LMFDB - Embedded newform 45.14.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,14,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, 14, names="a")

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

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 14 $
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:	$48.2539180284$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 24774x^{2} - 86616x + 52534656 $ x^4 - x^3 - 24774x^2 - 86616x + 52534656
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4}\cdot 5^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$152.425$ 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)

$f(q)$	$=$	\(q+148.425 q^{2} +13838.1 q^{4} +15625.0 q^{5} +215324. q^{7} +838027. q^{8} +2.31915e6 q^{10} +3.50194e6 q^{11} -1.22382e6 q^{13} +3.19596e7 q^{14} +1.10227e7 q^{16} +1.14596e8 q^{17} +1.26403e8 q^{19} +2.16221e8 q^{20} +5.19777e8 q^{22} -1.54080e8 q^{23} +2.44141e8 q^{25} -1.81646e8 q^{26} +2.97968e9 q^{28} -3.50622e9 q^{29} +6.23402e9 q^{31} -5.22907e9 q^{32} +1.70089e10 q^{34} +3.36444e9 q^{35} +2.11673e10 q^{37} +1.87614e10 q^{38} +1.30942e10 q^{40} +5.91552e10 q^{41} -1.84126e10 q^{43} +4.84603e10 q^{44} -2.28694e10 q^{46} -3.33828e10 q^{47} -5.05244e10 q^{49} +3.62367e10 q^{50} -1.69354e10 q^{52} -7.18621e10 q^{53} +5.47178e10 q^{55} +1.80448e11 q^{56} -5.20412e11 q^{58} +2.91405e10 q^{59} -5.52018e11 q^{61} +9.25288e11 q^{62} -8.66425e11 q^{64} -1.91222e10 q^{65} +9.87816e11 q^{67} +1.58579e12 q^{68} +4.99369e11 q^{70} +1.33203e11 q^{71} +1.18101e10 q^{73} +3.14177e12 q^{74} +1.74918e12 q^{76} +7.54053e11 q^{77} -3.39311e12 q^{79} +1.72230e11 q^{80} +8.78014e12 q^{82} -1.44794e12 q^{83} +1.79056e12 q^{85} -2.73290e12 q^{86} +2.93472e12 q^{88} -1.62879e11 q^{89} -2.63518e11 q^{91} -2.13218e12 q^{92} -4.95485e12 q^{94} +1.97504e12 q^{95} +3.58718e12 q^{97} -7.49911e12 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 15 q^{2} + 16837 q^{4} + 62500 q^{5} + 343040 q^{7} - 14865 q^{8} - 234375 q^{10} + 12697800 q^{11} + 34336040 q^{13} + 26944650 q^{14} + 66562801 q^{16} + 84377280 q^{17} - 131821144 q^{19} + 263078125 q^{20}+ \cdots - 17650752985395 q^{98}+O(q^{100}) $ 4 * q - 15 * q^2 + 16837 * q^4 + 62500 * q^5 + 343040 * q^7 - 14865 * q^8 - 234375 * q^10 + 12697800 * q^11 + 34336040 * q^13 + 26944650 * q^14 + 66562801 * q^16 + 84377280 * q^17 - 131821144 * q^19 + 263078125 * q^20 - 1016682450 * q^22 + 1046106360 * q^23 + 976562500 * q^25 - 2998448100 * q^26 - 2229409030 * q^28 - 1826812200 * q^29 + 3501222872 * q^31 - 1278404505 * q^32 + 2712309690 * q^34 + 5360000000 * q^35 + 8297642120 * q^37 + 50666879340 * q^38 - 232265625 * q^40 + 54459122400 * q^41 - 48173425360 * q^43 + 141205041150 * q^44 - 131025090720 * q^46 + 202496682840 * q^47 + 116605824372 * q^49 - 3662109375 * q^50 + 183603958220 * q^52 + 116147122920 * q^53 + 198403125000 * q^55 + 985319673150 * q^56 - 209673904950 * q^58 + 885100702200 * q^59 + 467836185368 * q^61 + 1845870142380 * q^62 - 1316914666607 * q^64 + 536500625000 * q^65 + 819037749440 * q^67 + 4568941043490 * q^68 + 421010156250 * q^70 + 2886468559200 * q^71 - 1235535876280 * q^73 + 5480818576200 * q^74 - 1007297478532 * q^76 + 3590015832000 * q^77 - 3553486866616 * q^79 + 1040043765625 * q^80 + 9494817638400 * q^82 + 3441261762480 * q^83 + 1318395000000 * q^85 + 1418346623400 * q^86 + 3182744059050 * q^88 - 2322663606000 * q^89 - 4966358282000 * q^91 - 3469668771120 * q^92 - 19865804246880 * q^94 - 2059705375000 * q^95 + 8462989649240 * q^97 - 17650752985395 * 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 $

$2$	148.425	1.63989	0.819943	−	0.572446i	$-0.194005\pi$
$2$	148.425	1.63989	0.819943	+	0.572446i	$0.194005\pi$
$3$	0	0
$3$	0	0
$4$	13838.1	1.68922
$5$	15625.0	0.447214
$5$	15625.0	0.447214
$6$	0	0
$7$	215324.	0.691761	0.345880	−	0.938279i	$-0.387580\pi$
$7$	215324.	0.691761	0.345880	+	0.938279i	$0.387580\pi$
$8$	838027.	1.13025
$9$	0	0
$10$	2.31915e6	0.733379
$11$	3.50194e6	0.596014	0.298007	−	0.954564i	$-0.403678\pi$
$11$	3.50194e6	0.596014	0.298007	+	0.954564i	$0.403678\pi$
$12$	0	0
$13$	−1.22382e6	−0.0703211	−0.0351605	−	0.999382i	$-0.511194\pi$
$13$	−1.22382e6	−0.0703211	−0.0351605	+	0.999382i	$0.511194\pi$
$14$	3.19596e7	1.13441
$15$	0	0
$16$	1.10227e7	0.164252
$17$	1.14596e8	1.15146	0.575731	−	0.817639i	$-0.304717\pi$
$17$	1.14596e8	1.15146	0.575731	+	0.817639i	$0.304717\pi$
$18$	0	0
$19$	1.26403e8	0.616393	0.308197	−	0.951323i	$-0.400275\pi$
$19$	1.26403e8	0.616393	0.308197	+	0.951323i	$0.400275\pi$
$20$	2.16221e8	0.755444
$21$	0	0
$22$	5.19777e8	0.977394
$23$	−1.54080e8	−0.217028	−0.108514	−	0.994095i	$-0.534609\pi$
$23$	−1.54080e8	−0.217028	−0.108514	+	0.994095i	$0.534609\pi$
$24$	0	0
$25$	2.44141e8	0.200000
$26$	−1.81646e8	−0.115318
$27$	0	0
$28$	2.97968e9	1.16854
$29$	−3.50622e9	−1.09459	−0.547295	−	0.836940i	$-0.684343\pi$
$29$	−3.50622e9	−1.09459	−0.547295	+	0.836940i	$0.684343\pi$
$30$	0	0
$31$	6.23402e9	1.26159	0.630794	−	0.775950i	$-0.282729\pi$
$31$	6.23402e9	1.26159	0.630794	+	0.775950i	$0.282729\pi$
$32$	−5.22907e9	−0.860893
$33$	0	0
$34$	1.70089e10	1.88827
$35$	3.36444e9	0.309365
$36$	0	0
$37$	2.11673e10	1.35630	0.678148	−	0.734925i	$-0.262783\pi$
$37$	2.11673e10	1.35630	0.678148	+	0.734925i	$0.262783\pi$
$38$	1.87614e10	1.01081
$39$	0	0
$40$	1.30942e10	0.505462
$41$	5.91552e10	1.94490	0.972451	−	0.233107i	$-0.0748892\pi$
$41$	5.91552e10	1.94490	0.972451	+	0.233107i	$0.0748892\pi$
$42$	0	0
$43$	−1.84126e10	−0.444192	−0.222096	−	0.975025i	$-0.571290\pi$
$43$	−1.84126e10	−0.444192	−0.222096	+	0.975025i	$0.571290\pi$
$44$	4.84603e10	1.00680
$45$	0	0
$46$	−2.28694e10	−0.355901
$47$	−3.33828e10	−0.451737	−0.225869	−	0.974158i	$-0.572522\pi$
$47$	−3.33828e10	−0.451737	−0.225869	+	0.974158i	$0.572522\pi$
$48$	0	0
$49$	−5.05244e10	−0.521467
$50$	3.62367e10	0.327977
$51$	0	0
$52$	−1.69354e10	−0.118788
$53$	−7.18621e10	−0.445356	−0.222678	−	0.974892i	$-0.571480\pi$
$53$	−7.18621e10	−0.445356	−0.222678	+	0.974892i	$0.571480\pi$
$54$	0	0
$55$	5.47178e10	0.266546
$56$	1.80448e11	0.781860
$57$	0	0
$58$	−5.20412e11	−1.79500
$59$	2.91405e10	0.0899412	0.0449706	−	0.998988i	$-0.485681\pi$
$59$	2.91405e10	0.0899412	0.0449706	+	0.998988i	$0.485681\pi$
$60$	0	0
$61$	−5.52018e11	−1.37186	−0.685929	−	0.727669i	$-0.740604\pi$
$61$	−5.52018e11	−1.37186	−0.685929	+	0.727669i	$0.740604\pi$
$62$	9.25288e11	2.06886
$63$	0	0
$64$	−8.66425e11	−1.57602
$65$	−1.91222e10	−0.0314485
$66$	0	0
$67$	9.87816e11	1.33410	0.667052	−	0.745011i	$-0.267556\pi$
$67$	9.87816e11	1.33410	0.667052	+	0.745011i	$0.267556\pi$
$68$	1.58579e12	1.94508
$69$	0	0
$70$	4.99369e11	0.507323
$71$	1.33203e11	0.123406	0.0617030	−	0.998095i	$-0.480347\pi$
$71$	1.33203e11	0.123406	0.0617030	+	0.998095i	$0.480347\pi$
$72$	0	0
$73$	1.18101e10	0.00913385	0.00456693	−	0.999990i	$-0.498546\pi$
$73$	1.18101e10	0.00913385	0.00456693	+	0.999990i	$0.498546\pi$
$74$	3.14177e12	2.22417
$75$	0	0
$76$	1.74918e12	1.04123
$77$	7.54053e11	0.412299
$78$	0	0
$79$	−3.39311e12	−1.57044	−0.785222	−	0.619215i	$-0.787451\pi$
$79$	−3.39311e12	−1.57044	−0.785222	+	0.619215i	$0.787451\pi$
$80$	1.72230e11	0.0734555
$81$	0	0
$82$	8.78014e12	3.18942
$83$	−1.44794e12	−0.486118	−0.243059	−	0.970011i	$-0.578151\pi$
$83$	−1.44794e12	−0.486118	−0.243059	+	0.970011i	$0.578151\pi$
$84$	0	0
$85$	1.79056e12	0.514950
$86$	−2.73290e12	−0.728424
$87$	0	0
$88$	2.93472e12	0.673643
$89$	−1.62879e11	−0.0347400	−0.0173700	−	0.999849i	$-0.505529\pi$
$89$	−1.62879e11	−0.0347400	−0.0173700	+	0.999849i	$0.505529\pi$
$90$	0	0
$91$	−2.63518e11	−0.0486454
$92$	−2.13218e12	−0.366609
$93$	0	0
$94$	−4.95485e12	−0.740798
$95$	1.97504e12	0.275659
$96$	0	0
$97$	3.58718e12	0.437258	0.218629	−	0.975808i	$-0.429842\pi$
$97$	3.58718e12	0.437258	0.218629	+	0.975808i	$0.429842\pi$
$98$	−7.49911e12	−0.855146
$99$	0	0
$100$	3.37845e12	0.337845
$101$	1.32945e13	1.24619	0.623095	−	0.782146i	$-0.285875\pi$
$101$	1.32945e13	1.24619	0.623095	+	0.782146i	$0.285875\pi$
$102$	0	0
$103$	−2.14656e13	−1.77134	−0.885669	−	0.464317i	$-0.846300\pi$
$103$	−2.14656e13	−1.77134	−0.885669	+	0.464317i	$0.846300\pi$
$104$	−1.02559e12	−0.0794801
$105$	0	0
$106$	−1.06662e13	−0.730332
$107$	−4.53570e12	−0.292180	−0.146090	−	0.989271i	$-0.546669\pi$
$107$	−4.53570e12	−0.292180	−0.146090	+	0.989271i	$0.546669\pi$
$108$	0	0
$109$	−2.75975e13	−1.57615	−0.788074	−	0.615580i	$-0.788922\pi$
$109$	−2.75975e13	−1.57615	−0.788074	+	0.615580i	$0.788922\pi$
$110$	8.12152e12	0.437104
$111$	0	0
$112$	2.37346e12	0.113623
$113$	−3.59293e13	−1.62345	−0.811725	−	0.584040i	$-0.801471\pi$
$113$	−3.59293e13	−1.62345	−0.811725	+	0.584040i	$0.801471\pi$
$114$	0	0
$115$	−2.40751e12	−0.0970580
$116$	−4.85194e13	−1.84901
$117$	0	0
$118$	4.32519e12	0.147493
$119$	2.46752e13	0.796537
$120$	0	0
$121$	−2.22591e13	−0.644767
$122$	−8.19335e13	−2.24969
$123$	0	0
$124$	8.62671e13	2.13110
$125$	3.81470e12	0.0894427
$126$	0	0
$127$	−8.24154e13	−1.74295	−0.871473	−	0.490443i	$-0.836835\pi$
$127$	−8.24154e13	−1.74295	−0.871473	+	0.490443i	$0.836835\pi$
$128$	−8.57630e13	−1.72359
$129$	0	0
$130$	−2.83822e12	−0.0515720
$131$	−9.61500e13	−1.66221	−0.831105	−	0.556115i	$-0.812291\pi$
$131$	−9.61500e13	−1.66221	−0.831105	+	0.556115i	$0.812291\pi$
$132$	0	0
$133$	2.72176e13	0.426397
$134$	1.46617e14	2.18778
$135$	0	0
$136$	9.60342e13	1.30144
$137$	1.15756e14	1.49575	0.747877	−	0.663837i	$-0.231073\pi$
$137$	1.15756e14	1.49575	0.747877	+	0.663837i	$0.231073\pi$
$138$	0	0
$139$	8.85611e13	1.04147	0.520735	−	0.853718i	$-0.325658\pi$
$139$	8.85611e13	1.04147	0.520735	+	0.853718i	$0.325658\pi$
$140$	4.65576e13	0.522586
$141$	0	0
$142$	1.97708e13	0.202372
$143$	−4.28574e12	−0.0419123
$144$	0	0
$145$	−5.47846e13	−0.489516
$146$	1.75292e12	0.0149785
$147$	0	0
$148$	2.92916e14	2.29109
$149$	1.45878e14	1.09215	0.546073	−	0.837738i	$-0.316122\pi$
$149$	1.45878e14	1.09215	0.546073	+	0.837738i	$0.316122\pi$
$150$	0	0
$151$	1.41584e13	0.0971994	0.0485997	−	0.998818i	$-0.484524\pi$
$151$	1.41584e13	0.0971994	0.0485997	+	0.998818i	$0.484524\pi$
$152$	1.05929e14	0.696676
$153$	0	0
$154$	1.11921e14	0.676123
$155$	9.74066e13	0.564199
$156$	0	0
$157$	−3.21755e14	−1.71466	−0.857329	−	0.514768i	$-0.827878\pi$
$157$	−3.21755e14	−1.71466	−0.857329	+	0.514768i	$0.827878\pi$
$158$	−5.03624e14	−2.57535
$159$	0	0
$160$	−8.17041e13	−0.385003
$161$	−3.31773e13	−0.150132
$162$	0	0
$163$	−1.93760e14	−0.809181	−0.404591	−	0.914498i	$-0.632586\pi$
$163$	−1.93760e14	−0.809181	−0.404591	+	0.914498i	$0.632586\pi$
$164$	8.18596e14	3.28537
$165$	0	0
$166$	−2.14911e14	−0.797178
$167$	−3.35780e14	−1.19784	−0.598919	−	0.800809i	$-0.704403\pi$
$167$	−3.35780e14	−1.19784	−0.598919	+	0.800809i	$0.704403\pi$
$168$	0	0
$169$	−3.01377e14	−0.995055
$170$	2.65764e14	0.844458
$171$	0	0
$172$	−2.54796e14	−0.750340
$173$	−1.10123e14	−0.312306	−0.156153	−	0.987733i	$-0.549909\pi$
$173$	−1.10123e14	−0.312306	−0.156153	+	0.987733i	$0.549909\pi$
$174$	0	0
$175$	5.25694e13	0.138352
$176$	3.86010e13	0.0978962
$177$	0	0
$178$	−2.41754e13	−0.0569697
$179$	7.53350e14	1.71180	0.855898	−	0.517145i	$-0.173005\pi$
$179$	7.53350e14	1.71180	0.855898	+	0.517145i	$0.173005\pi$
$180$	0	0
$181$	−1.93468e14	−0.408977	−0.204488	−	0.978869i	$-0.565553\pi$
$181$	−1.93468e14	−0.408977	−0.204488	+	0.978869i	$0.565553\pi$
$182$	−3.91128e13	−0.0797728
$183$	0	0
$184$	−1.29124e14	−0.245296
$185$	3.30739e14	0.606554
$186$	0	0
$187$	4.01307e14	0.686288
$188$	−4.61954e14	−0.763085
$189$	0	0
$190$	2.93147e14	0.452050
$191$	3.06017e14	0.456067	0.228033	−	0.973653i	$-0.426770\pi$
$191$	3.06017e14	0.456067	0.228033	+	0.973653i	$0.426770\pi$
$192$	0	0
$193$	2.32828e14	0.324274	0.162137	−	0.986768i	$-0.448161\pi$
$193$	2.32828e14	0.324274	0.162137	+	0.986768i	$0.448161\pi$
$194$	5.32429e14	0.717052
$195$	0	0
$196$	−6.99163e14	−0.880874
$197$	3.64429e14	0.444204	0.222102	−	0.975023i	$-0.428708\pi$
$197$	3.64429e14	0.444204	0.222102	+	0.975023i	$0.428708\pi$
$198$	0	0
$199$	−2.43854e14	−0.278346	−0.139173	−	0.990268i	$-0.544444\pi$
$199$	−2.43854e14	−0.278346	−0.139173	+	0.990268i	$0.544444\pi$
$200$	2.04596e14	0.226049
$201$	0	0
$202$	1.97325e15	2.04361
$203$	−7.54974e14	−0.757195
$204$	0	0
$205$	9.24300e14	0.869787
$206$	−3.18604e15	−2.90479
$207$	0	0
$208$	−1.34898e13	−0.0115503
$209$	4.42655e14	0.367379
$210$	0	0
$211$	1.86277e15	1.45319	0.726596	−	0.687065i	$-0.241101\pi$
$211$	1.86277e15	1.45319	0.726596	+	0.687065i	$0.241101\pi$
$212$	−9.94436e14	−0.752305
$213$	0	0
$214$	−6.73213e14	−0.479141
$215$	−2.87697e14	−0.198649
$216$	0	0
$217$	1.34234e15	0.872717
$218$	−4.09617e15	−2.58470
$219$	0	0
$220$	7.57192e14	0.450255
$221$	−1.40244e14	−0.0809721
$222$	0	0
$223$	8.44142e14	0.459657	0.229829	−	0.973231i	$-0.426183\pi$
$223$	8.44142e14	0.459657	0.229829	+	0.973231i	$0.426183\pi$
$224$	−1.12595e15	−0.595532
$225$	0	0
$226$	−5.33282e15	−2.66227
$227$	7.84425e14	0.380525	0.190263	−	0.981733i	$-0.439066\pi$
$227$	7.84425e14	0.380525	0.190263	+	0.981733i	$0.439066\pi$
$228$	0	0
$229$	2.50019e14	0.114563	0.0572813	−	0.998358i	$-0.481757\pi$
$229$	2.50019e14	0.114563	0.0572813	+	0.998358i	$0.481757\pi$
$230$	−3.57335e14	−0.159164
$231$	0	0
$232$	−2.93830e15	−1.23716
$233$	2.82252e15	1.15564	0.577822	−	0.816163i	$-0.303903\pi$
$233$	2.82252e15	1.15564	0.577822	+	0.816163i	$0.303903\pi$
$234$	0	0
$235$	−5.21606e14	−0.202023
$236$	4.03249e14	0.151931
$237$	0	0
$238$	3.66243e15	1.30623
$239$	3.19985e15	1.11056	0.555281	−	0.831663i	$-0.312611\pi$
$239$	3.19985e15	1.11056	0.555281	+	0.831663i	$0.312611\pi$
$240$	0	0
$241$	1.14929e15	0.377848	0.188924	−	0.981992i	$-0.439500\pi$
$241$	1.14929e15	0.377848	0.188924	+	0.981992i	$0.439500\pi$
$242$	−3.30382e15	−1.05734
$243$	0	0
$244$	−7.63888e15	−2.31737
$245$	−7.89444e14	−0.233207
$246$	0	0
$247$	−1.54694e14	−0.0433454
$248$	5.22428e15	1.42591
$249$	0	0
$250$	5.66198e14	0.146676
$251$	4.56021e15	1.15108	0.575540	−	0.817773i	$-0.304792\pi$
$251$	4.56021e15	1.15108	0.575540	+	0.817773i	$0.304792\pi$
$252$	0	0
$253$	−5.39580e14	−0.129352
$254$	−1.22325e16	−2.85823
$255$	0	0
$256$	−5.63166e15	−1.25048
$257$	2.41893e15	0.523671	0.261835	−	0.965113i	$-0.415672\pi$
$257$	2.41893e15	0.523671	0.261835	+	0.965113i	$0.415672\pi$
$258$	0	0
$259$	4.55784e15	0.938233
$260$	−2.64615e14	−0.0531236
$261$	0	0
$262$	−1.42711e16	−2.72583
$263$	1.39889e15	0.260657	0.130329	−	0.991471i	$-0.458397\pi$
$263$	1.39889e15	0.260657	0.130329	+	0.991471i	$0.458397\pi$
$264$	0	0
$265$	−1.12285e15	−0.199169
$266$	4.03978e15	0.699242
$267$	0	0
$268$	1.36695e16	2.25360
$269$	−7.53557e15	−1.21262	−0.606312	−	0.795227i	$-0.707352\pi$
$269$	−7.53557e15	−1.21262	−0.606312	+	0.795227i	$0.707352\pi$
$270$	0	0
$271$	−3.68276e15	−0.564772	−0.282386	−	0.959301i	$-0.591126\pi$
$271$	−3.68276e15	−0.564772	−0.282386	+	0.959301i	$0.591126\pi$
$272$	1.26316e15	0.189130
$273$	0	0
$274$	1.71812e16	2.45287
$275$	8.54966e14	0.119203
$276$	0	0
$277$	−2.90631e15	−0.386566	−0.193283	−	0.981143i	$-0.561914\pi$
$277$	−2.90631e15	−0.386566	−0.193283	+	0.981143i	$0.561914\pi$
$278$	1.31447e16	1.70789
$279$	0	0
$280$	2.81950e15	0.349659
$281$	4.98905e15	0.604542	0.302271	−	0.953222i	$-0.402255\pi$
$281$	4.98905e15	0.604542	0.302271	+	0.953222i	$0.402255\pi$
$282$	0	0
$283$	1.44585e16	1.67306	0.836529	−	0.547922i	$-0.184581\pi$
$283$	1.44585e16	1.67306	0.836529	+	0.547922i	$0.184581\pi$
$284$	1.84329e15	0.208460
$285$	0	0
$286$	−6.36113e14	−0.0687314
$287$	1.27376e16	1.34541
$288$	0	0
$289$	3.22757e15	0.325866
$290$	−8.13143e15	−0.802750
$291$	0	0
$292$	1.63429e14	0.0154291
$293$	1.80867e16	1.67001	0.835007	−	0.550239i	$-0.185463\pi$
$293$	1.80867e16	1.67001	0.835007	+	0.550239i	$0.185463\pi$
$294$	0	0
$295$	4.55320e14	0.0402229
$296$	1.77388e16	1.53295
$297$	0	0
$298$	2.16521e16	1.79099
$299$	1.88567e14	0.0152617
$300$	0	0
$301$	−3.96469e15	−0.307275
$302$	2.10147e15	0.159396
$303$	0	0
$304$	1.39330e15	0.101244
$305$	−8.62528e15	−0.613513
$306$	0	0
$307$	−2.08974e16	−1.42460	−0.712301	−	0.701874i	$-0.752347\pi$
$307$	−2.08974e16	−1.42460	−0.712301	+	0.701874i	$0.752347\pi$
$308$	1.04347e16	0.696465
$309$	0	0
$310$	1.44576e16	0.925222
$311$	−7.80809e15	−0.489330	−0.244665	−	0.969608i	$-0.578678\pi$
$311$	−7.80809e15	−0.489330	−0.244665	+	0.969608i	$0.578678\pi$
$312$	0	0
$313$	−1.63797e16	−0.984620	−0.492310	−	0.870420i	$-0.663847\pi$
$313$	−1.63797e16	−0.984620	−0.492310	+	0.870420i	$0.663847\pi$
$314$	−4.77566e16	−2.81184
$315$	0	0
$316$	−4.69543e16	−2.65283
$317$	3.59584e15	0.199028	0.0995141	−	0.995036i	$-0.468271\pi$
$317$	3.59584e15	0.199028	0.0995141	+	0.995036i	$0.468271\pi$
$318$	0	0
$319$	−1.22786e16	−0.652391
$320$	−1.35379e16	−0.704816
$321$	0	0
$322$	−4.92435e15	−0.246199
$323$	1.44852e16	0.709754
$324$	0	0
$325$	−2.98784e14	−0.0140642
$326$	−2.87590e16	−1.32696
$327$	0	0
$328$	4.95737e16	2.19822
$329$	−7.18812e15	−0.312494
$330$	0	0
$331$	−2.21791e16	−0.926961	−0.463480	−	0.886107i	$-0.653400\pi$
$331$	−2.21791e16	−0.926961	−0.463480	+	0.886107i	$0.653400\pi$
$332$	−2.00367e16	−0.821162
$333$	0	0
$334$	−4.98384e16	−1.96432
$335$	1.54346e16	0.596629
$336$	0	0
$337$	3.17420e16	1.18043	0.590215	−	0.807246i	$-0.299043\pi$
$337$	3.17420e16	1.18043	0.590215	+	0.807246i	$0.299043\pi$
$338$	−4.47321e16	−1.63178
$339$	0	0
$340$	2.47779e16	0.869865
$341$	2.18312e16	0.751924
$342$	0	0
$343$	−3.17417e16	−1.05249
$344$	−1.54303e16	−0.502047
$345$	0	0
$346$	−1.63451e16	−0.512146
$347$	−1.06948e16	−0.328876	−0.164438	−	0.986387i	$-0.552581\pi$
$347$	−1.06948e16	−0.328876	−0.164438	+	0.986387i	$0.552581\pi$
$348$	0	0
$349$	−4.43687e16	−1.31435	−0.657176	−	0.753737i	$-0.728249\pi$
$349$	−4.43687e16	−1.31435	−0.657176	+	0.753737i	$0.728249\pi$
$350$	7.80264e15	0.226882
$351$	0	0
$352$	−1.83119e16	−0.513104
$353$	1.05902e16	0.291319	0.145660	−	0.989335i	$-0.453470\pi$
$353$	1.05902e16	0.291319	0.145660	+	0.989335i	$0.453470\pi$
$354$	0	0
$355$	2.08130e15	0.0551888
$356$	−2.25394e15	−0.0586837
$357$	0	0
$358$	1.11816e17	2.80715
$359$	5.10726e15	0.125914	0.0629570	−	0.998016i	$-0.479947\pi$
$359$	5.10726e15	0.125914	0.0629570	+	0.998016i	$0.479947\pi$
$360$	0	0
$361$	−2.60753e16	−0.620059
$362$	−2.87156e16	−0.670675
$363$	0	0
$364$	−3.64659e15	−0.0821729
$365$	1.84532e14	0.00408478
$366$	0	0
$367$	5.49713e16	1.17437	0.587187	−	0.809452i	$-0.300235\pi$
$367$	5.49713e16	1.17437	0.587187	+	0.809452i	$0.300235\pi$
$368$	−1.69839e15	−0.0356472
$369$	0	0
$370$	4.90902e16	0.994679
$371$	−1.54737e16	−0.308080
$372$	0	0
$373$	4.60597e16	0.885551	0.442775	−	0.896633i	$-0.353994\pi$
$373$	4.60597e16	0.885551	0.442775	+	0.896633i	$0.353994\pi$
$374$	5.95642e16	1.12543
$375$	0	0
$376$	−2.79757e16	−0.510575
$377$	4.29097e15	0.0769728
$378$	0	0
$379$	1.36123e16	0.235926	0.117963	−	0.993018i	$-0.462364\pi$
$379$	1.36123e16	0.235926	0.117963	+	0.993018i	$0.462364\pi$
$380$	2.73309e16	0.465650
$381$	0	0
$382$	4.54207e16	0.747897
$383$	−1.14040e17	−1.84615	−0.923074	−	0.384623i	$-0.874331\pi$
$383$	−1.14040e17	−1.84615	−0.923074	+	0.384623i	$0.874331\pi$
$384$	0	0
$385$	1.17821e16	0.184386
$386$	3.45576e16	0.531772
$387$	0	0
$388$	4.96399e16	0.738626
$389$	5.53953e16	0.810589	0.405295	−	0.914186i	$-0.367169\pi$
$389$	5.53953e16	0.810589	0.405295	+	0.914186i	$0.367169\pi$
$390$	0	0
$391$	−1.76569e16	−0.249900
$392$	−4.23408e16	−0.589386
$393$	0	0
$394$	5.40905e16	0.728443
$395$	−5.30174e16	−0.702324
$396$	0	0
$397$	−7.12560e16	−0.913447	−0.456723	−	0.889609i	$-0.650977\pi$
$397$	−7.12560e16	−0.913447	−0.456723	+	0.889609i	$0.650977\pi$
$398$	−3.61941e16	−0.456455
$399$	0	0
$400$	2.69110e15	0.0328503
$401$	−1.25835e17	−1.51135	−0.755674	−	0.654948i	$-0.772691\pi$
$401$	−1.25835e17	−1.51135	−0.755674	+	0.654948i	$0.772691\pi$
$402$	0	0
$403$	−7.62932e15	−0.0887162
$404$	1.83971e17	2.10509
$405$	0	0
$406$	−1.12057e17	−1.24171
$407$	7.41267e16	0.808372
$408$	0	0
$409$	9.42268e16	0.995343	0.497672	−	0.867365i	$-0.334188\pi$
$409$	9.42268e16	0.995343	0.497672	+	0.867365i	$0.334188\pi$
$410$	1.37190e17	1.42635
$411$	0	0
$412$	−2.97044e17	−2.99219
$413$	6.27466e15	0.0622178
$414$	0	0
$415$	−2.26240e16	−0.217399
$416$	6.39943e15	0.0605389
$417$	0	0
$418$	6.57012e16	0.602459
$419$	2.06207e17	1.86171	0.930857	−	0.365383i	$-0.119062\pi$
$419$	2.06207e17	1.86171	0.930857	+	0.365383i	$0.119062\pi$
$420$	0	0
$421$	8.51583e16	0.745406	0.372703	−	0.927951i	$-0.378431\pi$
$421$	8.51583e16	0.745406	0.372703	+	0.927951i	$0.378431\pi$
$422$	2.76482e17	2.38307
$423$	0	0
$424$	−6.02224e16	−0.503362
$425$	2.79774e16	0.230293
$426$	0	0
$427$	−1.18863e17	−0.948997
$428$	−6.27656e16	−0.493556
$429$	0	0
$430$	−4.27016e16	−0.325761
$431$	−8.69860e16	−0.653653	−0.326826	−	0.945084i	$-0.605979\pi$
$431$	−8.69860e16	−0.653653	−0.326826	+	0.945084i	$0.605979\pi$
$432$	0	0
$433$	1.26824e17	0.924759	0.462380	−	0.886682i	$-0.346996\pi$
$433$	1.26824e17	0.924759	0.462380	+	0.886682i	$0.346996\pi$
$434$	1.99237e17	1.43116
$435$	0	0
$436$	−3.81897e17	−2.66247
$437$	−1.94762e16	−0.133775
$438$	0	0
$439$	4.81345e16	0.320950	0.160475	−	0.987040i	$-0.448697\pi$
$439$	4.81345e16	0.320950	0.160475	+	0.987040i	$0.448697\pi$
$440$	4.58550e16	0.301262
$441$	0	0
$442$	−2.08158e16	−0.132785
$443$	−1.59797e17	−1.00449	−0.502244	−	0.864726i	$-0.667492\pi$
$443$	−1.59797e17	−1.00449	−0.502244	+	0.864726i	$0.667492\pi$
$444$	0	0
$445$	−2.54499e15	−0.0155362
$446$	1.25292e17	0.753785
$447$	0	0
$448$	−1.86562e17	−1.09023
$449$	−8.88441e16	−0.511714	−0.255857	−	0.966715i	$-0.582358\pi$
$449$	−8.88441e16	−0.511714	−0.255857	+	0.966715i	$0.582358\pi$
$450$	0	0
$451$	2.07158e17	1.15919
$452$	−4.97194e17	−2.74237
$453$	0	0
$454$	1.16429e17	0.624017
$455$	−4.11747e15	−0.0217549
$456$	0	0
$457$	1.18218e16	0.0607056	0.0303528	−	0.999539i	$-0.490337\pi$
$457$	1.18218e16	0.0607056	0.0303528	+	0.999539i	$0.490337\pi$
$458$	3.71092e16	0.187869
$459$	0	0
$460$	−3.33153e16	−0.163953
$461$	−2.74119e17	−1.33010	−0.665049	−	0.746799i	$-0.731590\pi$
$461$	−2.74119e17	−1.33010	−0.665049	+	0.746799i	$0.731590\pi$
$462$	0	0
$463$	1.93108e17	0.911011	0.455506	−	0.890233i	$-0.349458\pi$
$463$	1.93108e17	0.911011	0.455506	+	0.890233i	$0.349458\pi$
$464$	−3.86481e16	−0.179788
$465$	0	0
$466$	4.18934e17	1.89512
$467$	−1.32213e17	−0.589812	−0.294906	−	0.955526i	$-0.595288\pi$
$467$	−1.32213e17	−0.589812	−0.294906	+	0.955526i	$0.595288\pi$
$468$	0	0
$469$	2.12701e17	0.922881
$470$	−7.74195e16	−0.331295
$471$	0	0
$472$	2.44205e16	0.101656
$473$	−6.44800e16	−0.264745
$474$	0	0
$475$	3.08600e16	0.123279
$476$	3.41459e17	1.34553
$477$	0	0
$478$	4.74939e17	1.82119
$479$	−2.43680e17	−0.921804	−0.460902	−	0.887451i	$-0.652474\pi$
$479$	−2.43680e17	−0.921804	−0.460902	+	0.887451i	$0.652474\pi$
$480$	0	0
$481$	−2.59050e16	−0.0953762
$482$	1.70583e17	0.619628
$483$	0	0
$484$	−3.08024e17	−1.08916
$485$	5.60497e16	0.195548
$486$	0	0
$487$	3.83909e17	1.30404	0.652018	−	0.758203i	$-0.273922\pi$
$487$	3.83909e17	1.30404	0.652018	+	0.758203i	$0.273922\pi$
$488$	−4.62606e17	−1.55054
$489$	0	0
$490$	−1.17174e17	−0.382433
$491$	4.57681e15	0.0147412	0.00737060	−	0.999973i	$-0.497654\pi$
$491$	4.57681e15	0.0147412	0.00737060	+	0.999973i	$0.497654\pi$
$492$	0	0
$493$	−4.01797e17	−1.26038
$494$	−2.29605e16	−0.0710815
$495$	0	0
$496$	6.87160e16	0.207218
$497$	2.86820e16	0.0853674
$498$	0	0
$499$	−5.27519e17	−1.52962	−0.764812	−	0.644254i	$-0.777168\pi$
$499$	−5.27519e17	−1.52962	−0.764812	+	0.644254i	$0.777168\pi$
$500$	5.27882e16	0.151089
$501$	0	0
$502$	6.76851e17	1.88764
$503$	3.47045e17	0.955418	0.477709	−	0.878518i	$-0.341467\pi$
$503$	3.47045e17	0.955418	0.477709	+	0.878518i	$0.341467\pi$
$504$	0	0
$505$	2.07727e17	0.557313
$506$	−8.00875e16	−0.212122
$507$	0	0
$508$	−1.14047e18	−2.94423
$509$	−1.22973e17	−0.313433	−0.156717	−	0.987644i	$-0.550091\pi$
$509$	−1.22973e17	−0.313433	−0.156717	+	0.987644i	$0.550091\pi$
$510$	0	0
$511$	2.54300e15	0.00631844
$512$	−1.33311e17	−0.327048
$513$	0	0
$514$	3.59031e17	0.858760
$515$	−3.35400e17	−0.792167
$516$	0	0
$517$	−1.16904e17	−0.269242
$518$	6.76500e17	1.53859
$519$	0	0
$520$	−1.60249e16	−0.0355446
$521$	−6.98024e17	−1.52906	−0.764531	−	0.644587i	$-0.777029\pi$
$521$	−6.98024e17	−1.52906	−0.764531	+	0.644587i	$0.777029\pi$
$522$	0	0
$523$	−1.17031e17	−0.250057	−0.125028	−	0.992153i	$-0.539902\pi$
$523$	−1.17031e17	−0.250057	−0.125028	+	0.992153i	$0.539902\pi$
$524$	−1.33053e18	−2.80784
$525$	0	0
$526$	2.07630e17	0.427448
$527$	7.14392e17	1.45267
$528$	0	0
$529$	−4.80296e17	−0.952899
$530$	−1.66659e17	−0.326614
$531$	0	0
$532$	3.76640e17	0.720279
$533$	−7.23953e16	−0.136768
$534$	0	0
$535$	−7.08703e16	−0.130667
$536$	8.27816e17	1.50787
$537$	0	0
$538$	−1.11847e18	−1.98857
$539$	−1.76933e17	−0.310801
$540$	0	0
$541$	−4.80930e17	−0.824707	−0.412353	−	0.911024i	$-0.635293\pi$
$541$	−4.80930e17	−0.824707	−0.412353	+	0.911024i	$0.635293\pi$
$542$	−5.46616e17	−0.926162
$543$	0	0
$544$	−5.99228e17	−0.991286
$545$	−4.31210e17	−0.704875
$546$	0	0
$547$	−1.58434e16	−0.0252889	−0.0126445	−	0.999920i	$-0.504025\pi$
$547$	−1.58434e16	−0.0252889	−0.0126445	+	0.999920i	$0.504025\pi$
$548$	1.60185e18	2.52666
$549$	0	0
$550$	1.26899e17	0.195479
$551$	−4.43195e17	−0.674698
$552$	0	0
$553$	−7.30620e17	−1.08637
$554$	−4.31370e17	−0.633923
$555$	0	0
$556$	1.22552e18	1.75928
$557$	7.17753e17	1.01840	0.509198	−	0.860650i	$-0.329942\pi$
$557$	7.17753e17	1.01840	0.509198	+	0.860650i	$0.329942\pi$
$558$	0	0
$559$	2.25337e16	0.0312361
$560$	3.70854e16	0.0508137
$561$	0	0
$562$	7.40502e17	0.991380
$563$	8.56529e17	1.13354	0.566771	−	0.823876i	$-0.308193\pi$
$563$	8.56529e17	1.13354	0.566771	+	0.823876i	$0.308193\pi$
$564$	0	0
$565$	−5.61395e17	−0.726029
$566$	2.14601e18	2.74362
$567$	0	0
$568$	1.11628e17	0.139479
$569$	4.50162e17	0.556083	0.278041	−	0.960569i	$-0.410315\pi$
$569$	4.50162e17	0.556083	0.278041	+	0.960569i	$0.410315\pi$
$570$	0	0
$571$	−8.83282e16	−0.106651	−0.0533255	−	0.998577i	$-0.516982\pi$
$571$	−8.83282e16	−0.106651	−0.0533255	+	0.998577i	$0.516982\pi$
$572$	−5.93066e16	−0.0707993
$573$	0	0
$574$	1.89058e18	2.20631
$575$	−3.76173e16	−0.0434057
$576$	0	0
$577$	−2.23416e17	−0.252042	−0.126021	−	0.992028i	$-0.540221\pi$
$577$	−2.23416e17	−0.252042	−0.126021	+	0.992028i	$0.540221\pi$
$578$	4.79054e17	0.534384
$579$	0	0
$580$	−7.58116e17	−0.826901
$581$	−3.11776e17	−0.336278
$582$	0	0
$583$	−2.51657e17	−0.265438
$584$	9.89717e15	0.0103235
$585$	0	0
$586$	2.68453e18	2.73863
$587$	9.27969e17	0.936237	0.468119	−	0.883666i	$-0.344932\pi$
$587$	9.27969e17	0.936237	0.468119	+	0.883666i	$0.344932\pi$
$588$	0	0
$589$	7.87997e17	0.777634
$590$	6.75811e16	0.0659610
$591$	0	0
$592$	2.33322e17	0.222774
$593$	−7.24111e16	−0.0683832	−0.0341916	−	0.999415i	$-0.510886\pi$
$593$	−7.24111e16	−0.0683832	−0.0341916	+	0.999415i	$0.510886\pi$
$594$	0	0
$595$	3.85550e17	0.356222
$596$	2.01868e18	1.84488
$597$	0	0
$598$	2.79881e16	0.0250274
$599$	−7.61377e17	−0.673481	−0.336740	−	0.941597i	$-0.609325\pi$
$599$	−7.61377e17	−0.673481	−0.336740	+	0.941597i	$0.609325\pi$
$600$	0	0
$601$	8.87128e17	0.767896	0.383948	−	0.923355i	$-0.374564\pi$
$601$	8.87128e17	0.767896	0.383948	+	0.923355i	$0.374564\pi$
$602$	−5.88461e17	−0.503895
$603$	0	0
$604$	1.95925e17	0.164192
$605$	−3.47799e17	−0.288349
$606$	0	0
$607$	−7.33626e17	−0.595317	−0.297658	−	0.954672i	$-0.596206\pi$
$607$	−7.33626e17	−0.595317	−0.297658	+	0.954672i	$0.596206\pi$
$608$	−6.60968e17	−0.530649
$609$	0	0
$610$	−1.28021e18	−1.00609
$611$	4.08545e16	0.0317667
$612$	0	0
$613$	1.49204e18	1.13576	0.567880	−	0.823111i	$-0.307764\pi$
$613$	1.49204e18	1.13576	0.567880	+	0.823111i	$0.307764\pi$
$614$	−3.10171e18	−2.33618
$615$	0	0
$616$	6.31917e17	0.466000
$617$	1.83926e17	0.134211	0.0671057	−	0.997746i	$-0.478624\pi$
$617$	1.83926e17	0.134211	0.0671057	+	0.997746i	$0.478624\pi$
$618$	0	0
$619$	1.24283e16	0.00888019	0.00444009	−	0.999990i	$-0.498587\pi$
$619$	1.24283e16	0.00888019	0.00444009	+	0.999990i	$0.498587\pi$
$620$	1.34792e18	0.953058
$621$	0	0
$622$	−1.15892e18	−0.802445
$623$	−3.50719e16	−0.0240318
$624$	0	0
$625$	5.96046e16	0.0400000
$626$	−2.43117e18	−1.61466
$627$	0	0
$628$	−4.45248e18	−2.89644
$629$	2.42568e18	1.56173
$630$	0	0
$631$	−1.68478e18	−1.06256	−0.531279	−	0.847197i	$-0.678289\pi$
$631$	−1.68478e18	−1.06256	−0.531279	+	0.847197i	$0.678289\pi$
$632$	−2.84352e18	−1.77499
$633$	0	0
$634$	5.33714e17	0.326383
$635$	−1.28774e18	−0.779469
$636$	0	0
$637$	6.18327e16	0.0366701
$638$	−1.82245e18	−1.06985
$639$	0	0
$640$	−1.34005e18	−0.770815
$641$	−2.25951e18	−1.28658	−0.643290	−	0.765623i	$-0.722431\pi$
$641$	−2.25951e18	−1.28658	−0.643290	+	0.765623i	$0.722431\pi$
$642$	0	0
$643$	1.14956e18	0.641449	0.320725	−	0.947173i	$-0.396074\pi$
$643$	1.14956e18	0.641449	0.320725	+	0.947173i	$0.396074\pi$
$644$	−4.59111e17	−0.253606
$645$	0	0
$646$	2.14997e18	1.16391
$647$	2.91569e18	1.56266	0.781328	−	0.624121i	$-0.214543\pi$
$647$	2.91569e18	1.56266	0.781328	+	0.624121i	$0.214543\pi$
$648$	0	0
$649$	1.02048e17	0.0536062
$650$	−4.43471e16	−0.0230637
$651$	0	0
$652$	−2.68128e18	−1.36689
$653$	2.16676e18	1.09364	0.546821	−	0.837249i	$-0.315838\pi$
$653$	2.16676e18	1.09364	0.546821	+	0.837249i	$0.315838\pi$
$654$	0	0
$655$	−1.50234e18	−0.743363
$656$	6.52052e17	0.319453
$657$	0	0
$658$	−1.06690e18	−0.512455
$659$	4.11523e18	1.95722	0.978608	−	0.205732i	$-0.0659574\pi$
$659$	4.11523e18	1.95722	0.978608	+	0.205732i	$0.0659574\pi$
$660$	0	0
$661$	1.51002e17	0.0704162	0.0352081	−	0.999380i	$-0.488791\pi$
$661$	1.51002e17	0.0704162	0.0352081	+	0.999380i	$0.488791\pi$
$662$	−3.29194e18	−1.52011
$663$	0	0
$664$	−1.21341e18	−0.549434
$665$	4.25275e17	0.190690
$666$	0	0
$667$	5.40239e17	0.237557
$668$	−4.64657e18	−2.02342
$669$	0	0
$670$	2.29089e18	0.978404
$671$	−1.93313e18	−0.817646
$672$	0	0
$673$	−4.17178e18	−1.73071	−0.865354	−	0.501161i	$-0.832906\pi$
$673$	−4.17178e18	−1.73071	−0.865354	+	0.501161i	$0.832906\pi$
$674$	4.71133e18	1.93577
$675$	0	0
$676$	−4.17049e18	−1.68087
$677$	1.27061e18	0.507206	0.253603	−	0.967308i	$-0.418384\pi$
$677$	1.27061e18	0.507206	0.253603	+	0.967308i	$0.418384\pi$
$678$	0	0
$679$	7.72408e17	0.302478
$680$	1.50053e18	0.582020
$681$	0	0
$682$	3.24030e18	1.23307
$683$	−2.64932e18	−0.998619	−0.499310	−	0.866424i	$-0.666413\pi$
$683$	−2.64932e18	−0.998619	−0.499310	+	0.866424i	$0.666413\pi$
$684$	0	0
$685$	1.80869e18	0.668922
$686$	−4.71128e18	−1.72596
$687$	0	0
$688$	−2.02958e17	−0.0729593
$689$	8.79463e16	0.0313179
$690$	0	0
$691$	4.37627e18	1.52931	0.764657	−	0.644438i	$-0.222909\pi$
$691$	4.37627e18	1.52931	0.764657	+	0.644438i	$0.222909\pi$
$692$	−1.52390e18	−0.527554
$693$	0	0
$694$	−1.58738e18	−0.539319
$695$	1.38377e18	0.465760
$696$	0	0
$697$	6.77892e18	2.23948
$698$	−6.58545e18	−2.15539
$699$	0	0
$700$	7.27462e17	0.233708
$701$	−3.76541e18	−1.19852	−0.599260	−	0.800555i	$-0.704538\pi$
$701$	−3.76541e18	−1.19852	−0.599260	+	0.800555i	$0.704538\pi$
$702$	0	0
$703$	2.67561e18	0.836012
$704$	−3.03417e18	−0.939328
$705$	0	0
$706$	1.57186e18	0.477730
$707$	2.86264e18	0.862066
$708$	0	0
$709$	1.74388e18	0.515604	0.257802	−	0.966198i	$-0.417002\pi$
$709$	1.74388e18	0.515604	0.257802	+	0.966198i	$0.417002\pi$
$710$	3.08919e17	0.0905034
$711$	0	0
$712$	−1.36497e17	−0.0392648
$713$	−9.60541e17	−0.273800
$714$	0	0
$715$	−6.69647e16	−0.0187438
$716$	1.04249e19	2.89160
$717$	0	0
$718$	7.58048e17	0.206485
$719$	4.92987e18	1.33075	0.665377	−	0.746507i	$-0.268271\pi$
$719$	4.92987e18	1.33075	0.665377	+	0.746507i	$0.268271\pi$
$720$	0	0
$721$	−4.62207e18	−1.22534
$722$	−3.87024e18	−1.01683
$723$	0	0
$724$	−2.67724e18	−0.690853
$725$	−8.56010e17	−0.218918
$726$	0	0
$727$	5.68333e18	1.42767	0.713837	−	0.700312i	$-0.246956\pi$
$727$	5.68333e18	1.42767	0.713837	+	0.700312i	$0.246956\pi$
$728$	−2.20835e17	−0.0549813
$729$	0	0
$730$	2.73893e16	0.00669858
$731$	−2.11001e18	−0.511471
$732$	0	0
$733$	2.88616e18	0.687298	0.343649	−	0.939098i	$-0.388337\pi$
$733$	2.88616e18	0.687298	0.343649	+	0.939098i	$0.388337\pi$
$734$	8.15913e18	1.92584
$735$	0	0
$736$	8.05696e17	0.186838
$737$	3.45927e18	0.795145
$738$	0	0
$739$	−7.62132e18	−1.72124	−0.860620	−	0.509248i	$-0.829924\pi$
$739$	−7.62132e18	−1.72124	−0.860620	+	0.509248i	$0.829924\pi$
$740$	4.57681e18	1.02461
$741$	0	0
$742$	−2.29669e18	−0.505215
$743$	6.95681e18	1.51699	0.758495	−	0.651679i	$-0.225935\pi$
$743$	6.95681e18	1.51699	0.758495	+	0.651679i	$0.225935\pi$
$744$	0	0
$745$	2.27935e18	0.488422
$746$	6.83643e18	1.45220
$747$	0	0
$748$	5.55333e18	1.15929
$749$	−9.76647e17	−0.202118
$750$	0	0
$751$	3.81292e18	0.775528	0.387764	−	0.921759i	$-0.373248\pi$
$751$	3.81292e18	0.775528	0.387764	+	0.921759i	$0.373248\pi$
$752$	−3.67969e17	−0.0741986
$753$	0	0
$754$	6.36890e17	0.126227
$755$	2.21225e17	0.0434689
$756$	0	0
$757$	−7.07019e18	−1.36555	−0.682775	−	0.730629i	$-0.739227\pi$
$757$	−7.07019e18	−1.36555	−0.682775	+	0.730629i	$0.739227\pi$
$758$	2.02041e18	0.386892
$759$	0	0
$760$	1.65514e18	0.311563
$761$	−4.26743e18	−0.796465	−0.398232	−	0.917285i	$-0.630376\pi$
$761$	−4.26743e18	−0.796465	−0.398232	+	0.917285i	$0.630376\pi$
$762$	0	0
$763$	−5.94241e18	−1.09032
$764$	4.23469e18	0.770398
$765$	0	0
$766$	−1.69265e19	−3.02747
$767$	−3.56627e16	−0.00632476
$768$	0	0
$769$	−5.55109e16	−0.00967959	−0.00483980	−	0.999988i	$-0.501541\pi$
$769$	−5.55109e16	−0.00967959	−0.00483980	+	0.999988i	$0.501541\pi$
$770$	1.74876e18	0.302371
$771$	0	0
$772$	3.22190e18	0.547771
$773$	−6.47032e18	−1.09083	−0.545417	−	0.838165i	$-0.683629\pi$
$773$	−6.47032e18	−1.09083	−0.545417	+	0.838165i	$0.683629\pi$
$774$	0	0
$775$	1.52198e18	0.252318
$776$	3.00616e18	0.494209
$777$	0	0
$778$	8.22208e18	1.32927
$779$	7.47738e18	1.19882
$780$	0	0
$781$	4.66471e17	0.0735517
$782$	−2.62074e18	−0.409807
$783$	0	0
$784$	−5.56917e17	−0.0856517
$785$	−5.02742e18	−0.766819
$786$	0	0
$787$	−8.41309e18	−1.26218	−0.631088	−	0.775712i	$-0.717391\pi$
$787$	−8.41309e18	−1.26218	−0.631088	+	0.775712i	$0.717391\pi$
$788$	5.04301e18	0.750360
$789$	0	0
$790$	−7.86913e18	−1.15173
$791$	−7.73645e18	−1.12304
$792$	0	0
$793$	6.75570e17	0.0964705
$794$	−1.05762e19	−1.49795
$795$	0	0
$796$	−3.37448e18	−0.470188
$797$	7.81826e18	1.08052	0.540258	−	0.841499i	$-0.318327\pi$
$797$	7.81826e18	1.08052	0.540258	+	0.841499i	$0.318327\pi$
$798$	0	0
$799$	−3.82552e18	−0.520159
$800$	−1.27663e18	−0.172179
$801$	0	0
$802$	−1.86772e19	−2.47844
$803$	4.13582e16	0.00544390
$804$	0	0
$805$	−5.18395e17	−0.0671409
$806$	−1.13238e18	−0.145484
$807$	0	0
$808$	1.11412e19	1.40850
$809$	−1.29209e19	−1.62041	−0.810207	−	0.586144i	$-0.800645\pi$
$809$	−1.29209e19	−1.62041	−0.810207	+	0.586144i	$0.800645\pi$
$810$	0	0
$811$	−7.85766e18	−0.969745	−0.484872	−	0.874585i	$-0.661134\pi$
$811$	−7.85766e18	−0.969745	−0.484872	+	0.874585i	$0.661134\pi$
$812$	−1.04474e19	−1.27907
$813$	0	0
$814$	1.10023e19	1.32564
$815$	−3.02751e18	−0.361877
$816$	0	0
$817$	−2.32741e18	−0.273797
$818$	1.39857e19	1.63225
$819$	0	0
$820$	1.27906e19	1.46926
$821$	−3.45810e17	−0.0394101	−0.0197051	−	0.999806i	$-0.506273\pi$
$821$	−3.45810e17	−0.0394101	−0.0197051	+	0.999806i	$0.506273\pi$
$822$	0	0
$823$	8.61836e18	0.966776	0.483388	−	0.875406i	$-0.339406\pi$
$823$	8.61836e18	0.966776	0.483388	+	0.875406i	$0.339406\pi$
$824$	−1.79888e19	−2.00205
$825$	0	0
$826$	9.31319e17	0.102030
$827$	1.58749e19	1.72554	0.862768	−	0.505600i	$-0.168729\pi$
$827$	1.58749e19	1.72554	0.862768	+	0.505600i	$0.168729\pi$
$828$	0	0
$829$	6.99647e18	0.748643	0.374321	−	0.927299i	$-0.377876\pi$
$829$	6.99647e18	0.748643	0.374321	+	0.927299i	$0.377876\pi$
$830$	−3.35798e18	−0.356509
$831$	0	0
$832$	1.06035e18	0.110827
$833$	−5.78987e18	−0.600450
$834$	0	0
$835$	−5.24657e18	−0.535690
$836$	6.12551e18	0.620585
$837$	0	0
$838$	3.06064e19	3.05300
$839$	1.67940e18	0.166227	0.0831134	−	0.996540i	$-0.473514\pi$
$839$	1.67940e18	0.166227	0.0831134	+	0.996540i	$0.473514\pi$
$840$	0	0
$841$	2.03292e18	0.198128
$842$	1.26397e19	1.22238
$843$	0	0
$844$	2.57772e19	2.45476
$845$	−4.70902e18	−0.445002
$846$	0	0
$847$	−4.79293e18	−0.446025
$848$	−7.92117e17	−0.0731503
$849$	0	0
$850$	4.15256e18	0.377653
$851$	−3.26147e18	−0.294355
$852$	0	0
$853$	−1.88404e19	−1.67464	−0.837322	−	0.546711i	$-0.815880\pi$
$853$	−1.88404e19	−1.67464	−0.837322	+	0.546711i	$0.815880\pi$
$854$	−1.76423e19	−1.55625
$855$	0	0
$856$	−3.80104e18	−0.330235
$857$	−1.46107e19	−1.25978	−0.629892	−	0.776683i	$-0.716901\pi$
$857$	−1.46107e19	−1.25978	−0.629892	+	0.776683i	$0.716901\pi$
$858$	0	0
$859$	8.75747e17	0.0743743	0.0371872	−	0.999308i	$-0.488160\pi$
$859$	8.75747e17	0.0743743	0.0371872	+	0.999308i	$0.488160\pi$
$860$	−3.98119e18	−0.335562
$861$	0	0
$862$	−1.29109e19	−1.07192
$863$	1.80636e19	1.48845	0.744226	−	0.667928i	$-0.232819\pi$
$863$	1.80636e19	1.48845	0.744226	+	0.667928i	$0.232819\pi$
$864$	0	0
$865$	−1.72068e18	−0.139667
$866$	1.88238e19	1.51650
$867$	0	0
$868$	1.85754e19	1.47421
$869$	−1.18825e19	−0.936006
$870$	0	0
$871$	−1.20891e18	−0.0938156
$872$	−2.31274e19	−1.78144
$873$	0	0
$874$	−2.89076e18	−0.219375
$875$	8.21397e17	0.0618730
$876$	0	0
$877$	3.65611e18	0.271345	0.135673	−	0.990754i	$-0.456680\pi$
$877$	3.65611e18	0.271345	0.135673	+	0.990754i	$0.456680\pi$
$878$	7.14439e18	0.526321
$879$	0	0
$880$	6.03140e17	0.0437805
$881$	1.10129e18	0.0793521	0.0396761	−	0.999213i	$-0.487367\pi$
$881$	1.10129e18	0.0793521	0.0396761	+	0.999213i	$0.487367\pi$
$882$	0	0
$883$	8.49972e18	0.603476	0.301738	−	0.953391i	$-0.402433\pi$
$883$	8.49972e18	0.603476	0.301738	+	0.953391i	$0.402433\pi$
$884$	−1.94072e18	−0.136780
$885$	0	0
$886$	−2.37180e19	−1.64725
$887$	−2.42544e19	−1.67219	−0.836096	−	0.548583i	$-0.815168\pi$
$887$	−2.42544e19	−1.67219	−0.836096	+	0.548583i	$0.815168\pi$
$888$	0	0
$889$	−1.77460e19	−1.20570
$890$	−3.77741e17	−0.0254776
$891$	0	0
$892$	1.16813e19	0.776463
$893$	−4.21967e18	−0.278448
$894$	0	0
$895$	1.17711e19	0.765538
$896$	−1.84669e19	−1.19231
$897$	0	0
$898$	−1.31867e19	−0.839152
$899$	−2.18578e19	−1.38092
$900$	0	0
$901$	−8.23508e18	−0.512810
$902$	3.07475e19	1.90094
$903$	0	0
$904$	−3.01097e19	−1.83490
$905$	−3.02294e18	−0.182900
$906$	0	0
$907$	2.28334e19	1.36183	0.680915	−	0.732362i	$-0.261582\pi$
$907$	2.28334e19	1.36183	0.680915	+	0.732362i	$0.261582\pi$
$908$	1.08550e19	0.642792
$909$	0	0
$910$	−6.11137e17	−0.0356755
$911$	1.92508e19	1.11578	0.557890	−	0.829915i	$-0.311611\pi$
$911$	1.92508e19	1.11578	0.557890	+	0.829915i	$0.311611\pi$
$912$	0	0
$913$	−5.07059e18	−0.289733
$914$	1.75466e18	0.0995502
$915$	0	0
$916$	3.45979e18	0.193522
$917$	−2.07034e19	−1.14985
$918$	0	0
$919$	−2.09489e19	−1.14712	−0.573561	−	0.819163i	$-0.694438\pi$
$919$	−2.09489e19	−1.14712	−0.573561	+	0.819163i	$0.694438\pi$
$920$	−2.01756e18	−0.109699
$921$	0	0
$922$	−4.06863e19	−2.18121
$923$	−1.63017e17	−0.00867804
$924$	0	0
$925$	5.16780e18	0.271259
$926$	2.86621e19	1.49395
$927$	0	0
$928$	1.83342e19	0.942325
$929$	3.06906e18	0.156640	0.0783201	−	0.996928i	$-0.475044\pi$
$929$	3.06906e18	0.156640	0.0783201	+	0.996928i	$0.475044\pi$
$930$	0	0
$931$	−6.38642e18	−0.321429
$932$	3.90584e19	1.95214
$933$	0	0
$934$	−1.96237e19	−0.967224
$935$	6.27042e18	0.306917
$936$	0	0
$937$	1.98628e19	0.958809	0.479405	−	0.877594i	$-0.340853\pi$
$937$	1.98628e19	0.958809	0.479405	+	0.877594i	$0.340853\pi$
$938$	3.15702e19	1.51342
$939$	0	0
$940$	−7.21804e18	−0.341262
$941$	−3.52700e19	−1.65605	−0.828023	−	0.560694i	$-0.810534\pi$
$941$	−3.52700e19	−1.65605	−0.828023	+	0.560694i	$0.810534\pi$
$942$	0	0
$943$	−9.11465e18	−0.422099
$944$	3.21208e17	0.0147730
$945$	0	0
$946$	−9.57047e18	−0.434151
$947$	4.03654e19	1.81859	0.909295	−	0.416153i	$-0.136622\pi$
$947$	4.03654e19	1.81859	0.909295	+	0.416153i	$0.136622\pi$
$948$	0	0
$949$	−1.44534e16	−0.000642302 0
$950$	4.58041e18	0.202163
$951$	0	0
$952$	2.06785e19	0.900283
$953$	6.85871e18	0.296578	0.148289	−	0.988944i	$-0.452623\pi$
$953$	6.85871e18	0.296578	0.148289	+	0.988944i	$0.452623\pi$
$954$	0	0
$955$	4.78151e18	0.203959
$956$	4.42799e19	1.87599
$957$	0	0
$958$	−3.61683e19	−1.51165
$959$	2.49251e19	1.03470
$960$	0	0
$961$	1.44455e19	0.591603
$962$	−3.84496e18	−0.156406
$963$	0	0
$964$	1.59040e19	0.638270
$965$	3.63793e18	0.145020
$966$	0	0
$967$	4.77789e19	1.87916	0.939581	−	0.342328i	$-0.111215\pi$
$967$	4.77789e19	1.87916	0.939581	+	0.342328i	$0.111215\pi$
$968$	−1.86538e19	−0.728746
$969$	0	0
$970$	8.31921e18	0.320675
$971$	−5.51002e18	−0.210974	−0.105487	−	0.994421i	$-0.533640\pi$
$971$	−5.51002e18	−0.210974	−0.105487	+	0.994421i	$0.533640\pi$
$972$	0	0
$973$	1.90694e19	0.720448
$974$	5.69818e19	2.13847
$975$	0	0
$976$	−6.08474e18	−0.225330
$977$	3.66274e18	0.134738	0.0673691	−	0.997728i	$-0.478539\pi$
$977$	3.66274e18	0.134738	0.0673691	+	0.997728i	$0.478539\pi$
$978$	0	0
$979$	−5.70393e17	−0.0207055
$980$	−1.09244e19	−0.393939
$981$	0	0
$982$	6.79315e17	0.0241739
$983$	−1.33604e19	−0.472303	−0.236151	−	0.971716i	$-0.575886\pi$
$983$	−1.33604e19	−0.472303	−0.236151	+	0.971716i	$0.575886\pi$
$984$	0	0
$985$	5.69420e18	0.198654
$986$	−5.96369e19	−2.06688
$987$	0	0
$988$	−2.14067e18	−0.0732201
$989$	2.83703e18	0.0964023
$990$	0	0
$991$	3.12719e19	1.04876	0.524379	−	0.851485i	$-0.324297\pi$
$991$	3.12719e19	1.04876	0.524379	+	0.851485i	$0.324297\pi$
$992$	−3.25981e19	−1.08609
$993$	0	0
$994$	4.25713e18	0.139993
$995$	−3.81022e18	−0.124480
$996$	0	0
$997$	5.94349e18	0.191656	0.0958282	−	0.995398i	$-0.469450\pi$
$997$	5.94349e18	0.191656	0.0958282	+	0.995398i	$0.469450\pi$
$998$	−7.82973e19	−2.50841
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.14.a.g.1.4	✓	4
3.2	odd	2		45.14.a.h.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.14.a.g.1.4	✓	4	1.1	even	1	trivial
45.14.a.h.1.1	yes	4	3.2	odd	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:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q+148.425 q^{2} +13838.1 q^{4} +15625.0 q^{5} +215324. q^{7} +838027. q^{8} +2.31915e6 q^{10} +3.50194e6 q^{11} -1.22382e6 q^{13} +3.19596e7 q^{14} +1.10227e7 q^{16} +1.14596e8 q^{17} +1.26403e8 q^{19} +2.16221e8 q^{20} +5.19777e8 q^{22} -1.54080e8 q^{23} +2.44141e8 q^{25} -1.81646e8 q^{26} +2.97968e9 q^{28} -3.50622e9 q^{29} +6.23402e9 q^{31} -5.22907e9 q^{32} +1.70089e10 q^{34} +3.36444e9 q^{35} +2.11673e10 q^{37} +1.87614e10 q^{38} +1.30942e10 q^{40} +5.91552e10 q^{41} -1.84126e10 q^{43} +4.84603e10 q^{44} -2.28694e10 q^{46} -3.33828e10 q^{47} -5.05244e10 q^{49} +3.62367e10 q^{50} -1.69354e10 q^{52} -7.18621e10 q^{53} +5.47178e10 q^{55} +1.80448e11 q^{56} -5.20412e11 q^{58} +2.91405e10 q^{59} -5.52018e11 q^{61} +9.25288e11 q^{62} -8.66425e11 q^{64} -1.91222e10 q^{65} +9.87816e11 q^{67} +1.58579e12 q^{68} +4.99369e11 q^{70} +1.33203e11 q^{71} +1.18101e10 q^{73} +3.14177e12 q^{74} +1.74918e12 q^{76} +7.54053e11 q^{77} -3.39311e12 q^{79} +1.72230e11 q^{80} +8.78014e12 q^{82} -1.44794e12 q^{83} +1.79056e12 q^{85} -2.73290e12 q^{86} +2.93472e12 q^{88} -1.62879e11 q^{89} -2.63518e11 q^{91} -2.13218e12 q^{92} -4.95485e12 q^{94} +1.97504e12 q^{95} +3.58718e12 q^{97} -7.49911e12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 15 q^{2} + 16837 q^{4} + 62500 q^{5} + 343040 q^{7} - 14865 q^{8} - 234375 q^{10} + 12697800 q^{11} + 34336040 q^{13} + 26944650 q^{14} + 66562801 q^{16} + 84377280 q^{17} - 131821144 q^{19} + 263078125 q^{20}+ \cdots - 17650752985395 q^{98}+O(q^{100}) \) 4 * q - 15 * q^2 + 16837 * q^4 + 62500 * q^5 + 343040 * q^7 - 14865 * q^8 - 234375 * q^10 + 12697800 * q^11 + 34336040 * q^13 + 26944650 * q^14 + 66562801 * q^16 + 84377280 * q^17 - 131821144 * q^19 + 263078125 * q^20 - 1016682450 * q^22 + 1046106360 * q^23 + 976562500 * q^25 - 2998448100 * q^26 - 2229409030 * q^28 - 1826812200 * q^29 + 3501222872 * q^31 - 1278404505 * q^32 + 2712309690 * q^34 + 5360000000 * q^35 + 8297642120 * q^37 + 50666879340 * q^38 - 232265625 * q^40 + 54459122400 * q^41 - 48173425360 * q^43 + 141205041150 * q^44 - 131025090720 * q^46 + 202496682840 * q^47 + 116605824372 * q^49 - 3662109375 * q^50 + 183603958220 * q^52 + 116147122920 * q^53 + 198403125000 * q^55 + 985319673150 * q^56 - 209673904950 * q^58 + 885100702200 * q^59 + 467836185368 * q^61 + 1845870142380 * q^62 - 1316914666607 * q^64 + 536500625000 * q^65 + 819037749440 * q^67 + 4568941043490 * q^68 + 421010156250 * q^70 + 2886468559200 * q^71 - 1235535876280 * q^73 + 5480818576200 * q^74 - 1007297478532 * q^76 + 3590015832000 * q^77 - 3553486866616 * q^79 + 1040043765625 * q^80 + 9494817638400 * q^82 + 3441261762480 * q^83 + 1318395000000 * q^85 + 1418346623400 * q^86 + 3182744059050 * q^88 - 2322663606000 * q^89 - 4966358282000 * q^91 - 3469668771120 * q^92 - 19865804246880 * q^94 - 2059705375000 * q^95 + 8462989649240 * q^97 - 17650752985395 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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