LMFDB - Embedded newform 128.8.b.e.65.3

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,-3028] 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:	$\Q(i, \sqrt{46})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 529 $ x^4 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.3
Root			$3.39116 - 3.39116i$ of defining polynomial
Character	$\chi$	$=$	128.65
Dual form			128.8.b.e.65.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+54.2586i q^{3} -180.000i q^{5} +217.035 q^{7} -757.000 q^{9} +2658.67i q^{11} -3820.00i q^{13} +9766.56 q^{15} -15842.0 q^{17} +35865.0i q^{19} +11776.0i q^{21} +103960. q^{23} +45725.0 q^{25} +77589.9i q^{27} +97748.0i q^{29} -123276. q^{31} -144256. q^{33} -39066.2i q^{35} +238748. i q^{37} +207268. q^{39} -455622. q^{41} -401677. i q^{43} +136260. i q^{45} -1.35213e6 q^{47} -776439. q^{49} -859565. i q^{51} +1.28436e6i q^{53} +478561. q^{55} -1.94598e6 q^{57} -109548. i q^{59} +1.60256e6i q^{61} -164295. q^{63} -687600. q^{65} +2.34772e6i q^{67} +5.64070e6i q^{69} +1.54463e6 q^{71} -2.74276e6 q^{73} +2.48098e6i q^{75} +577024. i q^{77} +6.27186e6 q^{79} -5.86548e6 q^{81} +549586. i q^{83} +2.85156e6i q^{85} -5.30367e6 q^{87} -346458. q^{89} -829072. i q^{91} -6.68877e6i q^{93} +6.45569e6 q^{95} -6.84130e6 q^{97} -2.01262e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3028 q^{9} - 63368 q^{17} + 182900 q^{25} - 577024 q^{33} - 1822488 q^{41} - 3105756 q^{49} - 7783936 q^{57} - 2750400 q^{65} - 10971048 q^{73} - 23461916 q^{81} - 1385832 q^{89} - 27365192 q^{97}+O(q^{100}) $ 4 * q - 3028 * q^9 - 63368 * q^17 + 182900 * q^25 - 577024 * q^33 - 1822488 * q^41 - 3105756 * q^49 - 7783936 * q^57 - 2750400 * q^65 - 10971048 * q^73 - 23461916 * q^81 - 1385832 * q^89 - 27365192 * 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
$3$	54.2586i	1.16023i	0.814534	+	0.580116i	$0.196993\pi$
$3$	54.2586i	1.16023i	−0.814534	+	0.580116i	$0.803007\pi$
$4$	0	0
$5$	− 180.000i	− 0.643988i	−0.946742	−	0.321994i	$-0.895647\pi$
$5$	− 180.000i	− 0.643988i	0.946742	−	0.321994i	$-0.104353\pi$
$6$	0	0
$7$	217.035	0.239158	0.119579	−	0.992825i	$-0.461845\pi$
$7$	217.035	0.239158	0.119579	+	0.992825i	$0.461845\pi$
$8$	0	0
$9$	−757.000	−0.346136
$10$	0	0
$11$	2658.67i	0.602269i	0.953582	+	0.301135i	$0.0973653\pi$
$11$	2658.67i	0.602269i	−0.953582	+	0.301135i	$0.902635\pi$
$12$	0	0
$13$	− 3820.00i	− 0.482238i	−0.970496	−	0.241119i	$-0.922486\pi$
$13$	− 3820.00i	− 0.482238i	0.970496	−	0.241119i	$-0.0775144\pi$
$14$	0	0
$15$	9766.56	0.747174
$16$	0	0
$17$	−15842.0	−0.782058	−0.391029	−	0.920378i	$-0.627881\pi$
$17$	−15842.0	−0.782058	−0.391029	+	0.920378i	$0.627881\pi$
$18$	0	0
$19$	35865.0i	1.19959i	0.800154	+	0.599795i	$0.204751\pi$
$19$	35865.0i	1.19959i	−0.800154	+	0.599795i	$0.795249\pi$
$20$	0	0
$21$	11776.0i	0.277479i
$22$	0	0
$23$	103960.	1.78163	0.890814	−	0.454368i	$-0.150135\pi$
$23$	103960.	1.78163	0.890814	+	0.454368i	$0.150135\pi$
$24$	0	0
$25$	45725.0	0.585280
$26$	0	0
$27$	77589.9i	0.758633i
$28$	0	0
$29$	97748.0i	0.744243i	0.928184	+	0.372122i	$0.121370\pi$
$29$	97748.0i	0.744243i	−0.928184	+	0.372122i	$0.878630\pi$
$30$	0	0
$31$	−123276.	−0.743210	−0.371605	−	0.928391i	$-0.621192\pi$
$31$	−123276.	−0.743210	−0.371605	+	0.928391i	$0.621192\pi$
$32$	0	0
$33$	−144256.	−0.698771
$34$	0	0
$35$	− 39066.2i	− 0.154015i
$36$	0	0
$37$	238748.i	0.774879i	0.921895	+	0.387439i	$0.126640\pi$
$37$	238748.i	0.774879i	−0.921895	+	0.387439i	$0.873360\pi$
$38$	0	0
$39$	207268.	0.559508
$40$	0	0
$41$	−455622.	−1.03243	−0.516216	−	0.856459i	$-0.672660\pi$
$41$	−455622.	−1.03243	−0.516216	+	0.856459i	$0.672660\pi$
$42$	0	0
$43$	− 401677.i	− 0.770437i	−0.922825	−	0.385218i	$-0.874126\pi$
$43$	− 401677.i	− 0.770437i	0.922825	−	0.385218i	$-0.125874\pi$
$44$	0	0
$45$	136260.i	0.222907i
$46$	0	0
$47$	−1.35213e6	−1.89965	−0.949827	−	0.312776i	$-0.898741\pi$
$47$	−1.35213e6	−1.89965	−0.949827	+	0.312776i	$0.898741\pi$
$48$	0	0
$49$	−776439.	−0.942803
$50$	0	0
$51$	− 859565.i	− 0.907368i
$52$	0	0
$53$	1.28436e6i	1.18501i	0.805566	+	0.592506i	$0.201861\pi$
$53$	1.28436e6i	1.18501i	−0.805566	+	0.592506i	$0.798139\pi$
$54$	0	0
$55$	478561.	0.387854
$56$	0	0
$57$	−1.94598e6	−1.39180
$58$	0	0
$59$	− 109548.i	− 0.0694422i	−0.999397	−	0.0347211i	$-0.988946\pi$
$59$	− 109548.i	− 0.0694422i	0.999397	−	0.0347211i	$-0.0110543\pi$
$60$	0	0
$61$	1.60256e6i	0.903984i	0.892022	+	0.451992i	$0.149287\pi$
$61$	1.60256e6i	0.903984i	−0.892022	+	0.451992i	$0.850713\pi$
$62$	0	0
$63$	−164295.	−0.0827814
$64$	0	0
$65$	−687600.	−0.310555
$66$	0	0
$67$	2.34772e6i	0.953639i	0.879001	+	0.476819i	$0.158210\pi$
$67$	2.34772e6i	0.953639i	−0.879001	+	0.476819i	$0.841790\pi$
$68$	0	0
$69$	5.64070e6i	2.06710i
$70$	0	0
$71$	1.54463e6	0.512179	0.256089	−	0.966653i	$-0.417566\pi$
$71$	1.54463e6	0.512179	0.256089	+	0.966653i	$0.417566\pi$
$72$	0	0
$73$	−2.74276e6	−0.825198	−0.412599	−	0.910913i	$-0.635379\pi$
$73$	−2.74276e6	−0.825198	−0.412599	+	0.910913i	$0.635379\pi$
$74$	0	0
$75$	2.48098e6i	0.679060i
$76$	0	0
$77$	577024.i	0.144038i
$78$	0	0
$79$	6.27186e6	1.43121	0.715603	−	0.698508i	$-0.246152\pi$
$79$	6.27186e6	1.43121	0.715603	+	0.698508i	$0.246152\pi$
$80$	0	0
$81$	−5.86548e6	−1.22633
$82$	0	0
$83$	549586.i	0.105502i	0.998608	+	0.0527512i	$0.0167990\pi$
$83$	549586.i	0.105502i	−0.998608	+	0.0527512i	$0.983201\pi$
$84$	0	0
$85$	2.85156e6i	0.503635i
$86$	0	0
$87$	−5.30367e6	−0.863494
$88$	0	0
$89$	−346458.	−0.0520937	−0.0260469	−	0.999661i	$-0.508292\pi$
$89$	−346458.	−0.0520937	−0.0260469	+	0.999661i	$0.508292\pi$
$90$	0	0
$91$	− 829072.i	− 0.115331i
$92$	0	0
$93$	− 6.68877e6i	− 0.862295i
$94$	0	0
$95$	6.45569e6	0.772521
$96$	0	0
$97$	−6.84130e6	−0.761092	−0.380546	−	0.924762i	$-0.624264\pi$
$97$	−6.84130e6	−0.761092	−0.380546	+	0.924762i	$0.624264\pi$
$98$	0	0
$99$	− 2.01262e6i	− 0.208467i
$100$	0	0
$101$	1.81944e7i	1.75717i	0.477587	+	0.878585i	$0.341512\pi$
$101$	1.81944e7i	1.75717i	−0.477587	+	0.878585i	$0.658488\pi$
$102$	0	0
$103$	−1.20912e7	−1.09028	−0.545142	−	0.838344i	$-0.683524\pi$
$103$	−1.20912e7	−1.09028	−0.545142	+	0.838344i	$0.683524\pi$
$104$	0	0
$105$	2.11968e6	0.178693
$106$	0	0
$107$	− 712253.i	− 0.0562071i	−0.999605	−	0.0281035i	$-0.991053\pi$
$107$	− 712253.i	− 0.0562071i	0.999605	−	0.0281035i	$-0.00894682\pi$
$108$	0	0
$109$	− 2.08543e7i	− 1.54242i	−0.636580	−	0.771210i	$-0.719652\pi$
$109$	− 2.08543e7i	− 1.54242i	0.636580	−	0.771210i	$-0.280348\pi$
$110$	0	0
$111$	−1.29541e7	−0.899038
$112$	0	0
$113$	1.86270e7	1.21442	0.607208	−	0.794543i	$-0.292289\pi$
$113$	1.86270e7	1.21442	0.607208	+	0.794543i	$0.292289\pi$
$114$	0	0
$115$	− 1.87127e7i	− 1.14735i
$116$	0	0
$117$	2.89174e6i	0.166920i
$118$	0	0
$119$	−3.43826e6	−0.187036
$120$	0	0
$121$	1.24186e7	0.637272
$122$	0	0
$123$	− 2.47214e7i	− 1.19786i
$124$	0	0
$125$	− 2.22930e7i	− 1.02090i
$126$	0	0
$127$	2.92823e6	0.126851	0.0634253	−	0.997987i	$-0.479798\pi$
$127$	2.92823e6	0.126851	0.0634253	+	0.997987i	$0.479798\pi$
$128$	0	0
$129$	2.17944e7	0.893885
$130$	0	0
$131$	3.95817e7i	1.53831i	0.639059	+	0.769157i	$0.279324\pi$
$131$	3.95817e7i	1.53831i	−0.639059	+	0.769157i	$0.720676\pi$
$132$	0	0
$133$	7.78394e6i	0.286892i
$134$	0	0
$135$	1.39662e7	0.488550
$136$	0	0
$137$	1.03205e7	0.342908	0.171454	−	0.985192i	$-0.445154\pi$
$137$	1.03205e7	0.342908	0.171454	+	0.985192i	$0.445154\pi$
$138$	0	0
$139$	3.81107e7i	1.20363i	0.798634	+	0.601817i	$0.205557\pi$
$139$	3.81107e7i	1.20363i	−0.798634	+	0.601817i	$0.794443\pi$
$140$	0	0
$141$	− 7.33645e7i	− 2.20404i
$142$	0	0
$143$	1.01561e7	0.290437
$144$	0	0
$145$	1.75946e7	0.479283
$146$	0	0
$147$	− 4.21285e7i	− 1.09387i
$148$	0	0
$149$	2.43731e6i	0.0603613i	0.999544	+	0.0301806i	$0.00960826\pi$
$149$	2.43731e6i	0.0603613i	−0.999544	+	0.0301806i	$0.990392\pi$
$150$	0	0
$151$	6.59470e7	1.55875	0.779374	−	0.626559i	$-0.215537\pi$
$151$	6.59470e7	1.55875	0.779374	+	0.626559i	$0.215537\pi$
$152$	0	0
$153$	1.19924e7	0.270699
$154$	0	0
$155$	2.21896e7i	0.478618i
$156$	0	0
$157$	− 7.55101e7i	− 1.55724i	−0.627494	−	0.778622i	$-0.715919\pi$
$157$	− 7.55101e7i	− 1.55724i	0.627494	−	0.778622i	$-0.284081\pi$
$158$	0	0
$159$	−6.96878e7	−1.37489
$160$	0	0
$161$	2.25628e7	0.426091
$162$	0	0
$163$	9.30710e7i	1.68329i	0.540035	+	0.841643i	$0.318411\pi$
$163$	9.30710e7i	1.68329i	−0.540035	+	0.841643i	$0.681589\pi$
$164$	0	0
$165$	2.59661e7i	0.450000i
$166$	0	0
$167$	5.10615e7	0.848371	0.424186	−	0.905575i	$-0.360560\pi$
$167$	5.10615e7	0.848371	0.424186	+	0.905575i	$0.360560\pi$
$168$	0	0
$169$	4.81561e7	0.767446
$170$	0	0
$171$	− 2.71498e7i	− 0.415222i
$172$	0	0
$173$	− 1.42642e7i	− 0.209452i	−0.994501	−	0.104726i	$-0.966603\pi$
$173$	− 1.42642e7i	− 0.209452i	0.994501	−	0.104726i	$-0.0333966\pi$
$174$	0	0
$175$	9.92391e6	0.139975
$176$	0	0
$177$	5.94394e6	0.0805689
$178$	0	0
$179$	− 9.64245e7i	− 1.25661i	−0.777966	−	0.628307i	$-0.783748\pi$
$179$	− 9.64245e7i	− 1.25661i	0.777966	−	0.628307i	$-0.216252\pi$
$180$	0	0
$181$	5.35944e7i	0.671807i	0.941896	+	0.335904i	$0.109042\pi$
$181$	5.35944e7i	0.671807i	−0.941896	+	0.335904i	$0.890958\pi$
$182$	0	0
$183$	−8.69529e7	−1.04883
$184$	0	0
$185$	4.29746e7	0.499012
$186$	0	0
$187$	− 4.21187e7i	− 0.471009i
$188$	0	0
$189$	1.68397e7i	0.181434i
$190$	0	0
$191$	2.81442e7	0.292262	0.146131	−	0.989265i	$-0.453318\pi$
$191$	2.81442e7	0.292262	0.146131	+	0.989265i	$0.453318\pi$
$192$	0	0
$193$	−7.65870e7	−0.766839	−0.383420	−	0.923574i	$-0.625254\pi$
$193$	−7.65870e7	−0.766839	−0.383420	+	0.923574i	$0.625254\pi$
$194$	0	0
$195$	− 3.73082e7i	− 0.360316i
$196$	0	0
$197$	1.63881e7i	0.152720i	0.997080	+	0.0763600i	$0.0243298\pi$
$197$	1.63881e7i	0.152720i	−0.997080	+	0.0763600i	$0.975670\pi$
$198$	0	0
$199$	−1.07038e8	−0.962834	−0.481417	−	0.876492i	$-0.659878\pi$
$199$	−1.07038e8	−0.962834	−0.481417	+	0.876492i	$0.659878\pi$
$200$	0	0
$201$	−1.27384e8	−1.10644
$202$	0	0
$203$	2.12147e7i	0.177992i
$204$	0	0
$205$	8.20120e7i	0.664873i
$206$	0	0
$207$	−7.86974e7	−0.616686
$208$	0	0
$209$	−9.53532e7	−0.722476
$210$	0	0
$211$	− 5.66316e7i	− 0.415021i	−0.978233	−	0.207511i	$-0.933464\pi$
$211$	− 5.66316e7i	− 0.415021i	0.978233	−	0.207511i	$-0.0665362\pi$
$212$	0	0
$213$	8.38098e7i	0.594246i
$214$	0	0
$215$	−7.23018e7	−0.496152
$216$	0	0
$217$	−2.67551e7	−0.177745
$218$	0	0
$219$	− 1.48819e8i	− 0.957420i
$220$	0	0
$221$	6.05164e7i	0.377138i
$222$	0	0
$223$	2.19794e8	1.32724	0.663618	−	0.748072i	$-0.269020\pi$
$223$	2.19794e8	1.32724	0.663618	+	0.748072i	$0.269020\pi$
$224$	0	0
$225$	−3.46138e7	−0.202587
$226$	0	0
$227$	2.78334e8i	1.57934i	0.613532	+	0.789670i	$0.289748\pi$
$227$	2.78334e8i	1.57934i	−0.613532	+	0.789670i	$0.710252\pi$
$228$	0	0
$229$	− 1.11568e8i	− 0.613923i	−0.951722	−	0.306962i	$-0.900688\pi$
$229$	− 1.11568e8i	− 0.613923i	0.951722	−	0.306962i	$-0.0993123\pi$
$230$	0	0
$231$	−3.13085e7	−0.167117
$232$	0	0
$233$	−3.70241e8	−1.91751	−0.958757	−	0.284225i	$-0.908264\pi$
$233$	−3.70241e8	−1.91751	−0.958757	+	0.284225i	$0.908264\pi$
$234$	0	0
$235$	2.43383e8i	1.22335i
$236$	0	0
$237$	3.40303e8i	1.66053i
$238$	0	0
$239$	2.74632e8	1.30124	0.650621	−	0.759403i	$-0.274509\pi$
$239$	2.74632e8	1.30124	0.650621	+	0.759403i	$0.274509\pi$
$240$	0	0
$241$	−4.14029e6	−0.0190533	−0.00952667	−	0.999955i	$-0.503032\pi$
$241$	−4.14029e6	−0.0190533	−0.00952667	+	0.999955i	$0.503032\pi$
$242$	0	0
$243$	− 1.48564e8i	− 0.664188i
$244$	0	0
$245$	1.39759e8i	0.607154i
$246$	0	0
$247$	1.37004e8	0.578488
$248$	0	0
$249$	−2.98198e7	−0.122407
$250$	0	0
$251$	9.61262e7i	0.383693i	0.981425	+	0.191846i	$0.0614475\pi$
$251$	9.61262e7i	0.383693i	−0.981425	+	0.191846i	$0.938552\pi$
$252$	0	0
$253$	2.76394e8i	1.07302i
$254$	0	0
$255$	−1.54722e8	−0.584334
$256$	0	0
$257$	−1.34766e8	−0.495237	−0.247618	−	0.968858i	$-0.579648\pi$
$257$	−1.34766e8	−0.495237	−0.247618	+	0.968858i	$0.579648\pi$
$258$	0	0
$259$	5.18166e7i	0.185319i
$260$	0	0
$261$	− 7.39952e7i	− 0.257610i
$262$	0	0
$263$	3.85690e7	0.130735	0.0653677	−	0.997861i	$-0.479178\pi$
$263$	3.85690e7	0.130735	0.0653677	+	0.997861i	$0.479178\pi$
$264$	0	0
$265$	2.31186e8	0.763133
$266$	0	0
$267$	− 1.87983e7i	− 0.0604408i
$268$	0	0
$269$	1.67903e8i	0.525927i	0.964806	+	0.262964i	$0.0846999\pi$
$269$	1.67903e8i	0.525927i	−0.964806	+	0.262964i	$0.915300\pi$
$270$	0	0
$271$	−5.56602e7	−0.169884	−0.0849420	−	0.996386i	$-0.527071\pi$
$271$	−5.56602e7	−0.169884	−0.0849420	+	0.996386i	$0.527071\pi$
$272$	0	0
$273$	4.49843e7	0.133811
$274$	0	0
$275$	1.21568e8i	0.352496i
$276$	0	0
$277$	− 4.17551e8i	− 1.18040i	−0.807256	−	0.590201i	$-0.799048\pi$
$277$	− 4.17551e8i	− 1.18040i	0.807256	−	0.590201i	$-0.200952\pi$
$278$	0	0
$279$	9.33197e7	0.257252
$280$	0	0
$281$	7.19842e7	0.193537	0.0967687	−	0.995307i	$-0.469149\pi$
$281$	7.19842e7	0.193537	0.0967687	+	0.995307i	$0.469149\pi$
$282$	0	0
$283$	− 7.18753e7i	− 0.188507i	−0.995548	−	0.0942534i	$-0.969954\pi$
$283$	− 7.18753e7i	− 0.188507i	0.995548	−	0.0942534i	$-0.0300464\pi$
$284$	0	0
$285$	3.50277e8i	0.896303i
$286$	0	0
$287$	−9.88857e7	−0.246915
$288$	0	0
$289$	−1.59370e8	−0.388386
$290$	0	0
$291$	− 3.71200e8i	− 0.883043i
$292$	0	0
$293$	− 6.78092e8i	− 1.57490i	−0.616380	−	0.787449i	$-0.711402\pi$
$293$	− 6.78092e8i	− 1.57490i	0.616380	−	0.787449i	$-0.288598\pi$
$294$	0	0
$295$	−1.97187e7	−0.0447199
$296$	0	0
$297$	−2.06286e8	−0.456901
$298$	0	0
$299$	− 3.97125e8i	− 0.859169i
$300$	0	0
$301$	− 8.71777e7i	− 0.184256i
$302$	0	0
$303$	−9.87205e8	−2.03872
$304$	0	0
$305$	2.88462e8	0.582155
$306$	0	0
$307$	9.75677e7i	0.192452i	0.995360	+	0.0962259i	$0.0306771\pi$
$307$	9.75677e7i	0.192452i	−0.995360	+	0.0962259i	$0.969323\pi$
$308$	0	0
$309$	− 6.56053e8i	− 1.26498i
$310$	0	0
$311$	−1.38783e8	−0.261622	−0.130811	−	0.991407i	$-0.541758\pi$
$311$	−1.38783e8	−0.261622	−0.130811	+	0.991407i	$0.541758\pi$
$312$	0	0
$313$	9.77285e8	1.80142	0.900712	−	0.434417i	$-0.143045\pi$
$313$	9.77285e8	1.80142	0.900712	+	0.434417i	$0.143045\pi$
$314$	0	0
$315$	2.95731e7i	0.0533102i
$316$	0	0
$317$	− 8.52052e6i	− 0.0150231i	−0.999972	−	0.00751154i	$-0.997609\pi$
$317$	− 8.52052e6i	− 0.0150231i	0.999972	−	0.00751154i	$-0.00239102\pi$
$318$	0	0
$319$	−2.59880e8	−0.448235
$320$	0	0
$321$	3.86459e7	0.0652132
$322$	0	0
$323$	− 5.68173e8i	− 0.938148i
$324$	0	0
$325$	− 1.74670e8i	− 0.282244i
$326$	0	0
$327$	1.13153e9	1.78956
$328$	0	0
$329$	−2.93458e8	−0.454318
$330$	0	0
$331$	− 4.69843e8i	− 0.712122i	−0.934463	−	0.356061i	$-0.884119\pi$
$331$	− 4.69843e8i	− 0.712122i	0.934463	−	0.356061i	$-0.115881\pi$
$332$	0	0
$333$	− 1.80732e8i	− 0.268214i
$334$	0	0
$335$	4.22589e8	0.614132
$336$	0	0
$337$	1.01378e9	1.44291	0.721453	−	0.692464i	$-0.243475\pi$
$337$	1.01378e9	1.44291	0.721453	+	0.692464i	$0.243475\pi$
$338$	0	0
$339$	1.01067e9i	1.40900i
$340$	0	0
$341$	− 3.27750e8i	− 0.447612i
$342$	0	0
$343$	−3.47251e8	−0.464638
$344$	0	0
$345$	1.01533e9	1.33119
$346$	0	0
$347$	1.12846e9i	1.44989i	0.688808	+	0.724943i	$0.258134\pi$
$347$	1.12846e9i	1.44989i	−0.688808	+	0.724943i	$0.741866\pi$
$348$	0	0
$349$	− 6.48422e8i	− 0.816523i	−0.912865	−	0.408262i	$-0.866135\pi$
$349$	− 6.48422e8i	− 0.816523i	0.912865	−	0.408262i	$-0.133865\pi$
$350$	0	0
$351$	2.96393e8	0.365842
$352$	0	0
$353$	3.46140e8	0.418832	0.209416	−	0.977827i	$-0.432844\pi$
$353$	3.46140e8	0.418832	0.209416	+	0.977827i	$0.432844\pi$
$354$	0	0
$355$	− 2.78034e8i	− 0.329837i
$356$	0	0
$357$	− 1.86555e8i	− 0.217005i
$358$	0	0
$359$	−1.10066e9	−1.25552	−0.627759	−	0.778408i	$-0.716028\pi$
$359$	−1.10066e9	−1.25552	−0.627759	+	0.778408i	$0.716028\pi$
$360$	0	0
$361$	−3.92424e8	−0.439016
$362$	0	0
$363$	6.73818e8i	0.739383i
$364$	0	0
$365$	4.93697e8i	0.531417i
$366$	0	0
$367$	1.66008e9	1.75307	0.876534	−	0.481340i	$-0.159850\pi$
$367$	1.66008e9	1.75307	0.876534	+	0.481340i	$0.159850\pi$
$368$	0	0
$369$	3.44906e8	0.357362
$370$	0	0
$371$	2.78751e8i	0.283406i
$372$	0	0
$373$	− 1.18885e9i	− 1.18617i	−0.805141	−	0.593083i	$-0.797911\pi$
$373$	− 1.18885e9i	− 1.18617i	0.805141	−	0.593083i	$-0.202089\pi$
$374$	0	0
$375$	1.20959e9	1.18448
$376$	0	0
$377$	3.73397e8	0.358903
$378$	0	0
$379$	− 3.90447e8i	− 0.368404i	−0.982888	−	0.184202i	$-0.941030\pi$
$379$	− 3.90447e8i	− 0.368404i	0.982888	−	0.184202i	$-0.0589701\pi$
$380$	0	0
$381$	1.58882e8i	0.147176i
$382$	0	0
$383$	3.11693e8	0.283486	0.141743	−	0.989904i	$-0.454729\pi$
$383$	3.11693e8	0.283486	0.141743	+	0.989904i	$0.454729\pi$
$384$	0	0
$385$	1.03864e8	0.0927585
$386$	0	0
$387$	3.04069e8i	0.266676i
$388$	0	0
$389$	1.80781e9i	1.55715i	0.627553	+	0.778573i	$0.284056\pi$
$389$	1.80781e9i	1.55715i	−0.627553	+	0.778573i	$0.715944\pi$
$390$	0	0
$391$	−1.64693e9	−1.39334
$392$	0	0
$393$	−2.14765e9	−1.78480
$394$	0	0
$395$	− 1.12894e9i	− 0.921678i
$396$	0	0
$397$	− 1.90692e9i	− 1.52956i	−0.644293	−	0.764779i	$-0.722848\pi$
$397$	− 1.90692e9i	− 1.52956i	0.644293	−	0.764779i	$-0.277152\pi$
$398$	0	0
$399$	−4.22346e8	−0.332861
$400$	0	0
$401$	−7.73206e8	−0.598811	−0.299405	−	0.954126i	$-0.596788\pi$
$401$	−7.73206e8	−0.598811	−0.299405	+	0.954126i	$0.596788\pi$
$402$	0	0
$403$	4.70913e8i	0.358404i
$404$	0	0
$405$	1.05579e9i	0.789739i
$406$	0	0
$407$	−6.34753e8	−0.466685
$408$	0	0
$409$	1.38156e9	0.998480	0.499240	−	0.866464i	$-0.333613\pi$
$409$	1.38156e9	0.998480	0.499240	+	0.866464i	$0.333613\pi$
$410$	0	0
$411$	5.59975e8i	0.397853i
$412$	0	0
$413$	− 2.37757e7i	− 0.0166077i
$414$	0	0
$415$	9.89254e7	0.0679422
$416$	0	0
$417$	−2.06783e9	−1.39649
$418$	0	0
$419$	− 2.82274e9i	− 1.87466i	−0.348447	−	0.937328i	$-0.613291\pi$
$419$	− 2.82274e9i	− 1.87466i	0.348447	−	0.937328i	$-0.386709\pi$
$420$	0	0
$421$	5.59886e7i	0.0365689i	0.999833	+	0.0182845i	$0.00582045\pi$
$421$	5.59886e7i	0.0365689i	−0.999833	+	0.0182845i	$0.994180\pi$
$422$	0	0
$423$	1.02356e9	0.657539
$424$	0	0
$425$	−7.24375e8	−0.457723
$426$	0	0
$427$	3.47812e8i	0.216196i
$428$	0	0
$429$	5.51058e8i	0.336974i
$430$	0	0
$431$	−7.29079e8	−0.438636	−0.219318	−	0.975653i	$-0.570383\pi$
$431$	−7.29079e8	−0.438636	−0.219318	+	0.975653i	$0.570383\pi$
$432$	0	0
$433$	7.51244e8	0.444706	0.222353	−	0.974966i	$-0.428626\pi$
$433$	7.51244e8	0.444706	0.222353	+	0.974966i	$0.428626\pi$
$434$	0	0
$435$	9.54661e8i	0.556080i
$436$	0	0
$437$	3.72851e9i	2.13722i
$438$	0	0
$439$	−3.14004e9	−1.77137	−0.885685	−	0.464286i	$-0.846311\pi$
$439$	−3.14004e9	−1.77137	−0.885685	+	0.464286i	$0.846311\pi$
$440$	0	0
$441$	5.87764e8	0.326338
$442$	0	0
$443$	− 1.89553e9i	− 1.03590i	−0.855411	−	0.517949i	$-0.826696\pi$
$443$	− 1.89553e9i	− 1.03590i	0.855411	−	0.517949i	$-0.173304\pi$
$444$	0	0
$445$	6.23624e7i	0.0335477i
$446$	0	0
$447$	−1.32245e8	−0.0700330
$448$	0	0
$449$	−2.27689e9	−1.18708	−0.593540	−	0.804804i	$-0.702270\pi$
$449$	−2.27689e9	−1.18708	−0.593540	+	0.804804i	$0.702270\pi$
$450$	0	0
$451$	− 1.21135e9i	− 0.621802i
$452$	0	0
$453$	3.57820e9i	1.80851i
$454$	0	0
$455$	−1.49233e8	−0.0742720
$456$	0	0
$457$	−2.29182e9	−1.12324	−0.561621	−	0.827395i	$-0.689822\pi$
$457$	−2.29182e9	−1.12324	−0.561621	+	0.827395i	$0.689822\pi$
$458$	0	0
$459$	− 1.22918e9i	− 0.593295i
$460$	0	0
$461$	− 2.14064e8i	− 0.101763i	−0.998705	−	0.0508816i	$-0.983797\pi$
$461$	− 2.14064e8i	− 0.101763i	0.998705	−	0.0508816i	$-0.0162031\pi$
$462$	0	0
$463$	−1.03421e9	−0.484258	−0.242129	−	0.970244i	$-0.577846\pi$
$463$	−1.03421e9	−0.484258	−0.242129	+	0.970244i	$0.577846\pi$
$464$	0	0
$465$	−1.20398e9	−0.555307
$466$	0	0
$467$	2.93319e9i	1.33269i	0.745642	+	0.666347i	$0.232143\pi$
$467$	2.93319e9i	1.33269i	−0.745642	+	0.666347i	$0.767857\pi$
$468$	0	0
$469$	5.09536e8i	0.228071i
$470$	0	0
$471$	4.09708e9	1.80676
$472$	0	0
$473$	1.06793e9	0.464010
$474$	0	0
$475$	1.63993e9i	0.702096i
$476$	0	0
$477$	− 9.72264e8i	− 0.410175i
$478$	0	0
$479$	−4.96997e8	−0.206623	−0.103312	−	0.994649i	$-0.532944\pi$
$479$	−4.96997e8	−0.206623	−0.103312	+	0.994649i	$0.532944\pi$
$480$	0	0
$481$	9.12017e8	0.373676
$482$	0	0
$483$	1.22423e9i	0.494364i
$484$	0	0
$485$	1.23143e9i	0.490134i
$486$	0	0
$487$	3.67557e9	1.44203	0.721013	−	0.692921i	$-0.243677\pi$
$487$	3.67557e9	1.44203	0.721013	+	0.692921i	$0.243677\pi$
$488$	0	0
$489$	−5.04991e9	−1.95300
$490$	0	0
$491$	3.00322e9i	1.14499i	0.819909	+	0.572494i	$0.194024\pi$
$491$	3.00322e9i	1.14499i	−0.819909	+	0.572494i	$0.805976\pi$
$492$	0	0
$493$	− 1.54852e9i	− 0.582041i
$494$	0	0
$495$	−3.62271e8	−0.134250
$496$	0	0
$497$	3.35239e8	0.122492
$498$	0	0
$499$	− 3.95565e9i	− 1.42517i	−0.701587	−	0.712583i	$-0.747525\pi$
$499$	− 3.95565e9i	− 1.42517i	0.701587	−	0.712583i	$-0.252475\pi$
$500$	0	0
$501$	2.77053e9i	0.984307i
$502$	0	0
$503$	8.20917e8	0.287615	0.143807	−	0.989606i	$-0.454065\pi$
$503$	8.20917e8	0.287615	0.143807	+	0.989606i	$0.454065\pi$
$504$	0	0
$505$	3.27500e9	1.13160
$506$	0	0
$507$	2.61289e9i	0.890415i
$508$	0	0
$509$	− 1.25294e9i	− 0.421130i	−0.977580	−	0.210565i	$-0.932470\pi$
$509$	− 1.25294e9i	− 0.421130i	0.977580	−	0.210565i	$-0.0675304\pi$
$510$	0	0
$511$	−5.95274e8	−0.197353
$512$	0	0
$513$	−2.78276e9	−0.910048
$514$	0	0
$515$	2.17642e9i	0.702129i
$516$	0	0
$517$	− 3.59486e9i	− 1.14410i
$518$	0	0
$519$	7.73955e8	0.243013
$520$	0	0
$521$	−4.46501e9	−1.38322	−0.691609	−	0.722272i	$-0.743098\pi$
$521$	−4.46501e9	−1.38322	−0.691609	+	0.722272i	$0.743098\pi$
$522$	0	0
$523$	− 3.16654e9i	− 0.967898i	−0.875096	−	0.483949i	$-0.839202\pi$
$523$	− 3.16654e9i	− 0.967898i	0.875096	−	0.483949i	$-0.160798\pi$
$524$	0	0
$525$	5.38458e8i	0.162403i
$526$	0	0
$527$	1.95293e9	0.581233
$528$	0	0
$529$	7.40276e9	2.17420
$530$	0	0
$531$	8.29280e7i	0.0240364i
$532$	0	0
$533$	1.74048e9i	0.497878i
$534$	0	0
$535$	−1.28206e8	−0.0361967
$536$	0	0
$537$	5.23186e9	1.45796
$538$	0	0
$539$	− 2.06430e9i	− 0.567821i
$540$	0	0
$541$	− 3.93146e9i	− 1.06749i	−0.845646	−	0.533745i	$-0.820784\pi$
$541$	− 3.93146e9i	− 1.06749i	0.845646	−	0.533745i	$-0.179216\pi$
$542$	0	0
$543$	−2.90796e9	−0.779452
$544$	0	0
$545$	−3.75377e9	−0.993300
$546$	0	0
$547$	6.36970e9i	1.66404i	0.554747	+	0.832019i	$0.312815\pi$
$547$	6.36970e9i	1.66404i	−0.554747	+	0.832019i	$0.687185\pi$
$548$	0	0
$549$	− 1.21314e9i	− 0.312902i
$550$	0	0
$551$	−3.50573e9	−0.892787
$552$	0	0
$553$	1.36121e9	0.342285
$554$	0	0
$555$	2.33175e9i	0.578970i
$556$	0	0
$557$	4.19132e8i	0.102768i	0.998679	+	0.0513840i	$0.0163632\pi$
$557$	4.19132e8i	0.102768i	−0.998679	+	0.0513840i	$0.983637\pi$
$558$	0	0
$559$	−1.53441e9	−0.371534
$560$	0	0
$561$	2.28530e9	0.546480
$562$	0	0
$563$	− 7.94827e9i	− 1.87712i	−0.345109	−	0.938562i	$-0.612158\pi$
$563$	− 7.94827e9i	− 1.87712i	0.345109	−	0.938562i	$-0.387842\pi$
$564$	0	0
$565$	− 3.35285e9i	− 0.782069i
$566$	0	0
$567$	−1.27301e9	−0.293286
$568$	0	0
$569$	3.46382e9	0.788247	0.394123	−	0.919058i	$-0.371048\pi$
$569$	3.46382e9	0.788247	0.394123	+	0.919058i	$0.371048\pi$
$570$	0	0
$571$	− 4.37465e9i	− 0.983370i	−0.870773	−	0.491685i	$-0.836381\pi$
$571$	− 4.37465e9i	− 0.983370i	0.870773	−	0.491685i	$-0.163619\pi$
$572$	0	0
$573$	1.52706e9i	0.339091i
$574$	0	0
$575$	4.75355e9	1.04275
$576$	0	0
$577$	6.14946e9	1.33267	0.666333	−	0.745654i	$-0.267863\pi$
$577$	6.14946e9	1.33267	0.666333	+	0.745654i	$0.267863\pi$
$578$	0	0
$579$	− 4.15551e9i	− 0.889711i
$580$	0	0
$581$	1.19279e8i	0.0252318i
$582$	0	0
$583$	−3.41470e9	−0.713696
$584$	0	0
$585$	5.20513e8	0.107494
$586$	0	0
$587$	− 1.09870e9i	− 0.224206i	−0.993697	−	0.112103i	$-0.964241\pi$
$587$	− 1.09870e9i	− 0.224206i	0.993697	−	0.112103i	$-0.0357587\pi$
$588$	0	0
$589$	− 4.42128e9i	− 0.891547i
$590$	0	0
$591$	−8.89195e8	−0.177191
$592$	0	0
$593$	4.28953e8	0.0844731	0.0422366	−	0.999108i	$-0.486552\pi$
$593$	4.28953e8	0.0844731	0.0422366	+	0.999108i	$0.486552\pi$
$594$	0	0
$595$	6.18887e8i	0.120449i
$596$	0	0
$597$	− 5.80772e9i	− 1.11711i
$598$	0	0
$599$	−7.24934e9	−1.37817	−0.689087	−	0.724678i	$-0.741988\pi$
$599$	−7.24934e9	−1.37817	−0.689087	+	0.724678i	$0.741988\pi$
$600$	0	0
$601$	9.22234e9	1.73293	0.866464	−	0.499240i	$-0.166388\pi$
$601$	9.22234e9	1.73293	0.866464	+	0.499240i	$0.166388\pi$
$602$	0	0
$603$	− 1.77722e9i	− 0.330089i
$604$	0	0
$605$	− 2.23535e9i	− 0.410395i
$606$	0	0
$607$	5.75921e9	1.04521	0.522604	−	0.852576i	$-0.324961\pi$
$607$	5.75921e9	1.04521	0.522604	+	0.852576i	$0.324961\pi$
$608$	0	0
$609$	−1.15108e9	−0.206512
$610$	0	0
$611$	5.16512e9i	0.916086i
$612$	0	0
$613$	− 1.07539e10i	− 1.88563i	−0.333322	−	0.942813i	$-0.608170\pi$
$613$	− 1.07539e10i	− 1.88563i	0.333322	−	0.942813i	$-0.391830\pi$
$614$	0	0
$615$	−4.44986e9	−0.771406
$616$	0	0
$617$	−5.82077e9	−0.997660	−0.498830	−	0.866700i	$-0.666237\pi$
$617$	−5.82077e9	−0.997660	−0.498830	+	0.866700i	$0.666237\pi$
$618$	0	0
$619$	5.98336e9i	1.01398i	0.861953	+	0.506988i	$0.169241\pi$
$619$	5.98336e9i	1.01398i	−0.861953	+	0.506988i	$0.830759\pi$
$620$	0	0
$621$	8.06621e9i	1.35160i
$622$	0	0
$623$	−7.51934e7	−0.0124587
$624$	0	0
$625$	−4.40474e8	−0.0721673
$626$	0	0
$627$	− 5.17374e9i	− 0.838239i
$628$	0	0
$629$	− 3.78225e9i	− 0.606000i
$630$	0	0
$631$	6.68543e9	1.05932	0.529659	−	0.848211i	$-0.322320\pi$
$631$	6.68543e9	1.05932	0.529659	+	0.848211i	$0.322320\pi$
$632$	0	0
$633$	3.07276e9	0.481521
$634$	0	0
$635$	− 5.27081e8i	− 0.0816901i
$636$	0	0
$637$	2.96600e9i	0.454656i
$638$	0	0
$639$	−1.16929e9	−0.177284
$640$	0	0
$641$	−5.46365e9	−0.819370	−0.409685	−	0.912227i	$-0.634361\pi$
$641$	−5.46365e9	−0.819370	−0.409685	+	0.912227i	$0.634361\pi$
$642$	0	0
$643$	3.42325e9i	0.507808i	0.967229	+	0.253904i	$0.0817148\pi$
$643$	3.42325e9i	0.507808i	−0.967229	+	0.253904i	$0.918285\pi$
$644$	0	0
$645$	− 3.92300e9i	− 0.575651i
$646$	0	0
$647$	1.04327e10	1.51437	0.757185	−	0.653201i	$-0.226574\pi$
$647$	1.04327e10	1.51437	0.757185	+	0.653201i	$0.226574\pi$
$648$	0	0
$649$	2.91253e8	0.0418229
$650$	0	0
$651$	− 1.45169e9i	− 0.206225i
$652$	0	0
$653$	1.21062e9i	0.170142i	0.996375	+	0.0850712i	$0.0271118\pi$
$653$	1.21062e9i	0.170142i	−0.996375	+	0.0850712i	$0.972888\pi$
$654$	0	0
$655$	7.12471e9	0.990656
$656$	0	0
$657$	2.07627e9	0.285631
$658$	0	0
$659$	7.92203e9i	1.07830i	0.842211	+	0.539148i	$0.181254\pi$
$659$	7.92203e9i	1.07830i	−0.842211	+	0.539148i	$0.818746\pi$
$660$	0	0
$661$	− 4.62495e9i	− 0.622876i	−0.950267	−	0.311438i	$-0.899189\pi$
$661$	− 4.62495e9i	− 0.622876i	0.950267	−	0.311438i	$-0.100811\pi$
$662$	0	0
$663$	−3.28354e9	−0.437567
$664$	0	0
$665$	1.40111e9	0.184755
$666$	0	0
$667$	1.01618e10i	1.32596i
$668$	0	0
$669$	1.19257e10i	1.53990i
$670$	0	0
$671$	−4.26069e9	−0.544442
$672$	0	0
$673$	7.57112e9	0.957430	0.478715	−	0.877970i	$-0.341103\pi$
$673$	7.57112e9	0.957430	0.478715	+	0.877970i	$0.341103\pi$
$674$	0	0
$675$	3.54780e9i	0.444013i
$676$	0	0
$677$	8.36067e9i	1.03557i	0.855510	+	0.517787i	$0.173244\pi$
$677$	8.36067e9i	1.03557i	−0.855510	+	0.517787i	$0.826756\pi$
$678$	0	0
$679$	−1.48480e9	−0.182022
$680$	0	0
$681$	−1.51020e10	−1.83240
$682$	0	0
$683$	7.57602e9i	0.909847i	0.890530	+	0.454924i	$0.150333\pi$
$683$	7.57602e9i	0.909847i	−0.890530	+	0.454924i	$0.849667\pi$
$684$	0	0
$685$	− 1.85769e9i	− 0.220829i
$686$	0	0
$687$	6.05351e9	0.712293
$688$	0	0
$689$	4.90627e9	0.571458
$690$	0	0
$691$	− 1.16658e10i	− 1.34506i	−0.740069	−	0.672531i	$-0.765207\pi$
$691$	− 1.16658e10i	− 1.34506i	0.740069	−	0.672531i	$-0.234793\pi$
$692$	0	0
$693$	− 4.36807e8i	− 0.0498567i
$694$	0	0
$695$	6.85992e9	0.775126
$696$	0	0
$697$	7.21796e9	0.807421
$698$	0	0
$699$	− 2.00888e10i	− 2.22476i
$700$	0	0
$701$	8.61744e9i	0.944855i	0.881370	+	0.472427i	$0.156622\pi$
$701$	8.61744e9i	0.944855i	−0.881370	+	0.472427i	$0.843378\pi$
$702$	0	0
$703$	−8.56269e9	−0.929537
$704$	0	0
$705$	−1.32056e10	−1.41937
$706$	0	0
$707$	3.94882e9i	0.420242i
$708$	0	0
$709$	− 5.63807e9i	− 0.594112i	−0.954860	−	0.297056i	$-0.903995\pi$
$709$	− 5.63807e9i	− 0.594112i	0.954860	−	0.297056i	$-0.0960049\pi$
$710$	0	0
$711$	−4.74780e9	−0.495392
$712$	0	0
$713$	−1.28157e10	−1.32412
$714$	0	0
$715$	− 1.82810e9i	− 0.187038i
$716$	0	0
$717$	1.49011e10i	1.50974i
$718$	0	0
$719$	−7.95313e9	−0.797971	−0.398986	−	0.916957i	$-0.630638\pi$
$719$	−7.95313e9	−0.797971	−0.398986	+	0.916957i	$0.630638\pi$
$720$	0	0
$721$	−2.62421e9	−0.260751
$722$	0	0
$723$	− 2.24647e8i	− 0.0221063i
$724$	0	0
$725$	4.46953e9i	0.435591i
$726$	0	0
$727$	6.10429e9	0.589202	0.294601	−	0.955620i	$-0.404813\pi$
$727$	6.10429e9	0.589202	0.294601	+	0.955620i	$0.404813\pi$
$728$	0	0
$729$	−4.76693e9	−0.455714
$730$	0	0
$731$	6.36336e9i	0.602526i
$732$	0	0
$733$	9.35527e9i	0.877389i	0.898636	+	0.438695i	$0.144559\pi$
$733$	9.35527e9i	0.877389i	−0.898636	+	0.438695i	$0.855441\pi$
$734$	0	0
$735$	−7.58313e9	−0.704438
$736$	0	0
$737$	−6.24181e9	−0.574347
$738$	0	0
$739$	− 1.08015e9i	− 0.0984528i	−0.998788	−	0.0492264i	$-0.984324\pi$
$739$	− 1.08015e9i	− 0.0984528i	0.998788	−	0.0492264i	$-0.0156756\pi$
$740$	0	0
$741$	7.43366e9i	0.671180i
$742$	0	0
$743$	1.64759e10	1.47363	0.736813	−	0.676097i	$-0.236330\pi$
$743$	1.64759e10	1.47363	0.736813	+	0.676097i	$0.236330\pi$
$744$	0	0
$745$	4.38715e8	0.0388719
$746$	0	0
$747$	− 4.16036e8i	− 0.0365182i
$748$	0	0
$749$	− 1.54584e8i	− 0.0134424i
$750$	0	0
$751$	−1.57676e10	−1.35839	−0.679196	−	0.733957i	$-0.737671\pi$
$751$	−1.57676e10	−1.35839	−0.679196	+	0.733957i	$0.737671\pi$
$752$	0	0
$753$	−5.21568e9	−0.445172
$754$	0	0
$755$	− 1.18705e10i	− 1.00381i
$756$	0	0
$757$	− 1.54352e10i	− 1.29323i	−0.762817	−	0.646614i	$-0.776184\pi$
$757$	− 1.54352e10i	− 1.29323i	0.762817	−	0.646614i	$-0.223816\pi$
$758$	0	0
$759$	−1.49968e10	−1.24495
$760$	0	0
$761$	−7.00289e9	−0.576012	−0.288006	−	0.957629i	$-0.592992\pi$
$761$	−7.00289e9	−0.576012	−0.288006	+	0.957629i	$0.592992\pi$
$762$	0	0
$763$	− 4.52610e9i	− 0.368883i
$764$	0	0
$765$	− 2.15863e9i	− 0.174326i
$766$	0	0
$767$	−4.18474e8	−0.0334877
$768$	0	0
$769$	−7.66268e9	−0.607629	−0.303814	−	0.952731i	$-0.598260\pi$
$769$	−7.66268e9	−0.607629	−0.303814	+	0.952731i	$0.598260\pi$
$770$	0	0
$771$	− 7.31220e9i	− 0.574589i
$772$	0	0
$773$	1.92438e10i	1.49852i	0.662275	+	0.749261i	$0.269591\pi$
$773$	1.92438e10i	1.49852i	−0.662275	+	0.749261i	$0.730409\pi$
$774$	0	0
$775$	−5.63678e9	−0.434986
$776$	0	0
$777$	−2.81150e9	−0.215013
$778$	0	0
$779$	− 1.63409e10i	− 1.23849i
$780$	0	0
$781$	4.10668e9i	0.308469i
$782$	0	0
$783$	−7.58425e9	−0.564608
$784$	0	0
$785$	−1.35918e10	−1.00285
$786$	0	0
$787$	2.37916e10i	1.73985i	0.493184	+	0.869925i	$0.335833\pi$
$787$	2.37916e10i	1.73985i	−0.493184	+	0.869925i	$0.664167\pi$
$788$	0	0
$789$	2.09270e9i	0.151683i
$790$	0	0
$791$	4.04269e9	0.290438
$792$	0	0
$793$	6.12179e9	0.435936
$794$	0	0
$795$	1.25438e10i	0.885410i
$796$	0	0
$797$	− 8.06044e9i	− 0.563968i	−0.959419	−	0.281984i	$-0.909007\pi$
$797$	− 8.06044e9i	− 0.563968i	0.959419	−	0.281984i	$-0.0909925\pi$
$798$	0	0
$799$	2.14204e10	1.48564
$800$	0	0
$801$	2.62269e8	0.0180315
$802$	0	0
$803$	− 7.29211e9i	− 0.496991i
$804$	0	0
$805$	− 4.06131e9i	− 0.274398i
$806$	0	0
$807$	−9.11019e9	−0.610197
$808$	0	0
$809$	1.19696e10	0.794805	0.397403	−	0.917644i	$-0.369912\pi$
$809$	1.19696e10	0.794805	0.397403	+	0.917644i	$0.369912\pi$
$810$	0	0
$811$	1.76446e9i	0.116156i	0.998312	+	0.0580778i	$0.0184971\pi$
$811$	1.76446e9i	0.116156i	−0.998312	+	0.0580778i	$0.981503\pi$
$812$	0	0
$813$	− 3.02005e9i	− 0.197105i
$814$	0	0
$815$	1.67528e10	1.08401
$816$	0	0
$817$	1.44061e10	0.924208
$818$	0	0
$819$	6.27608e8i	0.0399204i
$820$	0	0
$821$	2.29211e10i	1.44556i	0.691080	+	0.722778i	$0.257135\pi$
$821$	2.29211e10i	1.44556i	−0.691080	+	0.722778i	$0.742865\pi$
$822$	0	0
$823$	1.36475e10	0.853404	0.426702	−	0.904392i	$-0.359675\pi$
$823$	1.36475e10	0.853404	0.426702	+	0.904392i	$0.359675\pi$
$824$	0	0
$825$	−6.59611e9	−0.408977
$826$	0	0
$827$	1.92551e10i	1.18379i	0.806014	+	0.591897i	$0.201621\pi$
$827$	1.92551e10i	1.18379i	−0.806014	+	0.591897i	$0.798379\pi$
$828$	0	0
$829$	− 8.10248e9i	− 0.493943i	−0.969023	−	0.246972i	$-0.920565\pi$
$829$	− 8.10248e9i	− 0.493943i	0.969023	−	0.246972i	$-0.0794354\pi$
$830$	0	0
$831$	2.26557e10	1.36954
$832$	0	0
$833$	1.23003e10	0.737327
$834$	0	0
$835$	− 9.19107e9i	− 0.546341i
$836$	0	0
$837$	− 9.56494e9i	− 0.563823i
$838$	0	0
$839$	−1.36587e10	−0.798438	−0.399219	−	0.916856i	$-0.630719\pi$
$839$	−1.36587e10	−0.798438	−0.399219	+	0.916856i	$0.630719\pi$
$840$	0	0
$841$	7.69520e9	0.446102
$842$	0	0
$843$	3.90576e9i	0.224548i
$844$	0	0
$845$	− 8.66810e9i	− 0.494226i
$846$	0	0
$847$	2.69527e9	0.152409
$848$	0	0
$849$	3.89985e9	0.218711
$850$	0	0
$851$	2.48201e10i	1.38055i
$852$	0	0
$853$	3.83840e9i	0.211753i	0.994379	+	0.105876i	$0.0337648\pi$
$853$	3.83840e9i	0.211753i	−0.994379	+	0.105876i	$0.966235\pi$
$854$	0	0
$855$	−4.88696e9	−0.267398
$856$	0	0
$857$	6.18270e9	0.335541	0.167770	−	0.985826i	$-0.446343\pi$
$857$	6.18270e9	0.335541	0.167770	+	0.985826i	$0.446343\pi$
$858$	0	0
$859$	7.81827e9i	0.420857i	0.977609	+	0.210429i	$0.0674859\pi$
$859$	7.81827e9i	0.420857i	−0.977609	+	0.210429i	$0.932514\pi$
$860$	0	0
$861$	− 5.36540e9i	− 0.286478i
$862$	0	0
$863$	−1.74642e10	−0.924934	−0.462467	−	0.886636i	$-0.653036\pi$
$863$	−1.74642e10	−0.924934	−0.462467	+	0.886636i	$0.653036\pi$
$864$	0	0
$865$	−2.56755e9	−0.134885
$866$	0	0
$867$	− 8.64718e9i	− 0.450617i
$868$	0	0
$869$	1.66748e10i	0.861971i
$870$	0	0
$871$	8.96828e9	0.459881
$872$	0	0
$873$	5.17886e9	0.263442
$874$	0	0
$875$	− 4.83835e9i	− 0.244157i
$876$	0	0
$877$	− 3.68179e10i	− 1.84315i	−0.388203	−	0.921574i	$-0.626904\pi$
$877$	− 3.68179e10i	− 1.84315i	0.388203	−	0.921574i	$-0.373096\pi$
$878$	0	0
$879$	3.67924e10	1.82724
$880$	0	0
$881$	1.15013e10	0.566671	0.283336	−	0.959021i	$-0.408559\pi$
$881$	1.15013e10	0.566671	0.283336	+	0.959021i	$0.408559\pi$
$882$	0	0
$883$	1.54418e10i	0.754806i	0.926049	+	0.377403i	$0.123183\pi$
$883$	1.54418e10i	0.754806i	−0.926049	+	0.377403i	$0.876817\pi$
$884$	0	0
$885$	− 1.06991e9i	− 0.0518854i
$886$	0	0
$887$	−1.97963e10	−0.952471	−0.476235	−	0.879318i	$-0.657999\pi$
$887$	−1.97963e10	−0.952471	−0.476235	+	0.879318i	$0.657999\pi$
$888$	0	0
$889$	6.35527e8	0.0303374
$890$	0	0
$891$	− 1.55944e10i	− 0.738578i
$892$	0	0
$893$	− 4.84939e10i	− 2.27881i
$894$	0	0
$895$	−1.73564e10	−0.809244
$896$	0	0
$897$	2.15475e10	0.996835
$898$	0	0
$899$	− 1.20499e10i	− 0.553129i
$900$	0	0
$901$	− 2.03469e10i	− 0.926747i
$902$	0	0
$903$	4.73014e9	0.213780
$904$	0	0
$905$	9.64700e9	0.432635
$906$	0	0
$907$	− 3.23910e10i	− 1.44145i	−0.693222	−	0.720724i	$-0.743809\pi$
$907$	− 3.23910e10i	− 1.44145i	0.693222	−	0.720724i	$-0.256191\pi$
$908$	0	0
$909$	− 1.37732e10i	− 0.608220i
$910$	0	0
$911$	−9.19954e9	−0.403136	−0.201568	−	0.979475i	$-0.564604\pi$
$911$	−9.19954e9	−0.403136	−0.201568	+	0.979475i	$0.564604\pi$
$912$	0	0
$913$	−1.46117e9	−0.0635408
$914$	0	0
$915$	1.56515e10i	0.675434i
$916$	0	0
$917$	8.59060e9i	0.367901i
$918$	0	0
$919$	−1.73860e10	−0.738917	−0.369458	−	0.929247i	$-0.620457\pi$
$919$	−1.73860e10	−0.738917	−0.369458	+	0.929247i	$0.620457\pi$
$920$	0	0
$921$	−5.29389e9	−0.223289
$922$	0	0
$923$	− 5.90051e9i	− 0.246992i
$924$	0	0
$925$	1.09168e10i	0.453521i
$926$	0	0
$927$	9.15305e9	0.377387
$928$	0	0
$929$	2.41112e10	0.986653	0.493326	−	0.869844i	$-0.335781\pi$
$929$	2.41112e10	0.986653	0.493326	+	0.869844i	$0.335781\pi$
$930$	0	0
$931$	− 2.78470e10i	− 1.13098i
$932$	0	0
$933$	− 7.53015e9i	− 0.303541i
$934$	0	0
$935$	−7.58137e9	−0.303324
$936$	0	0
$937$	1.58105e10	0.627850	0.313925	−	0.949448i	$-0.398356\pi$
$937$	1.58105e10	0.627850	0.313925	+	0.949448i	$0.398356\pi$
$938$	0	0
$939$	5.30261e10i	2.09007i
$940$	0	0
$941$	− 1.50751e10i	− 0.589788i	−0.955530	−	0.294894i	$-0.904716\pi$
$941$	− 1.50751e10i	− 0.589788i	0.955530	−	0.294894i	$-0.0952844\pi$
$942$	0	0
$943$	−4.73663e10	−1.83941
$944$	0	0
$945$	3.03114e9	0.116841
$946$	0	0
$947$	− 3.51432e10i	− 1.34467i	−0.740245	−	0.672337i	$-0.765291\pi$
$947$	− 3.51432e10i	− 1.34467i	0.740245	−	0.672337i	$-0.234709\pi$
$948$	0	0
$949$	1.04774e10i	0.397942i
$950$	0	0
$951$	4.62312e8	0.0174302
$952$	0	0
$953$	−1.61528e10	−0.604538	−0.302269	−	0.953223i	$-0.597744\pi$
$953$	−1.61528e10	−0.604538	−0.302269	+	0.953223i	$0.597744\pi$
$954$	0	0
$955$	− 5.06595e9i	− 0.188213i
$956$	0	0
$957$	− 1.41007e10i	− 0.520056i
$958$	0	0
$959$	2.23990e9	0.0820094
$960$	0	0
$961$	−1.23157e10	−0.447640
$962$	0	0
$963$	5.39176e8i	0.0194553i
$964$	0	0
$965$	1.37857e10i	0.493835i
$966$	0	0
$967$	2.27594e10	0.809408	0.404704	−	0.914448i	$-0.367375\pi$
$967$	2.27594e10	0.809408	0.404704	+	0.914448i	$0.367375\pi$
$968$	0	0
$969$	3.08283e10	1.08847
$970$	0	0
$971$	− 3.40071e10i	− 1.19207i	−0.802958	−	0.596036i	$-0.796742\pi$
$971$	− 3.40071e10i	− 1.19207i	0.802958	−	0.596036i	$-0.203258\pi$
$972$	0	0
$973$	8.27133e9i	0.287859i
$974$	0	0
$975$	9.47733e9	0.327469
$976$	0	0
$977$	−8.64935e9	−0.296724	−0.148362	−	0.988933i	$-0.547400\pi$
$977$	−8.64935e9	−0.296724	−0.148362	+	0.988933i	$0.547400\pi$
$978$	0	0
$979$	− 9.21119e8i	− 0.0313745i
$980$	0	0
$981$	1.57867e10i	0.533888i
$982$	0	0
$983$	4.00601e10	1.34516	0.672581	−	0.740024i	$-0.265186\pi$
$983$	4.00601e10	1.34516	0.672581	+	0.740024i	$0.265186\pi$
$984$	0	0
$985$	2.94985e9	0.0983498
$986$	0	0
$987$	− 1.59226e10i	− 0.527114i
$988$	0	0
$989$	− 4.17581e10i	− 1.37263i
$990$	0	0
$991$	−2.06848e10	−0.675139	−0.337570	−	0.941301i	$-0.609605\pi$
$991$	−2.06848e10	−0.675139	−0.337570	+	0.941301i	$0.609605\pi$
$992$	0	0
$993$	2.54930e10	0.826227
$994$	0	0
$995$	1.92668e10i	0.620053i
$996$	0	0
$997$	1.01849e10i	0.325480i	0.986669	+	0.162740i	$0.0520331\pi$
$997$	1.01849e10i	0.325480i	−0.986669	+	0.162740i	$0.947967\pi$
$998$	0	0
$999$	−1.85244e10	−0.587849

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.8.b.e.65.1	✓	4	4.3	odd	2	inner
128.8.b.e.65.2	yes	4	8.5	even	2	inner
128.8.b.e.65.3	yes	4	1.1	even	1	trivial
128.8.b.e.65.4	yes	4	8.3	odd	2	inner
256.8.a.f.1.1		2	16.13	even	4
256.8.a.f.1.2		2	16.3	odd	4
256.8.a.h.1.1		2	16.11	odd	4
256.8.a.h.1.2		2	16.5	even	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)

\(f(q)\)	\(=\)	\(q+54.2586i q^{3} -180.000i q^{5} +217.035 q^{7} -757.000 q^{9} +2658.67i q^{11} -3820.00i q^{13} +9766.56 q^{15} -15842.0 q^{17} +35865.0i q^{19} +11776.0i q^{21} +103960. q^{23} +45725.0 q^{25} +77589.9i q^{27} +97748.0i q^{29} -123276. q^{31} -144256. q^{33} -39066.2i q^{35} +238748. i q^{37} +207268. q^{39} -455622. q^{41} -401677. i q^{43} +136260. i q^{45} -1.35213e6 q^{47} -776439. q^{49} -859565. i q^{51} +1.28436e6i q^{53} +478561. q^{55} -1.94598e6 q^{57} -109548. i q^{59} +1.60256e6i q^{61} -164295. q^{63} -687600. q^{65} +2.34772e6i q^{67} +5.64070e6i q^{69} +1.54463e6 q^{71} -2.74276e6 q^{73} +2.48098e6i q^{75} +577024. i q^{77} +6.27186e6 q^{79} -5.86548e6 q^{81} +549586. i q^{83} +2.85156e6i q^{85} -5.30367e6 q^{87} -346458. q^{89} -829072. i q^{91} -6.68877e6i q^{93} +6.45569e6 q^{95} -6.84130e6 q^{97} -2.01262e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3028 q^{9} - 63368 q^{17} + 182900 q^{25} - 577024 q^{33} - 1822488 q^{41} - 3105756 q^{49} - 7783936 q^{57} - 2750400 q^{65} - 10971048 q^{73} - 23461916 q^{81} - 1385832 q^{89} - 27365192 q^{97}+O(q^{100}) \) 4 * q - 3028 * q^9 - 63368 * q^17 + 182900 * q^25 - 577024 * q^33 - 1822488 * q^41 - 3105756 * q^49 - 7783936 * q^57 - 2750400 * q^65 - 10971048 * q^73 - 23461916 * q^81 - 1385832 * q^89 - 27365192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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