LMFDB - Embedded newform 224.8.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [224,8,Mod(1,224)] 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("224.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(224, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	224.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-54,0,84,0,1715] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$69.9742457084$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 1809x^{3} + 6482x^{2} + 488753x + 1733184 $ x^5 - 2x^4 - 1809x^3 + 6482x^2 + 488753x + 1733184
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$23.3450$ of defining polynomial
Character	$\chi$	$=$	224.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-56.6900 q^{3} -323.003 q^{5} +343.000 q^{7} +1026.75 q^{9} -2501.08 q^{11} -3411.88 q^{13} +18311.1 q^{15} +23601.1 q^{17} -3303.67 q^{19} -19444.7 q^{21} +29436.8 q^{23} +26206.1 q^{25} +65774.3 q^{27} +51663.4 q^{29} +132080. q^{31} +141786. q^{33} -110790. q^{35} +513068. q^{37} +193419. q^{39} +772583. q^{41} -714511. q^{43} -331645. q^{45} -999769. q^{47} +117649. q^{49} -1.33795e6 q^{51} -639463. q^{53} +807856. q^{55} +187285. q^{57} -71586.9 q^{59} -946191. q^{61} +352177. q^{63} +1.10205e6 q^{65} +208714. q^{67} -1.66877e6 q^{69} -2.79631e6 q^{71} +2.80937e6 q^{73} -1.48563e6 q^{75} -857869. q^{77} +218185. q^{79} -5.97426e6 q^{81} -2.47145e6 q^{83} -7.62324e6 q^{85} -2.92880e6 q^{87} +408724. q^{89} -1.17027e6 q^{91} -7.48764e6 q^{93} +1.06710e6 q^{95} +2.09904e6 q^{97} -2.56799e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 54 q^{3} + 84 q^{5} + 1715 q^{7} + 4133 q^{9} - 5324 q^{11} - 9880 q^{13} - 30128 q^{15} - 44322 q^{17} - 22898 q^{19} - 18522 q^{21} + 141016 q^{23} + 150679 q^{25} - 135972 q^{27} - 18998 q^{29}+ \cdots - 58451180 q^{99}+O(q^{100}) $ 5 * q - 54 * q^3 + 84 * q^5 + 1715 * q^7 + 4133 * q^9 - 5324 * q^11 - 9880 * q^13 - 30128 * q^15 - 44322 * q^17 - 22898 * q^19 - 18522 * q^21 + 141016 * q^23 + 150679 * q^25 - 135972 * q^27 - 18998 * q^29 + 167668 * q^31 + 830696 * q^33 + 28812 * q^35 - 434206 * q^37 - 1100728 * q^39 + 44278 * q^41 - 465076 * q^43 + 1823220 * q^45 - 7508 * q^47 + 588245 * q^49 - 65012 * q^51 - 1901986 * q^53 - 4619216 * q^55 - 3116556 * q^57 - 2342994 * q^59 - 553300 * q^61 + 1417619 * q^63 + 1749208 * q^65 - 1908152 * q^67 - 554944 * q^69 - 1721104 * q^71 + 1237634 * q^73 - 19901682 * q^75 - 1826132 * q^77 - 9787808 * q^79 + 4138481 * q^81 - 17717898 * q^83 - 13248184 * q^85 - 25003324 * q^87 + 7408138 * q^89 - 3388840 * q^91 + 13173560 * q^93 - 16659216 * q^95 - 17027554 * q^97 - 58451180 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−56.6900	−1.21222	−0.606111	−	0.795380i	$-0.707271\pi$
$3$	−56.6900	−1.21222	−0.606111	+	0.795380i	$0.707271\pi$
$4$	0	0
$5$	−323.003	−1.15561	−0.577806	−	0.816174i	$-0.696091\pi$
$5$	−323.003	−1.15561	−0.577806	+	0.816174i	$0.696091\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	1026.75	0.469481
$10$	0	0
$11$	−2501.08	−0.566569	−0.283284	−	0.959036i	$-0.591424\pi$
$11$	−2501.08	−0.566569	−0.283284	+	0.959036i	$0.591424\pi$
$12$	0	0
$13$	−3411.88	−0.430717	−0.215358	−	0.976535i	$-0.569092\pi$
$13$	−3411.88	−0.430717	−0.215358	+	0.976535i	$0.569092\pi$
$14$	0	0
$15$	18311.1	1.40086
$16$	0	0
$17$	23601.1	1.16510	0.582548	−	0.812796i	$-0.302056\pi$
$17$	23601.1	1.16510	0.582548	+	0.812796i	$0.302056\pi$
$18$	0	0
$19$	−3303.67	−0.110499	−0.0552495	−	0.998473i	$-0.517595\pi$
$19$	−3303.67	−0.110499	−0.0552495	+	0.998473i	$0.517595\pi$
$20$	0	0
$21$	−19444.7	−0.458177
$22$	0	0
$23$	29436.8	0.504479	0.252240	−	0.967665i	$-0.418833\pi$
$23$	29436.8	0.504479	0.252240	+	0.967665i	$0.418833\pi$
$24$	0	0
$25$	26206.1	0.335438
$26$	0	0
$27$	65774.3	0.643107
$28$	0	0
$29$	51663.4	0.393360	0.196680	−	0.980468i	$-0.436984\pi$
$29$	51663.4	0.393360	0.196680	+	0.980468i	$0.436984\pi$
$30$	0	0
$31$	132080.	0.796293	0.398146	−	0.917322i	$-0.369654\pi$
$31$	132080.	0.796293	0.398146	+	0.917322i	$0.369654\pi$
$32$	0	0
$33$	141786.	0.686807
$34$	0	0
$35$	−110790.	−0.436780
$36$	0	0
$37$	513068.	1.66521	0.832605	−	0.553867i	$-0.186848\pi$
$37$	513068.	1.66521	0.832605	+	0.553867i	$0.186848\pi$
$38$	0	0
$39$	193419.	0.522124
$40$	0	0
$41$	772583.	1.75066	0.875329	−	0.483527i	$-0.160645\pi$
$41$	772583.	1.75066	0.875329	+	0.483527i	$0.160645\pi$
$42$	0	0
$43$	−714511.	−1.37047	−0.685234	−	0.728323i	$-0.740300\pi$
$43$	−714511.	−1.37047	−0.685234	+	0.728323i	$0.740300\pi$
$44$	0	0
$45$	−331645.	−0.542538
$46$	0	0
$47$	−999769.	−1.40462	−0.702308	−	0.711874i	$-0.747847\pi$
$47$	−999769.	−1.40462	−0.702308	+	0.711874i	$0.747847\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	−1.33795e6	−1.41235
$52$	0	0
$53$	−639463.	−0.589997	−0.294999	−	0.955498i	$-0.595319\pi$
$53$	−639463.	−0.589997	−0.294999	+	0.955498i	$0.595319\pi$
$54$	0	0
$55$	807856.	0.654734
$56$	0	0
$57$	187285.	0.133949
$58$	0	0
$59$	−71586.9	−0.0453786	−0.0226893	−	0.999743i	$-0.507223\pi$
$59$	−71586.9	−0.0453786	−0.0226893	+	0.999743i	$0.507223\pi$
$60$	0	0
$61$	−946191.	−0.533733	−0.266867	−	0.963733i	$-0.585988\pi$
$61$	−946191.	−0.533733	−0.266867	+	0.963733i	$0.585988\pi$
$62$	0	0
$63$	352177.	0.177447
$64$	0	0
$65$	1.10205e6	0.497741
$66$	0	0
$67$	208714.	0.0847792	0.0423896	−	0.999101i	$-0.486503\pi$
$67$	208714.	0.0847792	0.0423896	+	0.999101i	$0.486503\pi$
$68$	0	0
$69$	−1.66877e6	−0.611541
$70$	0	0
$71$	−2.79631e6	−0.927217	−0.463609	−	0.886040i	$-0.653446\pi$
$71$	−2.79631e6	−0.927217	−0.463609	+	0.886040i	$0.653446\pi$
$72$	0	0
$73$	2.80937e6	0.845238	0.422619	−	0.906307i	$-0.361111\pi$
$73$	2.80937e6	0.845238	0.422619	+	0.906307i	$0.361111\pi$
$74$	0	0
$75$	−1.48563e6	−0.406626
$76$	0	0
$77$	−857869.	−0.214143
$78$	0	0
$79$	218185.	0.0497887	0.0248943	−	0.999690i	$-0.492075\pi$
$79$	218185.	0.0497887	0.0248943	+	0.999690i	$0.492075\pi$
$80$	0	0
$81$	−5.97426e6	−1.24907
$82$	0	0
$83$	−2.47145e6	−0.474437	−0.237218	−	0.971456i	$-0.576236\pi$
$83$	−2.47145e6	−0.474437	−0.237218	+	0.971456i	$0.576236\pi$
$84$	0	0
$85$	−7.62324e6	−1.34640
$86$	0	0
$87$	−2.92880e6	−0.476839
$88$	0	0
$89$	408724.	0.0614562	0.0307281	−	0.999528i	$-0.490217\pi$
$89$	408724.	0.0614562	0.0307281	+	0.999528i	$0.490217\pi$
$90$	0	0
$91$	−1.17027e6	−0.162796
$92$	0	0
$93$	−7.48764e6	−0.965283
$94$	0	0
$95$	1.06710e6	0.127694
$96$	0	0
$97$	2.09904e6	0.233518	0.116759	−	0.993160i	$-0.462750\pi$
$97$	2.09904e6	0.233518	0.116759	+	0.993160i	$0.462750\pi$
$98$	0	0
$99$	−2.56799e6	−0.265993
$100$	0	0
$101$	3.79759e6	0.366761	0.183381	−	0.983042i	$-0.441296\pi$
$101$	3.79759e6	0.366761	0.183381	+	0.983042i	$0.441296\pi$
$102$	0	0
$103$	−1.39792e7	−1.26052	−0.630262	−	0.776382i	$-0.717053\pi$
$103$	−1.39792e7	−1.26052	−0.630262	+	0.776382i	$0.717053\pi$
$104$	0	0
$105$	6.28069e6	0.529474
$106$	0	0
$107$	−1.13178e7	−0.893138	−0.446569	−	0.894749i	$-0.647354\pi$
$107$	−1.13178e7	−0.893138	−0.446569	+	0.894749i	$0.647354\pi$
$108$	0	0
$109$	−4.92125e6	−0.363984	−0.181992	−	0.983300i	$-0.558255\pi$
$109$	−4.92125e6	−0.363984	−0.181992	+	0.983300i	$0.558255\pi$
$110$	0	0
$111$	−2.90858e7	−2.01860
$112$	0	0
$113$	1.16256e7	0.757949	0.378975	−	0.925407i	$-0.376277\pi$
$113$	1.16256e7	0.757949	0.378975	+	0.925407i	$0.376277\pi$
$114$	0	0
$115$	−9.50819e6	−0.582982
$116$	0	0
$117$	−3.50316e6	−0.202213
$118$	0	0
$119$	8.09519e6	0.440365
$120$	0	0
$121$	−1.32318e7	−0.679000
$122$	0	0
$123$	−4.37977e7	−2.12219
$124$	0	0
$125$	1.67700e7	0.767975
$126$	0	0
$127$	−7.51642e6	−0.325610	−0.162805	−	0.986658i	$-0.552054\pi$
$127$	−7.51642e6	−0.325610	−0.162805	+	0.986658i	$0.552054\pi$
$128$	0	0
$129$	4.05056e7	1.66131
$130$	0	0
$131$	2.43636e7	0.946875	0.473437	−	0.880827i	$-0.343013\pi$
$131$	2.43636e7	0.946875	0.473437	+	0.880827i	$0.343013\pi$
$132$	0	0
$133$	−1.13316e6	−0.0417647
$134$	0	0
$135$	−2.12453e7	−0.743181
$136$	0	0
$137$	−3.78381e7	−1.25721	−0.628604	−	0.777725i	$-0.716373\pi$
$137$	−3.78381e7	−1.25721	−0.628604	+	0.777725i	$0.716373\pi$
$138$	0	0
$139$	−4.78675e7	−1.51178	−0.755890	−	0.654698i	$-0.772796\pi$
$139$	−4.78675e7	−1.51178	−0.755890	+	0.654698i	$0.772796\pi$
$140$	0	0
$141$	5.66769e7	1.70270
$142$	0	0
$143$	8.53337e6	0.244031
$144$	0	0
$145$	−1.66874e7	−0.454571
$146$	0	0
$147$	−6.66952e6	−0.173175
$148$	0	0
$149$	3.66023e7	0.906476	0.453238	−	0.891390i	$-0.350269\pi$
$149$	3.66023e7	0.906476	0.453238	+	0.891390i	$0.350269\pi$
$150$	0	0
$151$	3.32557e7	0.786045	0.393022	−	0.919529i	$-0.371429\pi$
$151$	3.32557e7	0.786045	0.393022	+	0.919529i	$0.371429\pi$
$152$	0	0
$153$	2.42326e7	0.546990
$154$	0	0
$155$	−4.26624e7	−0.920205
$156$	0	0
$157$	3.08012e7	0.635212	0.317606	−	0.948223i	$-0.397121\pi$
$157$	3.08012e7	0.635212	0.317606	+	0.948223i	$0.397121\pi$
$158$	0	0
$159$	3.62512e7	0.715208
$160$	0	0
$161$	1.00968e7	0.190675
$162$	0	0
$163$	−8.36295e7	−1.51253	−0.756263	−	0.654268i	$-0.772977\pi$
$163$	−8.36295e7	−1.51253	−0.756263	+	0.654268i	$0.772977\pi$
$164$	0	0
$165$	−4.57974e7	−0.793682
$166$	0	0
$167$	1.08373e8	1.80059	0.900293	−	0.435285i	$-0.143352\pi$
$167$	1.08373e8	1.80059	0.900293	+	0.435285i	$0.143352\pi$
$168$	0	0
$169$	−5.11076e7	−0.814483
$170$	0	0
$171$	−3.39206e6	−0.0518772
$172$	0	0
$173$	−9.39808e7	−1.38000	−0.689998	−	0.723812i	$-0.742388\pi$
$173$	−9.39808e7	−1.38000	−0.689998	+	0.723812i	$0.742388\pi$
$174$	0	0
$175$	8.98870e6	0.126784
$176$	0	0
$177$	4.05826e6	0.0550090
$178$	0	0
$179$	4.24193e7	0.552813	0.276406	−	0.961041i	$-0.410856\pi$
$179$	4.24193e7	0.552813	0.276406	+	0.961041i	$0.410856\pi$
$180$	0	0
$181$	1.04604e8	1.31121	0.655605	−	0.755104i	$-0.272414\pi$
$181$	1.04604e8	1.31121	0.655605	+	0.755104i	$0.272414\pi$
$182$	0	0
$183$	5.36396e7	0.647003
$184$	0	0
$185$	−1.65723e8	−1.92434
$186$	0	0
$187$	−5.90282e7	−0.660107
$188$	0	0
$189$	2.25606e7	0.243071
$190$	0	0
$191$	−1.16015e8	−1.20475	−0.602375	−	0.798213i	$-0.705779\pi$
$191$	−1.16015e8	−1.20475	−0.602375	+	0.798213i	$0.705779\pi$
$192$	0	0
$193$	−9.68602e7	−0.969828	−0.484914	−	0.874562i	$-0.661149\pi$
$193$	−9.68602e7	−0.969828	−0.484914	+	0.874562i	$0.661149\pi$
$194$	0	0
$195$	−6.24751e7	−0.603373
$196$	0	0
$197$	6.49031e7	0.604831	0.302415	−	0.953176i	$-0.402207\pi$
$197$	6.49031e7	0.604831	0.302415	+	0.953176i	$0.402207\pi$
$198$	0	0
$199$	1.26289e8	1.13601	0.568003	−	0.823026i	$-0.307716\pi$
$199$	1.26289e8	1.13601	0.568003	+	0.823026i	$0.307716\pi$
$200$	0	0
$201$	−1.18320e7	−0.102771
$202$	0	0
$203$	1.77205e7	0.148676
$204$	0	0
$205$	−2.49547e8	−2.02308
$206$	0	0
$207$	3.02244e7	0.236843
$208$	0	0
$209$	8.26273e6	0.0626053
$210$	0	0
$211$	−1.63928e8	−1.20134	−0.600669	−	0.799498i	$-0.705099\pi$
$211$	−1.63928e8	−1.20134	−0.600669	+	0.799498i	$0.705099\pi$
$212$	0	0
$213$	1.58523e8	1.12399
$214$	0	0
$215$	2.30789e8	1.58373
$216$	0	0
$217$	4.53036e7	0.300970
$218$	0	0
$219$	−1.59263e8	−1.02462
$220$	0	0
$221$	−8.05242e7	−0.501826
$222$	0	0
$223$	−8.21405e7	−0.496010	−0.248005	−	0.968759i	$-0.579775\pi$
$223$	−8.21405e7	−0.496010	−0.248005	+	0.968759i	$0.579775\pi$
$224$	0	0
$225$	2.69073e7	0.157482
$226$	0	0
$227$	2.38169e8	1.35143	0.675716	−	0.737162i	$-0.263834\pi$
$227$	2.38169e8	1.35143	0.675716	+	0.737162i	$0.263834\pi$
$228$	0	0
$229$	−7.12129e7	−0.391863	−0.195932	−	0.980618i	$-0.562773\pi$
$229$	−7.12129e7	−0.391863	−0.195932	+	0.980618i	$0.562773\pi$
$230$	0	0
$231$	4.86326e7	0.259589
$232$	0	0
$233$	8.88503e7	0.460164	0.230082	−	0.973171i	$-0.426100\pi$
$233$	8.88503e7	0.460164	0.230082	+	0.973171i	$0.426100\pi$
$234$	0	0
$235$	3.22929e8	1.62319
$236$	0	0
$237$	−1.23689e7	−0.0603549
$238$	0	0
$239$	1.94459e8	0.921372	0.460686	−	0.887563i	$-0.347603\pi$
$239$	1.94459e8	0.921372	0.460686	+	0.887563i	$0.347603\pi$
$240$	0	0
$241$	1.68472e8	0.775297	0.387648	−	0.921807i	$-0.373287\pi$
$241$	1.68472e8	0.775297	0.387648	+	0.921807i	$0.373287\pi$
$242$	0	0
$243$	1.94832e8	0.871041
$244$	0	0
$245$	−3.80010e7	−0.165087
$246$	0	0
$247$	1.12717e7	0.0475938
$248$	0	0
$249$	1.40106e8	0.575123
$250$	0	0
$251$	−1.40081e8	−0.559139	−0.279570	−	0.960125i	$-0.590192\pi$
$251$	−1.40081e8	−0.559139	−0.279570	+	0.960125i	$0.590192\pi$
$252$	0	0
$253$	−7.36237e7	−0.285822
$254$	0	0
$255$	4.32162e8	1.63213
$256$	0	0
$257$	4.30416e8	1.58169	0.790846	−	0.612015i	$-0.209641\pi$
$257$	4.30416e8	1.58169	0.790846	+	0.612015i	$0.209641\pi$
$258$	0	0
$259$	1.75982e8	0.629390
$260$	0	0
$261$	5.30456e7	0.184675
$262$	0	0
$263$	2.50030e8	0.847515	0.423758	−	0.905776i	$-0.360711\pi$
$263$	2.50030e8	0.847515	0.423758	+	0.905776i	$0.360711\pi$
$264$	0	0
$265$	2.06549e8	0.681808
$266$	0	0
$267$	−2.31706e7	−0.0744985
$268$	0	0
$269$	−3.45437e7	−0.108202	−0.0541011	−	0.998535i	$-0.517229\pi$
$269$	−3.45437e7	−0.108202	−0.0541011	+	0.998535i	$0.517229\pi$
$270$	0	0
$271$	4.29786e8	1.31178	0.655888	−	0.754858i	$-0.272294\pi$
$271$	4.29786e8	1.31178	0.655888	+	0.754858i	$0.272294\pi$
$272$	0	0
$273$	6.63428e7	0.197344
$274$	0	0
$275$	−6.55435e7	−0.190049
$276$	0	0
$277$	3.24682e8	0.917867	0.458933	−	0.888471i	$-0.348232\pi$
$277$	3.24682e8	0.917867	0.458933	+	0.888471i	$0.348232\pi$
$278$	0	0
$279$	1.35614e8	0.373844
$280$	0	0
$281$	−4.42085e8	−1.18859	−0.594297	−	0.804246i	$-0.702570\pi$
$281$	−4.42085e8	−1.18859	−0.594297	+	0.804246i	$0.702570\pi$
$282$	0	0
$283$	−4.74540e7	−0.124457	−0.0622287	−	0.998062i	$-0.519821\pi$
$283$	−4.74540e7	−0.124457	−0.0622287	+	0.998062i	$0.519821\pi$
$284$	0	0
$285$	−6.04936e7	−0.154793
$286$	0	0
$287$	2.64996e8	0.661687
$288$	0	0
$289$	1.46675e8	0.357448
$290$	0	0
$291$	−1.18995e8	−0.283075
$292$	0	0
$293$	−4.67253e8	−1.08521	−0.542607	−	0.839987i	$-0.682563\pi$
$293$	−4.67253e8	−1.08521	−0.542607	+	0.839987i	$0.682563\pi$
$294$	0	0
$295$	2.31228e7	0.0524401
$296$	0	0
$297$	−1.64507e8	−0.364364
$298$	0	0
$299$	−1.00435e8	−0.217288
$300$	0	0
$301$	−2.45077e8	−0.517989
$302$	0	0
$303$	−2.15285e8	−0.444596
$304$	0	0
$305$	3.05623e8	0.616789
$306$	0	0
$307$	−9.74878e8	−1.92294	−0.961471	−	0.274907i	$-0.911353\pi$
$307$	−9.74878e8	−1.92294	−0.961471	+	0.274907i	$0.911353\pi$
$308$	0	0
$309$	7.92479e8	1.52803
$310$	0	0
$311$	6.62439e8	1.24878	0.624388	−	0.781114i	$-0.285348\pi$
$311$	6.62439e8	1.24878	0.624388	+	0.781114i	$0.285348\pi$
$312$	0	0
$313$	−4.37407e8	−0.806269	−0.403135	−	0.915141i	$-0.632079\pi$
$313$	−4.37407e8	−0.806269	−0.403135	+	0.915141i	$0.632079\pi$
$314$	0	0
$315$	−1.13754e8	−0.205060
$316$	0	0
$317$	−2.74360e8	−0.483741	−0.241871	−	0.970309i	$-0.577761\pi$
$317$	−2.74360e8	−0.483741	−0.241871	+	0.970309i	$0.577761\pi$
$318$	0	0
$319$	−1.29214e8	−0.222865
$320$	0	0
$321$	6.41605e8	1.08268
$322$	0	0
$323$	−7.79703e7	−0.128742
$324$	0	0
$325$	−8.94121e7	−0.144479
$326$	0	0
$327$	2.78985e8	0.441229
$328$	0	0
$329$	−3.42921e8	−0.530895
$330$	0	0
$331$	−1.17981e9	−1.78819	−0.894096	−	0.447875i	$-0.852181\pi$
$331$	−1.17981e9	−1.78819	−0.894096	+	0.447875i	$0.852181\pi$
$332$	0	0
$333$	5.26795e8	0.781785
$334$	0	0
$335$	−6.74152e7	−0.0979718
$336$	0	0
$337$	−9.51938e8	−1.35489	−0.677445	−	0.735573i	$-0.736913\pi$
$337$	−9.51938e8	−1.35489	−0.677445	+	0.735573i	$0.736913\pi$
$338$	0	0
$339$	−6.59054e8	−0.918803
$340$	0	0
$341$	−3.30343e8	−0.451155
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	5.39019e8	0.706704
$346$	0	0
$347$	1.28195e9	1.64710	0.823549	−	0.567245i	$-0.191991\pi$
$347$	1.28195e9	1.64710	0.823549	+	0.567245i	$0.191991\pi$
$348$	0	0
$349$	−1.18173e9	−1.48809	−0.744043	−	0.668132i	$-0.767094\pi$
$349$	−1.18173e9	−1.48809	−0.744043	+	0.668132i	$0.767094\pi$
$350$	0	0
$351$	−2.24414e8	−0.276997
$352$	0	0
$353$	−6.07808e8	−0.735453	−0.367726	−	0.929934i	$-0.619864\pi$
$353$	−6.07808e8	−0.735453	−0.367726	+	0.929934i	$0.619864\pi$
$354$	0	0
$355$	9.03218e8	1.07150
$356$	0	0
$357$	−4.58916e8	−0.533820
$358$	0	0
$359$	−1.63977e9	−1.87048	−0.935238	−	0.354019i	$-0.884815\pi$
$359$	−1.63977e9	−1.87048	−0.935238	+	0.354019i	$0.884815\pi$
$360$	0	0
$361$	−8.82958e8	−0.987790
$362$	0	0
$363$	7.50110e8	0.823098
$364$	0	0
$365$	−9.07436e8	−0.976767
$366$	0	0
$367$	9.53936e8	1.00737	0.503684	−	0.863888i	$-0.331978\pi$
$367$	9.53936e8	1.00737	0.503684	+	0.863888i	$0.331978\pi$
$368$	0	0
$369$	7.93253e8	0.821901
$370$	0	0
$371$	−2.19336e8	−0.222998
$372$	0	0
$373$	1.82404e8	0.181992	0.0909960	−	0.995851i	$-0.470995\pi$
$373$	1.82404e8	0.181992	0.0909960	+	0.995851i	$0.470995\pi$
$374$	0	0
$375$	−9.50689e8	−0.930956
$376$	0	0
$377$	−1.76269e8	−0.169427
$378$	0	0
$379$	−1.07991e9	−1.01895	−0.509473	−	0.860486i	$-0.670160\pi$
$379$	−1.07991e9	−1.01895	−0.509473	+	0.860486i	$0.670160\pi$
$380$	0	0
$381$	4.26106e8	0.394712
$382$	0	0
$383$	−1.91481e9	−1.74153	−0.870764	−	0.491701i	$-0.836375\pi$
$383$	−1.91481e9	−1.74153	−0.870764	+	0.491701i	$0.836375\pi$
$384$	0	0
$385$	2.77095e8	0.247466
$386$	0	0
$387$	−7.33628e8	−0.643409
$388$	0	0
$389$	2.07520e9	1.78746	0.893728	−	0.448608i	$-0.148080\pi$
$389$	2.07520e9	1.78746	0.893728	+	0.448608i	$0.148080\pi$
$390$	0	0
$391$	6.94742e8	0.587767
$392$	0	0
$393$	−1.38117e9	−1.14782
$394$	0	0
$395$	−7.04745e7	−0.0575364
$396$	0	0
$397$	9.57462e8	0.767988	0.383994	−	0.923335i	$-0.374548\pi$
$397$	9.57462e8	0.767988	0.383994	+	0.923335i	$0.374548\pi$
$398$	0	0
$399$	6.42387e7	0.0506281
$400$	0	0
$401$	−1.49768e9	−1.15988	−0.579940	−	0.814659i	$-0.696924\pi$
$401$	−1.49768e9	−1.15988	−0.579940	+	0.814659i	$0.696924\pi$
$402$	0	0
$403$	−4.50643e8	−0.342977
$404$	0	0
$405$	1.92970e9	1.44344
$406$	0	0
$407$	−1.28322e9	−0.943456
$408$	0	0
$409$	−7.84971e8	−0.567312	−0.283656	−	0.958926i	$-0.591547\pi$
$409$	−7.84971e8	−0.567312	−0.283656	+	0.958926i	$0.591547\pi$
$410$	0	0
$411$	2.14504e9	1.52402
$412$	0	0
$413$	−2.45543e7	−0.0171515
$414$	0	0
$415$	7.98286e8	0.548265
$416$	0	0
$417$	2.71361e9	1.83261
$418$	0	0
$419$	−1.85449e9	−1.23162	−0.615809	−	0.787895i	$-0.711171\pi$
$419$	−1.85449e9	−1.23162	−0.615809	+	0.787895i	$0.711171\pi$
$420$	0	0
$421$	2.06696e9	1.35004	0.675018	−	0.737801i	$-0.264136\pi$
$421$	2.06696e9	1.35004	0.675018	+	0.737801i	$0.264136\pi$
$422$	0	0
$423$	−1.02652e9	−0.659440
$424$	0	0
$425$	6.18494e8	0.390818
$426$	0	0
$427$	−3.24543e8	−0.201732
$428$	0	0
$429$	−4.83757e8	−0.295819
$430$	0	0
$431$	−2.21175e9	−1.33065	−0.665327	−	0.746552i	$-0.731708\pi$
$431$	−2.21175e9	−1.33065	−0.665327	+	0.746552i	$0.731708\pi$
$432$	0	0
$433$	5.96992e8	0.353396	0.176698	−	0.984265i	$-0.443458\pi$
$433$	5.96992e8	0.353396	0.176698	+	0.984265i	$0.443458\pi$
$434$	0	0
$435$	9.46011e8	0.551041
$436$	0	0
$437$	−9.72494e7	−0.0557445
$438$	0	0
$439$	−2.76468e9	−1.55962	−0.779810	−	0.626016i	$-0.784684\pi$
$439$	−2.76468e9	−1.55962	−0.779810	+	0.626016i	$0.784684\pi$
$440$	0	0
$441$	1.20797e8	0.0670687
$442$	0	0
$443$	1.73383e9	0.947533	0.473766	−	0.880651i	$-0.342894\pi$
$443$	1.73383e9	0.947533	0.473766	+	0.880651i	$0.342894\pi$
$444$	0	0
$445$	−1.32019e8	−0.0710195
$446$	0	0
$447$	−2.07498e9	−1.09885
$448$	0	0
$449$	−1.06188e9	−0.553624	−0.276812	−	0.960924i	$-0.589278\pi$
$449$	−1.06188e9	−0.553624	−0.276812	+	0.960924i	$0.589278\pi$
$450$	0	0
$451$	−1.93229e9	−0.991869
$452$	0	0
$453$	−1.88527e9	−0.952860
$454$	0	0
$455$	3.78002e8	0.188129
$456$	0	0
$457$	−1.28026e9	−0.627469	−0.313735	−	0.949511i	$-0.601580\pi$
$457$	−1.28026e9	−0.627469	−0.313735	+	0.949511i	$0.601580\pi$
$458$	0	0
$459$	1.55235e9	0.749281
$460$	0	0
$461$	8.56855e8	0.407337	0.203669	−	0.979040i	$-0.434713\pi$
$461$	8.56855e8	0.407337	0.203669	+	0.979040i	$0.434713\pi$
$462$	0	0
$463$	6.50388e8	0.304536	0.152268	−	0.988339i	$-0.451342\pi$
$463$	6.50388e8	0.304536	0.152268	+	0.988339i	$0.451342\pi$
$464$	0	0
$465$	2.41853e9	1.11549
$466$	0	0
$467$	−1.04577e9	−0.475147	−0.237574	−	0.971370i	$-0.576352\pi$
$467$	−1.04577e9	−0.475147	−0.237574	+	0.971370i	$0.576352\pi$
$468$	0	0
$469$	7.15888e7	0.0320435
$470$	0	0
$471$	−1.74612e9	−0.770018
$472$	0	0
$473$	1.78705e9	0.776465
$474$	0	0
$475$	−8.65763e7	−0.0370656
$476$	0	0
$477$	−6.56572e8	−0.276993
$478$	0	0
$479$	−3.12323e9	−1.29846	−0.649232	−	0.760591i	$-0.724909\pi$
$479$	−3.12323e9	−1.29846	−0.649232	+	0.760591i	$0.724909\pi$
$480$	0	0
$481$	−1.75053e9	−0.717234
$482$	0	0
$483$	−5.72389e8	−0.231141
$484$	0	0
$485$	−6.77998e8	−0.269856
$486$	0	0
$487$	−2.76854e9	−1.08617	−0.543087	−	0.839677i	$-0.682745\pi$
$487$	−2.76854e9	−1.08617	−0.543087	+	0.839677i	$0.682745\pi$
$488$	0	0
$489$	4.74096e9	1.83352
$490$	0	0
$491$	−1.44692e8	−0.0551644	−0.0275822	−	0.999620i	$-0.508781\pi$
$491$	−1.44692e8	−0.0551644	−0.0275822	+	0.999620i	$0.508781\pi$
$492$	0	0
$493$	1.21931e9	0.458302
$494$	0	0
$495$	8.29470e8	0.307385
$496$	0	0
$497$	−9.59135e8	−0.350455
$498$	0	0
$499$	4.62813e9	1.66745	0.833727	−	0.552177i	$-0.186203\pi$
$499$	4.62813e9	1.66745	0.833727	+	0.552177i	$0.186203\pi$
$500$	0	0
$501$	−6.14367e9	−2.18271
$502$	0	0
$503$	3.59709e9	1.26027	0.630135	−	0.776486i	$-0.283000\pi$
$503$	3.59709e9	1.26027	0.630135	+	0.776486i	$0.283000\pi$
$504$	0	0
$505$	−1.22663e9	−0.423834
$506$	0	0
$507$	2.89729e9	0.987334
$508$	0	0
$509$	−1.38454e9	−0.465366	−0.232683	−	0.972553i	$-0.574750\pi$
$509$	−1.38454e9	−0.465366	−0.232683	+	0.972553i	$0.574750\pi$
$510$	0	0
$511$	9.63614e8	0.319470
$512$	0	0
$513$	−2.17296e8	−0.0710627
$514$	0	0
$515$	4.51532e9	1.45668
$516$	0	0
$517$	2.50050e9	0.795811
$518$	0	0
$519$	5.32777e9	1.67286
$520$	0	0
$521$	−2.64846e9	−0.820466	−0.410233	−	0.911981i	$-0.634553\pi$
$521$	−2.64846e9	−0.820466	−0.410233	+	0.911981i	$0.634553\pi$
$522$	0	0
$523$	−3.23641e9	−0.989253	−0.494626	−	0.869106i	$-0.664695\pi$
$523$	−3.23641e9	−0.989253	−0.494626	+	0.869106i	$0.664695\pi$
$524$	0	0
$525$	−5.09569e8	−0.153690
$526$	0	0
$527$	3.11725e9	0.927757
$528$	0	0
$529$	−2.53830e9	−0.745501
$530$	0	0
$531$	−7.35022e7	−0.0213044
$532$	0	0
$533$	−2.63596e9	−0.754038
$534$	0	0
$535$	3.65568e9	1.03212
$536$	0	0
$537$	−2.40475e9	−0.670131
$538$	0	0
$539$	−2.94249e8	−0.0809384
$540$	0	0
$541$	−5.31661e9	−1.44359	−0.721795	−	0.692107i	$-0.756683\pi$
$541$	−5.31661e9	−1.44359	−0.721795	+	0.692107i	$0.756683\pi$
$542$	0	0
$543$	−5.92999e9	−1.58948
$544$	0	0
$545$	1.58958e9	0.420624
$546$	0	0
$547$	−1.74355e9	−0.455491	−0.227746	−	0.973721i	$-0.573135\pi$
$547$	−1.74355e9	−0.455491	−0.227746	+	0.973721i	$0.573135\pi$
$548$	0	0
$549$	−9.71506e8	−0.250578
$550$	0	0
$551$	−1.70679e8	−0.0434659
$552$	0	0
$553$	7.48375e7	0.0188183
$554$	0	0
$555$	9.39482e9	2.33272
$556$	0	0
$557$	2.00876e9	0.492534	0.246267	−	0.969202i	$-0.420796\pi$
$557$	2.00876e9	0.492534	0.246267	+	0.969202i	$0.420796\pi$
$558$	0	0
$559$	2.43782e9	0.590284
$560$	0	0
$561$	3.34631e9	0.800196
$562$	0	0
$563$	−4.22735e9	−0.998364	−0.499182	−	0.866497i	$-0.666366\pi$
$563$	−4.22735e9	−0.998364	−0.499182	+	0.866497i	$0.666366\pi$
$564$	0	0
$565$	−3.75510e9	−0.875895
$566$	0	0
$567$	−2.04917e9	−0.472104
$568$	0	0
$569$	2.48508e9	0.565520	0.282760	−	0.959191i	$-0.408750\pi$
$569$	2.48508e9	0.565520	0.282760	+	0.959191i	$0.408750\pi$
$570$	0	0
$571$	6.85001e9	1.53980	0.769900	−	0.638164i	$-0.220306\pi$
$571$	6.85001e9	1.53980	0.769900	+	0.638164i	$0.220306\pi$
$572$	0	0
$573$	6.57688e9	1.46042
$574$	0	0
$575$	7.71425e8	0.169222
$576$	0	0
$577$	7.31129e9	1.58445	0.792225	−	0.610229i	$-0.208923\pi$
$577$	7.31129e9	1.58445	0.792225	+	0.610229i	$0.208923\pi$
$578$	0	0
$579$	5.49100e9	1.17565
$580$	0	0
$581$	−8.47707e8	−0.179320
$582$	0	0
$583$	1.59935e9	0.334274
$584$	0	0
$585$	1.13153e9	0.233680
$586$	0	0
$587$	4.80786e9	0.981112	0.490556	−	0.871410i	$-0.336794\pi$
$587$	4.80786e9	0.981112	0.490556	+	0.871410i	$0.336794\pi$
$588$	0	0
$589$	−4.36350e8	−0.0879896
$590$	0	0
$591$	−3.67936e9	−0.733189
$592$	0	0
$593$	−6.52956e9	−1.28586	−0.642928	−	0.765927i	$-0.722281\pi$
$593$	−6.52956e9	−1.28586	−0.642928	+	0.765927i	$0.722281\pi$
$594$	0	0
$595$	−2.61477e9	−0.508891
$596$	0	0
$597$	−7.15934e9	−1.37709
$598$	0	0
$599$	−1.59789e9	−0.303775	−0.151888	−	0.988398i	$-0.548535\pi$
$599$	−1.59789e9	−0.303775	−0.151888	+	0.988398i	$0.548535\pi$
$600$	0	0
$601$	4.18772e9	0.786896	0.393448	−	0.919347i	$-0.371282\pi$
$601$	4.18772e9	0.786896	0.393448	+	0.919347i	$0.371282\pi$
$602$	0	0
$603$	2.14298e8	0.0398022
$604$	0	0
$605$	4.27391e9	0.784660
$606$	0	0
$607$	−1.09570e10	−1.98853	−0.994267	−	0.106930i	$-0.965898\pi$
$607$	−1.09570e10	−1.98853	−0.994267	+	0.106930i	$0.965898\pi$
$608$	0	0
$609$	−1.00458e9	−0.180228
$610$	0	0
$611$	3.41109e9	0.604991
$612$	0	0
$613$	3.27020e9	0.573407	0.286703	−	0.958019i	$-0.407441\pi$
$613$	3.27020e9	0.573407	0.286703	+	0.958019i	$0.407441\pi$
$614$	0	0
$615$	1.41468e10	2.45242
$616$	0	0
$617$	2.47828e9	0.424769	0.212384	−	0.977186i	$-0.431877\pi$
$617$	2.47828e9	0.424769	0.212384	+	0.977186i	$0.431877\pi$
$618$	0	0
$619$	−4.01244e9	−0.679972	−0.339986	−	0.940430i	$-0.610422\pi$
$619$	−4.01244e9	−0.679972	−0.339986	+	0.940430i	$0.610422\pi$
$620$	0	0
$621$	1.93619e9	0.324434
$622$	0	0
$623$	1.40192e8	0.0232283
$624$	0	0
$625$	−7.46411e9	−1.22292
$626$	0	0
$627$	−4.68414e8	−0.0758915
$628$	0	0
$629$	1.21090e10	1.94013
$630$	0	0
$631$	−1.44173e9	−0.228444	−0.114222	−	0.993455i	$-0.536438\pi$
$631$	−1.44173e9	−0.228444	−0.114222	+	0.993455i	$0.536438\pi$
$632$	0	0
$633$	9.29309e9	1.45629
$634$	0	0
$635$	2.42783e9	0.376279
$636$	0	0
$637$	−4.01404e8	−0.0615310
$638$	0	0
$639$	−2.87113e9	−0.435311
$640$	0	0
$641$	−8.16565e9	−1.22458	−0.612291	−	0.790633i	$-0.709752\pi$
$641$	−8.16565e9	−1.22458	−0.612291	+	0.790633i	$0.709752\pi$
$642$	0	0
$643$	−1.00694e9	−0.149370	−0.0746851	−	0.997207i	$-0.523795\pi$
$643$	−1.00694e9	−0.149370	−0.0746851	+	0.997207i	$0.523795\pi$
$644$	0	0
$645$	−1.30834e10	−1.91983
$646$	0	0
$647$	4.35521e9	0.632185	0.316093	−	0.948728i	$-0.397629\pi$
$647$	4.35521e9	0.632185	0.316093	+	0.948728i	$0.397629\pi$
$648$	0	0
$649$	1.79044e8	0.0257101
$650$	0	0
$651$	−2.56826e9	−0.364843
$652$	0	0
$653$	8.12364e9	1.14171	0.570854	−	0.821052i	$-0.306612\pi$
$653$	8.12364e9	1.14171	0.570854	+	0.821052i	$0.306612\pi$
$654$	0	0
$655$	−7.86953e9	−1.09422
$656$	0	0
$657$	2.88453e9	0.396823
$658$	0	0
$659$	9.13242e9	1.24304	0.621522	−	0.783396i	$-0.286514\pi$
$659$	9.13242e9	1.24304	0.621522	+	0.783396i	$0.286514\pi$
$660$	0	0
$661$	6.45495e9	0.869336	0.434668	−	0.900591i	$-0.356866\pi$
$661$	6.45495e9	0.869336	0.434668	+	0.900591i	$0.356866\pi$
$662$	0	0
$663$	4.56491e9	0.608325
$664$	0	0
$665$	3.66014e8	0.0482638
$666$	0	0
$667$	1.52081e9	0.198442
$668$	0	0
$669$	4.65654e9	0.601274
$670$	0	0
$671$	2.36650e9	0.302397
$672$	0	0
$673$	−7.64368e9	−0.966607	−0.483303	−	0.875453i	$-0.660563\pi$
$673$	−7.64368e9	−0.966607	−0.483303	+	0.875453i	$0.660563\pi$
$674$	0	0
$675$	1.72369e9	0.215723
$676$	0	0
$677$	1.28545e10	1.59219	0.796097	−	0.605168i	$-0.206894\pi$
$677$	1.28545e10	1.59219	0.796097	+	0.605168i	$0.206894\pi$
$678$	0	0
$679$	7.19971e8	0.0882614
$680$	0	0
$681$	−1.35018e10	−1.63824
$682$	0	0
$683$	5.81192e9	0.697987	0.348993	−	0.937125i	$-0.386524\pi$
$683$	5.81192e9	0.697987	0.348993	+	0.937125i	$0.386524\pi$
$684$	0	0
$685$	1.22218e10	1.45284
$686$	0	0
$687$	4.03706e9	0.475025
$688$	0	0
$689$	2.18177e9	0.254122
$690$	0	0
$691$	−2.18590e8	−0.0252033	−0.0126016	−	0.999921i	$-0.504011\pi$
$691$	−2.18590e8	−0.0252033	−0.0126016	+	0.999921i	$0.504011\pi$
$692$	0	0
$693$	−8.80822e8	−0.100536
$694$	0	0
$695$	1.54614e10	1.74703
$696$	0	0
$697$	1.82338e10	2.03968
$698$	0	0
$699$	−5.03692e9	−0.557821
$700$	0	0
$701$	−1.21054e10	−1.32729	−0.663643	−	0.748049i	$-0.730991\pi$
$701$	−1.21054e10	−1.32729	−0.663643	+	0.748049i	$0.730991\pi$
$702$	0	0
$703$	−1.69501e9	−0.184004
$704$	0	0
$705$	−1.83068e10	−1.96767
$706$	0	0
$707$	1.30257e9	0.138623
$708$	0	0
$709$	−9.37487e9	−0.987878	−0.493939	−	0.869497i	$-0.664443\pi$
$709$	−9.37487e9	−0.987878	−0.493939	+	0.869497i	$0.664443\pi$
$710$	0	0
$711$	2.24023e8	0.0233748
$712$	0	0
$713$	3.88803e9	0.401713
$714$	0	0
$715$	−2.75631e9	−0.282005
$716$	0	0
$717$	−1.10239e10	−1.11691
$718$	0	0
$719$	−4.68162e9	−0.469727	−0.234863	−	0.972028i	$-0.575464\pi$
$719$	−4.68162e9	−0.469727	−0.234863	+	0.972028i	$0.575464\pi$
$720$	0	0
$721$	−4.79486e9	−0.476433
$722$	0	0
$723$	−9.55067e9	−0.939831
$724$	0	0
$725$	1.35390e9	0.131948
$726$	0	0
$727$	−1.62406e10	−1.56759	−0.783795	−	0.621020i	$-0.786719\pi$
$727$	−1.62406e10	−1.56759	−0.783795	+	0.621020i	$0.786719\pi$
$728$	0	0
$729$	2.02066e9	0.193174
$730$	0	0
$731$	−1.68633e10	−1.59673
$732$	0	0
$733$	3.98413e9	0.373654	0.186827	−	0.982393i	$-0.440180\pi$
$733$	3.98413e9	0.373654	0.186827	+	0.982393i	$0.440180\pi$
$734$	0	0
$735$	2.15428e9	0.200122
$736$	0	0
$737$	−5.22009e8	−0.0480332
$738$	0	0
$739$	−1.30330e10	−1.18793	−0.593964	−	0.804492i	$-0.702438\pi$
$739$	−1.30330e10	−1.18793	−0.593964	+	0.804492i	$0.702438\pi$
$740$	0	0
$741$	−6.38993e8	−0.0576942
$742$	0	0
$743$	1.28600e10	1.15021	0.575107	−	0.818078i	$-0.304960\pi$
$743$	1.28600e10	1.15021	0.575107	+	0.818078i	$0.304960\pi$
$744$	0	0
$745$	−1.18227e10	−1.04753
$746$	0	0
$747$	−2.53757e9	−0.222739
$748$	0	0
$749$	−3.88200e9	−0.337574
$750$	0	0
$751$	−1.23588e9	−0.106472	−0.0532361	−	0.998582i	$-0.516954\pi$
$751$	−1.23588e9	−0.106472	−0.0532361	+	0.998582i	$0.516954\pi$
$752$	0	0
$753$	7.94117e9	0.677801
$754$	0	0
$755$	−1.07417e10	−0.908363
$756$	0	0
$757$	5.43517e9	0.455384	0.227692	−	0.973733i	$-0.426882\pi$
$757$	5.43517e9	0.455384	0.227692	+	0.973733i	$0.426882\pi$
$758$	0	0
$759$	4.17373e9	0.346480
$760$	0	0
$761$	−1.22572e10	−1.00820	−0.504098	−	0.863647i	$-0.668175\pi$
$761$	−1.22572e10	−1.00820	−0.504098	+	0.863647i	$0.668175\pi$
$762$	0	0
$763$	−1.68799e9	−0.137573
$764$	0	0
$765$	−7.82720e9	−0.632108
$766$	0	0
$767$	2.44246e8	0.0195453
$768$	0	0
$769$	9.46621e9	0.750644	0.375322	−	0.926895i	$-0.377532\pi$
$769$	9.46621e9	0.750644	0.375322	+	0.926895i	$0.377532\pi$
$770$	0	0
$771$	−2.44003e10	−1.91736
$772$	0	0
$773$	−1.24927e10	−0.972807	−0.486403	−	0.873734i	$-0.661691\pi$
$773$	−1.24927e10	−0.972807	−0.486403	+	0.873734i	$0.661691\pi$
$774$	0	0
$775$	3.46132e9	0.267107
$776$	0	0
$777$	−9.97644e9	−0.762961
$778$	0	0
$779$	−2.55236e9	−0.193446
$780$	0	0
$781$	6.99379e9	0.525332
$782$	0	0
$783$	3.39812e9	0.252972
$784$	0	0
$785$	−9.94889e9	−0.734059
$786$	0	0
$787$	−2.40259e10	−1.75699	−0.878494	−	0.477753i	$-0.841451\pi$
$787$	−2.40259e10	−1.75699	−0.878494	+	0.477753i	$0.841451\pi$
$788$	0	0
$789$	−1.41742e10	−1.02738
$790$	0	0
$791$	3.98758e9	0.286478
$792$	0	0
$793$	3.22829e9	0.229888
$794$	0	0
$795$	−1.17092e10	−0.826502
$796$	0	0
$797$	2.14754e10	1.50258	0.751291	−	0.659972i	$-0.229432\pi$
$797$	2.14754e10	1.50258	0.751291	+	0.659972i	$0.229432\pi$
$798$	0	0
$799$	−2.35957e10	−1.63651
$800$	0	0
$801$	4.19660e8	0.0288525
$802$	0	0
$803$	−7.02645e9	−0.478885
$804$	0	0
$805$	−3.26131e9	−0.220347
$806$	0	0
$807$	1.95828e9	0.131165
$808$	0	0
$809$	2.47675e10	1.64460	0.822302	−	0.569051i	$-0.192689\pi$
$809$	2.47675e10	1.64460	0.822302	+	0.569051i	$0.192689\pi$
$810$	0	0
$811$	−7.32852e9	−0.482440	−0.241220	−	0.970471i	$-0.577547\pi$
$811$	−7.32852e9	−0.482440	−0.241220	+	0.970471i	$0.577547\pi$
$812$	0	0
$813$	−2.43646e10	−1.59016
$814$	0	0
$815$	2.70126e10	1.74789
$816$	0	0
$817$	2.36051e9	0.151436
$818$	0	0
$819$	−1.20158e9	−0.0764295
$820$	0	0
$821$	−1.60909e10	−1.01480	−0.507398	−	0.861712i	$-0.669393\pi$
$821$	−1.60909e10	−1.01480	−0.507398	+	0.861712i	$0.669393\pi$
$822$	0	0
$823$	−2.68161e9	−0.167685	−0.0838427	−	0.996479i	$-0.526719\pi$
$823$	−2.68161e9	−0.167685	−0.0838427	+	0.996479i	$0.526719\pi$
$824$	0	0
$825$	3.71566e9	0.230381
$826$	0	0
$827$	6.50971e9	0.400214	0.200107	−	0.979774i	$-0.435871\pi$
$827$	6.50971e9	0.400214	0.200107	+	0.979774i	$0.435871\pi$
$828$	0	0
$829$	5.32803e9	0.324807	0.162404	−	0.986724i	$-0.448075\pi$
$829$	5.32803e9	0.324807	0.162404	+	0.986724i	$0.448075\pi$
$830$	0	0
$831$	−1.84062e10	−1.11266
$832$	0	0
$833$	2.77665e9	0.166442
$834$	0	0
$835$	−3.50049e10	−2.08078
$836$	0	0
$837$	8.68750e9	0.512101
$838$	0	0
$839$	4.50948e9	0.263608	0.131804	−	0.991276i	$-0.457923\pi$
$839$	4.50948e9	0.263608	0.131804	+	0.991276i	$0.457923\pi$
$840$	0	0
$841$	−1.45808e10	−0.845268
$842$	0	0
$843$	2.50618e10	1.44084
$844$	0	0
$845$	1.65079e10	0.941226
$846$	0	0
$847$	−4.53850e9	−0.256638
$848$	0	0
$849$	2.69017e9	0.150870
$850$	0	0
$851$	1.51031e10	0.840064
$852$	0	0
$853$	2.18437e10	1.20505	0.602524	−	0.798101i	$-0.294162\pi$
$853$	2.18437e10	1.20505	0.602524	+	0.798101i	$0.294162\pi$
$854$	0	0
$855$	1.09565e9	0.0599499
$856$	0	0
$857$	2.66458e10	1.44609	0.723047	−	0.690799i	$-0.242741\pi$
$857$	2.66458e10	1.44609	0.723047	+	0.690799i	$0.242741\pi$
$858$	0	0
$859$	5.09522e9	0.274276	0.137138	−	0.990552i	$-0.456210\pi$
$859$	5.09522e9	0.274276	0.137138	+	0.990552i	$0.456210\pi$
$860$	0	0
$861$	−1.50226e10	−0.802111
$862$	0	0
$863$	−2.62987e10	−1.39282	−0.696412	−	0.717642i	$-0.745221\pi$
$863$	−2.62987e10	−1.39282	−0.696412	+	0.717642i	$0.745221\pi$
$864$	0	0
$865$	3.03561e10	1.59474
$866$	0	0
$867$	−8.31499e9	−0.433306
$868$	0	0
$869$	−5.45698e8	−0.0282087
$870$	0	0
$871$	−7.12106e8	−0.0365158
$872$	0	0
$873$	2.15520e9	0.109632
$874$	0	0
$875$	5.75210e9	0.290267
$876$	0	0
$877$	9.15180e9	0.458150	0.229075	−	0.973409i	$-0.426430\pi$
$877$	9.15180e9	0.458150	0.229075	+	0.973409i	$0.426430\pi$
$878$	0	0
$879$	2.64886e10	1.31552
$880$	0	0
$881$	2.00826e10	0.989472	0.494736	−	0.869043i	$-0.335265\pi$
$881$	2.00826e10	0.989472	0.494736	+	0.869043i	$0.335265\pi$
$882$	0	0
$883$	−1.83002e10	−0.894525	−0.447262	−	0.894403i	$-0.647601\pi$
$883$	−1.83002e10	−0.894525	−0.447262	+	0.894403i	$0.647601\pi$
$884$	0	0
$885$	−1.31083e9	−0.0635690
$886$	0	0
$887$	3.45826e10	1.66389	0.831945	−	0.554858i	$-0.187227\pi$
$887$	3.45826e10	1.66389	0.831945	+	0.554858i	$0.187227\pi$
$888$	0	0
$889$	−2.57813e9	−0.123069
$890$	0	0
$891$	1.49421e10	0.707683
$892$	0	0
$893$	3.30291e9	0.155209
$894$	0	0
$895$	−1.37016e10	−0.638837
$896$	0	0
$897$	5.69365e9	0.263401
$898$	0	0
$899$	6.82373e9	0.313229
$900$	0	0
$901$	−1.50921e10	−0.687403
$902$	0	0
$903$	1.38934e10	0.627917
$904$	0	0
$905$	−3.37874e10	−1.51525
$906$	0	0
$907$	1.80576e10	0.803590	0.401795	−	0.915730i	$-0.368387\pi$
$907$	1.80576e10	0.803590	0.401795	+	0.915730i	$0.368387\pi$
$908$	0	0
$909$	3.89920e9	0.172187
$910$	0	0
$911$	−2.82439e9	−0.123769	−0.0618844	−	0.998083i	$-0.519711\pi$
$911$	−2.82439e9	−0.123769	−0.0618844	+	0.998083i	$0.519711\pi$
$912$	0	0
$913$	6.18129e9	0.268801
$914$	0	0
$915$	−1.73258e10	−0.747684
$916$	0	0
$917$	8.35673e9	0.357885
$918$	0	0
$919$	−3.32469e10	−1.41302	−0.706508	−	0.707705i	$-0.749730\pi$
$919$	−3.32469e10	−1.41302	−0.706508	+	0.707705i	$0.749730\pi$
$920$	0	0
$921$	5.52658e10	2.33103
$922$	0	0
$923$	9.54068e9	0.399368
$924$	0	0
$925$	1.34455e10	0.558576
$926$	0	0
$927$	−1.43532e10	−0.591792
$928$	0	0
$929$	−1.01517e10	−0.415415	−0.207708	−	0.978191i	$-0.566600\pi$
$929$	−1.01517e10	−0.415415	−0.207708	+	0.978191i	$0.566600\pi$
$930$	0	0
$931$	−3.88673e8	−0.0157856
$932$	0	0
$933$	−3.75537e10	−1.51379
$934$	0	0
$935$	1.90663e10	0.762827
$936$	0	0
$937$	−1.45235e10	−0.576745	−0.288373	−	0.957518i	$-0.593114\pi$
$937$	−1.45235e10	−0.576745	−0.288373	+	0.957518i	$0.593114\pi$
$938$	0	0
$939$	2.47966e10	0.977377
$940$	0	0
$941$	−1.44927e10	−0.567002	−0.283501	−	0.958972i	$-0.591496\pi$
$941$	−1.44927e10	−0.567002	−0.283501	+	0.958972i	$0.591496\pi$
$942$	0	0
$943$	2.27424e10	0.883171
$944$	0	0
$945$	−7.28714e9	−0.280896
$946$	0	0
$947$	−4.05814e10	−1.55275	−0.776376	−	0.630270i	$-0.782944\pi$
$947$	−4.05814e10	−1.55275	−0.776376	+	0.630270i	$0.782944\pi$
$948$	0	0
$949$	−9.58523e9	−0.364058
$950$	0	0
$951$	1.55535e10	0.586402
$952$	0	0
$953$	−4.58321e10	−1.71532	−0.857659	−	0.514219i	$-0.828082\pi$
$953$	−4.58321e10	−1.71532	−0.857659	+	0.514219i	$0.828082\pi$
$954$	0	0
$955$	3.74732e10	1.39222
$956$	0	0
$957$	7.32515e9	0.270162
$958$	0	0
$959$	−1.29785e10	−0.475180
$960$	0	0
$961$	−1.00674e10	−0.365918
$962$	0	0
$963$	−1.16206e10	−0.419311
$964$	0	0
$965$	3.12862e10	1.12074
$966$	0	0
$967$	1.05053e10	0.373608	0.186804	−	0.982397i	$-0.440187\pi$
$967$	1.05053e10	0.373608	0.186804	+	0.982397i	$0.440187\pi$
$968$	0	0
$969$	4.42013e9	0.156064
$970$	0	0
$971$	1.92547e10	0.674948	0.337474	−	0.941335i	$-0.390427\pi$
$971$	1.92547e10	0.674948	0.337474	+	0.941335i	$0.390427\pi$
$972$	0	0
$973$	−1.64186e10	−0.571400
$974$	0	0
$975$	5.06877e9	0.175141
$976$	0	0
$977$	−3.58166e10	−1.22872	−0.614360	−	0.789026i	$-0.710586\pi$
$977$	−3.58166e10	−1.22872	−0.614360	+	0.789026i	$0.710586\pi$
$978$	0	0
$979$	−1.02225e9	−0.0348192
$980$	0	0
$981$	−5.05291e9	−0.170884
$982$	0	0
$983$	1.04773e10	0.351812	0.175906	−	0.984407i	$-0.443714\pi$
$983$	1.04773e10	0.351812	0.175906	+	0.984407i	$0.443714\pi$
$984$	0	0
$985$	−2.09639e10	−0.698950
$986$	0	0
$987$	1.94402e10	0.643562
$988$	0	0
$989$	−2.10329e10	−0.691373
$990$	0	0
$991$	−2.33526e10	−0.762215	−0.381108	−	0.924531i	$-0.624457\pi$
$991$	−2.33526e10	−0.762215	−0.381108	+	0.924531i	$0.624457\pi$
$992$	0	0
$993$	6.68835e10	2.16769
$994$	0	0
$995$	−4.07918e10	−1.31278
$996$	0	0
$997$	3.35975e10	1.07368	0.536838	−	0.843685i	$-0.319619\pi$
$997$	3.35975e10	1.07368	0.536838	+	0.843685i	$0.319619\pi$
$998$	0	0
$999$	3.37467e10	1.07091

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.8.a.c.1.2	✓	5
4.3	odd	2		224.8.a.d.1.4	yes	5
8.3	odd	2		448.8.a.ba.1.2		5
8.5	even	2		448.8.a.bb.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.8.a.c.1.2	✓	5	1.1	even	1	trivial
224.8.a.d.1.4	yes	5	4.3	odd	2
448.8.a.ba.1.2		5	8.3	odd	2
448.8.a.bb.1.4		5	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	224.a (trivial)

\(f(q)\)	\(=\)	\(q-56.6900 q^{3} -323.003 q^{5} +343.000 q^{7} +1026.75 q^{9} -2501.08 q^{11} -3411.88 q^{13} +18311.1 q^{15} +23601.1 q^{17} -3303.67 q^{19} -19444.7 q^{21} +29436.8 q^{23} +26206.1 q^{25} +65774.3 q^{27} +51663.4 q^{29} +132080. q^{31} +141786. q^{33} -110790. q^{35} +513068. q^{37} +193419. q^{39} +772583. q^{41} -714511. q^{43} -331645. q^{45} -999769. q^{47} +117649. q^{49} -1.33795e6 q^{51} -639463. q^{53} +807856. q^{55} +187285. q^{57} -71586.9 q^{59} -946191. q^{61} +352177. q^{63} +1.10205e6 q^{65} +208714. q^{67} -1.66877e6 q^{69} -2.79631e6 q^{71} +2.80937e6 q^{73} -1.48563e6 q^{75} -857869. q^{77} +218185. q^{79} -5.97426e6 q^{81} -2.47145e6 q^{83} -7.62324e6 q^{85} -2.92880e6 q^{87} +408724. q^{89} -1.17027e6 q^{91} -7.48764e6 q^{93} +1.06710e6 q^{95} +2.09904e6 q^{97} -2.56799e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 54 q^{3} + 84 q^{5} + 1715 q^{7} + 4133 q^{9} - 5324 q^{11} - 9880 q^{13} - 30128 q^{15} - 44322 q^{17} - 22898 q^{19} - 18522 q^{21} + 141016 q^{23} + 150679 q^{25} - 135972 q^{27} - 18998 q^{29}+ \cdots - 58451180 q^{99}+O(q^{100}) \) 5 * q - 54 * q^3 + 84 * q^5 + 1715 * q^7 + 4133 * q^9 - 5324 * q^11 - 9880 * q^13 - 30128 * q^15 - 44322 * q^17 - 22898 * q^19 - 18522 * q^21 + 141016 * q^23 + 150679 * q^25 - 135972 * q^27 - 18998 * q^29 + 167668 * q^31 + 830696 * q^33 + 28812 * q^35 - 434206 * q^37 - 1100728 * q^39 + 44278 * q^41 - 465076 * q^43 + 1823220 * q^45 - 7508 * q^47 + 588245 * q^49 - 65012 * q^51 - 1901986 * q^53 - 4619216 * q^55 - 3116556 * q^57 - 2342994 * q^59 - 553300 * q^61 + 1417619 * q^63 + 1749208 * q^65 - 1908152 * q^67 - 554944 * q^69 - 1721104 * q^71 + 1237634 * q^73 - 19901682 * q^75 - 1826132 * q^77 - 9787808 * q^79 + 4138481 * q^81 - 17717898 * q^83 - 13248184 * q^85 - 25003324 * q^87 + 7408138 * q^89 - 3388840 * q^91 + 13173560 * q^93 - 16659216 * q^95 - 17027554 * q^97 - 58451180 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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