LMFDB - Embedded newform 45.14.a.g.1.3

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			1.3
Root			$46.2561$ 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)

$f(q)$	$=$	\(q+42.2561 q^{2} -6406.42 q^{4} +15625.0 q^{5} -220964. q^{7} -616873. q^{8} +660252. q^{10} -3.84349e6 q^{11} +1.10025e7 q^{13} -9.33707e6 q^{14} +2.64147e7 q^{16} -2.42586e7 q^{17} -2.35835e7 q^{19} -1.00100e8 q^{20} -1.62411e8 q^{22} -2.04982e8 q^{23} +2.44141e8 q^{25} +4.64921e8 q^{26} +1.41559e9 q^{28} +2.15625e9 q^{29} +4.61294e9 q^{31} +6.16961e9 q^{32} -1.02507e9 q^{34} -3.45256e9 q^{35} +7.30589e9 q^{37} -9.96549e8 q^{38} -9.63864e9 q^{40} -1.85342e10 q^{41} +3.47937e10 q^{43} +2.46230e10 q^{44} -8.66174e9 q^{46} +8.28883e10 q^{47} -4.80641e10 q^{49} +1.03164e10 q^{50} -7.04864e10 q^{52} +2.52519e11 q^{53} -6.00545e10 q^{55} +1.36306e11 q^{56} +9.11149e10 q^{58} +3.88204e11 q^{59} +1.04691e11 q^{61} +1.94925e11 q^{62} +4.43141e10 q^{64} +1.71913e11 q^{65} -8.49553e11 q^{67} +1.55411e11 q^{68} -1.45892e11 q^{70} +1.84180e12 q^{71} -1.70795e11 q^{73} +3.08718e11 q^{74} +1.51086e11 q^{76} +8.49271e11 q^{77} +4.05791e12 q^{79} +4.12730e11 q^{80} -7.83185e11 q^{82} +1.14420e12 q^{83} -3.79040e11 q^{85} +1.47024e12 q^{86} +2.37094e12 q^{88} +3.13388e12 q^{89} -2.43114e12 q^{91} +1.31320e12 q^{92} +3.50254e12 q^{94} -3.68493e11 q^{95} -7.59534e12 q^{97} -2.03100e12 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	42.2561	0.466869	0.233434	−	0.972373i	$-0.425004\pi$
$2$	42.2561	0.466869	0.233434	+	0.972373i	$0.425004\pi$
$3$	0	0
$3$	0	0
$4$	−6406.42	−0.782034
$5$	15625.0	0.447214
$5$	15625.0	0.447214
$6$	0	0
$7$	−220964.	−0.709878	−0.354939	−	0.934890i	$-0.615498\pi$
$7$	−220964.	−0.709878	−0.354939	+	0.934890i	$0.615498\pi$
$8$	−616873.	−0.831976
$9$	0	0
$10$	660252.	0.208790
$11$	−3.84349e6	−0.654144	−0.327072	−	0.944999i	$-0.606062\pi$
$11$	−3.84349e6	−0.654144	−0.327072	+	0.944999i	$0.606062\pi$
$12$	0	0
$13$	1.10025e7	0.632205	0.316103	−	0.948725i	$-0.397626\pi$
$13$	1.10025e7	0.632205	0.316103	+	0.948725i	$0.397626\pi$
$14$	−9.33707e6	−0.331420
$15$	0	0
$16$	2.64147e7	0.393610
$17$	−2.42586e7	−0.243751	−0.121876	−	0.992545i	$-0.538891\pi$
$17$	−2.42586e7	−0.243751	−0.121876	+	0.992545i	$0.538891\pi$
$18$	0	0
$19$	−2.35835e7	−0.115003	−0.0575017	−	0.998345i	$-0.518313\pi$
$19$	−2.35835e7	−0.115003	−0.0575017	+	0.998345i	$0.518313\pi$
$20$	−1.00100e8	−0.349736
$21$	0	0
$22$	−1.62411e8	−0.305399
$23$	−2.04982e8	−0.288725	−0.144363	−	0.989525i	$-0.546113\pi$
$23$	−2.04982e8	−0.288725	−0.144363	+	0.989525i	$0.546113\pi$
$24$	0	0
$25$	2.44141e8	0.200000
$26$	4.64921e8	0.295157
$27$	0	0
$28$	1.41559e9	0.555148
$29$	2.15625e9	0.673151	0.336576	−	0.941656i	$-0.390731\pi$
$29$	2.15625e9	0.673151	0.336576	+	0.941656i	$0.390731\pi$
$30$	0	0
$31$	4.61294e9	0.933528	0.466764	−	0.884382i	$-0.345420\pi$
$31$	4.61294e9	0.933528	0.466764	+	0.884382i	$0.345420\pi$
$32$	6.16961e9	1.01574
$33$	0	0
$34$	−1.02507e9	−0.113800
$35$	−3.45256e9	−0.317467
$36$	0	0
$37$	7.30589e9	0.468125	0.234062	−	0.972222i	$-0.424798\pi$
$37$	7.30589e9	0.468125	0.234062	+	0.972222i	$0.424798\pi$
$38$	−9.96549e8	−0.0536915
$39$	0	0
$40$	−9.63864e9	−0.372071
$41$	−1.85342e10	−0.609368	−0.304684	−	0.952453i	$-0.598551\pi$
$41$	−1.85342e10	−0.609368	−0.304684	+	0.952453i	$0.598551\pi$
$42$	0	0
$43$	3.47937e10	0.839373	0.419687	−	0.907669i	$-0.362140\pi$
$43$	3.47937e10	0.839373	0.419687	+	0.907669i	$0.362140\pi$
$44$	2.46230e10	0.511562
$45$	0	0
$46$	−8.66174e9	−0.134797
$47$	8.28883e10	1.12165	0.560825	−	0.827935i	$-0.310484\pi$
$47$	8.28883e10	1.12165	0.560825	+	0.827935i	$0.310484\pi$
$48$	0	0
$49$	−4.80641e10	−0.496074
$50$	1.03164e10	0.0933737
$51$	0	0
$52$	−7.04864e10	−0.494406
$53$	2.52519e11	1.56495	0.782475	−	0.622681i	$-0.213957\pi$
$53$	2.52519e11	1.56495	0.782475	+	0.622681i	$0.213957\pi$
$54$	0	0
$55$	−6.00545e10	−0.292542
$56$	1.36306e11	0.590601
$57$	0	0
$58$	9.11149e10	0.314273
$59$	3.88204e11	1.19818	0.599090	−	0.800682i	$-0.295529\pi$
$59$	3.88204e11	1.19818	0.599090	+	0.800682i	$0.295529\pi$
$60$	0	0
$61$	1.04691e11	0.260176	0.130088	−	0.991502i	$-0.458474\pi$
$61$	1.04691e11	0.260176	0.130088	+	0.991502i	$0.458474\pi$
$62$	1.94925e11	0.435835
$63$	0	0
$64$	4.43141e10	0.0806069
$65$	1.71913e11	0.282731
$66$	0	0
$67$	−8.49553e11	−1.14737	−0.573686	−	0.819075i	$-0.694487\pi$
$67$	−8.49553e11	−1.14737	−0.573686	+	0.819075i	$0.694487\pi$
$68$	1.55411e11	0.190622
$69$	0	0
$70$	−1.45892e11	−0.148215
$71$	1.84180e12	1.70633	0.853167	−	0.521638i	$-0.174679\pi$
$71$	1.84180e12	1.70633	0.853167	+	0.521638i	$0.174679\pi$
$72$	0	0
$73$	−1.70795e11	−0.132092	−0.0660461	−	0.997817i	$-0.521038\pi$
$73$	−1.70795e11	−0.132092	−0.0660461	+	0.997817i	$0.521038\pi$
$74$	3.08718e11	0.218553
$75$	0	0
$76$	1.51086e11	0.0899365
$77$	8.49271e11	0.464362
$78$	0	0
$79$	4.05791e12	1.87813	0.939067	−	0.343735i	$-0.111692\pi$
$79$	4.05791e12	1.87813	0.939067	+	0.343735i	$0.111692\pi$
$80$	4.12730e11	0.176028
$81$	0	0
$82$	−7.83185e11	−0.284495
$83$	1.14420e12	0.384143	0.192072	−	0.981381i	$-0.438479\pi$
$83$	1.14420e12	0.384143	0.192072	+	0.981381i	$0.438479\pi$
$84$	0	0
$85$	−3.79040e11	−0.109009
$86$	1.47024e12	0.391877
$87$	0	0
$88$	2.37094e12	0.544232
$89$	3.13388e12	0.668417	0.334208	−	0.942499i	$-0.391531\pi$
$89$	3.13388e12	0.668417	0.334208	+	0.942499i	$0.391531\pi$
$90$	0	0
$91$	−2.43114e12	−0.448788
$92$	1.31320e12	0.225793
$93$	0	0
$94$	3.50254e12	0.523663
$95$	−3.68493e11	−0.0514311
$96$	0	0
$97$	−7.59534e12	−0.925830	−0.462915	−	0.886403i	$-0.653196\pi$
$97$	−7.59534e12	−0.925830	−0.462915	+	0.886403i	$0.653196\pi$
$98$	−2.03100e12	−0.231601
$99$	0	0
$100$	−1.56407e12	−0.156407
$101$	−1.86842e13	−1.75140	−0.875698	−	0.482859i	$-0.839598\pi$
$101$	−1.86842e13	−1.75140	−0.875698	+	0.482859i	$0.839598\pi$
$102$	0	0
$103$	−3.72543e12	−0.307422	−0.153711	−	0.988116i	$-0.549122\pi$
$103$	−3.72543e12	−0.307422	−0.153711	+	0.988116i	$0.549122\pi$
$104$	−6.78712e12	−0.525979
$105$	0	0
$106$	1.06705e13	0.730627
$107$	−1.35442e13	−0.872485	−0.436242	−	0.899829i	$-0.643691\pi$
$107$	−1.35442e13	−0.872485	−0.436242	+	0.899829i	$0.643691\pi$
$108$	0	0
$109$	−7.62252e12	−0.435338	−0.217669	−	0.976023i	$-0.569845\pi$
$109$	−7.62252e12	−0.435338	−0.217669	+	0.976023i	$0.569845\pi$
$110$	−2.53767e12	−0.136579
$111$	0	0
$112$	−5.83670e12	−0.279415
$113$	−2.08597e13	−0.942535	−0.471268	−	0.881990i	$-0.656203\pi$
$113$	−2.08597e13	−0.942535	−0.471268	+	0.881990i	$0.656203\pi$
$114$	0	0
$115$	−3.20284e12	−0.129122
$116$	−1.38139e13	−0.526427
$117$	0	0
$118$	1.64040e13	0.559393
$119$	5.36026e12	0.173034
$120$	0	0
$121$	−1.97503e13	−0.572096
$122$	4.42385e12	0.121468
$123$	0	0
$124$	−2.95525e13	−0.730050
$125$	3.81470e12	0.0894427
$126$	0	0
$127$	3.63166e13	0.768034	0.384017	−	0.923326i	$-0.374540\pi$
$127$	3.63166e13	0.768034	0.384017	+	0.923326i	$0.374540\pi$
$128$	−4.86689e13	−0.978107
$129$	0	0
$130$	7.26440e12	0.131998
$131$	2.06728e13	0.357384	0.178692	−	0.983905i	$-0.442813\pi$
$131$	2.06728e13	0.357384	0.178692	+	0.983905i	$0.442813\pi$
$132$	0	0
$133$	5.21110e12	0.0816383
$134$	−3.58988e13	−0.535672
$135$	0	0
$136$	1.49644e13	0.202795
$137$	−7.62379e13	−0.985115	−0.492558	−	0.870280i	$-0.663938\pi$
$137$	−7.62379e13	−0.985115	−0.492558	+	0.870280i	$0.663938\pi$
$138$	0	0
$139$	−1.07031e13	−0.125868	−0.0629338	−	0.998018i	$-0.520046\pi$
$139$	−1.07031e13	−0.125868	−0.0629338	+	0.998018i	$0.520046\pi$
$140$	2.21185e13	0.248270
$141$	0	0
$142$	7.78275e13	0.796634
$143$	−4.22878e13	−0.413553
$144$	0	0
$145$	3.36915e13	0.301042
$146$	−7.21715e12	−0.0616697
$147$	0	0
$148$	−4.68046e13	−0.366089
$149$	2.24869e14	1.68352	0.841762	−	0.539848i	$-0.181518\pi$
$149$	2.24869e14	1.68352	0.841762	+	0.539848i	$0.181518\pi$
$150$	0	0
$151$	−6.45080e13	−0.442857	−0.221428	−	0.975177i	$-0.571072\pi$
$151$	−6.45080e13	−0.442857	−0.221428	+	0.975177i	$0.571072\pi$
$152$	1.45480e13	0.0956800
$153$	0	0
$154$	3.58869e13	0.216796
$155$	7.20773e13	0.417486
$156$	0	0
$157$	2.87990e13	0.153472	0.0767362	−	0.997051i	$-0.475550\pi$
$157$	2.87990e13	0.153472	0.0767362	+	0.997051i	$0.475550\pi$
$158$	1.71472e14	0.876842
$159$	0	0
$160$	9.64001e13	0.454253
$161$	4.52935e13	0.204959
$162$	0	0
$163$	−1.02289e14	−0.427180	−0.213590	−	0.976923i	$-0.568516\pi$
$163$	−1.02289e14	−0.427180	−0.213590	+	0.976923i	$0.568516\pi$
$164$	1.18738e14	0.476546
$165$	0	0
$166$	4.83493e13	0.179344
$167$	4.53040e14	1.61614	0.808072	−	0.589084i	$-0.200511\pi$
$167$	4.53040e14	1.61614	0.808072	+	0.589084i	$0.200511\pi$
$168$	0	0
$169$	−1.81821e14	−0.600317
$170$	−1.60168e13	−0.0508929
$171$	0	0
$172$	−2.22903e14	−0.656418
$173$	−3.35208e13	−0.0950637	−0.0475318	−	0.998870i	$-0.515136\pi$
$173$	−3.35208e13	−0.0950637	−0.0475318	+	0.998870i	$0.515136\pi$
$174$	0	0
$175$	−5.39462e13	−0.141976
$176$	−1.01525e14	−0.257478
$177$	0	0
$178$	1.32426e14	0.312063
$179$	6.84985e14	1.55645	0.778227	−	0.627983i	$-0.216119\pi$
$179$	6.84985e14	1.55645	0.778227	+	0.627983i	$0.216119\pi$
$180$	0	0
$181$	−5.60034e14	−1.18387	−0.591935	−	0.805986i	$-0.701636\pi$
$181$	−5.60034e14	−1.18387	−0.591935	+	0.805986i	$0.701636\pi$
$182$	−1.02731e14	−0.209525
$183$	0	0
$184$	1.26448e14	0.240212
$185$	1.14154e14	0.209352
$186$	0	0
$187$	9.32375e13	0.159448
$188$	−5.31017e14	−0.877168
$189$	0	0
$190$	−1.55711e13	−0.0240116
$191$	−5.62402e13	−0.0838166	−0.0419083	−	0.999121i	$-0.513344\pi$
$191$	−5.62402e13	−0.0838166	−0.0419083	+	0.999121i	$0.513344\pi$
$192$	0	0
$193$	1.09259e15	1.52172	0.760860	−	0.648916i	$-0.224777\pi$
$193$	1.09259e15	1.52172	0.760860	+	0.648916i	$0.224777\pi$
$194$	−3.20950e14	−0.432241
$195$	0	0
$196$	3.07919e14	0.387946
$197$	−3.41280e13	−0.0415988	−0.0207994	−	0.999784i	$-0.506621\pi$
$197$	−3.41280e13	−0.0415988	−0.0207994	+	0.999784i	$0.506621\pi$
$198$	0	0
$199$	4.04423e14	0.461627	0.230813	−	0.972998i	$-0.425861\pi$
$199$	4.04423e14	0.461627	0.230813	+	0.972998i	$0.425861\pi$
$200$	−1.50604e14	−0.166395
$201$	0	0
$202$	−7.89520e14	−0.817672
$203$	−4.76453e14	−0.477855
$204$	0	0
$205$	−2.89597e14	−0.272518
$206$	−1.57422e14	−0.143526
$207$	0	0
$208$	2.90627e14	0.248842
$209$	9.06431e13	0.0752287
$210$	0	0
$211$	4.90229e13	0.0382440	0.0191220	−	0.999817i	$-0.493913\pi$
$211$	4.90229e13	0.0382440	0.0191220	+	0.999817i	$0.493913\pi$
$212$	−1.61774e15	−1.22384
$213$	0	0
$214$	−5.72324e14	−0.407336
$215$	5.43651e14	0.375379
$216$	0	0
$217$	−1.01929e15	−0.662690
$218$	−3.22098e14	−0.203246
$219$	0	0
$220$	3.84734e14	0.228778
$221$	−2.66904e14	−0.154101
$222$	0	0
$223$	2.45638e15	1.33756	0.668782	−	0.743459i	$-0.266816\pi$
$223$	2.45638e15	1.33756	0.668782	+	0.743459i	$0.266816\pi$
$224$	−1.36326e15	−0.721051
$225$	0	0
$226$	−8.81449e14	−0.440040
$227$	−8.64191e14	−0.419219	−0.209610	−	0.977785i	$-0.567219\pi$
$227$	−8.64191e14	−0.419219	−0.209610	+	0.977785i	$0.567219\pi$
$228$	0	0
$229$	1.66749e15	0.764068	0.382034	−	0.924148i	$-0.375224\pi$
$229$	1.66749e15	0.764068	0.382034	+	0.924148i	$0.375224\pi$
$230$	−1.35340e14	−0.0602829
$231$	0	0
$232$	−1.33013e15	−0.560046
$233$	−1.18763e14	−0.0486258	−0.0243129	−	0.999704i	$-0.507740\pi$
$233$	−1.18763e14	−0.0486258	−0.0243129	+	0.999704i	$0.507740\pi$
$234$	0	0
$235$	1.29513e15	0.501617
$236$	−2.48700e15	−0.937017
$237$	0	0
$238$	2.26504e14	0.0807840
$239$	−2.04502e15	−0.709759	−0.354880	−	0.934912i	$-0.615478\pi$
$239$	−2.04502e15	−0.709759	−0.354880	+	0.934912i	$0.615478\pi$
$240$	0	0
$241$	−4.84836e15	−1.59398	−0.796992	−	0.603989i	$-0.793577\pi$
$241$	−4.84836e15	−1.59398	−0.796992	+	0.603989i	$0.793577\pi$
$242$	−8.34571e14	−0.267094
$243$	0	0
$244$	−6.70697e14	−0.203466
$245$	−7.51002e14	−0.221851
$246$	0	0
$247$	−2.59477e14	−0.0727057
$248$	−2.84560e15	−0.776672
$249$	0	0
$250$	1.61194e14	0.0417580
$251$	2.42607e15	0.612385	0.306193	−	0.951970i	$-0.400945\pi$
$251$	2.42607e15	0.612385	0.306193	+	0.951970i	$0.400945\pi$
$252$	0	0
$253$	7.87846e14	0.188868
$254$	1.53460e15	0.358571
$255$	0	0
$256$	−2.41958e15	−0.537254
$257$	7.66198e15	1.65873	0.829365	−	0.558708i	$-0.188703\pi$
$257$	7.66198e15	1.65873	0.829365	+	0.558708i	$0.188703\pi$
$258$	0	0
$259$	−1.61433e15	−0.332311
$260$	−1.10135e15	−0.221105
$261$	0	0
$262$	8.73551e14	0.166851
$263$	9.69259e15	1.80604	0.903020	−	0.429598i	$-0.141345\pi$
$263$	9.69259e15	1.80604	0.903020	+	0.429598i	$0.141345\pi$
$264$	0	0
$265$	3.94561e15	0.699867
$266$	2.20201e14	0.0381144
$267$	0	0
$268$	5.44260e15	0.897284
$269$	7.59320e15	1.22190	0.610949	−	0.791670i	$-0.290788\pi$
$269$	7.59320e15	1.22190	0.610949	+	0.791670i	$0.290788\pi$
$270$	0	0
$271$	−6.60189e15	−1.01244	−0.506218	−	0.862405i	$-0.668957\pi$
$271$	−6.60189e15	−1.01244	−0.506218	+	0.862405i	$0.668957\pi$
$272$	−6.40784e14	−0.0959431
$273$	0	0
$274$	−3.22152e15	−0.459919
$275$	−9.38352e14	−0.130829
$276$	0	0
$277$	1.88392e15	0.250579	0.125289	−	0.992120i	$-0.460014\pi$
$277$	1.88392e15	0.250579	0.125289	+	0.992120i	$0.460014\pi$
$278$	−4.52272e14	−0.0587637
$279$	0	0
$280$	2.12979e15	0.264125
$281$	1.07715e16	1.30523	0.652615	−	0.757690i	$-0.273672\pi$
$281$	1.07715e16	1.30523	0.652615	+	0.757690i	$0.273672\pi$
$282$	0	0
$283$	1.49270e16	1.72728	0.863638	−	0.504113i	$-0.168181\pi$
$283$	1.49270e16	1.72728	0.863638	+	0.504113i	$0.168181\pi$
$284$	−1.17994e16	−1.33441
$285$	0	0
$286$	−1.78692e15	−0.193075
$287$	4.09539e15	0.432577
$288$	0	0
$289$	−9.31610e15	−0.940585
$290$	1.42367e15	0.140547
$291$	0	0
$292$	1.09419e15	0.103301
$293$	1.52009e16	1.40356	0.701779	−	0.712394i	$-0.252389\pi$
$293$	1.52009e16	1.40356	0.701779	+	0.712394i	$0.252389\pi$
$294$	0	0
$295$	6.06569e15	0.535842
$296$	−4.50680e15	−0.389468
$297$	0	0
$298$	9.50211e15	0.785985
$299$	−2.25531e15	−0.182534
$300$	0	0
$301$	−7.68813e15	−0.595852
$302$	−2.72586e15	−0.206756
$303$	0	0
$304$	−6.22953e14	−0.0452665
$305$	1.63580e15	0.116354
$306$	0	0
$307$	−1.82244e16	−1.24238	−0.621189	−	0.783661i	$-0.713350\pi$
$307$	−1.82244e16	−1.24238	−0.621189	+	0.783661i	$0.713350\pi$
$308$	−5.44079e15	−0.363147
$309$	0	0
$310$	3.04571e15	0.194911
$311$	1.21632e16	0.762261	0.381131	−	0.924521i	$-0.375535\pi$
$311$	1.21632e16	0.762261	0.381131	+	0.924521i	$0.375535\pi$
$312$	0	0
$313$	2.99383e16	1.79965	0.899827	−	0.436247i	$-0.143693\pi$
$313$	2.99383e16	1.79965	0.899827	+	0.436247i	$0.143693\pi$
$314$	1.21694e15	0.0716514
$315$	0	0
$316$	−2.59967e16	−1.46876
$317$	−8.07461e15	−0.446927	−0.223463	−	0.974712i	$-0.571736\pi$
$317$	−8.07461e15	−0.446927	−0.223463	+	0.974712i	$0.571736\pi$
$318$	0	0
$319$	−8.28754e15	−0.440338
$320$	6.92408e14	0.0360485
$321$	0	0
$322$	1.91393e15	0.0956892
$323$	5.72103e14	0.0280322
$324$	0	0
$325$	2.68615e15	0.126441
$326$	−4.32235e15	−0.199437
$327$	0	0
$328$	1.14333e16	0.506979
$329$	−1.83153e16	−0.796234
$330$	0	0
$331$	−3.66870e16	−1.53331	−0.766656	−	0.642058i	$-0.778081\pi$
$331$	−3.66870e16	−1.53331	−0.766656	+	0.642058i	$0.778081\pi$
$332$	−7.33021e15	−0.300413
$333$	0	0
$334$	1.91437e16	0.754527
$335$	−1.32743e16	−0.513121
$336$	0	0
$337$	−3.63316e16	−1.35111	−0.675554	−	0.737311i	$-0.736095\pi$
$337$	−3.63316e16	−1.35111	−0.675554	+	0.737311i	$0.736095\pi$
$338$	−7.68305e15	−0.280269
$339$	0	0
$340$	2.42829e15	0.0852487
$341$	−1.77298e16	−0.610661
$342$	0	0
$343$	3.20294e16	1.06203
$344$	−2.14633e16	−0.698338
$345$	0	0
$346$	−1.41646e15	−0.0443822
$347$	5.25833e15	0.161698	0.0808492	−	0.996726i	$-0.474237\pi$
$347$	5.25833e15	0.161698	0.0808492	+	0.996726i	$0.474237\pi$
$348$	0	0
$349$	2.07739e16	0.615392	0.307696	−	0.951485i	$-0.400442\pi$
$349$	2.07739e16	0.615392	0.307696	+	0.951485i	$0.400442\pi$
$350$	−2.27956e15	−0.0662839
$351$	0	0
$352$	−2.37128e16	−0.664440
$353$	−4.19887e16	−1.15504	−0.577519	−	0.816377i	$-0.695979\pi$
$353$	−4.19887e16	−1.15504	−0.577519	+	0.816377i	$0.695979\pi$
$354$	0	0
$355$	2.87782e16	0.763095
$356$	−2.00770e16	−0.522725
$357$	0	0
$358$	2.89448e16	0.726659
$359$	2.44302e16	0.602301	0.301150	−	0.953577i	$-0.402629\pi$
$359$	2.44302e16	0.602301	0.301150	+	0.953577i	$0.402629\pi$
$360$	0	0
$361$	−4.14968e16	−0.986774
$362$	−2.36649e16	−0.552712
$363$	0	0
$364$	1.55749e16	0.350967
$365$	−2.66868e15	−0.0590734
$366$	0	0
$367$	−5.26281e16	−1.12432	−0.562158	−	0.827030i	$-0.690029\pi$
$367$	−5.26281e16	−1.12432	−0.562158	+	0.827030i	$0.690029\pi$
$368$	−5.41454e15	−0.113645
$369$	0	0
$370$	4.82373e15	0.0977398
$371$	−5.57975e16	−1.11092
$372$	0	0
$373$	9.12086e16	1.75359	0.876796	−	0.480863i	$-0.159676\pi$
$373$	9.12086e16	1.75359	0.876796	+	0.480863i	$0.159676\pi$
$374$	3.93986e15	0.0744415
$375$	0	0
$376$	−5.11315e16	−0.933185
$377$	2.37241e16	0.425570
$378$	0	0
$379$	−2.64638e16	−0.458667	−0.229333	−	0.973348i	$-0.573655\pi$
$379$	−2.64638e16	−0.458667	−0.229333	+	0.973348i	$0.573655\pi$
$380$	2.36072e15	0.0402208
$381$	0	0
$382$	−2.37649e15	−0.0391313
$383$	5.05139e16	0.817746	0.408873	−	0.912591i	$-0.365922\pi$
$383$	5.05139e16	0.817746	0.408873	+	0.912591i	$0.365922\pi$
$384$	0	0
$385$	1.32699e16	0.207669
$386$	4.61686e16	0.710444
$387$	0	0
$388$	4.86589e16	0.724030
$389$	7.19909e15	0.105343	0.0526715	−	0.998612i	$-0.483226\pi$
$389$	7.19909e15	0.105343	0.0526715	+	0.998612i	$0.483226\pi$
$390$	0	0
$391$	4.97257e15	0.0703772
$392$	2.96494e16	0.412721
$393$	0	0
$394$	−1.44212e15	−0.0194212
$395$	6.34049e16	0.839927
$396$	0	0
$397$	3.99263e16	0.511825	0.255912	−	0.966700i	$-0.417624\pi$
$397$	3.99263e16	0.511825	0.255912	+	0.966700i	$0.417624\pi$
$398$	1.70894e16	0.215519
$399$	0	0
$400$	6.44891e15	0.0787221
$401$	6.26829e16	0.752854	0.376427	−	0.926446i	$-0.377153\pi$
$401$	6.26829e16	0.752854	0.376427	+	0.926446i	$0.377153\pi$
$402$	0	0
$403$	5.07537e16	0.590181
$404$	1.19699e17	1.36965
$405$	0	0
$406$	−2.01331e16	−0.223096
$407$	−2.80801e16	−0.306221
$408$	0	0
$409$	−6.76780e16	−0.714901	−0.357451	−	0.933932i	$-0.616354\pi$
$409$	−6.76780e16	−0.714901	−0.357451	+	0.933932i	$0.616354\pi$
$410$	−1.22373e16	−0.127230
$411$	0	0
$412$	2.38667e16	0.240414
$413$	−8.57789e16	−0.850561
$414$	0	0
$415$	1.78781e16	0.171794
$416$	6.78808e16	0.642156
$417$	0	0
$418$	3.83022e15	0.0351219
$419$	5.22210e16	0.471470	0.235735	−	0.971817i	$-0.424250\pi$
$419$	5.22210e16	0.471470	0.235735	+	0.971817i	$0.424250\pi$
$420$	0	0
$421$	−1.95031e17	−1.70714	−0.853571	−	0.520976i	$-0.825568\pi$
$421$	−1.95031e17	−1.70714	−0.853571	+	0.520976i	$0.825568\pi$
$422$	2.07152e15	0.0178549
$423$	0	0
$424$	−1.55772e17	−1.30200
$425$	−5.92250e15	−0.0487503
$426$	0	0
$427$	−2.31330e16	−0.184693
$428$	8.67696e16	0.682312
$429$	0	0
$430$	2.29726e16	0.175253
$431$	−2.88277e16	−0.216625	−0.108312	−	0.994117i	$-0.534545\pi$
$431$	−2.88277e16	−0.216625	−0.108312	+	0.994117i	$0.534545\pi$
$432$	0	0
$433$	4.15115e16	0.302689	0.151345	−	0.988481i	$-0.451640\pi$
$433$	4.15115e16	0.302689	0.151345	+	0.988481i	$0.451640\pi$
$434$	−4.30714e16	−0.309389
$435$	0	0
$436$	4.88331e16	0.340449
$437$	4.83420e15	0.0332044
$438$	0	0
$439$	4.78741e16	0.319214	0.159607	−	0.987181i	$-0.448977\pi$
$439$	4.78741e16	0.319214	0.159607	+	0.987181i	$0.448977\pi$
$440$	3.70460e16	0.243388
$441$	0	0
$442$	−1.12783e16	−0.0719449
$443$	−1.38438e17	−0.870227	−0.435113	−	0.900376i	$-0.643292\pi$
$443$	−1.38438e17	−0.870227	−0.435113	+	0.900376i	$0.643292\pi$
$444$	0	0
$445$	4.89669e16	0.298925
$446$	1.03797e17	0.624466
$447$	0	0
$448$	−9.79180e15	−0.0572210
$449$	1.55000e17	0.892749	0.446375	−	0.894846i	$-0.352715\pi$
$449$	1.55000e17	0.892749	0.446375	+	0.894846i	$0.352715\pi$
$450$	0	0
$451$	7.12361e16	0.398614
$452$	1.33636e17	0.737094
$453$	0	0
$454$	−3.65173e16	−0.195720
$455$	−3.79866e16	−0.200704
$456$	0	0
$457$	1.98612e17	1.01988	0.509942	−	0.860209i	$-0.329667\pi$
$457$	1.98612e17	1.01988	0.509942	+	0.860209i	$0.329667\pi$
$458$	7.04615e16	0.356719
$459$	0	0
$460$	2.05188e16	0.100978
$461$	−2.98129e17	−1.44660	−0.723299	−	0.690535i	$-0.757375\pi$
$461$	−2.98129e17	−1.44660	−0.723299	+	0.690535i	$0.757375\pi$
$462$	0	0
$463$	−1.19526e17	−0.563880	−0.281940	−	0.959432i	$-0.590978\pi$
$463$	−1.19526e17	−0.563880	−0.281940	+	0.959432i	$0.590978\pi$
$464$	5.69569e16	0.264959
$465$	0	0
$466$	−5.01845e15	−0.0227018
$467$	2.04536e16	0.0912450	0.0456225	−	0.998959i	$-0.485473\pi$
$467$	2.04536e16	0.0912450	0.0456225	+	0.998959i	$0.485473\pi$
$468$	0	0
$469$	1.87720e17	0.814494
$470$	5.47272e16	0.234189
$471$	0	0
$472$	−2.39472e17	−0.996856
$473$	−1.33729e17	−0.549071
$474$	0	0
$475$	−5.75770e15	−0.0230007
$476$	−3.43401e16	−0.135318
$477$	0	0
$478$	−8.64146e16	−0.331364
$479$	−3.74017e17	−1.41485	−0.707425	−	0.706789i	$-0.750143\pi$
$479$	−3.74017e17	−1.41485	−0.707425	+	0.706789i	$0.750143\pi$
$480$	0	0
$481$	8.03827e16	0.295951
$482$	−2.04873e17	−0.744182
$483$	0	0
$484$	1.26529e17	0.447398
$485$	−1.18677e17	−0.414044
$486$	0	0
$487$	1.68497e17	0.572341	0.286170	−	0.958179i	$-0.407618\pi$
$487$	1.68497e17	0.572341	0.286170	+	0.958179i	$0.407618\pi$
$488$	−6.45812e16	−0.216460
$489$	0	0
$490$	−3.17344e16	−0.103575
$491$	−4.12975e17	−1.33013	−0.665065	−	0.746785i	$-0.731596\pi$
$491$	−4.12975e17	−1.33013	−0.665065	+	0.746785i	$0.731596\pi$
$492$	0	0
$493$	−5.23076e16	−0.164082
$494$	−1.09645e16	−0.0339440
$495$	0	0
$496$	1.21850e17	0.367446
$497$	−4.06971e17	−1.21129
$498$	0	0
$499$	7.46128e16	0.216351	0.108176	−	0.994132i	$-0.465499\pi$
$499$	7.46128e16	0.216351	0.108176	+	0.994132i	$0.465499\pi$
$500$	−2.44386e16	−0.0699472
$501$	0	0
$502$	1.02516e17	0.285903
$503$	1.41500e17	0.389551	0.194775	−	0.980848i	$-0.437602\pi$
$503$	1.41500e17	0.389551	0.194775	+	0.980848i	$0.437602\pi$
$504$	0	0
$505$	−2.91940e17	−0.783248
$506$	3.32913e16	0.0881764
$507$	0	0
$508$	−2.32659e17	−0.600629
$509$	−3.14891e17	−0.802591	−0.401295	−	0.915949i	$-0.631440\pi$
$509$	−3.14891e17	−0.802591	−0.401295	+	0.915949i	$0.631440\pi$
$510$	0	0
$511$	3.77395e16	0.0937693
$512$	2.96453e17	0.727280
$513$	0	0
$514$	3.23765e17	0.774409
$515$	−5.82099e16	−0.137483
$516$	0	0
$517$	−3.18580e17	−0.733720
$518$	−6.82155e16	−0.155146
$519$	0	0
$520$	−1.06049e17	−0.235225
$521$	5.54510e17	1.21469	0.607343	−	0.794440i	$-0.292235\pi$
$521$	5.54510e17	1.21469	0.607343	+	0.794440i	$0.292235\pi$
$522$	0	0
$523$	−3.39258e17	−0.724886	−0.362443	−	0.932006i	$-0.618057\pi$
$523$	−3.39258e17	−0.724886	−0.362443	+	0.932006i	$0.618057\pi$
$524$	−1.32438e17	−0.279486
$525$	0	0
$526$	4.09571e17	0.843184
$527$	−1.11903e17	−0.227549
$528$	0	0
$529$	−4.62019e17	−0.916638
$530$	1.66726e17	0.326746
$531$	0	0
$532$	−3.33845e16	−0.0638439
$533$	−2.03922e17	−0.385246
$534$	0	0
$535$	−2.11628e17	−0.390187
$536$	5.24066e17	0.954586
$537$	0	0
$538$	3.20859e17	0.570466
$539$	1.84734e17	0.324504
$540$	0	0
$541$	8.10224e17	1.38939	0.694693	−	0.719306i	$-0.255540\pi$
$541$	8.10224e17	1.38939	0.694693	+	0.719306i	$0.255540\pi$
$542$	−2.78970e17	−0.472675
$543$	0	0
$544$	−1.49666e17	−0.247588
$545$	−1.19102e17	−0.194689
$546$	0	0
$547$	−2.02195e17	−0.322740	−0.161370	−	0.986894i	$-0.551591\pi$
$547$	−2.02195e17	−0.322740	−0.161370	+	0.986894i	$0.551591\pi$
$548$	4.88412e17	0.770393
$549$	0	0
$550$	−3.96511e16	−0.0610798
$551$	−5.08521e16	−0.0774147
$552$	0	0
$553$	−8.96651e17	−1.33324
$554$	7.96073e16	0.116987
$555$	0	0
$556$	6.85687e16	0.0984327
$557$	9.60827e17	1.36328	0.681642	−	0.731686i	$-0.261266\pi$
$557$	9.60827e17	1.36328	0.681642	+	0.731686i	$0.261266\pi$
$558$	0	0
$559$	3.82816e17	0.530656
$560$	−9.11984e16	−0.124958
$561$	0	0
$562$	4.55164e17	0.609371
$563$	1.00972e18	1.33627	0.668137	−	0.744038i	$-0.267092\pi$
$563$	1.00972e18	1.33627	0.668137	+	0.744038i	$0.267092\pi$
$564$	0	0
$565$	−3.25932e17	−0.421515
$566$	6.30758e17	0.806411
$567$	0	0
$568$	−1.13616e18	−1.41963
$569$	−4.86979e17	−0.601562	−0.300781	−	0.953693i	$-0.597247\pi$
$569$	−4.86979e17	−0.601562	−0.300781	+	0.953693i	$0.597247\pi$
$570$	0	0
$571$	7.63962e17	0.922438	0.461219	−	0.887286i	$-0.347412\pi$
$571$	7.63962e17	0.922438	0.461219	+	0.887286i	$0.347412\pi$
$572$	2.70914e17	0.323412
$573$	0	0
$574$	1.73055e17	0.201957
$575$	−5.00444e16	−0.0577450
$576$	0	0
$577$	1.24108e18	1.40009	0.700046	−	0.714098i	$-0.253163\pi$
$577$	1.24108e18	1.40009	0.700046	+	0.714098i	$0.253163\pi$
$578$	−3.93662e17	−0.439130
$579$	0	0
$580$	−2.15842e17	−0.235425
$581$	−2.52826e17	−0.272695
$582$	0	0
$583$	−9.70554e17	−1.02370
$584$	1.05359e17	0.109898
$585$	0	0
$586$	6.42332e17	0.655278
$587$	2.75161e17	0.277612	0.138806	−	0.990320i	$-0.455673\pi$
$587$	2.75161e17	0.277612	0.138806	+	0.990320i	$0.455673\pi$
$588$	0	0
$589$	−1.08790e17	−0.107359
$590$	2.56312e17	0.250168
$591$	0	0
$592$	1.92983e17	0.184259
$593$	1.41667e18	1.33787	0.668936	−	0.743320i	$-0.266750\pi$
$593$	1.41667e18	1.33787	0.668936	+	0.743320i	$0.266750\pi$
$594$	0	0
$595$	8.37541e16	0.0773830
$596$	−1.44061e18	−1.31657
$597$	0	0
$598$	−9.53005e16	−0.0852192
$599$	1.35996e18	1.20296	0.601480	−	0.798888i	$-0.294578\pi$
$599$	1.35996e18	1.20296	0.601480	+	0.798888i	$0.294578\pi$
$600$	0	0
$601$	7.71001e17	0.667377	0.333688	−	0.942683i	$-0.391707\pi$
$601$	7.71001e17	0.667377	0.333688	+	0.942683i	$0.391707\pi$
$602$	−3.24871e17	−0.278185
$603$	0	0
$604$	4.13265e17	0.346329
$605$	−3.08599e17	−0.255849
$606$	0	0
$607$	1.45183e18	1.17812	0.589060	−	0.808089i	$-0.299498\pi$
$607$	1.45183e18	1.17812	0.589060	+	0.808089i	$0.299498\pi$
$608$	−1.45501e17	−0.116814
$609$	0	0
$610$	6.91227e16	0.0543221
$611$	9.11975e17	0.709113
$612$	0	0
$613$	−2.05392e18	−1.56347	−0.781735	−	0.623611i	$-0.785665\pi$
$613$	−2.05392e18	−1.56347	−0.781735	+	0.623611i	$0.785665\pi$
$614$	−7.70093e17	−0.580027
$615$	0	0
$616$	−5.23892e17	−0.386338
$617$	1.33334e18	0.972942	0.486471	−	0.873697i	$-0.338284\pi$
$617$	1.33334e18	0.972942	0.486471	+	0.873697i	$0.338284\pi$
$618$	0	0
$619$	9.38644e16	0.0670674	0.0335337	−	0.999438i	$-0.489324\pi$
$619$	9.38644e16	0.0670674	0.0335337	+	0.999438i	$0.489324\pi$
$620$	−4.61757e17	−0.326488
$621$	0	0
$622$	5.13968e17	0.355876
$623$	−6.92474e17	−0.474494
$624$	0	0
$625$	5.96046e16	0.0400000
$626$	1.26508e18	0.840202
$627$	0	0
$628$	−1.84499e17	−0.120021
$629$	−1.77230e17	−0.114106
$630$	0	0
$631$	9.75436e17	0.615188	0.307594	−	0.951518i	$-0.400476\pi$
$631$	9.75436e17	0.615188	0.307594	+	0.951518i	$0.400476\pi$
$632$	−2.50321e18	−1.56256
$633$	0	0
$634$	−3.41202e17	−0.208656
$635$	5.67446e17	0.343475
$636$	0	0
$637$	−5.28823e17	−0.313620
$638$	−3.50199e17	−0.205580
$639$	0	0
$640$	−7.60451e17	−0.437423
$641$	−2.15983e18	−1.22982	−0.614911	−	0.788597i	$-0.710808\pi$
$641$	−2.15983e18	−1.22982	−0.614911	+	0.788597i	$0.710808\pi$
$642$	0	0
$643$	−2.90222e18	−1.61942	−0.809709	−	0.586832i	$-0.800375\pi$
$643$	−2.90222e18	−1.61942	−0.809709	+	0.586832i	$0.800375\pi$
$644$	−2.90169e17	−0.160285
$645$	0	0
$646$	2.41748e16	0.0130874
$647$	8.16394e17	0.437544	0.218772	−	0.975776i	$-0.429795\pi$
$647$	8.16394e17	0.437544	0.218772	+	0.975776i	$0.429795\pi$
$648$	0	0
$649$	−1.49206e18	−0.783782
$650$	1.13506e17	0.0590314
$651$	0	0
$652$	6.55308e17	0.334069
$653$	−3.16421e18	−1.59709	−0.798546	−	0.601934i	$-0.794397\pi$
$653$	−3.16421e18	−1.59709	−0.798546	+	0.601934i	$0.794397\pi$
$654$	0	0
$655$	3.23012e17	0.159827
$656$	−4.89577e17	−0.239854
$657$	0	0
$658$	−7.73934e17	−0.371737
$659$	−1.12170e18	−0.533484	−0.266742	−	0.963768i	$-0.585947\pi$
$659$	−1.12170e18	−0.533484	−0.266742	+	0.963768i	$0.585947\pi$
$660$	0	0
$661$	2.30711e18	1.07587	0.537934	−	0.842987i	$-0.319205\pi$
$661$	2.30711e18	1.07587	0.537934	+	0.842987i	$0.319205\pi$
$662$	−1.55025e18	−0.715855
$663$	0	0
$664$	−7.05824e17	−0.319598
$665$	8.14235e16	0.0365098
$666$	0	0
$667$	−4.41993e17	−0.194356
$668$	−2.90237e18	−1.26388
$669$	0	0
$670$	−5.60919e17	−0.239560
$671$	−4.02380e17	−0.170192
$672$	0	0
$673$	3.15520e18	1.30897	0.654483	−	0.756076i	$-0.272886\pi$
$673$	3.15520e18	1.30897	0.654483	+	0.756076i	$0.272886\pi$
$674$	−1.53523e18	−0.630790
$675$	0	0
$676$	1.16482e18	0.469468
$677$	−3.06808e18	−1.22473	−0.612366	−	0.790575i	$-0.709782\pi$
$677$	−3.06808e18	−1.22473	−0.612366	+	0.790575i	$0.709782\pi$
$678$	0	0
$679$	1.67829e18	0.657226
$680$	2.33819e17	0.0906928
$681$	0	0
$682$	−7.49193e17	−0.285099
$683$	2.64042e18	0.995265	0.497633	−	0.867388i	$-0.334203\pi$
$683$	2.64042e18	0.995265	0.497633	+	0.867388i	$0.334203\pi$
$684$	0	0
$685$	−1.19122e18	−0.440557
$686$	1.35344e18	0.495828
$687$	0	0
$688$	9.19065e17	0.330386
$689$	2.77833e18	0.989370
$690$	0	0
$691$	−1.32965e18	−0.464653	−0.232326	−	0.972638i	$-0.574634\pi$
$691$	−1.32965e18	−0.464653	−0.232326	+	0.972638i	$0.574634\pi$
$692$	2.14748e17	0.0743430
$693$	0	0
$694$	2.22196e17	0.0754919
$695$	−1.67236e17	−0.0562897
$696$	0	0
$697$	4.49614e17	0.148534
$698$	8.77823e17	0.287307
$699$	0	0
$700$	3.45602e17	0.111030
$701$	3.55951e17	0.113298	0.0566492	−	0.998394i	$-0.481958\pi$
$701$	3.55951e17	0.113298	0.0566492	+	0.998394i	$0.481958\pi$
$702$	0	0
$703$	−1.72299e17	−0.0538359
$704$	−1.70321e17	−0.0527285
$705$	0	0
$706$	−1.77428e18	−0.539251
$707$	4.12852e18	1.24328
$708$	0	0
$709$	1.33087e17	0.0393490	0.0196745	−	0.999806i	$-0.493737\pi$
$709$	1.33087e17	0.0393490	0.0196745	+	0.999806i	$0.493737\pi$
$710$	1.21605e18	0.356265
$711$	0	0
$712$	−1.93321e18	−0.556107
$713$	−9.45570e17	−0.269533
$714$	0	0
$715$	−6.60747e17	−0.184947
$716$	−4.38830e18	−1.21720
$717$	0	0
$718$	1.03233e18	0.281195
$719$	7.23572e18	1.95319	0.976594	−	0.215091i	$-0.0690049\pi$
$719$	7.23572e18	1.95319	0.976594	+	0.215091i	$0.0690049\pi$
$720$	0	0
$721$	8.23185e17	0.218232
$722$	−1.75349e18	−0.460694
$723$	0	0
$724$	3.58781e18	0.925826
$725$	5.26429e17	0.134630
$726$	0	0
$727$	−7.32402e17	−0.183982	−0.0919911	−	0.995760i	$-0.529323\pi$
$727$	−7.32402e17	−0.183982	−0.0919911	+	0.995760i	$0.529323\pi$
$728$	1.49971e18	0.373381
$729$	0	0
$730$	−1.12768e17	−0.0275795
$731$	−8.44044e17	−0.204598
$732$	0	0
$733$	5.01160e18	1.19344	0.596719	−	0.802450i	$-0.296471\pi$
$733$	5.01160e18	1.19344	0.596719	+	0.802450i	$0.296471\pi$
$734$	−2.22386e18	−0.524908
$735$	0	0
$736$	−1.26466e18	−0.293270
$737$	3.26525e18	0.750547
$738$	0	0
$739$	1.16075e18	0.262149	0.131075	−	0.991372i	$-0.458157\pi$
$739$	1.16075e18	0.262149	0.131075	+	0.991372i	$0.458157\pi$
$740$	−7.31321e17	−0.163720
$741$	0	0
$742$	−2.35779e18	−0.518655
$743$	−1.15591e17	−0.0252056	−0.0126028	−	0.999921i	$-0.504012\pi$
$743$	−1.15591e17	−0.0252056	−0.0126028	+	0.999921i	$0.504012\pi$
$744$	0	0
$745$	3.51358e18	0.752895
$746$	3.85412e18	0.818697
$747$	0	0
$748$	−5.97319e17	−0.124694
$749$	2.99277e18	0.619357
$750$	0	0
$751$	−5.19563e18	−1.05676	−0.528382	−	0.849006i	$-0.677201\pi$
$751$	−5.19563e18	−1.05676	−0.528382	+	0.849006i	$0.677201\pi$
$752$	2.18947e18	0.441493
$753$	0	0
$754$	1.00249e18	0.198685
$755$	−1.00794e18	−0.198052
$756$	0	0
$757$	−1.48614e18	−0.287037	−0.143518	−	0.989648i	$-0.545842\pi$
$757$	−1.48614e18	−0.287037	−0.143518	+	0.989648i	$0.545842\pi$
$758$	−1.11826e18	−0.214137
$759$	0	0
$760$	2.27313e17	0.0427894
$761$	−2.23421e17	−0.0416988	−0.0208494	−	0.999783i	$-0.506637\pi$
$761$	−2.23421e17	−0.0416988	−0.0208494	+	0.999783i	$0.506637\pi$
$762$	0	0
$763$	1.68430e18	0.309037
$764$	3.60298e17	0.0655474
$765$	0	0
$766$	2.13452e18	0.381780
$767$	4.27120e18	0.757495
$768$	0	0
$769$	−3.89476e18	−0.679140	−0.339570	−	0.940581i	$-0.610281\pi$
$769$	−3.89476e18	−0.679140	−0.339570	+	0.940581i	$0.610281\pi$
$770$	5.60733e17	0.0969541
$771$	0	0
$772$	−6.99959e18	−1.19004
$773$	3.27604e18	0.552310	0.276155	−	0.961113i	$-0.410940\pi$
$773$	3.27604e18	0.552310	0.276155	+	0.961113i	$0.410940\pi$
$774$	0	0
$775$	1.12621e18	0.186706
$776$	4.68536e18	0.770268
$777$	0	0
$778$	3.04206e17	0.0491813
$779$	4.37103e17	0.0700794
$780$	0	0
$781$	−7.07895e18	−1.11619
$782$	2.10121e17	0.0328569
$783$	0	0
$784$	−1.26960e18	−0.195260
$785$	4.49985e17	0.0686349
$786$	0	0
$787$	5.00117e18	0.750302	0.375151	−	0.926964i	$-0.377591\pi$
$787$	5.00117e18	0.750302	0.375151	+	0.926964i	$0.377591\pi$
$788$	2.18638e17	0.0325316
$789$	0	0
$790$	2.67924e18	0.392136
$791$	4.60923e18	0.669085
$792$	0	0
$793$	1.15186e18	0.164484
$794$	1.68713e18	0.238955
$795$	0	0
$796$	−2.59090e18	−0.361008
$797$	−8.18062e18	−1.13060	−0.565298	−	0.824887i	$-0.691239\pi$
$797$	−8.18062e18	−1.13060	−0.565298	+	0.824887i	$0.691239\pi$
$798$	0	0
$799$	−2.01075e18	−0.273404
$800$	1.50625e18	0.203148
$801$	0	0
$802$	2.64874e18	0.351484
$803$	6.56450e17	0.0864073
$804$	0	0
$805$	7.07712e17	0.0916607
$806$	2.14466e18	0.275537
$807$	0	0
$808$	1.15257e19	1.45712
$809$	−1.09399e19	−1.37198	−0.685992	−	0.727609i	$-0.740632\pi$
$809$	−1.09399e19	−1.37198	−0.685992	+	0.727609i	$0.740632\pi$
$810$	0	0
$811$	−1.19169e19	−1.47071	−0.735353	−	0.677684i	$-0.762984\pi$
$811$	−1.19169e19	−1.47071	−0.735353	+	0.677684i	$0.762984\pi$
$812$	3.05236e18	0.373699
$813$	0	0
$814$	−1.18656e18	−0.142965
$815$	−1.59827e18	−0.191041
$816$	0	0
$817$	−8.20557e17	−0.0965307
$818$	−2.85981e18	−0.333765
$819$	0	0
$820$	1.85528e18	0.213118
$821$	−1.08272e19	−1.23392	−0.616960	−	0.786995i	$-0.711636\pi$
$821$	−1.08272e19	−1.23392	−0.616960	+	0.786995i	$0.711636\pi$
$822$	0	0
$823$	1.07514e19	1.20605	0.603024	−	0.797723i	$-0.293962\pi$
$823$	1.07514e19	1.20605	0.603024	+	0.797723i	$0.293962\pi$
$824$	2.29812e18	0.255768
$825$	0	0
$826$	−3.62469e18	−0.397100
$827$	−7.44229e18	−0.808949	−0.404474	−	0.914549i	$-0.632546\pi$
$827$	−7.44229e18	−0.808949	−0.404474	+	0.914549i	$0.632546\pi$
$828$	0	0
$829$	−1.14183e19	−1.22180	−0.610898	−	0.791709i	$-0.709191\pi$
$829$	−1.14183e19	−1.22180	−0.610898	+	0.791709i	$0.709191\pi$
$830$	7.55458e17	0.0802053
$831$	0	0
$832$	4.87564e17	0.0509601
$833$	1.16597e18	0.120919
$834$	0	0
$835$	7.07876e18	0.722761
$836$	−5.80697e17	−0.0588314
$837$	0	0
$838$	2.20666e18	0.220115
$839$	−1.28197e19	−1.26889	−0.634445	−	0.772968i	$-0.718771\pi$
$839$	−1.28197e19	−1.26889	−0.634445	+	0.772968i	$0.718771\pi$
$840$	0	0
$841$	−5.61120e18	−0.546867
$842$	−8.24125e18	−0.797011
$843$	0	0
$844$	−3.14061e17	−0.0299081
$845$	−2.84095e18	−0.268470
$846$	0	0
$847$	4.36410e18	0.406118
$848$	6.67022e18	0.615981
$849$	0	0
$850$	−2.50262e17	−0.0227600
$851$	−1.49757e18	−0.135159
$852$	0	0
$853$	1.69312e19	1.50494	0.752470	−	0.658626i	$-0.228862\pi$
$853$	1.69312e19	1.50494	0.752470	+	0.658626i	$0.228862\pi$
$854$	−9.77510e17	−0.0862274
$855$	0	0
$856$	8.35503e18	0.725886
$857$	−5.86073e18	−0.505332	−0.252666	−	0.967554i	$-0.581307\pi$
$857$	−5.86073e18	−0.505332	−0.252666	+	0.967554i	$0.581307\pi$
$858$	0	0
$859$	−8.99580e17	−0.0763984	−0.0381992	−	0.999270i	$-0.512162\pi$
$859$	−8.99580e17	−0.0763984	−0.0381992	+	0.999270i	$0.512162\pi$
$860$	−3.48286e18	−0.293559
$861$	0	0
$862$	−1.21815e18	−0.101135
$863$	−1.06834e19	−0.880321	−0.440161	−	0.897919i	$-0.645078\pi$
$863$	−1.06834e19	−0.880321	−0.440161	+	0.897919i	$0.645078\pi$
$864$	0	0
$865$	−5.23762e17	−0.0425138
$866$	1.75412e18	0.141316
$867$	0	0
$868$	6.53002e18	0.518246
$869$	−1.55965e19	−1.22857
$870$	0	0
$871$	−9.34718e18	−0.725375
$872$	4.70213e18	0.362190
$873$	0	0
$874$	2.04274e17	0.0155021
$875$	−8.42909e17	−0.0634934
$876$	0	0
$877$	−5.33360e17	−0.0395843	−0.0197921	−	0.999804i	$-0.506300\pi$
$877$	−5.33360e17	−0.0395843	−0.0197921	+	0.999804i	$0.506300\pi$
$878$	2.02298e18	0.149031
$879$	0	0
$880$	−1.58632e18	−0.115148
$881$	−2.10202e19	−1.51459	−0.757293	−	0.653075i	$-0.773479\pi$
$881$	−2.10202e19	−1.51459	−0.757293	+	0.653075i	$0.773479\pi$
$882$	0	0
$883$	−4.99763e18	−0.354829	−0.177415	−	0.984136i	$-0.556773\pi$
$883$	−4.99763e18	−0.354829	−0.177415	+	0.984136i	$0.556773\pi$
$884$	1.70990e18	0.120512
$885$	0	0
$886$	−5.84987e18	−0.406282
$887$	2.22662e19	1.53512	0.767559	−	0.640978i	$-0.221471\pi$
$887$	2.22662e19	1.53512	0.767559	+	0.640978i	$0.221471\pi$
$888$	0	0
$889$	−8.02464e18	−0.545210
$890$	2.06915e18	0.139559
$891$	0	0
$892$	−1.57366e19	−1.04602
$893$	−1.95480e18	−0.128993
$894$	0	0
$895$	1.07029e19	0.696067
$896$	1.07540e19	0.694336
$897$	0	0
$898$	6.54968e18	0.416797
$899$	9.94668e18	0.628405
$900$	0	0
$901$	−6.12575e18	−0.381459
$902$	3.01016e18	0.186101
$903$	0	0
$904$	1.28678e19	0.784167
$905$	−8.75054e18	−0.529443
$906$	0	0
$907$	−1.83122e19	−1.09218	−0.546089	−	0.837727i	$-0.683884\pi$
$907$	−1.83122e19	−1.09218	−0.546089	+	0.837727i	$0.683884\pi$
$908$	5.53637e18	0.327844
$909$	0	0
$910$	−1.60517e18	−0.0937025
$911$	2.18746e19	1.26786	0.633930	−	0.773391i	$-0.281441\pi$
$911$	2.18746e19	1.26786	0.633930	+	0.773391i	$0.281441\pi$
$912$	0	0
$913$	−4.39771e18	−0.251285
$914$	8.39259e18	0.476152
$915$	0	0
$916$	−1.06826e19	−0.597527
$917$	−4.56793e18	−0.253699
$918$	0	0
$919$	1.68210e19	0.921086	0.460543	−	0.887637i	$-0.347655\pi$
$919$	1.68210e19	0.921086	0.460543	+	0.887637i	$0.347655\pi$
$920$	1.97575e18	0.107426
$921$	0	0
$922$	−1.25978e19	−0.675371
$923$	2.02644e19	1.07875
$924$	0	0
$925$	1.78366e18	0.0936250
$926$	−5.05072e18	−0.263258
$927$	0	0
$928$	1.33032e19	0.683747
$929$	−9.07747e18	−0.463300	−0.231650	−	0.972799i	$-0.574412\pi$
$929$	−9.07747e18	−0.463300	−0.231650	+	0.972799i	$0.574412\pi$
$930$	0	0
$931$	1.13352e18	0.0570502
$932$	7.60843e17	0.0380270
$933$	0	0
$934$	8.64288e17	0.0425995
$935$	1.45684e18	0.0713075
$936$	0	0
$937$	1.67978e19	0.810859	0.405430	−	0.914126i	$-0.367122\pi$
$937$	1.67978e19	0.810859	0.405430	+	0.914126i	$0.367122\pi$
$938$	7.93233e18	0.380262
$939$	0	0
$940$	−8.29714e18	−0.392281
$941$	−3.44047e18	−0.161542	−0.0807710	−	0.996733i	$-0.525738\pi$
$941$	−3.44047e18	−0.161542	−0.0807710	+	0.996733i	$0.525738\pi$
$942$	0	0
$943$	3.79918e18	0.175940
$944$	1.02543e19	0.471616
$945$	0	0
$946$	−5.65087e18	−0.256344
$947$	1.58887e18	0.0715836	0.0357918	−	0.999359i	$-0.488605\pi$
$947$	1.58887e18	0.0715836	0.0357918	+	0.999359i	$0.488605\pi$
$948$	0	0
$949$	−1.87917e18	−0.0835094
$950$	−2.43298e17	−0.0107383
$951$	0	0
$952$	−3.30660e18	−0.143960
$953$	−3.43986e19	−1.48743	−0.743715	−	0.668496i	$-0.766938\pi$
$953$	−3.43986e19	−1.48743	−0.743715	+	0.668496i	$0.766938\pi$
$954$	0	0
$955$	−8.78753e17	−0.0374839
$956$	1.31013e19	0.555056
$957$	0	0
$958$	−1.58045e19	−0.660549
$959$	1.68458e19	0.699311
$960$	0	0
$961$	−3.13829e18	−0.128526
$962$	3.39666e18	0.138170
$963$	0	0
$964$	3.10606e19	1.24655
$965$	1.70717e19	0.680534
$966$	0	0
$967$	−2.45546e18	−0.0965740	−0.0482870	−	0.998834i	$-0.515376\pi$
$967$	−2.45546e18	−0.0965740	−0.0482870	+	0.998834i	$0.515376\pi$
$968$	1.21834e19	0.475970
$969$	0	0
$970$	−5.01484e18	−0.193304
$971$	1.90983e19	0.731256	0.365628	−	0.930761i	$-0.380854\pi$
$971$	1.90983e19	0.731256	0.365628	+	0.930761i	$0.380854\pi$
$972$	0	0
$973$	2.36500e18	0.0893506
$974$	7.12005e18	0.267208
$975$	0	0
$976$	2.76540e18	0.102408
$977$	−2.24677e19	−0.826504	−0.413252	−	0.910617i	$-0.635607\pi$
$977$	−2.24677e19	−0.826504	−0.413252	+	0.910617i	$0.635607\pi$
$978$	0	0
$979$	−1.20450e19	−0.437241
$980$	4.81123e18	0.173495
$981$	0	0
$982$	−1.74507e19	−0.620996
$983$	4.38754e19	1.55104	0.775521	−	0.631322i	$-0.217487\pi$
$983$	4.38754e19	1.55104	0.775521	+	0.631322i	$0.217487\pi$
$984$	0	0
$985$	−5.33250e17	−0.0186035
$986$	−2.21032e18	−0.0766046
$987$	0	0
$988$	1.66232e18	0.0568583
$989$	−7.13207e18	−0.242348
$990$	0	0
$991$	4.35327e19	1.45994	0.729972	−	0.683477i	$-0.239533\pi$
$991$	4.35327e19	1.45994	0.729972	+	0.683477i	$0.239533\pi$
$992$	2.84600e19	0.948221
$993$	0	0
$994$	−1.71970e19	−0.565512
$995$	6.31911e18	0.206446
$996$	0	0
$997$	2.87888e19	0.928335	0.464168	−	0.885747i	$-0.346354\pi$
$997$	2.87888e19	0.928335	0.464168	+	0.885747i	$0.346354\pi$
$998$	3.15285e18	0.101008
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.14.a.g.1.3	✓	4	1.1	even	1	trivial
45.14.a.h.1.2	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)

\(f(q)\)	\(=\)	\(q+42.2561 q^{2} -6406.42 q^{4} +15625.0 q^{5} -220964. q^{7} -616873. q^{8} +660252. q^{10} -3.84349e6 q^{11} +1.10025e7 q^{13} -9.33707e6 q^{14} +2.64147e7 q^{16} -2.42586e7 q^{17} -2.35835e7 q^{19} -1.00100e8 q^{20} -1.62411e8 q^{22} -2.04982e8 q^{23} +2.44141e8 q^{25} +4.64921e8 q^{26} +1.41559e9 q^{28} +2.15625e9 q^{29} +4.61294e9 q^{31} +6.16961e9 q^{32} -1.02507e9 q^{34} -3.45256e9 q^{35} +7.30589e9 q^{37} -9.96549e8 q^{38} -9.63864e9 q^{40} -1.85342e10 q^{41} +3.47937e10 q^{43} +2.46230e10 q^{44} -8.66174e9 q^{46} +8.28883e10 q^{47} -4.80641e10 q^{49} +1.03164e10 q^{50} -7.04864e10 q^{52} +2.52519e11 q^{53} -6.00545e10 q^{55} +1.36306e11 q^{56} +9.11149e10 q^{58} +3.88204e11 q^{59} +1.04691e11 q^{61} +1.94925e11 q^{62} +4.43141e10 q^{64} +1.71913e11 q^{65} -8.49553e11 q^{67} +1.55411e11 q^{68} -1.45892e11 q^{70} +1.84180e12 q^{71} -1.70795e11 q^{73} +3.08718e11 q^{74} +1.51086e11 q^{76} +8.49271e11 q^{77} +4.05791e12 q^{79} +4.12730e11 q^{80} -7.83185e11 q^{82} +1.14420e12 q^{83} -3.79040e11 q^{85} +1.47024e12 q^{86} +2.37094e12 q^{88} +3.13388e12 q^{89} -2.43114e12 q^{91} +1.31320e12 q^{92} +3.50254e12 q^{94} -3.68493e11 q^{95} -7.59534e12 q^{97} -2.03100e12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 15 q^{2} + 16837 q^{4} + 62500 q^{5} + 343040 q^{7} - 14865 q^{8} - 234375 q^{10} + 12697800 q^{11} + 34336040 q^{13} + 26944650 q^{14} + 66562801 q^{16} + 84377280 q^{17} - 131821144 q^{19} + 263078125 q^{20}+ \cdots - 17650752985395 q^{98}+O(q^{100}) \) 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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