LMFDB - Embedded newform 378.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,8,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("378.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	378.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:	$118.081539633$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	378.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +64.0000 q^{4} -40.0000 q^{5} -343.000 q^{7} +512.000 q^{8} +O(q^{10})$

\(q+8.00000 q^{2} +64.0000 q^{4} -40.0000 q^{5} -343.000 q^{7} +512.000 q^{8} -320.000 q^{10} +7096.00 q^{11} -699.000 q^{13} -2744.00 q^{14} +4096.00 q^{16} -38657.0 q^{17} -3454.00 q^{19} -2560.00 q^{20} +56768.0 q^{22} -92789.0 q^{23} -76525.0 q^{25} -5592.00 q^{26} -21952.0 q^{28} +100961. q^{29} +232677. q^{31} +32768.0 q^{32} -309256. q^{34} +13720.0 q^{35} -63286.0 q^{37} -27632.0 q^{38} -20480.0 q^{40} +798762. q^{41} +189911. q^{43} +454144. q^{44} -742312. q^{46} -548526. q^{47} +117649. q^{49} -612200. q^{50} -44736.0 q^{52} -1.12357e6 q^{53} -283840. q^{55} -175616. q^{56} +807688. q^{58} -948703. q^{59} -2.26855e6 q^{61} +1.86142e6 q^{62} +262144. q^{64} +27960.0 q^{65} -1.75128e6 q^{67} -2.47405e6 q^{68} +109760. q^{70} -5.42456e6 q^{71} +1.32579e6 q^{73} -506288. q^{74} -221056. q^{76} -2.43393e6 q^{77} -7.74436e6 q^{79} -163840. q^{80} +6.39010e6 q^{82} +7.85621e6 q^{83} +1.54628e6 q^{85} +1.51929e6 q^{86} +3.63315e6 q^{88} +2.46953e6 q^{89} +239757. q^{91} -5.93850e6 q^{92} -4.38821e6 q^{94} +138160. q^{95} +1.11623e7 q^{97} +941192. q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	−40.0000	−0.143108	−0.0715542	−	0.997437i	$-0.522796\pi$
$5$	−40.0000	−0.143108	−0.0715542	+	0.997437i	$0.522796\pi$
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	512.000	0.353553
$9$	0	0
$10$	−320.000	−0.101193
$11$	7096.00	1.60746	0.803728	−	0.594997i	$-0.202847\pi$
$11$	7096.00	1.60746	0.803728	+	0.594997i	$0.202847\pi$
$12$	0	0
$13$	−699.000	−0.0882420	−0.0441210	−	0.999026i	$-0.514049\pi$
$13$	−699.000	−0.0882420	−0.0441210	+	0.999026i	$0.514049\pi$
$14$	−2744.00	−0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	−38657.0	−1.90835	−0.954173	−	0.299257i	$-0.903261\pi$
$17$	−38657.0	−1.90835	−0.954173	+	0.299257i	$0.903261\pi$
$18$	0	0
$19$	−3454.00	−0.115527	−0.0577637	−	0.998330i	$-0.518397\pi$
$19$	−3454.00	−0.115527	−0.0577637	+	0.998330i	$0.518397\pi$
$20$	−2560.00	−0.0715542
$21$	0	0
$22$	56768.0	1.13664
$23$	−92789.0	−1.59019	−0.795095	−	0.606485i	$-0.792579\pi$
$23$	−92789.0	−1.59019	−0.795095	+	0.606485i	$0.792579\pi$
$24$	0	0
$25$	−76525.0	−0.979520
$26$	−5592.00	−0.0623965
$27$	0	0
$28$	−21952.0	−0.188982
$29$	100961.	0.768707	0.384353	−	0.923186i	$-0.374424\pi$
$29$	100961.	0.768707	0.384353	+	0.923186i	$0.374424\pi$
$30$	0	0
$31$	232677.	1.40277	0.701387	−	0.712781i	$-0.252565\pi$
$31$	232677.	1.40277	0.701387	+	0.712781i	$0.252565\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	−309256.	−1.34940
$35$	13720.0	0.0540899
$36$	0	0
$37$	−63286.0	−0.205401	−0.102700	−	0.994712i	$-0.532748\pi$
$37$	−63286.0	−0.205401	−0.102700	+	0.994712i	$0.532748\pi$
$38$	−27632.0	−0.0816902
$39$	0	0
$40$	−20480.0	−0.0505964
$41$	798762.	1.80998	0.904990	−	0.425432i	$-0.139878\pi$
$41$	798762.	1.80998	0.904990	+	0.425432i	$0.139878\pi$
$42$	0	0
$43$	189911.	0.364259	0.182130	−	0.983275i	$-0.441701\pi$
$43$	189911.	0.364259	0.182130	+	0.983275i	$0.441701\pi$
$44$	454144.	0.803728
$45$	0	0
$46$	−742312.	−1.12443
$47$	−548526.	−0.770646	−0.385323	−	0.922782i	$-0.625910\pi$
$47$	−548526.	−0.770646	−0.385323	+	0.922782i	$0.625910\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	−612200.	−0.692625
$51$	0	0
$52$	−44736.0	−0.0441210
$53$	−1.12357e6	−1.03665	−0.518327	−	0.855182i	$-0.673445\pi$
$53$	−1.12357e6	−1.03665	−0.518327	+	0.855182i	$0.673445\pi$
$54$	0	0
$55$	−283840.	−0.230040
$56$	−175616.	−0.133631
$57$	0	0
$58$	807688.	0.543558
$59$	−948703.	−0.601379	−0.300689	−	0.953722i	$-0.597217\pi$
$59$	−948703.	−0.601379	−0.300689	+	0.953722i	$0.597217\pi$
$60$	0	0
$61$	−2.26855e6	−1.27966	−0.639828	−	0.768518i	$-0.720994\pi$
$61$	−2.26855e6	−1.27966	−0.639828	+	0.768518i	$0.720994\pi$
$62$	1.86142e6	0.991911
$63$	0	0
$64$	262144.	0.125000
$65$	27960.0	0.0126282
$66$	0	0
$67$	−1.75128e6	−0.711369	−0.355685	−	0.934606i	$-0.615752\pi$
$67$	−1.75128e6	−0.711369	−0.355685	+	0.934606i	$0.615752\pi$
$68$	−2.47405e6	−0.954173
$69$	0	0
$70$	109760.	0.0382473
$71$	−5.42456e6	−1.79871	−0.899354	−	0.437222i	$-0.855962\pi$
$71$	−5.42456e6	−1.79871	−0.899354	+	0.437222i	$0.855962\pi$
$72$	0	0
$73$	1.32579e6	0.398883	0.199442	−	0.979910i	$-0.436087\pi$
$73$	1.32579e6	0.398883	0.199442	+	0.979910i	$0.436087\pi$
$74$	−506288.	−0.145240
$75$	0	0
$76$	−221056.	−0.0577637
$77$	−2.43393e6	−0.607561
$78$	0	0
$79$	−7.74436e6	−1.76722	−0.883610	−	0.468223i	$-0.844894\pi$
$79$	−7.74436e6	−1.76722	−0.883610	+	0.468223i	$0.844894\pi$
$80$	−163840.	−0.0357771
$81$	0	0
$82$	6.39010e6	1.27985
$83$	7.85621e6	1.50813	0.754067	−	0.656798i	$-0.228089\pi$
$83$	7.85621e6	1.50813	0.754067	+	0.656798i	$0.228089\pi$
$84$	0	0
$85$	1.54628e6	0.273100
$86$	1.51929e6	0.257570
$87$	0	0
$88$	3.63315e6	0.568322
$89$	2.46953e6	0.371321	0.185660	−	0.982614i	$-0.440558\pi$
$89$	2.46953e6	0.371321	0.185660	+	0.982614i	$0.440558\pi$
$90$	0	0
$91$	239757.	0.0333523
$92$	−5.93850e6	−0.795095
$93$	0	0
$94$	−4.38821e6	−0.544929
$95$	138160.	0.0165329
$96$	0	0
$97$	1.11623e7	1.24180	0.620901	−	0.783889i	$-0.286767\pi$
$97$	1.11623e7	1.24180	0.620901	+	0.783889i	$0.286767\pi$
$98$	941192.	0.101015
$99$	0	0
$100$	−4.89760e6	−0.489760
$101$	−3.68623e6	−0.356006	−0.178003	−	0.984030i	$-0.556964\pi$
$101$	−3.68623e6	−0.356006	−0.178003	+	0.984030i	$0.556964\pi$
$102$	0	0
$103$	−1.62613e7	−1.46631	−0.733154	−	0.680063i	$-0.761953\pi$
$103$	−1.62613e7	−1.46631	−0.733154	+	0.680063i	$0.761953\pi$
$104$	−357888.	−0.0311983
$105$	0	0
$106$	−8.98855e6	−0.733026
$107$	−2.50137e7	−1.97394	−0.986971	−	0.160899i	$-0.948561\pi$
$107$	−2.50137e7	−1.97394	−0.986971	+	0.160899i	$0.948561\pi$
$108$	0	0
$109$	1.27010e6	0.0939385	0.0469693	−	0.998896i	$-0.485044\pi$
$109$	1.27010e6	0.0939385	0.0469693	+	0.998896i	$0.485044\pi$
$110$	−2.27072e6	−0.162663
$111$	0	0
$112$	−1.40493e6	−0.0944911
$113$	−2.62674e7	−1.71254	−0.856272	−	0.516525i	$-0.827225\pi$
$113$	−2.62674e7	−1.71254	−0.856272	+	0.516525i	$0.827225\pi$
$114$	0	0
$115$	3.71156e6	0.227569
$116$	6.46150e6	0.384353
$117$	0	0
$118$	−7.58962e6	−0.425239
$119$	1.32594e7	0.721287
$120$	0	0
$121$	3.08660e7	1.58392
$122$	−1.81484e7	−0.904853
$123$	0	0
$124$	1.48913e7	0.701387
$125$	6.18600e6	0.283286
$126$	0	0
$127$	−2.90160e7	−1.25697	−0.628485	−	0.777822i	$-0.716325\pi$
$127$	−2.90160e7	−1.25697	−0.628485	+	0.777822i	$0.716325\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	223680.	0.00892946
$131$	4.44204e7	1.72636	0.863182	−	0.504892i	$-0.168468\pi$
$131$	4.44204e7	1.72636	0.863182	+	0.504892i	$0.168468\pi$
$132$	0	0
$133$	1.18472e6	0.0436652
$134$	−1.40103e7	−0.503014
$135$	0	0
$136$	−1.97924e7	−0.674702
$137$	−2.16206e7	−0.718366	−0.359183	−	0.933267i	$-0.616945\pi$
$137$	−2.16206e7	−0.718366	−0.359183	+	0.933267i	$0.616945\pi$
$138$	0	0
$139$	−4.67382e7	−1.47612	−0.738058	−	0.674737i	$-0.764257\pi$
$139$	−4.67382e7	−1.47612	−0.738058	+	0.674737i	$0.764257\pi$
$140$	878080.	0.0270449
$141$	0	0
$142$	−4.33965e7	−1.27188
$143$	−4.96010e6	−0.141845
$144$	0	0
$145$	−4.03844e6	−0.110008
$146$	1.06064e7	0.282053
$147$	0	0
$148$	−4.05030e6	−0.102700
$149$	3.43699e7	0.851189	0.425594	−	0.904914i	$-0.360065\pi$
$149$	3.43699e7	0.851189	0.425594	+	0.904914i	$0.360065\pi$
$150$	0	0
$151$	−5.72690e7	−1.35363	−0.676815	−	0.736153i	$-0.736640\pi$
$151$	−5.72690e7	−1.35363	−0.676815	+	0.736153i	$0.736640\pi$
$152$	−1.76845e6	−0.0408451
$153$	0	0
$154$	−1.94714e7	−0.429611
$155$	−9.30708e6	−0.200749
$156$	0	0
$157$	−4.92768e7	−1.01623	−0.508117	−	0.861288i	$-0.669658\pi$
$157$	−4.92768e7	−1.01623	−0.508117	+	0.861288i	$0.669658\pi$
$158$	−6.19549e7	−1.24961
$159$	0	0
$160$	−1.31072e6	−0.0252982
$161$	3.18266e7	0.601035
$162$	0	0
$163$	1.60394e7	0.290089	0.145045	−	0.989425i	$-0.453667\pi$
$163$	1.60394e7	0.290089	0.145045	+	0.989425i	$0.453667\pi$
$164$	5.11208e7	0.904990
$165$	0	0
$166$	6.28497e7	1.06641
$167$	−2.29643e7	−0.381545	−0.190772	−	0.981634i	$-0.561099\pi$
$167$	−2.29643e7	−0.381545	−0.190772	+	0.981634i	$0.561099\pi$
$168$	0	0
$169$	−6.22599e7	−0.992213
$170$	1.23702e7	0.193111
$171$	0	0
$172$	1.21543e7	0.182130
$173$	−1.63070e7	−0.239449	−0.119724	−	0.992807i	$-0.538201\pi$
$173$	−1.63070e7	−0.239449	−0.119724	+	0.992807i	$0.538201\pi$
$174$	0	0
$175$	2.62481e7	0.370224
$176$	2.90652e7	0.401864
$177$	0	0
$178$	1.97562e7	0.262564
$179$	−3.58384e7	−0.467049	−0.233525	−	0.972351i	$-0.575026\pi$
$179$	−3.58384e7	−0.467049	−0.233525	+	0.972351i	$0.575026\pi$
$180$	0	0
$181$	−5.61542e7	−0.703894	−0.351947	−	0.936020i	$-0.614480\pi$
$181$	−5.61542e7	−0.703894	−0.351947	+	0.936020i	$0.614480\pi$
$182$	1.91806e6	0.0235837
$183$	0	0
$184$	−4.75080e7	−0.562217
$185$	2.53144e6	0.0293945
$186$	0	0
$187$	−2.74310e8	−3.06758
$188$	−3.51057e7	−0.385323
$189$	0	0
$190$	1.10528e6	0.0116905
$191$	1.36632e8	1.41884	0.709422	−	0.704784i	$-0.248956\pi$
$191$	1.36632e8	1.41884	0.709422	+	0.704784i	$0.248956\pi$
$192$	0	0
$193$	1.98641e6	0.0198893	0.00994464	−	0.999951i	$-0.496834\pi$
$193$	1.98641e6	0.0198893	0.00994464	+	0.999951i	$0.496834\pi$
$194$	8.92984e7	0.878087
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	1.26523e8	1.17907	0.589533	−	0.807744i	$-0.299312\pi$
$197$	1.26523e8	1.17907	0.589533	+	0.807744i	$0.299312\pi$
$198$	0	0
$199$	−8.79220e7	−0.790882	−0.395441	−	0.918491i	$-0.629408\pi$
$199$	−8.79220e7	−0.790882	−0.395441	+	0.918491i	$0.629408\pi$
$200$	−3.91808e7	−0.346313
$201$	0	0
$202$	−2.94898e7	−0.251734
$203$	−3.46296e7	−0.290544
$204$	0	0
$205$	−3.19505e7	−0.259023
$206$	−1.30090e8	−1.03684
$207$	0	0
$208$	−2.86310e6	−0.0220605
$209$	−2.45096e7	−0.185705
$210$	0	0
$211$	7.93361e7	0.581410	0.290705	−	0.956813i	$-0.406110\pi$
$211$	7.93361e7	0.581410	0.290705	+	0.956813i	$0.406110\pi$
$212$	−7.19084e7	−0.518327
$213$	0	0
$214$	−2.00109e8	−1.39579
$215$	−7.59644e6	−0.0521285
$216$	0	0
$217$	−7.98082e7	−0.530199
$218$	1.01608e7	0.0664246
$219$	0	0
$220$	−1.81658e7	−0.115020
$221$	2.70212e7	0.168396
$222$	0	0
$223$	2.89440e8	1.74780	0.873900	−	0.486106i	$-0.161583\pi$
$223$	2.89440e8	1.74780	0.873900	+	0.486106i	$0.161583\pi$
$224$	−1.12394e7	−0.0668153
$225$	0	0
$226$	−2.10139e8	−1.21095
$227$	5.25994e7	0.298463	0.149231	−	0.988802i	$-0.452320\pi$
$227$	5.25994e7	0.298463	0.149231	+	0.988802i	$0.452320\pi$
$228$	0	0
$229$	60070.0	0.000330547 0	0.000165274	−	1.00000i	$-0.499947\pi$
$229$	60070.0	0.000330547 0	0.000165274		1.00000i	$0.499947\pi$
$230$	2.96925e7	0.160916
$231$	0	0
$232$	5.16920e7	0.271779
$233$	6.35792e7	0.329283	0.164641	−	0.986353i	$-0.447353\pi$
$233$	6.35792e7	0.329283	0.164641	+	0.986353i	$0.447353\pi$
$234$	0	0
$235$	2.19410e7	0.110286
$236$	−6.07170e7	−0.300689
$237$	0	0
$238$	1.06075e8	0.510027
$239$	2.73533e8	1.29604	0.648019	−	0.761624i	$-0.275598\pi$
$239$	2.73533e8	1.29604	0.648019	+	0.761624i	$0.275598\pi$
$240$	0	0
$241$	−1.74294e8	−0.802090	−0.401045	−	0.916058i	$-0.631353\pi$
$241$	−1.74294e8	−0.802090	−0.401045	+	0.916058i	$0.631353\pi$
$242$	2.46928e8	1.12000
$243$	0	0
$244$	−1.45187e8	−0.639828
$245$	−4.70596e6	−0.0204441
$246$	0	0
$247$	2.41435e6	0.0101944
$248$	1.19131e8	0.495955
$249$	0	0
$250$	4.94880e7	0.200313
$251$	3.68747e8	1.47187	0.735936	−	0.677051i	$-0.236742\pi$
$251$	3.68747e8	1.47187	0.735936	+	0.677051i	$0.236742\pi$
$252$	0	0
$253$	−6.58431e8	−2.55616
$254$	−2.32128e8	−0.888812
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−1.32848e8	−0.488190	−0.244095	−	0.969751i	$-0.578491\pi$
$257$	−1.32848e8	−0.488190	−0.244095	+	0.969751i	$0.578491\pi$
$258$	0	0
$259$	2.17071e7	0.0776341
$260$	1.78944e6	0.00631408
$261$	0	0
$262$	3.55363e8	1.22072
$263$	3.43431e8	1.16411	0.582055	−	0.813149i	$-0.302249\pi$
$263$	3.43431e8	1.16411	0.582055	+	0.813149i	$0.302249\pi$
$264$	0	0
$265$	4.49428e7	0.148354
$266$	9.47778e6	0.0308760
$267$	0	0
$268$	−1.12082e8	−0.355685
$269$	3.95732e8	1.23956	0.619781	−	0.784775i	$-0.287222\pi$
$269$	3.95732e8	1.23956	0.619781	+	0.784775i	$0.287222\pi$
$270$	0	0
$271$	3.01378e8	0.919855	0.459927	−	0.887957i	$-0.347875\pi$
$271$	3.01378e8	0.919855	0.459927	+	0.887957i	$0.347875\pi$
$272$	−1.58339e8	−0.477086
$273$	0	0
$274$	−1.72965e8	−0.507962
$275$	−5.43021e8	−1.57454
$276$	0	0
$277$	−6.08511e8	−1.72024	−0.860121	−	0.510091i	$-0.829612\pi$
$277$	−6.08511e8	−1.72024	−0.860121	+	0.510091i	$0.829612\pi$
$278$	−3.73906e8	−1.04377
$279$	0	0
$280$	7.02464e6	0.0191237
$281$	2.72492e7	0.0732626	0.0366313	−	0.999329i	$-0.488337\pi$
$281$	2.72492e7	0.0732626	0.0366313	+	0.999329i	$0.488337\pi$
$282$	0	0
$283$	−5.70038e8	−1.49503	−0.747517	−	0.664243i	$-0.768754\pi$
$283$	−5.70038e8	−1.49503	−0.747517	+	0.664243i	$0.768754\pi$
$284$	−3.47172e8	−0.899354
$285$	0	0
$286$	−3.96808e7	−0.100300
$287$	−2.73975e8	−0.684108
$288$	0	0
$289$	1.08402e9	2.64178
$290$	−3.23075e7	−0.0777876
$291$	0	0
$292$	8.48508e7	0.199442
$293$	−3.28673e8	−0.763355	−0.381678	−	0.924296i	$-0.624654\pi$
$293$	−3.28673e8	−0.763355	−0.381678	+	0.924296i	$0.624654\pi$
$294$	0	0
$295$	3.79481e7	0.0860624
$296$	−3.24024e7	−0.0726201
$297$	0	0
$298$	2.74959e8	0.601881
$299$	6.48595e7	0.140322
$300$	0	0
$301$	−6.51395e7	−0.137677
$302$	−4.58152e8	−0.957161
$303$	0	0
$304$	−1.41476e7	−0.0288818
$305$	9.07418e7	0.183129
$306$	0	0
$307$	−6.57283e8	−1.29649	−0.648244	−	0.761433i	$-0.724496\pi$
$307$	−6.57283e8	−1.29649	−0.648244	+	0.761433i	$0.724496\pi$
$308$	−1.55771e8	−0.303781
$309$	0	0
$310$	−7.44566e7	−0.141951
$311$	5.74934e8	1.08382	0.541910	−	0.840437i	$-0.317701\pi$
$311$	5.74934e8	1.08382	0.541910	+	0.840437i	$0.317701\pi$
$312$	0	0
$313$	3.24099e8	0.597410	0.298705	−	0.954346i	$-0.403445\pi$
$313$	3.24099e8	0.597410	0.298705	+	0.954346i	$0.403445\pi$
$314$	−3.94214e8	−0.718586
$315$	0	0
$316$	−4.95639e8	−0.883610
$317$	−6.54816e8	−1.15455	−0.577274	−	0.816551i	$-0.695883\pi$
$317$	−6.54816e8	−1.15455	−0.577274	+	0.816551i	$0.695883\pi$
$318$	0	0
$319$	7.16419e8	1.23566
$320$	−1.04858e7	−0.0178885
$321$	0	0
$322$	2.54613e8	0.424996
$323$	1.33521e8	0.220466
$324$	0	0
$325$	5.34910e7	0.0864348
$326$	1.28315e8	0.205124
$327$	0	0
$328$	4.08966e8	0.639925
$329$	1.88144e8	0.291277
$330$	0	0
$331$	−1.04424e9	−1.58271	−0.791353	−	0.611359i	$-0.790623\pi$
$331$	−1.04424e9	−1.58271	−0.791353	+	0.611359i	$0.790623\pi$
$332$	5.02798e8	0.754067
$333$	0	0
$334$	−1.83714e8	−0.269793
$335$	7.00514e7	0.101803
$336$	0	0
$337$	1.16958e9	1.66466	0.832332	−	0.554278i	$-0.187006\pi$
$337$	1.16958e9	1.66466	0.832332	+	0.554278i	$0.187006\pi$
$338$	−4.98079e8	−0.701601
$339$	0	0
$340$	9.89619e7	0.136550
$341$	1.65108e9	2.25490
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	9.72344e7	0.128785
$345$	0	0
$346$	−1.30456e8	−0.169316
$347$	4.31399e8	0.554276	0.277138	−	0.960830i	$-0.410614\pi$
$347$	4.31399e8	0.554276	0.277138	+	0.960830i	$0.410614\pi$
$348$	0	0
$349$	−3.98778e7	−0.0502160	−0.0251080	−	0.999685i	$-0.507993\pi$
$349$	−3.98778e7	−0.0502160	−0.0251080	+	0.999685i	$0.507993\pi$
$350$	2.09985e8	0.261788
$351$	0	0
$352$	2.32522e8	0.284161
$353$	−9.78061e8	−1.18346	−0.591731	−	0.806136i	$-0.701555\pi$
$353$	−9.78061e8	−1.18346	−0.591731	+	0.806136i	$0.701555\pi$
$354$	0	0
$355$	2.16983e8	0.257410
$356$	1.58050e8	0.185660
$357$	0	0
$358$	−2.86707e8	−0.330254
$359$	−2.87425e6	−0.00327864	−0.00163932	−	0.999999i	$-0.500522\pi$
$359$	−2.87425e6	−0.00327864	−0.00163932	+	0.999999i	$0.500522\pi$
$360$	0	0
$361$	−8.81942e8	−0.986653
$362$	−4.49234e8	−0.497728
$363$	0	0
$364$	1.53444e7	0.0166762
$365$	−5.30318e7	−0.0570836
$366$	0	0
$367$	2.00520e8	0.211752	0.105876	−	0.994379i	$-0.466235\pi$
$367$	2.00520e8	0.211752	0.105876	+	0.994379i	$0.466235\pi$
$368$	−3.80064e8	−0.397548
$369$	0	0
$370$	2.02515e7	0.0207851
$371$	3.85384e8	0.391819
$372$	0	0
$373$	3.69643e8	0.368809	0.184404	−	0.982850i	$-0.440964\pi$
$373$	3.69643e8	0.368809	0.184404	+	0.982850i	$0.440964\pi$
$374$	−2.19448e9	−2.16911
$375$	0	0
$376$	−2.80845e8	−0.272464
$377$	−7.05717e7	−0.0678322
$378$	0	0
$379$	−2.22241e8	−0.209694	−0.104847	−	0.994488i	$-0.533435\pi$
$379$	−2.22241e8	−0.209694	−0.104847	+	0.994488i	$0.533435\pi$
$380$	8.84224e6	0.00826646
$381$	0	0
$382$	1.09305e9	1.00327
$383$	−1.49919e9	−1.36352	−0.681760	−	0.731576i	$-0.738785\pi$
$383$	−1.49919e9	−1.36352	−0.681760	+	0.731576i	$0.738785\pi$
$384$	0	0
$385$	9.73571e7	0.0869471
$386$	1.58913e7	0.0140638
$387$	0	0
$388$	7.14387e8	0.620901
$389$	−1.11498e9	−0.960381	−0.480191	−	0.877164i	$-0.659433\pi$
$389$	−1.11498e9	−0.960381	−0.480191	+	0.877164i	$0.659433\pi$
$390$	0	0
$391$	3.58694e9	3.03463
$392$	6.02363e7	0.0505076
$393$	0	0
$394$	1.01219e9	0.833726
$395$	3.09774e8	0.252904
$396$	0	0
$397$	7.47135e8	0.599283	0.299642	−	0.954052i	$-0.403133\pi$
$397$	7.47135e8	0.599283	0.299642	+	0.954052i	$0.403133\pi$
$398$	−7.03376e8	−0.559238
$399$	0	0
$400$	−3.13446e8	−0.244880
$401$	1.82963e9	1.41696	0.708482	−	0.705728i	$-0.249380\pi$
$401$	1.82963e9	1.41696	0.708482	+	0.705728i	$0.249380\pi$
$402$	0	0
$403$	−1.62641e8	−0.123784
$404$	−2.35919e8	−0.178003
$405$	0	0
$406$	−2.77037e8	−0.205445
$407$	−4.49077e8	−0.330172
$408$	0	0
$409$	−4.54872e8	−0.328744	−0.164372	−	0.986398i	$-0.552560\pi$
$409$	−4.54872e8	−0.328744	−0.164372	+	0.986398i	$0.552560\pi$
$410$	−2.55604e8	−0.183157
$411$	0	0
$412$	−1.04072e9	−0.733154
$413$	3.25405e8	0.227300
$414$	0	0
$415$	−3.14248e8	−0.215827
$416$	−2.29048e7	−0.0155991
$417$	0	0
$418$	−1.96077e8	−0.131313
$419$	−1.60248e9	−1.06425	−0.532126	−	0.846665i	$-0.678607\pi$
$419$	−1.60248e9	−1.06425	−0.532126	+	0.846665i	$0.678607\pi$
$420$	0	0
$421$	7.78830e8	0.508693	0.254346	−	0.967113i	$-0.418140\pi$
$421$	7.78830e8	0.508693	0.254346	+	0.967113i	$0.418140\pi$
$422$	6.34689e8	0.411119
$423$	0	0
$424$	−5.75267e8	−0.366513
$425$	2.95823e9	1.86926
$426$	0	0
$427$	7.78111e8	0.483664
$428$	−1.60088e9	−0.986971
$429$	0	0
$430$	−6.07715e7	−0.0368604
$431$	2.14156e9	1.28843	0.644214	−	0.764846i	$-0.277185\pi$
$431$	2.14156e9	1.28843	0.644214	+	0.764846i	$0.277185\pi$
$432$	0	0
$433$	1.60576e9	0.950547	0.475274	−	0.879838i	$-0.342349\pi$
$433$	1.60576e9	0.950547	0.475274	+	0.879838i	$0.342349\pi$
$434$	−6.38466e8	−0.374907
$435$	0	0
$436$	8.12861e7	0.0469693
$437$	3.20493e8	0.183710
$438$	0	0
$439$	1.36049e9	0.767482	0.383741	−	0.923441i	$-0.374636\pi$
$439$	1.36049e9	0.767482	0.383741	+	0.923441i	$0.374636\pi$
$440$	−1.45326e8	−0.0813316
$441$	0	0
$442$	2.16170e8	0.119074
$443$	1.68103e9	0.918679	0.459339	−	0.888261i	$-0.348086\pi$
$443$	1.68103e9	0.918679	0.459339	+	0.888261i	$0.348086\pi$
$444$	0	0
$445$	−9.87812e7	−0.0531391
$446$	2.31552e9	1.23588
$447$	0	0
$448$	−8.99154e7	−0.0472456
$449$	−7.86169e8	−0.409877	−0.204939	−	0.978775i	$-0.565699\pi$
$449$	−7.86169e8	−0.409877	−0.204939	+	0.978775i	$0.565699\pi$
$450$	0	0
$451$	5.66802e9	2.90946
$452$	−1.68111e9	−0.856272
$453$	0	0
$454$	4.20795e8	0.211045
$455$	−9.59028e6	−0.00477300
$456$	0	0
$457$	3.10170e9	1.52017	0.760086	−	0.649823i	$-0.225157\pi$
$457$	3.10170e9	1.52017	0.760086	+	0.649823i	$0.225157\pi$
$458$	480560.	0.000233732 0
$459$	0	0
$460$	2.37540e8	0.113785
$461$	6.74273e8	0.320540	0.160270	−	0.987073i	$-0.448763\pi$
$461$	6.74273e8	0.320540	0.160270	+	0.987073i	$0.448763\pi$
$462$	0	0
$463$	−1.42545e9	−0.667450	−0.333725	−	0.942670i	$-0.608306\pi$
$463$	−1.42545e9	−0.667450	−0.333725	+	0.942670i	$0.608306\pi$
$464$	4.13536e8	0.192177
$465$	0	0
$466$	5.08633e8	0.232838
$467$	3.58693e9	1.62972	0.814861	−	0.579656i	$-0.196813\pi$
$467$	3.58693e9	1.62972	0.814861	+	0.579656i	$0.196813\pi$
$468$	0	0
$469$	6.00691e8	0.268872
$470$	1.75528e8	0.0779839
$471$	0	0
$472$	−4.85736e8	−0.212620
$473$	1.34761e9	0.585531
$474$	0	0
$475$	2.64317e8	0.113161
$476$	8.48598e8	0.360643
$477$	0	0
$478$	2.18827e9	0.916437
$479$	2.00251e9	0.832530	0.416265	−	0.909243i	$-0.363339\pi$
$479$	2.00251e9	0.832530	0.416265	+	0.909243i	$0.363339\pi$
$480$	0	0
$481$	4.42369e7	0.0181250
$482$	−1.39435e9	−0.567163
$483$	0	0
$484$	1.97543e9	0.791958
$485$	−4.46492e8	−0.177712
$486$	0	0
$487$	−1.76697e9	−0.693232	−0.346616	−	0.938007i	$-0.612669\pi$
$487$	−1.76697e9	−0.693232	−0.346616	+	0.938007i	$0.612669\pi$
$488$	−1.16150e9	−0.452427
$489$	0	0
$490$	−3.76477e7	−0.0144561
$491$	−1.49601e9	−0.570359	−0.285179	−	0.958474i	$-0.592053\pi$
$491$	−1.49601e9	−0.570359	−0.285179	+	0.958474i	$0.592053\pi$
$492$	0	0
$493$	−3.90285e9	−1.46696
$494$	1.93148e7	0.00720850
$495$	0	0
$496$	9.53045e8	0.350693
$497$	1.86063e9	0.679848
$498$	0	0
$499$	−3.83651e9	−1.38224	−0.691121	−	0.722739i	$-0.742883\pi$
$499$	−3.83651e9	−1.38224	−0.691121	+	0.722739i	$0.742883\pi$
$500$	3.95904e8	0.141643
$501$	0	0
$502$	2.94997e9	1.04077
$503$	5.18996e9	1.81834	0.909172	−	0.416421i	$-0.136716\pi$
$503$	5.18996e9	1.81834	0.909172	+	0.416421i	$0.136716\pi$
$504$	0	0
$505$	1.47449e8	0.0509475
$506$	−5.26745e9	−1.80748
$507$	0	0
$508$	−1.85703e9	−0.628485
$509$	−2.07311e9	−0.696804	−0.348402	−	0.937345i	$-0.613276\pi$
$509$	−2.07311e9	−0.696804	−0.348402	+	0.937345i	$0.613276\pi$
$510$	0	0
$511$	−4.54747e8	−0.150764
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	−1.06278e9	−0.345202
$515$	6.50452e8	0.209841
$516$	0	0
$517$	−3.89234e9	−1.23878
$518$	1.73657e8	0.0548956
$519$	0	0
$520$	1.43155e7	0.00446473
$521$	3.01384e9	0.933658	0.466829	−	0.884348i	$-0.345396\pi$
$521$	3.01384e9	0.933658	0.466829	+	0.884348i	$0.345396\pi$
$522$	0	0
$523$	−1.74968e9	−0.534815	−0.267407	−	0.963584i	$-0.586167\pi$
$523$	−1.74968e9	−0.534815	−0.267407	+	0.963584i	$0.586167\pi$
$524$	2.84290e9	0.863182
$525$	0	0
$526$	2.74745e9	0.823150
$527$	−8.99459e9	−2.67698
$528$	0	0
$529$	5.20497e9	1.52870
$530$	3.59542e8	0.104902
$531$	0	0
$532$	7.58222e7	0.0218326
$533$	−5.58335e8	−0.159716
$534$	0	0
$535$	1.00055e9	0.282488
$536$	−8.96658e8	−0.251507
$537$	0	0
$538$	3.16585e9	0.876502
$539$	8.34837e8	0.229637
$540$	0	0
$541$	5.87936e8	0.159639	0.0798196	−	0.996809i	$-0.474566\pi$
$541$	5.87936e8	0.159639	0.0798196	+	0.996809i	$0.474566\pi$
$542$	2.41103e9	0.650436
$543$	0	0
$544$	−1.26671e9	−0.337351
$545$	−5.08038e7	−0.0134434
$546$	0	0
$547$	−2.70831e9	−0.707527	−0.353764	−	0.935335i	$-0.615098\pi$
$547$	−2.70831e9	−0.707527	−0.353764	+	0.935335i	$0.615098\pi$
$548$	−1.38372e9	−0.359183
$549$	0	0
$550$	−4.34417e9	−1.11336
$551$	−3.48719e8	−0.0888066
$552$	0	0
$553$	2.65632e9	0.667947
$554$	−4.86809e9	−1.21639
$555$	0	0
$556$	−2.99125e9	−0.738058
$557$	5.15002e9	1.26275	0.631373	−	0.775479i	$-0.282492\pi$
$557$	5.15002e9	1.26275	0.631373	+	0.775479i	$0.282492\pi$
$558$	0	0
$559$	−1.32748e8	−0.0321430
$560$	5.61971e7	0.0135225
$561$	0	0
$562$	2.17994e8	0.0518045
$563$	4.53480e9	1.07097	0.535487	−	0.844543i	$-0.320128\pi$
$563$	4.53480e9	1.07097	0.535487	+	0.844543i	$0.320128\pi$
$564$	0	0
$565$	1.05069e9	0.245079
$566$	−4.56030e9	−1.05715
$567$	0	0
$568$	−2.77738e9	−0.635939
$569$	−2.29980e9	−0.523355	−0.261678	−	0.965155i	$-0.584276\pi$
$569$	−2.29980e9	−0.523355	−0.261678	+	0.965155i	$0.584276\pi$
$570$	0	0
$571$	2.94581e9	0.662183	0.331092	−	0.943599i	$-0.392583\pi$
$571$	2.94581e9	0.662183	0.331092	+	0.943599i	$0.392583\pi$
$572$	−3.17447e8	−0.0709226
$573$	0	0
$574$	−2.19180e9	−0.483738
$575$	7.10068e9	1.55762
$576$	0	0
$577$	2.33877e7	0.00506842	0.00253421	−	0.999997i	$-0.499193\pi$
$577$	2.33877e7	0.00506842	0.00253421	+	0.999997i	$0.499193\pi$
$578$	8.67220e9	1.86802
$579$	0	0
$580$	−2.58460e8	−0.0550042
$581$	−2.69468e9	−0.570021
$582$	0	0
$583$	−7.97285e9	−1.66638
$584$	6.78807e8	0.141027
$585$	0	0
$586$	−2.62938e9	−0.539774
$587$	−4.00481e9	−0.817239	−0.408619	−	0.912705i	$-0.633990\pi$
$587$	−4.00481e9	−0.817239	−0.408619	+	0.912705i	$0.633990\pi$
$588$	0	0
$589$	−8.03666e8	−0.162059
$590$	3.03585e8	0.0608553
$591$	0	0
$592$	−2.59219e8	−0.0513501
$593$	−2.54801e9	−0.501776	−0.250888	−	0.968016i	$-0.580723\pi$
$593$	−2.54801e9	−0.501776	−0.250888	+	0.968016i	$0.580723\pi$
$594$	0	0
$595$	−5.30374e8	−0.103222
$596$	2.19967e9	0.425594
$597$	0	0
$598$	5.18876e8	0.0992224
$599$	−4.87050e9	−0.925933	−0.462966	−	0.886376i	$-0.653215\pi$
$599$	−4.87050e9	−0.925933	−0.462966	+	0.886376i	$0.653215\pi$
$600$	0	0
$601$	−3.57578e9	−0.671909	−0.335954	−	0.941878i	$-0.609059\pi$
$601$	−3.57578e9	−0.671909	−0.335954	+	0.941878i	$0.609059\pi$
$602$	−5.21116e8	−0.0973524
$603$	0	0
$604$	−3.66521e9	−0.676815
$605$	−1.23464e9	−0.226672
$606$	0	0
$607$	−4.53661e9	−0.823324	−0.411662	−	0.911337i	$-0.635052\pi$
$607$	−4.53661e9	−0.823324	−0.411662	+	0.911337i	$0.635052\pi$
$608$	−1.13181e8	−0.0204225
$609$	0	0
$610$	7.25935e8	0.129492
$611$	3.83420e8	0.0680033
$612$	0	0
$613$	8.80999e8	0.154477	0.0772385	−	0.997013i	$-0.475390\pi$
$613$	8.80999e8	0.154477	0.0772385	+	0.997013i	$0.475390\pi$
$614$	−5.25827e9	−0.916755
$615$	0	0
$616$	−1.24617e9	−0.214805
$617$	−2.35843e8	−0.0404227	−0.0202114	−	0.999796i	$-0.506434\pi$
$617$	−2.35843e8	−0.0404227	−0.0202114	+	0.999796i	$0.506434\pi$
$618$	0	0
$619$	7.63247e9	1.29344	0.646722	−	0.762725i	$-0.276139\pi$
$619$	7.63247e9	1.29344	0.646722	+	0.762725i	$0.276139\pi$
$620$	−5.95653e8	−0.100374
$621$	0	0
$622$	4.59948e9	0.766376
$623$	−8.47049e8	−0.140346
$624$	0	0
$625$	5.73108e9	0.938979
$626$	2.59279e9	0.422432
$627$	0	0
$628$	−3.15372e9	−0.508117
$629$	2.44645e9	0.391975
$630$	0	0
$631$	5.73772e9	0.909152	0.454576	−	0.890708i	$-0.349791\pi$
$631$	5.73772e9	0.909152	0.454576	+	0.890708i	$0.349791\pi$
$632$	−3.96511e9	−0.624807
$633$	0	0
$634$	−5.23853e9	−0.816388
$635$	1.16064e9	0.179883
$636$	0	0
$637$	−8.22367e7	−0.0126060
$638$	5.73135e9	0.873745
$639$	0	0
$640$	−8.38861e7	−0.0126491
$641$	−1.81838e9	−0.272698	−0.136349	−	0.990661i	$-0.543537\pi$
$641$	−1.81838e9	−0.272698	−0.136349	+	0.990661i	$0.543537\pi$
$642$	0	0
$643$	−6.60593e9	−0.979931	−0.489966	−	0.871742i	$-0.662991\pi$
$643$	−6.60593e9	−0.979931	−0.489966	+	0.871742i	$0.662991\pi$
$644$	2.03690e9	0.300518
$645$	0	0
$646$	1.06817e9	0.155893
$647$	4.44769e9	0.645609	0.322805	−	0.946466i	$-0.395374\pi$
$647$	4.44769e9	0.645609	0.322805	+	0.946466i	$0.395374\pi$
$648$	0	0
$649$	−6.73200e9	−0.966690
$650$	4.27928e8	0.0611186
$651$	0	0
$652$	1.02652e9	0.145045
$653$	−9.44622e9	−1.32758	−0.663792	−	0.747917i	$-0.731054\pi$
$653$	−9.44622e9	−1.32758	−0.663792	+	0.747917i	$0.731054\pi$
$654$	0	0
$655$	−1.77681e9	−0.247057
$656$	3.27173e9	0.452495
$657$	0	0
$658$	1.50516e9	0.205964
$659$	−9.98550e8	−0.135916	−0.0679580	−	0.997688i	$-0.521648\pi$
$659$	−9.98550e8	−0.135916	−0.0679580	+	0.997688i	$0.521648\pi$
$660$	0	0
$661$	−4.71275e9	−0.634702	−0.317351	−	0.948308i	$-0.602793\pi$
$661$	−4.71275e9	−0.634702	−0.317351	+	0.948308i	$0.602793\pi$
$662$	−8.35388e9	−1.11914
$663$	0	0
$664$	4.02238e9	0.533206
$665$	−4.73889e7	−0.00624886
$666$	0	0
$667$	−9.36807e9	−1.22239
$668$	−1.46971e9	−0.190772
$669$	0	0
$670$	5.60411e8	0.0719855
$671$	−1.60976e10	−2.05699
$672$	0	0
$673$	1.64781e9	0.208379	0.104189	−	0.994557i	$-0.466775\pi$
$673$	1.64781e9	0.208379	0.104189	+	0.994557i	$0.466775\pi$
$674$	9.35666e9	1.17709
$675$	0	0
$676$	−3.98463e9	−0.496107
$677$	5.73134e8	0.0709898	0.0354949	−	0.999370i	$-0.488699\pi$
$677$	5.73134e8	0.0709898	0.0354949	+	0.999370i	$0.488699\pi$
$678$	0	0
$679$	−3.82867e9	−0.469357
$680$	7.91695e8	0.0965555
$681$	0	0
$682$	1.32086e10	1.59445
$683$	5.80651e9	0.697338	0.348669	−	0.937246i	$-0.386634\pi$
$683$	5.80651e9	0.697338	0.348669	+	0.937246i	$0.386634\pi$
$684$	0	0
$685$	8.64825e8	0.102804
$686$	−3.22829e8	−0.0381802
$687$	0	0
$688$	7.77875e8	0.0910648
$689$	7.85375e8	0.0914765
$690$	0	0
$691$	−6.59043e9	−0.759872	−0.379936	−	0.925013i	$-0.624054\pi$
$691$	−6.59043e9	−0.759872	−0.379936	+	0.925013i	$0.624054\pi$
$692$	−1.04365e9	−0.119724
$693$	0	0
$694$	3.45119e9	0.391932
$695$	1.86953e9	0.211245
$696$	0	0
$697$	−3.08777e10	−3.45407
$698$	−3.19022e8	−0.0355081
$699$	0	0
$700$	1.67988e9	0.185112
$701$	−2.89242e9	−0.317138	−0.158569	−	0.987348i	$-0.550688\pi$
$701$	−2.89242e9	−0.317138	−0.158569	+	0.987348i	$0.550688\pi$
$702$	0	0
$703$	2.18590e8	0.0237294
$704$	1.86017e9	0.200932
$705$	0	0
$706$	−7.82448e9	−0.836834
$707$	1.26438e9	0.134558
$708$	0	0
$709$	−4.52890e9	−0.477234	−0.238617	−	0.971114i	$-0.576694\pi$
$709$	−4.52890e9	−0.477234	−0.238617	+	0.971114i	$0.576694\pi$
$710$	1.73586e9	0.182016
$711$	0	0
$712$	1.26440e9	0.131282
$713$	−2.15899e10	−2.23068
$714$	0	0
$715$	1.98404e8	0.0202992
$716$	−2.29366e9	−0.233525
$717$	0	0
$718$	−2.29940e7	−0.00231835
$719$	−1.73617e9	−0.174197	−0.0870985	−	0.996200i	$-0.527759\pi$
$719$	−1.73617e9	−0.174197	−0.0870985	+	0.996200i	$0.527759\pi$
$720$	0	0
$721$	5.57763e9	0.554212
$722$	−7.05553e9	−0.697669
$723$	0	0
$724$	−3.59387e9	−0.351947
$725$	−7.72604e9	−0.752964
$726$	0	0
$727$	4.03705e9	0.389667	0.194833	−	0.980836i	$-0.437583\pi$
$727$	4.03705e9	0.389667	0.194833	+	0.980836i	$0.437583\pi$
$728$	1.22756e8	0.0117918
$729$	0	0
$730$	−4.24254e8	−0.0403642
$731$	−7.34139e9	−0.695132
$732$	0	0
$733$	1.50393e9	0.141047	0.0705237	−	0.997510i	$-0.477533\pi$
$733$	1.50393e9	0.141047	0.0705237	+	0.997510i	$0.477533\pi$
$734$	1.60416e9	0.149731
$735$	0	0
$736$	−3.04051e9	−0.281109
$737$	−1.24271e10	−1.14349
$738$	0	0
$739$	1.27254e10	1.15989	0.579943	−	0.814657i	$-0.303075\pi$
$739$	1.27254e10	1.15989	0.579943	+	0.814657i	$0.303075\pi$
$740$	1.62012e8	0.0146973
$741$	0	0
$742$	3.08307e9	0.277058
$743$	6.07905e9	0.543720	0.271860	−	0.962337i	$-0.412361\pi$
$743$	6.07905e9	0.543720	0.271860	+	0.962337i	$0.412361\pi$
$744$	0	0
$745$	−1.37479e9	−0.121812
$746$	2.95714e9	0.260787
$747$	0	0
$748$	−1.75558e10	−1.53379
$749$	8.57969e9	0.746080
$750$	0	0
$751$	−2.95621e9	−0.254680	−0.127340	−	0.991859i	$-0.540644\pi$
$751$	−2.95621e9	−0.254680	−0.127340	+	0.991859i	$0.540644\pi$
$752$	−2.24676e9	−0.192661
$753$	0	0
$754$	−5.64574e8	−0.0479646
$755$	2.29076e9	0.193716
$756$	0	0
$757$	1.90445e9	0.159563	0.0797817	−	0.996812i	$-0.474578\pi$
$757$	1.90445e9	0.159563	0.0797817	+	0.996812i	$0.474578\pi$
$758$	−1.77793e9	−0.148276
$759$	0	0
$760$	7.07379e7	0.00584527
$761$	1.37415e9	0.113028	0.0565141	−	0.998402i	$-0.482001\pi$
$761$	1.37415e9	0.113028	0.0565141	+	0.998402i	$0.482001\pi$
$762$	0	0
$763$	−4.35643e8	−0.0355054
$764$	8.74443e9	0.709422
$765$	0	0
$766$	−1.19935e10	−0.964154
$767$	6.63143e8	0.0530669
$768$	0	0
$769$	1.05373e10	0.835580	0.417790	−	0.908544i	$-0.362805\pi$
$769$	1.05373e10	0.835580	0.417790	+	0.908544i	$0.362805\pi$
$770$	7.78857e8	0.0614809
$771$	0	0
$772$	1.27130e8	0.00994464
$773$	1.34661e10	1.04861	0.524306	−	0.851530i	$-0.324325\pi$
$773$	1.34661e10	1.04861	0.524306	+	0.851530i	$0.324325\pi$
$774$	0	0
$775$	−1.78056e10	−1.37404
$776$	5.71510e9	0.439043
$777$	0	0
$778$	−8.91984e9	−0.679092
$779$	−2.75892e9	−0.209102
$780$	0	0
$781$	−3.84927e10	−2.89134
$782$	2.86956e10	2.14581
$783$	0	0
$784$	4.81890e8	0.0357143
$785$	1.97107e9	0.145432
$786$	0	0
$787$	1.22462e10	0.895553	0.447777	−	0.894145i	$-0.352216\pi$
$787$	1.22462e10	0.895553	0.447777	+	0.894145i	$0.352216\pi$
$788$	8.09748e9	0.589533
$789$	0	0
$790$	2.47820e9	0.178830
$791$	9.00970e9	0.647281
$792$	0	0
$793$	1.58571e9	0.112919
$794$	5.97708e9	0.423757
$795$	0	0
$796$	−5.62701e9	−0.395441
$797$	3.84363e9	0.268929	0.134464	−	0.990918i	$-0.457069\pi$
$797$	3.84363e9	0.268929	0.134464	+	0.990918i	$0.457069\pi$
$798$	0	0
$799$	2.12044e10	1.47066
$800$	−2.50757e9	−0.173156
$801$	0	0
$802$	1.46371e10	1.00195
$803$	9.40783e9	0.641188
$804$	0	0
$805$	−1.27307e9	−0.0860132
$806$	−1.30113e9	−0.0875282
$807$	0	0
$808$	−1.88735e9	−0.125867
$809$	9.13110e9	0.606322	0.303161	−	0.952939i	$-0.401958\pi$
$809$	9.13110e9	0.606322	0.303161	+	0.952939i	$0.401958\pi$
$810$	0	0
$811$	2.60530e10	1.71508	0.857540	−	0.514417i	$-0.171992\pi$
$811$	2.60530e10	1.71508	0.857540	+	0.514417i	$0.171992\pi$
$812$	−2.21630e9	−0.145272
$813$	0	0
$814$	−3.59262e9	−0.233467
$815$	−6.41576e8	−0.0415142
$816$	0	0
$817$	−6.55953e8	−0.0420819
$818$	−3.63898e9	−0.232457
$819$	0	0
$820$	−2.04483e9	−0.129512
$821$	−2.30676e10	−1.45479	−0.727397	−	0.686216i	$-0.759270\pi$
$821$	−2.30676e10	−1.45479	−0.727397	+	0.686216i	$0.759270\pi$
$822$	0	0
$823$	5.82195e9	0.364057	0.182028	−	0.983293i	$-0.441734\pi$
$823$	5.82195e9	0.364057	0.182028	+	0.983293i	$0.441734\pi$
$824$	−8.32579e9	−0.518418
$825$	0	0
$826$	2.60324e9	0.160725
$827$	−3.02540e10	−1.86000	−0.930000	−	0.367559i	$-0.880194\pi$
$827$	−3.02540e10	−1.86000	−0.930000	+	0.367559i	$0.880194\pi$
$828$	0	0
$829$	−2.23803e10	−1.36435	−0.682174	−	0.731190i	$-0.738966\pi$
$829$	−2.23803e10	−1.36435	−0.682174	+	0.731190i	$0.738966\pi$
$830$	−2.51399e9	−0.152612
$831$	0	0
$832$	−1.83239e8	−0.0110303
$833$	−4.54796e9	−0.272621
$834$	0	0
$835$	9.18571e8	0.0546022
$836$	−1.56861e9	−0.0928526
$837$	0	0
$838$	−1.28199e10	−0.752539
$839$	1.08094e9	0.0631877	0.0315939	−	0.999501i	$-0.489942\pi$
$839$	1.08094e9	0.0631877	0.0315939	+	0.999501i	$0.489942\pi$
$840$	0	0
$841$	−7.05675e9	−0.409090
$842$	6.23064e9	0.359700
$843$	0	0
$844$	5.07751e9	0.290705
$845$	2.49040e9	0.141994
$846$	0	0
$847$	−1.05871e10	−0.598664
$848$	−4.60214e9	−0.259164
$849$	0	0
$850$	2.36658e10	1.32177
$851$	5.87224e9	0.326626
$852$	0	0
$853$	−5.55532e9	−0.306470	−0.153235	−	0.988190i	$-0.548969\pi$
$853$	−5.55532e9	−0.306470	−0.153235	+	0.988190i	$0.548969\pi$
$854$	6.22489e9	0.342002
$855$	0	0
$856$	−1.28070e10	−0.697894
$857$	−3.50309e9	−0.190116	−0.0950580	−	0.995472i	$-0.530304\pi$
$857$	−3.50309e9	−0.190116	−0.0950580	+	0.995472i	$0.530304\pi$
$858$	0	0
$859$	4.51369e9	0.242972	0.121486	−	0.992593i	$-0.461234\pi$
$859$	4.51369e9	0.242972	0.121486	+	0.992593i	$0.461234\pi$
$860$	−4.86172e8	−0.0260643
$861$	0	0
$862$	1.71325e10	0.911056
$863$	−1.14771e9	−0.0607848	−0.0303924	−	0.999538i	$-0.509676\pi$
$863$	−1.14771e9	−0.0607848	−0.0303924	+	0.999538i	$0.509676\pi$
$864$	0	0
$865$	6.52279e8	0.0342671
$866$	1.28461e10	0.672138
$867$	0	0
$868$	−5.10773e9	−0.265099
$869$	−5.49540e10	−2.84073
$870$	0	0
$871$	1.22415e9	0.0627726
$872$	6.50289e8	0.0332123
$873$	0	0
$874$	2.56395e9	0.129903
$875$	−2.12180e9	−0.107072
$876$	0	0
$877$	−2.81739e9	−0.141042	−0.0705209	−	0.997510i	$-0.522466\pi$
$877$	−2.81739e9	−0.141042	−0.0705209	+	0.997510i	$0.522466\pi$
$878$	1.08839e10	0.542692
$879$	0	0
$880$	−1.16261e9	−0.0575101
$881$	1.28725e10	0.634230	0.317115	−	0.948387i	$-0.397286\pi$
$881$	1.28725e10	0.634230	0.317115	+	0.948387i	$0.397286\pi$
$882$	0	0
$883$	2.46306e10	1.20396	0.601980	−	0.798511i	$-0.294379\pi$
$883$	2.46306e10	1.20396	0.601980	+	0.798511i	$0.294379\pi$
$884$	1.72936e9	0.0841981
$885$	0	0
$886$	1.34483e10	0.649604
$887$	3.54413e10	1.70520	0.852602	−	0.522561i	$-0.175023\pi$
$887$	3.54413e10	1.70520	0.852602	+	0.522561i	$0.175023\pi$
$888$	0	0
$889$	9.95250e9	0.475090
$890$	−7.90250e8	−0.0375750
$891$	0	0
$892$	1.85242e10	0.873900
$893$	1.89461e9	0.0890306
$894$	0	0
$895$	1.43354e9	0.0668387
$896$	−7.19323e8	−0.0334077
$897$	0	0
$898$	−6.28935e9	−0.289827
$899$	2.34913e10	1.07832
$900$	0	0
$901$	4.34338e10	1.97830
$902$	4.53441e10	2.05730
$903$	0	0
$904$	−1.34489e10	−0.605476
$905$	2.24617e9	0.100733
$906$	0	0
$907$	−4.41130e10	−1.96310	−0.981548	−	0.191216i	$-0.938757\pi$
$907$	−4.41130e10	−1.96310	−0.981548	+	0.191216i	$0.938757\pi$
$908$	3.36636e9	0.149231
$909$	0	0
$910$	−7.67222e7	−0.00337502
$911$	9.36825e9	0.410529	0.205265	−	0.978707i	$-0.434194\pi$
$911$	9.36825e9	0.410529	0.205265	+	0.978707i	$0.434194\pi$
$912$	0	0
$913$	5.57477e10	2.42426
$914$	2.48136e10	1.07492
$915$	0	0
$916$	3.84448e6	0.000165274 0
$917$	−1.52362e10	−0.652505
$918$	0	0
$919$	−9.46004e8	−0.0402058	−0.0201029	−	0.999798i	$-0.506399\pi$
$919$	−9.46004e8	−0.0402058	−0.0201029	+	0.999798i	$0.506399\pi$
$920$	1.90032e9	0.0804580
$921$	0	0
$922$	5.39419e9	0.226656
$923$	3.79177e9	0.158722
$924$	0	0
$925$	4.84296e9	0.201194
$926$	−1.14036e10	−0.471958
$927$	0	0
$928$	3.30829e9	0.135889
$929$	4.28644e9	0.175405	0.0877025	−	0.996147i	$-0.472048\pi$
$929$	4.28644e9	0.175405	0.0877025	+	0.996147i	$0.472048\pi$
$930$	0	0
$931$	−4.06360e8	−0.0165039
$932$	4.06907e9	0.164641
$933$	0	0
$934$	2.86954e10	1.15239
$935$	1.09724e10	0.438997
$936$	0	0
$937$	1.95718e10	0.777218	0.388609	−	0.921403i	$-0.372956\pi$
$937$	1.95718e10	0.777218	0.388609	+	0.921403i	$0.372956\pi$
$938$	4.80553e9	0.190121
$939$	0	0
$940$	1.40423e9	0.0551429
$941$	−1.74637e10	−0.683240	−0.341620	−	0.939838i	$-0.610976\pi$
$941$	−1.74637e10	−0.683240	−0.341620	+	0.939838i	$0.610976\pi$
$942$	0	0
$943$	−7.41163e10	−2.87821
$944$	−3.88589e9	−0.150345
$945$	0	0
$946$	1.07809e10	0.414033
$947$	−9.34036e9	−0.357387	−0.178693	−	0.983905i	$-0.557187\pi$
$947$	−9.34036e9	−0.357387	−0.178693	+	0.983905i	$0.557187\pi$
$948$	0	0
$949$	−9.26730e8	−0.0351983
$950$	2.11454e9	0.0800171
$951$	0	0
$952$	6.78879e9	0.255013
$953$	2.15672e10	0.807176	0.403588	−	0.914941i	$-0.367763\pi$
$953$	2.15672e10	0.807176	0.403588	+	0.914941i	$0.367763\pi$
$954$	0	0
$955$	−5.46527e9	−0.203048
$956$	1.75061e10	0.648019
$957$	0	0
$958$	1.60201e10	0.588688
$959$	7.41587e9	0.271517
$960$	0	0
$961$	2.66260e10	0.967773
$962$	3.53895e8	0.0128163
$963$	0	0
$964$	−1.11548e10	−0.401045
$965$	−7.94565e7	−0.00284632
$966$	0	0
$967$	4.26880e10	1.51814	0.759072	−	0.651007i	$-0.225653\pi$
$967$	4.26880e10	1.51814	0.759072	+	0.651007i	$0.225653\pi$
$968$	1.58034e10	0.559999
$969$	0	0
$970$	−3.57193e9	−0.125662
$971$	−2.54462e10	−0.891981	−0.445991	−	0.895038i	$-0.647149\pi$
$971$	−2.54462e10	−0.891981	−0.445991	+	0.895038i	$0.647149\pi$
$972$	0	0
$973$	1.60312e10	0.557920
$974$	−1.41358e10	−0.490189
$975$	0	0
$976$	−9.29196e9	−0.319914
$977$	−1.32964e10	−0.456145	−0.228072	−	0.973644i	$-0.573242\pi$
$977$	−1.32964e10	−0.456145	−0.228072	+	0.973644i	$0.573242\pi$
$978$	0	0
$979$	1.75238e10	0.596882
$980$	−3.01181e8	−0.0102220
$981$	0	0
$982$	−1.19680e10	−0.403304
$983$	−3.62175e10	−1.21613	−0.608067	−	0.793886i	$-0.708055\pi$
$983$	−3.62175e10	−1.21613	−0.608067	+	0.793886i	$0.708055\pi$
$984$	0	0
$985$	−5.06093e9	−0.168734
$986$	−3.12228e10	−1.03730
$987$	0	0
$988$	1.54518e8	0.00509718
$989$	−1.76217e10	−0.579241
$990$	0	0
$991$	−1.57195e10	−0.513074	−0.256537	−	0.966534i	$-0.582582\pi$
$991$	−1.57195e10	−0.513074	−0.256537	+	0.966534i	$0.582582\pi$
$992$	7.62436e9	0.247978
$993$	0	0
$994$	1.48850e10	0.480725
$995$	3.51688e9	0.113182
$996$	0	0
$997$	2.70651e10	0.864921	0.432461	−	0.901653i	$-0.357645\pi$
$997$	2.70651e10	0.864921	0.432461	+	0.901653i	$0.357645\pi$
$998$	−3.06921e10	−0.977393
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.8.a.c.1.1	yes	1
3.2	odd	2		378.8.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.8.a.b.1.1	✓	1	3.2	odd	2
378.8.a.c.1.1	yes	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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