LMFDB - Embedded newform 6036.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.1977026600$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 12 x^{12} + 112 x^{11} + 7 x^{10} - 710 x^{9} + 281 x^{8} + 1850 x^{7} - 830 x^{6} + \cdots + 1 $ x^14 - 6x^13 - 12x^12 + 112x^11 + 7x^10 - 710x^9 + 281x^8 + 1850x^7 - 830x^6 - 1918x^5 + 623x^4 + 480x^3 - 29x^2 - 17*x + 1
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			1.4
Root			$1.79209$ of defining polynomial
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.79209 q^{5} +1.22660 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.79209 q^{5} +1.22660 q^{7} +1.00000 q^{9} +0.167826 q^{11} -3.59756 q^{13} +1.79209 q^{15} +4.82219 q^{17} +4.33438 q^{19} -1.22660 q^{21} -3.47282 q^{23} -1.78841 q^{25} -1.00000 q^{27} +1.92174 q^{29} -6.99752 q^{31} -0.167826 q^{33} -2.19818 q^{35} -4.07816 q^{37} +3.59756 q^{39} +0.729110 q^{41} +9.76299 q^{43} -1.79209 q^{45} -8.14600 q^{47} -5.49545 q^{49} -4.82219 q^{51} +11.4350 q^{53} -0.300759 q^{55} -4.33438 q^{57} -12.7351 q^{59} +0.467969 q^{61} +1.22660 q^{63} +6.44716 q^{65} +8.46265 q^{67} +3.47282 q^{69} +12.7332 q^{71} +0.468869 q^{73} +1.78841 q^{75} +0.205856 q^{77} +15.9731 q^{79} +1.00000 q^{81} -3.31328 q^{83} -8.64181 q^{85} -1.92174 q^{87} +5.58054 q^{89} -4.41278 q^{91} +6.99752 q^{93} -7.76760 q^{95} +8.64165 q^{97} +0.167826 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * 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
$5$	−1.79209	−0.801447	−0.400724	−	0.916199i	$-0.631241\pi$
$5$	−1.79209	−0.801447	−0.400724	+	0.916199i	$0.631241\pi$
$6$	0	0
$7$	1.22660	0.463612	0.231806	−	0.972762i	$-0.425537\pi$
$7$	1.22660	0.463612	0.231806	+	0.972762i	$0.425537\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.167826	0.0506014	0.0253007	−	0.999680i	$-0.491946\pi$
$11$	0.167826	0.0506014	0.0253007	+	0.999680i	$0.491946\pi$
$12$	0	0
$13$	−3.59756	−0.997784	−0.498892	−	0.866664i	$-0.666260\pi$
$13$	−3.59756	−0.997784	−0.498892	+	0.866664i	$0.666260\pi$
$14$	0	0
$15$	1.79209	0.462716
$16$	0	0
$17$	4.82219	1.16955	0.584777	−	0.811194i	$-0.301182\pi$
$17$	4.82219	1.16955	0.584777	+	0.811194i	$0.301182\pi$
$18$	0	0
$19$	4.33438	0.994375	0.497187	−	0.867643i	$-0.334366\pi$
$19$	4.33438	0.994375	0.497187	+	0.867643i	$0.334366\pi$
$20$	0	0
$21$	−1.22660	−0.267667
$22$	0	0
$23$	−3.47282	−0.724132	−0.362066	−	0.932152i	$-0.617929\pi$
$23$	−3.47282	−0.724132	−0.362066	+	0.932152i	$0.617929\pi$
$24$	0	0
$25$	−1.78841	−0.357682
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.92174	0.356858	0.178429	−	0.983953i	$-0.442899\pi$
$29$	1.92174	0.356858	0.178429	+	0.983953i	$0.442899\pi$
$30$	0	0
$31$	−6.99752	−1.25679	−0.628396	−	0.777894i	$-0.716288\pi$
$31$	−6.99752	−1.25679	−0.628396	+	0.777894i	$0.716288\pi$
$32$	0	0
$33$	−0.167826	−0.0292147
$34$	0	0
$35$	−2.19818	−0.371561
$36$	0	0
$37$	−4.07816	−0.670445	−0.335223	−	0.942139i	$-0.608811\pi$
$37$	−4.07816	−0.670445	−0.335223	+	0.942139i	$0.608811\pi$
$38$	0	0
$39$	3.59756	0.576071
$40$	0	0
$41$	0.729110	0.113868	0.0569339	−	0.998378i	$-0.481868\pi$
$41$	0.729110	0.113868	0.0569339	+	0.998378i	$0.481868\pi$
$42$	0	0
$43$	9.76299	1.48884	0.744421	−	0.667710i	$-0.232726\pi$
$43$	9.76299	1.48884	0.744421	+	0.667710i	$0.232726\pi$
$44$	0	0
$45$	−1.79209	−0.267149
$46$	0	0
$47$	−8.14600	−1.18822	−0.594108	−	0.804385i	$-0.702495\pi$
$47$	−8.14600	−1.18822	−0.594108	+	0.804385i	$0.702495\pi$
$48$	0	0
$49$	−5.49545	−0.785064
$50$	0	0
$51$	−4.82219	−0.675242
$52$	0	0
$53$	11.4350	1.57072	0.785359	−	0.619041i	$-0.212479\pi$
$53$	11.4350	1.57072	0.785359	+	0.619041i	$0.212479\pi$
$54$	0	0
$55$	−0.300759	−0.0405543
$56$	0	0
$57$	−4.33438	−0.574103
$58$	0	0
$59$	−12.7351	−1.65797	−0.828983	−	0.559273i	$-0.811080\pi$
$59$	−12.7351	−1.65797	−0.828983	+	0.559273i	$0.811080\pi$
$60$	0	0
$61$	0.467969	0.0599173	0.0299587	−	0.999551i	$-0.490462\pi$
$61$	0.467969	0.0599173	0.0299587	+	0.999551i	$0.490462\pi$
$62$	0	0
$63$	1.22660	0.154537
$64$	0	0
$65$	6.44716	0.799671
$66$	0	0
$67$	8.46265	1.03388	0.516939	−	0.856022i	$-0.327072\pi$
$67$	8.46265	1.03388	0.516939	+	0.856022i	$0.327072\pi$
$68$	0	0
$69$	3.47282	0.418078
$70$	0	0
$71$	12.7332	1.51115	0.755577	−	0.655059i	$-0.227356\pi$
$71$	12.7332	1.51115	0.755577	+	0.655059i	$0.227356\pi$
$72$	0	0
$73$	0.468869	0.0548769	0.0274385	−	0.999623i	$-0.491265\pi$
$73$	0.468869	0.0548769	0.0274385	+	0.999623i	$0.491265\pi$
$74$	0	0
$75$	1.78841	0.206508
$76$	0	0
$77$	0.205856	0.0234594
$78$	0	0
$79$	15.9731	1.79712	0.898558	−	0.438855i	$-0.144616\pi$
$79$	15.9731	1.79712	0.898558	+	0.438855i	$0.144616\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.31328	−0.363680	−0.181840	−	0.983328i	$-0.558205\pi$
$83$	−3.31328	−0.363680	−0.181840	+	0.983328i	$0.558205\pi$
$84$	0	0
$85$	−8.64181	−0.937336
$86$	0	0
$87$	−1.92174	−0.206032
$88$	0	0
$89$	5.58054	0.591537	0.295768	−	0.955260i	$-0.404424\pi$
$89$	5.58054	0.591537	0.295768	+	0.955260i	$0.404424\pi$
$90$	0	0
$91$	−4.41278	−0.462585
$92$	0	0
$93$	6.99752	0.725609
$94$	0	0
$95$	−7.76760	−0.796939
$96$	0	0
$97$	8.64165	0.877427	0.438713	−	0.898627i	$-0.355434\pi$
$97$	8.64165	0.877427	0.438713	+	0.898627i	$0.355434\pi$
$98$	0	0
$99$	0.167826	0.0168671
$100$	0	0
$101$	8.70101	0.865783	0.432892	−	0.901446i	$-0.357493\pi$
$101$	8.70101	0.865783	0.432892	+	0.901446i	$0.357493\pi$
$102$	0	0
$103$	−16.5030	−1.62609	−0.813047	−	0.582198i	$-0.802193\pi$
$103$	−16.5030	−1.62609	−0.813047	+	0.582198i	$0.802193\pi$
$104$	0	0
$105$	2.19818	0.214521
$106$	0	0
$107$	−0.513286	−0.0496212	−0.0248106	−	0.999692i	$-0.507898\pi$
$107$	−0.513286	−0.0496212	−0.0248106	+	0.999692i	$0.507898\pi$
$108$	0	0
$109$	−14.5138	−1.39017	−0.695084	−	0.718929i	$-0.744633\pi$
$109$	−14.5138	−1.39017	−0.695084	+	0.718929i	$0.744633\pi$
$110$	0	0
$111$	4.07816	0.387082
$112$	0	0
$113$	−8.31921	−0.782606	−0.391303	−	0.920262i	$-0.627976\pi$
$113$	−8.31921	−0.782606	−0.391303	+	0.920262i	$0.627976\pi$
$114$	0	0
$115$	6.22360	0.580354
$116$	0	0
$117$	−3.59756	−0.332595
$118$	0	0
$119$	5.91491	0.542219
$120$	0	0
$121$	−10.9718	−0.997439
$122$	0	0
$123$	−0.729110	−0.0657416
$124$	0	0
$125$	12.1654	1.08811
$126$	0	0
$127$	−10.3432	−0.917808	−0.458904	−	0.888486i	$-0.651758\pi$
$127$	−10.3432	−0.917808	−0.458904	+	0.888486i	$0.651758\pi$
$128$	0	0
$129$	−9.76299	−0.859584
$130$	0	0
$131$	−6.62853	−0.579138	−0.289569	−	0.957157i	$-0.593512\pi$
$131$	−6.62853	−0.579138	−0.289569	+	0.957157i	$0.593512\pi$
$132$	0	0
$133$	5.31656	0.461004
$134$	0	0
$135$	1.79209	0.154239
$136$	0	0
$137$	−7.20509	−0.615572	−0.307786	−	0.951456i	$-0.599588\pi$
$137$	−7.20509	−0.615572	−0.307786	+	0.951456i	$0.599588\pi$
$138$	0	0
$139$	1.24778	0.105835	0.0529175	−	0.998599i	$-0.483148\pi$
$139$	1.24778	0.105835	0.0529175	+	0.998599i	$0.483148\pi$
$140$	0	0
$141$	8.14600	0.686017
$142$	0	0
$143$	−0.603764	−0.0504893
$144$	0	0
$145$	−3.44393	−0.286003
$146$	0	0
$147$	5.49545	0.453257
$148$	0	0
$149$	−0.440996	−0.0361278	−0.0180639	−	0.999837i	$-0.505750\pi$
$149$	−0.440996	−0.0361278	−0.0180639	+	0.999837i	$0.505750\pi$
$150$	0	0
$151$	−13.0409	−1.06125	−0.530626	−	0.847606i	$-0.678043\pi$
$151$	−13.0409	−1.06125	−0.530626	+	0.847606i	$0.678043\pi$
$152$	0	0
$153$	4.82219	0.389851
$154$	0	0
$155$	12.5402	1.00725
$156$	0	0
$157$	−6.66963	−0.532294	−0.266147	−	0.963932i	$-0.585751\pi$
$157$	−6.66963	−0.532294	−0.266147	+	0.963932i	$0.585751\pi$
$158$	0	0
$159$	−11.4350	−0.906854
$160$	0	0
$161$	−4.25977	−0.335717
$162$	0	0
$163$	−4.39227	−0.344029	−0.172015	−	0.985094i	$-0.555028\pi$
$163$	−4.39227	−0.344029	−0.172015	+	0.985094i	$0.555028\pi$
$164$	0	0
$165$	0.300759	0.0234141
$166$	0	0
$167$	−5.12358	−0.396475	−0.198237	−	0.980154i	$-0.563522\pi$
$167$	−5.12358	−0.396475	−0.198237	+	0.980154i	$0.563522\pi$
$168$	0	0
$169$	−0.0575485	−0.00442681
$170$	0	0
$171$	4.33438	0.331458
$172$	0	0
$173$	2.00142	0.152165	0.0760827	−	0.997102i	$-0.475759\pi$
$173$	2.00142	0.152165	0.0760827	+	0.997102i	$0.475759\pi$
$174$	0	0
$175$	−2.19367	−0.165826
$176$	0	0
$177$	12.7351	0.957228
$178$	0	0
$179$	6.13810	0.458783	0.229392	−	0.973334i	$-0.426326\pi$
$179$	6.13810	0.458783	0.229392	+	0.973334i	$0.426326\pi$
$180$	0	0
$181$	−20.3774	−1.51464	−0.757320	−	0.653044i	$-0.773492\pi$
$181$	−20.3774	−1.51464	−0.757320	+	0.653044i	$0.773492\pi$
$182$	0	0
$183$	−0.467969	−0.0345933
$184$	0	0
$185$	7.30843	0.537326
$186$	0	0
$187$	0.809289	0.0591810
$188$	0	0
$189$	−1.22660	−0.0892222
$190$	0	0
$191$	4.35677	0.315245	0.157622	−	0.987499i	$-0.449617\pi$
$191$	4.35677	0.315245	0.157622	+	0.987499i	$0.449617\pi$
$192$	0	0
$193$	5.63415	0.405555	0.202777	−	0.979225i	$-0.435003\pi$
$193$	5.63415	0.405555	0.202777	+	0.979225i	$0.435003\pi$
$194$	0	0
$195$	−6.44716	−0.461690
$196$	0	0
$197$	−9.15529	−0.652288	−0.326144	−	0.945320i	$-0.605749\pi$
$197$	−9.15529	−0.652288	−0.326144	+	0.945320i	$0.605749\pi$
$198$	0	0
$199$	−21.3685	−1.51477	−0.757386	−	0.652967i	$-0.773524\pi$
$199$	−21.3685	−1.51477	−0.757386	+	0.652967i	$0.773524\pi$
$200$	0	0
$201$	−8.46265	−0.596909
$202$	0	0
$203$	2.35721	0.165444
$204$	0	0
$205$	−1.30663	−0.0912590
$206$	0	0
$207$	−3.47282	−0.241377
$208$	0	0
$209$	0.727421	0.0503167
$210$	0	0
$211$	−2.00554	−0.138067	−0.0690336	−	0.997614i	$-0.521992\pi$
$211$	−2.00554	−0.138067	−0.0690336	+	0.997614i	$0.521992\pi$
$212$	0	0
$213$	−12.7332	−0.872466
$214$	0	0
$215$	−17.4962	−1.19323
$216$	0	0
$217$	−8.58317	−0.582664
$218$	0	0
$219$	−0.468869	−0.0316832
$220$	0	0
$221$	−17.3481	−1.16696
$222$	0	0
$223$	−6.76475	−0.453001	−0.226501	−	0.974011i	$-0.572729\pi$
$223$	−6.76475	−0.453001	−0.226501	+	0.974011i	$0.572729\pi$
$224$	0	0
$225$	−1.78841	−0.119227
$226$	0	0
$227$	21.8273	1.44873	0.724366	−	0.689416i	$-0.242133\pi$
$227$	21.8273	1.44873	0.724366	+	0.689416i	$0.242133\pi$
$228$	0	0
$229$	6.80311	0.449562	0.224781	−	0.974409i	$-0.427833\pi$
$229$	6.80311	0.449562	0.224781	+	0.974409i	$0.427833\pi$
$230$	0	0
$231$	−0.205856	−0.0135443
$232$	0	0
$233$	−1.05122	−0.0688679	−0.0344339	−	0.999407i	$-0.510963\pi$
$233$	−1.05122	−0.0688679	−0.0344339	+	0.999407i	$0.510963\pi$
$234$	0	0
$235$	14.5984	0.952293
$236$	0	0
$237$	−15.9731	−1.03757
$238$	0	0
$239$	−14.4774	−0.936465	−0.468233	−	0.883605i	$-0.655109\pi$
$239$	−14.4774	−0.936465	−0.468233	+	0.883605i	$0.655109\pi$
$240$	0	0
$241$	−1.65494	−0.106604	−0.0533020	−	0.998578i	$-0.516975\pi$
$241$	−1.65494	−0.106604	−0.0533020	+	0.998578i	$0.516975\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	9.84834	0.629187
$246$	0	0
$247$	−15.5932	−0.992171
$248$	0	0
$249$	3.31328	0.209971
$250$	0	0
$251$	−5.61587	−0.354471	−0.177235	−	0.984169i	$-0.556715\pi$
$251$	−5.61587	−0.354471	−0.177235	+	0.984169i	$0.556715\pi$
$252$	0	0
$253$	−0.582828	−0.0366421
$254$	0	0
$255$	8.64181	0.541171
$256$	0	0
$257$	−5.39589	−0.336586	−0.168293	−	0.985737i	$-0.553826\pi$
$257$	−5.39589	−0.336586	−0.168293	+	0.985737i	$0.553826\pi$
$258$	0	0
$259$	−5.00228	−0.310826
$260$	0	0
$261$	1.92174	0.118953
$262$	0	0
$263$	15.5309	0.957675	0.478837	−	0.877904i	$-0.341058\pi$
$263$	15.5309	0.957675	0.478837	+	0.877904i	$0.341058\pi$
$264$	0	0
$265$	−20.4925	−1.25885
$266$	0	0
$267$	−5.58054	−0.341524
$268$	0	0
$269$	−25.6680	−1.56500	−0.782502	−	0.622648i	$-0.786057\pi$
$269$	−25.6680	−1.56500	−0.782502	+	0.622648i	$0.786057\pi$
$270$	0	0
$271$	−24.8166	−1.50750	−0.753751	−	0.657160i	$-0.771758\pi$
$271$	−24.8166	−1.50750	−0.753751	+	0.657160i	$0.771758\pi$
$272$	0	0
$273$	4.41278	0.267073
$274$	0	0
$275$	−0.300142	−0.0180992
$276$	0	0
$277$	−30.9560	−1.85997	−0.929983	−	0.367601i	$-0.880179\pi$
$277$	−30.9560	−1.85997	−0.929983	+	0.367601i	$0.880179\pi$
$278$	0	0
$279$	−6.99752	−0.418930
$280$	0	0
$281$	25.2183	1.50440	0.752199	−	0.658936i	$-0.228993\pi$
$281$	25.2183	1.50440	0.752199	+	0.658936i	$0.228993\pi$
$282$	0	0
$283$	1.38297	0.0822089	0.0411044	−	0.999155i	$-0.486912\pi$
$283$	1.38297	0.0822089	0.0411044	+	0.999155i	$0.486912\pi$
$284$	0	0
$285$	7.76760	0.460113
$286$	0	0
$287$	0.894328	0.0527905
$288$	0	0
$289$	6.25355	0.367856
$290$	0	0
$291$	−8.64165	−0.506583
$292$	0	0
$293$	6.12940	0.358083	0.179042	−	0.983842i	$-0.442700\pi$
$293$	6.12940	0.358083	0.179042	+	0.983842i	$0.442700\pi$
$294$	0	0
$295$	22.8224	1.32877
$296$	0	0
$297$	−0.167826	−0.00973824
$298$	0	0
$299$	12.4937	0.722528
$300$	0	0
$301$	11.9753	0.690245
$302$	0	0
$303$	−8.70101	−0.499860
$304$	0	0
$305$	−0.838643	−0.0480205
$306$	0	0
$307$	18.2337	1.04065	0.520327	−	0.853967i	$-0.325810\pi$
$307$	18.2337	1.04065	0.520327	+	0.853967i	$0.325810\pi$
$308$	0	0
$309$	16.5030	0.938826
$310$	0	0
$311$	18.2705	1.03603	0.518014	−	0.855372i	$-0.326672\pi$
$311$	18.2705	1.03603	0.518014	+	0.855372i	$0.326672\pi$
$312$	0	0
$313$	15.8065	0.893434	0.446717	−	0.894675i	$-0.352593\pi$
$313$	15.8065	0.893434	0.446717	+	0.894675i	$0.352593\pi$
$314$	0	0
$315$	−2.19818	−0.123854
$316$	0	0
$317$	−33.0785	−1.85787	−0.928936	−	0.370241i	$-0.879275\pi$
$317$	−33.0785	−1.85787	−0.928936	+	0.370241i	$0.879275\pi$
$318$	0	0
$319$	0.322517	0.0180575
$320$	0	0
$321$	0.513286	0.0286488
$322$	0	0
$323$	20.9012	1.16297
$324$	0	0
$325$	6.43392	0.356890
$326$	0	0
$327$	14.5138	0.802614
$328$	0	0
$329$	−9.99190	−0.550872
$330$	0	0
$331$	−16.7567	−0.921031	−0.460516	−	0.887652i	$-0.652335\pi$
$331$	−16.7567	−0.921031	−0.460516	+	0.887652i	$0.652335\pi$
$332$	0	0
$333$	−4.07816	−0.223482
$334$	0	0
$335$	−15.1658	−0.828598
$336$	0	0
$337$	6.28938	0.342604	0.171302	−	0.985219i	$-0.445203\pi$
$337$	6.28938	0.342604	0.171302	+	0.985219i	$0.445203\pi$
$338$	0	0
$339$	8.31921	0.451838
$340$	0	0
$341$	−1.17436	−0.0635954
$342$	0	0
$343$	−15.3269	−0.827577
$344$	0	0
$345$	−6.22360	−0.335067
$346$	0	0
$347$	2.59570	0.139344	0.0696721	−	0.997570i	$-0.477805\pi$
$347$	2.59570	0.139344	0.0696721	+	0.997570i	$0.477805\pi$
$348$	0	0
$349$	−11.6020	−0.621038	−0.310519	−	0.950567i	$-0.600503\pi$
$349$	−11.6020	−0.621038	−0.310519	+	0.950567i	$0.600503\pi$
$350$	0	0
$351$	3.59756	0.192024
$352$	0	0
$353$	−24.7323	−1.31637	−0.658184	−	0.752857i	$-0.728675\pi$
$353$	−24.7323	−1.31637	−0.658184	+	0.752857i	$0.728675\pi$
$354$	0	0
$355$	−22.8191	−1.21111
$356$	0	0
$357$	−5.91491	−0.313050
$358$	0	0
$359$	11.8719	0.626576	0.313288	−	0.949658i	$-0.398569\pi$
$359$	11.8719	0.626576	0.313288	+	0.949658i	$0.398569\pi$
$360$	0	0
$361$	−0.213160	−0.0112189
$362$	0	0
$363$	10.9718	0.575872
$364$	0	0
$365$	−0.840255	−0.0439810
$366$	0	0
$367$	−17.1130	−0.893291	−0.446645	−	0.894711i	$-0.647381\pi$
$367$	−17.1130	−0.893291	−0.446645	+	0.894711i	$0.647381\pi$
$368$	0	0
$369$	0.729110	0.0379559
$370$	0	0
$371$	14.0262	0.728204
$372$	0	0
$373$	−22.5908	−1.16971	−0.584855	−	0.811138i	$-0.698848\pi$
$373$	−22.5908	−1.16971	−0.584855	+	0.811138i	$0.698848\pi$
$374$	0	0
$375$	−12.1654	−0.628221
$376$	0	0
$377$	−6.91358	−0.356067
$378$	0	0
$379$	5.01385	0.257544	0.128772	−	0.991674i	$-0.458896\pi$
$379$	5.01385	0.257544	0.128772	+	0.991674i	$0.458896\pi$
$380$	0	0
$381$	10.3432	0.529897
$382$	0	0
$383$	30.5020	1.55858	0.779288	−	0.626665i	$-0.215581\pi$
$383$	30.5020	1.55858	0.779288	+	0.626665i	$0.215581\pi$
$384$	0	0
$385$	−0.368912	−0.0188015
$386$	0	0
$387$	9.76299	0.496281
$388$	0	0
$389$	28.1008	1.42477	0.712383	−	0.701791i	$-0.247616\pi$
$389$	28.1008	1.42477	0.712383	+	0.701791i	$0.247616\pi$
$390$	0	0
$391$	−16.7466	−0.846912
$392$	0	0
$393$	6.62853	0.334365
$394$	0	0
$395$	−28.6253	−1.44029
$396$	0	0
$397$	−5.17767	−0.259860	−0.129930	−	0.991523i	$-0.541475\pi$
$397$	−5.17767	−0.259860	−0.129930	+	0.991523i	$0.541475\pi$
$398$	0	0
$399$	−5.31656	−0.266161
$400$	0	0
$401$	−12.8273	−0.640565	−0.320282	−	0.947322i	$-0.603778\pi$
$401$	−12.8273	−0.640565	−0.320282	+	0.947322i	$0.603778\pi$
$402$	0	0
$403$	25.1740	1.25401
$404$	0	0
$405$	−1.79209	−0.0890497
$406$	0	0
$407$	−0.684420	−0.0339254
$408$	0	0
$409$	−31.9374	−1.57920	−0.789601	−	0.613621i	$-0.789712\pi$
$409$	−31.9374	−1.57920	−0.789601	+	0.613621i	$0.789712\pi$
$410$	0	0
$411$	7.20509	0.355401
$412$	0	0
$413$	−15.6209	−0.768654
$414$	0	0
$415$	5.93770	0.291470
$416$	0	0
$417$	−1.24778	−0.0611039
$418$	0	0
$419$	−26.4743	−1.29335	−0.646677	−	0.762764i	$-0.723842\pi$
$419$	−26.4743	−1.29335	−0.646677	+	0.762764i	$0.723842\pi$
$420$	0	0
$421$	−15.8282	−0.771417	−0.385709	−	0.922621i	$-0.626043\pi$
$421$	−15.8282	−0.771417	−0.385709	+	0.922621i	$0.626043\pi$
$422$	0	0
$423$	−8.14600	−0.396072
$424$	0	0
$425$	−8.62407	−0.418329
$426$	0	0
$427$	0.574012	0.0277784
$428$	0	0
$429$	0.603764	0.0291500
$430$	0	0
$431$	0.250263	0.0120547	0.00602737	−	0.999982i	$-0.498081\pi$
$431$	0.250263	0.0120547	0.00602737	+	0.999982i	$0.498081\pi$
$432$	0	0
$433$	−27.6934	−1.33086	−0.665430	−	0.746460i	$-0.731752\pi$
$433$	−27.6934	−1.33086	−0.665430	+	0.746460i	$0.731752\pi$
$434$	0	0
$435$	3.44393	0.165124
$436$	0	0
$437$	−15.0525	−0.720059
$438$	0	0
$439$	−28.2656	−1.34905	−0.674523	−	0.738254i	$-0.735651\pi$
$439$	−28.2656	−1.34905	−0.674523	+	0.738254i	$0.735651\pi$
$440$	0	0
$441$	−5.49545	−0.261688
$442$	0	0
$443$	−10.1825	−0.483785	−0.241893	−	0.970303i	$-0.577768\pi$
$443$	−10.1825	−0.483785	−0.241893	+	0.970303i	$0.577768\pi$
$444$	0	0
$445$	−10.0008	−0.474085
$446$	0	0
$447$	0.440996	0.0208584
$448$	0	0
$449$	−17.1365	−0.808724	−0.404362	−	0.914599i	$-0.632506\pi$
$449$	−17.1365	−0.808724	−0.404362	+	0.914599i	$0.632506\pi$
$450$	0	0
$451$	0.122363	0.00576187
$452$	0	0
$453$	13.0409	0.612714
$454$	0	0
$455$	7.90810	0.370737
$456$	0	0
$457$	−41.2517	−1.92967	−0.964837	−	0.262848i	$-0.915338\pi$
$457$	−41.2517	−1.92967	−0.964837	+	0.262848i	$0.915338\pi$
$458$	0	0
$459$	−4.82219	−0.225081
$460$	0	0
$461$	7.15240	0.333121	0.166560	−	0.986031i	$-0.446734\pi$
$461$	7.15240	0.333121	0.166560	+	0.986031i	$0.446734\pi$
$462$	0	0
$463$	−11.2571	−0.523164	−0.261582	−	0.965181i	$-0.584244\pi$
$463$	−11.2571	−0.523164	−0.261582	+	0.965181i	$0.584244\pi$
$464$	0	0
$465$	−12.5402	−0.581537
$466$	0	0
$467$	−4.30744	−0.199325	−0.0996623	−	0.995021i	$-0.531776\pi$
$467$	−4.30744	−0.199325	−0.0996623	+	0.995021i	$0.531776\pi$
$468$	0	0
$469$	10.3803	0.479318
$470$	0	0
$471$	6.66963	0.307320
$472$	0	0
$473$	1.63848	0.0753375
$474$	0	0
$475$	−7.75166	−0.355670
$476$	0	0
$477$	11.4350	0.523572
$478$	0	0
$479$	−38.1095	−1.74127	−0.870634	−	0.491932i	$-0.836291\pi$
$479$	−38.1095	−1.74127	−0.870634	+	0.491932i	$0.836291\pi$
$480$	0	0
$481$	14.6714	0.668959
$482$	0	0
$483$	4.25977	0.193826
$484$	0	0
$485$	−15.4866	−0.703211
$486$	0	0
$487$	−0.291938	−0.0132290	−0.00661449	−	0.999978i	$-0.502105\pi$
$487$	−0.291938	−0.0132290	−0.00661449	+	0.999978i	$0.502105\pi$
$488$	0	0
$489$	4.39227	0.198625
$490$	0	0
$491$	23.8488	1.07628	0.538141	−	0.842855i	$-0.319127\pi$
$491$	23.8488	1.07628	0.538141	+	0.842855i	$0.319127\pi$
$492$	0	0
$493$	9.26700	0.417365
$494$	0	0
$495$	−0.300759	−0.0135181
$496$	0	0
$497$	15.6186	0.700590
$498$	0	0
$499$	24.8394	1.11196	0.555982	−	0.831195i	$-0.312342\pi$
$499$	24.8394	1.11196	0.555982	+	0.831195i	$0.312342\pi$
$500$	0	0
$501$	5.12358	0.228905
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−15.5930	−0.693879
$506$	0	0
$507$	0.0575485	0.00255582
$508$	0	0
$509$	−4.58612	−0.203276	−0.101638	−	0.994821i	$-0.532408\pi$
$509$	−4.58612	−0.203276	−0.101638	+	0.994821i	$0.532408\pi$
$510$	0	0
$511$	0.575116	0.0254416
$512$	0	0
$513$	−4.33438	−0.191368
$514$	0	0
$515$	29.5750	1.30323
$516$	0	0
$517$	−1.36711	−0.0601254
$518$	0	0
$519$	−2.00142	−0.0878527
$520$	0	0
$521$	−29.8412	−1.30737	−0.653683	−	0.756768i	$-0.726777\pi$
$521$	−29.8412	−1.30737	−0.653683	+	0.756768i	$0.726777\pi$
$522$	0	0
$523$	38.3765	1.67809	0.839044	−	0.544064i	$-0.183115\pi$
$523$	38.3765	1.67809	0.839044	+	0.544064i	$0.183115\pi$
$524$	0	0
$525$	2.19367	0.0957397
$526$	0	0
$527$	−33.7434	−1.46988
$528$	0	0
$529$	−10.9395	−0.475632
$530$	0	0
$531$	−12.7351	−0.552656
$532$	0	0
$533$	−2.62302	−0.113615
$534$	0	0
$535$	0.919854	0.0397688
$536$	0	0
$537$	−6.13810	−0.264879
$538$	0	0
$539$	−0.922278	−0.0397253
$540$	0	0
$541$	45.4329	1.95331	0.976657	−	0.214804i	$-0.0689113\pi$
$541$	45.4329	1.95331	0.976657	+	0.214804i	$0.0689113\pi$
$542$	0	0
$543$	20.3774	0.874478
$544$	0	0
$545$	26.0100	1.11415
$546$	0	0
$547$	−0.772790	−0.0330421	−0.0165211	−	0.999864i	$-0.505259\pi$
$547$	−0.772790	−0.0330421	−0.0165211	+	0.999864i	$0.505259\pi$
$548$	0	0
$549$	0.467969	0.0199724
$550$	0	0
$551$	8.32955	0.354851
$552$	0	0
$553$	19.5927	0.833165
$554$	0	0
$555$	−7.30843	−0.310225
$556$	0	0
$557$	−14.6155	−0.619278	−0.309639	−	0.950854i	$-0.600208\pi$
$557$	−14.6155	−0.619278	−0.309639	+	0.950854i	$0.600208\pi$
$558$	0	0
$559$	−35.1230	−1.48554
$560$	0	0
$561$	−0.809289	−0.0341682
$562$	0	0
$563$	2.79231	0.117682	0.0588409	−	0.998267i	$-0.481260\pi$
$563$	2.79231	0.117682	0.0588409	+	0.998267i	$0.481260\pi$
$564$	0	0
$565$	14.9088	0.627217
$566$	0	0
$567$	1.22660	0.0515125
$568$	0	0
$569$	−24.1424	−1.01210	−0.506051	−	0.862504i	$-0.668895\pi$
$569$	−24.1424	−1.01210	−0.506051	+	0.862504i	$0.668895\pi$
$570$	0	0
$571$	−7.23368	−0.302720	−0.151360	−	0.988479i	$-0.548365\pi$
$571$	−7.23368	−0.302720	−0.151360	+	0.988479i	$0.548365\pi$
$572$	0	0
$573$	−4.35677	−0.182007
$574$	0	0
$575$	6.21083	0.259009
$576$	0	0
$577$	−31.4079	−1.30753	−0.653763	−	0.756699i	$-0.726811\pi$
$577$	−31.4079	−1.30753	−0.653763	+	0.756699i	$0.726811\pi$
$578$	0	0
$579$	−5.63415	−0.234147
$580$	0	0
$581$	−4.06408	−0.168607
$582$	0	0
$583$	1.91909	0.0794805
$584$	0	0
$585$	6.44716	0.266557
$586$	0	0
$587$	48.0452	1.98304	0.991518	−	0.129970i	$-0.0414881\pi$
$587$	48.0452	1.98304	0.991518	+	0.129970i	$0.0414881\pi$
$588$	0	0
$589$	−30.3299	−1.24972
$590$	0	0
$591$	9.15529	0.376598
$592$	0	0
$593$	40.9814	1.68290	0.841452	−	0.540333i	$-0.181702\pi$
$593$	40.9814	1.68290	0.841452	+	0.540333i	$0.181702\pi$
$594$	0	0
$595$	−10.6001	−0.434560
$596$	0	0
$597$	21.3685	0.874554
$598$	0	0
$599$	−2.87303	−0.117389	−0.0586943	−	0.998276i	$-0.518694\pi$
$599$	−2.87303	−0.117389	−0.0586943	+	0.998276i	$0.518694\pi$
$600$	0	0
$601$	−22.8932	−0.933833	−0.466917	−	0.884301i	$-0.654635\pi$
$601$	−22.8932	−0.933833	−0.466917	+	0.884301i	$0.654635\pi$
$602$	0	0
$603$	8.46265	0.344626
$604$	0	0
$605$	19.6625	0.799395
$606$	0	0
$607$	−5.61404	−0.227867	−0.113933	−	0.993488i	$-0.536345\pi$
$607$	−5.61404	−0.227867	−0.113933	+	0.993488i	$0.536345\pi$
$608$	0	0
$609$	−2.35721	−0.0955190
$610$	0	0
$611$	29.3057	1.18558
$612$	0	0
$613$	−27.8081	−1.12316	−0.561579	−	0.827423i	$-0.689806\pi$
$613$	−27.8081	−1.12316	−0.561579	+	0.827423i	$0.689806\pi$
$614$	0	0
$615$	1.30663	0.0526884
$616$	0	0
$617$	−13.8028	−0.555681	−0.277841	−	0.960627i	$-0.589619\pi$
$617$	−13.8028	−0.555681	−0.277841	+	0.960627i	$0.589619\pi$
$618$	0	0
$619$	−23.4742	−0.943508	−0.471754	−	0.881730i	$-0.656379\pi$
$619$	−23.4742	−0.943508	−0.471754	+	0.881730i	$0.656379\pi$
$620$	0	0
$621$	3.47282	0.139359
$622$	0	0
$623$	6.84511	0.274244
$624$	0	0
$625$	−12.8595	−0.514381
$626$	0	0
$627$	−0.727421	−0.0290504
$628$	0	0
$629$	−19.6657	−0.784122
$630$	0	0
$631$	30.4048	1.21040	0.605198	−	0.796075i	$-0.293094\pi$
$631$	30.4048	1.21040	0.605198	+	0.796075i	$0.293094\pi$
$632$	0	0
$633$	2.00554	0.0797132
$634$	0	0
$635$	18.5359	0.735575
$636$	0	0
$637$	19.7702	0.783324
$638$	0	0
$639$	12.7332	0.503718
$640$	0	0
$641$	−20.5406	−0.811303	−0.405651	−	0.914028i	$-0.632955\pi$
$641$	−20.5406	−0.811303	−0.405651	+	0.914028i	$0.632955\pi$
$642$	0	0
$643$	−1.26364	−0.0498332	−0.0249166	−	0.999690i	$-0.507932\pi$
$643$	−1.26364	−0.0498332	−0.0249166	+	0.999690i	$0.507932\pi$
$644$	0	0
$645$	17.4962	0.688911
$646$	0	0
$647$	−18.2284	−0.716631	−0.358316	−	0.933600i	$-0.616649\pi$
$647$	−18.2284	−0.716631	−0.358316	+	0.933600i	$0.616649\pi$
$648$	0	0
$649$	−2.13728	−0.0838954
$650$	0	0
$651$	8.58317	0.336401
$652$	0	0
$653$	−2.09514	−0.0819892	−0.0409946	−	0.999159i	$-0.513053\pi$
$653$	−2.09514	−0.0819892	−0.0409946	+	0.999159i	$0.513053\pi$
$654$	0	0
$655$	11.8789	0.464148
$656$	0	0
$657$	0.468869	0.0182923
$658$	0	0
$659$	−5.54392	−0.215961	−0.107980	−	0.994153i	$-0.534438\pi$
$659$	−5.54392	−0.215961	−0.107980	+	0.994153i	$0.534438\pi$
$660$	0	0
$661$	7.55072	0.293689	0.146844	−	0.989160i	$-0.453088\pi$
$661$	7.55072	0.293689	0.146844	+	0.989160i	$0.453088\pi$
$662$	0	0
$663$	17.3481	0.673746
$664$	0	0
$665$	−9.52775	−0.369470
$666$	0	0
$667$	−6.67385	−0.258413
$668$	0	0
$669$	6.76475	0.261540
$670$	0	0
$671$	0.0785373	0.00303190
$672$	0	0
$673$	35.3569	1.36291	0.681454	−	0.731861i	$-0.261348\pi$
$673$	35.3569	1.36291	0.681454	+	0.731861i	$0.261348\pi$
$674$	0	0
$675$	1.78841	0.0688360
$676$	0	0
$677$	−21.6292	−0.831278	−0.415639	−	0.909530i	$-0.636442\pi$
$677$	−21.6292	−0.831278	−0.415639	+	0.909530i	$0.636442\pi$
$678$	0	0
$679$	10.5999	0.406786
$680$	0	0
$681$	−21.8273	−0.836425
$682$	0	0
$683$	42.9056	1.64174	0.820869	−	0.571116i	$-0.193489\pi$
$683$	42.9056	1.64174	0.820869	+	0.571116i	$0.193489\pi$
$684$	0	0
$685$	12.9122	0.493349
$686$	0	0
$687$	−6.80311	−0.259555
$688$	0	0
$689$	−41.1381	−1.56724
$690$	0	0
$691$	33.0579	1.25758	0.628790	−	0.777575i	$-0.283550\pi$
$691$	33.0579	1.25758	0.628790	+	0.777575i	$0.283550\pi$
$692$	0	0
$693$	0.205856	0.00781981
$694$	0	0
$695$	−2.23613	−0.0848211
$696$	0	0
$697$	3.51591	0.133175
$698$	0	0
$699$	1.05122	0.0397609
$700$	0	0
$701$	6.59831	0.249215	0.124607	−	0.992206i	$-0.460233\pi$
$701$	6.59831	0.249215	0.124607	+	0.992206i	$0.460233\pi$
$702$	0	0
$703$	−17.6763	−0.666674
$704$	0	0
$705$	−14.5984	−0.549806
$706$	0	0
$707$	10.6727	0.401388
$708$	0	0
$709$	22.9505	0.861925	0.430963	−	0.902370i	$-0.358174\pi$
$709$	22.9505	0.861925	0.430963	+	0.902370i	$0.358174\pi$
$710$	0	0
$711$	15.9731	0.599039
$712$	0	0
$713$	24.3011	0.910083
$714$	0	0
$715$	1.08200	0.0404645
$716$	0	0
$717$	14.4774	0.540668
$718$	0	0
$719$	−22.2754	−0.830731	−0.415365	−	0.909655i	$-0.636346\pi$
$719$	−22.2754	−0.830731	−0.415365	+	0.909655i	$0.636346\pi$
$720$	0	0
$721$	−20.2427	−0.753877
$722$	0	0
$723$	1.65494	0.0615478
$724$	0	0
$725$	−3.43686	−0.127642
$726$	0	0
$727$	46.2551	1.71551	0.857754	−	0.514061i	$-0.171859\pi$
$727$	46.2551	1.71551	0.857754	+	0.514061i	$0.171859\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	47.0790	1.74128
$732$	0	0
$733$	−2.79739	−0.103324	−0.0516620	−	0.998665i	$-0.516452\pi$
$733$	−2.79739	−0.103324	−0.0516620	+	0.998665i	$0.516452\pi$
$734$	0	0
$735$	−9.84834	−0.363261
$736$	0	0
$737$	1.42025	0.0523156
$738$	0	0
$739$	26.0000	0.956426	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956426	0.478213	+	0.878244i	$0.341285\pi$
$740$	0	0
$741$	15.5932	0.572830
$742$	0	0
$743$	−29.5433	−1.08384	−0.541919	−	0.840431i	$-0.682302\pi$
$743$	−29.5433	−1.08384	−0.541919	+	0.840431i	$0.682302\pi$
$744$	0	0
$745$	0.790304	0.0289545
$746$	0	0
$747$	−3.31328	−0.121227
$748$	0	0
$749$	−0.629597	−0.0230050
$750$	0	0
$751$	8.62456	0.314715	0.157357	−	0.987542i	$-0.449703\pi$
$751$	8.62456	0.314715	0.157357	+	0.987542i	$0.449703\pi$
$752$	0	0
$753$	5.61587	0.204654
$754$	0	0
$755$	23.3704	0.850537
$756$	0	0
$757$	18.6456	0.677688	0.338844	−	0.940843i	$-0.389964\pi$
$757$	18.6456	0.677688	0.338844	+	0.940843i	$0.389964\pi$
$758$	0	0
$759$	0.582828	0.0211553
$760$	0	0
$761$	−20.9769	−0.760411	−0.380206	−	0.924902i	$-0.624147\pi$
$761$	−20.9769	−0.760411	−0.380206	+	0.924902i	$0.624147\pi$
$762$	0	0
$763$	−17.8026	−0.644499
$764$	0	0
$765$	−8.64181	−0.312445
$766$	0	0
$767$	45.8153	1.65429
$768$	0	0
$769$	−20.7583	−0.748562	−0.374281	−	0.927315i	$-0.622111\pi$
$769$	−20.7583	−0.748562	−0.374281	+	0.927315i	$0.622111\pi$
$770$	0	0
$771$	5.39589	0.194328
$772$	0	0
$773$	42.3826	1.52440	0.762198	−	0.647344i	$-0.224120\pi$
$773$	42.3826	1.52440	0.762198	+	0.647344i	$0.224120\pi$
$774$	0	0
$775$	12.5144	0.449532
$776$	0	0
$777$	5.00228	0.179456
$778$	0	0
$779$	3.16024	0.113227
$780$	0	0
$781$	2.13696	0.0764665
$782$	0	0
$783$	−1.92174	−0.0686774
$784$	0	0
$785$	11.9526	0.426606
$786$	0	0
$787$	27.6534	0.985737	0.492868	−	0.870104i	$-0.335948\pi$
$787$	27.6534	0.985737	0.492868	+	0.870104i	$0.335948\pi$
$788$	0	0
$789$	−15.5309	−0.552914
$790$	0	0
$791$	−10.2044	−0.362826
$792$	0	0
$793$	−1.68355	−0.0597845
$794$	0	0
$795$	20.4925	0.726796
$796$	0	0
$797$	15.1906	0.538079	0.269040	−	0.963129i	$-0.413294\pi$
$797$	15.1906	0.538079	0.269040	+	0.963129i	$0.413294\pi$
$798$	0	0
$799$	−39.2816	−1.38968
$800$	0	0
$801$	5.58054	0.197179
$802$	0	0
$803$	0.0786883	0.00277685
$804$	0	0
$805$	7.63389	0.269059
$806$	0	0
$807$	25.6680	0.903555
$808$	0	0
$809$	1.67803	0.0589963	0.0294981	−	0.999565i	$-0.490609\pi$
$809$	1.67803	0.0589963	0.0294981	+	0.999565i	$0.490609\pi$
$810$	0	0
$811$	−3.41902	−0.120058	−0.0600291	−	0.998197i	$-0.519119\pi$
$811$	−3.41902	−0.120058	−0.0600291	+	0.998197i	$0.519119\pi$
$812$	0	0
$813$	24.8166	0.870357
$814$	0	0
$815$	7.87135	0.275721
$816$	0	0
$817$	42.3165	1.48047
$818$	0	0
$819$	−4.41278	−0.154195
$820$	0	0
$821$	−15.4492	−0.539182	−0.269591	−	0.962975i	$-0.586889\pi$
$821$	−15.4492	−0.539182	−0.269591	+	0.962975i	$0.586889\pi$
$822$	0	0
$823$	34.7016	1.20962	0.604811	−	0.796369i	$-0.293249\pi$
$823$	34.7016	1.20962	0.604811	+	0.796369i	$0.293249\pi$
$824$	0	0
$825$	0.300142	0.0104496
$826$	0	0
$827$	−40.7180	−1.41590	−0.707952	−	0.706261i	$-0.750381\pi$
$827$	−40.7180	−1.41590	−0.707952	+	0.706261i	$0.750381\pi$
$828$	0	0
$829$	1.45298	0.0504641	0.0252321	−	0.999682i	$-0.491968\pi$
$829$	1.45298	0.0504641	0.0252321	+	0.999682i	$0.491968\pi$
$830$	0	0
$831$	30.9560	1.07385
$832$	0	0
$833$	−26.5001	−0.918174
$834$	0	0
$835$	9.18192	0.317754
$836$	0	0
$837$	6.99752	0.241870
$838$	0	0
$839$	−25.8428	−0.892192	−0.446096	−	0.894985i	$-0.647186\pi$
$839$	−25.8428	−0.892192	−0.446096	+	0.894985i	$0.647186\pi$
$840$	0	0
$841$	−25.3069	−0.872652
$842$	0	0
$843$	−25.2183	−0.868565
$844$	0	0
$845$	0.103132	0.00354785
$846$	0	0
$847$	−13.4581	−0.462425
$848$	0	0
$849$	−1.38297	−0.0474633
$850$	0	0
$851$	14.1627	0.485491
$852$	0	0
$853$	40.0670	1.37187	0.685933	−	0.727664i	$-0.259394\pi$
$853$	40.0670	1.37187	0.685933	+	0.727664i	$0.259394\pi$
$854$	0	0
$855$	−7.76760	−0.265646
$856$	0	0
$857$	26.0739	0.890668	0.445334	−	0.895365i	$-0.353085\pi$
$857$	26.0739	0.890668	0.445334	+	0.895365i	$0.353085\pi$
$858$	0	0
$859$	32.9545	1.12439	0.562196	−	0.827004i	$-0.309957\pi$
$859$	32.9545	1.12439	0.562196	+	0.827004i	$0.309957\pi$
$860$	0	0
$861$	−0.894328	−0.0304786
$862$	0	0
$863$	35.7863	1.21818	0.609090	−	0.793101i	$-0.291535\pi$
$863$	35.7863	1.21818	0.609090	+	0.793101i	$0.291535\pi$
$864$	0	0
$865$	−3.58673	−0.121952
$866$	0	0
$867$	−6.25355	−0.212382
$868$	0	0
$869$	2.68070	0.0909366
$870$	0	0
$871$	−30.4449	−1.03159
$872$	0	0
$873$	8.64165	0.292476
$874$	0	0
$875$	14.9222	0.504461
$876$	0	0
$877$	−1.16445	−0.0393207	−0.0196604	−	0.999807i	$-0.506258\pi$
$877$	−1.16445	−0.0393207	−0.0196604	+	0.999807i	$0.506258\pi$
$878$	0	0
$879$	−6.12940	−0.206739
$880$	0	0
$881$	−4.07448	−0.137273	−0.0686363	−	0.997642i	$-0.521865\pi$
$881$	−4.07448	−0.137273	−0.0686363	+	0.997642i	$0.521865\pi$
$882$	0	0
$883$	22.0672	0.742621	0.371311	−	0.928509i	$-0.378908\pi$
$883$	22.0672	0.742621	0.371311	+	0.928509i	$0.378908\pi$
$884$	0	0
$885$	−22.8224	−0.767167
$886$	0	0
$887$	−32.4298	−1.08889	−0.544443	−	0.838798i	$-0.683259\pi$
$887$	−32.4298	−1.08889	−0.544443	+	0.838798i	$0.683259\pi$
$888$	0	0
$889$	−12.6870	−0.425507
$890$	0	0
$891$	0.167826	0.00562238
$892$	0	0
$893$	−35.3079	−1.18153
$894$	0	0
$895$	−11.0000	−0.367690
$896$	0	0
$897$	−12.4937	−0.417152
$898$	0	0
$899$	−13.4474	−0.448496
$900$	0	0
$901$	55.1418	1.83704
$902$	0	0
$903$	−11.9753	−0.398513
$904$	0	0
$905$	36.5181	1.21390
$906$	0	0
$907$	10.3276	0.342921	0.171461	−	0.985191i	$-0.445151\pi$
$907$	10.3276	0.342921	0.171461	+	0.985191i	$0.445151\pi$
$908$	0	0
$909$	8.70101	0.288594
$910$	0	0
$911$	26.3194	0.872002	0.436001	−	0.899946i	$-0.356394\pi$
$911$	26.3194	0.872002	0.436001	+	0.899946i	$0.356394\pi$
$912$	0	0
$913$	−0.556054	−0.0184027
$914$	0	0
$915$	0.838643	0.0277247
$916$	0	0
$917$	−8.13057	−0.268495
$918$	0	0
$919$	−4.68745	−0.154625	−0.0773123	−	0.997007i	$-0.524634\pi$
$919$	−4.68745	−0.154625	−0.0773123	+	0.997007i	$0.524634\pi$
$920$	0	0
$921$	−18.2337	−0.600821
$922$	0	0
$923$	−45.8085	−1.50781
$924$	0	0
$925$	7.29343	0.239806
$926$	0	0
$927$	−16.5030	−0.542031
$928$	0	0
$929$	−17.8047	−0.584154	−0.292077	−	0.956395i	$-0.594346\pi$
$929$	−17.8047	−0.584154	−0.292077	+	0.956395i	$0.594346\pi$
$930$	0	0
$931$	−23.8193	−0.780648
$932$	0	0
$933$	−18.2705	−0.598151
$934$	0	0
$935$	−1.45032	−0.0474305
$936$	0	0
$937$	8.68641	0.283773	0.141886	−	0.989883i	$-0.454683\pi$
$937$	8.68641	0.283773	0.141886	+	0.989883i	$0.454683\pi$
$938$	0	0
$939$	−15.8065	−0.515825
$940$	0	0
$941$	43.9205	1.43177	0.715883	−	0.698220i	$-0.246024\pi$
$941$	43.9205	1.43177	0.715883	+	0.698220i	$0.246024\pi$
$942$	0	0
$943$	−2.53206	−0.0824554
$944$	0	0
$945$	2.19818	0.0715069
$946$	0	0
$947$	29.7868	0.967940	0.483970	−	0.875085i	$-0.339194\pi$
$947$	29.7868	0.967940	0.483970	+	0.875085i	$0.339194\pi$
$948$	0	0
$949$	−1.68678	−0.0547553
$950$	0	0
$951$	33.0785	1.07264
$952$	0	0
$953$	−0.520357	−0.0168560	−0.00842801	−	0.999964i	$-0.502683\pi$
$953$	−0.520357	−0.0168560	−0.00842801	+	0.999964i	$0.502683\pi$
$954$	0	0
$955$	−7.80773	−0.252652
$956$	0	0
$957$	−0.322517	−0.0104255
$958$	0	0
$959$	−8.83778	−0.285387
$960$	0	0
$961$	17.9652	0.579524
$962$	0	0
$963$	−0.513286	−0.0165404
$964$	0	0
$965$	−10.0969	−0.325031
$966$	0	0
$967$	−52.2381	−1.67987	−0.839933	−	0.542690i	$-0.817406\pi$
$967$	−52.2381	−1.67987	−0.839933	+	0.542690i	$0.817406\pi$
$968$	0	0
$969$	−20.9012	−0.671444
$970$	0	0
$971$	−26.0570	−0.836209	−0.418105	−	0.908399i	$-0.637305\pi$
$971$	−26.0570	−0.836209	−0.418105	+	0.908399i	$0.637305\pi$
$972$	0	0
$973$	1.53053	0.0490664
$974$	0	0
$975$	−6.43392	−0.206050
$976$	0	0
$977$	7.78286	0.248996	0.124498	−	0.992220i	$-0.460268\pi$
$977$	7.78286	0.248996	0.124498	+	0.992220i	$0.460268\pi$
$978$	0	0
$979$	0.936559	0.0299326
$980$	0	0
$981$	−14.5138	−0.463389
$982$	0	0
$983$	30.8071	0.982592	0.491296	−	0.870993i	$-0.336523\pi$
$983$	30.8071	0.982592	0.491296	+	0.870993i	$0.336523\pi$
$984$	0	0
$985$	16.4071	0.522774
$986$	0	0
$987$	9.99190	0.318046
$988$	0	0
$989$	−33.9051	−1.07812
$990$	0	0
$991$	61.7023	1.96004	0.980019	−	0.198903i	$-0.0637379\pi$
$991$	61.7023	1.96004	0.980019	+	0.198903i	$0.0637379\pi$
$992$	0	0
$993$	16.7567	0.531758
$994$	0	0
$995$	38.2943	1.21401
$996$	0	0
$997$	27.6195	0.874720	0.437360	−	0.899287i	$-0.355914\pi$
$997$	27.6195	0.874720	0.437360	+	0.899287i	$0.355914\pi$
$998$	0	0
$999$	4.07816	0.129027

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.f.1.4	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.f.1.4	✓	14	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.79209 q^{5} +1.22660 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.79209 q^{5} +1.22660 q^{7} +1.00000 q^{9} +0.167826 q^{11} -3.59756 q^{13} +1.79209 q^{15} +4.82219 q^{17} +4.33438 q^{19} -1.22660 q^{21} -3.47282 q^{23} -1.78841 q^{25} -1.00000 q^{27} +1.92174 q^{29} -6.99752 q^{31} -0.167826 q^{33} -2.19818 q^{35} -4.07816 q^{37} +3.59756 q^{39} +0.729110 q^{41} +9.76299 q^{43} -1.79209 q^{45} -8.14600 q^{47} -5.49545 q^{49} -4.82219 q^{51} +11.4350 q^{53} -0.300759 q^{55} -4.33438 q^{57} -12.7351 q^{59} +0.467969 q^{61} +1.22660 q^{63} +6.44716 q^{65} +8.46265 q^{67} +3.47282 q^{69} +12.7332 q^{71} +0.468869 q^{73} +1.78841 q^{75} +0.205856 q^{77} +15.9731 q^{79} +1.00000 q^{81} -3.31328 q^{83} -8.64181 q^{85} -1.92174 q^{87} +5.58054 q^{89} -4.41278 q^{91} +6.99752 q^{93} -7.76760 q^{95} +8.64165 q^{97} +0.167826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + q^99

Introduction

L-functions

Modular forms

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Database

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