LMFDB - Embedded newform 45.14.a.e.1.1

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 = [3,-142] 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:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4466x - 18720 $ x^3 - x^2 - 4466*x - 18720
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-64.1084$ 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)

$f(q)$	$=$	\(q-176.217 q^{2} +22860.4 q^{4} -15625.0 q^{5} -201493. q^{7} -2.58481e6 q^{8} +2.75339e6 q^{10} +3.34359e6 q^{11} -7.80359e6 q^{13} +3.55065e7 q^{14} +2.68215e8 q^{16} +8.71750e7 q^{17} -1.66766e7 q^{19} -3.57193e8 q^{20} -5.89196e8 q^{22} -1.13518e9 q^{23} +2.44141e8 q^{25} +1.37512e9 q^{26} -4.60620e9 q^{28} -2.60673e9 q^{29} +8.33139e8 q^{31} -2.60892e10 q^{32} -1.53617e10 q^{34} +3.14833e9 q^{35} -1.05494e10 q^{37} +2.93870e9 q^{38} +4.03877e10 q^{40} +4.33030e9 q^{41} +1.93854e9 q^{43} +7.64356e10 q^{44} +2.00038e11 q^{46} -2.85468e10 q^{47} -5.62896e10 q^{49} -4.30217e10 q^{50} -1.78393e11 q^{52} -1.23249e11 q^{53} -5.22435e10 q^{55} +5.20821e11 q^{56} +4.59349e11 q^{58} +5.55404e11 q^{59} -4.10476e11 q^{61} -1.46813e11 q^{62} +2.40014e12 q^{64} +1.21931e11 q^{65} +3.36861e11 q^{67} +1.99285e12 q^{68} -5.54788e11 q^{70} -1.57323e12 q^{71} +2.05372e12 q^{73} +1.85898e12 q^{74} -3.81234e11 q^{76} -6.73709e11 q^{77} -6.93000e11 q^{79} -4.19086e12 q^{80} -7.63071e11 q^{82} -2.01116e12 q^{83} -1.36211e12 q^{85} -3.41604e11 q^{86} -8.64253e12 q^{88} +8.51832e12 q^{89} +1.57237e12 q^{91} -2.59507e13 q^{92} +5.03042e12 q^{94} +2.60572e11 q^{95} -7.99814e12 q^{97} +9.91916e12 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 142 q^{2} + 17876 q^{4} - 46875 q^{5} + 448292 q^{7} - 2580360 q^{8} + 2218750 q^{10} + 6604004 q^{11} - 33501974 q^{13} + 46562928 q^{14} + 199912208 q^{16} - 83129542 q^{17} + 97491100 q^{19} - 279312500 q^{20}+ \cdots + 10176508990306 q^{98}+O(q^{100}) $ 3 * q - 142 * q^2 + 17876 * q^4 - 46875 * q^5 + 448292 * q^7 - 2580360 * q^8 + 2218750 * q^10 + 6604004 * q^11 - 33501974 * q^13 + 46562928 * q^14 + 199912208 * q^16 - 83129542 * q^17 + 97491100 * q^19 - 279312500 * q^20 - 530907256 * q^22 - 316255836 * q^23 + 732421875 * q^25 + 3746814044 * q^26 - 6227646976 * q^28 - 2236171850 * q^29 + 7482994376 * q^31 - 32169857312 * q^32 - 30236073988 * q^34 - 7004562500 * q^35 + 31447174242 * q^37 + 42518132360 * q^38 + 40318125000 * q^40 + 10752884434 * q^41 + 16930554856 * q^43 + 68395825168 * q^44 + 246828204336 * q^46 - 31934201692 * q^47 - 38956926629 * q^49 - 34667968750 * q^50 - 18152597928 * q^52 + 221149123934 * q^53 - 103187562500 * q^55 + 522876451200 * q^56 + 854269976260 * q^58 + 55436423900 * q^59 + 496161392746 * q^61 + 598069940736 * q^62 + 3297565494336 * q^64 + 523468343750 * q^65 + 459297824792 * q^67 + 2008021178776 * q^68 - 727545750000 * q^70 - 521997878336 * q^71 + 2505025571086 * q^73 + 2418323586508 * q^74 + 621688050000 * q^76 + 385562457456 * q^77 + 2990636883200 * q^79 - 3123628250000 * q^80 + 5784848206924 * q^82 - 5137135467696 * q^83 + 1298899093750 * q^85 - 2343313645096 * q^86 - 8660607864480 * q^88 + 19423025958450 * q^89 - 6792522370184 * q^91 - 26870022340992 * q^92 - 7003557875968 * q^94 - 1523298437500 * q^95 - 11088325396458 * q^97 + 10176508990306 * 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 $

$2$	−176.217	−1.94694	−0.973469	−	0.228817i	$-0.926514\pi$
$2$	−176.217	−1.94694	−0.973469	+	0.228817i	$0.926514\pi$
$3$	0	0
$3$	0	0
$4$	22860.4	2.79057
$5$	−15625.0	−0.447214
$5$	−15625.0	−0.447214
$6$	0	0
$7$	−201493.	−0.647326	−0.323663	−	0.946172i	$-0.604914\pi$
$7$	−201493.	−0.647326	−0.323663	+	0.946172i	$0.604914\pi$
$8$	−2.58481e6	−3.48613
$9$	0	0
$10$	2.75339e6	0.870698
$11$	3.34359e6	0.569062	0.284531	−	0.958667i	$-0.408162\pi$
$11$	3.34359e6	0.569062	0.284531	+	0.958667i	$0.408162\pi$
$12$	0	0
$13$	−7.80359e6	−0.448397	−0.224198	−	0.974544i	$-0.571976\pi$
$13$	−7.80359e6	−0.448397	−0.224198	+	0.974544i	$0.571976\pi$
$14$	3.55065e7	1.26030
$15$	0	0
$16$	2.68215e8	3.99671
$17$	8.71750e7	0.875939	0.437969	−	0.898990i	$-0.355698\pi$
$17$	8.71750e7	0.875939	0.437969	+	0.898990i	$0.355698\pi$
$18$	0	0
$19$	−1.66766e7	−0.0813223	−0.0406612	−	0.999173i	$-0.512946\pi$
$19$	−1.66766e7	−0.0813223	−0.0406612	+	0.999173i	$0.512946\pi$
$20$	−3.57193e8	−1.24798
$21$	0	0
$22$	−5.89196e8	−1.10793
$23$	−1.13518e9	−1.59895	−0.799476	−	0.600698i	$-0.794889\pi$
$23$	−1.13518e9	−1.59895	−0.799476	+	0.600698i	$0.794889\pi$
$24$	0	0
$25$	2.44141e8	0.200000
$26$	1.37512e9	0.873002
$27$	0	0
$28$	−4.60620e9	−1.80641
$29$	−2.60673e9	−0.813783	−0.406891	−	0.913477i	$-0.633387\pi$
$29$	−2.60673e9	−0.813783	−0.406891	+	0.913477i	$0.633387\pi$
$30$	0	0
$31$	8.33139e8	0.168604	0.0843018	−	0.996440i	$-0.473134\pi$
$31$	8.33139e8	0.168604	0.0843018	+	0.996440i	$0.473134\pi$
$32$	−2.60892e10	−4.29523
$33$	0	0
$34$	−1.53617e10	−1.70540
$35$	3.14833e9	0.289493
$36$	0	0
$37$	−1.05494e10	−0.675953	−0.337976	−	0.941155i	$-0.609742\pi$
$37$	−1.05494e10	−0.675953	−0.337976	+	0.941155i	$0.609742\pi$
$38$	2.93870e9	0.158330
$39$	0	0
$40$	4.03877e10	1.55905
$41$	4.33030e9	0.142371	0.0711857	−	0.997463i	$-0.477322\pi$
$41$	4.33030e9	0.142371	0.0711857	+	0.997463i	$0.477322\pi$
$42$	0	0
$43$	1.93854e9	0.0467660	0.0233830	−	0.999727i	$-0.492556\pi$
$43$	1.93854e9	0.0467660	0.0233830	+	0.999727i	$0.492556\pi$
$44$	7.64356e10	1.58801
$45$	0	0
$46$	2.00038e11	3.11306
$47$	−2.85468e10	−0.386296	−0.193148	−	0.981170i	$-0.561870\pi$
$47$	−2.85468e10	−0.386296	−0.193148	+	0.981170i	$0.561870\pi$
$48$	0	0
$49$	−5.62896e10	−0.580969
$50$	−4.30217e10	−0.389388
$51$	0	0
$52$	−1.78393e11	−1.25128
$53$	−1.23249e11	−0.763819	−0.381910	−	0.924200i	$-0.624733\pi$
$53$	−1.23249e11	−0.763819	−0.381910	+	0.924200i	$0.624733\pi$
$54$	0	0
$55$	−5.22435e10	−0.254492
$56$	5.20821e11	2.25666
$57$	0	0
$58$	4.59349e11	1.58439
$59$	5.55404e11	1.71424	0.857118	−	0.515120i	$-0.172253\pi$
$59$	5.55404e11	1.71424	0.857118	+	0.515120i	$0.172253\pi$
$60$	0	0
$61$	−4.10476e11	−1.02010	−0.510051	−	0.860144i	$-0.670373\pi$
$61$	−4.10476e11	−1.02010	−0.510051	+	0.860144i	$0.670373\pi$
$62$	−1.46813e11	−0.328261
$63$	0	0
$64$	2.40014e12	4.36583
$65$	1.21931e11	0.200529
$66$	0	0
$67$	3.36861e11	0.454951	0.227475	−	0.973784i	$-0.426953\pi$
$67$	3.36861e11	0.454951	0.227475	+	0.973784i	$0.426953\pi$
$68$	1.99285e12	2.44437
$69$	0	0
$70$	−5.54788e11	−0.563625
$71$	−1.57323e12	−1.45751	−0.728757	−	0.684772i	$-0.759902\pi$
$71$	−1.57323e12	−1.45751	−0.728757	+	0.684772i	$0.759902\pi$
$72$	0	0
$73$	2.05372e12	1.58834	0.794170	−	0.607695i	$-0.207906\pi$
$73$	2.05372e12	1.58834	0.794170	+	0.607695i	$0.207906\pi$
$74$	1.85898e12	1.31604
$75$	0	0
$76$	−3.81234e11	−0.226936
$77$	−6.73709e11	−0.368369
$78$	0	0
$79$	−6.93000e11	−0.320743	−0.160372	−	0.987057i	$-0.551269\pi$
$79$	−6.93000e11	−0.320743	−0.160372	+	0.987057i	$0.551269\pi$
$80$	−4.19086e12	−1.78739
$81$	0	0
$82$	−7.63071e11	−0.277188
$83$	−2.01116e12	−0.675212	−0.337606	−	0.941288i	$-0.609617\pi$
$83$	−2.01116e12	−0.675212	−0.337606	+	0.941288i	$0.609617\pi$
$84$	0	0
$85$	−1.36211e12	−0.391732
$86$	−3.41604e11	−0.0910505
$87$	0	0
$88$	−8.64253e12	−1.98383
$89$	8.51832e12	1.81685	0.908424	−	0.418050i	$-0.137286\pi$
$89$	8.51832e12	1.81685	0.908424	+	0.418050i	$0.137286\pi$
$90$	0	0
$91$	1.57237e12	0.290259
$92$	−2.59507e13	−4.46199
$93$	0	0
$94$	5.03042e12	0.752096
$95$	2.60572e11	0.0363684
$96$	0	0
$97$	−7.99814e12	−0.974929	−0.487464	−	0.873143i	$-0.662078\pi$
$97$	−7.99814e12	−0.974929	−0.487464	+	0.873143i	$0.662078\pi$
$98$	9.91916e12	1.13111
$99$	0	0
$100$	5.58114e12	0.558114
$101$	1.60387e13	1.50342	0.751710	−	0.659494i	$-0.229229\pi$
$101$	1.60387e13	1.50342	0.751710	+	0.659494i	$0.229229\pi$
$102$	0	0
$103$	1.72566e13	1.42401	0.712004	−	0.702176i	$-0.247788\pi$
$103$	1.72566e13	1.42401	0.712004	+	0.702176i	$0.247788\pi$
$104$	2.01708e13	1.56317
$105$	0	0
$106$	2.17186e13	1.48711
$107$	3.98196e11	0.0256509	0.0128254	−	0.999918i	$-0.495917\pi$
$107$	3.98196e11	0.0256509	0.0128254	+	0.999918i	$0.495917\pi$
$108$	0	0
$109$	1.09886e13	0.627582	0.313791	−	0.949492i	$-0.398401\pi$
$109$	1.09886e13	0.627582	0.313791	+	0.949492i	$0.398401\pi$
$110$	9.20619e12	0.495481
$111$	0	0
$112$	−5.40435e13	−2.58718
$113$	2.06154e13	0.931497	0.465748	−	0.884917i	$-0.345785\pi$
$113$	2.06154e13	0.931497	0.465748	+	0.884917i	$0.345785\pi$
$114$	0	0
$115$	1.77372e13	0.715073
$116$	−5.95907e13	−2.27092
$117$	0	0
$118$	−9.78714e13	−3.33751
$119$	−1.75651e13	−0.567018
$120$	0	0
$121$	−2.33431e13	−0.676168
$122$	7.23327e13	1.98608
$123$	0	0
$124$	1.90459e13	0.470500
$125$	−3.81470e12	−0.0894427
$126$	0	0
$127$	3.03181e13	0.641177	0.320588	−	0.947219i	$-0.396119\pi$
$127$	3.03181e13	0.641177	0.320588	+	0.947219i	$0.396119\pi$
$128$	−2.09222e14	−4.20478
$129$	0	0
$130$	−2.14863e13	−0.390418
$131$	7.79100e13	1.34688	0.673441	−	0.739241i	$-0.264815\pi$
$131$	7.79100e13	1.34688	0.673441	+	0.739241i	$0.264815\pi$
$132$	0	0
$133$	3.36022e12	0.0526420
$134$	−5.93605e13	−0.885761
$135$	0	0
$136$	−2.25331e14	−3.05364
$137$	1.01051e14	1.30574	0.652871	−	0.757469i	$-0.273564\pi$
$137$	1.01051e14	1.30574	0.652871	+	0.757469i	$0.273564\pi$
$138$	0	0
$139$	1.42505e14	1.67585	0.837924	−	0.545787i	$-0.183769\pi$
$139$	1.42505e14	1.67585	0.837924	+	0.545787i	$0.183769\pi$
$140$	7.19719e13	0.807850
$141$	0	0
$142$	2.77230e14	2.83769
$143$	−2.60920e13	−0.255166
$144$	0	0
$145$	4.07301e13	0.363935
$146$	−3.61901e14	−3.09240
$147$	0	0
$148$	−2.41163e14	−1.88629
$149$	6.15555e13	0.460846	0.230423	−	0.973091i	$-0.425989\pi$
$149$	6.15555e13	0.460846	0.230423	+	0.973091i	$0.425989\pi$
$150$	0	0
$151$	−4.43226e13	−0.304281	−0.152141	−	0.988359i	$-0.548617\pi$
$151$	−4.43226e13	−0.304281	−0.152141	+	0.988359i	$0.548617\pi$
$152$	4.31059e13	0.283500
$153$	0	0
$154$	1.18719e14	0.717191
$155$	−1.30178e13	−0.0754018
$156$	0	0
$157$	−1.81701e14	−0.968301	−0.484150	−	0.874985i	$-0.660871\pi$
$157$	−1.81701e14	−0.968301	−0.484150	+	0.874985i	$0.660871\pi$
$158$	1.22118e14	0.624467
$159$	0	0
$160$	4.07644e14	1.92088
$161$	2.28732e14	1.03504
$162$	0	0
$163$	9.09336e13	0.379756	0.189878	−	0.981808i	$-0.439191\pi$
$163$	9.09336e13	0.379756	0.189878	+	0.981808i	$0.439191\pi$
$164$	9.89921e13	0.397297
$165$	0	0
$166$	3.54401e14	1.31460
$167$	2.40022e14	0.856237	0.428118	−	0.903723i	$-0.359177\pi$
$167$	2.40022e14	0.856237	0.428118	+	0.903723i	$0.359177\pi$
$168$	0	0
$169$	−2.41979e14	−0.798940
$170$	2.40026e14	0.762678
$171$	0	0
$172$	4.43158e13	0.130504
$173$	−4.10274e14	−1.16352	−0.581760	−	0.813360i	$-0.697636\pi$
$173$	−4.10274e14	−1.16352	−0.581760	+	0.813360i	$0.697636\pi$
$174$	0	0
$175$	−4.91926e13	−0.129465
$176$	8.96800e14	2.27438
$177$	0	0
$178$	−1.50107e15	−3.53729
$179$	2.37588e14	0.539858	0.269929	−	0.962880i	$-0.413000\pi$
$179$	2.37588e14	0.539858	0.269929	+	0.962880i	$0.413000\pi$
$180$	0	0
$181$	5.41620e14	1.14494	0.572472	−	0.819924i	$-0.305985\pi$
$181$	5.41620e14	1.14494	0.572472	+	0.819924i	$0.305985\pi$
$182$	−2.77078e14	−0.565116
$183$	0	0
$184$	2.93424e15	5.57416
$185$	1.64834e14	0.302295
$186$	0	0
$187$	2.91477e14	0.498464
$188$	−6.52589e14	−1.07799
$189$	0	0
$190$	−4.59172e13	−0.0708071
$191$	2.56027e12	0.00381565	0.00190782	−	0.999998i	$-0.499393\pi$
$191$	2.56027e12	0.00381565	0.00190782	+	0.999998i	$0.499393\pi$
$192$	0	0
$193$	1.10496e15	1.53895	0.769475	−	0.638677i	$-0.220518\pi$
$193$	1.10496e15	1.53895	0.769475	+	0.638677i	$0.220518\pi$
$194$	1.40941e15	1.89813
$195$	0	0
$196$	−1.28680e15	−1.62124
$197$	−7.90145e14	−0.963110	−0.481555	−	0.876416i	$-0.659928\pi$
$197$	−7.90145e14	−0.963110	−0.481555	+	0.876416i	$0.659928\pi$
$198$	0	0
$199$	−2.97088e14	−0.339109	−0.169555	−	0.985521i	$-0.554233\pi$
$199$	−2.97088e14	−0.339109	−0.169555	+	0.985521i	$0.554233\pi$
$200$	−6.31057e14	−0.697226
$201$	0	0
$202$	−2.82629e15	−2.92707
$203$	5.25237e14	0.526782
$204$	0	0
$205$	−6.76609e13	−0.0636704
$206$	−3.04090e15	−2.77246
$207$	0	0
$208$	−2.09304e15	−1.79211
$209$	−5.57597e13	−0.0462775
$210$	0	0
$211$	−1.25602e15	−0.979851	−0.489926	−	0.871764i	$-0.662976\pi$
$211$	−1.25602e15	−0.979851	−0.489926	+	0.871764i	$0.662976\pi$
$212$	−2.81752e15	−2.13149
$213$	0	0
$214$	−7.01688e13	−0.0499407
$215$	−3.02897e13	−0.0209144
$216$	0	0
$217$	−1.67872e14	−0.109141
$218$	−1.93638e15	−1.22186
$219$	0	0
$220$	−1.19431e15	−0.710179
$221$	−6.80278e14	−0.392768
$222$	0	0
$223$	−1.17841e15	−0.641676	−0.320838	−	0.947134i	$-0.603964\pi$
$223$	−1.17841e15	−0.641676	−0.320838	+	0.947134i	$0.603964\pi$
$224$	5.25680e15	2.78041
$225$	0	0
$226$	−3.63278e15	−1.81357
$227$	−4.91498e14	−0.238426	−0.119213	−	0.992869i	$-0.538037\pi$
$227$	−4.91498e14	−0.238426	−0.119213	+	0.992869i	$0.538037\pi$
$228$	0	0
$229$	3.17657e15	1.45555	0.727777	−	0.685814i	$-0.240554\pi$
$229$	3.17657e15	1.45555	0.727777	+	0.685814i	$0.240554\pi$
$230$	−3.12560e15	−1.39220
$231$	0	0
$232$	6.73789e15	2.83695
$233$	2.05201e14	0.0840168	0.0420084	−	0.999117i	$-0.486624\pi$
$233$	2.05201e14	0.0840168	0.0420084	+	0.999117i	$0.486624\pi$
$234$	0	0
$235$	4.46043e14	0.172757
$236$	1.26967e16	4.78370
$237$	0	0
$238$	3.09527e15	1.10395
$239$	−4.78414e15	−1.66042	−0.830209	−	0.557453i	$-0.811779\pi$
$239$	−4.78414e15	−1.66042	−0.830209	+	0.557453i	$0.811779\pi$
$240$	0	0
$241$	2.83675e15	0.932633	0.466316	−	0.884618i	$-0.345581\pi$
$241$	2.83675e15	0.932633	0.466316	+	0.884618i	$0.345581\pi$
$242$	4.11345e15	1.31646
$243$	0	0
$244$	−9.38362e15	−2.84667
$245$	8.79524e14	0.259817
$246$	0	0
$247$	1.30138e14	0.0364647
$248$	−2.15351e15	−0.587774
$249$	0	0
$250$	6.72214e14	0.174140
$251$	−3.50075e15	−0.883654	−0.441827	−	0.897100i	$-0.645669\pi$
$251$	−3.50075e15	−0.883654	−0.441827	+	0.897100i	$0.645669\pi$
$252$	0	0
$253$	−3.79558e15	−0.909903
$254$	−5.34256e15	−1.24833
$255$	0	0
$256$	1.72065e16	3.82061
$257$	7.14451e14	0.154670	0.0773352	−	0.997005i	$-0.475359\pi$
$257$	7.14451e14	0.154670	0.0773352	+	0.997005i	$0.475359\pi$
$258$	0	0
$259$	2.12563e15	0.437562
$260$	2.78739e15	0.559591
$261$	0	0
$262$	−1.37290e16	−2.62230
$263$	6.41873e15	1.19601	0.598007	−	0.801491i	$-0.295959\pi$
$263$	6.41873e15	1.19601	0.598007	+	0.801491i	$0.295959\pi$
$264$	0	0
$265$	1.92577e15	0.341590
$266$	−5.92128e14	−0.102491
$267$	0	0
$268$	7.70076e15	1.26957
$269$	1.09876e15	0.176813	0.0884063	−	0.996084i	$-0.471823\pi$
$269$	1.09876e15	0.176813	0.0884063	+	0.996084i	$0.471823\pi$
$270$	0	0
$271$	1.60206e15	0.245684	0.122842	−	0.992426i	$-0.460799\pi$
$271$	1.60206e15	0.245684	0.122842	+	0.992426i	$0.460799\pi$
$272$	2.33816e16	3.50088
$273$	0	0
$274$	−1.78069e16	−2.54220
$275$	8.16305e14	0.113812
$276$	0	0
$277$	−6.91414e14	−0.0919644	−0.0459822	−	0.998942i	$-0.514642\pi$
$277$	−6.91414e14	−0.0919644	−0.0459822	+	0.998942i	$0.514642\pi$
$278$	−2.51118e16	−3.26277
$279$	0	0
$280$	−8.13783e15	−1.00921
$281$	−2.20744e15	−0.267484	−0.133742	−	0.991016i	$-0.542699\pi$
$281$	−2.20744e15	−0.267484	−0.133742	+	0.991016i	$0.542699\pi$
$282$	0	0
$283$	8.67634e15	1.00398	0.501990	−	0.864873i	$-0.332601\pi$
$283$	8.67634e15	1.00398	0.501990	+	0.864873i	$0.332601\pi$
$284$	−3.59646e16	−4.06730
$285$	0	0
$286$	4.59784e15	0.496792
$287$	−8.72525e14	−0.0921606
$288$	0	0
$289$	−2.30511e15	−0.232731
$290$	−7.17733e15	−0.708559
$291$	0	0
$292$	4.69489e16	4.43238
$293$	7.81345e15	0.721445	0.360723	−	0.932673i	$-0.382530\pi$
$293$	7.81345e15	0.721445	0.360723	+	0.932673i	$0.382530\pi$
$294$	0	0
$295$	−8.67818e15	−0.766630
$296$	2.72682e16	2.35646
$297$	0	0
$298$	−1.08471e16	−0.897239
$299$	8.85851e15	0.716965
$300$	0	0
$301$	−3.90603e14	−0.0302728
$302$	7.81039e15	0.592417
$303$	0	0
$304$	−4.47292e15	−0.325022
$305$	6.41368e15	0.456203
$306$	0	0
$307$	1.73583e16	1.18334	0.591668	−	0.806182i	$-0.298470\pi$
$307$	1.73583e16	1.18334	0.591668	+	0.806182i	$0.298470\pi$
$308$	−1.54012e16	−1.02796
$309$	0	0
$310$	2.29396e15	0.146803
$311$	2.64097e15	0.165509	0.0827543	−	0.996570i	$-0.473628\pi$
$311$	2.64097e15	0.165509	0.0827543	+	0.996570i	$0.473628\pi$
$312$	0	0
$313$	1.04071e16	0.625594	0.312797	−	0.949820i	$-0.398734\pi$
$313$	1.04071e16	0.625594	0.312797	+	0.949820i	$0.398734\pi$
$314$	3.20188e16	1.88522
$315$	0	0
$316$	−1.58422e16	−0.895057
$317$	1.64664e16	0.911410	0.455705	−	0.890131i	$-0.349387\pi$
$317$	1.64664e16	0.911410	0.455705	+	0.890131i	$0.349387\pi$
$318$	0	0
$319$	−8.71581e15	−0.463093
$320$	−3.75022e16	−1.95246
$321$	0	0
$322$	−4.03064e16	−2.01516
$323$	−1.45378e15	−0.0712334
$324$	0	0
$325$	−1.90517e15	−0.0896794
$326$	−1.60240e16	−0.739362
$327$	0	0
$328$	−1.11930e16	−0.496325
$329$	5.75197e15	0.250060
$330$	0	0
$331$	1.40904e16	0.588900	0.294450	−	0.955667i	$-0.404863\pi$
$331$	1.40904e16	0.588900	0.294450	+	0.955667i	$0.404863\pi$
$332$	−4.59759e16	−1.88423
$333$	0	0
$334$	−4.22959e16	−1.66704
$335$	−5.26345e15	−0.203460
$336$	0	0
$337$	1.51673e16	0.564044	0.282022	−	0.959408i	$-0.408995\pi$
$337$	1.51673e16	0.564044	0.282022	+	0.959408i	$0.408995\pi$
$338$	4.26408e16	1.55549
$339$	0	0
$340$	−3.11383e16	−1.09316
$341$	2.78567e15	0.0959460
$342$	0	0
$343$	3.08644e16	1.02340
$344$	−5.01076e15	−0.163032
$345$	0	0
$346$	7.22971e16	2.26530
$347$	−3.87222e15	−0.119074	−0.0595372	−	0.998226i	$-0.518962\pi$
$347$	−3.87222e15	−0.119074	−0.0595372	+	0.998226i	$0.518962\pi$
$348$	0	0
$349$	5.50673e16	1.63128	0.815640	−	0.578560i	$-0.196385\pi$
$349$	5.50673e16	1.63128	0.815640	+	0.578560i	$0.196385\pi$
$350$	8.66857e15	0.252061
$351$	0	0
$352$	−8.72315e16	−2.44425
$353$	4.84424e15	0.133257	0.0666285	−	0.997778i	$-0.478776\pi$
$353$	4.84424e15	0.133257	0.0666285	+	0.997778i	$0.478776\pi$
$354$	0	0
$355$	2.45817e16	0.651820
$356$	1.94732e17	5.07004
$357$	0	0
$358$	−4.18670e16	−1.05107
$359$	5.71730e16	1.40954	0.704770	−	0.709436i	$-0.251050\pi$
$359$	5.71730e16	1.40954	0.704770	+	0.709436i	$0.251050\pi$
$360$	0	0
$361$	−4.17749e16	−0.993387
$362$	−9.54426e16	−2.22914
$363$	0	0
$364$	3.59449e16	0.809988
$365$	−3.20894e16	−0.710328
$366$	0	0
$367$	−3.48286e16	−0.744057	−0.372028	−	0.928221i	$-0.621338\pi$
$367$	−3.48286e16	−0.744057	−0.372028	+	0.928221i	$0.621338\pi$
$368$	−3.04473e17	−6.39055
$369$	0	0
$370$	−2.90466e16	−0.588551
$371$	2.48338e16	0.494440
$372$	0	0
$373$	−2.14588e16	−0.412570	−0.206285	−	0.978492i	$-0.566137\pi$
$373$	−2.14588e16	−0.412570	−0.206285	+	0.978492i	$0.566137\pi$
$374$	−5.13631e16	−0.970479
$375$	0	0
$376$	7.37880e16	1.34668
$377$	2.03418e16	0.364898
$378$	0	0
$379$	−1.90361e16	−0.329930	−0.164965	−	0.986299i	$-0.552751\pi$
$379$	−1.90361e16	−0.329930	−0.164965	+	0.986299i	$0.552751\pi$
$380$	5.95677e15	0.101489
$381$	0	0
$382$	−4.51162e14	−0.00742884
$383$	1.03414e17	1.67412	0.837059	−	0.547112i	$-0.184273\pi$
$383$	1.03414e17	1.67412	0.837059	+	0.547112i	$0.184273\pi$
$384$	0	0
$385$	1.05267e16	0.164740
$386$	−1.94713e17	−2.99624
$387$	0	0
$388$	−1.82840e17	−2.72061
$389$	−9.11608e16	−1.33394	−0.666969	−	0.745085i	$-0.732409\pi$
$389$	−9.11608e16	−1.33394	−0.666969	+	0.745085i	$0.732409\pi$
$390$	0	0
$391$	−9.89596e16	−1.40058
$392$	1.45498e17	2.02534
$393$	0	0
$394$	1.39237e17	1.87512
$395$	1.08281e16	0.143441
$396$	0	0
$397$	7.19989e15	0.0922970	0.0461485	−	0.998935i	$-0.485305\pi$
$397$	7.19989e15	0.0922970	0.0461485	+	0.998935i	$0.485305\pi$
$398$	5.23518e16	0.660225
$399$	0	0
$400$	6.54822e16	0.799343
$401$	−1.06602e17	−1.28034	−0.640172	−	0.768231i	$-0.721137\pi$
$401$	−1.06602e17	−1.28034	−0.640172	+	0.768231i	$0.721137\pi$
$402$	0	0
$403$	−6.50148e15	−0.0756013
$404$	3.66650e17	4.19540
$405$	0	0
$406$	−9.25556e16	−1.02561
$407$	−3.52728e16	−0.384659
$408$	0	0
$409$	6.00154e16	0.633959	0.316980	−	0.948432i	$-0.397331\pi$
$409$	6.00154e16	0.633959	0.316980	+	0.948432i	$0.397331\pi$
$410$	1.19230e16	0.123962
$411$	0	0
$412$	3.94491e17	3.97379
$413$	−1.11910e17	−1.10967
$414$	0	0
$415$	3.14244e16	0.301964
$416$	2.03590e17	1.92597
$417$	0	0
$418$	9.82580e15	0.0900994
$419$	−1.22532e17	−1.10626	−0.553130	−	0.833095i	$-0.686567\pi$
$419$	−1.22532e17	−1.10626	−0.553130	+	0.833095i	$0.686567\pi$
$420$	0	0
$421$	−1.19508e17	−1.04607	−0.523037	−	0.852310i	$-0.675201\pi$
$421$	−1.19508e17	−1.04607	−0.523037	+	0.852310i	$0.675201\pi$
$422$	2.21332e17	1.90771
$423$	0	0
$424$	3.18576e17	2.66277
$425$	2.12829e16	0.175188
$426$	0	0
$427$	8.27080e16	0.660338
$428$	9.10290e15	0.0715806
$429$	0	0
$430$	5.33756e15	0.0407190
$431$	3.18425e16	0.239279	0.119640	−	0.992817i	$-0.461826\pi$
$431$	3.18425e16	0.239279	0.119640	+	0.992817i	$0.461826\pi$
$432$	0	0
$433$	1.24850e16	0.0910366	0.0455183	−	0.998964i	$-0.485506\pi$
$433$	1.24850e16	0.0910366	0.0455183	+	0.998964i	$0.485506\pi$
$434$	2.95818e16	0.212492
$435$	0	0
$436$	2.51204e17	1.75131
$437$	1.89310e16	0.130030
$438$	0	0
$439$	−2.58876e17	−1.72613	−0.863063	−	0.505095i	$-0.831457\pi$
$439$	−2.58876e17	−1.72613	−0.863063	+	0.505095i	$0.831457\pi$
$440$	1.35040e17	0.887194
$441$	0	0
$442$	1.19876e17	0.764696
$443$	−1.53844e17	−0.967068	−0.483534	−	0.875326i	$-0.660647\pi$
$443$	−1.53844e17	−0.967068	−0.483534	+	0.875326i	$0.660647\pi$
$444$	0	0
$445$	−1.33099e17	−0.812519
$446$	2.07656e17	1.24930
$447$	0	0
$448$	−4.83612e17	−2.82611
$449$	9.02887e16	0.520034	0.260017	−	0.965604i	$-0.416272\pi$
$449$	9.02887e16	0.520034	0.260017	+	0.965604i	$0.416272\pi$
$450$	0	0
$451$	1.44787e16	0.0810182
$452$	4.71275e17	2.59941
$453$	0	0
$454$	8.66102e16	0.464201
$455$	−2.45683e16	−0.129808
$456$	0	0
$457$	−9.78643e16	−0.502538	−0.251269	−	0.967917i	$-0.580848\pi$
$457$	−9.78643e16	−0.502538	−0.251269	+	0.967917i	$0.580848\pi$
$458$	−5.59765e17	−2.83387
$459$	0	0
$460$	4.05480e17	1.99546
$461$	−2.08140e17	−1.00995	−0.504976	−	0.863134i	$-0.668499\pi$
$461$	−2.08140e17	−1.00995	−0.504976	+	0.863134i	$0.668499\pi$
$462$	0	0
$463$	−2.44823e17	−1.15498	−0.577491	−	0.816397i	$-0.695968\pi$
$463$	−2.44823e17	−1.15498	−0.577491	+	0.816397i	$0.695968\pi$
$464$	−6.99163e17	−3.25246
$465$	0	0
$466$	−3.61599e16	−0.163576
$467$	−3.54401e17	−1.58101	−0.790506	−	0.612454i	$-0.790182\pi$
$467$	−3.54401e17	−1.58101	−0.790506	+	0.612454i	$0.790182\pi$
$468$	0	0
$469$	−6.78751e16	−0.294501
$470$	−7.86003e16	−0.336347
$471$	0	0
$472$	−1.43561e18	−5.97605
$473$	6.48168e15	0.0266128
$474$	0	0
$475$	−4.07144e15	−0.0162645
$476$	−4.01546e17	−1.58230
$477$	0	0
$478$	8.43045e17	3.23273
$479$	1.54040e17	0.582712	0.291356	−	0.956615i	$-0.405894\pi$
$479$	1.54040e17	0.582712	0.291356	+	0.956615i	$0.405894\pi$
$480$	0	0
$481$	8.23232e16	0.303095
$482$	−4.99883e17	−1.81578
$483$	0	0
$484$	−5.33633e17	−1.88689
$485$	1.24971e17	0.436001
$486$	0	0
$487$	2.11302e17	0.717736	0.358868	−	0.933388i	$-0.383163\pi$
$487$	2.11302e17	0.717736	0.358868	+	0.933388i	$0.383163\pi$
$488$	1.06100e18	3.55621
$489$	0	0
$490$	−1.54987e17	−0.505849
$491$	−2.61474e17	−0.842167	−0.421083	−	0.907022i	$-0.638350\pi$
$491$	−2.61474e17	−0.842167	−0.421083	+	0.907022i	$0.638350\pi$
$492$	0	0
$493$	−2.27241e17	−0.712824
$494$	−2.29324e16	−0.0709945
$495$	0	0
$496$	2.23460e17	0.673860
$497$	3.16995e17	0.943487
$498$	0	0
$499$	3.87524e17	1.12369	0.561843	−	0.827244i	$-0.310092\pi$
$499$	3.87524e17	1.12369	0.561843	+	0.827244i	$0.310092\pi$
$500$	−8.72053e16	−0.249596
$501$	0	0
$502$	6.16891e17	1.72042
$503$	4.12680e17	1.13611	0.568056	−	0.822990i	$-0.307696\pi$
$503$	4.12680e17	1.13611	0.568056	+	0.822990i	$0.307696\pi$
$504$	0	0
$505$	−2.50605e17	−0.672350
$506$	6.68846e17	1.77153
$507$	0	0
$508$	6.93083e17	1.78925
$509$	−8.09532e16	−0.206333	−0.103166	−	0.994664i	$-0.532897\pi$
$509$	−8.09532e16	−0.206333	−0.103166	+	0.994664i	$0.532897\pi$
$510$	0	0
$511$	−4.13811e17	−1.02817
$512$	−1.31813e18	−3.23372
$513$	0	0
$514$	−1.25898e17	−0.301134
$515$	−2.69634e17	−0.636836
$516$	0	0
$517$	−9.54485e16	−0.219827
$518$	−3.74572e17	−0.851906
$519$	0	0
$520$	−3.15169e17	−0.699071
$521$	2.79201e17	0.611607	0.305803	−	0.952095i	$-0.401075\pi$
$521$	2.79201e17	0.611607	0.305803	+	0.952095i	$0.401075\pi$
$522$	0	0
$523$	9.97401e16	0.213112	0.106556	−	0.994307i	$-0.466018\pi$
$523$	9.97401e16	0.213112	0.106556	+	0.994307i	$0.466018\pi$
$524$	1.78105e18	3.75857
$525$	0	0
$526$	−1.13109e18	−2.32857
$527$	7.26289e16	0.147686
$528$	0	0
$529$	7.84606e17	1.55665
$530$	−3.39353e17	−0.665056
$531$	0	0
$532$	7.68159e16	0.146901
$533$	−3.37919e16	−0.0638389
$534$	0	0
$535$	−6.22181e15	−0.0114714
$536$	−8.70721e17	−1.58602
$537$	0	0
$538$	−1.93620e17	−0.344243
$539$	−1.88209e17	−0.330608
$540$	0	0
$541$	−2.15125e17	−0.368899	−0.184450	−	0.982842i	$-0.559050\pi$
$541$	−2.15125e17	−0.368899	−0.184450	+	0.982842i	$0.559050\pi$
$542$	−2.82309e17	−0.478332
$543$	0	0
$544$	−2.27433e18	−3.76236
$545$	−1.71697e17	−0.280663
$546$	0	0
$547$	9.04541e17	1.44381	0.721906	−	0.691991i	$-0.243266\pi$
$547$	9.04541e17	1.44381	0.721906	+	0.691991i	$0.243266\pi$
$548$	2.31007e18	3.64377
$549$	0	0
$550$	−1.43847e17	−0.221586
$551$	4.34714e16	0.0661787
$552$	0	0
$553$	1.39635e17	0.207625
$554$	1.21839e17	0.179049
$555$	0	0
$556$	3.25772e18	4.67657
$557$	9.25130e17	1.31263	0.656317	−	0.754485i	$-0.272113\pi$
$557$	9.25130e17	1.31263	0.656317	+	0.754485i	$0.272113\pi$
$558$	0	0
$559$	−1.51276e16	−0.0209697
$560$	8.44429e17	1.15702
$561$	0	0
$562$	3.88987e17	0.520774
$563$	7.77258e17	1.02863	0.514317	−	0.857600i	$-0.328046\pi$
$563$	7.77258e17	1.02863	0.514317	+	0.857600i	$0.328046\pi$
$564$	0	0
$565$	−3.22115e17	−0.416578
$566$	−1.52892e18	−1.95469
$567$	0	0
$568$	4.06650e18	5.08109
$569$	−5.27222e17	−0.651274	−0.325637	−	0.945495i	$-0.605579\pi$
$569$	−5.27222e17	−0.651274	−0.325637	+	0.945495i	$0.605579\pi$
$570$	0	0
$571$	−8.57655e17	−1.03557	−0.517783	−	0.855512i	$-0.673243\pi$
$571$	−8.57655e17	−1.03557	−0.517783	+	0.855512i	$0.673243\pi$
$572$	−5.96472e17	−0.712059
$573$	0	0
$574$	1.53753e17	0.179431
$575$	−2.77144e17	−0.319790
$576$	0	0
$577$	−6.01001e17	−0.678004	−0.339002	−	0.940786i	$-0.610089\pi$
$577$	−6.01001e17	−0.678004	−0.339002	+	0.940786i	$0.610089\pi$
$578$	4.06198e17	0.453114
$579$	0	0
$580$	9.31105e17	1.01559
$581$	4.05236e17	0.437082
$582$	0	0
$583$	−4.12094e17	−0.434661
$584$	−5.30849e18	−5.53716
$585$	0	0
$586$	−1.37686e18	−1.40461
$587$	−2.25235e17	−0.227241	−0.113621	−	0.993524i	$-0.536245\pi$
$587$	−2.25235e17	−0.227241	−0.113621	+	0.993524i	$0.536245\pi$
$588$	0	0
$589$	−1.38940e16	−0.0137112
$590$	1.52924e18	1.49258
$591$	0	0
$592$	−2.82951e18	−2.70159
$593$	−8.69033e17	−0.820693	−0.410347	−	0.911930i	$-0.634592\pi$
$593$	−8.69033e17	−0.820693	−0.410347	+	0.911930i	$0.634592\pi$
$594$	0	0
$595$	2.74455e17	0.253578
$596$	1.40718e18	1.28602
$597$	0	0
$598$	−1.56102e18	−1.39589
$599$	1.94567e18	1.72105	0.860527	−	0.509405i	$-0.170134\pi$
$599$	1.94567e18	1.72105	0.860527	+	0.509405i	$0.170134\pi$
$600$	0	0
$601$	2.96725e17	0.256844	0.128422	−	0.991720i	$-0.459009\pi$
$601$	2.96725e17	0.256844	0.128422	+	0.991720i	$0.459009\pi$
$602$	6.88307e16	0.0589393
$603$	0	0
$604$	−1.01323e18	−0.849118
$605$	3.64737e17	0.302391
$606$	0	0
$607$	2.06953e18	1.67937	0.839683	−	0.543076i	$-0.182741\pi$
$607$	2.06953e18	1.67937	0.839683	+	0.543076i	$0.182741\pi$
$608$	4.35080e17	0.349298
$609$	0	0
$610$	−1.13020e18	−0.888200
$611$	2.22767e17	0.173214
$612$	0	0
$613$	2.27988e18	1.73548	0.867738	−	0.497022i	$-0.165573\pi$
$613$	2.27988e18	1.73548	0.867738	+	0.497022i	$0.165573\pi$
$614$	−3.05883e18	−2.30388
$615$	0	0
$616$	1.74141e18	1.28418
$617$	2.03478e18	1.48479	0.742394	−	0.669964i	$-0.233690\pi$
$617$	2.03478e18	1.48479	0.742394	+	0.669964i	$0.233690\pi$
$618$	0	0
$619$	−7.79671e17	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$619$	−7.79671e17	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$620$	−2.97592e17	−0.210414
$621$	0	0
$622$	−4.65383e17	−0.322235
$623$	−1.71638e18	−1.17609
$624$	0	0
$625$	5.96046e16	0.0400000
$626$	−1.83391e18	−1.21799
$627$	0	0
$628$	−4.15376e18	−2.70211
$629$	−9.19643e17	−0.592093
$630$	0	0
$631$	−1.75763e18	−1.10850	−0.554252	−	0.832349i	$-0.686995\pi$
$631$	−1.75763e18	−1.10850	−0.554252	+	0.832349i	$0.686995\pi$
$632$	1.79128e18	1.11815
$633$	0	0
$634$	−2.90166e18	−1.77446
$635$	−4.73720e17	−0.286743
$636$	0	0
$637$	4.39261e17	0.260505
$638$	1.53587e18	0.901614
$639$	0	0
$640$	3.26910e18	1.88043
$641$	−2.32621e18	−1.32456	−0.662279	−	0.749257i	$-0.730411\pi$
$641$	−2.32621e18	−1.32456	−0.662279	+	0.749257i	$0.730411\pi$
$642$	0	0
$643$	2.60316e18	1.45254	0.726272	−	0.687408i	$-0.241251\pi$
$643$	2.60316e18	1.45254	0.726272	+	0.687408i	$0.241251\pi$
$644$	5.22889e18	2.88836
$645$	0	0
$646$	2.56181e17	0.138687
$647$	1.20524e18	0.645947	0.322974	−	0.946408i	$-0.395318\pi$
$647$	1.20524e18	0.645947	0.322974	+	0.946408i	$0.395318\pi$
$648$	0	0
$649$	1.85704e18	0.975507
$650$	3.35724e17	0.174600
$651$	0	0
$652$	2.07877e18	1.05974
$653$	−1.72181e18	−0.869061	−0.434531	−	0.900657i	$-0.643086\pi$
$653$	−1.72181e18	−0.869061	−0.434531	+	0.900657i	$0.643086\pi$
$654$	0	0
$655$	−1.21734e18	−0.602344
$656$	1.16145e18	0.569018
$657$	0	0
$658$	−1.01359e18	−0.486851
$659$	3.66491e18	1.74304	0.871521	−	0.490358i	$-0.163134\pi$
$659$	3.66491e18	1.74304	0.871521	+	0.490358i	$0.163134\pi$
$660$	0	0
$661$	−3.66523e18	−1.70920	−0.854599	−	0.519289i	$-0.826197\pi$
$661$	−3.66523e18	−1.70920	−0.854599	+	0.519289i	$0.826197\pi$
$662$	−2.48297e18	−1.14655
$663$	0	0
$664$	5.19848e18	2.35388
$665$	−5.25035e16	−0.0235422
$666$	0	0
$667$	2.95911e18	1.30120
$668$	5.48699e18	2.38939
$669$	0	0
$670$	9.27508e17	0.396124
$671$	−1.37246e18	−0.580502
$672$	0	0
$673$	−3.60071e18	−1.49379	−0.746897	−	0.664940i	$-0.768457\pi$
$673$	−3.60071e18	−1.49379	−0.746897	+	0.664940i	$0.768457\pi$
$674$	−2.67273e18	−1.09816
$675$	0	0
$676$	−5.53173e18	−2.22950
$677$	−1.46169e17	−0.0583484	−0.0291742	−	0.999574i	$-0.509288\pi$
$677$	−1.46169e17	−0.0583484	−0.0291742	+	0.999574i	$0.509288\pi$
$678$	0	0
$679$	1.61157e18	0.631097
$680$	3.52079e18	1.36563
$681$	0	0
$682$	−4.90882e17	−0.186801
$683$	2.32270e18	0.875505	0.437753	−	0.899095i	$-0.355775\pi$
$683$	2.32270e18	0.875505	0.437753	+	0.899095i	$0.355775\pi$
$684$	0	0
$685$	−1.57892e18	−0.583946
$686$	−5.43883e18	−1.99250
$687$	0	0
$688$	5.19946e17	0.186910
$689$	9.61786e17	0.342494
$690$	0	0
$691$	−6.92473e16	−0.0241989	−0.0120994	−	0.999927i	$-0.503851\pi$
$691$	−6.92473e16	−0.0241989	−0.0120994	+	0.999927i	$0.503851\pi$
$692$	−9.37900e18	−3.24688
$693$	0	0
$694$	6.82350e17	0.231831
$695$	−2.22664e18	−0.749462
$696$	0	0
$697$	3.77493e17	0.124709
$698$	−9.70378e18	−3.17600
$699$	0	0
$700$	−1.12456e18	−0.361282
$701$	6.22383e16	0.0198103	0.00990514	−	0.999951i	$-0.496847\pi$
$701$	6.22383e16	0.0198103	0.00990514	+	0.999951i	$0.496847\pi$
$702$	0	0
$703$	1.75928e17	0.0549700
$704$	8.02507e18	2.48443
$705$	0	0
$706$	−8.53637e17	−0.259443
$707$	−3.23169e18	−0.973202
$708$	0	0
$709$	2.94114e18	0.869592	0.434796	−	0.900529i	$-0.356820\pi$
$709$	2.94114e18	0.869592	0.434796	+	0.900529i	$0.356820\pi$
$710$	−4.33171e18	−1.26905
$711$	0	0
$712$	−2.20182e19	−6.33377
$713$	−9.45766e17	−0.269589
$714$	0	0
$715$	4.07687e17	0.114114
$716$	5.43134e18	1.50651
$717$	0	0
$718$	−1.00748e19	−2.74429
$719$	−4.39694e18	−1.18690	−0.593448	−	0.804872i	$-0.702234\pi$
$719$	−4.39694e18	−1.18690	−0.593448	+	0.804872i	$0.702234\pi$
$720$	0	0
$721$	−3.47708e18	−0.921797
$722$	7.36143e18	1.93406
$723$	0	0
$724$	1.23816e19	3.19505
$725$	−6.36408e17	−0.162757
$726$	0	0
$727$	6.21513e18	1.56126	0.780632	−	0.624991i	$-0.214897\pi$
$727$	6.21513e18	1.56126	0.780632	+	0.624991i	$0.214897\pi$
$728$	−4.06428e18	−1.01188
$729$	0	0
$730$	5.65470e18	1.38296
$731$	1.68992e17	0.0409641
$732$	0	0
$733$	−1.54714e18	−0.368429	−0.184215	−	0.982886i	$-0.558974\pi$
$733$	−1.54714e18	−0.368429	−0.184215	+	0.982886i	$0.558974\pi$
$734$	6.13738e18	1.44863
$735$	0	0
$736$	2.96161e19	6.86786
$737$	1.12632e18	0.258895
$738$	0	0
$739$	−1.96669e18	−0.444167	−0.222084	−	0.975028i	$-0.571286\pi$
$739$	−1.96669e18	−0.444167	−0.222084	+	0.975028i	$0.571286\pi$
$740$	3.76817e18	0.843577
$741$	0	0
$742$	−4.37614e18	−0.962644
$743$	5.66492e18	1.23528	0.617641	−	0.786460i	$-0.288088\pi$
$743$	5.66492e18	1.23528	0.617641	+	0.786460i	$0.288088\pi$
$744$	0	0
$745$	−9.61805e17	−0.206097
$746$	3.78140e18	0.803249
$747$	0	0
$748$	6.66327e18	1.39100
$749$	−8.02337e16	−0.0166045
$750$	0	0
$751$	4.65838e18	0.947491	0.473745	−	0.880662i	$-0.342902\pi$
$751$	4.65838e18	0.947491	0.473745	+	0.880662i	$0.342902\pi$
$752$	−7.65667e18	−1.54392
$753$	0	0
$754$	−3.58457e18	−0.710434
$755$	6.92541e17	0.136079
$756$	0	0
$757$	−3.51348e18	−0.678602	−0.339301	−	0.940678i	$-0.610190\pi$
$757$	−3.51348e18	−0.678602	−0.339301	+	0.940678i	$0.610190\pi$
$758$	3.35447e18	0.642354
$759$	0	0
$760$	−6.73530e17	−0.126785
$761$	−4.55489e18	−0.850116	−0.425058	−	0.905166i	$-0.639746\pi$
$761$	−4.55489e18	−0.850116	−0.425058	+	0.905166i	$0.639746\pi$
$762$	0	0
$763$	−2.21413e18	−0.406250
$764$	5.85286e16	0.0106478
$765$	0	0
$766$	−1.82232e19	−3.25941
$767$	−4.33414e18	−0.768658
$768$	0	0
$769$	5.81779e18	1.01446	0.507232	−	0.861810i	$-0.330669\pi$
$769$	5.81779e18	1.01446	0.507232	+	0.861810i	$0.330669\pi$
$770$	−1.85498e18	−0.320738
$771$	0	0
$772$	2.52598e19	4.29455
$773$	3.52257e18	0.593871	0.296936	−	0.954897i	$-0.404035\pi$
$773$	3.52257e18	0.593871	0.296936	+	0.954897i	$0.404035\pi$
$774$	0	0
$775$	2.03403e17	0.0337207
$776$	2.06737e19	3.39873
$777$	0	0
$778$	1.60641e19	2.59710
$779$	−7.22147e16	−0.0115780
$780$	0	0
$781$	−5.26023e18	−0.829417
$782$	1.74383e19	2.72685
$783$	0	0
$784$	−1.50977e19	−2.32197
$785$	2.83908e18	0.433037
$786$	0	0
$787$	1.48350e18	0.222562	0.111281	−	0.993789i	$-0.464505\pi$
$787$	1.48350e18	0.222562	0.111281	+	0.993789i	$0.464505\pi$
$788$	−1.80630e19	−2.68763
$789$	0	0
$790$	−1.90810e18	−0.279270
$791$	−4.15386e18	−0.602982
$792$	0	0
$793$	3.20318e18	0.457411
$794$	−1.26874e18	−0.179697
$795$	0	0
$796$	−6.79153e18	−0.946309
$797$	−3.72381e18	−0.514645	−0.257323	−	0.966326i	$-0.582840\pi$
$797$	−3.72381e18	−0.514645	−0.257323	+	0.966326i	$0.582840\pi$
$798$	0	0
$799$	−2.48856e18	−0.338372
$800$	−6.36944e18	−0.859045
$801$	0	0
$802$	1.87851e19	2.49275
$803$	6.86680e18	0.903865
$804$	0	0
$805$	−3.57393e18	−0.462885
$806$	1.14567e18	0.147191
$807$	0	0
$808$	−4.14570e19	−5.24112
$809$	1.30799e19	1.64036	0.820181	−	0.572104i	$-0.193873\pi$
$809$	1.30799e19	1.64036	0.820181	+	0.572104i	$0.193873\pi$
$810$	0	0
$811$	−4.17968e18	−0.515832	−0.257916	−	0.966167i	$-0.583036\pi$
$811$	−4.17968e18	−0.515832	−0.257916	+	0.966167i	$0.583036\pi$
$812$	1.20071e19	1.47002
$813$	0	0
$814$	6.21566e18	0.748908
$815$	−1.42084e18	−0.169832
$816$	0	0
$817$	−3.23283e16	−0.00380312
$818$	−1.05757e19	−1.23428
$819$	0	0
$820$	−1.54675e18	−0.177677
$821$	−5.28864e18	−0.602717	−0.301358	−	0.953511i	$-0.597440\pi$
$821$	−5.28864e18	−0.602717	−0.301358	+	0.953511i	$0.597440\pi$
$822$	0	0
$823$	7.96616e18	0.893614	0.446807	−	0.894630i	$-0.352561\pi$
$823$	7.96616e18	0.893614	0.446807	+	0.894630i	$0.352561\pi$
$824$	−4.46049e19	−4.96428
$825$	0	0
$826$	1.97204e19	2.16046
$827$	1.49178e19	1.62150	0.810752	−	0.585389i	$-0.199058\pi$
$827$	1.49178e19	1.62150	0.810752	+	0.585389i	$0.199058\pi$
$828$	0	0
$829$	−8.50781e18	−0.910360	−0.455180	−	0.890399i	$-0.650425\pi$
$829$	−8.50781e18	−0.910360	−0.455180	+	0.890399i	$0.650425\pi$
$830$	−5.53752e18	−0.587905
$831$	0	0
$832$	−1.87297e19	−1.95763
$833$	−4.90704e18	−0.508894
$834$	0	0
$835$	−3.75034e18	−0.382921
$836$	−1.27469e18	−0.129141
$837$	0	0
$838$	2.15921e19	2.15382
$839$	6.82788e18	0.675823	0.337912	−	0.941178i	$-0.390280\pi$
$839$	6.82788e18	0.675823	0.337912	+	0.941178i	$0.390280\pi$
$840$	0	0
$841$	−3.46561e18	−0.337758
$842$	2.10593e19	2.03664
$843$	0	0
$844$	−2.87130e19	−2.73434
$845$	3.78092e18	0.357297
$846$	0	0
$847$	4.70348e18	0.437701
$848$	−3.30573e19	−3.05277
$849$	0	0
$850$	−3.75041e18	−0.341080
$851$	1.19755e19	1.08082
$852$	0	0
$853$	6.17395e18	0.548775	0.274387	−	0.961619i	$-0.411525\pi$
$853$	6.17395e18	0.548775	0.274387	+	0.961619i	$0.411525\pi$
$854$	−1.45745e19	−1.28564
$855$	0	0
$856$	−1.02926e18	−0.0894224
$857$	−1.69937e18	−0.146525	−0.0732625	−	0.997313i	$-0.523341\pi$
$857$	−1.69937e18	−0.146525	−0.0732625	+	0.997313i	$0.523341\pi$
$858$	0	0
$859$	3.92966e18	0.333734	0.166867	−	0.985979i	$-0.446635\pi$
$859$	3.92966e18	0.333734	0.166867	+	0.985979i	$0.446635\pi$
$860$	−6.92434e17	−0.0583631
$861$	0	0
$862$	−5.61118e18	−0.465862
$863$	−6.77611e18	−0.558354	−0.279177	−	0.960240i	$-0.590062\pi$
$863$	−6.77611e18	−0.558354	−0.279177	+	0.960240i	$0.590062\pi$
$864$	0	0
$865$	6.41052e18	0.520342
$866$	−2.20006e18	−0.177243
$867$	0	0
$868$	−3.83761e18	−0.304567
$869$	−2.31711e18	−0.182523
$870$	0	0
$871$	−2.62872e18	−0.203998
$872$	−2.84035e19	−2.18783
$873$	0	0
$874$	−3.33597e18	−0.253161
$875$	7.68635e17	0.0578986
$876$	0	0
$877$	1.66517e19	1.23583	0.617917	−	0.786243i	$-0.287977\pi$
$877$	1.66517e19	1.23583	0.617917	+	0.786243i	$0.287977\pi$
$878$	4.56183e19	3.36066
$879$	0	0
$880$	−1.40125e19	−1.01713
$881$	−1.72426e19	−1.24239	−0.621195	−	0.783656i	$-0.713353\pi$
$881$	−1.72426e19	−1.24239	−0.621195	+	0.783656i	$0.713353\pi$
$882$	0	0
$883$	3.03788e18	0.215688	0.107844	−	0.994168i	$-0.465605\pi$
$883$	3.03788e18	0.215688	0.107844	+	0.994168i	$0.465605\pi$
$884$	−1.55514e19	−1.09605
$885$	0	0
$886$	2.71099e19	1.88282
$887$	−6.51405e18	−0.449105	−0.224552	−	0.974462i	$-0.572092\pi$
$887$	−6.51405e18	−0.449105	−0.224552	+	0.974462i	$0.572092\pi$
$888$	0	0
$889$	−6.10889e18	−0.415050
$890$	2.34542e19	1.58193
$891$	0	0
$892$	−2.69389e19	−1.79064
$893$	4.76063e17	0.0314145
$894$	0	0
$895$	−3.71231e18	−0.241432
$896$	4.21568e19	2.72186
$897$	0	0
$898$	−1.59104e19	−1.01248
$899$	−2.17177e18	−0.137207
$900$	0	0
$901$	−1.07442e19	−0.669059
$902$	−2.55139e18	−0.157737
$903$	0	0
$904$	−5.32868e19	−3.24732
$905$	−8.46282e18	−0.512035
$906$	0	0
$907$	−2.62324e19	−1.56455	−0.782277	−	0.622931i	$-0.785942\pi$
$907$	−2.62324e19	−1.56455	−0.782277	+	0.622931i	$0.785942\pi$
$908$	−1.12358e19	−0.665345
$909$	0	0
$910$	4.32934e18	0.252728
$911$	9.93847e18	0.576037	0.288018	−	0.957625i	$-0.407004\pi$
$911$	9.93847e18	0.576037	0.288018	+	0.957625i	$0.407004\pi$
$912$	0	0
$913$	−6.72450e18	−0.384238
$914$	1.72453e19	0.978411
$915$	0	0
$916$	7.26176e19	4.06182
$917$	−1.56983e19	−0.871872
$918$	0	0
$919$	1.61779e19	0.885874	0.442937	−	0.896553i	$-0.353937\pi$
$919$	1.61779e19	0.885874	0.442937	+	0.896553i	$0.353937\pi$
$920$	−4.58474e19	−2.49284
$921$	0	0
$922$	3.66778e19	1.96631
$923$	1.22768e19	0.653545
$924$	0	0
$925$	−2.57554e18	−0.135191
$926$	4.31419e19	2.24868
$927$	0	0
$928$	6.80074e19	3.49538
$929$	−2.14431e18	−0.109442	−0.0547211	−	0.998502i	$-0.517427\pi$
$929$	−2.14431e18	−0.109442	−0.0547211	+	0.998502i	$0.517427\pi$
$930$	0	0
$931$	9.38720e17	0.0472458
$932$	4.69097e18	0.234455
$933$	0	0
$934$	6.24514e19	3.07813
$935$	−4.55433e18	−0.222920
$936$	0	0
$937$	−3.39502e19	−1.63883	−0.819417	−	0.573199i	$-0.805702\pi$
$937$	−3.39502e19	−1.63883	−0.819417	+	0.573199i	$0.805702\pi$
$938$	1.19607e19	0.573376
$939$	0	0
$940$	1.01967e19	0.482091
$941$	1.95970e19	0.920149	0.460074	−	0.887880i	$-0.347823\pi$
$941$	1.95970e19	0.920149	0.460074	+	0.887880i	$0.347823\pi$
$942$	0	0
$943$	−4.91568e18	−0.227645
$944$	1.48968e20	6.85131
$945$	0	0
$946$	−1.14218e18	−0.0518134
$947$	1.67818e19	0.756072	0.378036	−	0.925791i	$-0.376600\pi$
$947$	1.67818e19	0.756072	0.378036	+	0.925791i	$0.376600\pi$
$948$	0	0
$949$	−1.60264e19	−0.712207
$950$	7.17456e17	0.0316659
$951$	0	0
$952$	4.54026e19	1.97670
$953$	−9.68702e18	−0.418877	−0.209438	−	0.977822i	$-0.567164\pi$
$953$	−9.68702e18	−0.418877	−0.209438	+	0.977822i	$0.567164\pi$
$954$	0	0
$955$	−4.00042e16	−0.00170641
$956$	−1.09367e20	−4.63351
$957$	0	0
$958$	−2.71445e19	−1.13450
$959$	−2.03611e19	−0.845241
$960$	0	0
$961$	−2.37234e19	−0.971573
$962$	−1.45067e19	−0.590108
$963$	0	0
$964$	6.48492e19	2.60258
$965$	−1.72650e19	−0.688239
$966$	0	0
$967$	2.34907e18	0.0923899	0.0461949	−	0.998932i	$-0.485290\pi$
$967$	2.34907e18	0.0923899	0.0461949	+	0.998932i	$0.485290\pi$
$968$	6.03376e19	2.35721
$969$	0	0
$970$	−2.20220e19	−0.848868
$971$	−2.00271e19	−0.766818	−0.383409	−	0.923579i	$-0.625250\pi$
$971$	−2.00271e19	−0.766818	−0.383409	+	0.923579i	$0.625250\pi$
$972$	0	0
$973$	−2.87138e19	−1.08482
$974$	−3.72349e19	−1.39739
$975$	0	0
$976$	−1.10096e20	−4.07706
$977$	−3.98189e19	−1.46479	−0.732394	−	0.680881i	$-0.761597\pi$
$977$	−3.98189e19	−1.46479	−0.732394	+	0.680881i	$0.761597\pi$
$978$	0	0
$979$	2.84817e19	1.03390
$980$	2.01062e19	0.725039
$981$	0	0
$982$	4.60760e19	1.63965
$983$	−2.29985e19	−0.813022	−0.406511	−	0.913646i	$-0.633255\pi$
$983$	−2.29985e19	−0.813022	−0.406511	+	0.913646i	$0.633255\pi$
$984$	0	0
$985$	1.23460e19	0.430716
$986$	4.00437e19	1.38782
$987$	0	0
$988$	2.97499e18	0.101757
$989$	−2.20060e18	−0.0747765
$990$	0	0
$991$	8.09114e18	0.271351	0.135675	−	0.990753i	$-0.456680\pi$
$991$	8.09114e18	0.271351	0.135675	+	0.990753i	$0.456680\pi$
$992$	−2.17360e19	−0.724191
$993$	0	0
$994$	−5.58598e19	−1.83691
$995$	4.64200e18	0.151654
$996$	0	0
$997$	−2.26704e19	−0.731039	−0.365519	−	0.930804i	$-0.619109\pi$
$997$	−2.26704e19	−0.731039	−0.365519	+	0.930804i	$0.619109\pi$
$998$	−6.82883e19	−2.18775
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.14.a.e.1.1		3
3.2	odd	2		5.14.a.b.1.3	✓	3
12.11	even	2		80.14.a.g.1.2		3
15.2	even	4		25.14.b.b.24.6		6
15.8	even	4		25.14.b.b.24.1		6
15.14	odd	2		25.14.a.b.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.14.a.b.1.3	✓	3	3.2	odd	2
25.14.a.b.1.1		3	15.14	odd	2
25.14.b.b.24.1		6	15.8	even	4
25.14.b.b.24.6		6	15.2	even	4
45.14.a.e.1.1		3	1.1	even	1	trivial
80.14.a.g.1.2		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-176.217 q^{2} +22860.4 q^{4} -15625.0 q^{5} -201493. q^{7} -2.58481e6 q^{8} +2.75339e6 q^{10} +3.34359e6 q^{11} -7.80359e6 q^{13} +3.55065e7 q^{14} +2.68215e8 q^{16} +8.71750e7 q^{17} -1.66766e7 q^{19} -3.57193e8 q^{20} -5.89196e8 q^{22} -1.13518e9 q^{23} +2.44141e8 q^{25} +1.37512e9 q^{26} -4.60620e9 q^{28} -2.60673e9 q^{29} +8.33139e8 q^{31} -2.60892e10 q^{32} -1.53617e10 q^{34} +3.14833e9 q^{35} -1.05494e10 q^{37} +2.93870e9 q^{38} +4.03877e10 q^{40} +4.33030e9 q^{41} +1.93854e9 q^{43} +7.64356e10 q^{44} +2.00038e11 q^{46} -2.85468e10 q^{47} -5.62896e10 q^{49} -4.30217e10 q^{50} -1.78393e11 q^{52} -1.23249e11 q^{53} -5.22435e10 q^{55} +5.20821e11 q^{56} +4.59349e11 q^{58} +5.55404e11 q^{59} -4.10476e11 q^{61} -1.46813e11 q^{62} +2.40014e12 q^{64} +1.21931e11 q^{65} +3.36861e11 q^{67} +1.99285e12 q^{68} -5.54788e11 q^{70} -1.57323e12 q^{71} +2.05372e12 q^{73} +1.85898e12 q^{74} -3.81234e11 q^{76} -6.73709e11 q^{77} -6.93000e11 q^{79} -4.19086e12 q^{80} -7.63071e11 q^{82} -2.01116e12 q^{83} -1.36211e12 q^{85} -3.41604e11 q^{86} -8.64253e12 q^{88} +8.51832e12 q^{89} +1.57237e12 q^{91} -2.59507e13 q^{92} +5.03042e12 q^{94} +2.60572e11 q^{95} -7.99814e12 q^{97} +9.91916e12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 142 q^{2} + 17876 q^{4} - 46875 q^{5} + 448292 q^{7} - 2580360 q^{8} + 2218750 q^{10} + 6604004 q^{11} - 33501974 q^{13} + 46562928 q^{14} + 199912208 q^{16} - 83129542 q^{17} + 97491100 q^{19} - 279312500 q^{20}+ \cdots + 10176508990306 q^{98}+O(q^{100}) \) 3 * q - 142 * q^2 + 17876 * q^4 - 46875 * q^5 + 448292 * q^7 - 2580360 * q^8 + 2218750 * q^10 + 6604004 * q^11 - 33501974 * q^13 + 46562928 * q^14 + 199912208 * q^16 - 83129542 * q^17 + 97491100 * q^19 - 279312500 * q^20 - 530907256 * q^22 - 316255836 * q^23 + 732421875 * q^25 + 3746814044 * q^26 - 6227646976 * q^28 - 2236171850 * q^29 + 7482994376 * q^31 - 32169857312 * q^32 - 30236073988 * q^34 - 7004562500 * q^35 + 31447174242 * q^37 + 42518132360 * q^38 + 40318125000 * q^40 + 10752884434 * q^41 + 16930554856 * q^43 + 68395825168 * q^44 + 246828204336 * q^46 - 31934201692 * q^47 - 38956926629 * q^49 - 34667968750 * q^50 - 18152597928 * q^52 + 221149123934 * q^53 - 103187562500 * q^55 + 522876451200 * q^56 + 854269976260 * q^58 + 55436423900 * q^59 + 496161392746 * q^61 + 598069940736 * q^62 + 3297565494336 * q^64 + 523468343750 * q^65 + 459297824792 * q^67 + 2008021178776 * q^68 - 727545750000 * q^70 - 521997878336 * q^71 + 2505025571086 * q^73 + 2418323586508 * q^74 + 621688050000 * q^76 + 385562457456 * q^77 + 2990636883200 * q^79 - 3123628250000 * q^80 + 5784848206924 * q^82 - 5137135467696 * q^83 + 1298899093750 * q^85 - 2343313645096 * q^86 - 8660607864480 * q^88 + 19423025958450 * q^89 - 6792522370184 * q^91 - 26870022340992 * q^92 - 7003557875968 * q^94 - 1523298437500 * q^95 - 11088325396458 * q^97 + 10176508990306 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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