LMFDB - Embedded newform 48.10.a.a.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 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)

$f(q)$

$=$

$q-81.0000 q^{3} -1530.00 q^{5} -9128.00 q^{7} +6561.00 q^{9} +O(q^{10})$

$q-81.0000 q^{3} -1530.00 q^{5} -9128.00 q^{7} +6561.00 q^{9} -21132.0 q^{11} +31214.0 q^{13} +123930. q^{15} -279342. q^{17} -144020. q^{19} +739368. q^{21} +1.76350e6 q^{23} +387775. q^{25} -531441. q^{27} +4.69251e6 q^{29} +369088. q^{31} +1.71169e6 q^{33} +1.39658e7 q^{35} +9.34708e6 q^{37} -2.52833e6 q^{39} -7.22684e6 q^{41} +2.31475e7 q^{43} -1.00383e7 q^{45} -2.29719e7 q^{47} +4.29668e7 q^{49} +2.26267e7 q^{51} +7.84772e7 q^{53} +3.23320e7 q^{55} +1.16656e7 q^{57} +2.03107e7 q^{59} -1.79340e8 q^{61} -5.98888e7 q^{63} -4.77574e7 q^{65} -2.74528e8 q^{67} -1.42843e8 q^{69} +3.63426e7 q^{71} -2.47090e8 q^{73} -3.14098e7 q^{75} +1.92893e8 q^{77} -1.91875e8 q^{79} +4.30467e7 q^{81} +2.76159e8 q^{83} +4.27393e8 q^{85} -3.80093e8 q^{87} -6.78997e8 q^{89} -2.84921e8 q^{91} -2.98961e7 q^{93} +2.20351e8 q^{95} -5.67658e8 q^{97} -1.38647e8 q^{99} +O(q^{100})$

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
$5$	−1530.00	−1.09478	−0.547389	−	0.836878i	$-0.684378\pi$
$5$	−1530.00	−1.09478	−0.547389	+	0.836878i	$0.684378\pi$
$6$	0	0
$7$	−9128.00	−1.43693	−0.718463	−	0.695565i	$-0.755154\pi$
$7$	−9128.00	−1.43693	−0.718463	+	0.695565i	$0.755154\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	−21132.0	−0.435185	−0.217592	−	0.976040i	$-0.569820\pi$
$11$	−21132.0	−0.435185	−0.217592	+	0.976040i	$0.569820\pi$
$12$	0	0
$13$	31214.0	0.303113	0.151556	−	0.988449i	$-0.451571\pi$
$13$	31214.0	0.303113	0.151556	+	0.988449i	$0.451571\pi$
$14$	0	0
$15$	123930.	0.632071
$16$	0	0
$17$	−279342.	−0.811178	−0.405589	−	0.914056i	$-0.632934\pi$
$17$	−279342.	−0.811178	−0.405589	+	0.914056i	$0.632934\pi$
$18$	0	0
$19$	−144020.	−0.253531	−0.126766	−	0.991933i	$-0.540460\pi$
$19$	−144020.	−0.253531	−0.126766	+	0.991933i	$0.540460\pi$
$20$	0	0
$21$	739368.	0.829610
$22$	0	0
$23$	1.76350e6	1.31401	0.657006	−	0.753885i	$-0.271823\pi$
$23$	1.76350e6	1.31401	0.657006	+	0.753885i	$0.271823\pi$
$24$	0	0
$25$	387775.	0.198541
$26$	0	0
$27$	−531441.	−0.192450
$28$	0	0
$29$	4.69251e6	1.23201	0.616005	−	0.787742i	$-0.288750\pi$
$29$	4.69251e6	1.23201	0.616005	+	0.787742i	$0.288750\pi$
$30$	0	0
$31$	369088.	0.0717798	0.0358899	−	0.999356i	$-0.488573\pi$
$31$	369088.	0.0717798	0.0358899	+	0.999356i	$0.488573\pi$
$32$	0	0
$33$	1.71169e6	0.251254
$34$	0	0
$35$	1.39658e7	1.57312
$36$	0	0
$37$	9.34708e6	0.819914	0.409957	−	0.912105i	$-0.365544\pi$
$37$	9.34708e6	0.819914	0.409957	+	0.912105i	$0.365544\pi$
$38$	0	0
$39$	−2.52833e6	−0.175002
$40$	0	0
$41$	−7.22684e6	−0.399412	−0.199706	−	0.979856i	$-0.563999\pi$
$41$	−7.22684e6	−0.399412	−0.199706	+	0.979856i	$0.563999\pi$
$42$	0	0
$43$	2.31475e7	1.03251	0.516257	−	0.856434i	$-0.327325\pi$
$43$	2.31475e7	1.03251	0.516257	+	0.856434i	$0.327325\pi$
$44$	0	0
$45$	−1.00383e7	−0.364926
$46$	0	0
$47$	−2.29719e7	−0.686683	−0.343342	−	0.939211i	$-0.611559\pi$
$47$	−2.29719e7	−0.686683	−0.343342	+	0.939211i	$0.611559\pi$
$48$	0	0
$49$	4.29668e7	1.06476
$50$	0	0
$51$	2.26267e7	0.468334
$52$	0	0
$53$	7.84772e7	1.36616	0.683081	−	0.730343i	$-0.260640\pi$
$53$	7.84772e7	1.36616	0.683081	+	0.730343i	$0.260640\pi$
$54$	0	0
$55$	3.23320e7	0.476431
$56$	0	0
$57$	1.16656e7	0.146376
$58$	0	0
$59$	2.03107e7	0.218218	0.109109	−	0.994030i	$-0.465200\pi$
$59$	2.03107e7	0.218218	0.109109	+	0.994030i	$0.465200\pi$
$60$	0	0
$61$	−1.79340e8	−1.65841	−0.829207	−	0.558942i	$-0.811207\pi$
$61$	−1.79340e8	−1.65841	−0.829207	+	0.558942i	$0.811207\pi$
$62$	0	0
$63$	−5.98888e7	−0.478975
$64$	0	0
$65$	−4.77574e7	−0.331842
$66$	0	0
$67$	−2.74528e8	−1.66437	−0.832186	−	0.554496i	$-0.812911\pi$
$67$	−2.74528e8	−1.66437	−0.832186	+	0.554496i	$0.812911\pi$
$68$	0	0
$69$	−1.42843e8	−0.758645
$70$	0	0
$71$	3.63426e7	0.169728	0.0848641	−	0.996393i	$-0.472954\pi$
$71$	3.63426e7	0.169728	0.0848641	+	0.996393i	$0.472954\pi$
$72$	0	0
$73$	−2.47090e8	−1.01836	−0.509180	−	0.860660i	$-0.670051\pi$
$73$	−2.47090e8	−1.01836	−0.509180	+	0.860660i	$0.670051\pi$
$74$	0	0
$75$	−3.14098e7	−0.114628
$76$	0	0
$77$	1.92893e8	0.625328
$78$	0	0
$79$	−1.91875e8	−0.554238	−0.277119	−	0.960836i	$-0.589380\pi$
$79$	−1.91875e8	−0.554238	−0.277119	+	0.960836i	$0.589380\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	2.76159e8	0.638717	0.319358	−	0.947634i	$-0.396533\pi$
$83$	2.76159e8	0.638717	0.319358	+	0.947634i	$0.396533\pi$
$84$	0	0
$85$	4.27393e8	0.888060
$86$	0	0
$87$	−3.80093e8	−0.711301
$88$	0	0
$89$	−6.78997e8	−1.14713	−0.573566	−	0.819160i	$-0.694440\pi$
$89$	−6.78997e8	−1.14713	−0.573566	+	0.819160i	$0.694440\pi$
$90$	0	0
$91$	−2.84921e8	−0.435551
$92$	0	0
$93$	−2.98961e7	−0.0414421
$94$	0	0
$95$	2.20351e8	0.277561
$96$	0	0
$97$	−5.67658e8	−0.651049	−0.325524	−	0.945534i	$-0.605541\pi$
$97$	−5.67658e8	−0.651049	−0.325524	+	0.945534i	$0.605541\pi$
$98$	0	0
$99$	−1.38647e8	−0.145062
$100$	0	0
$101$	1.62282e9	1.55176	0.775881	−	0.630879i	$-0.217306\pi$
$101$	1.62282e9	1.55176	0.775881	+	0.630879i	$0.217306\pi$
$102$	0	0
$103$	1.75103e9	1.53294	0.766470	−	0.642280i	$-0.222011\pi$
$103$	1.75103e9	1.53294	0.766470	+	0.642280i	$0.222011\pi$
$104$	0	0
$105$	−1.13123e9	−0.908239
$106$	0	0
$107$	1.54296e9	1.13796	0.568980	−	0.822352i	$-0.307338\pi$
$107$	1.54296e9	1.13796	0.568980	+	0.822352i	$0.307338\pi$
$108$	0	0
$109$	4.57665e8	0.310548	0.155274	−	0.987871i	$-0.450374\pi$
$109$	4.57665e8	0.310548	0.155274	+	0.987871i	$0.450374\pi$
$110$	0	0
$111$	−7.57113e8	−0.473377
$112$	0	0
$113$	3.26794e9	1.88548	0.942739	−	0.333531i	$-0.108240\pi$
$113$	3.26794e9	1.88548	0.942739	+	0.333531i	$0.108240\pi$
$114$	0	0
$115$	−2.69815e9	−1.43855
$116$	0	0
$117$	2.04795e8	0.101038
$118$	0	0
$119$	2.54983e9	1.16560
$120$	0	0
$121$	−1.91139e9	−0.810614
$122$	0	0
$123$	5.85374e8	0.230601
$124$	0	0
$125$	2.39499e9	0.877421
$126$	0	0
$127$	−9.28879e8	−0.316842	−0.158421	−	0.987372i	$-0.550640\pi$
$127$	−9.28879e8	−0.316842	−0.158421	+	0.987372i	$0.550640\pi$
$128$	0	0
$129$	−1.87495e9	−0.596122
$130$	0	0
$131$	9.88659e8	0.293309	0.146655	−	0.989188i	$-0.453149\pi$
$131$	9.88659e8	0.293309	0.146655	+	0.989188i	$0.453149\pi$
$132$	0	0
$133$	1.31461e9	0.364306
$134$	0	0
$135$	8.13105e8	0.210690
$136$	0	0
$137$	−5.73253e7	−0.0139028	−0.00695142	−	0.999976i	$-0.502213\pi$
$137$	−5.73253e7	−0.0139028	−0.00695142	+	0.999976i	$0.502213\pi$
$138$	0	0
$139$	4.65052e9	1.05666	0.528330	−	0.849039i	$-0.322818\pi$
$139$	4.65052e9	1.05666	0.528330	+	0.849039i	$0.322818\pi$
$140$	0	0
$141$	1.86072e9	0.396457
$142$	0	0
$143$	−6.59614e8	−0.131910
$144$	0	0
$145$	−7.17954e9	−1.34878
$146$	0	0
$147$	−3.48031e9	−0.614738
$148$	0	0
$149$	−1.40236e9	−0.233089	−0.116545	−	0.993185i	$-0.537182\pi$
$149$	−1.40236e9	−0.233089	−0.116545	+	0.993185i	$0.537182\pi$
$150$	0	0
$151$	−1.01548e10	−1.58955	−0.794773	−	0.606907i	$-0.792410\pi$
$151$	−1.01548e10	−1.58955	−0.794773	+	0.606907i	$0.792410\pi$
$152$	0	0
$153$	−1.83276e9	−0.270393
$154$	0	0
$155$	−5.64705e8	−0.0785830
$156$	0	0
$157$	9.36605e9	1.23029	0.615146	−	0.788413i	$-0.289097\pi$
$157$	9.36605e9	1.23029	0.615146	+	0.788413i	$0.289097\pi$
$158$	0	0
$159$	−6.35665e9	−0.788754
$160$	0	0
$161$	−1.60972e10	−1.88814
$162$	0	0
$163$	7.34780e8	0.0815292	0.0407646	−	0.999169i	$-0.487021\pi$
$163$	7.34780e8	0.0815292	0.0407646	+	0.999169i	$0.487021\pi$
$164$	0	0
$165$	−2.61889e9	−0.275068
$166$	0	0
$167$	−1.55584e9	−0.154789	−0.0773947	−	0.997001i	$-0.524660\pi$
$167$	−1.55584e9	−0.154789	−0.0773947	+	0.997001i	$0.524660\pi$
$168$	0	0
$169$	−9.63019e9	−0.908123
$170$	0	0
$171$	−9.44915e8	−0.0845104
$172$	0	0
$173$	1.90448e10	1.61648	0.808238	−	0.588856i	$-0.200421\pi$
$173$	1.90448e10	1.61648	0.808238	+	0.588856i	$0.200421\pi$
$174$	0	0
$175$	−3.53961e9	−0.285288
$176$	0	0
$177$	−1.64516e9	−0.125988
$178$	0	0
$179$	−4.32852e9	−0.315138	−0.157569	−	0.987508i	$-0.550366\pi$
$179$	−4.32852e9	−0.315138	−0.157569	+	0.987508i	$0.550366\pi$
$180$	0	0
$181$	−9.56757e8	−0.0662595	−0.0331298	−	0.999451i	$-0.510547\pi$
$181$	−9.56757e8	−0.0662595	−0.0331298	+	0.999451i	$0.510547\pi$
$182$	0	0
$183$	1.45265e10	0.957485
$184$	0	0
$185$	−1.43010e10	−0.897624
$186$	0	0
$187$	5.90306e9	0.353012
$188$	0	0
$189$	4.85099e9	0.276537
$190$	0	0
$191$	1.76438e10	0.959274	0.479637	−	0.877467i	$-0.340768\pi$
$191$	1.76438e10	0.959274	0.479637	+	0.877467i	$0.340768\pi$
$192$	0	0
$193$	1.30993e10	0.679581	0.339790	−	0.940501i	$-0.389644\pi$
$193$	1.30993e10	0.679581	0.339790	+	0.940501i	$0.389644\pi$
$194$	0	0
$195$	3.86835e9	0.191589
$196$	0	0
$197$	4.74589e9	0.224502	0.112251	−	0.993680i	$-0.464194\pi$
$197$	4.74589e9	0.224502	0.112251	+	0.993680i	$0.464194\pi$
$198$	0	0
$199$	1.92102e10	0.868348	0.434174	−	0.900829i	$-0.357040\pi$
$199$	1.92102e10	0.868348	0.434174	+	0.900829i	$0.357040\pi$
$200$	0	0
$201$	2.22368e10	0.960926
$202$	0	0
$203$	−4.28332e10	−1.77031
$204$	0	0
$205$	1.10571e10	0.437268
$206$	0	0
$207$	1.15703e10	0.438004
$208$	0	0
$209$	3.04343e9	0.110333
$210$	0	0
$211$	1.81273e10	0.629597	0.314798	−	0.949159i	$-0.398063\pi$
$211$	1.81273e10	0.629597	0.314798	+	0.949159i	$0.398063\pi$
$212$	0	0
$213$	−2.94375e9	−0.0979926
$214$	0	0
$215$	−3.54156e10	−1.13037
$216$	0	0
$217$	−3.36904e9	−0.103142
$218$	0	0
$219$	2.00143e10	0.587951
$220$	0	0
$221$	−8.71938e9	−0.245878
$222$	0	0
$223$	−3.91081e9	−0.105900	−0.0529499	−	0.998597i	$-0.516862\pi$
$223$	−3.91081e9	−0.105900	−0.0529499	+	0.998597i	$0.516862\pi$
$224$	0	0
$225$	2.54419e9	0.0661803
$226$	0	0
$227$	3.58126e10	0.895199	0.447599	−	0.894234i	$-0.352279\pi$
$227$	3.58126e10	0.895199	0.447599	+	0.894234i	$0.352279\pi$
$228$	0	0
$229$	5.30445e9	0.127462	0.0637310	−	0.997967i	$-0.479700\pi$
$229$	5.30445e9	0.127462	0.0637310	+	0.997967i	$0.479700\pi$
$230$	0	0
$231$	−1.56243e10	−0.361033
$232$	0	0
$233$	−2.54172e10	−0.564970	−0.282485	−	0.959272i	$-0.591159\pi$
$233$	−2.54172e10	−0.564970	−0.282485	+	0.959272i	$0.591159\pi$
$234$	0	0
$235$	3.51470e10	0.751766
$236$	0	0
$237$	1.55419e10	0.319989
$238$	0	0
$239$	6.33494e10	1.25589	0.627946	−	0.778257i	$-0.283896\pi$
$239$	6.33494e10	1.25589	0.627946	+	0.778257i	$0.283896\pi$
$240$	0	0
$241$	1.01929e11	1.94635	0.973173	−	0.230073i	$-0.0738965\pi$
$241$	1.01929e11	1.94635	0.973173	+	0.230073i	$0.0738965\pi$
$242$	0	0
$243$	−3.48678e9	−0.0641500
$244$	0	0
$245$	−6.57392e10	−1.16567
$246$	0	0
$247$	−4.49544e9	−0.0768486
$248$	0	0
$249$	−2.23689e10	−0.368763
$250$	0	0
$251$	−9.64745e10	−1.53420	−0.767098	−	0.641530i	$-0.778300\pi$
$251$	−9.64745e10	−1.53420	−0.767098	+	0.641530i	$0.778300\pi$
$252$	0	0
$253$	−3.72662e10	−0.571838
$254$	0	0
$255$	−3.46189e10	−0.512722
$256$	0	0
$257$	−7.99591e9	−0.114332	−0.0571661	−	0.998365i	$-0.518206\pi$
$257$	−7.99591e9	−0.114332	−0.0571661	+	0.998365i	$0.518206\pi$
$258$	0	0
$259$	−8.53201e10	−1.17816
$260$	0	0
$261$	3.07876e10	0.410670
$262$	0	0
$263$	−1.06269e11	−1.36964	−0.684821	−	0.728712i	$-0.740119\pi$
$263$	−1.06269e11	−1.36964	−0.684821	+	0.728712i	$0.740119\pi$
$264$	0	0
$265$	−1.20070e11	−1.49564
$266$	0	0
$267$	5.49988e10	0.662296
$268$	0	0
$269$	−1.00886e11	−1.17475	−0.587377	−	0.809313i	$-0.699840\pi$
$269$	−1.00886e11	−1.17475	−0.587377	+	0.809313i	$0.699840\pi$
$270$	0	0
$271$	2.86345e10	0.322498	0.161249	−	0.986914i	$-0.448448\pi$
$271$	2.86345e10	0.322498	0.161249	+	0.986914i	$0.448448\pi$
$272$	0	0
$273$	2.30786e10	0.251465
$274$	0	0
$275$	−8.19446e9	−0.0864019
$276$	0	0
$277$	4.12251e10	0.420729	0.210365	−	0.977623i	$-0.432535\pi$
$277$	4.12251e10	0.420729	0.210365	+	0.977623i	$0.432535\pi$
$278$	0	0
$279$	2.42159e9	0.0239266
$280$	0	0
$281$	1.21800e11	1.16538	0.582692	−	0.812693i	$-0.301999\pi$
$281$	1.21800e11	1.16538	0.582692	+	0.812693i	$0.301999\pi$
$282$	0	0
$283$	−2.32820e10	−0.215765	−0.107883	−	0.994164i	$-0.534407\pi$
$283$	−2.32820e10	−0.215765	−0.107883	+	0.994164i	$0.534407\pi$
$284$	0	0
$285$	−1.78484e10	−0.160250
$286$	0	0
$287$	6.59666e10	0.573925
$288$	0	0
$289$	−4.05559e10	−0.341990
$290$	0	0
$291$	4.59803e10	0.375883
$292$	0	0
$293$	−9.12801e10	−0.723555	−0.361778	−	0.932264i	$-0.617830\pi$
$293$	−9.12801e10	−0.723555	−0.361778	+	0.932264i	$0.617830\pi$
$294$	0	0
$295$	−3.10753e10	−0.238900
$296$	0	0
$297$	1.12304e10	0.0837513
$298$	0	0
$299$	5.50458e10	0.398294
$300$	0	0
$301$	−2.11290e11	−1.48365
$302$	0	0
$303$	−1.31449e11	−0.895910
$304$	0	0
$305$	2.74390e11	1.81560
$306$	0	0
$307$	4.39070e10	0.282105	0.141053	−	0.990002i	$-0.454951\pi$
$307$	4.39070e10	0.282105	0.141053	+	0.990002i	$0.454951\pi$
$308$	0	0
$309$	−1.41833e11	−0.885043
$310$	0	0
$311$	7.64835e10	0.463603	0.231801	−	0.972763i	$-0.425538\pi$
$311$	7.64835e10	0.463603	0.231801	+	0.972763i	$0.425538\pi$
$312$	0	0
$313$	2.68488e11	1.58116	0.790580	−	0.612359i	$-0.209779\pi$
$313$	2.68488e11	1.58116	0.790580	+	0.612359i	$0.209779\pi$
$314$	0	0
$315$	9.16299e10	0.524372
$316$	0	0
$317$	1.52056e11	0.845738	0.422869	−	0.906191i	$-0.361023\pi$
$317$	1.52056e11	0.845738	0.422869	+	0.906191i	$0.361023\pi$
$318$	0	0
$319$	−9.91621e10	−0.536152
$320$	0	0
$321$	−1.24980e11	−0.657001
$322$	0	0
$323$	4.02308e10	0.205659
$324$	0	0
$325$	1.21040e10	0.0601803
$326$	0	0
$327$	−3.70709e10	−0.179295
$328$	0	0
$329$	2.09687e11	0.986713
$330$	0	0
$331$	2.63057e11	1.20455	0.602274	−	0.798289i	$-0.294261\pi$
$331$	2.63057e11	1.20455	0.602274	+	0.798289i	$0.294261\pi$
$332$	0	0
$333$	6.13262e10	0.273305
$334$	0	0
$335$	4.20028e11	1.82212
$336$	0	0
$337$	2.97183e11	1.25513	0.627566	−	0.778564i	$-0.284051\pi$
$337$	2.97183e11	1.25513	0.627566	+	0.778564i	$0.284051\pi$
$338$	0	0
$339$	−2.64703e11	−1.08858
$340$	0	0
$341$	−7.79957e9	−0.0312375
$342$	0	0
$343$	−2.38530e10	−0.0930507
$344$	0	0
$345$	2.18550e11	0.830549
$346$	0	0
$347$	−1.40219e11	−0.519186	−0.259593	−	0.965718i	$-0.583588\pi$
$347$	−1.40219e11	−0.519186	−0.259593	+	0.965718i	$0.583588\pi$
$348$	0	0
$349$	−3.14891e11	−1.13618	−0.568088	−	0.822968i	$-0.692317\pi$
$349$	−3.14891e11	−1.13618	−0.568088	+	0.822968i	$0.692317\pi$
$350$	0	0
$351$	−1.65884e10	−0.0583341
$352$	0	0
$353$	−5.31195e11	−1.82082	−0.910411	−	0.413704i	$-0.864235\pi$
$353$	−5.31195e11	−1.82082	−0.910411	+	0.413704i	$0.864235\pi$
$354$	0	0
$355$	−5.56043e10	−0.185815
$356$	0	0
$357$	−2.06537e11	−0.672961
$358$	0	0
$359$	4.97684e11	1.58135	0.790676	−	0.612235i	$-0.209729\pi$
$359$	4.97684e11	1.58135	0.790676	+	0.612235i	$0.209729\pi$
$360$	0	0
$361$	−3.01946e11	−0.935722
$362$	0	0
$363$	1.54822e11	0.468008
$364$	0	0
$365$	3.78047e11	1.11488
$366$	0	0
$367$	−2.77743e11	−0.799183	−0.399591	−	0.916693i	$-0.630848\pi$
$367$	−2.77743e11	−0.799183	−0.399591	+	0.916693i	$0.630848\pi$
$368$	0	0
$369$	−4.74153e10	−0.133137
$370$	0	0
$371$	−7.16340e11	−1.96307
$372$	0	0
$373$	2.55319e11	0.682957	0.341478	−	0.939890i	$-0.389072\pi$
$373$	2.55319e11	0.682957	0.341478	+	0.939890i	$0.389072\pi$
$374$	0	0
$375$	−1.93994e11	−0.506579
$376$	0	0
$377$	1.46472e11	0.373438
$378$	0	0
$379$	−8.12103e10	−0.202178	−0.101089	−	0.994877i	$-0.532233\pi$
$379$	−8.12103e10	−0.202178	−0.101089	+	0.994877i	$0.532233\pi$
$380$	0	0
$381$	7.52392e10	0.182929
$382$	0	0
$383$	4.13123e11	0.981036	0.490518	−	0.871431i	$-0.336808\pi$
$383$	4.13123e11	0.981036	0.490518	+	0.871431i	$0.336808\pi$
$384$	0	0
$385$	−2.95126e11	−0.684596
$386$	0	0
$387$	1.51871e11	0.344171
$388$	0	0
$389$	2.53052e11	0.560321	0.280160	−	0.959953i	$-0.409612\pi$
$389$	2.53052e11	0.560321	0.280160	+	0.959953i	$0.409612\pi$
$390$	0	0
$391$	−4.92618e11	−1.06590
$392$	0	0
$393$	−8.00814e10	−0.169342
$394$	0	0
$395$	2.93568e11	0.606768
$396$	0	0
$397$	−6.53562e10	−0.132047	−0.0660236	−	0.997818i	$-0.521031\pi$
$397$	−6.53562e10	−0.132047	−0.0660236	+	0.997818i	$0.521031\pi$
$398$	0	0
$399$	−1.06484e11	−0.210332
$400$	0	0
$401$	−2.40886e11	−0.465225	−0.232612	−	0.972570i	$-0.574727\pi$
$401$	−2.40886e11	−0.465225	−0.232612	+	0.972570i	$0.574727\pi$
$402$	0	0
$403$	1.15207e10	0.0217574
$404$	0	0
$405$	−6.58615e10	−0.121642
$406$	0	0
$407$	−1.97522e11	−0.356814
$408$	0	0
$409$	−3.78340e11	−0.668540	−0.334270	−	0.942477i	$-0.608490\pi$
$409$	−3.78340e11	−0.668540	−0.334270	+	0.942477i	$0.608490\pi$
$410$	0	0
$411$	4.64335e9	0.00802681
$412$	0	0
$413$	−1.85396e11	−0.313563
$414$	0	0
$415$	−4.22524e11	−0.699253
$416$	0	0
$417$	−3.76693e11	−0.610063
$418$	0	0
$419$	−9.68766e11	−1.53552	−0.767760	−	0.640737i	$-0.778629\pi$
$419$	−9.68766e11	−1.53552	−0.767760	+	0.640737i	$0.778629\pi$
$420$	0	0
$421$	−1.43789e11	−0.223077	−0.111539	−	0.993760i	$-0.535578\pi$
$421$	−1.43789e11	−0.223077	−0.111539	+	0.993760i	$0.535578\pi$
$422$	0	0
$423$	−1.50719e11	−0.228894
$424$	0	0
$425$	−1.08322e11	−0.161052
$426$	0	0
$427$	1.63701e12	2.38302
$428$	0	0
$429$	5.34288e10	0.0761583
$430$	0	0
$431$	−2.06890e11	−0.288796	−0.144398	−	0.989520i	$-0.546125\pi$
$431$	−2.06890e11	−0.288796	−0.144398	+	0.989520i	$0.546125\pi$
$432$	0	0
$433$	−4.81883e10	−0.0658789	−0.0329394	−	0.999457i	$-0.510487\pi$
$433$	−4.81883e10	−0.0658789	−0.0329394	+	0.999457i	$0.510487\pi$
$434$	0	0
$435$	5.81543e11	0.778718
$436$	0	0
$437$	−2.53979e11	−0.333143
$438$	0	0
$439$	−4.37941e11	−0.562763	−0.281381	−	0.959596i	$-0.590793\pi$
$439$	−4.37941e11	−0.562763	−0.281381	+	0.959596i	$0.590793\pi$
$440$	0	0
$441$	2.81905e11	0.354919
$442$	0	0
$443$	−7.58665e11	−0.935908	−0.467954	−	0.883753i	$-0.655009\pi$
$443$	−7.58665e11	−0.935908	−0.467954	+	0.883753i	$0.655009\pi$
$444$	0	0
$445$	1.03887e12	1.25585
$446$	0	0
$447$	1.13591e11	0.134574
$448$	0	0
$449$	−1.37280e11	−0.159403	−0.0797017	−	0.996819i	$-0.525397\pi$
$449$	−1.37280e11	−0.159403	−0.0797017	+	0.996819i	$0.525397\pi$
$450$	0	0
$451$	1.52718e11	0.173818
$452$	0	0
$453$	8.22535e11	0.917725
$454$	0	0
$455$	4.35930e11	0.476832
$456$	0	0
$457$	1.31091e11	0.140589	0.0702944	−	0.997526i	$-0.477606\pi$
$457$	1.31091e11	0.140589	0.0702944	+	0.997526i	$0.477606\pi$
$458$	0	0
$459$	1.48454e11	0.156111
$460$	0	0
$461$	−8.03508e11	−0.828583	−0.414291	−	0.910144i	$-0.635971\pi$
$461$	−8.03508e11	−0.828583	−0.414291	+	0.910144i	$0.635971\pi$
$462$	0	0
$463$	1.78582e12	1.80602	0.903009	−	0.429621i	$-0.141353\pi$
$463$	1.78582e12	1.80602	0.903009	+	0.429621i	$0.141353\pi$
$464$	0	0
$465$	4.57411e10	0.0453699
$466$	0	0
$467$	−6.44366e11	−0.626912	−0.313456	−	0.949603i	$-0.601487\pi$
$467$	−6.44366e11	−0.626912	−0.313456	+	0.949603i	$0.601487\pi$
$468$	0	0
$469$	2.50590e12	2.39158
$470$	0	0
$471$	−7.58650e11	−0.710309
$472$	0	0
$473$	−4.89152e11	−0.449334
$474$	0	0
$475$	−5.58474e10	−0.0503363
$476$	0	0
$477$	5.14889e11	0.455387
$478$	0	0
$479$	1.87701e12	1.62913	0.814565	−	0.580073i	$-0.196976\pi$
$479$	1.87701e12	1.62913	0.814565	+	0.580073i	$0.196976\pi$
$480$	0	0
$481$	2.91760e11	0.248526
$482$	0	0
$483$	1.30387e12	1.09012
$484$	0	0
$485$	8.68516e11	0.712755
$486$	0	0
$487$	−1.49355e12	−1.20321	−0.601603	−	0.798795i	$-0.705471\pi$
$487$	−1.49355e12	−1.20321	−0.601603	+	0.798795i	$0.705471\pi$
$488$	0	0
$489$	−5.95172e10	−0.0470709
$490$	0	0
$491$	7.47313e11	0.580278	0.290139	−	0.956985i	$-0.406298\pi$
$491$	7.47313e11	0.580278	0.290139	+	0.956985i	$0.406298\pi$
$492$	0	0
$493$	−1.31082e12	−0.999379
$494$	0	0
$495$	2.12130e11	0.158810
$496$	0	0
$497$	−3.31736e11	−0.243887
$498$	0	0
$499$	−1.01300e12	−0.731402	−0.365701	−	0.930732i	$-0.619171\pi$
$499$	−1.01300e12	−0.731402	−0.365701	+	0.930732i	$0.619171\pi$
$500$	0	0
$501$	1.26023e11	0.0893677
$502$	0	0
$503$	−2.03399e11	−0.141675	−0.0708373	−	0.997488i	$-0.522567\pi$
$503$	−2.03399e11	−0.141675	−0.0708373	+	0.997488i	$0.522567\pi$
$504$	0	0
$505$	−2.48292e12	−1.69884
$506$	0	0
$507$	7.80045e11	0.524305
$508$	0	0
$509$	5.93699e11	0.392045	0.196023	−	0.980599i	$-0.437197\pi$
$509$	5.93699e11	0.392045	0.196023	+	0.980599i	$0.437197\pi$
$510$	0	0
$511$	2.25543e12	1.46331
$512$	0	0
$513$	7.65381e10	0.0487921
$514$	0	0
$515$	−2.67907e12	−1.67823
$516$	0	0
$517$	4.85442e11	0.298834
$518$	0	0
$519$	−1.54263e12	−0.933273
$520$	0	0
$521$	−1.74950e12	−1.04027	−0.520133	−	0.854085i	$-0.674118\pi$
$521$	−1.74950e12	−1.04027	−0.520133	+	0.854085i	$0.674118\pi$
$522$	0	0
$523$	−3.20555e11	−0.187346	−0.0936731	−	0.995603i	$-0.529861\pi$
$523$	−3.20555e11	−0.187346	−0.0936731	+	0.995603i	$0.529861\pi$
$524$	0	0
$525$	2.86708e11	0.164711
$526$	0	0
$527$	−1.03102e11	−0.0582262
$528$	0	0
$529$	1.30877e12	0.726627
$530$	0	0
$531$	1.33258e11	0.0727392
$532$	0	0
$533$	−2.25579e11	−0.121067
$534$	0	0
$535$	−2.36072e12	−1.24581
$536$	0	0
$537$	3.50610e11	0.181945
$538$	0	0
$539$	−9.07974e11	−0.463366
$540$	0	0
$541$	3.50229e11	0.175778	0.0878889	−	0.996130i	$-0.471988\pi$
$541$	3.50229e11	0.175778	0.0878889	+	0.996130i	$0.471988\pi$
$542$	0	0
$543$	7.74974e10	0.0382549
$544$	0	0
$545$	−7.00228e11	−0.339981
$546$	0	0
$547$	2.46037e12	1.17505	0.587527	−	0.809205i	$-0.300102\pi$
$547$	2.46037e12	1.17505	0.587527	+	0.809205i	$0.300102\pi$
$548$	0	0
$549$	−1.17665e12	−0.552804
$550$	0	0
$551$	−6.75815e11	−0.312353
$552$	0	0
$553$	1.75143e12	0.796399
$554$	0	0
$555$	1.15838e12	0.518244
$556$	0	0
$557$	1.04356e12	0.459376	0.229688	−	0.973264i	$-0.426229\pi$
$557$	1.04356e12	0.459376	0.229688	+	0.973264i	$0.426229\pi$
$558$	0	0
$559$	7.22525e11	0.312968
$560$	0	0
$561$	−4.78147e11	−0.203812
$562$	0	0
$563$	−3.81849e12	−1.60179	−0.800893	−	0.598808i	$-0.795641\pi$
$563$	−3.81849e12	−1.60179	−0.800893	+	0.598808i	$0.795641\pi$
$564$	0	0
$565$	−4.99995e12	−2.06418
$566$	0	0
$567$	−3.92930e11	−0.159658
$568$	0	0
$569$	4.84062e11	0.193596	0.0967979	−	0.995304i	$-0.469140\pi$
$569$	4.84062e11	0.193596	0.0967979	+	0.995304i	$0.469140\pi$
$570$	0	0
$571$	−1.77044e10	−0.00696978	−0.00348489	−	0.999994i	$-0.501109\pi$
$571$	−1.77044e10	−0.00696978	−0.00348489	+	0.999994i	$0.501109\pi$
$572$	0	0
$573$	−1.42915e12	−0.553837
$574$	0	0
$575$	6.83840e11	0.260885
$576$	0	0
$577$	−7.46985e11	−0.280557	−0.140278	−	0.990112i	$-0.544800\pi$
$577$	−7.46985e11	−0.280557	−0.140278	+	0.990112i	$0.544800\pi$
$578$	0	0
$579$	−1.06105e12	−0.392356
$580$	0	0
$581$	−2.52078e12	−0.917789
$582$	0	0
$583$	−1.65838e12	−0.594532
$584$	0	0
$585$	−3.13336e11	−0.110614
$586$	0	0
$587$	4.33948e12	1.50857	0.754286	−	0.656546i	$-0.227983\pi$
$587$	4.33948e12	1.50857	0.754286	+	0.656546i	$0.227983\pi$
$588$	0	0
$589$	−5.31561e10	−0.0181984
$590$	0	0
$591$	−3.84417e11	−0.129616
$592$	0	0
$593$	1.90120e12	0.631366	0.315683	−	0.948865i	$-0.397766\pi$
$593$	1.90120e12	0.631366	0.315683	+	0.948865i	$0.397766\pi$
$594$	0	0
$595$	−3.90125e12	−1.27608
$596$	0	0
$597$	−1.55603e12	−0.501341
$598$	0	0
$599$	5.49649e12	1.74447	0.872237	−	0.489083i	$-0.162668\pi$
$599$	5.49649e12	1.74447	0.872237	+	0.489083i	$0.162668\pi$
$600$	0	0
$601$	4.77698e12	1.49355	0.746773	−	0.665079i	$-0.231602\pi$
$601$	4.77698e12	1.49355	0.746773	+	0.665079i	$0.231602\pi$
$602$	0	0
$603$	−1.80118e12	−0.554791
$604$	0	0
$605$	2.92442e12	0.887443
$606$	0	0
$607$	−8.25000e11	−0.246663	−0.123332	−	0.992366i	$-0.539358\pi$
$607$	−8.25000e11	−0.246663	−0.123332	+	0.992366i	$0.539358\pi$
$608$	0	0
$609$	3.46949e12	1.02209
$610$	0	0
$611$	−7.17045e11	−0.208142
$612$	0	0
$613$	−1.46985e12	−0.420438	−0.210219	−	0.977654i	$-0.567418\pi$
$613$	−1.46985e12	−0.420438	−0.210219	+	0.977654i	$0.567418\pi$
$614$	0	0
$615$	−8.95622e11	−0.252457
$616$	0	0
$617$	5.17373e12	1.43721	0.718606	−	0.695418i	$-0.244781\pi$
$617$	5.17373e12	1.43721	0.718606	+	0.695418i	$0.244781\pi$
$618$	0	0
$619$	4.80274e12	1.31487	0.657433	−	0.753513i	$-0.271642\pi$
$619$	4.80274e12	1.31487	0.657433	+	0.753513i	$0.271642\pi$
$620$	0	0
$621$	−9.37194e11	−0.252882
$622$	0	0
$623$	6.19789e12	1.64834
$624$	0	0
$625$	−4.42170e12	−1.15912
$626$	0	0
$627$	−2.46518e11	−0.0637007
$628$	0	0
$629$	−2.61103e12	−0.665096
$630$	0	0
$631$	−5.93992e12	−1.49159	−0.745794	−	0.666177i	$-0.767929\pi$
$631$	−5.93992e12	−1.49159	−0.745794	+	0.666177i	$0.767929\pi$
$632$	0	0
$633$	−1.46831e12	−0.363498
$634$	0	0
$635$	1.42119e12	0.346872
$636$	0	0
$637$	1.34116e12	0.322741
$638$	0	0
$639$	2.38444e11	0.0565761
$640$	0	0
$641$	6.47738e12	1.51544	0.757720	−	0.652580i	$-0.226313\pi$
$641$	6.47738e12	1.51544	0.757720	+	0.652580i	$0.226313\pi$
$642$	0	0
$643$	1.74519e12	0.402619	0.201309	−	0.979528i	$-0.435480\pi$
$643$	1.74519e12	0.402619	0.201309	+	0.979528i	$0.435480\pi$
$644$	0	0
$645$	2.86867e12	0.652621
$646$	0	0
$647$	−4.97244e12	−1.11558	−0.557790	−	0.829982i	$-0.688350\pi$
$647$	−4.97244e12	−1.11558	−0.557790	+	0.829982i	$0.688350\pi$
$648$	0	0
$649$	−4.29205e11	−0.0949650
$650$	0	0
$651$	2.72892e11	0.0595492
$652$	0	0
$653$	−1.45057e12	−0.312198	−0.156099	−	0.987741i	$-0.549892\pi$
$653$	−1.45057e12	−0.312198	−0.156099	+	0.987741i	$0.549892\pi$
$654$	0	0
$655$	−1.51265e12	−0.321109
$656$	0	0
$657$	−1.62115e12	−0.339453
$658$	0	0
$659$	3.05175e12	0.630325	0.315162	−	0.949038i	$-0.397941\pi$
$659$	3.05175e12	0.630325	0.315162	+	0.949038i	$0.397941\pi$
$660$	0	0
$661$	−1.88315e12	−0.383688	−0.191844	−	0.981425i	$-0.561447\pi$
$661$	−1.88315e12	−0.383688	−0.191844	+	0.981425i	$0.561447\pi$
$662$	0	0
$663$	7.06270e11	0.141958
$664$	0	0
$665$	−2.01136e12	−0.398834
$666$	0	0
$667$	8.27522e12	1.61888
$668$	0	0
$669$	3.16776e11	0.0611412
$670$	0	0
$671$	3.78981e12	0.721716
$672$	0	0
$673$	7.84359e12	1.47383	0.736914	−	0.675986i	$-0.236282\pi$
$673$	7.84359e12	1.47383	0.736914	+	0.675986i	$0.236282\pi$
$674$	0	0
$675$	−2.06080e11	−0.0382092
$676$	0	0
$677$	2.32420e12	0.425230	0.212615	−	0.977136i	$-0.431802\pi$
$677$	2.32420e12	0.425230	0.212615	+	0.977136i	$0.431802\pi$
$678$	0	0
$679$	5.18158e12	0.935509
$680$	0	0
$681$	−2.90082e12	−0.516843
$682$	0	0
$683$	6.89265e12	1.21197	0.605986	−	0.795475i	$-0.292779\pi$
$683$	6.89265e12	1.21197	0.605986	+	0.795475i	$0.292779\pi$
$684$	0	0
$685$	8.77077e10	0.0152205
$686$	0	0
$687$	−4.29660e11	−0.0735902
$688$	0	0
$689$	2.44959e12	0.414101
$690$	0	0
$691$	−1.11430e13	−1.85931	−0.929654	−	0.368434i	$-0.879894\pi$
$691$	−1.11430e13	−1.85931	−0.929654	+	0.368434i	$0.879894\pi$
$692$	0	0
$693$	1.26557e12	0.208443
$694$	0	0
$695$	−7.11530e12	−1.15681
$696$	0	0
$697$	2.01876e12	0.323994
$698$	0	0
$699$	2.05879e12	0.326186
$700$	0	0
$701$	−5.78015e12	−0.904082	−0.452041	−	0.891997i	$-0.649304\pi$
$701$	−5.78015e12	−0.904082	−0.452041	+	0.891997i	$0.649304\pi$
$702$	0	0
$703$	−1.34617e12	−0.207874
$704$	0	0
$705$	−2.84691e12	−0.434032
$706$	0	0
$707$	−1.48131e13	−2.22977
$708$	0	0
$709$	−3.39902e12	−0.505180	−0.252590	−	0.967573i	$-0.581282\pi$
$709$	−3.39902e12	−0.505180	−0.252590	+	0.967573i	$0.581282\pi$
$710$	0	0
$711$	−1.25889e12	−0.184746
$712$	0	0
$713$	6.50885e11	0.0943195
$714$	0	0
$715$	1.00921e12	0.144412
$716$	0	0
$717$	−5.13130e12	−0.725089
$718$	0	0
$719$	−2.71272e11	−0.0378551	−0.0189276	−	0.999821i	$-0.506025\pi$
$719$	−2.71272e11	−0.0378551	−0.0189276	+	0.999821i	$0.506025\pi$
$720$	0	0
$721$	−1.59834e13	−2.20272
$722$	0	0
$723$	−8.25623e12	−1.12372
$724$	0	0
$725$	1.81964e12	0.244604
$726$	0	0
$727$	2.13863e12	0.283943	0.141972	−	0.989871i	$-0.454656\pi$
$727$	2.13863e12	0.283943	0.141972	+	0.989871i	$0.454656\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−6.46606e12	−0.837552
$732$	0	0
$733$	4.60033e12	0.588601	0.294301	−	0.955713i	$-0.404913\pi$
$733$	4.60033e12	0.588601	0.294301	+	0.955713i	$0.404913\pi$
$734$	0	0
$735$	5.32487e12	0.673002
$736$	0	0
$737$	5.80133e12	0.724309
$738$	0	0
$739$	−8.60245e12	−1.06102	−0.530508	−	0.847680i	$-0.677999\pi$
$739$	−8.60245e12	−1.06102	−0.530508	+	0.847680i	$0.677999\pi$
$740$	0	0
$741$	3.64131e11	0.0443686
$742$	0	0
$743$	−1.31407e12	−0.158187	−0.0790934	−	0.996867i	$-0.525203\pi$
$743$	−1.31407e12	−0.158187	−0.0790934	+	0.996867i	$0.525203\pi$
$744$	0	0
$745$	2.14561e12	0.255181
$746$	0	0
$747$	1.81188e12	0.212906
$748$	0	0
$749$	−1.40841e13	−1.63516
$750$	0	0
$751$	7.95810e12	0.912913	0.456457	−	0.889746i	$-0.349118\pi$
$751$	7.95810e12	0.912913	0.456457	+	0.889746i	$0.349118\pi$
$752$	0	0
$753$	7.81443e12	0.885768
$754$	0	0
$755$	1.55368e13	1.74020
$756$	0	0
$757$	−6.95936e12	−0.770261	−0.385131	−	0.922862i	$-0.625844\pi$
$757$	−6.95936e12	−0.770261	−0.385131	+	0.922862i	$0.625844\pi$
$758$	0	0
$759$	3.01856e12	0.330151
$760$	0	0
$761$	−1.38632e13	−1.49841	−0.749207	−	0.662336i	$-0.769565\pi$
$761$	−1.38632e13	−1.49841	−0.749207	+	0.662336i	$0.769565\pi$
$762$	0	0
$763$	−4.17757e12	−0.446234
$764$	0	0
$765$	2.80413e12	0.296020
$766$	0	0
$767$	6.33977e11	0.0661446
$768$	0	0
$769$	1.35604e13	1.39832	0.699158	−	0.714967i	$-0.253558\pi$
$769$	1.35604e13	1.39832	0.699158	+	0.714967i	$0.253558\pi$
$770$	0	0
$771$	6.47669e11	0.0660098
$772$	0	0
$773$	4.96740e12	0.500405	0.250203	−	0.968194i	$-0.419503\pi$
$773$	4.96740e12	0.500405	0.250203	+	0.968194i	$0.419503\pi$
$774$	0	0
$775$	1.43123e11	0.0142512
$776$	0	0
$777$	6.91093e12	0.680208
$778$	0	0
$779$	1.04081e12	0.101263
$780$	0	0
$781$	−7.67993e11	−0.0738631
$782$	0	0
$783$	−2.49379e12	−0.237100
$784$	0	0
$785$	−1.43301e13	−1.34690
$786$	0	0
$787$	5.25809e12	0.488587	0.244293	−	0.969701i	$-0.421444\pi$
$787$	5.25809e12	0.488587	0.244293	+	0.969701i	$0.421444\pi$
$788$	0	0
$789$	8.60781e12	0.790763
$790$	0	0
$791$	−2.98298e13	−2.70929
$792$	0	0
$793$	−5.59792e12	−0.502686
$794$	0	0
$795$	9.72568e12	0.863511
$796$	0	0
$797$	1.61816e13	1.42056	0.710281	−	0.703918i	$-0.248568\pi$
$797$	1.61816e13	1.42056	0.710281	+	0.703918i	$0.248568\pi$
$798$	0	0
$799$	6.41701e12	0.557022
$800$	0	0
$801$	−4.45490e12	−0.382377
$802$	0	0
$803$	5.22150e12	0.443175
$804$	0	0
$805$	2.46287e13	2.06709
$806$	0	0
$807$	8.17180e12	0.678245
$808$	0	0
$809$	−1.39276e13	−1.14317	−0.571583	−	0.820544i	$-0.693671\pi$
$809$	−1.39276e13	−1.14317	−0.571583	+	0.820544i	$0.693671\pi$
$810$	0	0
$811$	−1.38190e13	−1.12171	−0.560857	−	0.827913i	$-0.689528\pi$
$811$	−1.38190e13	−1.12171	−0.560857	+	0.827913i	$0.689528\pi$
$812$	0	0
$813$	−2.31939e12	−0.186194
$814$	0	0
$815$	−1.12421e12	−0.0892564
$816$	0	0
$817$	−3.33370e12	−0.261774
$818$	0	0
$819$	−1.86937e12	−0.145184
$820$	0	0
$821$	3.48578e12	0.267766	0.133883	−	0.990997i	$-0.457255\pi$
$821$	3.48578e12	0.267766	0.133883	+	0.990997i	$0.457255\pi$
$822$	0	0
$823$	1.21472e13	0.922948	0.461474	−	0.887154i	$-0.347321\pi$
$823$	1.21472e13	0.922948	0.461474	+	0.887154i	$0.347321\pi$
$824$	0	0
$825$	6.63751e11	0.0498842
$826$	0	0
$827$	1.70702e13	1.26901	0.634504	−	0.772919i	$-0.281204\pi$
$827$	1.70702e13	1.26901	0.634504	+	0.772919i	$0.281204\pi$
$828$	0	0
$829$	5.58071e12	0.410387	0.205194	−	0.978721i	$-0.434218\pi$
$829$	5.58071e12	0.410387	0.205194	+	0.978721i	$0.434218\pi$
$830$	0	0
$831$	−3.33923e12	−0.242908
$832$	0	0
$833$	−1.20024e13	−0.863707
$834$	0	0
$835$	2.38044e12	0.169460
$836$	0	0
$837$	−1.96148e11	−0.0138140
$838$	0	0
$839$	−1.28642e13	−0.896304	−0.448152	−	0.893957i	$-0.647918\pi$
$839$	−1.28642e13	−0.896304	−0.448152	+	0.893957i	$0.647918\pi$
$840$	0	0
$841$	7.51250e12	0.517849
$842$	0	0
$843$	−9.86580e12	−0.672835
$844$	0	0
$845$	1.47342e13	0.994193
$846$	0	0
$847$	1.74471e13	1.16479
$848$	0	0
$849$	1.88584e12	0.124572
$850$	0	0
$851$	1.64835e13	1.07738
$852$	0	0
$853$	1.03465e13	0.669147	0.334574	−	0.942370i	$-0.391408\pi$
$853$	1.03465e13	0.669147	0.334574	+	0.942370i	$0.391408\pi$
$854$	0	0
$855$	1.44572e12	0.0925202
$856$	0	0
$857$	5.63216e12	0.356666	0.178333	−	0.983970i	$-0.442930\pi$
$857$	5.63216e12	0.356666	0.178333	+	0.983970i	$0.442930\pi$
$858$	0	0
$859$	2.28289e13	1.43059	0.715295	−	0.698822i	$-0.246292\pi$
$859$	2.28289e13	1.43059	0.715295	+	0.698822i	$0.246292\pi$
$860$	0	0
$861$	−5.34329e12	−0.331356
$862$	0	0
$863$	−4.62918e12	−0.284090	−0.142045	−	0.989860i	$-0.545368\pi$
$863$	−4.62918e12	−0.284090	−0.142045	+	0.989860i	$0.545368\pi$
$864$	0	0
$865$	−2.91386e13	−1.76968
$866$	0	0
$867$	3.28503e12	0.197448
$868$	0	0
$869$	4.05470e12	0.241196
$870$	0	0
$871$	−8.56913e12	−0.504493
$872$	0	0
$873$	−3.72440e12	−0.217016
$874$	0	0
$875$	−2.18614e13	−1.26079
$876$	0	0
$877$	5.33611e12	0.304598	0.152299	−	0.988334i	$-0.451332\pi$
$877$	5.33611e12	0.304598	0.152299	+	0.988334i	$0.451332\pi$
$878$	0	0
$879$	7.39369e12	0.417745
$880$	0	0
$881$	7.87890e12	0.440630	0.220315	−	0.975429i	$-0.429291\pi$
$881$	7.87890e12	0.440630	0.220315	+	0.975429i	$0.429291\pi$
$882$	0	0
$883$	−1.38973e13	−0.769320	−0.384660	−	0.923058i	$-0.625681\pi$
$883$	−1.38973e13	−0.769320	−0.384660	+	0.923058i	$0.625681\pi$
$884$	0	0
$885$	2.51710e12	0.137929
$886$	0	0
$887$	5.86607e12	0.318193	0.159097	−	0.987263i	$-0.449142\pi$
$887$	5.86607e12	0.318193	0.159097	+	0.987263i	$0.449142\pi$
$888$	0	0
$889$	8.47881e12	0.455278
$890$	0	0
$891$	−9.09663e11	−0.0483538
$892$	0	0
$893$	3.30841e12	0.174096
$894$	0	0
$895$	6.62263e12	0.345006
$896$	0	0
$897$	−4.45871e12	−0.229955
$898$	0	0
$899$	1.73195e12	0.0884334
$900$	0	0
$901$	−2.19220e13	−1.10820
$902$	0	0
$903$	1.71145e13	0.856583
$904$	0	0
$905$	1.46384e12	0.0725395
$906$	0	0
$907$	−8.52250e12	−0.418152	−0.209076	−	0.977899i	$-0.567046\pi$
$907$	−8.52250e12	−0.418152	−0.209076	+	0.977899i	$0.567046\pi$
$908$	0	0
$909$	1.06473e13	0.517254
$910$	0	0
$911$	−1.63369e13	−0.785843	−0.392921	−	0.919572i	$-0.628536\pi$
$911$	−1.63369e13	−0.785843	−0.392921	+	0.919572i	$0.628536\pi$
$912$	0	0
$913$	−5.83580e12	−0.277960
$914$	0	0
$915$	−2.22256e13	−1.04823
$916$	0	0
$917$	−9.02448e12	−0.421464
$918$	0	0
$919$	−3.24712e13	−1.50169	−0.750843	−	0.660481i	$-0.770352\pi$
$919$	−3.24712e13	−1.50169	−0.750843	+	0.660481i	$0.770352\pi$
$920$	0	0
$921$	−3.55647e12	−0.162873
$922$	0	0
$923$	1.13440e12	0.0514468
$924$	0	0
$925$	3.62456e12	0.162786
$926$	0	0
$927$	1.14885e13	0.510980
$928$	0	0
$929$	−1.24571e13	−0.548716	−0.274358	−	0.961628i	$-0.588465\pi$
$929$	−1.24571e13	−0.548716	−0.274358	+	0.961628i	$0.588465\pi$
$930$	0	0
$931$	−6.18808e12	−0.269949
$932$	0	0
$933$	−6.19516e12	−0.267661
$934$	0	0
$935$	−9.03167e12	−0.386470
$936$	0	0
$937$	9.29012e12	0.393725	0.196863	−	0.980431i	$-0.436925\pi$
$937$	9.29012e12	0.393725	0.196863	+	0.980431i	$0.436925\pi$
$938$	0	0
$939$	−2.17475e13	−0.912883
$940$	0	0
$941$	−3.24831e13	−1.35053	−0.675265	−	0.737575i	$-0.735971\pi$
$941$	−3.24831e13	−1.35053	−0.675265	+	0.737575i	$0.735971\pi$
$942$	0	0
$943$	−1.27445e13	−0.524832
$944$	0	0
$945$	−7.42202e12	−0.302746
$946$	0	0
$947$	4.16258e13	1.68185	0.840926	−	0.541151i	$-0.182011\pi$
$947$	4.16258e13	1.68185	0.840926	+	0.541151i	$0.182011\pi$
$948$	0	0
$949$	−7.71265e12	−0.308678
$950$	0	0
$951$	−1.23165e13	−0.488287
$952$	0	0
$953$	1.86479e13	0.732339	0.366169	−	0.930548i	$-0.380669\pi$
$953$	1.86479e13	0.732339	0.366169	+	0.930548i	$0.380669\pi$
$954$	0	0
$955$	−2.69950e13	−1.05019
$956$	0	0
$957$	8.03213e12	0.309547
$958$	0	0
$959$	5.23265e11	0.0199774
$960$	0	0
$961$	−2.63034e13	−0.994848
$962$	0	0
$963$	1.01233e13	0.379320
$964$	0	0
$965$	−2.00420e13	−0.743991
$966$	0	0
$967$	2.32990e13	0.856875	0.428437	−	0.903572i	$-0.359064\pi$
$967$	2.32990e13	0.856875	0.428437	+	0.903572i	$0.359064\pi$
$968$	0	0
$969$	−3.25870e12	−0.118737
$970$	0	0
$971$	3.63628e13	1.31272	0.656358	−	0.754450i	$-0.272096\pi$
$971$	3.63628e13	1.31272	0.656358	+	0.754450i	$0.272096\pi$
$972$	0	0
$973$	−4.24500e13	−1.51834
$974$	0	0
$975$	−9.80425e11	−0.0347451
$976$	0	0
$977$	5.18141e13	1.81938	0.909688	−	0.415291i	$-0.136320\pi$
$977$	5.18141e13	1.81938	0.909688	+	0.415291i	$0.136320\pi$
$978$	0	0
$979$	1.43486e13	0.499214
$980$	0	0
$981$	3.00274e12	0.103516
$982$	0	0
$983$	2.54862e13	0.870592	0.435296	−	0.900287i	$-0.356644\pi$
$983$	2.54862e13	0.870592	0.435296	+	0.900287i	$0.356644\pi$
$984$	0	0
$985$	−7.26121e12	−0.245780
$986$	0	0
$987$	−1.69847e13	−0.569679
$988$	0	0
$989$	4.08205e13	1.35673
$990$	0	0
$991$	−3.92830e13	−1.29382	−0.646910	−	0.762567i	$-0.723939\pi$
$991$	−3.92830e13	−1.29382	−0.646910	+	0.762567i	$0.723939\pi$
$992$	0	0
$993$	−2.13076e13	−0.695447
$994$	0	0
$995$	−2.93917e13	−0.950649
$996$	0	0
$997$	4.98389e13	1.59750	0.798748	−	0.601665i	$-0.205496\pi$
$997$	4.98389e13	1.59750	0.798748	+	0.601665i	$0.205496\pi$
$998$	0	0
$999$	−4.96742e12	−0.157792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.10.a.a.1.1		1
3.2	odd	2		144.10.a.m.1.1		1
4.3	odd	2		3.10.a.b.1.1	✓	1
8.3	odd	2		192.10.a.g.1.1		1
8.5	even	2		192.10.a.n.1.1		1
12.11	even	2		9.10.a.a.1.1		1
20.3	even	4		75.10.b.c.49.1		2
20.7	even	4		75.10.b.c.49.2		2
20.19	odd	2		75.10.a.b.1.1		1
28.27	even	2		147.10.a.c.1.1		1
36.7	odd	6		81.10.c.b.28.1		2
36.11	even	6		81.10.c.d.28.1		2
36.23	even	6		81.10.c.d.55.1		2
36.31	odd	6		81.10.c.b.55.1		2
44.43	even	2		363.10.a.a.1.1		1
60.23	odd	4		225.10.b.c.199.2		2
60.47	odd	4		225.10.b.c.199.1		2
60.59	even	2		225.10.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.10.a.b.1.1	✓	1	4.3	odd	2
9.10.a.a.1.1		1	12.11	even	2
48.10.a.a.1.1		1	1.1	even	1	trivial
75.10.a.b.1.1		1	20.19	odd	2
75.10.b.c.49.1		2	20.3	even	4
75.10.b.c.49.2		2	20.7	even	4
81.10.c.b.28.1		2	36.7	odd	6
81.10.c.b.55.1		2	36.31	odd	6
81.10.c.d.28.1		2	36.11	even	6
81.10.c.d.55.1		2	36.23	even	6
144.10.a.m.1.1		1	3.2	odd	2
147.10.a.c.1.1		1	28.27	even	2
192.10.a.g.1.1		1	8.3	odd	2
192.10.a.n.1.1		1	8.5	even	2
225.10.a.e.1.1		1	60.59	even	2
225.10.b.c.199.1		2	60.47	odd	4
225.10.b.c.199.2		2	60.23	odd	4
363.10.a.a.1.1		1	44.43	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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