LMFDB - Embedded newform 6240.2.a.bs.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:	$0$
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			$-3.27492$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -3.27492 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -3.27492 q^{7} +1.00000 q^{9} +5.27492 q^{11} +1.00000 q^{13} -1.00000 q^{15} +1.27492 q^{17} -2.00000 q^{19} -3.27492 q^{21} +7.27492 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{31} +5.27492 q^{33} +3.27492 q^{35} +0.725083 q^{37} +1.00000 q^{39} +0.725083 q^{41} -6.54983 q^{43} -1.00000 q^{45} -4.00000 q^{47} +3.72508 q^{49} +1.27492 q^{51} +0.725083 q^{53} -5.27492 q^{55} -2.00000 q^{57} +4.00000 q^{59} -13.8248 q^{61} -3.27492 q^{63} -1.00000 q^{65} +8.00000 q^{67} +7.27492 q^{69} +9.27492 q^{71} +2.54983 q^{73} +1.00000 q^{75} -17.2749 q^{77} +6.72508 q^{79} +1.00000 q^{81} +12.0000 q^{83} -1.27492 q^{85} -0.725083 q^{89} -3.27492 q^{91} -4.00000 q^{93} +2.00000 q^{95} +6.72508 q^{97} +5.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−3.27492	−1.23780	−0.618901	−	0.785469i	$-0.712422\pi$
$7$	−3.27492	−1.23780	−0.618901	+	0.785469i	$0.712422\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.27492	1.59045	0.795224	−	0.606316i	$-0.207353\pi$
$11$	5.27492	1.59045	0.795224	+	0.606316i	$0.207353\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.27492	0.309213	0.154606	−	0.987976i	$-0.450589\pi$
$17$	1.27492	0.309213	0.154606	+	0.987976i	$0.450589\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−3.27492	−0.714646
$22$	0	0
$23$	7.27492	1.51693	0.758463	−	0.651717i	$-0.225951\pi$
$23$	7.27492	1.51693	0.758463	+	0.651717i	$0.225951\pi$
$24$	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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	5.27492	0.918245
$34$	0	0
$35$	3.27492	0.553562
$36$	0	0
$37$	0.725083	0.119203	0.0596014	−	0.998222i	$-0.481017\pi$
$37$	0.725083	0.119203	0.0596014	+	0.998222i	$0.481017\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	0.725083	0.113239	0.0566195	−	0.998396i	$-0.481968\pi$
$41$	0.725083	0.113239	0.0566195	+	0.998396i	$0.481968\pi$
$42$	0	0
$43$	−6.54983	−0.998840	−0.499420	−	0.866360i	$-0.666454\pi$
$43$	−6.54983	−0.998840	−0.499420	+	0.866360i	$0.666454\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	3.72508	0.532155
$50$	0	0
$51$	1.27492	0.178524
$52$	0	0
$53$	0.725083	0.0995978	0.0497989	−	0.998759i	$-0.484142\pi$
$53$	0.725083	0.0995978	0.0497989	+	0.998759i	$0.484142\pi$
$54$	0	0
$55$	−5.27492	−0.711270
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−13.8248	−1.77008	−0.885039	−	0.465517i	$-0.845868\pi$
$61$	−13.8248	−1.77008	−0.885039	+	0.465517i	$0.845868\pi$
$62$	0	0
$63$	−3.27492	−0.412601
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	7.27492	0.875797
$70$	0	0
$71$	9.27492	1.10073	0.550365	−	0.834924i	$-0.314489\pi$
$71$	9.27492	1.10073	0.550365	+	0.834924i	$0.314489\pi$
$72$	0	0
$73$	2.54983	0.298436	0.149218	−	0.988804i	$-0.452324\pi$
$73$	2.54983	0.298436	0.149218	+	0.988804i	$0.452324\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−17.2749	−1.96866
$78$	0	0
$79$	6.72508	0.756631	0.378315	−	0.925677i	$-0.376503\pi$
$79$	6.72508	0.756631	0.378315	+	0.925677i	$0.376503\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−1.27492	−0.138284
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.725083	−0.0768586	−0.0384293	−	0.999261i	$-0.512235\pi$
$89$	−0.725083	−0.0768586	−0.0384293	+	0.999261i	$0.512235\pi$
$90$	0	0
$91$	−3.27492	−0.343305
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	6.72508	0.682829	0.341414	−	0.939913i	$-0.389094\pi$
$97$	6.72508	0.682829	0.341414	+	0.939913i	$0.389094\pi$
$98$	0	0
$99$	5.27492	0.530149
$100$	0	0
$101$	−17.0997	−1.70148	−0.850740	−	0.525586i	$-0.823846\pi$
$101$	−17.0997	−1.70148	−0.850740	+	0.525586i	$0.823846\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	3.27492	0.319599
$106$	0	0
$107$	2.72508	0.263444	0.131722	−	0.991287i	$-0.457949\pi$
$107$	2.72508	0.263444	0.131722	+	0.991287i	$0.457949\pi$
$108$	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
$111$	0.725083	0.0688218
$112$	0	0
$113$	14.5498	1.36873	0.684367	−	0.729138i	$-0.260079\pi$
$113$	14.5498	1.36873	0.684367	+	0.729138i	$0.260079\pi$
$114$	0	0
$115$	−7.27492	−0.678390
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−4.17525	−0.382744
$120$	0	0
$121$	16.8248	1.52952
$122$	0	0
$123$	0.725083	0.0653785
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.09967	0.807465	0.403733	−	0.914877i	$-0.367713\pi$
$127$	9.09967	0.807465	0.403733	+	0.914877i	$0.367713\pi$
$128$	0	0
$129$	−6.54983	−0.576681
$130$	0	0
$131$	−3.45017	−0.301442	−0.150721	−	0.988576i	$-0.548160\pi$
$131$	−3.45017	−0.301442	−0.150721	+	0.988576i	$0.548160\pi$
$132$	0	0
$133$	6.54983	0.567943
$134$	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
$139$	0.175248	0.0148644	0.00743219	−	0.999972i	$-0.497634\pi$
$139$	0.175248	0.0148644	0.00743219	+	0.999972i	$0.497634\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	5.27492	0.441111
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.72508	0.307240
$148$	0	0
$149$	20.3746	1.66915	0.834576	−	0.550893i	$-0.185713\pi$
$149$	20.3746	1.66915	0.834576	+	0.550893i	$0.185713\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	1.27492	0.103071
$154$	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
$159$	0.725083	0.0575028
$160$	0	0
$161$	−23.8248	−1.87765
$162$	0	0
$163$	21.2749	1.66638	0.833190	−	0.552987i	$-0.186512\pi$
$163$	21.2749	1.66638	0.833190	+	0.552987i	$0.186512\pi$
$164$	0	0
$165$	−5.27492	−0.410652
$166$	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
$173$	7.09967	0.539778	0.269889	−	0.962891i	$-0.413013\pi$
$173$	7.09967	0.539778	0.269889	+	0.962891i	$0.413013\pi$
$174$	0	0
$175$	−3.27492	−0.247560
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−9.82475	−0.730268	−0.365134	−	0.930955i	$-0.618977\pi$
$181$	−9.82475	−0.730268	−0.365134	+	0.930955i	$0.618977\pi$
$182$	0	0
$183$	−13.8248	−1.02196
$184$	0	0
$185$	−0.725083	−0.0533091
$186$	0	0
$187$	6.72508	0.491787
$188$	0	0
$189$	−3.27492	−0.238215
$190$	0	0
$191$	−10.5498	−0.763359	−0.381680	−	0.924295i	$-0.624654\pi$
$191$	−10.5498	−0.763359	−0.381680	+	0.924295i	$0.624654\pi$
$192$	0	0
$193$	5.27492	0.379697	0.189848	−	0.981813i	$-0.439200\pi$
$193$	5.27492	0.379697	0.189848	+	0.981813i	$0.439200\pi$
$194$	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
$205$	−0.725083	−0.0506420
$206$	0	0
$207$	7.27492	0.505642
$208$	0	0
$209$	−10.5498	−0.729747
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	9.27492	0.635507
$214$	0	0
$215$	6.54983	0.446695
$216$	0	0
$217$	13.0997	0.889263
$218$	0	0
$219$	2.54983	0.172302
$220$	0	0
$221$	1.27492	0.0857602
$222$	0	0
$223$	20.5498	1.37612	0.688059	−	0.725654i	$-0.258463\pi$
$223$	20.5498	1.37612	0.688059	+	0.725654i	$0.258463\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	22.5498	1.49669	0.748343	−	0.663312i	$-0.230850\pi$
$227$	22.5498	1.49669	0.748343	+	0.663312i	$0.230850\pi$
$228$	0	0
$229$	5.09967	0.336996	0.168498	−	0.985702i	$-0.446108\pi$
$229$	5.09967	0.336996	0.168498	+	0.985702i	$0.446108\pi$
$230$	0	0
$231$	−17.2749	−1.13661
$232$	0	0
$233$	−9.27492	−0.607620	−0.303810	−	0.952733i	$-0.598259\pi$
$233$	−9.27492	−0.607620	−0.303810	+	0.952733i	$0.598259\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	6.72508	0.436841
$238$	0	0
$239$	9.27492	0.599945	0.299972	−	0.953948i	$-0.403023\pi$
$239$	9.27492	0.599945	0.299972	+	0.953948i	$0.403023\pi$
$240$	0	0
$241$	−16.5498	−1.06607	−0.533034	−	0.846094i	$-0.678948\pi$
$241$	−16.5498	−1.06607	−0.533034	+	0.846094i	$0.678948\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.72508	−0.237987
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	8.54983	0.539661	0.269830	−	0.962908i	$-0.413032\pi$
$251$	8.54983	0.539661	0.269830	+	0.962908i	$0.413032\pi$
$252$	0	0
$253$	38.3746	2.41259
$254$	0	0
$255$	−1.27492	−0.0798384
$256$	0	0
$257$	−14.5498	−0.907594	−0.453797	−	0.891105i	$-0.649931\pi$
$257$	−14.5498	−0.907594	−0.453797	+	0.891105i	$0.649931\pi$
$258$	0	0
$259$	−2.37459	−0.147550
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.5498	1.76046	0.880229	−	0.474549i	$-0.157389\pi$
$263$	28.5498	1.76046	0.880229	+	0.474549i	$0.157389\pi$
$264$	0	0
$265$	−0.725083	−0.0445415
$266$	0	0
$267$	−0.725083	−0.0443743
$268$	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
$271$	19.6495	1.19362	0.596811	−	0.802382i	$-0.296434\pi$
$271$	19.6495	1.19362	0.596811	+	0.802382i	$0.296434\pi$
$272$	0	0
$273$	−3.27492	−0.198207
$274$	0	0
$275$	5.27492	0.318089
$276$	0	0
$277$	17.6495	1.06046	0.530228	−	0.847855i	$-0.322106\pi$
$277$	17.6495	1.06046	0.530228	+	0.847855i	$0.322106\pi$
$278$	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
$283$	17.0997	1.01647	0.508235	−	0.861218i	$-0.330298\pi$
$283$	17.0997	1.01647	0.508235	+	0.861218i	$0.330298\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	−2.37459	−0.140167
$288$	0	0
$289$	−15.3746	−0.904387
$290$	0	0
$291$	6.72508	0.394231
$292$	0	0
$293$	−7.45017	−0.435243	−0.217622	−	0.976033i	$-0.569830\pi$
$293$	−7.45017	−0.435243	−0.217622	+	0.976033i	$0.569830\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	5.27492	0.306082
$298$	0	0
$299$	7.27492	0.420719
$300$	0	0
$301$	21.4502	1.23637
$302$	0	0
$303$	−17.0997	−0.982350
$304$	0	0
$305$	13.8248	0.791603
$306$	0	0
$307$	30.3746	1.73357	0.866785	−	0.498683i	$-0.166183\pi$
$307$	30.3746	1.73357	0.866785	+	0.498683i	$0.166183\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−23.6495	−1.34104	−0.670520	−	0.741891i	$-0.733929\pi$
$311$	−23.6495	−1.34104	−0.670520	+	0.741891i	$0.733929\pi$
$312$	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
$315$	3.27492	0.184521
$316$	0	0
$317$	7.09967	0.398757	0.199379	−	0.979923i	$-0.436108\pi$
$317$	7.09967	0.398757	0.199379	+	0.979923i	$0.436108\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.72508	0.152099
$322$	0	0
$323$	−2.54983	−0.141877
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	16.0000	0.884802
$328$	0	0
$329$	13.0997	0.722208
$330$	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
$333$	0.725083	0.0397343
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−23.0997	−1.25832	−0.629160	−	0.777276i	$-0.716601\pi$
$337$	−23.0997	−1.25832	−0.629160	+	0.777276i	$0.716601\pi$
$338$	0	0
$339$	14.5498	0.790239
$340$	0	0
$341$	−21.0997	−1.14261
$342$	0	0
$343$	10.7251	0.579100
$344$	0	0
$345$	−7.27492	−0.391668
$346$	0	0
$347$	6.72508	0.361021	0.180511	−	0.983573i	$-0.442225\pi$
$347$	6.72508	0.361021	0.180511	+	0.983573i	$0.442225\pi$
$348$	0	0
$349$	2.54983	0.136490	0.0682448	−	0.997669i	$-0.478260\pi$
$349$	2.54983	0.136490	0.0682448	+	0.997669i	$0.478260\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−29.6495	−1.57808	−0.789042	−	0.614339i	$-0.789423\pi$
$353$	−29.6495	−1.57808	−0.789042	+	0.614339i	$0.789423\pi$
$354$	0	0
$355$	−9.27492	−0.492261
$356$	0	0
$357$	−4.17525	−0.220978
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	16.8248	0.883070
$364$	0	0
$365$	−2.54983	−0.133464
$366$	0	0
$367$	−6.54983	−0.341899	−0.170949	−	0.985280i	$-0.554683\pi$
$367$	−6.54983	−0.341899	−0.170949	+	0.985280i	$0.554683\pi$
$368$	0	0
$369$	0.725083	0.0377463
$370$	0	0
$371$	−2.37459	−0.123282
$372$	0	0
$373$	31.0997	1.61028	0.805140	−	0.593085i	$-0.202090\pi$
$373$	31.0997	1.61028	0.805140	+	0.593085i	$0.202090\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3.45017	−0.177223	−0.0886116	−	0.996066i	$-0.528243\pi$
$379$	−3.45017	−0.177223	−0.0886116	+	0.996066i	$0.528243\pi$
$380$	0	0
$381$	9.09967	0.466190
$382$	0	0
$383$	−13.4502	−0.687271	−0.343636	−	0.939103i	$-0.611659\pi$
$383$	−13.4502	−0.687271	−0.343636	+	0.939103i	$0.611659\pi$
$384$	0	0
$385$	17.2749	0.880411
$386$	0	0
$387$	−6.54983	−0.332947
$388$	0	0
$389$	−9.09967	−0.461372	−0.230686	−	0.973028i	$-0.574097\pi$
$389$	−9.09967	−0.461372	−0.230686	+	0.973028i	$0.574097\pi$
$390$	0	0
$391$	9.27492	0.469053
$392$	0	0
$393$	−3.45017	−0.174038
$394$	0	0
$395$	−6.72508	−0.338376
$396$	0	0
$397$	0.725083	0.0363909	0.0181954	−	0.999834i	$-0.494208\pi$
$397$	0.725083	0.0363909	0.0181954	+	0.999834i	$0.494208\pi$
$398$	0	0
$399$	6.54983	0.327902
$400$	0	0
$401$	31.0997	1.55304	0.776522	−	0.630091i	$-0.216982\pi$
$401$	31.0997	1.55304	0.776522	+	0.630091i	$0.216982\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	3.82475	0.189586
$408$	0	0
$409$	−17.6495	−0.872712	−0.436356	−	0.899774i	$-0.643731\pi$
$409$	−17.6495	−0.872712	−0.436356	+	0.899774i	$0.643731\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	0	0
$413$	−13.0997	−0.644593
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	0.175248	0.00858195
$418$	0	0
$419$	−3.09967	−0.151429	−0.0757144	−	0.997130i	$-0.524124\pi$
$419$	−3.09967	−0.151429	−0.0757144	+	0.997130i	$0.524124\pi$
$420$	0	0
$421$	−25.0997	−1.22328	−0.611642	−	0.791135i	$-0.709490\pi$
$421$	−25.0997	−1.22328	−0.611642	+	0.791135i	$0.709490\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	1.27492	0.0618426
$426$	0	0
$427$	45.2749	2.19101
$428$	0	0
$429$	5.27492	0.254675
$430$	0	0
$431$	−34.1993	−1.64732	−0.823662	−	0.567081i	$-0.808073\pi$
$431$	−34.1993	−1.64732	−0.823662	+	0.567081i	$0.808073\pi$
$432$	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
$437$	−14.5498	−0.696013
$438$	0	0
$439$	25.2749	1.20631	0.603153	−	0.797626i	$-0.293911\pi$
$439$	25.2749	1.20631	0.603153	+	0.797626i	$0.293911\pi$
$440$	0	0
$441$	3.72508	0.177385
$442$	0	0
$443$	6.72508	0.319518	0.159759	−	0.987156i	$-0.448928\pi$
$443$	6.72508	0.319518	0.159759	+	0.987156i	$0.448928\pi$
$444$	0	0
$445$	0.725083	0.0343722
$446$	0	0
$447$	20.3746	0.963685
$448$	0	0
$449$	−37.4743	−1.76852	−0.884260	−	0.466995i	$-0.845336\pi$
$449$	−37.4743	−1.76852	−0.884260	+	0.466995i	$0.845336\pi$
$450$	0	0
$451$	3.82475	0.180101
$452$	0	0
$453$	4.00000	0.187936
$454$	0	0
$455$	3.27492	0.153530
$456$	0	0
$457$	−38.3746	−1.79509	−0.897544	−	0.440925i	$-0.854650\pi$
$457$	−38.3746	−1.79509	−0.897544	+	0.440925i	$0.854650\pi$
$458$	0	0
$459$	1.27492	0.0595080
$460$	0	0
$461$	34.9244	1.62659	0.813296	−	0.581850i	$-0.197671\pi$
$461$	34.9244	1.62659	0.813296	+	0.581850i	$0.197671\pi$
$462$	0	0
$463$	−2.17525	−0.101092	−0.0505462	−	0.998722i	$-0.516096\pi$
$463$	−2.17525	−0.101092	−0.0505462	+	0.998722i	$0.516096\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	−10.3746	−0.480079	−0.240039	−	0.970763i	$-0.577160\pi$
$467$	−10.3746	−0.480079	−0.240039	+	0.970763i	$0.577160\pi$
$468$	0	0
$469$	−26.1993	−1.20977
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	−34.5498	−1.58860
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	0.725083	0.0331993
$478$	0	0
$479$	1.62541	0.0742671	0.0371335	−	0.999310i	$-0.488177\pi$
$479$	1.62541	0.0742671	0.0371335	+	0.999310i	$0.488177\pi$
$480$	0	0
$481$	0.725083	0.0330609
$482$	0	0
$483$	−23.8248	−1.08406
$484$	0	0
$485$	−6.72508	−0.305370
$486$	0	0
$487$	−10.9244	−0.495033	−0.247516	−	0.968884i	$-0.579614\pi$
$487$	−10.9244	−0.495033	−0.247516	+	0.968884i	$0.579614\pi$
$488$	0	0
$489$	21.2749	0.962085
$490$	0	0
$491$	−27.0997	−1.22299	−0.611495	−	0.791248i	$-0.709432\pi$
$491$	−27.0997	−1.22299	−0.611495	+	0.791248i	$0.709432\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−5.27492	−0.237090
$496$	0	0
$497$	−30.3746	−1.36249
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.45017	−0.153835	−0.0769176	−	0.997037i	$-0.524508\pi$
$503$	−3.45017	−0.153835	−0.0769176	+	0.997037i	$0.524508\pi$
$504$	0	0
$505$	17.0997	0.760925
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−36.7251	−1.62781	−0.813905	−	0.580998i	$-0.802663\pi$
$509$	−36.7251	−1.62781	−0.813905	+	0.580998i	$0.802663\pi$
$510$	0	0
$511$	−8.35050	−0.369404
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	12.0000	0.528783
$516$	0	0
$517$	−21.0997	−0.927962
$518$	0	0
$519$	7.09967	0.311641
$520$	0	0
$521$	3.09967	0.135799	0.0678995	−	0.997692i	$-0.478370\pi$
$521$	3.09967	0.135799	0.0678995	+	0.997692i	$0.478370\pi$
$522$	0	0
$523$	11.6495	0.509397	0.254699	−	0.967020i	$-0.418024\pi$
$523$	11.6495	0.509397	0.254699	+	0.967020i	$0.418024\pi$
$524$	0	0
$525$	−3.27492	−0.142929
$526$	0	0
$527$	−5.09967	−0.222145
$528$	0	0
$529$	29.9244	1.30106
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	0.725083	0.0314068
$534$	0	0
$535$	−2.72508	−0.117816
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	19.6495	0.846364
$540$	0	0
$541$	6.54983	0.281599	0.140800	−	0.990038i	$-0.455033\pi$
$541$	6.54983	0.281599	0.140800	+	0.990038i	$0.455033\pi$
$542$	0	0
$543$	−9.82475	−0.421620
$544$	0	0
$545$	−16.0000	−0.685365
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	−13.8248	−0.590026
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−22.0241	−0.936560
$554$	0	0
$555$	−0.725083	−0.0307781
$556$	0	0
$557$	−40.5498	−1.71815	−0.859076	−	0.511848i	$-0.828961\pi$
$557$	−40.5498	−1.71815	−0.859076	+	0.511848i	$0.828961\pi$
$558$	0	0
$559$	−6.54983	−0.277028
$560$	0	0
$561$	6.72508	0.283933
$562$	0	0
$563$	19.8248	0.835514	0.417757	−	0.908559i	$-0.362816\pi$
$563$	19.8248	0.835514	0.417757	+	0.908559i	$0.362816\pi$
$564$	0	0
$565$	−14.5498	−0.612116
$566$	0	0
$567$	−3.27492	−0.137534
$568$	0	0
$569$	27.0997	1.13608	0.568039	−	0.823002i	$-0.307702\pi$
$569$	27.0997	1.13608	0.568039	+	0.823002i	$0.307702\pi$
$570$	0	0
$571$	−24.1752	−1.01170	−0.505851	−	0.862621i	$-0.668822\pi$
$571$	−24.1752	−1.01170	−0.505851	+	0.862621i	$0.668822\pi$
$572$	0	0
$573$	−10.5498	−0.440726
$574$	0	0
$575$	7.27492	0.303385
$576$	0	0
$577$	−42.3746	−1.76408	−0.882039	−	0.471177i	$-0.843829\pi$
$577$	−42.3746	−1.76408	−0.882039	+	0.471177i	$0.843829\pi$
$578$	0	0
$579$	5.27492	0.219218
$580$	0	0
$581$	−39.2990	−1.63040
$582$	0	0
$583$	3.82475	0.158405
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−9.45017	−0.390050	−0.195025	−	0.980798i	$-0.562479\pi$
$587$	−9.45017	−0.390050	−0.195025	+	0.980798i	$0.562479\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	42.7492	1.75550	0.877749	−	0.479121i	$-0.159044\pi$
$593$	42.7492	1.75550	0.877749	+	0.479121i	$0.159044\pi$
$594$	0	0
$595$	4.17525	0.171168
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	22.9244	0.935107	0.467553	−	0.883965i	$-0.345136\pi$
$601$	22.9244	0.935107	0.467553	+	0.883965i	$0.345136\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	−16.8248	−0.684023
$606$	0	0
$607$	19.6495	0.797549	0.398774	−	0.917049i	$-0.369436\pi$
$607$	19.6495	0.797549	0.398774	+	0.917049i	$0.369436\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	−38.9244	−1.57214	−0.786071	−	0.618136i	$-0.787888\pi$
$613$	−38.9244	−1.57214	−0.786071	+	0.618136i	$0.787888\pi$
$614$	0	0
$615$	−0.725083	−0.0292382
$616$	0	0
$617$	−43.0997	−1.73513	−0.867564	−	0.497326i	$-0.834315\pi$
$617$	−43.0997	−1.73513	−0.867564	+	0.497326i	$0.834315\pi$
$618$	0	0
$619$	−21.6495	−0.870167	−0.435084	−	0.900390i	$-0.643281\pi$
$619$	−21.6495	−0.870167	−0.435084	+	0.900390i	$0.643281\pi$
$620$	0	0
$621$	7.27492	0.291932
$622$	0	0
$623$	2.37459	0.0951358
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.5498	−0.421320
$628$	0	0
$629$	0.924421	0.0368591
$630$	0	0
$631$	26.1993	1.04298	0.521490	−	0.853258i	$-0.325377\pi$
$631$	26.1993	1.04298	0.521490	+	0.853258i	$0.325377\pi$
$632$	0	0
$633$	16.0000	0.635943
$634$	0	0
$635$	−9.09967	−0.361109
$636$	0	0
$637$	3.72508	0.147593
$638$	0	0
$639$	9.27492	0.366910
$640$	0	0
$641$	−0.900331	−0.0355609	−0.0177805	−	0.999842i	$-0.505660\pi$
$641$	−0.900331	−0.0355609	−0.0177805	+	0.999842i	$0.505660\pi$
$642$	0	0
$643$	43.4743	1.71446	0.857229	−	0.514936i	$-0.172184\pi$
$643$	43.4743	1.71446	0.857229	+	0.514936i	$0.172184\pi$
$644$	0	0
$645$	6.54983	0.257899
$646$	0	0
$647$	−15.2749	−0.600519	−0.300259	−	0.953858i	$-0.597073\pi$
$647$	−15.2749	−0.600519	−0.300259	+	0.953858i	$0.597073\pi$
$648$	0	0
$649$	21.0997	0.828234
$650$	0	0
$651$	13.0997	0.513417
$652$	0	0
$653$	11.0997	0.434364	0.217182	−	0.976131i	$-0.430314\pi$
$653$	11.0997	0.434364	0.217182	+	0.976131i	$0.430314\pi$
$654$	0	0
$655$	3.45017	0.134809
$656$	0	0
$657$	2.54983	0.0994785
$658$	0	0
$659$	25.6495	0.999163	0.499581	−	0.866267i	$-0.333487\pi$
$659$	25.6495	0.999163	0.499581	+	0.866267i	$0.333487\pi$
$660$	0	0
$661$	22.5498	0.877087	0.438543	−	0.898710i	$-0.355495\pi$
$661$	22.5498	0.877087	0.438543	+	0.898710i	$0.355495\pi$
$662$	0	0
$663$	1.27492	0.0495137
$664$	0	0
$665$	−6.54983	−0.253992
$666$	0	0
$667$	0	0
$668$	0	0
$669$	20.5498	0.794503
$670$	0	0
$671$	−72.9244	−2.81522
$672$	0	0
$673$	−13.6495	−0.526150	−0.263075	−	0.964775i	$-0.584737\pi$
$673$	−13.6495	−0.526150	−0.263075	+	0.964775i	$0.584737\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−45.4743	−1.74772	−0.873859	−	0.486180i	$-0.838390\pi$
$677$	−45.4743	−1.74772	−0.873859	+	0.486180i	$0.838390\pi$
$678$	0	0
$679$	−22.0241	−0.845207
$680$	0	0
$681$	22.5498	0.864112
$682$	0	0
$683$	−24.7492	−0.947001	−0.473500	−	0.880794i	$-0.657010\pi$
$683$	−24.7492	−0.947001	−0.473500	+	0.880794i	$0.657010\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	5.09967	0.194565
$688$	0	0
$689$	0.725083	0.0276235
$690$	0	0
$691$	−31.0997	−1.18309	−0.591544	−	0.806273i	$-0.701481\pi$
$691$	−31.0997	−1.18309	−0.591544	+	0.806273i	$0.701481\pi$
$692$	0	0
$693$	−17.2749	−0.656220
$694$	0	0
$695$	−0.175248	−0.00664755
$696$	0	0
$697$	0.924421	0.0350149
$698$	0	0
$699$	−9.27492	−0.350810
$700$	0	0
$701$	33.0997	1.25016	0.625079	−	0.780562i	$-0.285067\pi$
$701$	33.0997	1.25016	0.625079	+	0.780562i	$0.285067\pi$
$702$	0	0
$703$	−1.45017	−0.0546940
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	56.0000	2.10610
$708$	0	0
$709$	29.0997	1.09286	0.546431	−	0.837504i	$-0.315986\pi$
$709$	29.0997	1.09286	0.546431	+	0.837504i	$0.315986\pi$
$710$	0	0
$711$	6.72508	0.252210
$712$	0	0
$713$	−29.0997	−1.08979
$714$	0	0
$715$	−5.27492	−0.197271
$716$	0	0
$717$	9.27492	0.346378
$718$	0	0
$719$	−14.9003	−0.555689	−0.277844	−	0.960626i	$-0.589620\pi$
$719$	−14.9003	−0.555689	−0.277844	+	0.960626i	$0.589620\pi$
$720$	0	0
$721$	39.2990	1.46357
$722$	0	0
$723$	−16.5498	−0.615495
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−42.5498	−1.57809	−0.789043	−	0.614338i	$-0.789423\pi$
$727$	−42.5498	−1.57809	−0.789043	+	0.614338i	$0.789423\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.35050	−0.308854
$732$	0	0
$733$	8.72508	0.322268	0.161134	−	0.986933i	$-0.448485\pi$
$733$	8.72508	0.322268	0.161134	+	0.986933i	$0.448485\pi$
$734$	0	0
$735$	−3.72508	−0.137402
$736$	0	0
$737$	42.1993	1.55443
$738$	0	0
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	−7.64950	−0.280633	−0.140317	−	0.990107i	$-0.544812\pi$
$743$	−7.64950	−0.280633	−0.140317	+	0.990107i	$0.544812\pi$
$744$	0	0
$745$	−20.3746	−0.746467
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	−8.92442	−0.326091
$750$	0	0
$751$	27.4743	1.00255	0.501275	−	0.865288i	$-0.332865\pi$
$751$	27.4743	1.00255	0.501275	+	0.865288i	$0.332865\pi$
$752$	0	0
$753$	8.54983	0.311573
$754$	0	0
$755$	−4.00000	−0.145575
$756$	0	0
$757$	27.4502	0.997693	0.498847	−	0.866690i	$-0.333757\pi$
$757$	27.4502	0.997693	0.498847	+	0.866690i	$0.333757\pi$
$758$	0	0
$759$	38.3746	1.39291
$760$	0	0
$761$	15.0997	0.547363	0.273681	−	0.961820i	$-0.411759\pi$
$761$	15.0997	0.547363	0.273681	+	0.961820i	$0.411759\pi$
$762$	0	0
$763$	−52.3987	−1.89696
$764$	0	0
$765$	−1.27492	−0.0460947
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	12.9003	0.465198	0.232599	−	0.972573i	$-0.425277\pi$
$769$	12.9003	0.465198	0.232599	+	0.972573i	$0.425277\pi$
$770$	0	0
$771$	−14.5498	−0.523999
$772$	0	0
$773$	−33.6495	−1.21029	−0.605144	−	0.796116i	$-0.706885\pi$
$773$	−33.6495	−1.21029	−0.605144	+	0.796116i	$0.706885\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−2.37459	−0.0851878
$778$	0	0
$779$	−1.45017	−0.0519576
$780$	0	0
$781$	48.9244	1.75065
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	41.0997	1.46504	0.732522	−	0.680743i	$-0.238343\pi$
$787$	41.0997	1.46504	0.732522	+	0.680743i	$0.238343\pi$
$788$	0	0
$789$	28.5498	1.01640
$790$	0	0
$791$	−47.6495	−1.69422
$792$	0	0
$793$	−13.8248	−0.490931
$794$	0	0
$795$	−0.725083	−0.0257160
$796$	0	0
$797$	53.4743	1.89416	0.947078	−	0.321005i	$-0.104020\pi$
$797$	53.4743	1.89416	0.947078	+	0.321005i	$0.104020\pi$
$798$	0	0
$799$	−5.09967	−0.180413
$800$	0	0
$801$	−0.725083	−0.0256195
$802$	0	0
$803$	13.4502	0.474646
$804$	0	0
$805$	23.8248	0.839712
$806$	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
$813$	19.6495	0.689138
$814$	0	0
$815$	−21.2749	−0.745228
$816$	0	0
$817$	13.0997	0.458299
$818$	0	0
$819$	−3.27492	−0.114435
$820$	0	0
$821$	14.9244	0.520866	0.260433	−	0.965492i	$-0.416135\pi$
$821$	14.9244	0.520866	0.260433	+	0.965492i	$0.416135\pi$
$822$	0	0
$823$	−22.1993	−0.773820	−0.386910	−	0.922117i	$-0.626458\pi$
$823$	−22.1993	−0.773820	−0.386910	+	0.922117i	$0.626458\pi$
$824$	0	0
$825$	5.27492	0.183649
$826$	0	0
$827$	−6.19934	−0.215572	−0.107786	−	0.994174i	$-0.534376\pi$
$827$	−6.19934	−0.215572	−0.107786	+	0.994174i	$0.534376\pi$
$828$	0	0
$829$	27.0997	0.941210	0.470605	−	0.882344i	$-0.344036\pi$
$829$	27.0997	0.941210	0.470605	+	0.882344i	$0.344036\pi$
$830$	0	0
$831$	17.6495	0.612254
$832$	0	0
$833$	4.74917	0.164549
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−3.82475	−0.132045	−0.0660225	−	0.997818i	$-0.521031\pi$
$839$	−3.82475	−0.132045	−0.0660225	+	0.997818i	$0.521031\pi$
$840$	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
$847$	−55.0997	−1.89325
$848$	0	0
$849$	17.0997	0.586859
$850$	0	0
$851$	5.27492	0.180822
$852$	0	0
$853$	56.3746	1.93023	0.965115	−	0.261828i	$-0.0843253\pi$
$853$	56.3746	1.93023	0.965115	+	0.261828i	$0.0843253\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	−5.27492	−0.180188	−0.0900939	−	0.995933i	$-0.528717\pi$
$857$	−5.27492	−0.180188	−0.0900939	+	0.995933i	$0.528717\pi$
$858$	0	0
$859$	−35.8248	−1.22232	−0.611162	−	0.791505i	$-0.709298\pi$
$859$	−35.8248	−1.22232	−0.611162	+	0.791505i	$0.709298\pi$
$860$	0	0
$861$	−2.37459	−0.0809257
$862$	0	0
$863$	−36.7492	−1.25096	−0.625478	−	0.780242i	$-0.715096\pi$
$863$	−36.7492	−1.25096	−0.625478	+	0.780242i	$0.715096\pi$
$864$	0	0
$865$	−7.09967	−0.241396
$866$	0	0
$867$	−15.3746	−0.522148
$868$	0	0
$869$	35.4743	1.20338
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	6.72508	0.227610
$874$	0	0
$875$	3.27492	0.110712
$876$	0	0
$877$	27.0997	0.915091	0.457545	−	0.889186i	$-0.348729\pi$
$877$	27.0997	0.915091	0.457545	+	0.889186i	$0.348729\pi$
$878$	0	0
$879$	−7.45017	−0.251288
$880$	0	0
$881$	−7.45017	−0.251002	−0.125501	−	0.992093i	$-0.540054\pi$
$881$	−7.45017	−0.251002	−0.125501	+	0.992093i	$0.540054\pi$
$882$	0	0
$883$	6.19934	0.208624	0.104312	−	0.994545i	$-0.466736\pi$
$883$	6.19934	0.208624	0.104312	+	0.994545i	$0.466736\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	30.9244	1.03834	0.519170	−	0.854671i	$-0.326241\pi$
$887$	30.9244	1.03834	0.519170	+	0.854671i	$0.326241\pi$
$888$	0	0
$889$	−29.8007	−0.999482
$890$	0	0
$891$	5.27492	0.176716
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	7.27492	0.242902
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0.924421	0.0307969
$902$	0	0
$903$	21.4502	0.713817
$904$	0	0
$905$	9.82475	0.326586
$906$	0	0
$907$	6.54983	0.217484	0.108742	−	0.994070i	$-0.465318\pi$
$907$	6.54983	0.217484	0.108742	+	0.994070i	$0.465318\pi$
$908$	0	0
$909$	−17.0997	−0.567160
$910$	0	0
$911$	11.6495	0.385965	0.192983	−	0.981202i	$-0.438184\pi$
$911$	11.6495	0.385965	0.192983	+	0.981202i	$0.438184\pi$
$912$	0	0
$913$	63.2990	2.09489
$914$	0	0
$915$	13.8248	0.457032
$916$	0	0
$917$	11.2990	0.373126
$918$	0	0
$919$	−4.17525	−0.137729	−0.0688644	−	0.997626i	$-0.521938\pi$
$919$	−4.17525	−0.137729	−0.0688644	+	0.997626i	$0.521938\pi$
$920$	0	0
$921$	30.3746	1.00088
$922$	0	0
$923$	9.27492	0.305288
$924$	0	0
$925$	0.725083	0.0238406
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	34.1752	1.12125	0.560627	−	0.828069i	$-0.310560\pi$
$929$	34.1752	1.12125	0.560627	+	0.828069i	$0.310560\pi$
$930$	0	0
$931$	−7.45017	−0.244169
$932$	0	0
$933$	−23.6495	−0.774250
$934$	0	0
$935$	−6.72508	−0.219934
$936$	0	0
$937$	−19.4502	−0.635409	−0.317705	−	0.948190i	$-0.602912\pi$
$937$	−19.4502	−0.635409	−0.317705	+	0.948190i	$0.602912\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	0	0
$941$	15.2749	0.497948	0.248974	−	0.968510i	$-0.419907\pi$
$941$	15.2749	0.497948	0.248974	+	0.968510i	$0.419907\pi$
$942$	0	0
$943$	5.27492	0.171775
$944$	0	0
$945$	3.27492	0.106533
$946$	0	0
$947$	25.0997	0.815630	0.407815	−	0.913065i	$-0.366291\pi$
$947$	25.0997	0.815630	0.407815	+	0.913065i	$0.366291\pi$
$948$	0	0
$949$	2.54983	0.0827711
$950$	0	0
$951$	7.09967	0.230223
$952$	0	0
$953$	−51.8248	−1.67877	−0.839384	−	0.543539i	$-0.817084\pi$
$953$	−51.8248	−1.67877	−0.839384	+	0.543539i	$0.817084\pi$
$954$	0	0
$955$	10.5498	0.341385
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−32.7492	−1.05753
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	2.72508	0.0878146
$964$	0	0
$965$	−5.27492	−0.169806
$966$	0	0
$967$	3.45017	0.110950	0.0554749	−	0.998460i	$-0.482333\pi$
$967$	3.45017	0.110950	0.0554749	+	0.998460i	$0.482333\pi$
$968$	0	0
$969$	−2.54983	−0.0819125
$970$	0	0
$971$	37.2990	1.19698	0.598491	−	0.801129i	$-0.295767\pi$
$971$	37.2990	1.19698	0.598491	+	0.801129i	$0.295767\pi$
$972$	0	0
$973$	−0.573924	−0.0183992
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	8.90033	0.284747	0.142373	−	0.989813i	$-0.454527\pi$
$977$	8.90033	0.284747	0.142373	+	0.989813i	$0.454527\pi$
$978$	0	0
$979$	−3.82475	−0.122240
$980$	0	0
$981$	16.0000	0.510841
$982$	0	0
$983$	58.1993	1.85627	0.928135	−	0.372243i	$-0.121411\pi$
$983$	58.1993	1.85627	0.928135	+	0.372243i	$0.121411\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	13.0997	0.416967
$988$	0	0
$989$	−47.6495	−1.51517
$990$	0	0
$991$	−1.62541	−0.0516330	−0.0258165	−	0.999667i	$-0.508219\pi$
$991$	−1.62541	−0.0516330	−0.0258165	+	0.999667i	$0.508219\pi$
$992$	0	0
$993$	−15.0997	−0.479174
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−21.6495	−0.685647	−0.342823	−	0.939400i	$-0.611383\pi$
$997$	−21.6495	−0.685647	−0.342823	+	0.939400i	$0.611383\pi$
$998$	0	0
$999$	0.725083	0.0229406

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bg.1.2	✓	2	4.3	odd	2
6240.2.a.bs.1.1	yes	2	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -3.27492 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -3.27492 q^{7} +1.00000 q^{9} +5.27492 q^{11} +1.00000 q^{13} -1.00000 q^{15} +1.27492 q^{17} -2.00000 q^{19} -3.27492 q^{21} +7.27492 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{31} +5.27492 q^{33} +3.27492 q^{35} +0.725083 q^{37} +1.00000 q^{39} +0.725083 q^{41} -6.54983 q^{43} -1.00000 q^{45} -4.00000 q^{47} +3.72508 q^{49} +1.27492 q^{51} +0.725083 q^{53} -5.27492 q^{55} -2.00000 q^{57} +4.00000 q^{59} -13.8248 q^{61} -3.27492 q^{63} -1.00000 q^{65} +8.00000 q^{67} +7.27492 q^{69} +9.27492 q^{71} +2.54983 q^{73} +1.00000 q^{75} -17.2749 q^{77} +6.72508 q^{79} +1.00000 q^{81} +12.0000 q^{83} -1.27492 q^{85} -0.725083 q^{89} -3.27492 q^{91} -4.00000 q^{93} +2.00000 q^{95} +6.72508 q^{97} +5.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 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

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