LMFDB - Embedded newform 48.24.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,24,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	48.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:	$160.897937926$
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	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-177147. q^{3} -4.88637e7 q^{5} +1.72369e9 q^{7} +3.13811e10 q^{9} +O(q^{10})$

$q-177147. q^{3} -4.88637e7 q^{5} +1.72369e9 q^{7} +3.13811e10 q^{9} +1.42826e12 q^{11} -8.22096e12 q^{13} +8.65606e12 q^{15} -5.98921e12 q^{17} -6.80005e14 q^{19} -3.05346e14 q^{21} -1.54406e13 q^{23} -9.53326e15 q^{25} -5.55906e15 q^{27} +1.15094e17 q^{29} +9.08297e16 q^{31} -2.53013e17 q^{33} -8.42259e16 q^{35} -1.29787e18 q^{37} +1.45632e18 q^{39} +5.21404e18 q^{41} +2.41043e18 q^{43} -1.53340e18 q^{45} +2.31327e19 q^{47} -2.43976e19 q^{49} +1.06097e18 q^{51} -4.45126e19 q^{53} -6.97903e19 q^{55} +1.20461e20 q^{57} +3.23974e20 q^{59} -1.99406e20 q^{61} +5.40912e19 q^{63} +4.01707e20 q^{65} +6.46393e20 q^{67} +2.73526e18 q^{69} -3.55146e21 q^{71} +3.35319e21 q^{73} +1.68879e21 q^{75} +2.46188e21 q^{77} +6.87213e21 q^{79} +9.84771e20 q^{81} +1.16977e21 q^{83} +2.92655e20 q^{85} -2.03886e22 q^{87} -2.34572e22 q^{89} -1.41704e22 q^{91} -1.60902e22 q^{93} +3.32276e22 q^{95} -3.06039e22 q^{97} +4.48204e22 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−177147.	−0.577350
$3$	−177147.	−0.577350
$4$	0	0
$5$	−4.88637e7	−0.447540	−0.223770	−	0.974642i	$-0.571836\pi$
$5$	−4.88637e7	−0.447540	−0.223770	+	0.974642i	$0.571836\pi$
$6$	0	0
$7$	1.72369e9	0.329482	0.164741	−	0.986337i	$-0.447321\pi$
$7$	1.72369e9	0.329482	0.164741	+	0.986337i	$0.447321\pi$
$8$	0	0
$9$	3.13811e10	0.333333
$10$	0	0
$11$	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	$0.719465\pi$
$14$	0	0
$15$	8.65606e12	0.258387
$16$	0	0
$17$	−5.98921e12	−0.0423845	−0.0211922	−	0.999775i	$-0.506746\pi$
$17$	−5.98921e12	−0.0423845	−0.0211922	+	0.999775i	$0.506746\pi$
$18$	0	0
$19$	−6.80005e14	−1.33920	−0.669601	−	0.742721i	$-0.733535\pi$
$19$	−6.80005e14	−1.33920	−0.669601	+	0.742721i	$0.733535\pi$
$20$	0	0
$21$	−3.05346e14	−0.190226
$22$	0	0
$23$	−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
$26$	0	0
$27$	−5.55906e15	−0.192450
$28$	0	0
$29$	1.15094e17	1.75177	0.875884	−	0.482522i	$-0.160279\pi$
$29$	1.15094e17	1.75177	0.875884	+	0.482522i	$0.160279\pi$
$30$	0	0
$31$	9.08297e16	0.642050	0.321025	−	0.947071i	$-0.395973\pi$
$31$	9.08297e16	0.642050	0.321025	+	0.947071i	$0.395973\pi$
$32$	0	0
$33$	−2.53013e17	−0.871428
$34$	0	0
$35$	−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	$0.704685\pi$
$38$	0	0
$39$	1.45632e18	0.734537
$40$	0	0
$41$	5.21404e18	1.47965	0.739826	−	0.672798i	$-0.234908\pi$
$41$	5.21404e18	1.47965	0.739826	+	0.672798i	$0.234908\pi$
$42$	0	0
$43$	2.41043e18	0.395556	0.197778	−	0.980247i	$-0.436627\pi$
$43$	2.41043e18	0.395556	0.197778	+	0.980247i	$0.436627\pi$
$44$	0	0
$45$	−1.53340e18	−0.149180
$46$	0	0
$47$	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
$50$	0	0
$51$	1.06097e18	0.0244707
$52$	0	0
$53$	−4.45126e19	−0.659646	−0.329823	−	0.944043i	$-0.606989\pi$
$53$	−4.45126e19	−0.659646	−0.329823	+	0.944043i	$0.606989\pi$
$54$	0	0
$55$	−6.97903e19	−0.675498
$56$	0	0
$57$	1.20461e20	0.773188
$58$	0	0
$59$	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	$0.594777\pi$
$62$	0	0
$63$	5.40912e19	0.109827
$64$	0	0
$65$	4.01707e20	0.569385
$66$	0	0
$67$	6.46393e20	0.646601	0.323300	−	0.946296i	$-0.395208\pi$
$67$	6.46393e20	0.646601	0.323300	+	0.946296i	$0.395208\pi$
$68$	0	0
$69$	2.73526e18	0.00195090
$70$	0	0
$71$	−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	$0.284902\pi$
$74$	0	0
$75$	1.68879e21	0.461712
$76$	0	0
$77$	2.46188e21	0.497306
$78$	0	0
$79$	6.87213e21	1.03367	0.516833	−	0.856086i	$-0.327111\pi$
$79$	6.87213e21	1.03367	0.516833	+	0.856086i	$0.327111\pi$
$80$	0	0
$81$	9.84771e20	0.111111
$82$	0	0
$83$	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
$86$	0	0
$87$	−2.03886e22	−1.01138
$88$	0	0
$89$	−2.34572e22	−0.895966	−0.447983	−	0.894042i	$-0.647857\pi$
$89$	−2.34572e22	−0.895966	−0.447983	+	0.894042i	$0.647857\pi$
$90$	0	0
$91$	−1.41704e22	−0.419185
$92$	0	0
$93$	−1.60902e22	−0.370687
$94$	0	0
$95$	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	$0.569694\pi$
$98$	0	0
$99$	4.48204e22	0.503119
$100$	0	0
$101$	−2.39411e21	−0.0213525	−0.0106762	−	0.999943i	$-0.503398\pi$
$101$	−2.39411e21	−0.0213525	−0.0106762	+	0.999943i	$0.503398\pi$
$102$	0	0
$103$	2.98735e22	0.212646	0.106323	−	0.994332i	$-0.466092\pi$
$103$	2.98735e22	0.212646	0.106323	+	0.994332i	$0.466092\pi$
$104$	0	0
$105$	1.49204e22	0.0851339
$106$	0	0
$107$	−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	$0.591079\pi$
$110$	0	0
$111$	2.29914e23	0.692392
$112$	0	0
$113$	5.18685e22	0.127204	0.0636021	−	0.997975i	$-0.479741\pi$
$113$	5.18685e22	0.127204	0.0636021	+	0.997975i	$0.479741\pi$
$114$	0	0
$115$	7.54488e20	0.00151226
$116$	0	0
$117$	−2.57983e23	−0.424085
$118$	0	0
$119$	−1.03235e22	−0.0139649
$120$	0	0
$121$	1.14451e24	1.27816
$122$	0	0
$123$	−9.23651e23	−0.854278
$124$	0	0
$125$	1.04833e24	0.805441
$126$	0	0
$127$	3.34992e23	0.214433	0.107217	−	0.994236i	$-0.465806\pi$
$127$	3.34992e23	0.214433	0.107217	+	0.994236i	$0.465806\pi$
$128$	0	0
$129$	−4.27001e23	−0.228374
$130$	0	0
$131$	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
$134$	0	0
$135$	2.71636e23	0.0861291
$136$	0	0
$137$	−1.52550e24	−0.408437	−0.204219	−	0.978925i	$-0.565465\pi$
$137$	−1.52550e24	−0.408437	−0.204219	+	0.978925i	$0.565465\pi$
$138$	0	0
$139$	2.87052e24	0.650565	0.325283	−	0.945617i	$-0.394541\pi$
$139$	2.87052e24	0.650565	0.325283	+	0.945617i	$0.394541\pi$
$140$	0	0
$141$	−4.09788e24	−0.788025
$142$	0	0
$143$	−1.17417e25	−1.92029
$144$	0	0
$145$	−5.62393e24	−0.783985
$146$	0	0
$147$	4.32197e24	0.514674
$148$	0	0
$149$	7.06708e24	0.720440	0.360220	−	0.932867i	$-0.382702\pi$
$149$	7.06708e24	0.720440	0.360220	+	0.932867i	$0.382702\pi$
$150$	0	0
$151$	5.44882e24	0.476505	0.238253	−	0.971203i	$-0.423425\pi$
$151$	5.44882e24	0.476505	0.238253	+	0.971203i	$0.423425\pi$
$152$	0	0
$153$	−1.87948e23	−0.0141282
$154$	0	0
$155$	−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	$0.784700\pi$
$158$	0	0
$159$	7.88528e24	0.380847
$160$	0	0
$161$	−2.66149e22	−0.00111334
$162$	0	0
$163$	−4.83707e25	−1.75560	−0.877799	−	0.479029i	$-0.840989\pi$
$163$	−4.83707e25	−1.75560	−0.877799	+	0.479029i	$0.840989\pi$
$164$	0	0
$165$	1.23631e25	0.389999
$166$	0	0
$167$	−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
$170$	0	0
$171$	−2.13393e25	−0.446401
$172$	0	0
$173$	−6.18040e25	−1.13106	−0.565531	−	0.824727i	$-0.691329\pi$
$173$	−6.18040e25	−1.13106	−0.565531	+	0.824727i	$0.691329\pi$
$174$	0	0
$175$	−1.64324e25	−0.263489
$176$	0	0
$177$	−5.73911e25	−0.807518
$178$	0	0
$179$	−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	$0.692822\pi$
$182$	0	0
$183$	3.53242e25	0.338754
$184$	0	0
$185$	6.34189e25	0.536715
$186$	0	0
$187$	−8.55417e24	−0.0639734
$188$	0	0
$189$	−9.58209e24	−0.0634088
$190$	0	0
$191$	−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	$0.571407\pi$
$194$	0	0
$195$	−7.11612e25	−0.328734
$196$	0	0
$197$	−9.41370e25	−0.386722	−0.193361	−	0.981128i	$-0.561939\pi$
$197$	−9.41370e25	−0.386722	−0.193361	+	0.981128i	$0.561939\pi$
$198$	0	0
$199$	−7.46484e25	−0.273030	−0.136515	−	0.990638i	$-0.543590\pi$
$199$	−7.46484e25	−0.273030	−0.136515	+	0.990638i	$0.543590\pi$
$200$	0	0
$201$	−1.14506e26	−0.373315
$202$	0	0
$203$	1.98387e26	0.577175
$204$	0	0
$205$	−2.54777e26	−0.662203
$206$	0	0
$207$	−4.84544e23	−0.00112635
$208$	0	0
$209$	−9.71227e26	−2.02134
$210$	0	0
$211$	−6.91338e26	−1.28956	−0.644781	−	0.764368i	$-0.723051\pi$
$211$	−6.91338e26	−1.28956	−0.644781	+	0.764368i	$0.723051\pi$
$212$	0	0
$213$	6.29131e26	1.05287
$214$	0	0
$215$	−1.17783e26	−0.177027
$216$	0	0
$217$	1.56562e26	0.211544
$218$	0	0
$219$	−5.94007e26	−0.722245
$220$	0	0
$221$	4.92371e25	0.0539239
$222$	0	0
$223$	−7.98521e26	−0.788462	−0.394231	−	0.919011i	$-0.628989\pi$
$223$	−7.98521e26	−0.788462	−0.394231	+	0.919011i	$0.628989\pi$
$224$	0	0
$225$	−2.99164e26	−0.266569
$226$	0	0
$227$	−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	$0.703585\pi$
$230$	0	0
$231$	−4.36115e26	−0.287120
$232$	0	0
$233$	−1.47808e27	−0.881263	−0.440632	−	0.897688i	$-0.645246\pi$
$233$	−1.47808e27	−0.881263	−0.440632	+	0.897688i	$0.645246\pi$
$234$	0	0
$235$	−1.13035e27	−0.610846
$236$	0	0
$237$	−1.21738e27	−0.596787
$238$	0	0
$239$	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	$0.301297\pi$
$242$	0	0
$243$	−1.74449e26	−0.0641500
$244$	0	0
$245$	1.19216e27	0.398956
$246$	0	0
$247$	5.59030e27	1.70381
$248$	0	0
$249$	−2.07221e26	−0.0575628
$250$	0	0
$251$	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
$254$	0	0
$255$	−5.18430e25	−0.0109516
$256$	0	0
$257$	−5.74761e27	−1.10983	−0.554915	−	0.831907i	$-0.687249\pi$
$257$	−5.74761e27	−1.10983	−0.554915	+	0.831907i	$0.687249\pi$
$258$	0	0
$259$	−2.23713e27	−0.395134
$260$	0	0
$261$	3.61178e27	0.583922
$262$	0	0
$263$	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
$266$	0	0
$267$	4.15537e27	0.517286
$268$	0	0
$269$	2.45445e27	0.280416	0.140208	−	0.990122i	$-0.455223\pi$
$269$	2.45445e27	0.280416	0.140208	+	0.990122i	$0.455223\pi$
$270$	0	0
$271$	−5.65329e27	−0.593136	−0.296568	−	0.955012i	$-0.595842\pi$
$271$	−5.65329e27	−0.593136	−0.296568	+	0.955012i	$0.595842\pi$
$272$	0	0
$273$	2.51024e27	0.242016
$274$	0	0
$275$	−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	$0.453686\pi$
$278$	0	0
$279$	2.85033e27	0.214017
$280$	0	0
$281$	2.83906e28	1.96359	0.981797	−	0.189931i	$-0.0608265\pi$
$281$	2.83906e28	1.96359	0.981797	+	0.189931i	$0.0608265\pi$
$282$	0	0
$283$	2.29202e28	1.46108	0.730542	−	0.682868i	$-0.239268\pi$
$283$	2.29202e28	1.46108	0.730542	+	0.682868i	$0.239268\pi$
$284$	0	0
$285$	−5.88617e27	−0.346033
$286$	0	0
$287$	8.98738e27	0.487518
$288$	0	0
$289$	−1.99317e28	−0.998204
$290$	0	0
$291$	5.42139e27	0.250808
$292$	0	0
$293$	3.43109e28	1.46708	0.733542	−	0.679644i	$-0.237866\pi$
$293$	3.43109e28	1.46708	0.733542	+	0.679644i	$0.237866\pi$
$294$	0	0
$295$	−1.58306e28	−0.625957
$296$	0	0
$297$	−7.93980e27	−0.290476
$298$	0	0
$299$	1.26937e26	0.00429902
$300$	0	0
$301$	4.15484e27	0.130329
$302$	0	0
$303$	4.24110e26	0.0123279
$304$	0	0
$305$	9.74373e27	0.262589
$306$	0	0
$307$	1.16886e28	0.292195	0.146097	−	0.989270i	$-0.453329\pi$
$307$	1.16886e28	0.292195	0.146097	+	0.989270i	$0.453329\pi$
$308$	0	0
$309$	−5.29200e27	−0.122771
$310$	0	0
$311$	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	$0.293502\pi$
$314$	0	0
$315$	−2.64310e27	−0.0491521
$316$	0	0
$317$	−1.04255e29	−1.80266	−0.901330	−	0.433133i	$-0.857408\pi$
$317$	−1.04255e29	−1.80266	−0.901330	+	0.433133i	$0.857408\pi$
$318$	0	0
$319$	1.64385e29	2.64404
$320$	0	0
$321$	6.24689e28	0.935096
$322$	0	0
$323$	4.07270e27	0.0567614
$324$	0	0
$325$	7.83726e28	1.01743
$326$	0	0
$327$	2.69398e28	0.325908
$328$	0	0
$329$	3.98735e28	0.449709
$330$	0	0
$331$	−1.38759e29	−1.45962	−0.729809	−	0.683651i	$-0.760391\pi$
$331$	−1.38759e29	−1.45962	−0.729809	+	0.683651i	$0.760391\pi$
$332$	0	0
$333$	−4.07286e28	−0.399753
$334$	0	0
$335$	−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	$0.723175\pi$
$338$	0	0
$339$	−9.18835e27	−0.0734414
$340$	0	0
$341$	1.29729e29	0.969083
$342$	0	0
$343$	−8.92291e28	−0.623196
$344$	0	0
$345$	−1.33655e26	−0.000873105 0
$346$	0	0
$347$	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	$0.404582\pi$
$350$	0	0
$351$	4.57008e28	0.244846
$352$	0	0
$353$	−9.96681e28	−0.500204	−0.250102	−	0.968220i	$-0.580464\pi$
$353$	−9.96681e28	−0.500204	−0.250102	+	0.968220i	$0.580464\pi$
$354$	0	0
$355$	1.73538e29	0.816146
$356$	0	0
$357$	1.82878e27	0.00806265
$358$	0	0
$359$	−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
$362$	0	0
$363$	−2.02746e29	−0.737948
$364$	0	0
$365$	−1.63849e29	−0.559856
$366$	0	0
$367$	−5.60764e29	−1.79937	−0.899684	−	0.436541i	$-0.856203\pi$
$367$	−5.60764e29	−1.79937	−0.899684	+	0.436541i	$0.856203\pi$
$368$	0	0
$369$	1.63622e29	0.493217
$370$	0	0
$371$	−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	$0.702870\pi$
$374$	0	0
$375$	−1.85709e29	−0.465022
$376$	0	0
$377$	−9.46185e29	−2.22869
$378$	0	0
$379$	3.66574e29	0.812477	0.406239	−	0.913767i	$-0.366840\pi$
$379$	3.66574e29	0.812477	0.406239	+	0.913767i	$0.366840\pi$
$380$	0	0
$381$	−5.93429e28	−0.123803
$382$	0	0
$383$	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
$386$	0	0
$387$	7.56420e28	0.131852
$388$	0	0
$389$	−1.97534e29	−0.324506	−0.162253	−	0.986749i	$-0.551876\pi$
$389$	−1.97534e29	−0.324506	−0.162253	+	0.986749i	$0.551876\pi$
$390$	0	0
$391$	9.24773e25	0.000143220 0
$392$	0	0
$393$	−1.58505e29	−0.231488
$394$	0	0
$395$	−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	$0.505429\pi$
$398$	0	0
$399$	2.07637e29	0.254751
$400$	0	0
$401$	4.34375e29	0.503159	0.251579	−	0.967837i	$-0.419050\pi$
$401$	4.34375e29	0.503159	0.251579	+	0.967837i	$0.419050\pi$
$402$	0	0
$403$	−7.46708e29	−0.816851
$404$	0	0
$405$	−4.81196e28	−0.0497266
$406$	0	0
$407$	−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	$0.622672\pi$
$410$	0	0
$411$	2.70238e29	0.235811
$412$	0	0
$413$	5.58431e29	0.460833
$414$	0	0
$415$	−5.71593e28	−0.0446204
$416$	0	0
$417$	−5.08505e29	−0.375604
$418$	0	0
$419$	−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	$0.658974\pi$
$422$	0	0
$423$	7.25928e29	0.454966
$424$	0	0
$425$	5.70967e28	0.0338952
$426$	0	0
$427$	−3.43714e29	−0.193320
$428$	0	0
$429$	2.08001e30	1.10868
$430$	0	0
$431$	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	$0.154843\pi$
$434$	0	0
$435$	9.96263e29	0.452634
$436$	0	0
$437$	1.04997e28	0.00452524
$438$	0	0
$439$	−4.37828e29	−0.179045	−0.0895223	−	0.995985i	$-0.528534\pi$
$439$	−4.37828e29	−0.179045	−0.0895223	+	0.995985i	$0.528534\pi$
$440$	0	0
$441$	−7.65624e29	−0.297147
$442$	0	0
$443$	−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
$446$	0	0
$447$	−1.25191e30	−0.415946
$448$	0	0
$449$	−2.17842e29	−0.0687558	−0.0343779	−	0.999409i	$-0.510945\pi$
$449$	−2.17842e29	−0.0687558	−0.0343779	+	0.999409i	$0.510945\pi$
$450$	0	0
$451$	7.44702e30	2.23333
$452$	0	0
$453$	−9.65243e29	−0.275111
$454$	0	0
$455$	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	$0.578862\pi$
$458$	0	0
$459$	3.32944e28	0.00815690
$460$	0	0
$461$	−4.96919e30	−1.15804	−0.579022	−	0.815312i	$-0.696566\pi$
$461$	−4.96919e30	−1.15804	−0.579022	+	0.815312i	$0.696566\pi$
$462$	0	0
$463$	−1.11956e30	−0.248237	−0.124118	−	0.992267i	$-0.539610\pi$
$463$	−1.11956e30	−0.248237	−0.124118	+	0.992267i	$0.539610\pi$
$464$	0	0
$465$	7.86228e29	0.165897
$466$	0	0
$467$	−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
$470$	0	0
$471$	4.94557e30	0.900483
$472$	0	0
$473$	3.44273e30	0.597036
$474$	0	0
$475$	6.48267e30	1.07097
$476$	0	0
$477$	−1.39685e30	−0.219882
$478$	0	0
$479$	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
$482$	0	0
$483$	4.71474e27	0.000642786 0
$484$	0	0
$485$	1.49542e30	0.194417
$486$	0	0
$487$	4.30348e30	0.533626	0.266813	−	0.963748i	$-0.414029\pi$
$487$	4.30348e30	0.533626	0.266813	+	0.963748i	$0.414029\pi$
$488$	0	0
$489$	8.56873e30	1.01360
$490$	0	0
$491$	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
$494$	0	0
$495$	−2.19009e30	−0.225166
$496$	0	0
$497$	−6.12161e30	−0.600852
$498$	0	0
$499$	−9.99815e30	−0.937052	−0.468526	−	0.883450i	$-0.655215\pi$
$499$	−9.99815e30	−0.937052	−0.468526	+	0.883450i	$0.655215\pi$
$500$	0	0
$501$	6.37137e30	0.570295
$502$	0	0
$503$	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
$506$	0	0
$507$	−4.57577e30	−0.357168
$508$	0	0
$509$	2.98637e30	0.222786	0.111393	−	0.993776i	$-0.464469\pi$
$509$	2.98637e30	0.222786	0.111393	+	0.993776i	$0.464469\pi$
$510$	0	0
$511$	5.77985e30	0.412170
$512$	0	0
$513$	3.78019e30	0.257729
$514$	0	0
$515$	−1.45973e30	−0.0951677
$516$	0	0
$517$	3.30395e31	2.06012
$518$	0	0
$519$	1.09484e31	0.653019
$520$	0	0
$521$	−8.51705e30	−0.486022	−0.243011	−	0.970024i	$-0.578135\pi$
$521$	−8.51705e30	−0.486022	−0.243011	+	0.970024i	$0.578135\pi$
$522$	0	0
$523$	−1.27472e31	−0.696059	−0.348029	−	0.937484i	$-0.613149\pi$
$523$	−1.27472e31	−0.696059	−0.348029	+	0.937484i	$0.613149\pi$
$524$	0	0
$525$	2.91095e30	0.152126
$526$	0	0
$527$	−5.43998e29	−0.0272129
$528$	0	0
$529$	−2.08802e31	−0.999989
$530$	0	0
$531$	1.01667e31	0.466221
$532$	0	0
$533$	−4.28644e31	−1.88250
$534$	0	0
$535$	1.72312e31	0.724850
$536$	0	0
$537$	7.40796e30	0.298535
$538$	0	0
$539$	−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	$0.559608\pi$
$542$	0	0
$543$	1.85387e31	0.657479
$544$	0	0
$545$	7.43099e30	0.252631
$546$	0	0
$547$	4.27216e31	1.39249	0.696247	−	0.717802i	$-0.254852\pi$
$547$	4.27216e31	1.39249	0.696247	+	0.717802i	$0.254852\pi$
$548$	0	0
$549$	−6.25758e30	−0.195580
$550$	0	0
$551$	−7.82647e31	−2.34597
$552$	0	0
$553$	1.18454e31	0.340574
$554$	0	0
$555$	−1.12345e31	−0.309873
$556$	0	0
$557$	−6.81696e31	−1.80408	−0.902040	−	0.431652i	$-0.857931\pi$
$557$	−6.81696e31	−1.80408	−0.902040	+	0.431652i	$0.857931\pi$
$558$	0	0
$559$	−1.98161e31	−0.503248
$560$	0	0
$561$	1.51535e30	0.0369351
$562$	0	0
$563$	−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
$566$	0	0
$567$	1.69744e30	0.0366091
$568$	0	0
$569$	−1.87031e31	−0.387366	−0.193683	−	0.981064i	$-0.562043\pi$
$569$	−1.87031e31	−0.387366	−0.193683	+	0.981064i	$0.562043\pi$
$570$	0	0
$571$	−2.56599e31	−0.510433	−0.255217	−	0.966884i	$-0.582147\pi$
$571$	−2.56599e31	−0.510433	−0.255217	+	0.966884i	$0.582147\pi$
$572$	0	0
$573$	2.22659e31	0.425463
$574$	0	0
$575$	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	$0.566592\pi$
$578$	0	0
$579$	1.51535e31	0.256867
$580$	0	0
$581$	2.01632e30	0.0328499
$582$	0	0
$583$	−6.35758e31	−0.995642
$584$	0	0
$585$	1.26060e31	0.189795
$586$	0	0
$587$	−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
$590$	0	0
$591$	1.66761e31	0.223274
$592$	0	0
$593$	3.26412e31	0.420376	0.210188	−	0.977661i	$-0.432592\pi$
$593$	3.26412e31	0.420376	0.210188	+	0.977661i	$0.432592\pi$
$594$	0	0
$595$	5.04446e29	0.00624985
$596$	0	0
$597$	1.32237e31	0.157634
$598$	0	0
$599$	−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	$0.211515\pi$
$602$	0	0
$603$	2.02845e31	0.215534
$604$	0	0
$605$	−5.59248e31	−0.572029
$606$	0	0
$607$	1.51222e32	1.48917	0.744586	−	0.667526i	$-0.232647\pi$
$607$	1.51222e32	1.48917	0.744586	+	0.667526i	$0.232647\pi$
$608$	0	0
$609$	−3.51436e31	−0.333232
$610$	0	0
$611$	−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	$0.683786\pi$
$614$	0	0
$615$	4.51330e31	0.382323
$616$	0	0
$617$	−1.89291e32	−1.54472	−0.772361	−	0.635184i	$-0.780924\pi$
$617$	−1.89291e32	−1.54472	−0.772361	+	0.635184i	$0.780924\pi$
$618$	0	0
$619$	1.33375e32	1.04865	0.524327	−	0.851517i	$-0.324317\pi$
$619$	1.33375e32	1.04865	0.524327	+	0.851517i	$0.324317\pi$
$620$	0	0
$621$	8.58355e28	0.000650300 0
$622$	0	0
$623$	−4.04329e31	−0.295204
$624$	0	0
$625$	6.24200e31	0.439241
$626$	0	0
$627$	1.72050e32	1.16702
$628$	0	0
$629$	7.77324e30	0.0508299
$630$	0	0
$631$	−2.80913e31	−0.177107	−0.0885533	−	0.996071i	$-0.528224\pi$
$631$	−2.80913e31	−0.177107	−0.0885533	+	0.996071i	$0.528224\pi$
$632$	0	0
$633$	1.22468e32	0.744529
$634$	0	0
$635$	−1.63690e31	−0.0959674
$636$	0	0
$637$	2.00572e32	1.13414
$638$	0	0
$639$	−1.11449e32	−0.607876
$640$	0	0
$641$	−3.78628e31	−0.199225	−0.0996127	−	0.995026i	$-0.531760\pi$
$641$	−3.78628e31	−0.199225	−0.0996127	+	0.995026i	$0.531760\pi$
$642$	0	0
$643$	1.43262e32	0.727283	0.363641	−	0.931539i	$-0.381533\pi$
$643$	1.43262e32	0.727283	0.363641	+	0.931539i	$0.381533\pi$
$644$	0	0
$645$	2.08649e31	0.102207
$646$	0	0
$647$	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
$650$	0	0
$651$	−2.77345e31	−0.122135
$652$	0	0
$653$	−4.07291e32	−1.73142	−0.865712	−	0.500543i	$-0.833134\pi$
$653$	−4.07291e32	−1.73142	−0.865712	+	0.500543i	$0.833134\pi$
$654$	0	0
$655$	−4.37216e31	−0.179441
$656$	0	0
$657$	1.05227e32	0.416988
$658$	0	0
$659$	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	$0.784990\pi$
$662$	0	0
$663$	−8.72220e30	−0.0311330
$664$	0	0
$665$	5.72740e31	0.197474
$666$	0	0
$667$	−1.77713e30	−0.00591932
$668$	0	0
$669$	1.41456e32	0.455218
$670$	0	0
$671$	−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	$0.336176\pi$
$674$	0	0
$675$	5.29960e31	0.153904
$676$	0	0
$677$	1.65241e32	0.463820	0.231910	−	0.972737i	$-0.425503\pi$
$677$	1.65241e32	0.463820	0.231910	+	0.972737i	$0.425503\pi$
$678$	0	0
$679$	−5.27516e31	−0.143131
$680$	0	0
$681$	2.11311e32	0.554281
$682$	0	0
$683$	−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
$686$	0	0
$687$	2.90633e32	0.689194
$688$	0	0
$689$	3.65937e32	0.839238
$690$	0	0
$691$	−2.43790e32	−0.540778	−0.270389	−	0.962751i	$-0.587152\pi$
$691$	−2.43790e32	−0.540778	−0.270389	+	0.962751i	$0.587152\pi$
$692$	0	0
$693$	7.72564e31	0.165769
$694$	0	0
$695$	−1.40264e32	−0.291154
$696$	0	0
$697$	−3.12280e31	−0.0627143
$698$	0	0
$699$	2.61838e32	0.508798
$700$	0	0
$701$	−3.79931e32	−0.714409	−0.357205	−	0.934026i	$-0.616270\pi$
$701$	−3.79931e32	−0.714409	−0.357205	+	0.934026i	$0.616270\pi$
$702$	0	0
$703$	8.82561e32	1.60605
$704$	0	0
$705$	2.00238e32	0.352672
$706$	0	0
$707$	−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	$0.503797\pi$
$710$	0	0
$711$	2.15655e32	0.344555
$712$	0	0
$713$	−1.40247e30	−0.00216952
$714$	0	0
$715$	5.73743e32	0.859405
$716$	0	0
$717$	−1.78085e31	−0.0258319
$718$	0	0
$719$	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
$722$	0	0
$723$	−5.12073e32	−0.674903
$724$	0	0
$725$	−1.09722e33	−1.40090
$726$	0	0
$727$	−4.93859e31	−0.0610882	−0.0305441	−	0.999533i	$-0.509724\pi$
$727$	−4.93859e31	−0.0610882	−0.0305441	+	0.999533i	$0.509724\pi$
$728$	0	0
$729$	3.09032e31	0.0370370
$730$	0	0
$731$	−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	$0.297936\pi$
$734$	0	0
$735$	−2.11188e32	−0.230337
$736$	0	0
$737$	9.23219e32	0.975952
$738$	0	0
$739$	−1.09776e33	−1.12486	−0.562428	−	0.826846i	$-0.690133\pi$
$739$	−1.09776e33	−1.12486	−0.562428	+	0.826846i	$0.690133\pi$
$740$	0	0
$741$	−9.90305e32	−0.983693
$742$	0	0
$743$	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
$746$	0	0
$747$	3.67086e31	0.0332339
$748$	0	0
$749$	−6.07840e32	−0.533640
$750$	0	0
$751$	1.01215e33	0.861759	0.430880	−	0.902409i	$-0.358203\pi$
$751$	1.01215e33	0.861759	0.430880	+	0.902409i	$0.358203\pi$
$752$	0	0
$753$	−5.38258e31	−0.0444476
$754$	0	0
$755$	−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	$0.620112\pi$
$758$	0	0
$759$	3.90668e30	0.00294461
$760$	0	0
$761$	9.81408e32	0.717672	0.358836	−	0.933401i	$-0.383174\pi$
$761$	9.81408e32	0.717672	0.358836	+	0.933401i	$0.383174\pi$
$762$	0	0
$763$	−2.62131e32	−0.185989
$764$	0	0
$765$	9.18383e30	0.00632291
$766$	0	0
$767$	−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	$0.410364\pi$
$770$	0	0
$771$	1.01817e33	0.640761
$772$	0	0
$773$	−9.41593e31	−0.0575174	−0.0287587	−	0.999586i	$-0.509155\pi$
$773$	−9.41593e31	−0.0575174	−0.0287587	+	0.999586i	$0.509155\pi$
$774$	0	0
$775$	−8.65904e32	−0.513452
$776$	0	0
$777$	3.96301e32	0.228130
$778$	0	0
$779$	−3.54557e33	−1.98155
$780$	0	0
$781$	−5.07242e33	−2.75251
$782$	0	0
$783$	−6.39816e32	−0.337128
$784$	0	0
$785$	1.36417e33	0.698020
$786$	0	0
$787$	−2.72161e33	−1.35243	−0.676216	−	0.736703i	$-0.736382\pi$
$787$	−2.72161e33	−1.35243	−0.676216	+	0.736703i	$0.736382\pi$
$788$	0	0
$789$	−1.60933e33	−0.776710
$790$	0	0
$791$	8.94051e31	0.0419115
$792$	0	0
$793$	1.63931e33	0.746483
$794$	0	0
$795$	−3.85304e32	−0.170444
$796$	0	0
$797$	−3.49544e33	−1.50221	−0.751106	−	0.660182i	$-0.770479\pi$
$797$	−3.49544e33	−1.50221	−0.751106	+	0.660182i	$0.770479\pi$
$798$	0	0
$799$	−1.38546e32	−0.0578505
$800$	0	0
$801$	−7.36112e32	−0.298655
$802$	0	0
$803$	4.78923e33	1.88815
$804$	0	0
$805$	1.30050e30	0.000498263 0
$806$	0	0
$807$	−4.34799e32	−0.161898
$808$	0	0
$809$	1.04226e33	0.377196	0.188598	−	0.982054i	$-0.439606\pi$
$809$	1.04226e33	0.377196	0.188598	+	0.982054i	$0.439606\pi$
$810$	0	0
$811$	2.22771e33	0.783643	0.391821	−	0.920041i	$-0.371845\pi$
$811$	2.22771e33	0.783643	0.391821	+	0.920041i	$0.371845\pi$
$812$	0	0
$813$	1.00146e33	0.342447
$814$	0	0
$815$	2.36357e33	0.785700
$816$	0	0
$817$	−1.63911e33	−0.529729
$818$	0	0
$819$	−4.44682e32	−0.139728
$820$	0	0
$821$	−7.36864e31	−0.0225134	−0.0112567	−	0.999937i	$-0.503583\pi$
$821$	−7.36864e31	−0.0225134	−0.0112567	+	0.999937i	$0.503583\pi$
$822$	0	0
$823$	−4.08369e33	−1.21326	−0.606631	−	0.794984i	$-0.707479\pi$
$823$	−4.08369e33	−1.21326	−0.606631	+	0.794984i	$0.707479\pi$
$824$	0	0
$825$	2.41204e33	0.696888
$826$	0	0
$827$	−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	$0.672308\pi$
$830$	0	0
$831$	−6.29813e32	−0.167417
$832$	0	0
$833$	1.46123e32	0.0377833
$834$	0	0
$835$	1.75746e33	0.442070
$836$	0	0
$837$	−5.04928e32	−0.123562
$838$	0	0
$839$	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
$842$	0	0
$843$	−5.02931e33	−1.13368
$844$	0	0
$845$	−1.26217e33	−0.276863
$846$	0	0
$847$	1.97277e33	0.421131
$848$	0	0
$849$	−4.06025e33	−0.843557
$850$	0	0
$851$	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	$0.580041\pi$
$854$	0	0
$855$	1.04272e33	0.199782
$856$	0	0
$857$	7.45015e33	1.38959	0.694793	−	0.719209i	$-0.255496\pi$
$857$	7.45015e33	1.38959	0.694793	+	0.719209i	$0.255496\pi$
$858$	0	0
$859$	5.27466e32	0.0957797	0.0478898	−	0.998853i	$-0.484750\pi$
$859$	5.27466e32	0.0957797	0.0478898	+	0.998853i	$0.484750\pi$
$860$	0	0
$861$	−1.59209e33	−0.281469
$862$	0	0
$863$	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
$866$	0	0
$867$	3.53084e33	0.576313
$868$	0	0
$869$	9.81522e33	1.56017
$870$	0	0
$871$	−5.31397e33	−0.822641
$872$	0	0
$873$	−9.60382e32	−0.144804
$874$	0	0
$875$	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	$0.678924\pi$
$878$	0	0
$879$	−6.07808e33	−0.847021
$880$	0	0
$881$	1.17245e34	1.59174	0.795870	−	0.605468i	$-0.207014\pi$
$881$	1.17245e34	1.59174	0.795870	+	0.605468i	$0.207014\pi$
$882$	0	0
$883$	−1.05295e34	−1.39271	−0.696353	−	0.717699i	$-0.745195\pi$
$883$	−1.05295e34	−1.39271	−0.696353	+	0.717699i	$0.745195\pi$
$884$	0	0
$885$	2.80434e33	0.361396
$886$	0	0
$887$	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
$890$	0	0
$891$	1.40651e33	0.167706
$892$	0	0
$893$	−1.57303e34	−1.82787
$894$	0	0
$895$	2.04339e33	0.231412
$896$	0	0
$897$	−2.24865e31	−0.00248204
$898$	0	0
$899$	1.04540e34	1.12472
$900$	0	0
$901$	2.66596e32	0.0279587
$902$	0	0
$903$	−7.36017e32	−0.0752452
$904$	0	0
$905$	5.11367e33	0.509653
$906$	0	0
$907$	9.01545e33	0.875999	0.438000	−	0.898975i	$-0.355687\pi$
$907$	9.01545e33	0.875999	0.438000	+	0.898975i	$0.355687\pi$
$908$	0	0
$909$	−7.51298e31	−0.00711750
$910$	0	0
$911$	−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
$914$	0	0
$915$	−1.72607e33	−0.151606
$916$	0	0
$917$	1.54230e33	0.132106
$918$	0	0
$919$	−3.21779e32	−0.0268800	−0.0134400	−	0.999910i	$-0.504278\pi$
$919$	−3.21779e32	−0.0268800	−0.0134400	+	0.999910i	$0.504278\pi$
$920$	0	0
$921$	−2.07060e33	−0.168699
$922$	0	0
$923$	2.91964e34	2.32012
$924$	0	0
$925$	1.23730e34	0.959056
$926$	0	0
$927$	9.37462e32	0.0708821
$928$	0	0
$929$	6.89145e33	0.508311	0.254156	−	0.967163i	$-0.418202\pi$
$929$	6.89145e33	0.508311	0.254156	+	0.967163i	$0.418202\pi$
$930$	0	0
$931$	1.65905e34	1.19382
$932$	0	0
$933$	−4.42557e33	−0.310692
$934$	0	0
$935$	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	$0.493344\pi$
$938$	0	0
$939$	−1.06976e34	−0.697644
$940$	0	0
$941$	2.43281e34	1.54820	0.774101	−	0.633062i	$-0.218202\pi$
$941$	2.43281e34	1.54820	0.774101	+	0.633062i	$0.218202\pi$
$942$	0	0
$943$	−8.05081e31	−0.00499983
$944$	0	0
$945$	4.68217e32	0.0283780
$946$	0	0
$947$	−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
$950$	0	0
$951$	1.84684e34	1.04077
$952$	0	0
$953$	2.99818e33	0.164926	0.0824632	−	0.996594i	$-0.473721\pi$
$953$	2.99818e33	0.164926	0.0824632	+	0.996594i	$0.473721\pi$
$954$	0	0
$955$	6.14175e33	0.329802
$956$	0	0
$957$	−2.91203e34	−1.52654
$958$	0	0
$959$	−2.62948e33	−0.134573
$960$	0	0
$961$	−1.17633e34	−0.587772
$962$	0	0
$963$	−1.10662e34	−0.539878
$964$	0	0
$965$	4.17989e33	0.199114
$966$	0	0
$967$	−5.37477e33	−0.250009	−0.125004	−	0.992156i	$-0.539894\pi$
$967$	−5.37477e33	−0.250009	−0.125004	+	0.992156i	$0.539894\pi$
$968$	0	0
$969$	−7.21466e32	−0.0327712
$970$	0	0
$971$	−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
$974$	0	0
$975$	−1.38835e34	−0.587415
$976$	0	0
$977$	2.41460e34	0.997831	0.498915	−	0.866651i	$-0.333732\pi$
$977$	2.41460e34	0.997831	0.498915	+	0.866651i	$0.333732\pi$
$978$	0	0
$979$	−3.35031e34	−1.35233
$980$	0	0
$981$	−4.77230e33	−0.188163
$982$	0	0
$983$	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
$986$	0	0
$987$	−7.06347e33	−0.259640
$988$	0	0
$989$	−3.72187e31	−0.00133661
$990$	0	0
$991$	−1.83350e34	−0.643329	−0.321664	−	0.946854i	$-0.604242\pi$
$991$	−1.83350e34	−0.643329	−0.321664	+	0.946854i	$0.604242\pi$
$992$	0	0
$993$	2.45807e34	0.842711
$994$	0	0
$995$	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	$0.416076\pi$
$998$	0	0
$999$	7.21496e33	0.230797

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.24.a.a.1.1	✓	1	4.3	odd	2
9.24.a.a.1.1		1	12.11	even	2
48.24.a.a.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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