LMFDB - Embedded newform 128.12.b.f.65.1

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,-316732] 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} + 3286 x^{10} + 3725205 x^{8} + 1773266980 x^{6} + 401838244180 x^{4} + 42969249696816 x^{2} + 17\!\cdots\!96 $ x^12 + 3286x^10 + 3725205x^8 + 1773266980x^6 + 401838244180x^4 + 42969249696816*x^2 + 1736910980997696
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{120}\cdot 3^{2}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.1
Root			$36.4873i$ of defining polynomial
Character	$\chi$	$=$	128.65
Dual form			128.12.b.f.65.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-580.453i q^{3} -6948.91i q^{5} +45098.0 q^{7} -159779. q^{9} +289556. i q^{11} +1.75734e6i q^{13} -4.03352e6 q^{15} +1.03003e7 q^{17} -1.82205e7i q^{19} -2.61773e7i q^{21} +7.69390e6 q^{23} +540832. q^{25} -1.00813e7i q^{27} -5.73199e7i q^{29} +1.12000e8 q^{31} +1.68073e8 q^{33} -3.13382e8i q^{35} -3.18512e8i q^{37} +1.02005e9 q^{39} -4.00743e7 q^{41} -2.22418e8i q^{43} +1.11029e9i q^{45} +1.84115e9 q^{47} +5.65042e7 q^{49} -5.97887e9i q^{51} +2.89667e9i q^{53} +2.01209e9 q^{55} -1.05762e10 q^{57} -5.07972e9i q^{59} -8.50164e9i q^{61} -7.20572e9 q^{63} +1.22116e10 q^{65} +1.69897e10i q^{67} -4.46595e9i q^{69} +2.80844e10 q^{71} +1.17976e10 q^{73} -3.13928e8i q^{75} +1.30584e10i q^{77} -5.01575e10 q^{79} -3.41561e10 q^{81} +3.23224e10i q^{83} -7.15761e10i q^{85} -3.32715e10 q^{87} -7.05746e9 q^{89} +7.92523e10i q^{91} -6.50105e10i q^{93} -1.26613e11 q^{95} -6.57214e10 q^{97} -4.62649e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 316732 q^{9} + 11301736 q^{17} - 100402276 q^{25} - 370222336 q^{33} + 1458786872 q^{41} - 2425030420 q^{49} - 8422115584 q^{57} - 18057782080 q^{65} - 41779508088 q^{73} - 213089686484 q^{81} - 331639752632 q^{89}+ \cdots - 396775590616 q^{97}+O(q^{100}) $ 12 * q - 316732 * q^9 + 11301736 * q^17 - 100402276 * q^25 - 370222336 * q^33 + 1458786872 * q^41 - 2425030420 * q^49 - 8422115584 * q^57 - 18057782080 * q^65 - 41779508088 * q^73 - 213089686484 * q^81 - 331639752632 * q^89 - 396775590616 * 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
$3$	− 580.453i	− 1.37911i	−0.724231	−	0.689557i	$-0.757805\pi$
$3$	− 580.453i	− 1.37911i	0.724231	−	0.689557i	$-0.242195\pi$
$4$	0	0
$5$	− 6948.91i	− 0.994446i	−0.867623	−	0.497223i	$-0.834353\pi$
$5$	− 6948.91i	− 0.994446i	0.867623	−	0.497223i	$-0.165647\pi$
$6$	0	0
$7$	45098.0	1.01419	0.507094	−	0.861891i	$-0.330720\pi$
$7$	45098.0	1.01419	0.507094	+	0.861891i	$0.330720\pi$
$8$	0	0
$9$	−159779.	−0.901957
$10$	0	0
$11$	289556.i	0.542091i	0.962567	+	0.271046i	$0.0873694\pi$
$11$	289556.i	0.542091i	−0.962567	+	0.271046i	$0.912631\pi$
$12$	0	0
$13$	1.75734e6i	1.31270i	0.754456	+	0.656351i	$0.227901\pi$
$13$	1.75734e6i	1.31270i	−0.754456	+	0.656351i	$0.772099\pi$
$14$	0	0
$15$	−4.03352e6	−1.37146
$16$	0	0
$17$	1.03003e7	1.75947	0.879737	−	0.475461i	$-0.157719\pi$
$17$	1.03003e7	1.75947	0.879737	+	0.475461i	$0.157719\pi$
$18$	0	0
$19$	− 1.82205e7i	− 1.68817i	−0.536211	−	0.844084i	$-0.680145\pi$
$19$	− 1.82205e7i	− 1.68817i	0.536211	−	0.844084i	$-0.319855\pi$
$20$	0	0
$21$	− 2.61773e7i	− 1.39868i
$22$	0	0
$23$	7.69390e6	0.249255	0.124627	−	0.992204i	$-0.460226\pi$
$23$	7.69390e6	0.249255	0.124627	+	0.992204i	$0.460226\pi$
$24$	0	0
$25$	540832.	0.0110762
$26$	0	0
$27$	− 1.00813e7i	− 0.135212i
$28$	0	0
$29$	− 5.73199e7i	− 0.518939i	−0.965751	−	0.259469i	$-0.916452\pi$
$29$	− 5.73199e7i	− 0.518939i	0.965751	−	0.259469i	$-0.0835477\pi$
$30$	0	0
$31$	1.12000e8	0.702630	0.351315	−	0.936257i	$-0.385735\pi$
$31$	1.12000e8	0.702630	0.351315	+	0.936257i	$0.385735\pi$
$32$	0	0
$33$	1.68073e8	0.747606
$34$	0	0
$35$	− 3.13382e8i	− 1.00856i
$36$	0	0
$37$	− 3.18512e8i	− 0.755121i	−0.925985	−	0.377560i	$-0.876763\pi$
$37$	− 3.18512e8i	− 0.755121i	0.925985	−	0.377560i	$-0.123237\pi$
$38$	0	0
$39$	1.02005e9	1.81037
$40$	0	0
$41$	−4.00743e7	−0.0540200	−0.0270100	−	0.999635i	$-0.508599\pi$
$41$	−4.00743e7	−0.0540200	−0.0270100	+	0.999635i	$0.508599\pi$
$42$	0	0
$43$	− 2.22418e8i	− 0.230724i	−0.993323	−	0.115362i	$-0.963197\pi$
$43$	− 2.22418e8i	− 0.230724i	0.993323	−	0.115362i	$-0.0368029\pi$
$44$	0	0
$45$	1.11029e9i	0.896948i
$46$	0	0
$47$	1.84115e9	1.17098	0.585491	−	0.810679i	$-0.300902\pi$
$47$	1.84115e9	1.17098	0.585491	+	0.810679i	$0.300902\pi$
$48$	0	0
$49$	5.65042e7	0.0285761
$50$	0	0
$51$	− 5.97887e9i	− 2.42652i
$52$	0	0
$53$	2.89667e9i	0.951440i	0.879597	+	0.475720i	$0.157813\pi$
$53$	2.89667e9i	0.951440i	−0.879597	+	0.475720i	$0.842187\pi$
$54$	0	0
$55$	2.01209e9	0.539080
$56$	0	0
$57$	−1.05762e10	−2.32818
$58$	0	0
$59$	− 5.07972e9i	− 0.925026i	−0.886612	−	0.462513i	$-0.846948\pi$
$59$	− 5.07972e9i	− 0.925026i	0.886612	−	0.462513i	$-0.153052\pi$
$60$	0	0
$61$	− 8.50164e9i	− 1.28881i	−0.764685	−	0.644404i	$-0.777105\pi$
$61$	− 8.50164e9i	− 1.28881i	0.764685	−	0.644404i	$-0.222895\pi$
$62$	0	0
$63$	−7.20572e9	−0.914754
$64$	0	0
$65$	1.22116e10	1.30541
$66$	0	0
$67$	1.69897e10i	1.53736i	0.639634	+	0.768680i	$0.279086\pi$
$67$	1.69897e10i	1.53736i	−0.639634	+	0.768680i	$0.720914\pi$
$68$	0	0
$69$	− 4.46595e9i	− 0.343751i
$70$	0	0
$71$	2.80844e10	1.84733	0.923665	−	0.383201i	$-0.125178\pi$
$71$	2.80844e10	1.84733	0.923665	+	0.383201i	$0.125178\pi$
$72$	0	0
$73$	1.17976e10	0.666068	0.333034	−	0.942915i	$-0.391928\pi$
$73$	1.17976e10	0.666068	0.333034	+	0.942915i	$0.391928\pi$
$74$	0	0
$75$	− 3.13928e8i	− 0.0152754i
$76$	0	0
$77$	1.30584e10i	0.549782i
$78$	0	0
$79$	−5.01575e10	−1.83395	−0.916973	−	0.398948i	$-0.869375\pi$
$79$	−5.01575e10	−1.83395	−0.916973	+	0.398948i	$0.869375\pi$
$80$	0	0
$81$	−3.41561e10	−1.08843
$82$	0	0
$83$	3.23224e10i	0.900688i	0.892855	+	0.450344i	$0.148699\pi$
$83$	3.23224e10i	0.900688i	−0.892855	+	0.450344i	$0.851301\pi$
$84$	0	0
$85$	− 7.15761e10i	− 1.74970i
$86$	0	0
$87$	−3.32715e10	−0.715676
$88$	0	0
$89$	−7.05746e9	−0.133969	−0.0669843	−	0.997754i	$-0.521338\pi$
$89$	−7.05746e9	−0.133969	−0.0669843	+	0.997754i	$0.521338\pi$
$90$	0	0
$91$	7.92523e10i	1.33133i
$92$	0	0
$93$	− 6.50105e10i	− 0.969008i
$94$	0	0
$95$	−1.26613e11	−1.67879
$96$	0	0
$97$	−6.57214e10	−0.777074	−0.388537	−	0.921433i	$-0.627019\pi$
$97$	−6.57214e10	−0.777074	−0.388537	+	0.921433i	$0.627019\pi$
$98$	0	0
$99$	− 4.62649e10i	− 0.488943i
$100$	0	0
$101$	7.41643e10i	0.702146i	0.936348	+	0.351073i	$0.114183\pi$
$101$	7.41643e10i	0.702146i	−0.936348	+	0.351073i	$0.885817\pi$
$102$	0	0
$103$	1.57349e11	1.33739	0.668696	−	0.743536i	$-0.266853\pi$
$103$	1.57349e11	1.33739	0.668696	+	0.743536i	$0.266853\pi$
$104$	0	0
$105$	−1.81904e11	−1.39091
$106$	0	0
$107$	− 6.47704e10i	− 0.446443i	−0.974768	−	0.223221i	$-0.928343\pi$
$107$	− 6.47704e10i	− 0.446443i	0.974768	−	0.223221i	$-0.0716573\pi$
$108$	0	0
$109$	− 3.21051e10i	− 0.199861i	−0.994994	−	0.0999304i	$-0.968138\pi$
$109$	− 3.21051e10i	− 0.199861i	0.994994	−	0.0999304i	$-0.0318620\pi$
$110$	0	0
$111$	−1.84881e11	−1.04140
$112$	0	0
$113$	−3.59862e11	−1.83740	−0.918701	−	0.394954i	$-0.870761\pi$
$113$	−3.59862e11	−1.83740	−0.918701	+	0.394954i	$0.870761\pi$
$114$	0	0
$115$	− 5.34642e10i	− 0.247870i
$116$	0	0
$117$	− 2.80785e11i	− 1.18400i
$118$	0	0
$119$	4.64525e11	1.78444
$120$	0	0
$121$	2.01469e11	0.706137
$122$	0	0
$123$	2.32613e10i	0.0744998i
$124$	0	0
$125$	− 3.43060e11i	− 1.00546i
$126$	0	0
$127$	6.32223e11	1.69805	0.849024	−	0.528355i	$-0.177191\pi$
$127$	6.32223e11	1.69805	0.849024	+	0.528355i	$0.177191\pi$
$128$	0	0
$129$	−1.29103e11	−0.318195
$130$	0	0
$131$	− 3.83538e11i	− 0.868593i	−0.900770	−	0.434297i	$-0.856997\pi$
$131$	− 3.83538e11i	− 0.868593i	0.900770	−	0.434297i	$-0.143003\pi$
$132$	0	0
$133$	− 8.21709e11i	− 1.71212i
$134$	0	0
$135$	−7.00541e10	−0.134461
$136$	0	0
$137$	−9.59331e11	−1.69826	−0.849132	−	0.528180i	$-0.822874\pi$
$137$	−9.59331e11	−1.69826	−0.849132	+	0.528180i	$0.822874\pi$
$138$	0	0
$139$	− 5.48647e11i	− 0.896832i	−0.893825	−	0.448416i	$-0.851988\pi$
$139$	− 5.48647e11i	− 0.896832i	0.893825	−	0.448416i	$-0.148012\pi$
$140$	0	0
$141$	− 1.06870e12i	− 1.61492i
$142$	0	0
$143$	−5.08846e11	−0.711604
$144$	0	0
$145$	−3.98310e11	−0.516057
$146$	0	0
$147$	− 3.27981e10i	− 0.0394097i
$148$	0	0
$149$	− 9.63639e11i	− 1.07495i	−0.843278	−	0.537477i	$-0.819377\pi$
$149$	− 9.63639e11i	− 1.07495i	0.843278	−	0.537477i	$-0.180623\pi$
$150$	0	0
$151$	−9.05063e11	−0.938222	−0.469111	−	0.883139i	$-0.655426\pi$
$151$	−9.05063e11	−0.938222	−0.469111	+	0.883139i	$0.655426\pi$
$152$	0	0
$153$	−1.64578e12	−1.58697
$154$	0	0
$155$	− 7.78274e11i	− 0.698728i
$156$	0	0
$157$	1.53256e12i	1.28224i	0.767440	+	0.641121i	$0.221530\pi$
$157$	1.53256e12i	1.28224i	−0.767440	+	0.641121i	$0.778470\pi$
$158$	0	0
$159$	1.68138e12	1.31215
$160$	0	0
$161$	3.46980e11	0.252791
$162$	0	0
$163$	2.46200e12i	1.67593i	0.545722	+	0.837966i	$0.316255\pi$
$163$	2.46200e12i	1.67593i	−0.545722	+	0.837966i	$0.683745\pi$
$164$	0	0
$165$	− 1.16793e12i	− 0.743454i
$166$	0	0
$167$	−2.23163e12	−1.32948	−0.664741	−	0.747074i	$-0.731458\pi$
$167$	−2.23163e12	−1.32948	−0.664741	+	0.747074i	$0.731458\pi$
$168$	0	0
$169$	−1.29607e12	−0.723187
$170$	0	0
$171$	2.91126e12i	1.52266i
$172$	0	0
$173$	− 8.86199e11i	− 0.434788i	−0.976084	−	0.217394i	$-0.930244\pi$
$173$	− 8.86199e11i	− 0.434788i	0.976084	−	0.217394i	$-0.0697557\pi$
$174$	0	0
$175$	2.43904e10	0.0112334
$176$	0	0
$177$	−2.94854e12	−1.27572
$178$	0	0
$179$	8.75197e11i	0.355971i	0.984033	+	0.177985i	$0.0569579\pi$
$179$	8.75197e11i	0.355971i	−0.984033	+	0.177985i	$0.943042\pi$
$180$	0	0
$181$	− 3.55818e12i	− 1.36143i	−0.732547	−	0.680716i	$-0.761669\pi$
$181$	− 3.55818e12i	− 1.36143i	0.732547	−	0.680716i	$-0.238331\pi$
$182$	0	0
$183$	−4.93480e12	−1.77741
$184$	0	0
$185$	−2.21331e12	−0.750927
$186$	0	0
$187$	2.98252e12i	0.953795i
$188$	0	0
$189$	− 4.54647e11i	− 0.137131i
$190$	0	0
$191$	1.67284e12	0.476179	0.238090	−	0.971243i	$-0.423479\pi$
$191$	1.67284e12	0.476179	0.238090	+	0.971243i	$0.423479\pi$
$192$	0	0
$193$	−2.82751e12	−0.760045	−0.380023	−	0.924977i	$-0.624084\pi$
$193$	−2.82751e12	−0.760045	−0.380023	+	0.924977i	$0.624084\pi$
$194$	0	0
$195$	− 7.08824e12i	− 1.80031i
$196$	0	0
$197$	4.26195e12i	1.02340i	0.859165	+	0.511699i	$0.170984\pi$
$197$	4.26195e12i	1.02340i	−0.859165	+	0.511699i	$0.829016\pi$
$198$	0	0
$199$	4.84379e12	1.10026	0.550128	−	0.835080i	$-0.314579\pi$
$199$	4.84379e12	1.10026	0.550128	+	0.835080i	$0.314579\pi$
$200$	0	0
$201$	9.86175e12	2.12019
$202$	0	0
$203$	− 2.58501e12i	− 0.526301i
$204$	0	0
$205$	2.78473e11i	0.0537200i
$206$	0	0
$207$	−1.22932e12	−0.224817
$208$	0	0
$209$	5.27585e12	0.915141
$210$	0	0
$211$	3.10465e12i	0.511045i	0.966803	+	0.255522i	$0.0822474\pi$
$211$	3.10465e12i	0.511045i	−0.966803	+	0.255522i	$0.917753\pi$
$212$	0	0
$213$	− 1.63017e13i	− 2.54768i
$214$	0	0
$215$	−1.54556e12	−0.229443
$216$	0	0
$217$	5.05096e12	0.712599
$218$	0	0
$219$	− 6.84796e12i	− 0.918584i
$220$	0	0
$221$	1.81012e13i	2.30966i
$222$	0	0
$223$	6.22755e12	0.756207	0.378104	−	0.925763i	$-0.376576\pi$
$223$	6.22755e12	0.756207	0.378104	+	0.925763i	$0.376576\pi$
$224$	0	0
$225$	−8.64136e10	−0.00999029
$226$	0	0
$227$	− 6.95550e12i	− 0.765924i	−0.923764	−	0.382962i	$-0.874904\pi$
$227$	− 6.95550e12i	− 0.765924i	0.923764	−	0.382962i	$-0.125096\pi$
$228$	0	0
$229$	2.14549e12i	0.225129i	0.993644	+	0.112565i	$0.0359066\pi$
$229$	2.14549e12i	0.225129i	−0.993644	+	0.112565i	$0.964093\pi$
$230$	0	0
$231$	7.57978e12	0.758212
$232$	0	0
$233$	−1.30485e13	−1.24481	−0.622407	−	0.782694i	$-0.713845\pi$
$233$	−1.30485e13	−1.24481	−0.622407	+	0.782694i	$0.713845\pi$
$234$	0	0
$235$	− 1.27940e13i	− 1.16448i
$236$	0	0
$237$	2.91141e13i	2.52922i
$238$	0	0
$239$	6.34372e12	0.526206	0.263103	−	0.964768i	$-0.415254\pi$
$239$	6.34372e12	0.526206	0.263103	+	0.964768i	$0.415254\pi$
$240$	0	0
$241$	1.18515e13	0.939027	0.469514	−	0.882925i	$-0.344429\pi$
$241$	1.18515e13	0.939027	0.469514	+	0.882925i	$0.344429\pi$
$242$	0	0
$243$	1.80401e13i	1.36586i
$244$	0	0
$245$	− 3.92643e11i	− 0.0284174i
$246$	0	0
$247$	3.20195e13	2.21606
$248$	0	0
$249$	1.87617e13	1.24215
$250$	0	0
$251$	1.60684e13i	1.01804i	0.860754	+	0.509021i	$0.169993\pi$
$251$	1.60684e13i	1.01804i	−0.860754	+	0.509021i	$0.830007\pi$
$252$	0	0
$253$	2.22781e12i	0.135119i
$254$	0	0
$255$	−4.15466e13	−2.41304
$256$	0	0
$257$	−7.37440e12	−0.410293	−0.205147	−	0.978731i	$-0.565767\pi$
$257$	−7.37440e12	−0.410293	−0.205147	+	0.978731i	$0.565767\pi$
$258$	0	0
$259$	− 1.43643e13i	− 0.765834i
$260$	0	0
$261$	9.15851e12i	0.468061i
$262$	0	0
$263$	−6.23304e12	−0.305452	−0.152726	−	0.988269i	$-0.548805\pi$
$263$	−6.23304e12	−0.305452	−0.152726	+	0.988269i	$0.548805\pi$
$264$	0	0
$265$	2.01287e13	0.946157
$266$	0	0
$267$	4.09652e12i	0.184758i
$268$	0	0
$269$	3.04654e13i	1.31877i	0.751806	+	0.659384i	$0.229183\pi$
$269$	3.04654e13i	1.31877i	−0.751806	+	0.659384i	$0.770817\pi$
$270$	0	0
$271$	−3.51737e13	−1.46180	−0.730899	−	0.682486i	$-0.760899\pi$
$271$	−3.51737e13	−1.46180	−0.730899	+	0.682486i	$0.760899\pi$
$272$	0	0
$273$	4.60023e13	1.83605
$274$	0	0
$275$	1.56601e11i	0.00600433i
$276$	0	0
$277$	− 1.82082e13i	− 0.670854i	−0.942066	−	0.335427i	$-0.891119\pi$
$277$	− 1.82082e13i	− 0.670854i	0.942066	−	0.335427i	$-0.108881\pi$
$278$	0	0
$279$	−1.78952e13	−0.633743
$280$	0	0
$281$	−1.12053e13	−0.381539	−0.190770	−	0.981635i	$-0.561098\pi$
$281$	−1.12053e13	−0.381539	−0.190770	+	0.981635i	$0.561098\pi$
$282$	0	0
$283$	1.55481e13i	0.509157i	0.967052	+	0.254579i	$0.0819368\pi$
$283$	1.55481e13i	0.509157i	−0.967052	+	0.254579i	$0.918063\pi$
$284$	0	0
$285$	7.34927e13i	2.31525i
$286$	0	0
$287$	−1.80727e12	−0.0547864
$288$	0	0
$289$	7.18252e13	2.09575
$290$	0	0
$291$	3.81482e13i	1.07167i
$292$	0	0
$293$	− 3.12722e13i	− 0.846030i	−0.906123	−	0.423015i	$-0.860972\pi$
$293$	− 3.12722e13i	− 0.846030i	0.906123	−	0.423015i	$-0.139028\pi$
$294$	0	0
$295$	−3.52985e13	−0.919889
$296$	0	0
$297$	2.91910e12	0.0732974
$298$	0	0
$299$	1.35208e13i	0.327197i
$300$	0	0
$301$	− 1.00306e13i	− 0.233998i
$302$	0	0
$303$	4.30489e13	0.968340
$304$	0	0
$305$	−5.90771e13	−1.28165
$306$	0	0
$307$	1.92015e13i	0.401859i	0.979606	+	0.200930i	$0.0643962\pi$
$307$	1.92015e13i	0.401859i	−0.979606	+	0.200930i	$0.935604\pi$
$308$	0	0
$309$	− 9.13336e13i	− 1.84442i
$310$	0	0
$311$	5.30647e13	1.03425	0.517123	−	0.855911i	$-0.327003\pi$
$311$	5.30647e13	1.03425	0.517123	+	0.855911i	$0.327003\pi$
$312$	0	0
$313$	−1.49736e11	−0.00281730	−0.00140865	−	0.999999i	$-0.500448\pi$
$313$	−1.49736e11	−0.00281730	−0.00140865	+	0.999999i	$0.500448\pi$
$314$	0	0
$315$	5.00718e13i	0.909673i
$316$	0	0
$317$	7.50072e13i	1.31606i	0.752990	+	0.658032i	$0.228611\pi$
$317$	7.50072e13i	1.31606i	−0.752990	+	0.658032i	$0.771389\pi$
$318$	0	0
$319$	1.65973e13	0.281312
$320$	0	0
$321$	−3.75962e13	−0.615696
$322$	0	0
$323$	− 1.87678e14i	− 2.97029i
$324$	0	0
$325$	9.50423e11i	0.0145398i
$326$	0	0
$327$	−1.86355e13	−0.275631
$328$	0	0
$329$	8.30321e13	1.18760
$330$	0	0
$331$	− 1.27975e14i	− 1.77040i	−0.465207	−	0.885202i	$-0.654020\pi$
$331$	− 1.27975e14i	− 1.77040i	0.465207	−	0.885202i	$-0.345980\pi$
$332$	0	0
$333$	5.08915e13i	0.681086i
$334$	0	0
$335$	1.18060e14	1.52882
$336$	0	0
$337$	−1.12134e14	−1.40531	−0.702654	−	0.711531i	$-0.748002\pi$
$337$	−1.12134e14	−1.40531	−0.702654	+	0.711531i	$0.748002\pi$
$338$	0	0
$339$	2.08883e14i	2.53399i
$340$	0	0
$341$	3.24301e13i	0.380890i
$342$	0	0
$343$	−8.66253e13	−0.985206
$344$	0	0
$345$	−3.10335e13	−0.341842
$346$	0	0
$347$	− 3.56620e13i	− 0.380534i	−0.981732	−	0.190267i	$-0.939065\pi$
$347$	− 3.56620e13i	− 0.380534i	0.981732	−	0.190267i	$-0.0609354\pi$
$348$	0	0
$349$	5.99525e13i	0.619823i	0.950765	+	0.309911i	$0.100299\pi$
$349$	5.99525e13i	0.619823i	−0.950765	+	0.309911i	$0.899701\pi$
$350$	0	0
$351$	1.77162e13	0.177493
$352$	0	0
$353$	−6.42007e13	−0.623417	−0.311709	−	0.950178i	$-0.600901\pi$
$353$	−6.42007e13	−0.623417	−0.311709	+	0.950178i	$0.600901\pi$
$354$	0	0
$355$	− 1.95156e14i	− 1.83707i
$356$	0	0
$357$	− 2.69635e14i	− 2.46094i
$358$	0	0
$359$	4.25028e13	0.376182	0.188091	−	0.982152i	$-0.439770\pi$
$359$	4.25028e13	0.376182	0.188091	+	0.982152i	$0.439770\pi$
$360$	0	0
$361$	−2.15497e14	−1.84991
$362$	0	0
$363$	− 1.16943e14i	− 0.973844i
$364$	0	0
$365$	− 8.19805e13i	− 0.662369i
$366$	0	0
$367$	1.70912e14	1.34001	0.670007	−	0.742354i	$-0.266291\pi$
$367$	1.70912e14	1.34001	0.670007	+	0.742354i	$0.266291\pi$
$368$	0	0
$369$	6.40303e12	0.0487238
$370$	0	0
$371$	1.30634e14i	0.964939i
$372$	0	0
$373$	− 1.09528e14i	− 0.785464i	−0.919653	−	0.392732i	$-0.871530\pi$
$373$	− 1.09528e14i	− 0.785464i	0.919653	−	0.392732i	$-0.128470\pi$
$374$	0	0
$375$	−1.99130e14	−1.38665
$376$	0	0
$377$	1.00730e14	0.681212
$378$	0	0
$379$	− 1.99830e13i	− 0.131264i	−0.997844	−	0.0656318i	$-0.979094\pi$
$379$	− 1.99830e13i	− 0.131264i	0.997844	−	0.0656318i	$-0.0209063\pi$
$380$	0	0
$381$	− 3.66976e14i	− 2.34180i
$382$	0	0
$383$	1.17178e13	0.0726528	0.0363264	−	0.999340i	$-0.488434\pi$
$383$	1.17178e13	0.0726528	0.0363264	+	0.999340i	$0.488434\pi$
$384$	0	0
$385$	9.07415e13	0.546729
$386$	0	0
$387$	3.55377e13i	0.208103i
$388$	0	0
$389$	− 1.21767e14i	− 0.693119i	−0.938028	−	0.346559i	$-0.887350\pi$
$389$	− 1.21767e14i	− 0.693119i	0.938028	−	0.346559i	$-0.112650\pi$
$390$	0	0
$391$	7.92498e13	0.438557
$392$	0	0
$393$	−2.22626e14	−1.19789
$394$	0	0
$395$	3.48540e14i	1.82376i
$396$	0	0
$397$	1.21271e14i	0.617178i	0.951195	+	0.308589i	$0.0998568\pi$
$397$	1.21271e14i	0.617178i	−0.951195	+	0.308589i	$0.900143\pi$
$398$	0	0
$399$	−4.76964e14	−2.36121
$400$	0	0
$401$	−7.44203e13	−0.358424	−0.179212	−	0.983810i	$-0.557355\pi$
$401$	−7.44203e13	−0.358424	−0.179212	+	0.983810i	$0.557355\pi$
$402$	0	0
$403$	1.96821e14i	0.922344i
$404$	0	0
$405$	2.37348e14i	1.08239i
$406$	0	0
$407$	9.22269e13	0.409344
$408$	0	0
$409$	−2.43794e14	−1.05328	−0.526640	−	0.850088i	$-0.676549\pi$
$409$	−2.43794e14	−1.05328	−0.526640	+	0.850088i	$0.676549\pi$
$410$	0	0
$411$	5.56847e14i	2.34210i
$412$	0	0
$413$	− 2.29085e14i	− 0.938150i
$414$	0	0
$415$	2.24605e14	0.895686
$416$	0	0
$417$	−3.18464e14	−1.23683
$418$	0	0
$419$	− 4.28661e14i	− 1.62158i	−0.585340	−	0.810788i	$-0.699039\pi$
$419$	− 4.28661e14i	− 1.62158i	0.585340	−	0.810788i	$-0.300961\pi$
$420$	0	0
$421$	− 2.58105e14i	− 0.951142i	−0.879677	−	0.475571i	$-0.842241\pi$
$421$	− 2.58105e14i	− 0.951142i	0.879677	−	0.475571i	$-0.157759\pi$
$422$	0	0
$423$	−2.94177e14	−1.05618
$424$	0	0
$425$	5.57076e12	0.0194883
$426$	0	0
$427$	− 3.83407e14i	− 1.30709i
$428$	0	0
$429$	2.95361e14i	0.981383i
$430$	0	0
$431$	−3.60983e14	−1.16913	−0.584564	−	0.811348i	$-0.698734\pi$
$431$	−3.60983e14	−1.16913	−0.584564	+	0.811348i	$0.698734\pi$
$432$	0	0
$433$	3.86281e14	1.21961	0.609804	−	0.792552i	$-0.291248\pi$
$433$	3.86281e14	1.21961	0.609804	+	0.792552i	$0.291248\pi$
$434$	0	0
$435$	2.31201e14i	0.711702i
$436$	0	0
$437$	− 1.40187e14i	− 0.420784i
$438$	0	0
$439$	2.77212e14	0.811442	0.405721	−	0.913997i	$-0.367021\pi$
$439$	2.77212e14	0.811442	0.405721	+	0.913997i	$0.367021\pi$
$440$	0	0
$441$	−9.02819e12	−0.0257744
$442$	0	0
$443$	− 4.15207e14i	− 1.15623i	−0.815955	−	0.578115i	$-0.803789\pi$
$443$	− 4.15207e14i	− 1.15623i	0.815955	−	0.578115i	$-0.196211\pi$
$444$	0	0
$445$	4.90416e13i	0.133225i
$446$	0	0
$447$	−5.59348e14	−1.48249
$448$	0	0
$449$	−2.13366e14	−0.551786	−0.275893	−	0.961188i	$-0.588974\pi$
$449$	−2.13366e14	−0.551786	−0.275893	+	0.961188i	$0.588974\pi$
$450$	0	0
$451$	− 1.16037e13i	− 0.0292838i
$452$	0	0
$453$	5.25347e14i	1.29392i
$454$	0	0
$455$	5.50717e14	1.32393
$456$	0	0
$457$	−6.15476e13	−0.144435	−0.0722174	−	0.997389i	$-0.523008\pi$
$457$	−6.15476e13	−0.144435	−0.0722174	+	0.997389i	$0.523008\pi$
$458$	0	0
$459$	− 1.03841e14i	− 0.237902i
$460$	0	0
$461$	− 5.98791e14i	− 1.33943i	−0.742618	−	0.669715i	$-0.766416\pi$
$461$	− 5.98791e14i	− 1.33943i	0.742618	−	0.669715i	$-0.233584\pi$
$462$	0	0
$463$	2.94203e11	0.000642617 0	0.000321308	−	1.00000i	$-0.499898\pi$
$463$	2.94203e11	0.000642617 0	0.000321308		1.00000i	$0.499898\pi$
$464$	0	0
$465$	−4.51752e14	−0.963627
$466$	0	0
$467$	1.06508e14i	0.221891i	0.993827	+	0.110945i	$0.0353878\pi$
$467$	1.06508e14i	0.221891i	−0.993827	+	0.110945i	$0.964612\pi$
$468$	0	0
$469$	7.66204e14i	1.55917i
$470$	0	0
$471$	8.89580e14	1.76836
$472$	0	0
$473$	6.44024e13	0.125074
$474$	0	0
$475$	− 9.85423e12i	− 0.0186986i
$476$	0	0
$477$	− 4.62827e14i	− 0.858159i
$478$	0	0
$479$	8.44379e14	1.53000	0.765001	−	0.644029i	$-0.222738\pi$
$479$	8.44379e14	1.53000	0.765001	+	0.644029i	$0.222738\pi$
$480$	0	0
$481$	5.59732e14	0.991248
$482$	0	0
$483$	− 2.01405e14i	− 0.348628i
$484$	0	0
$485$	4.56692e14i	0.772759i
$486$	0	0
$487$	4.82693e14	0.798475	0.399238	−	0.916848i	$-0.369275\pi$
$487$	4.82693e14	0.798475	0.399238	+	0.916848i	$0.369275\pi$
$488$	0	0
$489$	1.42908e15	2.31130
$490$	0	0
$491$	− 7.65357e14i	− 1.21036i	−0.796087	−	0.605182i	$-0.793100\pi$
$491$	− 7.65357e14i	− 1.21036i	0.796087	−	0.605182i	$-0.206900\pi$
$492$	0	0
$493$	− 5.90414e14i	− 0.913059i
$494$	0	0
$495$	−3.21490e14	−0.486227
$496$	0	0
$497$	1.26655e15	1.87354
$498$	0	0
$499$	1.12419e14i	0.162663i	0.996687	+	0.0813314i	$0.0259172\pi$
$499$	1.12419e14i	0.162663i	−0.996687	+	0.0813314i	$0.974083\pi$
$500$	0	0
$501$	1.29536e15i	1.83351i
$502$	0	0
$503$	−1.22185e15	−1.69197	−0.845985	−	0.533207i	$-0.820987\pi$
$503$	−1.22185e15	−1.69197	−0.845985	+	0.533207i	$0.820987\pi$
$504$	0	0
$505$	5.15361e14	0.698247
$506$	0	0
$507$	7.52306e14i	0.997358i
$508$	0	0
$509$	4.96425e14i	0.644030i	0.946735	+	0.322015i	$0.104360\pi$
$509$	4.96425e14i	0.644030i	−0.946735	+	0.322015i	$0.895640\pi$
$510$	0	0
$511$	5.32049e14	0.675517
$512$	0	0
$513$	−1.83687e14	−0.228261
$514$	0	0
$515$	− 1.09340e15i	− 1.32996i
$516$	0	0
$517$	5.33114e14i	0.634779i
$518$	0	0
$519$	−5.14397e14	−0.599623
$520$	0	0
$521$	5.24045e14	0.598082	0.299041	−	0.954240i	$-0.403333\pi$
$521$	5.24045e14	0.598082	0.299041	+	0.954240i	$0.403333\pi$
$522$	0	0
$523$	− 5.29678e13i	− 0.0591906i	−0.999562	−	0.0295953i	$-0.990578\pi$
$523$	− 5.29678e13i	− 0.0591906i	0.999562	−	0.0295953i	$-0.00942185\pi$
$524$	0	0
$525$	− 1.41575e13i	− 0.0154921i
$526$	0	0
$527$	1.15363e15	1.23626
$528$	0	0
$529$	−8.93614e14	−0.937872
$530$	0	0
$531$	8.11633e14i	0.834334i
$532$	0	0
$533$	− 7.04240e13i	− 0.0709122i
$534$	0	0
$535$	−4.50084e14	−0.443964
$536$	0	0
$537$	5.08011e14	0.490924
$538$	0	0
$539$	1.63611e13i	0.0154908i
$540$	0	0
$541$	5.26351e14i	0.488305i	0.969737	+	0.244152i	$0.0785097\pi$
$541$	5.26351e14i	0.488305i	−0.969737	+	0.244152i	$0.921490\pi$
$542$	0	0
$543$	−2.06536e15	−1.87757
$544$	0	0
$545$	−2.23095e14	−0.198751
$546$	0	0
$547$	5.02844e14i	0.439038i	0.975608	+	0.219519i	$0.0704489\pi$
$547$	5.02844e14i	0.439038i	−0.975608	+	0.219519i	$0.929551\pi$
$548$	0	0
$549$	1.35838e15i	1.16245i
$550$	0	0
$551$	−1.04440e15	−0.876056
$552$	0	0
$553$	−2.26200e15	−1.85997
$554$	0	0
$555$	1.28472e15i	1.03561i
$556$	0	0
$557$	2.18116e14i	0.172379i	0.996279	+	0.0861896i	$0.0274691\pi$
$557$	2.18116e14i	0.172379i	−0.996279	+	0.0861896i	$0.972531\pi$
$558$	0	0
$559$	3.90863e14	0.302872
$560$	0	0
$561$	1.73122e15	1.31539
$562$	0	0
$563$	1.31875e14i	0.0982576i	0.998792	+	0.0491288i	$0.0156445\pi$
$563$	1.31875e14i	0.0982576i	−0.998792	+	0.0491288i	$0.984356\pi$
$564$	0	0
$565$	2.50065e15i	1.82720i
$566$	0	0
$567$	−1.54037e15	−1.10387
$568$	0	0
$569$	1.26560e15	0.889568	0.444784	−	0.895638i	$-0.353280\pi$
$569$	1.26560e15	0.889568	0.444784	+	0.895638i	$0.353280\pi$
$570$	0	0
$571$	8.30574e14i	0.572638i	0.958134	+	0.286319i	$0.0924317\pi$
$571$	8.30574e14i	0.572638i	−0.958134	+	0.286319i	$0.907568\pi$
$572$	0	0
$573$	− 9.71005e14i	− 0.656706i
$574$	0	0
$575$	4.16111e12	0.00276080
$576$	0	0
$577$	3.00156e15	1.95380	0.976900	−	0.213698i	$-0.0685509\pi$
$577$	3.00156e15	1.95380	0.976900	+	0.213698i	$0.0685509\pi$
$578$	0	0
$579$	1.64124e15i	1.04819i
$580$	0	0
$581$	1.45768e15i	0.913466i
$582$	0	0
$583$	−8.38747e14	−0.515767
$584$	0	0
$585$	−1.95115e15	−1.17743
$586$	0	0
$587$	1.25889e15i	0.745555i	0.927921	+	0.372778i	$0.121595\pi$
$587$	1.25889e15i	0.745555i	−0.927921	+	0.372778i	$0.878405\pi$
$588$	0	0
$589$	− 2.04069e15i	− 1.18616i
$590$	0	0
$591$	2.47386e15	1.41138
$592$	0	0
$593$	1.15002e14	0.0644026	0.0322013	−	0.999481i	$-0.489748\pi$
$593$	1.15002e14	0.0644026	0.0322013	+	0.999481i	$0.489748\pi$
$594$	0	0
$595$	− 3.22794e15i	− 1.77453i
$596$	0	0
$597$	− 2.81160e15i	− 1.51738i
$598$	0	0
$599$	1.32935e15	0.704353	0.352176	−	0.935934i	$-0.385442\pi$
$599$	1.32935e15	0.704353	0.352176	+	0.935934i	$0.385442\pi$
$600$	0	0
$601$	8.19476e14	0.426311	0.213155	−	0.977018i	$-0.431626\pi$
$601$	8.19476e14	0.426311	0.213155	+	0.977018i	$0.431626\pi$
$602$	0	0
$603$	− 2.71460e15i	− 1.38663i
$604$	0	0
$605$	− 1.39999e15i	− 0.702216i
$606$	0	0
$607$	2.77138e15	1.36508	0.682541	−	0.730848i	$-0.260875\pi$
$607$	2.77138e15	1.36508	0.682541	+	0.730848i	$0.260875\pi$
$608$	0	0
$609$	−1.50048e15	−0.725830
$610$	0	0
$611$	3.23551e15i	1.53715i
$612$	0	0
$613$	9.59996e14i	0.447957i	0.974594	+	0.223979i	$0.0719046\pi$
$613$	9.59996e14i	0.447957i	−0.974594	+	0.223979i	$0.928095\pi$
$614$	0	0
$615$	1.61640e14	0.0740861
$616$	0	0
$617$	−7.29160e14	−0.328288	−0.164144	−	0.986436i	$-0.552486\pi$
$617$	−7.29160e14	−0.328288	−0.164144	+	0.986436i	$0.552486\pi$
$618$	0	0
$619$	3.06883e14i	0.135729i	0.997695	+	0.0678646i	$0.0216186\pi$
$619$	3.06883e14i	0.135729i	−0.997695	+	0.0678646i	$0.978381\pi$
$620$	0	0
$621$	− 7.75646e13i	− 0.0337023i
$622$	0	0
$623$	−3.18277e14	−0.135869
$624$	0	0
$625$	−2.35749e15	−0.988801
$626$	0	0
$627$	− 3.06238e15i	− 1.26208i
$628$	0	0
$629$	− 3.28078e15i	− 1.32861i
$630$	0	0
$631$	−2.59220e14	−0.103159	−0.0515795	−	0.998669i	$-0.516426\pi$
$631$	−2.59220e14	−0.103159	−0.0515795	+	0.998669i	$0.516426\pi$
$632$	0	0
$633$	1.80210e15	0.704789
$634$	0	0
$635$	− 4.39326e15i	− 1.68862i
$636$	0	0
$637$	9.92969e13i	0.0375119i
$638$	0	0
$639$	−4.48730e15	−1.66621
$640$	0	0
$641$	−4.45616e15	−1.62645	−0.813227	−	0.581946i	$-0.802291\pi$
$641$	−4.45616e15	−1.62645	−0.813227	+	0.581946i	$0.802291\pi$
$642$	0	0
$643$	4.16268e14i	0.149353i	0.997208	+	0.0746763i	$0.0237924\pi$
$643$	4.16268e14i	0.149353i	−0.997208	+	0.0746763i	$0.976208\pi$
$644$	0	0
$645$	8.97126e14i	0.316428i
$646$	0	0
$647$	1.88622e15	0.654061	0.327030	−	0.945014i	$-0.393952\pi$
$647$	1.88622e15	0.654061	0.327030	+	0.945014i	$0.393952\pi$
$648$	0	0
$649$	1.47086e15	0.501448
$650$	0	0
$651$	− 2.93184e15i	− 0.982756i
$652$	0	0
$653$	− 2.47844e15i	− 0.816876i	−0.912786	−	0.408438i	$-0.866074\pi$
$653$	− 2.47844e15i	− 0.816876i	0.912786	−	0.408438i	$-0.133926\pi$
$654$	0	0
$655$	−2.66517e15	−0.863769
$656$	0	0
$657$	−1.88501e15	−0.600765
$658$	0	0
$659$	− 5.70088e14i	− 0.178678i	−0.996001	−	0.0893392i	$-0.971524\pi$
$659$	− 5.70088e14i	− 0.178678i	0.996001	−	0.0893392i	$-0.0284755\pi$
$660$	0	0
$661$	− 5.29360e15i	− 1.63171i	−0.578256	−	0.815855i	$-0.696267\pi$
$661$	− 5.29360e15i	− 1.63171i	0.578256	−	0.815855i	$-0.303733\pi$
$662$	0	0
$663$	1.05069e16	3.18529
$664$	0	0
$665$	−5.70998e15	−1.70261
$666$	0	0
$667$	− 4.41013e14i	− 0.129348i
$668$	0	0
$669$	− 3.61480e15i	− 1.04290i
$670$	0	0
$671$	2.46170e15	0.698652
$672$	0	0
$673$	3.40210e15	0.949871	0.474935	−	0.880021i	$-0.342471\pi$
$673$	3.40210e15	0.949871	0.474935	+	0.880021i	$0.342471\pi$
$674$	0	0
$675$	− 5.45229e12i	− 0.00149764i
$676$	0	0
$677$	9.87821e14i	0.266957i	0.991052	+	0.133478i	$0.0426147\pi$
$677$	9.87821e14i	0.266957i	−0.991052	+	0.133478i	$0.957385\pi$
$678$	0	0
$679$	−2.96391e15	−0.788099
$680$	0	0
$681$	−4.03734e15	−1.05630
$682$	0	0
$683$	5.81162e15i	1.49618i	0.663598	+	0.748090i	$0.269029\pi$
$683$	5.81162e15i	1.49618i	−0.663598	+	0.748090i	$0.730971\pi$
$684$	0	0
$685$	6.66630e15i	1.68883i
$686$	0	0
$687$	1.24536e15	0.310479
$688$	0	0
$689$	−5.09042e15	−1.24896
$690$	0	0
$691$	− 1.37529e15i	− 0.332097i	−0.986118	−	0.166049i	$-0.946899\pi$
$691$	− 1.37529e15i	− 0.332097i	0.986118	−	0.166049i	$-0.0531009\pi$
$692$	0	0
$693$	− 2.08646e15i	− 0.495880i
$694$	0	0
$695$	−3.81249e15	−0.891852
$696$	0	0
$697$	−4.12779e14	−0.0950468
$698$	0	0
$699$	7.57407e15i	1.71674i
$700$	0	0
$701$	8.41846e15i	1.87838i	0.343396	+	0.939191i	$0.388423\pi$
$701$	8.41846e15i	1.87838i	−0.343396	+	0.939191i	$0.611577\pi$
$702$	0	0
$703$	−5.80345e15	−1.27477
$704$	0	0
$705$	−7.42629e15	−1.60595
$706$	0	0
$707$	3.34466e15i	0.712108i
$708$	0	0
$709$	− 8.56874e15i	− 1.79623i	−0.439758	−	0.898116i	$-0.644936\pi$
$709$	− 8.56874e15i	− 1.79623i	0.439758	−	0.898116i	$-0.355064\pi$
$710$	0	0
$711$	8.01411e15	1.65414
$712$	0	0
$713$	8.61713e14	0.175134
$714$	0	0
$715$	3.53592e15i	0.707652i
$716$	0	0
$717$	− 3.68224e15i	− 0.725698i
$718$	0	0
$719$	−8.53831e15	−1.65715	−0.828577	−	0.559874i	$-0.810849\pi$
$719$	−8.53831e15	−1.65715	−0.828577	+	0.559874i	$0.810849\pi$
$720$	0	0
$721$	7.09612e15	1.35637
$722$	0	0
$723$	− 6.87922e15i	− 1.29503i
$724$	0	0
$725$	− 3.10004e13i	− 0.00574789i
$726$	0	0
$727$	−7.28132e15	−1.32975	−0.664876	−	0.746953i	$-0.731516\pi$
$727$	−7.28132e15	−1.32975	−0.664876	+	0.746953i	$0.731516\pi$
$728$	0	0
$729$	4.42081e15	0.795244
$730$	0	0
$731$	− 2.29098e15i	− 0.405953i
$732$	0	0
$733$	− 3.98737e15i	− 0.696009i	−0.937493	−	0.348004i	$-0.886859\pi$
$733$	− 3.98737e15i	− 0.696009i	0.937493	−	0.348004i	$-0.113141\pi$
$734$	0	0
$735$	−2.27911e14	−0.0391908
$736$	0	0
$737$	−4.91947e15	−0.833388
$738$	0	0
$739$	8.32659e15i	1.38971i	0.719152	+	0.694853i	$0.244531\pi$
$739$	8.32659e15i	1.38971i	−0.719152	+	0.694853i	$0.755469\pi$
$740$	0	0
$741$	− 1.85858e16i	− 3.05620i
$742$	0	0
$743$	−5.71778e15	−0.926380	−0.463190	−	0.886259i	$-0.653295\pi$
$743$	−5.71778e15	−0.926380	−0.463190	+	0.886259i	$0.653295\pi$
$744$	0	0
$745$	−6.69624e15	−1.06898
$746$	0	0
$747$	− 5.16444e15i	− 0.812382i
$748$	0	0
$749$	− 2.92102e15i	− 0.452777i
$750$	0	0
$751$	−1.40138e15	−0.214060	−0.107030	−	0.994256i	$-0.534134\pi$
$751$	−1.40138e15	−0.214060	−0.107030	+	0.994256i	$0.534134\pi$
$752$	0	0
$753$	9.32693e15	1.40400
$754$	0	0
$755$	6.28920e15i	0.933011i
$756$	0	0
$757$	6.08851e15i	0.890192i	0.895483	+	0.445096i	$0.146830\pi$
$757$	6.08851e15i	0.890192i	−0.895483	+	0.445096i	$0.853170\pi$
$758$	0	0
$759$	1.29314e15	0.186344
$760$	0	0
$761$	5.37439e15	0.763332	0.381666	−	0.924300i	$-0.375350\pi$
$761$	5.37439e15	0.763332	0.381666	+	0.924300i	$0.375350\pi$
$762$	0	0
$763$	− 1.44787e15i	− 0.202696i
$764$	0	0
$765$	1.14364e16i	1.57816i
$766$	0	0
$767$	8.92678e15	1.21428
$768$	0	0
$769$	2.80951e15	0.376734	0.188367	−	0.982099i	$-0.439681\pi$
$769$	2.80951e15	0.376734	0.188367	+	0.982099i	$0.439681\pi$
$770$	0	0
$771$	4.28049e15i	0.565841i
$772$	0	0
$773$	9.58543e15i	1.24918i	0.780954	+	0.624589i	$0.214733\pi$
$773$	9.58543e15i	1.24918i	−0.780954	+	0.624589i	$0.785267\pi$
$774$	0	0
$775$	6.05729e13	0.00778250
$776$	0	0
$777$	−8.33778e15	−1.05617
$778$	0	0
$779$	7.30174e14i	0.0911949i
$780$	0	0
$781$	8.13200e15i	1.00142i
$782$	0	0
$783$	−5.77859e14	−0.0701669
$784$	0	0
$785$	1.06496e16	1.27512
$786$	0	0
$787$	− 1.40930e16i	− 1.66395i	−0.554810	−	0.831977i	$-0.687209\pi$
$787$	− 1.40930e16i	− 1.66395i	0.554810	−	0.831977i	$-0.312791\pi$
$788$	0	0
$789$	3.61799e15i	0.421254i
$790$	0	0
$791$	−1.62290e16	−1.86347
$792$	0	0
$793$	1.49402e16	1.69182
$794$	0	0
$795$	− 1.16838e16i	− 1.30486i
$796$	0	0
$797$	5.39857e15i	0.594645i	0.954777	+	0.297322i	$0.0960936\pi$
$797$	5.39857e15i	0.594645i	−0.954777	+	0.297322i	$0.903906\pi$
$798$	0	0
$799$	1.89645e16	2.06031
$800$	0	0
$801$	1.12763e15	0.120834
$802$	0	0
$803$	3.41606e15i	0.361069i
$804$	0	0
$805$	− 2.41113e15i	− 0.251387i
$806$	0	0
$807$	1.76837e16	1.81873
$808$	0	0
$809$	−1.05031e16	−1.06561	−0.532806	−	0.846237i	$-0.678863\pi$
$809$	−1.05031e16	−1.06561	−0.532806	+	0.846237i	$0.678863\pi$
$810$	0	0
$811$	2.25918e15i	0.226119i	0.993588	+	0.113059i	$0.0360650\pi$
$811$	2.25918e15i	0.226119i	−0.993588	+	0.113059i	$0.963935\pi$
$812$	0	0
$813$	2.04167e16i	2.01599i
$814$	0	0
$815$	1.71082e16	1.66662
$816$	0	0
$817$	−4.05257e15	−0.389502
$818$	0	0
$819$	− 1.26629e16i	− 1.20080i
$820$	0	0
$821$	− 2.03003e16i	− 1.89939i	−0.313171	−	0.949697i	$-0.601391\pi$
$821$	− 2.03003e16i	− 1.89939i	0.313171	−	0.949697i	$-0.398609\pi$
$822$	0	0
$823$	−1.87497e14	−0.0173099	−0.00865497	−	0.999963i	$-0.502755\pi$
$823$	−1.87497e14	−0.0173099	−0.00865497	+	0.999963i	$0.502755\pi$
$824$	0	0
$825$	9.08995e13	0.00828066
$826$	0	0
$827$	− 2.19523e15i	− 0.197333i	−0.995121	−	0.0986666i	$-0.968542\pi$
$827$	− 2.19523e15i	− 0.197333i	0.995121	−	0.0986666i	$-0.0314578\pi$
$828$	0	0
$829$	1.25652e16i	1.11460i	0.830310	+	0.557302i	$0.188163\pi$
$829$	1.25652e16i	1.11460i	−0.830310	+	0.557302i	$0.811837\pi$
$830$	0	0
$831$	−1.05690e16	−0.925185
$832$	0	0
$833$	5.82013e14	0.0502789
$834$	0	0
$835$	1.55074e16i	1.32210i
$836$	0	0
$837$	− 1.12910e15i	− 0.0950043i
$838$	0	0
$839$	−1.16843e16	−0.970309	−0.485155	−	0.874428i	$-0.661237\pi$
$839$	−1.16843e16	−0.970309	−0.485155	+	0.874428i	$0.661237\pi$
$840$	0	0
$841$	8.91494e15	0.730702
$842$	0	0
$843$	6.50416e15i	0.526186i
$844$	0	0
$845$	9.00625e15i	0.719171i
$846$	0	0
$847$	9.08586e15	0.716156
$848$	0	0
$849$	9.02495e15	0.702186
$850$	0	0
$851$	− 2.45060e15i	− 0.188217i
$852$	0	0
$853$	2.18789e16i	1.65884i	0.558624	+	0.829421i	$0.311330\pi$
$853$	2.18789e16i	1.65884i	−0.558624	+	0.829421i	$0.688670\pi$
$854$	0	0
$855$	2.02300e16	1.51420
$856$	0	0
$857$	2.10716e16	1.55705	0.778524	−	0.627614i	$-0.215968\pi$
$857$	2.10716e16	1.55705	0.778524	+	0.627614i	$0.215968\pi$
$858$	0	0
$859$	8.11417e15i	0.591946i	0.955196	+	0.295973i	$0.0956438\pi$
$859$	8.11417e15i	0.591946i	−0.955196	+	0.295973i	$0.904356\pi$
$860$	0	0
$861$	1.04904e15i	0.0755568i
$862$	0	0
$863$	3.04030e15	0.216201	0.108100	−	0.994140i	$-0.465523\pi$
$863$	3.04030e15	0.216201	0.108100	+	0.994140i	$0.465523\pi$
$864$	0	0
$865$	−6.15811e15	−0.432373
$866$	0	0
$867$	− 4.16912e16i	− 2.89028i
$868$	0	0
$869$	− 1.45234e16i	− 0.994166i
$870$	0	0
$871$	−2.98567e16	−2.01809
$872$	0	0
$873$	1.05009e16	0.700888
$874$	0	0
$875$	− 1.54713e16i	− 1.01973i
$876$	0	0
$877$	1.29894e16i	0.845455i	0.906257	+	0.422727i	$0.138927\pi$
$877$	1.29894e16i	0.845455i	−0.906257	+	0.422727i	$0.861073\pi$
$878$	0	0
$879$	−1.81520e16	−1.16677
$880$	0	0
$881$	−5.15273e14	−0.0327092	−0.0163546	−	0.999866i	$-0.505206\pi$
$881$	−5.15273e14	−0.0327092	−0.0163546	+	0.999866i	$0.505206\pi$
$882$	0	0
$883$	3.14075e16i	1.96901i	0.175344	+	0.984507i	$0.443896\pi$
$883$	3.14075e16i	1.96901i	−0.175344	+	0.984507i	$0.556104\pi$
$884$	0	0
$885$	2.04891e16i	1.26863i
$886$	0	0
$887$	−1.12736e15	−0.0689420	−0.0344710	−	0.999406i	$-0.510975\pi$
$887$	−1.12736e15	−0.0689420	−0.0344710	+	0.999406i	$0.510975\pi$
$888$	0	0
$889$	2.85120e16	1.72214
$890$	0	0
$891$	− 9.89009e15i	− 0.590028i
$892$	0	0
$893$	− 3.35466e16i	− 1.97681i
$894$	0	0
$895$	6.08166e15	0.353994
$896$	0	0
$897$	7.84817e15	0.451242
$898$	0	0
$899$	− 6.41980e15i	− 0.364622i
$900$	0	0
$901$	2.98367e16i	1.67403i
$902$	0	0
$903$	−5.82230e15	−0.322710
$904$	0	0
$905$	−2.47255e16	−1.35387
$906$	0	0
$907$	1.89493e16i	1.02507i	0.858666	+	0.512536i	$0.171294\pi$
$907$	1.89493e16i	1.02507i	−0.858666	+	0.512536i	$0.828706\pi$
$908$	0	0
$909$	− 1.18499e16i	− 0.633306i
$910$	0	0
$911$	−1.45138e16	−0.766353	−0.383177	−	0.923675i	$-0.625170\pi$
$911$	−1.45138e16	−0.766353	−0.383177	+	0.923675i	$0.625170\pi$
$912$	0	0
$913$	−9.35914e15	−0.488255
$914$	0	0
$915$	3.42915e16i	1.76754i
$916$	0	0
$917$	− 1.72968e16i	− 0.880916i
$918$	0	0
$919$	−1.00596e16	−0.506226	−0.253113	−	0.967437i	$-0.581454\pi$
$919$	−1.00596e16	−0.506226	−0.253113	+	0.967437i	$0.581454\pi$
$920$	0	0
$921$	1.11456e16	0.554210
$922$	0	0
$923$	4.93537e16i	2.42499i
$924$	0	0
$925$	− 1.72261e14i	− 0.00836390i
$926$	0	0
$927$	−2.51410e16	−1.20627
$928$	0	0
$929$	2.09205e16	0.991940	0.495970	−	0.868340i	$-0.334813\pi$
$929$	2.09205e16	0.991940	0.495970	+	0.868340i	$0.334813\pi$
$930$	0	0
$931$	− 1.02954e15i	− 0.0482412i
$932$	0	0
$933$	− 3.08016e16i	− 1.42634i
$934$	0	0
$935$	2.07253e16	0.948498
$936$	0	0
$937$	9.87940e15	0.446851	0.223426	−	0.974721i	$-0.428276\pi$
$937$	9.87940e15	0.446851	0.223426	+	0.974721i	$0.428276\pi$
$938$	0	0
$939$	8.69150e13i	0.00388538i
$940$	0	0
$941$	8.95836e15i	0.395809i	0.980221	+	0.197904i	$0.0634136\pi$
$941$	8.95836e15i	0.395809i	−0.980221	+	0.197904i	$0.936586\pi$
$942$	0	0
$943$	−3.08328e14	−0.0134647
$944$	0	0
$945$	−3.15930e15	−0.136369
$946$	0	0
$947$	− 2.63888e15i	− 0.112589i	−0.998414	−	0.0562944i	$-0.982071\pi$
$947$	− 2.63888e15i	− 0.112589i	0.998414	−	0.0562944i	$-0.0179285\pi$
$948$	0	0
$949$	2.07324e16i	0.874348i
$950$	0	0
$951$	4.35381e16	1.81500
$952$	0	0
$953$	−3.22131e16	−1.32746	−0.663729	−	0.747973i	$-0.731027\pi$
$953$	−3.22131e16	−1.32746	−0.663729	+	0.747973i	$0.731027\pi$
$954$	0	0
$955$	− 1.16244e16i	− 0.473535i
$956$	0	0
$957$	− 9.63395e15i	− 0.387962i
$958$	0	0
$959$	−4.32639e16	−1.72236
$960$	0	0
$961$	−1.28646e16	−0.506310
$962$	0	0
$963$	1.03490e16i	0.402672i
$964$	0	0
$965$	1.96481e16i	0.755824i
$966$	0	0
$967$	3.32975e16	1.26638	0.633192	−	0.773995i	$-0.281744\pi$
$967$	3.32975e16	1.26638	0.633192	+	0.773995i	$0.281744\pi$
$968$	0	0
$969$	−1.08938e17	−4.09637
$970$	0	0
$971$	− 2.03551e16i	− 0.756777i	−0.925647	−	0.378388i	$-0.876478\pi$
$971$	− 2.03551e16i	− 0.756777i	0.925647	−	0.378388i	$-0.123522\pi$
$972$	0	0
$973$	− 2.47429e16i	− 0.909556i
$974$	0	0
$975$	5.51676e14	0.0200521
$976$	0	0
$977$	1.14008e16	0.409746	0.204873	−	0.978789i	$-0.434322\pi$
$977$	1.14008e16	0.409746	0.204873	+	0.978789i	$0.434322\pi$
$978$	0	0
$979$	− 2.04353e15i	− 0.0726232i
$980$	0	0
$981$	5.12971e15i	0.180266i
$982$	0	0
$983$	−8.12549e15	−0.282361	−0.141181	−	0.989984i	$-0.545090\pi$
$983$	−8.12549e15	−0.282361	−0.141181	+	0.989984i	$0.545090\pi$
$984$	0	0
$985$	2.96159e16	1.01771
$986$	0	0
$987$	− 4.81962e16i	− 1.63783i
$988$	0	0
$989$	− 1.71126e15i	− 0.0575091i
$990$	0	0
$991$	−1.87482e16	−0.623095	−0.311547	−	0.950231i	$-0.600847\pi$
$991$	−1.87482e16	−0.623095	−0.311547	+	0.950231i	$0.600847\pi$
$992$	0	0
$993$	−7.42837e16	−2.44159
$994$	0	0
$995$	− 3.36591e16i	− 1.09415i
$996$	0	0
$997$	− 2.16114e15i	− 0.0694800i	−0.999396	−	0.0347400i	$-0.988940\pi$
$997$	− 2.16114e15i	− 0.0694800i	0.999396	−	0.0347400i	$-0.0110603\pi$
$998$	0	0
$999$	−3.21102e15	−0.102102

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.12.b.f.65.1	✓	12
4.3	odd	2	inner	128.12.b.f.65.11	yes	12
8.3	odd	2	inner	128.12.b.f.65.2	yes	12
8.5	even	2	inner	128.12.b.f.65.12	yes	12
16.3	odd	4		256.12.a.k.1.1		6
16.5	even	4		256.12.a.j.1.1		6
16.11	odd	4		256.12.a.j.1.6		6
16.13	even	4		256.12.a.k.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.12.b.f.65.1	✓	12	1.1	even	1	trivial
128.12.b.f.65.2	yes	12	8.3	odd	2	inner
128.12.b.f.65.11	yes	12	4.3	odd	2	inner
128.12.b.f.65.12	yes	12	8.5	even	2	inner
256.12.a.j.1.1		6	16.5	even	4
256.12.a.j.1.6		6	16.11	odd	4
256.12.a.k.1.1		6	16.3	odd	4
256.12.a.k.1.6		6	16.13	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)

\(f(q)\)	\(=\)	\(q-580.453i q^{3} -6948.91i q^{5} +45098.0 q^{7} -159779. q^{9} +289556. i q^{11} +1.75734e6i q^{13} -4.03352e6 q^{15} +1.03003e7 q^{17} -1.82205e7i q^{19} -2.61773e7i q^{21} +7.69390e6 q^{23} +540832. q^{25} -1.00813e7i q^{27} -5.73199e7i q^{29} +1.12000e8 q^{31} +1.68073e8 q^{33} -3.13382e8i q^{35} -3.18512e8i q^{37} +1.02005e9 q^{39} -4.00743e7 q^{41} -2.22418e8i q^{43} +1.11029e9i q^{45} +1.84115e9 q^{47} +5.65042e7 q^{49} -5.97887e9i q^{51} +2.89667e9i q^{53} +2.01209e9 q^{55} -1.05762e10 q^{57} -5.07972e9i q^{59} -8.50164e9i q^{61} -7.20572e9 q^{63} +1.22116e10 q^{65} +1.69897e10i q^{67} -4.46595e9i q^{69} +2.80844e10 q^{71} +1.17976e10 q^{73} -3.13928e8i q^{75} +1.30584e10i q^{77} -5.01575e10 q^{79} -3.41561e10 q^{81} +3.23224e10i q^{83} -7.15761e10i q^{85} -3.32715e10 q^{87} -7.05746e9 q^{89} +7.92523e10i q^{91} -6.50105e10i q^{93} -1.26613e11 q^{95} -6.57214e10 q^{97} -4.62649e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 316732 q^{9} + 11301736 q^{17} - 100402276 q^{25} - 370222336 q^{33} + 1458786872 q^{41} - 2425030420 q^{49} - 8422115584 q^{57} - 18057782080 q^{65} - 41779508088 q^{73} - 213089686484 q^{81} - 331639752632 q^{89}+ \cdots - 396775590616 q^{97}+O(q^{100}) \) 12 * q - 316732 * q^9 + 11301736 * q^17 - 100402276 * q^25 - 370222336 * q^33 + 1458786872 * q^41 - 2425030420 * q^49 - 8422115584 * q^57 - 18057782080 * q^65 - 41779508088 * q^73 - 213089686484 * q^81 - 331639752632 * q^89 - 396775590616 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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