LMFDB - Embedded newform 6240.2.a.cb.1.3

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:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86081$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +4.32340 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +4.32340 q^{7} +1.00000 q^{9} -1.39821 q^{11} +1.00000 q^{13} -1.00000 q^{15} +5.11982 q^{17} +2.92520 q^{19} +4.32340 q^{21} +8.97021 q^{23} +1.00000 q^{25} +1.00000 q^{27} -8.36842 q^{29} +4.00000 q^{31} -1.39821 q^{33} -4.32340 q^{35} +3.39821 q^{37} +1.00000 q^{39} -0.601793 q^{41} -2.79641 q^{43} -1.00000 q^{45} -9.29362 q^{47} +11.6918 q^{49} +5.11982 q^{51} -2.19462 q^{53} +1.39821 q^{55} +2.92520 q^{57} +7.44322 q^{59} -5.24860 q^{61} +4.32340 q^{63} -1.00000 q^{65} +8.97021 q^{69} +10.0450 q^{71} -13.0152 q^{73} +1.00000 q^{75} -6.04502 q^{77} +1.95498 q^{79} +1.00000 q^{81} -17.2936 q^{83} -5.11982 q^{85} -8.36842 q^{87} +4.04502 q^{89} +4.32340 q^{91} +4.00000 q^{93} -2.92520 q^{95} +10.3234 q^{97} -1.39821 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} + q^{17} + 4 q^{19} + 5 q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} + 2 q^{29} + 12 q^{31} - q^{33} - 5 q^{35} + 7 q^{37} + 3 q^{39} - 5 q^{41} - 2 q^{43} - 3 q^{45} + 4 q^{47} + q^{51} + 3 q^{53} + q^{55} + 4 q^{57} - 3 q^{61} + 5 q^{63} - 3 q^{65} + 3 q^{69} + 11 q^{71} + 4 q^{73} + 3 q^{75} + q^{77} + 25 q^{79} + 3 q^{81} - 20 q^{83} - q^{85} + 2 q^{87} - 7 q^{89} + 5 q^{91} + 12 q^{93} - 4 q^{95} + 23 q^{97} - q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9 - q^11 + 3 * q^13 - 3 * q^15 + q^17 + 4 * q^19 + 5 * q^21 + 3 * q^23 + 3 * q^25 + 3 * q^27 + 2 * q^29 + 12 * q^31 - q^33 - 5 * q^35 + 7 * q^37 + 3 * q^39 - 5 * q^41 - 2 * q^43 - 3 * q^45 + 4 * q^47 + q^51 + 3 * q^53 + q^55 + 4 * q^57 - 3 * q^61 + 5 * q^63 - 3 * q^65 + 3 * q^69 + 11 * q^71 + 4 * q^73 + 3 * q^75 + q^77 + 25 * q^79 + 3 * q^81 - 20 * q^83 - q^85 + 2 * q^87 - 7 * q^89 + 5 * q^91 + 12 * q^93 - 4 * q^95 + 23 * q^97 - 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
$7$	4.32340	1.63409	0.817047	−	0.576572i	$-0.195610\pi$
$7$	4.32340	1.63409	0.817047	+	0.576572i	$0.195610\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.39821	−0.421575	−0.210788	−	0.977532i	$-0.567603\pi$
$11$	−1.39821	−0.421575	−0.210788	+	0.977532i	$0.567603\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.11982	1.24174	0.620869	−	0.783914i	$-0.286780\pi$
$17$	5.11982	1.24174	0.620869	+	0.783914i	$0.286780\pi$
$18$	0	0
$19$	2.92520	0.671086	0.335543	−	0.942025i	$-0.391080\pi$
$19$	2.92520	0.671086	0.335543	+	0.942025i	$0.391080\pi$
$20$	0	0
$21$	4.32340	0.943444
$22$	0	0
$23$	8.97021	1.87042	0.935209	−	0.354095i	$-0.115211\pi$
$23$	8.97021	1.87042	0.935209	+	0.354095i	$0.115211\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.36842	−1.55398	−0.776988	−	0.629515i	$-0.783254\pi$
$29$	−8.36842	−1.55398	−0.776988	+	0.629515i	$0.783254\pi$
$30$	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
$33$	−1.39821	−0.243397
$34$	0	0
$35$	−4.32340	−0.730789
$36$	0	0
$37$	3.39821	0.558662	0.279331	−	0.960195i	$-0.409887\pi$
$37$	3.39821	0.558662	0.279331	+	0.960195i	$0.409887\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−0.601793	−0.0939842	−0.0469921	−	0.998895i	$-0.514964\pi$
$41$	−0.601793	−0.0939842	−0.0469921	+	0.998895i	$0.514964\pi$
$42$	0	0
$43$	−2.79641	−0.426449	−0.213225	−	0.977003i	$-0.568397\pi$
$43$	−2.79641	−0.426449	−0.213225	+	0.977003i	$0.568397\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−9.29362	−1.35561	−0.677807	−	0.735240i	$-0.737069\pi$
$47$	−9.29362	−1.35561	−0.677807	+	0.735240i	$0.737069\pi$
$48$	0	0
$49$	11.6918	1.67026
$50$	0	0
$51$	5.11982	0.716918
$52$	0	0
$53$	−2.19462	−0.301455	−0.150727	−	0.988575i	$-0.548162\pi$
$53$	−2.19462	−0.301455	−0.150727	+	0.988575i	$0.548162\pi$
$54$	0	0
$55$	1.39821	0.188534
$56$	0	0
$57$	2.92520	0.387452
$58$	0	0
$59$	7.44322	0.969025	0.484513	−	0.874784i	$-0.338997\pi$
$59$	7.44322	0.969025	0.484513	+	0.874784i	$0.338997\pi$
$60$	0	0
$61$	−5.24860	−0.672015	−0.336007	−	0.941859i	$-0.609077\pi$
$61$	−5.24860	−0.672015	−0.336007	+	0.941859i	$0.609077\pi$
$62$	0	0
$63$	4.32340	0.544698
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	8.97021	1.07989
$70$	0	0
$71$	10.0450	1.19212	0.596062	−	0.802938i	$-0.296731\pi$
$71$	10.0450	1.19212	0.596062	+	0.802938i	$0.296731\pi$
$72$	0	0
$73$	−13.0152	−1.52332	−0.761659	−	0.647978i	$-0.775615\pi$
$73$	−13.0152	−1.52332	−0.761659	+	0.647978i	$0.775615\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−6.04502	−0.688894
$78$	0	0
$79$	1.95498	0.219953	0.109976	−	0.993934i	$-0.464922\pi$
$79$	1.95498	0.219953	0.109976	+	0.993934i	$0.464922\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−17.2936	−1.89822	−0.949111	−	0.314943i	$-0.898015\pi$
$83$	−17.2936	−1.89822	−0.949111	+	0.314943i	$0.898015\pi$
$84$	0	0
$85$	−5.11982	−0.555322
$86$	0	0
$87$	−8.36842	−0.897189
$88$	0	0
$89$	4.04502	0.428771	0.214385	−	0.976749i	$-0.431225\pi$
$89$	4.04502	0.428771	0.214385	+	0.976749i	$0.431225\pi$
$90$	0	0
$91$	4.32340	0.453216
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−2.92520	−0.300119
$96$	0	0
$97$	10.3234	1.04818	0.524091	−	0.851662i	$-0.324405\pi$
$97$	10.3234	1.04818	0.524091	+	0.851662i	$0.324405\pi$
$98$	0	0
$99$	−1.39821	−0.140525
$100$	0	0
$101$	−4.92520	−0.490075	−0.245038	−	0.969514i	$-0.578800\pi$
$101$	−4.92520	−0.490075	−0.245038	+	0.969514i	$0.578800\pi$
$102$	0	0
$103$	−1.29362	−0.127464	−0.0637319	−	0.997967i	$-0.520300\pi$
$103$	−1.29362	−0.127464	−0.0637319	+	0.997967i	$0.520300\pi$
$104$	0	0
$105$	−4.32340	−0.421921
$106$	0	0
$107$	−0.751399	−0.0726405	−0.0363202	−	0.999340i	$-0.511564\pi$
$107$	−0.751399	−0.0726405	−0.0363202	+	0.999340i	$0.511564\pi$
$108$	0	0
$109$	14.5180	1.39057	0.695287	−	0.718732i	$-0.255277\pi$
$109$	14.5180	1.39057	0.695287	+	0.718732i	$0.255277\pi$
$110$	0	0
$111$	3.39821	0.322544
$112$	0	0
$113$	−7.16484	−0.674011	−0.337005	−	0.941503i	$-0.609414\pi$
$113$	−7.16484	−0.674011	−0.337005	+	0.941503i	$0.609414\pi$
$114$	0	0
$115$	−8.97021	−0.836477
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	22.1350	2.02912
$120$	0	0
$121$	−9.04502	−0.822274
$122$	0	0
$123$	−0.601793	−0.0542618
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.7008	1.03828	0.519138	−	0.854690i	$-0.326253\pi$
$127$	11.7008	1.03828	0.519138	+	0.854690i	$0.326253\pi$
$128$	0	0
$129$	−2.79641	−0.246211
$130$	0	0
$131$	19.0152	1.66137	0.830684	−	0.556744i	$-0.187950\pi$
$131$	19.0152	1.66137	0.830684	+	0.556744i	$0.187950\pi$
$132$	0	0
$133$	12.6468	1.09662
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−14.7368	−1.25905	−0.629527	−	0.776979i	$-0.716751\pi$
$137$	−14.7368	−1.25905	−0.629527	+	0.776979i	$0.716751\pi$
$138$	0	0
$139$	−5.09899	−0.432491	−0.216246	−	0.976339i	$-0.569381\pi$
$139$	−5.09899	−0.432491	−0.216246	+	0.976339i	$0.569381\pi$
$140$	0	0
$141$	−9.29362	−0.782664
$142$	0	0
$143$	−1.39821	−0.116924
$144$	0	0
$145$	8.36842	0.694959
$146$	0	0
$147$	11.6918	0.964325
$148$	0	0
$149$	−4.69182	−0.384369	−0.192185	−	0.981359i	$-0.561557\pi$
$149$	−4.69182	−0.384369	−0.192185	+	0.981359i	$0.561557\pi$
$150$	0	0
$151$	18.8864	1.53696	0.768479	−	0.639875i	$-0.221014\pi$
$151$	18.8864	1.53696	0.768479	+	0.639875i	$0.221014\pi$
$152$	0	0
$153$	5.11982	0.413913
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	17.1440	1.36824	0.684121	−	0.729369i	$-0.260186\pi$
$157$	17.1440	1.36824	0.684121	+	0.729369i	$0.260186\pi$
$158$	0	0
$159$	−2.19462	−0.174045
$160$	0	0
$161$	38.7819	3.05644
$162$	0	0
$163$	0.194622	0.0152440	0.00762200	−	0.999971i	$-0.497574\pi$
$163$	0.194622	0.0152440	0.00762200	+	0.999971i	$0.497574\pi$
$164$	0	0
$165$	1.39821	0.108850
$166$	0	0
$167$	−11.4432	−0.885503	−0.442752	−	0.896644i	$-0.645998\pi$
$167$	−11.4432	−0.885503	−0.442752	+	0.896644i	$0.645998\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.92520	0.223695
$172$	0	0
$173$	9.44322	0.717955	0.358977	−	0.933346i	$-0.383125\pi$
$173$	9.44322	0.717955	0.358977	+	0.933346i	$0.383125\pi$
$174$	0	0
$175$	4.32340	0.326819
$176$	0	0
$177$	7.44322	0.559467
$178$	0	0
$179$	−6.92520	−0.517614	−0.258807	−	0.965929i	$-0.583329\pi$
$179$	−6.92520	−0.517614	−0.258807	+	0.965929i	$0.583329\pi$
$180$	0	0
$181$	7.48824	0.556596	0.278298	−	0.960495i	$-0.410230\pi$
$181$	7.48824	0.556596	0.278298	+	0.960495i	$0.410230\pi$
$182$	0	0
$183$	−5.24860	−0.387988
$184$	0	0
$185$	−3.39821	−0.249841
$186$	0	0
$187$	−7.15857	−0.523486
$188$	0	0
$189$	4.32340	0.314481
$190$	0	0
$191$	−12.0900	−0.874804	−0.437402	−	0.899266i	$-0.644101\pi$
$191$	−12.0900	−0.874804	−0.437402	+	0.899266i	$0.644101\pi$
$192$	0	0
$193$	24.5630	1.76809	0.884043	−	0.467405i	$-0.154811\pi$
$193$	24.5630	1.76809	0.884043	+	0.467405i	$0.154811\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	17.4432	1.24278	0.621389	−	0.783502i	$-0.286569\pi$
$197$	17.4432	1.24278	0.621389	+	0.783502i	$0.286569\pi$
$198$	0	0
$199$	−14.8864	−1.05527	−0.527636	−	0.849470i	$-0.676922\pi$
$199$	−14.8864	−1.05527	−0.527636	+	0.849470i	$0.676922\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−36.1801	−2.53934
$204$	0	0
$205$	0.601793	0.0420310
$206$	0	0
$207$	8.97021	0.623473
$208$	0	0
$209$	−4.09003	−0.282913
$210$	0	0
$211$	−5.29362	−0.364428	−0.182214	−	0.983259i	$-0.558326\pi$
$211$	−5.29362	−0.364428	−0.182214	+	0.983259i	$0.558326\pi$
$212$	0	0
$213$	10.0450	0.688273
$214$	0	0
$215$	2.79641	0.190714
$216$	0	0
$217$	17.2936	1.17397
$218$	0	0
$219$	−13.0152	−0.879488
$220$	0	0
$221$	5.11982	0.344396
$222$	0	0
$223$	0.427995	0.0286606	0.0143303	−	0.999897i	$-0.495438\pi$
$223$	0.427995	0.0286606	0.0143303	+	0.999897i	$0.495438\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−21.6829	−1.43914	−0.719571	−	0.694419i	$-0.755662\pi$
$227$	−21.6829	−1.43914	−0.719571	+	0.694419i	$0.755662\pi$
$228$	0	0
$229$	5.22441	0.345239	0.172619	−	0.984989i	$-0.444777\pi$
$229$	5.22441	0.345239	0.172619	+	0.984989i	$0.444777\pi$
$230$	0	0
$231$	−6.04502	−0.397733
$232$	0	0
$233$	11.6170	0.761056	0.380528	−	0.924769i	$-0.375742\pi$
$233$	11.6170	0.761056	0.380528	+	0.924769i	$0.375742\pi$
$234$	0	0
$235$	9.29362	0.606249
$236$	0	0
$237$	1.95498	0.126990
$238$	0	0
$239$	−11.2486	−0.727612	−0.363806	−	0.931475i	$-0.618523\pi$
$239$	−11.2486	−0.727612	−0.363806	+	0.931475i	$0.618523\pi$
$240$	0	0
$241$	−6.38924	−0.411567	−0.205784	−	0.978597i	$-0.565974\pi$
$241$	−6.38924	−0.411567	−0.205784	+	0.978597i	$0.565974\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−11.6918	−0.746963
$246$	0	0
$247$	2.92520	0.186126
$248$	0	0
$249$	−17.2936	−1.09594
$250$	0	0
$251$	−25.7216	−1.62353	−0.811767	−	0.583982i	$-0.801494\pi$
$251$	−25.7216	−1.62353	−0.811767	+	0.583982i	$0.801494\pi$
$252$	0	0
$253$	−12.5422	−0.788523
$254$	0	0
$255$	−5.11982	−0.320616
$256$	0	0
$257$	−13.2728	−0.827934	−0.413967	−	0.910292i	$-0.635857\pi$
$257$	−13.2728	−0.827934	−0.413967	+	0.910292i	$0.635857\pi$
$258$	0	0
$259$	14.6918	0.912906
$260$	0	0
$261$	−8.36842	−0.517992
$262$	0	0
$263$	−28.6081	−1.76405	−0.882024	−	0.471204i	$-0.843820\pi$
$263$	−28.6081	−1.76405	−0.882024	+	0.471204i	$0.843820\pi$
$264$	0	0
$265$	2.19462	0.134815
$266$	0	0
$267$	4.04502	0.247551
$268$	0	0
$269$	8.66763	0.528475	0.264237	−	0.964458i	$-0.414880\pi$
$269$	8.66763	0.528475	0.264237	+	0.964458i	$0.414880\pi$
$270$	0	0
$271$	29.6829	1.80311	0.901553	−	0.432669i	$-0.142428\pi$
$271$	29.6829	1.80311	0.901553	+	0.432669i	$0.142428\pi$
$272$	0	0
$273$	4.32340	0.261664
$274$	0	0
$275$	−1.39821	−0.0843151
$276$	0	0
$277$	13.0540	0.784338	0.392169	−	0.919893i	$-0.371725\pi$
$277$	13.0540	0.784338	0.392169	+	0.919893i	$0.371725\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	9.70079	0.578700	0.289350	−	0.957223i	$-0.406561\pi$
$281$	9.70079	0.578700	0.289350	+	0.957223i	$0.406561\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	−2.92520	−0.173274
$286$	0	0
$287$	−2.60179	−0.153579
$288$	0	0
$289$	9.21255	0.541915
$290$	0	0
$291$	10.3234	0.605169
$292$	0	0
$293$	22.6468	1.32304	0.661520	−	0.749927i	$-0.269912\pi$
$293$	22.6468	1.32304	0.661520	+	0.749927i	$0.269912\pi$
$294$	0	0
$295$	−7.44322	−0.433361
$296$	0	0
$297$	−1.39821	−0.0811322
$298$	0	0
$299$	8.97021	0.518761
$300$	0	0
$301$	−12.0900	−0.696858
$302$	0	0
$303$	−4.92520	−0.282945
$304$	0	0
$305$	5.24860	0.300534
$306$	0	0
$307$	−14.9494	−0.853207	−0.426603	−	0.904439i	$-0.640290\pi$
$307$	−14.9494	−0.853207	−0.426603	+	0.904439i	$0.640290\pi$
$308$	0	0
$309$	−1.29362	−0.0735913
$310$	0	0
$311$	−6.49720	−0.368423	−0.184211	−	0.982887i	$-0.558973\pi$
$311$	−6.49720	−0.368423	−0.184211	+	0.982887i	$0.558973\pi$
$312$	0	0
$313$	19.0361	1.07598	0.537991	−	0.842951i	$-0.319184\pi$
$313$	19.0361	1.07598	0.537991	+	0.842951i	$0.319184\pi$
$314$	0	0
$315$	−4.32340	−0.243596
$316$	0	0
$317$	12.4072	0.696856	0.348428	−	0.937336i	$-0.386716\pi$
$317$	12.4072	0.696856	0.348428	+	0.937336i	$0.386716\pi$
$318$	0	0
$319$	11.7008	0.655118
$320$	0	0
$321$	−0.751399	−0.0419390
$322$	0	0
$323$	14.9765	0.833314
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	14.5180	0.802849
$328$	0	0
$329$	−40.1801	−2.21520
$330$	0	0
$331$	−25.5541	−1.40458	−0.702290	−	0.711891i	$-0.747839\pi$
$331$	−25.5541	−1.40458	−0.702290	+	0.711891i	$0.747839\pi$
$332$	0	0
$333$	3.39821	0.186221
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.0305	−1.09113	−0.545564	−	0.838069i	$-0.683685\pi$
$337$	−20.0305	−1.09113	−0.545564	+	0.838069i	$0.683685\pi$
$338$	0	0
$339$	−7.16484	−0.389140
$340$	0	0
$341$	−5.59283	−0.302869
$342$	0	0
$343$	20.2847	1.09527
$344$	0	0
$345$	−8.97021	−0.482940
$346$	0	0
$347$	26.8719	1.44256	0.721279	−	0.692644i	$-0.243554\pi$
$347$	26.8719	1.44256	0.721279	+	0.692644i	$0.243554\pi$
$348$	0	0
$349$	23.4224	1.25377	0.626886	−	0.779111i	$-0.284329\pi$
$349$	23.4224	1.25377	0.626886	+	0.779111i	$0.284329\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−17.5333	−0.933201	−0.466601	−	0.884468i	$-0.654521\pi$
$353$	−17.5333	−0.933201	−0.466601	+	0.884468i	$0.654521\pi$
$354$	0	0
$355$	−10.0450	−0.533134
$356$	0	0
$357$	22.1350	1.17151
$358$	0	0
$359$	−13.2936	−0.701610	−0.350805	−	0.936448i	$-0.614092\pi$
$359$	−13.2936	−0.701610	−0.350805	+	0.936448i	$0.614092\pi$
$360$	0	0
$361$	−10.4432	−0.549643
$362$	0	0
$363$	−9.04502	−0.474740
$364$	0	0
$365$	13.0152	0.681248
$366$	0	0
$367$	12.0900	0.631095	0.315547	−	0.948910i	$-0.397812\pi$
$367$	12.0900	0.631095	0.315547	+	0.948910i	$0.397812\pi$
$368$	0	0
$369$	−0.601793	−0.0313281
$370$	0	0
$371$	−9.48824	−0.492605
$372$	0	0
$373$	3.55118	0.183873	0.0919366	−	0.995765i	$-0.470694\pi$
$373$	3.55118	0.183873	0.0919366	+	0.995765i	$0.470694\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−8.36842	−0.430996
$378$	0	0
$379$	−12.6081	−0.647632	−0.323816	−	0.946120i	$-0.604966\pi$
$379$	−12.6081	−0.647632	−0.323816	+	0.946120i	$0.604966\pi$
$380$	0	0
$381$	11.7008	0.599449
$382$	0	0
$383$	4.94602	0.252730	0.126365	−	0.991984i	$-0.459669\pi$
$383$	4.94602	0.252730	0.126365	+	0.991984i	$0.459669\pi$
$384$	0	0
$385$	6.04502	0.308083
$386$	0	0
$387$	−2.79641	−0.142150
$388$	0	0
$389$	33.3628	1.69156	0.845781	−	0.533530i	$-0.179135\pi$
$389$	33.3628	1.69156	0.845781	+	0.533530i	$0.179135\pi$
$390$	0	0
$391$	45.9259	2.32257
$392$	0	0
$393$	19.0152	0.959191
$394$	0	0
$395$	−1.95498	−0.0983659
$396$	0	0
$397$	−7.18903	−0.360807	−0.180403	−	0.983593i	$-0.557740\pi$
$397$	−7.18903	−0.360807	−0.180403	+	0.983593i	$0.557740\pi$
$398$	0	0
$399$	12.6468	0.633132
$400$	0	0
$401$	19.0361	0.950615	0.475308	−	0.879820i	$-0.342337\pi$
$401$	19.0361	0.950615	0.475308	+	0.879820i	$0.342337\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−4.75140	−0.235518
$408$	0	0
$409$	−21.5333	−1.06475	−0.532375	−	0.846508i	$-0.678701\pi$
$409$	−21.5333	−1.06475	−0.532375	+	0.846508i	$0.678701\pi$
$410$	0	0
$411$	−14.7368	−0.726915
$412$	0	0
$413$	32.1801	1.58348
$414$	0	0
$415$	17.2936	0.848910
$416$	0	0
$417$	−5.09899	−0.249699
$418$	0	0
$419$	35.9196	1.75479	0.877394	−	0.479771i	$-0.159280\pi$
$419$	35.9196	1.75479	0.877394	+	0.479771i	$0.159280\pi$
$420$	0	0
$421$	−23.5124	−1.14593	−0.572963	−	0.819581i	$-0.694206\pi$
$421$	−23.5124	−1.14593	−0.572963	+	0.819581i	$0.694206\pi$
$422$	0	0
$423$	−9.29362	−0.451871
$424$	0	0
$425$	5.11982	0.248348
$426$	0	0
$427$	−22.6918	−1.09813
$428$	0	0
$429$	−1.39821	−0.0675061
$430$	0	0
$431$	10.5872	0.509969	0.254985	−	0.966945i	$-0.417930\pi$
$431$	10.5872	0.509969	0.254985	+	0.966945i	$0.417930\pi$
$432$	0	0
$433$	−38.4376	−1.84719	−0.923597	−	0.383364i	$-0.874765\pi$
$433$	−38.4376	−1.84719	−0.923597	+	0.383364i	$0.874765\pi$
$434$	0	0
$435$	8.36842	0.401235
$436$	0	0
$437$	26.2396	1.25521
$438$	0	0
$439$	15.6378	0.746354	0.373177	−	0.927760i	$-0.378268\pi$
$439$	15.6378	0.746354	0.373177	+	0.927760i	$0.378268\pi$
$440$	0	0
$441$	11.6918	0.556754
$442$	0	0
$443$	−17.7875	−0.845107	−0.422554	−	0.906338i	$-0.638866\pi$
$443$	−17.7875	−0.845107	−0.422554	+	0.906338i	$0.638866\pi$
$444$	0	0
$445$	−4.04502	−0.191752
$446$	0	0
$447$	−4.69182	−0.221916
$448$	0	0
$449$	29.7279	1.40295	0.701473	−	0.712696i	$-0.252526\pi$
$449$	29.7279	1.40295	0.701473	+	0.712696i	$0.252526\pi$
$450$	0	0
$451$	0.841431	0.0396214
$452$	0	0
$453$	18.8864	0.887363
$454$	0	0
$455$	−4.32340	−0.202684
$456$	0	0
$457$	3.61702	0.169197	0.0845986	−	0.996415i	$-0.473039\pi$
$457$	3.61702	0.169197	0.0845986	+	0.996415i	$0.473039\pi$
$458$	0	0
$459$	5.11982	0.238973
$460$	0	0
$461$	12.1350	0.565186	0.282593	−	0.959240i	$-0.408805\pi$
$461$	12.1350	0.565186	0.282593	+	0.959240i	$0.408805\pi$
$462$	0	0
$463$	−7.11982	−0.330886	−0.165443	−	0.986219i	$-0.552905\pi$
$463$	−7.11982	−0.330886	−0.165443	+	0.986219i	$0.552905\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	0.542218	0.0250909	0.0125454	−	0.999921i	$-0.496007\pi$
$467$	0.542218	0.0250909	0.0125454	+	0.999921i	$0.496007\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	17.1440	0.789954
$472$	0	0
$473$	3.90997	0.179781
$474$	0	0
$475$	2.92520	0.134217
$476$	0	0
$477$	−2.19462	−0.100485
$478$	0	0
$479$	−17.7458	−0.810826	−0.405413	−	0.914134i	$-0.632872\pi$
$479$	−17.7458	−0.810826	−0.405413	+	0.914134i	$0.632872\pi$
$480$	0	0
$481$	3.39821	0.154945
$482$	0	0
$483$	38.7819	1.76464
$484$	0	0
$485$	−10.3234	−0.468762
$486$	0	0
$487$	17.5270	0.794224	0.397112	−	0.917770i	$-0.370012\pi$
$487$	17.5270	0.794224	0.397112	+	0.917770i	$0.370012\pi$
$488$	0	0
$489$	0.194622	0.00880112
$490$	0	0
$491$	19.4045	0.875712	0.437856	−	0.899045i	$-0.355738\pi$
$491$	19.4045	0.875712	0.437856	+	0.899045i	$0.355738\pi$
$492$	0	0
$493$	−42.8448	−1.92963
$494$	0	0
$495$	1.39821	0.0628448
$496$	0	0
$497$	43.4287	1.94804
$498$	0	0
$499$	15.6620	0.701129	0.350565	−	0.936539i	$-0.385990\pi$
$499$	15.6620	0.701129	0.350565	+	0.936539i	$0.385990\pi$
$500$	0	0
$501$	−11.4432	−0.511246
$502$	0	0
$503$	−8.90727	−0.397156	−0.198578	−	0.980085i	$-0.563632\pi$
$503$	−8.90727	−0.397156	−0.198578	+	0.980085i	$0.563632\pi$
$504$	0	0
$505$	4.92520	0.219168
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	40.2251	1.78295	0.891473	−	0.453074i	$-0.149673\pi$
$509$	40.2251	1.78295	0.891473	+	0.453074i	$0.149673\pi$
$510$	0	0
$511$	−56.2701	−2.48924
$512$	0	0
$513$	2.92520	0.129151
$514$	0	0
$515$	1.29362	0.0570036
$516$	0	0
$517$	12.9944	0.571493
$518$	0	0
$519$	9.44322	0.414512
$520$	0	0
$521$	−36.0305	−1.57852	−0.789262	−	0.614057i	$-0.789536\pi$
$521$	−36.0305	−1.57852	−0.789262	+	0.614057i	$0.789536\pi$
$522$	0	0
$523$	41.3836	1.80958	0.904790	−	0.425857i	$-0.140027\pi$
$523$	41.3836	1.80958	0.904790	+	0.425857i	$0.140027\pi$
$524$	0	0
$525$	4.32340	0.188689
$526$	0	0
$527$	20.4793	0.892091
$528$	0	0
$529$	57.4647	2.49847
$530$	0	0
$531$	7.44322	0.323008
$532$	0	0
$533$	−0.601793	−0.0260665
$534$	0	0
$535$	0.751399	0.0324858
$536$	0	0
$537$	−6.92520	−0.298844
$538$	0	0
$539$	−16.3476	−0.704141
$540$	0	0
$541$	−29.3144	−1.26033	−0.630163	−	0.776463i	$-0.717012\pi$
$541$	−29.3144	−1.26033	−0.630163	+	0.776463i	$0.717012\pi$
$542$	0	0
$543$	7.48824	0.321351
$544$	0	0
$545$	−14.5180	−0.621884
$546$	0	0
$547$	−30.5872	−1.30782	−0.653908	−	0.756574i	$-0.726872\pi$
$547$	−30.5872	−1.30782	−0.653908	+	0.756574i	$0.726872\pi$
$548$	0	0
$549$	−5.24860	−0.224005
$550$	0	0
$551$	−24.4793	−1.04285
$552$	0	0
$553$	8.45219	0.359424
$554$	0	0
$555$	−3.39821	−0.144246
$556$	0	0
$557$	13.9100	0.589384	0.294692	−	0.955592i	$-0.404783\pi$
$557$	13.9100	0.589384	0.294692	+	0.955592i	$0.404783\pi$
$558$	0	0
$559$	−2.79641	−0.118276
$560$	0	0
$561$	−7.15857	−0.302235
$562$	0	0
$563$	−35.5962	−1.50020	−0.750100	−	0.661324i	$-0.769995\pi$
$563$	−35.5962	−1.50020	−0.750100	+	0.661324i	$0.769995\pi$
$564$	0	0
$565$	7.16484	0.301427
$566$	0	0
$567$	4.32340	0.181566
$568$	0	0
$569$	−9.44322	−0.395881	−0.197940	−	0.980214i	$-0.563425\pi$
$569$	−9.44322	−0.395881	−0.197940	+	0.980214i	$0.563425\pi$
$570$	0	0
$571$	−18.3026	−0.765939	−0.382970	−	0.923761i	$-0.625099\pi$
$571$	−18.3026	−0.765939	−0.382970	+	0.923761i	$0.625099\pi$
$572$	0	0
$573$	−12.0900	−0.505068
$574$	0	0
$575$	8.97021	0.374084
$576$	0	0
$577$	4.29295	0.178718	0.0893589	−	0.995999i	$-0.471518\pi$
$577$	4.29295	0.178718	0.0893589	+	0.995999i	$0.471518\pi$
$578$	0	0
$579$	24.5630	1.02081
$580$	0	0
$581$	−74.7673	−3.10187
$582$	0	0
$583$	3.06854	0.127086
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−30.1980	−1.24640	−0.623202	−	0.782061i	$-0.714169\pi$
$587$	−30.1980	−1.24640	−0.623202	+	0.782061i	$0.714169\pi$
$588$	0	0
$589$	11.7008	0.482123
$590$	0	0
$591$	17.4432	0.717518
$592$	0	0
$593$	21.7908	0.894842	0.447421	−	0.894324i	$-0.352343\pi$
$593$	21.7908	0.894842	0.447421	+	0.894324i	$0.352343\pi$
$594$	0	0
$595$	−22.1350	−0.907448
$596$	0	0
$597$	−14.8864	−0.609262
$598$	0	0
$599$	−4.81434	−0.196709	−0.0983543	−	0.995151i	$-0.531358\pi$
$599$	−4.81434	−0.196709	−0.0983543	+	0.995151i	$0.531358\pi$
$600$	0	0
$601$	0.991037	0.0404252	0.0202126	−	0.999796i	$-0.493566\pi$
$601$	0.991037	0.0404252	0.0202126	+	0.999796i	$0.493566\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.04502	0.367732
$606$	0	0
$607$	28.0900	1.14014	0.570070	−	0.821596i	$-0.306916\pi$
$607$	28.0900	1.14014	0.570070	+	0.821596i	$0.306916\pi$
$608$	0	0
$609$	−36.1801	−1.46609
$610$	0	0
$611$	−9.29362	−0.375980
$612$	0	0
$613$	−23.3982	−0.945045	−0.472522	−	0.881319i	$-0.656656\pi$
$613$	−23.3982	−0.945045	−0.472522	+	0.881319i	$0.656656\pi$
$614$	0	0
$615$	0.601793	0.0242666
$616$	0	0
$617$	28.0305	1.12846	0.564232	−	0.825616i	$-0.309172\pi$
$617$	28.0305	1.12846	0.564232	+	0.825616i	$0.309172\pi$
$618$	0	0
$619$	27.7521	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$619$	27.7521	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$620$	0	0
$621$	8.97021	0.359962
$622$	0	0
$623$	17.4882	0.700652
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.09003	−0.163340
$628$	0	0
$629$	17.3982	0.693712
$630$	0	0
$631$	−50.2105	−1.99885	−0.999425	−	0.0339171i	$-0.989202\pi$
$631$	−50.2105	−1.99885	−0.999425	+	0.0339171i	$0.989202\pi$
$632$	0	0
$633$	−5.29362	−0.210402
$634$	0	0
$635$	−11.7008	−0.464332
$636$	0	0
$637$	11.6918	0.463247
$638$	0	0
$639$	10.0450	0.397375
$640$	0	0
$641$	−23.2936	−0.920043	−0.460021	−	0.887908i	$-0.652158\pi$
$641$	−23.2936	−0.920043	−0.460021	+	0.887908i	$0.652158\pi$
$642$	0	0
$643$	−4.54222	−0.179128	−0.0895638	−	0.995981i	$-0.528547\pi$
$643$	−4.54222	−0.179128	−0.0895638	+	0.995981i	$0.528547\pi$
$644$	0	0
$645$	2.79641	0.110109
$646$	0	0
$647$	2.73057	0.107350	0.0536750	−	0.998558i	$-0.482907\pi$
$647$	2.73057	0.107350	0.0536750	+	0.998558i	$0.482907\pi$
$648$	0	0
$649$	−10.4072	−0.408517
$650$	0	0
$651$	17.2936	0.677790
$652$	0	0
$653$	−49.5816	−1.94028	−0.970140	−	0.242547i	$-0.922017\pi$
$653$	−49.5816	−1.94028	−0.970140	+	0.242547i	$0.922017\pi$
$654$	0	0
$655$	−19.0152	−0.742986
$656$	0	0
$657$	−13.0152	−0.507772
$658$	0	0
$659$	−20.0513	−0.781087	−0.390544	−	0.920584i	$-0.627713\pi$
$659$	−20.0513	−0.781087	−0.390544	+	0.920584i	$0.627713\pi$
$660$	0	0
$661$	−15.5415	−0.604496	−0.302248	−	0.953229i	$-0.597737\pi$
$661$	−15.5415	−0.604496	−0.302248	+	0.953229i	$0.597737\pi$
$662$	0	0
$663$	5.11982	0.198837
$664$	0	0
$665$	−12.6468	−0.490422
$666$	0	0
$667$	−75.0665	−2.90659
$668$	0	0
$669$	0.427995	0.0165472
$670$	0	0
$671$	7.33863	0.283305
$672$	0	0
$673$	15.2036	0.586055	0.293028	−	0.956104i	$-0.405337\pi$
$673$	15.2036	0.586055	0.293028	+	0.956104i	$0.405337\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	26.5422	1.02010	0.510050	−	0.860145i	$-0.329627\pi$
$677$	26.5422	1.02010	0.510050	+	0.860145i	$0.329627\pi$
$678$	0	0
$679$	44.6323	1.71283
$680$	0	0
$681$	−21.6829	−0.830889
$682$	0	0
$683$	−34.1980	−1.30855	−0.654275	−	0.756257i	$-0.727026\pi$
$683$	−34.1980	−1.30855	−0.654275	+	0.756257i	$0.727026\pi$
$684$	0	0
$685$	14.7368	0.563066
$686$	0	0
$687$	5.22441	0.199324
$688$	0	0
$689$	−2.19462	−0.0836085
$690$	0	0
$691$	35.6204	1.35506	0.677532	−	0.735494i	$-0.263050\pi$
$691$	35.6204	1.35506	0.677532	+	0.735494i	$0.263050\pi$
$692$	0	0
$693$	−6.04502	−0.229631
$694$	0	0
$695$	5.09899	0.193416
$696$	0	0
$697$	−3.08107	−0.116704
$698$	0	0
$699$	11.6170	0.439396
$700$	0	0
$701$	−32.6260	−1.23227	−0.616133	−	0.787642i	$-0.711302\pi$
$701$	−32.6260	−1.23227	−0.616133	+	0.787642i	$0.711302\pi$
$702$	0	0
$703$	9.94043	0.374910
$704$	0	0
$705$	9.29362	0.350018
$706$	0	0
$707$	−21.2936	−0.800829
$708$	0	0
$709$	42.3989	1.59232	0.796162	−	0.605084i	$-0.206860\pi$
$709$	42.3989	1.59232	0.796162	+	0.605084i	$0.206860\pi$
$710$	0	0
$711$	1.95498	0.0733176
$712$	0	0
$713$	35.8809	1.34375
$714$	0	0
$715$	1.39821	0.0522900
$716$	0	0
$717$	−11.2486	−0.420087
$718$	0	0
$719$	−0.855989	−0.0319230	−0.0159615	−	0.999873i	$-0.505081\pi$
$719$	−0.855989	−0.0319230	−0.0159615	+	0.999873i	$0.505081\pi$
$720$	0	0
$721$	−5.59283	−0.208288
$722$	0	0
$723$	−6.38924	−0.237619
$724$	0	0
$725$	−8.36842	−0.310795
$726$	0	0
$727$	−12.3476	−0.457947	−0.228973	−	0.973433i	$-0.573537\pi$
$727$	−12.3476	−0.457947	−0.228973	+	0.973433i	$0.573537\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.3171	−0.529538
$732$	0	0
$733$	−27.4882	−1.01530	−0.507651	−	0.861563i	$-0.669486\pi$
$733$	−27.4882	−1.01530	−0.507651	+	0.861563i	$0.669486\pi$
$734$	0	0
$735$	−11.6918	−0.431259
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−36.3989	−1.33895	−0.669477	−	0.742833i	$-0.733482\pi$
$739$	−36.3989	−1.33895	−0.669477	+	0.742833i	$0.733482\pi$
$740$	0	0
$741$	2.92520	0.107460
$742$	0	0
$743$	35.2340	1.29261	0.646306	−	0.763078i	$-0.276313\pi$
$743$	35.2340	1.29261	0.646306	+	0.763078i	$0.276313\pi$
$744$	0	0
$745$	4.69182	0.171895
$746$	0	0
$747$	−17.2936	−0.632740
$748$	0	0
$749$	−3.24860	−0.118701
$750$	0	0
$751$	−3.54781	−0.129462	−0.0647308	−	0.997903i	$-0.520619\pi$
$751$	−3.54781	−0.129462	−0.0647308	+	0.997903i	$0.520619\pi$
$752$	0	0
$753$	−25.7216	−0.937348
$754$	0	0
$755$	−18.8864	−0.687348
$756$	0	0
$757$	−22.6052	−0.821599	−0.410799	−	0.911726i	$-0.634750\pi$
$757$	−22.6052	−0.821599	−0.410799	+	0.911726i	$0.634750\pi$
$758$	0	0
$759$	−12.5422	−0.455254
$760$	0	0
$761$	29.6233	1.07384	0.536922	−	0.843632i	$-0.319587\pi$
$761$	29.6233	1.07384	0.536922	+	0.843632i	$0.319587\pi$
$762$	0	0
$763$	62.7673	2.27233
$764$	0	0
$765$	−5.11982	−0.185107
$766$	0	0
$767$	7.44322	0.268759
$768$	0	0
$769$	3.89204	0.140351	0.0701753	−	0.997535i	$-0.477644\pi$
$769$	3.89204	0.140351	0.0701753	+	0.997535i	$0.477644\pi$
$770$	0	0
$771$	−13.2728	−0.478008
$772$	0	0
$773$	−15.1261	−0.544047	−0.272024	−	0.962291i	$-0.587693\pi$
$773$	−15.1261	−0.544047	−0.272024	+	0.962291i	$0.587693\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	14.6918	0.527066
$778$	0	0
$779$	−1.76036	−0.0630715
$780$	0	0
$781$	−14.0450	−0.502570
$782$	0	0
$783$	−8.36842	−0.299063
$784$	0	0
$785$	−17.1440	−0.611896
$786$	0	0
$787$	7.14401	0.254656	0.127328	−	0.991861i	$-0.459360\pi$
$787$	7.14401	0.254656	0.127328	+	0.991861i	$0.459360\pi$
$788$	0	0
$789$	−28.6081	−1.01847
$790$	0	0
$791$	−30.9765	−1.10140
$792$	0	0
$793$	−5.24860	−0.186383
$794$	0	0
$795$	2.19462	0.0778352
$796$	0	0
$797$	−24.6918	−0.874629	−0.437315	−	0.899309i	$-0.644070\pi$
$797$	−24.6918	−0.874629	−0.437315	+	0.899309i	$0.644070\pi$
$798$	0	0
$799$	−47.5816	−1.68332
$800$	0	0
$801$	4.04502	0.142924
$802$	0	0
$803$	18.1980	0.642193
$804$	0	0
$805$	−38.7819	−1.36688
$806$	0	0
$807$	8.66763	0.305115
$808$	0	0
$809$	−16.5872	−0.583176	−0.291588	−	0.956544i	$-0.594184\pi$
$809$	−16.5872	−0.583176	−0.291588	+	0.956544i	$0.594184\pi$
$810$	0	0
$811$	−30.3684	−1.06638	−0.533190	−	0.845996i	$-0.679007\pi$
$811$	−30.3684	−1.06638	−0.533190	+	0.845996i	$0.679007\pi$
$812$	0	0
$813$	29.6829	1.04102
$814$	0	0
$815$	−0.194622	−0.00681732
$816$	0	0
$817$	−8.18006	−0.286184
$818$	0	0
$819$	4.32340	0.151072
$820$	0	0
$821$	−12.9010	−0.450248	−0.225124	−	0.974330i	$-0.572279\pi$
$821$	−12.9010	−0.450248	−0.225124	+	0.974330i	$0.572279\pi$
$822$	0	0
$823$	14.7064	0.512632	0.256316	−	0.966593i	$-0.417491\pi$
$823$	14.7064	0.512632	0.256316	+	0.966593i	$0.417491\pi$
$824$	0	0
$825$	−1.39821	−0.0486793
$826$	0	0
$827$	−37.5928	−1.30723	−0.653615	−	0.756827i	$-0.726749\pi$
$827$	−37.5928	−1.30723	−0.653615	+	0.756827i	$0.726749\pi$
$828$	0	0
$829$	−29.3657	−1.01991	−0.509957	−	0.860200i	$-0.670339\pi$
$829$	−29.3657	−1.01991	−0.509957	+	0.860200i	$0.670339\pi$
$830$	0	0
$831$	13.0540	0.452838
$832$	0	0
$833$	59.8600	2.07403
$834$	0	0
$835$	11.4432	0.396009
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	18.8594	0.651097	0.325549	−	0.945525i	$-0.394451\pi$
$839$	18.8594	0.651097	0.325549	+	0.945525i	$0.394451\pi$
$840$	0	0
$841$	41.0305	1.41484
$842$	0	0
$843$	9.70079	0.334113
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−39.1053	−1.34367
$848$	0	0
$849$	0	0
$850$	0	0
$851$	30.4826	1.04493
$852$	0	0
$853$	−39.5783	−1.35513	−0.677567	−	0.735461i	$-0.736966\pi$
$853$	−39.5783	−1.35513	−0.677567	+	0.735461i	$0.736966\pi$
$854$	0	0
$855$	−2.92520	−0.100040
$856$	0	0
$857$	36.0546	1.23160	0.615802	−	0.787901i	$-0.288832\pi$
$857$	36.0546	1.23160	0.615802	+	0.787901i	$0.288832\pi$
$858$	0	0
$859$	8.75140	0.298594	0.149297	−	0.988792i	$-0.452299\pi$
$859$	8.75140	0.298594	0.149297	+	0.988792i	$0.452299\pi$
$860$	0	0
$861$	−2.60179	−0.0886689
$862$	0	0
$863$	35.4141	1.20551	0.602755	−	0.797926i	$-0.294070\pi$
$863$	35.4141	1.20551	0.602755	+	0.797926i	$0.294070\pi$
$864$	0	0
$865$	−9.44322	−0.321079
$866$	0	0
$867$	9.21255	0.312875
$868$	0	0
$869$	−2.73347	−0.0927267
$870$	0	0
$871$	0	0
$872$	0	0
$873$	10.3234	0.349394
$874$	0	0
$875$	−4.32340	−0.146158
$876$	0	0
$877$	53.0665	1.79193	0.895964	−	0.444126i	$-0.146486\pi$
$877$	53.0665	1.79193	0.895964	+	0.444126i	$0.146486\pi$
$878$	0	0
$879$	22.6468	0.763858
$880$	0	0
$881$	−38.6468	−1.30204	−0.651022	−	0.759059i	$-0.725659\pi$
$881$	−38.6468	−1.30204	−0.651022	+	0.759059i	$0.725659\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	−7.44322	−0.250201
$886$	0	0
$887$	−38.2222	−1.28338	−0.641688	−	0.766966i	$-0.721765\pi$
$887$	−38.2222	−1.28338	−0.641688	+	0.766966i	$0.721765\pi$
$888$	0	0
$889$	50.5872	1.69664
$890$	0	0
$891$	−1.39821	−0.0468417
$892$	0	0
$893$	−27.1857	−0.909733
$894$	0	0
$895$	6.92520	0.231484
$896$	0	0
$897$	8.97021	0.299507
$898$	0	0
$899$	−33.4737	−1.11641
$900$	0	0
$901$	−11.2361	−0.374328
$902$	0	0
$903$	−12.0900	−0.402331
$904$	0	0
$905$	−7.48824	−0.248917
$906$	0	0
$907$	38.8573	1.29024	0.645118	−	0.764083i	$-0.276808\pi$
$907$	38.8573	1.29024	0.645118	+	0.764083i	$0.276808\pi$
$908$	0	0
$909$	−4.92520	−0.163358
$910$	0	0
$911$	−32.8269	−1.08760	−0.543801	−	0.839214i	$-0.683016\pi$
$911$	−32.8269	−1.08760	−0.543801	+	0.839214i	$0.683016\pi$
$912$	0	0
$913$	24.1801	0.800243
$914$	0	0
$915$	5.24860	0.173513
$916$	0	0
$917$	82.2105	2.71483
$918$	0	0
$919$	46.3151	1.52779	0.763897	−	0.645338i	$-0.223283\pi$
$919$	46.3151	1.52779	0.763897	+	0.645338i	$0.223283\pi$
$920$	0	0
$921$	−14.9494	−0.492599
$922$	0	0
$923$	10.0450	0.330636
$924$	0	0
$925$	3.39821	0.111732
$926$	0	0
$927$	−1.29362	−0.0424880
$928$	0	0
$929$	−24.4343	−0.801662	−0.400831	−	0.916152i	$-0.631279\pi$
$929$	−24.4343	−0.801662	−0.400831	+	0.916152i	$0.631279\pi$
$930$	0	0
$931$	34.2009	1.12089
$932$	0	0
$933$	−6.49720	−0.212709
$934$	0	0
$935$	7.15857	0.234110
$936$	0	0
$937$	14.6885	0.479851	0.239925	−	0.970791i	$-0.422877\pi$
$937$	14.6885	0.479851	0.239925	+	0.970791i	$0.422877\pi$
$938$	0	0
$939$	19.0361	0.621218
$940$	0	0
$941$	12.0034	0.391299	0.195649	−	0.980674i	$-0.437319\pi$
$941$	12.0034	0.391299	0.195649	+	0.980674i	$0.437319\pi$
$942$	0	0
$943$	−5.39821	−0.175790
$944$	0	0
$945$	−4.32340	−0.140640
$946$	0	0
$947$	−35.7008	−1.16012	−0.580060	−	0.814574i	$-0.696971\pi$
$947$	−35.7008	−1.16012	−0.580060	+	0.814574i	$0.696971\pi$
$948$	0	0
$949$	−13.0152	−0.422492
$950$	0	0
$951$	12.4072	0.402330
$952$	0	0
$953$	−28.1738	−0.912639	−0.456319	−	0.889816i	$-0.650833\pi$
$953$	−28.1738	−0.912639	−0.456319	+	0.889816i	$0.650833\pi$
$954$	0	0
$955$	12.0900	0.391224
$956$	0	0
$957$	11.7008	0.378233
$958$	0	0
$959$	−63.7133	−2.05741
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−0.751399	−0.0242135
$964$	0	0
$965$	−24.5630	−0.790712
$966$	0	0
$967$	0.685559	0.0220461	0.0110230	−	0.999939i	$-0.496491\pi$
$967$	0.685559	0.0220461	0.0110230	+	0.999939i	$0.496491\pi$
$968$	0	0
$969$	14.9765	0.481114
$970$	0	0
$971$	−33.6925	−1.08124	−0.540622	−	0.841266i	$-0.681811\pi$
$971$	−33.6925	−1.08124	−0.540622	+	0.841266i	$0.681811\pi$
$972$	0	0
$973$	−22.0450	−0.706731
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−38.4793	−1.23106	−0.615531	−	0.788113i	$-0.711058\pi$
$977$	−38.4793	−1.23106	−0.615531	+	0.788113i	$0.711058\pi$
$978$	0	0
$979$	−5.65577	−0.180759
$980$	0	0
$981$	14.5180	0.463525
$982$	0	0
$983$	33.7312	1.07586	0.537930	−	0.842990i	$-0.319207\pi$
$983$	33.7312	1.07586	0.537930	+	0.842990i	$0.319207\pi$
$984$	0	0
$985$	−17.4432	−0.555787
$986$	0	0
$987$	−40.1801	−1.27895
$988$	0	0
$989$	−25.0844	−0.797639
$990$	0	0
$991$	−33.5366	−1.06533	−0.532663	−	0.846327i	$-0.678809\pi$
$991$	−33.5366	−1.06533	−0.532663	+	0.846327i	$0.678809\pi$
$992$	0	0
$993$	−25.5541	−0.810934
$994$	0	0
$995$	14.8864	0.471932
$996$	0	0
$997$	−54.2284	−1.71743	−0.858716	−	0.512452i	$-0.828737\pi$
$997$	−54.2284	−1.71743	−0.858716	+	0.512452i	$0.828737\pi$
$998$	0	0
$999$	3.39821	0.107515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.cb.1.3	yes	3
4.3	odd	2		6240.2.a.bw.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bw.1.1	✓	3	4.3	odd	2
6240.2.a.cb.1.3	yes	3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +4.32340 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +4.32340 q^{7} +1.00000 q^{9} -1.39821 q^{11} +1.00000 q^{13} -1.00000 q^{15} +5.11982 q^{17} +2.92520 q^{19} +4.32340 q^{21} +8.97021 q^{23} +1.00000 q^{25} +1.00000 q^{27} -8.36842 q^{29} +4.00000 q^{31} -1.39821 q^{33} -4.32340 q^{35} +3.39821 q^{37} +1.00000 q^{39} -0.601793 q^{41} -2.79641 q^{43} -1.00000 q^{45} -9.29362 q^{47} +11.6918 q^{49} +5.11982 q^{51} -2.19462 q^{53} +1.39821 q^{55} +2.92520 q^{57} +7.44322 q^{59} -5.24860 q^{61} +4.32340 q^{63} -1.00000 q^{65} +8.97021 q^{69} +10.0450 q^{71} -13.0152 q^{73} +1.00000 q^{75} -6.04502 q^{77} +1.95498 q^{79} +1.00000 q^{81} -17.2936 q^{83} -5.11982 q^{85} -8.36842 q^{87} +4.04502 q^{89} +4.32340 q^{91} +4.00000 q^{93} -2.92520 q^{95} +10.3234 q^{97} -1.39821 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} + q^{17} + 4 q^{19} + 5 q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} + 2 q^{29} + 12 q^{31} - q^{33} - 5 q^{35} + 7 q^{37} + 3 q^{39} - 5 q^{41} - 2 q^{43} - 3 q^{45} + 4 q^{47} + q^{51} + 3 q^{53} + q^{55} + 4 q^{57} - 3 q^{61} + 5 q^{63} - 3 q^{65} + 3 q^{69} + 11 q^{71} + 4 q^{73} + 3 q^{75} + q^{77} + 25 q^{79} + 3 q^{81} - 20 q^{83} - q^{85} + 2 q^{87} - 7 q^{89} + 5 q^{91} + 12 q^{93} - 4 q^{95} + 23 q^{97} - q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9 - q^11 + 3 * q^13 - 3 * q^15 + q^17 + 4 * q^19 + 5 * q^21 + 3 * q^23 + 3 * q^25 + 3 * q^27 + 2 * q^29 + 12 * q^31 - q^33 - 5 * q^35 + 7 * q^37 + 3 * q^39 - 5 * q^41 - 2 * q^43 - 3 * q^45 + 4 * q^47 + q^51 + 3 * q^53 + q^55 + 4 * q^57 - 3 * q^61 + 5 * q^63 - 3 * q^65 + 3 * q^69 + 11 * q^71 + 4 * q^73 + 3 * q^75 + q^77 + 25 * q^79 + 3 * q^81 - 20 * q^83 - q^85 + 2 * q^87 - 7 * q^89 + 5 * q^91 + 12 * q^93 - 4 * q^95 + 23 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more