LMFDB - Embedded newform 64.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,8,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, 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("64.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	64.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:	$19.9926416310$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-12.0000 q^{3} +210.000 q^{5} +1016.00 q^{7} -2043.00 q^{9} +O(q^{10})$

$q-12.0000 q^{3} +210.000 q^{5} +1016.00 q^{7} -2043.00 q^{9} -1092.00 q^{11} -1382.00 q^{13} -2520.00 q^{15} +14706.0 q^{17} +39940.0 q^{19} -12192.0 q^{21} +68712.0 q^{23} -34025.0 q^{25} +50760.0 q^{27} +102570. q^{29} +227552. q^{31} +13104.0 q^{33} +213360. q^{35} -160526. q^{37} +16584.0 q^{39} +10842.0 q^{41} +630748. q^{43} -429030. q^{45} +472656. q^{47} +208713. q^{49} -176472. q^{51} +1.49402e6 q^{53} -229320. q^{55} -479280. q^{57} -2.64066e6 q^{59} -827702. q^{61} -2.07569e6 q^{63} -290220. q^{65} +126004. q^{67} -824544. q^{69} -1.41473e6 q^{71} +980282. q^{73} +408300. q^{75} -1.10947e6 q^{77} -3.56680e6 q^{79} +3.85892e6 q^{81} -5.67289e6 q^{83} +3.08826e6 q^{85} -1.23084e6 q^{87} -1.19512e7 q^{89} -1.40411e6 q^{91} -2.73062e6 q^{93} +8.38740e6 q^{95} +8.68215e6 q^{97} +2.23096e6 q^{99} +O(q^{100})$

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
$3$	−12.0000	−0.256600	−0.128300	−	0.991735i	$-0.540952\pi$
$3$	−12.0000	−0.256600	−0.128300	+	0.991735i	$0.540952\pi$
$4$	0	0
$5$	210.000	0.751319	0.375659	−	0.926758i	$-0.377416\pi$
$5$	210.000	0.751319	0.375659	+	0.926758i	$0.377416\pi$
$6$	0	0
$7$	1016.00	1.11957	0.559784	−	0.828638i	$-0.310884\pi$
$7$	1016.00	1.11957	0.559784	+	0.828638i	$0.310884\pi$
$8$	0	0
$9$	−2043.00	−0.934156
$10$	0	0
$11$	−1092.00	−0.247371	−0.123685	−	0.992321i	$-0.539471\pi$
$11$	−1092.00	−0.247371	−0.123685	+	0.992321i	$0.539471\pi$
$12$	0	0
$13$	−1382.00	−0.174464	−0.0872321	−	0.996188i	$-0.527802\pi$
$13$	−1382.00	−0.174464	−0.0872321	+	0.996188i	$0.527802\pi$
$14$	0	0
$15$	−2520.00	−0.192789
$16$	0	0
$17$	14706.0	0.725978	0.362989	−	0.931793i	$-0.381756\pi$
$17$	14706.0	0.725978	0.362989	+	0.931793i	$0.381756\pi$
$18$	0	0
$19$	39940.0	1.33589	0.667945	−	0.744211i	$-0.267174\pi$
$19$	39940.0	1.33589	0.667945	+	0.744211i	$0.267174\pi$
$20$	0	0
$21$	−12192.0	−0.287281
$22$	0	0
$23$	68712.0	1.17757	0.588783	−	0.808291i	$-0.299607\pi$
$23$	68712.0	1.17757	0.588783	+	0.808291i	$0.299607\pi$
$24$	0	0
$25$	−34025.0	−0.435520
$26$	0	0
$27$	50760.0	0.496305
$28$	0	0
$29$	102570.	0.780957	0.390479	−	0.920612i	$-0.372310\pi$
$29$	102570.	0.780957	0.390479	+	0.920612i	$0.372310\pi$
$30$	0	0
$31$	227552.	1.37188	0.685938	−	0.727660i	$-0.259392\pi$
$31$	227552.	1.37188	0.685938	+	0.727660i	$0.259392\pi$
$32$	0	0
$33$	13104.0	0.0634753
$34$	0	0
$35$	213360.	0.841153
$36$	0	0
$37$	−160526.	−0.521002	−0.260501	−	0.965474i	$-0.583888\pi$
$37$	−160526.	−0.521002	−0.260501	+	0.965474i	$0.583888\pi$
$38$	0	0
$39$	16584.0	0.0447675
$40$	0	0
$41$	10842.0	0.0245678	0.0122839	−	0.999925i	$-0.496090\pi$
$41$	10842.0	0.0245678	0.0122839	+	0.999925i	$0.496090\pi$
$42$	0	0
$43$	630748.	1.20981	0.604904	−	0.796299i	$-0.293212\pi$
$43$	630748.	1.20981	0.604904	+	0.796299i	$0.293212\pi$
$44$	0	0
$45$	−429030.	−0.701849
$46$	0	0
$47$	472656.	0.664053	0.332026	−	0.943270i	$-0.392268\pi$
$47$	472656.	0.664053	0.332026	+	0.943270i	$0.392268\pi$
$48$	0	0
$49$	208713.	0.253433
$50$	0	0
$51$	−176472.	−0.186286
$52$	0	0
$53$	1.49402e6	1.37845	0.689224	−	0.724548i	$-0.257952\pi$
$53$	1.49402e6	1.37845	0.689224	+	0.724548i	$0.257952\pi$
$54$	0	0
$55$	−229320.	−0.185854
$56$	0	0
$57$	−479280.	−0.342789
$58$	0	0
$59$	−2.64066e6	−1.67390	−0.836952	−	0.547277i	$-0.815665\pi$
$59$	−2.64066e6	−1.67390	−0.836952	+	0.547277i	$0.815665\pi$
$60$	0	0
$61$	−827702.	−0.466895	−0.233448	−	0.972369i	$-0.575001\pi$
$61$	−827702.	−0.466895	−0.233448	+	0.972369i	$0.575001\pi$
$62$	0	0
$63$	−2.07569e6	−1.04585
$64$	0	0
$65$	−290220.	−0.131078
$66$	0	0
$67$	126004.	0.0511826	0.0255913	−	0.999672i	$-0.491853\pi$
$67$	126004.	0.0511826	0.0255913	+	0.999672i	$0.491853\pi$
$68$	0	0
$69$	−824544.	−0.302164
$70$	0	0
$71$	−1.41473e6	−0.469104	−0.234552	−	0.972104i	$-0.575362\pi$
$71$	−1.41473e6	−0.469104	−0.234552	+	0.972104i	$0.575362\pi$
$72$	0	0
$73$	980282.	0.294931	0.147466	−	0.989067i	$-0.452888\pi$
$73$	980282.	0.294931	0.147466	+	0.989067i	$0.452888\pi$
$74$	0	0
$75$	408300.	0.111754
$76$	0	0
$77$	−1.10947e6	−0.276948
$78$	0	0
$79$	−3.56680e6	−0.813924	−0.406962	−	0.913445i	$-0.633412\pi$
$79$	−3.56680e6	−0.813924	−0.406962	+	0.913445i	$0.633412\pi$
$80$	0	0
$81$	3.85892e6	0.806805
$82$	0	0
$83$	−5.67289e6	−1.08901	−0.544504	−	0.838758i	$-0.683282\pi$
$83$	−5.67289e6	−1.08901	−0.544504	+	0.838758i	$0.683282\pi$
$84$	0	0
$85$	3.08826e6	0.545441
$86$	0	0
$87$	−1.23084e6	−0.200394
$88$	0	0
$89$	−1.19512e7	−1.79699	−0.898496	−	0.438982i	$-0.855339\pi$
$89$	−1.19512e7	−1.79699	−0.898496	+	0.438982i	$0.855339\pi$
$90$	0	0
$91$	−1.40411e6	−0.195325
$92$	0	0
$93$	−2.73062e6	−0.352023
$94$	0	0
$95$	8.38740e6	1.00368
$96$	0	0
$97$	8.68215e6	0.965886	0.482943	−	0.875652i	$-0.339568\pi$
$97$	8.68215e6	0.965886	0.482943	+	0.875652i	$0.339568\pi$
$98$	0	0
$99$	2.23096e6	0.231083
$100$	0	0
$101$	1.00795e7	0.973455	0.486727	−	0.873554i	$-0.338190\pi$
$101$	1.00795e7	0.973455	0.486727	+	0.873554i	$0.338190\pi$
$102$	0	0
$103$	3.74799e6	0.337962	0.168981	−	0.985619i	$-0.445952\pi$
$103$	3.74799e6	0.337962	0.168981	+	0.985619i	$0.445952\pi$
$104$	0	0
$105$	−2.56032e6	−0.215840
$106$	0	0
$107$	1.79856e7	1.41932	0.709661	−	0.704543i	$-0.248848\pi$
$107$	1.79856e7	1.41932	0.709661	+	0.704543i	$0.248848\pi$
$108$	0	0
$109$	−1.22570e7	−0.906552	−0.453276	−	0.891370i	$-0.649745\pi$
$109$	−1.22570e7	−0.906552	−0.453276	+	0.891370i	$0.649745\pi$
$110$	0	0
$111$	1.92631e6	0.133689
$112$	0	0
$113$	1.65950e7	1.08194	0.540968	−	0.841043i	$-0.318058\pi$
$113$	1.65950e7	1.08194	0.540968	+	0.841043i	$0.318058\pi$
$114$	0	0
$115$	1.44295e7	0.884727
$116$	0	0
$117$	2.82343e6	0.162977
$118$	0	0
$119$	1.49413e7	0.812782
$120$	0	0
$121$	−1.82947e7	−0.938808
$122$	0	0
$123$	−130104.	−0.00630410
$124$	0	0
$125$	−2.35515e7	−1.07853
$126$	0	0
$127$	1.16826e6	0.0506087	0.0253043	−	0.999680i	$-0.491945\pi$
$127$	1.16826e6	0.0506087	0.0253043	+	0.999680i	$0.491945\pi$
$128$	0	0
$129$	−7.56898e6	−0.310437
$130$	0	0
$131$	7.92383e6	0.307954	0.153977	−	0.988074i	$-0.450792\pi$
$131$	7.92383e6	0.307954	0.153977	+	0.988074i	$0.450792\pi$
$132$	0	0
$133$	4.05790e7	1.49562
$134$	0	0
$135$	1.06596e7	0.372883
$136$	0	0
$137$	−315654.	−0.0104879	−0.00524396	−	0.999986i	$-0.501669\pi$
$137$	−315654.	−0.0104879	−0.00524396	+	0.999986i	$0.501669\pi$
$138$	0	0
$139$	−3.92038e7	−1.23816	−0.619079	−	0.785329i	$-0.712494\pi$
$139$	−3.92038e7	−1.23816	−0.619079	+	0.785329i	$0.712494\pi$
$140$	0	0
$141$	−5.67187e6	−0.170396
$142$	0	0
$143$	1.50914e6	0.0431573
$144$	0	0
$145$	2.15397e7	0.586748
$146$	0	0
$147$	−2.50456e6	−0.0650309
$148$	0	0
$149$	2.18860e7	0.542020	0.271010	−	0.962577i	$-0.412642\pi$
$149$	2.18860e7	0.542020	0.271010	+	0.962577i	$0.412642\pi$
$150$	0	0
$151$	−2.94154e7	−0.695274	−0.347637	−	0.937629i	$-0.613016\pi$
$151$	−2.94154e7	−0.695274	−0.347637	+	0.937629i	$0.613016\pi$
$152$	0	0
$153$	−3.00444e7	−0.678177
$154$	0	0
$155$	4.77859e7	1.03072
$156$	0	0
$157$	−6.05550e7	−1.24882	−0.624412	−	0.781095i	$-0.714661\pi$
$157$	−6.05550e7	−1.24882	−0.624412	+	0.781095i	$0.714661\pi$
$158$	0	0
$159$	−1.79282e7	−0.353710
$160$	0	0
$161$	6.98114e7	1.31837
$162$	0	0
$163$	−5.70853e7	−1.03245	−0.516223	−	0.856454i	$-0.672663\pi$
$163$	−5.70853e7	−1.03245	−0.516223	+	0.856454i	$0.672663\pi$
$164$	0	0
$165$	2.75184e6	0.0476902
$166$	0	0
$167$	−8.77265e7	−1.45755	−0.728775	−	0.684754i	$-0.759910\pi$
$167$	−8.77265e7	−1.45755	−0.728775	+	0.684754i	$0.759910\pi$
$168$	0	0
$169$	−6.08386e7	−0.969562
$170$	0	0
$171$	−8.15974e7	−1.24793
$172$	0	0
$173$	−8.56954e6	−0.125833	−0.0629167	−	0.998019i	$-0.520040\pi$
$173$	−8.56954e6	−0.125833	−0.0629167	+	0.998019i	$0.520040\pi$
$174$	0	0
$175$	−3.45694e7	−0.487594
$176$	0	0
$177$	3.16879e7	0.429524
$178$	0	0
$179$	−1.88041e7	−0.245056	−0.122528	−	0.992465i	$-0.539100\pi$
$179$	−1.88041e7	−0.245056	−0.122528	+	0.992465i	$0.539100\pi$
$180$	0	0
$181$	5.99625e7	0.751631	0.375816	−	0.926694i	$-0.377363\pi$
$181$	5.99625e7	0.751631	0.375816	+	0.926694i	$0.377363\pi$
$182$	0	0
$183$	9.93242e6	0.119805
$184$	0	0
$185$	−3.37105e7	−0.391439
$186$	0	0
$187$	−1.60590e7	−0.179586
$188$	0	0
$189$	5.15722e7	0.555647
$190$	0	0
$191$	9.39861e7	0.975993	0.487997	−	0.872845i	$-0.337728\pi$
$191$	9.39861e7	0.975993	0.487997	+	0.872845i	$0.337728\pi$
$192$	0	0
$193$	−3.51946e7	−0.352391	−0.176196	−	0.984355i	$-0.556379\pi$
$193$	−3.51946e7	−0.352391	−0.176196	+	0.984355i	$0.556379\pi$
$194$	0	0
$195$	3.48264e6	0.0336347
$196$	0	0
$197$	−1.02985e8	−0.959718	−0.479859	−	0.877346i	$-0.659312\pi$
$197$	−1.02985e8	−0.959718	−0.479859	+	0.877346i	$0.659312\pi$
$198$	0	0
$199$	8.36376e7	0.752342	0.376171	−	0.926550i	$-0.377240\pi$
$199$	8.36376e7	0.752342	0.376171	+	0.926550i	$0.377240\pi$
$200$	0	0
$201$	−1.51205e6	−0.0131335
$202$	0	0
$203$	1.04211e8	0.874335
$204$	0	0
$205$	2.27682e6	0.0184582
$206$	0	0
$207$	−1.40379e8	−1.10003
$208$	0	0
$209$	−4.36145e7	−0.330460
$210$	0	0
$211$	9.74010e7	0.713797	0.356899	−	0.934143i	$-0.383834\pi$
$211$	9.74010e7	0.713797	0.356899	+	0.934143i	$0.383834\pi$
$212$	0	0
$213$	1.69767e7	0.120372
$214$	0	0
$215$	1.32457e8	0.908951
$216$	0	0
$217$	2.31193e8	1.53591
$218$	0	0
$219$	−1.17634e7	−0.0756794
$220$	0	0
$221$	−2.03237e7	−0.126657
$222$	0	0
$223$	−1.46457e7	−0.0884390	−0.0442195	−	0.999022i	$-0.514080\pi$
$223$	−1.46457e7	−0.0884390	−0.0442195	+	0.999022i	$0.514080\pi$
$224$	0	0
$225$	6.95131e7	0.406844
$226$	0	0
$227$	1.84541e8	1.04713	0.523567	−	0.851985i	$-0.324601\pi$
$227$	1.84541e8	1.04713	0.523567	+	0.851985i	$0.324601\pi$
$228$	0	0
$229$	8.75461e6	0.0481740	0.0240870	−	0.999710i	$-0.492332\pi$
$229$	8.75461e6	0.0481740	0.0240870	+	0.999710i	$0.492332\pi$
$230$	0	0
$231$	1.33137e7	0.0710650
$232$	0	0
$233$	−1.19556e8	−0.619193	−0.309597	−	0.950868i	$-0.600194\pi$
$233$	−1.19556e8	−0.619193	−0.309597	+	0.950868i	$0.600194\pi$
$234$	0	0
$235$	9.92578e7	0.498915
$236$	0	0
$237$	4.28016e7	0.208853
$238$	0	0
$239$	3.96209e8	1.87729	0.938646	−	0.344883i	$-0.112081\pi$
$239$	3.96209e8	1.87729	0.938646	+	0.344883i	$0.112081\pi$
$240$	0	0
$241$	−2.56606e8	−1.18089	−0.590443	−	0.807080i	$-0.701047\pi$
$241$	−2.56606e8	−1.18089	−0.590443	+	0.807080i	$0.701047\pi$
$242$	0	0
$243$	−1.57319e8	−0.703331
$244$	0	0
$245$	4.38297e7	0.190409
$246$	0	0
$247$	−5.51971e7	−0.233065
$248$	0	0
$249$	6.80747e7	0.279440
$250$	0	0
$251$	7.34775e7	0.293290	0.146645	−	0.989189i	$-0.453153\pi$
$251$	7.34775e7	0.293290	0.146645	+	0.989189i	$0.453153\pi$
$252$	0	0
$253$	−7.50335e7	−0.291295
$254$	0	0
$255$	−3.70591e7	−0.139960
$256$	0	0
$257$	−2.02701e8	−0.744886	−0.372443	−	0.928055i	$-0.621480\pi$
$257$	−2.02701e8	−0.744886	−0.372443	+	0.928055i	$0.621480\pi$
$258$	0	0
$259$	−1.63094e8	−0.583297
$260$	0	0
$261$	−2.09551e8	−0.729536
$262$	0	0
$263$	1.54254e8	0.522867	0.261434	−	0.965221i	$-0.415805\pi$
$263$	1.54254e8	0.522867	0.261434	+	0.965221i	$0.415805\pi$
$264$	0	0
$265$	3.13744e8	1.03565
$266$	0	0
$267$	1.43414e8	0.461108
$268$	0	0
$269$	6.24018e8	1.95463	0.977315	−	0.211793i	$-0.0679302\pi$
$269$	6.24018e8	1.95463	0.977315	+	0.211793i	$0.0679302\pi$
$270$	0	0
$271$	−3.87983e8	−1.18419	−0.592094	−	0.805869i	$-0.701698\pi$
$271$	−3.87983e8	−1.18419	−0.592094	+	0.805869i	$0.701698\pi$
$272$	0	0
$273$	1.68493e7	0.0501203
$274$	0	0
$275$	3.71553e7	0.107735
$276$	0	0
$277$	−4.53952e8	−1.28331	−0.641654	−	0.766994i	$-0.721752\pi$
$277$	−4.53952e8	−1.28331	−0.641654	+	0.766994i	$0.721752\pi$
$278$	0	0
$279$	−4.64889e8	−1.28155
$280$	0	0
$281$	3.33770e8	0.897377	0.448689	−	0.893688i	$-0.351891\pi$
$281$	3.33770e8	0.897377	0.448689	+	0.893688i	$0.351891\pi$
$282$	0	0
$283$	−5.37695e8	−1.41021	−0.705104	−	0.709104i	$-0.749100\pi$
$283$	−5.37695e8	−1.41021	−0.705104	+	0.709104i	$0.749100\pi$
$284$	0	0
$285$	−1.00649e8	−0.257544
$286$	0	0
$287$	1.10155e7	0.0275053
$288$	0	0
$289$	−1.94072e8	−0.472956
$290$	0	0
$291$	−1.04186e8	−0.247847
$292$	0	0
$293$	−3.35600e8	−0.779445	−0.389722	−	0.920932i	$-0.627429\pi$
$293$	−3.35600e8	−0.779445	−0.389722	+	0.920932i	$0.627429\pi$
$294$	0	0
$295$	−5.54539e8	−1.25764
$296$	0	0
$297$	−5.54299e7	−0.122771
$298$	0	0
$299$	−9.49600e7	−0.205443
$300$	0	0
$301$	6.40840e8	1.35446
$302$	0	0
$303$	−1.20954e8	−0.249789
$304$	0	0
$305$	−1.73817e8	−0.350787
$306$	0	0
$307$	−2.15029e8	−0.424143	−0.212072	−	0.977254i	$-0.568021\pi$
$307$	−2.15029e8	−0.424143	−0.212072	+	0.977254i	$0.568021\pi$
$308$	0	0
$309$	−4.49759e7	−0.0867212
$310$	0	0
$311$	7.92062e8	1.49313	0.746565	−	0.665313i	$-0.231702\pi$
$311$	7.92062e8	1.49313	0.746565	+	0.665313i	$0.231702\pi$
$312$	0	0
$313$	−1.18457e8	−0.218352	−0.109176	−	0.994022i	$-0.534821\pi$
$313$	−1.18457e8	−0.218352	−0.109176	+	0.994022i	$0.534821\pi$
$314$	0	0
$315$	−4.35894e8	−0.785768
$316$	0	0
$317$	5.07310e7	0.0894470	0.0447235	−	0.998999i	$-0.485759\pi$
$317$	5.07310e7	0.0894470	0.0447235	+	0.998999i	$0.485759\pi$
$318$	0	0
$319$	−1.12006e8	−0.193186
$320$	0	0
$321$	−2.15827e8	−0.364198
$322$	0	0
$323$	5.87358e8	0.969826
$324$	0	0
$325$	4.70226e7	0.0759826
$326$	0	0
$327$	1.47084e8	0.232621
$328$	0	0
$329$	4.80218e8	0.743453
$330$	0	0
$331$	−2.73757e8	−0.414923	−0.207461	−	0.978243i	$-0.566520\pi$
$331$	−2.73757e8	−0.414923	−0.207461	+	0.978243i	$0.566520\pi$
$332$	0	0
$333$	3.27955e8	0.486697
$334$	0	0
$335$	2.64608e7	0.0384545
$336$	0	0
$337$	−9.18512e7	−0.130732	−0.0653658	−	0.997861i	$-0.520821\pi$
$337$	−9.18512e7	−0.130732	−0.0653658	+	0.997861i	$0.520821\pi$
$338$	0	0
$339$	−1.99140e8	−0.277625
$340$	0	0
$341$	−2.48487e8	−0.339362
$342$	0	0
$343$	−6.24667e8	−0.835833
$344$	0	0
$345$	−1.73154e8	−0.227021
$346$	0	0
$347$	1.36700e9	1.75637	0.878187	−	0.478318i	$-0.158753\pi$
$347$	1.36700e9	1.75637	0.878187	+	0.478318i	$0.158753\pi$
$348$	0	0
$349$	−1.13143e9	−1.42475	−0.712377	−	0.701797i	$-0.752381\pi$
$349$	−1.13143e9	−1.42475	−0.712377	+	0.701797i	$0.752381\pi$
$350$	0	0
$351$	−7.01503e7	−0.0865874
$352$	0	0
$353$	−4.48395e7	−0.0542562	−0.0271281	−	0.999632i	$-0.508636\pi$
$353$	−4.48395e7	−0.0542562	−0.0271281	+	0.999632i	$0.508636\pi$
$354$	0	0
$355$	−2.97093e8	−0.352446
$356$	0	0
$357$	−1.79296e8	−0.208560
$358$	0	0
$359$	3.98281e8	0.454317	0.227158	−	0.973858i	$-0.427057\pi$
$359$	3.98281e8	0.454317	0.227158	+	0.973858i	$0.427057\pi$
$360$	0	0
$361$	7.01332e8	0.784600
$362$	0	0
$363$	2.19536e8	0.240898
$364$	0	0
$365$	2.05859e8	0.221588
$366$	0	0
$367$	1.63472e9	1.72628	0.863140	−	0.504964i	$-0.168494\pi$
$367$	1.63472e9	1.72628	0.863140	+	0.504964i	$0.168494\pi$
$368$	0	0
$369$	−2.21502e7	−0.0229501
$370$	0	0
$371$	1.51792e9	1.54327
$372$	0	0
$373$	1.54633e9	1.54284	0.771421	−	0.636325i	$-0.219546\pi$
$373$	1.54633e9	1.54284	0.771421	+	0.636325i	$0.219546\pi$
$374$	0	0
$375$	2.82618e8	0.276752
$376$	0	0
$377$	−1.41752e8	−0.136249
$378$	0	0
$379$	1.05688e9	0.997216	0.498608	−	0.866828i	$-0.333845\pi$
$379$	1.05688e9	0.997216	0.498608	+	0.866828i	$0.333845\pi$
$380$	0	0
$381$	−1.40191e7	−0.0129862
$382$	0	0
$383$	2.24910e8	0.204556	0.102278	−	0.994756i	$-0.467387\pi$
$383$	2.24910e8	0.204556	0.102278	+	0.994756i	$0.467387\pi$
$384$	0	0
$385$	−2.32989e8	−0.208077
$386$	0	0
$387$	−1.28862e9	−1.13015
$388$	0	0
$389$	−1.01788e9	−0.876746	−0.438373	−	0.898793i	$-0.644445\pi$
$389$	−1.01788e9	−0.876746	−0.438373	+	0.898793i	$0.644445\pi$
$390$	0	0
$391$	1.01048e9	0.854887
$392$	0	0
$393$	−9.50859e7	−0.0790210
$394$	0	0
$395$	−7.49028e8	−0.611517
$396$	0	0
$397$	1.47565e9	1.18363	0.591817	−	0.806072i	$-0.298411\pi$
$397$	1.47565e9	1.18363	0.591817	+	0.806072i	$0.298411\pi$
$398$	0	0
$399$	−4.86948e8	−0.383776
$400$	0	0
$401$	2.74912e8	0.212906	0.106453	−	0.994318i	$-0.466051\pi$
$401$	2.74912e8	0.212906	0.106453	+	0.994318i	$0.466051\pi$
$402$	0	0
$403$	−3.14477e8	−0.239343
$404$	0	0
$405$	8.10373e8	0.606167
$406$	0	0
$407$	1.75294e8	0.128881
$408$	0	0
$409$	−1.63427e9	−1.18112	−0.590558	−	0.806995i	$-0.701092\pi$
$409$	−1.63427e9	−1.18112	−0.590558	+	0.806995i	$0.701092\pi$
$410$	0	0
$411$	3.78785e6	0.00269120
$412$	0	0
$413$	−2.68291e9	−1.87405
$414$	0	0
$415$	−1.19131e9	−0.818192
$416$	0	0
$417$	4.70445e8	0.317712
$418$	0	0
$419$	1.11280e9	0.739039	0.369519	−	0.929223i	$-0.379522\pi$
$419$	1.11280e9	0.739039	0.369519	+	0.929223i	$0.379522\pi$
$420$	0	0
$421$	−9.22528e8	−0.602549	−0.301274	−	0.953537i	$-0.597412\pi$
$421$	−9.22528e8	−0.602549	−0.301274	+	0.953537i	$0.597412\pi$
$422$	0	0
$423$	−9.65636e8	−0.620329
$424$	0	0
$425$	−5.00372e8	−0.316178
$426$	0	0
$427$	−8.40945e8	−0.522721
$428$	0	0
$429$	−1.81097e7	−0.0110742
$430$	0	0
$431$	−9.81508e8	−0.590505	−0.295252	−	0.955419i	$-0.595404\pi$
$431$	−9.81508e8	−0.590505	−0.295252	+	0.955419i	$0.595404\pi$
$432$	0	0
$433$	2.84998e9	1.68707	0.843537	−	0.537071i	$-0.180469\pi$
$433$	2.84998e9	1.68707	0.843537	+	0.537071i	$0.180469\pi$
$434$	0	0
$435$	−2.58476e8	−0.150560
$436$	0	0
$437$	2.74436e9	1.57310
$438$	0	0
$439$	−1.05622e9	−0.595838	−0.297919	−	0.954591i	$-0.596292\pi$
$439$	−1.05622e9	−0.595838	−0.297919	+	0.954591i	$0.596292\pi$
$440$	0	0
$441$	−4.26401e8	−0.236746
$442$	0	0
$443$	−1.82325e9	−0.996401	−0.498201	−	0.867062i	$-0.666006\pi$
$443$	−1.82325e9	−0.996401	−0.498201	+	0.867062i	$0.666006\pi$
$444$	0	0
$445$	−2.50975e9	−1.35011
$446$	0	0
$447$	−2.62633e8	−0.139082
$448$	0	0
$449$	1.84846e9	0.963713	0.481856	−	0.876250i	$-0.339963\pi$
$449$	1.84846e9	0.963713	0.481856	+	0.876250i	$0.339963\pi$
$450$	0	0
$451$	−1.18395e7	−0.00607735
$452$	0	0
$453$	3.52985e8	0.178407
$454$	0	0
$455$	−2.94864e8	−0.146751
$456$	0	0
$457$	−2.98066e9	−1.46085	−0.730425	−	0.682993i	$-0.760678\pi$
$457$	−2.98066e9	−1.46085	−0.730425	+	0.682993i	$0.760678\pi$
$458$	0	0
$459$	7.46477e8	0.360306
$460$	0	0
$461$	2.52781e9	1.20169	0.600843	−	0.799367i	$-0.294832\pi$
$461$	2.52781e9	1.20169	0.600843	+	0.799367i	$0.294832\pi$
$462$	0	0
$463$	−8.90291e8	−0.416868	−0.208434	−	0.978036i	$-0.566837\pi$
$463$	−8.90291e8	−0.416868	−0.208434	+	0.978036i	$0.566837\pi$
$464$	0	0
$465$	−5.73431e8	−0.264482
$466$	0	0
$467$	−2.65667e9	−1.20706	−0.603529	−	0.797341i	$-0.706239\pi$
$467$	−2.65667e9	−1.20706	−0.603529	+	0.797341i	$0.706239\pi$
$468$	0	0
$469$	1.28020e8	0.0573024
$470$	0	0
$471$	7.26660e8	0.320448
$472$	0	0
$473$	−6.88777e8	−0.299271
$474$	0	0
$475$	−1.35896e9	−0.581806
$476$	0	0
$477$	−3.05228e9	−1.28769
$478$	0	0
$479$	1.30093e9	0.540855	0.270428	−	0.962740i	$-0.412835\pi$
$479$	1.30093e9	0.540855	0.270428	+	0.962740i	$0.412835\pi$
$480$	0	0
$481$	2.21847e8	0.0908962
$482$	0	0
$483$	−8.37737e8	−0.338293
$484$	0	0
$485$	1.82325e9	0.725689
$486$	0	0
$487$	−1.07447e9	−0.421542	−0.210771	−	0.977535i	$-0.567598\pi$
$487$	−1.07447e9	−0.421542	−0.210771	+	0.977535i	$0.567598\pi$
$488$	0	0
$489$	6.85024e8	0.264926
$490$	0	0
$491$	7.83344e8	0.298653	0.149327	−	0.988788i	$-0.452289\pi$
$491$	7.83344e8	0.298653	0.149327	+	0.988788i	$0.452289\pi$
$492$	0	0
$493$	1.50839e9	0.566958
$494$	0	0
$495$	4.68501e8	0.173617
$496$	0	0
$497$	−1.43736e9	−0.525193
$498$	0	0
$499$	6.23188e8	0.224526	0.112263	−	0.993679i	$-0.464190\pi$
$499$	6.23188e8	0.224526	0.112263	+	0.993679i	$0.464190\pi$
$500$	0	0
$501$	1.05272e9	0.374007
$502$	0	0
$503$	−2.70927e9	−0.949215	−0.474607	−	0.880198i	$-0.657410\pi$
$503$	−2.70927e9	−0.949215	−0.474607	+	0.880198i	$0.657410\pi$
$504$	0	0
$505$	2.11670e9	0.731375
$506$	0	0
$507$	7.30063e8	0.248790
$508$	0	0
$509$	−3.49943e9	−1.17621	−0.588106	−	0.808784i	$-0.700126\pi$
$509$	−3.49943e9	−1.17621	−0.588106	+	0.808784i	$0.700126\pi$
$510$	0	0
$511$	9.95967e8	0.330196
$512$	0	0
$513$	2.02735e9	0.663008
$514$	0	0
$515$	7.87078e8	0.253918
$516$	0	0
$517$	−5.16140e8	−0.164267
$518$	0	0
$519$	1.02835e8	0.0322889
$520$	0	0
$521$	−1.37683e9	−0.426530	−0.213265	−	0.976994i	$-0.568410\pi$
$521$	−1.37683e9	−0.426530	−0.213265	+	0.976994i	$0.568410\pi$
$522$	0	0
$523$	2.86154e9	0.874669	0.437334	−	0.899299i	$-0.355923\pi$
$523$	2.86154e9	0.874669	0.437334	+	0.899299i	$0.355923\pi$
$524$	0	0
$525$	4.14833e8	0.125117
$526$	0	0
$527$	3.34638e9	0.995951
$528$	0	0
$529$	1.31651e9	0.386661
$530$	0	0
$531$	5.39487e9	1.56369
$532$	0	0
$533$	−1.49836e7	−0.00428620
$534$	0	0
$535$	3.77697e9	1.06636
$536$	0	0
$537$	2.25649e8	0.0628815
$538$	0	0
$539$	−2.27915e8	−0.0626919
$540$	0	0
$541$	−5.34467e9	−1.45121	−0.725605	−	0.688111i	$-0.758440\pi$
$541$	−5.34467e9	−1.45121	−0.725605	+	0.688111i	$0.758440\pi$
$542$	0	0
$543$	−7.19550e8	−0.192869
$544$	0	0
$545$	−2.57398e9	−0.681109
$546$	0	0
$547$	3.37135e9	0.880740	0.440370	−	0.897816i	$-0.354847\pi$
$547$	3.37135e9	0.880740	0.440370	+	0.897816i	$0.354847\pi$
$548$	0	0
$549$	1.69100e9	0.436153
$550$	0	0
$551$	4.09665e9	1.04327
$552$	0	0
$553$	−3.62387e9	−0.911244
$554$	0	0
$555$	4.04526e8	0.100443
$556$	0	0
$557$	5.61106e9	1.37579	0.687894	−	0.725811i	$-0.258535\pi$
$557$	5.61106e9	1.37579	0.687894	+	0.725811i	$0.258535\pi$
$558$	0	0
$559$	−8.71694e8	−0.211068
$560$	0	0
$561$	1.92707e8	0.0460817
$562$	0	0
$563$	−6.69690e9	−1.58159	−0.790795	−	0.612081i	$-0.790333\pi$
$563$	−6.69690e9	−1.58159	−0.790795	+	0.612081i	$0.790333\pi$
$564$	0	0
$565$	3.48494e9	0.812879
$566$	0	0
$567$	3.92066e9	0.903273
$568$	0	0
$569$	1.96850e9	0.447964	0.223982	−	0.974593i	$-0.428094\pi$
$569$	1.96850e9	0.447964	0.223982	+	0.974593i	$0.428094\pi$
$570$	0	0
$571$	−1.02926e9	−0.231365	−0.115682	−	0.993286i	$-0.536906\pi$
$571$	−1.02926e9	−0.231365	−0.115682	+	0.993286i	$0.536906\pi$
$572$	0	0
$573$	−1.12783e9	−0.250440
$574$	0	0
$575$	−2.33793e9	−0.512853
$576$	0	0
$577$	3.31179e9	0.717708	0.358854	−	0.933394i	$-0.383168\pi$
$577$	3.31179e9	0.717708	0.358854	+	0.933394i	$0.383168\pi$
$578$	0	0
$579$	4.22335e8	0.0904236
$580$	0	0
$581$	−5.76366e9	−1.21922
$582$	0	0
$583$	−1.63147e9	−0.340988
$584$	0	0
$585$	5.92919e8	0.122448
$586$	0	0
$587$	5.59411e8	0.114156	0.0570778	−	0.998370i	$-0.481822\pi$
$587$	5.59411e8	0.114156	0.0570778	+	0.998370i	$0.481822\pi$
$588$	0	0
$589$	9.08843e9	1.83267
$590$	0	0
$591$	1.23582e9	0.246264
$592$	0	0
$593$	−3.02459e9	−0.595628	−0.297814	−	0.954624i	$-0.596258\pi$
$593$	−3.02459e9	−0.595628	−0.297814	+	0.954624i	$0.596258\pi$
$594$	0	0
$595$	3.13767e9	0.610658
$596$	0	0
$597$	−1.00365e9	−0.193051
$598$	0	0
$599$	−5.63246e9	−1.07079	−0.535395	−	0.844602i	$-0.679837\pi$
$599$	−5.63246e9	−1.07079	−0.535395	+	0.844602i	$0.679837\pi$
$600$	0	0
$601$	3.40792e8	0.0640366	0.0320183	−	0.999487i	$-0.489807\pi$
$601$	3.40792e8	0.0640366	0.0320183	+	0.999487i	$0.489807\pi$
$602$	0	0
$603$	−2.57426e8	−0.0478126
$604$	0	0
$605$	−3.84189e9	−0.705344
$606$	0	0
$607$	3.85420e9	0.699477	0.349739	−	0.936847i	$-0.386270\pi$
$607$	3.85420e9	0.699477	0.349739	+	0.936847i	$0.386270\pi$
$608$	0	0
$609$	−1.25053e9	−0.224355
$610$	0	0
$611$	−6.53211e8	−0.115853
$612$	0	0
$613$	−9.22245e9	−1.61709	−0.808545	−	0.588434i	$-0.799745\pi$
$613$	−9.22245e9	−1.61709	−0.808545	+	0.588434i	$0.799745\pi$
$614$	0	0
$615$	−2.73218e7	−0.00473639
$616$	0	0
$617$	6.53611e9	1.12027	0.560133	−	0.828402i	$-0.310750\pi$
$617$	6.53611e9	1.12027	0.560133	+	0.828402i	$0.310750\pi$
$618$	0	0
$619$	−1.36559e9	−0.231420	−0.115710	−	0.993283i	$-0.536914\pi$
$619$	−1.36559e9	−0.231420	−0.115710	+	0.993283i	$0.536914\pi$
$620$	0	0
$621$	3.48782e9	0.584431
$622$	0	0
$623$	−1.21424e10	−2.01186
$624$	0	0
$625$	−2.28761e9	−0.374802
$626$	0	0
$627$	5.23374e8	0.0847960
$628$	0	0
$629$	−2.36070e9	−0.378236
$630$	0	0
$631$	1.54079e9	0.244141	0.122070	−	0.992521i	$-0.461047\pi$
$631$	1.54079e9	0.244141	0.122070	+	0.992521i	$0.461047\pi$
$632$	0	0
$633$	−1.16881e9	−0.183160
$634$	0	0
$635$	2.45334e8	0.0380233
$636$	0	0
$637$	−2.88441e8	−0.0442150
$638$	0	0
$639$	2.89029e9	0.438216
$640$	0	0
$641$	−4.54018e9	−0.680879	−0.340440	−	0.940266i	$-0.610576\pi$
$641$	−4.54018e9	−0.680879	−0.340440	+	0.940266i	$0.610576\pi$
$642$	0	0
$643$	−1.14054e10	−1.69189	−0.845944	−	0.533272i	$-0.820962\pi$
$643$	−1.14054e10	−1.69189	−0.845944	+	0.533272i	$0.820962\pi$
$644$	0	0
$645$	−1.58948e9	−0.233237
$646$	0	0
$647$	−1.26393e10	−1.83468	−0.917338	−	0.398109i	$-0.869666\pi$
$647$	−1.26393e10	−1.83468	−0.917338	+	0.398109i	$0.869666\pi$
$648$	0	0
$649$	2.88360e9	0.414075
$650$	0	0
$651$	−2.77431e9	−0.394114
$652$	0	0
$653$	1.05004e10	1.47575	0.737873	−	0.674940i	$-0.235830\pi$
$653$	1.05004e10	1.47575	0.737873	+	0.674940i	$0.235830\pi$
$654$	0	0
$655$	1.66400e9	0.231371
$656$	0	0
$657$	−2.00272e9	−0.275512
$658$	0	0
$659$	−9.64818e9	−1.31325	−0.656624	−	0.754219i	$-0.728016\pi$
$659$	−9.64818e9	−1.31325	−0.656624	+	0.754219i	$0.728016\pi$
$660$	0	0
$661$	6.58299e9	0.886580	0.443290	−	0.896378i	$-0.353811\pi$
$661$	6.58299e9	0.886580	0.443290	+	0.896378i	$0.353811\pi$
$662$	0	0
$663$	2.43884e8	0.0325002
$664$	0	0
$665$	8.52160e9	1.12369
$666$	0	0
$667$	7.04779e9	0.919629
$668$	0	0
$669$	1.75749e8	0.0226935
$670$	0	0
$671$	9.03851e8	0.115496
$672$	0	0
$673$	−8.54649e9	−1.08077	−0.540387	−	0.841416i	$-0.681722\pi$
$673$	−8.54649e9	−1.08077	−0.540387	+	0.841416i	$0.681722\pi$
$674$	0	0
$675$	−1.72711e9	−0.216151
$676$	0	0
$677$	−8.71305e9	−1.07922	−0.539610	−	0.841915i	$-0.681428\pi$
$677$	−8.71305e9	−1.07922	−0.539610	+	0.841915i	$0.681428\pi$
$678$	0	0
$679$	8.82106e9	1.08138
$680$	0	0
$681$	−2.21449e9	−0.268695
$682$	0	0
$683$	−1.46109e10	−1.75470	−0.877351	−	0.479849i	$-0.840692\pi$
$683$	−1.46109e10	−1.75470	−0.877351	+	0.479849i	$0.840692\pi$
$684$	0	0
$685$	−6.62873e7	−0.00787977
$686$	0	0
$687$	−1.05055e8	−0.0123615
$688$	0	0
$689$	−2.06473e9	−0.240490
$690$	0	0
$691$	1.47348e10	1.69891	0.849454	−	0.527662i	$-0.176931\pi$
$691$	1.47348e10	1.69891	0.849454	+	0.527662i	$0.176931\pi$
$692$	0	0
$693$	2.26665e9	0.258713
$694$	0	0
$695$	−8.23279e9	−0.930252
$696$	0	0
$697$	1.59442e8	0.0178357
$698$	0	0
$699$	1.43467e9	0.158885
$700$	0	0
$701$	−1.31502e9	−0.144185	−0.0720923	−	0.997398i	$-0.522968\pi$
$701$	−1.31502e9	−0.144185	−0.0720923	+	0.997398i	$0.522968\pi$
$702$	0	0
$703$	−6.41141e9	−0.696001
$704$	0	0
$705$	−1.19109e9	−0.128022
$706$	0	0
$707$	1.02408e10	1.08985
$708$	0	0
$709$	−6.64028e8	−0.0699721	−0.0349860	−	0.999388i	$-0.511139\pi$
$709$	−6.64028e8	−0.0699721	−0.0349860	+	0.999388i	$0.511139\pi$
$710$	0	0
$711$	7.28697e9	0.760332
$712$	0	0
$713$	1.56356e10	1.61547
$714$	0	0
$715$	3.16920e8	0.0324249
$716$	0	0
$717$	−4.75451e9	−0.481713
$718$	0	0
$719$	4.95034e9	0.496689	0.248344	−	0.968672i	$-0.420114\pi$
$719$	4.95034e9	0.496689	0.248344	+	0.968672i	$0.420114\pi$
$720$	0	0
$721$	3.80796e9	0.378372
$722$	0	0
$723$	3.07928e9	0.303015
$724$	0	0
$725$	−3.48994e9	−0.340123
$726$	0	0
$727$	8.81101e9	0.850463	0.425231	−	0.905085i	$-0.360193\pi$
$727$	8.81101e9	0.850463	0.425231	+	0.905085i	$0.360193\pi$
$728$	0	0
$729$	−6.55163e9	−0.626330
$730$	0	0
$731$	9.27578e9	0.878293
$732$	0	0
$733$	1.49414e8	0.0140129	0.00700643	−	0.999975i	$-0.497770\pi$
$733$	1.49414e8	0.0140129	0.00700643	+	0.999975i	$0.497770\pi$
$734$	0	0
$735$	−5.25957e8	−0.0488590
$736$	0	0
$737$	−1.37596e8	−0.0126611
$738$	0	0
$739$	4.70806e9	0.429127	0.214564	−	0.976710i	$-0.431167\pi$
$739$	4.70806e9	0.429127	0.214564	+	0.976710i	$0.431167\pi$
$740$	0	0
$741$	6.62365e8	0.0598045
$742$	0	0
$743$	1.69676e9	0.151761	0.0758805	−	0.997117i	$-0.475823\pi$
$743$	1.69676e9	0.151761	0.0758805	+	0.997117i	$0.475823\pi$
$744$	0	0
$745$	4.59607e9	0.407230
$746$	0	0
$747$	1.15897e10	1.01730
$748$	0	0
$749$	1.82733e10	1.58903
$750$	0	0
$751$	1.06650e10	0.918800	0.459400	−	0.888229i	$-0.348064\pi$
$751$	1.06650e10	0.918800	0.459400	+	0.888229i	$0.348064\pi$
$752$	0	0
$753$	−8.81731e8	−0.0752581
$754$	0	0
$755$	−6.17724e9	−0.522373
$756$	0	0
$757$	−6.22876e9	−0.521874	−0.260937	−	0.965356i	$-0.584032\pi$
$757$	−6.22876e9	−0.521874	−0.260937	+	0.965356i	$0.584032\pi$
$758$	0	0
$759$	9.00402e8	0.0747464
$760$	0	0
$761$	−8.38334e9	−0.689558	−0.344779	−	0.938684i	$-0.612046\pi$
$761$	−8.38334e9	−0.689558	−0.344779	+	0.938684i	$0.612046\pi$
$762$	0	0
$763$	−1.24531e10	−1.01495
$764$	0	0
$765$	−6.30932e9	−0.509527
$766$	0	0
$767$	3.64939e9	0.292036
$768$	0	0
$769$	−1.18649e10	−0.940852	−0.470426	−	0.882439i	$-0.655900\pi$
$769$	−1.18649e10	−0.940852	−0.470426	+	0.882439i	$0.655900\pi$
$770$	0	0
$771$	2.43241e9	0.191138
$772$	0	0
$773$	−5.56680e9	−0.433488	−0.216744	−	0.976228i	$-0.569544\pi$
$773$	−5.56680e9	−0.433488	−0.216744	+	0.976228i	$0.569544\pi$
$774$	0	0
$775$	−7.74246e9	−0.597479
$776$	0	0
$777$	1.95713e9	0.149674
$778$	0	0
$779$	4.33029e8	0.0328198
$780$	0	0
$781$	1.54488e9	0.116042
$782$	0	0
$783$	5.20645e9	0.387593
$784$	0	0
$785$	−1.27165e10	−0.938264
$786$	0	0
$787$	−1.34611e8	−0.00984395	−0.00492198	−	0.999988i	$-0.501567\pi$
$787$	−1.34611e8	−0.00984395	−0.00492198	+	0.999988i	$0.501567\pi$
$788$	0	0
$789$	−1.85105e9	−0.134168
$790$	0	0
$791$	1.68605e10	1.21130
$792$	0	0
$793$	1.14388e9	0.0814565
$794$	0	0
$795$	−3.76493e9	−0.265749
$796$	0	0
$797$	7.41548e9	0.518842	0.259421	−	0.965764i	$-0.416468\pi$
$797$	7.41548e9	0.518842	0.259421	+	0.965764i	$0.416468\pi$
$798$	0	0
$799$	6.95088e9	0.482088
$800$	0	0
$801$	2.44163e10	1.67867
$802$	0	0
$803$	−1.07047e9	−0.0729574
$804$	0	0
$805$	1.46604e10	0.990513
$806$	0	0
$807$	−7.48822e9	−0.501558
$808$	0	0
$809$	−1.41542e10	−0.939863	−0.469932	−	0.882703i	$-0.655721\pi$
$809$	−1.41542e10	−0.939863	−0.469932	+	0.882703i	$0.655721\pi$
$810$	0	0
$811$	2.63708e10	1.73600	0.868001	−	0.496563i	$-0.165405\pi$
$811$	2.63708e10	1.73600	0.868001	+	0.496563i	$0.165405\pi$
$812$	0	0
$813$	4.65580e9	0.303863
$814$	0	0
$815$	−1.19879e10	−0.775697
$816$	0	0
$817$	2.51921e10	1.61617
$818$	0	0
$819$	2.86860e9	0.182464
$820$	0	0
$821$	−8.06264e9	−0.508483	−0.254241	−	0.967141i	$-0.581826\pi$
$821$	−8.06264e9	−0.508483	−0.254241	+	0.967141i	$0.581826\pi$
$822$	0	0
$823$	−2.34202e10	−1.46451	−0.732253	−	0.681033i	$-0.761531\pi$
$823$	−2.34202e10	−1.46451	−0.732253	+	0.681033i	$0.761531\pi$
$824$	0	0
$825$	−4.45864e8	−0.0276448
$826$	0	0
$827$	−5.55722e9	−0.341655	−0.170828	−	0.985301i	$-0.554644\pi$
$827$	−5.55722e9	−0.341655	−0.170828	+	0.985301i	$0.554644\pi$
$828$	0	0
$829$	−2.84256e10	−1.73288	−0.866440	−	0.499281i	$-0.833597\pi$
$829$	−2.84256e10	−1.73288	−0.866440	+	0.499281i	$0.833597\pi$
$830$	0	0
$831$	5.44743e9	0.329297
$832$	0	0
$833$	3.06933e9	0.183987
$834$	0	0
$835$	−1.84226e10	−1.09508
$836$	0	0
$837$	1.15505e10	0.680868
$838$	0	0
$839$	1.04036e10	0.608156	0.304078	−	0.952647i	$-0.401652\pi$
$839$	1.04036e10	0.608156	0.304078	+	0.952647i	$0.401652\pi$
$840$	0	0
$841$	−6.72927e9	−0.390105
$842$	0	0
$843$	−4.00524e9	−0.230267
$844$	0	0
$845$	−1.27761e10	−0.728450
$846$	0	0
$847$	−1.85874e10	−1.05106
$848$	0	0
$849$	6.45234e9	0.361860
$850$	0	0
$851$	−1.10301e10	−0.613514
$852$	0	0
$853$	1.80580e10	0.996205	0.498102	−	0.867118i	$-0.334030\pi$
$853$	1.80580e10	0.996205	0.498102	+	0.867118i	$0.334030\pi$
$854$	0	0
$855$	−1.71355e10	−0.937593
$856$	0	0
$857$	−6.34034e9	−0.344096	−0.172048	−	0.985089i	$-0.555038\pi$
$857$	−6.34034e9	−0.344096	−0.172048	+	0.985089i	$0.555038\pi$
$858$	0	0
$859$	−1.21489e10	−0.653973	−0.326987	−	0.945029i	$-0.606033\pi$
$859$	−1.21489e10	−0.653973	−0.326987	+	0.945029i	$0.606033\pi$
$860$	0	0
$861$	−1.32186e8	−0.00705786
$862$	0	0
$863$	−2.87111e10	−1.52059	−0.760295	−	0.649578i	$-0.774946\pi$
$863$	−2.87111e10	−1.52059	−0.760295	+	0.649578i	$0.774946\pi$
$864$	0	0
$865$	−1.79960e9	−0.0945411
$866$	0	0
$867$	2.32887e9	0.121361
$868$	0	0
$869$	3.89495e9	0.201341
$870$	0	0
$871$	−1.74138e8	−0.00892953
$872$	0	0
$873$	−1.77376e10	−0.902289
$874$	0	0
$875$	−2.39283e10	−1.20749
$876$	0	0
$877$	−2.46021e10	−1.23161	−0.615806	−	0.787898i	$-0.711169\pi$
$877$	−2.46021e10	−1.23161	−0.615806	+	0.787898i	$0.711169\pi$
$878$	0	0
$879$	4.02720e9	0.200006
$880$	0	0
$881$	−1.25378e10	−0.617738	−0.308869	−	0.951105i	$-0.599951\pi$
$881$	−1.25378e10	−0.617738	−0.308869	+	0.951105i	$0.599951\pi$
$882$	0	0
$883$	−1.93097e10	−0.943873	−0.471937	−	0.881633i	$-0.656445\pi$
$883$	−1.93097e10	−0.943873	−0.471937	+	0.881633i	$0.656445\pi$
$884$	0	0
$885$	6.65446e9	0.322709
$886$	0	0
$887$	3.20268e10	1.54092	0.770462	−	0.637486i	$-0.220026\pi$
$887$	3.20268e10	1.54092	0.770462	+	0.637486i	$0.220026\pi$
$888$	0	0
$889$	1.18695e9	0.0566599
$890$	0	0
$891$	−4.21394e9	−0.199580
$892$	0	0
$893$	1.88779e10	0.887101
$894$	0	0
$895$	−3.94885e9	−0.184115
$896$	0	0
$897$	1.13952e9	0.0527167
$898$	0	0
$899$	2.33400e10	1.07138
$900$	0	0
$901$	2.19710e10	1.00072
$902$	0	0
$903$	−7.69008e9	−0.347555
$904$	0	0
$905$	1.25921e10	0.564715
$906$	0	0
$907$	−2.33703e9	−0.104002	−0.0520008	−	0.998647i	$-0.516560\pi$
$907$	−2.33703e9	−0.104002	−0.0520008	+	0.998647i	$0.516560\pi$
$908$	0	0
$909$	−2.05925e10	−0.909359
$910$	0	0
$911$	2.20343e10	0.965573	0.482786	−	0.875738i	$-0.339625\pi$
$911$	2.20343e10	0.965573	0.482786	+	0.875738i	$0.339625\pi$
$912$	0	0
$913$	6.19480e9	0.269389
$914$	0	0
$915$	2.08581e9	0.0900121
$916$	0	0
$917$	8.05061e9	0.344775
$918$	0	0
$919$	−1.43277e10	−0.608938	−0.304469	−	0.952522i	$-0.598479\pi$
$919$	−1.43277e10	−0.608938	−0.304469	+	0.952522i	$0.598479\pi$
$920$	0	0
$921$	2.58035e9	0.108835
$922$	0	0
$923$	1.95515e9	0.0818418
$924$	0	0
$925$	5.46190e9	0.226907
$926$	0	0
$927$	−7.65715e9	−0.315710
$928$	0	0
$929$	1.31280e10	0.537208	0.268604	−	0.963251i	$-0.413438\pi$
$929$	1.31280e10	0.537208	0.268604	+	0.963251i	$0.413438\pi$
$930$	0	0
$931$	8.33600e9	0.338558
$932$	0	0
$933$	−9.50474e9	−0.383137
$934$	0	0
$935$	−3.37238e9	−0.134926
$936$	0	0
$937$	−3.87626e10	−1.53930	−0.769652	−	0.638463i	$-0.779571\pi$
$937$	−3.87626e10	−1.53930	−0.769652	+	0.638463i	$0.779571\pi$
$938$	0	0
$939$	1.42149e9	0.0560291
$940$	0	0
$941$	−2.06279e10	−0.807035	−0.403517	−	0.914972i	$-0.632212\pi$
$941$	−2.06279e10	−0.807035	−0.403517	+	0.914972i	$0.632212\pi$
$942$	0	0
$943$	7.44976e8	0.0289302
$944$	0	0
$945$	1.08302e10	0.417468
$946$	0	0
$947$	2.11705e10	0.810040	0.405020	−	0.914308i	$-0.367264\pi$
$947$	2.11705e10	0.810040	0.405020	+	0.914308i	$0.367264\pi$
$948$	0	0
$949$	−1.35475e9	−0.0514550
$950$	0	0
$951$	−6.08771e8	−0.0229521
$952$	0	0
$953$	2.14876e10	0.804196	0.402098	−	0.915597i	$-0.368281\pi$
$953$	2.14876e10	0.804196	0.402098	+	0.915597i	$0.368281\pi$
$954$	0	0
$955$	1.97371e10	0.733282
$956$	0	0
$957$	1.34408e9	0.0495715
$958$	0	0
$959$	−3.20704e8	−0.0117419
$960$	0	0
$961$	2.42673e10	0.882043
$962$	0	0
$963$	−3.67445e10	−1.32587
$964$	0	0
$965$	−7.39086e9	−0.264758
$966$	0	0
$967$	3.92625e10	1.39632	0.698161	−	0.715941i	$-0.254002\pi$
$967$	3.92625e10	1.39632	0.698161	+	0.715941i	$0.254002\pi$
$968$	0	0
$969$	−7.04829e9	−0.248857
$970$	0	0
$971$	5.62647e10	1.97228	0.986140	−	0.165917i	$-0.0530585\pi$
$971$	5.62647e10	1.97228	0.986140	+	0.165917i	$0.0530585\pi$
$972$	0	0
$973$	−3.98310e10	−1.38620
$974$	0	0
$975$	−5.64271e8	−0.0194972
$976$	0	0
$977$	−8.43437e9	−0.289349	−0.144674	−	0.989479i	$-0.546213\pi$
$977$	−8.43437e9	−0.289349	−0.144674	+	0.989479i	$0.546213\pi$
$978$	0	0
$979$	1.30507e10	0.444523
$980$	0	0
$981$	2.50411e10	0.846861
$982$	0	0
$983$	−2.24230e10	−0.752932	−0.376466	−	0.926430i	$-0.622861\pi$
$983$	−2.24230e10	−0.752932	−0.376466	+	0.926430i	$0.622861\pi$
$984$	0	0
$985$	−2.16269e10	−0.721054
$986$	0	0
$987$	−5.76262e9	−0.190770
$988$	0	0
$989$	4.33400e10	1.42463
$990$	0	0
$991$	3.46728e10	1.13170	0.565849	−	0.824509i	$-0.308548\pi$
$991$	3.46728e10	1.13170	0.565849	+	0.824509i	$0.308548\pi$
$992$	0	0
$993$	3.28508e9	0.106469
$994$	0	0
$995$	1.75639e10	0.565249
$996$	0	0
$997$	2.96474e10	0.947444	0.473722	−	0.880674i	$-0.342910\pi$
$997$	2.96474e10	0.947444	0.473722	+	0.880674i	$0.342910\pi$
$998$	0	0
$999$	−8.14830e9	−0.258576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.8.a.c.1.1		1
3.2	odd	2		576.8.a.g.1.1		1
4.3	odd	2		64.8.a.e.1.1		1
8.3	odd	2		16.8.a.b.1.1		1
8.5	even	2		2.8.a.a.1.1	✓	1
12.11	even	2		576.8.a.f.1.1		1
16.3	odd	4		256.8.b.f.129.2		2
16.5	even	4		256.8.b.b.129.2		2
16.11	odd	4		256.8.b.f.129.1		2
16.13	even	4		256.8.b.b.129.1		2
24.5	odd	2		18.8.a.b.1.1		1
24.11	even	2		144.8.a.i.1.1		1
40.3	even	4		400.8.c.j.49.1		2
40.13	odd	4		50.8.b.c.49.2		2
40.19	odd	2		400.8.a.l.1.1		1
40.27	even	4		400.8.c.j.49.2		2
40.29	even	2		50.8.a.g.1.1		1
40.37	odd	4		50.8.b.c.49.1		2
56.5	odd	6		98.8.c.e.67.1		2
56.13	odd	2		98.8.a.a.1.1		1
56.37	even	6		98.8.c.d.67.1		2
56.45	odd	6		98.8.c.e.79.1		2
56.53	even	6		98.8.c.d.79.1		2
72.5	odd	6		162.8.c.a.55.1		2
72.13	even	6		162.8.c.l.55.1		2
72.29	odd	6		162.8.c.a.109.1		2
72.61	even	6		162.8.c.l.109.1		2
88.21	odd	2		242.8.a.e.1.1		1
104.5	odd	4		338.8.b.d.337.2		2
104.21	odd	4		338.8.b.d.337.1		2
104.77	even	2		338.8.a.d.1.1		1
120.29	odd	2		450.8.a.c.1.1		1
120.53	even	4		450.8.c.g.199.1		2
120.77	even	4		450.8.c.g.199.2		2
136.101	even	2		578.8.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.8.a.a.1.1	✓	1	8.5	even	2
16.8.a.b.1.1		1	8.3	odd	2
18.8.a.b.1.1		1	24.5	odd	2
50.8.a.g.1.1		1	40.29	even	2
50.8.b.c.49.1		2	40.37	odd	4
50.8.b.c.49.2		2	40.13	odd	4
64.8.a.c.1.1		1	1.1	even	1	trivial
64.8.a.e.1.1		1	4.3	odd	2
98.8.a.a.1.1		1	56.13	odd	2
98.8.c.d.67.1		2	56.37	even	6
98.8.c.d.79.1		2	56.53	even	6
98.8.c.e.67.1		2	56.5	odd	6
98.8.c.e.79.1		2	56.45	odd	6
144.8.a.i.1.1		1	24.11	even	2
162.8.c.a.55.1		2	72.5	odd	6
162.8.c.a.109.1		2	72.29	odd	6
162.8.c.l.55.1		2	72.13	even	6
162.8.c.l.109.1		2	72.61	even	6
242.8.a.e.1.1		1	88.21	odd	2
256.8.b.b.129.1		2	16.13	even	4
256.8.b.b.129.2		2	16.5	even	4
256.8.b.f.129.1		2	16.11	odd	4
256.8.b.f.129.2		2	16.3	odd	4
338.8.a.d.1.1		1	104.77	even	2
338.8.b.d.337.1		2	104.21	odd	4
338.8.b.d.337.2		2	104.5	odd	4
400.8.a.l.1.1		1	40.19	odd	2
400.8.c.j.49.1		2	40.3	even	4
400.8.c.j.49.2		2	40.27	even	4
450.8.a.c.1.1		1	120.29	odd	2
450.8.c.g.199.1		2	120.53	even	4
450.8.c.g.199.2		2	120.77	even	4
576.8.a.f.1.1		1	12.11	even	2
576.8.a.g.1.1		1	3.2	odd	2
578.8.a.b.1.1		1	136.101	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:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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