LMFDB - Embedded newform 162.8.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,8,Mod(1,162)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$50.6063741284$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.43103376.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 383x^{2} + 384x + 18612 $ x^4 - 2x^3 - 383x^2 + 384*x + 18612
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			1.1
Root			$-17.5928$ 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)

$f(q)$	$=$	\(q-8.00000 q^{2} +64.0000 q^{4} -471.679 q^{5} +735.308 q^{7} -512.000 q^{8} +3773.43 q^{10} -5321.11 q^{11} +8865.59 q^{13} -5882.46 q^{14} +4096.00 q^{16} +34175.5 q^{17} -8704.81 q^{19} -30187.5 q^{20} +42568.9 q^{22} -52105.1 q^{23} +144356. q^{25} -70924.8 q^{26} +47059.7 q^{28} +213409. q^{29} -233086. q^{31} -32768.0 q^{32} -273404. q^{34} -346829. q^{35} +67328.2 q^{37} +69638.5 q^{38} +241500. q^{40} +45100.8 q^{41} +214634. q^{43} -340551. q^{44} +416841. q^{46} +560459. q^{47} -282865. q^{49} -1.15485e6 q^{50} +567398. q^{52} +244520. q^{53} +2.50985e6 q^{55} -376478. q^{56} -1.70728e6 q^{58} -1.93366e6 q^{59} +2.12899e6 q^{61} +1.86469e6 q^{62} +262144. q^{64} -4.18172e6 q^{65} -1.62940e6 q^{67} +2.18723e6 q^{68} +2.77463e6 q^{70} -5.38016e6 q^{71} -1.01989e6 q^{73} -538625. q^{74} -557108. q^{76} -3.91265e6 q^{77} -7.74151e6 q^{79} -1.93200e6 q^{80} -360806. q^{82} -2.09394e6 q^{83} -1.61199e7 q^{85} -1.71707e6 q^{86} +2.72441e6 q^{88} -1.17927e7 q^{89} +6.51894e6 q^{91} -3.33473e6 q^{92} -4.48367e6 q^{94} +4.10588e6 q^{95} +1.92229e6 q^{97} +2.26292e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} + 256 q^{4} - 528 q^{5} + 560 q^{7} - 2048 q^{8} + 4224 q^{10} - 2160 q^{11} + 13460 q^{13} - 4480 q^{14} + 16384 q^{16} - 22560 q^{17} + 36704 q^{19} - 33792 q^{20} + 17280 q^{22} - 62640 q^{23}+ \cdots - 11595168 q^{98}+O(q^{100}) $ 4 * q - 32 * q^2 + 256 * q^4 - 528 * q^5 + 560 * q^7 - 2048 * q^8 + 4224 * q^10 - 2160 * q^11 + 13460 * q^13 - 4480 * q^14 + 16384 * q^16 - 22560 * q^17 + 36704 * q^19 - 33792 * q^20 + 17280 * q^22 - 62640 * q^23 + 94696 * q^25 - 107680 * q^26 + 35840 * q^28 - 68400 * q^29 + 227504 * q^31 - 131072 * q^32 + 180480 * q^34 + 63024 * q^35 + 523580 * q^37 - 293632 * q^38 + 270336 * q^40 - 67200 * q^41 + 562640 * q^43 - 138240 * q^44 + 501120 * q^46 - 515328 * q^47 + 1449396 * q^49 - 757568 * q^50 + 861440 * q^52 - 2498016 * q^53 + 1322784 * q^55 - 286720 * q^56 + 547200 * q^58 - 4155840 * q^59 + 2130764 * q^61 - 1820032 * q^62 + 1048576 * q^64 - 8966496 * q^65 - 1205440 * q^67 - 1443840 * q^68 - 504192 * q^70 - 12486480 * q^71 - 4820860 * q^73 - 4188640 * q^74 + 2349056 * q^76 - 18686592 * q^77 - 11471680 * q^79 - 2162688 * q^80 + 537600 * q^82 - 16811232 * q^83 - 21006396 * q^85 - 4501120 * q^86 + 1105920 * q^88 - 16857600 * q^89 - 23200544 * q^91 - 4008960 * q^92 + 4122624 * q^94 - 14090160 * q^95 - 27078040 * q^97 - 11595168 * 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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	−471.679	−1.68753	−0.843765	−	0.536713i	$-0.819666\pi$
$5$	−471.679	−1.68753	−0.843765	+	0.536713i	$0.819666\pi$
$6$	0	0
$7$	735.308	0.810263	0.405132	−	0.914258i	$-0.367226\pi$
$7$	735.308	0.810263	0.405132	+	0.914258i	$0.367226\pi$
$8$	−512.000	−0.353553
$9$	0	0
$10$	3773.43	1.19326
$11$	−5321.11	−1.20539	−0.602695	−	0.797972i	$-0.705906\pi$
$11$	−5321.11	−1.20539	−0.602695	+	0.797972i	$0.705906\pi$
$12$	0	0
$13$	8865.59	1.11920	0.559598	−	0.828764i	$-0.310956\pi$
$13$	8865.59	1.11920	0.559598	+	0.828764i	$0.310956\pi$
$14$	−5882.46	−0.572943
$15$	0	0
$16$	4096.00	0.250000
$17$	34175.5	1.68711	0.843555	−	0.537042i	$-0.180458\pi$
$17$	34175.5	1.68711	0.843555	+	0.537042i	$0.180458\pi$
$18$	0	0
$19$	−8704.81	−0.291153	−0.145577	−	0.989347i	$-0.546504\pi$
$19$	−8704.81	−0.291153	−0.145577	+	0.989347i	$0.546504\pi$
$20$	−30187.5	−0.843765
$21$	0	0
$22$	42568.9	0.852340
$23$	−52105.1	−0.892961	−0.446481	−	0.894793i	$-0.647323\pi$
$23$	−52105.1	−0.892961	−0.446481	+	0.894793i	$0.647323\pi$
$24$	0	0
$25$	144356.	1.84776
$26$	−70924.8	−0.791391
$27$	0	0
$28$	47059.7	0.405132
$29$	213409.	1.62488	0.812439	−	0.583046i	$-0.198139\pi$
$29$	213409.	1.62488	0.812439	+	0.583046i	$0.198139\pi$
$30$	0	0
$31$	−233086.	−1.40524	−0.702620	−	0.711566i	$-0.747986\pi$
$31$	−233086.	−1.40524	−0.702620	+	0.711566i	$0.747986\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	−273404.	−1.19297
$35$	−346829.	−1.36734
$36$	0	0
$37$	67328.2	0.218520	0.109260	−	0.994013i	$-0.465152\pi$
$37$	67328.2	0.218520	0.109260	+	0.994013i	$0.465152\pi$
$38$	69638.5	0.205876
$39$	0	0
$40$	241500.	0.596632
$41$	45100.8	0.102198	0.0510988	−	0.998694i	$-0.483728\pi$
$41$	45100.8	0.102198	0.0510988	+	0.998694i	$0.483728\pi$
$42$	0	0
$43$	214634.	0.411679	0.205839	−	0.978586i	$-0.434008\pi$
$43$	214634.	0.411679	0.205839	+	0.978586i	$0.434008\pi$
$44$	−340551.	−0.602695
$45$	0	0
$46$	416841.	0.631419
$47$	560459.	0.787411	0.393705	−	0.919237i	$-0.371193\pi$
$47$	560459.	0.787411	0.393705	+	0.919237i	$0.371193\pi$
$48$	0	0
$49$	−282865.	−0.343474
$50$	−1.15485e6	−1.30656
$51$	0	0
$52$	567398.	0.559598
$53$	244520.	0.225606	0.112803	−	0.993617i	$-0.464017\pi$
$53$	244520.	0.225606	0.112803	+	0.993617i	$0.464017\pi$
$54$	0	0
$55$	2.50985e6	2.03413
$56$	−376478.	−0.286471
$57$	0	0
$58$	−1.70728e6	−1.14896
$59$	−1.93366e6	−1.22574	−0.612871	−	0.790183i	$-0.709985\pi$
$59$	−1.93366e6	−1.22574	−0.612871	+	0.790183i	$0.709985\pi$
$60$	0	0
$61$	2.12899e6	1.20093	0.600467	−	0.799650i	$-0.294981\pi$
$61$	2.12899e6	1.20093	0.600467	+	0.799650i	$0.294981\pi$
$62$	1.86469e6	0.993654
$63$	0	0
$64$	262144.	0.125000
$65$	−4.18172e6	−1.88868
$66$	0	0
$67$	−1.62940e6	−0.661860	−0.330930	−	0.943655i	$-0.607362\pi$
$67$	−1.62940e6	−0.661860	−0.330930	+	0.943655i	$0.607362\pi$
$68$	2.18723e6	0.843555
$69$	0	0
$70$	2.77463e6	0.966858
$71$	−5.38016e6	−1.78398	−0.891992	−	0.452052i	$-0.850692\pi$
$71$	−5.38016e6	−1.78398	−0.891992	+	0.452052i	$0.850692\pi$
$72$	0	0
$73$	−1.01989e6	−0.306849	−0.153424	−	0.988160i	$-0.549030\pi$
$73$	−1.01989e6	−0.306849	−0.153424	+	0.988160i	$0.549030\pi$
$74$	−538625.	−0.154517
$75$	0	0
$76$	−557108.	−0.145577
$77$	−3.91265e6	−0.976683
$78$	0	0
$79$	−7.74151e6	−1.76657	−0.883286	−	0.468835i	$-0.844674\pi$
$79$	−7.74151e6	−1.76657	−0.883286	+	0.468835i	$0.844674\pi$
$80$	−1.93200e6	−0.421883
$81$	0	0
$82$	−360806.	−0.0722646
$83$	−2.09394e6	−0.401968	−0.200984	−	0.979594i	$-0.564414\pi$
$83$	−2.09394e6	−0.401968	−0.200984	+	0.979594i	$0.564414\pi$
$84$	0	0
$85$	−1.61199e7	−2.84705
$86$	−1.71707e6	−0.291101
$87$	0	0
$88$	2.72441e6	0.426170
$89$	−1.17927e7	−1.77316	−0.886580	−	0.462574i	$-0.846926\pi$
$89$	−1.17927e7	−1.77316	−0.886580	+	0.462574i	$0.846926\pi$
$90$	0	0
$91$	6.51894e6	0.906843
$92$	−3.33473e6	−0.446481
$93$	0	0
$94$	−4.48367e6	−0.556783
$95$	4.10588e6	0.491330
$96$	0	0
$97$	1.92229e6	0.213854	0.106927	−	0.994267i	$-0.465899\pi$
$97$	1.92229e6	0.213854	0.106927	+	0.994267i	$0.465899\pi$
$98$	2.26292e6	0.242873
$99$	0	0
$100$	9.23879e6	0.923879
$101$	−1.09908e7	−1.06146	−0.530731	−	0.847540i	$-0.678083\pi$
$101$	−1.09908e7	−1.06146	−0.530731	+	0.847540i	$0.678083\pi$
$102$	0	0
$103$	−1.49478e7	−1.34787	−0.673933	−	0.738792i	$-0.735396\pi$
$103$	−1.49478e7	−1.34787	−0.673933	+	0.738792i	$0.735396\pi$
$104$	−4.53918e6	−0.395696
$105$	0	0
$106$	−1.95616e6	−0.159527
$107$	1.72961e7	1.36491	0.682457	−	0.730926i	$-0.260912\pi$
$107$	1.72961e7	1.36491	0.682457	+	0.730926i	$0.260912\pi$
$108$	0	0
$109$	−1.14307e7	−0.845434	−0.422717	−	0.906262i	$-0.638924\pi$
$109$	−1.14307e7	−0.845434	−0.422717	+	0.906262i	$0.638924\pi$
$110$	−2.00788e7	−1.43835
$111$	0	0
$112$	3.01182e6	0.202566
$113$	3.44984e6	0.224918	0.112459	−	0.993656i	$-0.464127\pi$
$113$	3.44984e6	0.224918	0.112459	+	0.993656i	$0.464127\pi$
$114$	0	0
$115$	2.45769e7	1.50690
$116$	1.36582e7	0.812439
$117$	0	0
$118$	1.54693e7	0.866730
$119$	2.51295e7	1.36700
$120$	0	0
$121$	8.82701e6	0.452965
$122$	−1.70319e7	−0.849188
$123$	0	0
$124$	−1.49175e7	−0.702620
$125$	−3.12398e7	−1.43062
$126$	0	0
$127$	4.26026e7	1.84554	0.922770	−	0.385352i	$-0.125920\pi$
$127$	4.26026e7	1.84554	0.922770	+	0.385352i	$0.125920\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	3.34537e7	1.33550
$131$	4.21661e6	0.163876	0.0819378	−	0.996637i	$-0.473889\pi$
$131$	4.21661e6	0.163876	0.0819378	+	0.996637i	$0.473889\pi$
$132$	0	0
$133$	−6.40072e6	−0.235911
$134$	1.30352e7	0.468005
$135$	0	0
$136$	−1.74979e7	−0.596484
$137$	1.40660e6	0.0467358	0.0233679	−	0.999727i	$-0.492561\pi$
$137$	1.40660e6	0.0467358	0.0233679	+	0.999727i	$0.492561\pi$
$138$	0	0
$139$	−3.95211e7	−1.24818	−0.624090	−	0.781352i	$-0.714530\pi$
$139$	−3.95211e7	−1.24818	−0.624090	+	0.781352i	$0.714530\pi$
$140$	−2.21971e7	−0.683672
$141$	0	0
$142$	4.30413e7	1.26147
$143$	−4.71748e7	−1.34907
$144$	0	0
$145$	−1.00661e8	−2.74203
$146$	8.15914e6	0.216975
$147$	0	0
$148$	4.30900e6	0.109260
$149$	−1.51798e7	−0.375936	−0.187968	−	0.982175i	$-0.560190\pi$
$149$	−1.51798e7	−0.375936	−0.187968	+	0.982175i	$0.560190\pi$
$150$	0	0
$151$	−7.94865e7	−1.87877	−0.939386	−	0.342861i	$-0.888604\pi$
$151$	−7.94865e7	−1.87877	−0.939386	+	0.342861i	$0.888604\pi$
$152$	4.45686e6	0.102938
$153$	0	0
$154$	3.13012e7	0.690619
$155$	1.09942e8	2.37138
$156$	0	0
$157$	8.34353e7	1.72068	0.860342	−	0.509718i	$-0.170250\pi$
$157$	8.34353e7	1.72068	0.860342	+	0.509718i	$0.170250\pi$
$158$	6.19321e7	1.24915
$159$	0	0
$160$	1.54560e7	0.298316
$161$	−3.83133e7	−0.723534
$162$	0	0
$163$	−4.02444e7	−0.727862	−0.363931	−	0.931426i	$-0.618566\pi$
$163$	−4.02444e7	−0.727862	−0.363931	+	0.931426i	$0.618566\pi$
$164$	2.88645e6	0.0510988
$165$	0	0
$166$	1.67516e7	0.284235
$167$	7.35725e7	1.22238	0.611192	−	0.791482i	$-0.290690\pi$
$167$	7.35725e7	1.22238	0.611192	+	0.791482i	$0.290690\pi$
$168$	0	0
$169$	1.58503e7	0.252600
$170$	1.28959e8	2.01317
$171$	0	0
$172$	1.37366e7	0.205839
$173$	6.67747e7	0.980507	0.490254	−	0.871580i	$-0.336904\pi$
$173$	6.67747e7	0.980507	0.490254	+	0.871580i	$0.336904\pi$
$174$	0	0
$175$	1.06146e8	1.49717
$176$	−2.17953e7	−0.301348
$177$	0	0
$178$	9.43416e7	1.25381
$179$	8.00165e7	1.04278	0.521392	−	0.853317i	$-0.325413\pi$
$179$	8.00165e7	1.04278	0.521392	+	0.853317i	$0.325413\pi$
$180$	0	0
$181$	4.68357e7	0.587086	0.293543	−	0.955946i	$-0.405166\pi$
$181$	4.68357e7	0.587086	0.293543	+	0.955946i	$0.405166\pi$
$182$	−5.21515e7	−0.641235
$183$	0	0
$184$	2.66778e7	0.315710
$185$	−3.17573e7	−0.368759
$186$	0	0
$187$	−1.81851e8	−2.03363
$188$	3.58694e7	0.393705
$189$	0	0
$190$	−3.28470e7	−0.347423
$191$	−1.03268e8	−1.07238	−0.536191	−	0.844096i	$-0.680138\pi$
$191$	−1.03268e8	−1.07238	−0.536191	+	0.844096i	$0.680138\pi$
$192$	0	0
$193$	1.13874e7	0.114018	0.0570089	−	0.998374i	$-0.481844\pi$
$193$	1.13874e7	0.114018	0.0570089	+	0.998374i	$0.481844\pi$
$194$	−1.53783e7	−0.151218
$195$	0	0
$196$	−1.81034e7	−0.171737
$197$	3.46478e7	0.322882	0.161441	−	0.986882i	$-0.448386\pi$
$197$	3.46478e7	0.322882	0.161441	+	0.986882i	$0.448386\pi$
$198$	0	0
$199$	−1.21715e8	−1.09486	−0.547432	−	0.836850i	$-0.684394\pi$
$199$	−1.21715e8	−1.09486	−0.547432	+	0.836850i	$0.684394\pi$
$200$	−7.39103e7	−0.653281
$201$	0	0
$202$	8.79265e7	0.750568
$203$	1.56922e8	1.31658
$204$	0	0
$205$	−2.12731e7	−0.172461
$206$	1.19582e8	0.953085
$207$	0	0
$208$	3.63135e7	0.279799
$209$	4.63192e7	0.350953
$210$	0	0
$211$	−7.29077e7	−0.534300	−0.267150	−	0.963655i	$-0.586082\pi$
$211$	−7.29077e7	−0.534300	−0.267150	+	0.963655i	$0.586082\pi$
$212$	1.56493e7	0.112803
$213$	0	0
$214$	−1.38369e8	−0.965139
$215$	−1.01238e8	−0.694720
$216$	0	0
$217$	−1.71390e8	−1.13861
$218$	9.14456e7	0.597812
$219$	0	0
$220$	1.60631e8	1.01707
$221$	3.02986e8	1.88821
$222$	0	0
$223$	8.62944e7	0.521093	0.260547	−	0.965461i	$-0.416097\pi$
$223$	8.62944e7	0.521093	0.260547	+	0.965461i	$0.416097\pi$
$224$	−2.40946e7	−0.143236
$225$	0	0
$226$	−2.75987e7	−0.159041
$227$	1.41831e8	0.804785	0.402393	−	0.915467i	$-0.368179\pi$
$227$	1.41831e8	0.804785	0.402393	+	0.915467i	$0.368179\pi$
$228$	0	0
$229$	−1.78349e8	−0.981399	−0.490700	−	0.871329i	$-0.663259\pi$
$229$	−1.78349e8	−0.981399	−0.490700	+	0.871329i	$0.663259\pi$
$230$	−1.96615e8	−1.06554
$231$	0	0
$232$	−1.09266e8	−0.574481
$233$	−1.65012e8	−0.854613	−0.427306	−	0.904107i	$-0.640537\pi$
$233$	−1.65012e8	−0.854613	−0.427306	+	0.904107i	$0.640537\pi$
$234$	0	0
$235$	−2.64357e8	−1.32878
$236$	−1.23754e8	−0.612871
$237$	0	0
$238$	−2.01036e8	−0.966617
$239$	−1.86834e8	−0.885247	−0.442623	−	0.896708i	$-0.645952\pi$
$239$	−1.86834e8	−0.885247	−0.442623	+	0.896708i	$0.645952\pi$
$240$	0	0
$241$	1.30787e8	0.601871	0.300936	−	0.953644i	$-0.402701\pi$
$241$	1.30787e8	0.601871	0.300936	+	0.953644i	$0.402701\pi$
$242$	−7.06161e7	−0.320295
$243$	0	0
$244$	1.36255e8	0.600467
$245$	1.33422e8	0.579622
$246$	0	0
$247$	−7.71733e7	−0.325858
$248$	1.19340e8	0.496827
$249$	0	0
$250$	2.49919e8	1.01160
$251$	−7.33350e7	−0.292720	−0.146360	−	0.989231i	$-0.546756\pi$
$251$	−7.33350e7	−0.292720	−0.146360	+	0.989231i	$0.546756\pi$
$252$	0	0
$253$	2.77257e8	1.07637
$254$	−3.40821e8	−1.30499
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−4.08853e7	−0.150245	−0.0751227	−	0.997174i	$-0.523935\pi$
$257$	−4.08853e7	−0.150245	−0.0751227	+	0.997174i	$0.523935\pi$
$258$	0	0
$259$	4.95069e7	0.177059
$260$	−2.67630e8	−0.944339
$261$	0	0
$262$	−3.37329e7	−0.115878
$263$	−4.01148e8	−1.35975	−0.679875	−	0.733328i	$-0.737966\pi$
$263$	−4.01148e8	−1.35975	−0.679875	+	0.733328i	$0.737966\pi$
$264$	0	0
$265$	−1.15335e8	−0.380716
$266$	5.12057e7	0.166814
$267$	0	0
$268$	−1.04282e8	−0.330930
$269$	3.65089e8	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$269$	3.65089e8	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$270$	0	0
$271$	5.78472e8	1.76559	0.882794	−	0.469760i	$-0.155659\pi$
$271$	5.78472e8	1.76559	0.882794	+	0.469760i	$0.155659\pi$
$272$	1.39983e8	0.421778
$273$	0	0
$274$	−1.12528e7	−0.0330472
$275$	−7.68134e8	−2.22727
$276$	0	0
$277$	−2.88645e8	−0.815989	−0.407995	−	0.912984i	$-0.633772\pi$
$277$	−2.88645e8	−0.815989	−0.407995	+	0.912984i	$0.633772\pi$
$278$	3.16169e8	0.882597
$279$	0	0
$280$	1.77577e8	0.483429
$281$	5.45701e7	0.146718	0.0733588	−	0.997306i	$-0.476628\pi$
$281$	5.45701e7	0.146718	0.0733588	+	0.997306i	$0.476628\pi$
$282$	0	0
$283$	3.40620e7	0.0893342	0.0446671	−	0.999002i	$-0.485777\pi$
$283$	3.40620e7	0.0893342	0.0446671	+	0.999002i	$0.485777\pi$
$284$	−3.44330e8	−0.891992
$285$	0	0
$286$	3.77398e8	0.953935
$287$	3.31630e7	0.0828069
$288$	0	0
$289$	7.57626e8	1.84634
$290$	8.05286e8	1.93891
$291$	0	0
$292$	−6.52731e7	−0.153424
$293$	−2.61727e8	−0.607871	−0.303935	−	0.952693i	$-0.598301\pi$
$293$	−2.61727e8	−0.607871	−0.303935	+	0.952693i	$0.598301\pi$
$294$	0	0
$295$	9.12068e8	2.06848
$296$	−3.44720e7	−0.0772584
$297$	0	0
$298$	1.21438e8	0.265827
$299$	−4.61943e8	−0.999399
$300$	0	0
$301$	1.57822e8	0.333568
$302$	6.35892e8	1.32849
$303$	0	0
$304$	−3.56549e7	−0.0727883
$305$	−1.00420e9	−2.02661
$306$	0	0
$307$	−4.92725e8	−0.971898	−0.485949	−	0.873987i	$-0.661526\pi$
$307$	−4.92725e8	−0.971898	−0.485949	+	0.873987i	$0.661526\pi$
$308$	−2.50410e8	−0.488342
$309$	0	0
$310$	−8.79534e8	−1.67682
$311$	−2.46322e8	−0.464347	−0.232173	−	0.972674i	$-0.574584\pi$
$311$	−2.46322e8	−0.464347	−0.232173	+	0.972674i	$0.574584\pi$
$312$	0	0
$313$	−5.40387e8	−0.996093	−0.498047	−	0.867150i	$-0.665949\pi$
$313$	−5.40387e8	−0.996093	−0.498047	+	0.867150i	$0.665949\pi$
$314$	−6.67482e8	−1.21671
$315$	0	0
$316$	−4.95457e8	−0.883286
$317$	−3.81493e8	−0.672635	−0.336318	−	0.941749i	$-0.609182\pi$
$317$	−3.81493e8	−0.672635	−0.336318	+	0.941749i	$0.609182\pi$
$318$	0	0
$319$	−1.13557e9	−1.95861
$320$	−1.23648e8	−0.210941
$321$	0	0
$322$	3.06506e8	0.511616
$323$	−2.97491e8	−0.491208
$324$	0	0
$325$	1.27980e9	2.06800
$326$	3.21955e8	0.514676
$327$	0	0
$328$	−2.30916e7	−0.0361323
$329$	4.12110e8	0.638010
$330$	0	0
$331$	2.11612e8	0.320732	0.160366	−	0.987058i	$-0.448733\pi$
$331$	2.11612e8	0.320732	0.160366	+	0.987058i	$0.448733\pi$
$332$	−1.34012e8	−0.200984
$333$	0	0
$334$	−5.88580e8	−0.864357
$335$	7.68554e8	1.11691
$336$	0	0
$337$	−2.97235e8	−0.423054	−0.211527	−	0.977372i	$-0.567844\pi$
$337$	−2.97235e8	−0.423054	−0.211527	+	0.977372i	$0.567844\pi$
$338$	−1.26802e8	−0.178615
$339$	0	0
$340$	−1.03167e9	−1.42353
$341$	1.24028e9	1.69386
$342$	0	0
$343$	−8.13551e8	−1.08857
$344$	−1.09892e8	−0.145550
$345$	0	0
$346$	−5.34198e8	−0.693323
$347$	−9.05189e7	−0.116302	−0.0581508	−	0.998308i	$-0.518520\pi$
$347$	−9.05189e7	−0.116302	−0.0581508	+	0.998308i	$0.518520\pi$
$348$	0	0
$349$	6.00104e8	0.755680	0.377840	−	0.925871i	$-0.376667\pi$
$349$	6.00104e8	0.755680	0.377840	+	0.925871i	$0.376667\pi$
$350$	−8.49170e8	−1.05866
$351$	0	0
$352$	1.74362e8	0.213085
$353$	−1.47428e9	−1.78390	−0.891949	−	0.452137i	$-0.850662\pi$
$353$	−1.47428e9	−1.78390	−0.891949	+	0.452137i	$0.850662\pi$
$354$	0	0
$355$	2.53771e9	3.01053
$356$	−7.54733e8	−0.886580
$357$	0	0
$358$	−6.40132e8	−0.737359
$359$	1.77364e8	0.202318	0.101159	−	0.994870i	$-0.467745\pi$
$359$	1.77364e8	0.202318	0.101159	+	0.994870i	$0.467745\pi$
$360$	0	0
$361$	−8.18098e8	−0.915230
$362$	−3.74686e8	−0.415133
$363$	0	0
$364$	4.17212e8	0.453422
$365$	4.81062e8	0.517817
$366$	0	0
$367$	−1.96281e8	−0.207275	−0.103638	−	0.994615i	$-0.533048\pi$
$367$	−1.96281e8	−0.207275	−0.103638	+	0.994615i	$0.533048\pi$
$368$	−2.13422e8	−0.223240
$369$	0	0
$370$	2.54058e8	0.260752
$371$	1.79798e8	0.182800
$372$	0	0
$373$	−2.69799e8	−0.269190	−0.134595	−	0.990901i	$-0.542973\pi$
$373$	−2.69799e8	−0.269190	−0.134595	+	0.990901i	$0.542973\pi$
$374$	1.45481e9	1.43799
$375$	0	0
$376$	−2.86955e8	−0.278392
$377$	1.89200e9	1.81856
$378$	0	0
$379$	2.20916e8	0.208444	0.104222	−	0.994554i	$-0.466765\pi$
$379$	2.20916e8	0.208444	0.104222	+	0.994554i	$0.466765\pi$
$380$	2.62776e8	0.245665
$381$	0	0
$382$	8.26146e8	0.758289
$383$	−8.31130e8	−0.755916	−0.377958	−	0.925823i	$-0.623374\pi$
$383$	−8.31130e8	−0.755916	−0.377958	+	0.925823i	$0.623374\pi$
$384$	0	0
$385$	1.84552e9	1.64818
$386$	−9.10990e7	−0.0806228
$387$	0	0
$388$	1.23027e8	0.106927
$389$	−1.65749e9	−1.42767	−0.713833	−	0.700316i	$-0.753042\pi$
$389$	−1.65749e9	−1.42767	−0.713833	+	0.700316i	$0.753042\pi$
$390$	0	0
$391$	−1.78072e9	−1.50652
$392$	1.44827e8	0.121436
$393$	0	0
$394$	−2.77182e8	−0.228312
$395$	3.65151e9	2.98114
$396$	0	0
$397$	−3.00261e8	−0.240842	−0.120421	−	0.992723i	$-0.538424\pi$
$397$	−3.00261e8	−0.240842	−0.120421	+	0.992723i	$0.538424\pi$
$398$	9.73723e8	0.774185
$399$	0	0
$400$	5.91283e8	0.461940
$401$	1.70634e9	1.32148	0.660739	−	0.750615i	$-0.270243\pi$
$401$	1.70634e9	1.32148	0.660739	+	0.750615i	$0.270243\pi$
$402$	0	0
$403$	−2.06645e9	−1.57274
$404$	−7.03412e8	−0.530731
$405$	0	0
$406$	−1.25537e9	−0.930962
$407$	−3.58260e8	−0.263402
$408$	0	0
$409$	1.81636e9	1.31271	0.656356	−	0.754451i	$-0.272097\pi$
$409$	1.81636e9	1.31271	0.656356	+	0.754451i	$0.272097\pi$
$410$	1.70185e8	0.121949
$411$	0	0
$412$	−9.56659e8	−0.673933
$413$	−1.42184e9	−0.993173
$414$	0	0
$415$	9.87670e8	0.678334
$416$	−2.90508e8	−0.197848
$417$	0	0
$418$	−3.70554e8	−0.248161
$419$	9.93132e8	0.659566	0.329783	−	0.944057i	$-0.393024\pi$
$419$	9.93132e8	0.659566	0.329783	+	0.944057i	$0.393024\pi$
$420$	0	0
$421$	5.17027e8	0.337696	0.168848	−	0.985642i	$-0.445995\pi$
$421$	5.17027e8	0.337696	0.168848	+	0.985642i	$0.445995\pi$
$422$	5.83262e8	0.377807
$423$	0	0
$424$	−1.25194e8	−0.0797636
$425$	4.93344e9	3.11737
$426$	0	0
$427$	1.56546e9	0.973072
$428$	1.10695e9	0.682457
$429$	0	0
$430$	8.09906e8	0.491241
$431$	−3.62445e7	−0.0218058	−0.0109029	−	0.999941i	$-0.503471\pi$
$431$	−3.62445e7	−0.0218058	−0.0109029	+	0.999941i	$0.503471\pi$
$432$	0	0
$433$	−2.26569e9	−1.34120	−0.670598	−	0.741821i	$-0.733962\pi$
$433$	−2.26569e9	−1.34120	−0.670598	+	0.741821i	$0.733962\pi$
$434$	1.37112e9	0.805121
$435$	0	0
$436$	−7.31564e8	−0.422717
$437$	4.53565e8	0.259989
$438$	0	0
$439$	−1.49751e9	−0.844780	−0.422390	−	0.906414i	$-0.638809\pi$
$439$	−1.49751e9	−0.844780	−0.422390	+	0.906414i	$0.638809\pi$
$440$	−1.28505e9	−0.719174
$441$	0	0
$442$	−2.42389e9	−1.33516
$443$	1.18355e9	0.646804	0.323402	−	0.946262i	$-0.395173\pi$
$443$	1.18355e9	0.646804	0.323402	+	0.946262i	$0.395173\pi$
$444$	0	0
$445$	5.56237e9	2.99226
$446$	−6.90355e8	−0.368469
$447$	0	0
$448$	1.92757e8	0.101283
$449$	−2.91053e9	−1.51743	−0.758717	−	0.651420i	$-0.774174\pi$
$449$	−2.91053e9	−1.51743	−0.758717	+	0.651420i	$0.774174\pi$
$450$	0	0
$451$	−2.39986e8	−0.123188
$452$	2.20790e8	0.112459
$453$	0	0
$454$	−1.13465e9	−0.569069
$455$	−3.07485e9	−1.53033
$456$	0	0
$457$	−2.21819e9	−1.08716	−0.543578	−	0.839358i	$-0.682931\pi$
$457$	−2.21819e9	−1.08716	−0.543578	+	0.839358i	$0.682931\pi$
$458$	1.42679e9	0.693954
$459$	0	0
$460$	1.57292e9	0.753450
$461$	−8.69031e8	−0.413126	−0.206563	−	0.978433i	$-0.566228\pi$
$461$	−8.69031e8	−0.413126	−0.206563	+	0.978433i	$0.566228\pi$
$462$	0	0
$463$	−1.31990e9	−0.618027	−0.309013	−	0.951058i	$-0.599999\pi$
$463$	−1.31990e9	−0.618027	−0.309013	+	0.951058i	$0.599999\pi$
$464$	8.74125e8	0.406219
$465$	0	0
$466$	1.32009e9	0.604302
$467$	4.68391e8	0.212814	0.106407	−	0.994323i	$-0.466065\pi$
$467$	4.68391e8	0.212814	0.106407	+	0.994323i	$0.466065\pi$
$468$	0	0
$469$	−1.19811e9	−0.536280
$470$	2.11485e9	0.939589
$471$	0	0
$472$	9.90036e8	0.433365
$473$	−1.14209e9	−0.496234
$474$	0	0
$475$	−1.25659e9	−0.537981
$476$	1.60829e9	0.683502
$477$	0	0
$478$	1.49468e9	0.625964
$479$	−1.65794e9	−0.689280	−0.344640	−	0.938735i	$-0.611999\pi$
$479$	−1.65794e9	−0.689280	−0.344640	+	0.938735i	$0.611999\pi$
$480$	0	0
$481$	5.96904e8	0.244567
$482$	−1.04629e9	−0.425587
$483$	0	0
$484$	5.64929e8	0.226483
$485$	−9.06704e8	−0.360886
$486$	0	0
$487$	1.71816e9	0.674079	0.337040	−	0.941490i	$-0.390574\pi$
$487$	1.71816e9	0.674079	0.337040	+	0.941490i	$0.390574\pi$
$488$	−1.09004e9	−0.424594
$489$	0	0
$490$	−1.06737e9	−0.409855
$491$	4.05932e9	1.54763	0.773817	−	0.633410i	$-0.218345\pi$
$491$	4.05932e9	1.54763	0.773817	+	0.633410i	$0.218345\pi$
$492$	0	0
$493$	7.29337e9	2.74135
$494$	6.17386e8	0.230416
$495$	0	0
$496$	−9.54720e8	−0.351310
$497$	−3.95607e9	−1.44550
$498$	0	0
$499$	9.74300e8	0.351027	0.175514	−	0.984477i	$-0.443841\pi$
$499$	9.74300e8	0.351027	0.175514	+	0.984477i	$0.443841\pi$
$500$	−1.99935e9	−0.715309
$501$	0	0
$502$	5.86680e8	0.206985
$503$	3.68171e9	1.28992	0.644958	−	0.764218i	$-0.276875\pi$
$503$	3.68171e9	1.28992	0.644958	+	0.764218i	$0.276875\pi$
$504$	0	0
$505$	5.18413e9	1.79125
$506$	−2.21805e9	−0.761106
$507$	0	0
$508$	2.72657e9	0.922770
$509$	−8.31362e7	−0.0279433	−0.0139717	−	0.999902i	$-0.504447\pi$
$509$	−8.31362e7	−0.0279433	−0.0139717	+	0.999902i	$0.504447\pi$
$510$	0	0
$511$	−7.49935e8	−0.248628
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	3.27082e8	0.106240
$515$	7.05056e9	2.27456
$516$	0	0
$517$	−2.98226e9	−0.949137
$518$	−3.96056e8	−0.125199
$519$	0	0
$520$	2.14104e9	0.667748
$521$	−3.54439e9	−1.09802	−0.549009	−	0.835816i	$-0.684995\pi$
$521$	−3.54439e9	−1.09802	−0.549009	+	0.835816i	$0.684995\pi$
$522$	0	0
$523$	−3.60605e9	−1.10224	−0.551120	−	0.834426i	$-0.685799\pi$
$523$	−3.60605e9	−1.10224	−0.551120	+	0.834426i	$0.685799\pi$
$524$	2.69863e8	0.0819378
$525$	0	0
$526$	3.20918e9	0.961488
$527$	−7.96583e9	−2.37079
$528$	0	0
$529$	−6.89886e8	−0.202620
$530$	9.22681e8	0.269207
$531$	0	0
$532$	−4.09646e8	−0.117955
$533$	3.99845e8	0.114379
$534$	0	0
$535$	−8.15821e9	−2.30333
$536$	8.34253e8	0.234003
$537$	0	0
$538$	−2.92071e9	−0.808631
$539$	1.50516e9	0.414020
$540$	0	0
$541$	−2.67387e9	−0.726023	−0.363012	−	0.931785i	$-0.618252\pi$
$541$	−2.67387e9	−0.726023	−0.363012	+	0.931785i	$0.618252\pi$
$542$	−4.62777e9	−1.24846
$543$	0	0
$544$	−1.11986e9	−0.298242
$545$	5.39162e9	1.42670
$546$	0	0
$547$	−2.31958e9	−0.605972	−0.302986	−	0.952995i	$-0.597984\pi$
$547$	−2.31958e9	−0.605972	−0.302986	+	0.952995i	$0.597984\pi$
$548$	9.00227e7	0.0233679
$549$	0	0
$550$	6.14507e9	1.57492
$551$	−1.85769e9	−0.473088
$552$	0	0
$553$	−5.69240e9	−1.43139
$554$	2.30916e9	0.576992
$555$	0	0
$556$	−2.52935e9	−0.624090
$557$	−6.21643e9	−1.52422	−0.762110	−	0.647447i	$-0.775837\pi$
$557$	−6.21643e9	−1.52422	−0.762110	+	0.647447i	$0.775837\pi$
$558$	0	0
$559$	1.90286e9	0.460749
$560$	−1.42061e9	−0.341836
$561$	0	0
$562$	−4.36561e8	−0.103745
$563$	4.43510e9	1.04743	0.523714	−	0.851894i	$-0.324546\pi$
$563$	4.43510e9	1.04743	0.523714	+	0.851894i	$0.324546\pi$
$564$	0	0
$565$	−1.62722e9	−0.379556
$566$	−2.72496e8	−0.0631688
$567$	0	0
$568$	2.75464e9	0.630733
$569$	−3.89464e9	−0.886286	−0.443143	−	0.896451i	$-0.646137\pi$
$569$	−3.89464e9	−0.886286	−0.443143	+	0.896451i	$0.646137\pi$
$570$	0	0
$571$	−8.04118e9	−1.80756	−0.903782	−	0.427994i	$-0.859221\pi$
$571$	−8.04118e9	−1.80756	−0.903782	+	0.427994i	$0.859221\pi$
$572$	−3.01919e9	−0.674534
$573$	0	0
$574$	−2.65304e8	−0.0585533
$575$	−7.52169e9	−1.64998
$576$	0	0
$577$	3.17866e9	0.688856	0.344428	−	0.938813i	$-0.388073\pi$
$577$	3.17866e9	0.688856	0.344428	+	0.938813i	$0.388073\pi$
$578$	−6.06100e9	−1.30556
$579$	0	0
$580$	−6.44229e9	−1.37102
$581$	−1.53969e9	−0.325700
$582$	0	0
$583$	−1.30112e9	−0.271943
$584$	5.22185e8	0.108487
$585$	0	0
$586$	2.09381e9	0.429830
$587$	5.58549e9	1.13980	0.569899	−	0.821714i	$-0.306982\pi$
$587$	5.58549e9	1.13980	0.569899	+	0.821714i	$0.306982\pi$
$588$	0	0
$589$	2.02897e9	0.409140
$590$	−7.29655e9	−1.46263
$591$	0	0
$592$	2.75776e8	0.0546300
$593$	−5.31470e9	−1.04662	−0.523308	−	0.852143i	$-0.675302\pi$
$593$	−5.31470e9	−1.04662	−0.523308	+	0.852143i	$0.675302\pi$
$594$	0	0
$595$	−1.18531e10	−2.30686
$596$	−9.71506e8	−0.187968
$597$	0	0
$598$	3.69554e9	0.706682
$599$	−1.24066e9	−0.235862	−0.117931	−	0.993022i	$-0.537626\pi$
$599$	−1.24066e9	−0.235862	−0.117931	+	0.993022i	$0.537626\pi$
$600$	0	0
$601$	2.42921e9	0.456461	0.228231	−	0.973607i	$-0.426706\pi$
$601$	2.42921e9	0.456461	0.228231	+	0.973607i	$0.426706\pi$
$602$	−1.26258e9	−0.235868
$603$	0	0
$604$	−5.08714e9	−0.939386
$605$	−4.16352e9	−0.764393
$606$	0	0
$607$	−8.23090e9	−1.49378	−0.746890	−	0.664948i	$-0.768454\pi$
$607$	−8.23090e9	−1.49378	−0.746890	+	0.664948i	$0.768454\pi$
$608$	2.85239e8	0.0514691
$609$	0	0
$610$	8.03360e9	1.43303
$611$	4.96880e9	0.881267
$612$	0	0
$613$	6.14959e9	1.07829	0.539144	−	0.842214i	$-0.318748\pi$
$613$	6.14959e9	1.07829	0.539144	+	0.842214i	$0.318748\pi$
$614$	3.94180e9	0.687236
$615$	0	0
$616$	2.00328e9	0.345310
$617$	9.32737e9	1.59868	0.799339	−	0.600880i	$-0.205183\pi$
$617$	9.32737e9	1.59868	0.799339	+	0.600880i	$0.205183\pi$
$618$	0	0
$619$	2.11704e9	0.358767	0.179383	−	0.983779i	$-0.442590\pi$
$619$	2.11704e9	0.358767	0.179383	+	0.983779i	$0.442590\pi$
$620$	7.03627e9	1.18569
$621$	0	0
$622$	1.97058e9	0.328343
$623$	−8.67126e9	−1.43673
$624$	0	0
$625$	3.45735e9	0.566452
$626$	4.32310e9	0.704344
$627$	0	0
$628$	5.33986e9	0.860342
$629$	2.30097e9	0.368667
$630$	0	0
$631$	4.96507e9	0.786725	0.393362	−	0.919383i	$-0.371312\pi$
$631$	4.96507e9	0.786725	0.393362	+	0.919383i	$0.371312\pi$
$632$	3.96366e9	0.624577
$633$	0	0
$634$	3.05195e9	0.475625
$635$	−2.00948e10	−3.11440
$636$	0	0
$637$	−2.50777e9	−0.384414
$638$	9.08460e9	1.38495
$639$	0	0
$640$	9.89183e8	0.149158
$641$	5.46682e9	0.819845	0.409923	−	0.912120i	$-0.365556\pi$
$641$	5.46682e9	0.819845	0.409923	+	0.912120i	$0.365556\pi$
$642$	0	0
$643$	9.76161e9	1.44805	0.724024	−	0.689774i	$-0.242290\pi$
$643$	9.76161e9	1.44805	0.724024	+	0.689774i	$0.242290\pi$
$644$	−2.45205e9	−0.361767
$645$	0	0
$646$	2.37993e9	0.347336
$647$	−8.78713e9	−1.27551	−0.637753	−	0.770241i	$-0.720136\pi$
$647$	−8.78713e9	−1.27551	−0.637753	+	0.770241i	$0.720136\pi$
$648$	0	0
$649$	1.02892e10	1.47750
$650$	−1.02384e10	−1.46230
$651$	0	0
$652$	−2.57564e9	−0.363931
$653$	1.17186e10	1.64694	0.823472	−	0.567357i	$-0.192034\pi$
$653$	1.17186e10	1.64694	0.823472	+	0.567357i	$0.192034\pi$
$654$	0	0
$655$	−1.98889e9	−0.276545
$656$	1.84733e8	0.0255494
$657$	0	0
$658$	−3.29688e9	−0.451141
$659$	−2.65559e9	−0.361461	−0.180731	−	0.983533i	$-0.557846\pi$
$659$	−2.65559e9	−0.361461	−0.180731	+	0.983533i	$0.557846\pi$
$660$	0	0
$661$	−2.43661e8	−0.0328156	−0.0164078	−	0.999865i	$-0.505223\pi$
$661$	−2.43661e8	−0.0328156	−0.0164078	+	0.999865i	$0.505223\pi$
$662$	−1.69290e9	−0.226792
$663$	0	0
$664$	1.07210e9	0.142117
$665$	3.01908e9	0.398107
$666$	0	0
$667$	−1.11197e10	−1.45095
$668$	4.70864e9	0.611192
$669$	0	0
$670$	−6.14843e9	−0.789773
$671$	−1.13286e10	−1.44759
$672$	0	0
$673$	2.78437e9	0.352107	0.176053	−	0.984381i	$-0.443667\pi$
$673$	2.78437e9	0.352107	0.176053	+	0.984381i	$0.443667\pi$
$674$	2.37788e9	0.299144
$675$	0	0
$676$	1.01442e9	0.126300
$677$	1.09692e9	0.135867	0.0679336	−	0.997690i	$-0.478359\pi$
$677$	1.09692e9	0.135867	0.0679336	+	0.997690i	$0.478359\pi$
$678$	0	0
$679$	1.41348e9	0.173278
$680$	8.25337e9	1.00658
$681$	0	0
$682$	−9.92221e9	−1.19774
$683$	−6.42315e9	−0.771393	−0.385696	−	0.922626i	$-0.626039\pi$
$683$	−6.42315e9	−0.771393	−0.385696	+	0.922626i	$0.626039\pi$
$684$	0	0
$685$	−6.63466e8	−0.0788681
$686$	6.50841e9	0.769733
$687$	0	0
$688$	8.79140e8	0.102920
$689$	2.16782e9	0.252497
$690$	0	0
$691$	−5.37346e9	−0.619556	−0.309778	−	0.950809i	$-0.600255\pi$
$691$	−5.37346e9	−0.619556	−0.309778	+	0.950809i	$0.600255\pi$
$692$	4.27358e9	0.490254
$693$	0	0
$694$	7.24151e8	0.0822377
$695$	1.86413e10	2.10634
$696$	0	0
$697$	1.54134e9	0.172419
$698$	−4.80083e9	−0.534346
$699$	0	0
$700$	6.79336e9	0.748585
$701$	−7.91279e9	−0.867594	−0.433797	−	0.901011i	$-0.642826\pi$
$701$	−7.91279e9	−0.867594	−0.433797	+	0.901011i	$0.642826\pi$
$702$	0	0
$703$	−5.86079e8	−0.0636228
$704$	−1.39490e9	−0.150674
$705$	0	0
$706$	1.17943e10	1.26141
$707$	−8.08163e9	−0.860064
$708$	0	0
$709$	−1.40765e10	−1.48331	−0.741654	−	0.670782i	$-0.765958\pi$
$709$	−1.40765e10	−1.48331	−0.741654	+	0.670782i	$0.765958\pi$
$710$	−2.03017e10	−2.12876
$711$	0	0
$712$	6.03786e9	0.626907
$713$	1.21450e10	1.25482
$714$	0	0
$715$	2.22514e10	2.27659
$716$	5.12106e9	0.521392
$717$	0	0
$718$	−1.41891e9	−0.143061
$719$	1.11759e10	1.12132	0.560661	−	0.828045i	$-0.310547\pi$
$719$	1.11759e10	1.12132	0.560661	+	0.828045i	$0.310547\pi$
$720$	0	0
$721$	−1.09912e10	−1.09213
$722$	6.54478e9	0.647165
$723$	0	0
$724$	2.99749e9	0.293543
$725$	3.08070e10	3.00238
$726$	0	0
$727$	4.01633e9	0.387667	0.193834	−	0.981034i	$-0.437908\pi$
$727$	4.01633e9	0.387667	0.193834	+	0.981034i	$0.437908\pi$
$728$	−3.33770e9	−0.320618
$729$	0	0
$730$	−3.84850e9	−0.366152
$731$	7.33521e9	0.694548
$732$	0	0
$733$	−9.95769e8	−0.0933887	−0.0466944	−	0.998909i	$-0.514869\pi$
$733$	−9.95769e8	−0.0933887	−0.0466944	+	0.998909i	$0.514869\pi$
$734$	1.57025e9	0.146566
$735$	0	0
$736$	1.70738e9	0.157855
$737$	8.67021e9	0.797799
$738$	0	0
$739$	−3.54656e9	−0.323260	−0.161630	−	0.986851i	$-0.551675\pi$
$739$	−3.54656e9	−0.323260	−0.161630	+	0.986851i	$0.551675\pi$
$740$	−2.03247e9	−0.184379
$741$	0	0
$742$	−1.43838e9	−0.129259
$743$	9.71329e8	0.0868771	0.0434386	−	0.999056i	$-0.486169\pi$
$743$	9.71329e8	0.0868771	0.0434386	+	0.999056i	$0.486169\pi$
$744$	0	0
$745$	7.15999e9	0.634403
$746$	2.15839e9	0.190346
$747$	0	0
$748$	−1.16385e10	−1.01681
$749$	1.27180e10	1.10594
$750$	0	0
$751$	−1.06185e10	−0.914795	−0.457398	−	0.889262i	$-0.651218\pi$
$751$	−1.06185e10	−0.914795	−0.457398	+	0.889262i	$0.651218\pi$
$752$	2.29564e9	0.196853
$753$	0	0
$754$	−1.51360e10	−1.28591
$755$	3.74921e10	3.17049
$756$	0	0
$757$	7.16479e9	0.600299	0.300149	−	0.953892i	$-0.402963\pi$
$757$	7.16479e9	0.600299	0.300149	+	0.953892i	$0.402963\pi$
$758$	−1.76733e9	−0.147392
$759$	0	0
$760$	−2.10221e9	−0.173711
$761$	7.91291e9	0.650863	0.325432	−	0.945566i	$-0.394490\pi$
$761$	7.91291e9	0.650863	0.325432	+	0.945566i	$0.394490\pi$
$762$	0	0
$763$	−8.40508e9	−0.685024
$764$	−6.60917e9	−0.536191
$765$	0	0
$766$	6.64904e9	0.534513
$767$	−1.71431e10	−1.37184
$768$	0	0
$769$	1.50800e10	1.19580	0.597902	−	0.801569i	$-0.296001\pi$
$769$	1.50800e10	1.19580	0.597902	+	0.801569i	$0.296001\pi$
$770$	−1.47641e10	−1.16544
$771$	0	0
$772$	7.28792e8	0.0570089
$773$	−5.21596e9	−0.406168	−0.203084	−	0.979161i	$-0.565097\pi$
$773$	−5.21596e9	−0.406168	−0.203084	+	0.979161i	$0.565097\pi$
$774$	0	0
$775$	−3.36474e10	−2.59654
$776$	−9.84213e8	−0.0756089
$777$	0	0
$778$	1.32599e10	1.00951
$779$	−3.92594e8	−0.0297551
$780$	0	0
$781$	2.86284e10	2.15040
$782$	1.42457e10	1.06527
$783$	0	0
$784$	−1.15862e9	−0.0858684
$785$	−3.93547e10	−2.90371
$786$	0	0
$787$	−5.02319e9	−0.367340	−0.183670	−	0.982988i	$-0.558798\pi$
$787$	−5.02319e9	−0.367340	−0.183670	+	0.982988i	$0.558798\pi$
$788$	2.21746e9	0.161441
$789$	0	0
$790$	−2.92121e10	−2.10799
$791$	2.53669e9	0.182243
$792$	0	0
$793$	1.88748e10	1.34408
$794$	2.40209e9	0.170301
$795$	0	0
$796$	−7.78979e9	−0.547432
$797$	−2.72965e10	−1.90986	−0.954932	−	0.296823i	$-0.904073\pi$
$797$	−2.72965e10	−1.90986	−0.954932	+	0.296823i	$0.904073\pi$
$798$	0	0
$799$	1.91540e10	1.32845
$800$	−4.73026e9	−0.326641
$801$	0	0
$802$	−1.36507e10	−0.934426
$803$	5.42696e9	0.369873
$804$	0	0
$805$	1.80716e10	1.22098
$806$	1.65316e10	1.11209
$807$	0	0
$808$	5.62729e9	0.375284
$809$	1.86800e9	0.124039	0.0620194	−	0.998075i	$-0.480246\pi$
$809$	1.86800e9	0.124039	0.0620194	+	0.998075i	$0.480246\pi$
$810$	0	0
$811$	2.75954e9	0.181662	0.0908308	−	0.995866i	$-0.471048\pi$
$811$	2.75954e9	0.181662	0.0908308	+	0.995866i	$0.471048\pi$
$812$	1.00430e10	0.658289
$813$	0	0
$814$	2.86608e9	0.186253
$815$	1.89824e10	1.22829
$816$	0	0
$817$	−1.86835e9	−0.119862
$818$	−1.45308e10	−0.928228
$819$	0	0
$820$	−1.36148e9	−0.0862307
$821$	−1.16476e10	−0.734573	−0.367287	−	0.930108i	$-0.619713\pi$
$821$	−1.16476e10	−0.734573	−0.367287	+	0.930108i	$0.619713\pi$
$822$	0	0
$823$	4.22891e9	0.264441	0.132221	−	0.991220i	$-0.457789\pi$
$823$	4.22891e9	0.264441	0.132221	+	0.991220i	$0.457789\pi$
$824$	7.65327e9	0.476543
$825$	0	0
$826$	1.13747e10	0.702279
$827$	2.14248e10	1.31719	0.658595	−	0.752498i	$-0.271151\pi$
$827$	2.14248e10	1.31719	0.658595	+	0.752498i	$0.271151\pi$
$828$	0	0
$829$	2.03553e10	1.24090	0.620451	−	0.784246i	$-0.286950\pi$
$829$	2.03553e10	1.24090	0.620451	+	0.784246i	$0.286950\pi$
$830$	−7.90136e9	−0.479654
$831$	0	0
$832$	2.32406e9	0.139899
$833$	−9.66706e9	−0.579478
$834$	0	0
$835$	−3.47026e10	−2.06281
$836$	2.96443e9	0.175477
$837$	0	0
$838$	−7.94506e9	−0.466383
$839$	5.87651e9	0.343520	0.171760	−	0.985139i	$-0.445055\pi$
$839$	5.87651e9	0.343520	0.171760	+	0.985139i	$0.445055\pi$
$840$	0	0
$841$	2.82937e10	1.64023
$842$	−4.13622e9	−0.238787
$843$	0	0
$844$	−4.66609e9	−0.267150
$845$	−7.47623e9	−0.426269
$846$	0	0
$847$	6.49057e9	0.367021
$848$	1.00156e9	0.0564014
$849$	0	0
$850$	−3.94675e10	−2.20432
$851$	−3.50814e9	−0.195130
$852$	0	0
$853$	−1.41237e10	−0.779160	−0.389580	−	0.920993i	$-0.627380\pi$
$853$	−1.41237e10	−0.779160	−0.389580	+	0.920993i	$0.627380\pi$
$854$	−1.25237e10	−0.688066
$855$	0	0
$856$	−8.85561e9	−0.482570
$857$	−1.85850e8	−0.0100863	−0.00504313	−	0.999987i	$-0.501605\pi$
$857$	−1.85850e8	−0.0100863	−0.00504313	+	0.999987i	$0.501605\pi$
$858$	0	0
$859$	−3.07008e10	−1.65262	−0.826312	−	0.563213i	$-0.809565\pi$
$859$	−3.07008e10	−1.65262	−0.826312	+	0.563213i	$0.809565\pi$
$860$	−6.47925e9	−0.347360
$861$	0	0
$862$	2.89956e8	0.0154190
$863$	−3.25081e9	−0.172168	−0.0860842	−	0.996288i	$-0.527435\pi$
$863$	−3.25081e9	−0.172168	−0.0860842	+	0.996288i	$0.527435\pi$
$864$	0	0
$865$	−3.14962e10	−1.65464
$866$	1.81255e10	0.948369
$867$	0	0
$868$	−1.09690e10	−0.569307
$869$	4.11934e10	2.12941
$870$	0	0
$871$	−1.44456e10	−0.740751
$872$	5.85252e9	0.298906
$873$	0	0
$874$	−3.62852e9	−0.183840
$875$	−2.29709e10	−1.15918
$876$	0	0
$877$	−1.68627e10	−0.844169	−0.422085	−	0.906556i	$-0.638701\pi$
$877$	−1.68627e10	−0.844169	−0.422085	+	0.906556i	$0.638701\pi$
$878$	1.19801e10	0.597350
$879$	0	0
$880$	1.02804e10	0.508533
$881$	−2.46279e9	−0.121342	−0.0606710	−	0.998158i	$-0.519324\pi$
$881$	−2.46279e9	−0.121342	−0.0606710	+	0.998158i	$0.519324\pi$
$882$	0	0
$883$	2.67375e10	1.30695	0.653475	−	0.756948i	$-0.273311\pi$
$883$	2.67375e10	1.30695	0.653475	+	0.756948i	$0.273311\pi$
$884$	1.93911e10	0.944104
$885$	0	0
$886$	−9.46838e9	−0.457360
$887$	−2.84416e10	−1.36843	−0.684214	−	0.729282i	$-0.739854\pi$
$887$	−2.84416e10	−1.36843	−0.684214	+	0.729282i	$0.739854\pi$
$888$	0	0
$889$	3.13260e10	1.49537
$890$	−4.44989e10	−2.11585
$891$	0	0
$892$	5.52284e9	0.260547
$893$	−4.87869e9	−0.229257
$894$	0	0
$895$	−3.77421e10	−1.75973
$896$	−1.54205e9	−0.0716178
$897$	0	0
$898$	2.32842e10	1.07299
$899$	−4.97428e10	−2.28334
$900$	0	0
$901$	8.35661e9	0.380621
$902$	1.91989e9	0.0871070
$903$	0	0
$904$	−1.76632e9	−0.0795205
$905$	−2.20914e10	−0.990726
$906$	0	0
$907$	8.08689e9	0.359879	0.179939	−	0.983678i	$-0.442410\pi$
$907$	8.08689e9	0.359879	0.179939	+	0.983678i	$0.442410\pi$
$908$	9.07717e9	0.402393
$909$	0	0
$910$	2.45988e10	1.08210
$911$	6.47512e9	0.283748	0.141874	−	0.989885i	$-0.454687\pi$
$911$	6.47512e9	0.283748	0.141874	+	0.989885i	$0.454687\pi$
$912$	0	0
$913$	1.11421e10	0.484529
$914$	1.77455e10	0.768736
$915$	0	0
$916$	−1.14143e10	−0.490700
$917$	3.10051e9	0.132782
$918$	0	0
$919$	3.18121e10	1.35204	0.676018	−	0.736885i	$-0.263704\pi$
$919$	3.18121e10	1.35204	0.676018	+	0.736885i	$0.263704\pi$
$920$	−1.25834e10	−0.532769
$921$	0	0
$922$	6.95225e9	0.292124
$923$	−4.76983e10	−1.99663
$924$	0	0
$925$	9.71923e9	0.403772
$926$	1.05592e10	0.437011
$927$	0	0
$928$	−6.99300e9	−0.287241
$929$	−1.41549e10	−0.579232	−0.289616	−	0.957143i	$-0.593528\pi$
$929$	−1.41549e10	−0.579232	−0.289616	+	0.957143i	$0.593528\pi$
$930$	0	0
$931$	2.46229e9	0.100003
$932$	−1.05608e10	−0.427306
$933$	0	0
$934$	−3.74713e9	−0.150482
$935$	8.57755e10	3.43181
$936$	0	0
$937$	1.51843e10	0.602984	0.301492	−	0.953469i	$-0.402515\pi$
$937$	1.51843e10	0.602984	0.301492	+	0.953469i	$0.402515\pi$
$938$	9.58489e9	0.379208
$939$	0	0
$940$	−1.69188e10	−0.664390
$941$	−4.08371e10	−1.59768	−0.798842	−	0.601541i	$-0.794554\pi$
$941$	−4.08371e10	−1.59768	−0.798842	+	0.601541i	$0.794554\pi$
$942$	0	0
$943$	−2.34998e9	−0.0912584
$944$	−7.92029e9	−0.306435
$945$	0	0
$946$	9.13671e9	0.350890
$947$	2.04381e10	0.782015	0.391008	−	0.920387i	$-0.372127\pi$
$947$	2.04381e10	0.782015	0.391008	+	0.920387i	$0.372127\pi$
$948$	0	0
$949$	−9.04196e9	−0.343424
$950$	1.00527e10	0.380410
$951$	0	0
$952$	−1.28663e10	−0.483309
$953$	−4.34656e10	−1.62675	−0.813374	−	0.581741i	$-0.802372\pi$
$953$	−4.34656e10	−1.62675	−0.813374	+	0.581741i	$0.802372\pi$
$954$	0	0
$955$	4.87095e10	1.80968
$956$	−1.19574e10	−0.442623
$957$	0	0
$958$	1.32636e10	0.487395
$959$	1.03429e9	0.0378683
$960$	0	0
$961$	2.68165e10	0.974697
$962$	−4.77523e9	−0.172935
$963$	0	0
$964$	8.37034e9	0.300936
$965$	−5.37118e9	−0.192409
$966$	0	0
$967$	3.84118e10	1.36607	0.683034	−	0.730386i	$-0.260660\pi$
$967$	3.84118e10	1.36607	0.683034	+	0.730386i	$0.260660\pi$
$968$	−4.51943e9	−0.160147
$969$	0	0
$970$	7.25363e9	0.255185
$971$	−1.63481e10	−0.573061	−0.286531	−	0.958071i	$-0.592502\pi$
$971$	−1.63481e10	−0.573061	−0.286531	+	0.958071i	$0.592502\pi$
$972$	0	0
$973$	−2.90602e10	−1.01135
$974$	−1.37452e10	−0.476646
$975$	0	0
$976$	8.72034e9	0.300233
$977$	1.41494e9	0.0485409	0.0242704	−	0.999705i	$-0.492274\pi$
$977$	1.41494e9	0.0485409	0.0242704	+	0.999705i	$0.492274\pi$
$978$	0	0
$979$	6.27502e10	2.13735
$980$	8.53898e9	0.289811
$981$	0	0
$982$	−3.24746e10	−1.09434
$983$	−2.19251e10	−0.736213	−0.368107	−	0.929784i	$-0.619994\pi$
$983$	−2.19251e10	−0.736213	−0.368107	+	0.929784i	$0.619994\pi$
$984$	0	0
$985$	−1.63426e10	−0.544873
$986$	−5.83470e10	−1.93843
$987$	0	0
$988$	−4.93909e9	−0.162929
$989$	−1.11835e10	−0.367613
$990$	0	0
$991$	1.50750e10	0.492040	0.246020	−	0.969265i	$-0.420877\pi$
$991$	1.50750e10	0.492040	0.246020	+	0.969265i	$0.420877\pi$
$992$	7.63776e9	0.248414
$993$	0	0
$994$	3.16486e10	1.02212
$995$	5.74106e10	1.84761
$996$	0	0
$997$	2.01110e10	0.642687	0.321344	−	0.946963i	$-0.395866\pi$
$997$	2.01110e10	0.642687	0.321344	+	0.946963i	$0.395866\pi$
$998$	−7.79440e9	−0.248214
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.8.a.g.1.1	✓	4
3.2	odd	2		162.8.a.j.1.4	yes	4
9.2	odd	6		162.8.c.q.109.1		8
9.4	even	3		162.8.c.r.55.4		8
9.5	odd	6		162.8.c.q.55.1		8
9.7	even	3		162.8.c.r.109.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.8.a.g.1.1	✓	4	1.1	even	1	trivial
162.8.a.j.1.4	yes	4	3.2	odd	2
162.8.c.q.55.1		8	9.5	odd	6
162.8.c.q.109.1		8	9.2	odd	6
162.8.c.r.55.4		8	9.4	even	3
162.8.c.r.109.4		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +64.0000 q^{4} -471.679 q^{5} +735.308 q^{7} -512.000 q^{8} +3773.43 q^{10} -5321.11 q^{11} +8865.59 q^{13} -5882.46 q^{14} +4096.00 q^{16} +34175.5 q^{17} -8704.81 q^{19} -30187.5 q^{20} +42568.9 q^{22} -52105.1 q^{23} +144356. q^{25} -70924.8 q^{26} +47059.7 q^{28} +213409. q^{29} -233086. q^{31} -32768.0 q^{32} -273404. q^{34} -346829. q^{35} +67328.2 q^{37} +69638.5 q^{38} +241500. q^{40} +45100.8 q^{41} +214634. q^{43} -340551. q^{44} +416841. q^{46} +560459. q^{47} -282865. q^{49} -1.15485e6 q^{50} +567398. q^{52} +244520. q^{53} +2.50985e6 q^{55} -376478. q^{56} -1.70728e6 q^{58} -1.93366e6 q^{59} +2.12899e6 q^{61} +1.86469e6 q^{62} +262144. q^{64} -4.18172e6 q^{65} -1.62940e6 q^{67} +2.18723e6 q^{68} +2.77463e6 q^{70} -5.38016e6 q^{71} -1.01989e6 q^{73} -538625. q^{74} -557108. q^{76} -3.91265e6 q^{77} -7.74151e6 q^{79} -1.93200e6 q^{80} -360806. q^{82} -2.09394e6 q^{83} -1.61199e7 q^{85} -1.71707e6 q^{86} +2.72441e6 q^{88} -1.17927e7 q^{89} +6.51894e6 q^{91} -3.33473e6 q^{92} -4.48367e6 q^{94} +4.10588e6 q^{95} +1.92229e6 q^{97} +2.26292e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} + 256 q^{4} - 528 q^{5} + 560 q^{7} - 2048 q^{8} + 4224 q^{10} - 2160 q^{11} + 13460 q^{13} - 4480 q^{14} + 16384 q^{16} - 22560 q^{17} + 36704 q^{19} - 33792 q^{20} + 17280 q^{22} - 62640 q^{23}+ \cdots - 11595168 q^{98}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 - 528 * q^5 + 560 * q^7 - 2048 * q^8 + 4224 * q^10 - 2160 * q^11 + 13460 * q^13 - 4480 * q^14 + 16384 * q^16 - 22560 * q^17 + 36704 * q^19 - 33792 * q^20 + 17280 * q^22 - 62640 * q^23 + 94696 * q^25 - 107680 * q^26 + 35840 * q^28 - 68400 * q^29 + 227504 * q^31 - 131072 * q^32 + 180480 * q^34 + 63024 * q^35 + 523580 * q^37 - 293632 * q^38 + 270336 * q^40 - 67200 * q^41 + 562640 * q^43 - 138240 * q^44 + 501120 * q^46 - 515328 * q^47 + 1449396 * q^49 - 757568 * q^50 + 861440 * q^52 - 2498016 * q^53 + 1322784 * q^55 - 286720 * q^56 + 547200 * q^58 - 4155840 * q^59 + 2130764 * q^61 - 1820032 * q^62 + 1048576 * q^64 - 8966496 * q^65 - 1205440 * q^67 - 1443840 * q^68 - 504192 * q^70 - 12486480 * q^71 - 4820860 * q^73 - 4188640 * q^74 + 2349056 * q^76 - 18686592 * q^77 - 11471680 * q^79 - 2162688 * q^80 + 537600 * q^82 - 16811232 * q^83 - 21006396 * q^85 - 4501120 * q^86 + 1105920 * q^88 - 16857600 * q^89 - 23200544 * q^91 - 4008960 * q^92 + 4122624 * q^94 - 14090160 * q^95 - 27078040 * q^97 - 11595168 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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