LMFDB - Embedded newform 162.10.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,10,Mod(1,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-64,0,1024,1968] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$83.4358054585$
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} - 2x^{3} - 953x^{2} + 954x + 195702 $ x^4 - 2x^3 - 953x^2 + 954*x + 195702
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-25.1057$ of defining polynomial
Character	$\chi$	$=$	162.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-16.0000 q^{2} +256.000 q^{4} -637.540 q^{5} +3781.54 q^{7} -4096.00 q^{8} +10200.6 q^{10} +89240.4 q^{11} +21274.0 q^{13} -60504.7 q^{14} +65536.0 q^{16} +132912. q^{17} -268177. q^{19} -163210. q^{20} -1.42785e6 q^{22} +1.64033e6 q^{23} -1.54667e6 q^{25} -340384. q^{26} +968075. q^{28} +999678. q^{29} +1.71956e6 q^{31} -1.04858e6 q^{32} -2.12659e6 q^{34} -2.41088e6 q^{35} +7.62670e6 q^{37} +4.29083e6 q^{38} +2.61136e6 q^{40} +8.18478e6 q^{41} -1.48027e7 q^{43} +2.28455e7 q^{44} -2.62453e7 q^{46} -5.49249e7 q^{47} -2.60536e7 q^{49} +2.47467e7 q^{50} +5.44614e6 q^{52} +3.84097e7 q^{53} -5.68943e7 q^{55} -1.54892e7 q^{56} -1.59948e7 q^{58} +2.83312e7 q^{59} -6.13813e6 q^{61} -2.75130e7 q^{62} +1.67772e7 q^{64} -1.35630e7 q^{65} +2.02943e8 q^{67} +3.40254e7 q^{68} +3.85741e7 q^{70} +6.46176e7 q^{71} -4.04658e8 q^{73} -1.22027e8 q^{74} -6.86533e7 q^{76} +3.37466e8 q^{77} -6.50469e8 q^{79} -4.17818e7 q^{80} -1.30957e8 q^{82} -1.35342e8 q^{83} -8.47365e7 q^{85} +2.36844e8 q^{86} -3.65528e8 q^{88} +6.44360e8 q^{89} +8.04484e7 q^{91} +4.19925e8 q^{92} +8.78798e8 q^{94} +1.70974e8 q^{95} +1.13571e9 q^{97} +4.16857e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{2} + 1024 q^{4} + 1968 q^{5} + 4496 q^{7} - 16384 q^{8} - 31488 q^{10} - 8784 q^{11} - 162556 q^{13} - 71936 q^{14} + 262144 q^{16} + 538080 q^{17} - 8224 q^{19} + 503808 q^{20} + 140544 q^{22}+ \cdots + 2038788288 q^{98}+O(q^{100}) $ 4 * q - 64 * q^2 + 1024 * q^4 + 1968 * q^5 + 4496 * q^7 - 16384 * q^8 - 31488 * q^10 - 8784 * q^11 - 162556 * q^13 - 71936 * q^14 + 262144 * q^16 + 538080 * q^17 - 8224 * q^19 + 503808 * q^20 + 140544 * q^22 + 2594736 * q^23 + 1873192 * q^25 + 2600896 * q^26 + 1150976 * q^28 + 3232656 * q^29 + 3482576 * q^31 - 4194304 * q^32 - 8609280 * q^34 + 7969296 * q^35 - 2487892 * q^37 + 131584 * q^38 - 8060928 * q^40 + 34657152 * q^41 - 41410000 * q^43 - 2248704 * q^44 - 41515776 * q^46 + 40558848 * q^47 - 127424268 * q^49 - 29971072 * q^50 - 41614336 * q^52 + 8421024 * q^53 - 229068000 * q^55 - 18415616 * q^56 - 51722496 * q^58 + 26843328 * q^59 - 111504484 * q^61 - 55721216 * q^62 + 67108864 * q^64 + 23507616 * q^65 - 208064512 * q^67 + 137748480 * q^68 - 127508736 * q^70 + 356008272 * q^71 - 37126108 * q^73 + 39806272 * q^74 - 2105344 * q^76 + 33010560 * q^77 - 645134848 * q^79 + 128974848 * q^80 - 554514432 * q^82 + 1019036256 * q^83 + 80955540 * q^85 + 662560000 * q^86 + 35979264 * q^88 + 1548192768 * q^89 - 130776608 * q^91 + 664252416 * q^92 - 648941568 * q^94 + 3445933872 * q^95 + 1112014568 * q^97 + 2038788288 * 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 $

$2$	−16.0000	−0.707107
$2$	−16.0000	−0.707107
$3$	0	0
$3$	0	0
$4$	256.000	0.500000
$5$	−637.540	−0.456186	−0.228093	−	0.973639i	$-0.573249\pi$
$5$	−637.540	−0.456186	−0.228093	+	0.973639i	$0.573249\pi$
$6$	0	0
$7$	3781.54	0.595289	0.297644	−	0.954677i	$-0.403799\pi$
$7$	3781.54	0.595289	0.297644	+	0.954677i	$0.403799\pi$
$8$	−4096.00	−0.353553
$9$	0	0
$10$	10200.6	0.322573
$11$	89240.4	1.83778	0.918891	−	0.394510i	$-0.129086\pi$
$11$	89240.4	1.83778	0.918891	+	0.394510i	$0.129086\pi$
$12$	0	0
$13$	21274.0	0.206587	0.103294	−	0.994651i	$-0.467062\pi$
$13$	21274.0	0.206587	0.103294	+	0.994651i	$0.467062\pi$
$14$	−60504.7	−0.420933
$15$	0	0
$16$	65536.0	0.250000
$17$	132912.	0.385960	0.192980	−	0.981203i	$-0.438185\pi$
$17$	132912.	0.385960	0.192980	+	0.981203i	$0.438185\pi$
$18$	0	0
$19$	−268177.	−0.472096	−0.236048	−	0.971741i	$-0.575852\pi$
$19$	−268177.	−0.472096	−0.236048	+	0.971741i	$0.575852\pi$
$20$	−163210.	−0.228093
$21$	0	0
$22$	−1.42785e6	−1.29951
$23$	1.64033e6	1.22224	0.611121	−	0.791537i	$-0.290719\pi$
$23$	1.64033e6	1.22224	0.611121	+	0.791537i	$0.290719\pi$
$24$	0	0
$25$	−1.54667e6	−0.791894
$26$	−340384.	−0.146079
$27$	0	0
$28$	968075.	0.297644
$29$	999678.	0.262464	0.131232	−	0.991352i	$-0.458107\pi$
$29$	999678.	0.262464	0.131232	+	0.991352i	$0.458107\pi$
$30$	0	0
$31$	1.71956e6	0.334419	0.167209	−	0.985921i	$-0.446524\pi$
$31$	1.71956e6	0.334419	0.167209	+	0.985921i	$0.446524\pi$
$32$	−1.04858e6	−0.176777
$33$	0	0
$34$	−2.12659e6	−0.272915
$35$	−2.41088e6	−0.271563
$36$	0	0
$37$	7.62670e6	0.669004	0.334502	−	0.942395i	$-0.391432\pi$
$37$	7.62670e6	0.669004	0.334502	+	0.942395i	$0.391432\pi$
$38$	4.29083e6	0.333822
$39$	0	0
$40$	2.61136e6	0.161286
$41$	8.18478e6	0.452355	0.226178	−	0.974086i	$-0.427377\pi$
$41$	8.18478e6	0.452355	0.226178	+	0.974086i	$0.427377\pi$
$42$	0	0
$43$	−1.48027e7	−0.660289	−0.330145	−	0.943930i	$-0.607097\pi$
$43$	−1.48027e7	−0.660289	−0.330145	+	0.943930i	$0.607097\pi$
$44$	2.28455e7	0.918891
$45$	0	0
$46$	−2.62453e7	−0.864255
$47$	−5.49249e7	−1.64183	−0.820916	−	0.571049i	$-0.806536\pi$
$47$	−5.49249e7	−1.64183	−0.820916	+	0.571049i	$0.806536\pi$
$48$	0	0
$49$	−2.60536e7	−0.645631
$50$	2.47467e7	0.559954
$51$	0	0
$52$	5.44614e6	0.103294
$53$	3.84097e7	0.668650	0.334325	−	0.942458i	$-0.391492\pi$
$53$	3.84097e7	0.668650	0.334325	+	0.942458i	$0.391492\pi$
$54$	0	0
$55$	−5.68943e7	−0.838372
$56$	−1.54892e7	−0.210466
$57$	0	0
$58$	−1.59948e7	−0.185590
$59$	2.83312e7	0.304390	0.152195	−	0.988350i	$-0.451366\pi$
$59$	2.83312e7	0.304390	0.152195	+	0.988350i	$0.451366\pi$
$60$	0	0
$61$	−6.13813e6	−0.0567612	−0.0283806	−	0.999597i	$-0.509035\pi$
$61$	−6.13813e6	−0.0567612	−0.0283806	+	0.999597i	$0.509035\pi$
$62$	−2.75130e7	−0.236470
$63$	0	0
$64$	1.67772e7	0.125000
$65$	−1.35630e7	−0.0942423
$66$	0	0
$67$	2.02943e8	1.23038	0.615188	−	0.788380i	$-0.289080\pi$
$67$	2.02943e8	1.23038	0.615188	+	0.788380i	$0.289080\pi$
$68$	3.40254e7	0.192980
$69$	0	0
$70$	3.85741e7	0.192024
$71$	6.46176e7	0.301778	0.150889	−	0.988551i	$-0.451786\pi$
$71$	6.46176e7	0.301778	0.150889	+	0.988551i	$0.451786\pi$
$72$	0	0
$73$	−4.04658e8	−1.66777	−0.833884	−	0.551940i	$-0.813888\pi$
$73$	−4.04658e8	−1.66777	−0.833884	+	0.551940i	$0.813888\pi$
$74$	−1.22027e8	−0.473057
$75$	0	0
$76$	−6.86533e7	−0.236048
$77$	3.37466e8	1.09401
$78$	0	0
$79$	−6.50469e8	−1.87890	−0.939452	−	0.342679i	$-0.888666\pi$
$79$	−6.50469e8	−1.87890	−0.939452	+	0.342679i	$0.888666\pi$
$80$	−4.17818e7	−0.114047
$81$	0	0
$82$	−1.30957e8	−0.319864
$83$	−1.35342e8	−0.313028	−0.156514	−	0.987676i	$-0.550026\pi$
$83$	−1.35342e8	−0.313028	−0.156514	+	0.987676i	$0.550026\pi$
$84$	0	0
$85$	−8.47365e7	−0.176070
$86$	2.36844e8	0.466895
$87$	0	0
$88$	−3.65528e8	−0.649754
$89$	6.44360e8	1.08861	0.544307	−	0.838886i	$-0.316793\pi$
$89$	6.44360e8	1.08861	0.544307	+	0.838886i	$0.316793\pi$
$90$	0	0
$91$	8.04484e7	0.122979
$92$	4.19925e8	0.611121
$93$	0	0
$94$	8.78798e8	1.16095
$95$	1.70974e8	0.215364
$96$	0	0
$97$	1.13571e9	1.30255	0.651274	−	0.758842i	$-0.274235\pi$
$97$	1.13571e9	1.30255	0.651274	+	0.758842i	$0.274235\pi$
$98$	4.16857e8	0.456530
$99$	0	0
$100$	−3.95947e8	−0.395947
$101$	1.74490e9	1.66849	0.834245	−	0.551394i	$-0.185904\pi$
$101$	1.74490e9	1.66849	0.834245	+	0.551394i	$0.185904\pi$
$102$	0	0
$103$	1.75960e9	1.54044	0.770222	−	0.637776i	$-0.220145\pi$
$103$	1.75960e9	1.54044	0.770222	+	0.637776i	$0.220145\pi$
$104$	−8.71382e7	−0.0730396
$105$	0	0
$106$	−6.14554e8	−0.472807
$107$	7.07625e8	0.521887	0.260943	−	0.965354i	$-0.415966\pi$
$107$	7.07625e8	0.521887	0.260943	+	0.965354i	$0.415966\pi$
$108$	0	0
$109$	−1.68385e8	−0.114258	−0.0571288	−	0.998367i	$-0.518195\pi$
$109$	−1.68385e8	−0.114258	−0.0571288	+	0.998367i	$0.518195\pi$
$110$	9.10309e8	0.592818
$111$	0	0
$112$	2.47827e8	0.148822
$113$	1.51673e9	0.875093	0.437547	−	0.899196i	$-0.355848\pi$
$113$	1.51673e9	0.875093	0.437547	+	0.899196i	$0.355848\pi$
$114$	0	0
$115$	−1.04578e9	−0.557570
$116$	2.55918e8	0.131232
$117$	0	0
$118$	−4.53299e8	−0.215236
$119$	5.02611e8	0.229758
$120$	0	0
$121$	5.60589e9	2.37745
$122$	9.82101e7	0.0401363
$123$	0	0
$124$	4.40208e8	0.167209
$125$	2.23126e9	0.817438
$126$	0	0
$127$	−4.26548e9	−1.45496	−0.727480	−	0.686129i	$-0.759309\pi$
$127$	−4.26548e9	−1.45496	−0.727480	+	0.686129i	$0.759309\pi$
$128$	−2.68435e8	−0.0883883
$129$	0	0
$130$	2.17008e8	0.0666394
$131$	−1.17007e8	−0.0347128	−0.0173564	−	0.999849i	$-0.505525\pi$
$131$	−1.17007e8	−0.0347128	−0.0173564	+	0.999849i	$0.505525\pi$
$132$	0	0
$133$	−1.01412e9	−0.281033
$134$	−3.24709e9	−0.870008
$135$	0	0
$136$	−5.44406e8	−0.136458
$137$	7.28209e9	1.76609	0.883047	−	0.469285i	$-0.155488\pi$
$137$	7.28209e9	1.76609	0.883047	+	0.469285i	$0.155488\pi$
$138$	0	0
$139$	7.27338e9	1.65261	0.826303	−	0.563226i	$-0.190440\pi$
$139$	7.27338e9	1.65261	0.826303	+	0.563226i	$0.190440\pi$
$140$	−6.17186e8	−0.135781
$141$	0	0
$142$	−1.03388e9	−0.213389
$143$	1.89850e9	0.379662
$144$	0	0
$145$	−6.37335e8	−0.119732
$146$	6.47453e9	1.17929
$147$	0	0
$148$	1.95243e9	0.334502
$149$	−2.85456e9	−0.474461	−0.237231	−	0.971453i	$-0.576240\pi$
$149$	−2.85456e9	−0.474461	−0.237231	+	0.971453i	$0.576240\pi$
$150$	0	0
$151$	2.03027e9	0.317803	0.158901	−	0.987294i	$-0.449205\pi$
$151$	2.03027e9	0.317803	0.158901	+	0.987294i	$0.449205\pi$
$152$	1.09845e9	0.166911
$153$	0	0
$154$	−5.39946e9	−0.773583
$155$	−1.09629e9	−0.152557
$156$	0	0
$157$	4.25662e9	0.559134	0.279567	−	0.960126i	$-0.409809\pi$
$157$	4.25662e9	0.559134	0.279567	+	0.960126i	$0.409809\pi$
$158$	1.04075e10	1.32859
$159$	0	0
$160$	6.68509e8	0.0806431
$161$	6.20299e9	0.727587
$162$	0	0
$163$	−4.16936e9	−0.462621	−0.231311	−	0.972880i	$-0.574301\pi$
$163$	−4.16936e9	−0.462621	−0.231311	+	0.972880i	$0.574301\pi$
$164$	2.09530e9	0.226178
$165$	0	0
$166$	2.16548e9	0.221344
$167$	1.07024e10	1.06477	0.532386	−	0.846502i	$-0.321295\pi$
$167$	1.07024e10	1.06477	0.532386	+	0.846502i	$0.321295\pi$
$168$	0	0
$169$	−1.01519e10	−0.957322
$170$	1.35578e9	0.124500
$171$	0	0
$172$	−3.78950e9	−0.330145
$173$	−1.59012e10	−1.34966	−0.674828	−	0.737975i	$-0.735782\pi$
$173$	−1.59012e10	−1.34966	−0.674828	+	0.737975i	$0.735782\pi$
$174$	0	0
$175$	−5.84879e9	−0.471406
$176$	5.84846e9	0.459446
$177$	0	0
$178$	−1.03098e10	−0.769766
$179$	2.05365e10	1.49516	0.747579	−	0.664173i	$-0.231216\pi$
$179$	2.05365e10	1.49516	0.747579	+	0.664173i	$0.231216\pi$
$180$	0	0
$181$	1.50797e10	1.04433	0.522167	−	0.852843i	$-0.325124\pi$
$181$	1.50797e10	1.04433	0.522167	+	0.852843i	$0.325124\pi$
$182$	−1.28717e9	−0.0869593
$183$	0	0
$184$	−6.71881e9	−0.432128
$185$	−4.86232e9	−0.305191
$186$	0	0
$187$	1.18611e10	0.709312
$188$	−1.40608e10	−0.820916
$189$	0	0
$190$	−2.73558e9	−0.152285
$191$	−4.78381e9	−0.260090	−0.130045	−	0.991508i	$-0.541512\pi$
$191$	−4.78381e9	−0.260090	−0.130045	+	0.991508i	$0.541512\pi$
$192$	0	0
$193$	1.94088e10	1.00691	0.503456	−	0.864021i	$-0.332062\pi$
$193$	1.94088e10	1.00691	0.503456	+	0.864021i	$0.332062\pi$
$194$	−1.81713e10	−0.921041
$195$	0	0
$196$	−6.66971e9	−0.322816
$197$	3.64545e9	0.172446	0.0862230	−	0.996276i	$-0.472520\pi$
$197$	3.64545e9	0.172446	0.0862230	+	0.996276i	$0.472520\pi$
$198$	0	0
$199$	−6.13755e9	−0.277432	−0.138716	−	0.990332i	$-0.544297\pi$
$199$	−6.13755e9	−0.277432	−0.138716	+	0.990332i	$0.544297\pi$
$200$	6.33515e9	0.279977
$201$	0	0
$202$	−2.79184e10	−1.17980
$203$	3.78032e9	0.156242
$204$	0	0
$205$	−5.21813e9	−0.206358
$206$	−2.81536e10	−1.08926
$207$	0	0
$208$	1.39421e9	0.0516468
$209$	−2.39322e10	−0.867610
$210$	0	0
$211$	−1.57584e10	−0.547321	−0.273660	−	0.961826i	$-0.588234\pi$
$211$	−1.57584e10	−0.547321	−0.273660	+	0.961826i	$0.588234\pi$
$212$	9.83287e9	0.334325
$213$	0	0
$214$	−1.13220e10	−0.369030
$215$	9.43734e9	0.301215
$216$	0	0
$217$	6.50260e9	0.199076
$218$	2.69416e9	0.0807923
$219$	0	0
$220$	−1.45649e10	−0.419186
$221$	2.82756e9	0.0797345
$222$	0	0
$223$	4.21334e10	1.14092	0.570459	−	0.821326i	$-0.306765\pi$
$223$	4.21334e10	1.14092	0.570459	+	0.821326i	$0.306765\pi$
$224$	−3.96523e9	−0.105233
$225$	0	0
$226$	−2.42676e10	−0.618784
$227$	−6.76606e10	−1.69129	−0.845647	−	0.533742i	$-0.820785\pi$
$227$	−6.76606e10	−1.69129	−0.845647	+	0.533742i	$0.820785\pi$
$228$	0	0
$229$	−6.88813e10	−1.65517	−0.827583	−	0.561343i	$-0.810285\pi$
$229$	−6.88813e10	−1.65517	−0.827583	+	0.561343i	$0.810285\pi$
$230$	1.67325e10	0.394261
$231$	0	0
$232$	−4.09468e9	−0.0927949
$233$	2.23441e9	0.0496662	0.0248331	−	0.999692i	$-0.492095\pi$
$233$	2.23441e9	0.0496662	0.0248331	+	0.999692i	$0.492095\pi$
$234$	0	0
$235$	3.50168e10	0.748981
$236$	7.25278e9	0.152195
$237$	0	0
$238$	−8.04177e9	−0.162463
$239$	−9.21624e10	−1.82710	−0.913552	−	0.406723i	$-0.866671\pi$
$239$	−9.21624e10	−1.82710	−0.913552	+	0.406723i	$0.866671\pi$
$240$	0	0
$241$	7.86149e10	1.50116	0.750582	−	0.660777i	$-0.229773\pi$
$241$	7.86149e10	1.50116	0.750582	+	0.660777i	$0.229773\pi$
$242$	−8.96943e10	−1.68111
$243$	0	0
$244$	−1.57136e9	−0.0283806
$245$	1.66102e10	0.294528
$246$	0	0
$247$	−5.70519e9	−0.0975290
$248$	−7.04333e9	−0.118235
$249$	0	0
$250$	−3.57001e10	−0.578016
$251$	−5.33618e10	−0.848591	−0.424296	−	0.905524i	$-0.639478\pi$
$251$	−5.33618e10	−0.848591	−0.424296	+	0.905524i	$0.639478\pi$
$252$	0	0
$253$	1.46384e11	2.24621
$254$	6.82477e10	1.02881
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	−8.11180e10	−1.15989	−0.579947	−	0.814654i	$-0.696927\pi$
$257$	−8.11180e10	−1.15989	−0.579947	+	0.814654i	$0.696927\pi$
$258$	0	0
$259$	2.88407e10	0.398251
$260$	−3.47213e9	−0.0471211
$261$	0	0
$262$	1.87211e9	0.0245457
$263$	−2.55619e10	−0.329452	−0.164726	−	0.986339i	$-0.552674\pi$
$263$	−2.55619e10	−0.329452	−0.164726	+	0.986339i	$0.552674\pi$
$264$	0	0
$265$	−2.44877e10	−0.305029
$266$	1.62260e10	0.198721
$267$	0	0
$268$	5.19535e10	0.615188
$269$	1.19720e11	1.39406	0.697028	−	0.717044i	$-0.254505\pi$
$269$	1.19720e11	1.39406	0.697028	+	0.717044i	$0.254505\pi$
$270$	0	0
$271$	1.32991e11	1.49783	0.748914	−	0.662667i	$-0.230576\pi$
$271$	1.32991e11	1.49783	0.748914	+	0.662667i	$0.230576\pi$
$272$	8.71050e9	0.0964901
$273$	0	0
$274$	−1.16514e11	−1.24882
$275$	−1.38025e11	−1.45533
$276$	0	0
$277$	−1.27005e10	−0.129617	−0.0648084	−	0.997898i	$-0.520644\pi$
$277$	−1.27005e10	−0.129617	−0.0648084	+	0.997898i	$0.520644\pi$
$278$	−1.16374e11	−1.16857
$279$	0	0
$280$	9.87498e9	0.0960119
$281$	−1.00052e11	−0.957302	−0.478651	−	0.878005i	$-0.658874\pi$
$281$	−1.00052e11	−0.957302	−0.478651	+	0.878005i	$0.658874\pi$
$282$	0	0
$283$	9.22944e10	0.855336	0.427668	−	0.903936i	$-0.359335\pi$
$283$	9.22944e10	0.855336	0.427668	+	0.903936i	$0.359335\pi$
$284$	1.65421e10	0.150889
$285$	0	0
$286$	−3.03759e10	−0.268462
$287$	3.09511e10	0.269282
$288$	0	0
$289$	−1.00922e11	−0.851035
$290$	1.01974e10	0.0846636
$291$	0	0
$292$	−1.03593e11	−0.833884
$293$	1.21880e11	0.966114	0.483057	−	0.875589i	$-0.339526\pi$
$293$	1.21880e11	0.966114	0.483057	+	0.875589i	$0.339526\pi$
$294$	0	0
$295$	−1.80622e10	−0.138859
$296$	−3.12390e10	−0.236529
$297$	0	0
$298$	4.56730e10	0.335495
$299$	3.48964e10	0.252499
$300$	0	0
$301$	−5.59772e10	−0.393063
$302$	−3.24843e10	−0.224720
$303$	0	0
$304$	−1.75752e10	−0.118024
$305$	3.91330e9	0.0258937
$306$	0	0
$307$	1.46439e11	0.940881	0.470441	−	0.882432i	$-0.344095\pi$
$307$	1.46439e11	0.940881	0.470441	+	0.882432i	$0.344095\pi$
$308$	8.63913e10	0.547006
$309$	0	0
$310$	1.75407e10	0.107874
$311$	6.86279e10	0.415986	0.207993	−	0.978130i	$-0.433307\pi$
$311$	6.86279e10	0.415986	0.207993	+	0.978130i	$0.433307\pi$
$312$	0	0
$313$	2.72147e11	1.60270	0.801352	−	0.598193i	$-0.204114\pi$
$313$	2.72147e11	1.60270	0.801352	+	0.598193i	$0.204114\pi$
$314$	−6.81059e10	−0.395368
$315$	0	0
$316$	−1.66520e11	−0.939452
$317$	3.26443e9	0.0181569	0.00907844	−	0.999959i	$-0.497110\pi$
$317$	3.26443e9	0.0181569	0.00907844	+	0.999959i	$0.497110\pi$
$318$	0	0
$319$	8.92116e10	0.482351
$320$	−1.06961e10	−0.0570233
$321$	0	0
$322$	−9.92478e10	−0.514481
$323$	−3.56438e10	−0.182210
$324$	0	0
$325$	−3.29038e10	−0.163595
$326$	6.67098e10	0.327123
$327$	0	0
$328$	−3.35249e10	−0.159932
$329$	−2.07701e11	−0.977364
$330$	0	0
$331$	−5.33327e10	−0.244212	−0.122106	−	0.992517i	$-0.538965\pi$
$331$	−5.33327e10	−0.244212	−0.122106	+	0.992517i	$0.538965\pi$
$332$	−3.46477e10	−0.156514
$333$	0	0
$334$	−1.71238e11	−0.752908
$335$	−1.29384e11	−0.561281
$336$	0	0
$337$	9.17605e10	0.387544	0.193772	−	0.981047i	$-0.437928\pi$
$337$	9.17605e10	0.387544	0.193772	+	0.981047i	$0.437928\pi$
$338$	1.62431e11	0.676929
$339$	0	0
$340$	−2.16925e10	−0.0880350
$341$	1.53454e11	0.614589
$342$	0	0
$343$	−2.51121e11	−0.979626
$344$	6.06320e10	0.233447
$345$	0	0
$346$	2.54420e11	0.954351
$347$	2.81551e11	1.04250	0.521248	−	0.853405i	$-0.325467\pi$
$347$	2.81551e11	1.04250	0.521248	+	0.853405i	$0.325467\pi$
$348$	0	0
$349$	−4.04395e11	−1.45912	−0.729561	−	0.683916i	$-0.760275\pi$
$349$	−4.04395e11	−1.45912	−0.729561	+	0.683916i	$0.760275\pi$
$350$	9.35806e10	0.333334
$351$	0	0
$352$	−9.35753e10	−0.324877
$353$	−3.56621e11	−1.22242	−0.611211	−	0.791468i	$-0.709317\pi$
$353$	−3.56621e11	−1.22242	−0.611211	+	0.791468i	$0.709317\pi$
$354$	0	0
$355$	−4.11963e10	−0.137667
$356$	1.64956e11	0.544307
$357$	0	0
$358$	−3.28583e11	−1.05724
$359$	4.93016e11	1.56652	0.783261	−	0.621693i	$-0.213555\pi$
$359$	4.93016e11	1.56652	0.783261	+	0.621693i	$0.213555\pi$
$360$	0	0
$361$	−2.50769e11	−0.777125
$362$	−2.41275e11	−0.738456
$363$	0	0
$364$	2.05948e10	0.0614895
$365$	2.57986e11	0.760813
$366$	0	0
$367$	1.06005e11	0.305021	0.152510	−	0.988302i	$-0.451264\pi$
$367$	1.06005e11	0.305021	0.152510	+	0.988302i	$0.451264\pi$
$368$	1.07501e11	0.305560
$369$	0	0
$370$	7.77972e10	0.215802
$371$	1.45248e11	0.398040
$372$	0	0
$373$	5.40318e11	1.44531	0.722653	−	0.691211i	$-0.242923\pi$
$373$	5.40318e11	1.44531	0.722653	+	0.691211i	$0.242923\pi$
$374$	−1.89777e11	−0.501559
$375$	0	0
$376$	2.24972e11	0.580475
$377$	2.12671e10	0.0542216
$378$	0	0
$379$	−4.89358e11	−1.21829	−0.609145	−	0.793059i	$-0.708487\pi$
$379$	−4.89358e11	−1.21829	−0.609145	+	0.793059i	$0.708487\pi$
$380$	4.37692e10	0.107682
$381$	0	0
$382$	7.65409e10	0.183911
$383$	2.95665e11	0.702110	0.351055	−	0.936355i	$-0.385823\pi$
$383$	2.95665e11	0.702110	0.351055	+	0.936355i	$0.385823\pi$
$384$	0	0
$385$	−2.15148e11	−0.499073
$386$	−3.10541e11	−0.711994
$387$	0	0
$388$	2.90741e11	0.651274
$389$	7.57118e11	1.67645	0.838224	−	0.545326i	$-0.183594\pi$
$389$	7.57118e11	1.67645	0.838224	+	0.545326i	$0.183594\pi$
$390$	0	0
$391$	2.18019e11	0.471737
$392$	1.06715e11	0.228265
$393$	0	0
$394$	−5.83272e10	−0.121938
$395$	4.14700e11	0.857131
$396$	0	0
$397$	1.91762e11	0.387441	0.193721	−	0.981057i	$-0.437944\pi$
$397$	1.91762e11	0.387441	0.193721	+	0.981057i	$0.437944\pi$
$398$	9.82008e10	0.196174
$399$	0	0
$400$	−1.01362e11	−0.197973
$401$	3.59283e10	0.0693884	0.0346942	−	0.999398i	$-0.488954\pi$
$401$	3.59283e10	0.0693884	0.0346942	+	0.999398i	$0.488954\pi$
$402$	0	0
$403$	3.65820e10	0.0690867
$404$	4.46694e11	0.834245
$405$	0	0
$406$	−6.04852e10	−0.110480
$407$	6.80609e11	1.22948
$408$	0	0
$409$	5.56662e11	0.983642	0.491821	−	0.870696i	$-0.336331\pi$
$409$	5.56662e11	0.983642	0.491821	+	0.870696i	$0.336331\pi$
$410$	8.34900e10	0.145917
$411$	0	0
$412$	4.50457e11	0.770222
$413$	1.07135e11	0.181200
$414$	0	0
$415$	8.62862e10	0.142799
$416$	−2.23074e10	−0.0365198
$417$	0	0
$418$	3.82915e11	0.613493
$419$	4.50049e11	0.713341	0.356670	−	0.934230i	$-0.383912\pi$
$419$	4.50049e11	0.713341	0.356670	+	0.934230i	$0.383912\pi$
$420$	0	0
$421$	−6.42393e11	−0.996625	−0.498312	−	0.866998i	$-0.666047\pi$
$421$	−6.42393e11	−0.996625	−0.498312	+	0.866998i	$0.666047\pi$
$422$	2.52135e11	0.387014
$423$	0	0
$424$	−1.57326e11	−0.236404
$425$	−2.05570e11	−0.305640
$426$	0	0
$427$	−2.32116e10	−0.0337893
$428$	1.81152e11	0.260943
$429$	0	0
$430$	−1.50997e11	−0.212991
$431$	−8.75529e11	−1.22215	−0.611073	−	0.791574i	$-0.709262\pi$
$431$	−8.75529e11	−1.22215	−0.611073	+	0.791574i	$0.709262\pi$
$432$	0	0
$433$	−1.13112e12	−1.54637	−0.773185	−	0.634180i	$-0.781338\pi$
$433$	−1.13112e12	−1.54637	−0.773185	+	0.634180i	$0.781338\pi$
$434$	−1.04042e11	−0.140768
$435$	0	0
$436$	−4.31066e10	−0.0571288
$437$	−4.39900e11	−0.577015
$438$	0	0
$439$	−4.62516e11	−0.594343	−0.297171	−	0.954824i	$-0.596043\pi$
$439$	−4.62516e11	−0.594343	−0.297171	+	0.954824i	$0.596043\pi$
$440$	2.33039e11	0.296409
$441$	0	0
$442$	−4.52409e10	−0.0563808
$443$	1.40914e11	0.173834	0.0869172	−	0.996216i	$-0.472298\pi$
$443$	1.40914e11	0.173834	0.0869172	+	0.996216i	$0.472298\pi$
$444$	0	0
$445$	−4.10805e11	−0.496611
$446$	−6.74134e11	−0.806751
$447$	0	0
$448$	6.34437e10	0.0744111
$449$	−1.31651e12	−1.52867	−0.764337	−	0.644817i	$-0.776934\pi$
$449$	−1.31651e12	−1.52867	−0.764337	+	0.644817i	$0.776934\pi$
$450$	0	0
$451$	7.30413e11	0.831331
$452$	3.88282e11	0.437547
$453$	0	0
$454$	1.08257e12	1.19593
$455$	−5.12891e10	−0.0561014
$456$	0	0
$457$	1.00287e12	1.07553	0.537766	−	0.843094i	$-0.319269\pi$
$457$	1.00287e12	1.07553	0.537766	+	0.843094i	$0.319269\pi$
$458$	1.10210e12	1.17038
$459$	0	0
$460$	−2.67719e11	−0.278785
$461$	1.22220e12	1.26034	0.630171	−	0.776457i	$-0.282985\pi$
$461$	1.22220e12	1.26034	0.630171	+	0.776457i	$0.282985\pi$
$462$	0	0
$463$	−2.35538e11	−0.238203	−0.119102	−	0.992882i	$-0.538001\pi$
$463$	−2.35538e11	−0.238203	−0.119102	+	0.992882i	$0.538001\pi$
$464$	6.55149e10	0.0656159
$465$	0	0
$466$	−3.57505e10	−0.0351193
$467$	−6.25942e11	−0.608987	−0.304493	−	0.952514i	$-0.598487\pi$
$467$	−6.25942e11	−0.608987	−0.304493	+	0.952514i	$0.598487\pi$
$468$	0	0
$469$	7.67438e11	0.732429
$470$	−5.60269e11	−0.529610
$471$	0	0
$472$	−1.16044e11	−0.107618
$473$	−1.32100e12	−1.21347
$474$	0	0
$475$	4.14781e11	0.373850
$476$	1.28668e11	0.114879
$477$	0	0
$478$	1.47460e12	1.29196
$479$	−5.29764e11	−0.459804	−0.229902	−	0.973214i	$-0.573841\pi$
$479$	−5.29764e11	−0.459804	−0.229902	+	0.973214i	$0.573841\pi$
$480$	0	0
$481$	1.62250e11	0.138208
$482$	−1.25784e12	−1.06148
$483$	0	0
$484$	1.43511e12	1.18872
$485$	−7.24059e11	−0.594205
$486$	0	0
$487$	−5.57418e10	−0.0449056	−0.0224528	−	0.999748i	$-0.507148\pi$
$487$	−5.57418e10	−0.0449056	−0.0224528	+	0.999748i	$0.507148\pi$
$488$	2.51418e10	0.0200681
$489$	0	0
$490$	−2.65763e11	−0.208263
$491$	3.73586e11	0.290084	0.145042	−	0.989426i	$-0.453668\pi$
$491$	3.73586e11	0.290084	0.145042	+	0.989426i	$0.453668\pi$
$492$	0	0
$493$	1.32869e11	0.101301
$494$	9.12830e10	0.0689634
$495$	0	0
$496$	1.12693e11	0.0836047
$497$	2.44354e11	0.179645
$498$	0	0
$499$	−3.44539e11	−0.248763	−0.124381	−	0.992234i	$-0.539695\pi$
$499$	−3.44539e11	−0.248763	−0.124381	+	0.992234i	$0.539695\pi$
$500$	5.71202e11	0.408719
$501$	0	0
$502$	8.53789e11	0.600045
$503$	−1.14060e12	−0.794468	−0.397234	−	0.917717i	$-0.630030\pi$
$503$	−1.14060e12	−0.794468	−0.397234	+	0.917717i	$0.630030\pi$
$504$	0	0
$505$	−1.11244e12	−0.761143
$506$	−2.34214e12	−1.58831
$507$	0	0
$508$	−1.09196e12	−0.727480
$509$	1.14345e12	0.755068	0.377534	−	0.925996i	$-0.376772\pi$
$509$	1.14345e12	0.755068	0.377534	+	0.925996i	$0.376772\pi$
$510$	0	0
$511$	−1.53023e12	−0.992803
$512$	−6.87195e10	−0.0441942
$513$	0	0
$514$	1.29789e12	0.820169
$515$	−1.12181e12	−0.702730
$516$	0	0
$517$	−4.90151e12	−3.01733
$518$	−4.61451e11	−0.281606
$519$	0	0
$520$	5.55541e10	0.0333197
$521$	2.26626e12	1.34753	0.673767	−	0.738944i	$-0.264675\pi$
$521$	2.26626e12	1.34753	0.673767	+	0.738944i	$0.264675\pi$
$522$	0	0
$523$	5.87333e11	0.343263	0.171631	−	0.985161i	$-0.445096\pi$
$523$	5.87333e11	0.343263	0.171631	+	0.985161i	$0.445096\pi$
$524$	−2.99537e10	−0.0173564
$525$	0	0
$526$	4.08990e11	0.232958
$527$	2.28550e11	0.129072
$528$	0	0
$529$	8.89542e11	0.493874
$530$	3.91803e11	0.215688
$531$	0	0
$532$	−2.59615e11	−0.140517
$533$	1.74123e11	0.0934509
$534$	0	0
$535$	−4.51139e11	−0.238078
$536$	−8.31256e11	−0.435004
$537$	0	0
$538$	−1.91552e12	−0.985747
$539$	−2.32503e12	−1.18653
$540$	0	0
$541$	5.23918e11	0.262951	0.131476	−	0.991319i	$-0.458028\pi$
$541$	5.23918e11	0.262951	0.131476	+	0.991319i	$0.458028\pi$
$542$	−2.12786e12	−1.05912
$543$	0	0
$544$	−1.39368e11	−0.0682288
$545$	1.07352e11	0.0521228
$546$	0	0
$547$	−1.45512e12	−0.694955	−0.347477	−	0.937688i	$-0.612962\pi$
$547$	−1.45512e12	−0.694955	−0.347477	+	0.937688i	$0.612962\pi$
$548$	1.86422e12	0.883047
$549$	0	0
$550$	2.20840e12	1.02907
$551$	−2.68091e11	−0.123908
$552$	0	0
$553$	−2.45978e12	−1.11849
$554$	2.03208e11	0.0916529
$555$	0	0
$556$	1.86198e12	0.826303
$557$	−3.34803e12	−1.47381	−0.736903	−	0.675999i	$-0.763713\pi$
$557$	−3.34803e12	−1.47381	−0.736903	+	0.675999i	$0.763713\pi$
$558$	0	0
$559$	−3.14913e11	−0.136407
$560$	−1.58000e11	−0.0678907
$561$	0	0
$562$	1.60084e12	0.676915
$563$	−2.74036e12	−1.14953	−0.574764	−	0.818319i	$-0.694906\pi$
$563$	−2.74036e12	−1.14953	−0.574764	+	0.818319i	$0.694906\pi$
$564$	0	0
$565$	−9.66974e11	−0.399206
$566$	−1.47671e12	−0.604814
$567$	0	0
$568$	−2.64674e11	−0.106695
$569$	−7.62690e11	−0.305030	−0.152515	−	0.988301i	$-0.548737\pi$
$569$	−7.62690e11	−0.305030	−0.152515	+	0.988301i	$0.548737\pi$
$570$	0	0
$571$	−3.98085e12	−1.56716	−0.783581	−	0.621290i	$-0.786609\pi$
$571$	−3.98085e12	−1.56716	−0.783581	+	0.621290i	$0.786609\pi$
$572$	4.86015e11	0.189831
$573$	0	0
$574$	−4.95218e11	−0.190411
$575$	−2.53705e12	−0.967886
$576$	0	0
$577$	3.34927e12	1.25794	0.628969	−	0.777430i	$-0.283477\pi$
$577$	3.34927e12	1.25794	0.628969	+	0.777430i	$0.283477\pi$
$578$	1.61476e12	0.601772
$579$	0	0
$580$	−1.63158e11	−0.0598662
$581$	−5.11803e11	−0.186342
$582$	0	0
$583$	3.42769e12	1.22883
$584$	1.65748e12	0.589645
$585$	0	0
$586$	−1.95008e12	−0.683145
$587$	−1.78876e12	−0.621844	−0.310922	−	0.950435i	$-0.600638\pi$
$587$	−1.78876e12	−0.621844	−0.310922	+	0.950435i	$0.600638\pi$
$588$	0	0
$589$	−4.61147e11	−0.157878
$590$	2.88996e11	0.0981878
$591$	0	0
$592$	4.99823e11	0.167251
$593$	2.66988e12	0.886638	0.443319	−	0.896364i	$-0.353801\pi$
$593$	2.66988e12	0.886638	0.443319	+	0.896364i	$0.353801\pi$
$594$	0	0
$595$	−3.20434e11	−0.104812
$596$	−7.30767e11	−0.237231
$597$	0	0
$598$	−5.58343e11	−0.178544
$599$	3.81780e12	1.21169	0.605847	−	0.795581i	$-0.292834\pi$
$599$	3.81780e12	1.21169	0.605847	+	0.795581i	$0.292834\pi$
$600$	0	0
$601$	−5.09322e12	−1.59242	−0.796210	−	0.605020i	$-0.793165\pi$
$601$	−5.09322e12	−1.59242	−0.796210	+	0.605020i	$0.793165\pi$
$602$	8.95635e11	0.277937
$603$	0	0
$604$	5.19749e11	0.158901
$605$	−3.57398e12	−1.08456
$606$	0	0
$607$	2.26983e12	0.678649	0.339324	−	0.940669i	$-0.389802\pi$
$607$	2.26983e12	0.678649	0.339324	+	0.940669i	$0.389802\pi$
$608$	2.81204e11	0.0834556
$609$	0	0
$610$	−6.26128e10	−0.0183096
$611$	−1.16847e12	−0.339181
$612$	0	0
$613$	−6.42677e12	−1.83832	−0.919158	−	0.393888i	$-0.871130\pi$
$613$	−6.42677e12	−1.83832	−0.919158	+	0.393888i	$0.871130\pi$
$614$	−2.34303e12	−0.665303
$615$	0	0
$616$	−1.38226e12	−0.386791
$617$	−1.45823e11	−0.0405082	−0.0202541	−	0.999795i	$-0.506448\pi$
$617$	−1.45823e11	−0.0405082	−0.0202541	+	0.999795i	$0.506448\pi$
$618$	0	0
$619$	6.02775e12	1.65024	0.825120	−	0.564957i	$-0.191107\pi$
$619$	6.02775e12	1.65024	0.825120	+	0.564957i	$0.191107\pi$
$620$	−2.80650e11	−0.0762787
$621$	0	0
$622$	−1.09805e12	−0.294147
$623$	2.43668e12	0.648039
$624$	0	0
$625$	1.59832e12	0.418990
$626$	−4.35435e12	−1.13328
$627$	0	0
$628$	1.08969e12	0.279567
$629$	1.01368e12	0.258209
$630$	0	0
$631$	1.57062e12	0.394401	0.197201	−	0.980363i	$-0.436815\pi$
$631$	1.57062e12	0.394401	0.197201	+	0.980363i	$0.436815\pi$
$632$	2.66432e12	0.664293
$633$	0	0
$634$	−5.22309e10	−0.0128389
$635$	2.71941e12	0.663733
$636$	0	0
$637$	−5.54262e11	−0.133379
$638$	−1.42739e12	−0.341074
$639$	0	0
$640$	1.71138e11	0.0403216
$641$	3.74112e12	0.875266	0.437633	−	0.899154i	$-0.355817\pi$
$641$	3.74112e12	0.875266	0.437633	+	0.899154i	$0.355817\pi$
$642$	0	0
$643$	6.57372e12	1.51657	0.758284	−	0.651924i	$-0.226038\pi$
$643$	6.57372e12	1.51657	0.758284	+	0.651924i	$0.226038\pi$
$644$	1.58797e12	0.363793
$645$	0	0
$646$	5.70301e11	0.128842
$647$	4.02950e12	0.904029	0.452014	−	0.892011i	$-0.350706\pi$
$647$	4.02950e12	0.904029	0.452014	+	0.892011i	$0.350706\pi$
$648$	0	0
$649$	2.52828e12	0.559403
$650$	5.26460e11	0.115679
$651$	0	0
$652$	−1.06736e12	−0.231311
$653$	−7.61308e12	−1.63852	−0.819259	−	0.573424i	$-0.805615\pi$
$653$	−7.61308e12	−1.63852	−0.819259	+	0.573424i	$0.805615\pi$
$654$	0	0
$655$	7.45964e10	0.0158355
$656$	5.36398e11	0.113089
$657$	0	0
$658$	3.32321e12	0.691101
$659$	6.20851e12	1.28234	0.641170	−	0.767399i	$-0.278449\pi$
$659$	6.20851e12	1.28234	0.641170	+	0.767399i	$0.278449\pi$
$660$	0	0
$661$	7.26145e12	1.47951	0.739753	−	0.672878i	$-0.234942\pi$
$661$	7.26145e12	1.47951	0.739753	+	0.672878i	$0.234942\pi$
$662$	8.53323e11	0.172684
$663$	0	0
$664$	5.54363e11	0.110672
$665$	6.46544e11	0.128204
$666$	0	0
$667$	1.63981e12	0.320794
$668$	2.73981e12	0.532386
$669$	0	0
$670$	2.07015e12	0.396886
$671$	−5.47769e11	−0.104315
$672$	0	0
$673$	−3.46324e12	−0.650750	−0.325375	−	0.945585i	$-0.605491\pi$
$673$	−3.46324e12	−0.650750	−0.325375	+	0.945585i	$0.605491\pi$
$674$	−1.46817e12	−0.274035
$675$	0	0
$676$	−2.59889e12	−0.478661
$677$	3.52792e12	0.645461	0.322731	−	0.946491i	$-0.395399\pi$
$677$	3.52792e12	0.645461	0.322731	+	0.946491i	$0.395399\pi$
$678$	0	0
$679$	4.29473e12	0.775393
$680$	3.47081e11	0.0622501
$681$	0	0
$682$	−2.45527e12	−0.434580
$683$	2.51223e11	0.0441739	0.0220869	−	0.999756i	$-0.492969\pi$
$683$	2.51223e11	0.0441739	0.0220869	+	0.999756i	$0.492969\pi$
$684$	0	0
$685$	−4.64263e12	−0.805668
$686$	4.01794e12	0.692700
$687$	0	0
$688$	−9.70113e11	−0.165072
$689$	8.17126e11	0.138135
$690$	0	0
$691$	−6.74721e11	−0.112583	−0.0562915	−	0.998414i	$-0.517928\pi$
$691$	−6.74721e11	−0.112583	−0.0562915	+	0.998414i	$0.517928\pi$
$692$	−4.07071e12	−0.674828
$693$	0	0
$694$	−4.50482e12	−0.737157
$695$	−4.63707e12	−0.753897
$696$	0	0
$697$	1.08785e12	0.174591
$698$	6.47032e12	1.03175
$699$	0	0
$700$	−1.49729e12	−0.235703
$701$	4.08357e12	0.638718	0.319359	−	0.947634i	$-0.396532\pi$
$701$	4.08357e12	0.638718	0.319359	+	0.947634i	$0.396532\pi$
$702$	0	0
$703$	−2.04531e12	−0.315834
$704$	1.49720e12	0.229723
$705$	0	0
$706$	5.70594e12	0.864383
$707$	6.59840e12	0.993233
$708$	0	0
$709$	−4.13746e12	−0.614930	−0.307465	−	0.951559i	$-0.599481\pi$
$709$	−4.13746e12	−0.614930	−0.307465	+	0.951559i	$0.599481\pi$
$710$	6.59140e11	0.0973454
$711$	0	0
$712$	−2.63930e12	−0.384883
$713$	2.82066e12	0.408741
$714$	0	0
$715$	−1.21037e12	−0.173197
$716$	5.25733e12	0.747579
$717$	0	0
$718$	−7.88826e12	−1.10770
$719$	3.39080e12	0.473175	0.236588	−	0.971610i	$-0.423971\pi$
$719$	3.39080e12	0.473175	0.236588	+	0.971610i	$0.423971\pi$
$720$	0	0
$721$	6.65399e12	0.917009
$722$	4.01230e12	0.549511
$723$	0	0
$724$	3.86041e12	0.522167
$725$	−1.54617e12	−0.207843
$726$	0	0
$727$	8.11665e12	1.07764	0.538818	−	0.842422i	$-0.318871\pi$
$727$	8.11665e12	1.07764	0.538818	+	0.842422i	$0.318871\pi$
$728$	−3.29517e11	−0.0434797
$729$	0	0
$730$	−4.12777e12	−0.537976
$731$	−1.96746e12	−0.254845
$732$	0	0
$733$	2.21754e12	0.283729	0.141864	−	0.989886i	$-0.454690\pi$
$733$	2.21754e12	0.283729	0.141864	+	0.989886i	$0.454690\pi$
$734$	−1.69608e12	−0.215682
$735$	0	0
$736$	−1.72001e12	−0.216064
$737$	1.81107e13	2.26117
$738$	0	0
$739$	−2.27864e11	−0.0281045	−0.0140522	−	0.999901i	$-0.504473\pi$
$739$	−2.27864e11	−0.0281045	−0.0140522	+	0.999901i	$0.504473\pi$
$740$	−1.24476e12	−0.152595
$741$	0	0
$742$	−2.32396e12	−0.281457
$743$	1.15018e13	1.38457	0.692284	−	0.721625i	$-0.256604\pi$
$743$	1.15018e13	1.38457	0.692284	+	0.721625i	$0.256604\pi$
$744$	0	0
$745$	1.81990e12	0.216443
$746$	−8.64509e12	−1.02199
$747$	0	0
$748$	3.03644e12	0.354656
$749$	2.67591e12	0.310673
$750$	0	0
$751$	1.42337e13	1.63282	0.816410	−	0.577472i	$-0.195961\pi$
$751$	1.42337e13	1.63282	0.816410	+	0.577472i	$0.195961\pi$
$752$	−3.59956e12	−0.410458
$753$	0	0
$754$	−3.40274e11	−0.0383405
$755$	−1.29438e12	−0.144977
$756$	0	0
$757$	−1.25855e13	−1.39296	−0.696482	−	0.717574i	$-0.745252\pi$
$757$	−1.25855e13	−1.39296	−0.696482	+	0.717574i	$0.745252\pi$
$758$	7.82974e12	0.861461
$759$	0	0
$760$	−7.00308e11	−0.0761426
$761$	−1.19352e13	−1.29003	−0.645015	−	0.764170i	$-0.723149\pi$
$761$	−1.19352e13	−1.29003	−0.645015	+	0.764170i	$0.723149\pi$
$762$	0	0
$763$	−6.36756e11	−0.0680162
$764$	−1.22465e12	−0.130045
$765$	0	0
$766$	−4.73064e12	−0.496467
$767$	6.02716e11	0.0628831
$768$	0	0
$769$	−4.81732e12	−0.496749	−0.248375	−	0.968664i	$-0.579896\pi$
$769$	−4.81732e12	−0.496749	−0.248375	+	0.968664i	$0.579896\pi$
$770$	3.44237e12	0.352898
$771$	0	0
$772$	4.96866e12	0.503456
$773$	2.50535e12	0.252383	0.126192	−	0.992006i	$-0.459725\pi$
$773$	2.50535e12	0.252383	0.126192	+	0.992006i	$0.459725\pi$
$774$	0	0
$775$	−2.65959e12	−0.264824
$776$	−4.65186e12	−0.460520
$777$	0	0
$778$	−1.21139e13	−1.18543
$779$	−2.19497e12	−0.213555
$780$	0	0
$781$	5.76649e12	0.554603
$782$	−3.48831e12	−0.333568
$783$	0	0
$784$	−1.70745e12	−0.161408
$785$	−2.71376e12	−0.255069
$786$	0	0
$787$	1.17409e13	1.09097	0.545487	−	0.838120i	$-0.316345\pi$
$787$	1.17409e13	1.09097	0.545487	+	0.838120i	$0.316345\pi$
$788$	9.33235e11	0.0862230
$789$	0	0
$790$	−6.63520e12	−0.606083
$791$	5.73556e12	0.520933
$792$	0	0
$793$	−1.30582e11	−0.0117261
$794$	−3.06820e12	−0.273962
$795$	0	0
$796$	−1.57121e12	−0.138716
$797$	9.51802e12	0.835573	0.417786	−	0.908545i	$-0.362806\pi$
$797$	9.51802e12	0.835573	0.417786	+	0.908545i	$0.362806\pi$
$798$	0	0
$799$	−7.30015e12	−0.633682
$800$	1.62180e12	0.139988
$801$	0	0
$802$	−5.74853e11	−0.0490650
$803$	−3.61118e13	−3.06499
$804$	0	0
$805$	−3.95465e12	−0.331915
$806$	−5.85311e11	−0.0488516
$807$	0	0
$808$	−7.14710e12	−0.589900
$809$	1.01787e13	0.835457	0.417728	−	0.908572i	$-0.362826\pi$
$809$	1.01787e13	0.835457	0.417728	+	0.908572i	$0.362826\pi$
$810$	0	0
$811$	1.40364e13	1.13937	0.569683	−	0.821864i	$-0.307066\pi$
$811$	1.40364e13	1.13937	0.569683	+	0.821864i	$0.307066\pi$
$812$	9.67763e11	0.0781208
$813$	0	0
$814$	−1.08897e13	−0.869377
$815$	2.65814e12	0.211042
$816$	0	0
$817$	3.96976e12	0.311720
$818$	−8.90660e12	−0.695540
$819$	0	0
$820$	−1.33584e12	−0.103179
$821$	−8.80683e12	−0.676512	−0.338256	−	0.941054i	$-0.609837\pi$
$821$	−8.80683e12	−0.676512	−0.338256	+	0.941054i	$0.609837\pi$
$822$	0	0
$823$	2.92677e12	0.222377	0.111188	−	0.993799i	$-0.464534\pi$
$823$	2.92677e12	0.222377	0.111188	+	0.993799i	$0.464534\pi$
$824$	−7.20731e12	−0.544629
$825$	0	0
$826$	−1.71417e12	−0.128128
$827$	−7.82793e12	−0.581932	−0.290966	−	0.956733i	$-0.593977\pi$
$827$	−7.82793e12	−0.581932	−0.290966	+	0.956733i	$0.593977\pi$
$828$	0	0
$829$	−1.26014e13	−0.926669	−0.463334	−	0.886183i	$-0.653347\pi$
$829$	−1.26014e13	−0.926669	−0.463334	+	0.886183i	$0.653347\pi$
$830$	−1.38058e12	−0.100974
$831$	0	0
$832$	3.56918e11	0.0258234
$833$	−3.46282e12	−0.249188
$834$	0	0
$835$	−6.82320e12	−0.485735
$836$	−6.12665e12	−0.433805
$837$	0	0
$838$	−7.20079e12	−0.504408
$839$	3.58798e12	0.249989	0.124995	−	0.992157i	$-0.460109\pi$
$839$	3.58798e12	0.249989	0.124995	+	0.992157i	$0.460109\pi$
$840$	0	0
$841$	−1.35078e13	−0.931113
$842$	1.02783e13	0.704720
$843$	0	0
$844$	−4.03416e12	−0.273660
$845$	6.47225e12	0.436717
$846$	0	0
$847$	2.11989e13	1.41527
$848$	2.51722e12	0.167163
$849$	0	0
$850$	3.28912e12	0.216120
$851$	1.25103e13	0.817685
$852$	0	0
$853$	1.74992e13	1.13174	0.565872	−	0.824493i	$-0.308539\pi$
$853$	1.74992e13	1.13174	0.565872	+	0.824493i	$0.308539\pi$
$854$	3.71385e11	0.0238927
$855$	0	0
$856$	−2.89843e12	−0.184515
$857$	2.49952e11	0.0158286	0.00791431	−	0.999969i	$-0.497481\pi$
$857$	2.49952e11	0.0158286	0.00791431	+	0.999969i	$0.497481\pi$
$858$	0	0
$859$	−2.48864e13	−1.55953	−0.779764	−	0.626074i	$-0.784661\pi$
$859$	−2.48864e13	−1.55953	−0.779764	+	0.626074i	$0.784661\pi$
$860$	2.41596e12	0.150607
$861$	0	0
$862$	1.40085e13	0.864188
$863$	−2.38153e13	−1.46153	−0.730764	−	0.682630i	$-0.760836\pi$
$863$	−2.38153e13	−1.46153	−0.730764	+	0.682630i	$0.760836\pi$
$864$	0	0
$865$	1.01377e13	0.615695
$866$	1.80979e13	1.09345
$867$	0	0
$868$	1.66467e12	0.0995379
$869$	−5.80481e13	−3.45302
$870$	0	0
$871$	4.31741e12	0.254180
$872$	6.89706e11	0.0403961
$873$	0	0
$874$	7.03840e12	0.408011
$875$	8.43759e12	0.486611
$876$	0	0
$877$	−9.77854e12	−0.558182	−0.279091	−	0.960265i	$-0.590033\pi$
$877$	−9.77854e12	−0.558182	−0.279091	+	0.960265i	$0.590033\pi$
$878$	7.40026e12	0.420264
$879$	0	0
$880$	−3.72862e12	−0.209593
$881$	−1.16336e13	−0.650615	−0.325307	−	0.945608i	$-0.605468\pi$
$881$	−1.16336e13	−0.650615	−0.325307	+	0.945608i	$0.605468\pi$
$882$	0	0
$883$	2.99837e13	1.65982	0.829912	−	0.557895i	$-0.188391\pi$
$883$	2.99837e13	1.65982	0.829912	+	0.557895i	$0.188391\pi$
$884$	7.23855e11	0.0398673
$885$	0	0
$886$	−2.25462e12	−0.122920
$887$	1.01486e13	0.550488	0.275244	−	0.961374i	$-0.411241\pi$
$887$	1.01486e13	0.550488	0.275244	+	0.961374i	$0.411241\pi$
$888$	0	0
$889$	−1.61301e13	−0.866121
$890$	6.57289e12	0.351157
$891$	0	0
$892$	1.07861e13	0.570459
$893$	1.47296e13	0.775102
$894$	0	0
$895$	−1.30928e13	−0.682071
$896$	−1.01510e12	−0.0526166
$897$	0	0
$898$	2.10641e13	1.08094
$899$	1.71901e12	0.0877728
$900$	0	0
$901$	5.10509e12	0.258073
$902$	−1.16866e13	−0.587840
$903$	0	0
$904$	−6.21251e12	−0.309392
$905$	−9.61392e12	−0.476411
$906$	0	0
$907$	3.02546e13	1.48443	0.742213	−	0.670164i	$-0.233776\pi$
$907$	3.02546e13	1.48443	0.742213	+	0.670164i	$0.233776\pi$
$908$	−1.73211e13	−0.845647
$909$	0	0
$910$	8.20625e11	0.0396697
$911$	−3.44509e13	−1.65717	−0.828587	−	0.559860i	$-0.810855\pi$
$911$	−3.44509e13	−1.65717	−0.828587	+	0.559860i	$0.810855\pi$
$912$	0	0
$913$	−1.20780e13	−0.575277
$914$	−1.60460e13	−0.760515
$915$	0	0
$916$	−1.76336e13	−0.827583
$917$	−4.42466e11	−0.0206641
$918$	0	0
$919$	−1.95520e12	−0.0904216	−0.0452108	−	0.998977i	$-0.514396\pi$
$919$	−1.95520e12	−0.0904216	−0.0452108	+	0.998977i	$0.514396\pi$
$920$	4.28351e12	0.197131
$921$	0	0
$922$	−1.95552e13	−0.891196
$923$	1.37467e12	0.0623435
$924$	0	0
$925$	−1.17960e13	−0.529780
$926$	3.76862e12	0.168435
$927$	0	0
$928$	−1.04824e12	−0.0463975
$929$	7.83914e12	0.345301	0.172651	−	0.984983i	$-0.444767\pi$
$929$	7.83914e12	0.345301	0.172651	+	0.984983i	$0.444767\pi$
$930$	0	0
$931$	6.98696e12	0.304800
$932$	5.72008e11	0.0248331
$933$	0	0
$934$	1.00151e13	0.430619
$935$	−7.56191e12	−0.323578
$936$	0	0
$937$	−1.69235e13	−0.717236	−0.358618	−	0.933484i	$-0.616752\pi$
$937$	−1.69235e13	−0.717236	−0.358618	+	0.933484i	$0.616752\pi$
$938$	−1.22790e13	−0.517906
$939$	0	0
$940$	8.96430e12	0.374491
$941$	−2.11604e13	−0.879775	−0.439887	−	0.898053i	$-0.644982\pi$
$941$	−2.11604e13	−0.879775	−0.439887	+	0.898053i	$0.644982\pi$
$942$	0	0
$943$	1.34258e13	0.552888
$944$	1.85671e12	0.0760975
$945$	0	0
$946$	2.11360e13	0.858051
$947$	1.81201e13	0.732125	0.366063	−	0.930590i	$-0.380706\pi$
$947$	1.81201e13	0.732125	0.366063	+	0.930590i	$0.380706\pi$
$948$	0	0
$949$	−8.60869e12	−0.344539
$950$	−6.63649e12	−0.264352
$951$	0	0
$952$	−2.05869e12	−0.0812317
$953$	2.04420e13	0.802798	0.401399	−	0.915903i	$-0.368524\pi$
$953$	2.04420e13	0.802798	0.401399	+	0.915903i	$0.368524\pi$
$954$	0	0
$955$	3.04987e12	0.118649
$956$	−2.35936e13	−0.913552
$957$	0	0
$958$	8.47623e12	0.325131
$959$	2.75375e13	1.05134
$960$	0	0
$961$	−2.34827e13	−0.888164
$962$	−2.59600e12	−0.0977276
$963$	0	0
$964$	2.01254e13	0.750582
$965$	−1.23739e13	−0.459339
$966$	0	0
$967$	3.07852e13	1.13220	0.566099	−	0.824337i	$-0.308452\pi$
$967$	3.07852e13	1.13220	0.566099	+	0.824337i	$0.308452\pi$
$968$	−2.29617e13	−0.840554
$969$	0	0
$970$	1.15849e13	0.420166
$971$	−4.17249e13	−1.50629	−0.753145	−	0.657854i	$-0.771464\pi$
$971$	−4.17249e13	−1.50629	−0.753145	+	0.657854i	$0.771464\pi$
$972$	0	0
$973$	2.75046e13	0.983778
$974$	8.91868e11	0.0317530
$975$	0	0
$976$	−4.02268e11	−0.0141903
$977$	−3.41065e13	−1.19760	−0.598799	−	0.800899i	$-0.704355\pi$
$977$	−3.41065e13	−1.19760	−0.598799	+	0.800899i	$0.704355\pi$
$978$	0	0
$979$	5.75029e13	2.00064
$980$	4.25221e12	0.147264
$981$	0	0
$982$	−5.97737e12	−0.205120
$983$	4.91805e13	1.67997	0.839987	−	0.542607i	$-0.182563\pi$
$983$	4.91805e13	1.67997	0.839987	+	0.542607i	$0.182563\pi$
$984$	0	0
$985$	−2.32412e12	−0.0786675
$986$	−2.12590e12	−0.0716303
$987$	0	0
$988$	−1.46053e12	−0.0487645
$989$	−2.42814e13	−0.807033
$990$	0	0
$991$	5.97858e12	0.196909	0.0984547	−	0.995142i	$-0.468610\pi$
$991$	5.97858e12	0.196909	0.0984547	+	0.995142i	$0.468610\pi$
$992$	−1.80309e12	−0.0591175
$993$	0	0
$994$	−3.90966e12	−0.127028
$995$	3.91293e12	0.126561
$996$	0	0
$997$	−2.84771e13	−0.912784	−0.456392	−	0.889779i	$-0.650859\pi$
$997$	−2.84771e13	−0.912784	−0.456392	+	0.889779i	$0.650859\pi$
$998$	5.51262e12	0.175902
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.10.a.f.1.2	✓	4
3.2	odd	2		162.10.a.g.1.3	yes	4
9.2	odd	6		162.10.c.s.109.2		8
9.4	even	3		162.10.c.t.55.3		8
9.5	odd	6		162.10.c.s.55.2		8
9.7	even	3		162.10.c.t.109.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.10.a.f.1.2	✓	4	1.1	even	1	trivial
162.10.a.g.1.3	yes	4	3.2	odd	2
162.10.c.s.55.2		8	9.5	odd	6
162.10.c.s.109.2		8	9.2	odd	6
162.10.c.t.55.3		8	9.4	even	3
162.10.c.t.109.3		8	9.7	even	3

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q-16.0000 q^{2} +256.000 q^{4} -637.540 q^{5} +3781.54 q^{7} -4096.00 q^{8} +10200.6 q^{10} +89240.4 q^{11} +21274.0 q^{13} -60504.7 q^{14} +65536.0 q^{16} +132912. q^{17} -268177. q^{19} -163210. q^{20} -1.42785e6 q^{22} +1.64033e6 q^{23} -1.54667e6 q^{25} -340384. q^{26} +968075. q^{28} +999678. q^{29} +1.71956e6 q^{31} -1.04858e6 q^{32} -2.12659e6 q^{34} -2.41088e6 q^{35} +7.62670e6 q^{37} +4.29083e6 q^{38} +2.61136e6 q^{40} +8.18478e6 q^{41} -1.48027e7 q^{43} +2.28455e7 q^{44} -2.62453e7 q^{46} -5.49249e7 q^{47} -2.60536e7 q^{49} +2.47467e7 q^{50} +5.44614e6 q^{52} +3.84097e7 q^{53} -5.68943e7 q^{55} -1.54892e7 q^{56} -1.59948e7 q^{58} +2.83312e7 q^{59} -6.13813e6 q^{61} -2.75130e7 q^{62} +1.67772e7 q^{64} -1.35630e7 q^{65} +2.02943e8 q^{67} +3.40254e7 q^{68} +3.85741e7 q^{70} +6.46176e7 q^{71} -4.04658e8 q^{73} -1.22027e8 q^{74} -6.86533e7 q^{76} +3.37466e8 q^{77} -6.50469e8 q^{79} -4.17818e7 q^{80} -1.30957e8 q^{82} -1.35342e8 q^{83} -8.47365e7 q^{85} +2.36844e8 q^{86} -3.65528e8 q^{88} +6.44360e8 q^{89} +8.04484e7 q^{91} +4.19925e8 q^{92} +8.78798e8 q^{94} +1.70974e8 q^{95} +1.13571e9 q^{97} +4.16857e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{2} + 1024 q^{4} + 1968 q^{5} + 4496 q^{7} - 16384 q^{8} - 31488 q^{10} - 8784 q^{11} - 162556 q^{13} - 71936 q^{14} + 262144 q^{16} + 538080 q^{17} - 8224 q^{19} + 503808 q^{20} + 140544 q^{22}+ \cdots + 2038788288 q^{98}+O(q^{100}) \) 4 * q - 64 * q^2 + 1024 * q^4 + 1968 * q^5 + 4496 * q^7 - 16384 * q^8 - 31488 * q^10 - 8784 * q^11 - 162556 * q^13 - 71936 * q^14 + 262144 * q^16 + 538080 * q^17 - 8224 * q^19 + 503808 * q^20 + 140544 * q^22 + 2594736 * q^23 + 1873192 * q^25 + 2600896 * q^26 + 1150976 * q^28 + 3232656 * q^29 + 3482576 * q^31 - 4194304 * q^32 - 8609280 * q^34 + 7969296 * q^35 - 2487892 * q^37 + 131584 * q^38 - 8060928 * q^40 + 34657152 * q^41 - 41410000 * q^43 - 2248704 * q^44 - 41515776 * q^46 + 40558848 * q^47 - 127424268 * q^49 - 29971072 * q^50 - 41614336 * q^52 + 8421024 * q^53 - 229068000 * q^55 - 18415616 * q^56 - 51722496 * q^58 + 26843328 * q^59 - 111504484 * q^61 - 55721216 * q^62 + 67108864 * q^64 + 23507616 * q^65 - 208064512 * q^67 + 137748480 * q^68 - 127508736 * q^70 + 356008272 * q^71 - 37126108 * q^73 + 39806272 * q^74 - 2105344 * q^76 + 33010560 * q^77 - 645134848 * q^79 + 128974848 * q^80 - 554514432 * q^82 + 1019036256 * q^83 + 80955540 * q^85 + 662560000 * q^86 + 35979264 * q^88 + 1548192768 * q^89 - 130776608 * q^91 + 664252416 * q^92 - 648941568 * q^94 + 3445933872 * q^95 + 1112014568 * q^97 + 2038788288 * q^98

Introduction

L-functions

Modular forms

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Database

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