LMFDB - Embedded newform 45.14.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,14,Mod(1,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	45.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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			1.2
Root			$-50.6908$ 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)

$f(q)$	$=$	\(q-54.6908 q^{2} -5200.92 q^{4} +15625.0 q^{5} +591583. q^{7} +732469. q^{8} -854543. q^{10} +6.17625e6 q^{11} +4.42594e6 q^{13} -3.23541e7 q^{14} +2.54662e6 q^{16} -1.47159e8 q^{17} -2.62355e7 q^{19} -8.12644e7 q^{20} -3.37784e8 q^{22} +1.17003e9 q^{23} +2.44141e8 q^{25} -2.42058e8 q^{26} -3.07677e9 q^{28} +1.53295e9 q^{29} -3.98217e9 q^{31} -6.13966e9 q^{32} +8.04822e9 q^{34} +9.24348e9 q^{35} -1.05503e10 q^{37} +1.43484e9 q^{38} +1.14448e10 q^{40} +3.72513e10 q^{41} -7.33761e10 q^{43} -3.21222e10 q^{44} -6.39897e10 q^{46} +4.86683e10 q^{47} +2.53081e11 q^{49} -1.33522e10 q^{50} -2.30189e10 q^{52} +8.04270e10 q^{53} +9.65039e10 q^{55} +4.33316e11 q^{56} -8.38382e10 q^{58} +5.93559e11 q^{59} +1.83725e11 q^{61} +2.17788e11 q^{62} +3.14921e11 q^{64} +6.91553e10 q^{65} -4.49402e11 q^{67} +7.65360e11 q^{68} -5.05533e11 q^{70} -1.20467e12 q^{71} +1.31386e12 q^{73} +5.77004e11 q^{74} +1.36449e11 q^{76} +3.65376e12 q^{77} -2.33024e12 q^{79} +3.97909e10 q^{80} -2.03730e12 q^{82} +4.35985e12 q^{83} -2.29935e12 q^{85} +4.01300e12 q^{86} +4.52391e12 q^{88} -2.38600e12 q^{89} +2.61831e12 q^{91} -6.08522e12 q^{92} -2.66171e12 q^{94} -4.09929e11 q^{95} -3.60565e12 q^{97} -1.38412e13 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−54.6908	−0.604253	−0.302127	−	0.953268i	$-0.597697\pi$
$2$	−54.6908	−0.604253	−0.302127	+	0.953268i	$0.597697\pi$
$3$	0	0
$3$	0	0
$4$	−5200.92	−0.634878
$5$	15625.0	0.447214
$5$	15625.0	0.447214
$6$	0	0
$7$	591583.	1.90055	0.950273	−	0.311419i	$-0.100804\pi$
$7$	591583.	1.90055	0.950273	+	0.311419i	$0.100804\pi$
$8$	732469.	0.987881
$9$	0	0
$10$	−854543.	−0.270230
$11$	6.17625e6	1.05117	0.525584	−	0.850742i	$-0.323847\pi$
$11$	6.17625e6	1.05117	0.525584	+	0.850742i	$0.323847\pi$
$12$	0	0
$13$	4.42594e6	0.254316	0.127158	−	0.991882i	$-0.459415\pi$
$13$	4.42594e6	0.254316	0.127158	+	0.991882i	$0.459415\pi$
$14$	−3.23541e7	−1.14841
$15$	0	0
$16$	2.54662e6	0.0379476
$17$	−1.47159e8	−1.47866	−0.739329	−	0.673344i	$-0.764857\pi$
$17$	−1.47159e8	−1.47866	−0.739329	+	0.673344i	$0.764857\pi$
$18$	0	0
$19$	−2.62355e7	−0.127935	−0.0639677	−	0.997952i	$-0.520375\pi$
$19$	−2.62355e7	−0.127935	−0.0639677	+	0.997952i	$0.520375\pi$
$20$	−8.12644e7	−0.283926
$21$	0	0
$22$	−3.37784e8	−0.635172
$23$	1.17003e9	1.64803	0.824015	−	0.566568i	$-0.191729\pi$
$23$	1.17003e9	1.64803	0.824015	+	0.566568i	$0.191729\pi$
$24$	0	0
$25$	2.44141e8	0.200000
$26$	−2.42058e8	−0.153671
$27$	0	0
$28$	−3.07677e9	−1.20661
$29$	1.53295e9	0.478565	0.239282	−	0.970950i	$-0.423088\pi$
$29$	1.53295e9	0.478565	0.239282	+	0.970950i	$0.423088\pi$
$30$	0	0
$31$	−3.98217e9	−0.805876	−0.402938	−	0.915227i	$-0.632011\pi$
$31$	−3.98217e9	−0.805876	−0.402938	+	0.915227i	$0.632011\pi$
$32$	−6.13966e9	−1.01081
$33$	0	0
$34$	8.04822e9	0.893484
$35$	9.24348e9	0.849950
$36$	0	0
$37$	−1.05503e10	−0.676011	−0.338006	−	0.941144i	$-0.609752\pi$
$37$	−1.05503e10	−0.676011	−0.338006	+	0.941144i	$0.609752\pi$
$38$	1.43484e9	0.0773054
$39$	0	0
$40$	1.14448e10	0.441794
$41$	3.72513e10	1.22475	0.612374	−	0.790568i	$-0.290215\pi$
$41$	3.72513e10	1.22475	0.612374	+	0.790568i	$0.290215\pi$
$42$	0	0
$43$	−7.33761e10	−1.77015	−0.885074	−	0.465450i	$-0.845893\pi$
$43$	−7.33761e10	−1.77015	−0.885074	+	0.465450i	$0.845893\pi$
$44$	−3.21222e10	−0.667364
$45$	0	0
$46$	−6.39897e10	−0.995828
$47$	4.86683e10	0.658583	0.329291	−	0.944228i	$-0.393190\pi$
$47$	4.86683e10	0.658583	0.329291	+	0.944228i	$0.393190\pi$
$48$	0	0
$49$	2.53081e11	2.61207
$50$	−1.33522e10	−0.120851
$51$	0	0
$52$	−2.30189e10	−0.161459
$53$	8.04270e10	0.498435	0.249218	−	0.968448i	$-0.419827\pi$
$53$	8.04270e10	0.498435	0.249218	+	0.968448i	$0.419827\pi$
$54$	0	0
$55$	9.65039e10	0.470097
$56$	4.33316e11	1.87751
$57$	0	0
$58$	−8.38382e10	−0.289174
$59$	5.93559e11	1.83200	0.916001	−	0.401177i	$-0.131399\pi$
$59$	5.93559e11	1.83200	0.916001	+	0.401177i	$0.131399\pi$
$60$	0	0
$61$	1.83725e11	0.456589	0.228294	−	0.973592i	$-0.426685\pi$
$61$	1.83725e11	0.456589	0.228294	+	0.973592i	$0.426685\pi$
$62$	2.17788e11	0.486954
$63$	0	0
$64$	3.14921e11	0.572838
$65$	6.91553e10	0.113733
$66$	0	0
$67$	−4.49402e11	−0.606944	−0.303472	−	0.952840i	$-0.598146\pi$
$67$	−4.49402e11	−0.606944	−0.303472	+	0.952840i	$0.598146\pi$
$68$	7.65360e11	0.938767
$69$	0	0
$70$	−5.05533e11	−0.513585
$71$	−1.20467e12	−1.11607	−0.558033	−	0.829819i	$-0.688444\pi$
$71$	−1.20467e12	−1.11607	−0.558033	+	0.829819i	$0.688444\pi$
$72$	0	0
$73$	1.31386e12	1.01613	0.508066	−	0.861318i	$-0.330361\pi$
$73$	1.31386e12	1.01613	0.508066	+	0.861318i	$0.330361\pi$
$74$	5.77004e11	0.408482
$75$	0	0
$76$	1.36449e11	0.0812233
$77$	3.65376e12	1.99779
$78$	0	0
$79$	−2.33024e12	−1.07851	−0.539255	−	0.842143i	$-0.681294\pi$
$79$	−2.33024e12	−1.07851	−0.539255	+	0.842143i	$0.681294\pi$
$80$	3.97909e10	0.0169707
$81$	0	0
$82$	−2.03730e12	−0.740058
$83$	4.35985e12	1.46374	0.731870	−	0.681444i	$-0.238648\pi$
$83$	4.35985e12	1.46374	0.731870	+	0.681444i	$0.238648\pi$
$84$	0	0
$85$	−2.29935e12	−0.661276
$86$	4.01300e12	1.06962
$87$	0	0
$88$	4.52391e12	1.03843
$89$	−2.38600e12	−0.508904	−0.254452	−	0.967085i	$-0.581895\pi$
$89$	−2.38600e12	−0.508904	−0.254452	+	0.967085i	$0.581895\pi$
$90$	0	0
$91$	2.61831e12	0.483339
$92$	−6.08522e12	−1.04630
$93$	0	0
$94$	−2.66171e12	−0.397951
$95$	−4.09929e11	−0.0572145
$96$	0	0
$97$	−3.60565e12	−0.439509	−0.219754	−	0.975555i	$-0.570526\pi$
$97$	−3.60565e12	−0.439509	−0.219754	+	0.975555i	$0.570526\pi$
$98$	−1.38412e13	−1.57835
$99$	0	0
$100$	−1.26976e12	−0.126976
$101$	1.31398e12	0.123168	0.0615842	−	0.998102i	$-0.480385\pi$
$101$	1.31398e12	0.123168	0.0615842	+	0.998102i	$0.480385\pi$
$102$	0	0
$103$	−3.88674e11	−0.0320733	−0.0160366	−	0.999871i	$-0.505105\pi$
$103$	−3.88674e11	−0.0320733	−0.0160366	+	0.999871i	$0.505105\pi$
$104$	3.24186e12	0.251234
$105$	0	0
$106$	−4.39861e12	−0.301181
$107$	−1.22080e13	−0.786409	−0.393205	−	0.919451i	$-0.628634\pi$
$107$	−1.22080e13	−0.786409	−0.393205	+	0.919451i	$0.628634\pi$
$108$	0	0
$109$	3.26020e13	1.86197	0.930984	−	0.365059i	$-0.118951\pi$
$109$	3.26020e13	1.86197	0.930984	+	0.365059i	$0.118951\pi$
$110$	−5.27787e12	−0.284058
$111$	0	0
$112$	1.50654e12	0.0721211
$113$	3.76809e12	0.170259	0.0851296	−	0.996370i	$-0.472870\pi$
$113$	3.76809e12	0.170259	0.0851296	+	0.996370i	$0.472870\pi$
$114$	0	0
$115$	1.82817e13	0.737021
$116$	−7.97274e12	−0.303830
$117$	0	0
$118$	−3.24622e13	−1.10699
$119$	−8.70565e13	−2.81026
$120$	0	0
$121$	3.62335e12	0.104955
$122$	−1.00481e13	−0.275895
$123$	0	0
$124$	2.07109e13	0.511633
$125$	3.81470e12	0.0894427
$126$	0	0
$127$	2.97915e12	0.0630039	0.0315020	−	0.999504i	$-0.489971\pi$
$127$	2.97915e12	0.0630039	0.0315020	+	0.999504i	$0.489971\pi$
$128$	3.30728e13	0.664671
$129$	0	0
$130$	−3.78215e12	−0.0687239
$131$	−4.09882e13	−0.708591	−0.354296	−	0.935133i	$-0.615279\pi$
$131$	−4.09882e13	−0.708591	−0.354296	+	0.935133i	$0.615279\pi$
$132$	0	0
$133$	−1.55205e13	−0.243147
$134$	2.45781e13	0.366748
$135$	0	0
$136$	−1.07789e14	−1.46074
$137$	5.07093e13	0.655245	0.327623	−	0.944809i	$-0.393753\pi$
$137$	5.07093e13	0.655245	0.327623	+	0.944809i	$0.393753\pi$
$138$	0	0
$139$	−1.03534e14	−1.21755	−0.608773	−	0.793345i	$-0.708338\pi$
$139$	−1.03534e14	−1.21755	−0.608773	+	0.793345i	$0.708338\pi$
$140$	−4.80746e13	−0.539614
$141$	0	0
$142$	6.58845e13	0.674387
$143$	2.73357e13	0.267329
$144$	0	0
$145$	2.39523e13	0.214021
$146$	−7.18559e13	−0.614001
$147$	0	0
$148$	5.48713e13	0.429184
$149$	3.40713e13	0.255081	0.127541	−	0.991833i	$-0.459292\pi$
$149$	3.40713e13	0.255081	0.127541	+	0.991833i	$0.459292\pi$
$150$	0	0
$151$	2.30769e14	1.58426	0.792132	−	0.610349i	$-0.208971\pi$
$151$	2.30769e14	1.58426	0.792132	+	0.610349i	$0.208971\pi$
$152$	−1.92167e13	−0.126385
$153$	0	0
$154$	−1.99827e14	−1.20717
$155$	−6.22213e13	−0.360399
$156$	0	0
$157$	−3.76735e13	−0.200765	−0.100383	−	0.994949i	$-0.532007\pi$
$157$	−3.76735e13	−0.200765	−0.100383	+	0.994949i	$0.532007\pi$
$158$	1.27442e14	0.651693
$159$	0	0
$160$	−9.59322e13	−0.452048
$161$	6.92168e14	3.13216
$162$	0	0
$163$	3.14661e14	1.31408	0.657042	−	0.753854i	$-0.271808\pi$
$163$	3.14661e14	1.31408	0.657042	+	0.753854i	$0.271808\pi$
$164$	−1.93741e14	−0.777565
$165$	0	0
$166$	−2.38444e14	−0.884470
$167$	3.43206e14	1.22433	0.612165	−	0.790730i	$-0.290299\pi$
$167$	3.43206e14	1.22433	0.612165	+	0.790730i	$0.290299\pi$
$168$	0	0
$169$	−2.83286e14	−0.935323
$170$	1.25753e14	0.399578
$171$	0	0
$172$	3.81623e14	1.12383
$173$	6.52149e14	1.84947	0.924735	−	0.380613i	$-0.124287\pi$
$173$	6.52149e14	1.84947	0.924735	+	0.380613i	$0.124287\pi$
$174$	0	0
$175$	1.44429e14	0.380109
$176$	1.57286e13	0.0398893
$177$	0	0
$178$	1.30492e14	0.307507
$179$	−4.63473e14	−1.05312	−0.526562	−	0.850137i	$-0.676519\pi$
$179$	−4.63473e14	−1.05312	−0.526562	+	0.850137i	$0.676519\pi$
$180$	0	0
$181$	3.15366e14	0.666660	0.333330	−	0.942810i	$-0.391828\pi$
$181$	3.15366e14	0.666660	0.333330	+	0.942810i	$0.391828\pi$
$182$	−1.43197e14	−0.292059
$183$	0	0
$184$	8.57009e14	1.62806
$185$	−1.64849e14	−0.302321
$186$	0	0
$187$	−9.08889e14	−1.55432
$188$	−2.53120e14	−0.418120
$189$	0	0
$190$	2.24194e13	0.0345720
$191$	−4.14815e14	−0.618212	−0.309106	−	0.951028i	$-0.600030\pi$
$191$	−4.14815e14	−0.618212	−0.309106	+	0.951028i	$0.600030\pi$
$192$	0	0
$193$	−5.12018e14	−0.713121	−0.356560	−	0.934272i	$-0.616051\pi$
$193$	−5.12018e14	−0.713121	−0.356560	+	0.934272i	$0.616051\pi$
$194$	1.97196e14	0.265575
$195$	0	0
$196$	−1.31625e15	−1.65835
$197$	−3.78535e14	−0.461398	−0.230699	−	0.973025i	$-0.574101\pi$
$197$	−3.78535e14	−0.461398	−0.230699	+	0.973025i	$0.574101\pi$
$198$	0	0
$199$	8.18109e14	0.933827	0.466913	−	0.884303i	$-0.345366\pi$
$199$	8.18109e14	0.933827	0.466913	+	0.884303i	$0.345366\pi$
$200$	1.78825e14	0.197576
$201$	0	0
$202$	−7.18625e13	−0.0744249
$203$	9.06866e14	0.909534
$204$	0	0
$205$	5.82052e14	0.547724
$206$	2.12569e13	0.0193804
$207$	0	0
$208$	1.12712e13	0.00965068
$209$	−1.62037e14	−0.134482
$210$	0	0
$211$	−6.74377e14	−0.526098	−0.263049	−	0.964782i	$-0.584728\pi$
$211$	−6.74377e14	−0.526098	−0.263049	+	0.964782i	$0.584728\pi$
$212$	−4.18294e14	−0.316445
$213$	0	0
$214$	6.67663e14	0.475191
$215$	−1.14650e15	−0.791635
$216$	0	0
$217$	−2.35578e15	−1.53160
$218$	−1.78303e15	−1.12510
$219$	0	0
$220$	−5.01909e14	−0.298454
$221$	−6.51315e14	−0.376046
$222$	0	0
$223$	1.87655e15	1.02183	0.510915	−	0.859632i	$-0.329307\pi$
$223$	1.87655e15	1.02183	0.510915	+	0.859632i	$0.329307\pi$
$224$	−3.63212e15	−1.92109
$225$	0	0
$226$	−2.06080e14	−0.102880
$227$	1.31763e15	0.639182	0.319591	−	0.947556i	$-0.396455\pi$
$227$	1.31763e15	0.639182	0.319591	+	0.947556i	$0.396455\pi$
$228$	0	0
$229$	−2.17538e15	−0.996794	−0.498397	−	0.866949i	$-0.666078\pi$
$229$	−2.17538e15	−0.996794	−0.498397	+	0.866949i	$0.666078\pi$
$230$	−9.99839e14	−0.445348
$231$	0	0
$232$	1.12284e15	0.472765
$233$	−7.79147e14	−0.319011	−0.159506	−	0.987197i	$-0.550990\pi$
$233$	−7.79147e14	−0.319011	−0.159506	+	0.987197i	$0.550990\pi$
$234$	0	0
$235$	7.60443e14	0.294527
$236$	−3.08705e15	−1.16310
$237$	0	0
$238$	4.76119e15	1.69811
$239$	3.17607e15	1.10231	0.551154	−	0.834403i	$-0.314188\pi$
$239$	3.17607e15	1.10231	0.551154	+	0.834403i	$0.314188\pi$
$240$	0	0
$241$	4.61354e14	0.151678	0.0758392	−	0.997120i	$-0.475836\pi$
$241$	4.61354e14	0.151678	0.0758392	+	0.997120i	$0.475836\pi$
$242$	−1.98164e14	−0.0634197
$243$	0	0
$244$	−9.55540e14	−0.289878
$245$	3.95439e15	1.16815
$246$	0	0
$247$	−1.16117e14	−0.0325360
$248$	−2.91681e15	−0.796109
$249$	0	0
$250$	−2.08629e14	−0.0540461
$251$	4.47854e15	1.13047	0.565233	−	0.824932i	$-0.308786\pi$
$251$	4.47854e15	1.13047	0.565233	+	0.824932i	$0.308786\pi$
$252$	0	0
$253$	7.22638e15	1.73236
$254$	−1.62932e14	−0.0380703
$255$	0	0
$256$	−4.38861e15	−0.974468
$257$	−6.05693e15	−1.31126	−0.655628	−	0.755084i	$-0.727596\pi$
$257$	−6.05693e15	−1.31126	−0.655628	+	0.755084i	$0.727596\pi$
$258$	0	0
$259$	−6.24138e15	−1.28479
$260$	−3.59671e14	−0.0722069
$261$	0	0
$262$	2.24168e15	0.428169
$263$	5.60011e15	1.04348	0.521740	−	0.853104i	$-0.325283\pi$
$263$	5.60011e15	1.04348	0.521740	+	0.853104i	$0.325283\pi$
$264$	0	0
$265$	1.25667e15	0.222907
$266$	8.48826e14	0.146922
$267$	0	0
$268$	2.33730e15	0.385335
$269$	8.57287e15	1.37955	0.689773	−	0.724025i	$-0.257710\pi$
$269$	8.57287e15	1.37955	0.689773	+	0.724025i	$0.257710\pi$
$270$	0	0
$271$	−7.28893e15	−1.11780	−0.558899	−	0.829236i	$-0.688776\pi$
$271$	−7.28893e15	−1.11780	−0.558899	+	0.829236i	$0.688776\pi$
$272$	−3.74757e14	−0.0561115
$273$	0	0
$274$	−2.77333e15	−0.395934
$275$	1.50787e15	0.210234
$276$	0	0
$277$	−4.91465e15	−0.653693	−0.326846	−	0.945077i	$-0.605986\pi$
$277$	−4.91465e15	−0.653693	−0.326846	+	0.945077i	$0.605986\pi$
$278$	5.66233e15	0.735706
$279$	0	0
$280$	6.77056e15	0.839649
$281$	4.38583e15	0.531448	0.265724	−	0.964049i	$-0.414389\pi$
$281$	4.38583e15	0.531448	0.265724	+	0.964049i	$0.414389\pi$
$282$	0	0
$283$	1.03886e15	0.120211	0.0601055	−	0.998192i	$-0.480856\pi$
$283$	1.03886e15	0.120211	0.0601055	+	0.998192i	$0.480856\pi$
$284$	6.26540e15	0.708565
$285$	0	0
$286$	−1.49501e15	−0.161534
$287$	2.20372e16	2.32769
$288$	0	0
$289$	1.17511e16	1.18643
$290$	−1.30997e15	−0.129323
$291$	0	0
$292$	−6.83327e15	−0.645119
$293$	−1.27273e16	−1.17516	−0.587582	−	0.809165i	$-0.699920\pi$
$293$	−1.27273e16	−1.17516	−0.587582	+	0.809165i	$0.699920\pi$
$294$	0	0
$295$	9.27436e15	0.819296
$296$	−7.72777e15	−0.667818
$297$	0	0
$298$	−1.86339e15	−0.154134
$299$	5.17847e15	0.419120
$300$	0	0
$301$	−4.34080e16	−3.36425
$302$	−1.26210e16	−0.957298
$303$	0	0
$304$	−6.68118e13	−0.00485484
$305$	2.87071e15	0.204193
$306$	0	0
$307$	6.03206e15	0.411213	0.205606	−	0.978635i	$-0.434083\pi$
$307$	6.03206e15	0.411213	0.205606	+	0.978635i	$0.434083\pi$
$308$	−1.90029e16	−1.26835
$309$	0	0
$310$	3.40293e15	0.217772
$311$	6.95161e14	0.0435655	0.0217828	−	0.999763i	$-0.493066\pi$
$311$	6.95161e14	0.0435655	0.0217828	+	0.999763i	$0.493066\pi$
$312$	0	0
$313$	2.87272e16	1.72685	0.863426	−	0.504475i	$-0.168314\pi$
$313$	2.87272e16	1.72685	0.863426	+	0.504475i	$0.168314\pi$
$314$	2.06039e15	0.121313
$315$	0	0
$316$	1.21194e16	0.684722
$317$	1.60837e16	0.890226	0.445113	−	0.895474i	$-0.353163\pi$
$317$	1.60837e16	0.890226	0.445113	+	0.895474i	$0.353163\pi$
$318$	0	0
$319$	9.46788e15	0.503052
$320$	4.92064e15	0.256181
$321$	0	0
$322$	−3.78552e16	−1.89262
$323$	3.86078e15	0.189173
$324$	0	0
$325$	1.08055e15	0.0508632
$326$	−1.72090e16	−0.794039
$327$	0	0
$328$	2.72855e16	1.20990
$329$	2.87913e16	1.25167
$330$	0	0
$331$	−8.72833e15	−0.364795	−0.182398	−	0.983225i	$-0.558386\pi$
$331$	−8.72833e15	−0.364795	−0.182398	+	0.983225i	$0.558386\pi$
$332$	−2.26752e16	−0.929296
$333$	0	0
$334$	−1.87702e16	−0.739805
$335$	−7.02191e15	−0.271434
$336$	0	0
$337$	2.46150e16	0.915389	0.457695	−	0.889109i	$-0.348675\pi$
$337$	2.46150e16	0.915389	0.457695	+	0.889109i	$0.348675\pi$
$338$	1.54931e16	0.565172
$339$	0	0
$340$	1.19588e16	0.419830
$341$	−2.45949e16	−0.847112
$342$	0	0
$343$	9.24006e16	3.06382
$344$	−5.37457e16	−1.74870
$345$	0	0
$346$	−3.56665e16	−1.11755
$347$	−4.78204e16	−1.47052	−0.735261	−	0.677784i	$-0.762940\pi$
$347$	−4.78204e16	−1.47052	−0.735261	+	0.677784i	$0.762940\pi$
$348$	0	0
$349$	−2.28390e16	−0.676569	−0.338284	−	0.941044i	$-0.609847\pi$
$349$	−2.28390e16	−0.676569	−0.338284	+	0.941044i	$0.609847\pi$
$350$	−7.89896e15	−0.229682
$351$	0	0
$352$	−3.79201e16	−1.06253
$353$	−2.51503e16	−0.691844	−0.345922	−	0.938263i	$-0.612434\pi$
$353$	−2.51503e16	−0.691844	−0.345922	+	0.938263i	$0.612434\pi$
$354$	0	0
$355$	−1.88230e16	−0.499120
$356$	1.24094e16	0.323092
$357$	0	0
$358$	2.53477e16	0.636354
$359$	−6.05373e16	−1.49248	−0.746241	−	0.665676i	$-0.768143\pi$
$359$	−6.05373e16	−1.49248	−0.746241	+	0.665676i	$0.768143\pi$
$360$	0	0
$361$	−4.13647e16	−0.983633
$362$	−1.72476e16	−0.402832
$363$	0	0
$364$	−1.36176e16	−0.306861
$365$	2.05290e16	0.454428
$366$	0	0
$367$	−7.91650e16	−1.69124	−0.845618	−	0.533789i	$-0.820768\pi$
$367$	−7.91650e16	−1.69124	−0.845618	+	0.533789i	$0.820768\pi$
$368$	2.97961e15	0.0625388
$369$	0	0
$370$	9.01569e15	0.182679
$371$	4.75792e16	0.947299
$372$	0	0
$373$	5.10378e16	0.981261	0.490630	−	0.871368i	$-0.336767\pi$
$373$	5.10378e16	0.981261	0.490630	+	0.871368i	$0.336767\pi$
$374$	4.97078e16	0.939203
$375$	0	0
$376$	3.56480e16	0.650601
$377$	6.78474e15	0.121707
$378$	0	0
$379$	−1.52501e16	−0.264313	−0.132156	−	0.991229i	$-0.542190\pi$
$379$	−1.52501e16	−0.264313	−0.132156	+	0.991229i	$0.542190\pi$
$380$	2.13201e15	0.0363242
$381$	0	0
$382$	2.26865e16	0.373557
$383$	−1.28046e16	−0.207288	−0.103644	−	0.994614i	$-0.533050\pi$
$383$	−1.28046e16	−0.207288	−0.103644	+	0.994614i	$0.533050\pi$
$384$	0	0
$385$	5.70900e16	0.893441
$386$	2.80027e16	0.430906
$387$	0	0
$388$	1.87527e16	0.279034
$389$	5.63032e15	0.0823874	0.0411937	−	0.999151i	$-0.486884\pi$
$389$	5.63032e15	0.0823874	0.0411937	+	0.999151i	$0.486884\pi$
$390$	0	0
$391$	−1.72180e17	−2.43687
$392$	1.85374e17	2.58042
$393$	0	0
$394$	2.07024e16	0.278801
$395$	−3.64100e16	−0.482324
$396$	0	0
$397$	−2.10209e16	−0.269472	−0.134736	−	0.990882i	$-0.543019\pi$
$397$	−2.10209e16	−0.269472	−0.134736	+	0.990882i	$0.543019\pi$
$398$	−4.47430e16	−0.564268
$399$	0	0
$400$	6.21733e14	0.00758952
$401$	−3.20257e16	−0.384646	−0.192323	−	0.981332i	$-0.561602\pi$
$401$	−3.20257e16	−0.384646	−0.192323	+	0.981332i	$0.561602\pi$
$402$	0	0
$403$	−1.76248e16	−0.204947
$404$	−6.83390e15	−0.0781969
$405$	0	0
$406$	−4.95972e16	−0.549589
$407$	−6.51613e16	−0.710602
$408$	0	0
$409$	1.37925e17	1.45694	0.728470	−	0.685078i	$-0.240232\pi$
$409$	1.37925e17	1.45694	0.728470	+	0.685078i	$0.240232\pi$
$410$	−3.18329e16	−0.330964
$411$	0	0
$412$	2.02146e15	0.0203626
$413$	3.51139e17	3.48180
$414$	0	0
$415$	6.81227e16	0.654605
$416$	−2.71738e16	−0.257065
$417$	0	0
$418$	8.86193e15	0.0812610
$419$	−7.17187e16	−0.647502	−0.323751	−	0.946142i	$-0.604944\pi$
$419$	−7.17187e16	−0.647502	−0.323751	+	0.946142i	$0.604944\pi$
$420$	0	0
$421$	4.80799e16	0.420852	0.210426	−	0.977610i	$-0.432515\pi$
$421$	4.80799e16	0.420852	0.210426	+	0.977610i	$0.432515\pi$
$422$	3.68822e16	0.317897
$423$	0	0
$424$	5.89103e16	0.492394
$425$	−3.59274e16	−0.295732
$426$	0	0
$427$	1.08689e17	0.867767
$428$	6.34926e16	0.499274
$429$	0	0
$430$	6.27031e16	0.478348
$431$	2.26631e17	1.70301	0.851504	−	0.524347i	$-0.175691\pi$
$431$	2.26631e17	1.70301	0.851504	+	0.524347i	$0.175691\pi$
$432$	0	0
$433$	−5.46016e16	−0.398138	−0.199069	−	0.979985i	$-0.563792\pi$
$433$	−5.46016e16	−0.398138	−0.199069	+	0.979985i	$0.563792\pi$
$434$	1.28839e17	0.925477
$435$	0	0
$436$	−1.69560e17	−1.18212
$437$	−3.06962e16	−0.210841
$438$	0	0
$439$	−2.44779e17	−1.63213	−0.816065	−	0.577960i	$-0.803849\pi$
$439$	−2.44779e17	−1.63213	−0.816065	+	0.577960i	$0.803849\pi$
$440$	7.06861e16	0.464400
$441$	0	0
$442$	3.56209e16	0.227227
$443$	−1.34073e17	−0.842787	−0.421393	−	0.906878i	$-0.638459\pi$
$443$	−1.34073e17	−0.842787	−0.421393	+	0.906878i	$0.638459\pi$
$444$	0	0
$445$	−3.72813e16	−0.227589
$446$	−1.02630e17	−0.617444
$447$	0	0
$448$	1.86302e17	1.08870
$449$	−1.97634e17	−1.13831	−0.569155	−	0.822230i	$-0.692730\pi$
$449$	−1.97634e17	−1.13831	−0.569155	+	0.822230i	$0.692730\pi$
$450$	0	0
$451$	2.30074e17	1.28742
$452$	−1.95975e16	−0.108094
$453$	0	0
$454$	−7.20620e16	−0.386228
$455$	4.09111e16	0.216156
$456$	0	0
$457$	−2.37686e17	−1.22053	−0.610266	−	0.792197i	$-0.708937\pi$
$457$	−2.37686e17	−1.22053	−0.610266	+	0.792197i	$0.708937\pi$
$458$	1.18973e17	0.602316
$459$	0	0
$460$	−9.50815e16	−0.467918
$461$	−2.46272e17	−1.19498	−0.597488	−	0.801878i	$-0.703835\pi$
$461$	−2.46272e17	−1.19498	−0.597488	+	0.801878i	$0.703835\pi$
$462$	0	0
$463$	1.86589e17	0.880259	0.440130	−	0.897934i	$-0.354932\pi$
$463$	1.86589e17	0.880259	0.440130	+	0.897934i	$0.354932\pi$
$464$	3.90384e15	0.0181604
$465$	0	0
$466$	4.26121e16	0.192763
$467$	7.37264e16	0.328899	0.164450	−	0.986385i	$-0.447415\pi$
$467$	7.37264e16	0.328899	0.164450	+	0.986385i	$0.447415\pi$
$468$	0	0
$469$	−2.65858e17	−1.15353
$470$	−4.15892e16	−0.177969
$471$	0	0
$472$	4.34764e17	1.80980
$473$	−4.53189e17	−1.86072
$474$	0	0
$475$	−6.40515e15	−0.0255871
$476$	4.52774e17	1.78417
$477$	0	0
$478$	−1.73701e17	−0.666074
$479$	−2.15386e17	−0.814775	−0.407387	−	0.913255i	$-0.633560\pi$
$479$	−2.15386e17	−0.814775	−0.407387	+	0.913255i	$0.633560\pi$
$480$	0	0
$481$	−4.66950e16	−0.171920
$482$	−2.52318e16	−0.0916521
$483$	0	0
$484$	−1.88447e16	−0.0666339
$485$	−5.63383e16	−0.196554
$486$	0	0
$487$	−2.75094e17	−0.934421	−0.467211	−	0.884146i	$-0.654741\pi$
$487$	−2.75094e17	−0.934421	−0.467211	+	0.884146i	$0.654741\pi$
$488$	1.34573e17	0.451055
$489$	0	0
$490$	−2.16269e17	−0.705861
$491$	−3.15362e16	−0.101573	−0.0507867	−	0.998710i	$-0.516173\pi$
$491$	−3.15362e16	−0.101573	−0.0507867	+	0.998710i	$0.516173\pi$
$492$	0	0
$493$	−2.25587e17	−0.707634
$494$	6.35051e15	0.0196600
$495$	0	0
$496$	−1.01411e16	−0.0305811
$497$	−7.12664e17	−2.12113
$498$	0	0
$499$	1.08836e17	0.315586	0.157793	−	0.987472i	$-0.449562\pi$
$499$	1.08836e17	0.315586	0.157793	+	0.987472i	$0.449562\pi$
$500$	−1.98399e16	−0.0567852
$501$	0	0
$502$	−2.44935e17	−0.683088
$503$	−5.29080e17	−1.45656	−0.728282	−	0.685277i	$-0.759681\pi$
$503$	−5.29080e17	−1.45656	−0.728282	+	0.685277i	$0.759681\pi$
$504$	0	0
$505$	2.05309e16	0.0550826
$506$	−3.95216e17	−1.04678
$507$	0	0
$508$	−1.54943e16	−0.0399998
$509$	−1.52200e17	−0.387925	−0.193962	−	0.981009i	$-0.562134\pi$
$509$	−1.52200e17	−0.387925	−0.193962	+	0.981009i	$0.562134\pi$
$510$	0	0
$511$	7.77256e17	1.93120
$512$	−3.09161e16	−0.0758455
$513$	0	0
$514$	3.31258e17	0.792331
$515$	−6.07303e15	−0.0143436
$516$	0	0
$517$	3.00588e17	0.692282
$518$	3.41346e17	0.776339
$519$	0	0
$520$	5.06541e16	0.112355
$521$	−4.07718e17	−0.893130	−0.446565	−	0.894751i	$-0.647353\pi$
$521$	−4.07718e17	−0.893130	−0.446565	+	0.894751i	$0.647353\pi$
$522$	0	0
$523$	−5.11063e17	−1.09198	−0.545988	−	0.837793i	$-0.683846\pi$
$523$	−5.11063e17	−1.09198	−0.545988	+	0.837793i	$0.683846\pi$
$524$	2.13177e17	0.449869
$525$	0	0
$526$	−3.06275e17	−0.630527
$527$	5.86010e17	1.19162
$528$	0	0
$529$	8.64927e17	1.71600
$530$	−6.87283e16	−0.134692
$531$	0	0
$532$	8.07207e16	0.154369
$533$	1.64872e17	0.311473
$534$	0	0
$535$	−1.90749e17	−0.351693
$536$	−3.29173e17	−0.599588
$537$	0	0
$538$	−4.68857e17	−0.833596
$539$	1.56309e18	2.74573
$540$	0	0
$541$	3.70761e17	0.635787	0.317894	−	0.948126i	$-0.397025\pi$
$541$	3.70761e17	0.635787	0.317894	+	0.948126i	$0.397025\pi$
$542$	3.98637e17	0.675433
$543$	0	0
$544$	9.03505e17	1.49464
$545$	5.09407e17	0.832698
$546$	0	0
$547$	1.40125e17	0.223665	0.111832	−	0.993727i	$-0.464328\pi$
$547$	1.40125e17	0.223665	0.111832	+	0.993727i	$0.464328\pi$
$548$	−2.63735e17	−0.416001
$549$	0	0
$550$	−8.24668e16	−0.127034
$551$	−4.02177e16	−0.0612254
$552$	0	0
$553$	−1.37853e18	−2.04976
$554$	2.68786e17	0.394996
$555$	0	0
$556$	5.38470e17	0.772992
$557$	3.52292e17	0.499855	0.249927	−	0.968265i	$-0.419593\pi$
$557$	3.52292e17	0.499855	0.249927	+	0.968265i	$0.419593\pi$
$558$	0	0
$559$	−3.24758e17	−0.450177
$560$	2.35396e16	0.0322536
$561$	0	0
$562$	−2.39864e17	−0.321129
$563$	−5.08298e16	−0.0672688	−0.0336344	−	0.999434i	$-0.510708\pi$
$563$	−5.08298e16	−0.0672688	−0.0336344	+	0.999434i	$0.510708\pi$
$564$	0	0
$565$	5.88763e16	0.0761423
$566$	−5.68159e16	−0.0726379
$567$	0	0
$568$	−8.82386e17	−1.10254
$569$	−1.45320e18	−1.79512	−0.897562	−	0.440888i	$-0.854664\pi$
$569$	−1.45320e18	−1.79512	−0.897562	+	0.440888i	$0.854664\pi$
$570$	0	0
$571$	6.84455e17	0.826438	0.413219	−	0.910632i	$-0.364404\pi$
$571$	6.84455e17	0.826438	0.413219	+	0.910632i	$0.364404\pi$
$572$	−1.42171e17	−0.169721
$573$	0	0
$574$	−1.20523e18	−1.40651
$575$	2.85651e17	0.329606
$576$	0	0
$577$	8.39937e17	0.947554	0.473777	−	0.880645i	$-0.342890\pi$
$577$	8.39937e17	0.947554	0.473777	+	0.880645i	$0.342890\pi$
$578$	−6.42677e17	−0.716905
$579$	0	0
$580$	−1.24574e17	−0.135877
$581$	2.57921e18	2.78191
$582$	0	0
$583$	4.96737e17	0.523939
$584$	9.62360e17	1.00382
$585$	0	0
$586$	6.96068e17	0.710096
$587$	−1.30926e18	−1.32092	−0.660461	−	0.750860i	$-0.729639\pi$
$587$	−1.30926e18	−1.32092	−0.660461	+	0.750860i	$0.729639\pi$
$588$	0	0
$589$	1.04474e17	0.103100
$590$	−5.07222e17	−0.495062
$591$	0	0
$592$	−2.68676e16	−0.0256530
$593$	1.03390e18	0.976390	0.488195	−	0.872735i	$-0.337656\pi$
$593$	1.03390e18	0.976390	0.488195	+	0.872735i	$0.337656\pi$
$594$	0	0
$595$	−1.36026e18	−1.25679
$596$	−1.77202e17	−0.161945
$597$	0	0
$598$	−2.83214e17	−0.253255
$599$	1.12423e18	0.994445	0.497222	−	0.867623i	$-0.334353\pi$
$599$	1.12423e18	0.994445	0.497222	+	0.867623i	$0.334353\pi$
$600$	0	0
$601$	−4.70936e17	−0.407641	−0.203821	−	0.979008i	$-0.565336\pi$
$601$	−4.70936e17	−0.407641	−0.203821	+	0.979008i	$0.565336\pi$
$602$	2.37402e18	2.03286
$603$	0	0
$604$	−1.20021e18	−1.00581
$605$	5.66148e16	0.0469375
$606$	0	0
$607$	2.19282e18	1.77941	0.889704	−	0.456538i	$-0.150911\pi$
$607$	2.19282e18	1.77941	0.889704	+	0.456538i	$0.150911\pi$
$608$	1.61077e17	0.129318
$609$	0	0
$610$	−1.57001e17	−0.123384
$611$	2.15403e17	0.167488
$612$	0	0
$613$	−1.62215e18	−1.23480	−0.617401	−	0.786649i	$-0.711814\pi$
$613$	−1.62215e18	−1.23480	−0.617401	+	0.786649i	$0.711814\pi$
$614$	−3.29898e17	−0.248477
$615$	0	0
$616$	2.67627e18	1.97358
$617$	−1.78496e18	−1.30249	−0.651247	−	0.758866i	$-0.725754\pi$
$617$	−1.78496e18	−1.30249	−0.651247	+	0.758866i	$0.725754\pi$
$618$	0	0
$619$	−1.73312e18	−1.23834	−0.619171	−	0.785256i	$-0.712531\pi$
$619$	−1.73312e18	−1.23834	−0.619171	+	0.785256i	$0.712531\pi$
$620$	3.23608e17	0.228809
$621$	0	0
$622$	−3.80189e16	−0.0263246
$623$	−1.41152e18	−0.967196
$624$	0	0
$625$	5.96046e16	0.0400000
$626$	−1.57111e18	−1.04346
$627$	0	0
$628$	1.95937e17	0.127461
$629$	1.55257e18	0.999590
$630$	0	0
$631$	8.39081e17	0.529192	0.264596	−	0.964359i	$-0.414761\pi$
$631$	8.39081e17	0.529192	0.264596	+	0.964359i	$0.414761\pi$
$632$	−1.70683e18	−1.06544
$633$	0	0
$634$	−8.79629e17	−0.537922
$635$	4.65492e16	0.0281762
$636$	0	0
$637$	1.12012e18	0.664292
$638$	−5.17806e17	−0.303971
$639$	0	0
$640$	5.16763e17	0.297250
$641$	−2.22288e18	−1.26572	−0.632861	−	0.774266i	$-0.718119\pi$
$641$	−2.22288e18	−1.26572	−0.632861	+	0.774266i	$0.718119\pi$
$642$	0	0
$643$	−4.50718e17	−0.251497	−0.125749	−	0.992062i	$-0.540133\pi$
$643$	−4.50718e17	−0.251497	−0.125749	+	0.992062i	$0.540133\pi$
$644$	−3.59991e18	−1.98854
$645$	0	0
$646$	−2.11149e17	−0.114308
$647$	1.08565e18	0.581850	0.290925	−	0.956746i	$-0.406037\pi$
$647$	1.08565e18	0.581850	0.290925	+	0.956746i	$0.406037\pi$
$648$	0	0
$649$	3.66597e18	1.92574
$650$	−5.90962e16	−0.0307342
$651$	0	0
$652$	−1.63652e18	−0.834282
$653$	2.01270e18	1.01588	0.507940	−	0.861392i	$-0.330407\pi$
$653$	2.01270e18	1.01588	0.507940	+	0.861392i	$0.330407\pi$
$654$	0	0
$655$	−6.40441e17	−0.316892
$656$	9.48650e16	0.0464762
$657$	0	0
$658$	−1.57462e18	−0.756324
$659$	−3.00271e18	−1.42810	−0.714049	−	0.700096i	$-0.753140\pi$
$659$	−3.00271e18	−1.42810	−0.714049	+	0.700096i	$0.753140\pi$
$660$	0	0
$661$	2.02464e17	0.0944144	0.0472072	−	0.998885i	$-0.484968\pi$
$661$	2.02464e17	0.0944144	0.0472072	+	0.998885i	$0.484968\pi$
$662$	4.77359e17	0.220429
$663$	0	0
$664$	3.19346e18	1.44600
$665$	−2.42507e17	−0.108739
$666$	0	0
$667$	1.79359e18	0.788689
$668$	−1.78499e18	−0.777300
$669$	0	0
$670$	3.84034e17	0.164015
$671$	1.13473e18	0.479952
$672$	0	0
$673$	−3.15622e18	−1.30939	−0.654696	−	0.755892i	$-0.727203\pi$
$673$	−3.15622e18	−1.30939	−0.654696	+	0.755892i	$0.727203\pi$
$674$	−1.34621e18	−0.553127
$675$	0	0
$676$	1.47335e18	0.593816
$677$	1.94751e18	0.777416	0.388708	−	0.921361i	$-0.372921\pi$
$677$	1.94751e18	0.777416	0.388708	+	0.921361i	$0.372921\pi$
$678$	0	0
$679$	−2.13304e18	−0.835306
$680$	−1.68421e18	−0.653262
$681$	0	0
$682$	1.34511e18	0.511870
$683$	2.97871e18	1.12278	0.561389	−	0.827552i	$-0.310267\pi$
$683$	2.97871e18	1.12278	0.561389	+	0.827552i	$0.310267\pi$
$684$	0	0
$685$	7.92333e17	0.293035
$686$	−5.05346e18	−1.85132
$687$	0	0
$688$	−1.86861e17	−0.0671729
$689$	3.55965e17	0.126760
$690$	0	0
$691$	−4.44766e18	−1.55426	−0.777131	−	0.629338i	$-0.783326\pi$
$691$	−4.44766e18	−1.55426	−0.777131	+	0.629338i	$0.783326\pi$
$692$	−3.39177e18	−1.17419
$693$	0	0
$694$	2.61533e18	0.888568
$695$	−1.61771e18	−0.544503
$696$	0	0
$697$	−5.48186e18	−1.81098
$698$	1.24908e18	0.408819
$699$	0	0
$700$	−7.51166e17	−0.241323
$701$	4.25623e18	1.35475	0.677373	−	0.735640i	$-0.263118\pi$
$701$	4.25623e18	1.35475	0.677373	+	0.735640i	$0.263118\pi$
$702$	0	0
$703$	2.76792e17	0.0864858
$704$	1.94503e18	0.602149
$705$	0	0
$706$	1.37549e18	0.418049
$707$	7.77327e17	0.234087
$708$	0	0
$709$	3.93663e18	1.16392	0.581961	−	0.813216i	$-0.302285\pi$
$709$	3.93663e18	1.16392	0.581961	+	0.813216i	$0.302285\pi$
$710$	1.02945e18	0.301595
$711$	0	0
$712$	−1.74767e18	−0.502737
$713$	−4.65924e18	−1.32811
$714$	0	0
$715$	4.27120e17	0.119553
$716$	2.41048e18	0.668605
$717$	0	0
$718$	3.31083e18	0.901838
$719$	−1.42708e18	−0.385222	−0.192611	−	0.981275i	$-0.561696\pi$
$719$	−1.42708e18	−0.385222	−0.192611	+	0.981275i	$0.561696\pi$
$720$	0	0
$721$	−2.29933e17	−0.0609567
$722$	2.26227e18	0.594363
$723$	0	0
$724$	−1.64019e18	−0.423248
$725$	3.74255e17	0.0957130
$726$	0	0
$727$	−5.23752e17	−0.131569	−0.0657843	−	0.997834i	$-0.520955\pi$
$727$	−5.23752e17	−0.131569	−0.0657843	+	0.997834i	$0.520955\pi$
$728$	1.91783e18	0.477481
$729$	0	0
$730$	−1.12275e18	−0.274590
$731$	1.07979e19	2.61745
$732$	0	0
$733$	−1.24417e18	−0.296280	−0.148140	−	0.988966i	$-0.547329\pi$
$733$	−1.24417e18	−0.296280	−0.148140	+	0.988966i	$0.547329\pi$
$734$	4.32960e18	1.02193
$735$	0	0
$736$	−7.18357e18	−1.66585
$737$	−2.77562e18	−0.638001
$738$	0	0
$739$	−8.07413e18	−1.82350	−0.911752	−	0.410741i	$-0.865270\pi$
$739$	−8.07413e18	−1.82350	−0.911752	+	0.410741i	$0.865270\pi$
$740$	8.57364e17	0.191937
$741$	0	0
$742$	−2.60214e18	−0.572408
$743$	7.44712e18	1.62391	0.811954	−	0.583722i	$-0.198404\pi$
$743$	7.44712e18	1.62391	0.811954	+	0.583722i	$0.198404\pi$
$744$	0	0
$745$	5.32364e17	0.114076
$746$	−2.79130e18	−0.592930
$747$	0	0
$748$	4.72706e18	0.986803
$749$	−7.22202e18	−1.49461
$750$	0	0
$751$	7.86071e18	1.59883	0.799415	−	0.600780i	$-0.205143\pi$
$751$	7.86071e18	1.59883	0.799415	+	0.600780i	$0.205143\pi$
$752$	1.23940e17	0.0249916
$753$	0	0
$754$	−3.71062e17	−0.0735416
$755$	3.60577e18	0.708505
$756$	0	0
$757$	4.47616e18	0.864534	0.432267	−	0.901746i	$-0.357714\pi$
$757$	4.47616e18	0.864534	0.432267	+	0.901746i	$0.357714\pi$
$758$	8.34041e17	0.159712
$759$	0	0
$760$	−3.00261e17	−0.0565210
$761$	−4.15316e18	−0.775137	−0.387568	−	0.921841i	$-0.626685\pi$
$761$	−4.15316e18	−0.775137	−0.387568	+	0.921841i	$0.626685\pi$
$762$	0	0
$763$	1.92868e19	3.53876
$764$	2.15742e18	0.392489
$765$	0	0
$766$	7.00293e17	0.125254
$767$	2.62705e18	0.465907
$768$	0	0
$769$	−2.77331e18	−0.483590	−0.241795	−	0.970327i	$-0.577736\pi$
$769$	−2.77331e18	−0.483590	−0.241795	+	0.970327i	$0.577736\pi$
$770$	−3.12230e18	−0.539864
$771$	0	0
$772$	2.66297e18	0.452744
$773$	−3.45989e17	−0.0583304	−0.0291652	−	0.999575i	$-0.509285\pi$
$773$	−3.45989e17	−0.0583304	−0.0291652	+	0.999575i	$0.509285\pi$
$774$	0	0
$775$	−9.72209e17	−0.161175
$776$	−2.64103e18	−0.434182
$777$	0	0
$778$	−3.07927e17	−0.0497829
$779$	−9.77307e17	−0.156689
$780$	0	0
$781$	−7.44036e18	−1.17317
$782$	9.41664e18	1.47249
$783$	0	0
$784$	6.44502e17	0.0991219
$785$	−5.88648e17	−0.0897849
$786$	0	0
$787$	−9.43723e18	−1.41582	−0.707911	−	0.706302i	$-0.750362\pi$
$787$	−9.43723e18	−1.41582	−0.707911	+	0.706302i	$0.750362\pi$
$788$	1.96873e18	0.292931
$789$	0	0
$790$	1.99129e18	0.291446
$791$	2.22913e18	0.323586
$792$	0	0
$793$	8.13157e17	0.116118
$794$	1.14965e18	0.162829
$795$	0	0
$796$	−4.25492e18	−0.592866
$797$	−2.55893e17	−0.0353655	−0.0176827	−	0.999844i	$-0.505629\pi$
$797$	−2.55893e17	−0.0353655	−0.0176827	+	0.999844i	$0.505629\pi$
$798$	0	0
$799$	−7.16197e18	−0.973819
$800$	−1.49894e18	−0.202162
$801$	0	0
$802$	1.75151e18	0.232423
$803$	8.11471e18	1.06813
$804$	0	0
$805$	1.08151e19	1.40074
$806$	9.63915e17	0.123840
$807$	0	0
$808$	9.62449e17	0.121676
$809$	1.11512e19	1.39848	0.699242	−	0.714885i	$-0.253521\pi$
$809$	1.11512e19	1.39848	0.699242	+	0.714885i	$0.253521\pi$
$810$	0	0
$811$	−1.19792e18	−0.147840	−0.0739199	−	0.997264i	$-0.523551\pi$
$811$	−1.19792e18	−0.147840	−0.0739199	+	0.997264i	$0.523551\pi$
$812$	−4.71654e18	−0.577443
$813$	0	0
$814$	3.56372e18	0.429384
$815$	4.91657e18	0.587676
$816$	0	0
$817$	1.92506e18	0.226465
$818$	−7.54323e18	−0.880361
$819$	0	0
$820$	−3.02721e18	−0.347738
$821$	−1.05282e19	−1.19984	−0.599919	−	0.800060i	$-0.704801\pi$
$821$	−1.05282e19	−1.19984	−0.599919	+	0.800060i	$0.704801\pi$
$822$	0	0
$823$	8.89011e17	0.0997260	0.0498630	−	0.998756i	$-0.484122\pi$
$823$	8.89011e17	0.0997260	0.0498630	+	0.998756i	$0.484122\pi$
$824$	−2.84692e17	−0.0316846
$825$	0	0
$826$	−1.92041e19	−2.10389
$827$	−1.07764e19	−1.17135	−0.585675	−	0.810546i	$-0.699170\pi$
$827$	−1.07764e19	−1.17135	−0.585675	+	0.810546i	$0.699170\pi$
$828$	0	0
$829$	1.19525e19	1.27896	0.639478	−	0.768809i	$-0.279151\pi$
$829$	1.19525e19	1.27896	0.639478	+	0.768809i	$0.279151\pi$
$830$	−3.72568e18	−0.395547
$831$	0	0
$832$	1.39382e18	0.145682
$833$	−3.72431e19	−3.86236
$834$	0	0
$835$	5.36260e18	0.547537
$836$	8.42741e17	0.0853794
$837$	0	0
$838$	3.92235e18	0.391256
$839$	7.74966e18	0.767061	0.383531	−	0.923528i	$-0.374708\pi$
$839$	7.74966e18	0.767061	0.383531	+	0.923528i	$0.374708\pi$
$840$	0	0
$841$	−7.91070e18	−0.770976
$842$	−2.62952e18	−0.254301
$843$	0	0
$844$	3.50738e18	0.334008
$845$	−4.42635e18	−0.418289
$846$	0	0
$847$	2.14351e18	0.199473
$848$	2.04817e17	0.0189144
$849$	0	0
$850$	1.96490e18	0.178697
$851$	−1.23441e19	−1.11409
$852$	0	0
$853$	−9.68239e18	−0.860625	−0.430313	−	0.902680i	$-0.641597\pi$
$853$	−9.68239e18	−0.860625	−0.430313	+	0.902680i	$0.641597\pi$
$854$	−5.94427e18	−0.524351
$855$	0	0
$856$	−8.94196e18	−0.776878
$857$	−1.09234e19	−0.941851	−0.470926	−	0.882173i	$-0.656080\pi$
$857$	−1.09234e19	−0.941851	−0.470926	+	0.882173i	$0.656080\pi$
$858$	0	0
$859$	−1.30865e19	−1.11139	−0.555695	−	0.831386i	$-0.687548\pi$
$859$	−1.30865e19	−1.11139	−0.555695	+	0.831386i	$0.687548\pi$
$860$	5.96286e18	0.502591
$861$	0	0
$862$	−1.23946e19	−1.02905
$863$	2.64998e18	0.218359	0.109180	−	0.994022i	$-0.465178\pi$
$863$	2.64998e18	0.218359	0.109180	+	0.994022i	$0.465178\pi$
$864$	0	0
$865$	1.01898e19	0.827108
$866$	2.98620e18	0.240576
$867$	0	0
$868$	1.22522e19	0.972382
$869$	−1.43921e19	−1.13370
$870$	0	0
$871$	−1.98902e18	−0.154356
$872$	2.38800e19	1.83940
$873$	0	0
$874$	1.67880e18	0.127402
$875$	2.25671e18	0.169990
$876$	0	0
$877$	−1.23612e19	−0.917406	−0.458703	−	0.888590i	$-0.651686\pi$
$877$	−1.23612e19	−0.917406	−0.458703	+	0.888590i	$0.651686\pi$
$878$	1.33871e19	0.986220
$879$	0	0
$880$	2.45759e17	0.0178390
$881$	−4.79526e18	−0.345516	−0.172758	−	0.984964i	$-0.555268\pi$
$881$	−4.79526e18	−0.345516	−0.172758	+	0.984964i	$0.555268\pi$
$882$	0	0
$883$	1.23899e19	0.879680	0.439840	−	0.898076i	$-0.355035\pi$
$883$	1.23899e19	0.879680	0.439840	+	0.898076i	$0.355035\pi$
$884$	3.38744e18	0.238743
$885$	0	0
$886$	7.33257e18	0.509257
$887$	−3.48555e18	−0.240308	−0.120154	−	0.992755i	$-0.538339\pi$
$887$	−3.48555e18	−0.240308	−0.120154	+	0.992755i	$0.538339\pi$
$888$	0	0
$889$	1.76241e18	0.119742
$890$	2.03894e18	0.137521
$891$	0	0
$892$	−9.75978e18	−0.648737
$893$	−1.27684e18	−0.0842560
$894$	0	0
$895$	−7.24176e18	−0.470971
$896$	1.95653e19	1.26324
$897$	0	0
$898$	1.08088e19	0.687828
$899$	−6.10446e18	−0.385664
$900$	0	0
$901$	−1.18355e19	−0.737015
$902$	−1.25829e19	−0.777926
$903$	0	0
$904$	2.76001e18	0.168196
$905$	4.92760e18	0.298139
$906$	0	0
$907$	2.02823e19	1.20968	0.604838	−	0.796349i	$-0.293238\pi$
$907$	2.02823e19	1.20968	0.604838	+	0.796349i	$0.293238\pi$
$908$	−6.85287e18	−0.405802
$909$	0	0
$910$	−2.23746e18	−0.130613
$911$	−2.56082e18	−0.148426	−0.0742129	−	0.997242i	$-0.523644\pi$
$911$	−2.56082e18	−0.148426	−0.0742129	+	0.997242i	$0.523644\pi$
$912$	0	0
$913$	2.69275e19	1.53864
$914$	1.29993e19	0.737511
$915$	0	0
$916$	1.13140e19	0.632842
$917$	−2.42479e19	−1.34671
$918$	0	0
$919$	−1.75792e19	−0.962604	−0.481302	−	0.876555i	$-0.659836\pi$
$919$	−1.75792e19	−0.962604	−0.481302	+	0.876555i	$0.659836\pi$
$920$	1.33908e19	0.728089
$921$	0	0
$922$	1.34688e19	0.722069
$923$	−5.33180e18	−0.283833
$924$	0	0
$925$	−2.57576e18	−0.135202
$926$	−1.02047e19	−0.531900
$927$	0	0
$928$	−9.41179e18	−0.483738
$929$	2.58966e19	1.32172	0.660861	−	0.750508i	$-0.270191\pi$
$929$	2.58966e19	1.32172	0.660861	+	0.750508i	$0.270191\pi$
$930$	0	0
$931$	−6.63971e18	−0.334177
$932$	4.05228e18	0.202533
$933$	0	0
$934$	−4.03215e18	−0.198739
$935$	−1.42014e19	−0.695113
$936$	0	0
$937$	−9.46286e18	−0.456788	−0.228394	−	0.973569i	$-0.573347\pi$
$937$	−9.46286e18	−0.456788	−0.228394	+	0.973569i	$0.573347\pi$
$938$	1.45400e19	0.697022
$939$	0	0
$940$	−3.95500e18	−0.186989
$941$	2.16626e19	1.01713	0.508567	−	0.861022i	$-0.330175\pi$
$941$	2.16626e19	1.01713	0.508567	+	0.861022i	$0.330175\pi$
$942$	0	0
$943$	4.35851e19	2.01842
$944$	1.51157e18	0.0695201
$945$	0	0
$946$	2.47853e19	1.12435
$947$	4.28845e19	1.93208	0.966041	−	0.258390i	$-0.0831921\pi$
$947$	4.28845e19	1.93208	0.966041	+	0.258390i	$0.0831921\pi$
$948$	0	0
$949$	5.81505e18	0.258418
$950$	3.50303e17	0.0154611
$951$	0	0
$952$	−6.37662e19	−2.77620
$953$	1.37774e19	0.595750	0.297875	−	0.954605i	$-0.403722\pi$
$953$	1.37774e19	0.595750	0.297875	+	0.954605i	$0.403722\pi$
$954$	0	0
$955$	−6.48148e18	−0.276473
$956$	−1.65185e19	−0.699831
$957$	0	0
$958$	1.17797e19	0.492330
$959$	2.99987e19	1.24532
$960$	0	0
$961$	−8.55990e18	−0.350563
$962$	2.55379e18	0.103883
$963$	0	0
$964$	−2.39946e18	−0.0962972
$965$	−8.00028e18	−0.318917
$966$	0	0
$967$	4.15597e19	1.63456	0.817280	−	0.576241i	$-0.195481\pi$
$967$	4.15597e19	1.63456	0.817280	+	0.576241i	$0.195481\pi$
$968$	2.65399e18	0.103683
$969$	0	0
$970$	3.08119e18	0.118769
$971$	7.09991e18	0.271849	0.135925	−	0.990719i	$-0.456600\pi$
$971$	7.09991e18	0.271849	0.135925	+	0.990719i	$0.456600\pi$
$972$	0	0
$973$	−6.12487e19	−2.31400
$974$	1.50451e19	0.564627
$975$	0	0
$976$	4.67879e17	0.0173264
$977$	2.45976e19	0.904855	0.452427	−	0.891801i	$-0.350558\pi$
$977$	2.45976e19	0.904855	0.452427	+	0.891801i	$0.350558\pi$
$978$	0	0
$979$	−1.47366e19	−0.534944
$980$	−2.05665e19	−0.741635
$981$	0	0
$982$	1.72474e18	0.0613761
$983$	8.12973e18	0.287394	0.143697	−	0.989622i	$-0.454101\pi$
$983$	8.12973e18	0.287394	0.143697	+	0.989622i	$0.454101\pi$
$984$	0	0
$985$	−5.91461e18	−0.206343
$986$	1.23375e19	0.427590
$987$	0	0
$988$	6.03913e17	0.0206564
$989$	−8.58520e19	−2.91726
$990$	0	0
$991$	−1.85665e19	−0.622661	−0.311331	−	0.950302i	$-0.600775\pi$
$991$	−1.85665e19	−0.622661	−0.311331	+	0.950302i	$0.600775\pi$
$992$	2.44492e19	0.814588
$993$	0	0
$994$	3.89761e19	1.28170
$995$	1.27830e19	0.417620
$996$	0	0
$997$	2.74746e19	0.885958	0.442979	−	0.896532i	$-0.353922\pi$
$997$	2.74746e19	0.885958	0.442979	+	0.896532i	$0.353922\pi$
$998$	−5.95231e18	−0.190694
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.14.a.g.1.2	✓	4
3.2	odd	2		45.14.a.h.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.14.a.g.1.2	✓	4	1.1	even	1	trivial
45.14.a.h.1.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q-54.6908 q^{2} -5200.92 q^{4} +15625.0 q^{5} +591583. q^{7} +732469. q^{8} -854543. q^{10} +6.17625e6 q^{11} +4.42594e6 q^{13} -3.23541e7 q^{14} +2.54662e6 q^{16} -1.47159e8 q^{17} -2.62355e7 q^{19} -8.12644e7 q^{20} -3.37784e8 q^{22} +1.17003e9 q^{23} +2.44141e8 q^{25} -2.42058e8 q^{26} -3.07677e9 q^{28} +1.53295e9 q^{29} -3.98217e9 q^{31} -6.13966e9 q^{32} +8.04822e9 q^{34} +9.24348e9 q^{35} -1.05503e10 q^{37} +1.43484e9 q^{38} +1.14448e10 q^{40} +3.72513e10 q^{41} -7.33761e10 q^{43} -3.21222e10 q^{44} -6.39897e10 q^{46} +4.86683e10 q^{47} +2.53081e11 q^{49} -1.33522e10 q^{50} -2.30189e10 q^{52} +8.04270e10 q^{53} +9.65039e10 q^{55} +4.33316e11 q^{56} -8.38382e10 q^{58} +5.93559e11 q^{59} +1.83725e11 q^{61} +2.17788e11 q^{62} +3.14921e11 q^{64} +6.91553e10 q^{65} -4.49402e11 q^{67} +7.65360e11 q^{68} -5.05533e11 q^{70} -1.20467e12 q^{71} +1.31386e12 q^{73} +5.77004e11 q^{74} +1.36449e11 q^{76} +3.65376e12 q^{77} -2.33024e12 q^{79} +3.97909e10 q^{80} -2.03730e12 q^{82} +4.35985e12 q^{83} -2.29935e12 q^{85} +4.01300e12 q^{86} +4.52391e12 q^{88} -2.38600e12 q^{89} +2.61831e12 q^{91} -6.08522e12 q^{92} -2.66171e12 q^{94} -4.09929e11 q^{95} -3.60565e12 q^{97} -1.38412e13 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 15 q^{2} + 16837 q^{4} + 62500 q^{5} + 343040 q^{7} - 14865 q^{8} - 234375 q^{10} + 12697800 q^{11} + 34336040 q^{13} + 26944650 q^{14} + 66562801 q^{16} + 84377280 q^{17} - 131821144 q^{19} + 263078125 q^{20}+ \cdots - 17650752985395 q^{98}+O(q^{100}) \) 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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