LMFDB - Embedded newform 256.8.b.f.129.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,8,Mod(129,256)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256.129"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,2032] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.8.b.f.129.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-12.0000i q^{3} +210.000i q^{5} +1016.00 q^{7} +2043.00 q^{9} +1092.00i q^{11} +1382.00i q^{13} +2520.00 q^{15} +14706.0 q^{17} +39940.0i q^{19} -12192.0i q^{21} +68712.0 q^{23} +34025.0 q^{25} -50760.0i q^{27} -102570. i q^{29} -227552. q^{31} +13104.0 q^{33} +213360. i q^{35} -160526. i q^{37} +16584.0 q^{39} -10842.0 q^{41} -630748. i q^{43} +429030. i q^{45} -472656. q^{47} +208713. q^{49} -176472. i q^{51} +1.49402e6i q^{53} -229320. q^{55} +479280. q^{57} +2.64066e6i q^{59} +827702. i q^{61} +2.07569e6 q^{63} -290220. q^{65} +126004. i q^{67} -824544. i q^{69} -1.41473e6 q^{71} -980282. q^{73} -408300. i q^{75} +1.10947e6i q^{77} +3.56680e6 q^{79} +3.85892e6 q^{81} -5.67289e6i q^{83} +3.08826e6i q^{85} -1.23084e6 q^{87} +1.19512e7 q^{89} +1.40411e6i q^{91} +2.73062e6i q^{93} -8.38740e6 q^{95} +8.68215e6 q^{97} +2.23096e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2032 q^{7} + 4086 q^{9} + 5040 q^{15} + 29412 q^{17} + 137424 q^{23} + 68050 q^{25} - 455104 q^{31} + 26208 q^{33} + 33168 q^{39} - 21684 q^{41} - 945312 q^{47} + 417426 q^{49} - 458640 q^{55} + 958560 q^{57}+ \cdots + 17364292 q^{97}+O(q^{100}) $ 2 * q + 2032 * q^7 + 4086 * q^9 + 5040 * q^15 + 29412 * q^17 + 137424 * q^23 + 68050 * q^25 - 455104 * q^31 + 26208 * q^33 + 33168 * q^39 - 21684 * q^41 - 945312 * q^47 + 417426 * q^49 - 458640 * q^55 + 958560 * q^57 + 4151376 * q^63 - 580440 * q^65 - 2829456 * q^71 - 1960564 * q^73 + 7133600 * q^79 + 7717842 * q^81 - 2461680 * q^87 + 23902380 * q^89 - 16774800 * q^95 + 17364292 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

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
$3$	− 12.0000i	− 0.256600i	−0.991735	−	0.128300i	$-0.959048\pi$
$3$	− 12.0000i	− 0.256600i	0.991735	−	0.128300i	$-0.0409521\pi$
$4$	0	0
$5$	210.000i	0.751319i	0.926758	+	0.375659i	$0.122584\pi$
$5$	210.000i	0.751319i	−0.926758	+	0.375659i	$0.877416\pi$
$6$	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
$11$	1092.00i	0.247371i	0.992321	+	0.123685i	$0.0394713\pi$
$11$	1092.00i	0.247371i	−0.992321	+	0.123685i	$0.960529\pi$
$12$	0	0
$13$	1382.00i	0.174464i	0.996188	+	0.0872321i	$0.0278022\pi$
$13$	1382.00i	0.174464i	−0.996188	+	0.0872321i	$0.972198\pi$
$14$	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
$19$	39940.0i	1.33589i	0.744211	+	0.667945i	$0.232826\pi$
$19$	39940.0i	1.33589i	−0.744211	+	0.667945i	$0.767174\pi$
$20$	0	0
$21$	− 12192.0i	− 0.287281i
$22$	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
$27$	− 50760.0i	− 0.496305i
$28$	0	0
$29$	− 102570.i	− 0.780957i	−0.920612	−	0.390479i	$-0.872310\pi$
$29$	− 102570.i	− 0.780957i	0.920612	−	0.390479i	$-0.127690\pi$
$30$	0	0
$31$	−227552.	−1.37188	−0.685938	−	0.727660i	$-0.740608\pi$
$31$	−227552.	−1.37188	−0.685938	+	0.727660i	$0.740608\pi$
$32$	0	0
$33$	13104.0	0.0634753
$34$	0	0
$35$	213360.i	0.841153i
$36$	0	0
$37$	− 160526.i	− 0.521002i	−0.965474	−	0.260501i	$-0.916112\pi$
$37$	− 160526.i	− 0.521002i	0.965474	−	0.260501i	$-0.0838877\pi$
$38$	0	0
$39$	16584.0	0.0447675
$40$	0	0
$41$	−10842.0	−0.0245678	−0.0122839	−	0.999925i	$-0.503910\pi$
$41$	−10842.0	−0.0245678	−0.0122839	+	0.999925i	$0.503910\pi$
$42$	0	0
$43$	− 630748.i	− 1.20981i	−0.796299	−	0.604904i	$-0.793212\pi$
$43$	− 630748.i	− 1.20981i	0.796299	−	0.604904i	$-0.206788\pi$
$44$	0	0
$45$	429030.i	0.701849i
$46$	0	0
$47$	−472656.	−0.664053	−0.332026	−	0.943270i	$-0.607732\pi$
$47$	−472656.	−0.664053	−0.332026	+	0.943270i	$0.607732\pi$
$48$	0	0
$49$	208713.	0.253433
$50$	0	0
$51$	− 176472.i	− 0.186286i
$52$	0	0
$53$	1.49402e6i	1.37845i	0.724548	+	0.689224i	$0.242048\pi$
$53$	1.49402e6i	1.37845i	−0.724548	+	0.689224i	$0.757952\pi$
$54$	0	0
$55$	−229320.	−0.185854
$56$	0	0
$57$	479280.	0.342789
$58$	0	0
$59$	2.64066e6i	1.67390i	0.547277	+	0.836952i	$0.315665\pi$
$59$	2.64066e6i	1.67390i	−0.547277	+	0.836952i	$0.684335\pi$
$60$	0	0
$61$	827702.i	0.466895i	0.972369	+	0.233448i	$0.0750008\pi$
$61$	827702.i	0.466895i	−0.972369	+	0.233448i	$0.924999\pi$
$62$	0	0
$63$	2.07569e6	1.04585
$64$	0	0
$65$	−290220.	−0.131078
$66$	0	0
$67$	126004.i	0.0511826i	0.999672	+	0.0255913i	$0.00814686\pi$
$67$	126004.i	0.0511826i	−0.999672	+	0.0255913i	$0.991853\pi$
$68$	0	0
$69$	− 824544.i	− 0.302164i
$70$	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	$-0.547112\pi$
$73$	−980282.	−0.294931	−0.147466	+	0.989067i	$0.547112\pi$
$74$	0	0
$75$	− 408300.i	− 0.111754i
$76$	0	0
$77$	1.10947e6i	0.276948i
$78$	0	0
$79$	3.56680e6	0.813924	0.406962	−	0.913445i	$-0.366588\pi$
$79$	3.56680e6	0.813924	0.406962	+	0.913445i	$0.366588\pi$
$80$	0	0
$81$	3.85892e6	0.806805
$82$	0	0
$83$	− 5.67289e6i	− 1.08901i	−0.838758	−	0.544504i	$-0.816718\pi$
$83$	− 5.67289e6i	− 1.08901i	0.838758	−	0.544504i	$-0.183282\pi$
$84$	0	0
$85$	3.08826e6i	0.545441i
$86$	0	0
$87$	−1.23084e6	−0.200394
$88$	0	0
$89$	1.19512e7	1.79699	0.898496	−	0.438982i	$-0.144661\pi$
$89$	1.19512e7	1.79699	0.898496	+	0.438982i	$0.144661\pi$
$90$	0	0
$91$	1.40411e6i	0.195325i
$92$	0	0
$93$	2.73062e6i	0.352023i
$94$	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
$99$	2.23096e6i	0.231083i
$100$	0	0
$101$	1.00795e7i	0.973455i	0.873554	+	0.486727i	$0.161810\pi$
$101$	1.00795e7i	0.973455i	−0.873554	+	0.486727i	$0.838190\pi$
$102$	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
$107$	− 1.79856e7i	− 1.41932i	−0.704543	−	0.709661i	$-0.748848\pi$
$107$	− 1.79856e7i	− 1.41932i	0.704543	−	0.709661i	$-0.251152\pi$
$108$	0	0
$109$	1.22570e7i	0.906552i	0.891370	+	0.453276i	$0.149745\pi$
$109$	1.22570e7i	0.906552i	−0.891370	+	0.453276i	$0.850255\pi$
$110$	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
$115$	1.44295e7i	0.884727i
$116$	0	0
$117$	2.82343e6i	0.162977i
$118$	0	0
$119$	1.49413e7	0.812782
$120$	0	0
$121$	1.82947e7	0.938808
$122$	0	0
$123$	130104.i	0.00630410i
$124$	0	0
$125$	2.35515e7i	1.07853i
$126$	0	0
$127$	−1.16826e6	−0.0506087	−0.0253043	−	0.999680i	$-0.508055\pi$
$127$	−1.16826e6	−0.0506087	−0.0253043	+	0.999680i	$0.508055\pi$
$128$	0	0
$129$	−7.56898e6	−0.310437
$130$	0	0
$131$	7.92383e6i	0.307954i	0.988074	+	0.153977i	$0.0492081\pi$
$131$	7.92383e6i	0.307954i	−0.988074	+	0.153977i	$0.950792\pi$
$132$	0	0
$133$	4.05790e7i	1.49562i
$134$	0	0
$135$	1.06596e7	0.372883
$136$	0	0
$137$	315654.	0.0104879	0.00524396	−	0.999986i	$-0.498331\pi$
$137$	315654.	0.0104879	0.00524396	+	0.999986i	$0.498331\pi$
$138$	0	0
$139$	3.92038e7i	1.23816i	0.785329	+	0.619079i	$0.212494\pi$
$139$	3.92038e7i	1.23816i	−0.785329	+	0.619079i	$0.787506\pi$
$140$	0	0
$141$	5.67187e6i	0.170396i
$142$	0	0
$143$	−1.50914e6	−0.0431573
$144$	0	0
$145$	2.15397e7	0.586748
$146$	0	0
$147$	− 2.50456e6i	− 0.0650309i
$148$	0	0
$149$	2.18860e7i	0.542020i	0.962577	+	0.271010i	$0.0873577\pi$
$149$	2.18860e7i	0.542020i	−0.962577	+	0.271010i	$0.912642\pi$
$150$	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
$155$	− 4.77859e7i	− 1.03072i
$156$	0	0
$157$	6.05550e7i	1.24882i	0.781095	+	0.624412i	$0.214661\pi$
$157$	6.05550e7i	1.24882i	−0.781095	+	0.624412i	$0.785339\pi$
$158$	0	0
$159$	1.79282e7	0.353710
$160$	0	0
$161$	6.98114e7	1.31837
$162$	0	0
$163$	− 5.70853e7i	− 1.03245i	−0.856454	−	0.516223i	$-0.827337\pi$
$163$	− 5.70853e7i	− 1.03245i	0.856454	−	0.516223i	$-0.172663\pi$
$164$	0	0
$165$	2.75184e6i	0.0476902i
$166$	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
$171$	8.15974e7i	1.24793i
$172$	0	0
$173$	8.56954e6i	0.125833i	0.998019	+	0.0629167i	$0.0200403\pi$
$173$	8.56954e6i	0.125833i	−0.998019	+	0.0629167i	$0.979960\pi$
$174$	0	0
$175$	3.45694e7	0.487594
$176$	0	0
$177$	3.16879e7	0.429524
$178$	0	0
$179$	− 1.88041e7i	− 0.245056i	−0.992465	−	0.122528i	$-0.960900\pi$
$179$	− 1.88041e7i	− 0.245056i	0.992465	−	0.122528i	$-0.0391002\pi$
$180$	0	0
$181$	5.99625e7i	0.751631i	0.926694	+	0.375816i	$0.122637\pi$
$181$	5.99625e7i	0.751631i	−0.926694	+	0.375816i	$0.877363\pi$
$182$	0	0
$183$	9.93242e6	0.119805
$184$	0	0
$185$	3.37105e7	0.391439
$186$	0	0
$187$	1.60590e7i	0.179586i
$188$	0	0
$189$	− 5.15722e7i	− 0.555647i
$190$	0	0
$191$	−9.39861e7	−0.975993	−0.487997	−	0.872845i	$-0.662272\pi$
$191$	−9.39861e7	−0.975993	−0.487997	+	0.872845i	$0.662272\pi$
$192$	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
$195$	3.48264e6i	0.0336347i
$196$	0	0
$197$	− 1.02985e8i	− 0.959718i	−0.877346	−	0.479859i	$-0.840688\pi$
$197$	− 1.02985e8i	− 0.959718i	0.877346	−	0.479859i	$-0.159312\pi$
$198$	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
$203$	− 1.04211e8i	− 0.874335i
$204$	0	0
$205$	− 2.27682e6i	− 0.0184582i
$206$	0	0
$207$	1.40379e8	1.10003
$208$	0	0
$209$	−4.36145e7	−0.330460
$210$	0	0
$211$	9.74010e7i	0.713797i	0.934143	+	0.356899i	$0.116166\pi$
$211$	9.74010e7i	0.713797i	−0.934143	+	0.356899i	$0.883834\pi$
$212$	0	0
$213$	1.69767e7i	0.120372i
$214$	0	0
$215$	1.32457e8	0.908951
$216$	0	0
$217$	−2.31193e8	−1.53591
$218$	0	0
$219$	1.17634e7i	0.0756794i
$220$	0	0
$221$	2.03237e7i	0.126657i
$222$	0	0
$223$	1.46457e7	0.0884390	0.0442195	−	0.999022i	$-0.485920\pi$
$223$	1.46457e7	0.0884390	0.0442195	+	0.999022i	$0.485920\pi$
$224$	0	0
$225$	6.95131e7	0.406844
$226$	0	0
$227$	1.84541e8i	1.04713i	0.851985	+	0.523567i	$0.175399\pi$
$227$	1.84541e8i	1.04713i	−0.851985	+	0.523567i	$0.824601\pi$
$228$	0	0
$229$	8.75461e6i	0.0481740i	0.999710	+	0.0240870i	$0.00766787\pi$
$229$	8.75461e6i	0.0481740i	−0.999710	+	0.0240870i	$0.992332\pi$
$230$	0	0
$231$	1.33137e7	0.0710650
$232$	0	0
$233$	1.19556e8	0.619193	0.309597	−	0.950868i	$-0.399806\pi$
$233$	1.19556e8	0.619193	0.309597	+	0.950868i	$0.399806\pi$
$234$	0	0
$235$	− 9.92578e7i	− 0.498915i
$236$	0	0
$237$	− 4.28016e7i	− 0.208853i
$238$	0	0
$239$	−3.96209e8	−1.87729	−0.938646	−	0.344883i	$-0.887919\pi$
$239$	−3.96209e8	−1.87729	−0.938646	+	0.344883i	$0.887919\pi$
$240$	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
$243$	− 1.57319e8i	− 0.703331i
$244$	0	0
$245$	4.38297e7i	0.190409i
$246$	0	0
$247$	−5.51971e7	−0.233065
$248$	0	0
$249$	−6.80747e7	−0.279440
$250$	0	0
$251$	− 7.34775e7i	− 0.293290i	−0.989189	−	0.146645i	$-0.953153\pi$
$251$	− 7.34775e7i	− 0.293290i	0.989189	−	0.146645i	$-0.0468474\pi$
$252$	0	0
$253$	7.50335e7i	0.291295i
$254$	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
$259$	− 1.63094e8i	− 0.583297i
$260$	0	0
$261$	− 2.09551e8i	− 0.729536i
$262$	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
$267$	− 1.43414e8i	− 0.461108i
$268$	0	0
$269$	− 6.24018e8i	− 1.95463i	−0.211793	−	0.977315i	$-0.567930\pi$
$269$	− 6.24018e8i	− 1.95463i	0.211793	−	0.977315i	$-0.432070\pi$
$270$	0	0
$271$	3.87983e8	1.18419	0.592094	−	0.805869i	$-0.298302\pi$
$271$	3.87983e8	1.18419	0.592094	+	0.805869i	$0.298302\pi$
$272$	0	0
$273$	1.68493e7	0.0501203
$274$	0	0
$275$	3.71553e7i	0.107735i
$276$	0	0
$277$	− 4.53952e8i	− 1.28331i	−0.766994	−	0.641654i	$-0.778248\pi$
$277$	− 4.53952e8i	− 1.28331i	0.766994	−	0.641654i	$-0.221752\pi$
$278$	0	0
$279$	−4.64889e8	−1.28155
$280$	0	0
$281$	−3.33770e8	−0.897377	−0.448689	−	0.893688i	$-0.648109\pi$
$281$	−3.33770e8	−0.897377	−0.448689	+	0.893688i	$0.648109\pi$
$282$	0	0
$283$	5.37695e8i	1.41021i	0.709104	+	0.705104i	$0.249100\pi$
$283$	5.37695e8i	1.41021i	−0.709104	+	0.705104i	$0.750900\pi$
$284$	0	0
$285$	1.00649e8i	0.257544i
$286$	0	0
$287$	−1.10155e7	−0.0275053
$288$	0	0
$289$	−1.94072e8	−0.472956
$290$	0	0
$291$	− 1.04186e8i	− 0.247847i
$292$	0	0
$293$	− 3.35600e8i	− 0.779445i	−0.920932	−	0.389722i	$-0.872571\pi$
$293$	− 3.35600e8i	− 0.779445i	0.920932	−	0.389722i	$-0.127429\pi$
$294$	0	0
$295$	−5.54539e8	−1.25764
$296$	0	0
$297$	5.54299e7	0.122771
$298$	0	0
$299$	9.49600e7i	0.205443i
$300$	0	0
$301$	− 6.40840e8i	− 1.35446i
$302$	0	0
$303$	1.20954e8	0.249789
$304$	0	0
$305$	−1.73817e8	−0.350787
$306$	0	0
$307$	− 2.15029e8i	− 0.424143i	−0.977254	−	0.212072i	$-0.931979\pi$
$307$	− 2.15029e8i	− 0.424143i	0.977254	−	0.212072i	$-0.0680210\pi$
$308$	0	0
$309$	− 4.49759e7i	− 0.0867212i
$310$	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	$-0.465179\pi$
$313$	1.18457e8	0.218352	0.109176	+	0.994022i	$0.465179\pi$
$314$	0	0
$315$	4.35894e8i	0.785768i
$316$	0	0
$317$	− 5.07310e7i	− 0.0894470i	−0.998999	−	0.0447235i	$-0.985759\pi$
$317$	− 5.07310e7i	− 0.0894470i	0.998999	−	0.0447235i	$-0.0142407\pi$
$318$	0	0
$319$	1.12006e8	0.193186
$320$	0	0
$321$	−2.15827e8	−0.364198
$322$	0	0
$323$	5.87358e8i	0.969826i
$324$	0	0
$325$	4.70226e7i	0.0759826i
$326$	0	0
$327$	1.47084e8	0.232621
$328$	0	0
$329$	−4.80218e8	−0.743453
$330$	0	0
$331$	2.73757e8i	0.414923i	0.978243	+	0.207461i	$0.0665201\pi$
$331$	2.73757e8i	0.414923i	−0.978243	+	0.207461i	$0.933480\pi$
$332$	0	0
$333$	− 3.27955e8i	− 0.486697i
$334$	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
$339$	− 1.99140e8i	− 0.277625i
$340$	0	0
$341$	− 2.48487e8i	− 0.339362i
$342$	0	0
$343$	−6.24667e8	−0.835833
$344$	0	0
$345$	1.73154e8	0.227021
$346$	0	0
$347$	− 1.36700e9i	− 1.75637i	−0.478318	−	0.878187i	$-0.658753\pi$
$347$	− 1.36700e9i	− 1.75637i	0.478318	−	0.878187i	$-0.341247\pi$
$348$	0	0
$349$	1.13143e9i	1.42475i	0.701797	+	0.712377i	$0.252381\pi$
$349$	1.13143e9i	1.42475i	−0.701797	+	0.712377i	$0.747619\pi$
$350$	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
$355$	− 2.97093e8i	− 0.352446i
$356$	0	0
$357$	− 1.79296e8i	− 0.208560i
$358$	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
$363$	− 2.19536e8i	− 0.240898i
$364$	0	0
$365$	− 2.05859e8i	− 0.221588i
$366$	0	0
$367$	−1.63472e9	−1.72628	−0.863140	−	0.504964i	$-0.831506\pi$
$367$	−1.63472e9	−1.72628	−0.863140	+	0.504964i	$0.831506\pi$
$368$	0	0
$369$	−2.21502e7	−0.0229501
$370$	0	0
$371$	1.51792e9i	1.54327i
$372$	0	0
$373$	1.54633e9i	1.54284i	0.636325	+	0.771421i	$0.280454\pi$
$373$	1.54633e9i	1.54284i	−0.636325	+	0.771421i	$0.719546\pi$
$374$	0	0
$375$	2.82618e8	0.276752
$376$	0	0
$377$	1.41752e8	0.136249
$378$	0	0
$379$	− 1.05688e9i	− 0.997216i	−0.866828	−	0.498608i	$-0.833845\pi$
$379$	− 1.05688e9i	− 0.997216i	0.866828	−	0.498608i	$-0.166155\pi$
$380$	0	0
$381$	1.40191e7i	0.0129862i
$382$	0	0
$383$	−2.24910e8	−0.204556	−0.102278	−	0.994756i	$-0.532613\pi$
$383$	−2.24910e8	−0.204556	−0.102278	+	0.994756i	$0.532613\pi$
$384$	0	0
$385$	−2.32989e8	−0.208077
$386$	0	0
$387$	− 1.28862e9i	− 1.13015i
$388$	0	0
$389$	− 1.01788e9i	− 0.876746i	−0.898793	−	0.438373i	$-0.855555\pi$
$389$	− 1.01788e9i	− 0.876746i	0.898793	−	0.438373i	$-0.144445\pi$
$390$	0	0
$391$	1.01048e9	0.854887
$392$	0	0
$393$	9.50859e7	0.0790210
$394$	0	0
$395$	7.49028e8i	0.611517i
$396$	0	0
$397$	− 1.47565e9i	− 1.18363i	−0.806072	−	0.591817i	$-0.798411\pi$
$397$	− 1.47565e9i	− 1.18363i	0.806072	−	0.591817i	$-0.201589\pi$
$398$	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
$403$	− 3.14477e8i	− 0.239343i
$404$	0	0
$405$	8.10373e8i	0.606167i
$406$	0	0
$407$	1.75294e8	0.128881
$408$	0	0
$409$	1.63427e9	1.18112	0.590558	−	0.806995i	$-0.298908\pi$
$409$	1.63427e9	1.18112	0.590558	+	0.806995i	$0.298908\pi$
$410$	0	0
$411$	− 3.78785e6i	− 0.00269120i
$412$	0	0
$413$	2.68291e9i	1.87405i
$414$	0	0
$415$	1.19131e9	0.818192
$416$	0	0
$417$	4.70445e8	0.317712
$418$	0	0
$419$	1.11280e9i	0.739039i	0.929223	+	0.369519i	$0.120478\pi$
$419$	1.11280e9i	0.739039i	−0.929223	+	0.369519i	$0.879522\pi$
$420$	0	0
$421$	− 9.22528e8i	− 0.602549i	−0.953537	−	0.301274i	$-0.902588\pi$
$421$	− 9.22528e8i	− 0.602549i	0.953537	−	0.301274i	$-0.0974120\pi$
$422$	0	0
$423$	−9.65636e8	−0.620329
$424$	0	0
$425$	5.00372e8	0.316178
$426$	0	0
$427$	8.40945e8i	0.522721i
$428$	0	0
$429$	1.81097e7i	0.0110742i
$430$	0	0
$431$	9.81508e8	0.590505	0.295252	−	0.955419i	$-0.404596\pi$
$431$	9.81508e8	0.590505	0.295252	+	0.955419i	$0.404596\pi$
$432$	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
$435$	− 2.58476e8i	− 0.150560i
$436$	0	0
$437$	2.74436e9i	1.57310i
$438$	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
$443$	1.82325e9i	0.996401i	0.867062	+	0.498201i	$0.166006\pi$
$443$	1.82325e9i	0.996401i	−0.867062	+	0.498201i	$0.833994\pi$
$444$	0	0
$445$	2.50975e9i	1.35011i
$446$	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
$451$	− 1.18395e7i	− 0.00607735i
$452$	0	0
$453$	3.52985e8i	0.178407i
$454$	0	0
$455$	−2.94864e8	−0.146751
$456$	0	0
$457$	2.98066e9	1.46085	0.730425	−	0.682993i	$-0.239322\pi$
$457$	2.98066e9	1.46085	0.730425	+	0.682993i	$0.239322\pi$
$458$	0	0
$459$	− 7.46477e8i	− 0.360306i
$460$	0	0
$461$	− 2.52781e9i	− 1.20169i	−0.799367	−	0.600843i	$-0.794832\pi$
$461$	− 2.52781e9i	− 1.20169i	0.799367	−	0.600843i	$-0.205168\pi$
$462$	0	0
$463$	8.90291e8	0.416868	0.208434	−	0.978036i	$-0.433163\pi$
$463$	8.90291e8	0.416868	0.208434	+	0.978036i	$0.433163\pi$
$464$	0	0
$465$	−5.73431e8	−0.264482
$466$	0	0
$467$	− 2.65667e9i	− 1.20706i	−0.797341	−	0.603529i	$-0.793761\pi$
$467$	− 2.65667e9i	− 1.20706i	0.797341	−	0.603529i	$-0.206239\pi$
$468$	0	0
$469$	1.28020e8i	0.0573024i
$470$	0	0
$471$	7.26660e8	0.320448
$472$	0	0
$473$	6.88777e8	0.299271
$474$	0	0
$475$	1.35896e9i	0.581806i
$476$	0	0
$477$	3.05228e9i	1.28769i
$478$	0	0
$479$	−1.30093e9	−0.540855	−0.270428	−	0.962740i	$-0.587165\pi$
$479$	−1.30093e9	−0.540855	−0.270428	+	0.962740i	$0.587165\pi$
$480$	0	0
$481$	2.21847e8	0.0908962
$482$	0	0
$483$	− 8.37737e8i	− 0.338293i
$484$	0	0
$485$	1.82325e9i	0.725689i
$486$	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
$491$	− 7.83344e8i	− 0.298653i	−0.988788	−	0.149327i	$-0.952289\pi$
$491$	− 7.83344e8i	− 0.298653i	0.988788	−	0.149327i	$-0.0477106\pi$
$492$	0	0
$493$	− 1.50839e9i	− 0.566958i
$494$	0	0
$495$	−4.68501e8	−0.173617
$496$	0	0
$497$	−1.43736e9	−0.525193
$498$	0	0
$499$	6.23188e8i	0.224526i	0.993679	+	0.112263i	$0.0358100\pi$
$499$	6.23188e8i	0.224526i	−0.993679	+	0.112263i	$0.964190\pi$
$500$	0	0
$501$	1.05272e9i	0.374007i
$502$	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
$507$	− 7.30063e8i	− 0.248790i
$508$	0	0
$509$	3.49943e9i	1.17621i	0.808784	+	0.588106i	$0.200126\pi$
$509$	3.49943e9i	1.17621i	−0.808784	+	0.588106i	$0.799874\pi$
$510$	0	0
$511$	−9.95967e8	−0.330196
$512$	0	0
$513$	2.02735e9	0.663008
$514$	0	0
$515$	7.87078e8i	0.253918i
$516$	0	0
$517$	− 5.16140e8i	− 0.164267i
$518$	0	0
$519$	1.02835e8	0.0322889
$520$	0	0
$521$	1.37683e9	0.426530	0.213265	−	0.976994i	$-0.431590\pi$
$521$	1.37683e9	0.426530	0.213265	+	0.976994i	$0.431590\pi$
$522$	0	0
$523$	− 2.86154e9i	− 0.874669i	−0.899299	−	0.437334i	$-0.855923\pi$
$523$	− 2.86154e9i	− 0.874669i	0.899299	−	0.437334i	$-0.144077\pi$
$524$	0	0
$525$	− 4.14833e8i	− 0.125117i
$526$	0	0
$527$	−3.34638e9	−0.995951
$528$	0	0
$529$	1.31651e9	0.386661
$530$	0	0
$531$	5.39487e9i	1.56369i
$532$	0	0
$533$	− 1.49836e7i	− 0.00428620i
$534$	0	0
$535$	3.77697e9	1.06636
$536$	0	0
$537$	−2.25649e8	−0.0628815
$538$	0	0
$539$	2.27915e8i	0.0626919i
$540$	0	0
$541$	5.34467e9i	1.45121i	0.688111	+	0.725605i	$0.258440\pi$
$541$	5.34467e9i	1.45121i	−0.688111	+	0.725605i	$0.741560\pi$
$542$	0	0
$543$	7.19550e8	0.192869
$544$	0	0
$545$	−2.57398e9	−0.681109
$546$	0	0
$547$	3.37135e9i	0.880740i	0.897816	+	0.440370i	$0.145153\pi$
$547$	3.37135e9i	0.880740i	−0.897816	+	0.440370i	$0.854847\pi$
$548$	0	0
$549$	1.69100e9i	0.436153i
$550$	0	0
$551$	4.09665e9	1.04327
$552$	0	0
$553$	3.62387e9	0.911244
$554$	0	0
$555$	− 4.04526e8i	− 0.100443i
$556$	0	0
$557$	− 5.61106e9i	− 1.37579i	−0.725811	−	0.687894i	$-0.758535\pi$
$557$	− 5.61106e9i	− 1.37579i	0.725811	−	0.687894i	$-0.241465\pi$
$558$	0	0
$559$	8.71694e8	0.211068
$560$	0	0
$561$	1.92707e8	0.0460817
$562$	0	0
$563$	− 6.69690e9i	− 1.58159i	−0.612081	−	0.790795i	$-0.709667\pi$
$563$	− 6.69690e9i	− 1.58159i	0.612081	−	0.790795i	$-0.290333\pi$
$564$	0	0
$565$	3.48494e9i	0.812879i
$566$	0	0
$567$	3.92066e9	0.903273
$568$	0	0
$569$	−1.96850e9	−0.447964	−0.223982	−	0.974593i	$-0.571906\pi$
$569$	−1.96850e9	−0.447964	−0.223982	+	0.974593i	$0.571906\pi$
$570$	0	0
$571$	1.02926e9i	0.231365i	0.993286	+	0.115682i	$0.0369055\pi$
$571$	1.02926e9i	0.231365i	−0.993286	+	0.115682i	$0.963094\pi$
$572$	0	0
$573$	1.12783e9i	0.250440i
$574$	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
$579$	4.22335e8i	0.0904236i
$580$	0	0
$581$	− 5.76366e9i	− 1.21922i
$582$	0	0
$583$	−1.63147e9	−0.340988
$584$	0	0
$585$	−5.92919e8	−0.122448
$586$	0	0
$587$	− 5.59411e8i	− 0.114156i	−0.998370	−	0.0570778i	$-0.981822\pi$
$587$	− 5.59411e8i	− 0.114156i	0.998370	−	0.0570778i	$-0.0181783\pi$
$588$	0	0
$589$	− 9.08843e9i	− 1.83267i
$590$	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
$595$	3.13767e9i	0.610658i
$596$	0	0
$597$	− 1.00365e9i	− 0.193051i
$598$	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	$-0.510193\pi$
$601$	−3.40792e8	−0.0640366	−0.0320183	+	0.999487i	$0.510193\pi$
$602$	0	0
$603$	2.57426e8i	0.0478126i
$604$	0	0
$605$	3.84189e9i	0.705344i
$606$	0	0
$607$	−3.85420e9	−0.699477	−0.349739	−	0.936847i	$-0.613730\pi$
$607$	−3.85420e9	−0.699477	−0.349739	+	0.936847i	$0.613730\pi$
$608$	0	0
$609$	−1.25053e9	−0.224355
$610$	0	0
$611$	− 6.53211e8i	− 0.115853i
$612$	0	0
$613$	− 9.22245e9i	− 1.61709i	−0.588434	−	0.808545i	$-0.700255\pi$
$613$	− 9.22245e9i	− 1.61709i	0.588434	−	0.808545i	$-0.299745\pi$
$614$	0	0
$615$	−2.73218e7	−0.00473639
$616$	0	0
$617$	−6.53611e9	−1.12027	−0.560133	−	0.828402i	$-0.689250\pi$
$617$	−6.53611e9	−1.12027	−0.560133	+	0.828402i	$0.689250\pi$
$618$	0	0
$619$	1.36559e9i	0.231420i	0.993283	+	0.115710i	$0.0369144\pi$
$619$	1.36559e9i	0.231420i	−0.993283	+	0.115710i	$0.963086\pi$
$620$	0	0
$621$	− 3.48782e9i	− 0.584431i
$622$	0	0
$623$	1.21424e10	2.01186
$624$	0	0
$625$	−2.28761e9	−0.374802
$626$	0	0
$627$	5.23374e8i	0.0847960i
$628$	0	0
$629$	− 2.36070e9i	− 0.378236i
$630$	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
$635$	− 2.45334e8i	− 0.0380233i
$636$	0	0
$637$	2.88441e8i	0.0442150i
$638$	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
$643$	− 1.14054e10i	− 1.69189i	−0.533272	−	0.845944i	$-0.679038\pi$
$643$	− 1.14054e10i	− 1.69189i	0.533272	−	0.845944i	$-0.320962\pi$
$644$	0	0
$645$	− 1.58948e9i	− 0.233237i
$646$	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
$651$	2.77431e9i	0.394114i
$652$	0	0
$653$	− 1.05004e10i	− 1.47575i	−0.674940	−	0.737873i	$-0.735830\pi$
$653$	− 1.05004e10i	− 1.47575i	0.674940	−	0.737873i	$-0.264170\pi$
$654$	0	0
$655$	−1.66400e9	−0.231371
$656$	0	0
$657$	−2.00272e9	−0.275512
$658$	0	0
$659$	− 9.64818e9i	− 1.31325i	−0.754219	−	0.656624i	$-0.771984\pi$
$659$	− 9.64818e9i	− 1.31325i	0.754219	−	0.656624i	$-0.228016\pi$
$660$	0	0
$661$	6.58299e9i	0.886580i	0.896378	+	0.443290i	$0.146189\pi$
$661$	6.58299e9i	0.886580i	−0.896378	+	0.443290i	$0.853811\pi$
$662$	0	0
$663$	2.43884e8	0.0325002
$664$	0	0
$665$	−8.52160e9	−1.12369
$666$	0	0
$667$	− 7.04779e9i	− 0.919629i
$668$	0	0
$669$	− 1.75749e8i	− 0.0226935i
$670$	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
$675$	− 1.72711e9i	− 0.216151i
$676$	0	0
$677$	− 8.71305e9i	− 1.07922i	−0.841915	−	0.539610i	$-0.818572\pi$
$677$	− 8.71305e9i	− 1.07922i	0.841915	−	0.539610i	$-0.181428\pi$
$678$	0	0
$679$	8.82106e9	1.08138
$680$	0	0
$681$	2.21449e9	0.268695
$682$	0	0
$683$	1.46109e10i	1.75470i	0.479849	+	0.877351i	$0.340692\pi$
$683$	1.46109e10i	1.75470i	−0.479849	+	0.877351i	$0.659308\pi$
$684$	0	0
$685$	6.62873e7i	0.00787977i
$686$	0	0
$687$	1.05055e8	0.0123615
$688$	0	0
$689$	−2.06473e9	−0.240490
$690$	0	0
$691$	1.47348e10i	1.69891i	0.527662	+	0.849454i	$0.323069\pi$
$691$	1.47348e10i	1.69891i	−0.527662	+	0.849454i	$0.676931\pi$
$692$	0	0
$693$	2.26665e9i	0.258713i
$694$	0	0
$695$	−8.23279e9	−0.930252
$696$	0	0
$697$	−1.59442e8	−0.0178357
$698$	0	0
$699$	− 1.43467e9i	− 0.158885i
$700$	0	0
$701$	1.31502e9i	0.144185i	0.997398	+	0.0720923i	$0.0229676\pi$
$701$	1.31502e9i	0.144185i	−0.997398	+	0.0720923i	$0.977032\pi$
$702$	0	0
$703$	6.41141e9	0.696001
$704$	0	0
$705$	−1.19109e9	−0.128022
$706$	0	0
$707$	1.02408e10i	1.08985i
$708$	0	0
$709$	− 6.64028e8i	− 0.0699721i	−0.999388	−	0.0349860i	$-0.988861\pi$
$709$	− 6.64028e8i	− 0.0699721i	0.999388	−	0.0349860i	$-0.0111387\pi$
$710$	0	0
$711$	7.28697e9	0.760332
$712$	0	0
$713$	−1.56356e10	−1.61547
$714$	0	0
$715$	− 3.16920e8i	− 0.0324249i
$716$	0	0
$717$	4.75451e9i	0.481713i
$718$	0	0
$719$	−4.95034e9	−0.496689	−0.248344	−	0.968672i	$-0.579886\pi$
$719$	−4.95034e9	−0.496689	−0.248344	+	0.968672i	$0.579886\pi$
$720$	0	0
$721$	3.80796e9	0.378372
$722$	0	0
$723$	3.07928e9i	0.303015i
$724$	0	0
$725$	− 3.48994e9i	− 0.340123i
$726$	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
$731$	− 9.27578e9i	− 0.878293i
$732$	0	0
$733$	− 1.49414e8i	− 0.0140129i	−0.999975	−	0.00700643i	$-0.997770\pi$
$733$	− 1.49414e8i	− 0.0140129i	0.999975	−	0.00700643i	$-0.00223023\pi$
$734$	0	0
$735$	5.25957e8	0.0488590
$736$	0	0
$737$	−1.37596e8	−0.0126611
$738$	0	0
$739$	4.70806e9i	0.429127i	0.976710	+	0.214564i	$0.0688329\pi$
$739$	4.70806e9i	0.429127i	−0.976710	+	0.214564i	$0.931167\pi$
$740$	0	0
$741$	6.62365e8i	0.0598045i
$742$	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
$747$	− 1.15897e10i	− 1.01730i
$748$	0	0
$749$	− 1.82733e10i	− 1.58903i
$750$	0	0
$751$	−1.06650e10	−0.918800	−0.459400	−	0.888229i	$-0.651936\pi$
$751$	−1.06650e10	−0.918800	−0.459400	+	0.888229i	$0.651936\pi$
$752$	0	0
$753$	−8.81731e8	−0.0752581
$754$	0	0
$755$	− 6.17724e9i	− 0.522373i
$756$	0	0
$757$	− 6.22876e9i	− 0.521874i	−0.965356	−	0.260937i	$-0.915968\pi$
$757$	− 6.22876e9i	− 0.521874i	0.965356	−	0.260937i	$-0.0840315\pi$
$758$	0	0
$759$	9.00402e8	0.0747464
$760$	0	0
$761$	8.38334e9	0.689558	0.344779	−	0.938684i	$-0.387954\pi$
$761$	8.38334e9	0.689558	0.344779	+	0.938684i	$0.387954\pi$
$762$	0	0
$763$	1.24531e10i	1.01495i
$764$	0	0
$765$	6.30932e9i	0.509527i
$766$	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
$771$	2.43241e9i	0.191138i
$772$	0	0
$773$	− 5.56680e9i	− 0.433488i	−0.976228	−	0.216744i	$-0.930456\pi$
$773$	− 5.56680e9i	− 0.433488i	0.976228	−	0.216744i	$-0.0695438\pi$
$774$	0	0
$775$	−7.74246e9	−0.597479
$776$	0	0
$777$	−1.95713e9	−0.149674
$778$	0	0
$779$	− 4.33029e8i	− 0.0328198i
$780$	0	0
$781$	− 1.54488e9i	− 0.116042i
$782$	0	0
$783$	−5.20645e9	−0.387593
$784$	0	0
$785$	−1.27165e10	−0.938264
$786$	0	0
$787$	− 1.34611e8i	− 0.00984395i	−0.999988	−	0.00492198i	$-0.998433\pi$
$787$	− 1.34611e8i	− 0.00984395i	0.999988	−	0.00492198i	$-0.00156672\pi$
$788$	0	0
$789$	− 1.85105e9i	− 0.134168i
$790$	0	0
$791$	1.68605e10	1.21130
$792$	0	0
$793$	−1.14388e9	−0.0814565
$794$	0	0
$795$	3.76493e9i	0.265749i
$796$	0	0
$797$	− 7.41548e9i	− 0.518842i	−0.965764	−	0.259421i	$-0.916468\pi$
$797$	− 7.41548e9i	− 0.518842i	0.965764	−	0.259421i	$-0.0835317\pi$
$798$	0	0
$799$	−6.95088e9	−0.482088
$800$	0	0
$801$	2.44163e10	1.67867
$802$	0	0
$803$	− 1.07047e9i	− 0.0729574i
$804$	0	0
$805$	1.46604e10i	0.990513i
$806$	0	0
$807$	−7.48822e9	−0.501558
$808$	0	0
$809$	1.41542e10	0.939863	0.469932	−	0.882703i	$-0.344279\pi$
$809$	1.41542e10	0.939863	0.469932	+	0.882703i	$0.344279\pi$
$810$	0	0
$811$	− 2.63708e10i	− 1.73600i	−0.496563	−	0.868001i	$-0.665405\pi$
$811$	− 2.63708e10i	− 1.73600i	0.496563	−	0.868001i	$-0.334595\pi$
$812$	0	0
$813$	− 4.65580e9i	− 0.303863i
$814$	0	0
$815$	1.19879e10	0.775697
$816$	0	0
$817$	2.51921e10	1.61617
$818$	0	0
$819$	2.86860e9i	0.182464i
$820$	0	0
$821$	− 8.06264e9i	− 0.508483i	−0.967141	−	0.254241i	$-0.918174\pi$
$821$	− 8.06264e9i	− 0.508483i	0.967141	−	0.254241i	$-0.0818258\pi$
$822$	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
$827$	5.55722e9i	0.341655i	0.985301	+	0.170828i	$0.0546442\pi$
$827$	5.55722e9i	0.341655i	−0.985301	+	0.170828i	$0.945356\pi$
$828$	0	0
$829$	2.84256e10i	1.73288i	0.499281	+	0.866440i	$0.333597\pi$
$829$	2.84256e10i	1.73288i	−0.499281	+	0.866440i	$0.666403\pi$
$830$	0	0
$831$	−5.44743e9	−0.329297
$832$	0	0
$833$	3.06933e9	0.183987
$834$	0	0
$835$	− 1.84226e10i	− 1.09508i
$836$	0	0
$837$	1.15505e10i	0.680868i
$838$	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
$843$	4.00524e9i	0.230267i
$844$	0	0
$845$	1.27761e10i	0.728450i
$846$	0	0
$847$	1.85874e10	1.05106
$848$	0	0
$849$	6.45234e9	0.361860
$850$	0	0
$851$	− 1.10301e10i	− 0.613514i
$852$	0	0
$853$	1.80580e10i	0.996205i	0.867118	+	0.498102i	$0.165970\pi$
$853$	1.80580e10i	0.996205i	−0.867118	+	0.498102i	$0.834030\pi$
$854$	0	0
$855$	−1.71355e10	−0.937593
$856$	0	0
$857$	6.34034e9	0.344096	0.172048	−	0.985089i	$-0.444962\pi$
$857$	6.34034e9	0.344096	0.172048	+	0.985089i	$0.444962\pi$
$858$	0	0
$859$	1.21489e10i	0.653973i	0.945029	+	0.326987i	$0.106033\pi$
$859$	1.21489e10i	0.653973i	−0.945029	+	0.326987i	$0.893967\pi$
$860$	0	0
$861$	1.32186e8i	0.00705786i
$862$	0	0
$863$	2.87111e10	1.52059	0.760295	−	0.649578i	$-0.225054\pi$
$863$	2.87111e10	1.52059	0.760295	+	0.649578i	$0.225054\pi$
$864$	0	0
$865$	−1.79960e9	−0.0945411
$866$	0	0
$867$	2.32887e9i	0.121361i
$868$	0	0
$869$	3.89495e9i	0.201341i
$870$	0	0
$871$	−1.74138e8	−0.00892953
$872$	0	0
$873$	1.77376e10	0.902289
$874$	0	0
$875$	2.39283e10i	1.20749i
$876$	0	0
$877$	2.46021e10i	1.23161i	0.787898	+	0.615806i	$0.211169\pi$
$877$	2.46021e10i	1.23161i	−0.787898	+	0.615806i	$0.788831\pi$
$878$	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
$883$	− 1.93097e10i	− 0.943873i	−0.881633	−	0.471937i	$-0.843555\pi$
$883$	− 1.93097e10i	− 0.943873i	0.881633	−	0.471937i	$-0.156445\pi$
$884$	0	0
$885$	6.65446e9i	0.322709i
$886$	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
$891$	4.21394e9i	0.199580i
$892$	0	0
$893$	− 1.88779e10i	− 0.887101i
$894$	0	0
$895$	3.94885e9	0.184115
$896$	0	0
$897$	1.13952e9	0.0527167
$898$	0	0
$899$	2.33400e10i	1.07138i
$900$	0	0
$901$	2.19710e10i	1.00072i
$902$	0	0
$903$	−7.69008e9	−0.347555
$904$	0	0
$905$	−1.25921e10	−0.564715
$906$	0	0
$907$	2.33703e9i	0.104002i	0.998647	+	0.0520008i	$0.0165598\pi$
$907$	2.33703e9i	0.104002i	−0.998647	+	0.0520008i	$0.983440\pi$
$908$	0	0
$909$	2.05925e10i	0.909359i
$910$	0	0
$911$	−2.20343e10	−0.965573	−0.482786	−	0.875738i	$-0.660375\pi$
$911$	−2.20343e10	−0.965573	−0.482786	+	0.875738i	$0.660375\pi$
$912$	0	0
$913$	6.19480e9	0.269389
$914$	0	0
$915$	2.08581e9i	0.0900121i
$916$	0	0
$917$	8.05061e9i	0.344775i
$918$	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
$923$	− 1.95515e9i	− 0.0818418i
$924$	0	0
$925$	− 5.46190e9i	− 0.226907i
$926$	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
$931$	8.33600e9i	0.338558i
$932$	0	0
$933$	− 9.50474e9i	− 0.383137i
$934$	0	0
$935$	−3.37238e9	−0.134926
$936$	0	0
$937$	3.87626e10	1.53930	0.769652	−	0.638463i	$-0.220429\pi$
$937$	3.87626e10	1.53930	0.769652	+	0.638463i	$0.220429\pi$
$938$	0	0
$939$	− 1.42149e9i	− 0.0560291i
$940$	0	0
$941$	2.06279e10i	0.807035i	0.914972	+	0.403517i	$0.132212\pi$
$941$	2.06279e10i	0.807035i	−0.914972	+	0.403517i	$0.867788\pi$
$942$	0	0
$943$	−7.44976e8	−0.0289302
$944$	0	0
$945$	1.08302e10	0.417468
$946$	0	0
$947$	2.11705e10i	0.810040i	0.914308	+	0.405020i	$0.132736\pi$
$947$	2.11705e10i	0.810040i	−0.914308	+	0.405020i	$0.867264\pi$
$948$	0	0
$949$	− 1.35475e9i	− 0.0514550i
$950$	0	0
$951$	−6.08771e8	−0.0229521
$952$	0	0
$953$	−2.14876e10	−0.804196	−0.402098	−	0.915597i	$-0.631719\pi$
$953$	−2.14876e10	−0.804196	−0.402098	+	0.915597i	$0.631719\pi$
$954$	0	0
$955$	− 1.97371e10i	− 0.733282i
$956$	0	0
$957$	− 1.34408e9i	− 0.0495715i
$958$	0	0
$959$	3.20704e8	0.0117419
$960$	0	0
$961$	2.42673e10	0.882043
$962$	0	0
$963$	− 3.67445e10i	− 1.32587i
$964$	0	0
$965$	− 7.39086e9i	− 0.264758i
$966$	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
$971$	− 5.62647e10i	− 1.97228i	−0.165917	−	0.986140i	$-0.553059\pi$
$971$	− 5.62647e10i	− 1.97228i	0.165917	−	0.986140i	$-0.446941\pi$
$972$	0	0
$973$	3.98310e10i	1.38620i
$974$	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
$979$	1.30507e10i	0.444523i
$980$	0	0
$981$	2.50411e10i	0.846861i
$982$	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
$987$	5.76262e9i	0.190770i
$988$	0	0
$989$	− 4.33400e10i	− 1.42463i
$990$	0	0
$991$	−3.46728e10	−1.13170	−0.565849	−	0.824509i	$-0.691452\pi$
$991$	−3.46728e10	−1.13170	−0.565849	+	0.824509i	$0.691452\pi$
$992$	0	0
$993$	3.28508e9	0.106469
$994$	0	0
$995$	1.75639e10i	0.565249i
$996$	0	0
$997$	2.96474e10i	0.947444i	0.880674	+	0.473722i	$0.157090\pi$
$997$	2.96474e10i	0.947444i	−0.880674	+	0.473722i	$0.842910\pi$
$998$	0	0
$999$	−8.14830e9	−0.258576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.b.f.129.1		2
4.3	odd	2		256.8.b.b.129.2		2
8.3	odd	2		256.8.b.b.129.1		2
8.5	even	2	inner	256.8.b.f.129.2		2
16.3	odd	4		64.8.a.c.1.1		1
16.5	even	4		16.8.a.b.1.1		1
16.11	odd	4		2.8.a.a.1.1	✓	1
16.13	even	4		64.8.a.e.1.1		1
48.5	odd	4		144.8.a.i.1.1		1
48.11	even	4		18.8.a.b.1.1		1
48.29	odd	4		576.8.a.f.1.1		1
48.35	even	4		576.8.a.g.1.1		1
80.27	even	4		50.8.b.c.49.1		2
80.37	odd	4		400.8.c.j.49.2		2
80.43	even	4		50.8.b.c.49.2		2
80.53	odd	4		400.8.c.j.49.1		2
80.59	odd	4		50.8.a.g.1.1		1
80.69	even	4		400.8.a.l.1.1		1
112.11	odd	12		98.8.c.d.79.1		2
112.27	even	4		98.8.a.a.1.1		1
112.59	even	12		98.8.c.e.79.1		2
112.75	even	12		98.8.c.e.67.1		2
112.107	odd	12		98.8.c.d.67.1		2
144.11	even	12		162.8.c.a.109.1		2
144.43	odd	12		162.8.c.l.109.1		2
144.59	even	12		162.8.c.a.55.1		2
144.139	odd	12		162.8.c.l.55.1		2
176.43	even	4		242.8.a.e.1.1		1
208.155	odd	4		338.8.a.d.1.1		1
208.187	even	4		338.8.b.d.337.2		2
208.203	even	4		338.8.b.d.337.1		2
240.59	even	4		450.8.a.c.1.1		1
240.107	odd	4		450.8.c.g.199.2		2
240.203	odd	4		450.8.c.g.199.1		2
272.203	odd	4		578.8.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.8.a.a.1.1	✓	1	16.11	odd	4
16.8.a.b.1.1		1	16.5	even	4
18.8.a.b.1.1		1	48.11	even	4
50.8.a.g.1.1		1	80.59	odd	4
50.8.b.c.49.1		2	80.27	even	4
50.8.b.c.49.2		2	80.43	even	4
64.8.a.c.1.1		1	16.3	odd	4
64.8.a.e.1.1		1	16.13	even	4
98.8.a.a.1.1		1	112.27	even	4
98.8.c.d.67.1		2	112.107	odd	12
98.8.c.d.79.1		2	112.11	odd	12
98.8.c.e.67.1		2	112.75	even	12
98.8.c.e.79.1		2	112.59	even	12
144.8.a.i.1.1		1	48.5	odd	4
162.8.c.a.55.1		2	144.59	even	12
162.8.c.a.109.1		2	144.11	even	12
162.8.c.l.55.1		2	144.139	odd	12
162.8.c.l.109.1		2	144.43	odd	12
242.8.a.e.1.1		1	176.43	even	4
256.8.b.b.129.1		2	8.3	odd	2
256.8.b.b.129.2		2	4.3	odd	2
256.8.b.f.129.1		2	1.1	even	1	trivial
256.8.b.f.129.2		2	8.5	even	2	inner
338.8.a.d.1.1		1	208.155	odd	4
338.8.b.d.337.1		2	208.203	even	4
338.8.b.d.337.2		2	208.187	even	4
400.8.a.l.1.1		1	80.69	even	4
400.8.c.j.49.1		2	80.53	odd	4
400.8.c.j.49.2		2	80.37	odd	4
450.8.a.c.1.1		1	240.59	even	4
450.8.c.g.199.1		2	240.203	odd	4
450.8.c.g.199.2		2	240.107	odd	4
576.8.a.f.1.1		1	48.29	odd	4
576.8.a.g.1.1		1	48.35	even	4
578.8.a.b.1.1		1	272.203	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.0000i q^{3} +210.000i q^{5} +1016.00 q^{7} +2043.00 q^{9} +1092.00i q^{11} +1382.00i q^{13} +2520.00 q^{15} +14706.0 q^{17} +39940.0i q^{19} -12192.0i q^{21} +68712.0 q^{23} +34025.0 q^{25} -50760.0i q^{27} -102570. i q^{29} -227552. q^{31} +13104.0 q^{33} +213360. i q^{35} -160526. i q^{37} +16584.0 q^{39} -10842.0 q^{41} -630748. i q^{43} +429030. i q^{45} -472656. q^{47} +208713. q^{49} -176472. i q^{51} +1.49402e6i q^{53} -229320. q^{55} +479280. q^{57} +2.64066e6i q^{59} +827702. i q^{61} +2.07569e6 q^{63} -290220. q^{65} +126004. i q^{67} -824544. i q^{69} -1.41473e6 q^{71} -980282. q^{73} -408300. i q^{75} +1.10947e6i q^{77} +3.56680e6 q^{79} +3.85892e6 q^{81} -5.67289e6i q^{83} +3.08826e6i q^{85} -1.23084e6 q^{87} +1.19512e7 q^{89} +1.40411e6i q^{91} +2.73062e6i q^{93} -8.38740e6 q^{95} +8.68215e6 q^{97} +2.23096e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2032 q^{7} + 4086 q^{9} + 5040 q^{15} + 29412 q^{17} + 137424 q^{23} + 68050 q^{25} - 455104 q^{31} + 26208 q^{33} + 33168 q^{39} - 21684 q^{41} - 945312 q^{47} + 417426 q^{49} - 458640 q^{55} + 958560 q^{57}+ \cdots + 17364292 q^{97}+O(q^{100}) \) 2 * q + 2032 * q^7 + 4086 * q^9 + 5040 * q^15 + 29412 * q^17 + 137424 * q^23 + 68050 * q^25 - 455104 * q^31 + 26208 * q^33 + 33168 * q^39 - 21684 * q^41 - 945312 * q^47 + 417426 * q^49 - 458640 * q^55 + 958560 * q^57 + 4151376 * q^63 - 580440 * q^65 - 2829456 * q^71 - 1960564 * q^73 + 7133600 * q^79 + 7717842 * q^81 - 2461680 * q^87 + 23902380 * q^89 - 16774800 * q^95 + 17364292 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more