$847$	30.0945	1.03406
$848$	0	0
$849$	13.3285	0.457434
$850$	0	0
$851$	13.2637	0.454673
$852$	0	0
$853$	−24.8694	−0.851513	−0.425756	−	0.904838i	$-0.639992\pi$
$853$	−24.8694	−0.851513	−0.425756	+	0.904838i	$0.639992\pi$
$854$	0	0
$855$	0.610713	0.0208859
$856$	0	0
$857$	−2.91568	−0.0995977	−0.0497989	−	0.998759i	$-0.515858\pi$
$857$	−2.91568	−0.0995977	−0.0497989	+	0.998759i	$0.515858\pi$
$858$	0	0
$859$	49.2403	1.68006	0.840029	−	0.542541i	$-0.182538\pi$
$859$	49.2403	1.68006	0.840029	+	0.542541i	$0.182538\pi$
$860$	0	0
$861$	−12.3231	−0.419969
$862$	0	0
$863$	−9.69768	−0.330113	−0.165056	−	0.986284i	$-0.552781\pi$
$863$	−9.69768	−0.330113	−0.165056	+	0.986284i	$0.552781\pi$
$864$	0	0
$865$	−9.64055	−0.327789
$866$	0	0
$867$	−2.52781	−0.0858490
$868$	0	0
$869$	17.4534	0.592065
$870$	0	0
$871$	−42.2304	−1.43092
$872$	0	0
$873$	−13.0636	−0.442135
$874$	0	0
$875$	−20.6652	−0.698611
$876$	0	0
$877$	22.5508	0.761487	0.380743	−	0.924681i	$-0.375668\pi$
$877$	22.5508	0.761487	0.380743	+	0.924681i	$0.375668\pi$
$878$	0	0
$879$	3.87130	0.130576
$880$	0	0
$881$	17.4747	0.588738	0.294369	−	0.955692i	$-0.404891\pi$
$881$	17.4747	0.588738	0.294369	+	0.955692i	$0.404891\pi$
$882$	0	0
$883$	−30.1607	−1.01499	−0.507495	−	0.861655i	$-0.669428\pi$
$883$	−30.1607	−1.01499	−0.507495	+	0.861655i	$0.669428\pi$
$884$	0	0
$885$	−3.29922	−0.110902
$886$	0	0
$887$	4.28707	0.143946	0.0719729	−	0.997407i	$-0.477070\pi$
$887$	4.28707	0.143946	0.0719729	+	0.997407i	$0.477070\pi$
$888$	0	0
$889$	4.86780	0.163261
$890$	0	0
$891$	−1.99729	−0.0669118
$892$	0	0
$893$	−2.34669	−0.0785289
$894$	0	0
$895$	0.662616	0.0221488
$896$	0	0
$897$	−15.6883	−0.523819
$898$	0	0
$899$	13.9749	0.466088
$900$	0	0
$901$	−2.79007	−0.0929507
$902$	0	0
$903$	−24.3999	−0.811979
$904$	0	0
$905$	−5.31606	−0.176712
$906$	0	0
$907$	35.5897	1.18174	0.590869	−	0.806768i	$-0.298785\pi$
$907$	35.5897	1.18174	0.590869	+	0.806768i	$0.298785\pi$
$908$	0	0
$909$	−6.34553	−0.210468
$910$	0	0
$911$	16.6103	0.550323	0.275161	−	0.961398i	$-0.411269\pi$
$911$	16.6103	0.550323	0.275161	+	0.961398i	$0.411269\pi$
$912$	0	0
$913$	−19.1571	−0.634006
$914$	0	0
$915$	6.42909	0.212539
$916$	0	0
$917$	−60.3302	−1.99228
$918$	0	0
$919$	54.0968	1.78449	0.892244	−	0.451553i	$-0.149130\pi$
$919$	54.0968	1.78449	0.892244	+	0.451553i	$0.149130\pi$
$920$	0	0
$921$	23.1709	0.763507
$922$	0	0
$923$	−12.2266	−0.402443
$924$	0	0
$925$	−18.7703	−0.617165
$926$	0	0
$927$	19.7514	0.648721
$928$	0	0
$929$	27.0805	0.888482	0.444241	−	0.895907i	$-0.353473\pi$
$929$	27.0805	0.888482	0.444241	+	0.895907i	$0.353473\pi$
$930$	0	0
$931$	−14.1422	−0.463491
$932$	0	0
$933$	−10.5663	−0.345925
$934$	0	0
$935$	3.74916	0.122611
$936$	0	0
$937$	20.3163	0.663704	0.331852	−	0.943331i	$-0.392327\pi$
$937$	20.3163	0.663704	0.331852	+	0.943331i	$0.392327\pi$
$938$	0	0
$939$	−10.0366	−0.327532
$940$	0	0
$941$	32.2487	1.05128	0.525640	−	0.850707i	$-0.323826\pi$
$941$	32.2487	1.05128	0.525640	+	0.850707i	$0.323826\pi$
$942$	0	0
$943$	9.64900	0.314215
$944$	0	0
$945$	2.11809	0.0689014
$946$	0	0
$947$	39.9004	1.29659	0.648294	−	0.761390i	$-0.275483\pi$
$947$	39.9004	1.29659	0.648294	+	0.761390i	$0.275483\pi$
$948$	0	0
$949$	10.2226	0.331839
$950$	0	0
$951$	−16.5926	−0.538052
$952$	0	0
$953$	−3.87198	−0.125426	−0.0627128	−	0.998032i	$-0.519975\pi$
$953$	−3.87198	−0.125426	−0.0627128	+	0.998032i	$0.519975\pi$
$954$	0	0
$955$	−4.70300	−0.152185
$956$	0	0
$957$	8.96181	0.289694
$958$	0	0
$959$	63.0062	2.03458
$960$	0	0
$961$	−21.2997	−0.687086
$962$	0	0
$963$	15.9003	0.512381
$964$	0	0
$965$	−10.3559	−0.333368
$966$	0	0
$967$	−15.9427	−0.512681	−0.256341	−	0.966587i	$-0.582517\pi$
$967$	−15.9427	−0.512681	−0.256341	+	0.966587i	$0.582517\pi$
$968$	0	0
$969$	−4.70846	−0.151257
$970$	0	0
$971$	26.3927	0.846983	0.423491	−	0.905900i	$-0.360804\pi$
$971$	26.3927	0.846983	0.423491	+	0.905900i	$0.360804\pi$
$972$	0	0
$973$	92.5951	2.96846
$974$	0	0
$975$	22.2016	0.711022
$976$	0	0
$977$	−33.3034	−1.06547	−0.532735	−	0.846282i	$-0.678836\pi$
$977$	−33.3034	−1.06547	−0.532735	+	0.846282i	$0.678836\pi$
$978$	0	0
$979$	−35.8242	−1.14495
$980$	0	0
$981$	5.22996	0.166980
$982$	0	0
$983$	−0.590572	−0.0188363	−0.00941817	−	0.999956i	$-0.502998\pi$
$983$	−0.590572	−0.0188363	−0.00941817	+	0.999956i	$0.502998\pi$
$984$	0	0
$985$	0.525232	0.0167353
$986$	0	0
$987$	−8.13884	−0.259062
$988$	0	0
$989$	19.1052	0.607511
$990$	0	0
$991$	−0.965504	−0.0306703	−0.0153351	−	0.999882i	$-0.504882\pi$
$991$	−0.965504	−0.0306703	−0.0153351	+	0.999882i	$0.504882\pi$
$992$	0	0
$993$	12.6881	0.402646
$994$	0	0
$995$	−12.4406	−0.394392
$996$	0	0
$997$	−14.9313	−0.472880	−0.236440	−	0.971646i	$-0.575981\pi$
$997$	−14.9313	−0.472880	−0.236440	+	0.971646i	$0.575981\pi$
$998$	0	0
$999$	3.94623	0.124853

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.q.1.8	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.q.1.8	✓	18	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.493430 q^{5} -4.29258 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.493430 q^{5} -4.29258 q^{7} +1.00000 q^{9} -1.99729 q^{11} -4.66762 q^{13} -0.493430 q^{15} +3.80423 q^{17} -1.23769 q^{19} -4.29258 q^{21} +3.36111 q^{23} -4.75653 q^{25} +1.00000 q^{27} -4.48698 q^{29} -3.11454 q^{31} -1.99729 q^{33} +2.11809 q^{35} +3.94623 q^{37} -4.66762 q^{39} +2.87078 q^{41} +5.68421 q^{43} -0.493430 q^{45} +1.89602 q^{47} +11.4263 q^{49} +3.80423 q^{51} -0.733412 q^{53} +0.985524 q^{55} -1.23769 q^{57} +6.68629 q^{59} -13.0294 q^{61} -4.29258 q^{63} +2.30314 q^{65} +9.04753 q^{67} +3.36111 q^{69} +2.61945 q^{71} -2.19011 q^{73} -4.75653 q^{75} +8.57355 q^{77} -8.73852 q^{79} +1.00000 q^{81} +9.59151 q^{83} -1.87712 q^{85} -4.48698 q^{87} +17.9364 q^{89} +20.0361 q^{91} -3.11454 q^{93} +0.610713 q^{95} -13.0636 q^{97} -1.99729 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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