LMFDB - Embedded newform 48.10.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,10,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, 10, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
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:	$24.7217201359$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
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)

$f(q)$

$=$

$q+81.0000 q^{3} +830.000 q^{5} -672.000 q^{7} +6561.00 q^{9} +O(q^{10})$

$q+81.0000 q^{3} +830.000 q^{5} -672.000 q^{7} +6561.00 q^{9} +73468.0 q^{11} -78242.0 q^{13} +67230.0 q^{15} -161726. q^{17} +653572. q^{19} -54432.0 q^{21} +1.06670e6 q^{23} -1.26422e6 q^{25} +531441. q^{27} +3.82484e6 q^{29} +1.57948e6 q^{31} +5.95091e6 q^{33} -557760. q^{35} +1.60156e7 q^{37} -6.33760e6 q^{39} +2.62683e7 q^{41} +4.44952e7 q^{43} +5.44563e6 q^{45} -1.43242e7 q^{47} -3.99020e7 q^{49} -1.30998e7 q^{51} -2.43860e7 q^{53} +6.09784e7 q^{55} +5.29393e7 q^{57} -1.19421e7 q^{59} -1.89740e8 q^{61} -4.40899e6 q^{63} -6.49409e7 q^{65} +1.06710e8 q^{67} +8.64024e7 q^{69} -3.02754e8 q^{71} +8.17695e7 q^{73} -1.02402e8 q^{75} -4.93705e7 q^{77} -3.15315e8 q^{79} +4.30467e7 q^{81} -7.52833e8 q^{83} -1.34233e8 q^{85} +3.09812e8 q^{87} -4.33284e8 q^{89} +5.25786e7 q^{91} +1.27938e8 q^{93} +5.42465e8 q^{95} +1.28250e9 q^{97} +4.82024e8 q^{99} +O(q^{100})$

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
$3$	81.0000	0.577350
$3$	81.0000	0.577350
$4$	0	0
$5$	830.000	0.593900	0.296950	−	0.954893i	$-0.404031\pi$
$5$	830.000	0.593900	0.296950	+	0.954893i	$0.404031\pi$
$6$	0	0
$7$	−672.000	−0.105786	−0.0528930	−	0.998600i	$-0.516844\pi$
$7$	−672.000	−0.105786	−0.0528930	+	0.998600i	$0.516844\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	73468.0	1.51297	0.756486	−	0.654009i	$-0.226914\pi$
$11$	73468.0	1.51297	0.756486	+	0.654009i	$0.226914\pi$
$12$	0	0
$13$	−78242.0	−0.759792	−0.379896	−	0.925029i	$-0.624040\pi$
$13$	−78242.0	−0.759792	−0.379896	+	0.925029i	$0.624040\pi$
$14$	0	0
$15$	67230.0	0.342888
$16$	0	0
$17$	−161726.	−0.469634	−0.234817	−	0.972040i	$-0.575449\pi$
$17$	−161726.	−0.469634	−0.234817	+	0.972040i	$0.575449\pi$
$18$	0	0
$19$	653572.	1.15054	0.575271	−	0.817963i	$-0.304897\pi$
$19$	653572.	1.15054	0.575271	+	0.817963i	$0.304897\pi$
$20$	0	0
$21$	−54432.0	−0.0610756
$22$	0	0
$23$	1.06670e6	0.794814	0.397407	−	0.917642i	$-0.369910\pi$
$23$	1.06670e6	0.794814	0.397407	+	0.917642i	$0.369910\pi$
$24$	0	0
$25$	−1.26422e6	−0.647283
$26$	0	0
$27$	531441.	0.192450
$28$	0	0
$29$	3.82484e6	1.00420	0.502102	−	0.864808i	$-0.332560\pi$
$29$	3.82484e6	1.00420	0.502102	+	0.864808i	$0.332560\pi$
$30$	0	0
$31$	1.57948e6	0.307175	0.153588	−	0.988135i	$-0.450917\pi$
$31$	1.57948e6	0.307175	0.153588	+	0.988135i	$0.450917\pi$
$32$	0	0
$33$	5.95091e6	0.873515
$34$	0	0
$35$	−557760.	−0.0628263
$36$	0	0
$37$	1.60156e7	1.40487	0.702433	−	0.711749i	$-0.252097\pi$
$37$	1.60156e7	1.40487	0.702433	+	0.711749i	$0.252097\pi$
$38$	0	0
$39$	−6.33760e6	−0.438666
$40$	0	0
$41$	2.62683e7	1.45179	0.725896	−	0.687805i	$-0.241425\pi$
$41$	2.62683e7	1.45179	0.725896	+	0.687805i	$0.241425\pi$
$42$	0	0
$43$	4.44952e7	1.98475	0.992374	−	0.123263i	$-0.0393359\pi$
$43$	4.44952e7	1.98475	0.992374	+	0.123263i	$0.0393359\pi$
$44$	0	0
$45$	5.44563e6	0.197967
$46$	0	0
$47$	−1.43242e7	−0.428182	−0.214091	−	0.976814i	$-0.568679\pi$
$47$	−1.43242e7	−0.428182	−0.214091	+	0.976814i	$0.568679\pi$
$48$	0	0
$49$	−3.99020e7	−0.988809
$50$	0	0
$51$	−1.30998e7	−0.271143
$52$	0	0
$53$	−2.43860e7	−0.424522	−0.212261	−	0.977213i	$-0.568083\pi$
$53$	−2.43860e7	−0.424522	−0.212261	+	0.977213i	$0.568083\pi$
$54$	0	0
$55$	6.09784e7	0.898554
$56$	0	0
$57$	5.29393e7	0.664265
$58$	0	0
$59$	−1.19421e7	−0.128306	−0.0641529	−	0.997940i	$-0.520435\pi$
$59$	−1.19421e7	−0.128306	−0.0641529	+	0.997940i	$0.520435\pi$
$60$	0	0
$61$	−1.89740e8	−1.75459	−0.877294	−	0.479953i	$-0.840654\pi$
$61$	−1.89740e8	−1.75459	−0.877294	+	0.479953i	$0.840654\pi$
$62$	0	0
$63$	−4.40899e6	−0.0352620
$64$	0	0
$65$	−6.49409e7	−0.451240
$66$	0	0
$67$	1.06710e8	0.646944	0.323472	−	0.946238i	$-0.395150\pi$
$67$	1.06710e8	0.646944	0.323472	+	0.946238i	$0.395150\pi$
$68$	0	0
$69$	8.64024e7	0.458886
$70$	0	0
$71$	−3.02754e8	−1.41393	−0.706965	−	0.707249i	$-0.749936\pi$
$71$	−3.02754e8	−1.41393	−0.706965	+	0.707249i	$0.749936\pi$
$72$	0	0
$73$	8.17695e7	0.337007	0.168503	−	0.985701i	$-0.446107\pi$
$73$	8.17695e7	0.337007	0.168503	+	0.985701i	$0.446107\pi$
$74$	0	0
$75$	−1.02402e8	−0.373709
$76$	0	0
$77$	−4.93705e7	−0.160051
$78$	0	0
$79$	−3.15315e8	−0.910800	−0.455400	−	0.890287i	$-0.650504\pi$
$79$	−3.15315e8	−0.910800	−0.455400	+	0.890287i	$0.650504\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	−7.52833e8	−1.74119	−0.870597	−	0.491996i	$-0.836267\pi$
$83$	−7.52833e8	−1.74119	−0.870597	+	0.491996i	$0.836267\pi$
$84$	0	0
$85$	−1.34233e8	−0.278916
$86$	0	0
$87$	3.09812e8	0.579778
$88$	0	0
$89$	−4.33284e8	−0.732012	−0.366006	−	0.930613i	$-0.619275\pi$
$89$	−4.33284e8	−0.732012	−0.366006	+	0.930613i	$0.619275\pi$
$90$	0	0
$91$	5.25786e7	0.0803754
$92$	0	0
$93$	1.27938e8	0.177348
$94$	0	0
$95$	5.42465e8	0.683306
$96$	0	0
$97$	1.28250e9	1.47090	0.735450	−	0.677578i	$-0.236971\pi$
$97$	1.28250e9	1.47090	0.735450	+	0.677578i	$0.236971\pi$
$98$	0	0
$99$	4.82024e8	0.504324
$100$	0	0
$101$	−1.96115e9	−1.87528	−0.937639	−	0.347611i	$-0.886993\pi$
$101$	−1.96115e9	−1.87528	−0.937639	+	0.347611i	$0.886993\pi$
$102$	0	0
$103$	9.48061e8	0.829982	0.414991	−	0.909825i	$-0.363785\pi$
$103$	9.48061e8	0.829982	0.414991	+	0.909825i	$0.363785\pi$
$104$	0	0
$105$	−4.51786e7	−0.0362728
$106$	0	0
$107$	2.46613e8	0.181881	0.0909407	−	0.995856i	$-0.471013\pi$
$107$	2.46613e8	0.181881	0.0909407	+	0.995856i	$0.471013\pi$
$108$	0	0
$109$	1.47308e9	0.999556	0.499778	−	0.866154i	$-0.333415\pi$
$109$	1.47308e9	0.999556	0.499778	+	0.866154i	$0.333415\pi$
$110$	0	0
$111$	1.29726e9	0.811100
$112$	0	0
$113$	7.66579e8	0.442287	0.221143	−	0.975241i	$-0.429021\pi$
$113$	7.66579e8	0.442287	0.221143	+	0.975241i	$0.429021\pi$
$114$	0	0
$115$	8.85358e8	0.472040
$116$	0	0
$117$	−5.13346e8	−0.253264
$118$	0	0
$119$	1.08680e8	0.0496807
$120$	0	0
$121$	3.03960e9	1.28909
$122$	0	0
$123$	2.12773e9	0.838192
$124$	0	0
$125$	−2.67040e9	−0.978321
$126$	0	0
$127$	−3.08546e9	−1.05246	−0.526228	−	0.850344i	$-0.676394\pi$
$127$	−3.08546e9	−1.05246	−0.526228	+	0.850344i	$0.676394\pi$
$128$	0	0
$129$	3.60411e9	1.14589
$130$	0	0
$131$	−2.76317e9	−0.819759	−0.409880	−	0.912140i	$-0.634429\pi$
$131$	−2.76317e9	−0.819759	−0.409880	+	0.912140i	$0.634429\pi$
$132$	0	0
$133$	−4.39200e8	−0.121711
$134$	0	0
$135$	4.41096e8	0.114296
$136$	0	0
$137$	2.75041e9	0.667044	0.333522	−	0.942742i	$-0.391763\pi$
$137$	2.75041e9	0.667044	0.333522	+	0.942742i	$0.391763\pi$
$138$	0	0
$139$	−1.58704e9	−0.360597	−0.180299	−	0.983612i	$-0.557706\pi$
$139$	−1.58704e9	−0.360597	−0.180299	+	0.983612i	$0.557706\pi$
$140$	0	0
$141$	−1.16026e9	−0.247211
$142$	0	0
$143$	−5.74828e9	−1.14955
$144$	0	0
$145$	3.17462e9	0.596397
$146$	0	0
$147$	−3.23206e9	−0.570889
$148$	0	0
$149$	−9.16328e9	−1.52305	−0.761523	−	0.648138i	$-0.775548\pi$
$149$	−9.16328e9	−1.52305	−0.761523	+	0.648138i	$0.775548\pi$
$150$	0	0
$151$	6.46073e9	1.01131	0.505656	−	0.862735i	$-0.331250\pi$
$151$	6.46073e9	1.01131	0.505656	+	0.862735i	$0.331250\pi$
$152$	0	0
$153$	−1.06108e9	−0.156545
$154$	0	0
$155$	1.31097e9	0.182431
$156$	0	0
$157$	−1.34930e10	−1.77239	−0.886194	−	0.463315i	$-0.846660\pi$
$157$	−1.34930e10	−1.77239	−0.886194	+	0.463315i	$0.846660\pi$
$158$	0	0
$159$	−1.97527e9	−0.245098
$160$	0	0
$161$	−7.16820e8	−0.0840802
$162$	0	0
$163$	9.11445e9	1.01131	0.505657	−	0.862734i	$-0.331250\pi$
$163$	9.11445e9	1.01131	0.505657	+	0.862734i	$0.331250\pi$
$164$	0	0
$165$	4.93925e9	0.518780
$166$	0	0
$167$	−7.28739e9	−0.725016	−0.362508	−	0.931981i	$-0.618079\pi$
$167$	−7.28739e9	−0.725016	−0.362508	+	0.931981i	$0.618079\pi$
$168$	0	0
$169$	−4.48269e9	−0.422716
$170$	0	0
$171$	4.28809e9	0.383514
$172$	0	0
$173$	1.79271e9	0.152161	0.0760804	−	0.997102i	$-0.475759\pi$
$173$	1.79271e9	0.152161	0.0760804	+	0.997102i	$0.475759\pi$
$174$	0	0
$175$	8.49559e8	0.0684735
$176$	0	0
$177$	−9.67309e8	−0.0740774
$178$	0	0
$179$	−3.58921e9	−0.261313	−0.130656	−	0.991428i	$-0.541708\pi$
$179$	−3.58921e9	−0.261313	−0.130656	+	0.991428i	$0.541708\pi$
$180$	0	0
$181$	−2.34516e10	−1.62412	−0.812060	−	0.583574i	$-0.801654\pi$
$181$	−2.34516e10	−1.62412	−0.812060	+	0.583574i	$0.801654\pi$
$182$	0	0
$183$	−1.53690e10	−1.01301
$184$	0	0
$185$	1.32929e10	0.834350
$186$	0	0
$187$	−1.18817e10	−0.710544
$188$	0	0
$189$	−3.57128e8	−0.0203585
$190$	0	0
$191$	2.13072e10	1.15845	0.579224	−	0.815169i	$-0.303356\pi$
$191$	2.13072e10	1.15845	0.579224	+	0.815169i	$0.303356\pi$
$192$	0	0
$193$	2.02497e10	1.05054	0.525269	−	0.850936i	$-0.323965\pi$
$193$	2.02497e10	1.05054	0.525269	+	0.850936i	$0.323965\pi$
$194$	0	0
$195$	−5.26021e9	−0.260524
$196$	0	0
$197$	1.27339e10	0.602372	0.301186	−	0.953565i	$-0.402618\pi$
$197$	1.27339e10	0.602372	0.301186	+	0.953565i	$0.402618\pi$
$198$	0	0
$199$	5.66200e9	0.255936	0.127968	−	0.991778i	$-0.459155\pi$
$199$	5.66200e9	0.255936	0.127968	+	0.991778i	$0.459155\pi$
$200$	0	0
$201$	8.64348e9	0.373513
$202$	0	0
$203$	−2.57029e9	−0.106231
$204$	0	0
$205$	2.18027e10	0.862219
$206$	0	0
$207$	6.99859e9	0.264938
$208$	0	0
$209$	4.80166e10	1.74074
$210$	0	0
$211$	−5.34254e9	−0.185557	−0.0927783	−	0.995687i	$-0.529575\pi$
$211$	−5.34254e9	−0.185557	−0.0927783	+	0.995687i	$0.529575\pi$
$212$	0	0
$213$	−2.45231e10	−0.816333
$214$	0	0
$215$	3.69310e10	1.17874
$216$	0	0
$217$	−1.06141e9	−0.0324949
$218$	0	0
$219$	6.62333e9	0.194571
$220$	0	0
$221$	1.26538e10	0.356824
$222$	0	0
$223$	−1.53127e10	−0.414649	−0.207325	−	0.978272i	$-0.566476\pi$
$223$	−1.53127e10	−0.414649	−0.207325	+	0.978272i	$0.566476\pi$
$224$	0	0
$225$	−8.29458e9	−0.215761
$226$	0	0
$227$	5.24911e10	1.31211	0.656053	−	0.754714i	$-0.272225\pi$
$227$	5.24911e10	1.31211	0.656053	+	0.754714i	$0.272225\pi$
$228$	0	0
$229$	4.27719e9	0.102778	0.0513888	−	0.998679i	$-0.483635\pi$
$229$	4.27719e9	0.102778	0.0513888	+	0.998679i	$0.483635\pi$
$230$	0	0
$231$	−3.99901e9	−0.0924057
$232$	0	0
$233$	−4.40992e10	−0.980232	−0.490116	−	0.871657i	$-0.663046\pi$
$233$	−4.40992e10	−0.980232	−0.490116	+	0.871657i	$0.663046\pi$
$234$	0	0
$235$	−1.18891e10	−0.254297
$236$	0	0
$237$	−2.55405e10	−0.525851
$238$	0	0
$239$	3.95413e10	0.783899	0.391949	−	0.919987i	$-0.371801\pi$
$239$	3.95413e10	0.783899	0.391949	+	0.919987i	$0.371801\pi$
$240$	0	0
$241$	−4.42570e10	−0.845094	−0.422547	−	0.906341i	$-0.638864\pi$
$241$	−4.42570e10	−0.845094	−0.422547	+	0.906341i	$0.638864\pi$
$242$	0	0
$243$	3.48678e9	0.0641500
$244$	0	0
$245$	−3.31187e10	−0.587254
$246$	0	0
$247$	−5.11368e10	−0.874172
$248$	0	0
$249$	−6.09795e10	−1.00528
$250$	0	0
$251$	−2.93264e10	−0.466366	−0.233183	−	0.972433i	$-0.574914\pi$
$251$	−2.93264e10	−0.466366	−0.233183	+	0.972433i	$0.574914\pi$
$252$	0	0
$253$	7.83680e10	1.20253
$254$	0	0
$255$	−1.08728e10	−0.161032
$256$	0	0
$257$	−4.75455e9	−0.0679846	−0.0339923	−	0.999422i	$-0.510822\pi$
$257$	−4.75455e9	−0.0679846	−0.0339923	+	0.999422i	$0.510822\pi$
$258$	0	0
$259$	−1.07625e10	−0.148615
$260$	0	0
$261$	2.50948e10	0.334735
$262$	0	0
$263$	4.32650e7	0.000557617 0	0.000278809	−	1.00000i	$-0.499911\pi$
$263$	4.32650e7	0.000557617 0	0.000278809		1.00000i	$0.499911\pi$
$264$	0	0
$265$	−2.02404e10	−0.252123
$266$	0	0
$267$	−3.50960e10	−0.422627
$268$	0	0
$269$	6.74801e10	0.785761	0.392881	−	0.919590i	$-0.371479\pi$
$269$	6.74801e10	0.785761	0.392881	+	0.919590i	$0.371479\pi$
$270$	0	0
$271$	−1.17456e11	−1.32286	−0.661431	−	0.750006i	$-0.730050\pi$
$271$	−1.17456e11	−1.32286	−0.661431	+	0.750006i	$0.730050\pi$
$272$	0	0
$273$	4.25887e9	0.0464047
$274$	0	0
$275$	−9.28801e10	−0.979322
$276$	0	0
$277$	−4.12220e10	−0.420698	−0.210349	−	0.977626i	$-0.567460\pi$
$277$	−4.12220e10	−0.420698	−0.210349	+	0.977626i	$0.567460\pi$
$278$	0	0
$279$	1.03630e10	0.102392
$280$	0	0
$281$	−8.48899e10	−0.812227	−0.406114	−	0.913823i	$-0.633116\pi$
$281$	−8.48899e10	−0.812227	−0.406114	+	0.913823i	$0.633116\pi$
$282$	0	0
$283$	3.45640e10	0.320321	0.160160	−	0.987091i	$-0.448799\pi$
$283$	3.45640e10	0.320321	0.160160	+	0.987091i	$0.448799\pi$
$284$	0	0
$285$	4.39396e10	0.394507
$286$	0	0
$287$	−1.76523e10	−0.153579
$288$	0	0
$289$	−9.24326e10	−0.779444
$290$	0	0
$291$	1.03882e11	0.849225
$292$	0	0
$293$	2.21053e11	1.75223	0.876116	−	0.482100i	$-0.160126\pi$
$293$	2.21053e11	1.75223	0.876116	+	0.482100i	$0.160126\pi$
$294$	0	0
$295$	−9.91193e9	−0.0762007
$296$	0	0
$297$	3.90439e10	0.291172
$298$	0	0
$299$	−8.34604e10	−0.603893
$300$	0	0
$301$	−2.99008e10	−0.209959
$302$	0	0
$303$	−1.58854e11	−1.08269
$304$	0	0
$305$	−1.57484e11	−1.04205
$306$	0	0
$307$	2.52457e11	1.62205	0.811024	−	0.585012i	$-0.198910\pi$
$307$	2.52457e11	1.62205	0.811024	+	0.585012i	$0.198910\pi$
$308$	0	0
$309$	7.67930e10	0.479191
$310$	0	0
$311$	9.13082e10	0.553462	0.276731	−	0.960947i	$-0.410749\pi$
$311$	9.13082e10	0.553462	0.276731	+	0.960947i	$0.410749\pi$
$312$	0	0
$313$	4.06665e10	0.239490	0.119745	−	0.992805i	$-0.461792\pi$
$313$	4.06665e10	0.239490	0.119745	+	0.992805i	$0.461792\pi$
$314$	0	0
$315$	−3.65946e9	−0.0209421
$316$	0	0
$317$	−2.10345e11	−1.16995	−0.584974	−	0.811052i	$-0.698895\pi$
$317$	−2.10345e11	−1.16995	−0.584974	+	0.811052i	$0.698895\pi$
$318$	0	0
$319$	2.81003e11	1.51933
$320$	0	0
$321$	1.99756e10	0.105009
$322$	0	0
$323$	−1.05700e11	−0.540334
$324$	0	0
$325$	9.89155e10	0.491801
$326$	0	0
$327$	1.19319e11	0.577094
$328$	0	0
$329$	9.62584e9	0.0452957
$330$	0	0
$331$	3.07269e11	1.40700	0.703499	−	0.710697i	$-0.251620\pi$
$331$	3.07269e11	1.40700	0.703499	+	0.710697i	$0.251620\pi$
$332$	0	0
$333$	1.05078e11	0.468289
$334$	0	0
$335$	8.85689e10	0.384220
$336$	0	0
$337$	−1.81037e11	−0.764596	−0.382298	−	0.924039i	$-0.624867\pi$
$337$	−1.81037e11	−0.764596	−0.382298	+	0.924039i	$0.624867\pi$
$338$	0	0
$339$	6.20929e10	0.255354
$340$	0	0
$341$	1.16041e11	0.464748
$342$	0	0
$343$	5.39318e10	0.210388
$344$	0	0
$345$	7.17140e10	0.272532
$346$	0	0
$347$	−5.22927e11	−1.93624	−0.968119	−	0.250491i	$-0.919408\pi$
$347$	−5.22927e11	−1.93624	−0.968119	+	0.250491i	$0.919408\pi$
$348$	0	0
$349$	3.04567e11	1.09893	0.549463	−	0.835518i	$-0.314832\pi$
$349$	3.04567e11	1.09893	0.549463	+	0.835518i	$0.314832\pi$
$350$	0	0
$351$	−4.15810e10	−0.146222
$352$	0	0
$353$	7.04187e10	0.241380	0.120690	−	0.992690i	$-0.461489\pi$
$353$	7.04187e10	0.241380	0.120690	+	0.992690i	$0.461489\pi$
$354$	0	0
$355$	−2.51286e11	−0.839732
$356$	0	0
$357$	8.80307e9	0.0286832
$358$	0	0
$359$	2.91400e11	0.925900	0.462950	−	0.886384i	$-0.346791\pi$
$359$	2.91400e11	0.925900	0.462950	+	0.886384i	$0.346791\pi$
$360$	0	0
$361$	1.04469e11	0.323745
$362$	0	0
$363$	2.46208e11	0.744255
$364$	0	0
$365$	6.78687e10	0.200148
$366$	0	0
$367$	−2.65943e11	−0.765230	−0.382615	−	0.923908i	$-0.624976\pi$
$367$	−2.65943e11	−0.765230	−0.382615	+	0.923908i	$0.624976\pi$
$368$	0	0
$369$	1.72346e11	0.483931
$370$	0	0
$371$	1.63874e10	0.0449085
$372$	0	0
$373$	3.21656e11	0.860403	0.430201	−	0.902733i	$-0.358443\pi$
$373$	3.21656e11	0.860403	0.430201	+	0.902733i	$0.358443\pi$
$374$	0	0
$375$	−2.16302e11	−0.564834
$376$	0	0
$377$	−2.99263e11	−0.762987
$378$	0	0
$379$	−2.83177e11	−0.704988	−0.352494	−	0.935814i	$-0.614666\pi$
$379$	−2.83177e11	−0.704988	−0.352494	+	0.935814i	$0.614666\pi$
$380$	0	0
$381$	−2.49923e11	−0.607635
$382$	0	0
$383$	−6.70279e11	−1.59170	−0.795850	−	0.605494i	$-0.792975\pi$
$383$	−6.70279e11	−1.59170	−0.795850	+	0.605494i	$0.792975\pi$
$384$	0	0
$385$	−4.09775e10	−0.0950544
$386$	0	0
$387$	2.91933e11	0.661583
$388$	0	0
$389$	−8.67505e11	−1.92087	−0.960437	−	0.278498i	$-0.910163\pi$
$389$	−8.67505e11	−1.92087	−0.960437	+	0.278498i	$0.910163\pi$
$390$	0	0
$391$	−1.72512e11	−0.373272
$392$	0	0
$393$	−2.23817e11	−0.473288
$394$	0	0
$395$	−2.61712e11	−0.540924
$396$	0	0
$397$	−3.14512e11	−0.635449	−0.317724	−	0.948183i	$-0.602919\pi$
$397$	−3.14512e11	−0.635449	−0.317724	+	0.948183i	$0.602919\pi$
$398$	0	0
$399$	−3.55752e10	−0.0702700
$400$	0	0
$401$	−1.95667e11	−0.377892	−0.188946	−	0.981987i	$-0.560507\pi$
$401$	−1.95667e11	−0.377892	−0.188946	+	0.981987i	$0.560507\pi$
$402$	0	0
$403$	−1.23582e11	−0.233390
$404$	0	0
$405$	3.57288e10	0.0659889
$406$	0	0
$407$	1.17663e12	2.12553
$408$	0	0
$409$	3.63692e11	0.642656	0.321328	−	0.946968i	$-0.395871\pi$
$409$	3.63692e11	0.642656	0.321328	+	0.946968i	$0.395871\pi$
$410$	0	0
$411$	2.22783e11	0.385118
$412$	0	0
$413$	8.02508e9	0.0135729
$414$	0	0
$415$	−6.24852e11	−1.03410
$416$	0	0
$417$	−1.28551e11	−0.208191
$418$	0	0
$419$	−2.20311e11	−0.349199	−0.174600	−	0.984640i	$-0.555863\pi$
$419$	−2.20311e11	−0.349199	−0.174600	+	0.984640i	$0.555863\pi$
$420$	0	0
$421$	−9.96326e11	−1.54572	−0.772862	−	0.634574i	$-0.781175\pi$
$421$	−9.96326e11	−1.54572	−0.772862	+	0.634574i	$0.781175\pi$
$422$	0	0
$423$	−9.39808e10	−0.142727
$424$	0	0
$425$	2.04458e11	0.303986
$426$	0	0
$427$	1.27505e11	0.185611
$428$	0	0
$429$	−4.65611e11	−0.663690
$430$	0	0
$431$	9.96850e11	1.39150	0.695748	−	0.718286i	$-0.255073\pi$
$431$	9.96850e11	1.39150	0.695748	+	0.718286i	$0.255073\pi$
$432$	0	0
$433$	8.11011e11	1.10874	0.554372	−	0.832269i	$-0.312959\pi$
$433$	8.11011e11	1.10874	0.554372	+	0.832269i	$0.312959\pi$
$434$	0	0
$435$	2.57144e11	0.344330
$436$	0	0
$437$	6.97163e11	0.914466
$438$	0	0
$439$	1.24677e11	0.160212	0.0801062	−	0.996786i	$-0.474474\pi$
$439$	1.24677e11	0.160212	0.0801062	+	0.996786i	$0.474474\pi$
$440$	0	0
$441$	−2.61797e11	−0.329603
$442$	0	0
$443$	−5.13417e11	−0.633364	−0.316682	−	0.948532i	$-0.602569\pi$
$443$	−5.13417e11	−0.633364	−0.316682	+	0.948532i	$0.602569\pi$
$444$	0	0
$445$	−3.59626e11	−0.434741
$446$	0	0
$447$	−7.42226e11	−0.879330
$448$	0	0
$449$	7.29234e11	0.846756	0.423378	−	0.905953i	$-0.360844\pi$
$449$	7.29234e11	0.846756	0.423378	+	0.905953i	$0.360844\pi$
$450$	0	0
$451$	1.92988e12	2.19652
$452$	0	0
$453$	5.23319e11	0.583882
$454$	0	0
$455$	4.36403e10	0.0477349
$456$	0	0
$457$	1.10073e12	1.18048	0.590240	−	0.807227i	$-0.299033\pi$
$457$	1.10073e12	1.18048	0.590240	+	0.807227i	$0.299033\pi$
$458$	0	0
$459$	−8.59478e10	−0.0903811
$460$	0	0
$461$	−1.26400e12	−1.30345	−0.651723	−	0.758457i	$-0.725953\pi$
$461$	−1.26400e12	−1.30345	−0.651723	+	0.758457i	$0.725953\pi$
$462$	0	0
$463$	−1.82843e12	−1.84911	−0.924557	−	0.381045i	$-0.875564\pi$
$463$	−1.82843e12	−1.84911	−0.924557	+	0.381045i	$0.875564\pi$
$464$	0	0
$465$	1.06188e11	0.105327
$466$	0	0
$467$	9.95640e10	0.0968671	0.0484336	−	0.998826i	$-0.484577\pi$
$467$	9.95640e10	0.0968671	0.0484336	+	0.998826i	$0.484577\pi$
$468$	0	0
$469$	−7.17088e10	−0.0684376
$470$	0	0
$471$	−1.09293e12	−1.02329
$472$	0	0
$473$	3.26898e12	3.00287
$474$	0	0
$475$	−8.26262e11	−0.744726
$476$	0	0
$477$	−1.59997e11	−0.141507
$478$	0	0
$479$	1.32763e12	1.15230	0.576150	−	0.817344i	$-0.304554\pi$
$479$	1.32763e12	1.15230	0.576150	+	0.817344i	$0.304554\pi$
$480$	0	0
$481$	−1.25309e12	−1.06741
$482$	0	0
$483$	−5.80624e10	−0.0485437
$484$	0	0
$485$	1.06447e12	0.873568
$486$	0	0
$487$	1.37834e12	1.11039	0.555197	−	0.831719i	$-0.312643\pi$
$487$	1.37834e12	1.11039	0.555197	+	0.831719i	$0.312643\pi$
$488$	0	0
$489$	7.38270e11	0.583883
$490$	0	0
$491$	−1.13645e12	−0.882439	−0.441219	−	0.897399i	$-0.645454\pi$
$491$	−1.13645e12	−0.882439	−0.441219	+	0.897399i	$0.645454\pi$
$492$	0	0
$493$	−6.18576e11	−0.471609
$494$	0	0
$495$	4.00080e11	0.299518
$496$	0	0
$497$	2.03451e11	0.149574
$498$	0	0
$499$	1.91132e12	1.38001	0.690004	−	0.723806i	$-0.257609\pi$
$499$	1.91132e12	1.38001	0.690004	+	0.723806i	$0.257609\pi$
$500$	0	0
$501$	−5.90278e11	−0.418588
$502$	0	0
$503$	1.79892e12	1.25302	0.626509	−	0.779414i	$-0.284483\pi$
$503$	1.79892e12	1.25302	0.626509	+	0.779414i	$0.284483\pi$
$504$	0	0
$505$	−1.62776e12	−1.11373
$506$	0	0
$507$	−3.63098e11	−0.244055
$508$	0	0
$509$	1.65330e12	1.09174	0.545872	−	0.837869i	$-0.316199\pi$
$509$	1.65330e12	1.09174	0.545872	+	0.837869i	$0.316199\pi$
$510$	0	0
$511$	−5.49491e10	−0.0356506
$512$	0	0
$513$	3.47335e11	0.221422
$514$	0	0
$515$	7.86891e11	0.492926
$516$	0	0
$517$	−1.05237e12	−0.647828
$518$	0	0
$519$	1.45210e11	0.0878501
$520$	0	0
$521$	−2.43255e12	−1.44641	−0.723207	−	0.690632i	$-0.757333\pi$
$521$	−2.43255e12	−1.44641	−0.723207	+	0.690632i	$0.757333\pi$
$522$	0	0
$523$	−1.35462e12	−0.791697	−0.395849	−	0.918316i	$-0.629549\pi$
$523$	−1.35462e12	−0.791697	−0.395849	+	0.918316i	$0.629549\pi$
$524$	0	0
$525$	6.88143e10	0.0395332
$526$	0	0
$527$	−2.55443e11	−0.144260
$528$	0	0
$529$	−6.63312e11	−0.368271
$530$	0	0
$531$	−7.83520e10	−0.0427686
$532$	0	0
$533$	−2.05528e12	−1.10306
$534$	0	0
$535$	2.04689e11	0.108019
$536$	0	0
$537$	−2.90726e11	−0.150869
$538$	0	0
$539$	−2.93152e12	−1.49604
$540$	0	0
$541$	7.25578e11	0.364163	0.182082	−	0.983283i	$-0.441716\pi$
$541$	7.25578e11	0.364163	0.182082	+	0.983283i	$0.441716\pi$
$542$	0	0
$543$	−1.89958e12	−0.937686
$544$	0	0
$545$	1.22266e12	0.593636
$546$	0	0
$547$	2.32193e12	1.10893	0.554467	−	0.832206i	$-0.312922\pi$
$547$	2.32193e12	1.10893	0.554467	+	0.832206i	$0.312922\pi$
$548$	0	0
$549$	−1.24489e12	−0.584863
$550$	0	0
$551$	2.49981e12	1.15538
$552$	0	0
$553$	2.11892e11	0.0963499
$554$	0	0
$555$	1.07673e12	0.481712
$556$	0	0
$557$	−1.15050e12	−0.506452	−0.253226	−	0.967407i	$-0.581492\pi$
$557$	−1.15050e12	−0.506452	−0.253226	+	0.967407i	$0.581492\pi$
$558$	0	0
$559$	−3.48140e12	−1.50800
$560$	0	0
$561$	−9.62417e11	−0.410233
$562$	0	0
$563$	−3.26347e12	−1.36897	−0.684483	−	0.729029i	$-0.739972\pi$
$563$	−3.26347e12	−1.36897	−0.684483	+	0.729029i	$0.739972\pi$
$564$	0	0
$565$	6.36261e11	0.262674
$566$	0	0
$567$	−2.89274e10	−0.0117540
$568$	0	0
$569$	−3.14577e12	−1.25812	−0.629059	−	0.777357i	$-0.716560\pi$
$569$	−3.14577e12	−1.25812	−0.629059	+	0.777357i	$0.716560\pi$
$570$	0	0
$571$	−2.86591e12	−1.12824	−0.564119	−	0.825694i	$-0.690784\pi$
$571$	−2.86591e12	−1.12824	−0.564119	+	0.825694i	$0.690784\pi$
$572$	0	0
$573$	1.72588e12	0.668830
$574$	0	0
$575$	−1.34854e12	−0.514470
$576$	0	0
$577$	3.40532e12	1.27899	0.639494	−	0.768796i	$-0.279144\pi$
$577$	3.40532e12	1.27899	0.639494	+	0.768796i	$0.279144\pi$
$578$	0	0
$579$	1.64023e12	0.606528
$580$	0	0
$581$	5.05904e11	0.184194
$582$	0	0
$583$	−1.79159e12	−0.642290
$584$	0	0
$585$	−4.26077e11	−0.150413
$586$	0	0
$587$	4.19783e11	0.145933	0.0729665	−	0.997334i	$-0.476753\pi$
$587$	4.19783e11	0.145933	0.0729665	+	0.997334i	$0.476753\pi$
$588$	0	0
$589$	1.03230e12	0.353418
$590$	0	0
$591$	1.03145e12	0.347779
$592$	0	0
$593$	−3.91495e9	−0.00130011	−0.000650055	−	1.00000i	$-0.500207\pi$
$593$	−3.91495e9	−0.00130011	−0.000650055		1.00000i	$0.500207\pi$
$594$	0	0
$595$	9.02043e10	0.0295054
$596$	0	0
$597$	4.58622e11	0.147765
$598$	0	0
$599$	−9.79764e11	−0.310957	−0.155479	−	0.987839i	$-0.549692\pi$
$599$	−9.79764e11	−0.310957	−0.155479	+	0.987839i	$0.549692\pi$
$600$	0	0
$601$	−3.20416e11	−0.100179	−0.0500897	−	0.998745i	$-0.515951\pi$
$601$	−3.20416e11	−0.100179	−0.0500897	+	0.998745i	$0.515951\pi$
$602$	0	0
$603$	7.00122e11	0.215648
$604$	0	0
$605$	2.52287e12	0.765588
$606$	0	0
$607$	−4.50671e11	−0.134744	−0.0673721	−	0.997728i	$-0.521461\pi$
$607$	−4.50671e11	−0.134744	−0.0673721	+	0.997728i	$0.521461\pi$
$608$	0	0
$609$	−2.08194e11	−0.0613323
$610$	0	0
$611$	1.12075e12	0.325330
$612$	0	0
$613$	4.91821e12	1.40681	0.703404	−	0.710790i	$-0.251662\pi$
$613$	4.91821e12	1.40681	0.703404	+	0.710790i	$0.251662\pi$
$614$	0	0
$615$	1.76602e12	0.497802
$616$	0	0
$617$	−3.02063e12	−0.839102	−0.419551	−	0.907732i	$-0.637812\pi$
$617$	−3.02063e12	−0.839102	−0.419551	+	0.907732i	$0.637812\pi$
$618$	0	0
$619$	1.29184e12	0.353672	0.176836	−	0.984240i	$-0.443414\pi$
$619$	1.29184e12	0.353672	0.176836	+	0.984240i	$0.443414\pi$
$620$	0	0
$621$	5.66886e11	0.152962
$622$	0	0
$623$	2.91167e11	0.0774366
$624$	0	0
$625$	2.52757e11	0.0662587
$626$	0	0
$627$	3.88935e12	1.00502
$628$	0	0
$629$	−2.59014e12	−0.659774
$630$	0	0
$631$	7.29259e12	1.83126	0.915629	−	0.402024i	$-0.131693\pi$
$631$	7.29259e12	1.83126	0.915629	+	0.402024i	$0.131693\pi$
$632$	0	0
$633$	−4.32746e11	−0.107131
$634$	0	0
$635$	−2.56093e12	−0.625053
$636$	0	0
$637$	3.12201e12	0.751290
$638$	0	0
$639$	−1.98637e12	−0.471310
$640$	0	0
$641$	−6.42622e12	−1.50347	−0.751734	−	0.659466i	$-0.770782\pi$
$641$	−6.42622e12	−1.50347	−0.751734	+	0.659466i	$0.770782\pi$
$642$	0	0
$643$	1.85597e12	0.428175	0.214087	−	0.976814i	$-0.431322\pi$
$643$	1.85597e12	0.428175	0.214087	+	0.976814i	$0.431322\pi$
$644$	0	0
$645$	2.99141e12	0.680547
$646$	0	0
$647$	6.59916e12	1.48054	0.740269	−	0.672311i	$-0.234698\pi$
$647$	6.59916e12	1.48054	0.740269	+	0.672311i	$0.234698\pi$
$648$	0	0
$649$	−8.77361e11	−0.194123
$650$	0	0
$651$	−8.59743e10	−0.0187609
$652$	0	0
$653$	3.05288e12	0.657054	0.328527	−	0.944495i	$-0.393448\pi$
$653$	3.05288e12	0.657054	0.328527	+	0.944495i	$0.393448\pi$
$654$	0	0
$655$	−2.29343e12	−0.486855
$656$	0	0
$657$	5.36490e11	0.112336
$658$	0	0
$659$	9.04254e12	1.86770	0.933848	−	0.357669i	$-0.116429\pi$
$659$	9.04254e12	1.86770	0.933848	+	0.357669i	$0.116429\pi$
$660$	0	0
$661$	−2.42286e12	−0.493652	−0.246826	−	0.969060i	$-0.579388\pi$
$661$	−2.42286e12	−0.493652	−0.246826	+	0.969060i	$0.579388\pi$
$662$	0	0
$663$	1.02496e12	0.206013
$664$	0	0
$665$	−3.64536e11	−0.0722842
$666$	0	0
$667$	4.07994e12	0.798155
$668$	0	0
$669$	−1.24033e12	−0.239398
$670$	0	0
$671$	−1.39398e13	−2.65464
$672$	0	0
$673$	−5.42076e12	−1.01857	−0.509287	−	0.860597i	$-0.670091\pi$
$673$	−5.42076e12	−1.01857	−0.509287	+	0.860597i	$0.670091\pi$
$674$	0	0
$675$	−6.71861e11	−0.124570
$676$	0	0
$677$	5.70581e12	1.04392	0.521961	−	0.852969i	$-0.325201\pi$
$677$	5.70581e12	1.04392	0.521961	+	0.852969i	$0.325201\pi$
$678$	0	0
$679$	−8.61838e11	−0.155601
$680$	0	0
$681$	4.25178e12	0.757545
$682$	0	0
$683$	−9.01096e12	−1.58445	−0.792224	−	0.610231i	$-0.791077\pi$
$683$	−9.01096e12	−1.58445	−0.792224	+	0.610231i	$0.791077\pi$
$684$	0	0
$685$	2.28284e12	0.396157
$686$	0	0
$687$	3.46452e11	0.0593387
$688$	0	0
$689$	1.90801e12	0.322548
$690$	0	0
$691$	−3.93638e12	−0.656819	−0.328410	−	0.944535i	$-0.606513\pi$
$691$	−3.93638e12	−0.656819	−0.328410	+	0.944535i	$0.606513\pi$
$692$	0	0
$693$	−3.23920e11	−0.0533504
$694$	0	0
$695$	−1.31725e12	−0.214159
$696$	0	0
$697$	−4.24826e12	−0.681811
$698$	0	0
$699$	−3.57203e12	−0.565937
$700$	0	0
$701$	−2.61506e12	−0.409026	−0.204513	−	0.978864i	$-0.565561\pi$
$701$	−2.61506e12	−0.409026	−0.204513	+	0.978864i	$0.565561\pi$
$702$	0	0
$703$	1.04673e13	1.61636
$704$	0	0
$705$	−9.63013e11	−0.146819
$706$	0	0
$707$	1.31790e12	0.198378
$708$	0	0
$709$	6.29669e12	0.935845	0.467923	−	0.883769i	$-0.345003\pi$
$709$	6.29669e12	0.935845	0.467923	+	0.883769i	$0.345003\pi$
$710$	0	0
$711$	−2.06878e12	−0.303600
$712$	0	0
$713$	1.68482e12	0.244147
$714$	0	0
$715$	−4.77108e12	−0.682714
$716$	0	0
$717$	3.20284e12	0.452584
$718$	0	0
$719$	−3.23756e12	−0.451791	−0.225895	−	0.974152i	$-0.572531\pi$
$719$	−3.23756e12	−0.451791	−0.225895	+	0.974152i	$0.572531\pi$
$720$	0	0
$721$	−6.37097e11	−0.0878005
$722$	0	0
$723$	−3.58482e12	−0.487915
$724$	0	0
$725$	−4.83546e12	−0.650005
$726$	0	0
$727$	3.73928e12	0.496459	0.248230	−	0.968701i	$-0.420151\pi$
$727$	3.73928e12	0.496459	0.248230	+	0.968701i	$0.420151\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−7.19604e12	−0.932106
$732$	0	0
$733$	−8.14796e12	−1.04251	−0.521256	−	0.853400i	$-0.674536\pi$
$733$	−8.14796e12	−1.04251	−0.521256	+	0.853400i	$0.674536\pi$
$734$	0	0
$735$	−2.68261e12	−0.339051
$736$	0	0
$737$	7.83974e12	0.978809
$738$	0	0
$739$	1.90382e12	0.234815	0.117408	−	0.993084i	$-0.462542\pi$
$739$	1.90382e12	0.234815	0.117408	+	0.993084i	$0.462542\pi$
$740$	0	0
$741$	−4.14208e12	−0.504704
$742$	0	0
$743$	7.95774e12	0.957945	0.478972	−	0.877830i	$-0.341009\pi$
$743$	7.95774e12	0.957945	0.478972	+	0.877830i	$0.341009\pi$
$744$	0	0
$745$	−7.60552e12	−0.904536
$746$	0	0
$747$	−4.93934e12	−0.580398
$748$	0	0
$749$	−1.65724e11	−0.0192405
$750$	0	0
$751$	−1.55077e13	−1.77897	−0.889483	−	0.456969i	$-0.848935\pi$
$751$	−1.55077e13	−1.77897	−0.889483	+	0.456969i	$0.848935\pi$
$752$	0	0
$753$	−2.37544e12	−0.269256
$754$	0	0
$755$	5.36241e12	0.600618
$756$	0	0
$757$	−5.49750e12	−0.608463	−0.304231	−	0.952598i	$-0.598400\pi$
$757$	−5.49750e12	−0.608463	−0.304231	+	0.952598i	$0.598400\pi$
$758$	0	0
$759$	6.34781e12	0.694282
$760$	0	0
$761$	3.81389e12	0.412228	0.206114	−	0.978528i	$-0.433918\pi$
$761$	3.81389e12	0.412228	0.206114	+	0.978528i	$0.433918\pi$
$762$	0	0
$763$	−9.89910e11	−0.105739
$764$	0	0
$765$	−8.80700e11	−0.0929719
$766$	0	0
$767$	9.34373e11	0.0974857
$768$	0	0
$769$	1.01006e12	0.104155	0.0520775	−	0.998643i	$-0.483416\pi$
$769$	1.01006e12	0.104155	0.0520775	+	0.998643i	$0.483416\pi$
$770$	0	0
$771$	−3.85119e11	−0.0392509
$772$	0	0
$773$	−6.41222e12	−0.645953	−0.322976	−	0.946407i	$-0.604683\pi$
$773$	−6.41222e12	−0.645953	−0.322976	+	0.946407i	$0.604683\pi$
$774$	0	0
$775$	−1.99682e12	−0.198830
$776$	0	0
$777$	−8.71761e11	−0.0858030
$778$	0	0
$779$	1.71682e13	1.67035
$780$	0	0
$781$	−2.22428e13	−2.13924
$782$	0	0
$783$	2.03268e12	0.193259
$784$	0	0
$785$	−1.11992e13	−1.05262
$786$	0	0
$787$	1.11599e12	0.103699	0.0518494	−	0.998655i	$-0.483488\pi$
$787$	1.11599e12	0.103699	0.0518494	+	0.998655i	$0.483488\pi$
$788$	0	0
$789$	3.50447e9	0.000321940 0
$790$	0	0
$791$	−5.15141e11	−0.0467877
$792$	0	0
$793$	1.48457e13	1.33312
$794$	0	0
$795$	−1.63947e12	−0.145564
$796$	0	0
$797$	1.36680e13	1.19989	0.599947	−	0.800040i	$-0.295188\pi$
$797$	1.36680e13	1.19989	0.599947	+	0.800040i	$0.295188\pi$
$798$	0	0
$799$	2.31659e12	0.201089
$800$	0	0
$801$	−2.84278e12	−0.244004
$802$	0	0
$803$	6.00745e12	0.509882
$804$	0	0
$805$	−5.94960e11	−0.0499352
$806$	0	0
$807$	5.46589e12	0.453659
$808$	0	0
$809$	1.54771e13	1.27035	0.635174	−	0.772369i	$-0.280929\pi$
$809$	1.54771e13	1.27035	0.635174	+	0.772369i	$0.280929\pi$
$810$	0	0
$811$	−8.55212e12	−0.694192	−0.347096	−	0.937830i	$-0.612832\pi$
$811$	−8.55212e12	−0.694192	−0.347096	+	0.937830i	$0.612832\pi$
$812$	0	0
$813$	−9.51396e12	−0.763755
$814$	0	0
$815$	7.56499e12	0.600619
$816$	0	0
$817$	2.90808e13	2.28353
$818$	0	0
$819$	3.44968e11	0.0267918
$820$	0	0
$821$	8.29284e12	0.637029	0.318515	−	0.947918i	$-0.396816\pi$
$821$	8.29284e12	0.637029	0.318515	+	0.947918i	$0.396816\pi$
$822$	0	0
$823$	6.55958e12	0.498399	0.249199	−	0.968452i	$-0.419833\pi$
$823$	6.55958e12	0.498399	0.249199	+	0.968452i	$0.419833\pi$
$824$	0	0
$825$	−7.52329e12	−0.565412
$826$	0	0
$827$	1.63826e13	1.21789	0.608944	−	0.793214i	$-0.291594\pi$
$827$	1.63826e13	1.21789	0.608944	+	0.793214i	$0.291594\pi$
$828$	0	0
$829$	2.16258e13	1.59029	0.795144	−	0.606420i	$-0.207395\pi$
$829$	2.16258e13	1.59029	0.795144	+	0.606420i	$0.207395\pi$
$830$	0	0
$831$	−3.33898e12	−0.242890
$832$	0	0
$833$	6.45319e12	0.464379
$834$	0	0
$835$	−6.04853e12	−0.430587
$836$	0	0
$837$	8.39400e11	0.0591159
$838$	0	0
$839$	−1.73801e13	−1.21094	−0.605471	−	0.795868i	$-0.707015\pi$
$839$	−1.73801e13	−1.21094	−0.605471	+	0.795868i	$0.707015\pi$
$840$	0	0
$841$	1.22240e11	0.00842617
$842$	0	0
$843$	−6.87608e12	−0.468940
$844$	0	0
$845$	−3.72063e12	−0.251051
$846$	0	0
$847$	−2.04261e12	−0.136367
$848$	0	0
$849$	2.79969e12	0.184937
$850$	0	0
$851$	1.70838e13	1.11661
$852$	0	0
$853$	4.42386e12	0.286109	0.143054	−	0.989715i	$-0.454308\pi$
$853$	4.42386e12	0.286109	0.143054	+	0.989715i	$0.454308\pi$
$854$	0	0
$855$	3.55911e12	0.227769
$856$	0	0
$857$	1.01532e13	0.642966	0.321483	−	0.946915i	$-0.395819\pi$
$857$	1.01532e13	0.642966	0.321483	+	0.946915i	$0.395819\pi$
$858$	0	0
$859$	−2.64047e13	−1.65467	−0.827337	−	0.561706i	$-0.810145\pi$
$859$	−2.64047e13	−1.65467	−0.827337	+	0.561706i	$0.810145\pi$
$860$	0	0
$861$	−1.42984e12	−0.0886690
$862$	0	0
$863$	1.60032e12	0.0982104	0.0491052	−	0.998794i	$-0.484363\pi$
$863$	1.60032e12	0.0982104	0.0491052	+	0.998794i	$0.484363\pi$
$864$	0	0
$865$	1.48795e12	0.0903683
$866$	0	0
$867$	−7.48704e12	−0.450012
$868$	0	0
$869$	−2.31656e13	−1.37802
$870$	0	0
$871$	−8.34917e12	−0.491543
$872$	0	0
$873$	8.41446e12	0.490300
$874$	0	0
$875$	1.79451e12	0.103493
$876$	0	0
$877$	−1.44970e13	−0.827525	−0.413763	−	0.910385i	$-0.635786\pi$
$877$	−1.44970e13	−0.827525	−0.413763	+	0.910385i	$0.635786\pi$
$878$	0	0
$879$	1.79053e13	1.01165
$880$	0	0
$881$	1.26716e13	0.708661	0.354330	−	0.935120i	$-0.384709\pi$
$881$	1.26716e13	0.708661	0.354330	+	0.935120i	$0.384709\pi$
$882$	0	0
$883$	3.35703e13	1.85837	0.929183	−	0.369619i	$-0.120512\pi$
$883$	3.35703e13	1.85837	0.929183	+	0.369619i	$0.120512\pi$
$884$	0	0
$885$	−8.02866e11	−0.0439945
$886$	0	0
$887$	−1.82723e13	−0.991145	−0.495573	−	0.868567i	$-0.665042\pi$
$887$	−1.82723e13	−0.991145	−0.495573	+	0.868567i	$0.665042\pi$
$888$	0	0
$889$	2.07343e12	0.111335
$890$	0	0
$891$	3.16256e12	0.168108
$892$	0	0
$893$	−9.36187e12	−0.492642
$894$	0	0
$895$	−2.97905e12	−0.155194
$896$	0	0
$897$	−6.76029e12	−0.348658
$898$	0	0
$899$	6.04126e12	0.308467
$900$	0	0
$901$	3.94386e12	0.199370
$902$	0	0
$903$	−2.42196e12	−0.121220
$904$	0	0
$905$	−1.94648e13	−0.964564
$906$	0	0
$907$	−6.92303e12	−0.339675	−0.169837	−	0.985472i	$-0.554324\pi$
$907$	−6.92303e12	−0.339675	−0.169837	+	0.985472i	$0.554324\pi$
$908$	0	0
$909$	−1.28671e13	−0.625093
$910$	0	0
$911$	−3.10020e13	−1.49127	−0.745636	−	0.666353i	$-0.767854\pi$
$911$	−3.10020e13	−1.49127	−0.745636	+	0.666353i	$0.767854\pi$
$912$	0	0
$913$	−5.53092e13	−2.63438
$914$	0	0
$915$	−1.27562e13	−0.601627
$916$	0	0
$917$	1.85685e12	0.0867191
$918$	0	0
$919$	2.81550e13	1.30208	0.651038	−	0.759045i	$-0.274334\pi$
$919$	2.81550e13	1.30208	0.651038	+	0.759045i	$0.274334\pi$
$920$	0	0
$921$	2.04490e13	0.936490
$922$	0	0
$923$	2.36881e13	1.07429
$924$	0	0
$925$	−2.02473e13	−0.909347
$926$	0	0
$927$	6.22023e12	0.276661
$928$	0	0
$929$	2.28510e13	1.00655	0.503273	−	0.864127i	$-0.332129\pi$
$929$	2.28510e13	1.00655	0.503273	+	0.864127i	$0.332129\pi$
$930$	0	0
$931$	−2.60788e13	−1.13767
$932$	0	0
$933$	7.39596e12	0.319542
$934$	0	0
$935$	−9.86180e12	−0.421992
$936$	0	0
$937$	3.16648e13	1.34199	0.670995	−	0.741462i	$-0.265867\pi$
$937$	3.16648e13	1.34199	0.670995	+	0.741462i	$0.265867\pi$
$938$	0	0
$939$	3.29399e12	0.138270
$940$	0	0
$941$	−4.10217e13	−1.70553	−0.852767	−	0.522292i	$-0.825077\pi$
$941$	−4.10217e13	−1.70553	−0.852767	+	0.522292i	$0.825077\pi$
$942$	0	0
$943$	2.80203e13	1.15390
$944$	0	0
$945$	−2.96417e11	−0.0120909
$946$	0	0
$947$	−2.26511e13	−0.915196	−0.457598	−	0.889159i	$-0.651290\pi$
$947$	−2.26511e13	−0.915196	−0.457598	+	0.889159i	$0.651290\pi$
$948$	0	0
$949$	−6.39781e12	−0.256055
$950$	0	0
$951$	−1.70380e13	−0.675469
$952$	0	0
$953$	−3.06846e13	−1.20504	−0.602521	−	0.798103i	$-0.705837\pi$
$953$	−3.06846e13	−1.20504	−0.602521	+	0.798103i	$0.705837\pi$
$954$	0	0
$955$	1.76850e13	0.688002
$956$	0	0
$957$	2.27613e13	0.877188
$958$	0	0
$959$	−1.84827e12	−0.0705639
$960$	0	0
$961$	−2.39449e13	−0.905643
$962$	0	0
$963$	1.61803e12	0.0606272
$964$	0	0
$965$	1.68073e13	0.623914
$966$	0	0
$967$	1.91408e13	0.703949	0.351975	−	0.936010i	$-0.385510\pi$
$967$	1.91408e13	0.703949	0.351975	+	0.936010i	$0.385510\pi$
$968$	0	0
$969$	−8.56167e12	−0.311962
$970$	0	0
$971$	−4.98467e13	−1.79949	−0.899746	−	0.436414i	$-0.856248\pi$
$971$	−4.98467e13	−1.79949	−0.899746	+	0.436414i	$0.856248\pi$
$972$	0	0
$973$	1.06649e12	0.0381461
$974$	0	0
$975$	8.01215e12	0.283941
$976$	0	0
$977$	−1.74066e13	−0.611205	−0.305603	−	0.952159i	$-0.598858\pi$
$977$	−1.74066e13	−0.611205	−0.305603	+	0.952159i	$0.598858\pi$
$978$	0	0
$979$	−3.18325e13	−1.10751
$980$	0	0
$981$	9.66488e12	0.333185
$982$	0	0
$983$	−2.09397e13	−0.715286	−0.357643	−	0.933858i	$-0.616420\pi$
$983$	−2.09397e13	−0.715286	−0.357643	+	0.933858i	$0.616420\pi$
$984$	0	0
$985$	1.05692e13	0.357748
$986$	0	0
$987$	7.79693e11	0.0261515
$988$	0	0
$989$	4.74629e13	1.57751
$990$	0	0
$991$	9.68052e12	0.318836	0.159418	−	0.987211i	$-0.449038\pi$
$991$	9.68052e12	0.318836	0.159418	+	0.987211i	$0.449038\pi$
$992$	0	0
$993$	2.48888e13	0.812330
$994$	0	0
$995$	4.69946e12	0.152000
$996$	0	0
$997$	4.45555e12	0.142815	0.0714074	−	0.997447i	$-0.477251\pi$
$997$	4.45555e12	0.142815	0.0714074	+	0.997447i	$0.477251\pi$
$998$	0	0
$999$	8.51134e12	0.270367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.10.a.f.1.1		1
3.2	odd	2		144.10.a.e.1.1		1
4.3	odd	2		24.10.a.a.1.1	✓	1
8.3	odd	2		192.10.a.j.1.1		1
8.5	even	2		192.10.a.c.1.1		1
12.11	even	2		72.10.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.10.a.a.1.1	✓	1	4.3	odd	2
48.10.a.f.1.1		1	1.1	even	1	trivial
72.10.a.b.1.1		1	12.11	even	2
144.10.a.e.1.1		1	3.2	odd	2
192.10.a.c.1.1		1	8.5	even	2
192.10.a.j.1.1		1	8.3	odd	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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