LMFDB - Embedded newform 350.10.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,10,Mod(1,350)] 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("350.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	350.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-16,-170] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$180.262542657$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	350.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-16.0000 q^{2} -170.000 q^{3} +256.000 q^{4} +2720.00 q^{6} +2401.00 q^{7} -4096.00 q^{8} +9217.00 q^{9} +48824.0 q^{11} -43520.0 q^{12} +15876.0 q^{13} -38416.0 q^{14} +65536.0 q^{16} +21418.0 q^{17} -147472. q^{18} -716410. q^{19} -408170. q^{21} -781184. q^{22} +2.47000e6 q^{23} +696320. q^{24} -254016. q^{26} +1.77922e6 q^{27} +614656. q^{28} +5.55683e6 q^{29} +5.79935e6 q^{31} -1.04858e6 q^{32} -8.30008e6 q^{33} -342688. q^{34} +2.35955e6 q^{36} +3.89443e6 q^{37} +1.14626e7 q^{38} -2.69892e6 q^{39} -6.36086e6 q^{41} +6.53072e6 q^{42} +1.87013e7 q^{43} +1.24989e7 q^{44} -3.95200e7 q^{46} -5.65391e7 q^{47} -1.11411e7 q^{48} +5.76480e6 q^{49} -3.64106e6 q^{51} +4.06426e6 q^{52} +5.98947e7 q^{53} -2.84675e7 q^{54} -9.83450e6 q^{56} +1.21790e8 q^{57} -8.89092e7 q^{58} +1.65630e8 q^{59} +5.14190e7 q^{61} -9.27896e7 q^{62} +2.21300e7 q^{63} +1.67772e7 q^{64} +1.32801e8 q^{66} -9.35465e7 q^{67} +5.48301e6 q^{68} -4.19900e8 q^{69} -9.56335e7 q^{71} -3.77528e7 q^{72} -3.06496e8 q^{73} -6.23109e7 q^{74} -1.83401e8 q^{76} +1.17226e8 q^{77} +4.31827e7 q^{78} +4.96474e8 q^{79} -4.83886e8 q^{81} +1.01774e8 q^{82} +3.71487e8 q^{83} -1.04492e8 q^{84} -2.99221e8 q^{86} -9.44660e8 q^{87} -1.99983e8 q^{88} -1.65483e8 q^{89} +3.81183e7 q^{91} +6.32320e8 q^{92} -9.85889e8 q^{93} +9.04625e8 q^{94} +1.78258e8 q^{96} -7.58017e8 q^{97} -9.22368e7 q^{98} +4.50011e8 q^{99} +O(q^{100})\)

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
$3$	−170.000	−1.21172	−0.605861	−	0.795570i	$-0.707171\pi$
$3$	−170.000	−1.21172	−0.605861	+	0.795570i	$0.707171\pi$
$4$	256.000	0.500000
$5$	0	0
$5$	0	0
$6$	2720.00	0.856817
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	−4096.00	−0.353553
$9$	9217.00	0.468272
$10$	0	0
$11$	48824.0	1.00546	0.502732	−	0.864442i	$-0.332328\pi$
$11$	48824.0	1.00546	0.502732	+	0.864442i	$0.332328\pi$
$12$	−43520.0	−0.605861
$13$	15876.0	0.154169	0.0770843	−	0.997025i	$-0.475439\pi$
$13$	15876.0	0.154169	0.0770843	+	0.997025i	$0.475439\pi$
$14$	−38416.0	−0.267261
$15$	0	0
$16$	65536.0	0.250000
$17$	21418.0	0.0621955	0.0310977	−	0.999516i	$-0.490100\pi$
$17$	21418.0	0.0621955	0.0310977	+	0.999516i	$0.490100\pi$
$18$	−147472.	−0.331118
$19$	−716410.	−1.26116	−0.630580	−	0.776124i	$-0.717183\pi$
$19$	−716410.	−1.26116	−0.630580	+	0.776124i	$0.717183\pi$
$20$	0	0
$21$	−408170.	−0.457988
$22$	−781184.	−0.710970
$23$	2.47000e6	1.84044	0.920220	−	0.391401i	$-0.128010\pi$
$23$	2.47000e6	1.84044	0.920220	+	0.391401i	$0.128010\pi$
$24$	696320.	0.428409
$25$	0	0
$26$	−254016.	−0.109014
$27$	1.77922e6	0.644307
$28$	614656.	0.188982
$29$	5.55683e6	1.45893	0.729467	−	0.684016i	$-0.239768\pi$
$29$	5.55683e6	1.45893	0.729467	+	0.684016i	$0.239768\pi$
$30$	0	0
$31$	5.79935e6	1.12785	0.563925	−	0.825826i	$-0.309291\pi$
$31$	5.79935e6	1.12785	0.563925	+	0.825826i	$0.309291\pi$
$32$	−1.04858e6	−0.176777
$33$	−8.30008e6	−1.21834
$34$	−342688.	−0.0439788
$35$	0	0
$36$	2.35955e6	0.234136
$37$	3.89443e6	0.341614	0.170807	−	0.985304i	$-0.445362\pi$
$37$	3.89443e6	0.341614	0.170807	+	0.985304i	$0.445362\pi$
$38$	1.14626e7	0.891775
$39$	−2.69892e6	−0.186810
$40$	0	0
$41$	−6.36086e6	−0.351551	−0.175776	−	0.984430i	$-0.556243\pi$
$41$	−6.36086e6	−0.351551	−0.175776	+	0.984430i	$0.556243\pi$
$42$	6.53072e6	0.323847
$43$	1.87013e7	0.834187	0.417094	−	0.908863i	$-0.363049\pi$
$43$	1.87013e7	0.834187	0.417094	+	0.908863i	$0.363049\pi$
$44$	1.24989e7	0.502732
$45$	0	0
$46$	−3.95200e7	−1.30139
$47$	−5.65391e7	−1.69008	−0.845042	−	0.534700i	$-0.820425\pi$
$47$	−5.65391e7	−1.69008	−0.845042	+	0.534700i	$0.820425\pi$
$48$	−1.11411e7	−0.302931
$49$	5.76480e6	0.142857
$50$	0	0
$51$	−3.64106e6	−0.0753637
$52$	4.06426e6	0.0770843
$53$	5.98947e7	1.04267	0.521335	−	0.853352i	$-0.325434\pi$
$53$	5.98947e7	1.04267	0.521335	+	0.853352i	$0.325434\pi$
$54$	−2.84675e7	−0.455594
$55$	0	0
$56$	−9.83450e6	−0.133631
$57$	1.21790e8	1.52818
$58$	−8.89092e7	−1.03162
$59$	1.65630e8	1.77952	0.889762	−	0.456424i	$-0.150870\pi$
$59$	1.65630e8	1.77952	0.889762	+	0.456424i	$0.150870\pi$
$60$	0	0
$61$	5.14190e7	0.475488	0.237744	−	0.971328i	$-0.423592\pi$
$61$	5.14190e7	0.475488	0.237744	+	0.971328i	$0.423592\pi$
$62$	−9.27896e7	−0.797511
$63$	2.21300e7	0.176990
$64$	1.67772e7	0.125000
$65$	0	0
$66$	1.32801e8	0.861499
$67$	−9.35465e7	−0.567141	−0.283570	−	0.958951i	$-0.591519\pi$
$67$	−9.35465e7	−0.567141	−0.283570	+	0.958951i	$0.591519\pi$
$68$	5.48301e6	0.0310977
$69$	−4.19900e8	−2.23010
$70$	0	0
$71$	−9.56335e7	−0.446630	−0.223315	−	0.974746i	$-0.571688\pi$
$71$	−9.56335e7	−0.446630	−0.223315	+	0.974746i	$0.571688\pi$
$72$	−3.77528e7	−0.165559
$73$	−3.06496e8	−1.26320	−0.631601	−	0.775294i	$-0.717602\pi$
$73$	−3.06496e8	−1.26320	−0.631601	+	0.775294i	$0.717602\pi$
$74$	−6.23109e7	−0.241558
$75$	0	0
$76$	−1.83401e8	−0.630580
$77$	1.17226e8	0.380029
$78$	4.31827e7	0.132094
$79$	4.96474e8	1.43408	0.717042	−	0.697030i	$-0.245495\pi$
$79$	4.96474e8	1.43408	0.717042	+	0.697030i	$0.245495\pi$
$80$	0	0
$81$	−4.83886e8	−1.24899
$82$	1.01774e8	0.248584
$83$	3.71487e8	0.859196	0.429598	−	0.903020i	$-0.358655\pi$
$83$	3.71487e8	0.859196	0.429598	+	0.903020i	$0.358655\pi$
$84$	−1.04492e8	−0.228994
$85$	0	0
$86$	−2.99221e8	−0.589860
$87$	−9.44660e8	−1.76782
$88$	−1.99983e8	−0.355485
$89$	−1.65483e8	−0.279574	−0.139787	−	0.990182i	$-0.544642\pi$
$89$	−1.65483e8	−0.279574	−0.139787	+	0.990182i	$0.544642\pi$
$90$	0	0
$91$	3.81183e7	0.0582703
$92$	6.32320e8	0.920220
$93$	−9.85889e8	−1.36664
$94$	9.04625e8	1.19507
$95$	0	0
$96$	1.78258e8	0.214204
$97$	−7.58017e8	−0.869373	−0.434686	−	0.900582i	$-0.643141\pi$
$97$	−7.58017e8	−0.869373	−0.434686	+	0.900582i	$0.643141\pi$
$98$	−9.22368e7	−0.101015
$99$	4.50011e8	0.470830
$100$	0	0
$101$	−9.04212e8	−0.864618	−0.432309	−	0.901726i	$-0.642301\pi$
$101$	−9.04212e8	−0.864618	−0.432309	+	0.901726i	$0.642301\pi$
$102$	5.82570e7	0.0532902
$103$	−1.98157e9	−1.73477	−0.867384	−	0.497639i	$-0.834200\pi$
$103$	−1.98157e9	−1.73477	−0.867384	+	0.497639i	$0.834200\pi$
$104$	−6.50281e7	−0.0545068
$105$	0	0
$106$	−9.58315e8	−0.737279
$107$	−4.16379e8	−0.307087	−0.153544	−	0.988142i	$-0.549069\pi$
$107$	−4.16379e8	−0.307087	−0.153544	+	0.988142i	$0.549069\pi$
$108$	4.55480e8	0.322153
$109$	−1.26921e9	−0.861220	−0.430610	−	0.902538i	$-0.641701\pi$
$109$	−1.26921e9	−0.861220	−0.430610	+	0.902538i	$0.641701\pi$
$110$	0	0
$111$	−6.62053e8	−0.413942
$112$	1.57352e8	0.0944911
$113$	2.83528e9	1.63585	0.817923	−	0.575328i	$-0.195126\pi$
$113$	2.83528e9	1.63585	0.817923	+	0.575328i	$0.195126\pi$
$114$	−1.94864e9	−1.08058
$115$	0	0
$116$	1.42255e9	0.729467
$117$	1.46329e8	0.0721929
$118$	−2.65007e9	−1.25831
$119$	5.14246e7	0.0235077
$120$	0	0
$121$	2.58353e7	0.0109567
$122$	−8.22704e8	−0.336221
$123$	1.08135e9	0.425982
$124$	1.48463e9	0.563925
$125$	0	0
$126$	−3.54080e8	−0.125151
$127$	−5.44282e9	−1.85655	−0.928277	−	0.371889i	$-0.878710\pi$
$127$	−5.44282e9	−1.85655	−0.928277	+	0.371889i	$0.878710\pi$
$128$	−2.68435e8	−0.0883883
$129$	−3.17922e9	−1.01080
$130$	0	0
$131$	−6.44057e8	−0.191075	−0.0955374	−	0.995426i	$-0.530457\pi$
$131$	−6.44057e8	−0.191075	−0.0955374	+	0.995426i	$0.530457\pi$
$132$	−2.12482e9	−0.609171
$133$	−1.72010e9	−0.476674
$134$	1.49674e9	0.401029
$135$	0	0
$136$	−8.77281e7	−0.0219894
$137$	−1.67376e9	−0.405928	−0.202964	−	0.979186i	$-0.565058\pi$
$137$	−1.67376e9	−0.405928	−0.202964	+	0.979186i	$0.565058\pi$
$138$	6.71840e9	1.57692
$139$	−4.17330e9	−0.948229	−0.474115	−	0.880463i	$-0.657232\pi$
$139$	−4.17330e9	−0.948229	−0.474115	+	0.880463i	$0.657232\pi$
$140$	0	0
$141$	9.61164e9	2.04791
$142$	1.53014e9	0.315815
$143$	7.75130e8	0.155011
$144$	6.04045e8	0.117068
$145$	0	0
$146$	4.90394e9	0.893218
$147$	−9.80016e8	−0.173103
$148$	9.96974e8	0.170807
$149$	−4.64096e8	−0.0771382	−0.0385691	−	0.999256i	$-0.512280\pi$
$149$	−4.64096e8	−0.0771382	−0.0385691	+	0.999256i	$0.512280\pi$
$150$	0	0
$151$	7.31929e9	1.14571	0.572853	−	0.819658i	$-0.305837\pi$
$151$	7.31929e9	1.14571	0.572853	+	0.819658i	$0.305837\pi$
$152$	2.93442e9	0.445888
$153$	1.97410e8	0.0291244
$154$	−1.87562e9	−0.268721
$155$	0	0
$156$	−6.90924e8	−0.0934048
$157$	4.43050e9	0.581975	0.290987	−	0.956727i	$-0.406016\pi$
$157$	4.43050e9	0.581975	0.290987	+	0.956727i	$0.406016\pi$
$158$	−7.94359e9	−1.01405
$159$	−1.01821e10	−1.26343
$160$	0	0
$161$	5.93047e9	0.695621
$162$	7.74217e9	0.883172
$163$	1.33645e10	1.48289	0.741446	−	0.671013i	$-0.234140\pi$
$163$	1.33645e10	1.48289	0.741446	+	0.671013i	$0.234140\pi$
$164$	−1.62838e9	−0.175776
$165$	0	0
$166$	−5.94379e9	−0.607543
$167$	1.24456e10	1.23821	0.619103	−	0.785310i	$-0.287496\pi$
$167$	1.24456e10	1.23821	0.619103	+	0.785310i	$0.287496\pi$
$168$	1.67186e9	0.161923
$169$	−1.03525e10	−0.976232
$170$	0	0
$171$	−6.60315e9	−0.590566
$172$	4.78753e9	0.417094
$173$	−1.04544e10	−0.887345	−0.443672	−	0.896189i	$-0.646325\pi$
$173$	−1.04544e10	−0.887345	−0.443672	+	0.896189i	$0.646325\pi$
$174$	1.51146e10	1.25004
$175$	0	0
$176$	3.19973e9	0.251366
$177$	−2.81570e10	−2.15629
$178$	2.64772e9	0.197689
$179$	−4.04391e9	−0.294417	−0.147208	−	0.989105i	$-0.547029\pi$
$179$	−4.04391e9	−0.294417	−0.147208	+	0.989105i	$0.547029\pi$
$180$	0	0
$181$	1.24735e10	0.863843	0.431922	−	0.901911i	$-0.357836\pi$
$181$	1.24735e10	0.863843	0.431922	+	0.901911i	$0.357836\pi$
$182$	−6.09892e8	−0.0412033
$183$	−8.74123e9	−0.576160
$184$	−1.01171e10	−0.650694
$185$	0	0
$186$	1.57742e10	0.966362
$187$	1.04571e9	0.0625353
$188$	−1.44740e10	−0.845042
$189$	4.27191e9	0.243525
$190$	0	0
$191$	−3.81947e9	−0.207660	−0.103830	−	0.994595i	$-0.533110\pi$
$191$	−3.81947e9	−0.207660	−0.103830	+	0.994595i	$0.533110\pi$
$192$	−2.85213e9	−0.151465
$193$	2.41193e10	1.25129	0.625644	−	0.780109i	$-0.284836\pi$
$193$	2.41193e10	1.25129	0.625644	+	0.780109i	$0.284836\pi$
$194$	1.21283e10	0.614739
$195$	0	0
$196$	1.47579e9	0.0714286
$197$	1.24798e10	0.590351	0.295176	−	0.955443i	$-0.404622\pi$
$197$	1.24798e10	0.590351	0.295176	+	0.955443i	$0.404622\pi$
$198$	−7.20017e9	−0.332927
$199$	−2.93127e9	−0.132500	−0.0662502	−	0.997803i	$-0.521104\pi$
$199$	−2.93127e9	−0.132500	−0.0662502	+	0.997803i	$0.521104\pi$
$200$	0	0
$201$	1.59029e10	0.687218
$202$	1.44674e10	0.611377
$203$	1.33419e10	0.551425
$204$	−9.32111e8	−0.0376818
$205$	0	0
$206$	3.17051e10	1.22667
$207$	2.27660e10	0.861827
$208$	1.04045e9	0.0385422
$209$	−3.49780e10	−1.26805
$210$	0	0
$211$	3.36978e10	1.17039	0.585195	−	0.810892i	$-0.301018\pi$
$211$	3.36978e10	1.17039	0.585195	+	0.810892i	$0.301018\pi$
$212$	1.53330e10	0.521335
$213$	1.62577e10	0.541191
$214$	6.66206e9	0.217143
$215$	0	0
$216$	−7.28769e9	−0.227797
$217$	1.39242e10	0.426287
$218$	2.03073e10	0.608974
$219$	5.21044e10	1.53065
$220$	0	0
$221$	3.40032e8	0.00958859
$222$	1.05928e10	0.292701
$223$	3.87208e10	1.04851	0.524255	−	0.851561i	$-0.324344\pi$
$223$	3.87208e10	1.04851	0.524255	+	0.851561i	$0.324344\pi$
$224$	−2.51763e9	−0.0668153
$225$	0	0
$226$	−4.53644e10	−1.15672
$227$	−7.69011e10	−1.92228	−0.961139	−	0.276063i	$-0.910970\pi$
$227$	−7.69011e10	−1.92228	−0.961139	+	0.276063i	$0.910970\pi$
$228$	3.11782e10	0.764089
$229$	4.35114e10	1.04555	0.522773	−	0.852472i	$-0.324897\pi$
$229$	4.35114e10	1.04555	0.522773	+	0.852472i	$0.324897\pi$
$230$	0	0
$231$	−1.99285e10	−0.460490
$232$	−2.27608e10	−0.515811
$233$	2.07043e10	0.460213	0.230107	−	0.973165i	$-0.426093\pi$
$233$	2.07043e10	0.460213	0.230107	+	0.973165i	$0.426093\pi$
$234$	−2.34127e9	−0.0510481
$235$	0	0
$236$	4.24012e10	0.889762
$237$	−8.44006e10	−1.73771
$238$	−8.22794e8	−0.0166224
$239$	2.16220e10	0.428653	0.214326	−	0.976762i	$-0.431244\pi$
$239$	2.16220e10	0.428653	0.214326	+	0.976762i	$0.431244\pi$
$240$	0	0
$241$	6.77789e10	1.29425	0.647124	−	0.762385i	$-0.275971\pi$
$241$	6.77789e10	1.29425	0.647124	+	0.762385i	$0.275971\pi$
$242$	−4.13365e8	−0.00774754
$243$	4.72402e10	0.869127
$244$	1.31633e10	0.237744
$245$	0	0
$246$	−1.73015e10	−0.301215
$247$	−1.13737e10	−0.194431
$248$	−2.37541e10	−0.398755
$249$	−6.31528e10	−1.04111
$250$	0	0
$251$	4.87895e9	0.0775881	0.0387940	−	0.999247i	$-0.487648\pi$
$251$	4.87895e9	0.0775881	0.0387940	+	0.999247i	$0.487648\pi$
$252$	5.66528e9	0.0884951
$253$	1.20595e11	1.85050
$254$	8.70852e10	1.31278
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	2.75029e10	0.393259	0.196630	−	0.980478i	$-0.437000\pi$
$257$	2.75029e10	0.393259	0.196630	+	0.980478i	$0.437000\pi$
$258$	5.08675e10	0.714746
$259$	9.35053e9	0.129118
$260$	0	0
$261$	5.12173e10	0.683178
$262$	1.03049e10	0.135110
$263$	2.22595e10	0.286889	0.143445	−	0.989658i	$-0.454182\pi$
$263$	2.22595e10	0.286889	0.143445	+	0.989658i	$0.454182\pi$
$264$	3.39971e10	0.430749
$265$	0	0
$266$	2.75216e10	0.337059
$267$	2.81320e10	0.338766
$268$	−2.39479e10	−0.283570
$269$	1.73017e10	0.201466	0.100733	−	0.994913i	$-0.467881\pi$
$269$	1.73017e10	0.201466	0.100733	+	0.994913i	$0.467881\pi$
$270$	0	0
$271$	4.81901e10	0.542745	0.271372	−	0.962474i	$-0.412523\pi$
$271$	4.81901e10	0.542745	0.271372	+	0.962474i	$0.412523\pi$
$272$	1.40365e9	0.0155489
$273$	−6.48011e9	−0.0706074
$274$	2.67801e10	0.287035
$275$	0	0
$276$	−1.07494e11	−1.11505
$277$	−8.03834e10	−0.820365	−0.410183	−	0.912003i	$-0.634535\pi$
$277$	−8.03834e10	−0.820365	−0.410183	+	0.912003i	$0.634535\pi$
$278$	6.67728e10	0.670499
$279$	5.34526e10	0.528141
$280$	0	0
$281$	−1.95595e11	−1.87146	−0.935729	−	0.352719i	$-0.885257\pi$
$281$	−1.95595e11	−1.87146	−0.935729	+	0.352719i	$0.885257\pi$
$282$	−1.53786e11	−1.44809
$283$	6.02802e10	0.558645	0.279322	−	0.960197i	$-0.409890\pi$
$283$	6.02802e10	0.558645	0.279322	+	0.960197i	$0.409890\pi$
$284$	−2.44822e10	−0.223315
$285$	0	0
$286$	−1.24021e10	−0.109609
$287$	−1.52724e10	−0.132874
$288$	−9.66472e9	−0.0827796
$289$	−1.18129e11	−0.996132
$290$	0	0
$291$	1.28863e11	1.05344
$292$	−7.84631e10	−0.631601
$293$	4.86743e10	0.385830	0.192915	−	0.981216i	$-0.438206\pi$
$293$	4.86743e10	0.385830	0.192915	+	0.981216i	$0.438206\pi$
$294$	1.56803e10	0.122402
$295$	0	0
$296$	−1.59516e10	−0.120779
$297$	8.68686e10	0.647827
$298$	7.42553e9	0.0545449
$299$	3.92137e10	0.283738
$300$	0	0
$301$	4.49018e10	0.315293
$302$	−1.17109e11	−0.810136
$303$	1.53716e11	1.04768
$304$	−4.69506e10	−0.315290
$305$	0	0
$306$	−3.15856e9	−0.0205941
$307$	−2.75178e11	−1.76804	−0.884018	−	0.467453i	$-0.845172\pi$
$307$	−2.75178e11	−1.76804	−0.884018	+	0.467453i	$0.845172\pi$
$308$	3.00100e10	0.190015
$309$	3.36867e11	2.10206
$310$	0	0
$311$	1.12322e11	0.680835	0.340418	−	0.940274i	$-0.389432\pi$
$311$	1.12322e11	0.680835	0.340418	+	0.940274i	$0.389432\pi$
$312$	1.10548e10	0.0660472
$313$	1.06140e11	0.625069	0.312535	−	0.949906i	$-0.398822\pi$
$313$	1.06140e11	0.625069	0.312535	+	0.949906i	$0.398822\pi$
$314$	−7.08880e10	−0.411518
$315$	0	0
$316$	1.27097e11	0.717042
$317$	−2.31358e9	−0.0128682	−0.00643409	−	0.999979i	$-0.502048\pi$
$317$	−2.31358e9	−0.0128682	−0.00643409	+	0.999979i	$0.502048\pi$
$318$	1.62914e11	0.893378
$319$	2.71306e11	1.46691
$320$	0	0
$321$	7.07844e10	0.372105
$322$	−9.48875e10	−0.491878
$323$	−1.53441e10	−0.0784385
$324$	−1.23875e11	−0.624497
$325$	0	0
$326$	−2.13832e11	−1.04856
$327$	2.15766e11	1.04356
$328$	2.60541e10	0.124292
$329$	−1.35750e11	−0.638792
$330$	0	0
$331$	−2.16185e11	−0.989921	−0.494960	−	0.868916i	$-0.664817\pi$
$331$	−2.16185e11	−0.989921	−0.494960	+	0.868916i	$0.664817\pi$
$332$	9.51007e10	0.429598
$333$	3.58950e10	0.159968
$334$	−1.99130e11	−0.875544
$335$	0	0
$336$	−2.67498e10	−0.114497
$337$	−5.00291e10	−0.211294	−0.105647	−	0.994404i	$-0.533691\pi$
$337$	−5.00291e10	−0.211294	−0.105647	+	0.994404i	$0.533691\pi$
$338$	1.65639e11	0.690300
$339$	−4.81997e11	−1.98219
$340$	0	0
$341$	2.83147e11	1.13401
$342$	1.05650e11	0.417594
$343$	1.38413e10	0.0539949
$344$	−7.66005e10	−0.294930
$345$	0	0
$346$	1.67271e11	0.627448
$347$	1.84606e11	0.683541	0.341770	−	0.939784i	$-0.388974\pi$
$347$	1.84606e11	0.683541	0.341770	+	0.939784i	$0.388974\pi$
$348$	−2.41833e11	−0.883912
$349$	2.74666e11	0.991039	0.495520	−	0.868597i	$-0.334978\pi$
$349$	2.74666e11	0.991039	0.495520	+	0.868597i	$0.334978\pi$
$350$	0	0
$351$	2.82469e10	0.0993319
$352$	−5.11957e10	−0.177743
$353$	1.58053e11	0.541774	0.270887	−	0.962611i	$-0.412683\pi$
$353$	1.58053e11	0.541774	0.270887	+	0.962611i	$0.412683\pi$
$354$	4.50513e11	1.52473
$355$	0	0
$356$	−4.23635e10	−0.139787
$357$	−8.74219e9	−0.0284848
$358$	6.47025e10	0.208184
$359$	3.40759e11	1.08274	0.541368	−	0.840786i	$-0.317907\pi$
$359$	3.40759e11	1.08274	0.541368	+	0.840786i	$0.317907\pi$
$360$	0	0
$361$	1.90556e11	0.590526
$362$	−1.99576e11	−0.610829
$363$	−4.39200e9	−0.0132765
$364$	9.75828e9	0.0291351
$365$	0	0
$366$	1.39860e11	0.407406
$367$	6.10216e11	1.75584	0.877922	−	0.478803i	$-0.158929\pi$
$367$	6.10216e11	1.75584	0.877922	+	0.478803i	$0.158929\pi$
$368$	1.61874e11	0.460110
$369$	−5.86280e10	−0.164622
$370$	0	0
$371$	1.43807e11	0.394092
$372$	−2.52388e11	−0.683321
$373$	−4.34930e11	−1.16340	−0.581701	−	0.813402i	$-0.697613\pi$
$373$	−4.34930e11	−1.16340	−0.581701	+	0.813402i	$0.697613\pi$
$374$	−1.67314e10	−0.0442191
$375$	0	0
$376$	2.31584e11	0.597535
$377$	8.82202e10	0.224922
$378$	−6.83505e10	−0.172198
$379$	−7.30677e11	−1.81907	−0.909534	−	0.415630i	$-0.863561\pi$
$379$	−7.30677e11	−1.81907	−0.909534	+	0.415630i	$0.863561\pi$
$380$	0	0
$381$	9.25280e11	2.24963
$382$	6.11115e10	0.146838
$383$	−2.11074e11	−0.501233	−0.250617	−	0.968086i	$-0.580633\pi$
$383$	−2.11074e11	−0.501233	−0.250617	+	0.968086i	$0.580633\pi$
$384$	4.56340e10	0.107102
$385$	0	0
$386$	−3.85909e11	−0.884794
$387$	1.72370e11	0.390627
$388$	−1.94052e11	−0.434686
$389$	7.21857e9	0.0159837	0.00799186	−	0.999968i	$-0.497456\pi$
$389$	7.21857e9	0.0159837	0.00799186	+	0.999968i	$0.497456\pi$
$390$	0	0
$391$	5.29025e10	0.114467
$392$	−2.36126e10	−0.0505076
$393$	1.09490e11	0.231530
$394$	−1.99677e11	−0.417441
$395$	0	0
$396$	1.15203e11	0.235415
$397$	6.99387e11	1.41306	0.706529	−	0.707684i	$-0.250260\pi$
$397$	6.99387e11	1.41306	0.706529	+	0.707684i	$0.250260\pi$
$398$	4.69004e10	0.0936920
$399$	2.92417e11	0.577597
$400$	0	0
$401$	6.40644e11	1.23728	0.618638	−	0.785676i	$-0.287685\pi$
$401$	6.40644e11	1.23728	0.618638	+	0.785676i	$0.287685\pi$
$402$	−2.54447e11	−0.485936
$403$	9.20704e10	0.173879
$404$	−2.31478e11	−0.432309
$405$	0	0
$406$	−2.13471e11	−0.389917
$407$	1.90142e11	0.343481
$408$	1.49138e10	0.0266451
$409$	−1.31500e10	−0.0232365	−0.0116182	−	0.999933i	$-0.503698\pi$
$409$	−1.31500e10	−0.0232365	−0.0116182	+	0.999933i	$0.503698\pi$
$410$	0	0
$411$	2.84538e11	0.491873
$412$	−5.07281e11	−0.867384
$413$	3.97677e11	0.672597
$414$	−3.64256e11	−0.609404
$415$	0	0
$416$	−1.66472e10	−0.0272534
$417$	7.09461e11	1.14899
$418$	5.59648e11	0.896647
$419$	−5.79915e11	−0.919182	−0.459591	−	0.888131i	$-0.652004\pi$
$419$	−5.79915e11	−0.919182	−0.459591	+	0.888131i	$0.652004\pi$
$420$	0	0
$421$	1.66175e11	0.257808	0.128904	−	0.991657i	$-0.458854\pi$
$421$	1.66175e11	0.257808	0.128904	+	0.991657i	$0.458854\pi$
$422$	−5.39165e11	−0.827591
$423$	−5.21121e11	−0.791419
$424$	−2.45329e11	−0.368639
$425$	0	0
$426$	−2.60123e11	−0.382680
$427$	1.23457e11	0.179718
$428$	−1.06593e11	−0.153544
$429$	−1.31772e11	−0.187830
$430$	0	0
$431$	7.57723e11	1.05770	0.528850	−	0.848715i	$-0.322623\pi$
$431$	7.57723e11	1.05770	0.528850	+	0.848715i	$0.322623\pi$
$432$	1.16603e11	0.161077
$433$	1.07485e12	1.46944	0.734719	−	0.678371i	$-0.237314\pi$
$433$	1.07485e12	1.46944	0.734719	+	0.678371i	$0.237314\pi$
$434$	−2.22788e11	−0.301431
$435$	0	0
$436$	−3.24918e11	−0.430610
$437$	−1.76953e12	−2.32109
$438$	−8.33670e11	−1.08233
$439$	1.70418e11	0.218991	0.109496	−	0.993987i	$-0.465076\pi$
$439$	1.70418e11	0.218991	0.109496	+	0.993987i	$0.465076\pi$
$440$	0	0
$441$	5.31342e10	0.0668960
$442$	−5.44051e9	−0.00678016
$443$	−1.22937e12	−1.51658	−0.758290	−	0.651918i	$-0.773965\pi$
$443$	−1.22937e12	−1.51658	−0.758290	+	0.651918i	$0.773965\pi$
$444$	−1.69486e11	−0.206971
$445$	0	0
$446$	−6.19533e11	−0.741409
$447$	7.88963e10	0.0934701
$448$	4.02821e10	0.0472456
$449$	−7.25792e10	−0.0842759	−0.0421380	−	0.999112i	$-0.513417\pi$
$449$	−7.25792e10	−0.0842759	−0.0421380	+	0.999112i	$0.513417\pi$
$450$	0	0
$451$	−3.10563e11	−0.353472
$452$	7.25831e11	0.817923
$453$	−1.24428e12	−1.38828
$454$	1.23042e12	1.35926
$455$	0	0
$456$	−4.98851e11	−0.540292
$457$	6.64172e11	0.712291	0.356146	−	0.934431i	$-0.384091\pi$
$457$	6.64172e11	0.712291	0.356146	+	0.934431i	$0.384091\pi$
$458$	−6.96183e11	−0.739313
$459$	3.81073e10	0.0400730
$460$	0	0
$461$	−1.21501e12	−1.25293	−0.626463	−	0.779451i	$-0.715498\pi$
$461$	−1.21501e12	−1.25293	−0.626463	+	0.779451i	$0.715498\pi$
$462$	3.18856e11	0.325616
$463$	−2.93878e11	−0.297202	−0.148601	−	0.988897i	$-0.547477\pi$
$463$	−2.93878e11	−0.297202	−0.148601	+	0.988897i	$0.547477\pi$
$464$	3.64172e11	0.364734
$465$	0	0
$466$	−3.31269e11	−0.325420
$467$	−4.73112e11	−0.460297	−0.230149	−	0.973155i	$-0.573921\pi$
$467$	−4.73112e11	−0.460297	−0.230149	+	0.973155i	$0.573921\pi$
$468$	3.74602e10	0.0360964
$469$	−2.24605e11	−0.214359
$470$	0	0
$471$	−7.53185e11	−0.705192
$472$	−6.78419e11	−0.629157
$473$	9.13072e11	0.838745
$474$	1.35041e12	1.22875
$475$	0	0
$476$	1.31647e10	0.0117538
$477$	5.52049e11	0.488253
$478$	−3.45952e11	−0.303103
$479$	−2.05945e12	−1.78748	−0.893742	−	0.448582i	$-0.851929\pi$
$479$	−2.05945e12	−1.78748	−0.893742	+	0.448582i	$0.851929\pi$
$480$	0	0
$481$	6.18280e10	0.0526662
$482$	−1.08446e12	−0.915172
$483$	−1.00818e12	−0.842900
$484$	6.61383e9	0.00547834
$485$	0	0
$486$	−7.55843e11	−0.614566
$487$	1.22247e12	0.984821	0.492411	−	0.870363i	$-0.336116\pi$
$487$	1.22247e12	0.984821	0.492411	+	0.870363i	$0.336116\pi$
$488$	−2.10612e11	−0.168110
$489$	−2.27197e12	−1.79685
$490$	0	0
$491$	1.98225e11	0.153918	0.0769592	−	0.997034i	$-0.475479\pi$
$491$	1.98225e11	0.153918	0.0769592	+	0.997034i	$0.475479\pi$
$492$	2.76825e11	0.212991
$493$	1.19016e11	0.0907391
$494$	1.81980e11	0.137484
$495$	0	0
$496$	3.80066e11	0.281963
$497$	−2.29616e11	−0.168810
$498$	1.01044e12	0.736174
$499$	−3.00745e11	−0.217143	−0.108571	−	0.994089i	$-0.534628\pi$
$499$	−3.00745e11	−0.217143	−0.108571	+	0.994089i	$0.534628\pi$
$500$	0	0
$501$	−2.11576e12	−1.50036
$502$	−7.80633e10	−0.0548631
$503$	−3.30194e11	−0.229993	−0.114996	−	0.993366i	$-0.536686\pi$
$503$	−3.30194e11	−0.229993	−0.114996	+	0.993366i	$0.536686\pi$
$504$	−9.06445e10	−0.0625755
$505$	0	0
$506$	−1.92952e12	−1.30850
$507$	1.75992e12	1.18292
$508$	−1.39336e12	−0.928277
$509$	−6.32399e10	−0.0417600	−0.0208800	−	0.999782i	$-0.506647\pi$
$509$	−6.32399e10	−0.0417600	−0.0208800	+	0.999782i	$0.506647\pi$
$510$	0	0
$511$	−7.35898e11	−0.477445
$512$	−6.87195e10	−0.0441942
$513$	−1.27465e12	−0.812574
$514$	−4.40046e11	−0.278076
$515$	0	0
$516$	−8.13880e11	−0.505402
$517$	−2.76046e12	−1.69932
$518$	−1.49608e11	−0.0913003
$519$	1.77725e12	1.07522
$520$	0	0
$521$	−1.88994e12	−1.12377	−0.561886	−	0.827215i	$-0.689924\pi$
$521$	−1.88994e12	−1.12377	−0.561886	+	0.827215i	$0.689924\pi$
$522$	−8.19476e11	−0.483080
$523$	−8.95863e11	−0.523581	−0.261791	−	0.965125i	$-0.584313\pi$
$523$	−8.95863e11	−0.523581	−0.261791	+	0.965125i	$0.584313\pi$
$524$	−1.64879e11	−0.0955374
$525$	0	0
$526$	−3.56152e11	−0.202861
$527$	1.24210e11	0.0701472
$528$	−5.43954e11	−0.304586
$529$	4.29975e12	2.38722
$530$	0	0
$531$	1.52661e12	0.833302
$532$	−4.40346e11	−0.238337
$533$	−1.00985e11	−0.0541981
$534$	−4.50113e11	−0.239544
$535$	0	0
$536$	3.83166e11	0.200515
$537$	6.87464e11	0.356752
$538$	−2.76826e11	−0.142458
$539$	2.81461e11	0.143638
$540$	0	0
$541$	1.00221e12	0.503005	0.251502	−	0.967857i	$-0.419075\pi$
$541$	1.00221e12	0.503005	0.251502	+	0.967857i	$0.419075\pi$
$542$	−7.71041e11	−0.383778
$543$	−2.12050e12	−1.04674
$544$	−2.24584e10	−0.0109947
$545$	0	0
$546$	1.03682e11	0.0499270
$547$	−2.73436e12	−1.30591	−0.652955	−	0.757396i	$-0.726471\pi$
$547$	−2.73436e12	−1.30591	−0.652955	+	0.757396i	$0.726471\pi$
$548$	−4.28481e11	−0.202964
$549$	4.73929e11	0.222658
$550$	0	0
$551$	−3.98097e12	−1.83995
$552$	1.71991e12	0.788461
$553$	1.19203e12	0.542033
$554$	1.28613e12	0.580086
$555$	0	0
$556$	−1.06837e12	−0.474115
$557$	4.22359e12	1.85923	0.929614	−	0.368534i	$-0.120140\pi$
$557$	4.22359e12	1.85923	0.929614	+	0.368534i	$0.120140\pi$
$558$	−8.55241e11	−0.373452
$559$	2.96902e11	0.128606
$560$	0	0
$561$	−1.77771e11	−0.0757754
$562$	3.12953e12	1.32332
$563$	3.08311e12	1.29330	0.646652	−	0.762785i	$-0.276169\pi$
$563$	3.08311e12	1.29330	0.646652	+	0.762785i	$0.276169\pi$
$564$	2.46058e12	1.02396
$565$	0	0
$566$	−9.64483e11	−0.395021
$567$	−1.16181e12	−0.472075
$568$	3.91715e11	0.157907
$569$	−2.74669e11	−0.109851	−0.0549256	−	0.998490i	$-0.517492\pi$
$569$	−2.74669e11	−0.109851	−0.0549256	+	0.998490i	$0.517492\pi$
$570$	0	0
$571$	2.82499e12	1.11213	0.556064	−	0.831140i	$-0.312311\pi$
$571$	2.82499e12	1.11213	0.556064	+	0.831140i	$0.312311\pi$
$572$	1.98433e11	0.0775055
$573$	6.49310e11	0.251626
$574$	2.44359e11	0.0939560
$575$	0	0
$576$	1.54636e11	0.0585340
$577$	−3.76585e12	−1.41440	−0.707199	−	0.707014i	$-0.750042\pi$
$577$	−3.76585e12	−1.41440	−0.707199	+	0.707014i	$0.750042\pi$
$578$	1.89007e12	0.704371
$579$	−4.10028e12	−1.51621
$580$	0	0
$581$	8.91940e11	0.324745
$582$	−2.06181e12	−0.744894
$583$	2.92430e12	1.04837
$584$	1.25541e12	0.446609
$585$	0	0
$586$	−7.78789e11	−0.272823
$587$	3.00831e12	1.04580	0.522902	−	0.852393i	$-0.324849\pi$
$587$	3.00831e12	1.04580	0.522902	+	0.852393i	$0.324849\pi$
$588$	−2.50884e11	−0.0865516
$589$	−4.15471e12	−1.42240
$590$	0	0
$591$	−2.12157e12	−0.715342
$592$	2.55225e11	0.0854036
$593$	−3.64775e12	−1.21138	−0.605688	−	0.795703i	$-0.707102\pi$
$593$	−3.64775e12	−1.21138	−0.605688	+	0.795703i	$0.707102\pi$
$594$	−1.38990e12	−0.458083
$595$	0	0
$596$	−1.18809e11	−0.0385691
$597$	4.98316e11	0.160554
$598$	−6.27420e11	−0.200633
$599$	4.17778e12	1.32594	0.662972	−	0.748644i	$-0.269295\pi$
$599$	4.17778e12	1.32594	0.662972	+	0.748644i	$0.269295\pi$
$600$	0	0
$601$	4.84445e12	1.51464	0.757321	−	0.653043i	$-0.226508\pi$
$601$	4.84445e12	1.51464	0.757321	+	0.653043i	$0.226508\pi$
$602$	−7.18429e11	−0.222946
$603$	−8.62218e11	−0.265576
$604$	1.87374e12	0.572853
$605$	0	0
$606$	−2.45946e12	−0.740819
$607$	1.58444e12	0.473726	0.236863	−	0.971543i	$-0.423881\pi$
$607$	1.58444e12	0.473726	0.236863	+	0.971543i	$0.423881\pi$
$608$	7.51210e11	0.222944
$609$	−2.26813e12	−0.668175
$610$	0	0
$611$	−8.97614e11	−0.260558
$612$	5.05369e10	0.0145622
$613$	2.79203e11	0.0798635	0.0399318	−	0.999202i	$-0.487286\pi$
$613$	2.79203e11	0.0798635	0.0399318	+	0.999202i	$0.487286\pi$
$614$	4.40285e12	1.25019
$615$	0	0
$616$	−4.80159e11	−0.134361
$617$	5.55576e12	1.54333	0.771667	−	0.636027i	$-0.219423\pi$
$617$	5.55576e12	1.54333	0.771667	+	0.636027i	$0.219423\pi$
$618$	−5.38986e12	−1.48638
$619$	−2.70437e12	−0.740385	−0.370192	−	0.928955i	$-0.620708\pi$
$619$	−2.70437e12	−0.740385	−0.370192	+	0.928955i	$0.620708\pi$
$620$	0	0
$621$	4.39467e12	1.18581
$622$	−1.79715e12	−0.481423
$623$	−3.97324e11	−0.105669
$624$	−1.76876e11	−0.0467024
$625$	0	0
$626$	−1.69823e12	−0.441991
$627$	5.94626e12	1.53653
$628$	1.13421e12	0.290987
$629$	8.34109e10	0.0212469
$630$	0	0
$631$	−4.73158e12	−1.18816	−0.594078	−	0.804407i	$-0.702483\pi$
$631$	−4.73158e12	−1.18816	−0.594078	+	0.804407i	$0.702483\pi$
$632$	−2.03356e12	−0.507025
$633$	−5.72863e12	−1.41819
$634$	3.70172e10	0.00909918
$635$	0	0
$636$	−2.60662e12	−0.631713
$637$	9.15220e10	0.0220241
$638$	−4.34090e12	−1.03726
$639$	−8.81454e11	−0.209144
$640$	0	0
$641$	1.38865e12	0.324887	0.162444	−	0.986718i	$-0.448062\pi$
$641$	1.38865e12	0.324887	0.162444	+	0.986718i	$0.448062\pi$
$642$	−1.13255e12	−0.263118
$643$	−3.09398e12	−0.713786	−0.356893	−	0.934145i	$-0.616164\pi$
$643$	−3.09398e12	−0.713786	−0.356893	+	0.934145i	$0.616164\pi$
$644$	1.51820e12	0.347810
$645$	0	0
$646$	2.45505e11	0.0554644
$647$	−2.31453e12	−0.519270	−0.259635	−	0.965707i	$-0.583602\pi$
$647$	−2.31453e12	−0.519270	−0.259635	+	0.965707i	$0.583602\pi$
$648$	1.98200e12	0.441586
$649$	8.08670e12	1.78925
$650$	0	0
$651$	−2.36712e12	−0.516542
$652$	3.42132e12	0.741446
$653$	8.10575e11	0.174455	0.0872276	−	0.996188i	$-0.472199\pi$
$653$	8.10575e11	0.174455	0.0872276	+	0.996188i	$0.472199\pi$
$654$	−3.45225e12	−0.737908
$655$	0	0
$656$	−4.16865e11	−0.0878878
$657$	−2.82498e12	−0.591522
$658$	2.17200e12	0.451694
$659$	4.57355e11	0.0944645	0.0472323	−	0.998884i	$-0.484960\pi$
$659$	4.57355e11	0.0944645	0.0472323	+	0.998884i	$0.484960\pi$
$660$	0	0
$661$	7.94557e12	1.61889	0.809447	−	0.587193i	$-0.199767\pi$
$661$	7.94557e12	1.61889	0.809447	+	0.587193i	$0.199767\pi$
$662$	3.45897e12	0.699980
$663$	−5.78055e10	−0.0116187
$664$	−1.52161e12	−0.303772
$665$	0	0
$666$	−5.74319e11	−0.113115
$667$	1.37254e13	2.68508
$668$	3.18608e12	0.619103
$669$	−6.58254e12	−1.27050
$670$	0	0
$671$	2.51048e12	0.478086
$672$	4.27997e11	0.0809616
$673$	8.92805e12	1.67760	0.838801	−	0.544439i	$-0.183257\pi$
$673$	8.92805e12	1.67760	0.838801	+	0.544439i	$0.183257\pi$
$674$	8.00465e11	0.149408
$675$	0	0
$676$	−2.65023e12	−0.488116
$677$	8.01730e12	1.46683	0.733414	−	0.679782i	$-0.237926\pi$
$677$	8.01730e12	1.46683	0.733414	+	0.679782i	$0.237926\pi$
$678$	7.71195e12	1.40162
$679$	−1.82000e12	−0.328592
$680$	0	0
$681$	1.30732e13	2.32927
$682$	−4.53036e12	−0.801868
$683$	−4.14724e12	−0.729233	−0.364617	−	0.931158i	$-0.618800\pi$
$683$	−4.14724e12	−0.729233	−0.364617	+	0.931158i	$0.618800\pi$
$684$	−1.69041e12	−0.295283
$685$	0	0
$686$	−2.21461e11	−0.0381802
$687$	−7.39694e12	−1.26691
$688$	1.22561e12	0.208547
$689$	9.50888e11	0.160747
$690$	0	0
$691$	−6.05580e12	−1.01046	−0.505231	−	0.862984i	$-0.668593\pi$
$691$	−6.05580e12	−1.01046	−0.505231	+	0.862984i	$0.668593\pi$
$692$	−2.67633e12	−0.443672
$693$	1.08048e12	0.177957
$694$	−2.95370e12	−0.483336
$695$	0	0
$696$	3.86933e12	0.625020
$697$	−1.36237e11	−0.0218649
$698$	−4.39466e12	−0.700771
$699$	−3.51973e12	−0.557651
$700$	0	0
$701$	1.88599e12	0.294990	0.147495	−	0.989063i	$-0.452879\pi$
$701$	1.88599e12	0.294990	0.147495	+	0.989063i	$0.452879\pi$
$702$	−4.51950e11	−0.0702383
$703$	−2.79001e12	−0.430831
$704$	8.19131e11	0.125683
$705$	0	0
$706$	−2.52886e12	−0.383092
$707$	−2.17101e12	−0.326795
$708$	−7.20820e12	−1.07815
$709$	−4.76210e12	−0.707767	−0.353884	−	0.935289i	$-0.615139\pi$
$709$	−4.76210e12	−0.707767	−0.353884	+	0.935289i	$0.615139\pi$
$710$	0	0
$711$	4.57600e12	0.671542
$712$	6.77817e11	0.0988444
$713$	1.43244e13	2.07574
$714$	1.39875e11	0.0201418
$715$	0	0
$716$	−1.03524e12	−0.147208
$717$	−3.67574e12	−0.519408
$718$	−5.45215e12	−0.765610
$719$	5.34893e12	0.746427	0.373213	−	0.927746i	$-0.378256\pi$
$719$	5.34893e12	0.746427	0.373213	+	0.927746i	$0.378256\pi$
$720$	0	0
$721$	−4.75774e12	−0.655681
$722$	−3.04889e12	−0.417565
$723$	−1.15224e13	−1.56827
$724$	3.19322e12	0.431922
$725$	0	0
$726$	7.02720e10	0.00938788
$727$	−9.08222e12	−1.20583	−0.602917	−	0.797804i	$-0.705995\pi$
$727$	−9.08222e12	−1.20583	−0.602917	+	0.797804i	$0.705995\pi$
$728$	−1.56132e11	−0.0206016
$729$	1.49349e12	0.195852
$730$	0	0
$731$	4.00544e11	0.0518827
$732$	−2.23776e12	−0.288080
$733$	1.20547e13	1.54237	0.771185	−	0.636611i	$-0.219664\pi$
$733$	1.20547e13	1.54237	0.771185	+	0.636611i	$0.219664\pi$
$734$	−9.76345e12	−1.24157
$735$	0	0
$736$	−2.58998e12	−0.325347
$737$	−4.56731e12	−0.570239
$738$	9.38048e11	0.116405
$739$	1.08128e13	1.33364	0.666822	−	0.745217i	$-0.267654\pi$
$739$	1.08128e13	1.33364	0.666822	+	0.745217i	$0.267654\pi$
$740$	0	0
$741$	1.93353e12	0.235597
$742$	−2.30091e12	−0.278665
$743$	−6.39433e12	−0.769742	−0.384871	−	0.922970i	$-0.625754\pi$
$743$	−6.39433e12	−0.769742	−0.384871	+	0.922970i	$0.625754\pi$
$744$	4.03820e12	0.483181
$745$	0	0
$746$	6.95889e12	0.822650
$747$	3.42400e12	0.402337
$748$	2.67702e11	0.0312676
$749$	−9.99726e11	−0.116068
$750$	0	0
$751$	−2.30580e12	−0.264510	−0.132255	−	0.991216i	$-0.542222\pi$
$751$	−2.30580e12	−0.264510	−0.132255	+	0.991216i	$0.542222\pi$
$752$	−3.70534e12	−0.422521
$753$	−8.29422e11	−0.0940152
$754$	−1.41152e12	−0.159044
$755$	0	0
$756$	1.09361e12	0.121763
$757$	6.85316e12	0.758507	0.379253	−	0.925293i	$-0.376181\pi$
$757$	6.85316e12	0.758507	0.379253	+	0.925293i	$0.376181\pi$
$758$	1.16908e13	1.28628
$759$	−2.05012e13	−2.24229
$760$	0	0
$761$	−1.55520e12	−0.168095	−0.0840474	−	0.996462i	$-0.526785\pi$
$761$	−1.55520e12	−0.168095	−0.0840474	+	0.996462i	$0.526785\pi$
$762$	−1.48045e13	−1.59073
$763$	−3.04737e12	−0.325510
$764$	−9.77784e11	−0.103830
$765$	0	0
$766$	3.37718e12	0.354425
$767$	2.62954e12	0.274347
$768$	−7.30144e11	−0.0757327
$769$	−1.31148e12	−0.135236	−0.0676179	−	0.997711i	$-0.521540\pi$
$769$	−1.31148e12	−0.135236	−0.0676179	+	0.997711i	$0.521540\pi$
$770$	0	0
$771$	−4.67549e12	−0.476521
$772$	6.17454e12	0.625644
$773$	9.82010e12	0.989255	0.494627	−	0.869105i	$-0.335305\pi$
$773$	9.82010e12	0.989255	0.494627	+	0.869105i	$0.335305\pi$
$774$	−2.75792e12	−0.276215
$775$	0	0
$776$	3.10484e12	0.307370
$777$	−1.58959e12	−0.156455
$778$	−1.15497e11	−0.0113022
$779$	4.55698e12	0.443362
$780$	0	0
$781$	−4.66921e12	−0.449070
$782$	−8.46439e11	−0.0809404
$783$	9.88682e12	0.940001
$784$	3.77802e11	0.0357143
$785$	0	0
$786$	−1.75184e12	−0.163716
$787$	4.81658e12	0.447562	0.223781	−	0.974639i	$-0.428160\pi$
$787$	4.81658e12	0.447562	0.223781	+	0.974639i	$0.428160\pi$
$788$	3.19484e12	0.295176
$789$	−3.78411e12	−0.347630
$790$	0	0
$791$	6.80750e12	0.618292
$792$	−1.84324e12	−0.166464
$793$	8.16328e11	0.0733053
$794$	−1.11902e13	−0.999183
$795$	0	0
$796$	−7.50406e11	−0.0662502
$797$	−7.71344e12	−0.677151	−0.338575	−	0.940939i	$-0.609945\pi$
$797$	−7.71344e12	−0.677151	−0.338575	+	0.940939i	$0.609945\pi$
$798$	−4.67867e12	−0.408423
$799$	−1.21095e12	−0.105116
$800$	0	0
$801$	−1.52525e12	−0.130917
$802$	−1.02503e13	−0.874887
$803$	−1.49644e13	−1.27010
$804$	4.07114e12	0.343609
$805$	0	0
$806$	−1.47313e12	−0.122951
$807$	−2.94128e12	−0.244121
$808$	3.70365e12	0.305688
$809$	2.12869e13	1.74721	0.873604	−	0.486637i	$-0.161777\pi$
$809$	2.12869e13	1.74721	0.873604	+	0.486637i	$0.161777\pi$
$810$	0	0
$811$	2.45053e13	1.98914	0.994570	−	0.104067i	$-0.0331856\pi$
$811$	2.45053e13	1.98914	0.994570	+	0.104067i	$0.0331856\pi$
$812$	3.41554e12	0.275713
$813$	−8.19231e12	−0.657656
$814$	−3.04227e12	−0.242878
$815$	0	0
$816$	−2.38621e11	−0.0188409
$817$	−1.33978e13	−1.05204
$818$	2.10400e11	0.0164307
$819$	3.51336e11	0.0272863
$820$	0	0
$821$	−9.72826e12	−0.747293	−0.373646	−	0.927571i	$-0.621893\pi$
$821$	−9.72826e12	−0.747293	−0.373646	+	0.927571i	$0.621893\pi$
$822$	−4.55261e12	−0.347807
$823$	8.28745e11	0.0629683	0.0314841	−	0.999504i	$-0.489977\pi$
$823$	8.28745e11	0.0629683	0.0314841	+	0.999504i	$0.489977\pi$
$824$	8.11650e12	0.613333
$825$	0	0
$826$	−6.36283e12	−0.475598
$827$	−2.42842e13	−1.80530	−0.902649	−	0.430376i	$-0.858381\pi$
$827$	−2.42842e13	−1.80530	−0.902649	+	0.430376i	$0.858381\pi$
$828$	5.82809e12	0.430913
$829$	1.95570e13	1.43816	0.719078	−	0.694929i	$-0.244564\pi$
$829$	1.95570e13	1.43816	0.719078	+	0.694929i	$0.244564\pi$
$830$	0	0
$831$	1.36652e13	0.994055
$832$	2.66355e11	0.0192711
$833$	1.23471e11	0.00888507
$834$	−1.13514e13	−0.812459
$835$	0	0
$836$	−8.95437e12	−0.634025
$837$	1.03183e13	0.726682
$838$	9.27865e12	0.649960
$839$	−7.69577e12	−0.536196	−0.268098	−	0.963392i	$-0.586395\pi$
$839$	−7.69577e12	−0.536196	−0.268098	+	0.963392i	$0.586395\pi$
$840$	0	0
$841$	1.63712e13	1.12849
$842$	−2.65880e12	−0.182298
$843$	3.32512e13	2.26769
$844$	8.62664e12	0.585195
$845$	0	0
$846$	8.33793e12	0.559618
$847$	6.20305e10	0.00414124
$848$	3.92526e12	0.260667
$849$	−1.02476e13	−0.676923
$850$	0	0
$851$	9.61924e12	0.628721
$852$	4.16197e12	0.270596
$853$	2.79895e12	0.181019	0.0905097	−	0.995896i	$-0.471150\pi$
$853$	2.79895e12	0.181019	0.0905097	+	0.995896i	$0.471150\pi$
$854$	−1.97531e12	−0.127079
$855$	0	0
$856$	1.70549e12	0.108572
$857$	5.56141e12	0.352185	0.176093	−	0.984374i	$-0.443654\pi$
$857$	5.56141e12	0.352185	0.176093	+	0.984374i	$0.443654\pi$
$858$	2.10835e12	0.132816
$859$	3.51052e12	0.219989	0.109995	−	0.993932i	$-0.464917\pi$
$859$	3.51052e12	0.219989	0.109995	+	0.993932i	$0.464917\pi$
$860$	0	0
$861$	2.59631e12	0.161006
$862$	−1.21236e13	−0.747908
$863$	−1.43838e13	−0.882721	−0.441361	−	0.897330i	$-0.645504\pi$
$863$	−1.43838e13	−0.882721	−0.441361	+	0.897330i	$0.645504\pi$
$864$	−1.86565e12	−0.113898
$865$	0	0
$866$	−1.71976e13	−1.03905
$867$	2.00820e13	1.20704
$868$	3.56460e12	0.213144
$869$	2.42399e13	1.44192
$870$	0	0
$871$	−1.48514e12	−0.0874353
$872$	5.19868e12	0.304487
$873$	−6.98664e12	−0.407103
$874$	2.83125e13	1.64126
$875$	0	0
$876$	1.33387e13	0.765325
$877$	6.34278e12	0.362061	0.181030	−	0.983477i	$-0.442057\pi$
$877$	6.34278e12	0.362061	0.181030	+	0.983477i	$0.442057\pi$
$878$	−2.72670e12	−0.154850
$879$	−8.27463e12	−0.467519
$880$	0	0
$881$	−2.89282e13	−1.61782	−0.808910	−	0.587933i	$-0.799942\pi$
$881$	−2.89282e13	−1.61782	−0.808910	+	0.587933i	$0.799942\pi$
$882$	−8.50147e11	−0.0473026
$883$	7.17154e12	0.396999	0.198500	−	0.980101i	$-0.436393\pi$
$883$	7.17154e12	0.396999	0.198500	+	0.980101i	$0.436393\pi$
$884$	8.70482e10	0.00479430
$885$	0	0
$886$	1.96699e13	1.07238
$887$	1.68020e13	0.911389	0.455695	−	0.890136i	$-0.349391\pi$
$887$	1.68020e13	0.911389	0.455695	+	0.890136i	$0.349391\pi$
$888$	2.71177e12	0.146351
$889$	−1.30682e13	−0.701711
$890$	0	0
$891$	−2.36252e13	−1.25582
$892$	9.91254e12	0.524255
$893$	4.05052e13	2.13147
$894$	−1.26234e12	−0.0660933
$895$	0	0
$896$	−6.44514e11	−0.0334077
$897$	−6.66633e12	−0.343812
$898$	1.16127e12	0.0595921
$899$	3.22260e13	1.64546
$900$	0	0
$901$	1.28282e12	0.0648493
$902$	4.96900e12	0.249942
$903$	−7.63331e12	−0.382048
$904$	−1.16133e13	−0.578359
$905$	0	0
$906$	1.99085e13	0.981660
$907$	1.87924e13	0.922038	0.461019	−	0.887390i	$-0.347484\pi$
$907$	1.87924e13	0.922038	0.461019	+	0.887390i	$0.347484\pi$
$908$	−1.96867e13	−0.961139
$909$	−8.33412e12	−0.404876
$910$	0	0
$911$	−3.39496e13	−1.63306	−0.816529	−	0.577304i	$-0.804105\pi$
$911$	−3.39496e13	−1.63306	−0.816529	+	0.577304i	$0.804105\pi$
$912$	7.98161e12	0.382044
$913$	1.81375e13	0.863890
$914$	−1.06267e13	−0.503666
$915$	0	0
$916$	1.11389e13	0.522773
$917$	−1.54638e12	−0.0722195
$918$	−6.09717e11	−0.0283359
$919$	1.03287e13	0.477667	0.238833	−	0.971061i	$-0.423235\pi$
$919$	1.03287e13	0.477667	0.238833	+	0.971061i	$0.423235\pi$
$920$	0	0
$921$	4.67803e13	2.14237
$922$	1.94401e13	0.885952
$923$	−1.51828e12	−0.0688563
$924$	−5.10169e12	−0.230245
$925$	0	0
$926$	4.70205e12	0.210154
$927$	−1.82641e13	−0.812344
$928$	−5.82675e12	−0.257906
$929$	−2.79767e13	−1.23233	−0.616163	−	0.787619i	$-0.711314\pi$
$929$	−2.79767e13	−1.23233	−0.616163	+	0.787619i	$0.711314\pi$
$930$	0	0
$931$	−4.12996e12	−0.180166
$932$	5.30030e12	0.230107
$933$	−1.90947e13	−0.824984
$934$	7.56980e12	0.325479
$935$	0	0
$936$	−5.99364e11	−0.0255240
$937$	−3.31649e13	−1.40556	−0.702782	−	0.711406i	$-0.748059\pi$
$937$	−3.31649e13	−1.40556	−0.702782	+	0.711406i	$0.748059\pi$
$938$	3.59368e12	0.151575
$939$	−1.80437e13	−0.757411
$940$	0	0
$941$	3.19309e13	1.32757	0.663786	−	0.747923i	$-0.268949\pi$
$941$	3.19309e13	1.32757	0.663786	+	0.747923i	$0.268949\pi$
$942$	1.20510e13	0.498646
$943$	−1.57113e13	−0.647009
$944$	1.08547e13	0.444881
$945$	0	0
$946$	−1.46092e13	−0.593082
$947$	−3.56921e13	−1.44210	−0.721052	−	0.692881i	$-0.756341\pi$
$947$	−3.56921e13	−1.44210	−0.721052	+	0.692881i	$0.756341\pi$
$948$	−2.16066e13	−0.868856
$949$	−4.86594e12	−0.194746
$950$	0	0
$951$	3.93308e11	0.0155927
$952$	−2.10635e11	−0.00831122
$953$	3.04923e13	1.19749	0.598745	−	0.800940i	$-0.295666\pi$
$953$	3.04923e13	1.19749	0.598745	+	0.800940i	$0.295666\pi$
$954$	−8.83279e12	−0.345247
$955$	0	0
$956$	5.53524e12	0.214326
$957$	−4.61221e13	−1.77748
$958$	3.29512e13	1.26394
$959$	−4.01869e12	−0.153427
$960$	0	0
$961$	7.19282e12	0.272047
$962$	−9.89248e11	−0.0372406
$963$	−3.83776e12	−0.143800
$964$	1.73514e13	0.647124
$965$	0	0
$966$	1.61309e13	0.596020
$967$	3.45533e13	1.27078	0.635389	−	0.772192i	$-0.280840\pi$
$967$	3.45533e13	1.27078	0.635389	+	0.772192i	$0.280840\pi$
$968$	−1.05821e11	−0.00387377
$969$	2.60849e12	0.0950457
$970$	0	0
$971$	−2.06708e13	−0.746225	−0.373112	−	0.927786i	$-0.621709\pi$
$971$	−2.06708e13	−0.746225	−0.373112	+	0.927786i	$0.621709\pi$
$972$	1.20935e13	0.434563
$973$	−1.00201e13	−0.358397
$974$	−1.95595e13	−0.696374
$975$	0	0
$976$	3.36980e12	0.118872
$977$	−1.78789e11	−0.00627791	−0.00313896	−	0.999995i	$-0.500999\pi$
$977$	−1.78789e11	−0.00627791	−0.00313896	+	0.999995i	$0.500999\pi$
$978$	3.63515e13	1.27057
$979$	−8.07952e12	−0.281102
$980$	0	0
$981$	−1.16983e13	−0.403285
$982$	−3.17159e12	−0.108837
$983$	1.42657e13	0.487305	0.243652	−	0.969863i	$-0.421654\pi$
$983$	1.42657e13	0.487305	0.243652	+	0.969863i	$0.421654\pi$
$984$	−4.42919e12	−0.150608
$985$	0	0
$986$	−1.90426e12	−0.0641622
$987$	2.30776e13	0.774038
$988$	−2.91167e12	−0.0972157
$989$	4.61922e13	1.53527
$990$	0	0
$991$	−2.71296e13	−0.893537	−0.446768	−	0.894650i	$-0.647425\pi$
$991$	−2.71296e13	−0.893537	−0.446768	+	0.894650i	$0.647425\pi$
$992$	−6.08106e12	−0.199378
$993$	3.67515e13	1.19951
$994$	3.67386e12	0.119367
$995$	0	0
$996$	−1.61671e13	−0.520554
$997$	3.86723e12	0.123957	0.0619787	−	0.998077i	$-0.480259\pi$
$997$	3.86723e12	0.123957	0.0619787	+	0.998077i	$0.480259\pi$
$998$	4.81191e12	0.153543
$999$	6.92905e12	0.220104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.10.a.a.1.1		1
5.2	odd	4		350.10.c.d.99.1		2
5.3	odd	4		350.10.c.d.99.2		2
5.4	even	2		14.10.a.b.1.1	✓	1
15.14	odd	2		126.10.a.a.1.1		1
20.19	odd	2		112.10.a.a.1.1		1
35.4	even	6		98.10.c.a.79.1		2
35.9	even	6		98.10.c.a.67.1		2
35.19	odd	6		98.10.c.d.67.1		2
35.24	odd	6		98.10.c.d.79.1		2
35.34	odd	2		98.10.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.10.a.b.1.1	✓	1	5.4	even	2
98.10.a.b.1.1		1	35.34	odd	2
98.10.c.a.67.1		2	35.9	even	6
98.10.c.a.79.1		2	35.4	even	6
98.10.c.d.67.1		2	35.19	odd	6
98.10.c.d.79.1		2	35.24	odd	6
112.10.a.a.1.1		1	20.19	odd	2
126.10.a.a.1.1		1	15.14	odd	2
350.10.a.a.1.1		1	1.1	even	1	trivial
350.10.c.d.99.1		2	5.2	odd	4
350.10.c.d.99.2		2	5.3	odd	4

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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more