LMFDB - Embedded newform 128.12.b.e.65.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [128,12,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, 12); 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, 12, names="a")

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	128.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-1152028] 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:	$98.3479271116$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 102428 x^{10} + 3685567675 x^{8} + 57536665921720 x^{6} + \cdots + 24\!\cdots\!21 $ x^12 + 102428x^10 + 3685567675x^8 + 57536665921720x^6 + 441779443959540995x^4 + 1654485347892342540892*x^2 + 2424123860975397647704521
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{123}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.11
Root			$0.707107 - 196.084i$ of defining polynomial
Character	$\chi$	$=$	128.65
Dual form			128.12.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)

$f(q)$	$=$	$q+784.336i q^{3} -11907.7i q^{5} +21148.3 q^{7} -438037. q^{9} -561003. i q^{11} -1.10728e6i q^{13} +9.33962e6 q^{15} +5.37054e6 q^{17} -8.72557e6i q^{19} +1.65874e7i q^{21} -2.29090e7 q^{23} -9.29644e7 q^{25} -2.04625e8i q^{27} +1.74228e8i q^{29} -2.58795e8 q^{31} +4.40015e8 q^{33} -2.51827e8i q^{35} +3.35460e8i q^{37} +8.68483e8 q^{39} -4.92499e8 q^{41} +7.74649e8i q^{43} +5.21599e9i q^{45} +1.19986e9 q^{47} -1.53008e9 q^{49} +4.21231e9i q^{51} +1.79948e9i q^{53} -6.68023e9 q^{55} +6.84378e9 q^{57} -9.05069e8i q^{59} +9.99168e9i q^{61} -9.26374e9 q^{63} -1.31852e10 q^{65} +1.13658e10i q^{67} -1.79684e10i q^{69} +2.40027e10 q^{71} -5.92056e9 q^{73} -7.29153e10i q^{75} -1.18643e10i q^{77} +1.80146e10 q^{79} +8.28982e10 q^{81} +2.45012e10i q^{83} -6.39506e10i q^{85} -1.36654e11 q^{87} -8.73556e10 q^{89} -2.34172e10i q^{91} -2.02982e11i q^{93} -1.03901e11 q^{95} -1.35707e11 q^{97} +2.45740e11i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 1152028 q^{9} + 26470216 q^{17} - 207502500 q^{25} + 935847712 q^{33} - 2031735176 q^{41} - 2342764308 q^{49} + 23753487776 q^{57} - 39461105920 q^{65} - 43591631448 q^{73} + 232401225292 q^{81}+ \cdots - 285441034488 q^{97}+O(q^{100}) $ 12 * q - 1152028 * q^9 + 26470216 * q^17 - 207502500 * q^25 + 935847712 * q^33 - 2031735176 * q^41 - 2342764308 * q^49 + 23753487776 * q^57 - 39461105920 * q^65 - 43591631448 * q^73 + 232401225292 * q^81 - 209389352088 * q^89 - 285441034488 * 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
$3$	784.336i	1.86353i	0.363067	+	0.931763i	$0.381730\pi$
$3$	784.336i	1.86353i	−0.363067	+	0.931763i	$0.618270\pi$
$4$	0	0
$5$	− 11907.7i	− 1.70409i	−0.523472	−	0.852043i	$-0.675363\pi$
$5$	− 11907.7i	− 1.70409i	0.523472	−	0.852043i	$-0.324637\pi$
$6$	0	0
$7$	21148.3	0.475594	0.237797	−	0.971315i	$-0.423575\pi$
$7$	21148.3	0.475594	0.237797	+	0.971315i	$0.423575\pi$
$8$	0	0
$9$	−438037.	−2.47273
$10$	0	0
$11$	− 561003.i	− 1.05028i	−0.851016	−	0.525140i	$-0.824013\pi$
$11$	− 561003.i	− 1.05028i	0.851016	−	0.525140i	$-0.175987\pi$
$12$	0	0
$13$	− 1.10728e6i	− 0.827124i	−0.910476	−	0.413562i	$-0.864285\pi$
$13$	− 1.10728e6i	− 0.827124i	0.910476	−	0.413562i	$-0.135715\pi$
$14$	0	0
$15$	9.33962e6	3.17561
$16$	0	0
$17$	5.37054e6	0.917379	0.458689	−	0.888597i	$-0.348319\pi$
$17$	5.37054e6	0.917379	0.458689	+	0.888597i	$0.348319\pi$
$18$	0	0
$19$	− 8.72557e6i	− 0.808442i	−0.914661	−	0.404221i	$-0.867543\pi$
$19$	− 8.72557e6i	− 0.808442i	0.914661	−	0.404221i	$-0.132457\pi$
$20$	0	0
$21$	1.65874e7i	0.886282i
$22$	0	0
$23$	−2.29090e7	−0.742171	−0.371085	−	0.928599i	$-0.621014\pi$
$23$	−2.29090e7	−0.742171	−0.371085	+	0.928599i	$0.621014\pi$
$24$	0	0
$25$	−9.29644e7	−1.90391
$26$	0	0
$27$	− 2.04625e8i	− 2.74447i
$28$	0	0
$29$	1.74228e8i	1.57736i	0.614806	+	0.788678i	$0.289234\pi$
$29$	1.74228e8i	1.57736i	−0.614806	+	0.788678i	$0.710766\pi$
$30$	0	0
$31$	−2.58795e8	−1.62355	−0.811777	−	0.583968i	$-0.801499\pi$
$31$	−2.58795e8	−1.62355	−0.811777	+	0.583968i	$0.801499\pi$
$32$	0	0
$33$	4.40015e8	1.95723
$34$	0	0
$35$	− 2.51827e8i	− 0.810453i
$36$	0	0
$37$	3.35460e8i	0.795302i	0.917537	+	0.397651i	$0.130174\pi$
$37$	3.35460e8i	0.795302i	−0.917537	+	0.397651i	$0.869826\pi$
$38$	0	0
$39$	8.68483e8	1.54137
$40$	0	0
$41$	−4.92499e8	−0.663888	−0.331944	−	0.943299i	$-0.607704\pi$
$41$	−4.92499e8	−0.663888	−0.331944	+	0.943299i	$0.607704\pi$
$42$	0	0
$43$	7.74649e8i	0.803579i	0.915732	+	0.401789i	$0.131612\pi$
$43$	7.74649e8i	0.803579i	−0.915732	+	0.401789i	$0.868388\pi$
$44$	0	0
$45$	5.21599e9i	4.21375i
$46$	0	0
$47$	1.19986e9	0.763118	0.381559	−	0.924344i	$-0.375387\pi$
$47$	1.19986e9	0.763118	0.381559	+	0.924344i	$0.375387\pi$
$48$	0	0
$49$	−1.53008e9	−0.773810
$50$	0	0
$51$	4.21231e9i	1.70956i
$52$	0	0
$53$	1.79948e9i	0.591058i	0.955334	+	0.295529i	$0.0954958\pi$
$53$	1.79948e9i	0.591058i	−0.955334	+	0.295529i	$0.904504\pi$
$54$	0	0
$55$	−6.68023e9	−1.78977
$56$	0	0
$57$	6.84378e9	1.50655
$58$	0	0
$59$	− 9.05069e8i	− 0.164815i	−0.996599	−	0.0824073i	$-0.973739\pi$
$59$	− 9.05069e8i	− 0.164815i	0.996599	−	0.0824073i	$-0.0262608\pi$
$60$	0	0
$61$	9.99168e9i	1.51469i	0.653014	+	0.757346i	$0.273504\pi$
$61$	9.99168e9i	1.51469i	−0.653014	+	0.757346i	$0.726496\pi$
$62$	0	0
$63$	−9.26374e9	−1.17602
$64$	0	0
$65$	−1.31852e10	−1.40949
$66$	0	0
$67$	1.13658e10i	1.02846i	0.857652	+	0.514230i	$0.171922\pi$
$67$	1.13658e10i	1.02846i	−0.857652	+	0.514230i	$0.828078\pi$
$68$	0	0
$69$	− 1.79684e10i	− 1.38305i
$70$	0	0
$71$	2.40027e10	1.57884	0.789421	−	0.613852i	$-0.210381\pi$
$71$	2.40027e10	1.57884	0.789421	+	0.613852i	$0.210381\pi$
$72$	0	0
$73$	−5.92056e9	−0.334262	−0.167131	−	0.985935i	$-0.553450\pi$
$73$	−5.92056e9	−0.334262	−0.167131	+	0.985935i	$0.553450\pi$
$74$	0	0
$75$	− 7.29153e10i	− 3.54799i
$76$	0	0
$77$	− 1.18643e10i	− 0.499507i
$78$	0	0
$79$	1.80146e10	0.658682	0.329341	−	0.944211i	$-0.393173\pi$
$79$	1.80146e10	0.658682	0.329341	+	0.944211i	$0.393173\pi$
$80$	0	0
$81$	8.28982e10	2.64166
$82$	0	0
$83$	2.45012e10i	0.682743i	0.939928	+	0.341372i	$0.110891\pi$
$83$	2.45012e10i	0.682743i	−0.939928	+	0.341372i	$0.889109\pi$
$84$	0	0
$85$	− 6.39506e10i	− 1.56329i
$86$	0	0
$87$	−1.36654e11	−2.93944
$88$	0	0
$89$	−8.73556e10	−1.65823	−0.829117	−	0.559075i	$-0.811156\pi$
$89$	−8.73556e10	−1.65823	−0.829117	+	0.559075i	$0.811156\pi$
$90$	0	0
$91$	− 2.34172e10i	− 0.393375i
$92$	0	0
$93$	− 2.02982e11i	− 3.02553i
$94$	0	0
$95$	−1.03901e11	−1.37765
$96$	0	0
$97$	−1.35707e11	−1.60456	−0.802281	−	0.596947i	$-0.796380\pi$
$97$	−1.35707e11	−1.60456	−0.802281	+	0.596947i	$0.796380\pi$
$98$	0	0
$99$	2.45740e11i	2.59706i
$100$	0	0
$101$	1.17395e11i	1.11143i	0.831372	+	0.555716i	$0.187556\pi$
$101$	1.17395e11i	1.11143i	−0.831372	+	0.555716i	$0.812444\pi$
$102$	0	0
$103$	1.84106e10	0.156481	0.0782407	−	0.996934i	$-0.475070\pi$
$103$	1.84106e10	0.156481	0.0782407	+	0.996934i	$0.475070\pi$
$104$	0	0
$105$	1.97517e11	1.51030
$106$	0	0
$107$	− 1.08175e11i	− 0.745616i	−0.927909	−	0.372808i	$-0.878395\pi$
$107$	− 1.08175e11i	− 0.745616i	0.927909	−	0.372808i	$-0.121605\pi$
$108$	0	0
$109$	− 1.80324e9i	− 0.0112256i	−0.999984	−	0.00561279i	$-0.998213\pi$
$109$	− 1.80324e9i	− 0.0112256i	0.999984	−	0.00561279i	$-0.00178662\pi$
$110$	0	0
$111$	−2.63114e11	−1.48207
$112$	0	0
$113$	1.50400e11	0.767921	0.383960	−	0.923350i	$-0.374560\pi$
$113$	1.50400e11	0.767921	0.383960	+	0.923350i	$0.374560\pi$
$114$	0	0
$115$	2.72793e11i	1.26472i
$116$	0	0
$117$	4.85031e11i	2.04525i
$118$	0	0
$119$	1.13578e11	0.436300
$120$	0	0
$121$	−2.94126e10	−0.103089
$122$	0	0
$123$	− 3.86285e11i	− 1.23717i
$124$	0	0
$125$	5.25560e11i	1.54034i
$126$	0	0
$127$	−4.09141e11	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$127$	−4.09141e11	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$128$	0	0
$129$	−6.07585e11	−1.49749
$130$	0	0
$131$	− 2.82610e11i	− 0.640023i	−0.947414	−	0.320012i	$-0.896313\pi$
$131$	− 2.82610e11i	− 0.640023i	0.947414	−	0.320012i	$-0.103687\pi$
$132$	0	0
$133$	− 1.84531e11i	− 0.384490i
$134$	0	0
$135$	−2.43661e12	−4.67682
$136$	0	0
$137$	−5.04027e9	−0.00892258	−0.00446129	−	0.999990i	$-0.501420\pi$
$137$	−5.04027e9	−0.00892258	−0.00446129	+	0.999990i	$0.501420\pi$
$138$	0	0
$139$	− 5.20421e11i	− 0.850695i	−0.905030	−	0.425347i	$-0.860152\pi$
$139$	− 5.20421e11i	− 0.850695i	0.905030	−	0.425347i	$-0.139848\pi$
$140$	0	0
$141$	9.41093e11i	1.42209i
$142$	0	0
$143$	−6.21189e11	−0.868712
$144$	0	0
$145$	2.07465e12	2.68795
$146$	0	0
$147$	− 1.20009e12i	− 1.44202i
$148$	0	0
$149$	3.21703e11i	0.358865i	0.983770	+	0.179432i	$0.0574261\pi$
$149$	3.21703e11i	0.358865i	−0.983770	+	0.179432i	$0.942574\pi$
$150$	0	0
$151$	6.64644e11	0.688995	0.344497	−	0.938787i	$-0.388049\pi$
$151$	6.64644e11	0.688995	0.344497	+	0.938787i	$0.388049\pi$
$152$	0	0
$153$	−2.35249e12	−2.26843
$154$	0	0
$155$	3.08164e12i	2.76668i
$156$	0	0
$157$	− 1.91656e11i	− 0.160352i	−0.996781	−	0.0801760i	$-0.974452\pi$
$157$	− 1.91656e11i	− 0.160352i	0.996781	−	0.0801760i	$-0.0255482\pi$
$158$	0	0
$159$	−1.41140e12	−1.10145
$160$	0	0
$161$	−4.84488e11	−0.352972
$162$	0	0
$163$	− 1.19385e12i	− 0.812679i	−0.913722	−	0.406339i	$-0.866805\pi$
$163$	− 1.19385e12i	− 0.812679i	0.913722	−	0.406339i	$-0.133195\pi$
$164$	0	0
$165$	− 5.23955e12i	− 3.33528i
$166$	0	0
$167$	3.53708e11	0.210719	0.105360	−	0.994434i	$-0.466401\pi$
$167$	3.53708e11	0.210719	0.105360	+	0.994434i	$0.466401\pi$
$168$	0	0
$169$	5.66083e11	0.315866
$170$	0	0
$171$	3.82212e12i	1.99906i
$172$	0	0
$173$	− 2.41578e12i	− 1.18523i	−0.805484	−	0.592617i	$-0.798095\pi$
$173$	− 2.41578e12i	− 1.18523i	0.805484	−	0.592617i	$-0.201905\pi$
$174$	0	0
$175$	−1.96604e12	−0.905488
$176$	0	0
$177$	7.09879e11	0.307136
$178$	0	0
$179$	2.07221e12i	0.842834i	0.906867	+	0.421417i	$0.138467\pi$
$179$	2.07221e12i	0.842834i	−0.906867	+	0.421417i	$0.861533\pi$
$180$	0	0
$181$	8.77418e11i	0.335718i	0.985811	+	0.167859i	$0.0536853\pi$
$181$	8.77418e11i	0.335718i	−0.985811	+	0.167859i	$0.946315\pi$
$182$	0	0
$183$	−7.83684e12	−2.82267
$184$	0	0
$185$	3.99455e12	1.35526
$186$	0	0
$187$	− 3.01289e12i	− 0.963505i
$188$	0	0
$189$	− 4.32748e12i	− 1.30525i
$190$	0	0
$191$	−2.74196e12	−0.780510	−0.390255	−	0.920707i	$-0.627613\pi$
$191$	−2.74196e12	−0.780510	−0.390255	+	0.920707i	$0.627613\pi$
$192$	0	0
$193$	5.39488e11	0.145016	0.0725081	−	0.997368i	$-0.476900\pi$
$193$	5.39488e11	0.145016	0.0725081	+	0.997368i	$0.476900\pi$
$194$	0	0
$195$	− 1.03416e13i	− 2.62662i
$196$	0	0
$197$	6.83190e11i	0.164050i	0.996630	+	0.0820252i	$0.0261388\pi$
$197$	6.83190e11i	0.164050i	−0.996630	+	0.0820252i	$0.973861\pi$
$198$	0	0
$199$	4.65463e12	1.05729	0.528644	−	0.848843i	$-0.322701\pi$
$199$	4.65463e12	1.05729	0.528644	+	0.848843i	$0.322701\pi$
$200$	0	0
$201$	−8.91459e12	−1.91656
$202$	0	0
$203$	3.68463e12i	0.750181i
$204$	0	0
$205$	5.86452e12i	1.13132i
$206$	0	0
$207$	1.00350e13	1.83519
$208$	0	0
$209$	−4.89507e12	−0.849091
$210$	0	0
$211$	− 2.06896e12i	− 0.340565i	−0.985395	−	0.170282i	$-0.945532\pi$
$211$	− 2.06896e12i	− 0.340565i	0.985395	−	0.170282i	$-0.0544680\pi$
$212$	0	0
$213$	1.88262e13i	2.94221i
$214$	0	0
$215$	9.22426e12	1.36937
$216$	0	0
$217$	−5.47308e12	−0.772153
$218$	0	0
$219$	− 4.64371e12i	− 0.622906i
$220$	0	0
$221$	− 5.94671e12i	− 0.758786i
$222$	0	0
$223$	−4.03814e12	−0.490348	−0.245174	−	0.969479i	$-0.578845\pi$
$223$	−4.03814e12	−0.490348	−0.245174	+	0.969479i	$0.578845\pi$
$224$	0	0
$225$	4.07218e13	4.70786
$226$	0	0
$227$	1.10735e13i	1.21939i	0.792637	+	0.609694i	$0.208708\pi$
$227$	1.10735e13i	1.21939i	−0.792637	+	0.609694i	$0.791292\pi$
$228$	0	0
$229$	1.38640e12i	0.145477i	0.997351	+	0.0727383i	$0.0231738\pi$
$229$	1.38640e12i	0.145477i	−0.997351	+	0.0727383i	$0.976826\pi$
$230$	0	0
$231$	9.30557e12	0.930845
$232$	0	0
$233$	−1.76073e13	−1.67972	−0.839859	−	0.542805i	$-0.817362\pi$
$233$	−1.76073e13	−1.67972	−0.839859	+	0.542805i	$0.817362\pi$
$234$	0	0
$235$	− 1.42875e13i	− 1.30042i
$236$	0	0
$237$	1.41295e13i	1.22747i
$238$	0	0
$239$	4.02394e11	0.0333782	0.0166891	−	0.999861i	$-0.494687\pi$
$239$	4.02394e11	0.0333782	0.0166891	+	0.999861i	$0.494687\pi$
$240$	0	0
$241$	−6.87906e12	−0.545049	−0.272524	−	0.962149i	$-0.587859\pi$
$241$	−6.87906e12	−0.545049	−0.272524	+	0.962149i	$0.587859\pi$
$242$	0	0
$243$	2.87713e13i	2.17834i
$244$	0	0
$245$	1.82196e13i	1.31864i
$246$	0	0
$247$	−9.66168e12	−0.668681
$248$	0	0
$249$	−1.92172e13	−1.27231
$250$	0	0
$251$	2.50600e13i	1.58772i	0.608097	+	0.793862i	$0.291933\pi$
$251$	2.50600e13i	1.58772i	−0.608097	+	0.793862i	$0.708067\pi$
$252$	0	0
$253$	1.28520e13i	0.779488i
$254$	0	0
$255$	5.01588e13	2.91324
$256$	0	0
$257$	2.61595e12	0.145545	0.0727724	−	0.997349i	$-0.476815\pi$
$257$	2.61595e12	0.145545	0.0727724	+	0.997349i	$0.476815\pi$
$258$	0	0
$259$	7.09442e12i	0.378241i
$260$	0	0
$261$	− 7.63184e13i	− 3.90038i
$262$	0	0
$263$	−1.35684e12	−0.0664924	−0.0332462	−	0.999447i	$-0.510585\pi$
$263$	−1.35684e12	−0.0664924	−0.0332462	+	0.999447i	$0.510585\pi$
$264$	0	0
$265$	2.14276e13	1.00721
$266$	0	0
$267$	− 6.85162e13i	− 3.09016i
$268$	0	0
$269$	6.55358e12i	0.283688i	0.989889	+	0.141844i	$0.0453032\pi$
$269$	6.55358e12i	0.283688i	−0.989889	+	0.141844i	$0.954697\pi$
$270$	0	0
$271$	2.39710e13	0.996220	0.498110	−	0.867114i	$-0.334028\pi$
$271$	2.39710e13	0.996220	0.498110	+	0.867114i	$0.334028\pi$
$272$	0	0
$273$	1.83669e13	0.733065
$274$	0	0
$275$	5.21533e13i	1.99964i
$276$	0	0
$277$	− 2.54063e13i	− 0.936058i	−0.883713	−	0.468029i	$-0.844964\pi$
$277$	− 2.54063e13i	− 0.936058i	0.883713	−	0.468029i	$-0.155036\pi$
$278$	0	0
$279$	1.13362e14	4.01461
$280$	0	0
$281$	−9.10879e12	−0.310153	−0.155076	−	0.987902i	$-0.549562\pi$
$281$	−9.10879e12	−0.310153	−0.155076	+	0.987902i	$0.549562\pi$
$282$	0	0
$283$	− 3.57642e13i	− 1.17118i	−0.810608	−	0.585589i	$-0.800863\pi$
$283$	− 3.57642e13i	− 1.17118i	0.810608	−	0.585589i	$-0.199137\pi$
$284$	0	0
$285$	− 8.14934e13i	− 2.56730i
$286$	0	0
$287$	−1.04155e13	−0.315741
$288$	0	0
$289$	−5.42922e12	−0.158416
$290$	0	0
$291$	− 1.06440e14i	− 2.99014i
$292$	0	0
$293$	− 8.24513e11i	− 0.0223062i	−0.999938	−	0.0111531i	$-0.996450\pi$
$293$	− 8.24513e11i	− 0.0223062i	0.999938	−	0.0111531i	$-0.00355021\pi$
$294$	0	0
$295$	−1.07773e13	−0.280858
$296$	0	0
$297$	−1.14795e14	−2.88246
$298$	0	0
$299$	2.53668e13i	0.613867i
$300$	0	0
$301$	1.63825e13i	0.382177i
$302$	0	0
$303$	−9.20774e13	−2.07118
$304$	0	0
$305$	1.18978e14	2.58117
$306$	0	0
$307$	2.09611e13i	0.438685i	0.975648	+	0.219342i	$0.0703911\pi$
$307$	2.09611e13i	0.438685i	−0.975648	+	0.219342i	$0.929609\pi$
$308$	0	0
$309$	1.44401e13i	0.291607i
$310$	0	0
$311$	−7.82526e13	−1.52516	−0.762582	−	0.646892i	$-0.776069\pi$
$311$	−7.82526e13	−1.52516	−0.762582	+	0.646892i	$0.776069\pi$
$312$	0	0
$313$	−3.47387e13	−0.653611	−0.326806	−	0.945092i	$-0.605972\pi$
$313$	−3.47387e13	−0.653611	−0.326806	+	0.945092i	$0.605972\pi$
$314$	0	0
$315$	1.10309e14i	2.00403i
$316$	0	0
$317$	9.15418e13i	1.60618i	0.595860	+	0.803088i	$0.296811\pi$
$317$	9.15418e13i	1.60618i	−0.595860	+	0.803088i	$0.703189\pi$
$318$	0	0
$319$	9.77426e13	1.65667
$320$	0	0
$321$	8.48454e13	1.38947
$322$	0	0
$323$	− 4.68610e13i	− 0.741647i
$324$	0	0
$325$	1.02938e14i	1.57477i
$326$	0	0
$327$	1.41435e12	0.0209192
$328$	0	0
$329$	2.53750e13	0.362935
$330$	0	0
$331$	3.51219e13i	0.485875i	0.970042	+	0.242937i	$0.0781109\pi$
$331$	3.51219e13i	0.485875i	−0.970042	+	0.242937i	$0.921889\pi$
$332$	0	0
$333$	− 1.46944e14i	− 1.96657i
$334$	0	0
$335$	1.35340e14	1.75259
$336$	0	0
$337$	5.42414e13	0.679777	0.339889	−	0.940466i	$-0.389611\pi$
$337$	5.42414e13	0.679777	0.339889	+	0.940466i	$0.389611\pi$
$338$	0	0
$339$	1.17964e14i	1.43104i
$340$	0	0
$341$	1.45185e14i	1.70519i
$342$	0	0
$343$	−7.41756e13	−0.843614
$344$	0	0
$345$	−2.13962e14	−2.35684
$346$	0	0
$347$	1.38780e14i	1.48086i	0.672134	+	0.740430i	$0.265378\pi$
$347$	1.38780e14i	1.48086i	−0.672134	+	0.740430i	$0.734622\pi$
$348$	0	0
$349$	1.70059e14i	1.75816i	0.476674	+	0.879080i	$0.341842\pi$
$349$	1.70059e14i	1.75816i	−0.476674	+	0.879080i	$0.658158\pi$
$350$	0	0
$351$	−2.26578e14	−2.27002
$352$	0	0
$353$	−1.71901e14	−1.66924	−0.834620	−	0.550826i	$-0.814313\pi$
$353$	−1.71901e14	−1.66924	−0.834620	+	0.550826i	$0.814313\pi$
$354$	0	0
$355$	− 2.85816e14i	− 2.69048i
$356$	0	0
$357$	8.90832e13i	0.813056i
$358$	0	0
$359$	1.27135e14	1.12524	0.562619	−	0.826716i	$-0.309794\pi$
$359$	1.27135e14	1.12524	0.562619	+	0.826716i	$0.309794\pi$
$360$	0	0
$361$	4.03548e13	0.346422
$362$	0	0
$363$	− 2.30694e13i	− 0.192110i
$364$	0	0
$365$	7.05000e13i	0.569611i
$366$	0	0
$367$	−2.04239e14	−1.60131	−0.800656	−	0.599124i	$-0.795515\pi$
$367$	−2.04239e14	−1.60131	−0.800656	+	0.599124i	$0.795515\pi$
$368$	0	0
$369$	2.15733e14	1.64161
$370$	0	0
$371$	3.80560e13i	0.281104i
$372$	0	0
$373$	− 1.16113e14i	− 0.832688i	−0.909207	−	0.416344i	$-0.863311\pi$
$373$	− 1.16113e14i	− 0.832688i	0.909207	−	0.416344i	$-0.136689\pi$
$374$	0	0
$375$	−4.12216e14	−2.87047
$376$	0	0
$377$	1.92920e14	1.30467
$378$	0	0
$379$	3.86376e13i	0.253802i	0.991915	+	0.126901i	$0.0405030\pi$
$379$	3.86376e13i	0.253802i	−0.991915	+	0.126901i	$0.959497\pi$
$380$	0	0
$381$	− 3.20904e14i	− 2.04780i
$382$	0	0
$383$	−2.83128e14	−1.75545	−0.877727	−	0.479161i	$-0.840941\pi$
$383$	−2.83128e14	−1.75545	−0.877727	+	0.479161i	$0.840941\pi$
$384$	0	0
$385$	−1.41276e14	−0.851203
$386$	0	0
$387$	− 3.39325e14i	− 1.98703i
$388$	0	0
$389$	5.49647e13i	0.312868i	0.987688	+	0.156434i	$0.0499998\pi$
$389$	5.49647e13i	0.312868i	−0.987688	+	0.156434i	$0.950000\pi$
$390$	0	0
$391$	−1.23034e14	−0.680852
$392$	0	0
$393$	2.21661e14	1.19270
$394$	0	0
$395$	− 2.14512e14i	− 1.12245i
$396$	0	0
$397$	2.00590e14i	1.02085i	0.859923	+	0.510423i	$0.170511\pi$
$397$	2.00590e14i	1.02085i	−0.859923	+	0.510423i	$0.829489\pi$
$398$	0	0
$399$	1.44734e14	0.716508
$400$	0	0
$401$	−2.05103e14	−0.987819	−0.493910	−	0.869513i	$-0.664433\pi$
$401$	−2.05103e14	−0.987819	−0.493910	+	0.869513i	$0.664433\pi$
$402$	0	0
$403$	2.86560e14i	1.34288i
$404$	0	0
$405$	− 9.87124e14i	− 4.50162i
$406$	0	0
$407$	1.88194e14	0.835290
$408$	0	0
$409$	−4.18830e14	−1.80950	−0.904751	−	0.425940i	$-0.859943\pi$
$409$	−4.18830e14	−1.80950	−0.904751	+	0.425940i	$0.859943\pi$
$410$	0	0
$411$	− 3.95326e12i	− 0.0166275i
$412$	0	0
$413$	− 1.91407e13i	− 0.0783849i
$414$	0	0
$415$	2.91752e14	1.16345
$416$	0	0
$417$	4.08185e14	1.58529
$418$	0	0
$419$	− 2.95630e14i	− 1.11833i	−0.829056	−	0.559166i	$-0.811121\pi$
$419$	− 2.95630e14i	− 1.11833i	0.829056	−	0.559166i	$-0.188879\pi$
$420$	0	0
$421$	− 7.24565e13i	− 0.267009i	−0.991048	−	0.133504i	$-0.957377\pi$
$421$	− 7.24565e13i	− 0.267009i	0.991048	−	0.133504i	$-0.0426231\pi$
$422$	0	0
$423$	−5.25582e14	−1.88699
$424$	0	0
$425$	−4.99269e14	−1.74661
$426$	0	0
$427$	2.11307e14i	0.720379i
$428$	0	0
$429$	− 4.87221e14i	− 1.61887i
$430$	0	0
$431$	−4.24283e14	−1.37414	−0.687069	−	0.726592i	$-0.741103\pi$
$431$	−4.24283e14	−1.37414	−0.687069	+	0.726592i	$0.741103\pi$
$432$	0	0
$433$	1.62175e14	0.512038	0.256019	−	0.966672i	$-0.417589\pi$
$433$	1.62175e14	0.512038	0.256019	+	0.966672i	$0.417589\pi$
$434$	0	0
$435$	1.62723e15i	5.00907i
$436$	0	0
$437$	1.99894e14i	0.600002i
$438$	0	0
$439$	6.41896e14	1.87893	0.939464	−	0.342648i	$-0.111324\pi$
$439$	6.41896e14	1.87893	0.939464	+	0.342648i	$0.111324\pi$
$440$	0	0
$441$	6.70229e14	1.91342
$442$	0	0
$443$	− 4.43845e14i	− 1.23598i	−0.786187	−	0.617989i	$-0.787948\pi$
$443$	− 4.43845e14i	− 1.23598i	0.786187	−	0.617989i	$-0.212052\pi$
$444$	0	0
$445$	1.04020e15i	2.82577i
$446$	0	0
$447$	−2.52323e14	−0.668754
$448$	0	0
$449$	3.08017e14	0.796562	0.398281	−	0.917263i	$-0.369607\pi$
$449$	3.08017e14	0.796562	0.398281	+	0.917263i	$0.369607\pi$
$450$	0	0
$451$	2.76294e14i	0.697268i
$452$	0	0
$453$	5.21304e14i	1.28396i
$454$	0	0
$455$	−2.78844e14	−0.670345
$456$	0	0
$457$	2.49484e14	0.585469	0.292734	−	0.956194i	$-0.405435\pi$
$457$	2.49484e14	0.585469	0.292734	+	0.956194i	$0.405435\pi$
$458$	0	0
$459$	− 1.09895e15i	− 2.51772i
$460$	0	0
$461$	− 2.25843e14i	− 0.505186i	−0.967573	−	0.252593i	$-0.918717\pi$
$461$	− 2.25843e14i	− 0.505186i	0.967573	−	0.252593i	$-0.0812833\pi$
$462$	0	0
$463$	−4.88011e14	−1.06594	−0.532971	−	0.846133i	$-0.678925\pi$
$463$	−4.88011e14	−1.06594	−0.532971	+	0.846133i	$0.678925\pi$
$464$	0	0
$465$	−2.41705e15	−5.15577
$466$	0	0
$467$	− 4.26504e14i	− 0.888547i	−0.895891	−	0.444274i	$-0.853462\pi$
$467$	− 4.26504e14i	− 0.888547i	0.895891	−	0.444274i	$-0.146538\pi$
$468$	0	0
$469$	2.40367e14i	0.489130i
$470$	0	0
$471$	1.50323e14	0.298820
$472$	0	0
$473$	4.34580e14	0.843983
$474$	0	0
$475$	8.11167e14i	1.53920i
$476$	0	0
$477$	− 7.88238e14i	− 1.46153i
$478$	0	0
$479$	8.00213e14	1.44997	0.724987	−	0.688762i	$-0.241846\pi$
$479$	8.00213e14	1.44997	0.724987	+	0.688762i	$0.241846\pi$
$480$	0	0
$481$	3.71450e14	0.657813
$482$	0	0
$483$	− 3.80001e14i	− 0.657773i
$484$	0	0
$485$	1.61595e15i	2.73431i
$486$	0	0
$487$	2.70351e14	0.447217	0.223608	−	0.974679i	$-0.428216\pi$
$487$	2.70351e14	0.447217	0.223608	+	0.974679i	$0.428216\pi$
$488$	0	0
$489$	9.36382e14	1.51445
$490$	0	0
$491$	− 1.09128e15i	− 1.72578i	−0.505388	−	0.862892i	$-0.668651\pi$
$491$	− 1.09128e15i	− 1.72578i	0.505388	−	0.862892i	$-0.331349\pi$
$492$	0	0
$493$	9.35700e14i	1.44703i
$494$	0	0
$495$	2.92619e15	4.42561
$496$	0	0
$497$	5.07616e14	0.750888
$498$	0	0
$499$	− 5.65887e14i	− 0.818798i	−0.912355	−	0.409399i	$-0.865738\pi$
$499$	− 5.65887e14i	− 0.818798i	0.912355	−	0.409399i	$-0.134262\pi$
$500$	0	0
$501$	2.77426e14i	0.392681i
$502$	0	0
$503$	3.29370e14	0.456100	0.228050	−	0.973649i	$-0.426765\pi$
$503$	3.29370e14	0.456100	0.228050	+	0.973649i	$0.426765\pi$
$504$	0	0
$505$	1.39790e15	1.89398
$506$	0	0
$507$	4.44000e14i	0.588625i
$508$	0	0
$509$	1.64942e14i	0.213985i	0.994260	+	0.106992i	$0.0341221\pi$
$509$	1.64942e14i	0.213985i	−0.994260	+	0.106992i	$0.965878\pi$
$510$	0	0
$511$	−1.25210e14	−0.158973
$512$	0	0
$513$	−1.78547e15	−2.21874
$514$	0	0
$515$	− 2.19227e14i	− 0.266658i
$516$	0	0
$517$	− 6.73124e14i	− 0.801488i
$518$	0	0
$519$	1.89479e15	2.20871
$520$	0	0
$521$	−2.94308e14	−0.335888	−0.167944	−	0.985797i	$-0.553713\pi$
$521$	−2.94308e14	−0.335888	−0.167944	+	0.985797i	$0.553713\pi$
$522$	0	0
$523$	6.57551e14i	0.734802i	0.930063	+	0.367401i	$0.119752\pi$
$523$	6.57551e14i	0.734802i	−0.930063	+	0.367401i	$0.880248\pi$
$524$	0	0
$525$	− 1.54204e15i	− 1.68740i
$526$	0	0
$527$	−1.38987e15	−1.48941
$528$	0	0
$529$	−4.27985e14	−0.449182
$530$	0	0
$531$	3.96454e14i	0.407542i
$532$	0	0
$533$	5.45337e14i	0.549117i
$534$	0	0
$535$	−1.28811e15	−1.27059
$536$	0	0
$537$	−1.62531e15	−1.57064
$538$	0	0
$539$	8.58377e14i	0.812718i
$540$	0	0
$541$	4.39506e14i	0.407736i	0.978998	+	0.203868i	$0.0653514\pi$
$541$	4.39506e14i	0.407736i	−0.978998	+	0.203868i	$0.934649\pi$
$542$	0	0
$543$	−6.88191e14	−0.625619
$544$	0	0
$545$	−2.14724e13	−0.0191294
$546$	0	0
$547$	− 3.67457e14i	− 0.320831i	−0.987050	−	0.160416i	$-0.948717\pi$
$547$	− 3.67457e14i	− 0.320831i	0.987050	−	0.160416i	$-0.0512835\pi$
$548$	0	0
$549$	− 4.37672e15i	− 3.74543i
$550$	0	0
$551$	1.52024e15	1.27520
$552$	0	0
$553$	3.80979e14	0.313265
$554$	0	0
$555$	3.13307e15i	2.52557i
$556$	0	0
$557$	− 3.41815e13i	− 0.0270139i	−0.999909	−	0.0135070i	$-0.995700\pi$
$557$	− 3.41815e13i	− 0.0270139i	0.999909	−	0.0135070i	$-0.00429953\pi$
$558$	0	0
$559$	8.57756e14	0.664659
$560$	0	0
$561$	2.36312e15	1.79552
$562$	0	0
$563$	− 1.57295e15i	− 1.17198i	−0.810320	−	0.585988i	$-0.800707\pi$
$563$	− 1.57295e15i	− 1.17198i	0.810320	−	0.585988i	$-0.199293\pi$
$564$	0	0
$565$	− 1.79091e15i	− 1.30860i
$566$	0	0
$567$	1.75316e15	1.25636
$568$	0	0
$569$	4.74133e14	0.333260	0.166630	−	0.986020i	$-0.446711\pi$
$569$	4.74133e14	0.333260	0.166630	+	0.986020i	$0.446711\pi$
$570$	0	0
$571$	− 2.54823e14i	− 0.175687i	−0.996134	−	0.0878436i	$-0.972002\pi$
$571$	− 2.54823e14i	− 0.175687i	0.996134	−	0.0878436i	$-0.0279976\pi$
$572$	0	0
$573$	− 2.15062e15i	− 1.45450i
$574$	0	0
$575$	2.12972e15	1.41303
$576$	0	0
$577$	−1.70605e15	−1.11051	−0.555257	−	0.831679i	$-0.687380\pi$
$577$	−1.70605e15	−1.11051	−0.555257	+	0.831679i	$0.687380\pi$
$578$	0	0
$579$	4.23140e14i	0.270241i
$580$	0	0
$581$	5.18158e14i	0.324709i
$582$	0	0
$583$	1.00951e15	0.620776
$584$	0	0
$585$	5.77559e15	3.48529
$586$	0	0
$587$	2.17525e15i	1.28825i	0.764922	+	0.644123i	$0.222778\pi$
$587$	2.17525e15i	1.28825i	−0.764922	+	0.644123i	$0.777222\pi$
$588$	0	0
$589$	2.25813e15i	1.31255i
$590$	0	0
$591$	−5.35851e14	−0.305712
$592$	0	0
$593$	−3.32178e15	−1.86025	−0.930123	−	0.367249i	$-0.880300\pi$
$593$	−3.32178e15	−1.86025	−0.930123	+	0.367249i	$0.880300\pi$
$594$	0	0
$595$	− 1.35245e15i	− 0.743493i
$596$	0	0
$597$	3.65080e15i	1.97028i
$598$	0	0
$599$	−3.39864e15	−1.80077	−0.900384	−	0.435096i	$-0.856715\pi$
$599$	−3.39864e15	−1.80077	−0.900384	+	0.435096i	$0.856715\pi$
$600$	0	0
$601$	−2.62640e15	−1.36632	−0.683159	−	0.730269i	$-0.739395\pi$
$601$	−2.62640e15	−1.36632	−0.683159	+	0.730269i	$0.739395\pi$
$602$	0	0
$603$	− 4.97863e15i	− 2.54311i
$604$	0	0
$605$	3.50235e14i	0.175673i
$606$	0	0
$607$	3.25639e15	1.60398	0.801989	−	0.597339i	$-0.203775\pi$
$607$	3.25639e15	1.60398	0.801989	+	0.597339i	$0.203775\pi$
$608$	0	0
$609$	−2.88999e15	−1.39798
$610$	0	0
$611$	− 1.32858e15i	− 0.631193i
$612$	0	0
$613$	− 3.86533e15i	− 1.80366i	−0.432092	−	0.901829i	$-0.642224\pi$
$613$	− 3.86533e15i	− 1.80366i	0.432092	−	0.901829i	$-0.357776\pi$
$614$	0	0
$615$	−4.59976e15	−2.10825
$616$	0	0
$617$	2.76303e14	0.124399	0.0621994	−	0.998064i	$-0.480189\pi$
$617$	2.76303e14	0.124399	0.0621994	+	0.998064i	$0.480189\pi$
$618$	0	0
$619$	− 1.13832e15i	− 0.503463i	−0.967797	−	0.251731i	$-0.919000\pi$
$619$	− 1.13832e15i	− 0.503463i	0.967797	−	0.251731i	$-0.0810000\pi$
$620$	0	0
$621$	4.68777e15i	2.03687i
$622$	0	0
$623$	−1.84742e15	−0.788646
$624$	0	0
$625$	1.71891e15	0.720964
$626$	0	0
$627$	− 3.83938e15i	− 1.58230i
$628$	0	0
$629$	1.80160e15i	0.729593i
$630$	0	0
$631$	1.38413e15	0.550828	0.275414	−	0.961326i	$-0.411185\pi$
$631$	1.38413e15	0.550828	0.275414	+	0.961326i	$0.411185\pi$
$632$	0	0
$633$	1.62276e15	0.634651
$634$	0	0
$635$	4.87191e15i	1.87259i
$636$	0	0
$637$	1.69423e15i	0.640037i
$638$	0	0
$639$	−1.05141e16	−3.90405
$640$	0	0
$641$	1.75717e15	0.641351	0.320675	−	0.947189i	$-0.396090\pi$
$641$	1.75717e15	0.641351	0.320675	+	0.947189i	$0.396090\pi$
$642$	0	0
$643$	2.92244e15i	1.04854i	0.851552	+	0.524271i	$0.175662\pi$
$643$	2.92244e15i	1.04854i	−0.851552	+	0.524271i	$0.824338\pi$
$644$	0	0
$645$	7.23492e15i	2.55185i
$646$	0	0
$647$	−2.78639e15	−0.966203	−0.483102	−	0.875564i	$-0.660490\pi$
$647$	−2.78639e15	−0.966203	−0.483102	+	0.875564i	$0.660490\pi$
$648$	0	0
$649$	−5.07746e14	−0.173102
$650$	0	0
$651$	− 4.29273e15i	− 1.43893i
$652$	0	0
$653$	− 4.07215e15i	− 1.34215i	−0.741390	−	0.671075i	$-0.765833\pi$
$653$	− 4.07215e15i	− 1.34215i	0.741390	−	0.671075i	$-0.234167\pi$
$654$	0	0
$655$	−3.36523e15	−1.09065
$656$	0	0
$657$	2.59342e15	0.826539
$658$	0	0
$659$	3.54311e15i	1.11049i	0.831686	+	0.555246i	$0.187376\pi$
$659$	3.54311e15i	1.11049i	−0.831686	+	0.555246i	$0.812624\pi$
$660$	0	0
$661$	3.95683e15i	1.21966i	0.792531	+	0.609831i	$0.208763\pi$
$661$	3.95683e15i	1.21966i	−0.792531	+	0.609831i	$0.791237\pi$
$662$	0	0
$663$	4.66422e15	1.41402
$664$	0	0
$665$	−2.19733e15	−0.655204
$666$	0	0
$667$	− 3.99140e15i	− 1.17067i
$668$	0	0
$669$	− 3.16726e15i	− 0.913777i
$670$	0	0
$671$	5.60536e15	1.59085
$672$	0	0
$673$	−6.68109e14	−0.186537	−0.0932684	−	0.995641i	$-0.529731\pi$
$673$	−6.68109e14	−0.186537	−0.0932684	+	0.995641i	$0.529731\pi$
$674$	0	0
$675$	1.90229e16i	5.22523i
$676$	0	0
$677$	− 1.84247e15i	− 0.497924i	−0.968513	−	0.248962i	$-0.919911\pi$
$677$	− 1.84247e15i	− 0.497924i	0.968513	−	0.248962i	$-0.0800895\pi$
$678$	0	0
$679$	−2.86996e15	−0.763120
$680$	0	0
$681$	−8.68534e15	−2.27236
$682$	0	0
$683$	3.44924e15i	0.887993i	0.896028	+	0.443997i	$0.146440\pi$
$683$	3.44924e15i	0.887993i	−0.896028	+	0.443997i	$0.853560\pi$
$684$	0	0
$685$	6.00178e13i	0.0152048i
$686$	0	0
$687$	−1.08740e15	−0.271099
$688$	0	0
$689$	1.99254e15	0.488878
$690$	0	0
$691$	− 6.31976e15i	− 1.52606i	−0.646363	−	0.763030i	$-0.723711\pi$
$691$	− 6.31976e15i	− 1.52606i	0.646363	−	0.763030i	$-0.276289\pi$
$692$	0	0
$693$	5.19698e15i	1.23515i
$694$	0	0
$695$	−6.19700e15	−1.44966
$696$	0	0
$697$	−2.64499e15	−0.609036
$698$	0	0
$699$	− 1.38101e16i	− 3.13020i
$700$	0	0
$701$	− 4.54606e15i	− 1.01435i	−0.861844	−	0.507173i	$-0.830690\pi$
$701$	− 4.54606e15i	− 1.01435i	0.861844	−	0.507173i	$-0.169310\pi$
$702$	0	0
$703$	2.92708e15	0.642955
$704$	0	0
$705$	1.12062e16	2.42337
$706$	0	0
$707$	2.48271e15i	0.528591i
$708$	0	0
$709$	− 5.71404e15i	− 1.19781i	−0.800819	−	0.598906i	$-0.795602\pi$
$709$	− 5.71404e15i	− 1.19781i	0.800819	−	0.598906i	$-0.204398\pi$
$710$	0	0
$711$	−7.89106e15	−1.62874
$712$	0	0
$713$	5.92875e15	1.20495
$714$	0	0
$715$	7.39692e15i	1.48036i
$716$	0	0
$717$	3.15612e14i	0.0622012i
$718$	0	0
$719$	−3.36530e15	−0.653153	−0.326577	−	0.945171i	$-0.605895\pi$
$719$	−3.36530e15	−0.653153	−0.326577	+	0.945171i	$0.605895\pi$
$720$	0	0
$721$	3.89353e14	0.0744217
$722$	0	0
$723$	− 5.39550e15i	− 1.01571i
$724$	0	0
$725$	− 1.61970e16i	− 3.00314i
$726$	0	0
$727$	5.08263e14	0.0928216	0.0464108	−	0.998922i	$-0.485222\pi$
$727$	5.08263e14	0.0928216	0.0464108	+	0.998922i	$0.485222\pi$
$728$	0	0
$729$	−7.88123e15	−1.41773
$730$	0	0
$731$	4.16028e15i	0.737186i
$732$	0	0
$733$	− 8.76006e15i	− 1.52910i	−0.644566	−	0.764548i	$-0.722962\pi$
$733$	− 8.76006e15i	− 1.52910i	0.644566	−	0.764548i	$-0.277038\pi$
$734$	0	0
$735$	−1.42903e16	−2.45732
$736$	0	0
$737$	6.37623e15	1.08017
$738$	0	0
$739$	− 2.63072e15i	− 0.439066i	−0.975605	−	0.219533i	$-0.929547\pi$
$739$	− 2.63072e15i	− 0.439066i	0.975605	−	0.219533i	$-0.0704534\pi$
$740$	0	0
$741$	− 7.57801e15i	− 1.24611i
$742$	0	0
$743$	9.60892e15	1.55681	0.778406	−	0.627761i	$-0.216029\pi$
$743$	9.60892e15	1.55681	0.778406	+	0.627761i	$0.216029\pi$
$744$	0	0
$745$	3.83073e15	0.611536
$746$	0	0
$747$	− 1.07324e16i	− 1.68824i
$748$	0	0
$749$	− 2.28771e15i	− 0.354611i
$750$	0	0
$751$	−1.83390e15	−0.280128	−0.140064	−	0.990142i	$-0.544731\pi$
$751$	−1.83390e15	−0.280128	−0.140064	+	0.990142i	$0.544731\pi$
$752$	0	0
$753$	−1.96555e16	−2.95877
$754$	0	0
$755$	− 7.91436e15i	− 1.17411i
$756$	0	0
$757$	3.59036e14i	0.0524942i	0.999655	+	0.0262471i	$0.00835567\pi$
$757$	3.59036e14i	0.0524942i	−0.999655	+	0.0262471i	$0.991644\pi$
$758$	0	0
$759$	−1.00803e16	−1.45260
$760$	0	0
$761$	−6.83481e15	−0.970757	−0.485379	−	0.874304i	$-0.661318\pi$
$761$	−6.83481e15	−0.970757	−0.485379	+	0.874304i	$0.661318\pi$
$762$	0	0
$763$	− 3.81356e13i	− 0.00533882i
$764$	0	0
$765$	2.80127e16i	3.86560i
$766$	0	0
$767$	−1.00217e15	−0.136322
$768$	0	0
$769$	4.00775e15	0.537410	0.268705	−	0.963223i	$-0.413404\pi$
$769$	4.00775e15	0.537410	0.268705	+	0.963223i	$0.413404\pi$
$770$	0	0
$771$	2.05178e15i	0.271227i
$772$	0	0
$773$	6.76108e15i	0.881108i	0.897726	+	0.440554i	$0.145218\pi$
$773$	6.76108e15i	0.881108i	−0.897726	+	0.440554i	$0.854782\pi$
$774$	0	0
$775$	2.40587e16	3.09110
$776$	0	0
$777$	−5.56441e15	−0.704862
$778$	0	0
$779$	4.29734e15i	0.536715i
$780$	0	0
$781$	− 1.34656e16i	− 1.65823i
$782$	0	0
$783$	3.56515e16	4.32901
$784$	0	0
$785$	−2.28218e15	−0.273254
$786$	0	0
$787$	− 1.39564e16i	− 1.64783i	−0.566714	−	0.823915i	$-0.691785\pi$
$787$	− 1.39564e16i	− 1.64783i	0.566714	−	0.823915i	$-0.308215\pi$
$788$	0	0
$789$	− 1.06422e15i	− 0.123910i
$790$	0	0
$791$	3.18070e15	0.365218
$792$	0	0
$793$	1.10636e16	1.25284
$794$	0	0
$795$	1.68065e16i	1.87697i
$796$	0	0
$797$	1.39940e16i	1.54143i	0.637183	+	0.770713i	$0.280100\pi$
$797$	1.39940e16i	1.54143i	−0.637183	+	0.770713i	$0.719900\pi$
$798$	0	0
$799$	6.44389e15	0.700069
$800$	0	0
$801$	3.82650e16	4.10036
$802$	0	0
$803$	3.32145e15i	0.351069i
$804$	0	0
$805$	5.76912e15i	0.601495i
$806$	0	0
$807$	−5.14021e15	−0.528660
$808$	0	0
$809$	−7.03097e13	−0.00713343	−0.00356671	−	0.999994i	$-0.501135\pi$
$809$	−7.03097e13	−0.00713343	−0.00356671	+	0.999994i	$0.501135\pi$
$810$	0	0
$811$	1.76449e16i	1.76605i	0.469324	+	0.883026i	$0.344498\pi$
$811$	1.76449e16i	1.76605i	−0.469324	+	0.883026i	$0.655502\pi$
$812$	0	0
$813$	1.88013e16i	1.85648i
$814$	0	0
$815$	−1.42160e16	−1.38487
$816$	0	0
$817$	6.75925e15	0.649647
$818$	0	0
$819$	1.02576e16i	0.972711i
$820$	0	0
$821$	− 5.86886e15i	− 0.549118i	−0.961570	−	0.274559i	$-0.911468\pi$
$821$	− 5.86886e15i	− 0.549118i	0.961570	−	0.274559i	$-0.0885319\pi$
$822$	0	0
$823$	−9.75691e15	−0.900769	−0.450384	−	0.892835i	$-0.648713\pi$
$823$	−9.75691e15	−0.900769	−0.450384	+	0.892835i	$0.648713\pi$
$824$	0	0
$825$	−4.09057e16	−3.72638
$826$	0	0
$827$	4.32365e15i	0.388661i	0.980936	+	0.194330i	$0.0622534\pi$
$827$	4.32365e15i	0.388661i	−0.980936	+	0.194330i	$0.937747\pi$
$828$	0	0
$829$	− 1.50130e16i	− 1.33174i	−0.746069	−	0.665868i	$-0.768061\pi$
$829$	− 1.50130e16i	− 1.33174i	0.746069	−	0.665868i	$-0.231939\pi$
$830$	0	0
$831$	1.99271e16	1.74437
$832$	0	0
$833$	−8.21733e15	−0.709877
$834$	0	0
$835$	− 4.21184e15i	− 0.359084i
$836$	0	0
$837$	5.29560e16i	4.45580i
$838$	0	0
$839$	−1.30045e16	−1.07995	−0.539974	−	0.841681i	$-0.681566\pi$
$839$	−1.30045e16	−1.07995	−0.539974	+	0.841681i	$0.681566\pi$
$840$	0	0
$841$	−1.81550e16	−1.48805
$842$	0	0
$843$	− 7.14435e15i	− 0.577978i
$844$	0	0
$845$	− 6.74073e15i	− 0.538263i
$846$	0	0
$847$	−6.22026e14	−0.0490287
$848$	0	0
$849$	2.80512e16	2.18252
$850$	0	0
$851$	− 7.68508e15i	− 0.590250i
$852$	0	0
$853$	− 1.16924e16i	− 0.886510i	−0.896395	−	0.443255i	$-0.853824\pi$
$853$	− 1.16924e16i	− 0.886510i	0.896395	−	0.443255i	$-0.146176\pi$
$854$	0	0
$855$	4.55125e16	3.40657
$856$	0	0
$857$	−6.40223e15	−0.473083	−0.236541	−	0.971621i	$-0.576014\pi$
$857$	−6.40223e15	−0.473083	−0.236541	+	0.971621i	$0.576014\pi$
$858$	0	0
$859$	1.33379e16i	0.973030i	0.873672	+	0.486515i	$0.161732\pi$
$859$	1.33379e16i	0.973030i	−0.873672	+	0.486515i	$0.838268\pi$
$860$	0	0
$861$	− 8.16928e15i	− 0.588392i
$862$	0	0
$863$	−1.01919e16	−0.724759	−0.362380	−	0.932031i	$-0.618036\pi$
$863$	−1.01919e16	−0.724759	−0.362380	+	0.932031i	$0.618036\pi$
$864$	0	0
$865$	−2.87663e16	−2.01974
$866$	0	0
$867$	− 4.25833e15i	− 0.295212i
$868$	0	0
$869$	− 1.01062e16i	− 0.691801i
$870$	0	0
$871$	1.25851e16	0.850664
$872$	0	0
$873$	5.94445e16	3.96765
$874$	0	0
$875$	1.11147e16i	0.732577i
$876$	0	0
$877$	1.14838e16i	0.747460i	0.927537	+	0.373730i	$0.121921\pi$
$877$	1.14838e16i	0.747460i	−0.927537	+	0.373730i	$0.878079\pi$
$878$	0	0
$879$	6.46695e14	0.0415682
$880$	0	0
$881$	−1.96603e15	−0.124802	−0.0624011	−	0.998051i	$-0.519876\pi$
$881$	−1.96603e15	−0.124802	−0.0624011	+	0.998051i	$0.519876\pi$
$882$	0	0
$883$	− 2.77835e16i	− 1.74182i	−0.491442	−	0.870910i	$-0.663530\pi$
$883$	− 2.77835e16i	− 1.74182i	0.491442	−	0.870910i	$-0.336470\pi$
$884$	0	0
$885$	− 8.45300e15i	− 0.523387i
$886$	0	0
$887$	2.98288e16	1.82413	0.912064	−	0.410048i	$-0.134488\pi$
$887$	2.98288e16	1.82413	0.912064	+	0.410048i	$0.134488\pi$
$888$	0	0
$889$	−8.65263e15	−0.522623
$890$	0	0
$891$	− 4.65061e16i	− 2.77449i
$892$	0	0
$893$	− 1.04694e16i	− 0.616937i
$894$	0	0
$895$	2.46752e16	1.43626
$896$	0	0
$897$	−1.98961e16	−1.14396
$898$	0	0
$899$	− 4.50894e16i	− 2.56092i
$900$	0	0
$901$	9.66418e15i	0.542224i
$902$	0	0
$903$	−1.28494e16	−0.712197
$904$	0	0
$905$	1.04480e16	0.572092
$906$	0	0
$907$	6.55702e15i	0.354704i	0.984147	+	0.177352i	$0.0567531\pi$
$907$	6.55702e15i	0.354704i	−0.984147	+	0.177352i	$0.943247\pi$
$908$	0	0
$909$	− 5.14234e16i	− 2.74827i
$910$	0	0
$911$	6.38239e14	0.0337002	0.0168501	−	0.999858i	$-0.494636\pi$
$911$	6.38239e14	0.0337002	0.0168501	+	0.999858i	$0.494636\pi$
$912$	0	0
$913$	1.37452e16	0.717072
$914$	0	0
$915$	9.33185e16i	4.81007i
$916$	0	0
$917$	− 5.97673e15i	− 0.304391i
$918$	0	0
$919$	7.76358e15	0.390685	0.195343	−	0.980735i	$-0.437418\pi$
$919$	7.76358e15	0.390685	0.195343	+	0.980735i	$0.437418\pi$
$920$	0	0
$921$	−1.64405e16	−0.817500
$922$	0	0
$923$	− 2.65778e16i	− 1.30590i
$924$	0	0
$925$	− 3.11859e16i	− 1.51418i
$926$	0	0
$927$	−8.06451e15	−0.386936
$928$	0	0
$929$	−1.74081e16	−0.825399	−0.412699	−	0.910867i	$-0.635414\pi$
$929$	−1.74081e16	−0.825399	−0.412699	+	0.910867i	$0.635414\pi$
$930$	0	0
$931$	1.33508e16i	0.625581i
$932$	0	0
$933$	− 6.13763e16i	− 2.84218i
$934$	0	0
$935$	−3.58765e16	−1.64190
$936$	0	0
$937$	2.93626e16	1.32809	0.664043	−	0.747694i	$-0.268839\pi$
$937$	2.93626e16	1.32809	0.664043	+	0.747694i	$0.268839\pi$
$938$	0	0
$939$	− 2.72468e16i	− 1.21802i
$940$	0	0
$941$	8.19312e15i	0.361998i	0.983483	+	0.180999i	$0.0579331\pi$
$941$	8.19312e15i	0.361998i	−0.983483	+	0.180999i	$0.942067\pi$
$942$	0	0
$943$	1.12827e16	0.492718
$944$	0	0
$945$	−5.15302e16	−2.22427
$946$	0	0
$947$	2.85510e13i	0.00121814i	1.00000		0.000609070i	$0.000193873\pi$
$947$	2.85510e13i	0.00121814i	−1.00000		0.000609070i	$0.999806\pi$
$948$	0	0
$949$	6.55574e15i	0.276476i
$950$	0	0
$951$	−7.17996e16	−2.99315
$952$	0	0
$953$	−1.18686e16	−0.489090	−0.244545	−	0.969638i	$-0.578639\pi$
$953$	−1.18686e16	−0.489090	−0.244545	+	0.969638i	$0.578639\pi$
$954$	0	0
$955$	3.26504e16i	1.33006i
$956$	0	0
$957$	7.66631e16i	3.08724i
$958$	0	0
$959$	−1.06593e14	−0.00424353
$960$	0	0
$961$	4.15664e16	1.63593
$962$	0	0
$963$	4.73845e16i	1.84371i
$964$	0	0
$965$	− 6.42404e15i	− 0.247120i
$966$	0	0
$967$	1.82443e16	0.693877	0.346939	−	0.937888i	$-0.387221\pi$
$967$	1.82443e16	0.693877	0.346939	+	0.937888i	$0.387221\pi$
$968$	0	0
$969$	3.67548e16	1.38208
$970$	0	0
$971$	− 4.17627e16i	− 1.55268i	−0.630312	−	0.776342i	$-0.717073\pi$
$971$	− 4.17627e16i	− 1.55268i	0.630312	−	0.776342i	$-0.282927\pi$
$972$	0	0
$973$	− 1.10060e16i	− 0.404585i
$974$	0	0
$975$	−8.07380e16	−2.93462
$976$	0	0
$977$	3.92505e16	1.41067	0.705335	−	0.708874i	$-0.250797\pi$
$977$	3.92505e16	1.41067	0.705335	+	0.708874i	$0.250797\pi$
$978$	0	0
$979$	4.90068e16i	1.74161i
$980$	0	0
$981$	7.89887e14i	0.0277578i
$982$	0	0
$983$	−4.75481e16	−1.65230	−0.826150	−	0.563451i	$-0.809473\pi$
$983$	−4.75481e16	−1.65230	−0.826150	+	0.563451i	$0.809473\pi$
$984$	0	0
$985$	8.13519e15	0.279556
$986$	0	0
$987$	1.99025e16i	0.676338i
$988$	0	0
$989$	− 1.77465e16i	− 0.596393i
$990$	0	0
$991$	2.53768e16	0.843396	0.421698	−	0.906736i	$-0.361434\pi$
$991$	2.53768e16	0.843396	0.421698	+	0.906736i	$0.361434\pi$
$992$	0	0
$993$	−2.75474e16	−0.905440
$994$	0	0
$995$	− 5.54258e16i	− 1.80171i
$996$	0	0
$997$	4.92972e16i	1.58489i	0.609944	+	0.792444i	$0.291192\pi$
$997$	4.92972e16i	1.58489i	−0.609944	+	0.792444i	$0.708808\pi$
$998$	0	0
$999$	6.86437e16	2.18268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.12.b.e.65.11	yes	12
4.3	odd	2	inner	128.12.b.e.65.1	✓	12
8.3	odd	2	inner	128.12.b.e.65.12	yes	12
8.5	even	2	inner	128.12.b.e.65.2	yes	12
16.3	odd	4		256.12.a.q.1.11		12
16.5	even	4		256.12.a.q.1.12		12
16.11	odd	4		256.12.a.q.1.2		12
16.13	even	4		256.12.a.q.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.12.b.e.65.1	✓	12	4.3	odd	2	inner
128.12.b.e.65.2	yes	12	8.5	even	2	inner
128.12.b.e.65.11	yes	12	1.1	even	1	trivial
128.12.b.e.65.12	yes	12	8.3	odd	2	inner
256.12.a.q.1.1		12	16.13	even	4
256.12.a.q.1.2		12	16.11	odd	4
256.12.a.q.1.11		12	16.3	odd	4
256.12.a.q.1.12		12	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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	128.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+784.336i q^{3} -11907.7i q^{5} +21148.3 q^{7} -438037. q^{9} -561003. i q^{11} -1.10728e6i q^{13} +9.33962e6 q^{15} +5.37054e6 q^{17} -8.72557e6i q^{19} +1.65874e7i q^{21} -2.29090e7 q^{23} -9.29644e7 q^{25} -2.04625e8i q^{27} +1.74228e8i q^{29} -2.58795e8 q^{31} +4.40015e8 q^{33} -2.51827e8i q^{35} +3.35460e8i q^{37} +8.68483e8 q^{39} -4.92499e8 q^{41} +7.74649e8i q^{43} +5.21599e9i q^{45} +1.19986e9 q^{47} -1.53008e9 q^{49} +4.21231e9i q^{51} +1.79948e9i q^{53} -6.68023e9 q^{55} +6.84378e9 q^{57} -9.05069e8i q^{59} +9.99168e9i q^{61} -9.26374e9 q^{63} -1.31852e10 q^{65} +1.13658e10i q^{67} -1.79684e10i q^{69} +2.40027e10 q^{71} -5.92056e9 q^{73} -7.29153e10i q^{75} -1.18643e10i q^{77} +1.80146e10 q^{79} +8.28982e10 q^{81} +2.45012e10i q^{83} -6.39506e10i q^{85} -1.36654e11 q^{87} -8.73556e10 q^{89} -2.34172e10i q^{91} -2.02982e11i q^{93} -1.03901e11 q^{95} -1.35707e11 q^{97} +2.45740e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 1152028 q^{9} + 26470216 q^{17} - 207502500 q^{25} + 935847712 q^{33} - 2031735176 q^{41} - 2342764308 q^{49} + 23753487776 q^{57} - 39461105920 q^{65} - 43591631448 q^{73} + 232401225292 q^{81}+ \cdots - 285441034488 q^{97}+O(q^{100}) \) 12 * q - 1152028 * q^9 + 26470216 * q^17 - 207502500 * q^25 + 935847712 * q^33 - 2031735176 * q^41 - 2342764308 * q^49 + 23753487776 * q^57 - 39461105920 * q^65 - 43591631448 * q^73 + 232401225292 * q^81 - 209389352088 * q^89 - 285441034488 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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