LMFDB - Embedded newform 6240.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -4.27492 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -4.27492 q^{7} +1.00000 q^{9} +2.27492 q^{11} +1.00000 q^{13} +1.00000 q^{15} -6.27492 q^{17} +2.00000 q^{19} +4.27492 q^{21} +0.274917 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.00000 q^{31} -2.27492 q^{33} +4.27492 q^{35} +8.27492 q^{37} -1.00000 q^{39} +8.27492 q^{41} -8.54983 q^{43} -1.00000 q^{45} +4.00000 q^{47} +11.2749 q^{49} +6.27492 q^{51} +8.27492 q^{53} -2.27492 q^{55} -2.00000 q^{57} -4.00000 q^{59} +8.82475 q^{61} -4.27492 q^{63} -1.00000 q^{65} -8.00000 q^{67} -0.274917 q^{69} -1.72508 q^{71} -12.5498 q^{73} -1.00000 q^{75} -9.72508 q^{77} -14.2749 q^{79} +1.00000 q^{81} -12.0000 q^{83} +6.27492 q^{85} -8.27492 q^{89} -4.27492 q^{91} -4.00000 q^{93} -2.00000 q^{95} +14.2749 q^{97} +2.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - 3 q^{11} + 2 q^{13} + 2 q^{15} - 5 q^{17} + 4 q^{19} + q^{21} - 7 q^{23} + 2 q^{25} - 2 q^{27} + 8 q^{31} + 3 q^{33} + q^{35} + 9 q^{37} - 2 q^{39} + 9 q^{41} - 2 q^{43} - 2 q^{45} + 8 q^{47} + 15 q^{49} + 5 q^{51} + 9 q^{53} + 3 q^{55} - 4 q^{57} - 8 q^{59} - 5 q^{61} - q^{63} - 2 q^{65} - 16 q^{67} + 7 q^{69} - 11 q^{71} - 10 q^{73} - 2 q^{75} - 27 q^{77} - 21 q^{79} + 2 q^{81} - 24 q^{83} + 5 q^{85} - 9 q^{89} - q^{91} - 8 q^{93} - 4 q^{95} + 21 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - 3 * q^11 + 2 * q^13 + 2 * q^15 - 5 * q^17 + 4 * q^19 + q^21 - 7 * q^23 + 2 * q^25 - 2 * q^27 + 8 * q^31 + 3 * q^33 + q^35 + 9 * q^37 - 2 * q^39 + 9 * q^41 - 2 * q^43 - 2 * q^45 + 8 * q^47 + 15 * q^49 + 5 * q^51 + 9 * q^53 + 3 * q^55 - 4 * q^57 - 8 * q^59 - 5 * q^61 - q^63 - 2 * q^65 - 16 * q^67 + 7 * q^69 - 11 * q^71 - 10 * q^73 - 2 * q^75 - 27 * q^77 - 21 * q^79 + 2 * q^81 - 24 * q^83 + 5 * q^85 - 9 * q^89 - q^91 - 8 * q^93 - 4 * q^95 + 21 * q^97 - 3 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.27492	−1.61577	−0.807883	−	0.589342i	$-0.799387\pi$
$7$	−4.27492	−1.61577	−0.807883	+	0.589342i	$0.799387\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.27492	0.685913	0.342957	−	0.939351i	$-0.388572\pi$
$11$	2.27492	0.685913	0.342957	+	0.939351i	$0.388572\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.27492	−1.52189	−0.760945	−	0.648816i	$-0.775265\pi$
$17$	−6.27492	−1.52189	−0.760945	+	0.648816i	$0.775265\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	4.27492	0.932863
$22$	0	0
$23$	0.274917	0.0573242	0.0286621	−	0.999589i	$-0.490875\pi$
$23$	0.274917	0.0573242	0.0286621	+	0.999589i	$0.490875\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−2.27492	−0.396012
$34$	0	0
$35$	4.27492	0.722593
$36$	0	0
$37$	8.27492	1.36039	0.680194	−	0.733032i	$-0.261896\pi$
$37$	8.27492	1.36039	0.680194	+	0.733032i	$0.261896\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	8.27492	1.29232	0.646162	−	0.763200i	$-0.276373\pi$
$41$	8.27492	1.29232	0.646162	+	0.763200i	$0.276373\pi$
$42$	0	0
$43$	−8.54983	−1.30384	−0.651919	−	0.758289i	$-0.726036\pi$
$43$	−8.54983	−1.30384	−0.651919	+	0.758289i	$0.726036\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	11.2749	1.61070
$50$	0	0
$51$	6.27492	0.878664
$52$	0	0
$53$	8.27492	1.13665	0.568324	−	0.822805i	$-0.307592\pi$
$53$	8.27492	1.13665	0.568324	+	0.822805i	$0.307592\pi$
$54$	0	0
$55$	−2.27492	−0.306750
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	8.82475	1.12989	0.564947	−	0.825127i	$-0.308897\pi$
$61$	8.82475	1.12989	0.564947	+	0.825127i	$0.308897\pi$
$62$	0	0
$63$	−4.27492	−0.538589
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−0.274917	−0.0330961
$70$	0	0
$71$	−1.72508	−0.204730	−0.102365	−	0.994747i	$-0.532641\pi$
$71$	−1.72508	−0.204730	−0.102365	+	0.994747i	$0.532641\pi$
$72$	0	0
$73$	−12.5498	−1.46885	−0.734424	−	0.678691i	$-0.762547\pi$
$73$	−12.5498	−1.46885	−0.734424	+	0.678691i	$0.762547\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−9.72508	−1.10828
$78$	0	0
$79$	−14.2749	−1.60605	−0.803027	−	0.595943i	$-0.796778\pi$
$79$	−14.2749	−1.60605	−0.803027	+	0.595943i	$0.796778\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	6.27492	0.680610
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.27492	−0.877139	−0.438570	−	0.898697i	$-0.644515\pi$
$89$	−8.27492	−0.877139	−0.438570	+	0.898697i	$0.644515\pi$
$90$	0	0
$91$	−4.27492	−0.448133
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	14.2749	1.44940	0.724699	−	0.689065i	$-0.241979\pi$
$97$	14.2749	1.44940	0.724699	+	0.689065i	$0.241979\pi$
$98$	0	0
$99$	2.27492	0.228638
$100$	0	0
$101$	13.0997	1.30347	0.651733	−	0.758449i	$-0.274042\pi$
$101$	13.0997	1.30347	0.651733	+	0.758449i	$0.274042\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	−4.27492	−0.417189
$106$	0	0
$107$	−10.2749	−0.993314	−0.496657	−	0.867947i	$-0.665439\pi$
$107$	−10.2749	−0.993314	−0.496657	+	0.867947i	$0.665439\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−8.27492	−0.785420
$112$	0	0
$113$	−0.549834	−0.0517241	−0.0258620	−	0.999666i	$-0.508233\pi$
$113$	−0.549834	−0.0517241	−0.0258620	+	0.999666i	$0.508233\pi$
$114$	0	0
$115$	−0.274917	−0.0256362
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	26.8248	2.45902
$120$	0	0
$121$	−5.82475	−0.529523
$122$	0	0
$123$	−8.27492	−0.746124
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	21.0997	1.87229	0.936146	−	0.351610i	$-0.114366\pi$
$127$	21.0997	1.87229	0.936146	+	0.351610i	$0.114366\pi$
$128$	0	0
$129$	8.54983	0.752771
$130$	0	0
$131$	18.5498	1.62071	0.810353	−	0.585942i	$-0.199275\pi$
$131$	18.5498	1.62071	0.810353	+	0.585942i	$0.199275\pi$
$132$	0	0
$133$	−8.54983	−0.741365
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−22.8248	−1.93597	−0.967985	−	0.251008i	$-0.919238\pi$
$139$	−22.8248	−1.93597	−0.967985	+	0.251008i	$0.919238\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	2.27492	0.190238
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−11.2749	−0.929939
$148$	0	0
$149$	−17.3746	−1.42338	−0.711691	−	0.702493i	$-0.752070\pi$
$149$	−17.3746	−1.42338	−0.711691	+	0.702493i	$0.752070\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	−6.27492	−0.507297
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−8.27492	−0.656244
$160$	0	0
$161$	−1.17525	−0.0926225
$162$	0	0
$163$	−13.7251	−1.07503	−0.537516	−	0.843254i	$-0.680637\pi$
$163$	−13.7251	−1.07503	−0.537516	+	0.843254i	$0.680637\pi$
$164$	0	0
$165$	2.27492	0.177102
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	−23.0997	−1.75624	−0.878118	−	0.478445i	$-0.841201\pi$
$173$	−23.0997	−1.75624	−0.878118	+	0.478445i	$0.841201\pi$
$174$	0	0
$175$	−4.27492	−0.323153
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	12.8248	0.953256	0.476628	−	0.879105i	$-0.341859\pi$
$181$	12.8248	0.953256	0.476628	+	0.879105i	$0.341859\pi$
$182$	0	0
$183$	−8.82475	−0.652344
$184$	0	0
$185$	−8.27492	−0.608384
$186$	0	0
$187$	−14.2749	−1.04389
$188$	0	0
$189$	4.27492	0.310954
$190$	0	0
$191$	−4.54983	−0.329214	−0.164607	−	0.986359i	$-0.552636\pi$
$191$	−4.54983	−0.329214	−0.164607	+	0.986359i	$0.552636\pi$
$192$	0	0
$193$	−2.27492	−0.163752	−0.0818761	−	0.996643i	$-0.526091\pi$
$193$	−2.27492	−0.163752	−0.0818761	+	0.996643i	$0.526091\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−8.27492	−0.577945
$206$	0	0
$207$	0.274917	0.0191081
$208$	0	0
$209$	4.54983	0.314719
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	1.72508	0.118201
$214$	0	0
$215$	8.54983	0.583094
$216$	0	0
$217$	−17.0997	−1.16080
$218$	0	0
$219$	12.5498	0.848039
$220$	0	0
$221$	−6.27492	−0.422097
$222$	0	0
$223$	−5.45017	−0.364970	−0.182485	−	0.983209i	$-0.558414\pi$
$223$	−5.45017	−0.364970	−0.182485	+	0.983209i	$0.558414\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−7.45017	−0.494485	−0.247242	−	0.968954i	$-0.579524\pi$
$227$	−7.45017	−0.494485	−0.247242	+	0.968954i	$0.579524\pi$
$228$	0	0
$229$	−25.0997	−1.65863	−0.829316	−	0.558779i	$-0.811270\pi$
$229$	−25.0997	−1.65863	−0.829316	+	0.558779i	$0.811270\pi$
$230$	0	0
$231$	9.72508	0.639863
$232$	0	0
$233$	−1.72508	−0.113014	−0.0565070	−	0.998402i	$-0.517996\pi$
$233$	−1.72508	−0.113014	−0.0565070	+	0.998402i	$0.517996\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	14.2749	0.927256
$238$	0	0
$239$	−1.72508	−0.111586	−0.0557932	−	0.998442i	$-0.517769\pi$
$239$	−1.72508	−0.111586	−0.0557932	+	0.998442i	$0.517769\pi$
$240$	0	0
$241$	−1.45017	−0.0934134	−0.0467067	−	0.998909i	$-0.514873\pi$
$241$	−1.45017	−0.0934134	−0.0467067	+	0.998909i	$0.514873\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−11.2749	−0.720328
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	6.54983	0.413422	0.206711	−	0.978402i	$-0.433724\pi$
$251$	6.54983	0.413422	0.206711	+	0.978402i	$0.433724\pi$
$252$	0	0
$253$	0.625414	0.0393194
$254$	0	0
$255$	−6.27492	−0.392951
$256$	0	0
$257$	0.549834	0.0342977	0.0171489	−	0.999853i	$-0.494541\pi$
$257$	0.549834	0.0342977	0.0171489	+	0.999853i	$0.494541\pi$
$258$	0	0
$259$	−35.3746	−2.19807
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.4502	−0.829373	−0.414686	−	0.909964i	$-0.636109\pi$
$263$	−13.4502	−0.829373	−0.414686	+	0.909964i	$0.636109\pi$
$264$	0	0
$265$	−8.27492	−0.508324
$266$	0	0
$267$	8.27492	0.506417
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	25.6495	1.55810	0.779048	−	0.626964i	$-0.215703\pi$
$271$	25.6495	1.55810	0.779048	+	0.626964i	$0.215703\pi$
$272$	0	0
$273$	4.27492	0.258730
$274$	0	0
$275$	2.27492	0.137183
$276$	0	0
$277$	−27.6495	−1.66130	−0.830649	−	0.556797i	$-0.812030\pi$
$277$	−27.6495	−1.66130	−0.830649	+	0.556797i	$0.812030\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	13.0997	0.778694	0.389347	−	0.921091i	$-0.372701\pi$
$283$	13.0997	0.778694	0.389347	+	0.921091i	$0.372701\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	−35.3746	−2.08810
$288$	0	0
$289$	22.3746	1.31615
$290$	0	0
$291$	−14.2749	−0.836810
$292$	0	0
$293$	−22.5498	−1.31738	−0.658688	−	0.752416i	$-0.728888\pi$
$293$	−22.5498	−1.31738	−0.658688	+	0.752416i	$0.728888\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−2.27492	−0.132004
$298$	0	0
$299$	0.274917	0.0158989
$300$	0	0
$301$	36.5498	2.10670
$302$	0	0
$303$	−13.0997	−0.752556
$304$	0	0
$305$	−8.82475	−0.505304
$306$	0	0
$307$	7.37459	0.420890	0.210445	−	0.977606i	$-0.432509\pi$
$307$	7.37459	0.420890	0.210445	+	0.977606i	$0.432509\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−21.6495	−1.22763	−0.613815	−	0.789450i	$-0.710366\pi$
$311$	−21.6495	−1.22763	−0.613815	+	0.789450i	$0.710366\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	4.27492	0.240864
$316$	0	0
$317$	−23.0997	−1.29741	−0.648703	−	0.761041i	$-0.724688\pi$
$317$	−23.0997	−1.29741	−0.648703	+	0.761041i	$0.724688\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	10.2749	0.573490
$322$	0	0
$323$	−12.5498	−0.698291
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−16.0000	−0.884802
$328$	0	0
$329$	−17.0997	−0.942735
$330$	0	0
$331$	−15.0997	−0.829953	−0.414976	−	0.909832i	$-0.636210\pi$
$331$	−15.0997	−0.829953	−0.414976	+	0.909832i	$0.636210\pi$
$332$	0	0
$333$	8.27492	0.453463
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	7.09967	0.386744	0.193372	−	0.981126i	$-0.438058\pi$
$337$	7.09967	0.386744	0.193372	+	0.981126i	$0.438058\pi$
$338$	0	0
$339$	0.549834	0.0298629
$340$	0	0
$341$	9.09967	0.492775
$342$	0	0
$343$	−18.2749	−0.986753
$344$	0	0
$345$	0.274917	0.0148010
$346$	0	0
$347$	−14.2749	−0.766318	−0.383159	−	0.923682i	$-0.625164\pi$
$347$	−14.2749	−0.766318	−0.383159	+	0.923682i	$0.625164\pi$
$348$	0	0
$349$	−12.5498	−0.671777	−0.335889	−	0.941902i	$-0.609037\pi$
$349$	−12.5498	−0.671777	−0.335889	+	0.941902i	$0.609037\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	15.6495	0.832939	0.416470	−	0.909150i	$-0.363267\pi$
$353$	15.6495	0.832939	0.416470	+	0.909150i	$0.363267\pi$
$354$	0	0
$355$	1.72508	0.0915579
$356$	0	0
$357$	−26.8248	−1.41972
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	5.82475	0.305720
$364$	0	0
$365$	12.5498	0.656888
$366$	0	0
$367$	−8.54983	−0.446298	−0.223149	−	0.974784i	$-0.571634\pi$
$367$	−8.54983	−0.446298	−0.223149	+	0.974784i	$0.571634\pi$
$368$	0	0
$369$	8.27492	0.430775
$370$	0	0
$371$	−35.3746	−1.83656
$372$	0	0
$373$	0.900331	0.0466174	0.0233087	−	0.999728i	$-0.492580\pi$
$373$	0.900331	0.0466174	0.0233087	+	0.999728i	$0.492580\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	18.5498	0.952841	0.476420	−	0.879218i	$-0.341934\pi$
$379$	18.5498	0.952841	0.476420	+	0.879218i	$0.341934\pi$
$380$	0	0
$381$	−21.0997	−1.08097
$382$	0	0
$383$	28.5498	1.45883	0.729414	−	0.684072i	$-0.239793\pi$
$383$	28.5498	1.45883	0.729414	+	0.684072i	$0.239793\pi$
$384$	0	0
$385$	9.72508	0.495636
$386$	0	0
$387$	−8.54983	−0.434613
$388$	0	0
$389$	21.0997	1.06980	0.534898	−	0.844917i	$-0.320350\pi$
$389$	21.0997	1.06980	0.534898	+	0.844917i	$0.320350\pi$
$390$	0	0
$391$	−1.72508	−0.0872412
$392$	0	0
$393$	−18.5498	−0.935715
$394$	0	0
$395$	14.2749	0.718249
$396$	0	0
$397$	8.27492	0.415306	0.207653	−	0.978203i	$-0.433417\pi$
$397$	8.27492	0.415306	0.207653	+	0.978203i	$0.433417\pi$
$398$	0	0
$399$	8.54983	0.428027
$400$	0	0
$401$	0.900331	0.0449604	0.0224802	−	0.999747i	$-0.492844\pi$
$401$	0.900331	0.0449604	0.0224802	+	0.999747i	$0.492844\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	18.8248	0.933108
$408$	0	0
$409$	27.6495	1.36718	0.683590	−	0.729866i	$-0.260418\pi$
$409$	27.6495	1.36718	0.683590	+	0.729866i	$0.260418\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	0	0
$413$	17.0997	0.841420
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	22.8248	1.11773
$418$	0	0
$419$	−27.0997	−1.32391	−0.661953	−	0.749546i	$-0.730272\pi$
$419$	−27.0997	−1.32391	−0.661953	+	0.749546i	$0.730272\pi$
$420$	0	0
$421$	5.09967	0.248543	0.124271	−	0.992248i	$-0.460341\pi$
$421$	5.09967	0.248543	0.124271	+	0.992248i	$0.460341\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	−6.27492	−0.304378
$426$	0	0
$427$	−37.7251	−1.82564
$428$	0	0
$429$	−2.27492	−0.109834
$430$	0	0
$431$	−26.1993	−1.26198	−0.630989	−	0.775792i	$-0.717351\pi$
$431$	−26.1993	−1.26198	−0.630989	+	0.775792i	$0.717351\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.549834	0.0263021
$438$	0	0
$439$	−17.7251	−0.845972	−0.422986	−	0.906136i	$-0.639018\pi$
$439$	−17.7251	−0.845972	−0.422986	+	0.906136i	$0.639018\pi$
$440$	0	0
$441$	11.2749	0.536901
$442$	0	0
$443$	−14.2749	−0.678222	−0.339111	−	0.940746i	$-0.610126\pi$
$443$	−14.2749	−0.678222	−0.339111	+	0.940746i	$0.610126\pi$
$444$	0	0
$445$	8.27492	0.392269
$446$	0	0
$447$	17.3746	0.821790
$448$	0	0
$449$	30.4743	1.43817	0.719085	−	0.694923i	$-0.244561\pi$
$449$	30.4743	1.43817	0.719085	+	0.694923i	$0.244561\pi$
$450$	0	0
$451$	18.8248	0.886423
$452$	0	0
$453$	4.00000	0.187936
$454$	0	0
$455$	4.27492	0.200411
$456$	0	0
$457$	−0.625414	−0.0292556	−0.0146278	−	0.999893i	$-0.504656\pi$
$457$	−0.625414	−0.0292556	−0.0146278	+	0.999893i	$0.504656\pi$
$458$	0	0
$459$	6.27492	0.292888
$460$	0	0
$461$	−17.9244	−0.834823	−0.417412	−	0.908717i	$-0.637063\pi$
$461$	−17.9244	−0.834823	−0.417412	+	0.908717i	$0.637063\pi$
$462$	0	0
$463$	24.8248	1.15370	0.576852	−	0.816849i	$-0.304281\pi$
$463$	24.8248	1.15370	0.576852	+	0.816849i	$0.304281\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	−27.3746	−1.26674	−0.633372	−	0.773847i	$-0.718330\pi$
$467$	−27.3746	−1.26674	−0.633372	+	0.773847i	$0.718330\pi$
$468$	0	0
$469$	34.1993	1.57918
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	−19.4502	−0.894320
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	8.27492	0.378882
$478$	0	0
$479$	−39.3746	−1.79907	−0.899535	−	0.436848i	$-0.856095\pi$
$479$	−39.3746	−1.79907	−0.899535	+	0.436848i	$0.856095\pi$
$480$	0	0
$481$	8.27492	0.377304
$482$	0	0
$483$	1.17525	0.0534757
$484$	0	0
$485$	−14.2749	−0.648191
$486$	0	0
$487$	−41.9244	−1.89978	−0.949888	−	0.312589i	$-0.898804\pi$
$487$	−41.9244	−1.89978	−0.949888	+	0.312589i	$0.898804\pi$
$488$	0	0
$489$	13.7251	0.620670
$490$	0	0
$491$	−3.09967	−0.139886	−0.0699430	−	0.997551i	$-0.522282\pi$
$491$	−3.09967	−0.139886	−0.0699430	+	0.997551i	$0.522282\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−2.27492	−0.102250
$496$	0	0
$497$	7.37459	0.330795
$498$	0	0
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.5498	0.827096	0.413548	−	0.910482i	$-0.364289\pi$
$503$	18.5498	0.827096	0.413548	+	0.910482i	$0.364289\pi$
$504$	0	0
$505$	−13.0997	−0.582928
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−44.2749	−1.96245	−0.981226	−	0.192863i	$-0.938223\pi$
$509$	−44.2749	−1.96245	−0.981226	+	0.192863i	$0.938223\pi$
$510$	0	0
$511$	53.6495	2.37331
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	9.09967	0.400203
$518$	0	0
$519$	23.0997	1.01396
$520$	0	0
$521$	−27.0997	−1.18726	−0.593629	−	0.804739i	$-0.702305\pi$
$521$	−27.0997	−1.18726	−0.593629	+	0.804739i	$0.702305\pi$
$522$	0	0
$523$	33.6495	1.47139	0.735695	−	0.677313i	$-0.236856\pi$
$523$	33.6495	1.47139	0.735695	+	0.677313i	$0.236856\pi$
$524$	0	0
$525$	4.27492	0.186573
$526$	0	0
$527$	−25.0997	−1.09336
$528$	0	0
$529$	−22.9244	−0.996714
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	8.27492	0.358426
$534$	0	0
$535$	10.2749	0.444223
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	25.6495	1.10480
$540$	0	0
$541$	−8.54983	−0.367586	−0.183793	−	0.982965i	$-0.558838\pi$
$541$	−8.54983	−0.367586	−0.183793	+	0.982965i	$0.558838\pi$
$542$	0	0
$543$	−12.8248	−0.550363
$544$	0	0
$545$	−16.0000	−0.685365
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	8.82475	0.376631
$550$	0	0
$551$	0	0
$552$	0	0
$553$	61.0241	2.59501
$554$	0	0
$555$	8.27492	0.351251
$556$	0	0
$557$	−25.4502	−1.07836	−0.539179	−	0.842191i	$-0.681265\pi$
$557$	−25.4502	−1.07836	−0.539179	+	0.842191i	$0.681265\pi$
$558$	0	0
$559$	−8.54983	−0.361619
$560$	0	0
$561$	14.2749	0.602687
$562$	0	0
$563$	2.82475	0.119049	0.0595245	−	0.998227i	$-0.481042\pi$
$563$	2.82475	0.119049	0.0595245	+	0.998227i	$0.481042\pi$
$564$	0	0
$565$	0.549834	0.0231317
$566$	0	0
$567$	−4.27492	−0.179530
$568$	0	0
$569$	−3.09967	−0.129945	−0.0649724	−	0.997887i	$-0.520696\pi$
$569$	−3.09967	−0.129945	−0.0649724	+	0.997887i	$0.520696\pi$
$570$	0	0
$571$	46.8248	1.95955	0.979777	−	0.200090i	$-0.0641236\pi$
$571$	46.8248	1.95955	0.979777	+	0.200090i	$0.0641236\pi$
$572$	0	0
$573$	4.54983	0.190072
$574$	0	0
$575$	0.274917	0.0114648
$576$	0	0
$577$	−4.62541	−0.192559	−0.0962793	−	0.995354i	$-0.530694\pi$
$577$	−4.62541	−0.192559	−0.0962793	+	0.995354i	$0.530694\pi$
$578$	0	0
$579$	2.27492	0.0945423
$580$	0	0
$581$	51.2990	2.12824
$582$	0	0
$583$	18.8248	0.779642
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	24.5498	1.01328	0.506640	−	0.862158i	$-0.330887\pi$
$587$	24.5498	1.01328	0.506640	+	0.862158i	$0.330887\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	−32.7492	−1.34485	−0.672424	−	0.740166i	$-0.734747\pi$
$593$	−32.7492	−1.34485	−0.672424	+	0.740166i	$0.734747\pi$
$594$	0	0
$595$	−26.8248	−1.09971
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−29.9244	−1.22064	−0.610321	−	0.792154i	$-0.708960\pi$
$601$	−29.9244	−1.22064	−0.610321	+	0.792154i	$0.708960\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	5.82475	0.236810
$606$	0	0
$607$	25.6495	1.04108	0.520541	−	0.853837i	$-0.325730\pi$
$607$	25.6495	1.04108	0.520541	+	0.853837i	$0.325730\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	13.9244	0.562402	0.281201	−	0.959649i	$-0.409267\pi$
$613$	13.9244	0.562402	0.281201	+	0.959649i	$0.409267\pi$
$614$	0	0
$615$	8.27492	0.333677
$616$	0	0
$617$	−12.9003	−0.519348	−0.259674	−	0.965696i	$-0.583615\pi$
$617$	−12.9003	−0.519348	−0.259674	+	0.965696i	$0.583615\pi$
$618$	0	0
$619$	−23.6495	−0.950554	−0.475277	−	0.879836i	$-0.657652\pi$
$619$	−23.6495	−0.950554	−0.475277	+	0.879836i	$0.657652\pi$
$620$	0	0
$621$	−0.274917	−0.0110320
$622$	0	0
$623$	35.3746	1.41725
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.54983	−0.181703
$628$	0	0
$629$	−51.9244	−2.07036
$630$	0	0
$631$	34.1993	1.36145	0.680727	−	0.732537i	$-0.261664\pi$
$631$	34.1993	1.36145	0.680727	+	0.732537i	$0.261664\pi$
$632$	0	0
$633$	16.0000	0.635943
$634$	0	0
$635$	−21.0997	−0.837315
$636$	0	0
$637$	11.2749	0.446728
$638$	0	0
$639$	−1.72508	−0.0682432
$640$	0	0
$641$	−31.0997	−1.22836	−0.614182	−	0.789165i	$-0.710514\pi$
$641$	−31.0997	−1.22836	−0.614182	+	0.789165i	$0.710514\pi$
$642$	0	0
$643$	24.4743	0.965170	0.482585	−	0.875849i	$-0.339698\pi$
$643$	24.4743	0.965170	0.482585	+	0.875849i	$0.339698\pi$
$644$	0	0
$645$	−8.54983	−0.336649
$646$	0	0
$647$	7.72508	0.303704	0.151852	−	0.988403i	$-0.451476\pi$
$647$	7.72508	0.303704	0.151852	+	0.988403i	$0.451476\pi$
$648$	0	0
$649$	−9.09967	−0.357193
$650$	0	0
$651$	17.0997	0.670189
$652$	0	0
$653$	−19.0997	−0.747428	−0.373714	−	0.927544i	$-0.621916\pi$
$653$	−19.0997	−0.747428	−0.373714	+	0.927544i	$0.621916\pi$
$654$	0	0
$655$	−18.5498	−0.724802
$656$	0	0
$657$	−12.5498	−0.489616
$658$	0	0
$659$	19.6495	0.765436	0.382718	−	0.923865i	$-0.374988\pi$
$659$	19.6495	0.765436	0.382718	+	0.923865i	$0.374988\pi$
$660$	0	0
$661$	7.45017	0.289778	0.144889	−	0.989448i	$-0.453718\pi$
$661$	7.45017	0.289778	0.144889	+	0.989448i	$0.453718\pi$
$662$	0	0
$663$	6.27492	0.243698
$664$	0	0
$665$	8.54983	0.331548
$666$	0	0
$667$	0	0
$668$	0	0
$669$	5.45017	0.210716
$670$	0	0
$671$	20.0756	0.775009
$672$	0	0
$673$	31.6495	1.22000	0.609999	−	0.792402i	$-0.291170\pi$
$673$	31.6495	1.22000	0.609999	+	0.792402i	$0.291170\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	22.4743	0.863756	0.431878	−	0.901932i	$-0.357851\pi$
$677$	22.4743	0.863756	0.431878	+	0.901932i	$0.357851\pi$
$678$	0	0
$679$	−61.0241	−2.34189
$680$	0	0
$681$	7.45017	0.285491
$682$	0	0
$683$	−50.7492	−1.94186	−0.970932	−	0.239357i	$-0.923063\pi$
$683$	−50.7492	−1.94186	−0.970932	+	0.239357i	$0.923063\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	25.0997	0.957612
$688$	0	0
$689$	8.27492	0.315249
$690$	0	0
$691$	0.900331	0.0342502	0.0171251	−	0.999853i	$-0.494549\pi$
$691$	0.900331	0.0342502	0.0171251	+	0.999853i	$0.494549\pi$
$692$	0	0
$693$	−9.72508	−0.369425
$694$	0	0
$695$	22.8248	0.865792
$696$	0	0
$697$	−51.9244	−1.96678
$698$	0	0
$699$	1.72508	0.0652486
$700$	0	0
$701$	2.90033	0.109544	0.0547720	−	0.998499i	$-0.482557\pi$
$701$	2.90033	0.109544	0.0547720	+	0.998499i	$0.482557\pi$
$702$	0	0
$703$	16.5498	0.624189
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	−56.0000	−2.10610
$708$	0	0
$709$	−1.09967	−0.0412989	−0.0206495	−	0.999787i	$-0.506573\pi$
$709$	−1.09967	−0.0412989	−0.0206495	+	0.999787i	$0.506573\pi$
$710$	0	0
$711$	−14.2749	−0.535351
$712$	0	0
$713$	1.09967	0.0411829
$714$	0	0
$715$	−2.27492	−0.0850771
$716$	0	0
$717$	1.72508	0.0644244
$718$	0	0
$719$	45.0997	1.68193	0.840967	−	0.541087i	$-0.181987\pi$
$719$	45.0997	1.68193	0.840967	+	0.541087i	$0.181987\pi$
$720$	0	0
$721$	−51.2990	−1.91047
$722$	0	0
$723$	1.45017	0.0539322
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27.4502	1.01807	0.509035	−	0.860746i	$-0.330002\pi$
$727$	27.4502	1.01807	0.509035	+	0.860746i	$0.330002\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	53.6495	1.98430
$732$	0	0
$733$	16.2749	0.601128	0.300564	−	0.953762i	$-0.402825\pi$
$733$	16.2749	0.601128	0.300564	+	0.953762i	$0.402825\pi$
$734$	0	0
$735$	11.2749	0.415882
$736$	0	0
$737$	−18.1993	−0.670381
$738$	0	0
$739$	−26.0000	−0.956425	−0.478213	−	0.878244i	$-0.658715\pi$
$739$	−26.0000	−0.956425	−0.478213	+	0.878244i	$0.658715\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	−37.6495	−1.38123	−0.690613	−	0.723224i	$-0.742659\pi$
$743$	−37.6495	−1.38123	−0.690613	+	0.723224i	$0.742659\pi$
$744$	0	0
$745$	17.3746	0.636556
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	43.9244	1.60496
$750$	0	0
$751$	40.4743	1.47693	0.738463	−	0.674294i	$-0.235552\pi$
$751$	40.4743	1.47693	0.738463	+	0.674294i	$0.235552\pi$
$752$	0	0
$753$	−6.54983	−0.238689
$754$	0	0
$755$	4.00000	0.145575
$756$	0	0
$757$	42.5498	1.54650	0.773250	−	0.634101i	$-0.218630\pi$
$757$	42.5498	1.54650	0.773250	+	0.634101i	$0.218630\pi$
$758$	0	0
$759$	−0.625414	−0.0227011
$760$	0	0
$761$	−15.0997	−0.547363	−0.273681	−	0.961820i	$-0.588241\pi$
$761$	−15.0997	−0.547363	−0.273681	+	0.961820i	$0.588241\pi$
$762$	0	0
$763$	−68.3987	−2.47620
$764$	0	0
$765$	6.27492	0.226870
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	43.0997	1.55421	0.777107	−	0.629369i	$-0.216686\pi$
$769$	43.0997	1.55421	0.777107	+	0.629369i	$0.216686\pi$
$770$	0	0
$771$	−0.549834	−0.0198018
$772$	0	0
$773$	11.6495	0.419004	0.209502	−	0.977808i	$-0.432816\pi$
$773$	11.6495	0.419004	0.209502	+	0.977808i	$0.432816\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	35.3746	1.26906
$778$	0	0
$779$	16.5498	0.592959
$780$	0	0
$781$	−3.92442	−0.140427
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	−10.9003	−0.388555	−0.194277	−	0.980947i	$-0.562236\pi$
$787$	−10.9003	−0.388555	−0.194277	+	0.980947i	$0.562236\pi$
$788$	0	0
$789$	13.4502	0.478839
$790$	0	0
$791$	2.35050	0.0835740
$792$	0	0
$793$	8.82475	0.313376
$794$	0	0
$795$	8.27492	0.293481
$796$	0	0
$797$	−14.4743	−0.512704	−0.256352	−	0.966583i	$-0.582521\pi$
$797$	−14.4743	−0.512704	−0.256352	+	0.966583i	$0.582521\pi$
$798$	0	0
$799$	−25.0997	−0.887962
$800$	0	0
$801$	−8.27492	−0.292380
$802$	0	0
$803$	−28.5498	−1.00750
$804$	0	0
$805$	1.17525	0.0414221
$806$	0	0
$807$	4.00000	0.140807
$808$	0	0
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	−15.0997	−0.530221	−0.265111	−	0.964218i	$-0.585408\pi$
$811$	−15.0997	−0.530221	−0.265111	+	0.964218i	$0.585408\pi$
$812$	0	0
$813$	−25.6495	−0.899567
$814$	0	0
$815$	13.7251	0.480769
$816$	0	0
$817$	−17.0997	−0.598242
$818$	0	0
$819$	−4.27492	−0.149378
$820$	0	0
$821$	−37.9244	−1.32357	−0.661786	−	0.749693i	$-0.730201\pi$
$821$	−37.9244	−1.32357	−0.661786	+	0.749693i	$0.730201\pi$
$822$	0	0
$823$	−38.1993	−1.33155	−0.665773	−	0.746155i	$-0.731898\pi$
$823$	−38.1993	−1.33155	−0.665773	+	0.746155i	$0.731898\pi$
$824$	0	0
$825$	−2.27492	−0.0792025
$826$	0	0
$827$	−54.1993	−1.88470	−0.942348	−	0.334635i	$-0.891387\pi$
$827$	−54.1993	−1.88470	−0.942348	+	0.334635i	$0.891387\pi$
$828$	0	0
$829$	−3.09967	−0.107656	−0.0538280	−	0.998550i	$-0.517142\pi$
$829$	−3.09967	−0.107656	−0.0538280	+	0.998550i	$0.517142\pi$
$830$	0	0
$831$	27.6495	0.959151
$832$	0	0
$833$	−70.7492	−2.45131
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−18.8248	−0.649903	−0.324951	−	0.945731i	$-0.605348\pi$
$839$	−18.8248	−0.649903	−0.324951	+	0.945731i	$0.605348\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	10.0000	0.344418
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	24.9003	0.855586
$848$	0	0
$849$	−13.0997	−0.449579
$850$	0	0
$851$	2.27492	0.0779832
$852$	0	0
$853$	18.6254	0.637722	0.318861	−	0.947801i	$-0.396700\pi$
$853$	18.6254	0.637722	0.318861	+	0.947801i	$0.396700\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	0	0
$857$	2.27492	0.0777097	0.0388548	−	0.999245i	$-0.487629\pi$
$857$	2.27492	0.0777097	0.0388548	+	0.999245i	$0.487629\pi$
$858$	0	0
$859$	13.1752	0.449534	0.224767	−	0.974413i	$-0.427838\pi$
$859$	13.1752	0.449534	0.224767	+	0.974413i	$0.427838\pi$
$860$	0	0
$861$	35.3746	1.20556
$862$	0	0
$863$	−38.7492	−1.31904	−0.659519	−	0.751688i	$-0.729240\pi$
$863$	−38.7492	−1.31904	−0.659519	+	0.751688i	$0.729240\pi$
$864$	0	0
$865$	23.0997	0.785412
$866$	0	0
$867$	−22.3746	−0.759881
$868$	0	0
$869$	−32.4743	−1.10161
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	14.2749	0.483133
$874$	0	0
$875$	4.27492	0.144519
$876$	0	0
$877$	−3.09967	−0.104668	−0.0523342	−	0.998630i	$-0.516666\pi$
$877$	−3.09967	−0.104668	−0.0523342	+	0.998630i	$0.516666\pi$
$878$	0	0
$879$	22.5498	0.760587
$880$	0	0
$881$	−22.5498	−0.759723	−0.379862	−	0.925043i	$-0.624028\pi$
$881$	−22.5498	−0.759723	−0.379862	+	0.925043i	$0.624028\pi$
$882$	0	0
$883$	54.1993	1.82395	0.911976	−	0.410243i	$-0.134556\pi$
$883$	54.1993	1.82395	0.911976	+	0.410243i	$0.134556\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	21.9244	0.736150	0.368075	−	0.929796i	$-0.380017\pi$
$887$	21.9244	0.736150	0.368075	+	0.929796i	$0.380017\pi$
$888$	0	0
$889$	−90.1993	−3.02519
$890$	0	0
$891$	2.27492	0.0762126
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	6.00000	0.200558
$896$	0	0
$897$	−0.274917	−0.00917922
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−51.9244	−1.72985
$902$	0	0
$903$	−36.5498	−1.21630
$904$	0	0
$905$	−12.8248	−0.426309
$906$	0	0
$907$	8.54983	0.283893	0.141946	−	0.989874i	$-0.454664\pi$
$907$	8.54983	0.283893	0.141946	+	0.989874i	$0.454664\pi$
$908$	0	0
$909$	13.0997	0.434489
$910$	0	0
$911$	33.6495	1.11486	0.557429	−	0.830225i	$-0.311788\pi$
$911$	33.6495	1.11486	0.557429	+	0.830225i	$0.311788\pi$
$912$	0	0
$913$	−27.2990	−0.903465
$914$	0	0
$915$	8.82475	0.291737
$916$	0	0
$917$	−79.2990	−2.61868
$918$	0	0
$919$	26.8248	0.884867	0.442433	−	0.896801i	$-0.354115\pi$
$919$	26.8248	0.884867	0.442433	+	0.896801i	$0.354115\pi$
$920$	0	0
$921$	−7.37459	−0.243001
$922$	0	0
$923$	−1.72508	−0.0567818
$924$	0	0
$925$	8.27492	0.272078
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	56.8248	1.86436	0.932180	−	0.361996i	$-0.117904\pi$
$929$	56.8248	1.86436	0.932180	+	0.361996i	$0.117904\pi$
$930$	0	0
$931$	22.5498	0.739041
$932$	0	0
$933$	21.6495	0.708773
$934$	0	0
$935$	14.2749	0.466840
$936$	0	0
$937$	−34.5498	−1.12869	−0.564347	−	0.825538i	$-0.690872\pi$
$937$	−34.5498	−1.12869	−0.564347	+	0.825538i	$0.690872\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	7.72508	0.251831	0.125915	−	0.992041i	$-0.459813\pi$
$941$	7.72508	0.251831	0.125915	+	0.992041i	$0.459813\pi$
$942$	0	0
$943$	2.27492	0.0740815
$944$	0	0
$945$	−4.27492	−0.139063
$946$	0	0
$947$	5.09967	0.165717	0.0828585	−	0.996561i	$-0.473595\pi$
$947$	5.09967	0.165717	0.0828585	+	0.996561i	$0.473595\pi$
$948$	0	0
$949$	−12.5498	−0.407385
$950$	0	0
$951$	23.0997	0.749058
$952$	0	0
$953$	−29.1752	−0.945079	−0.472539	−	0.881309i	$-0.656663\pi$
$953$	−29.1752	−0.945079	−0.472539	+	0.881309i	$0.656663\pi$
$954$	0	0
$955$	4.54983	0.147229
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−42.7492	−1.38044
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−10.2749	−0.331105
$964$	0	0
$965$	2.27492	0.0732322
$966$	0	0
$967$	−18.5498	−0.596522	−0.298261	−	0.954484i	$-0.596407\pi$
$967$	−18.5498	−0.596522	−0.298261	+	0.954484i	$0.596407\pi$
$968$	0	0
$969$	12.5498	0.403159
$970$	0	0
$971$	53.2990	1.71045	0.855223	−	0.518260i	$-0.173420\pi$
$971$	53.2990	1.71045	0.855223	+	0.518260i	$0.173420\pi$
$972$	0	0
$973$	97.5739	3.12808
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	39.0997	1.25091	0.625455	−	0.780261i	$-0.284914\pi$
$977$	39.0997	1.25091	0.625455	+	0.780261i	$0.284914\pi$
$978$	0	0
$979$	−18.8248	−0.601642
$980$	0	0
$981$	16.0000	0.510841
$982$	0	0
$983$	2.19934	0.0701480	0.0350740	−	0.999385i	$-0.488833\pi$
$983$	2.19934	0.0701480	0.0350740	+	0.999385i	$0.488833\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	17.0997	0.544288
$988$	0	0
$989$	−2.35050	−0.0747414
$990$	0	0
$991$	39.3746	1.25077	0.625387	−	0.780314i	$-0.284941\pi$
$991$	39.3746	1.25077	0.625387	+	0.780314i	$0.284941\pi$
$992$	0	0
$993$	15.0997	0.479174
$994$	0	0
$995$	0	0
$996$	0	0
$997$	23.6495	0.748987	0.374494	−	0.927229i	$-0.377817\pi$
$997$	23.6495	0.748987	0.374494	+	0.927229i	$0.377817\pi$
$998$	0	0
$999$	−8.27492	−0.261807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bg.1.1	✓	2
4.3	odd	2		6240.2.a.bs.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bg.1.1	✓	2	1.1	even	1	trivial
6240.2.a.bs.1.2	yes	2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -4.27492 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -4.27492 q^{7} +1.00000 q^{9} +2.27492 q^{11} +1.00000 q^{13} +1.00000 q^{15} -6.27492 q^{17} +2.00000 q^{19} +4.27492 q^{21} +0.274917 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.00000 q^{31} -2.27492 q^{33} +4.27492 q^{35} +8.27492 q^{37} -1.00000 q^{39} +8.27492 q^{41} -8.54983 q^{43} -1.00000 q^{45} +4.00000 q^{47} +11.2749 q^{49} +6.27492 q^{51} +8.27492 q^{53} -2.27492 q^{55} -2.00000 q^{57} -4.00000 q^{59} +8.82475 q^{61} -4.27492 q^{63} -1.00000 q^{65} -8.00000 q^{67} -0.274917 q^{69} -1.72508 q^{71} -12.5498 q^{73} -1.00000 q^{75} -9.72508 q^{77} -14.2749 q^{79} +1.00000 q^{81} -12.0000 q^{83} +6.27492 q^{85} -8.27492 q^{89} -4.27492 q^{91} -4.00000 q^{93} -2.00000 q^{95} +14.2749 q^{97} +2.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - 3 q^{11} + 2 q^{13} + 2 q^{15} - 5 q^{17} + 4 q^{19} + q^{21} - 7 q^{23} + 2 q^{25} - 2 q^{27} + 8 q^{31} + 3 q^{33} + q^{35} + 9 q^{37} - 2 q^{39} + 9 q^{41} - 2 q^{43} - 2 q^{45} + 8 q^{47} + 15 q^{49} + 5 q^{51} + 9 q^{53} + 3 q^{55} - 4 q^{57} - 8 q^{59} - 5 q^{61} - q^{63} - 2 q^{65} - 16 q^{67} + 7 q^{69} - 11 q^{71} - 10 q^{73} - 2 q^{75} - 27 q^{77} - 21 q^{79} + 2 q^{81} - 24 q^{83} + 5 q^{85} - 9 q^{89} - q^{91} - 8 q^{93} - 4 q^{95} + 21 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - 3 * q^11 + 2 * q^13 + 2 * q^15 - 5 * q^17 + 4 * q^19 + q^21 - 7 * q^23 + 2 * q^25 - 2 * q^27 + 8 * q^31 + 3 * q^33 + q^35 + 9 * q^37 - 2 * q^39 + 9 * q^41 - 2 * q^43 - 2 * q^45 + 8 * q^47 + 15 * q^49 + 5 * q^51 + 9 * q^53 + 3 * q^55 - 4 * q^57 - 8 * q^59 - 5 * q^61 - q^63 - 2 * q^65 - 16 * q^67 + 7 * q^69 - 11 * q^71 - 10 * q^73 - 2 * q^75 - 27 * q^77 - 21 * q^79 + 2 * q^81 - 24 * q^83 + 5 * q^85 - 9 * q^89 - q^91 - 8 * q^93 - 4 * q^95 + 21 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more