LMFDB - Embedded newform 45.12.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,12,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	45.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:	$34.5754431252$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1801}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 450 $ x^2 - x - 450
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$21.7191$ of defining polynomial
Character	$\chi$	$=$	45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-14.7191 q^{2} -1831.35 q^{4} +3125.00 q^{5} +79941.2 q^{7} +57100.5 q^{8} -45997.2 q^{10} -805067. q^{11} -1.19767e6 q^{13} -1.17666e6 q^{14} +2.91013e6 q^{16} -2.63319e6 q^{17} +1.16061e7 q^{19} -5.72296e6 q^{20} +1.18499e7 q^{22} -1.84216e7 q^{23} +9.76562e6 q^{25} +1.76286e7 q^{26} -1.46400e8 q^{28} +1.90527e8 q^{29} +1.01127e8 q^{31} -1.59776e8 q^{32} +3.87581e7 q^{34} +2.49816e8 q^{35} +8.06675e7 q^{37} -1.70831e8 q^{38} +1.78439e8 q^{40} -2.26316e8 q^{41} +1.67149e9 q^{43} +1.47436e9 q^{44} +2.71150e8 q^{46} -8.58507e8 q^{47} +4.41327e9 q^{49} -1.43741e8 q^{50} +2.19335e9 q^{52} +3.52750e9 q^{53} -2.51583e9 q^{55} +4.56468e9 q^{56} -2.80438e9 q^{58} -4.35760e9 q^{59} -1.65393e9 q^{61} -1.48849e9 q^{62} -3.60819e9 q^{64} -3.74272e9 q^{65} +7.58610e9 q^{67} +4.82228e9 q^{68} -3.67707e9 q^{70} +2.75809e10 q^{71} +3.22368e10 q^{73} -1.18735e9 q^{74} -2.12548e10 q^{76} -6.43580e10 q^{77} -2.43149e9 q^{79} +9.09417e9 q^{80} +3.33116e9 q^{82} -1.20729e10 q^{83} -8.22871e9 q^{85} -2.46028e10 q^{86} -4.59697e10 q^{88} -4.44073e9 q^{89} -9.57433e10 q^{91} +3.37364e10 q^{92} +1.26364e10 q^{94} +3.62690e10 q^{95} -2.04453e10 q^{97} -6.49594e10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 13 q^{2} - 3111 q^{4} + 6250 q^{5} + 7784 q^{7} - 35139 q^{8} + 40625 q^{10} - 295568 q^{11} + 657492 q^{13} - 3176796 q^{14} + 2974065 q^{16} - 8579948 q^{17} + 17627976 q^{19} - 9721875 q^{20} + 25972696 q^{22}+ \cdots + 24554976677 q^{98}+O(q^{100}) $ 2 * q + 13 * q^2 - 3111 * q^4 + 6250 * q^5 + 7784 * q^7 - 35139 * q^8 + 40625 * q^10 - 295568 * q^11 + 657492 * q^13 - 3176796 * q^14 + 2974065 * q^16 - 8579948 * q^17 + 17627976 * q^19 - 9721875 * q^20 + 25972696 * q^22 + 29841072 * q^23 + 19531250 * q^25 + 69052066 * q^26 - 54064108 * q^28 + 201881948 * q^29 - 71057008 * q^31 + 30902261 * q^32 - 126080702 * q^34 + 24325000 * q^35 + 705858484 * q^37 - 3909548 * q^38 - 109809375 * q^40 + 327655148 * q^41 + 3192552120 * q^43 + 822376568 * q^44 + 1608947832 * q^46 - 2053064720 * q^47 + 7642614770 * q^49 + 126953125 * q^50 - 180610022 * q^52 + 2304299452 * q^53 - 923650000 * q^55 + 11220431460 * q^56 - 2489635114 * q^58 + 1478770576 * q^59 - 8264891460 * q^61 - 6261263448 * q^62 + 1546316681 * q^64 + 2054662500 * q^65 + 24212177528 * q^67 + 12432065594 * q^68 - 9927487500 * q^70 + 20218888256 * q^71 + 25879583268 * q^73 + 16142374034 * q^74 - 28960704764 * q^76 - 101122045248 * q^77 + 22324995440 * q^79 + 9293953125 * q^80 + 18686740478 * q^82 - 48014508984 * q^83 - 26812337500 * q^85 + 17559698092 * q^86 - 92965609896 * q^88 - 79209683076 * q^89 - 229606712048 * q^91 - 28023036264 * q^92 - 20475612776 * q^94 + 55087425000 * q^95 - 37075227452 * q^97 + 24554976677 * q^98

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 $

$2$	−14.7191	−0.325249	−0.162625	−	0.986688i	$-0.551996\pi$
$2$	−14.7191	−0.325249	−0.162625	+	0.986688i	$0.551996\pi$
$3$	0	0
$3$	0	0
$4$	−1831.35	−0.894213
$5$	3125.00	0.447214
$5$	3125.00	0.447214
$6$	0	0
$7$	79941.2	1.79776	0.898880	−	0.438195i	$-0.144382\pi$
$7$	79941.2	1.79776	0.898880	+	0.438195i	$0.144382\pi$
$8$	57100.5	0.616091
$9$	0	0
$10$	−45997.2	−0.145456
$11$	−805067.	−1.50720	−0.753602	−	0.657331i	$-0.771685\pi$
$11$	−805067.	−1.50720	−0.753602	+	0.657331i	$0.771685\pi$
$12$	0	0
$13$	−1.19767e6	−0.894641	−0.447321	−	0.894374i	$-0.647622\pi$
$13$	−1.19767e6	−0.894641	−0.447321	+	0.894374i	$0.647622\pi$
$14$	−1.17666e6	−0.584720
$15$	0	0
$16$	2.91013e6	0.693830
$17$	−2.63319e6	−0.449793	−0.224896	−	0.974383i	$-0.572204\pi$
$17$	−2.63319e6	−0.449793	−0.224896	+	0.974383i	$0.572204\pi$
$18$	0	0
$19$	1.16061e7	1.07533	0.537664	−	0.843159i	$-0.319307\pi$
$19$	1.16061e7	1.07533	0.537664	+	0.843159i	$0.319307\pi$
$20$	−5.72296e6	−0.399904
$21$	0	0
$22$	1.18499e7	0.490217
$23$	−1.84216e7	−0.596794	−0.298397	−	0.954442i	$-0.596452\pi$
$23$	−1.84216e7	−0.596794	−0.298397	+	0.954442i	$0.596452\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	1.76286e7	0.290981
$27$	0	0
$28$	−1.46400e8	−1.60758
$29$	1.90527e8	1.72491	0.862457	−	0.506130i	$-0.168924\pi$
$29$	1.90527e8	1.72491	0.862457	+	0.506130i	$0.168924\pi$
$30$	0	0
$31$	1.01127e8	0.634419	0.317209	−	0.948356i	$-0.397254\pi$
$31$	1.01127e8	0.634419	0.317209	+	0.948356i	$0.397254\pi$
$32$	−1.59776e8	−0.841759
$33$	0	0
$34$	3.87581e7	0.146295
$35$	2.49816e8	0.803983
$36$	0	0
$37$	8.06675e7	0.191245	0.0956223	−	0.995418i	$-0.469516\pi$
$37$	8.06675e7	0.191245	0.0956223	+	0.995418i	$0.469516\pi$
$38$	−1.70831e8	−0.349749
$39$	0	0
$40$	1.78439e8	0.275524
$41$	−2.26316e8	−0.305073	−0.152537	−	0.988298i	$-0.548744\pi$
$41$	−2.26316e8	−0.305073	−0.152537	+	0.988298i	$0.548744\pi$
$42$	0	0
$43$	1.67149e9	1.73391	0.866956	−	0.498385i	$-0.166073\pi$
$43$	1.67149e9	1.73391	0.866956	+	0.498385i	$0.166073\pi$
$44$	1.47436e9	1.34776
$45$	0	0
$46$	2.71150e8	0.194107
$47$	−8.58507e8	−0.546016	−0.273008	−	0.962012i	$-0.588019\pi$
$47$	−8.58507e8	−0.546016	−0.273008	+	0.962012i	$0.588019\pi$
$48$	0	0
$49$	4.41327e9	2.23194
$50$	−1.43741e8	−0.0650498
$51$	0	0
$52$	2.19335e9	0.800000
$53$	3.52750e9	1.15864	0.579322	−	0.815099i	$-0.303317\pi$
$53$	3.52750e9	1.15864	0.579322	+	0.815099i	$0.303317\pi$
$54$	0	0
$55$	−2.51583e9	−0.674042
$56$	4.56468e9	1.10758
$57$	0	0
$58$	−2.80438e9	−0.561027
$59$	−4.35760e9	−0.793527	−0.396763	−	0.917921i	$-0.629867\pi$
$59$	−4.35760e9	−0.793527	−0.396763	+	0.917921i	$0.629867\pi$
$60$	0	0
$61$	−1.65393e9	−0.250727	−0.125364	−	0.992111i	$-0.540010\pi$
$61$	−1.65393e9	−0.250727	−0.125364	+	0.992111i	$0.540010\pi$
$62$	−1.48849e9	−0.206344
$63$	0	0
$64$	−3.60819e9	−0.420049
$65$	−3.74272e9	−0.400096
$66$	0	0
$67$	7.58610e9	0.686447	0.343224	−	0.939254i	$-0.388481\pi$
$67$	7.58610e9	0.686447	0.343224	+	0.939254i	$0.388481\pi$
$68$	4.82228e9	0.402211
$69$	0	0
$70$	−3.67707e9	−0.261495
$71$	2.75809e10	1.81421	0.907104	−	0.420905i	$-0.138288\pi$
$71$	2.75809e10	1.81421	0.907104	+	0.420905i	$0.138288\pi$
$72$	0	0
$73$	3.22368e10	1.82002	0.910011	−	0.414584i	$-0.136073\pi$
$73$	3.22368e10	1.82002	0.910011	+	0.414584i	$0.136073\pi$
$74$	−1.18735e9	−0.0622022
$75$	0	0
$76$	−2.12548e10	−0.961572
$77$	−6.43580e10	−2.70959
$78$	0	0
$79$	−2.43149e9	−0.0889046	−0.0444523	−	0.999012i	$-0.514154\pi$
$79$	−2.43149e9	−0.0889046	−0.0444523	+	0.999012i	$0.514154\pi$
$80$	9.09417e9	0.310290
$81$	0	0
$82$	3.33116e9	0.0992247
$83$	−1.20729e10	−0.336421	−0.168210	−	0.985751i	$-0.553799\pi$
$83$	−1.20729e10	−0.336421	−0.168210	+	0.985751i	$0.553799\pi$
$84$	0	0
$85$	−8.22871e9	−0.201154
$86$	−2.46028e10	−0.563953
$87$	0	0
$88$	−4.59697e10	−0.928575
$89$	−4.44073e9	−0.0842964	−0.0421482	−	0.999111i	$-0.513420\pi$
$89$	−4.44073e9	−0.0842964	−0.0421482	+	0.999111i	$0.513420\pi$
$90$	0	0
$91$	−9.57433e10	−1.60835
$92$	3.37364e10	0.533661
$93$	0	0
$94$	1.26364e10	0.177591
$95$	3.62690e10	0.480901
$96$	0	0
$97$	−2.04453e10	−0.241740	−0.120870	−	0.992668i	$-0.538568\pi$
$97$	−2.04453e10	−0.241740	−0.120870	+	0.992668i	$0.538568\pi$
$98$	−6.49594e10	−0.725937
$99$	0	0
$100$	−1.78843e10	−0.178843
$101$	1.55947e11	1.47642	0.738209	−	0.674572i	$-0.235672\pi$
$101$	1.55947e11	1.47642	0.738209	+	0.674572i	$0.235672\pi$
$102$	0	0
$103$	−5.10325e10	−0.433752	−0.216876	−	0.976199i	$-0.569587\pi$
$103$	−5.10325e10	−0.433752	−0.216876	+	0.976199i	$0.569587\pi$
$104$	−6.83876e10	−0.551181
$105$	0	0
$106$	−5.19217e10	−0.376848
$107$	−1.06665e10	−0.0735207	−0.0367603	−	0.999324i	$-0.511704\pi$
$107$	−1.06665e10	−0.0735207	−0.0367603	+	0.999324i	$0.511704\pi$
$108$	0	0
$109$	−3.12513e10	−0.194546	−0.0972729	−	0.995258i	$-0.531012\pi$
$109$	−3.12513e10	−0.194546	−0.0972729	+	0.995258i	$0.531012\pi$
$110$	3.70308e10	0.219232
$111$	0	0
$112$	2.32640e11	1.24734
$113$	3.96346e10	0.202369	0.101184	−	0.994868i	$-0.467737\pi$
$113$	3.96346e10	0.202369	0.101184	+	0.994868i	$0.467737\pi$
$114$	0	0
$115$	−5.75676e10	−0.266895
$116$	−3.48921e11	−1.54244
$117$	0	0
$118$	6.41400e10	0.258094
$119$	−2.10500e11	−0.808620
$120$	0	0
$121$	3.62821e11	1.27166
$122$	2.43443e10	0.0815489
$123$	0	0
$124$	−1.85198e11	−0.567305
$125$	3.05176e10	0.0894427
$126$	0	0
$127$	2.63460e10	0.0707611	0.0353806	−	0.999374i	$-0.488736\pi$
$127$	2.63460e10	0.0707611	0.0353806	+	0.999374i	$0.488736\pi$
$128$	3.80331e11	0.978379
$129$	0	0
$130$	5.50895e10	0.130131
$131$	2.19917e11	0.498044	0.249022	−	0.968498i	$-0.419891\pi$
$131$	2.19917e11	0.498044	0.249022	+	0.968498i	$0.419891\pi$
$132$	0	0
$133$	9.27805e11	1.93318
$134$	−1.11661e11	−0.223266
$135$	0	0
$136$	−1.50356e11	−0.277113
$137$	5.54041e11	0.980795	0.490398	−	0.871499i	$-0.336852\pi$
$137$	5.54041e11	0.980795	0.490398	+	0.871499i	$0.336852\pi$
$138$	0	0
$139$	−6.17540e11	−1.00945	−0.504723	−	0.863281i	$-0.668406\pi$
$139$	−6.17540e11	−1.00945	−0.504723	+	0.863281i	$0.668406\pi$
$140$	−4.57501e11	−0.718932
$141$	0	0
$142$	−4.05966e11	−0.590070
$143$	9.64205e11	1.34841
$144$	0	0
$145$	5.95397e11	0.771405
$146$	−4.74497e11	−0.591961
$147$	0	0
$148$	−1.47730e11	−0.171013
$149$	6.68742e11	0.745992	0.372996	−	0.927833i	$-0.378330\pi$
$149$	6.68742e11	0.745992	0.372996	+	0.927833i	$0.378330\pi$
$150$	0	0
$151$	1.38243e12	1.43308	0.716541	−	0.697545i	$-0.245724\pi$
$151$	1.38243e12	1.43308	0.716541	+	0.697545i	$0.245724\pi$
$152$	6.62713e11	0.662500
$153$	0	0
$154$	9.47292e11	0.881292
$155$	3.16020e11	0.283721
$156$	0	0
$157$	−7.93661e11	−0.664029	−0.332014	−	0.943274i	$-0.607728\pi$
$157$	−7.93661e11	−0.664029	−0.332014	+	0.943274i	$0.607728\pi$
$158$	3.57894e10	0.0289161
$159$	0	0
$160$	−4.99301e11	−0.376446
$161$	−1.47265e12	−1.07289
$162$	0	0
$163$	−4.67399e11	−0.318168	−0.159084	−	0.987265i	$-0.550854\pi$
$163$	−4.67399e11	−0.318168	−0.159084	+	0.987265i	$0.550854\pi$
$164$	4.14463e11	0.272800
$165$	0	0
$166$	1.77702e11	0.109421
$167$	−2.87482e12	−1.71266	−0.856328	−	0.516432i	$-0.827260\pi$
$167$	−2.87482e12	−1.71266	−0.856328	+	0.516432i	$0.827260\pi$
$168$	0	0
$169$	−3.57745e11	−0.199617
$170$	1.21119e11	0.0654250
$171$	0	0
$172$	−3.06108e12	−1.55049
$173$	−1.53085e12	−0.751068	−0.375534	−	0.926809i	$-0.622541\pi$
$173$	−1.53085e12	−0.751068	−0.375534	+	0.926809i	$0.622541\pi$
$174$	0	0
$175$	7.80676e11	0.359552
$176$	−2.34285e12	−1.04574
$177$	0	0
$178$	6.53635e10	0.0274173
$179$	1.91337e12	0.778228	0.389114	−	0.921190i	$-0.372781\pi$
$179$	1.91337e12	0.778228	0.389114	+	0.921190i	$0.372781\pi$
$180$	0	0
$181$	−1.70819e12	−0.653587	−0.326794	−	0.945096i	$-0.605968\pi$
$181$	−1.70819e12	−0.653587	−0.326794	+	0.945096i	$0.605968\pi$
$182$	1.40925e12	0.523115
$183$	0	0
$184$	−1.05188e12	−0.367680
$185$	2.52086e11	0.0855272
$186$	0	0
$187$	2.11989e12	0.677930
$188$	1.57223e12	0.488255
$189$	0	0
$190$	−5.33847e11	−0.156413
$191$	−3.16020e12	−0.899562	−0.449781	−	0.893139i	$-0.648498\pi$
$191$	−3.16020e12	−0.899562	−0.449781	+	0.893139i	$0.648498\pi$
$192$	0	0
$193$	2.77296e12	0.745382	0.372691	−	0.927956i	$-0.378435\pi$
$193$	2.77296e12	0.745382	0.372691	+	0.927956i	$0.378435\pi$
$194$	3.00936e11	0.0786256
$195$	0	0
$196$	−8.08224e12	−1.99583
$197$	−3.86504e12	−0.928089	−0.464044	−	0.885812i	$-0.653602\pi$
$197$	−3.86504e12	−0.928089	−0.464044	+	0.885812i	$0.653602\pi$
$198$	0	0
$199$	−1.01839e12	−0.231325	−0.115663	−	0.993289i	$-0.536899\pi$
$199$	−1.01839e12	−0.231325	−0.115663	+	0.993289i	$0.536899\pi$
$200$	5.57622e11	0.123218
$201$	0	0
$202$	−2.29540e12	−0.480204
$203$	1.52310e13	3.10098
$204$	0	0
$205$	−7.07237e11	−0.136433
$206$	7.51152e11	0.141078
$207$	0	0
$208$	−3.48538e12	−0.620729
$209$	−9.34367e12	−1.62074
$210$	0	0
$211$	2.05668e12	0.338543	0.169271	−	0.985569i	$-0.445859\pi$
$211$	2.05668e12	0.338543	0.169271	+	0.985569i	$0.445859\pi$
$212$	−6.46009e12	−1.03608
$213$	0	0
$214$	1.57001e11	0.0239125
$215$	5.22340e12	0.775429
$216$	0	0
$217$	8.08418e12	1.14053
$218$	4.59990e11	0.0632759
$219$	0	0
$220$	4.60737e12	0.602737
$221$	3.15369e12	0.402403
$222$	0	0
$223$	9.99986e12	1.21428	0.607138	−	0.794597i	$-0.292318\pi$
$223$	9.99986e12	1.21428	0.607138	+	0.794597i	$0.292318\pi$
$224$	−1.27727e13	−1.51328
$225$	0	0
$226$	−5.83386e11	−0.0658202
$227$	6.77123e12	0.745634	0.372817	−	0.927905i	$-0.378392\pi$
$227$	6.77123e12	0.745634	0.372817	+	0.927905i	$0.378392\pi$
$228$	0	0
$229$	1.01933e13	1.06959	0.534796	−	0.844981i	$-0.320389\pi$
$229$	1.01933e13	1.06959	0.534796	+	0.844981i	$0.320389\pi$
$230$	8.47343e11	0.0868072
$231$	0	0
$232$	1.08792e13	1.06270
$233$	−6.74446e12	−0.643412	−0.321706	−	0.946840i	$-0.604256\pi$
$233$	−6.74446e12	−0.643412	−0.321706	+	0.946840i	$0.604256\pi$
$234$	0	0
$235$	−2.68283e12	−0.244186
$236$	7.98029e12	0.709582
$237$	0	0
$238$	3.09837e12	0.263003
$239$	1.76501e13	1.46406	0.732030	−	0.681272i	$-0.238573\pi$
$239$	1.76501e13	1.46406	0.732030	+	0.681272i	$0.238573\pi$
$240$	0	0
$241$	−1.37662e13	−1.09074	−0.545369	−	0.838196i	$-0.683611\pi$
$241$	−1.37662e13	−1.09074	−0.545369	+	0.838196i	$0.683611\pi$
$242$	−5.34039e12	−0.413608
$243$	0	0
$244$	3.02891e12	0.224204
$245$	1.37915e13	0.998154
$246$	0	0
$247$	−1.39003e13	−0.962033
$248$	5.77438e12	0.390860
$249$	0	0
$250$	−4.49191e11	−0.0290912
$251$	−3.02252e13	−1.91498	−0.957490	−	0.288467i	$-0.906855\pi$
$251$	−3.02252e13	−1.91498	−0.957490	+	0.288467i	$0.906855\pi$
$252$	0	0
$253$	1.48306e13	0.899491
$254$	−3.87790e11	−0.0230150
$255$	0	0
$256$	1.79144e12	0.101832
$257$	−6.88263e12	−0.382933	−0.191466	−	0.981499i	$-0.561324\pi$
$257$	−6.88263e12	−0.382933	−0.191466	+	0.981499i	$0.561324\pi$
$258$	0	0
$259$	6.44866e12	0.343812
$260$	6.85423e12	0.357771
$261$	0	0
$262$	−3.23698e12	−0.161988
$263$	3.29948e13	1.61692	0.808462	−	0.588549i	$-0.200301\pi$
$263$	3.29948e13	1.61692	0.808462	+	0.588549i	$0.200301\pi$
$264$	0	0
$265$	1.10235e13	0.518162
$266$	−1.36564e13	−0.628765
$267$	0	0
$268$	−1.38928e13	−0.613830
$269$	2.97478e13	1.28771	0.643855	−	0.765148i	$-0.277334\pi$
$269$	2.97478e13	1.28771	0.643855	+	0.765148i	$0.277334\pi$
$270$	0	0
$271$	−4.30410e13	−1.78876	−0.894378	−	0.447312i	$-0.852381\pi$
$271$	−4.30410e13	−1.78876	−0.894378	+	0.447312i	$0.852381\pi$
$272$	−7.66293e12	−0.312080
$273$	0	0
$274$	−8.15498e12	−0.319003
$275$	−7.86198e12	−0.301441
$276$	0	0
$277$	1.46706e13	0.540518	0.270259	−	0.962788i	$-0.412891\pi$
$277$	1.46706e13	0.540518	0.270259	+	0.962788i	$0.412891\pi$
$278$	9.08963e12	0.328322
$279$	0	0
$280$	1.42646e13	0.495327
$281$	−4.18378e13	−1.42457	−0.712286	−	0.701890i	$-0.752340\pi$
$281$	−4.18378e13	−1.42457	−0.712286	+	0.701890i	$0.752340\pi$
$282$	0	0
$283$	2.34242e13	0.767078	0.383539	−	0.923525i	$-0.374705\pi$
$283$	2.34242e13	0.767078	0.383539	+	0.923525i	$0.374705\pi$
$284$	−5.05102e13	−1.62229
$285$	0	0
$286$	−1.41922e13	−0.438568
$287$	−1.80920e13	−0.548448
$288$	0	0
$289$	−2.73382e13	−0.797686
$290$	−8.76370e12	−0.250899
$291$	0	0
$292$	−5.90369e13	−1.62749
$293$	7.31258e12	0.197833	0.0989166	−	0.995096i	$-0.468462\pi$
$293$	7.31258e12	0.197833	0.0989166	+	0.995096i	$0.468462\pi$
$294$	0	0
$295$	−1.36175e13	−0.354876
$296$	4.60616e12	0.117824
$297$	0	0
$298$	−9.84328e12	−0.242633
$299$	2.20630e13	0.533917
$300$	0	0
$301$	1.33621e14	3.11716
$302$	−2.03482e13	−0.466109
$303$	0	0
$304$	3.37752e13	0.746094
$305$	−5.16852e12	−0.112129
$306$	0	0
$307$	−5.37239e13	−1.12436	−0.562181	−	0.827014i	$-0.690038\pi$
$307$	−5.37239e13	−1.12436	−0.562181	+	0.827014i	$0.690038\pi$
$308$	1.17862e14	2.42295
$309$	0	0
$310$	−4.65154e12	−0.0922799
$311$	−5.41406e13	−1.05521	−0.527607	−	0.849489i	$-0.676911\pi$
$311$	−5.41406e13	−1.05521	−0.527607	+	0.849489i	$0.676911\pi$
$312$	0	0
$313$	−4.29721e12	−0.0808524	−0.0404262	−	0.999183i	$-0.512872\pi$
$313$	−4.29721e12	−0.0808524	−0.0404262	+	0.999183i	$0.512872\pi$
$314$	1.16820e13	0.215975
$315$	0	0
$316$	4.45291e12	0.0794996
$317$	2.81928e13	0.494666	0.247333	−	0.968931i	$-0.420446\pi$
$317$	2.81928e13	0.494666	0.247333	+	0.968931i	$0.420446\pi$
$318$	0	0
$319$	−1.53387e14	−2.59980
$320$	−1.12756e13	−0.187851
$321$	0	0
$322$	2.16760e13	0.348958
$323$	−3.05610e13	−0.483675
$324$	0	0
$325$	−1.16960e13	−0.178928
$326$	6.87969e12	0.103484
$327$	0	0
$328$	−1.29227e13	−0.187953
$329$	−6.86301e13	−0.981606
$330$	0	0
$331$	−9.95469e13	−1.37713	−0.688563	−	0.725176i	$-0.741758\pi$
$331$	−9.95469e13	−1.37713	−0.688563	+	0.725176i	$0.741758\pi$
$332$	2.21097e13	0.300832
$333$	0	0
$334$	4.23148e13	0.557040
$335$	2.37066e13	0.306989
$336$	0	0
$337$	−9.05175e13	−1.13440	−0.567202	−	0.823579i	$-0.691974\pi$
$337$	−9.05175e13	−1.13440	−0.567202	+	0.823579i	$0.691974\pi$
$338$	5.26568e12	0.0649251
$339$	0	0
$340$	1.50696e13	0.179874
$341$	−8.14136e13	−0.956198
$342$	0	0
$343$	1.94733e14	2.21473
$344$	9.54428e13	1.06825
$345$	0	0
$346$	2.25327e13	0.244284
$347$	6.36892e13	0.679601	0.339801	−	0.940498i	$-0.389640\pi$
$347$	6.36892e13	0.679601	0.339801	+	0.940498i	$0.389640\pi$
$348$	0	0
$349$	7.08598e13	0.732588	0.366294	−	0.930499i	$-0.380626\pi$
$349$	7.08598e13	0.732588	0.366294	+	0.930499i	$0.380626\pi$
$350$	−1.14908e13	−0.116944
$351$	0	0
$352$	1.28631e14	1.26870
$353$	−6.62893e13	−0.643699	−0.321849	−	0.946791i	$-0.604304\pi$
$353$	−6.62893e13	−0.643699	−0.321849	+	0.946791i	$0.604304\pi$
$354$	0	0
$355$	8.61903e13	0.811339
$356$	8.13252e12	0.0753790
$357$	0	0
$358$	−2.81630e13	−0.253118
$359$	−6.80217e13	−0.602044	−0.301022	−	0.953617i	$-0.597328\pi$
$359$	−6.80217e13	−0.602044	−0.301022	+	0.953617i	$0.597328\pi$
$360$	0	0
$361$	1.82109e13	0.156330
$362$	2.51430e13	0.212579
$363$	0	0
$364$	1.75339e14	1.43821
$365$	1.00740e14	0.813939
$366$	0	0
$367$	2.19771e14	1.72309	0.861543	−	0.507685i	$-0.169499\pi$
$367$	2.19771e14	1.72309	0.861543	+	0.507685i	$0.169499\pi$
$368$	−5.36094e13	−0.414074
$369$	0	0
$370$	−3.71048e12	−0.0278177
$371$	2.81993e14	2.08296
$372$	0	0
$373$	−1.67232e14	−1.19928	−0.599639	−	0.800271i	$-0.704689\pi$
$373$	−1.67232e14	−1.19928	−0.599639	+	0.800271i	$0.704689\pi$
$374$	−3.12029e13	−0.220496
$375$	0	0
$376$	−4.90212e13	−0.336396
$377$	−2.28189e14	−1.54318
$378$	0	0
$379$	−1.57778e14	−1.03641	−0.518205	−	0.855256i	$-0.673400\pi$
$379$	−1.57778e14	−1.03641	−0.518205	+	0.855256i	$0.673400\pi$
$380$	−6.64212e13	−0.430028
$381$	0	0
$382$	4.65153e13	0.292582
$383$	−5.49255e13	−0.340550	−0.170275	−	0.985397i	$-0.554466\pi$
$383$	−5.49255e13	−0.340550	−0.170275	+	0.985397i	$0.554466\pi$
$384$	0	0
$385$	−2.01119e14	−1.21177
$386$	−4.08155e13	−0.242435
$387$	0	0
$388$	3.74424e13	0.216167
$389$	2.32940e14	1.32593	0.662965	−	0.748650i	$-0.269298\pi$
$389$	2.32940e14	1.32593	0.662965	+	0.748650i	$0.269298\pi$
$390$	0	0
$391$	4.85076e13	0.268434
$392$	2.52000e14	1.37508
$393$	0	0
$394$	5.68899e13	0.301860
$395$	−7.59842e12	−0.0397593
$396$	0	0
$397$	−4.76134e13	−0.242316	−0.121158	−	0.992633i	$-0.538661\pi$
$397$	−4.76134e13	−0.242316	−0.121158	+	0.992633i	$0.538661\pi$
$398$	1.49898e13	0.0752383
$399$	0	0
$400$	2.84193e13	0.138766
$401$	2.99074e14	1.44040	0.720202	−	0.693764i	$-0.244049\pi$
$401$	2.99074e14	1.44040	0.720202	+	0.693764i	$0.244049\pi$
$402$	0	0
$403$	−1.21116e14	−0.567577
$404$	−2.85593e14	−1.32023
$405$	0	0
$406$	−2.24186e14	−1.00859
$407$	−6.49428e13	−0.288245
$408$	0	0
$409$	−2.55879e14	−1.10550	−0.552748	−	0.833349i	$-0.686421\pi$
$409$	−2.55879e14	−1.10550	−0.552748	+	0.833349i	$0.686421\pi$
$410$	1.04099e13	0.0443747
$411$	0	0
$412$	9.34582e13	0.387867
$413$	−3.48352e14	−1.42657
$414$	0	0
$415$	−3.77279e13	−0.150452
$416$	1.91359e14	0.753072
$417$	0	0
$418$	1.37530e14	0.527144
$419$	−2.21825e14	−0.839139	−0.419570	−	0.907723i	$-0.637819\pi$
$419$	−2.21825e14	−0.839139	−0.419570	+	0.907723i	$0.637819\pi$
$420$	0	0
$421$	9.78949e13	0.360752	0.180376	−	0.983598i	$-0.442269\pi$
$421$	9.78949e13	0.360752	0.180376	+	0.983598i	$0.442269\pi$
$422$	−3.02725e13	−0.110111
$423$	0	0
$424$	2.01422e14	0.713831
$425$	−2.57147e13	−0.0899586
$426$	0	0
$427$	−1.32217e14	−0.450748
$428$	1.95340e13	0.0657432
$429$	0	0
$430$	−7.68838e13	−0.252208
$431$	−1.11415e14	−0.360842	−0.180421	−	0.983589i	$-0.557746\pi$
$431$	−1.11415e14	−0.360842	−0.180421	+	0.983589i	$0.557746\pi$
$432$	0	0
$433$	−3.42045e14	−1.07994	−0.539970	−	0.841684i	$-0.681564\pi$
$433$	−3.42045e14	−1.07994	−0.539970	+	0.841684i	$0.681564\pi$
$434$	−1.18992e14	−0.370957
$435$	0	0
$436$	5.72320e13	0.173965
$437$	−2.13803e14	−0.641750
$438$	0	0
$439$	1.20947e14	0.354029	0.177015	−	0.984208i	$-0.443356\pi$
$439$	1.20947e14	0.354029	0.177015	+	0.984208i	$0.443356\pi$
$440$	−1.43655e14	−0.415271
$441$	0	0
$442$	−4.64195e13	−0.130881
$443$	1.27316e14	0.354539	0.177270	−	0.984162i	$-0.443274\pi$
$443$	1.27316e14	0.354539	0.177270	+	0.984162i	$0.443274\pi$
$444$	0	0
$445$	−1.38773e13	−0.0376985
$446$	−1.47189e14	−0.394942
$447$	0	0
$448$	−2.88443e14	−0.755147
$449$	1.61837e14	0.418526	0.209263	−	0.977859i	$-0.432894\pi$
$449$	1.61837e14	0.418526	0.209263	+	0.977859i	$0.432894\pi$
$450$	0	0
$451$	1.82199e14	0.459807
$452$	−7.25848e13	−0.180961
$453$	0	0
$454$	−9.96664e13	−0.242517
$455$	−2.99198e14	−0.719276
$456$	0	0
$457$	−3.49165e14	−0.819391	−0.409696	−	0.912222i	$-0.634365\pi$
$457$	−3.49165e14	−0.819391	−0.409696	+	0.912222i	$0.634365\pi$
$458$	−1.50036e14	−0.347884
$459$	0	0
$460$	1.05426e14	0.238661
$461$	4.49619e14	1.00575	0.502875	−	0.864359i	$-0.332276\pi$
$461$	4.49619e14	1.00575	0.502875	+	0.864359i	$0.332276\pi$
$462$	0	0
$463$	4.21098e13	0.0919787	0.0459894	−	0.998942i	$-0.485356\pi$
$463$	4.21098e13	0.0919787	0.0459894	+	0.998942i	$0.485356\pi$
$464$	5.54459e14	1.19680
$465$	0	0
$466$	9.92723e13	0.209269
$467$	7.91358e13	0.164866	0.0824328	−	0.996597i	$-0.473731\pi$
$467$	7.91358e13	0.164866	0.0824328	+	0.996597i	$0.473731\pi$
$468$	0	0
$469$	6.06442e14	1.23407
$470$	3.94889e13	0.0794213
$471$	0	0
$472$	−2.48821e14	−0.488885
$473$	−1.34566e15	−2.61336
$474$	0	0
$475$	1.13341e14	0.215066
$476$	3.85499e14	0.723078
$477$	0	0
$478$	−2.59794e14	−0.476184
$479$	9.62318e14	1.74371	0.871853	−	0.489768i	$-0.162919\pi$
$479$	9.62318e14	1.74371	0.871853	+	0.489768i	$0.162919\pi$
$480$	0	0
$481$	−9.66132e13	−0.171095
$482$	2.02626e14	0.354762
$483$	0	0
$484$	−6.64451e14	−1.13714
$485$	−6.38914e13	−0.108109
$486$	0	0
$487$	4.57117e14	0.756168	0.378084	−	0.925771i	$-0.376583\pi$
$487$	4.57117e14	0.756168	0.378084	+	0.925771i	$0.376583\pi$
$488$	−9.44399e13	−0.154471
$489$	0	0
$490$	−2.02998e14	−0.324649
$491$	−4.40122e14	−0.696025	−0.348013	−	0.937490i	$-0.613143\pi$
$491$	−4.40122e14	−0.696025	−0.348013	+	0.937490i	$0.613143\pi$
$492$	0	0
$493$	−5.01693e14	−0.775854
$494$	2.04599e14	0.312900
$495$	0	0
$496$	2.94292e14	0.440179
$497$	2.20485e15	3.26151
$498$	0	0
$499$	6.36085e14	0.920369	0.460185	−	0.887823i	$-0.347783\pi$
$499$	6.36085e14	0.920369	0.460185	+	0.887823i	$0.347783\pi$
$500$	−5.58883e13	−0.0799808
$501$	0	0
$502$	4.44888e14	0.622846
$503$	4.81639e12	0.00666957	0.00333478	−	0.999994i	$-0.498939\pi$
$503$	4.81639e12	0.00666957	0.00333478	+	0.999994i	$0.498939\pi$
$504$	0	0
$505$	4.87334e14	0.660274
$506$	−2.18294e14	−0.292559
$507$	0	0
$508$	−4.82488e13	−0.0632755
$509$	5.73829e14	0.744449	0.372224	−	0.928143i	$-0.378595\pi$
$509$	5.73829e14	0.744449	0.372224	+	0.928143i	$0.378595\pi$
$510$	0	0
$511$	2.57705e15	3.27196
$512$	−8.05287e14	−1.01150
$513$	0	0
$514$	1.01306e14	0.124548
$515$	−1.59476e14	−0.193980
$516$	0	0
$517$	6.91155e14	0.822958
$518$	−9.49185e13	−0.111825
$519$	0	0
$520$	−2.13711e14	−0.246495
$521$	−5.14705e14	−0.587422	−0.293711	−	0.955894i	$-0.594890\pi$
$521$	−5.14705e14	−0.587422	−0.293711	+	0.955894i	$0.594890\pi$
$522$	0	0
$523$	1.20146e15	1.34261	0.671304	−	0.741182i	$-0.265735\pi$
$523$	1.20146e15	1.34261	0.671304	+	0.741182i	$0.265735\pi$
$524$	−4.02745e14	−0.445357
$525$	0	0
$526$	−4.85654e14	−0.525903
$527$	−2.66285e14	−0.285357
$528$	0	0
$529$	−6.13454e14	−0.643836
$530$	−1.62255e14	−0.168532
$531$	0	0
$532$	−1.69913e15	−1.72868
$533$	2.71052e14	0.272931
$534$	0	0
$535$	−3.33327e13	−0.0328795
$536$	4.33170e14	0.422914
$537$	0	0
$538$	−4.37861e14	−0.418826
$539$	−3.55298e15	−3.36399
$540$	0	0
$541$	−1.89107e15	−1.75438	−0.877190	−	0.480143i	$-0.840585\pi$
$541$	−1.89107e15	−1.75438	−0.877190	+	0.480143i	$0.840585\pi$
$542$	6.33524e14	0.581791
$543$	0	0
$544$	4.20721e14	0.378617
$545$	−9.76602e13	−0.0870035
$546$	0	0
$547$	1.37400e14	0.119966	0.0599830	−	0.998199i	$-0.480895\pi$
$547$	1.37400e14	0.119966	0.0599830	+	0.998199i	$0.480895\pi$
$548$	−1.01464e15	−0.877040
$549$	0	0
$550$	1.15721e14	0.0980434
$551$	2.21127e15	1.85485
$552$	0	0
$553$	−1.94377e14	−0.159829
$554$	−2.15939e14	−0.175803
$555$	0	0
$556$	1.13093e15	0.902661
$557$	1.18622e15	0.937483	0.468742	−	0.883335i	$-0.344708\pi$
$557$	1.18622e15	0.937483	0.468742	+	0.883335i	$0.344708\pi$
$558$	0	0
$559$	−2.00189e15	−1.55123
$560$	7.26999e14	0.557827
$561$	0	0
$562$	6.15815e14	0.463340
$563$	2.58208e15	1.92386	0.961929	−	0.273298i	$-0.0881146\pi$
$563$	2.58208e15	1.92386	0.961929	+	0.273298i	$0.0881146\pi$
$564$	0	0
$565$	1.23858e14	0.0905020
$566$	−3.44783e14	−0.249491
$567$	0	0
$568$	1.57488e15	1.11772
$569$	−6.86455e13	−0.0482497	−0.0241249	−	0.999709i	$-0.507680\pi$
$569$	−6.86455e13	−0.0482497	−0.0241249	+	0.999709i	$0.507680\pi$
$570$	0	0
$571$	−1.48521e15	−1.02397	−0.511986	−	0.858994i	$-0.671090\pi$
$571$	−1.48521e15	−1.02397	−0.511986	+	0.858994i	$0.671090\pi$
$572$	−1.76579e15	−1.20576
$573$	0	0
$574$	2.66297e14	0.178382
$575$	−1.79899e14	−0.119359
$576$	0	0
$577$	−1.11682e14	−0.0726966	−0.0363483	−	0.999339i	$-0.511573\pi$
$577$	−1.11682e14	−0.0726966	−0.0363483	+	0.999339i	$0.511573\pi$
$578$	4.02394e14	0.259447
$579$	0	0
$580$	−1.09038e15	−0.689800
$581$	−9.65124e14	−0.604803
$582$	0	0
$583$	−2.83988e15	−1.74631
$584$	1.84074e15	1.12130
$585$	0	0
$586$	−1.07635e14	−0.0643450
$587$	−1.03228e15	−0.611347	−0.305673	−	0.952136i	$-0.598882\pi$
$587$	−1.03228e15	−0.611347	−0.305673	+	0.952136i	$0.598882\pi$
$588$	0	0
$589$	1.17368e15	0.682208
$590$	2.00437e14	0.115423
$591$	0	0
$592$	2.34753e14	0.132691
$593$	−5.05946e14	−0.283337	−0.141668	−	0.989914i	$-0.545247\pi$
$593$	−5.05946e14	−0.283337	−0.141668	+	0.989914i	$0.545247\pi$
$594$	0	0
$595$	−6.57813e14	−0.361626
$596$	−1.22470e15	−0.667076
$597$	0	0
$598$	−3.24748e14	−0.173656
$599$	1.86439e14	0.0987846	0.0493923	−	0.998779i	$-0.484272\pi$
$599$	1.86439e14	0.0987846	0.0493923	+	0.998779i	$0.484272\pi$
$600$	0	0
$601$	−6.39018e14	−0.332433	−0.166216	−	0.986089i	$-0.553155\pi$
$601$	−6.39018e14	−0.332433	−0.166216	+	0.986089i	$0.553155\pi$
$602$	−1.96678e15	−1.01385
$603$	0	0
$604$	−2.53172e15	−1.28148
$605$	1.13381e15	0.568706
$606$	0	0
$607$	3.07440e15	1.51434	0.757169	−	0.653219i	$-0.226582\pi$
$607$	3.07440e15	1.51434	0.757169	+	0.653219i	$0.226582\pi$
$608$	−1.85438e15	−0.905166
$609$	0	0
$610$	7.60759e13	0.0364698
$611$	1.02821e15	0.488489
$612$	0	0
$613$	1.45823e15	0.680447	0.340223	−	0.940345i	$-0.389497\pi$
$613$	1.45823e15	0.680447	0.340223	+	0.940345i	$0.389497\pi$
$614$	7.90767e14	0.365698
$615$	0	0
$616$	−3.67488e15	−1.66935
$617$	−2.52038e15	−1.13474	−0.567372	−	0.823462i	$-0.692040\pi$
$617$	−2.52038e15	−1.13474	−0.567372	+	0.823462i	$0.692040\pi$
$618$	0	0
$619$	2.76117e15	1.22122	0.610612	−	0.791930i	$-0.290924\pi$
$619$	2.76117e15	1.22122	0.610612	+	0.791930i	$0.290924\pi$
$620$	−5.78743e14	−0.253707
$621$	0	0
$622$	7.96900e14	0.343207
$623$	−3.54997e14	−0.151545
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	6.32511e13	0.0262972
$627$	0	0
$628$	1.45347e15	0.593783
$629$	−2.12413e14	−0.0860205
$630$	0	0
$631$	−5.89126e14	−0.234448	−0.117224	−	0.993105i	$-0.537400\pi$
$631$	−5.89126e14	−0.234448	−0.117224	+	0.993105i	$0.537400\pi$
$632$	−1.38839e14	−0.0547733
$633$	0	0
$634$	−4.14972e14	−0.160890
$635$	8.23314e13	0.0316453
$636$	0	0
$637$	−5.28565e15	−1.99679
$638$	2.25772e15	0.845582
$639$	0	0
$640$	1.18854e15	0.437544
$641$	−1.59787e15	−0.583208	−0.291604	−	0.956539i	$-0.594189\pi$
$641$	−1.59787e15	−0.583208	−0.291604	+	0.956539i	$0.594189\pi$
$642$	0	0
$643$	4.44445e15	1.59462	0.797310	−	0.603569i	$-0.206255\pi$
$643$	4.44445e15	1.59462	0.797310	+	0.603569i	$0.206255\pi$
$644$	2.69693e15	0.959395
$645$	0	0
$646$	4.49830e14	0.157315
$647$	8.54247e14	0.296217	0.148108	−	0.988971i	$-0.452682\pi$
$647$	8.54247e14	0.296217	0.148108	+	0.988971i	$0.452682\pi$
$648$	0	0
$649$	3.50816e15	1.19601
$650$	1.72155e14	0.0581963
$651$	0	0
$652$	8.55971e14	0.284510
$653$	2.86428e15	0.944045	0.472022	−	0.881587i	$-0.343524\pi$
$653$	2.86428e15	0.944045	0.472022	+	0.881587i	$0.343524\pi$
$654$	0	0
$655$	6.87242e14	0.222732
$656$	−6.58609e14	−0.211669
$657$	0	0
$658$	1.01017e15	0.319267
$659$	−4.78135e15	−1.49858	−0.749291	−	0.662241i	$-0.769606\pi$
$659$	−4.78135e15	−1.49858	−0.749291	+	0.662241i	$0.769606\pi$
$660$	0	0
$661$	−5.52275e14	−0.170234	−0.0851172	−	0.996371i	$-0.527126\pi$
$661$	−5.52275e14	−0.170234	−0.0851172	+	0.996371i	$0.527126\pi$
$662$	1.46524e15	0.447909
$663$	0	0
$664$	−6.89369e14	−0.207266
$665$	2.89939e15	0.864545
$666$	0	0
$667$	−3.50982e15	−1.02942
$668$	5.26480e15	1.53148
$669$	0	0
$670$	−3.48939e14	−0.0998478
$671$	1.33152e15	0.377897
$672$	0	0
$673$	−3.31175e15	−0.924645	−0.462323	−	0.886712i	$-0.652984\pi$
$673$	−3.31175e15	−0.924645	−0.462323	+	0.886712i	$0.652984\pi$
$674$	1.33234e15	0.368964
$675$	0	0
$676$	6.55156e14	0.178500
$677$	−6.69181e15	−1.80845	−0.904223	−	0.427060i	$-0.859549\pi$
$677$	−6.69181e15	−1.80845	−0.904223	+	0.427060i	$0.859549\pi$
$678$	0	0
$679$	−1.63442e15	−0.434590
$680$	−4.69863e14	−0.123929
$681$	0	0
$682$	1.19833e15	0.311003
$683$	7.20810e15	1.85570	0.927848	−	0.372959i	$-0.121657\pi$
$683$	7.20810e15	1.85570	0.927848	+	0.372959i	$0.121657\pi$
$684$	0	0
$685$	1.73138e15	0.438625
$686$	−2.86629e15	−0.720340
$687$	0	0
$688$	4.86426e15	1.20304
$689$	−4.22479e15	−1.03657
$690$	0	0
$691$	7.79467e15	1.88221	0.941106	−	0.338112i	$-0.109788\pi$
$691$	7.79467e15	1.88221	0.941106	+	0.338112i	$0.109788\pi$
$692$	2.80352e15	0.671615
$693$	0	0
$694$	−9.37448e14	−0.221040
$695$	−1.92981e15	−0.451438
$696$	0	0
$697$	5.95932e14	0.137220
$698$	−1.04299e15	−0.238274
$699$	0	0
$700$	−1.42969e15	−0.321516
$701$	−6.71290e15	−1.49782	−0.748912	−	0.662669i	$-0.769423\pi$
$701$	−6.71290e15	−1.49782	−0.748912	+	0.662669i	$0.769423\pi$
$702$	0	0
$703$	9.36234e14	0.205651
$704$	2.90483e15	0.633099
$705$	0	0
$706$	9.75719e14	0.209362
$707$	1.24666e16	2.65424
$708$	0	0
$709$	−4.10214e15	−0.859917	−0.429959	−	0.902849i	$-0.641472\pi$
$709$	−4.10214e15	−0.859917	−0.429959	+	0.902849i	$0.641472\pi$
$710$	−1.26864e15	−0.263887
$711$	0	0
$712$	−2.53568e14	−0.0519343
$713$	−1.86291e15	−0.378617
$714$	0	0
$715$	3.01314e15	0.603026
$716$	−3.50404e15	−0.695902
$717$	0	0
$718$	1.00122e15	0.195814
$719$	3.31818e15	0.644008	0.322004	−	0.946738i	$-0.395644\pi$
$719$	3.31818e15	0.644008	0.322004	+	0.946738i	$0.395644\pi$
$720$	0	0
$721$	−4.07960e15	−0.779783
$722$	−2.68047e14	−0.0508460
$723$	0	0
$724$	3.12829e15	0.584446
$725$	1.86061e15	0.344983
$726$	0	0
$727$	3.34273e15	0.610467	0.305233	−	0.952278i	$-0.401266\pi$
$727$	3.34273e15	0.610467	0.305233	+	0.952278i	$0.401266\pi$
$728$	−5.46699e15	−0.990890
$729$	0	0
$730$	−1.48280e15	−0.264733
$731$	−4.40134e15	−0.779901
$732$	0	0
$733$	−6.14966e15	−1.07344	−0.536722	−	0.843759i	$-0.680338\pi$
$733$	−6.14966e15	−1.07344	−0.536722	+	0.843759i	$0.680338\pi$
$734$	−3.23483e15	−0.560432
$735$	0	0
$736$	2.94334e15	0.502357
$737$	−6.10732e15	−1.03462
$738$	0	0
$739$	−4.15701e15	−0.693804	−0.346902	−	0.937901i	$-0.612766\pi$
$739$	−4.15701e15	−0.693804	−0.346902	+	0.937901i	$0.612766\pi$
$740$	−4.61657e14	−0.0764795
$741$	0	0
$742$	−4.15068e15	−0.677482
$743$	4.90364e15	0.794474	0.397237	−	0.917716i	$-0.369969\pi$
$743$	4.90364e15	0.794474	0.397237	+	0.917716i	$0.369969\pi$
$744$	0	0
$745$	2.08982e15	0.333618
$746$	2.46150e15	0.390064
$747$	0	0
$748$	−3.88226e15	−0.606214
$749$	−8.52690e14	−0.132173
$750$	0	0
$751$	−2.00580e15	−0.306385	−0.153192	−	0.988196i	$-0.548955\pi$
$751$	−2.00580e15	−0.306385	−0.153192	+	0.988196i	$0.548955\pi$
$752$	−2.49837e15	−0.378842
$753$	0	0
$754$	3.35873e15	0.501918
$755$	4.32010e15	0.640894
$756$	0	0
$757$	2.85016e15	0.416718	0.208359	−	0.978052i	$-0.433188\pi$
$757$	2.85016e15	0.416718	0.208359	+	0.978052i	$0.433188\pi$
$758$	2.32235e15	0.337091
$759$	0	0
$760$	2.07098e15	0.296279
$761$	−8.65962e15	−1.22994	−0.614969	−	0.788551i	$-0.710832\pi$
$761$	−8.65962e15	−1.22994	−0.614969	+	0.788551i	$0.710832\pi$
$762$	0	0
$763$	−2.49827e15	−0.349747
$764$	5.78743e15	0.804400
$765$	0	0
$766$	8.08453e14	0.110763
$767$	5.21897e15	0.709922
$768$	0	0
$769$	−7.49727e15	−1.00533	−0.502665	−	0.864481i	$-0.667647\pi$
$769$	−7.49727e15	−1.00533	−0.502665	+	0.864481i	$0.667647\pi$
$770$	2.96029e15	0.394126
$771$	0	0
$772$	−5.07826e15	−0.666530
$773$	−2.07867e15	−0.270894	−0.135447	−	0.990785i	$-0.543247\pi$
$773$	−2.07867e15	−0.270894	−0.135447	+	0.990785i	$0.543247\pi$
$774$	0	0
$775$	9.87564e14	0.126884
$776$	−1.16743e15	−0.148934
$777$	0	0
$778$	−3.42866e15	−0.431258
$779$	−2.62664e15	−0.328053
$780$	0	0
$781$	−2.22045e16	−2.73438
$782$	−7.13988e14	−0.0873079
$783$	0	0
$784$	1.28432e16	1.54859
$785$	−2.48019e15	−0.296963
$786$	0	0
$787$	−1.58421e16	−1.87047	−0.935235	−	0.354026i	$-0.884812\pi$
$787$	−1.58421e16	−1.87047	−0.935235	+	0.354026i	$0.884812\pi$
$788$	7.07823e15	0.829909
$789$	0	0
$790$	1.11842e14	0.0129317
$791$	3.16844e15	0.363810
$792$	0	0
$793$	1.98086e15	0.224311
$794$	7.00826e14	0.0788129
$795$	0	0
$796$	1.86503e15	0.206854
$797$	4.58469e15	0.504997	0.252499	−	0.967597i	$-0.418748\pi$
$797$	4.58469e15	0.504997	0.252499	+	0.967597i	$0.418748\pi$
$798$	0	0
$799$	2.26061e15	0.245594
$800$	−1.56032e15	−0.168352
$801$	0	0
$802$	−4.40210e15	−0.468490
$803$	−2.59528e16	−2.74315
$804$	0	0
$805$	−4.60202e15	−0.479812
$806$	1.78272e15	0.184604
$807$	0	0
$808$	8.90465e15	0.909608
$809$	1.72813e15	0.175331	0.0876656	−	0.996150i	$-0.472059\pi$
$809$	1.72813e15	0.175331	0.0876656	+	0.996150i	$0.472059\pi$
$810$	0	0
$811$	4.24638e14	0.0425015	0.0212507	−	0.999774i	$-0.493235\pi$
$811$	4.24638e14	0.0425015	0.0212507	+	0.999774i	$0.493235\pi$
$812$	−2.78932e16	−2.77294
$813$	0	0
$814$	9.55899e14	0.0937514
$815$	−1.46062e15	−0.142289
$816$	0	0
$817$	1.93994e16	1.86452
$818$	3.76631e15	0.359561
$819$	0	0
$820$	1.29520e15	0.122000
$821$	1.96517e16	1.83871	0.919354	−	0.393431i	$-0.128712\pi$
$821$	1.96517e16	1.83871	0.919354	+	0.393431i	$0.128712\pi$
$822$	0	0
$823$	5.95701e14	0.0549957	0.0274979	−	0.999622i	$-0.491246\pi$
$823$	5.95701e14	0.0549957	0.0274979	+	0.999622i	$0.491246\pi$
$824$	−2.91398e15	−0.267231
$825$	0	0
$826$	5.12743e15	0.463991
$827$	1.40975e16	1.26725	0.633626	−	0.773639i	$-0.281566\pi$
$827$	1.40975e16	1.26725	0.633626	+	0.773639i	$0.281566\pi$
$828$	0	0
$829$	−7.87187e15	−0.698277	−0.349139	−	0.937071i	$-0.613526\pi$
$829$	−7.87187e15	−0.698277	−0.349139	+	0.937071i	$0.613526\pi$
$830$	5.55320e14	0.0489343
$831$	0	0
$832$	4.32142e15	0.375793
$833$	−1.16210e16	−1.00391
$834$	0	0
$835$	−8.98381e15	−0.765923
$836$	1.71115e16	1.44929
$837$	0	0
$838$	3.26507e15	0.272929
$839$	−6.12117e15	−0.508327	−0.254164	−	0.967161i	$-0.581800\pi$
$839$	−6.12117e15	−0.508327	−0.254164	+	0.967161i	$0.581800\pi$
$840$	0	0
$841$	2.41000e16	1.97533
$842$	−1.44092e15	−0.117334
$843$	0	0
$844$	−3.76650e15	−0.302729
$845$	−1.11795e15	−0.0892713
$846$	0	0
$847$	2.90043e16	2.28615
$848$	1.02655e16	0.803902
$849$	0	0
$850$	3.78497e14	0.0292590
$851$	−1.48603e15	−0.114134
$852$	0	0
$853$	2.83032e15	0.214594	0.107297	−	0.994227i	$-0.465781\pi$
$853$	2.83032e15	0.214594	0.107297	+	0.994227i	$0.465781\pi$
$854$	1.94611e15	0.146605
$855$	0	0
$856$	−6.09060e14	−0.0452954
$857$	4.83193e14	0.0357048	0.0178524	−	0.999841i	$-0.494317\pi$
$857$	4.83193e14	0.0357048	0.0178524	+	0.999841i	$0.494317\pi$
$858$	0	0
$859$	1.23886e16	0.903775	0.451888	−	0.892075i	$-0.350751\pi$
$859$	1.23886e16	0.903775	0.451888	+	0.892075i	$0.350751\pi$
$860$	−9.56587e15	−0.693399
$861$	0	0
$862$	1.63992e15	0.117364
$863$	2.26458e16	1.61038	0.805191	−	0.593016i	$-0.202063\pi$
$863$	2.26458e16	1.61038	0.805191	+	0.593016i	$0.202063\pi$
$864$	0	0
$865$	−4.78391e15	−0.335888
$866$	5.03459e15	0.351249
$867$	0	0
$868$	−1.48050e16	−1.01988
$869$	1.95751e15	0.133997
$870$	0	0
$871$	−9.08565e15	−0.614124
$872$	−1.78446e15	−0.119858
$873$	0	0
$874$	3.14698e15	0.208728
$875$	2.43961e15	0.160797
$876$	0	0
$877$	−8.89622e15	−0.579039	−0.289519	−	0.957172i	$-0.593495\pi$
$877$	−8.89622e15	−0.579039	−0.289519	+	0.957172i	$0.593495\pi$
$878$	−1.78023e15	−0.115148
$879$	0	0
$880$	−7.32141e15	−0.467671
$881$	1.82720e16	1.15989	0.579946	−	0.814655i	$-0.303073\pi$
$881$	1.82720e16	1.15989	0.579946	+	0.814655i	$0.303073\pi$
$882$	0	0
$883$	8.75758e15	0.549035	0.274518	−	0.961582i	$-0.411482\pi$
$883$	8.75758e15	0.549035	0.274518	+	0.961582i	$0.411482\pi$
$884$	−5.77551e15	−0.359834
$885$	0	0
$886$	−1.87398e15	−0.115314
$887$	−8.52159e15	−0.521124	−0.260562	−	0.965457i	$-0.583908\pi$
$887$	−8.52159e15	−0.521124	−0.260562	+	0.965457i	$0.583908\pi$
$888$	0	0
$889$	2.10613e15	0.127212
$890$	2.04261e14	0.0122614
$891$	0	0
$892$	−1.83132e16	−1.08582
$893$	−9.96390e15	−0.587146
$894$	0	0
$895$	5.97928e15	0.348034
$896$	3.04042e16	1.75889
$897$	0	0
$898$	−2.38209e15	−0.136125
$899$	1.92673e16	1.09432
$900$	0	0
$901$	−9.28858e15	−0.521150
$902$	−2.68181e15	−0.149552
$903$	0	0
$904$	2.26316e15	0.124677
$905$	−5.33809e15	−0.292293
$906$	0	0
$907$	2.10550e16	1.13898	0.569490	−	0.821998i	$-0.307141\pi$
$907$	2.10550e16	1.13898	0.569490	+	0.821998i	$0.307141\pi$
$908$	−1.24005e16	−0.666756
$909$	0	0
$910$	4.40392e15	0.233944
$911$	−3.04479e16	−1.60771	−0.803854	−	0.594827i	$-0.797220\pi$
$911$	−3.04479e16	−1.60771	−0.803854	+	0.594827i	$0.797220\pi$
$912$	0	0
$913$	9.71950e15	0.507055
$914$	5.13939e15	0.266506
$915$	0	0
$916$	−1.86674e16	−0.956443
$917$	1.75805e16	0.895363
$918$	0	0
$919$	−4.47695e15	−0.225293	−0.112646	−	0.993635i	$-0.535933\pi$
$919$	−4.47695e15	−0.225293	−0.112646	+	0.993635i	$0.535933\pi$
$920$	−3.28714e15	−0.164431
$921$	0	0
$922$	−6.61799e15	−0.327119
$923$	−3.30328e16	−1.62307
$924$	0	0
$925$	7.87769e14	0.0382489
$926$	−6.19818e14	−0.0299160
$927$	0	0
$928$	−3.04417e16	−1.45196
$929$	−2.63310e16	−1.24848	−0.624238	−	0.781234i	$-0.714591\pi$
$929$	−2.63310e16	−1.24848	−0.624238	+	0.781234i	$0.714591\pi$
$930$	0	0
$931$	5.12208e16	2.40007
$932$	1.23515e16	0.575348
$933$	0	0
$934$	−1.16481e15	−0.0536224
$935$	6.62466e15	0.303179
$936$	0	0
$937$	1.18259e16	0.534894	0.267447	−	0.963573i	$-0.413820\pi$
$937$	1.18259e16	0.534894	0.267447	+	0.963573i	$0.413820\pi$
$938$	−8.92628e15	−0.401379
$939$	0	0
$940$	4.91320e15	0.218354
$941$	−6.96003e15	−0.307517	−0.153758	−	0.988108i	$-0.549138\pi$
$941$	−6.96003e15	−0.307517	−0.153758	+	0.988108i	$0.549138\pi$
$942$	0	0
$943$	4.16911e15	0.182066
$944$	−1.26812e16	−0.550573
$945$	0	0
$946$	1.98069e16	0.849993
$947$	3.38501e16	1.44423	0.722113	−	0.691775i	$-0.243171\pi$
$947$	3.38501e16	1.44423	0.722113	+	0.691775i	$0.243171\pi$
$948$	0	0
$949$	−3.86091e16	−1.62827
$950$	−1.66827e15	−0.0699499
$951$	0	0
$952$	−1.20197e16	−0.498183
$953$	−4.50062e16	−1.85465	−0.927323	−	0.374262i	$-0.877896\pi$
$953$	−4.50062e16	−1.85465	−0.927323	+	0.374262i	$0.877896\pi$
$954$	0	0
$955$	−9.87563e15	−0.402297
$956$	−3.23235e16	−1.30918
$957$	0	0
$958$	−1.41644e16	−0.567139
$959$	4.42907e16	1.76323
$960$	0	0
$961$	−1.51819e16	−0.597513
$962$	1.42206e15	0.0556486
$963$	0	0
$964$	2.52107e16	0.975353
$965$	8.66551e15	0.333345
$966$	0	0
$967$	−1.48359e16	−0.564246	−0.282123	−	0.959378i	$-0.591039\pi$
$967$	−1.48359e16	−0.564246	−0.282123	+	0.959378i	$0.591039\pi$
$968$	2.07172e16	0.783461
$969$	0	0
$970$	9.40424e14	0.0351624
$971$	−1.14414e16	−0.425378	−0.212689	−	0.977120i	$-0.568222\pi$
$971$	−1.14414e16	−0.425378	−0.212689	+	0.977120i	$0.568222\pi$
$972$	0	0
$973$	−4.93669e16	−1.81474
$974$	−6.72835e15	−0.245943
$975$	0	0
$976$	−4.81314e15	−0.173962
$977$	2.64437e16	0.950390	0.475195	−	0.879881i	$-0.342378\pi$
$977$	2.64437e16	0.950390	0.475195	+	0.879881i	$0.342378\pi$
$978$	0	0
$979$	3.57508e15	0.127052
$980$	−2.52570e16	−0.892562
$981$	0	0
$982$	6.47820e15	0.226382
$983$	−3.63656e16	−1.26371	−0.631854	−	0.775088i	$-0.717706\pi$
$983$	−3.63656e16	−1.26371	−0.631854	+	0.775088i	$0.717706\pi$
$984$	0	0
$985$	−1.20782e16	−0.415054
$986$	7.38447e15	0.252346
$987$	0	0
$988$	2.54562e16	0.860262
$989$	−3.07915e16	−1.03479
$990$	0	0
$991$	5.09067e16	1.69188	0.845941	−	0.533276i	$-0.179039\pi$
$991$	5.09067e16	1.69188	0.845941	+	0.533276i	$0.179039\pi$
$992$	−1.61576e16	−0.534027
$993$	0	0
$994$	−3.24534e16	−1.06080
$995$	−3.18247e15	−0.103452
$996$	0	0
$997$	−4.28291e16	−1.37694	−0.688471	−	0.725264i	$-0.741718\pi$
$997$	−4.28291e16	−1.37694	−0.688471	+	0.725264i	$0.741718\pi$
$998$	−9.36259e15	−0.299349
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.12.a.c.1.1		2
3.2	odd	2		15.12.a.c.1.2	✓	2
5.2	odd	4		225.12.b.i.199.2		4
5.3	odd	4		225.12.b.i.199.3		4
5.4	even	2		225.12.a.i.1.2		2
12.11	even	2		240.12.a.m.1.1		2
15.2	even	4		75.12.b.d.49.3		4
15.8	even	4		75.12.b.d.49.2		4
15.14	odd	2		75.12.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.c.1.2	✓	2	3.2	odd	2
45.12.a.c.1.1		2	1.1	even	1	trivial
75.12.a.c.1.1		2	15.14	odd	2
75.12.b.d.49.2		4	15.8	even	4
75.12.b.d.49.3		4	15.2	even	4
225.12.a.i.1.2		2	5.4	even	2
225.12.b.i.199.2		4	5.2	odd	4
225.12.b.i.199.3		4	5.3	odd	4
240.12.a.m.1.1		2	12.11	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q-14.7191 q^{2} -1831.35 q^{4} +3125.00 q^{5} +79941.2 q^{7} +57100.5 q^{8} -45997.2 q^{10} -805067. q^{11} -1.19767e6 q^{13} -1.17666e6 q^{14} +2.91013e6 q^{16} -2.63319e6 q^{17} +1.16061e7 q^{19} -5.72296e6 q^{20} +1.18499e7 q^{22} -1.84216e7 q^{23} +9.76562e6 q^{25} +1.76286e7 q^{26} -1.46400e8 q^{28} +1.90527e8 q^{29} +1.01127e8 q^{31} -1.59776e8 q^{32} +3.87581e7 q^{34} +2.49816e8 q^{35} +8.06675e7 q^{37} -1.70831e8 q^{38} +1.78439e8 q^{40} -2.26316e8 q^{41} +1.67149e9 q^{43} +1.47436e9 q^{44} +2.71150e8 q^{46} -8.58507e8 q^{47} +4.41327e9 q^{49} -1.43741e8 q^{50} +2.19335e9 q^{52} +3.52750e9 q^{53} -2.51583e9 q^{55} +4.56468e9 q^{56} -2.80438e9 q^{58} -4.35760e9 q^{59} -1.65393e9 q^{61} -1.48849e9 q^{62} -3.60819e9 q^{64} -3.74272e9 q^{65} +7.58610e9 q^{67} +4.82228e9 q^{68} -3.67707e9 q^{70} +2.75809e10 q^{71} +3.22368e10 q^{73} -1.18735e9 q^{74} -2.12548e10 q^{76} -6.43580e10 q^{77} -2.43149e9 q^{79} +9.09417e9 q^{80} +3.33116e9 q^{82} -1.20729e10 q^{83} -8.22871e9 q^{85} -2.46028e10 q^{86} -4.59697e10 q^{88} -4.44073e9 q^{89} -9.57433e10 q^{91} +3.37364e10 q^{92} +1.26364e10 q^{94} +3.62690e10 q^{95} -2.04453e10 q^{97} -6.49594e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 13 q^{2} - 3111 q^{4} + 6250 q^{5} + 7784 q^{7} - 35139 q^{8} + 40625 q^{10} - 295568 q^{11} + 657492 q^{13} - 3176796 q^{14} + 2974065 q^{16} - 8579948 q^{17} + 17627976 q^{19} - 9721875 q^{20} + 25972696 q^{22}+ \cdots + 24554976677 q^{98}+O(q^{100}) \) 2 * q + 13 * q^2 - 3111 * q^4 + 6250 * q^5 + 7784 * q^7 - 35139 * q^8 + 40625 * q^10 - 295568 * q^11 + 657492 * q^13 - 3176796 * q^14 + 2974065 * q^16 - 8579948 * q^17 + 17627976 * q^19 - 9721875 * q^20 + 25972696 * q^22 + 29841072 * q^23 + 19531250 * q^25 + 69052066 * q^26 - 54064108 * q^28 + 201881948 * q^29 - 71057008 * q^31 + 30902261 * q^32 - 126080702 * q^34 + 24325000 * q^35 + 705858484 * q^37 - 3909548 * q^38 - 109809375 * q^40 + 327655148 * q^41 + 3192552120 * q^43 + 822376568 * q^44 + 1608947832 * q^46 - 2053064720 * q^47 + 7642614770 * q^49 + 126953125 * q^50 - 180610022 * q^52 + 2304299452 * q^53 - 923650000 * q^55 + 11220431460 * q^56 - 2489635114 * q^58 + 1478770576 * q^59 - 8264891460 * q^61 - 6261263448 * q^62 + 1546316681 * q^64 + 2054662500 * q^65 + 24212177528 * q^67 + 12432065594 * q^68 - 9927487500 * q^70 + 20218888256 * q^71 + 25879583268 * q^73 + 16142374034 * q^74 - 28960704764 * q^76 - 101122045248 * q^77 + 22324995440 * q^79 + 9293953125 * q^80 + 18686740478 * q^82 - 48014508984 * q^83 - 26812337500 * q^85 + 17559698092 * q^86 - 92965609896 * q^88 - 79209683076 * q^89 - 229606712048 * q^91 - 28023036264 * q^92 - 20475612776 * q^94 + 55087425000 * q^95 - 37075227452 * q^97 + 24554976677 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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