LMFDB - Embedded newform 128.8.b.f.65.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [128,8,Mod(65,128)] 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("128.65"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	128.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,7308] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$39.9852832620$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 9 $ x^4 + 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.4
Root			$-0.707107 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	128.65
Dual form			128.8.b.f.65.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+18.9737i q^{3} +304.105i q^{5} +656.195 q^{7} +1827.00 q^{9} -7317.51i q^{11} -8604.39i q^{13} -5769.99 q^{15} -8890.00 q^{17} -2384.36i q^{19} +12450.4i q^{21} +44417.6 q^{23} -14355.0 q^{25} +76160.3i q^{27} -170460. i q^{29} +108612. q^{31} +138840. q^{33} +199552. i q^{35} +253677. i q^{37} +163257. q^{39} -122902. q^{41} -869354. i q^{43} +555600. i q^{45} +1.31225e6 q^{47} -392951. q^{49} -168676. i q^{51} -1.15787e6i q^{53} +2.22529e6 q^{55} +45240.0 q^{57} +1.53018e6i q^{59} +1.29461e6i q^{61} +1.19887e6 q^{63} +2.61664e6 q^{65} +798760. i q^{67} +842765. i q^{69} +4.47084e6 q^{71} -3.84917e6 q^{73} -272367. i q^{75} -4.80171e6i q^{77} +7.33332e6 q^{79} +2.55061e6 q^{81} +452452. i q^{83} -2.70350e6i q^{85} +3.23425e6 q^{87} -5.43559e6 q^{89} -5.64616e6i q^{91} +2.06076e6i q^{93} +725096. q^{95} +6.17903e6 q^{97} -1.33691e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 7308 q^{9} - 35560 q^{17} - 57420 q^{25} + 555360 q^{33} - 491608 q^{41} - 1571804 q^{49} + 180960 q^{57} + 10466560 q^{65} - 15396680 q^{73} + 10202436 q^{81} - 21742344 q^{89} + 24716120 q^{97}+O(q^{100}) $ 4 * q + 7308 * q^9 - 35560 * q^17 - 57420 * q^25 + 555360 * q^33 - 491608 * q^41 - 1571804 * q^49 + 180960 * q^57 + 10466560 * q^65 - 15396680 * q^73 + 10202436 * q^81 - 21742344 * q^89 + 24716120 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$-1$	$1$

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
$3$	18.9737i	0.405720i	0.979208	+	0.202860i	$0.0650237\pi$
$3$	18.9737i	0.405720i	−0.979208	+	0.202860i	$0.934976\pi$
$4$	0	0
$5$	304.105i	1.08800i	0.839085	+	0.544000i	$0.183091\pi$
$5$	304.105i	1.08800i	−0.839085	+	0.544000i	$0.816909\pi$
$6$	0	0
$7$	656.195	0.723086	0.361543	−	0.932355i	$-0.382250\pi$
$7$	656.195	0.723086	0.361543	+	0.932355i	$0.382250\pi$
$8$	0	0
$9$	1827.00	0.835391
$10$	0	0
$11$	− 7317.51i	− 1.65764i	−0.559519	−	0.828818i	$-0.689014\pi$
$11$	− 7317.51i	− 1.65764i	0.559519	−	0.828818i	$-0.310986\pi$
$12$	0	0
$13$	− 8604.39i	− 1.08622i	−0.839661	−	0.543111i	$-0.817246\pi$
$13$	− 8604.39i	− 1.08622i	0.839661	−	0.543111i	$-0.182754\pi$
$14$	0	0
$15$	−5769.99	−0.441424
$16$	0	0
$17$	−8890.00	−0.438865	−0.219432	−	0.975628i	$-0.570421\pi$
$17$	−8890.00	−0.438865	−0.219432	+	0.975628i	$0.570421\pi$
$18$	0	0
$19$	− 2384.36i	− 0.0797506i	−0.999205	−	0.0398753i	$-0.987304\pi$
$19$	− 2384.36i	− 0.0797506i	0.999205	−	0.0398753i	$-0.0126961\pi$
$20$	0	0
$21$	12450.4i	0.293371i
$22$	0	0
$23$	44417.6	0.761216	0.380608	−	0.924736i	$-0.375715\pi$
$23$	44417.6	0.761216	0.380608	+	0.924736i	$0.375715\pi$
$24$	0	0
$25$	−14355.0	−0.183744
$26$	0	0
$27$	76160.3i	0.744656i
$28$	0	0
$29$	− 170460.i	− 1.29786i	−0.760846	−	0.648932i	$-0.775216\pi$
$29$	− 170460.i	− 1.29786i	0.760846	−	0.648932i	$-0.224784\pi$
$30$	0	0
$31$	108612.	0.654802	0.327401	−	0.944885i	$-0.393827\pi$
$31$	108612.	0.654802	0.327401	+	0.944885i	$0.393827\pi$
$32$	0	0
$33$	138840.	0.672536
$34$	0	0
$35$	199552.i	0.786717i
$36$	0	0
$37$	253677.i	0.823334i	0.911334	+	0.411667i	$0.135053\pi$
$37$	253677.i	0.823334i	−0.911334	+	0.411667i	$0.864947\pi$
$38$	0	0
$39$	163257.	0.440702
$40$	0	0
$41$	−122902.	−0.278494	−0.139247	−	0.990258i	$-0.544468\pi$
$41$	−122902.	−0.278494	−0.139247	+	0.990258i	$0.544468\pi$
$42$	0	0
$43$	− 869354.i	− 1.66747i	−0.552167	−	0.833734i	$-0.686199\pi$
$43$	− 869354.i	− 1.66747i	0.552167	−	0.833734i	$-0.313801\pi$
$44$	0	0
$45$	555600.i	0.908905i
$46$	0	0
$47$	1.31225e6	1.84364	0.921819	−	0.387621i	$-0.126703\pi$
$47$	1.31225e6	1.84364	0.921819	+	0.387621i	$0.126703\pi$
$48$	0	0
$49$	−392951.	−0.477147
$50$	0	0
$51$	− 168676.i	− 0.178056i
$52$	0	0
$53$	− 1.15787e6i	− 1.06830i	−0.845388	−	0.534152i	$-0.820631\pi$
$53$	− 1.15787e6i	− 1.06830i	0.845388	−	0.534152i	$-0.179369\pi$
$54$	0	0
$55$	2.22529e6	1.80351
$56$	0	0
$57$	45240.0	0.0323564
$58$	0	0
$59$	1.53018e6i	0.969976i	0.874521	+	0.484988i	$0.161176\pi$
$59$	1.53018e6i	0.969976i	−0.874521	+	0.484988i	$0.838824\pi$
$60$	0	0
$61$	1.29461e6i	0.730273i	0.930954	+	0.365136i	$0.118978\pi$
$61$	1.29461e6i	0.730273i	−0.930954	+	0.365136i	$0.881022\pi$
$62$	0	0
$63$	1.19887e6	0.604059
$64$	0	0
$65$	2.61664e6	1.18181
$66$	0	0
$67$	798760.i	0.324455i	0.986753	+	0.162227i	$0.0518678\pi$
$67$	798760.i	0.324455i	−0.986753	+	0.162227i	$0.948132\pi$
$68$	0	0
$69$	842765.i	0.308841i
$70$	0	0
$71$	4.47084e6	1.48247	0.741233	−	0.671248i	$-0.234241\pi$
$71$	4.47084e6	1.48247	0.741233	+	0.671248i	$0.234241\pi$
$72$	0	0
$73$	−3.84917e6	−1.15808	−0.579038	−	0.815301i	$-0.696572\pi$
$73$	−3.84917e6	−1.15808	−0.579038	+	0.815301i	$0.696572\pi$
$74$	0	0
$75$	− 272367.i	− 0.0745487i
$76$	0	0
$77$	− 4.80171e6i	− 1.19861i
$78$	0	0
$79$	7.33332e6	1.67342	0.836712	−	0.547644i	$-0.184475\pi$
$79$	7.33332e6	1.67342	0.836712	+	0.547644i	$0.184475\pi$
$80$	0	0
$81$	2.55061e6	0.533269
$82$	0	0
$83$	452452.i	0.0868559i	0.999057	+	0.0434280i	$0.0138279\pi$
$83$	452452.i	0.0868559i	−0.999057	+	0.0434280i	$0.986172\pi$
$84$	0	0
$85$	− 2.70350e6i	− 0.477485i
$86$	0	0
$87$	3.23425e6	0.526570
$88$	0	0
$89$	−5.43559e6	−0.817300	−0.408650	−	0.912691i	$-0.634000\pi$
$89$	−5.43559e6	−0.817300	−0.408650	+	0.912691i	$0.634000\pi$
$90$	0	0
$91$	− 5.64616e6i	− 0.785431i
$92$	0	0
$93$	2.06076e6i	0.265667i
$94$	0	0
$95$	725096.	0.0867686
$96$	0	0
$97$	6.17903e6	0.687415	0.343708	−	0.939077i	$-0.388317\pi$
$97$	6.17903e6	0.687415	0.343708	+	0.939077i	$0.388317\pi$
$98$	0	0
$99$	− 1.33691e7i	− 1.38477i
$100$	0	0
$101$	1.64351e7i	1.58726i	0.608403	+	0.793628i	$0.291810\pi$
$101$	1.64351e7i	1.58726i	−0.608403	+	0.793628i	$0.708190\pi$
$102$	0	0
$103$	−6.86885e6	−0.619375	−0.309687	−	0.950838i	$-0.600224\pi$
$103$	−6.86885e6	−0.619375	−0.309687	+	0.950838i	$0.600224\pi$
$104$	0	0
$105$	−3.78624e6	−0.319187
$106$	0	0
$107$	2.06652e7i	1.63079i	0.578908	+	0.815393i	$0.303479\pi$
$107$	2.06652e7i	1.63079i	−0.578908	+	0.815393i	$0.696521\pi$
$108$	0	0
$109$	− 6.67996e6i	− 0.494061i	−0.969008	−	0.247031i	$-0.920545\pi$
$109$	− 6.67996e6i	− 0.494061i	0.969008	−	0.247031i	$-0.0794549\pi$
$110$	0	0
$111$	−4.81319e6	−0.334043
$112$	0	0
$113$	2.20276e7	1.43612	0.718062	−	0.695979i	$-0.245029\pi$
$113$	2.20276e7	1.43612	0.718062	+	0.695979i	$0.245029\pi$
$114$	0	0
$115$	1.35076e7i	0.828203i
$116$	0	0
$117$	− 1.57202e7i	− 0.907419i
$118$	0	0
$119$	−5.83357e6	−0.317337
$120$	0	0
$121$	−3.40588e7	−1.74775
$122$	0	0
$123$	− 2.33190e6i	− 0.112991i
$124$	0	0
$125$	1.93928e7i	0.888087i
$126$	0	0
$127$	−2.65129e7	−1.14854	−0.574268	−	0.818668i	$-0.694713\pi$
$127$	−2.65129e7	−1.14854	−0.574268	+	0.818668i	$0.694713\pi$
$128$	0	0
$129$	1.64948e7	0.676525
$130$	0	0
$131$	1.06575e7i	0.414197i	0.978320	+	0.207098i	$0.0664021\pi$
$131$	1.06575e7i	0.414197i	−0.978320	+	0.207098i	$0.933598\pi$
$132$	0	0
$133$	− 1.56460e6i	− 0.0576665i
$134$	0	0
$135$	−2.31607e7	−0.810185
$136$	0	0
$137$	−3.53739e7	−1.17533	−0.587667	−	0.809103i	$-0.699953\pi$
$137$	−3.53739e7	−1.17533	−0.587667	+	0.809103i	$0.699953\pi$
$138$	0	0
$139$	− 5.25690e7i	− 1.66027i	−0.557565	−	0.830133i	$-0.688264\pi$
$139$	− 5.25690e7i	− 1.66027i	0.557565	−	0.830133i	$-0.311736\pi$
$140$	0	0
$141$	2.48983e7i	0.748001i
$142$	0	0
$143$	−6.29627e7	−1.80056
$144$	0	0
$145$	5.18378e7	1.41208
$146$	0	0
$147$	− 7.45572e6i	− 0.193588i
$148$	0	0
$149$	− 4.86290e7i	− 1.20432i	−0.798374	−	0.602162i	$-0.794306\pi$
$149$	− 4.86290e7i	− 1.20432i	0.798374	−	0.602162i	$-0.205694\pi$
$150$	0	0
$151$	−5.89690e7	−1.39381	−0.696906	−	0.717162i	$-0.745441\pi$
$151$	−5.89690e7	−1.39381	−0.696906	+	0.717162i	$0.745441\pi$
$152$	0	0
$153$	−1.62420e7	−0.366624
$154$	0	0
$155$	3.30294e7i	0.712425i
$156$	0	0
$157$	− 5.93808e7i	− 1.22461i	−0.790623	−	0.612304i	$-0.790243\pi$
$157$	− 5.93808e7i	− 1.22461i	0.790623	−	0.612304i	$-0.209757\pi$
$158$	0	0
$159$	2.19691e7	0.433433
$160$	0	0
$161$	2.91466e7	0.550424
$162$	0	0
$163$	− 3.15637e7i	− 0.570862i	−0.958399	−	0.285431i	$-0.907863\pi$
$163$	− 3.15637e7i	− 0.570862i	0.958399	−	0.285431i	$-0.0921368\pi$
$164$	0	0
$165$	4.22220e7i	0.731720i
$166$	0	0
$167$	−1.04068e7	−0.172905	−0.0864527	−	0.996256i	$-0.527553\pi$
$167$	−1.04068e7	−0.172905	−0.0864527	+	0.996256i	$0.527553\pi$
$168$	0	0
$169$	−1.12870e7	−0.179877
$170$	0	0
$171$	− 4.35622e6i	− 0.0666229i
$172$	0	0
$173$	− 2.64971e7i	− 0.389078i	−0.980895	−	0.194539i	$-0.937679\pi$
$173$	− 2.64971e7i	− 0.389078i	0.980895	−	0.194539i	$-0.0623211\pi$
$174$	0	0
$175$	−9.41968e6	−0.132863
$176$	0	0
$177$	−2.90332e7	−0.393539
$178$	0	0
$179$	7.09835e7i	0.925065i	0.886602	+	0.462532i	$0.153059\pi$
$179$	7.09835e7i	0.925065i	−0.886602	+	0.462532i	$0.846941\pi$
$180$	0	0
$181$	1.34126e8i	1.68128i	0.541597	+	0.840638i	$0.317820\pi$
$181$	1.34126e8i	1.68128i	−0.541597	+	0.840638i	$0.682180\pi$
$182$	0	0
$183$	−2.45635e7	−0.296287
$184$	0	0
$185$	−7.71446e7	−0.895787
$186$	0	0
$187$	6.50527e7i	0.727477i
$188$	0	0
$189$	4.99760e7i	0.538450i
$190$	0	0
$191$	4.26527e6	0.0442924	0.0221462	−	0.999755i	$-0.492950\pi$
$191$	4.26527e6	0.0442924	0.0221462	+	0.999755i	$0.492950\pi$
$192$	0	0
$193$	9.80464e7	0.981705	0.490853	−	0.871243i	$-0.336685\pi$
$193$	9.80464e7	0.981705	0.490853	+	0.871243i	$0.336685\pi$
$194$	0	0
$195$	4.96473e7i	0.479484i
$196$	0	0
$197$	− 1.42504e8i	− 1.32799i	−0.747736	−	0.663996i	$-0.768859\pi$
$197$	− 1.42504e8i	− 1.32799i	0.747736	−	0.663996i	$-0.231141\pi$
$198$	0	0
$199$	8.80778e7	0.792284	0.396142	−	0.918189i	$-0.370349\pi$
$199$	8.80778e7	0.792284	0.396142	+	0.918189i	$0.370349\pi$
$200$	0	0
$201$	−1.51554e7	−0.131638
$202$	0	0
$203$	− 1.11855e8i	− 0.938467i
$204$	0	0
$205$	− 3.73751e7i	− 0.303001i
$206$	0	0
$207$	8.11510e7	0.635913
$208$	0	0
$209$	−1.74476e7	−0.132197
$210$	0	0
$211$	− 8.87198e7i	− 0.650177i	−0.945684	−	0.325089i	$-0.894606\pi$
$211$	− 8.87198e7i	− 0.650177i	0.945684	−	0.325089i	$-0.105394\pi$
$212$	0	0
$213$	8.48282e7i	0.601467i
$214$	0	0
$215$	2.64375e8	1.81420
$216$	0	0
$217$	7.12704e7	0.473478
$218$	0	0
$219$	− 7.30329e7i	− 0.469855i
$220$	0	0
$221$	7.64930e7i	0.476704i
$222$	0	0
$223$	1.29630e8	0.782781	0.391390	−	0.920225i	$-0.371994\pi$
$223$	1.29630e8	0.782781	0.391390	+	0.920225i	$0.371994\pi$
$224$	0	0
$225$	−2.62266e7	−0.153498
$226$	0	0
$227$	− 3.05546e8i	− 1.73375i	−0.498525	−	0.866875i	$-0.666125\pi$
$227$	− 3.05546e8i	− 1.73375i	0.498525	−	0.866875i	$-0.333875\pi$
$228$	0	0
$229$	− 7.82110e7i	− 0.430372i	−0.976573	−	0.215186i	$-0.930964\pi$
$229$	− 7.82110e7i	− 0.430372i	0.976573	−	0.215186i	$-0.0690357\pi$
$230$	0	0
$231$	9.11061e7	0.486302
$232$	0	0
$233$	2.26800e8	1.17462	0.587311	−	0.809362i	$-0.300187\pi$
$233$	2.26800e8	1.17462	0.587311	+	0.809362i	$0.300187\pi$
$234$	0	0
$235$	3.99063e8i	2.00588i
$236$	0	0
$237$	1.39140e8i	0.678942i
$238$	0	0
$239$	2.67813e8	1.26894	0.634468	−	0.772949i	$-0.281219\pi$
$239$	2.67813e8	1.26894	0.634468	+	0.772949i	$0.281219\pi$
$240$	0	0
$241$	−3.67784e8	−1.69252	−0.846259	−	0.532772i	$-0.821150\pi$
$241$	−3.67784e8	−1.69252	−0.846259	+	0.532772i	$0.821150\pi$
$242$	0	0
$243$	2.14957e8i	0.961014i
$244$	0	0
$245$	− 1.19498e8i	− 0.519136i
$246$	0	0
$247$	−2.05159e7	−0.0866268
$248$	0	0
$249$	−8.58468e6	−0.0352392
$250$	0	0
$251$	− 1.13407e8i	− 0.452670i	−0.974049	−	0.226335i	$-0.927326\pi$
$251$	− 1.13407e8i	− 0.452670i	0.974049	−	0.226335i	$-0.0726744\pi$
$252$	0	0
$253$	− 3.25026e8i	− 1.26182i
$254$	0	0
$255$	5.12952e7	0.193725
$256$	0	0
$257$	−2.12976e8	−0.782645	−0.391322	−	0.920254i	$-0.627982\pi$
$257$	−2.12976e8	−0.782645	−0.391322	+	0.920254i	$0.627982\pi$
$258$	0	0
$259$	1.66462e8i	0.595341i
$260$	0	0
$261$	− 3.11430e8i	− 1.08422i
$262$	0	0
$263$	6.19202e7	0.209888	0.104944	−	0.994478i	$-0.466534\pi$
$263$	6.19202e7	0.209888	0.104944	+	0.994478i	$0.466534\pi$
$264$	0	0
$265$	3.52115e8	1.16231
$266$	0	0
$267$	− 1.03133e8i	− 0.331595i
$268$	0	0
$269$	2.92571e8i	0.916429i	0.888842	+	0.458214i	$0.151511\pi$
$269$	2.92571e8i	0.916429i	−0.888842	+	0.458214i	$0.848489\pi$
$270$	0	0
$271$	6.34959e7	0.193800	0.0968999	−	0.995294i	$-0.469107\pi$
$271$	6.34959e7	0.193800	0.0968999	+	0.995294i	$0.469107\pi$
$272$	0	0
$273$	1.07128e8	0.318665
$274$	0	0
$275$	1.05043e8i	0.304581i
$276$	0	0
$277$	3.96328e8i	1.12041i	0.828356	+	0.560203i	$0.189277\pi$
$277$	3.96328e8i	1.12041i	−0.828356	+	0.560203i	$0.810723\pi$
$278$	0	0
$279$	1.98433e8	0.547016
$280$	0	0
$281$	1.17486e8	0.315873	0.157937	−	0.987449i	$-0.449516\pi$
$281$	1.17486e8	0.315873	0.157937	+	0.987449i	$0.449516\pi$
$282$	0	0
$283$	− 1.39179e8i	− 0.365025i	−0.983204	−	0.182513i	$-0.941577\pi$
$283$	− 1.39179e8i	− 0.365025i	0.983204	−	0.182513i	$-0.0584230\pi$
$284$	0	0
$285$	1.37577e7i	0.0352038i
$286$	0	0
$287$	−8.06477e7	−0.201375
$288$	0	0
$289$	−3.31307e8	−0.807398
$290$	0	0
$291$	1.17239e8i	0.278898i
$292$	0	0
$293$	6.05094e8i	1.40535i	0.711509	+	0.702677i	$0.248012\pi$
$293$	6.05094e8i	1.40535i	−0.711509	+	0.702677i	$0.751988\pi$
$294$	0	0
$295$	−4.65336e8	−1.05533
$296$	0	0
$297$	5.57304e8	1.23437
$298$	0	0
$299$	− 3.82187e8i	− 0.826849i
$300$	0	0
$301$	− 5.70466e8i	− 1.20572i
$302$	0	0
$303$	−3.11834e8	−0.643982
$304$	0	0
$305$	−3.93698e8	−0.794537
$306$	0	0
$307$	− 6.47350e8i	− 1.27689i	−0.769665	−	0.638447i	$-0.779577\pi$
$307$	− 6.47350e8i	− 1.27689i	0.769665	−	0.638447i	$-0.220423\pi$
$308$	0	0
$309$	− 1.30327e8i	− 0.251293i
$310$	0	0
$311$	−5.08804e8	−0.959155	−0.479578	−	0.877499i	$-0.659210\pi$
$311$	−5.08804e8	−0.959155	−0.479578	+	0.877499i	$0.659210\pi$
$312$	0	0
$313$	−2.71720e8	−0.500859	−0.250430	−	0.968135i	$-0.580572\pi$
$313$	−2.71720e8	−0.500859	−0.250430	+	0.968135i	$0.580572\pi$
$314$	0	0
$315$	3.64582e8i	0.657217i
$316$	0	0
$317$	− 1.61191e8i	− 0.284206i	−0.989852	−	0.142103i	$-0.954614\pi$
$317$	− 1.61191e8i	− 0.284206i	0.989852	−	0.142103i	$-0.0453863\pi$
$318$	0	0
$319$	−1.24734e9	−2.15139
$320$	0	0
$321$	−3.92095e8	−0.661643
$322$	0	0
$323$	2.11969e7i	0.0349997i
$324$	0	0
$325$	1.23516e8i	0.199587i
$326$	0	0
$327$	1.26743e8	0.200451
$328$	0	0
$329$	8.61095e8	1.33311
$330$	0	0
$331$	− 1.01812e8i	− 0.154313i	−0.997019	−	0.0771566i	$-0.975416\pi$
$331$	− 1.01812e8i	− 0.154313i	0.997019	−	0.0771566i	$-0.0245841\pi$
$332$	0	0
$333$	4.63469e8i	0.687805i
$334$	0	0
$335$	−2.42907e8	−0.353007
$336$	0	0
$337$	−5.88023e8	−0.836932	−0.418466	−	0.908233i	$-0.637432\pi$
$337$	−5.88023e8	−0.836932	−0.418466	+	0.908233i	$0.637432\pi$
$338$	0	0
$339$	4.17944e8i	0.582665i
$340$	0	0
$341$	− 7.94767e8i	− 1.08542i
$342$	0	0
$343$	−7.98257e8	−1.06810
$344$	0	0
$345$	−2.56289e8	−0.336019
$346$	0	0
$347$	8.09980e8i	1.04069i	0.853957	+	0.520344i	$0.174196\pi$
$347$	8.09980e8i	1.04069i	−0.853957	+	0.520344i	$0.825804\pi$
$348$	0	0
$349$	1.32146e9i	1.66405i	0.554739	+	0.832025i	$0.312818\pi$
$349$	1.32146e9i	1.66405i	−0.554739	+	0.832025i	$0.687182\pi$
$350$	0	0
$351$	6.55313e8	0.808861
$352$	0	0
$353$	6.44478e8	0.779824	0.389912	−	0.920852i	$-0.372505\pi$
$353$	6.44478e8	0.779824	0.389912	+	0.920852i	$0.372505\pi$
$354$	0	0
$355$	1.35961e9i	1.61292i
$356$	0	0
$357$	− 1.10684e8i	− 0.128750i
$358$	0	0
$359$	−6.64364e8	−0.757837	−0.378918	−	0.925430i	$-0.623704\pi$
$359$	−6.64364e8	−0.757837	−0.378918	+	0.925430i	$0.623704\pi$
$360$	0	0
$361$	8.88187e8	0.993640
$362$	0	0
$363$	− 6.46220e8i	− 0.709100i
$364$	0	0
$365$	− 1.17055e9i	− 1.25999i
$366$	0	0
$367$	−1.32224e9	−1.39631	−0.698153	−	0.715949i	$-0.745994\pi$
$367$	−1.32224e9	−1.39631	−0.698153	+	0.715949i	$0.745994\pi$
$368$	0	0
$369$	−2.24542e8	−0.232651
$370$	0	0
$371$	− 7.59790e8i	− 0.772476i
$372$	0	0
$373$	− 1.00411e9i	− 1.00185i	−0.865491	−	0.500924i	$-0.832994\pi$
$373$	− 1.00411e9i	− 1.00185i	0.865491	−	0.500924i	$-0.167006\pi$
$374$	0	0
$375$	−3.67952e8	−0.360315
$376$	0	0
$377$	−1.46670e9	−1.40977
$378$	0	0
$379$	− 1.18744e9i	− 1.12040i	−0.828356	−	0.560202i	$-0.810723\pi$
$379$	− 1.18744e9i	− 1.12040i	0.828356	−	0.560202i	$-0.189277\pi$
$380$	0	0
$381$	− 5.03047e8i	− 0.465984i
$382$	0	0
$383$	1.02632e9	0.933442	0.466721	−	0.884405i	$-0.345435\pi$
$383$	1.02632e9	0.933442	0.466721	+	0.884405i	$0.345435\pi$
$384$	0	0
$385$	1.46023e9	1.30409
$386$	0	0
$387$	− 1.58831e9i	− 1.39299i
$388$	0	0
$389$	3.97443e8i	0.342335i	0.985242	+	0.171167i	$0.0547539\pi$
$389$	3.97443e8i	0.342335i	−0.985242	+	0.171167i	$0.945246\pi$
$390$	0	0
$391$	−3.94873e8	−0.334071
$392$	0	0
$393$	−2.02212e8	−0.168048
$394$	0	0
$395$	2.23010e9i	1.82068i
$396$	0	0
$397$	5.63579e8i	0.452052i	0.974121	+	0.226026i	$0.0725734\pi$
$397$	5.63579e8i	0.452052i	−0.974121	+	0.226026i	$0.927427\pi$
$398$	0	0
$399$	2.96863e7	0.0233965
$400$	0	0
$401$	−4.84191e8	−0.374983	−0.187491	−	0.982266i	$-0.560036\pi$
$401$	−4.84191e8	−0.374983	−0.187491	+	0.982266i	$0.560036\pi$
$402$	0	0
$403$	− 9.34537e8i	− 0.711260i
$404$	0	0
$405$	7.75654e8i	0.580197i
$406$	0	0
$407$	1.85629e9	1.36479
$408$	0	0
$409$	−1.84969e9	−1.33680	−0.668402	−	0.743800i	$-0.733022\pi$
$409$	−1.84969e9	−1.33680	−0.668402	+	0.743800i	$0.733022\pi$
$410$	0	0
$411$	− 6.71173e8i	− 0.476857i
$412$	0	0
$413$	1.00410e9i	0.701376i
$414$	0	0
$415$	−1.37593e8	−0.0944993
$416$	0	0
$417$	9.97427e8	0.673604
$418$	0	0
$419$	2.85977e8i	0.189925i	0.995481	+	0.0949625i	$0.0302731\pi$
$419$	2.85977e8i	0.189925i	−0.995481	+	0.0949625i	$0.969727\pi$
$420$	0	0
$421$	1.65356e9i	1.08002i	0.841658	+	0.540011i	$0.181580\pi$
$421$	1.65356e9i	1.08002i	−0.841658	+	0.540011i	$0.818420\pi$
$422$	0	0
$423$	2.39749e9	1.54016
$424$	0	0
$425$	1.27616e8	0.0806387
$426$	0	0
$427$	8.49518e8i	0.528050i
$428$	0	0
$429$	− 1.19463e9i	− 0.730523i
$430$	0	0
$431$	2.38205e9	1.43311	0.716556	−	0.697530i	$-0.245718\pi$
$431$	2.38205e9	1.43311	0.716556	+	0.697530i	$0.245718\pi$
$432$	0	0
$433$	6.70100e8	0.396673	0.198336	−	0.980134i	$-0.436446\pi$
$433$	6.70100e8	0.396673	0.198336	+	0.980134i	$0.436446\pi$
$434$	0	0
$435$	9.83552e8i	0.572908i
$436$	0	0
$437$	− 1.05907e8i	− 0.0607074i
$438$	0	0
$439$	−2.38414e6	−0.00134495	−0.000672474	−	1.00000i	$-0.500214\pi$
$439$	−2.38414e6	−0.00134495	−0.000672474		1.00000i	$0.500214\pi$
$440$	0	0
$441$	−7.17921e8	−0.398604
$442$	0	0
$443$	− 4.55084e8i	− 0.248701i	−0.992238	−	0.124351i	$-0.960315\pi$
$443$	− 4.55084e8i	− 0.248701i	0.992238	−	0.124351i	$-0.0396848\pi$
$444$	0	0
$445$	− 1.65299e9i	− 0.889222i
$446$	0	0
$447$	9.22670e8	0.488619
$448$	0	0
$449$	2.40223e9	1.25242	0.626212	−	0.779652i	$-0.284604\pi$
$449$	2.40223e9	1.25242	0.626212	+	0.779652i	$0.284604\pi$
$450$	0	0
$451$	8.99337e8i	0.461641i
$452$	0	0
$453$	− 1.11886e9i	− 0.565498i
$454$	0	0
$455$	1.71703e9	0.854549
$456$	0	0
$457$	1.79624e9	0.880356	0.440178	−	0.897911i	$-0.354915\pi$
$457$	1.79624e9	0.880356	0.440178	+	0.897911i	$0.354915\pi$
$458$	0	0
$459$	− 6.77065e8i	− 0.326803i
$460$	0	0
$461$	2.10954e9i	1.00285i	0.865203	+	0.501423i	$0.167190\pi$
$461$	2.10954e9i	1.00285i	−0.865203	+	0.501423i	$0.832810\pi$
$462$	0	0
$463$	−1.01037e8	−0.0473093	−0.0236546	−	0.999720i	$-0.507530\pi$
$463$	−1.01037e8	−0.0473093	−0.0236546	+	0.999720i	$0.507530\pi$
$464$	0	0
$465$	−6.26688e8	−0.289045
$466$	0	0
$467$	2.58934e9i	1.17647i	0.808691	+	0.588233i	$0.200176\pi$
$467$	2.58934e9i	1.17647i	−0.808691	+	0.588233i	$0.799824\pi$
$468$	0	0
$469$	5.24142e8i	0.234609i
$470$	0	0
$471$	1.12667e9	0.496848
$472$	0	0
$473$	−6.36151e9	−2.76405
$474$	0	0
$475$	3.42274e7i	0.0146537i
$476$	0	0
$477$	− 2.11543e9i	− 0.892452i
$478$	0	0
$479$	1.80092e9	0.748723	0.374361	−	0.927283i	$-0.377862\pi$
$479$	1.80092e9	0.748723	0.374361	+	0.927283i	$0.377862\pi$
$480$	0	0
$481$	2.18274e9	0.894322
$482$	0	0
$483$	5.53018e8i	0.223318i
$484$	0	0
$485$	1.87908e9i	0.747908i
$486$	0	0
$487$	−1.95227e9	−0.765928	−0.382964	−	0.923763i	$-0.625097\pi$
$487$	−1.95227e9	−0.765928	−0.382964	+	0.923763i	$0.625097\pi$
$488$	0	0
$489$	5.98879e8	0.231611
$490$	0	0
$491$	− 1.74715e9i	− 0.666107i	−0.942908	−	0.333053i	$-0.891921\pi$
$491$	− 1.74715e9i	− 0.666107i	0.942908	−	0.333053i	$-0.108079\pi$
$492$	0	0
$493$	1.51539e9i	0.569587i
$494$	0	0
$495$	4.06561e9	1.50663
$496$	0	0
$497$	2.93374e9	1.07195
$498$	0	0
$499$	− 5.22157e9i	− 1.88126i	−0.339433	−	0.940630i	$-0.610235\pi$
$499$	− 5.22157e9i	− 1.88126i	0.339433	−	0.940630i	$-0.389765\pi$
$500$	0	0
$501$	− 1.97455e8i	− 0.0701513i
$502$	0	0
$503$	2.67104e9	0.935820	0.467910	−	0.883776i	$-0.345007\pi$
$503$	2.67104e9	0.935820	0.467910	+	0.883776i	$0.345007\pi$
$504$	0	0
$505$	−4.99799e9	−1.72693
$506$	0	0
$507$	− 2.14156e8i	− 0.0729797i
$508$	0	0
$509$	1.71394e8i	0.0576080i	0.999585	+	0.0288040i	$0.00916986\pi$
$509$	1.71394e8i	0.0576080i	−0.999585	+	0.0288040i	$0.990830\pi$
$510$	0	0
$511$	−2.52581e9	−0.837388
$512$	0	0
$513$	1.81593e8	0.0593867
$514$	0	0
$515$	− 2.08885e9i	− 0.673880i
$516$	0	0
$517$	− 9.60244e9i	− 3.05608i
$518$	0	0
$519$	5.02747e8	0.157857
$520$	0	0
$521$	3.79368e9	1.17525	0.587623	−	0.809135i	$-0.300064\pi$
$521$	3.79368e9	1.17525	0.587623	+	0.809135i	$0.300064\pi$
$522$	0	0
$523$	− 3.00670e8i	− 0.0919039i	−0.998944	−	0.0459520i	$-0.985368\pi$
$523$	− 3.00670e8i	− 0.0919039i	0.998944	−	0.0459520i	$-0.0146321\pi$
$524$	0	0
$525$	− 1.78726e8i	− 0.0539051i
$526$	0	0
$527$	−9.65557e8	−0.287370
$528$	0	0
$529$	−1.43190e9	−0.420550
$530$	0	0
$531$	2.79564e9i	0.810309i
$532$	0	0
$533$	1.05750e9i	0.302506i
$534$	0	0
$535$	−6.28440e9	−1.77429
$536$	0	0
$537$	−1.34682e9	−0.375318
$538$	0	0
$539$	2.87542e9i	0.790936i
$540$	0	0
$541$	5.26924e9i	1.43073i	0.698751	+	0.715365i	$0.253739\pi$
$541$	5.26924e9i	1.43073i	−0.698751	+	0.715365i	$0.746261\pi$
$542$	0	0
$543$	−2.54487e9	−0.682128
$544$	0	0
$545$	2.03141e9	0.537539
$546$	0	0
$547$	2.71695e9i	0.709783i	0.934908	+	0.354891i	$0.115482\pi$
$547$	2.71695e9i	0.709783i	−0.934908	+	0.354891i	$0.884518\pi$
$548$	0	0
$549$	2.36526e9i	0.610063i
$550$	0	0
$551$	−4.06437e8	−0.103505
$552$	0	0
$553$	4.81209e9	1.21003
$554$	0	0
$555$	− 1.46372e9i	− 0.363439i
$556$	0	0
$557$	− 3.09754e9i	− 0.759493i	−0.925091	−	0.379746i	$-0.876011\pi$
$557$	− 3.09754e9i	− 0.759493i	0.925091	−	0.379746i	$-0.123989\pi$
$558$	0	0
$559$	−7.48026e9	−1.81124
$560$	0	0
$561$	−1.23429e9	−0.295152
$562$	0	0
$563$	2.25890e9i	0.533478i	0.963769	+	0.266739i	$0.0859462\pi$
$563$	2.25890e9i	0.533478i	−0.963769	+	0.266739i	$0.914054\pi$
$564$	0	0
$565$	6.69870e9i	1.56250i
$566$	0	0
$567$	1.67370e9	0.385599
$568$	0	0
$569$	−3.12185e9	−0.710426	−0.355213	−	0.934785i	$-0.615592\pi$
$569$	−3.12185e9	−0.710426	−0.355213	+	0.934785i	$0.615592\pi$
$570$	0	0
$571$	4.08140e9i	0.917451i	0.888578	+	0.458726i	$0.151694\pi$
$571$	4.08140e9i	0.917451i	−0.888578	+	0.458726i	$0.848306\pi$
$572$	0	0
$573$	8.09278e7i	0.0179703i
$574$	0	0
$575$	−6.37615e8	−0.139869
$576$	0	0
$577$	−3.23584e9	−0.701247	−0.350624	−	0.936516i	$-0.614030\pi$
$577$	−3.23584e9	−0.701247	−0.350624	+	0.936516i	$0.614030\pi$
$578$	0	0
$579$	1.86030e9i	0.398298i
$580$	0	0
$581$	2.96897e8i	0.0628043i
$582$	0	0
$583$	−8.47274e9	−1.77086
$584$	0	0
$585$	4.78060e9	0.987272
$586$	0	0
$587$	− 6.86056e9i	− 1.40000i	−0.714145	−	0.699998i	$-0.753184\pi$
$587$	− 6.86056e9i	− 1.40000i	0.714145	−	0.699998i	$-0.246816\pi$
$588$	0	0
$589$	− 2.58969e8i	− 0.0522209i
$590$	0	0
$591$	2.70382e9	0.538793
$592$	0	0
$593$	−4.59082e9	−0.904063	−0.452032	−	0.892002i	$-0.649301\pi$
$593$	−4.59082e9	−0.904063	−0.452032	+	0.892002i	$0.649301\pi$
$594$	0	0
$595$	− 1.77402e9i	− 0.345262i
$596$	0	0
$597$	1.67116e9i	0.321446i
$598$	0	0
$599$	1.06090e9	0.201689	0.100844	−	0.994902i	$-0.467846\pi$
$599$	1.06090e9	0.201689	0.100844	+	0.994902i	$0.467846\pi$
$600$	0	0
$601$	−2.96035e9	−0.556266	−0.278133	−	0.960543i	$-0.589716\pi$
$601$	−2.96035e9	−0.556266	−0.278133	+	0.960543i	$0.589716\pi$
$602$	0	0
$603$	1.45933e9i	0.271047i
$604$	0	0
$605$	− 1.03575e10i	− 1.90156i
$606$	0	0
$607$	−9.64415e9	−1.75026	−0.875132	−	0.483885i	$-0.839225\pi$
$607$	−9.64415e9	−1.75026	−0.875132	+	0.483885i	$0.839225\pi$
$608$	0	0
$609$	2.12230e9	0.380755
$610$	0	0
$611$	− 1.12911e10i	− 2.00260i
$612$	0	0
$613$	− 5.49491e8i	− 0.0963494i	−0.998839	−	0.0481747i	$-0.984660\pi$
$613$	− 5.49491e8i	− 0.0963494i	0.998839	−	0.0481747i	$-0.0153404\pi$
$614$	0	0
$615$	7.09143e8	0.122934
$616$	0	0
$617$	4.08820e9	0.700703	0.350352	−	0.936618i	$-0.386062\pi$
$617$	4.08820e9	0.700703	0.350352	+	0.936618i	$0.386062\pi$
$618$	0	0
$619$	1.14901e10i	1.94718i	0.228310	+	0.973588i	$0.426680\pi$
$619$	1.14901e10i	1.94718i	−0.228310	+	0.973588i	$0.573320\pi$
$620$	0	0
$621$	3.38286e9i	0.566844i
$622$	0	0
$623$	−3.56680e9	−0.590978
$624$	0	0
$625$	−7.01893e9	−1.14998
$626$	0	0
$627$	− 3.31044e8i	− 0.0536352i
$628$	0	0
$629$	− 2.25519e9i	− 0.361332i
$630$	0	0
$631$	−1.02253e10	−1.62022	−0.810111	−	0.586277i	$-0.800593\pi$
$631$	−1.02253e10	−1.62022	−0.810111	+	0.586277i	$0.800593\pi$
$632$	0	0
$633$	1.68334e9	0.263790
$634$	0	0
$635$	− 8.06271e9i	− 1.24961i
$636$	0	0
$637$	3.38110e9i	0.518287i
$638$	0	0
$639$	8.16822e9	1.23844
$640$	0	0
$641$	4.63206e9	0.694658	0.347329	−	0.937743i	$-0.387089\pi$
$641$	4.63206e9	0.694658	0.347329	+	0.937743i	$0.387089\pi$
$642$	0	0
$643$	− 8.83206e8i	− 0.131016i	−0.997852	−	0.0655079i	$-0.979133\pi$
$643$	− 8.83206e8i	− 0.131016i	0.997852	−	0.0655079i	$-0.0208668\pi$
$644$	0	0
$645$	5.01617e9i	0.736060i
$646$	0	0
$647$	4.87252e8	0.0707275	0.0353637	−	0.999375i	$-0.488741\pi$
$647$	4.87252e8	0.0707275	0.0353637	+	0.999375i	$0.488741\pi$
$648$	0	0
$649$	1.11971e10	1.60787
$650$	0	0
$651$	1.35226e9i	0.192100i
$652$	0	0
$653$	3.94136e9i	0.553924i	0.960881	+	0.276962i	$0.0893277\pi$
$653$	3.94136e9i	0.553924i	−0.960881	+	0.276962i	$0.910672\pi$
$654$	0	0
$655$	−3.24101e9	−0.450646
$656$	0	0
$657$	−7.03243e9	−0.967446
$658$	0	0
$659$	2.22234e9i	0.302490i	0.988496	+	0.151245i	$0.0483283\pi$
$659$	2.22234e9i	0.302490i	−0.988496	+	0.151245i	$0.951672\pi$
$660$	0	0
$661$	− 1.01908e10i	− 1.37247i	−0.727378	−	0.686237i	$-0.759261\pi$
$661$	− 1.01908e10i	− 1.37247i	0.727378	−	0.686237i	$-0.240739\pi$
$662$	0	0
$663$	−1.45135e9	−0.193409
$664$	0	0
$665$	4.75804e8	0.0627412
$666$	0	0
$667$	− 7.57142e9i	− 0.987955i
$668$	0	0
$669$	2.45957e9i	0.317590i
$670$	0	0
$671$	9.47334e9	1.21053
$672$	0	0
$673$	−3.98658e9	−0.504136	−0.252068	−	0.967709i	$-0.581111\pi$
$673$	−3.98658e9	−0.504136	−0.252068	+	0.967709i	$0.581111\pi$
$674$	0	0
$675$	− 1.09328e9i	− 0.136826i
$676$	0	0
$677$	− 1.55337e9i	− 0.192404i	−0.995362	−	0.0962022i	$-0.969330\pi$
$677$	− 1.55337e9i	− 0.192404i	0.995362	−	0.0962022i	$-0.0306696\pi$
$678$	0	0
$679$	4.05465e9	0.497060
$680$	0	0
$681$	5.79734e9	0.703418
$682$	0	0
$683$	9.13483e9i	1.09705i	0.836133	+	0.548527i	$0.184811\pi$
$683$	9.13483e9i	1.09705i	−0.836133	+	0.548527i	$0.815189\pi$
$684$	0	0
$685$	− 1.07574e10i	− 1.27876i
$686$	0	0
$687$	1.48395e9	0.174611
$688$	0	0
$689$	−9.96278e9	−1.16041
$690$	0	0
$691$	1.66040e10i	1.91443i	0.289380	+	0.957214i	$0.406551\pi$
$691$	1.66040e10i	1.91443i	−0.289380	+	0.957214i	$0.593449\pi$
$692$	0	0
$693$	− 8.77273e9i	− 1.00131i
$694$	0	0
$695$	1.59865e10	1.80637
$696$	0	0
$697$	1.09260e9	0.122221
$698$	0	0
$699$	4.30323e9i	0.476568i
$700$	0	0
$701$	2.43703e9i	0.267207i	0.991035	+	0.133603i	$0.0426548\pi$
$701$	2.43703e9i	0.267207i	−0.991035	+	0.133603i	$0.957345\pi$
$702$	0	0
$703$	6.04858e8	0.0656613
$704$	0	0
$705$	−7.57170e9	−0.813826
$706$	0	0
$707$	1.07846e10i	1.14772i
$708$	0	0
$709$	8.96235e9i	0.944409i	0.881489	+	0.472205i	$0.156542\pi$
$709$	8.96235e9i	0.944409i	−0.881489	+	0.472205i	$0.843458\pi$
$710$	0	0
$711$	1.33980e10	1.39796
$712$	0	0
$713$	4.82427e9	0.498446
$714$	0	0
$715$	− 1.91473e10i	− 1.95901i
$716$	0	0
$717$	5.08140e9i	0.514833i
$718$	0	0
$719$	4.61294e9	0.462836	0.231418	−	0.972854i	$-0.425664\pi$
$719$	4.61294e9	0.462836	0.231418	+	0.972854i	$0.425664\pi$
$720$	0	0
$721$	−4.50730e9	−0.447861
$722$	0	0
$723$	− 6.97821e9i	− 0.686689i
$724$	0	0
$725$	2.44695e9i	0.238475i
$726$	0	0
$727$	−4.48188e9	−0.432604	−0.216302	−	0.976327i	$-0.569400\pi$
$727$	−4.48188e9	−0.432604	−0.216302	+	0.976327i	$0.569400\pi$
$728$	0	0
$729$	1.49966e9	0.143366
$730$	0	0
$731$	7.72856e9i	0.731792i
$732$	0	0
$733$	7.54299e9i	0.707424i	0.935354	+	0.353712i	$0.115081\pi$
$733$	7.54299e9i	0.707424i	−0.935354	+	0.353712i	$0.884919\pi$
$734$	0	0
$735$	2.26732e9	0.210624
$736$	0	0
$737$	5.84493e9	0.537828
$738$	0	0
$739$	− 8.64053e9i	− 0.787562i	−0.919204	−	0.393781i	$-0.871167\pi$
$739$	− 8.64053e9i	− 0.787562i	0.919204	−	0.393781i	$-0.128833\pi$
$740$	0	0
$741$	− 3.89263e8i	− 0.0351462i
$742$	0	0
$743$	5.33165e9	0.476871	0.238435	−	0.971158i	$-0.423366\pi$
$743$	5.33165e9	0.476871	0.238435	+	0.971158i	$0.423366\pi$
$744$	0	0
$745$	1.47883e10	1.31030
$746$	0	0
$747$	8.26630e8i	0.0725587i
$748$	0	0
$749$	1.35604e10i	1.17920i
$750$	0	0
$751$	5.55902e9	0.478915	0.239458	−	0.970907i	$-0.423030\pi$
$751$	5.55902e9	0.478915	0.239458	+	0.970907i	$0.423030\pi$
$752$	0	0
$753$	2.15175e9	0.183658
$754$	0	0
$755$	− 1.79328e10i	− 1.51647i
$756$	0	0
$757$	1.06555e9i	0.0892767i	0.999003	+	0.0446383i	$0.0142135\pi$
$757$	1.06555e9i	0.0892767i	−0.999003	+	0.0446383i	$0.985786\pi$
$758$	0	0
$759$	6.16694e9	0.511945
$760$	0	0
$761$	−2.06800e10	−1.70100	−0.850501	−	0.525974i	$-0.823701\pi$
$761$	−2.06800e10	−1.70100	−0.850501	+	0.525974i	$0.823701\pi$
$762$	0	0
$763$	− 4.38336e9i	− 0.357249i
$764$	0	0
$765$	− 4.93929e9i	− 0.398886i
$766$	0	0
$767$	1.31663e10	1.05361
$768$	0	0
$769$	1.49795e9	0.118784	0.0593918	−	0.998235i	$-0.481084\pi$
$769$	1.49795e9	0.118784	0.0593918	+	0.998235i	$0.481084\pi$
$770$	0	0
$771$	− 4.04094e9i	− 0.317535i
$772$	0	0
$773$	4.29468e9i	0.334428i	0.985921	+	0.167214i	$0.0534771\pi$
$773$	4.29468e9i	0.334428i	−0.985921	+	0.167214i	$0.946523\pi$
$774$	0	0
$775$	−1.55912e9	−0.120316
$776$	0	0
$777$	−3.15839e9	−0.241542
$778$	0	0
$779$	2.93042e8i	0.0222100i
$780$	0	0
$781$	− 3.27154e10i	− 2.45739i
$782$	0	0
$783$	1.29823e10	0.966462
$784$	0	0
$785$	1.80580e10	1.33237
$786$	0	0
$787$	8.50862e9i	0.622225i	0.950373	+	0.311112i	$0.100702\pi$
$787$	8.50862e9i	0.622225i	−0.950373	+	0.311112i	$0.899298\pi$
$788$	0	0
$789$	1.17485e9i	0.0851558i
$790$	0	0
$791$	1.44544e10	1.03844
$792$	0	0
$793$	1.11393e10	0.793238
$794$	0	0
$795$	6.68091e9i	0.471575i
$796$	0	0
$797$	5.17223e8i	0.0361888i	0.999836	+	0.0180944i	$0.00575994\pi$
$797$	5.17223e8i	0.0361888i	−0.999836	+	0.0180944i	$0.994240\pi$
$798$	0	0
$799$	−1.16659e10	−0.809107
$800$	0	0
$801$	−9.93082e9	−0.682765
$802$	0	0
$803$	2.81663e10i	1.91967i
$804$	0	0
$805$	8.86364e9i	0.598862i
$806$	0	0
$807$	−5.55115e9	−0.371814
$808$	0	0
$809$	−1.65016e10	−1.09574	−0.547868	−	0.836565i	$-0.684560\pi$
$809$	−1.65016e10	−1.09574	−0.547868	+	0.836565i	$0.684560\pi$
$810$	0	0
$811$	− 1.05694e10i	− 0.695792i	−0.937533	−	0.347896i	$-0.886896\pi$
$811$	− 1.05694e10i	− 0.695792i	0.937533	−	0.347896i	$-0.113104\pi$
$812$	0	0
$813$	1.20475e9i	0.0786285i
$814$	0	0
$815$	9.59869e9	0.621098
$816$	0	0
$817$	−2.07285e9	−0.132981
$818$	0	0
$819$	− 1.03155e10i	− 0.656142i
$820$	0	0
$821$	− 2.31068e9i	− 0.145726i	−0.997342	−	0.0728632i	$-0.976786\pi$
$821$	− 2.31068e9i	− 0.145726i	0.997342	−	0.0728632i	$-0.0232137\pi$
$822$	0	0
$823$	−8.90640e9	−0.556933	−0.278466	−	0.960446i	$-0.589826\pi$
$823$	−8.90640e9	−0.556933	−0.278466	+	0.960446i	$0.589826\pi$
$824$	0	0
$825$	−1.99305e9	−0.123575
$826$	0	0
$827$	7.73757e9i	0.475703i	0.971302	+	0.237851i	$0.0764431\pi$
$827$	7.73757e9i	0.475703i	−0.971302	+	0.237851i	$0.923557\pi$
$828$	0	0
$829$	4.54755e9i	0.277228i	0.990347	+	0.138614i	$0.0442647\pi$
$829$	4.54755e9i	0.277228i	−0.990347	+	0.138614i	$0.955735\pi$
$830$	0	0
$831$	−7.51979e9	−0.454571
$832$	0	0
$833$	3.49333e9	0.209403
$834$	0	0
$835$	− 3.16476e9i	− 0.188121i
$836$	0	0
$837$	8.27189e9i	0.487602i
$838$	0	0
$839$	−2.61818e10	−1.53050	−0.765249	−	0.643735i	$-0.777384\pi$
$839$	−2.61818e10	−1.53050	−0.765249	+	0.643735i	$0.777384\pi$
$840$	0	0
$841$	−1.18067e10	−0.684452
$842$	0	0
$843$	2.22914e9i	0.128156i
$844$	0	0
$845$	− 3.43244e9i	− 0.195706i
$846$	0	0
$847$	−2.23492e10	−1.26378
$848$	0	0
$849$	2.64074e9	0.148098
$850$	0	0
$851$	1.12677e10i	0.626735i
$852$	0	0
$853$	9.62659e9i	0.531069i	0.964101	+	0.265535i	$0.0855484\pi$
$853$	9.62659e9i	0.531069i	−0.964101	+	0.265535i	$0.914452\pi$
$854$	0	0
$855$	1.32475e9	0.0724857
$856$	0	0
$857$	−4.51155e9	−0.244846	−0.122423	−	0.992478i	$-0.539066\pi$
$857$	−4.51155e9	−0.244846	−0.122423	+	0.992478i	$0.539066\pi$
$858$	0	0
$859$	9.96875e9i	0.536617i	0.963333	+	0.268309i	$0.0864647\pi$
$859$	9.96875e9i	0.536617i	−0.963333	+	0.268309i	$0.913535\pi$
$860$	0	0
$861$	− 1.53018e9i	− 0.0817019i
$862$	0	0
$863$	−6.94171e9	−0.367645	−0.183823	−	0.982959i	$-0.558847\pi$
$863$	−6.94171e9	−0.367645	−0.183823	+	0.982959i	$0.558847\pi$
$864$	0	0
$865$	8.05791e9	0.423317
$866$	0	0
$867$	− 6.28610e9i	− 0.327578i
$868$	0	0
$869$	− 5.36616e10i	− 2.77393i
$870$	0	0
$871$	6.87284e9	0.352430
$872$	0	0
$873$	1.12891e10	0.574261
$874$	0	0
$875$	1.27255e10i	0.642163i
$876$	0	0
$877$	1.21630e10i	0.608894i	0.952529	+	0.304447i	$0.0984716\pi$
$877$	1.21630e10i	0.608894i	−0.952529	+	0.304447i	$0.901528\pi$
$878$	0	0
$879$	−1.14808e10	−0.570181
$880$	0	0
$881$	3.75232e10	1.84878	0.924388	−	0.381455i	$-0.124577\pi$
$881$	3.75232e10	1.84878	0.924388	+	0.381455i	$0.124577\pi$
$882$	0	0
$883$	− 2.19084e10i	− 1.07090i	−0.844568	−	0.535449i	$-0.820142\pi$
$883$	− 2.19084e10i	− 1.07090i	0.844568	−	0.535449i	$-0.179858\pi$
$884$	0	0
$885$	− 8.82914e9i	− 0.428171i
$886$	0	0
$887$	−2.56718e10	−1.23516	−0.617581	−	0.786507i	$-0.711887\pi$
$887$	−2.56718e10	−1.23516	−0.617581	+	0.786507i	$0.711887\pi$
$888$	0	0
$889$	−1.73976e10	−0.830490
$890$	0	0
$891$	− 1.86641e10i	− 0.883965i
$892$	0	0
$893$	− 3.12888e9i	− 0.147031i
$894$	0	0
$895$	−2.15865e10	−1.00647
$896$	0	0
$897$	7.25148e9	0.335470
$898$	0	0
$899$	− 1.85139e10i	− 0.849845i
$900$	0	0
$901$	1.02935e10i	0.468841i
$902$	0	0
$903$	1.08238e10	0.489186
$904$	0	0
$905$	−4.07885e10	−1.82923
$906$	0	0
$907$	2.87033e10i	1.27734i	0.769481	+	0.638670i	$0.220515\pi$
$907$	2.87033e10i	1.27734i	−0.769481	+	0.638670i	$0.779485\pi$
$908$	0	0
$909$	3.00269e10i	1.32598i
$910$	0	0
$911$	−3.97535e9	−0.174205	−0.0871026	−	0.996199i	$-0.527761\pi$
$911$	−3.97535e9	−0.174205	−0.0871026	+	0.996199i	$0.527761\pi$
$912$	0	0
$913$	3.31082e9	0.143975
$914$	0	0
$915$	− 7.46990e9i	− 0.322360i
$916$	0	0
$917$	6.99342e9i	0.299500i
$918$	0	0
$919$	−1.97480e10	−0.839304	−0.419652	−	0.907685i	$-0.637848\pi$
$919$	−1.97480e10	−0.839304	−0.419652	+	0.907685i	$0.637848\pi$
$920$	0	0
$921$	1.22826e10	0.518062
$922$	0	0
$923$	− 3.84688e10i	− 1.61029i
$924$	0	0
$925$	− 3.64154e9i	− 0.151283i
$926$	0	0
$927$	−1.25494e10	−0.517420
$928$	0	0
$929$	3.96457e10	1.62234	0.811168	−	0.584814i	$-0.198832\pi$
$929$	3.96457e10	1.62234	0.811168	+	0.584814i	$0.198832\pi$
$930$	0	0
$931$	9.36936e8i	0.0380527i
$932$	0	0
$933$	− 9.65387e9i	− 0.389149i
$934$	0	0
$935$	−1.97829e10	−0.791495
$936$	0	0
$937$	−3.57784e10	−1.42080	−0.710399	−	0.703799i	$-0.751486\pi$
$937$	−3.57784e10	−1.42080	−0.710399	+	0.703799i	$0.751486\pi$
$938$	0	0
$939$	− 5.15552e9i	− 0.203209i
$940$	0	0
$941$	3.31375e10i	1.29645i	0.761449	+	0.648225i	$0.224488\pi$
$941$	3.31375e10i	1.29645i	−0.761449	+	0.648225i	$0.775512\pi$
$942$	0	0
$943$	−5.45901e9	−0.211994
$944$	0	0
$945$	−1.51980e10	−0.585833
$946$	0	0
$947$	4.86644e9i	0.186203i	0.995657	+	0.0931014i	$0.0296781\pi$
$947$	4.86644e9i	0.186203i	−0.995657	+	0.0931014i	$0.970322\pi$
$948$	0	0
$949$	3.31198e10i	1.25793i
$950$	0	0
$951$	3.05838e9	0.115308
$952$	0	0
$953$	−1.66448e10	−0.622950	−0.311475	−	0.950254i	$-0.600823\pi$
$953$	−1.66448e10	−0.622950	−0.311475	+	0.950254i	$0.600823\pi$
$954$	0	0
$955$	1.29709e9i	0.0481902i
$956$	0	0
$957$	− 2.36667e10i	− 0.872861i
$958$	0	0
$959$	−2.32122e10	−0.849867
$960$	0	0
$961$	−1.57161e10	−0.571234
$962$	0	0
$963$	3.77554e10i	1.36234i
$964$	0	0
$965$	2.98164e10i	1.06810i
$966$	0	0
$967$	−1.87081e10	−0.665328	−0.332664	−	0.943045i	$-0.607948\pi$
$967$	−1.87081e10	−0.665328	−0.332664	+	0.943045i	$0.607948\pi$
$968$	0	0
$969$	−4.02184e8	−0.0142001
$970$	0	0
$971$	2.92891e9i	0.102669i	0.998682	+	0.0513345i	$0.0163475\pi$
$971$	2.92891e9i	0.102669i	−0.998682	+	0.0513345i	$0.983653\pi$
$972$	0	0
$973$	− 3.44955e10i	− 1.20052i
$974$	0	0
$975$	−2.34355e9	−0.0809764
$976$	0	0
$977$	1.93476e10	0.663737	0.331869	−	0.943326i	$-0.392321\pi$
$977$	1.93476e10	0.663737	0.331869	+	0.943326i	$0.392321\pi$
$978$	0	0
$979$	3.97750e10i	1.35478i
$980$	0	0
$981$	− 1.22043e10i	− 0.412734i
$982$	0	0
$983$	1.20601e9	0.0404961	0.0202480	−	0.999795i	$-0.493554\pi$
$983$	1.20601e9	0.0404961	0.0202480	+	0.999795i	$0.493554\pi$
$984$	0	0
$985$	4.33362e10	1.44485
$986$	0	0
$987$	1.63381e10i	0.540869i
$988$	0	0
$989$	− 3.86147e10i	− 1.26930i
$990$	0	0
$991$	5.78492e10	1.88816	0.944082	−	0.329712i	$-0.106952\pi$
$991$	5.78492e10	1.88816	0.944082	+	0.329712i	$0.106952\pi$
$992$	0	0
$993$	1.93176e9	0.0626080
$994$	0	0
$995$	2.67849e10i	0.862004i
$996$	0	0
$997$	− 4.07141e10i	− 1.30110i	−0.759462	−	0.650552i	$-0.774538\pi$
$997$	− 4.07141e10i	− 1.30110i	0.759462	−	0.650552i	$-0.225462\pi$
$998$	0	0
$999$	−1.93201e10	−0.613100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.8.b.f.65.4	yes	4
4.3	odd	2	inner	128.8.b.f.65.2	yes	4
8.3	odd	2	inner	128.8.b.f.65.3	yes	4
8.5	even	2	inner	128.8.b.f.65.1	✓	4
16.3	odd	4		256.8.a.l.1.4		4
16.5	even	4		256.8.a.l.1.3		4
16.11	odd	4		256.8.a.l.1.1		4
16.13	even	4		256.8.a.l.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.8.b.f.65.1	✓	4	8.5	even	2	inner
128.8.b.f.65.2	yes	4	4.3	odd	2	inner
128.8.b.f.65.3	yes	4	8.3	odd	2	inner
128.8.b.f.65.4	yes	4	1.1	even	1	trivial
256.8.a.l.1.1		4	16.11	odd	4
256.8.a.l.1.2		4	16.13	even	4
256.8.a.l.1.3		4	16.5	even	4
256.8.a.l.1.4		4	16.3	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:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	128.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+18.9737i q^{3} +304.105i q^{5} +656.195 q^{7} +1827.00 q^{9} -7317.51i q^{11} -8604.39i q^{13} -5769.99 q^{15} -8890.00 q^{17} -2384.36i q^{19} +12450.4i q^{21} +44417.6 q^{23} -14355.0 q^{25} +76160.3i q^{27} -170460. i q^{29} +108612. q^{31} +138840. q^{33} +199552. i q^{35} +253677. i q^{37} +163257. q^{39} -122902. q^{41} -869354. i q^{43} +555600. i q^{45} +1.31225e6 q^{47} -392951. q^{49} -168676. i q^{51} -1.15787e6i q^{53} +2.22529e6 q^{55} +45240.0 q^{57} +1.53018e6i q^{59} +1.29461e6i q^{61} +1.19887e6 q^{63} +2.61664e6 q^{65} +798760. i q^{67} +842765. i q^{69} +4.47084e6 q^{71} -3.84917e6 q^{73} -272367. i q^{75} -4.80171e6i q^{77} +7.33332e6 q^{79} +2.55061e6 q^{81} +452452. i q^{83} -2.70350e6i q^{85} +3.23425e6 q^{87} -5.43559e6 q^{89} -5.64616e6i q^{91} +2.06076e6i q^{93} +725096. q^{95} +6.17903e6 q^{97} -1.33691e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 7308 q^{9} - 35560 q^{17} - 57420 q^{25} + 555360 q^{33} - 491608 q^{41} - 1571804 q^{49} + 180960 q^{57} + 10466560 q^{65} - 15396680 q^{73} + 10202436 q^{81} - 21742344 q^{89} + 24716120 q^{97}+O(q^{100}) \) 4 * q + 7308 * q^9 - 35560 * q^17 - 57420 * q^25 + 555360 * q^33 - 491608 * q^41 - 1571804 * q^49 + 180960 * q^57 + 10466560 * q^65 - 15396680 * q^73 + 10202436 * q^81 - 21742344 * q^89 + 24716120 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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