LMFDB - Embedded newform 6840.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6840.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:	$54.6176749826$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	6840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +2.71982 q^{7} +5.55691 q^{11} +2.02760 q^{13} +3.77846 q^{17} +1.00000 q^{19} +5.77846 q^{23} +1.00000 q^{25} +5.66119 q^{29} +7.55691 q^{31} +2.71982 q^{35} -3.75086 q^{37} +12.6155 q^{41} -9.43965 q^{43} -11.1138 q^{47} +0.397442 q^{49} +8.85170 q^{53} +5.55691 q^{55} -11.4526 q^{59} -10.6155 q^{61} +2.02760 q^{65} -11.5845 q^{67} -9.45264 q^{73} +15.1138 q^{77} +8.94137 q^{79} +4.94137 q^{83} +3.77846 q^{85} -15.4948 q^{89} +5.51471 q^{91} +1.00000 q^{95} +10.8647 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - q^{7} - 11 q^{13} + 3 q^{17} + 3 q^{19} + 9 q^{23} + 3 q^{25} + 7 q^{29} + 6 q^{31} - q^{35} - 20 q^{37} + 22 q^{41} - 10 q^{43} + 12 q^{49} + 7 q^{53} - 11 q^{59} - 16 q^{61} - 11 q^{65}+ \cdots + 8 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - q^7 - 11 * q^13 + 3 * q^17 + 3 * q^19 + 9 * q^23 + 3 * q^25 + 7 * q^29 + 6 * q^31 - q^35 - 20 * q^37 + 22 * q^41 - 10 * q^43 + 12 * q^49 + 7 * q^53 - 11 * q^59 - 16 * q^61 - 11 * q^65 - q^67 - 5 * q^73 + 12 * q^77 + 26 * q^79 + 14 * q^83 + 3 * q^85 + 6 * q^89 + 29 * q^91 + 3 * q^95 + 8 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.71982	1.02800	0.513998	−	0.857791i	$-0.328164\pi$
$7$	2.71982	1.02800	0.513998	+	0.857791i	$0.328164\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.55691	1.67547	0.837736	−	0.546075i	$-0.183879\pi$
$11$	5.55691	1.67547	0.837736	+	0.546075i	$0.183879\pi$
$12$	0	0
$13$	2.02760	0.562354	0.281177	−	0.959656i	$-0.409275\pi$
$13$	2.02760	0.562354	0.281177	+	0.959656i	$0.409275\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.77846	0.916410	0.458205	−	0.888846i	$-0.348492\pi$
$17$	3.77846	0.916410	0.458205	+	0.888846i	$0.348492\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.77846	1.20489	0.602446	−	0.798160i	$-0.294193\pi$
$23$	5.77846	1.20489	0.602446	+	0.798160i	$0.294193\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.66119	1.05126	0.525628	−	0.850714i	$-0.323830\pi$
$29$	5.66119	1.05126	0.525628	+	0.850714i	$0.323830\pi$
$30$	0	0
$31$	7.55691	1.35726	0.678631	−	0.734479i	$-0.262574\pi$
$31$	7.55691	1.35726	0.678631	+	0.734479i	$0.262574\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.71982	0.459734
$36$	0	0
$37$	−3.75086	−0.616637	−0.308319	−	0.951283i	$-0.599766\pi$
$37$	−3.75086	−0.616637	−0.308319	+	0.951283i	$0.599766\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.6155	1.97022	0.985109	−	0.171932i	$-0.0550011\pi$
$41$	12.6155	1.97022	0.985109	+	0.171932i	$0.0550011\pi$
$42$	0	0
$43$	−9.43965	−1.43953	−0.719766	−	0.694216i	$-0.755751\pi$
$43$	−9.43965	−1.43953	−0.719766	+	0.694216i	$0.755751\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.1138	−1.62112	−0.810559	−	0.585657i	$-0.800837\pi$
$47$	−11.1138	−1.62112	−0.810559	+	0.585657i	$0.800837\pi$
$48$	0	0
$49$	0.397442	0.0567775
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.85170	1.21587	0.607937	−	0.793985i	$-0.291997\pi$
$53$	8.85170	1.21587	0.607937	+	0.793985i	$0.291997\pi$
$54$	0	0
$55$	5.55691	0.749294
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.4526	−1.49101	−0.745503	−	0.666502i	$-0.767791\pi$
$59$	−11.4526	−1.49101	−0.745503	+	0.666502i	$0.767791\pi$
$60$	0	0
$61$	−10.6155	−1.35918	−0.679591	−	0.733591i	$-0.737843\pi$
$61$	−10.6155	−1.35918	−0.679591	+	0.733591i	$0.737843\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.02760	0.251493
$66$	0	0
$67$	−11.5845	−1.41527	−0.707637	−	0.706576i	$-0.750239\pi$
$67$	−11.5845	−1.41527	−0.707637	+	0.706576i	$0.750239\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−9.45264	−1.10635	−0.553174	−	0.833066i	$-0.686583\pi$
$73$	−9.45264	−1.10635	−0.553174	+	0.833066i	$0.686583\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.1138	1.72238
$78$	0	0
$79$	8.94137	1.00598	0.502991	−	0.864292i	$-0.332233\pi$
$79$	8.94137	1.00598	0.502991	+	0.864292i	$0.332233\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.94137	0.542385	0.271193	−	0.962525i	$-0.412582\pi$
$83$	4.94137	0.542385	0.271193	+	0.962525i	$0.412582\pi$
$84$	0	0
$85$	3.77846	0.409831
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−15.4948	−1.64245	−0.821225	−	0.570604i	$-0.806709\pi$
$89$	−15.4948	−1.64245	−0.821225	+	0.570604i	$0.806709\pi$
$90$	0	0
$91$	5.51471	0.578099
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	10.8647	1.10314	0.551571	−	0.834128i	$-0.314029\pi$
$97$	10.8647	1.10314	0.551571	+	0.834128i	$0.314029\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.49828	−0.447596	−0.223798	−	0.974636i	$-0.571846\pi$
$101$	−4.49828	−0.447596	−0.223798	+	0.974636i	$0.571846\pi$
$102$	0	0
$103$	4.36641	0.430235	0.215117	−	0.976588i	$-0.430987\pi$
$103$	4.36641	0.430235	0.215117	+	0.976588i	$0.430987\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.64658	0.159181	0.0795906	−	0.996828i	$-0.474639\pi$
$107$	1.64658	0.159181	0.0795906	+	0.996828i	$0.474639\pi$
$108$	0	0
$109$	−0.954357	−0.0914108	−0.0457054	−	0.998955i	$-0.514554\pi$
$109$	−0.954357	−0.0914108	−0.0457054	+	0.998955i	$0.514554\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.68879	−0.535156	−0.267578	−	0.963536i	$-0.586223\pi$
$113$	−5.68879	−0.535156	−0.267578	+	0.963536i	$0.586223\pi$
$114$	0	0
$115$	5.77846	0.538844
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.2767	0.942067
$120$	0	0
$121$	19.8793	1.80721
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.3043	−0.914362	−0.457181	−	0.889374i	$-0.651141\pi$
$127$	−10.3043	−0.914362	−0.457181	+	0.889374i	$0.651141\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.11383	−0.272056	−0.136028	−	0.990705i	$-0.543434\pi$
$131$	−3.11383	−0.272056	−0.136028	+	0.990705i	$0.543434\pi$
$132$	0	0
$133$	2.71982	0.235839
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.4526	−1.14934	−0.574668	−	0.818386i	$-0.694869\pi$
$137$	−13.4526	−1.14934	−0.574668	+	0.818386i	$0.694869\pi$
$138$	0	0
$139$	9.55691	0.810607	0.405303	−	0.914182i	$-0.367166\pi$
$139$	9.55691	0.810607	0.405303	+	0.914182i	$0.367166\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.2672	0.942209
$144$	0	0
$145$	5.66119	0.470136
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.6121	−1.60669	−0.803343	−	0.595516i	$-0.796948\pi$
$149$	−19.6121	−1.60669	−0.803343	+	0.595516i	$0.796948\pi$
$150$	0	0
$151$	−17.0518	−1.38765	−0.693826	−	0.720143i	$-0.744076\pi$
$151$	−17.0518	−1.38765	−0.693826	+	0.720143i	$0.744076\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.55691	0.606986
$156$	0	0
$157$	−22.4983	−1.79556	−0.897779	−	0.440446i	$-0.854820\pi$
$157$	−22.4983	−1.79556	−0.897779	+	0.440446i	$0.854820\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	15.7164	1.23862
$162$	0	0
$163$	22.4362	1.75734	0.878670	−	0.477430i	$-0.158432\pi$
$163$	22.4362	1.75734	0.878670	+	0.477430i	$0.158432\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.69223	0.517860	0.258930	−	0.965896i	$-0.416630\pi$
$167$	6.69223	0.517860	0.258930	+	0.965896i	$0.416630\pi$
$168$	0	0
$169$	−8.88885	−0.683758
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.86469	0.217798	0.108899	−	0.994053i	$-0.465267\pi$
$173$	2.86469	0.217798	0.108899	+	0.994053i	$0.465267\pi$
$174$	0	0
$175$	2.71982	0.205599
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.38445	−0.103479	−0.0517394	−	0.998661i	$-0.516477\pi$
$179$	−1.38445	−0.103479	−0.0517394	+	0.998661i	$0.516477\pi$
$180$	0	0
$181$	−20.8793	−1.55195	−0.775973	−	0.630766i	$-0.782741\pi$
$181$	−20.8793	−1.55195	−0.775973	+	0.630766i	$0.782741\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.75086	−0.275769
$186$	0	0
$187$	20.9966	1.53542
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.2733	−1.39457	−0.697284	−	0.716795i	$-0.745608\pi$
$191$	−19.2733	−1.39457	−0.697284	+	0.716795i	$0.745608\pi$
$192$	0	0
$193$	−8.01461	−0.576904	−0.288452	−	0.957494i	$-0.593141\pi$
$193$	−8.01461	−0.576904	−0.288452	+	0.957494i	$0.593141\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.8793	−1.20260	−0.601300	−	0.799023i	$-0.705350\pi$
$197$	−16.8793	−1.20260	−0.601300	+	0.799023i	$0.705350\pi$
$198$	0	0
$199$	1.28018	0.0907493	0.0453746	−	0.998970i	$-0.485552\pi$
$199$	1.28018	0.0907493	0.0453746	+	0.998970i	$0.485552\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.3974	1.08069
$204$	0	0
$205$	12.6155	0.881108
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.55691	0.384380
$210$	0	0
$211$	−1.54392	−0.106288	−0.0531441	−	0.998587i	$-0.516924\pi$
$211$	−1.54392	−0.106288	−0.0531441	+	0.998587i	$0.516924\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.43965	−0.643779
$216$	0	0
$217$	20.5535	1.39526
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.66119	0.515347
$222$	0	0
$223$	−3.92332	−0.262725	−0.131363	−	0.991334i	$-0.541935\pi$
$223$	−3.92332	−0.262725	−0.131363	+	0.991334i	$0.541935\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.9069	−0.723916	−0.361958	−	0.932194i	$-0.617892\pi$
$227$	−10.9069	−0.723916	−0.361958	+	0.932194i	$0.617892\pi$
$228$	0	0
$229$	10.1725	0.672215	0.336108	−	0.941824i	$-0.390889\pi$
$229$	10.1725	0.672215	0.336108	+	0.941824i	$0.390889\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.11383	0.597067	0.298533	−	0.954399i	$-0.403503\pi$
$233$	9.11383	0.597067	0.298533	+	0.954399i	$0.403503\pi$
$234$	0	0
$235$	−11.1138	−0.724986
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.1595	−1.30401	−0.652004	−	0.758216i	$-0.726071\pi$
$239$	−20.1595	−1.30401	−0.652004	+	0.758216i	$0.726071\pi$
$240$	0	0
$241$	−4.82410	−0.310748	−0.155374	−	0.987856i	$-0.549658\pi$
$241$	−4.82410	−0.310748	−0.155374	+	0.987856i	$0.549658\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.397442	0.0253917
$246$	0	0
$247$	2.02760	0.129013
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.2277	1.15052	0.575260	−	0.817971i	$-0.304901\pi$
$251$	18.2277	1.15052	0.575260	+	0.817971i	$0.304901\pi$
$252$	0	0
$253$	32.1104	2.01876
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.5941	−1.78365	−0.891824	−	0.452382i	$-0.850574\pi$
$257$	−28.5941	−1.78365	−0.891824	+	0.452382i	$0.850574\pi$
$258$	0	0
$259$	−10.2017	−0.633901
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27.3776	1.68817	0.844087	−	0.536206i	$-0.180143\pi$
$263$	27.3776	1.68817	0.844087	+	0.536206i	$0.180143\pi$
$264$	0	0
$265$	8.85170	0.543755
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	15.8337	0.961826	0.480913	−	0.876768i	$-0.340305\pi$
$271$	15.8337	0.961826	0.480913	+	0.876768i	$0.340305\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.55691	0.335095
$276$	0	0
$277$	−6.70683	−0.402975	−0.201487	−	0.979491i	$-0.564577\pi$
$277$	−6.70683	−0.402975	−0.201487	+	0.979491i	$0.564577\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8.38101	−0.499969	−0.249985	−	0.968250i	$-0.580426\pi$
$281$	−8.38101	−0.499969	−0.249985	+	0.968250i	$0.580426\pi$
$282$	0	0
$283$	−20.2897	−1.20610	−0.603050	−	0.797704i	$-0.706048\pi$
$283$	−20.2897	−1.20610	−0.603050	+	0.797704i	$0.706048\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	34.3121	2.02538
$288$	0	0
$289$	−2.72326	−0.160192
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.76041	0.394947	0.197474	−	0.980308i	$-0.436726\pi$
$293$	6.76041	0.394947	0.197474	+	0.980308i	$0.436726\pi$
$294$	0	0
$295$	−11.4526	−0.666798
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.7164	0.677576
$300$	0	0
$301$	−25.6742	−1.47984
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−10.6155	−0.607844
$306$	0	0
$307$	7.74398	0.441972	0.220986	−	0.975277i	$-0.429072\pi$
$307$	7.74398	0.441972	0.220986	+	0.975277i	$0.429072\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.0456	0.626341	0.313170	−	0.949697i	$-0.398609\pi$
$311$	11.0456	0.626341	0.313170	+	0.949697i	$0.398609\pi$
$312$	0	0
$313$	1.16291	0.0657315	0.0328658	−	0.999460i	$-0.489537\pi$
$313$	1.16291	0.0657315	0.0328658	+	0.999460i	$0.489537\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.8742	1.22858	0.614290	−	0.789080i	$-0.289443\pi$
$317$	21.8742	1.22858	0.614290	+	0.789080i	$0.289443\pi$
$318$	0	0
$319$	31.4588	1.76135
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.77846	0.210239
$324$	0	0
$325$	2.02760	0.112471
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−30.2277	−1.66650
$330$	0	0
$331$	14.6646	0.806041	0.403020	−	0.915191i	$-0.367960\pi$
$331$	14.6646	0.806041	0.403020	+	0.915191i	$0.367960\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.5845	−0.632929
$336$	0	0
$337$	−20.0958	−1.09469	−0.547344	−	0.836908i	$-0.684361\pi$
$337$	−20.0958	−1.09469	−0.547344	+	0.836908i	$0.684361\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	41.9931	2.27406
$342$	0	0
$343$	−17.9578	−0.969630
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.0586	0.593659	0.296829	−	0.954931i	$-0.404071\pi$
$347$	11.0586	0.593659	0.296829	+	0.954931i	$0.404071\pi$
$348$	0	0
$349$	8.11727	0.434507	0.217254	−	0.976115i	$-0.430290\pi$
$349$	8.11727	0.434507	0.217254	+	0.976115i	$0.430290\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.0130	−0.639387	−0.319693	−	0.947521i	$-0.603580\pi$
$353$	−12.0130	−0.639387	−0.319693	+	0.947521i	$0.603580\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.9509	0.947413	0.473707	−	0.880683i	$-0.342916\pi$
$359$	17.9509	0.947413	0.473707	+	0.880683i	$0.342916\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.45264	−0.494774
$366$	0	0
$367$	−3.11383	−0.162541	−0.0812703	−	0.996692i	$-0.525898\pi$
$367$	−3.11383	−0.162541	−0.0812703	+	0.996692i	$0.525898\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	24.0751	1.24991
$372$	0	0
$373$	−21.8742	−1.13261	−0.566303	−	0.824197i	$-0.691627\pi$
$373$	−21.8742	−1.13261	−0.566303	+	0.824197i	$0.691627\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.4786	0.591179
$378$	0	0
$379$	−5.28018	−0.271224	−0.135612	−	0.990762i	$-0.543300\pi$
$379$	−5.28018	−0.271224	−0.135612	+	0.990762i	$0.543300\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.1319	0.824300	0.412150	−	0.911116i	$-0.364778\pi$
$383$	16.1319	0.824300	0.412150	+	0.911116i	$0.364778\pi$
$384$	0	0
$385$	15.1138	0.770272
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.43965	−0.174397	−0.0871985	−	0.996191i	$-0.527791\pi$
$389$	−3.43965	−0.174397	−0.0871985	+	0.996191i	$0.527791\pi$
$390$	0	0
$391$	21.8337	1.10418
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.94137	0.449889
$396$	0	0
$397$	−1.29317	−0.0649021	−0.0324511	−	0.999473i	$-0.510331\pi$
$397$	−1.29317	−0.0649021	−0.0324511	+	0.999473i	$0.510331\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.1070	1.75316	0.876579	−	0.481258i	$-0.159820\pi$
$401$	35.1070	1.75316	0.876579	+	0.481258i	$0.159820\pi$
$402$	0	0
$403$	15.3224	0.763262
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.8432	−1.03316
$408$	0	0
$409$	−24.8793	−1.23020	−0.615101	−	0.788448i	$-0.710885\pi$
$409$	−24.8793	−1.23020	−0.615101	+	0.788448i	$0.710885\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−31.1492	−1.53275
$414$	0	0
$415$	4.94137	0.242562
$416$	0	0
$417$	0	0
$418$	0	0
$419$	35.4328	1.73100	0.865502	−	0.500905i	$-0.167000\pi$
$419$	35.4328	1.73100	0.865502	+	0.500905i	$0.167000\pi$
$420$	0	0
$421$	14.2147	0.692780	0.346390	−	0.938091i	$-0.387407\pi$
$421$	14.2147	0.692780	0.346390	+	0.938091i	$0.387407\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.77846	0.183282
$426$	0	0
$427$	−28.8724	−1.39723
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.4983	1.18004	0.590020	−	0.807388i	$-0.299120\pi$
$431$	24.4983	1.18004	0.590020	+	0.807388i	$0.299120\pi$
$432$	0	0
$433$	9.45426	0.454343	0.227171	−	0.973855i	$-0.427052\pi$
$433$	9.45426	0.454343	0.227171	+	0.973855i	$0.427052\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.77846	0.276421
$438$	0	0
$439$	9.70340	0.463118	0.231559	−	0.972821i	$-0.425617\pi$
$439$	9.70340	0.463118	0.231559	+	0.972821i	$0.425617\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.85008	0.325457	0.162729	−	0.986671i	$-0.447971\pi$
$443$	6.85008	0.325457	0.162729	+	0.986671i	$0.447971\pi$
$444$	0	0
$445$	−15.4948	−0.734526
$446$	0	0
$447$	0	0
$448$	0	0
$449$	25.7655	1.21595	0.607974	−	0.793957i	$-0.291983\pi$
$449$	25.7655	1.21595	0.607974	+	0.793957i	$0.291983\pi$
$450$	0	0
$451$	70.1035	3.30105
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.51471	0.258534
$456$	0	0
$457$	0.400880	0.0187524	0.00937619	−	0.999956i	$-0.497015\pi$
$457$	0.400880	0.0187524	0.00937619	+	0.999956i	$0.497015\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.1070	−1.07620	−0.538099	−	0.842882i	$-0.680857\pi$
$461$	−23.1070	−1.07620	−0.538099	+	0.842882i	$0.680857\pi$
$462$	0	0
$463$	4.96735	0.230852	0.115426	−	0.993316i	$-0.463177\pi$
$463$	4.96735	0.230852	0.115426	+	0.993316i	$0.463177\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.6190	−1.00041	−0.500204	−	0.865908i	$-0.666742\pi$
$467$	−21.6190	−1.00041	−0.500204	+	0.865908i	$0.666742\pi$
$468$	0	0
$469$	−31.5078	−1.45490
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−52.4553	−2.41190
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.20855	−0.100911	−0.0504557	−	0.998726i	$-0.516067\pi$
$479$	−2.20855	−0.100911	−0.0504557	+	0.998726i	$0.516067\pi$
$480$	0	0
$481$	−7.60523	−0.346769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.8647	0.493340
$486$	0	0
$487$	−11.4250	−0.517718	−0.258859	−	0.965915i	$-0.583346\pi$
$487$	−11.4250	−0.517718	−0.258859	+	0.965915i	$0.583346\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−23.6673	−1.06809	−0.534045	−	0.845456i	$-0.679329\pi$
$491$	−23.6673	−1.06809	−0.534045	+	0.845456i	$0.679329\pi$
$492$	0	0
$493$	21.3906	0.963383
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3.99312	0.178757	0.0893784	−	0.995998i	$-0.471512\pi$
$499$	3.99312	0.178757	0.0893784	+	0.995998i	$0.471512\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.338809	−0.0151068	−0.00755338	−	0.999971i	$-0.502404\pi$
$503$	−0.338809	−0.0151068	−0.00755338	+	0.999971i	$0.502404\pi$
$504$	0	0
$505$	−4.49828	−0.200171
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.4914	1.26286	0.631430	−	0.775433i	$-0.282468\pi$
$509$	28.4914	1.26286	0.631430	+	0.775433i	$0.282468\pi$
$510$	0	0
$511$	−25.7095	−1.13732
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.36641	0.192407
$516$	0	0
$517$	−61.7586	−2.71614
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−41.4328	−1.81520	−0.907601	−	0.419833i	$-0.862089\pi$
$521$	−41.4328	−1.81520	−0.907601	+	0.419833i	$0.862089\pi$
$522$	0	0
$523$	−22.1741	−0.969605	−0.484802	−	0.874624i	$-0.661108\pi$
$523$	−22.1741	−0.969605	−0.484802	+	0.874624i	$0.661108\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.5535	1.24381
$528$	0	0
$529$	10.3906	0.451764
$530$	0	0
$531$	0	0
$532$	0	0
$533$	25.5793	1.10796
$534$	0	0
$535$	1.64658	0.0711880
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.20855	0.0951291
$540$	0	0
$541$	−7.61211	−0.327270	−0.163635	−	0.986521i	$-0.552322\pi$
$541$	−7.61211	−0.327270	−0.163635	+	0.986521i	$0.552322\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.954357	−0.0408801
$546$	0	0
$547$	−18.3303	−0.783748	−0.391874	−	0.920019i	$-0.628173\pi$
$547$	−18.3303	−0.783748	−0.391874	+	0.920019i	$0.628173\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.66119	0.241175
$552$	0	0
$553$	24.3189	1.03415
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.7000	−1.55503	−0.777514	−	0.628866i	$-0.783519\pi$
$557$	−36.7000	−1.55503	−0.777514	+	0.628866i	$0.783519\pi$
$558$	0	0
$559$	−19.1398	−0.809528
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.1319	0.679877	0.339939	−	0.940448i	$-0.389594\pi$
$563$	16.1319	0.679877	0.339939	+	0.940448i	$0.389594\pi$
$564$	0	0
$565$	−5.68879	−0.239329
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.2051	0.805120	0.402560	−	0.915394i	$-0.368120\pi$
$569$	19.2051	0.805120	0.402560	+	0.915394i	$0.368120\pi$
$570$	0	0
$571$	−2.46907	−0.103327	−0.0516636	−	0.998665i	$-0.516452\pi$
$571$	−2.46907	−0.103327	−0.0516636	+	0.998665i	$0.516452\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.77846	0.240978
$576$	0	0
$577$	11.7233	0.488046	0.244023	−	0.969769i	$-0.421533\pi$
$577$	11.7233	0.488046	0.244023	+	0.969769i	$0.421533\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.4396	0.557571
$582$	0	0
$583$	49.1881	2.03716
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.28973	0.342154	0.171077	−	0.985258i	$-0.445275\pi$
$587$	8.28973	0.342154	0.171077	+	0.985258i	$0.445275\pi$
$588$	0	0
$589$	7.55691	0.311377
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.5760	1.29667	0.648336	−	0.761354i	$-0.275465\pi$
$593$	31.5760	1.29667	0.648336	+	0.761354i	$0.275465\pi$
$594$	0	0
$595$	10.2767	0.421305
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.28629	−0.0525564	−0.0262782	−	0.999655i	$-0.508366\pi$
$599$	−1.28629	−0.0525564	−0.0262782	+	0.999655i	$0.508366\pi$
$600$	0	0
$601$	38.8432	1.58445	0.792224	−	0.610231i	$-0.208923\pi$
$601$	38.8432	1.58445	0.792224	+	0.610231i	$0.208923\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19.8793	0.808208
$606$	0	0
$607$	43.6888	1.77327	0.886637	−	0.462467i	$-0.153036\pi$
$607$	43.6888	1.77327	0.886637	+	0.462467i	$0.153036\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22.5344	−0.911643
$612$	0	0
$613$	−30.8172	−1.24470	−0.622348	−	0.782741i	$-0.713821\pi$
$613$	−30.8172	−1.24470	−0.622348	+	0.782741i	$0.713821\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.4983	−0.422645	−0.211322	−	0.977416i	$-0.567777\pi$
$617$	−10.4983	−0.422645	−0.211322	+	0.977416i	$0.567777\pi$
$618$	0	0
$619$	13.3224	0.535472	0.267736	−	0.963492i	$-0.413725\pi$
$619$	13.3224	0.535472	0.267736	+	0.963492i	$0.413725\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−42.1432	−1.68843
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14.1725	−0.565093
$630$	0	0
$631$	1.21199	0.0482486	0.0241243	−	0.999709i	$-0.492320\pi$
$631$	1.21199	0.0482486	0.0241243	+	0.999709i	$0.492320\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10.3043	−0.408915
$636$	0	0
$637$	0.805853	0.0319291
$638$	0	0
$639$	0	0
$640$	0	0
$641$	49.6965	1.96289	0.981447	−	0.191732i	$-0.0614105\pi$
$641$	49.6965	1.96289	0.981447	+	0.191732i	$0.0614105\pi$
$642$	0	0
$643$	−39.0449	−1.53978	−0.769890	−	0.638177i	$-0.779689\pi$
$643$	−39.0449	−1.53978	−0.769890	+	0.638177i	$0.779689\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.4232	1.11743	0.558716	−	0.829359i	$-0.311294\pi$
$647$	28.4232	1.11743	0.558716	+	0.829359i	$0.311294\pi$
$648$	0	0
$649$	−63.6413	−2.49814
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−29.0586	−1.13715	−0.568576	−	0.822631i	$-0.692506\pi$
$653$	−29.0586	−1.13715	−0.568576	+	0.822631i	$0.692506\pi$
$654$	0	0
$655$	−3.11383	−0.121667
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−11.8957	−0.463392	−0.231696	−	0.972788i	$-0.574427\pi$
$659$	−11.8957	−0.463392	−0.231696	+	0.972788i	$0.574427\pi$
$660$	0	0
$661$	30.8923	1.20157	0.600785	−	0.799410i	$-0.294855\pi$
$661$	30.8923	1.20157	0.600785	+	0.799410i	$0.294855\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.71982	0.105470
$666$	0	0
$667$	32.7129	1.26665
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−58.9897	−2.27727
$672$	0	0
$673$	37.5354	1.44688	0.723442	−	0.690385i	$-0.242559\pi$
$673$	37.5354	1.44688	0.723442	+	0.690385i	$0.242559\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.6466	1.21628	0.608138	−	0.793831i	$-0.291917\pi$
$677$	31.6466	1.21628	0.608138	+	0.793831i	$0.291917\pi$
$678$	0	0
$679$	29.5500	1.13403
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−22.6233	−0.865656	−0.432828	−	0.901477i	$-0.642484\pi$
$683$	−22.6233	−0.865656	−0.432828	+	0.901477i	$0.642484\pi$
$684$	0	0
$685$	−13.4526	−0.513999
$686$	0	0
$687$	0	0
$688$	0	0
$689$	17.9477	0.683752
$690$	0	0
$691$	17.7655	0.675830	0.337915	−	0.941177i	$-0.390278\pi$
$691$	17.7655	0.675830	0.337915	+	0.941177i	$0.390278\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.55691	0.362514
$696$	0	0
$697$	47.6673	1.80553
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−25.6052	−0.967096	−0.483548	−	0.875318i	$-0.660652\pi$
$701$	−25.6052	−0.967096	−0.483548	+	0.875318i	$0.660652\pi$
$702$	0	0
$703$	−3.75086	−0.141466
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.2345	−0.460127
$708$	0	0
$709$	23.2311	0.872462	0.436231	−	0.899835i	$-0.356313\pi$
$709$	23.2311	0.872462	0.436231	+	0.899835i	$0.356313\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	43.6673	1.63535
$714$	0	0
$715$	11.2672	0.421369
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.9544	−0.930640	−0.465320	−	0.885142i	$-0.654061\pi$
$719$	−24.9544	−0.930640	−0.465320	+	0.885142i	$0.654061\pi$
$720$	0	0
$721$	11.8759	0.442280
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.66119	0.210251
$726$	0	0
$727$	−3.39744	−0.126004	−0.0630021	−	0.998013i	$-0.520067\pi$
$727$	−3.39744	−0.126004	−0.0630021	+	0.998013i	$0.520067\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−35.6673	−1.31920
$732$	0	0
$733$	−9.66730	−0.357070	−0.178535	−	0.983934i	$-0.557136\pi$
$733$	−9.66730	−0.357070	−0.178535	+	0.983934i	$0.557136\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−64.3741	−2.37125
$738$	0	0
$739$	17.0225	0.626184	0.313092	−	0.949723i	$-0.398635\pi$
$739$	17.0225	0.626184	0.313092	+	0.949723i	$0.398635\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.57496	−0.314585	−0.157292	−	0.987552i	$-0.550277\pi$
$743$	−8.57496	−0.314585	−0.157292	+	0.987552i	$0.550277\pi$
$744$	0	0
$745$	−19.6121	−0.718532
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.47842	0.163638
$750$	0	0
$751$	12.8310	0.468209	0.234104	−	0.972211i	$-0.424784\pi$
$751$	12.8310	0.468209	0.234104	+	0.972211i	$0.424784\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−17.0518	−0.620577
$756$	0	0
$757$	−35.8827	−1.30418	−0.652090	−	0.758142i	$-0.726108\pi$
$757$	−35.8827	−1.30418	−0.652090	+	0.758142i	$0.726108\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.1234	1.09197	0.545986	−	0.837794i	$-0.316155\pi$
$761$	30.1234	1.09197	0.545986	+	0.837794i	$0.316155\pi$
$762$	0	0
$763$	−2.59568	−0.0939700
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23.2213	−0.838474
$768$	0	0
$769$	−6.22154	−0.224355	−0.112177	−	0.993688i	$-0.535782\pi$
$769$	−6.22154	−0.224355	−0.112177	+	0.993688i	$0.535782\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10.2362	−0.368169	−0.184084	−	0.982910i	$-0.558932\pi$
$773$	−10.2362	−0.368169	−0.184084	+	0.982910i	$0.558932\pi$
$774$	0	0
$775$	7.55691	0.271452
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.6155	0.451999
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22.4983	−0.802998
$786$	0	0
$787$	−4.34654	−0.154937	−0.0774687	−	0.996995i	$-0.524684\pi$
$787$	−4.34654	−0.154937	−0.0774687	+	0.996995i	$0.524684\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.4725	−0.550139
$792$	0	0
$793$	−21.5241	−0.764342
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.1932	1.06950	0.534749	−	0.845011i	$-0.320406\pi$
$797$	30.1932	1.06950	0.534749	+	0.845011i	$0.320406\pi$
$798$	0	0
$799$	−41.9931	−1.48561
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−52.5275	−1.85366
$804$	0	0
$805$	15.7164	0.553930
$806$	0	0
$807$	0	0
$808$	0	0
$809$	11.8077	0.415136	0.207568	−	0.978221i	$-0.433445\pi$
$809$	11.8077	0.415136	0.207568	+	0.978221i	$0.433445\pi$
$810$	0	0
$811$	0.811111	0.0284820	0.0142410	−	0.999899i	$-0.495467\pi$
$811$	0.811111	0.0284820	0.0142410	+	0.999899i	$0.495467\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	22.4362	0.785906
$816$	0	0
$817$	−9.43965	−0.330251
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−40.6707	−1.41942	−0.709709	−	0.704495i	$-0.751174\pi$
$821$	−40.6707	−1.41942	−0.709709	+	0.704495i	$0.751174\pi$
$822$	0	0
$823$	−7.48185	−0.260801	−0.130401	−	0.991461i	$-0.541626\pi$
$823$	−7.48185	−0.260801	−0.130401	+	0.991461i	$0.541626\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.4346	1.61469	0.807344	−	0.590080i	$-0.200904\pi$
$827$	46.4346	1.61469	0.807344	+	0.590080i	$0.200904\pi$
$828$	0	0
$829$	−10.7267	−0.372554	−0.186277	−	0.982497i	$-0.559642\pi$
$829$	−10.7267	−0.372554	−0.186277	+	0.982497i	$0.559642\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.50172	0.0520315
$834$	0	0
$835$	6.69223	0.231594
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10.1465	−0.350295	−0.175148	−	0.984542i	$-0.556040\pi$
$839$	−10.1465	−0.350295	−0.175148	+	0.984542i	$0.556040\pi$
$840$	0	0
$841$	3.04908	0.105141
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.88885	−0.305786
$846$	0	0
$847$	54.0682	1.85780
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−21.6742	−0.742981
$852$	0	0
$853$	−26.1104	−0.894003	−0.447001	−	0.894533i	$-0.647508\pi$
$853$	−26.1104	−0.894003	−0.447001	+	0.894533i	$0.647508\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.6922	−0.706833	−0.353416	−	0.935466i	$-0.614980\pi$
$857$	−20.6922	−0.706833	−0.353416	+	0.935466i	$0.614980\pi$
$858$	0	0
$859$	3.53093	0.120474	0.0602370	−	0.998184i	$-0.480814\pi$
$859$	3.53093	0.120474	0.0602370	+	0.998184i	$0.480814\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.39906	0.115705	0.0578527	−	0.998325i	$-0.481575\pi$
$863$	3.39906	0.115705	0.0578527	+	0.998325i	$0.481575\pi$
$864$	0	0
$865$	2.86469	0.0974023
$866$	0	0
$867$	0	0
$868$	0	0
$869$	49.6864	1.68550
$870$	0	0
$871$	−23.4887	−0.795885
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.71982	0.0919468
$876$	0	0
$877$	−0.422364	−0.0142622	−0.00713111	−	0.999975i	$-0.502270\pi$
$877$	−0.422364	−0.0142622	−0.00713111	+	0.999975i	$0.502270\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16.5243	−0.556716	−0.278358	−	0.960477i	$-0.589790\pi$
$881$	−16.5243	−0.556716	−0.278358	+	0.960477i	$0.589790\pi$
$882$	0	0
$883$	24.5535	0.826290	0.413145	−	0.910665i	$-0.364430\pi$
$883$	24.5535	0.826290	0.413145	+	0.910665i	$0.364430\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.5975	1.39671	0.698354	−	0.715753i	$-0.253916\pi$
$887$	41.5975	1.39671	0.698354	+	0.715753i	$0.253916\pi$
$888$	0	0
$889$	−28.0260	−0.939961
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.1138	−0.371910
$894$	0	0
$895$	−1.38445	−0.0462771
$896$	0	0
$897$	0	0
$898$	0	0
$899$	42.7811	1.42683
$900$	0	0
$901$	33.4458	1.11424
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−20.8793	−0.694051
$906$	0	0
$907$	31.9294	1.06020	0.530100	−	0.847935i	$-0.322154\pi$
$907$	31.9294	1.06020	0.530100	+	0.847935i	$0.322154\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.9605	−0.893240	−0.446620	−	0.894724i	$-0.647372\pi$
$911$	−26.9605	−0.893240	−0.446620	+	0.894724i	$0.647372\pi$
$912$	0	0
$913$	27.4588	0.908752
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.46907	−0.279673
$918$	0	0
$919$	−0.394005	−0.0129970	−0.00649850	−	0.999979i	$-0.502069\pi$
$919$	−0.394005	−0.0129970	−0.00649850	+	0.999979i	$0.502069\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−3.75086	−0.123327
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32.9284	1.08035	0.540173	−	0.841554i	$-0.318359\pi$
$929$	32.9284	1.08035	0.540173	+	0.841554i	$0.318359\pi$
$930$	0	0
$931$	0.397442	0.0130256
$932$	0	0
$933$	0	0
$934$	0	0
$935$	20.9966	0.686661
$936$	0	0
$937$	−26.3388	−0.860451	−0.430226	−	0.902721i	$-0.641566\pi$
$937$	−26.3388	−0.860451	−0.430226	+	0.902721i	$0.641566\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.71982	−0.153862	−0.0769309	−	0.997036i	$-0.524512\pi$
$941$	−4.71982	−0.153862	−0.0769309	+	0.997036i	$0.524512\pi$
$942$	0	0
$943$	72.8984	2.37390
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−53.9639	−1.75359	−0.876796	−	0.480863i	$-0.840323\pi$
$947$	−53.9639	−1.75359	−0.876796	+	0.480863i	$0.840323\pi$
$948$	0	0
$949$	−19.1661	−0.622159
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13.0801	0.423707	0.211853	−	0.977301i	$-0.432050\pi$
$953$	13.0801	0.423707	0.211853	+	0.977301i	$0.432050\pi$
$954$	0	0
$955$	−19.2733	−0.623669
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36.5888	−1.18151
$960$	0	0
$961$	26.1070	0.842160
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8.01461	−0.257999
$966$	0	0
$967$	10.4914	0.337381	0.168690	−	0.985669i	$-0.446046\pi$
$967$	10.4914	0.337381	0.168690	+	0.985669i	$0.446046\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.4691	0.400151	0.200076	−	0.979780i	$-0.435881\pi$
$971$	12.4691	0.400151	0.200076	+	0.979780i	$0.435881\pi$
$972$	0	0
$973$	25.9931	0.833301
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.4699	1.67866	0.839331	−	0.543621i	$-0.182947\pi$
$977$	52.4699	1.67866	0.839331	+	0.543621i	$0.182947\pi$
$978$	0	0
$979$	−86.1035	−2.75188
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−14.1939	−0.452717	−0.226358	−	0.974044i	$-0.572682\pi$
$983$	−14.1939	−0.452717	−0.226358	+	0.974044i	$0.572682\pi$
$984$	0	0
$985$	−16.8793	−0.537819
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−54.5466	−1.73448
$990$	0	0
$991$	19.4036	0.616374	0.308187	−	0.951326i	$-0.400278\pi$
$991$	19.4036	0.616374	0.308187	+	0.951326i	$0.400278\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.28018	0.0405843
$996$	0	0
$997$	−8.14648	−0.258002	−0.129001	−	0.991644i	$-0.541177\pi$
$997$	−8.14648	−0.258002	−0.129001	+	0.991644i	$0.541177\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bm.1.3		3
3.2	odd	2		760.2.a.i.1.2	✓	3
12.11	even	2		1520.2.a.q.1.2		3
15.2	even	4		3800.2.d.n.3649.4		6
15.8	even	4		3800.2.d.n.3649.3		6
15.14	odd	2		3800.2.a.w.1.2		3
24.5	odd	2		6080.2.a.bx.1.2		3
24.11	even	2		6080.2.a.br.1.2		3
60.59	even	2		7600.2.a.bp.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.i.1.2	✓	3	3.2	odd	2
1520.2.a.q.1.2		3	12.11	even	2
3800.2.a.w.1.2		3	15.14	odd	2
3800.2.d.n.3649.3		6	15.8	even	4
3800.2.d.n.3649.4		6	15.2	even	4
6080.2.a.br.1.2		3	24.11	even	2
6080.2.a.bx.1.2		3	24.5	odd	2
6840.2.a.bm.1.3		3	1.1	even	1	trivial
7600.2.a.bp.1.2		3	60.59	even	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:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +2.71982 q^{7} +5.55691 q^{11} +2.02760 q^{13} +3.77846 q^{17} +1.00000 q^{19} +5.77846 q^{23} +1.00000 q^{25} +5.66119 q^{29} +7.55691 q^{31} +2.71982 q^{35} -3.75086 q^{37} +12.6155 q^{41} -9.43965 q^{43} -11.1138 q^{47} +0.397442 q^{49} +8.85170 q^{53} +5.55691 q^{55} -11.4526 q^{59} -10.6155 q^{61} +2.02760 q^{65} -11.5845 q^{67} -9.45264 q^{73} +15.1138 q^{77} +8.94137 q^{79} +4.94137 q^{83} +3.77846 q^{85} -15.4948 q^{89} +5.51471 q^{91} +1.00000 q^{95} +10.8647 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - q^{7} - 11 q^{13} + 3 q^{17} + 3 q^{19} + 9 q^{23} + 3 q^{25} + 7 q^{29} + 6 q^{31} - q^{35} - 20 q^{37} + 22 q^{41} - 10 q^{43} + 12 q^{49} + 7 q^{53} - 11 q^{59} - 16 q^{61} - 11 q^{65}+ \cdots + 8 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - q^7 - 11 * q^13 + 3 * q^17 + 3 * q^19 + 9 * q^23 + 3 * q^25 + 7 * q^29 + 6 * q^31 - q^35 - 20 * q^37 + 22 * q^41 - 10 * q^43 + 12 * q^49 + 7 * q^53 - 11 * q^59 - 16 * q^61 - 11 * q^65 - q^67 - 5 * q^73 + 12 * q^77 + 26 * q^79 + 14 * q^83 + 3 * q^85 + 6 * q^89 + 29 * q^91 + 3 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more