LMFDB - Embedded newform 126.10.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,10,Mod(1,126)] 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("126.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	126.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-16,0,256,-544] 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:	$64.8945153566$
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	$\chi$	$=$	126.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-16.0000 q^{2} +256.000 q^{4} -544.000 q^{5} -2401.00 q^{7} -4096.00 q^{8} +8704.00 q^{10} -48824.0 q^{11} -15876.0 q^{13} +38416.0 q^{14} +65536.0 q^{16} +21418.0 q^{17} -716410. q^{19} -139264. q^{20} +781184. q^{22} +2.47000e6 q^{23} -1.65719e6 q^{25} +254016. q^{26} -614656. q^{28} -5.55683e6 q^{29} +5.79935e6 q^{31} -1.04858e6 q^{32} -342688. q^{34} +1.30614e6 q^{35} -3.89443e6 q^{37} +1.14626e7 q^{38} +2.22822e6 q^{40} +6.36086e6 q^{41} -1.87013e7 q^{43} -1.24989e7 q^{44} -3.95200e7 q^{46} -5.65391e7 q^{47} +5.76480e6 q^{49} +2.65150e7 q^{50} -4.06426e6 q^{52} +5.98947e7 q^{53} +2.65603e7 q^{55} +9.83450e6 q^{56} +8.89092e7 q^{58} -1.65630e8 q^{59} +5.14190e7 q^{61} -9.27896e7 q^{62} +1.67772e7 q^{64} +8.63654e6 q^{65} +9.35465e7 q^{67} +5.48301e6 q^{68} -2.08983e7 q^{70} +9.56335e7 q^{71} +3.06496e8 q^{73} +6.23109e7 q^{74} -1.83401e8 q^{76} +1.17226e8 q^{77} +4.96474e8 q^{79} -3.56516e7 q^{80} -1.01774e8 q^{82} +3.71487e8 q^{83} -1.16514e7 q^{85} +2.99221e8 q^{86} +1.99983e8 q^{88} +1.65483e8 q^{89} +3.81183e7 q^{91} +6.32320e8 q^{92} +9.04625e8 q^{94} +3.89727e8 q^{95} +7.58017e8 q^{97} -9.22368e7 q^{98} +O(q^{100})\)

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
$3$	0	0
$3$	0	0
$4$	256.000	0.500000
$5$	−544.000	−0.389255	−0.194627	−	0.980877i	$-0.562350\pi$
$5$	−544.000	−0.389255	−0.194627	+	0.980877i	$0.562350\pi$
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	−4096.00	−0.353553
$9$	0	0
$10$	8704.00	0.275245
$11$	−48824.0	−1.00546	−0.502732	−	0.864442i	$-0.667672\pi$
$11$	−48824.0	−1.00546	−0.502732	+	0.864442i	$0.667672\pi$
$12$	0	0
$13$	−15876.0	−0.154169	−0.0770843	−	0.997025i	$-0.524561\pi$
$13$	−15876.0	−0.154169	−0.0770843	+	0.997025i	$0.524561\pi$
$14$	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	$0.490100\pi$
$18$	0	0
$19$	−716410.	−1.26116	−0.630580	−	0.776124i	$-0.717183\pi$
$19$	−716410.	−1.26116	−0.630580	+	0.776124i	$0.717183\pi$
$20$	−139264.	−0.194627
$21$	0	0
$22$	781184.	0.710970
$23$	2.47000e6	1.84044	0.920220	−	0.391401i	$-0.128010\pi$
$23$	2.47000e6	1.84044	0.920220	+	0.391401i	$0.128010\pi$
$24$	0	0
$25$	−1.65719e6	−0.848481
$26$	254016.	0.109014
$27$	0	0
$28$	−614656.	−0.188982
$29$	−5.55683e6	−1.45893	−0.729467	−	0.684016i	$-0.760232\pi$
$29$	−5.55683e6	−1.45893	−0.729467	+	0.684016i	$0.760232\pi$
$30$	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
$33$	0	0
$34$	−342688.	−0.0439788
$35$	1.30614e6	0.147124
$36$	0	0
$37$	−3.89443e6	−0.341614	−0.170807	−	0.985304i	$-0.554638\pi$
$37$	−3.89443e6	−0.341614	−0.170807	+	0.985304i	$0.554638\pi$
$38$	1.14626e7	0.891775
$39$	0	0
$40$	2.22822e6	0.137622
$41$	6.36086e6	0.351551	0.175776	−	0.984430i	$-0.443757\pi$
$41$	6.36086e6	0.351551	0.175776	+	0.984430i	$0.443757\pi$
$42$	0	0
$43$	−1.87013e7	−0.834187	−0.417094	−	0.908863i	$-0.636951\pi$
$43$	−1.87013e7	−0.834187	−0.417094	+	0.908863i	$0.636951\pi$
$44$	−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	$0.820425\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	2.65150e7	0.599967
$51$	0	0
$52$	−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	$0.325434\pi$
$54$	0	0
$55$	2.65603e7	0.391381
$56$	9.83450e6	0.133631
$57$	0	0
$58$	8.89092e7	1.03162
$59$	−1.65630e8	−1.77952	−0.889762	−	0.456424i	$-0.849130\pi$
$59$	−1.65630e8	−1.77952	−0.889762	+	0.456424i	$0.849130\pi$
$60$	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
$63$	0	0
$64$	1.67772e7	0.125000
$65$	8.63654e6	0.0600109
$66$	0	0
$67$	9.35465e7	0.567141	0.283570	−	0.958951i	$-0.408481\pi$
$67$	9.35465e7	0.567141	0.283570	+	0.958951i	$0.408481\pi$
$68$	5.48301e6	0.0310977
$69$	0	0
$70$	−2.08983e7	−0.104033
$71$	9.56335e7	0.446630	0.223315	−	0.974746i	$-0.428312\pi$
$71$	9.56335e7	0.446630	0.223315	+	0.974746i	$0.428312\pi$
$72$	0	0
$73$	3.06496e8	1.26320	0.631601	−	0.775294i	$-0.282398\pi$
$73$	3.06496e8	1.26320	0.631601	+	0.775294i	$0.282398\pi$
$74$	6.23109e7	0.241558
$75$	0	0
$76$	−1.83401e8	−0.630580
$77$	1.17226e8	0.380029
$78$	0	0
$79$	4.96474e8	1.43408	0.717042	−	0.697030i	$-0.245495\pi$
$79$	4.96474e8	1.43408	0.717042	+	0.697030i	$0.245495\pi$
$80$	−3.56516e7	−0.0973137
$81$	0	0
$82$	−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	$0.358655\pi$
$84$	0	0
$85$	−1.16514e7	−0.0242099
$86$	2.99221e8	0.589860
$87$	0	0
$88$	1.99983e8	0.355485
$89$	1.65483e8	0.279574	0.139787	−	0.990182i	$-0.455358\pi$
$89$	1.65483e8	0.279574	0.139787	+	0.990182i	$0.455358\pi$
$90$	0	0
$91$	3.81183e7	0.0582703
$92$	6.32320e8	0.920220
$93$	0	0
$94$	9.04625e8	1.19507
$95$	3.89727e8	0.490913
$96$	0	0
$97$	7.58017e8	0.869373	0.434686	−	0.900582i	$-0.356859\pi$
$97$	7.58017e8	0.869373	0.434686	+	0.900582i	$0.356859\pi$
$98$	−9.22368e7	−0.101015
$99$	0	0
$100$	−4.24240e8	−0.424240
$101$	9.04212e8	0.864618	0.432309	−	0.901726i	$-0.357699\pi$
$101$	9.04212e8	0.864618	0.432309	+	0.901726i	$0.357699\pi$
$102$	0	0
$103$	1.98157e9	1.73477	0.867384	−	0.497639i	$-0.165800\pi$
$103$	1.98157e9	1.73477	0.867384	+	0.497639i	$0.165800\pi$
$104$	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	$0.549069\pi$
$108$	0	0
$109$	−1.26921e9	−0.861220	−0.430610	−	0.902538i	$-0.641701\pi$
$109$	−1.26921e9	−0.861220	−0.430610	+	0.902538i	$0.641701\pi$
$110$	−4.24964e8	−0.276748
$111$	0	0
$112$	−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	$0.195126\pi$
$114$	0	0
$115$	−1.34368e9	−0.716400
$116$	−1.42255e9	−0.729467
$117$	0	0
$118$	2.65007e9	1.25831
$119$	−5.14246e7	−0.0235077
$120$	0	0
$121$	2.58353e7	0.0109567
$122$	−8.22704e8	−0.336221
$123$	0	0
$124$	1.48463e9	0.563925
$125$	1.96401e9	0.719530
$126$	0	0
$127$	5.44282e9	1.85655	0.928277	−	0.371889i	$-0.121290\pi$
$127$	5.44282e9	1.85655	0.928277	+	0.371889i	$0.121290\pi$
$128$	−2.68435e8	−0.0883883
$129$	0	0
$130$	−1.38185e8	−0.0424341
$131$	6.44057e8	0.191075	0.0955374	−	0.995426i	$-0.469543\pi$
$131$	6.44057e8	0.191075	0.0955374	+	0.995426i	$0.469543\pi$
$132$	0	0
$133$	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	$0.565058\pi$
$138$	0	0
$139$	−4.17330e9	−0.948229	−0.474115	−	0.880463i	$-0.657232\pi$
$139$	−4.17330e9	−0.948229	−0.474115	+	0.880463i	$0.657232\pi$
$140$	3.34373e8	0.0735622
$141$	0	0
$142$	−1.53014e9	−0.315815
$143$	7.75130e8	0.155011
$144$	0	0
$145$	3.02291e9	0.567897
$146$	−4.90394e9	−0.893218
$147$	0	0
$148$	−9.96974e8	−0.170807
$149$	4.64096e8	0.0771382	0.0385691	−	0.999256i	$-0.487720\pi$
$149$	4.64096e8	0.0771382	0.0385691	+	0.999256i	$0.487720\pi$
$150$	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
$153$	0	0
$154$	−1.87562e9	−0.268721
$155$	−3.15485e9	−0.439021
$156$	0	0
$157$	−4.43050e9	−0.581975	−0.290987	−	0.956727i	$-0.593984\pi$
$157$	−4.43050e9	−0.581975	−0.290987	+	0.956727i	$0.593984\pi$
$158$	−7.94359e9	−1.01405
$159$	0	0
$160$	5.70425e8	0.0688112
$161$	−5.93047e9	−0.695621
$162$	0	0
$163$	−1.33645e10	−1.48289	−0.741446	−	0.671013i	$-0.765860\pi$
$163$	−1.33645e10	−1.48289	−0.741446	+	0.671013i	$0.765860\pi$
$164$	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	$0.287496\pi$
$168$	0	0
$169$	−1.03525e10	−0.976232
$170$	1.86422e8	0.0171190
$171$	0	0
$172$	−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	$0.646325\pi$
$174$	0	0
$175$	3.97891e9	0.320696
$176$	−3.19973e9	−0.251366
$177$	0	0
$178$	−2.64772e9	−0.197689
$179$	4.04391e9	0.294417	0.147208	−	0.989105i	$-0.452971\pi$
$179$	4.04391e9	0.294417	0.147208	+	0.989105i	$0.452971\pi$
$180$	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
$183$	0	0
$184$	−1.01171e10	−0.650694
$185$	2.11857e9	0.132975
$186$	0	0
$187$	−1.04571e9	−0.0625353
$188$	−1.44740e10	−0.845042
$189$	0	0
$190$	−6.23563e9	−0.347128
$191$	3.81947e9	0.207660	0.103830	−	0.994595i	$-0.466890\pi$
$191$	3.81947e9	0.207660	0.103830	+	0.994595i	$0.466890\pi$
$192$	0	0
$193$	−2.41193e10	−1.25129	−0.625644	−	0.780109i	$-0.715164\pi$
$193$	−2.41193e10	−1.25129	−0.625644	+	0.780109i	$0.715164\pi$
$194$	−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	$0.404622\pi$
$198$	0	0
$199$	−2.93127e9	−0.132500	−0.0662502	−	0.997803i	$-0.521104\pi$
$199$	−2.93127e9	−0.132500	−0.0662502	+	0.997803i	$0.521104\pi$
$200$	6.78785e9	0.299983
$201$	0	0
$202$	−1.44674e10	−0.611377
$203$	1.33419e10	0.551425
$204$	0	0
$205$	−3.46031e9	−0.136843
$206$	−3.17051e10	−1.22667
$207$	0	0
$208$	−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
$213$	0	0
$214$	6.66206e9	0.217143
$215$	1.01735e10	0.324711
$216$	0	0
$217$	−1.39242e10	−0.426287
$218$	2.03073e10	0.608974
$219$	0	0
$220$	6.79943e9	0.195691
$221$	−3.40032e8	−0.00958859
$222$	0	0
$223$	−3.87208e10	−1.04851	−0.524255	−	0.851561i	$-0.675656\pi$
$223$	−3.87208e10	−1.04851	−0.524255	+	0.851561i	$0.675656\pi$
$224$	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	$0.910970\pi$
$228$	0	0
$229$	4.35114e10	1.04555	0.522773	−	0.852472i	$-0.324897\pi$
$229$	4.35114e10	1.04555	0.522773	+	0.852472i	$0.324897\pi$
$230$	2.14989e10	0.506571
$231$	0	0
$232$	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	$0.426093\pi$
$234$	0	0
$235$	3.07573e10	0.657873
$236$	−4.24012e10	−0.889762
$237$	0	0
$238$	8.22794e8	0.0166224
$239$	−2.16220e10	−0.428653	−0.214326	−	0.976762i	$-0.568756\pi$
$239$	−2.16220e10	−0.428653	−0.214326	+	0.976762i	$0.568756\pi$
$240$	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
$243$	0	0
$244$	1.31633e10	0.237744
$245$	−3.13605e9	−0.0556078
$246$	0	0
$247$	1.13737e10	0.194431
$248$	−2.37541e10	−0.398755
$249$	0	0
$250$	−3.14242e10	−0.508784
$251$	−4.87895e9	−0.0775881	−0.0387940	−	0.999247i	$-0.512352\pi$
$251$	−4.87895e9	−0.0775881	−0.0387940	+	0.999247i	$0.512352\pi$
$252$	0	0
$253$	−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	$0.437000\pi$
$258$	0	0
$259$	9.35053e9	0.129118
$260$	2.21096e9	0.0300054
$261$	0	0
$262$	−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	$0.454182\pi$
$264$	0	0
$265$	−3.25827e10	−0.405864
$266$	−2.75216e10	−0.337059
$267$	0	0
$268$	2.39479e10	0.283570
$269$	−1.73017e10	−0.201466	−0.100733	−	0.994913i	$-0.532119\pi$
$269$	−1.73017e10	−0.201466	−0.100733	+	0.994913i	$0.532119\pi$
$270$	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
$273$	0	0
$274$	2.67801e10	0.287035
$275$	8.09106e10	0.853116
$276$	0	0
$277$	8.03834e10	0.820365	0.410183	−	0.912003i	$-0.365465\pi$
$277$	8.03834e10	0.820365	0.410183	+	0.912003i	$0.365465\pi$
$278$	6.67728e10	0.670499
$279$	0	0
$280$	−5.34997e9	−0.0520163
$281$	1.95595e11	1.87146	0.935729	−	0.352719i	$-0.114743\pi$
$281$	1.95595e11	1.87146	0.935729	+	0.352719i	$0.114743\pi$
$282$	0	0
$283$	−6.02802e10	−0.558645	−0.279322	−	0.960197i	$-0.590110\pi$
$283$	−6.02802e10	−0.558645	−0.279322	+	0.960197i	$0.590110\pi$
$284$	2.44822e10	0.223315
$285$	0	0
$286$	−1.24021e10	−0.109609
$287$	−1.52724e10	−0.132874
$288$	0	0
$289$	−1.18129e11	−0.996132
$290$	−4.83666e10	−0.401564
$291$	0	0
$292$	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	$0.438206\pi$
$294$	0	0
$295$	9.01025e10	0.692688
$296$	1.59516e10	0.120779
$297$	0	0
$298$	−7.42553e9	−0.0545449
$299$	−3.92137e10	−0.283738
$300$	0	0
$301$	4.49018e10	0.315293
$302$	−1.17109e11	−0.810136
$303$	0	0
$304$	−4.69506e10	−0.315290
$305$	−2.79719e10	−0.185086
$306$	0	0
$307$	2.75178e11	1.76804	0.884018	−	0.467453i	$-0.154828\pi$
$307$	2.75178e11	1.76804	0.884018	+	0.467453i	$0.154828\pi$
$308$	3.00100e10	0.190015
$309$	0	0
$310$	5.04775e10	0.310435
$311$	−1.12322e11	−0.680835	−0.340418	−	0.940274i	$-0.610568\pi$
$311$	−1.12322e11	−0.680835	−0.340418	+	0.940274i	$0.610568\pi$
$312$	0	0
$313$	−1.06140e11	−0.625069	−0.312535	−	0.949906i	$-0.601178\pi$
$313$	−1.06140e11	−0.625069	−0.312535	+	0.949906i	$0.601178\pi$
$314$	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	$0.502048\pi$
$318$	0	0
$319$	2.71306e11	1.46691
$320$	−9.12681e9	−0.0486568
$321$	0	0
$322$	9.48875e10	0.491878
$323$	−1.53441e10	−0.0784385
$324$	0	0
$325$	2.63095e10	0.130809
$326$	2.13832e11	1.04856
$327$	0	0
$328$	−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
$333$	0	0
$334$	−1.99130e11	−0.875544
$335$	−5.08893e10	−0.220762
$336$	0	0
$337$	5.00291e10	0.211294	0.105647	−	0.994404i	$-0.466309\pi$
$337$	5.00291e10	0.211294	0.105647	+	0.994404i	$0.466309\pi$
$338$	1.65639e11	0.690300
$339$	0	0
$340$	−2.98276e9	−0.0121049
$341$	−2.83147e11	−1.13401
$342$	0	0
$343$	−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	$0.388974\pi$
$348$	0	0
$349$	2.74666e11	0.991039	0.495520	−	0.868597i	$-0.334978\pi$
$349$	2.74666e11	0.991039	0.495520	+	0.868597i	$0.334978\pi$
$350$	−6.36626e10	−0.226766
$351$	0	0
$352$	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	$0.412683\pi$
$354$	0	0
$355$	−5.20246e10	−0.173853
$356$	4.23635e10	0.139787
$357$	0	0
$358$	−6.47025e10	−0.208184
$359$	−3.40759e11	−1.08274	−0.541368	−	0.840786i	$-0.682093\pi$
$359$	−3.40759e11	−1.08274	−0.541368	+	0.840786i	$0.682093\pi$
$360$	0	0
$361$	1.90556e11	0.590526
$362$	−1.99576e11	−0.610829
$363$	0	0
$364$	9.75828e9	0.0291351
$365$	−1.66734e11	−0.491707
$366$	0	0
$367$	−6.10216e11	−1.75584	−0.877922	−	0.478803i	$-0.841071\pi$
$367$	−6.10216e11	−1.75584	−0.877922	+	0.478803i	$0.841071\pi$
$368$	1.61874e11	0.460110
$369$	0	0
$370$	−3.38971e10	−0.0940275
$371$	−1.43807e11	−0.394092
$372$	0	0
$373$	4.34930e11	1.16340	0.581701	−	0.813402i	$-0.302387\pi$
$373$	4.34930e11	1.16340	0.581701	+	0.813402i	$0.302387\pi$
$374$	1.67314e10	0.0442191
$375$	0	0
$376$	2.31584e11	0.597535
$377$	8.82202e10	0.224922
$378$	0	0
$379$	−7.30677e11	−1.81907	−0.909534	−	0.415630i	$-0.863561\pi$
$379$	−7.30677e11	−1.81907	−0.909534	+	0.415630i	$0.863561\pi$
$380$	9.97701e10	0.245456
$381$	0	0
$382$	−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	$0.580633\pi$
$384$	0	0
$385$	−6.37712e10	−0.147928
$386$	3.85909e11	0.884794
$387$	0	0
$388$	1.94052e11	0.434686
$389$	−7.21857e9	−0.0159837	−0.00799186	−	0.999968i	$-0.502544\pi$
$389$	−7.21857e9	−0.0159837	−0.00799186	+	0.999968i	$0.502544\pi$
$390$	0	0
$391$	5.29025e10	0.114467
$392$	−2.36126e10	−0.0505076
$393$	0	0
$394$	−1.99677e11	−0.417441
$395$	−2.70082e11	−0.558224
$396$	0	0
$397$	−6.99387e11	−1.41306	−0.706529	−	0.707684i	$-0.749740\pi$
$397$	−6.99387e11	−1.41306	−0.706529	+	0.707684i	$0.749740\pi$
$398$	4.69004e10	0.0936920
$399$	0	0
$400$	−1.08606e11	−0.212120
$401$	−6.40644e11	−1.23728	−0.618638	−	0.785676i	$-0.712315\pi$
$401$	−6.40644e11	−1.23728	−0.618638	+	0.785676i	$0.712315\pi$
$402$	0	0
$403$	−9.20704e10	−0.173879
$404$	2.31478e11	0.432309
$405$	0	0
$406$	−2.13471e11	−0.389917
$407$	1.90142e11	0.343481
$408$	0	0
$409$	−1.31500e10	−0.0232365	−0.0116182	−	0.999933i	$-0.503698\pi$
$409$	−1.31500e10	−0.0232365	−0.0116182	+	0.999933i	$0.503698\pi$
$410$	5.53649e10	0.0967625
$411$	0	0
$412$	5.07281e11	0.867384
$413$	3.97677e11	0.672597
$414$	0	0
$415$	−2.02089e11	−0.334446
$416$	1.66472e10	0.0272534
$417$	0	0
$418$	−5.59648e11	−0.896647
$419$	5.79915e11	0.919182	0.459591	−	0.888131i	$-0.347996\pi$
$419$	5.79915e11	0.919182	0.459591	+	0.888131i	$0.347996\pi$
$420$	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
$423$	0	0
$424$	−2.45329e11	−0.368639
$425$	−3.54937e10	−0.0527717
$426$	0	0
$427$	−1.23457e11	−0.179718
$428$	−1.06593e11	−0.153544
$429$	0	0
$430$	−1.62776e11	−0.229606
$431$	−7.57723e11	−1.05770	−0.528850	−	0.848715i	$-0.677377\pi$
$431$	−7.57723e11	−1.05770	−0.528850	+	0.848715i	$0.677377\pi$
$432$	0	0
$433$	−1.07485e12	−1.46944	−0.734719	−	0.678371i	$-0.762686\pi$
$433$	−1.07485e12	−1.46944	−0.734719	+	0.678371i	$0.762686\pi$
$434$	2.22788e11	0.301431
$435$	0	0
$436$	−3.24918e11	−0.430610
$437$	−1.76953e12	−2.32109
$438$	0	0
$439$	1.70418e11	0.218991	0.109496	−	0.993987i	$-0.465076\pi$
$439$	1.70418e11	0.218991	0.109496	+	0.993987i	$0.465076\pi$
$440$	−1.08791e11	−0.138374
$441$	0	0
$442$	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	$0.773965\pi$
$444$	0	0
$445$	−9.00225e10	−0.108826
$446$	6.19533e11	0.741409
$447$	0	0
$448$	−4.02821e10	−0.0472456
$449$	7.25792e10	0.0842759	0.0421380	−	0.999112i	$-0.486583\pi$
$449$	7.25792e10	0.0842759	0.0421380	+	0.999112i	$0.486583\pi$
$450$	0	0
$451$	−3.10563e11	−0.353472
$452$	7.25831e11	0.817923
$453$	0	0
$454$	1.23042e12	1.35926
$455$	−2.07363e10	−0.0226820
$456$	0	0
$457$	−6.64172e11	−0.712291	−0.356146	−	0.934431i	$-0.615909\pi$
$457$	−6.64172e11	−0.712291	−0.356146	+	0.934431i	$0.615909\pi$
$458$	−6.96183e11	−0.739313
$459$	0	0
$460$	−3.43982e11	−0.358200
$461$	1.21501e12	1.25293	0.626463	−	0.779451i	$-0.284502\pi$
$461$	1.21501e12	1.25293	0.626463	+	0.779451i	$0.284502\pi$
$462$	0	0
$463$	2.93878e11	0.297202	0.148601	−	0.988897i	$-0.452523\pi$
$463$	2.93878e11	0.297202	0.148601	+	0.988897i	$0.452523\pi$
$464$	−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	$0.573921\pi$
$468$	0	0
$469$	−2.24605e11	−0.214359
$470$	−4.92116e11	−0.465187
$471$	0	0
$472$	6.78419e11	0.629157
$473$	9.13072e11	0.838745
$474$	0	0
$475$	1.18723e12	1.07007
$476$	−1.31647e10	−0.0117538
$477$	0	0
$478$	3.45952e11	0.303103
$479$	2.05945e12	1.78748	0.893742	−	0.448582i	$-0.148071\pi$
$479$	2.05945e12	1.78748	0.893742	+	0.448582i	$0.148071\pi$
$480$	0	0
$481$	6.18280e10	0.0526662
$482$	−1.08446e12	−0.915172
$483$	0	0
$484$	6.61383e9	0.00547834
$485$	−4.12361e11	−0.338407
$486$	0	0
$487$	−1.22247e12	−0.984821	−0.492411	−	0.870363i	$-0.663884\pi$
$487$	−1.22247e12	−0.984821	−0.492411	+	0.870363i	$0.663884\pi$
$488$	−2.10612e11	−0.168110
$489$	0	0
$490$	5.01768e10	0.0393207
$491$	−1.98225e11	−0.153918	−0.0769592	−	0.997034i	$-0.524521\pi$
$491$	−1.98225e11	−0.153918	−0.0769592	+	0.997034i	$0.524521\pi$
$492$	0	0
$493$	−1.19016e11	−0.0907391
$494$	−1.81980e11	−0.137484
$495$	0	0
$496$	3.80066e11	0.281963
$497$	−2.29616e11	−0.168810
$498$	0	0
$499$	−3.00745e11	−0.217143	−0.108571	−	0.994089i	$-0.534628\pi$
$499$	−3.00745e11	−0.217143	−0.108571	+	0.994089i	$0.534628\pi$
$500$	5.02787e11	0.359765
$501$	0	0
$502$	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	$0.536686\pi$
$504$	0	0
$505$	−4.91891e11	−0.336556
$506$	1.92952e12	1.30850
$507$	0	0
$508$	1.39336e12	0.928277
$509$	6.32399e10	0.0417600	0.0208800	−	0.999782i	$-0.493353\pi$
$509$	6.32399e10	0.0417600	0.0208800	+	0.999782i	$0.493353\pi$
$510$	0	0
$511$	−7.35898e11	−0.477445
$512$	−6.87195e10	−0.0441942
$513$	0	0
$514$	−4.40046e11	−0.278076
$515$	−1.07797e12	−0.675267
$516$	0	0
$517$	2.76046e12	1.69932
$518$	−1.49608e11	−0.0913003
$519$	0	0
$520$	−3.53753e10	−0.0212170
$521$	1.88994e12	1.12377	0.561886	−	0.827215i	$-0.310076\pi$
$521$	1.88994e12	1.12377	0.561886	+	0.827215i	$0.310076\pi$
$522$	0	0
$523$	8.95863e11	0.523581	0.261791	−	0.965125i	$-0.415687\pi$
$523$	8.95863e11	0.523581	0.261791	+	0.965125i	$0.415687\pi$
$524$	1.64879e11	0.0955374
$525$	0	0
$526$	−3.56152e11	−0.202861
$527$	1.24210e11	0.0701472
$528$	0	0
$529$	4.29975e12	2.38722
$530$	5.21323e11	0.286989
$531$	0	0
$532$	4.40346e11	0.238337
$533$	−1.00985e11	−0.0541981
$534$	0	0
$535$	2.26510e11	0.119535
$536$	−3.83166e11	−0.200515
$537$	0	0
$538$	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
$543$	0	0
$544$	−2.24584e10	−0.0109947
$545$	6.90450e11	0.335234
$546$	0	0
$547$	2.73436e12	1.30591	0.652955	−	0.757396i	$-0.273529\pi$
$547$	2.73436e12	1.30591	0.652955	+	0.757396i	$0.273529\pi$
$548$	−4.28481e11	−0.202964
$549$	0	0
$550$	−1.29457e12	−0.603244
$551$	3.98097e12	1.83995
$552$	0	0
$553$	−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	$0.120140\pi$
$558$	0	0
$559$	2.96902e11	0.128606
$560$	8.55995e10	0.0367811
$561$	0	0
$562$	−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	$0.276169\pi$
$564$	0	0
$565$	−1.54239e12	−0.636761
$566$	9.64483e11	0.395021
$567$	0	0
$568$	−3.91715e11	−0.157907
$569$	2.74669e11	0.109851	0.0549256	−	0.998490i	$-0.482508\pi$
$569$	2.74669e11	0.109851	0.0549256	+	0.998490i	$0.482508\pi$
$570$	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
$573$	0	0
$574$	2.44359e11	0.0939560
$575$	−4.09326e12	−1.56158
$576$	0	0
$577$	3.76585e12	1.41440	0.707199	−	0.707014i	$-0.249958\pi$
$577$	3.76585e12	1.41440	0.707199	+	0.707014i	$0.249958\pi$
$578$	1.89007e12	0.704371
$579$	0	0
$580$	7.73866e11	0.283949
$581$	−8.91940e11	−0.324745
$582$	0	0
$583$	−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	$0.324849\pi$
$588$	0	0
$589$	−4.15471e12	−1.42240
$590$	−1.44164e12	−0.489805
$591$	0	0
$592$	−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	$0.707102\pi$
$594$	0	0
$595$	2.79750e10	0.00915047
$596$	1.18809e11	0.0385691
$597$	0	0
$598$	6.27420e11	0.200633
$599$	−4.17778e12	−1.32594	−0.662972	−	0.748644i	$-0.730705\pi$
$599$	−4.17778e12	−1.32594	−0.662972	+	0.748644i	$0.730705\pi$
$600$	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
$603$	0	0
$604$	1.87374e12	0.572853
$605$	−1.40544e10	−0.00426494
$606$	0	0
$607$	−1.58444e12	−0.473726	−0.236863	−	0.971543i	$-0.576119\pi$
$607$	−1.58444e12	−0.473726	−0.236863	+	0.971543i	$0.576119\pi$
$608$	7.51210e11	0.222944
$609$	0	0
$610$	4.47551e11	0.130876
$611$	8.97614e11	0.260558
$612$	0	0
$613$	−2.79203e11	−0.0798635	−0.0399318	−	0.999202i	$-0.512714\pi$
$613$	−2.79203e11	−0.0798635	−0.0399318	+	0.999202i	$0.512714\pi$
$614$	−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	$0.219423\pi$
$618$	0	0
$619$	−2.70437e12	−0.740385	−0.370192	−	0.928955i	$-0.620708\pi$
$619$	−2.70437e12	−0.740385	−0.370192	+	0.928955i	$0.620708\pi$
$620$	−8.07640e11	−0.219511
$621$	0	0
$622$	1.79715e12	0.481423
$623$	−3.97324e11	−0.105669
$624$	0	0
$625$	2.16828e12	0.568400
$626$	1.69823e12	0.441991
$627$	0	0
$628$	−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
$633$	0	0
$634$	3.70172e10	0.00909918
$635$	−2.96090e12	−0.722672
$636$	0	0
$637$	−9.15220e10	−0.0220241
$638$	−4.34090e12	−1.03726
$639$	0	0
$640$	1.46029e11	0.0344056
$641$	−1.38865e12	−0.324887	−0.162444	−	0.986718i	$-0.551938\pi$
$641$	−1.38865e12	−0.324887	−0.162444	+	0.986718i	$0.551938\pi$
$642$	0	0
$643$	3.09398e12	0.713786	0.356893	−	0.934145i	$-0.383836\pi$
$643$	3.09398e12	0.713786	0.356893	+	0.934145i	$0.383836\pi$
$644$	−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	$0.583602\pi$
$648$	0	0
$649$	8.08670e12	1.78925
$650$	−4.20953e11	−0.0924960
$651$	0	0
$652$	−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	$0.472199\pi$
$654$	0	0
$655$	−3.50367e11	−0.0743768
$656$	4.16865e11	0.0878878
$657$	0	0
$658$	−2.17200e12	−0.451694
$659$	−4.57355e11	−0.0944645	−0.0472323	−	0.998884i	$-0.515040\pi$
$659$	−4.57355e11	−0.0944645	−0.0472323	+	0.998884i	$0.515040\pi$
$660$	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
$663$	0	0
$664$	−1.52161e12	−0.303772
$665$	−9.35735e11	−0.185548
$666$	0	0
$667$	−1.37254e13	−2.68508
$668$	3.18608e12	0.619103
$669$	0	0
$670$	8.14229e11	0.156102
$671$	−2.51048e12	−0.478086
$672$	0	0
$673$	−8.92805e12	−1.67760	−0.838801	−	0.544439i	$-0.816743\pi$
$673$	−8.92805e12	−1.67760	−0.838801	+	0.544439i	$0.816743\pi$
$674$	−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	$0.237926\pi$
$678$	0	0
$679$	−1.82000e12	−0.328592
$680$	4.77241e10	0.00855949
$681$	0	0
$682$	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	$0.618800\pi$
$684$	0	0
$685$	9.10523e11	0.158010
$686$	2.21461e11	0.0381802
$687$	0	0
$688$	−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
$693$	0	0
$694$	−2.95370e12	−0.483336
$695$	2.27028e12	0.369103
$696$	0	0
$697$	1.36237e11	0.0218649
$698$	−4.39466e12	−0.700771
$699$	0	0
$700$	1.01860e12	0.160348
$701$	−1.88599e12	−0.294990	−0.147495	−	0.989063i	$-0.547121\pi$
$701$	−1.88599e12	−0.294990	−0.147495	+	0.989063i	$0.547121\pi$
$702$	0	0
$703$	2.79001e12	0.430831
$704$	−8.19131e11	−0.125683
$705$	0	0
$706$	−2.52886e12	−0.383092
$707$	−2.17101e12	−0.326795
$708$	0	0
$709$	−4.76210e12	−0.707767	−0.353884	−	0.935289i	$-0.615139\pi$
$709$	−4.76210e12	−0.707767	−0.353884	+	0.935289i	$0.615139\pi$
$710$	8.32394e11	0.122932
$711$	0	0
$712$	−6.77817e11	−0.0988444
$713$	1.43244e13	2.07574
$714$	0	0
$715$	−4.21671e11	−0.0603387
$716$	1.03524e12	0.147208
$717$	0	0
$718$	5.45215e12	0.765610
$719$	−5.34893e12	−0.746427	−0.373213	−	0.927746i	$-0.621744\pi$
$719$	−5.34893e12	−0.746427	−0.373213	+	0.927746i	$0.621744\pi$
$720$	0	0
$721$	−4.75774e12	−0.655681
$722$	−3.04889e12	−0.417565
$723$	0	0
$724$	3.19322e12	0.431922
$725$	9.20871e12	1.23788
$726$	0	0
$727$	9.08222e12	1.20583	0.602917	−	0.797804i	$-0.294005\pi$
$727$	9.08222e12	1.20583	0.602917	+	0.797804i	$0.294005\pi$
$728$	−1.56132e11	−0.0206016
$729$	0	0
$730$	2.66774e12	0.347689
$731$	−4.00544e11	−0.0518827
$732$	0	0
$733$	−1.20547e13	−1.54237	−0.771185	−	0.636611i	$-0.780336\pi$
$733$	−1.20547e13	−1.54237	−0.771185	+	0.636611i	$0.780336\pi$
$734$	9.76345e12	1.24157
$735$	0	0
$736$	−2.58998e12	−0.325347
$737$	−4.56731e12	−0.570239
$738$	0	0
$739$	1.08128e13	1.33364	0.666822	−	0.745217i	$-0.267654\pi$
$739$	1.08128e13	1.33364	0.666822	+	0.745217i	$0.267654\pi$
$740$	5.42354e11	0.0664875
$741$	0	0
$742$	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	$0.625754\pi$
$744$	0	0
$745$	−2.52468e11	−0.0300264
$746$	−6.95889e12	−0.822650
$747$	0	0
$748$	−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
$753$	0	0
$754$	−1.41152e12	−0.159044
$755$	−3.98170e12	−0.445971
$756$	0	0
$757$	−6.85316e12	−0.758507	−0.379253	−	0.925293i	$-0.623819\pi$
$757$	−6.85316e12	−0.758507	−0.379253	+	0.925293i	$0.623819\pi$
$758$	1.16908e13	1.28628
$759$	0	0
$760$	−1.59632e12	−0.173564
$761$	1.55520e12	0.168095	0.0840474	−	0.996462i	$-0.473215\pi$
$761$	1.55520e12	0.168095	0.0840474	+	0.996462i	$0.473215\pi$
$762$	0	0
$763$	3.04737e12	0.325510
$764$	9.77784e11	0.103830
$765$	0	0
$766$	3.37718e12	0.354425
$767$	2.62954e12	0.274347
$768$	0	0
$769$	−1.31148e12	−0.135236	−0.0676179	−	0.997711i	$-0.521540\pi$
$769$	−1.31148e12	−0.135236	−0.0676179	+	0.997711i	$0.521540\pi$
$770$	1.02034e12	0.104601
$771$	0	0
$772$	−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	$0.335305\pi$
$774$	0	0
$775$	−9.61062e12	−0.956959
$776$	−3.10484e12	−0.307370
$777$	0	0
$778$	1.15497e11	0.0113022
$779$	−4.55698e12	−0.443362
$780$	0	0
$781$	−4.66921e12	−0.449070
$782$	−8.46439e11	−0.0809404
$783$	0	0
$784$	3.77802e11	0.0357143
$785$	2.41019e12	0.226536
$786$	0	0
$787$	−4.81658e12	−0.447562	−0.223781	−	0.974639i	$-0.571840\pi$
$787$	−4.81658e12	−0.447562	−0.223781	+	0.974639i	$0.571840\pi$
$788$	3.19484e12	0.295176
$789$	0	0
$790$	4.32131e12	0.394724
$791$	−6.80750e12	−0.618292
$792$	0	0
$793$	−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	$0.609945\pi$
$798$	0	0
$799$	−1.21095e12	−0.105116
$800$	1.73769e12	0.149992
$801$	0	0
$802$	1.02503e13	0.874887
$803$	−1.49644e13	−1.27010
$804$	0	0
$805$	3.22618e12	0.270774
$806$	1.47313e12	0.122951
$807$	0	0
$808$	−3.70365e12	−0.305688
$809$	−2.12869e13	−1.74721	−0.873604	−	0.486637i	$-0.838223\pi$
$809$	−2.12869e13	−1.74721	−0.873604	+	0.486637i	$0.838223\pi$
$810$	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
$813$	0	0
$814$	−3.04227e12	−0.242878
$815$	7.27030e12	0.577223
$816$	0	0
$817$	1.33978e13	1.05204
$818$	2.10400e11	0.0164307
$819$	0	0
$820$	−8.85839e11	−0.0684214
$821$	9.72826e12	0.747293	0.373646	−	0.927571i	$-0.378107\pi$
$821$	9.72826e12	0.747293	0.373646	+	0.927571i	$0.378107\pi$
$822$	0	0
$823$	−8.28745e11	−0.0629683	−0.0314841	−	0.999504i	$-0.510023\pi$
$823$	−8.28745e11	−0.0629683	−0.0314841	+	0.999504i	$0.510023\pi$
$824$	−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	$0.858381\pi$
$828$	0	0
$829$	1.95570e13	1.43816	0.719078	−	0.694929i	$-0.244564\pi$
$829$	1.95570e13	1.43816	0.719078	+	0.694929i	$0.244564\pi$
$830$	3.23342e12	0.236489
$831$	0	0
$832$	−2.66355e11	−0.0192711
$833$	1.23471e11	0.00888507
$834$	0	0
$835$	−6.77043e12	−0.481978
$836$	8.95437e12	0.634025
$837$	0	0
$838$	−9.27865e12	−0.649960
$839$	7.69577e12	0.536196	0.268098	−	0.963392i	$-0.413605\pi$
$839$	7.69577e12	0.536196	0.268098	+	0.963392i	$0.413605\pi$
$840$	0	0
$841$	1.63712e13	1.12849
$842$	−2.65880e12	−0.182298
$843$	0	0
$844$	8.62664e12	0.585195
$845$	5.63173e12	0.380003
$846$	0	0
$847$	−6.20305e10	−0.00414124
$848$	3.92526e12	0.260667
$849$	0	0
$850$	5.67899e11	0.0373152
$851$	−9.61924e12	−0.628721
$852$	0	0
$853$	−2.79895e12	−0.181019	−0.0905097	−	0.995896i	$-0.528850\pi$
$853$	−2.79895e12	−0.181019	−0.0905097	+	0.995896i	$0.528850\pi$
$854$	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	$0.443654\pi$
$858$	0	0
$859$	3.51052e12	0.219989	0.109995	−	0.993932i	$-0.464917\pi$
$859$	3.51052e12	0.219989	0.109995	+	0.993932i	$0.464917\pi$
$860$	2.60442e12	0.162356
$861$	0	0
$862$	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	$0.645504\pi$
$864$	0	0
$865$	5.68720e12	0.345403
$866$	1.71976e13	1.03905
$867$	0	0
$868$	−3.56460e12	−0.213144
$869$	−2.42399e13	−1.44192
$870$	0	0
$871$	−1.48514e12	−0.0874353
$872$	5.19868e12	0.304487
$873$	0	0
$874$	2.83125e13	1.64126
$875$	−4.71559e12	−0.271957
$876$	0	0
$877$	−6.34278e12	−0.362061	−0.181030	−	0.983477i	$-0.557943\pi$
$877$	−6.34278e12	−0.362061	−0.181030	+	0.983477i	$0.557943\pi$
$878$	−2.72670e12	−0.154850
$879$	0	0
$880$	1.74065e12	0.0978453
$881$	2.89282e13	1.61782	0.808910	−	0.587933i	$-0.200058\pi$
$881$	2.89282e13	1.61782	0.808910	+	0.587933i	$0.200058\pi$
$882$	0	0
$883$	−7.17154e12	−0.396999	−0.198500	−	0.980101i	$-0.563607\pi$
$883$	−7.17154e12	−0.396999	−0.198500	+	0.980101i	$0.563607\pi$
$884$	−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	$0.349391\pi$
$888$	0	0
$889$	−1.30682e13	−0.701711
$890$	1.44036e12	0.0769513
$891$	0	0
$892$	−9.91254e12	−0.524255
$893$	4.05052e13	2.13147
$894$	0	0
$895$	−2.19989e12	−0.114603
$896$	6.44514e11	0.0334077
$897$	0	0
$898$	−1.16127e12	−0.0595921
$899$	−3.22260e13	−1.64546
$900$	0	0
$901$	1.28282e12	0.0648493
$902$	4.96900e12	0.249942
$903$	0	0
$904$	−1.16133e13	−0.578359
$905$	−6.78559e12	−0.336255
$906$	0	0
$907$	−1.87924e13	−0.922038	−0.461019	−	0.887390i	$-0.652516\pi$
$907$	−1.87924e13	−0.922038	−0.461019	+	0.887390i	$0.652516\pi$
$908$	−1.96867e13	−0.961139
$909$	0	0
$910$	3.31781e11	0.0160386
$911$	3.39496e13	1.63306	0.816529	−	0.577304i	$-0.195895\pi$
$911$	3.39496e13	1.63306	0.816529	+	0.577304i	$0.195895\pi$
$912$	0	0
$913$	−1.81375e13	−0.863890
$914$	1.06267e13	0.503666
$915$	0	0
$916$	1.11389e13	0.522773
$917$	−1.54638e12	−0.0722195
$918$	0	0
$919$	1.03287e13	0.477667	0.238833	−	0.971061i	$-0.423235\pi$
$919$	1.03287e13	0.477667	0.238833	+	0.971061i	$0.423235\pi$
$920$	5.50371e12	0.253286
$921$	0	0
$922$	−1.94401e13	−0.885952
$923$	−1.51828e12	−0.0688563
$924$	0	0
$925$	6.45381e12	0.289853
$926$	−4.70205e12	−0.210154
$927$	0	0
$928$	5.82675e12	0.257906
$929$	2.79767e13	1.23233	0.616163	−	0.787619i	$-0.288686\pi$
$929$	2.79767e13	1.23233	0.616163	+	0.787619i	$0.288686\pi$
$930$	0	0
$931$	−4.12996e12	−0.180166
$932$	5.30030e12	0.230107
$933$	0	0
$934$	7.56980e12	0.325479
$935$	5.68868e11	0.0243421
$936$	0	0
$937$	3.31649e13	1.40556	0.702782	−	0.711406i	$-0.251941\pi$
$937$	3.31649e13	1.40556	0.702782	+	0.711406i	$0.251941\pi$
$938$	3.59368e12	0.151575
$939$	0	0
$940$	7.87386e12	0.328937
$941$	−3.19309e13	−1.32757	−0.663786	−	0.747923i	$-0.731051\pi$
$941$	−3.19309e13	−1.32757	−0.663786	+	0.747923i	$0.731051\pi$
$942$	0	0
$943$	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	$0.756341\pi$
$948$	0	0
$949$	−4.86594e12	−0.194746
$950$	−1.89956e13	−0.756654
$951$	0	0
$952$	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	$0.295666\pi$
$954$	0	0
$955$	−2.07779e12	−0.0808327
$956$	−5.53524e12	−0.214326
$957$	0	0
$958$	−3.29512e13	−1.26394
$959$	4.01869e12	0.153427
$960$	0	0
$961$	7.19282e12	0.272047
$962$	−9.89248e11	−0.0372406
$963$	0	0
$964$	1.73514e13	0.647124
$965$	1.31209e13	0.487069
$966$	0	0
$967$	−3.45533e13	−1.27078	−0.635389	−	0.772192i	$-0.719160\pi$
$967$	−3.45533e13	−1.27078	−0.635389	+	0.772192i	$0.719160\pi$
$968$	−1.05821e11	−0.00387377
$969$	0	0
$970$	6.59778e12	0.239290
$971$	2.06708e13	0.746225	0.373112	−	0.927786i	$-0.378291\pi$
$971$	2.06708e13	0.746225	0.373112	+	0.927786i	$0.378291\pi$
$972$	0	0
$973$	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	$0.500999\pi$
$978$	0	0
$979$	−8.07952e12	−0.281102
$980$	−8.02829e11	−0.0278039
$981$	0	0
$982$	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	$0.421654\pi$
$984$	0	0
$985$	−6.78902e12	−0.229797
$986$	1.90426e12	0.0641622
$987$	0	0
$988$	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
$993$	0	0
$994$	3.67386e12	0.119367
$995$	1.59461e12	0.0515764
$996$	0	0
$997$	−3.86723e12	−0.123957	−0.0619787	−	0.998077i	$-0.519741\pi$
$997$	−3.86723e12	−0.123957	−0.0619787	+	0.998077i	$0.519741\pi$
$998$	4.81191e12	0.153543
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.10.a.a.1.1		1
3.2	odd	2		14.10.a.b.1.1	✓	1
12.11	even	2		112.10.a.a.1.1		1
15.2	even	4		350.10.c.d.99.2		2
15.8	even	4		350.10.c.d.99.1		2
15.14	odd	2		350.10.a.a.1.1		1
21.2	odd	6		98.10.c.a.67.1		2
21.5	even	6		98.10.c.d.67.1		2
21.11	odd	6		98.10.c.a.79.1		2
21.17	even	6		98.10.c.d.79.1		2
21.20	even	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	3.2	odd	2
98.10.a.b.1.1		1	21.20	even	2
98.10.c.a.67.1		2	21.2	odd	6
98.10.c.a.79.1		2	21.11	odd	6
98.10.c.d.67.1		2	21.5	even	6
98.10.c.d.79.1		2	21.17	even	6
112.10.a.a.1.1		1	12.11	even	2
126.10.a.a.1.1		1	1.1	even	1	trivial
350.10.a.a.1.1		1	15.14	odd	2
350.10.c.d.99.1		2	15.8	even	4
350.10.c.d.99.2		2	15.2	even	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:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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