LMFDB - Embedded newform 9.24.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,24,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 24, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	9.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$30.1683633611$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1128.00 q^{2} -7.11622e6 q^{4} +4.88637e7 q^{5} -1.72369e9 q^{7} +1.74895e10 q^{8} +O(q^{10})$

\(q-1128.00 q^{2} -7.11622e6 q^{4} +4.88637e7 q^{5} -1.72369e9 q^{7} +1.74895e10 q^{8} -5.51183e10 q^{10} +1.42826e12 q^{11} -8.22096e12 q^{13} +1.94432e12 q^{14} +3.99671e13 q^{16} +5.98921e12 q^{17} +6.80005e14 q^{19} -3.47725e14 q^{20} -1.61108e15 q^{22} -1.54406e13 q^{23} -9.53326e15 q^{25} +9.27325e15 q^{26} +1.22662e16 q^{28} -1.15094e17 q^{29} -9.08297e16 q^{31} -1.91795e17 q^{32} -6.75583e15 q^{34} -8.42259e16 q^{35} -1.29787e18 q^{37} -7.67046e17 q^{38} +8.54600e17 q^{40} -5.21404e18 q^{41} -2.41043e18 q^{43} -1.01638e19 q^{44} +1.74171e16 q^{46} +2.31327e19 q^{47} -2.43976e19 q^{49} +1.07535e19 q^{50} +5.85022e19 q^{52} +4.45126e19 q^{53} +6.97903e19 q^{55} -3.01464e19 q^{56} +1.29826e20 q^{58} +3.23974e20 q^{59} -1.99406e20 q^{61} +1.02456e20 q^{62} -1.18924e20 q^{64} -4.01707e20 q^{65} -6.46393e20 q^{67} -4.26206e19 q^{68} +9.50068e19 q^{70} -3.55146e21 q^{71} +3.35319e21 q^{73} +1.46400e21 q^{74} -4.83907e21 q^{76} -2.46188e21 q^{77} -6.87213e21 q^{79} +1.95294e21 q^{80} +5.88143e21 q^{82} +1.16977e21 q^{83} +2.92655e20 q^{85} +2.71897e21 q^{86} +2.49795e22 q^{88} +2.34572e22 q^{89} +1.41704e22 q^{91} +1.09879e20 q^{92} -2.60937e22 q^{94} +3.32276e22 q^{95} -3.06039e22 q^{97} +2.75205e22 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−1128.00	−0.389461	−0.194731	−	0.980857i	$-0.562383\pi$
$2$	−1128.00	−0.389461	−0.194731	+	0.980857i	$0.562383\pi$
$3$	0	0
$3$	0	0
$4$	−7.11622e6	−0.848320
$5$	4.88637e7	0.447540	0.223770	−	0.974642i	$-0.428164\pi$
$5$	4.88637e7	0.447540	0.223770	+	0.974642i	$0.428164\pi$
$6$	0	0
$7$	−1.72369e9	−0.329482	−0.164741	−	0.986337i	$-0.552679\pi$
$7$	−1.72369e9	−0.329482	−0.164741	+	0.986337i	$0.552679\pi$
$8$	1.74895e10	0.719849
$9$	0	0
$10$	−5.51183e10	−0.174299
$11$	1.42826e12	1.50936	0.754679	−	0.656094i	$-0.227793\pi$
$11$	1.42826e12	1.50936	0.754679	+	0.656094i	$0.227793\pi$
$12$	0	0
$13$	−8.22096e12	−1.27226	−0.636128	−	0.771584i	$-0.719465\pi$
$13$	−8.22096e12	−1.27226	−0.636128	+	0.771584i	$0.719465\pi$
$14$	1.94432e12	0.128320
$15$	0	0
$16$	3.99671e13	0.567967
$17$	5.98921e12	0.0423845	0.0211922	−	0.999775i	$-0.493254\pi$
$17$	5.98921e12	0.0423845	0.0211922	+	0.999775i	$0.493254\pi$
$18$	0	0
$19$	6.80005e14	1.33920	0.669601	−	0.742721i	$-0.266465\pi$
$19$	6.80005e14	1.33920	0.669601	+	0.742721i	$0.266465\pi$
$20$	−3.47725e14	−0.379657
$21$	0	0
$22$	−1.61108e15	−0.587836
$23$	−1.54406e13	−0.00337906	−0.00168953	−	0.999999i	$-0.500538\pi$
$23$	−1.54406e13	−0.00337906	−0.00168953	+	0.999999i	$0.500538\pi$
$24$	0	0
$25$	−9.53326e15	−0.799708
$26$	9.27325e15	0.495494
$27$	0	0
$28$	1.22662e16	0.279506
$29$	−1.15094e17	−1.75177	−0.875884	−	0.482522i	$-0.839721\pi$
$29$	−1.15094e17	−1.75177	−0.875884	+	0.482522i	$0.839721\pi$
$30$	0	0
$31$	−9.08297e16	−0.642050	−0.321025	−	0.947071i	$-0.604027\pi$
$31$	−9.08297e16	−0.642050	−0.321025	+	0.947071i	$0.604027\pi$
$32$	−1.91795e17	−0.941050
$33$	0	0
$34$	−6.75583e15	−0.0165071
$35$	−8.42259e16	−0.147456
$36$	0	0
$37$	−1.29787e18	−1.19926	−0.599629	−	0.800278i	$-0.704685\pi$
$37$	−1.29787e18	−1.19926	−0.599629	+	0.800278i	$0.704685\pi$
$38$	−7.67046e17	−0.521567
$39$	0	0
$40$	8.54600e17	0.322161
$41$	−5.21404e18	−1.47965	−0.739826	−	0.672798i	$-0.765092\pi$
$41$	−5.21404e18	−1.47965	−0.739826	+	0.672798i	$0.765092\pi$
$42$	0	0
$43$	−2.41043e18	−0.395556	−0.197778	−	0.980247i	$-0.563373\pi$
$43$	−2.41043e18	−0.395556	−0.197778	+	0.980247i	$0.563373\pi$
$44$	−1.01638e19	−1.28042
$45$	0	0
$46$	1.74171e16	0.00131601
$47$	2.31327e19	1.36490	0.682449	−	0.730933i	$-0.260915\pi$
$47$	2.31327e19	1.36490	0.682449	+	0.730933i	$0.260915\pi$
$48$	0	0
$49$	−2.43976e19	−0.891442
$50$	1.07535e19	0.311455
$51$	0	0
$52$	5.85022e19	1.07928
$53$	4.45126e19	0.659646	0.329823	−	0.944043i	$-0.393011\pi$
$53$	4.45126e19	0.659646	0.329823	+	0.944043i	$0.393011\pi$
$54$	0	0
$55$	6.97903e19	0.675498
$56$	−3.01464e19	−0.237177
$57$	0	0
$58$	1.29826e20	0.682245
$59$	3.23974e20	1.39866	0.699331	−	0.714798i	$-0.253481\pi$
$59$	3.23974e20	1.39866	0.699331	+	0.714798i	$0.253481\pi$
$60$	0	0
$61$	−1.99406e20	−0.586740	−0.293370	−	0.955999i	$-0.594777\pi$
$61$	−1.99406e20	−0.586740	−0.293370	+	0.955999i	$0.594777\pi$
$62$	1.02456e20	0.250053
$63$	0	0
$64$	−1.18924e20	−0.201464
$65$	−4.01707e20	−0.569385
$66$	0	0
$67$	−6.46393e20	−0.646601	−0.323300	−	0.946296i	$-0.604792\pi$
$67$	−6.46393e20	−0.646601	−0.323300	+	0.946296i	$0.604792\pi$
$68$	−4.26206e19	−0.0359556
$69$	0	0
$70$	9.50068e19	0.0574284
$71$	−3.55146e21	−1.82363	−0.911814	−	0.410604i	$-0.865318\pi$
$71$	−3.55146e21	−1.82363	−0.911814	+	0.410604i	$0.865318\pi$
$72$	0	0
$73$	3.35319e21	1.25096	0.625482	−	0.780238i	$-0.284902\pi$
$73$	3.35319e21	1.25096	0.625482	+	0.780238i	$0.284902\pi$
$74$	1.46400e21	0.467064
$75$	0	0
$76$	−4.83907e21	−1.13607
$77$	−2.46188e21	−0.497306
$78$	0	0
$79$	−6.87213e21	−1.03367	−0.516833	−	0.856086i	$-0.672889\pi$
$79$	−6.87213e21	−1.03367	−0.516833	+	0.856086i	$0.672889\pi$
$80$	1.95294e21	0.254188
$81$	0	0
$82$	5.88143e21	0.576267
$83$	1.16977e21	0.0997016	0.0498508	−	0.998757i	$-0.484125\pi$
$83$	1.16977e21	0.0997016	0.0498508	+	0.998757i	$0.484125\pi$
$84$	0	0
$85$	2.92655e20	0.0189687
$86$	2.71897e21	0.154054
$87$	0	0
$88$	2.49795e22	1.08651
$89$	2.34572e22	0.895966	0.447983	−	0.894042i	$-0.352143\pi$
$89$	2.34572e22	0.895966	0.447983	+	0.894042i	$0.352143\pi$
$90$	0	0
$91$	1.41704e22	0.419185
$92$	1.09879e20	0.00286652
$93$	0	0
$94$	−2.60937e22	−0.531575
$95$	3.32276e22	0.599346
$96$	0	0
$97$	−3.06039e22	−0.434412	−0.217206	−	0.976126i	$-0.569694\pi$
$97$	−3.06039e22	−0.434412	−0.217206	+	0.976126i	$0.569694\pi$
$98$	2.75205e22	0.347182
$99$	0	0
$100$	6.78408e22	0.678408
$101$	2.39411e21	0.0213525	0.0106762	−	0.999943i	$-0.496602\pi$
$101$	2.39411e21	0.0213525	0.0106762	+	0.999943i	$0.496602\pi$
$102$	0	0
$103$	−2.98735e22	−0.212646	−0.106323	−	0.994332i	$-0.533908\pi$
$103$	−2.98735e22	−0.212646	−0.106323	+	0.994332i	$0.533908\pi$
$104$	−1.43780e23	−0.915831
$105$	0	0
$106$	−5.02102e22	−0.256906
$107$	−3.52639e23	−1.61963	−0.809817	−	0.586683i	$-0.800433\pi$
$107$	−3.52639e23	−1.61963	−0.809817	+	0.586683i	$0.800433\pi$
$108$	0	0
$109$	−1.52076e23	−0.564489	−0.282245	−	0.959342i	$-0.591079\pi$
$109$	−1.52076e23	−0.564489	−0.282245	+	0.959342i	$0.591079\pi$
$110$	−7.87234e22	−0.263080
$111$	0	0
$112$	−6.88909e22	−0.187135
$113$	−5.18685e22	−0.127204	−0.0636021	−	0.997975i	$-0.520259\pi$
$113$	−5.18685e22	−0.127204	−0.0636021	+	0.997975i	$0.520259\pi$
$114$	0	0
$115$	−7.54488e20	−0.00151226
$116$	8.19036e23	1.48606
$117$	0	0
$118$	−3.65443e23	−0.544724
$119$	−1.03235e22	−0.0139649
$120$	0	0
$121$	1.14451e24	1.27816
$122$	2.24930e23	0.228512
$123$	0	0
$124$	6.46365e23	0.544663
$125$	−1.04833e24	−0.805441
$126$	0	0
$127$	−3.34992e23	−0.214433	−0.107217	−	0.994236i	$-0.534194\pi$
$127$	−3.34992e23	−0.214433	−0.107217	+	0.994236i	$0.534194\pi$
$128$	1.74304e24	1.01951
$129$	0	0
$130$	4.53125e23	0.221753
$131$	8.94767e23	0.400950	0.200475	−	0.979699i	$-0.435752\pi$
$131$	8.94767e23	0.400950	0.200475	+	0.979699i	$0.435752\pi$
$132$	0	0
$133$	−1.17212e24	−0.441243
$134$	7.29131e23	0.251826
$135$	0	0
$136$	1.04748e23	0.0305104
$137$	1.52550e24	0.408437	0.204219	−	0.978925i	$-0.434535\pi$
$137$	1.52550e24	0.408437	0.204219	+	0.978925i	$0.434535\pi$
$138$	0	0
$139$	−2.87052e24	−0.650565	−0.325283	−	0.945617i	$-0.605459\pi$
$139$	−2.87052e24	−0.650565	−0.325283	+	0.945617i	$0.605459\pi$
$140$	5.99370e23	0.125090
$141$	0	0
$142$	4.00605e24	0.710232
$143$	−1.17417e25	−1.92029
$144$	0	0
$145$	−5.62393e24	−0.783985
$146$	−3.78240e24	−0.487202
$147$	0	0
$148$	9.23596e24	1.01735
$149$	−7.06708e24	−0.720440	−0.360220	−	0.932867i	$-0.617298\pi$
$149$	−7.06708e24	−0.720440	−0.360220	+	0.932867i	$0.617298\pi$
$150$	0	0
$151$	−5.44882e24	−0.476505	−0.238253	−	0.971203i	$-0.576575\pi$
$151$	−5.44882e24	−0.476505	−0.238253	+	0.971203i	$0.576575\pi$
$152$	1.18929e25	0.964023
$153$	0	0
$154$	2.77700e24	0.193681
$155$	−4.43828e24	−0.287343
$156$	0	0
$157$	−2.79179e25	−1.55968	−0.779841	−	0.625977i	$-0.784700\pi$
$157$	−2.79179e25	−1.55968	−0.779841	+	0.625977i	$0.784700\pi$
$158$	7.75177e24	0.402573
$159$	0	0
$160$	−9.37182e24	−0.421157
$161$	2.66149e22	0.00111334
$162$	0	0
$163$	4.83707e25	1.75560	0.877799	−	0.479029i	$-0.159011\pi$
$163$	4.83707e25	1.75560	0.877799	+	0.479029i	$0.159011\pi$
$164$	3.71042e25	1.25522
$165$	0	0
$166$	−1.31950e24	−0.0388299
$167$	−3.59666e25	−0.987779	−0.493890	−	0.869525i	$-0.664425\pi$
$167$	−3.59666e25	−0.987779	−0.493890	+	0.869525i	$0.664425\pi$
$168$	0	0
$169$	2.58303e25	0.618633
$170$	−3.30115e23	−0.00738759
$171$	0	0
$172$	1.71532e25	0.335558
$173$	6.18040e25	1.13106	0.565531	−	0.824727i	$-0.308671\pi$
$173$	6.18040e25	1.13106	0.565531	+	0.824727i	$0.308671\pi$
$174$	0	0
$175$	1.64324e25	0.263489
$176$	5.70836e25	0.857265
$177$	0	0
$178$	−2.64597e25	−0.348944
$179$	−4.18182e25	−0.517077	−0.258539	−	0.966001i	$-0.583241\pi$
$179$	−4.18182e25	−0.517077	−0.258539	+	0.966001i	$0.583241\pi$
$180$	0	0
$181$	−1.04652e26	−1.13879	−0.569394	−	0.822065i	$-0.692822\pi$
$181$	−1.04652e26	−1.13879	−0.569394	+	0.822065i	$0.692822\pi$
$182$	−1.59842e25	−0.163256
$183$	0	0
$184$	−2.70048e23	−0.00243241
$185$	−6.34189e25	−0.536715
$186$	0	0
$187$	8.55417e24	0.0639734
$188$	−1.64617e26	−1.15787
$189$	0	0
$190$	−3.74807e25	−0.233422
$191$	−1.25691e26	−0.736923	−0.368461	−	0.929643i	$-0.620115\pi$
$191$	−1.25691e26	−0.736923	−0.368461	+	0.929643i	$0.620115\pi$
$192$	0	0
$193$	−8.55418e25	−0.444907	−0.222453	−	0.974943i	$-0.571407\pi$
$193$	−8.55418e25	−0.444907	−0.222453	+	0.974943i	$0.571407\pi$
$194$	3.45212e25	0.169187
$195$	0	0
$196$	1.73619e26	0.756228
$197$	9.41370e25	0.386722	0.193361	−	0.981128i	$-0.438061\pi$
$197$	9.41370e25	0.386722	0.193361	+	0.981128i	$0.438061\pi$
$198$	0	0
$199$	7.46484e25	0.273030	0.136515	−	0.990638i	$-0.456410\pi$
$199$	7.46484e25	0.273030	0.136515	+	0.990638i	$0.456410\pi$
$200$	−1.66732e26	−0.575669
$201$	0	0
$202$	−2.70056e24	−0.00831597
$203$	1.98387e26	0.577175
$204$	0	0
$205$	−2.54777e26	−0.662203
$206$	3.36973e25	0.0828175
$207$	0	0
$208$	−3.28568e26	−0.722599
$209$	9.71227e26	2.02134
$210$	0	0
$211$	6.91338e26	1.28956	0.644781	−	0.764368i	$-0.276949\pi$
$211$	6.91338e26	1.28956	0.644781	+	0.764368i	$0.276949\pi$
$212$	−3.16762e26	−0.559591
$213$	0	0
$214$	3.97777e26	0.630784
$215$	−1.17783e26	−0.177027
$216$	0	0
$217$	1.56562e26	0.211544
$218$	1.71542e26	0.219847
$219$	0	0
$220$	−4.96643e26	−0.573038
$221$	−4.92371e25	−0.0539239
$222$	0	0
$223$	7.98521e26	0.788462	0.394231	−	0.919011i	$-0.371011\pi$
$223$	7.98521e26	0.788462	0.394231	+	0.919011i	$0.371011\pi$
$224$	3.30595e26	0.310059
$225$	0	0
$226$	5.85077e25	0.0495411
$227$	−1.19285e27	−0.960042	−0.480021	−	0.877257i	$-0.659371\pi$
$227$	−1.19285e27	−0.960042	−0.480021	+	0.877257i	$0.659371\pi$
$228$	0	0
$229$	−1.64063e27	−1.19372	−0.596860	−	0.802345i	$-0.703585\pi$
$229$	−1.64063e27	−1.19372	−0.596860	+	0.802345i	$0.703585\pi$
$230$	8.51062e23	0.000588967 0
$231$	0	0
$232$	−2.01293e27	−1.26101
$233$	1.47808e27	0.881263	0.440632	−	0.897688i	$-0.354754\pi$
$233$	1.47808e27	0.881263	0.440632	+	0.897688i	$0.354754\pi$
$234$	0	0
$235$	1.13035e27	0.610846
$236$	−2.30547e27	−1.18651
$237$	0	0
$238$	1.16449e25	0.00543879
$239$	1.00529e26	0.0447421	0.0223711	−	0.999750i	$-0.492878\pi$
$239$	1.00529e26	0.0447421	0.0223711	+	0.999750i	$0.492878\pi$
$240$	0	0
$241$	2.89067e27	1.16897	0.584483	−	0.811406i	$-0.301297\pi$
$241$	2.89067e27	1.16897	0.584483	+	0.811406i	$0.301297\pi$
$242$	−1.29100e27	−0.497795
$243$	0	0
$244$	1.41902e27	0.497743
$245$	−1.19216e27	−0.398956
$246$	0	0
$247$	−5.59030e27	−1.70381
$248$	−1.58856e27	−0.462179
$249$	0	0
$250$	1.18252e27	0.313688
$251$	3.03848e26	0.0769855	0.0384928	−	0.999259i	$-0.487744\pi$
$251$	3.03848e26	0.0769855	0.0384928	+	0.999259i	$0.487744\pi$
$252$	0	0
$253$	−2.20533e25	−0.00510021
$254$	3.77872e26	0.0835134
$255$	0	0
$256$	−9.68545e26	−0.195596
$257$	5.74761e27	1.10983	0.554915	−	0.831907i	$-0.312751\pi$
$257$	5.74761e27	1.10983	0.554915	+	0.831907i	$0.312751\pi$
$258$	0	0
$259$	2.23713e27	0.395134
$260$	2.85864e27	0.483020
$261$	0	0
$262$	−1.00930e27	−0.156154
$263$	9.08470e27	1.34530	0.672650	−	0.739961i	$-0.265156\pi$
$263$	9.08470e27	1.34530	0.672650	+	0.739961i	$0.265156\pi$
$264$	0	0
$265$	2.17505e27	0.295218
$266$	1.32215e27	0.171847
$267$	0	0
$268$	4.59987e27	0.548524
$269$	−2.45445e27	−0.280416	−0.140208	−	0.990122i	$-0.544777\pi$
$269$	−2.45445e27	−0.280416	−0.140208	+	0.990122i	$0.544777\pi$
$270$	0	0
$271$	5.65329e27	0.593136	0.296568	−	0.955012i	$-0.404158\pi$
$271$	5.65329e27	0.593136	0.296568	+	0.955012i	$0.404158\pi$
$272$	2.39371e26	0.0240730
$273$	0	0
$274$	−1.72076e27	−0.159070
$275$	−1.36160e28	−1.20705
$276$	0	0
$277$	3.55531e27	0.289975	0.144987	−	0.989433i	$-0.453686\pi$
$277$	3.55531e27	0.289975	0.144987	+	0.989433i	$0.453686\pi$
$278$	3.23795e27	0.253370
$279$	0	0
$280$	−1.47306e27	−0.106146
$281$	−2.83906e28	−1.96359	−0.981797	−	0.189931i	$-0.939173\pi$
$281$	−2.83906e28	−1.96359	−0.981797	+	0.189931i	$0.939173\pi$
$282$	0	0
$283$	−2.29202e28	−1.46108	−0.730542	−	0.682868i	$-0.760732\pi$
$283$	−2.29202e28	−1.46108	−0.730542	+	0.682868i	$0.760732\pi$
$284$	2.52730e28	1.54702
$285$	0	0
$286$	1.32446e28	0.747878
$287$	8.98738e27	0.487518
$288$	0	0
$289$	−1.99317e28	−0.998204
$290$	6.34379e27	0.305332
$291$	0	0
$292$	−2.38620e28	−1.06122
$293$	−3.43109e28	−1.46708	−0.733542	−	0.679644i	$-0.762134\pi$
$293$	−3.43109e28	−1.46708	−0.733542	+	0.679644i	$0.762134\pi$
$294$	0	0
$295$	1.58306e28	0.625957
$296$	−2.26991e28	−0.863284
$297$	0	0
$298$	7.97167e27	0.280584
$299$	1.26937e26	0.00429902
$300$	0	0
$301$	4.15484e27	0.130329
$302$	6.14627e27	0.185580
$303$	0	0
$304$	2.71779e28	0.760622
$305$	−9.74373e27	−0.262589
$306$	0	0
$307$	−1.16886e28	−0.292195	−0.146097	−	0.989270i	$-0.546671\pi$
$307$	−1.16886e28	−0.292195	−0.146097	+	0.989270i	$0.546671\pi$
$308$	1.75193e28	0.421875
$309$	0	0
$310$	5.00638e27	0.111909
$311$	2.49825e28	0.538135	0.269067	−	0.963121i	$-0.413285\pi$
$311$	2.49825e28	0.538135	0.269067	+	0.963121i	$0.413285\pi$
$312$	0	0
$313$	6.03885e28	1.20835	0.604177	−	0.796850i	$-0.293502\pi$
$313$	6.03885e28	1.20835	0.604177	+	0.796850i	$0.293502\pi$
$314$	3.14914e28	0.607436
$315$	0	0
$316$	4.89036e28	0.876879
$317$	1.04255e29	1.80266	0.901330	−	0.433133i	$-0.142592\pi$
$317$	1.04255e29	1.80266	0.901330	+	0.433133i	$0.142592\pi$
$318$	0	0
$319$	−1.64385e29	−2.64404
$320$	−5.81105e27	−0.0901633
$321$	0	0
$322$	−3.00216e25	−0.000433602 0
$323$	4.07270e27	0.0567614
$324$	0	0
$325$	7.83726e28	1.01743
$326$	−5.45622e28	−0.683737
$327$	0	0
$328$	−9.11906e28	−1.06513
$329$	−3.98735e28	−0.449709
$330$	0	0
$331$	1.38759e29	1.45962	0.729809	−	0.683651i	$-0.239609\pi$
$331$	1.38759e29	1.45962	0.729809	+	0.683651i	$0.239609\pi$
$332$	−8.32434e27	−0.0845789
$333$	0	0
$334$	4.05703e28	0.384702
$335$	−3.15851e28	−0.289379
$336$	0	0
$337$	−1.50795e29	−1.29016	−0.645078	−	0.764117i	$-0.723175\pi$
$337$	−1.50795e29	−1.29016	−0.645078	+	0.764117i	$0.723175\pi$
$338$	−2.91366e28	−0.240934
$339$	0	0
$340$	−2.08260e27	−0.0160916
$341$	−1.29729e29	−0.969083
$342$	0	0
$343$	8.92291e28	0.623196
$344$	−4.21572e28	−0.284741
$345$	0	0
$346$	−6.97149e28	−0.440504
$347$	1.45030e29	0.886482	0.443241	−	0.896403i	$-0.353829\pi$
$347$	1.45030e29	0.886482	0.443241	+	0.896403i	$0.353829\pi$
$348$	0	0
$349$	1.03224e29	0.590592	0.295296	−	0.955406i	$-0.404582\pi$
$349$	1.03224e29	0.590592	0.295296	+	0.955406i	$0.404582\pi$
$350$	−1.85357e28	−0.102619
$351$	0	0
$352$	−2.73934e29	−1.42038
$353$	9.96681e28	0.500204	0.250102	−	0.968220i	$-0.419536\pi$
$353$	9.96681e28	0.500204	0.250102	+	0.968220i	$0.419536\pi$
$354$	0	0
$355$	−1.73538e29	−0.816146
$356$	−1.66927e29	−0.760066
$357$	0	0
$358$	4.71709e28	0.201381
$359$	−1.99929e29	−0.826589	−0.413294	−	0.910597i	$-0.635622\pi$
$359$	−1.99929e29	−0.826589	−0.413294	+	0.910597i	$0.635622\pi$
$360$	0	0
$361$	2.04578e29	0.793461
$362$	1.18047e29	0.443514
$363$	0	0
$364$	−1.00840e29	−0.355603
$365$	1.63849e29	0.559856
$366$	0	0
$367$	5.60764e29	1.79937	0.899684	−	0.436541i	$-0.143797\pi$
$367$	5.60764e29	1.79937	0.899684	+	0.436541i	$0.143797\pi$
$368$	−6.17118e26	−0.00191919
$369$	0	0
$370$	7.15366e28	0.209030
$371$	−7.67259e28	−0.217341
$372$	0	0
$373$	−4.46930e29	−1.19011	−0.595056	−	0.803684i	$-0.702870\pi$
$373$	−4.46930e29	−1.19011	−0.595056	+	0.803684i	$0.702870\pi$
$374$	−9.64910e27	−0.0249151
$375$	0	0
$376$	4.04578e29	0.982521
$377$	9.46185e29	2.22869
$378$	0	0
$379$	−3.66574e29	−0.812477	−0.406239	−	0.913767i	$-0.633160\pi$
$379$	−3.66574e29	−0.812477	−0.406239	+	0.913767i	$0.633160\pi$
$380$	−2.36455e29	−0.508437
$381$	0	0
$382$	1.41780e29	0.287003
$383$	1.49291e29	0.293257	0.146628	−	0.989192i	$-0.453158\pi$
$383$	1.49291e29	0.293257	0.146628	+	0.989192i	$0.453158\pi$
$384$	0	0
$385$	−1.20297e29	−0.222564
$386$	9.64911e28	0.173274
$387$	0	0
$388$	2.17784e29	0.368520
$389$	1.97534e29	0.324506	0.162253	−	0.986749i	$-0.448124\pi$
$389$	1.97534e29	0.324506	0.162253	+	0.986749i	$0.448124\pi$
$390$	0	0
$391$	−9.24773e25	−0.000143220 0
$392$	−4.26701e29	−0.641703
$393$	0	0
$394$	−1.06187e29	−0.150613
$395$	−3.35798e29	−0.462606
$396$	0	0
$397$	−2.62393e28	−0.0341084	−0.0170542	−	0.999855i	$-0.505429\pi$
$397$	−2.62393e28	−0.0341084	−0.0170542	+	0.999855i	$0.505429\pi$
$398$	−8.42034e28	−0.106334
$399$	0	0
$400$	−3.81017e29	−0.454208
$401$	−4.34375e29	−0.503159	−0.251579	−	0.967837i	$-0.580950\pi$
$401$	−4.34375e29	−0.503159	−0.251579	+	0.967837i	$0.580950\pi$
$402$	0	0
$403$	7.46708e29	0.816851
$404$	−1.70370e28	−0.0181137
$405$	0	0
$406$	−2.23780e29	−0.224787
$407$	−1.85370e30	−1.81011
$408$	0	0
$409$	−8.14589e29	−0.751832	−0.375916	−	0.926654i	$-0.622672\pi$
$409$	−8.14589e29	−0.751832	−0.375916	+	0.926654i	$0.622672\pi$
$410$	2.87389e29	0.257902
$411$	0	0
$412$	2.12586e29	0.180392
$413$	−5.58431e29	−0.460833
$414$	0	0
$415$	5.71593e28	0.0446204
$416$	1.57674e30	1.19726
$417$	0	0
$418$	−1.09554e30	−0.787232
$419$	−1.28769e29	−0.0900221	−0.0450111	−	0.998986i	$-0.514332\pi$
$419$	−1.28769e29	−0.0900221	−0.0450111	+	0.998986i	$0.514332\pi$
$420$	0	0
$421$	−1.44725e30	−0.957852	−0.478926	−	0.877855i	$-0.658974\pi$
$421$	−1.44725e30	−0.957852	−0.478926	+	0.877855i	$0.658974\pi$
$422$	−7.79829e29	−0.502234
$423$	0	0
$424$	7.78501e29	0.474845
$425$	−5.70967e28	−0.0338952
$426$	0	0
$427$	3.43714e29	0.193320
$428$	2.50946e30	1.37397
$429$	0	0
$430$	1.32859e29	0.0689452
$431$	1.55229e30	0.784303	0.392152	−	0.919901i	$-0.371731\pi$
$431$	1.55229e30	0.784303	0.392152	+	0.919901i	$0.371731\pi$
$432$	0	0
$433$	3.69055e30	1.76799	0.883997	−	0.467493i	$-0.154843\pi$
$433$	3.69055e30	1.76799	0.883997	+	0.467493i	$0.154843\pi$
$434$	−1.76602e29	−0.0823880
$435$	0	0
$436$	1.08221e30	0.478867
$437$	−1.04997e28	−0.00452524
$438$	0	0
$439$	4.37828e29	0.179045	0.0895223	−	0.995985i	$-0.471466\pi$
$439$	4.37828e29	0.179045	0.0895223	+	0.995985i	$0.471466\pi$
$440$	1.22059e30	0.486256
$441$	0	0
$442$	5.55394e28	0.0210013
$443$	−1.47008e30	−0.541624	−0.270812	−	0.962632i	$-0.587292\pi$
$443$	−1.47008e30	−0.541624	−0.270812	+	0.962632i	$0.587292\pi$
$444$	0	0
$445$	1.14621e30	0.400980
$446$	−9.00732e29	−0.307075
$447$	0	0
$448$	2.04987e29	0.0663789
$449$	2.17842e29	0.0687558	0.0343779	−	0.999409i	$-0.489055\pi$
$449$	2.17842e29	0.0687558	0.0343779	+	0.999409i	$0.489055\pi$
$450$	0	0
$451$	−7.44702e30	−2.23333
$452$	3.69108e29	0.107910
$453$	0	0
$454$	1.34554e30	0.373899
$455$	6.92418e29	0.187602
$456$	0	0
$457$	−1.90386e30	−0.490454	−0.245227	−	0.969466i	$-0.578862\pi$
$457$	−1.90386e30	−0.490454	−0.245227	+	0.969466i	$0.578862\pi$
$458$	1.85063e30	0.464908
$459$	0	0
$460$	5.36910e27	0.00128288
$461$	4.96919e30	1.15804	0.579022	−	0.815312i	$-0.303434\pi$
$461$	4.96919e30	1.15804	0.579022	+	0.815312i	$0.303434\pi$
$462$	0	0
$463$	1.11956e30	0.248237	0.124118	−	0.992267i	$-0.460390\pi$
$463$	1.11956e30	0.248237	0.124118	+	0.992267i	$0.460390\pi$
$464$	−4.59998e30	−0.994946
$465$	0	0
$466$	−1.66728e30	−0.343218
$467$	−8.69961e30	−1.74725	−0.873626	−	0.486598i	$-0.838238\pi$
$467$	−8.69961e30	−1.74725	−0.873626	+	0.486598i	$0.838238\pi$
$468$	0	0
$469$	1.11418e30	0.213043
$470$	−1.27503e30	−0.237901
$471$	0	0
$472$	5.66614e30	1.00682
$473$	−3.44273e30	−0.597036
$474$	0	0
$475$	−6.48267e30	−1.07097
$476$	7.34646e28	0.0118467
$477$	0	0
$478$	−1.13397e29	−0.0174253
$479$	6.06140e29	0.0909315	0.0454658	−	0.998966i	$-0.485523\pi$
$479$	6.06140e29	0.0909315	0.0454658	+	0.998966i	$0.485523\pi$
$480$	0	0
$481$	1.06698e31	1.52576
$482$	−3.26067e30	−0.455267
$483$	0	0
$484$	−8.14456e30	−1.08429
$485$	−1.49542e30	−0.194417
$486$	0	0
$487$	−4.30348e30	−0.533626	−0.266813	−	0.963748i	$-0.585971\pi$
$487$	−4.30348e30	−0.533626	−0.266813	+	0.963748i	$0.585971\pi$
$488$	−3.48750e30	−0.422364
$489$	0	0
$490$	1.34476e30	0.155378
$491$	1.13566e31	1.28177	0.640885	−	0.767637i	$-0.278568\pi$
$491$	1.13566e31	1.28177	0.640885	+	0.767637i	$0.278568\pi$
$492$	0	0
$493$	−6.89323e29	−0.0742478
$494$	6.30586e30	0.663566
$495$	0	0
$496$	−3.63020e30	−0.364663
$497$	6.12161e30	0.600852
$498$	0	0
$499$	9.99815e30	0.937052	0.468526	−	0.883450i	$-0.344785\pi$
$499$	9.99815e30	0.937052	0.468526	+	0.883450i	$0.344785\pi$
$500$	7.46016e30	0.683272
$501$	0	0
$502$	−3.42741e29	−0.0299829
$503$	6.16806e30	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$503$	6.16806e30	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$504$	0	0
$505$	1.16985e29	0.00955609
$506$	2.48761e28	0.00198633
$507$	0	0
$508$	2.38388e30	0.181908
$509$	−2.98637e30	−0.222786	−0.111393	−	0.993776i	$-0.535531\pi$
$509$	−2.98637e30	−0.222786	−0.111393	+	0.993776i	$0.535531\pi$
$510$	0	0
$511$	−5.77985e30	−0.412170
$512$	−1.35292e31	−0.943335
$513$	0	0
$514$	−6.48331e30	−0.432236
$515$	−1.45973e30	−0.0951677
$516$	0	0
$517$	3.30395e31	2.06012
$518$	−2.52348e30	−0.153889
$519$	0	0
$520$	−7.02563e30	−0.409871
$521$	8.51705e30	0.486022	0.243011	−	0.970024i	$-0.421865\pi$
$521$	8.51705e30	0.486022	0.243011	+	0.970024i	$0.421865\pi$
$522$	0	0
$523$	1.27472e31	0.696059	0.348029	−	0.937484i	$-0.386851\pi$
$523$	1.27472e31	0.696059	0.348029	+	0.937484i	$0.386851\pi$
$524$	−6.36736e30	−0.340134
$525$	0	0
$526$	−1.02475e31	−0.523942
$527$	−5.43998e29	−0.0272129
$528$	0	0
$529$	−2.08802e31	−0.999989
$530$	−2.45346e30	−0.114976
$531$	0	0
$532$	8.34105e30	0.374315
$533$	4.28644e31	1.88250
$534$	0	0
$535$	−1.72312e31	−0.724850
$536$	−1.13050e31	−0.465455
$537$	0	0
$538$	2.76862e30	0.109211
$539$	−3.48463e31	−1.34551
$540$	0	0
$541$	−1.00626e31	−0.372342	−0.186171	−	0.982517i	$-0.559608\pi$
$541$	−1.00626e31	−0.372342	−0.186171	+	0.982517i	$0.559608\pi$
$542$	−6.37691e30	−0.231004
$543$	0	0
$544$	−1.14870e30	−0.0398859
$545$	−7.43099e30	−0.252631
$546$	0	0
$547$	−4.27216e31	−1.39249	−0.696247	−	0.717802i	$-0.745148\pi$
$547$	−4.27216e31	−1.39249	−0.696247	+	0.717802i	$0.745148\pi$
$548$	−1.08558e31	−0.346485
$549$	0	0
$550$	1.53589e31	0.470098
$551$	−7.82647e31	−2.34597
$552$	0	0
$553$	1.18454e31	0.340574
$554$	−4.01039e30	−0.112934
$555$	0	0
$556$	2.04273e31	0.551887
$557$	6.81696e31	1.80408	0.902040	−	0.431652i	$-0.142069\pi$
$557$	6.81696e31	1.80408	0.902040	+	0.431652i	$0.142069\pi$
$558$	0	0
$559$	1.98161e31	0.503248
$560$	−3.36626e30	−0.0837502
$561$	0	0
$562$	3.20246e31	0.764744
$563$	−6.17906e30	−0.144569	−0.0722846	−	0.997384i	$-0.523029\pi$
$563$	−6.17906e30	−0.144569	−0.0722846	+	0.997384i	$0.523029\pi$
$564$	0	0
$565$	−2.53449e30	−0.0569289
$566$	2.58540e31	0.569035
$567$	0	0
$568$	−6.21131e31	−1.31274
$569$	1.87031e31	0.387366	0.193683	−	0.981064i	$-0.437957\pi$
$569$	1.87031e31	0.387366	0.193683	+	0.981064i	$0.437957\pi$
$570$	0	0
$571$	2.56599e31	0.510433	0.255217	−	0.966884i	$-0.417853\pi$
$571$	2.56599e31	0.510433	0.255217	+	0.966884i	$0.417853\pi$
$572$	8.35566e31	1.62902
$573$	0	0
$574$	−1.01378e31	−0.189870
$575$	1.47200e29	0.00270226
$576$	0	0
$577$	−2.35477e31	−0.415362	−0.207681	−	0.978197i	$-0.566592\pi$
$577$	−2.35477e31	−0.415362	−0.207681	+	0.978197i	$0.566592\pi$
$578$	2.24830e31	0.388762
$579$	0	0
$580$	4.00212e31	0.665071
$581$	−2.01632e30	−0.0328499
$582$	0	0
$583$	6.35758e31	0.995642
$584$	5.86454e31	0.900505
$585$	0	0
$586$	3.87027e31	0.571372
$587$	−7.54955e31	−1.09291	−0.546453	−	0.837490i	$-0.684022\pi$
$587$	−7.54955e31	−1.09291	−0.546453	+	0.837490i	$0.684022\pi$
$588$	0	0
$589$	−6.17647e31	−0.859834
$590$	−1.78569e31	−0.243786
$591$	0	0
$592$	−5.18723e31	−0.681139
$593$	−3.26412e31	−0.420376	−0.210188	−	0.977661i	$-0.567408\pi$
$593$	−3.26412e31	−0.420376	−0.210188	+	0.977661i	$0.567408\pi$
$594$	0	0
$595$	−5.04446e29	−0.00624985
$596$	5.02909e31	0.611164
$597$	0	0
$598$	−1.43185e29	−0.00167430
$599$	−5.33038e31	−0.611434	−0.305717	−	0.952122i	$-0.598896\pi$
$599$	−5.33038e31	−0.611434	−0.305717	+	0.952122i	$0.598896\pi$
$600$	0	0
$601$	1.42622e32	1.57446	0.787229	−	0.616661i	$-0.211515\pi$
$601$	1.42622e32	1.57446	0.787229	+	0.616661i	$0.211515\pi$
$602$	−4.68666e30	−0.0507579
$603$	0	0
$604$	3.87750e31	0.404229
$605$	5.59248e31	0.572029
$606$	0	0
$607$	−1.51222e32	−1.48917	−0.744586	−	0.667526i	$-0.767353\pi$
$607$	−1.51222e32	−1.48917	−0.744586	+	0.667526i	$0.767353\pi$
$608$	−1.30422e32	−1.26026
$609$	0	0
$610$	1.09909e31	0.102268
$611$	−1.90173e32	−1.73650
$612$	0	0
$613$	−1.24132e32	−1.09166	−0.545831	−	0.837895i	$-0.683786\pi$
$613$	−1.24132e32	−1.09166	−0.545831	+	0.837895i	$0.683786\pi$
$614$	1.31848e31	0.113798
$615$	0	0
$616$	−4.30569e31	−0.357985
$617$	1.89291e32	1.54472	0.772361	−	0.635184i	$-0.219076\pi$
$617$	1.89291e32	1.54472	0.772361	+	0.635184i	$0.219076\pi$
$618$	0	0
$619$	−1.33375e32	−1.04865	−0.524327	−	0.851517i	$-0.675683\pi$
$619$	−1.33375e32	−1.04865	−0.524327	+	0.851517i	$0.675683\pi$
$620$	3.15838e31	0.243759
$621$	0	0
$622$	−2.81802e31	−0.209583
$623$	−4.04329e31	−0.295204
$624$	0	0
$625$	6.24200e31	0.439241
$626$	−6.81182e31	−0.470607
$627$	0	0
$628$	1.98670e32	1.32311
$629$	−7.77324e30	−0.0508299
$630$	0	0
$631$	2.80913e31	0.177107	0.0885533	−	0.996071i	$-0.471776\pi$
$631$	2.80913e31	0.177107	0.0885533	+	0.996071i	$0.471776\pi$
$632$	−1.20190e32	−0.744083
$633$	0	0
$634$	−1.17599e32	−0.702066
$635$	−1.63690e31	−0.0959674
$636$	0	0
$637$	2.00572e32	1.13414
$638$	1.85426e32	1.02975
$639$	0	0
$640$	8.51714e31	0.456272
$641$	3.78628e31	0.199225	0.0996127	−	0.995026i	$-0.468240\pi$
$641$	3.78628e31	0.199225	0.0996127	+	0.995026i	$0.468240\pi$
$642$	0	0
$643$	−1.43262e32	−0.727283	−0.363641	−	0.931539i	$-0.618467\pi$
$643$	−1.43262e32	−0.727283	−0.363641	+	0.931539i	$0.618467\pi$
$644$	−1.89397e29	−0.000944466 0
$645$	0	0
$646$	−4.59400e30	−0.0221064
$647$	2.32752e31	0.110026	0.0550129	−	0.998486i	$-0.482480\pi$
$647$	2.32752e31	0.110026	0.0550129	+	0.998486i	$0.482480\pi$
$648$	0	0
$649$	4.62721e32	2.11108
$650$	−8.84043e31	−0.396251
$651$	0	0
$652$	−3.44217e32	−1.48931
$653$	4.07291e32	1.73142	0.865712	−	0.500543i	$-0.166866\pi$
$653$	4.07291e32	1.73142	0.865712	+	0.500543i	$0.166866\pi$
$654$	0	0
$655$	4.37216e31	0.179441
$656$	−2.08390e32	−0.840393
$657$	0	0
$658$	4.49773e31	0.175144
$659$	4.35631e32	1.66700	0.833501	−	0.552518i	$-0.186333\pi$
$659$	4.35631e32	1.66700	0.833501	+	0.552518i	$0.186333\pi$
$660$	0	0
$661$	−4.22347e32	−1.56082	−0.780410	−	0.625268i	$-0.784990\pi$
$661$	−4.22347e32	−1.56082	−0.780410	+	0.625268i	$0.784990\pi$
$662$	−1.56520e32	−0.568465
$663$	0	0
$664$	2.04586e31	0.0717701
$665$	−5.72740e31	−0.197474
$666$	0	0
$667$	1.77713e30	0.00591932
$668$	2.55946e32	0.837953
$669$	0	0
$670$	3.56280e31	0.112702
$671$	−2.84805e32	−0.885601
$672$	0	0
$673$	3.27631e32	0.984491	0.492245	−	0.870456i	$-0.336176\pi$
$673$	3.27631e32	0.984491	0.492245	+	0.870456i	$0.336176\pi$
$674$	1.70096e32	0.502466
$675$	0	0
$676$	−1.83815e32	−0.524799
$677$	−1.65241e32	−0.463820	−0.231910	−	0.972737i	$-0.574497\pi$
$677$	−1.65241e32	−0.463820	−0.231910	+	0.972737i	$0.574497\pi$
$678$	0	0
$679$	5.27516e31	0.143131
$680$	5.11838e30	0.0136546
$681$	0	0
$682$	1.46334e32	0.377420
$683$	−1.17826e32	−0.298816	−0.149408	−	0.988776i	$-0.547737\pi$
$683$	−1.17826e32	−0.298816	−0.149408	+	0.988776i	$0.547737\pi$
$684$	0	0
$685$	7.45416e31	0.182792
$686$	−1.00650e32	−0.242710
$687$	0	0
$688$	−9.63381e31	−0.224663
$689$	−3.65937e32	−0.839238
$690$	0	0
$691$	2.43790e32	0.540778	0.270389	−	0.962751i	$-0.412848\pi$
$691$	2.43790e32	0.540778	0.270389	+	0.962751i	$0.412848\pi$
$692$	−4.39811e32	−0.959502
$693$	0	0
$694$	−1.63594e32	−0.345250
$695$	−1.40264e32	−0.291154
$696$	0	0
$697$	−3.12280e31	−0.0627143
$698$	−1.16437e32	−0.230013
$699$	0	0
$700$	−1.16937e32	−0.223523
$701$	3.79931e32	0.714409	0.357205	−	0.934026i	$-0.383730\pi$
$701$	3.79931e32	0.714409	0.357205	+	0.934026i	$0.383730\pi$
$702$	0	0
$703$	−8.82561e32	−1.60605
$704$	−1.69854e32	−0.304082
$705$	0	0
$706$	−1.12426e32	−0.194810
$707$	−4.12670e30	−0.00703526
$708$	0	0
$709$	−1.44569e31	−0.0238586	−0.0119293	−	0.999929i	$-0.503797\pi$
$709$	−1.44569e31	−0.0238586	−0.0119293	+	0.999929i	$0.503797\pi$
$710$	1.95750e32	0.317857
$711$	0	0
$712$	4.10254e32	0.644960
$713$	1.40247e30	0.00216952
$714$	0	0
$715$	−5.73743e32	−0.859405
$716$	2.97588e32	0.438647
$717$	0	0
$718$	2.25520e32	0.321924
$719$	1.33387e32	0.187383	0.0936915	−	0.995601i	$-0.470133\pi$
$719$	1.33387e32	0.187383	0.0936915	+	0.995601i	$0.470133\pi$
$720$	0	0
$721$	5.14926e31	0.0700631
$722$	−2.30764e32	−0.309022
$723$	0	0
$724$	7.44725e32	0.966057
$725$	1.09722e33	1.40090
$726$	0	0
$727$	4.93859e31	0.0610882	0.0305441	−	0.999533i	$-0.490276\pi$
$727$	4.93859e31	0.0610882	0.0305441	+	0.999533i	$0.490276\pi$
$728$	2.47832e32	0.301750
$729$	0	0
$730$	−1.84822e32	−0.218042
$731$	−1.44366e31	−0.0167654
$732$	0	0
$733$	1.05389e33	1.18604	0.593019	−	0.805189i	$-0.297936\pi$
$733$	1.05389e33	1.18604	0.593019	+	0.805189i	$0.297936\pi$
$734$	−6.32541e32	−0.700784
$735$	0	0
$736$	2.96144e30	0.00317986
$737$	−9.23219e32	−0.975952
$738$	0	0
$739$	1.09776e33	1.12486	0.562428	−	0.826846i	$-0.309867\pi$
$739$	1.09776e33	1.12486	0.562428	+	0.826846i	$0.309867\pi$
$740$	4.51303e32	0.455306
$741$	0	0
$742$	8.65468e31	0.0846460
$743$	5.60678e32	0.539936	0.269968	−	0.962869i	$-0.412987\pi$
$743$	5.60678e32	0.539936	0.269968	+	0.962869i	$0.412987\pi$
$744$	0	0
$745$	−3.45324e32	−0.322426
$746$	5.04137e32	0.463502
$747$	0	0
$748$	−6.08734e31	−0.0542699
$749$	6.07840e32	0.533640
$750$	0	0
$751$	−1.01215e33	−0.861759	−0.430880	−	0.902409i	$-0.641797\pi$
$751$	−1.01215e33	−0.861759	−0.430880	+	0.902409i	$0.641797\pi$
$752$	9.24546e32	0.775217
$753$	0	0
$754$	−1.06730e33	−0.867990
$755$	−2.66250e32	−0.213255
$756$	0	0
$757$	−9.48447e32	−0.736903	−0.368452	−	0.929647i	$-0.620112\pi$
$757$	−9.48447e32	−0.736903	−0.368452	+	0.929647i	$0.620112\pi$
$758$	4.13496e32	0.316428
$759$	0	0
$760$	5.81133e32	0.431438
$761$	−9.81408e32	−0.717672	−0.358836	−	0.933401i	$-0.616826\pi$
$761$	−9.81408e32	−0.717672	−0.358836	+	0.933401i	$0.616826\pi$
$762$	0	0
$763$	2.62131e32	0.185989
$764$	8.94448e32	0.625146
$765$	0	0
$766$	−1.68400e32	−0.114212
$767$	−2.66338e33	−1.77945
$768$	0	0
$769$	8.57155e32	0.555785	0.277892	−	0.960612i	$-0.410364\pi$
$769$	8.57155e32	0.555785	0.277892	+	0.960612i	$0.410364\pi$
$770$	1.35695e32	0.0866801
$771$	0	0
$772$	6.08734e32	0.377423
$773$	9.41593e31	0.0575174	0.0287587	−	0.999586i	$-0.490845\pi$
$773$	9.41593e31	0.0575174	0.0287587	+	0.999586i	$0.490845\pi$
$774$	0	0
$775$	8.65904e32	0.513452
$776$	−5.35245e32	−0.312711
$777$	0	0
$778$	−2.22819e32	−0.126382
$779$	−3.54557e33	−1.98155
$780$	0	0
$781$	−5.07242e33	−2.75251
$782$	1.04314e29	5.57785e−5 0
$783$	0	0
$784$	−9.75103e32	−0.506309
$785$	−1.36417e33	−0.698020
$786$	0	0
$787$	2.72161e33	1.35243	0.676216	−	0.736703i	$-0.263618\pi$
$787$	2.72161e33	1.35243	0.676216	+	0.736703i	$0.263618\pi$
$788$	−6.69900e32	−0.328064
$789$	0	0
$790$	3.78780e32	0.180167
$791$	8.94051e31	0.0419115
$792$	0	0
$793$	1.63931e33	0.746483
$794$	2.95979e31	0.0132839
$795$	0	0
$796$	−5.31215e32	−0.231617
$797$	3.49544e33	1.50221	0.751106	−	0.660182i	$-0.229521\pi$
$797$	3.49544e33	1.50221	0.751106	+	0.660182i	$0.229521\pi$
$798$	0	0
$799$	1.38546e32	0.0578505
$800$	1.82843e33	0.752565
$801$	0	0
$802$	4.89975e32	0.195961
$803$	4.78923e33	1.88815
$804$	0	0
$805$	1.30050e30	0.000498263 0
$806$	−8.42287e32	−0.318132
$807$	0	0
$808$	4.18717e31	0.0153706
$809$	−1.04226e33	−0.377196	−0.188598	−	0.982054i	$-0.560394\pi$
$809$	−1.04226e33	−0.377196	−0.188598	+	0.982054i	$0.560394\pi$
$810$	0	0
$811$	−2.22771e33	−0.783643	−0.391821	−	0.920041i	$-0.628155\pi$
$811$	−2.22771e33	−0.783643	−0.391821	+	0.920041i	$0.628155\pi$
$812$	−1.41176e33	−0.489629
$813$	0	0
$814$	2.09098e33	0.704967
$815$	2.36357e33	0.785700
$816$	0	0
$817$	−1.63911e33	−0.529729
$818$	9.18857e32	0.292809
$819$	0	0
$820$	1.81305e33	0.561760
$821$	7.36864e31	0.0225134	0.0112567	−	0.999937i	$-0.496417\pi$
$821$	7.36864e31	0.0225134	0.0112567	+	0.999937i	$0.496417\pi$
$822$	0	0
$823$	4.08369e33	1.21326	0.606631	−	0.794984i	$-0.292521\pi$
$823$	4.08369e33	1.21326	0.606631	+	0.794984i	$0.292521\pi$
$824$	−5.22471e32	−0.153073
$825$	0	0
$826$	6.29910e32	0.179477
$827$	−6.73455e32	−0.189232	−0.0946161	−	0.995514i	$-0.530162\pi$
$827$	−6.73455e32	−0.189232	−0.0946161	+	0.995514i	$0.530162\pi$
$828$	0	0
$829$	−3.77086e33	−1.03054	−0.515268	−	0.857029i	$-0.672308\pi$
$829$	−3.77086e33	−1.03054	−0.515268	+	0.857029i	$0.672308\pi$
$830$	−6.44757e31	−0.0173779
$831$	0	0
$832$	9.77667e32	0.256314
$833$	−1.46123e32	−0.0377833
$834$	0	0
$835$	−1.75746e33	−0.442070
$836$	−6.91147e33	−1.71474
$837$	0	0
$838$	1.45251e32	0.0350601
$839$	4.99200e33	1.18853	0.594267	−	0.804268i	$-0.297442\pi$
$839$	4.99200e33	1.18853	0.594267	+	0.804268i	$0.297442\pi$
$840$	0	0
$841$	8.92995e33	2.06869
$842$	1.63249e33	0.373046
$843$	0	0
$844$	−4.91971e33	−1.09396
$845$	1.26217e33	0.276863
$846$	0	0
$847$	−1.97277e33	−0.421131
$848$	1.77904e33	0.374657
$849$	0	0
$850$	6.44051e31	0.0132009
$851$	2.00400e31	0.00405236
$852$	0	0
$853$	−2.52826e33	−0.497631	−0.248815	−	0.968551i	$-0.580041\pi$
$853$	−2.52826e33	−0.497631	−0.248815	+	0.968551i	$0.580041\pi$
$854$	−3.87710e32	−0.0752907
$855$	0	0
$856$	−6.16746e33	−1.16589
$857$	−7.45015e33	−1.38959	−0.694793	−	0.719209i	$-0.744504\pi$
$857$	−7.45015e33	−1.38959	−0.694793	+	0.719209i	$0.744504\pi$
$858$	0	0
$859$	−5.27466e32	−0.0957797	−0.0478898	−	0.998853i	$-0.515250\pi$
$859$	−5.27466e32	−0.0957797	−0.0478898	+	0.998853i	$0.515250\pi$
$860$	8.38169e32	0.150176
$861$	0	0
$862$	−1.75098e33	−0.305456
$863$	6.72847e33	1.15822	0.579112	−	0.815248i	$-0.303400\pi$
$863$	6.72847e33	1.15822	0.579112	+	0.815248i	$0.303400\pi$
$864$	0	0
$865$	3.01997e33	0.506195
$866$	−4.16294e33	−0.688565
$867$	0	0
$868$	−1.11413e33	−0.179457
$869$	−9.81522e33	−1.56017
$870$	0	0
$871$	5.31397e33	0.822641
$872$	−2.65972e33	−0.406347
$873$	0	0
$874$	1.18437e31	0.00176240
$875$	1.80700e33	0.265378
$876$	0	0
$877$	−7.45124e33	−1.06594	−0.532970	−	0.846134i	$-0.678924\pi$
$877$	−7.45124e33	−1.06594	−0.532970	+	0.846134i	$0.678924\pi$
$878$	−4.93870e32	−0.0697310
$879$	0	0
$880$	2.78932e33	0.383660
$881$	−1.17245e34	−1.59174	−0.795870	−	0.605468i	$-0.792986\pi$
$881$	−1.17245e34	−1.59174	−0.795870	+	0.605468i	$0.792986\pi$
$882$	0	0
$883$	1.05295e34	1.39271	0.696353	−	0.717699i	$-0.254805\pi$
$883$	1.05295e34	1.39271	0.696353	+	0.717699i	$0.254805\pi$
$884$	3.50382e32	0.0457447
$885$	0	0
$886$	1.65825e33	0.210941
$887$	6.86475e33	0.861992	0.430996	−	0.902354i	$-0.358162\pi$
$887$	6.86475e33	0.861992	0.430996	+	0.902354i	$0.358162\pi$
$888$	0	0
$889$	5.77423e32	0.0706519
$890$	−1.29292e33	−0.156166
$891$	0	0
$892$	−5.68246e33	−0.668868
$893$	1.57303e34	1.82787
$894$	0	0
$895$	−2.04339e33	−0.231412
$896$	−3.00446e33	−0.335911
$897$	0	0
$898$	−2.45726e32	−0.0267777
$899$	1.04540e34	1.12472
$900$	0	0
$901$	2.66596e32	0.0279587
$902$	8.40023e33	0.869794
$903$	0	0
$904$	−9.07152e32	−0.0915678
$905$	−5.11367e33	−0.509653
$906$	0	0
$907$	−9.01545e33	−0.875999	−0.438000	−	0.898975i	$-0.644313\pi$
$907$	−9.01545e33	−0.875999	−0.438000	+	0.898975i	$0.644313\pi$
$908$	8.48862e33	0.814423
$909$	0	0
$910$	−7.81047e32	−0.0730636
$911$	−1.54259e34	−1.42492	−0.712459	−	0.701714i	$-0.752418\pi$
$911$	−1.54259e34	−1.42492	−0.712459	+	0.701714i	$0.752418\pi$
$912$	0	0
$913$	1.67074e33	0.150486
$914$	2.14755e33	0.191013
$915$	0	0
$916$	1.16751e34	1.01266
$917$	−1.54230e33	−0.132106
$918$	0	0
$919$	3.21779e32	0.0268800	0.0134400	−	0.999910i	$-0.495722\pi$
$919$	3.21779e32	0.0268800	0.0134400	+	0.999910i	$0.495722\pi$
$920$	−1.31956e31	−0.00108860
$921$	0	0
$922$	−5.60525e33	−0.451013
$923$	2.91964e34	2.32012
$924$	0	0
$925$	1.23730e34	0.959056
$926$	−1.26286e33	−0.0966786
$927$	0	0
$928$	2.20745e34	1.64850
$929$	−6.89145e33	−0.508311	−0.254156	−	0.967163i	$-0.581798\pi$
$929$	−6.89145e33	−0.508311	−0.254156	+	0.967163i	$0.581798\pi$
$930$	0	0
$931$	−1.65905e34	−1.19382
$932$	−1.05184e34	−0.747593
$933$	0	0
$934$	9.81316e33	0.680487
$935$	4.17989e32	0.0286306
$936$	0	0
$937$	6.25703e32	0.0418180	0.0209090	−	0.999781i	$-0.493344\pi$
$937$	6.25703e32	0.0418180	0.0209090	+	0.999781i	$0.493344\pi$
$938$	−1.25679e33	−0.0829720
$939$	0	0
$940$	−8.04381e33	−0.518193
$941$	−2.43281e34	−1.54820	−0.774101	−	0.633062i	$-0.781798\pi$
$941$	−2.43281e34	−1.54820	−0.774101	+	0.633062i	$0.781798\pi$
$942$	0	0
$943$	8.05081e31	0.00499983
$944$	1.29483e34	0.794393
$945$	0	0
$946$	3.88340e33	0.232522
$947$	−1.95908e34	−1.15885	−0.579424	−	0.815026i	$-0.696723\pi$
$947$	−1.95908e34	−1.15885	−0.579424	+	0.815026i	$0.696723\pi$
$948$	0	0
$949$	−2.75664e34	−1.59155
$950$	7.31245e33	0.417101
$951$	0	0
$952$	−1.80553e32	−0.0100526
$953$	−2.99818e33	−0.164926	−0.0824632	−	0.996594i	$-0.526279\pi$
$953$	−2.99818e33	−0.164926	−0.0824632	+	0.996594i	$0.526279\pi$
$954$	0	0
$955$	−6.14175e33	−0.329802
$956$	−7.15389e32	−0.0379556
$957$	0	0
$958$	−6.83726e32	−0.0354143
$959$	−2.62948e33	−0.134573
$960$	0	0
$961$	−1.17633e34	−0.587772
$962$	−1.20355e34	−0.594225
$963$	0	0
$964$	−2.05706e34	−0.991658
$965$	−4.17989e33	−0.199114
$966$	0	0
$967$	5.37477e33	0.250009	0.125004	−	0.992156i	$-0.460106\pi$
$967$	5.37477e33	0.250009	0.125004	+	0.992156i	$0.460106\pi$
$968$	2.00168e34	0.920084
$969$	0	0
$970$	1.68683e33	0.0757177
$971$	−1.12238e34	−0.497872	−0.248936	−	0.968520i	$-0.580081\pi$
$971$	−1.12238e34	−0.497872	−0.248936	+	0.968520i	$0.580081\pi$
$972$	0	0
$973$	4.94789e33	0.214349
$974$	4.85433e33	0.207827
$975$	0	0
$976$	−7.96969e33	−0.333249
$977$	−2.41460e34	−0.997831	−0.498915	−	0.866651i	$-0.666268\pi$
$977$	−2.41460e34	−0.997831	−0.498915	+	0.866651i	$0.666268\pi$
$978$	0	0
$979$	3.35031e34	1.35233
$980$	8.48368e33	0.338442
$981$	0	0
$982$	−1.28102e34	−0.499199
$983$	1.21263e34	0.467048	0.233524	−	0.972351i	$-0.424974\pi$
$983$	1.21263e34	0.467048	0.233524	+	0.972351i	$0.424974\pi$
$984$	0	0
$985$	4.59989e33	0.173073
$986$	7.77557e32	0.0289166
$987$	0	0
$988$	3.97818e34	1.44537
$989$	3.72187e31	0.00133661
$990$	0	0
$991$	1.83350e34	0.643329	0.321664	−	0.946854i	$-0.395758\pi$
$991$	1.83350e34	0.643329	0.321664	+	0.946854i	$0.395758\pi$
$992$	1.74207e34	0.604201
$993$	0	0
$994$	−6.90518e33	−0.234008
$995$	3.64760e33	0.122192
$996$	0	0
$997$	1.59227e34	0.521222	0.260611	−	0.965444i	$-0.416076\pi$
$997$	1.59227e34	0.521222	0.260611	+	0.965444i	$0.416076\pi$
$998$	−1.12779e34	−0.364945
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.24.a.a.1.1		1
3.2	odd	2		3.24.a.a.1.1	✓	1
12.11	even	2		48.24.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.24.a.a.1.1	✓	1	3.2	odd	2
9.24.a.a.1.1		1	1.1	even	1	trivial
48.24.a.a.1.1		1	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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