LMFDB - Embedded newform 384.9.g.a.127.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,9,Mod(127,384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("384.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	384.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$156.433386263$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.4
Character	$\chi$	$=$	384.127
Dual form			384.9.g.a.127.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} -230.157 q^{5} -3733.97i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} -230.157 q^{5} -3733.97i q^{7} -2187.00 q^{9} +4197.71i q^{11} -38949.6 q^{13} -10763.4i q^{15} -148205. q^{17} +34029.1i q^{19} +174621. q^{21} +388149. i q^{23} -337653. q^{25} -102276. i q^{27} -286859. q^{29} +605032. i q^{31} -196308. q^{33} +859400. i q^{35} +3.48168e6 q^{37} -1.82149e6i q^{39} -259520. q^{41} +4.88306e6i q^{43} +503353. q^{45} +5.00662e6i q^{47} -8.17776e6 q^{49} -6.93086e6i q^{51} -9.11327e6 q^{53} -966133. i q^{55} -1.59139e6 q^{57} -8.08424e6i q^{59} +1.17585e7 q^{61} +8.16620e6i q^{63} +8.96452e6 q^{65} -3.08031e7i q^{67} -1.81519e7 q^{69} -3.64486e7i q^{71} +2.37525e7 q^{73} -1.57905e7i q^{75} +1.56742e7 q^{77} -3.38140e7i q^{79} +4.78297e6 q^{81} -4.02794e7i q^{83} +3.41104e7 q^{85} -1.34151e7i q^{87} -1.10334e8 q^{89} +1.45437e8i q^{91} -2.82945e7 q^{93} -7.83204e6i q^{95} +1.13290e8 q^{97} -9.18040e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) $ 32 * q - 1344 * q^5 - 69984 * q^9	$ 32 q - 1344 q^{5} - 69984 q^{9} - 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} - 275520 q^{29} + 2421440 q^{37} - 4374720 q^{41} + 2939328 q^{45} - 14219104 q^{49} - 6224448 q^{53} + 3100032 q^{57} - 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} + 154517760 q^{77} + 153055008 q^{81} - 384830848 q^{85} - 182669760 q^{89} + 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) $ 32 * q - 1344 * q^5 - 69984 * q^9 - 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 - 275520 * q^29 + 2421440 * q^37 - 4374720 * q^41 + 2939328 * q^45 - 14219104 * q^49 - 6224448 * q^53 + 3100032 * q^57 - 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 + 154517760 * q^77 + 153055008 * q^81 - 384830848 * q^85 - 182669760 * q^89 + 149817600 * q^93 - 149408192 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	46.7654i	0.577350i
$3$	46.7654i	0.577350i
$4$	0	0
$5$	−230.157	−0.368251	−0.184126	−	0.982903i	$-0.558945\pi$
$5$	−230.157	−0.368251	−0.184126	+	0.982903i	$0.558945\pi$
$6$	0	0
$7$	− 3733.97i	− 1.55517i	−0.628775	−	0.777587i	$-0.716443\pi$
$7$	− 3733.97i	− 1.55517i	0.628775	−	0.777587i	$-0.283557\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	4197.71i	0.286710i	0.989671	+	0.143355i	$0.0457890\pi$
$11$	4197.71i	0.286710i	−0.989671	+	0.143355i	$0.954211\pi$
$12$	0	0
$13$	−38949.6	−1.36373	−0.681867	−	0.731476i	$-0.738832\pi$
$13$	−38949.6	−1.36373	−0.681867	+	0.731476i	$0.738832\pi$
$14$	0	0
$15$	− 10763.4i	− 0.212610i
$16$	0	0
$17$	−148205.	−1.77446	−0.887232	−	0.461324i	$-0.847375\pi$
$17$	−148205.	−1.77446	−0.887232	+	0.461324i	$0.847375\pi$
$18$	0	0
$19$	34029.1i	0.261118i	0.991441	+	0.130559i	$0.0416772\pi$
$19$	34029.1i	0.261118i	−0.991441	+	0.130559i	$0.958323\pi$
$20$	0	0
$21$	174621.	0.897880
$22$	0	0
$23$	388149.i	1.38703i	0.720441	+	0.693517i	$0.243940\pi$
$23$	388149.i	1.38703i	−0.720441	+	0.693517i	$0.756060\pi$
$24$	0	0
$25$	−337653.	−0.864391
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	−286859.	−0.405580	−0.202790	−	0.979222i	$-0.565001\pi$
$29$	−286859.	−0.405580	−0.202790	+	0.979222i	$0.565001\pi$
$30$	0	0
$31$	605032.i	0.655136i	0.944828	+	0.327568i	$0.106229\pi$
$31$	605032.i	0.655136i	−0.944828	+	0.327568i	$0.893771\pi$
$32$	0	0
$33$	−196308.	−0.165532
$34$	0	0
$35$	859400.i	0.572695i
$36$	0	0
$37$	3.48168e6	1.85773	0.928865	−	0.370419i	$-0.120786\pi$
$37$	3.48168e6	1.85773	0.928865	+	0.370419i	$0.120786\pi$
$38$	0	0
$39$	− 1.82149e6i	− 0.787353i
$40$	0	0
$41$	−259520.	−0.0918407	−0.0459204	−	0.998945i	$-0.514622\pi$
$41$	−259520.	−0.0918407	−0.0459204	+	0.998945i	$0.514622\pi$
$42$	0	0
$43$	4.88306e6i	1.42830i	0.699994	+	0.714149i	$0.253186\pi$
$43$	4.88306e6i	1.42830i	−0.699994	+	0.714149i	$0.746814\pi$
$44$	0	0
$45$	503353.	0.122750
$46$	0	0
$47$	5.00662e6i	1.02601i	0.858384	+	0.513007i	$0.171469\pi$
$47$	5.00662e6i	1.02601i	−0.858384	+	0.513007i	$0.828531\pi$
$48$	0	0
$49$	−8.17776e6	−1.41857
$50$	0	0
$51$	− 6.93086e6i	− 1.02449i
$52$	0	0
$53$	−9.11327e6	−1.15497	−0.577485	−	0.816401i	$-0.695966\pi$
$53$	−9.11327e6	−1.15497	−0.577485	+	0.816401i	$0.695966\pi$
$54$	0	0
$55$	− 966133.i	− 0.105581i
$56$	0	0
$57$	−1.59139e6	−0.150756
$58$	0	0
$59$	− 8.08424e6i	− 0.667162i	−0.942721	−	0.333581i	$-0.891743\pi$
$59$	− 8.08424e6i	− 0.667162i	0.942721	−	0.333581i	$-0.108257\pi$
$60$	0	0
$61$	1.17585e7	0.849244	0.424622	−	0.905371i	$-0.360407\pi$
$61$	1.17585e7	0.849244	0.424622	+	0.905371i	$0.360407\pi$
$62$	0	0
$63$	8.16620e6i	0.518391i
$64$	0	0
$65$	8.96452e6	0.502197
$66$	0	0
$67$	− 3.08031e7i	− 1.52861i	−0.644857	−	0.764303i	$-0.723083\pi$
$67$	− 3.08031e7i	− 1.52861i	0.644857	−	0.764303i	$-0.276917\pi$
$68$	0	0
$69$	−1.81519e7	−0.800804
$70$	0	0
$71$	− 3.64486e7i	− 1.43432i	−0.696907	−	0.717162i	$-0.745441\pi$
$71$	− 3.64486e7i	− 1.43432i	0.696907	−	0.717162i	$-0.254559\pi$
$72$	0	0
$73$	2.37525e7	0.836408	0.418204	−	0.908353i	$-0.362660\pi$
$73$	2.37525e7	0.836408	0.418204	+	0.908353i	$0.362660\pi$
$74$	0	0
$75$	− 1.57905e7i	− 0.499056i
$76$	0	0
$77$	1.56742e7	0.445883
$78$	0	0
$79$	− 3.38140e7i	− 0.868136i	−0.900880	−	0.434068i	$-0.857078\pi$
$79$	− 3.38140e7i	− 0.868136i	0.900880	−	0.434068i	$-0.142922\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 4.02794e7i	− 0.848733i	−0.905491	−	0.424366i	$-0.860497\pi$
$83$	− 4.02794e7i	− 0.848733i	0.905491	−	0.424366i	$-0.139503\pi$
$84$	0	0
$85$	3.41104e7	0.653448
$86$	0	0
$87$	− 1.34151e7i	− 0.234162i
$88$	0	0
$89$	−1.10334e8	−1.75853	−0.879263	−	0.476337i	$-0.841964\pi$
$89$	−1.10334e8	−1.75853	−0.879263	+	0.476337i	$0.841964\pi$
$90$	0	0
$91$	1.45437e8i	2.12085i
$92$	0	0
$93$	−2.82945e7	−0.378243
$94$	0	0
$95$	− 7.83204e6i	− 0.0961569i
$96$	0	0
$97$	1.13290e8	1.27969	0.639844	−	0.768505i	$-0.278999\pi$
$97$	1.13290e8	1.27969	0.639844	+	0.768505i	$0.278999\pi$
$98$	0	0
$99$	− 9.18040e6i	− 0.0955698i
$100$	0	0
$101$	5.12790e7	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$101$	5.12790e7	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$102$	0	0
$103$	− 8.64304e7i	− 0.767923i	−0.923349	−	0.383962i	$-0.874560\pi$
$103$	− 8.64304e7i	− 0.767923i	0.923349	−	0.383962i	$-0.125440\pi$
$104$	0	0
$105$	−4.01901e7	−0.330645
$106$	0	0
$107$	− 1.24867e8i	− 0.952606i	−0.879281	−	0.476303i	$-0.841977\pi$
$107$	− 1.24867e8i	− 0.952606i	0.879281	−	0.476303i	$-0.158023\pi$
$108$	0	0
$109$	1.53925e8	1.09044	0.545221	−	0.838293i	$-0.316446\pi$
$109$	1.53925e8	1.09044	0.545221	+	0.838293i	$0.316446\pi$
$110$	0	0
$111$	1.62822e8i	1.07256i
$112$	0	0
$113$	−1.07511e8	−0.659386	−0.329693	−	0.944088i	$-0.606945\pi$
$113$	−1.07511e8	−0.659386	−0.329693	+	0.944088i	$0.606945\pi$
$114$	0	0
$115$	− 8.93351e7i	− 0.510776i
$116$	0	0
$117$	8.51828e7	0.454578
$118$	0	0
$119$	5.53393e8i	2.75960i
$120$	0	0
$121$	1.96738e8	0.917798
$122$	0	0
$123$	− 1.21365e7i	− 0.0530243i
$124$	0	0
$125$	1.67618e8	0.686564
$126$	0	0
$127$	− 2.72055e8i	− 1.04578i	−0.852400	−	0.522891i	$-0.824853\pi$
$127$	− 2.72055e8i	− 1.04578i	0.852400	−	0.522891i	$-0.175147\pi$
$128$	0	0
$129$	−2.28358e8	−0.824628
$130$	0	0
$131$	3.51442e8i	1.19335i	0.802482	+	0.596676i	$0.203512\pi$
$131$	3.51442e8i	1.19335i	−0.802482	+	0.596676i	$0.796488\pi$
$132$	0	0
$133$	1.27064e8	0.406084
$134$	0	0
$135$	2.35395e7i	0.0708699i
$136$	0	0
$137$	1.41813e8	0.402564	0.201282	−	0.979533i	$-0.435489\pi$
$137$	1.41813e8	0.402564	0.201282	+	0.979533i	$0.435489\pi$
$138$	0	0
$139$	− 2.48852e8i	− 0.666625i	−0.942816	−	0.333312i	$-0.891834\pi$
$139$	− 2.48852e8i	− 0.666625i	0.942816	−	0.333312i	$-0.108166\pi$
$140$	0	0
$141$	−2.34137e8	−0.592370
$142$	0	0
$143$	− 1.63499e8i	− 0.390996i
$144$	0	0
$145$	6.60226e7	0.149355
$146$	0	0
$147$	− 3.82436e8i	− 0.819010i
$148$	0	0
$149$	−4.01840e8	−0.815281	−0.407641	−	0.913142i	$-0.633648\pi$
$149$	−4.01840e8	−0.815281	−0.407641	+	0.913142i	$0.633648\pi$
$150$	0	0
$151$	1.77091e8i	0.340634i	0.985389	+	0.170317i	$0.0544792\pi$
$151$	1.77091e8i	0.340634i	−0.985389	+	0.170317i	$0.945521\pi$
$152$	0	0
$153$	3.24124e8	0.591488
$154$	0	0
$155$	− 1.39252e8i	− 0.241254i
$156$	0	0
$157$	−1.18930e8	−0.195747	−0.0978733	−	0.995199i	$-0.531204\pi$
$157$	−1.18930e8	−0.195747	−0.0978733	+	0.995199i	$0.531204\pi$
$158$	0	0
$159$	− 4.26185e8i	− 0.666822i
$160$	0	0
$161$	1.44934e9	2.15708
$162$	0	0
$163$	− 3.51529e7i	− 0.0497979i	−0.999690	−	0.0248989i	$-0.992074\pi$
$163$	− 3.51529e7i	− 0.0497979i	0.999690	−	0.0248989i	$-0.00792639\pi$
$164$	0	0
$165$	4.51816e7	0.0609573
$166$	0	0
$167$	7.90565e8i	1.01642i	0.861234	+	0.508208i	$0.169692\pi$
$167$	7.90565e8i	1.01642i	−0.861234	+	0.508208i	$0.830308\pi$
$168$	0	0
$169$	7.01343e8	0.859772
$170$	0	0
$171$	− 7.44217e7i	− 0.0870393i
$172$	0	0
$173$	8.03573e8	0.897100	0.448550	−	0.893758i	$-0.351941\pi$
$173$	8.03573e8	0.897100	0.448550	+	0.893758i	$0.351941\pi$
$174$	0	0
$175$	1.26079e9i	1.34428i
$176$	0	0
$177$	3.78063e8	0.385186
$178$	0	0
$179$	− 1.60389e9i	− 1.56229i	−0.624347	−	0.781147i	$-0.714635\pi$
$179$	− 1.60389e9i	− 1.56229i	0.624347	−	0.781147i	$-0.285365\pi$
$180$	0	0
$181$	1.92174e7	0.0179053	0.00895265	−	0.999960i	$-0.497150\pi$
$181$	1.92174e7	0.0179053	0.00895265	+	0.999960i	$0.497150\pi$
$182$	0	0
$183$	5.49890e8i	0.490311i
$184$	0	0
$185$	−8.01334e8	−0.684111
$186$	0	0
$187$	− 6.22122e8i	− 0.508756i
$188$	0	0
$189$	−3.81895e8	−0.299293
$190$	0	0
$191$	3.96831e8i	0.298175i	0.988824	+	0.149088i	$0.0476337\pi$
$191$	3.96831e8i	0.298175i	−0.988824	+	0.149088i	$0.952366\pi$
$192$	0	0
$193$	−5.28535e8	−0.380929	−0.190465	−	0.981694i	$-0.560999\pi$
$193$	−5.28535e8	−0.380929	−0.190465	+	0.981694i	$0.560999\pi$
$194$	0	0
$195$	4.19229e8i	0.289943i
$196$	0	0
$197$	−1.58109e8	−0.104976	−0.0524881	−	0.998622i	$-0.516715\pi$
$197$	−1.58109e8	−0.104976	−0.0524881	+	0.998622i	$0.516715\pi$
$198$	0	0
$199$	2.38882e9i	1.52325i	0.648019	+	0.761624i	$0.275598\pi$
$199$	2.38882e9i	1.52325i	−0.648019	+	0.761624i	$0.724402\pi$
$200$	0	0
$201$	1.44052e9	0.882541
$202$	0	0
$203$	1.07112e9i	0.630747i
$204$	0	0
$205$	5.97303e7	0.0338204
$206$	0	0
$207$	− 8.48881e8i	− 0.462344i
$208$	0	0
$209$	−1.42845e8	−0.0748650
$210$	0	0
$211$	− 1.49198e9i	− 0.752720i	−0.926473	−	0.376360i	$-0.877176\pi$
$211$	− 1.49198e9i	− 0.752720i	0.926473	−	0.376360i	$-0.122824\pi$
$212$	0	0
$213$	1.70453e9	0.828107
$214$	0	0
$215$	− 1.12387e9i	− 0.525972i
$216$	0	0
$217$	2.25917e9	1.01885
$218$	0	0
$219$	1.11080e9i	0.482901i
$220$	0	0
$221$	5.77253e9	2.41990
$222$	0	0
$223$	3.26293e9i	1.31944i	0.751513	+	0.659718i	$0.229324\pi$
$223$	3.26293e9i	1.31944i	−0.751513	+	0.659718i	$0.770676\pi$
$224$	0	0
$225$	7.38447e8	0.288130
$226$	0	0
$227$	− 1.33563e9i	− 0.503016i	−0.967855	−	0.251508i	$-0.919073\pi$
$227$	− 1.33563e9i	− 0.503016i	0.967855	−	0.251508i	$-0.0809265\pi$
$228$	0	0
$229$	4.74937e9	1.72701	0.863503	−	0.504343i	$-0.168265\pi$
$229$	4.74937e9	1.72701	0.863503	+	0.504343i	$0.168265\pi$
$230$	0	0
$231$	7.33008e8i	0.257431i
$232$	0	0
$233$	3.71199e9	1.25946	0.629728	−	0.776816i	$-0.283166\pi$
$233$	3.71199e9	1.25946	0.629728	+	0.776816i	$0.283166\pi$
$234$	0	0
$235$	− 1.15231e9i	− 0.377831i
$236$	0	0
$237$	1.58132e9	0.501219
$238$	0	0
$239$	3.76972e9i	1.15536i	0.816264	+	0.577680i	$0.196042\pi$
$239$	3.76972e9i	1.15536i	−0.816264	+	0.577680i	$0.803958\pi$
$240$	0	0
$241$	−4.99148e9	−1.47966	−0.739829	−	0.672795i	$-0.765094\pi$
$241$	−4.99148e9	−1.47966	−0.739829	+	0.672795i	$0.765094\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	1.88217e9	0.522389
$246$	0	0
$247$	− 1.32542e9i	− 0.356095i
$248$	0	0
$249$	1.88368e9	0.490016
$250$	0	0
$251$	4.92725e8i	0.124139i	0.998072	+	0.0620697i	$0.0197701\pi$
$251$	4.92725e8i	0.124139i	−0.998072	+	0.0620697i	$0.980230\pi$
$252$	0	0
$253$	−1.62934e9	−0.397676
$254$	0	0
$255$	1.59519e9i	0.377268i
$256$	0	0
$257$	8.61069e9	1.97381	0.986906	−	0.161298i	$-0.0515681\pi$
$257$	8.61069e9	1.97381	0.986906	+	0.161298i	$0.0515681\pi$
$258$	0	0
$259$	− 1.30005e10i	− 2.88909i
$260$	0	0
$261$	6.27361e8	0.135193
$262$	0	0
$263$	1.00322e9i	0.209689i	0.994489	+	0.104844i	$0.0334345\pi$
$263$	1.00322e9i	0.209689i	−0.994489	+	0.104844i	$0.966566\pi$
$264$	0	0
$265$	2.09748e9	0.425319
$266$	0	0
$267$	− 5.15980e9i	− 1.01529i
$268$	0	0
$269$	4.53410e8	0.0865928	0.0432964	−	0.999062i	$-0.486214\pi$
$269$	4.53410e8	0.0865928	0.0432964	+	0.999062i	$0.486214\pi$
$270$	0	0
$271$	− 2.92332e9i	− 0.542000i	−0.962579	−	0.271000i	$-0.912646\pi$
$271$	− 2.92332e9i	− 0.542000i	0.962579	−	0.271000i	$-0.0873543\pi$
$272$	0	0
$273$	−6.80141e9	−1.22447
$274$	0	0
$275$	− 1.41737e9i	− 0.247829i
$276$	0	0
$277$	−5.36536e9	−0.911338	−0.455669	−	0.890149i	$-0.650600\pi$
$277$	−5.36536e9	−0.911338	−0.455669	+	0.890149i	$0.650600\pi$
$278$	0	0
$279$	− 1.32320e9i	− 0.218379i
$280$	0	0
$281$	−3.65426e9	−0.586104	−0.293052	−	0.956097i	$-0.594671\pi$
$281$	−3.65426e9	−0.586104	−0.293052	+	0.956097i	$0.594671\pi$
$282$	0	0
$283$	− 6.54721e9i	− 1.02073i	−0.859958	−	0.510365i	$-0.829510\pi$
$283$	− 6.54721e9i	− 1.02073i	0.859958	−	0.510365i	$-0.170490\pi$
$284$	0	0
$285$	3.66268e8	0.0555162
$286$	0	0
$287$	9.69040e8i	0.142828i
$288$	0	0
$289$	1.49890e10	2.14872
$290$	0	0
$291$	5.29804e9i	0.738828i
$292$	0	0
$293$	−8.67990e9	−1.17773	−0.588863	−	0.808233i	$-0.700424\pi$
$293$	−8.67990e9	−1.17773	−0.588863	+	0.808233i	$0.700424\pi$
$294$	0	0
$295$	1.86064e9i	0.245683i
$296$	0	0
$297$	4.29325e8	0.0551773
$298$	0	0
$299$	− 1.51182e10i	− 1.89155i
$300$	0	0
$301$	1.82332e10	2.22125
$302$	0	0
$303$	2.39808e9i	0.284507i
$304$	0	0
$305$	−2.70630e9	−0.312735
$306$	0	0
$307$	1.26295e10i	1.42179i	0.703300	+	0.710893i	$0.251709\pi$
$307$	1.26295e10i	1.42179i	−0.703300	+	0.710893i	$0.748291\pi$
$308$	0	0
$309$	4.04195e9	0.443361
$310$	0	0
$311$	6.60307e9i	0.705836i	0.935654	+	0.352918i	$0.114811\pi$
$311$	6.60307e9i	0.705836i	−0.935654	+	0.352918i	$0.885189\pi$
$312$	0	0
$313$	1.48273e9	0.154484	0.0772420	−	0.997012i	$-0.475389\pi$
$313$	1.48273e9	0.154484	0.0772420	+	0.997012i	$0.475389\pi$
$314$	0	0
$315$	− 1.87951e9i	− 0.190898i
$316$	0	0
$317$	1.22834e10	1.21641	0.608206	−	0.793779i	$-0.291889\pi$
$317$	1.22834e10	1.21641	0.608206	+	0.793779i	$0.291889\pi$
$318$	0	0
$319$	− 1.20415e9i	− 0.116284i
$320$	0	0
$321$	5.83946e9	0.549987
$322$	0	0
$323$	− 5.04329e9i	− 0.463344i
$324$	0	0
$325$	1.31514e10	1.17880
$326$	0	0
$327$	7.19835e9i	0.629567i
$328$	0	0
$329$	1.86946e10	1.59563
$330$	0	0
$331$	4.82881e9i	0.402280i	0.979562	+	0.201140i	$0.0644646\pi$
$331$	4.82881e9i	0.402280i	−0.979562	+	0.201140i	$0.935535\pi$
$332$	0	0
$333$	−7.61444e9	−0.619243
$334$	0	0
$335$	7.08955e9i	0.562911i
$336$	0	0
$337$	−2.01309e10	−1.56079	−0.780394	−	0.625288i	$-0.784981\pi$
$337$	−2.01309e10	−1.56079	−0.780394	+	0.625288i	$0.784981\pi$
$338$	0	0
$339$	− 5.02780e9i	− 0.380697i
$340$	0	0
$341$	−2.53975e9	−0.187834
$342$	0	0
$343$	9.00992e9i	0.650945i
$344$	0	0
$345$	4.17779e9	0.294897
$346$	0	0
$347$	− 1.51482e10i	− 1.04482i	−0.852694	−	0.522411i	$-0.825033\pi$
$347$	− 1.51482e10i	− 1.04482i	0.852694	−	0.522411i	$-0.174967\pi$
$348$	0	0
$349$	1.98926e10	1.34088	0.670440	−	0.741964i	$-0.266105\pi$
$349$	1.98926e10	1.34088	0.670440	+	0.741964i	$0.266105\pi$
$350$	0	0
$351$	3.98361e9i	0.262451i
$352$	0	0
$353$	−2.31216e10	−1.48908	−0.744542	−	0.667576i	$-0.767332\pi$
$353$	−2.31216e10	−1.48908	−0.744542	+	0.667576i	$0.767332\pi$
$354$	0	0
$355$	8.38889e9i	0.528191i
$356$	0	0
$357$	−2.58797e10	−1.59326
$358$	0	0
$359$	5.49496e9i	0.330816i	0.986225	+	0.165408i	$0.0528941\pi$
$359$	5.49496e9i	0.330816i	−0.986225	+	0.165408i	$0.947106\pi$
$360$	0	0
$361$	1.58256e10	0.931817
$362$	0	0
$363$	9.20053e9i	0.529891i
$364$	0	0
$365$	−5.46681e9	−0.308008
$366$	0	0
$367$	1.47594e10i	0.813589i	0.913520	+	0.406794i	$0.133353\pi$
$367$	1.47594e10i	0.813589i	−0.913520	+	0.406794i	$0.866647\pi$
$368$	0	0
$369$	5.67570e8	0.0306136
$370$	0	0
$371$	3.40287e10i	1.79618i
$372$	0	0
$373$	−1.31000e10	−0.676762	−0.338381	−	0.941009i	$-0.609879\pi$
$373$	−1.31000e10	−0.676762	−0.338381	+	0.941009i	$0.609879\pi$
$374$	0	0
$375$	7.83873e9i	0.396388i
$376$	0	0
$377$	1.11730e10	0.553103
$378$	0	0
$379$	1.98356e10i	0.961367i	0.876894	+	0.480684i	$0.159611\pi$
$379$	1.98356e10i	0.961367i	−0.876894	+	0.480684i	$0.840389\pi$
$380$	0	0
$381$	1.27227e10	0.603783
$382$	0	0
$383$	− 1.37283e10i	− 0.638002i	−0.947754	−	0.319001i	$-0.896653\pi$
$383$	− 1.37283e10i	− 0.638002i	0.947754	−	0.319001i	$-0.103347\pi$
$384$	0	0
$385$	−3.60751e9	−0.164197
$386$	0	0
$387$	− 1.06793e10i	− 0.476099i
$388$	0	0
$389$	−9.99543e9	−0.436519	−0.218259	−	0.975891i	$-0.570038\pi$
$389$	−9.99543e9	−0.436519	−0.218259	+	0.975891i	$0.570038\pi$
$390$	0	0
$391$	− 5.75256e10i	− 2.46124i
$392$	0	0
$393$	−1.64353e10	−0.688982
$394$	0	0
$395$	7.78252e9i	0.319692i
$396$	0	0
$397$	−3.70574e9	−0.149181	−0.0745904	−	0.997214i	$-0.523765\pi$
$397$	−3.70574e9	−0.149181	−0.0745904	+	0.997214i	$0.523765\pi$
$398$	0	0
$399$	5.94219e9i	0.234453i
$400$	0	0
$401$	4.95902e9	0.191787	0.0958934	−	0.995392i	$-0.469429\pi$
$401$	4.95902e9	0.191787	0.0958934	+	0.995392i	$0.469429\pi$
$402$	0	0
$403$	− 2.35658e10i	− 0.893431i
$404$	0	0
$405$	−1.10083e9	−0.0409168
$406$	0	0
$407$	1.46151e10i	0.532629i
$408$	0	0
$409$	1.17117e10	0.418529	0.209264	−	0.977859i	$-0.432893\pi$
$409$	1.17117e10	0.418529	0.209264	+	0.977859i	$0.432893\pi$
$410$	0	0
$411$	6.63195e9i	0.232420i
$412$	0	0
$413$	−3.01864e10	−1.03755
$414$	0	0
$415$	9.27059e9i	0.312547i
$416$	0	0
$417$	1.16376e10	0.384876
$418$	0	0
$419$	− 1.16189e9i	− 0.0376971i	−0.999822	−	0.0188485i	$-0.994000\pi$
$419$	− 1.16189e9i	− 0.0376971i	0.999822	−	0.0188485i	$-0.00600003\pi$
$420$	0	0
$421$	−1.12373e10	−0.357712	−0.178856	−	0.983875i	$-0.557240\pi$
$421$	−1.12373e10	−0.357712	−0.178856	+	0.983875i	$0.557240\pi$
$422$	0	0
$423$	− 1.09495e10i	− 0.342005i
$424$	0	0
$425$	5.00418e10	1.53383
$426$	0	0
$427$	− 4.39059e10i	− 1.32072i
$428$	0	0
$429$	7.64611e9	0.225741
$430$	0	0
$431$	− 1.10851e10i	− 0.321240i	−0.987016	−	0.160620i	$-0.948651\pi$
$431$	− 1.10851e10i	− 0.321240i	0.987016	−	0.160620i	$-0.0513493\pi$
$432$	0	0
$433$	−1.34289e10	−0.382023	−0.191011	−	0.981588i	$-0.561177\pi$
$433$	−1.34289e10	−0.382023	−0.191011	+	0.981588i	$0.561177\pi$
$434$	0	0
$435$	3.08757e9i	0.0862303i
$436$	0	0
$437$	−1.32084e10	−0.362179
$438$	0	0
$439$	− 4.62855e10i	− 1.24620i	−0.782143	−	0.623099i	$-0.785873\pi$
$439$	− 4.62855e10i	− 1.24620i	0.782143	−	0.623099i	$-0.214127\pi$
$440$	0	0
$441$	1.78848e10	0.472856
$442$	0	0
$443$	6.64464e10i	1.72527i	0.505829	+	0.862634i	$0.331187\pi$
$443$	6.64464e10i	1.72527i	−0.505829	+	0.862634i	$0.668813\pi$
$444$	0	0
$445$	2.53941e10	0.647579
$446$	0	0
$447$	− 1.87922e10i	− 0.470703i
$448$	0	0
$449$	−5.32942e10	−1.31128	−0.655638	−	0.755075i	$-0.727600\pi$
$449$	−5.32942e10	−1.31128	−0.655638	+	0.755075i	$0.727600\pi$
$450$	0	0
$451$	− 1.08939e9i	− 0.0263316i
$452$	0	0
$453$	−8.28172e9	−0.196665
$454$	0	0
$455$	− 3.34733e10i	− 0.781003i
$456$	0	0
$457$	−5.43272e10	−1.24552	−0.622762	−	0.782411i	$-0.713990\pi$
$457$	−5.43272e10	−1.24552	−0.622762	+	0.782411i	$0.713990\pi$
$458$	0	0
$459$	1.51578e10i	0.341496i
$460$	0	0
$461$	−6.99499e9	−0.154876	−0.0774379	−	0.996997i	$-0.524674\pi$
$461$	−6.99499e9	−0.154876	−0.0774379	+	0.996997i	$0.524674\pi$
$462$	0	0
$463$	− 5.48523e10i	− 1.19363i	−0.802378	−	0.596816i	$-0.796432\pi$
$463$	− 5.48523e10i	− 1.19363i	0.802378	−	0.596816i	$-0.203568\pi$
$464$	0	0
$465$	6.51218e9	0.139288
$466$	0	0
$467$	− 5.49982e9i	− 0.115633i	−0.998327	−	0.0578164i	$-0.981586\pi$
$467$	− 5.49982e9i	− 0.115633i	0.998327	−	0.0578164i	$-0.0184138\pi$
$468$	0	0
$469$	−1.15018e11	−2.37725
$470$	0	0
$471$	− 5.56182e9i	− 0.113014i
$472$	0	0
$473$	−2.04977e10	−0.409506
$474$	0	0
$475$	− 1.14900e10i	− 0.225708i
$476$	0	0
$477$	1.99307e10	0.384990
$478$	0	0
$479$	7.78175e10i	1.47821i	0.673592	+	0.739104i	$0.264751\pi$
$479$	7.78175e10i	1.47821i	−0.673592	+	0.739104i	$0.735249\pi$
$480$	0	0
$481$	−1.35610e11	−2.53345
$482$	0	0
$483$	6.77788e10i	1.24539i
$484$	0	0
$485$	−2.60744e10	−0.471246
$486$	0	0
$487$	7.21986e10i	1.28355i	0.766893	+	0.641775i	$0.221802\pi$
$487$	7.21986e10i	1.28355i	−0.766893	+	0.641775i	$0.778198\pi$
$488$	0	0
$489$	1.64394e9	0.0287508
$490$	0	0
$491$	7.74287e10i	1.33222i	0.745854	+	0.666110i	$0.232042\pi$
$491$	7.74287e10i	1.33222i	−0.745854	+	0.666110i	$0.767958\pi$
$492$	0	0
$493$	4.25139e10	0.719687
$494$	0	0
$495$	2.11293e9i	0.0351937i
$496$	0	0
$497$	−1.36098e11	−2.23062
$498$	0	0
$499$	− 9.13201e10i	− 1.47287i	−0.676509	−	0.736434i	$-0.736508\pi$
$499$	− 9.13201e10i	− 1.47287i	0.676509	−	0.736434i	$-0.263492\pi$
$500$	0	0
$501$	−3.69711e10	−0.586828
$502$	0	0
$503$	− 5.38258e10i	− 0.840850i	−0.907327	−	0.420425i	$-0.861881\pi$
$503$	− 5.38258e10i	− 0.840850i	0.907327	−	0.420425i	$-0.138119\pi$
$504$	0	0
$505$	−1.18022e10	−0.181467
$506$	0	0
$507$	3.27985e10i	0.496390i
$508$	0	0
$509$	8.42159e10	1.25465	0.627325	−	0.778757i	$-0.284150\pi$
$509$	8.42159e10	1.25465	0.627325	+	0.778757i	$0.284150\pi$
$510$	0	0
$511$	− 8.86913e10i	− 1.30076i
$512$	0	0
$513$	3.48036e9	0.0502521
$514$	0	0
$515$	1.98926e10i	0.282789i
$516$	0	0
$517$	−2.10164e10	−0.294168
$518$	0	0
$519$	3.75794e10i	0.517941i
$520$	0	0
$521$	9.42846e10	1.27965	0.639823	−	0.768522i	$-0.279008\pi$
$521$	9.42846e10	1.27965	0.639823	+	0.768522i	$0.279008\pi$
$522$	0	0
$523$	− 3.86076e9i	− 0.0516019i	−0.999667	−	0.0258010i	$-0.991786\pi$
$523$	− 3.86076e9i	− 0.0516019i	0.999667	−	0.0258010i	$-0.00821361\pi$
$524$	0	0
$525$	−5.89612e10	−0.776120
$526$	0	0
$527$	− 8.96687e10i	− 1.16251i
$528$	0	0
$529$	−7.23485e10	−0.923861
$530$	0	0
$531$	1.76802e10i	0.222387i
$532$	0	0
$533$	1.01082e10	0.125246
$534$	0	0
$535$	2.87390e10i	0.350798i
$536$	0	0
$537$	7.50066e10	0.901991
$538$	0	0
$539$	− 3.43279e10i	− 0.406717i
$540$	0	0
$541$	7.09980e10	0.828814	0.414407	−	0.910092i	$-0.363989\pi$
$541$	7.09980e10	0.828814	0.414407	+	0.910092i	$0.363989\pi$
$542$	0	0
$543$	8.98711e8i	0.0103376i
$544$	0	0
$545$	−3.54268e10	−0.401556
$546$	0	0
$547$	1.16336e11i	1.29947i	0.760163	+	0.649733i	$0.225119\pi$
$547$	1.16336e11i	1.29947i	−0.760163	+	0.649733i	$0.774881\pi$
$548$	0	0
$549$	−2.57158e10	−0.283081
$550$	0	0
$551$	− 9.76156e9i	− 0.105904i
$552$	0	0
$553$	−1.26260e11	−1.35010
$554$	0	0
$555$	− 3.74747e10i	− 0.394972i
$556$	0	0
$557$	−8.27096e10	−0.859281	−0.429640	−	0.903000i	$-0.641360\pi$
$557$	−8.27096e10	−0.859281	−0.429640	+	0.903000i	$0.641360\pi$
$558$	0	0
$559$	− 1.90194e11i	− 1.94782i
$560$	0	0
$561$	2.90938e10	0.293730
$562$	0	0
$563$	1.83781e11i	1.82922i	0.404333	+	0.914612i	$0.367504\pi$
$563$	1.83781e11i	1.82922i	−0.404333	+	0.914612i	$0.632496\pi$
$564$	0	0
$565$	2.47444e10	0.242820
$566$	0	0
$567$	− 1.78595e10i	− 0.172797i
$568$	0	0
$569$	1.00635e10	0.0960062	0.0480031	−	0.998847i	$-0.484714\pi$
$569$	1.00635e10	0.0960062	0.0480031	+	0.998847i	$0.484714\pi$
$570$	0	0
$571$	8.81816e10i	0.829533i	0.909928	+	0.414766i	$0.136137\pi$
$571$	8.81816e10i	0.829533i	−0.909928	+	0.414766i	$0.863863\pi$
$572$	0	0
$573$	−1.85579e10	−0.172152
$574$	0	0
$575$	− 1.31060e11i	− 1.19894i
$576$	0	0
$577$	−1.72778e11	−1.55878	−0.779391	−	0.626538i	$-0.784471\pi$
$577$	−1.72778e11	−1.55878	−0.779391	+	0.626538i	$0.784471\pi$
$578$	0	0
$579$	− 2.47171e10i	− 0.219929i
$580$	0	0
$581$	−1.50402e11	−1.31993
$582$	0	0
$583$	− 3.82549e10i	− 0.331141i
$584$	0	0
$585$	−1.96054e10	−0.167399
$586$	0	0
$587$	− 2.55221e10i	− 0.214963i	−0.994207	−	0.107482i	$-0.965721\pi$
$587$	− 2.55221e10i	− 0.214963i	0.994207	−	0.107482i	$-0.0342787\pi$
$588$	0	0
$589$	−2.05887e10	−0.171068
$590$	0	0
$591$	− 7.39401e9i	− 0.0606080i
$592$	0	0
$593$	2.13684e11	1.72803	0.864017	−	0.503463i	$-0.167941\pi$
$593$	2.13684e11	1.72803	0.864017	+	0.503463i	$0.167941\pi$
$594$	0	0
$595$	− 1.27367e11i	− 1.01623i
$596$	0	0
$597$	−1.11714e11	−0.879448
$598$	0	0
$599$	− 2.51463e9i	− 0.0195329i	−0.999952	−	0.00976644i	$-0.996891\pi$
$599$	− 2.51463e9i	− 0.0195329i	0.999952	−	0.00976644i	$-0.00310880\pi$
$600$	0	0
$601$	1.51985e11	1.16494	0.582470	−	0.812853i	$-0.302087\pi$
$601$	1.51985e11	1.16494	0.582470	+	0.812853i	$0.302087\pi$
$602$	0	0
$603$	6.73664e10i	0.509535i
$604$	0	0
$605$	−4.52806e10	−0.337980
$606$	0	0
$607$	− 7.40439e10i	− 0.545425i	−0.962096	−	0.272712i	$-0.912079\pi$
$607$	− 7.40439e10i	− 0.545425i	0.962096	−	0.272712i	$-0.0879207\pi$
$608$	0	0
$609$	−5.00915e10	−0.364162
$610$	0	0
$611$	− 1.95006e11i	− 1.39921i
$612$	0	0
$613$	4.60024e10	0.325790	0.162895	−	0.986643i	$-0.447917\pi$
$613$	4.60024e10	0.325790	0.162895	+	0.986643i	$0.447917\pi$
$614$	0	0
$615$	2.79331e9i	0.0195262i
$616$	0	0
$617$	5.09884e10	0.351828	0.175914	−	0.984406i	$-0.443712\pi$
$617$	5.09884e10	0.351828	0.175914	+	0.984406i	$0.443712\pi$
$618$	0	0
$619$	− 1.45541e11i	− 0.991339i	−0.868511	−	0.495670i	$-0.834923\pi$
$619$	− 1.45541e11i	− 0.991339i	0.868511	−	0.495670i	$-0.165077\pi$
$620$	0	0
$621$	3.96983e10	0.266935
$622$	0	0
$623$	4.11984e11i	2.73481i
$624$	0	0
$625$	9.33172e10	0.611563
$626$	0	0
$627$	− 6.68018e9i	− 0.0432233i
$628$	0	0
$629$	−5.16003e11	−3.29647
$630$	0	0
$631$	− 1.60148e11i	− 1.01019i	−0.863063	−	0.505096i	$-0.831457\pi$
$631$	− 1.60148e11i	− 1.01019i	0.863063	−	0.505096i	$-0.168543\pi$
$632$	0	0
$633$	6.97730e10	0.434583
$634$	0	0
$635$	6.26152e10i	0.385110i
$636$	0	0
$637$	3.18521e11	1.93455
$638$	0	0
$639$	7.97130e10i	0.478108i
$640$	0	0
$641$	−1.80516e11	−1.06926	−0.534629	−	0.845087i	$-0.679549\pi$
$641$	−1.80516e11	−1.06926	−0.534629	+	0.845087i	$0.679549\pi$
$642$	0	0
$643$	1.67871e10i	0.0982044i	0.998794	+	0.0491022i	$0.0156360\pi$
$643$	1.67871e10i	0.0982044i	−0.998794	+	0.0491022i	$0.984364\pi$
$644$	0	0
$645$	5.25582e10	0.303670
$646$	0	0
$647$	1.94306e11i	1.10884i	0.832238	+	0.554419i	$0.187060\pi$
$647$	1.94306e11i	1.10884i	−0.832238	+	0.554419i	$0.812940\pi$
$648$	0	0
$649$	3.39353e10	0.191282
$650$	0	0
$651$	1.05651e11i	0.588233i
$652$	0	0
$653$	2.46521e11	1.35582	0.677908	−	0.735147i	$-0.262887\pi$
$653$	2.46521e11	1.35582	0.677908	+	0.735147i	$0.262887\pi$
$654$	0	0
$655$	− 8.08868e10i	− 0.439453i
$656$	0	0
$657$	−5.19468e10	−0.278803
$658$	0	0
$659$	3.30534e11i	1.75257i	0.481796	+	0.876283i	$0.339984\pi$
$659$	3.30534e11i	1.75257i	−0.481796	+	0.876283i	$0.660016\pi$
$660$	0	0
$661$	2.90417e11	1.52131	0.760653	−	0.649158i	$-0.224879\pi$
$661$	2.90417e11	1.52131	0.760653	+	0.649158i	$0.224879\pi$
$662$	0	0
$663$	2.69954e11i	1.39713i
$664$	0	0
$665$	−2.92446e10	−0.149541
$666$	0	0
$667$	− 1.11344e11i	− 0.562553i
$668$	0	0
$669$	−1.52592e11	−0.761777
$670$	0	0
$671$	4.93588e10i	0.243486i
$672$	0	0
$673$	−1.34941e11	−0.657782	−0.328891	−	0.944368i	$-0.606675\pi$
$673$	−1.34941e11	−0.657782	−0.328891	+	0.944368i	$0.606675\pi$
$674$	0	0
$675$	3.45337e10i	0.166352i
$676$	0	0
$677$	−3.52912e10	−0.168001	−0.0840004	−	0.996466i	$-0.526770\pi$
$677$	−3.52912e10	−0.168001	−0.0840004	+	0.996466i	$0.526770\pi$
$678$	0	0
$679$	− 4.23021e11i	− 1.99014i
$680$	0	0
$681$	6.24612e10	0.290417
$682$	0	0
$683$	3.22338e11i	1.48125i	0.671918	+	0.740625i	$0.265471\pi$
$683$	3.22338e11i	1.48125i	−0.671918	+	0.740625i	$0.734529\pi$
$684$	0	0
$685$	−3.26393e10	−0.148244
$686$	0	0
$687$	2.22106e11i	0.997088i
$688$	0	0
$689$	3.54958e11	1.57507
$690$	0	0
$691$	4.40598e10i	0.193255i	0.995321	+	0.0966274i	$0.0308055\pi$
$691$	4.40598e10i	0.193255i	−0.995321	+	0.0966274i	$0.969194\pi$
$692$	0	0
$693$	−3.42794e10	−0.148628
$694$	0	0
$695$	5.72749e10i	0.245485i
$696$	0	0
$697$	3.84621e10	0.162968
$698$	0	0
$699$	1.73592e11i	0.727147i
$700$	0	0
$701$	2.17836e11	0.902107	0.451054	−	0.892497i	$-0.351048\pi$
$701$	2.17836e11	0.902107	0.451054	+	0.892497i	$0.351048\pi$
$702$	0	0
$703$	1.18479e11i	0.485086i
$704$	0	0
$705$	5.38882e10	0.218141
$706$	0	0
$707$	− 1.91474e11i	− 0.766360i
$708$	0	0
$709$	7.31626e10	0.289537	0.144768	−	0.989466i	$-0.453756\pi$
$709$	7.31626e10	0.289537	0.144768	+	0.989466i	$0.453756\pi$
$710$	0	0
$711$	7.39512e10i	0.289379i
$712$	0	0
$713$	−2.34842e11	−0.908695
$714$	0	0
$715$	3.76305e10i	0.143985i
$716$	0	0
$717$	−1.76292e11	−0.667047
$718$	0	0
$719$	− 5.06940e10i	− 0.189688i	−0.995492	−	0.0948442i	$-0.969765\pi$
$719$	− 5.06940e10i	− 0.189688i	0.995492	−	0.0948442i	$-0.0302353\pi$
$720$	0	0
$721$	−3.22729e11	−1.19425
$722$	0	0
$723$	− 2.33429e11i	− 0.854281i
$724$	0	0
$725$	9.68587e10	0.350580
$726$	0	0
$727$	2.06775e11i	0.740220i	0.928988	+	0.370110i	$0.120680\pi$
$727$	2.06775e11i	0.740220i	−0.928988	+	0.370110i	$0.879320\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	− 7.23694e11i	− 2.53446i
$732$	0	0
$733$	1.18165e11	0.409329	0.204665	−	0.978832i	$-0.434390\pi$
$733$	1.18165e11	0.409329	0.204665	+	0.978832i	$0.434390\pi$
$734$	0	0
$735$	8.80203e10i	0.301601i
$736$	0	0
$737$	1.29303e11	0.438266
$738$	0	0
$739$	− 4.44589e11i	− 1.49067i	−0.666691	−	0.745334i	$-0.732290\pi$
$739$	− 4.44589e11i	− 1.49067i	0.666691	−	0.745334i	$-0.267710\pi$
$740$	0	0
$741$	6.19838e10	0.205592
$742$	0	0
$743$	− 2.06083e11i	− 0.676217i	−0.941107	−	0.338109i	$-0.890213\pi$
$743$	− 2.06083e11i	− 0.676217i	0.941107	−	0.338109i	$-0.109787\pi$
$744$	0	0
$745$	9.24861e10	0.300228
$746$	0	0
$747$	8.80911e10i	0.282911i
$748$	0	0
$749$	−4.66251e11	−1.48147
$750$	0	0
$751$	5.07412e11i	1.59515i	0.603221	+	0.797574i	$0.293884\pi$
$751$	5.07412e11i	1.59515i	−0.603221	+	0.797574i	$0.706116\pi$
$752$	0	0
$753$	−2.30425e10	−0.0716719
$754$	0	0
$755$	− 4.07587e10i	− 0.125439i
$756$	0	0
$757$	4.97648e11	1.51544	0.757720	−	0.652580i	$-0.226313\pi$
$757$	4.97648e11	1.51544	0.757720	+	0.652580i	$0.226313\pi$
$758$	0	0
$759$	− 7.61966e10i	− 0.229598i
$760$	0	0
$761$	5.61910e11	1.67544	0.837719	−	0.546102i	$-0.183889\pi$
$761$	5.61910e11	1.67544	0.837719	+	0.546102i	$0.183889\pi$
$762$	0	0
$763$	− 5.74751e11i	− 1.69583i
$764$	0	0
$765$	−7.45994e10	−0.217816
$766$	0	0
$767$	3.14878e11i	0.909832i
$768$	0	0
$769$	1.02129e11	0.292040	0.146020	−	0.989282i	$-0.453354\pi$
$769$	1.02129e11	0.292040	0.146020	+	0.989282i	$0.453354\pi$
$770$	0	0
$771$	4.02682e11i	1.13958i
$772$	0	0
$773$	6.18597e11	1.73257	0.866284	−	0.499552i	$-0.166502\pi$
$773$	6.18597e11	1.73257	0.866284	+	0.499552i	$0.166502\pi$
$774$	0	0
$775$	− 2.04291e11i	− 0.566294i
$776$	0	0
$777$	6.07974e11	1.66802
$778$	0	0
$779$	− 8.83124e9i	− 0.0239812i
$780$	0	0
$781$	1.53001e11	0.411234
$782$	0	0
$783$	2.93388e10i	0.0780539i
$784$	0	0
$785$	2.73726e10	0.0720839
$786$	0	0
$787$	− 5.08685e11i	− 1.32602i	−0.748611	−	0.663009i	$-0.769279\pi$
$787$	− 5.08685e11i	− 1.32602i	0.748611	−	0.663009i	$-0.230721\pi$
$788$	0	0
$789$	−4.69162e10	−0.121064
$790$	0	0
$791$	4.01444e11i	1.02546i
$792$	0	0
$793$	−4.57989e11	−1.15814
$794$	0	0
$795$	9.80895e10i	0.245558i
$796$	0	0
$797$	3.29439e11	0.816473	0.408237	−	0.912876i	$-0.366144\pi$
$797$	3.29439e11	0.816473	0.408237	+	0.912876i	$0.366144\pi$
$798$	0	0
$799$	− 7.42007e11i	− 1.82063i
$800$	0	0
$801$	2.41300e11	0.586175
$802$	0	0
$803$	9.97063e10i	0.239806i
$804$	0	0
$805$	−3.33575e11	−0.794346
$806$	0	0
$807$	2.12039e10i	0.0499944i
$808$	0	0
$809$	3.29638e11	0.769561	0.384780	−	0.923008i	$-0.374277\pi$
$809$	3.29638e11	0.769561	0.384780	+	0.923008i	$0.374277\pi$
$810$	0	0
$811$	− 5.13155e11i	− 1.18622i	−0.805122	−	0.593110i	$-0.797900\pi$
$811$	− 5.13155e11i	− 1.18622i	0.805122	−	0.593110i	$-0.202100\pi$
$812$	0	0
$813$	1.36710e11	0.312924
$814$	0	0
$815$	8.09068e9i	0.0183381i
$816$	0	0
$817$	−1.66166e11	−0.372954
$818$	0	0
$819$	− 3.18070e11i	− 0.706948i
$820$	0	0
$821$	−4.30640e11	−0.947854	−0.473927	−	0.880564i	$-0.657164\pi$
$821$	−4.30640e11	−0.947854	−0.473927	+	0.880564i	$0.657164\pi$
$822$	0	0
$823$	2.16005e11i	0.470830i	0.971895	+	0.235415i	$0.0756449\pi$
$823$	2.16005e11i	0.470830i	−0.971895	+	0.235415i	$0.924355\pi$
$824$	0	0
$825$	6.62838e10	0.143084
$826$	0	0
$827$	− 3.29620e11i	− 0.704679i	−0.935872	−	0.352339i	$-0.885386\pi$
$827$	− 3.29620e11i	− 0.704679i	0.935872	−	0.352339i	$-0.114614\pi$
$828$	0	0
$829$	8.43310e11	1.78554	0.892769	−	0.450515i	$-0.148759\pi$
$829$	8.43310e11	1.78554	0.892769	+	0.450515i	$0.148759\pi$
$830$	0	0
$831$	− 2.50913e11i	− 0.526162i
$832$	0	0
$833$	1.21198e12	2.51720
$834$	0	0
$835$	− 1.81954e11i	− 0.374296i
$836$	0	0
$837$	6.18801e10	0.126081
$838$	0	0
$839$	− 2.15591e11i	− 0.435093i	−0.976050	−	0.217546i	$-0.930195\pi$
$839$	− 2.15591e11i	− 0.435093i	0.976050	−	0.217546i	$-0.0698053\pi$
$840$	0	0
$841$	−4.17958e11	−0.835505
$842$	0	0
$843$	− 1.70893e11i	− 0.338387i
$844$	0	0
$845$	−1.61419e11	−0.316612
$846$	0	0
$847$	− 7.34615e11i	− 1.42734i
$848$	0	0
$849$	3.06183e11	0.589318
$850$	0	0
$851$	1.35141e12i	2.57673i
$852$	0	0
$853$	6.99249e11	1.32080	0.660398	−	0.750915i	$-0.270387\pi$
$853$	6.99249e11	1.32080	0.660398	+	0.750915i	$0.270387\pi$
$854$	0	0
$855$	1.71287e10i	0.0320523i
$856$	0	0
$857$	5.49747e10	0.101915	0.0509577	−	0.998701i	$-0.483773\pi$
$857$	5.49747e10	0.101915	0.0509577	+	0.998701i	$0.483773\pi$
$858$	0	0
$859$	5.77556e10i	0.106077i	0.998592	+	0.0530385i	$0.0168906\pi$
$859$	5.77556e10i	0.106077i	−0.998592	+	0.0530385i	$0.983109\pi$
$860$	0	0
$861$	−4.53175e10	−0.0824620
$862$	0	0
$863$	9.16983e11i	1.65317i	0.562810	+	0.826586i	$0.309720\pi$
$863$	9.16983e11i	1.65317i	−0.562810	+	0.826586i	$0.690280\pi$
$864$	0	0
$865$	−1.84948e11	−0.330358
$866$	0	0
$867$	7.00964e11i	1.24056i
$868$	0	0
$869$	1.41941e11	0.248903
$870$	0	0
$871$	1.19977e12i	2.08461i
$872$	0	0
$873$	−2.47765e11	−0.426563
$874$	0	0
$875$	− 6.25882e11i	− 1.06773i
$876$	0	0
$877$	1.13696e11	0.192196	0.0960982	−	0.995372i	$-0.469364\pi$
$877$	1.13696e11	0.192196	0.0960982	+	0.995372i	$0.469364\pi$
$878$	0	0
$879$	− 4.05919e11i	− 0.679960i
$880$	0	0
$881$	−8.21734e11	−1.36404	−0.682021	−	0.731333i	$-0.738899\pi$
$881$	−8.21734e11	−1.36404	−0.682021	+	0.731333i	$0.738899\pi$
$882$	0	0
$883$	− 9.80897e11i	− 1.61354i	−0.590864	−	0.806771i	$-0.701213\pi$
$883$	− 9.80897e11i	− 1.61354i	0.590864	−	0.806771i	$-0.298787\pi$
$884$	0	0
$885$	−8.70137e10	−0.141845
$886$	0	0
$887$	− 3.84074e11i	− 0.620469i	−0.950660	−	0.310234i	$-0.899592\pi$
$887$	− 3.84074e11i	− 0.620469i	0.950660	−	0.310234i	$-0.100408\pi$
$888$	0	0
$889$	−1.01584e12	−1.62637
$890$	0	0
$891$	2.00775e10i	0.0318566i
$892$	0	0
$893$	−1.70371e11	−0.267911
$894$	0	0
$895$	3.69147e11i	0.575317i
$896$	0	0
$897$	7.07010e11	1.09208
$898$	0	0
$899$	− 1.73559e11i	− 0.265710i
$900$	0	0
$901$	1.35063e12	2.04945
$902$	0	0
$903$	8.52684e11i	1.28244i
$904$	0	0
$905$	−4.42303e9	−0.00659364
$906$	0	0
$907$	− 2.89290e11i	− 0.427468i	−0.976892	−	0.213734i	$-0.931437\pi$
$907$	− 2.89290e11i	− 0.427468i	0.976892	−	0.213734i	$-0.0685627\pi$
$908$	0	0
$909$	−1.12147e11	−0.164260
$910$	0	0
$911$	1.23817e12i	1.79766i	0.438300	+	0.898829i	$0.355581\pi$
$911$	1.23817e12i	1.79766i	−0.438300	+	0.898829i	$0.644419\pi$
$912$	0	0
$913$	1.69082e11	0.243340
$914$	0	0
$915$	− 1.26561e11i	− 0.180558i
$916$	0	0
$917$	1.31227e12	1.85587
$918$	0	0
$919$	− 4.17266e11i	− 0.584993i	−0.956267	−	0.292496i	$-0.905514\pi$
$919$	− 4.17266e11i	− 0.584993i	0.956267	−	0.292496i	$-0.0944859\pi$
$920$	0	0
$921$	−5.90625e11	−0.820868
$922$	0	0
$923$	1.41966e12i	1.95604i
$924$	0	0
$925$	−1.17560e12	−1.60581
$926$	0	0
$927$	1.89023e11i	0.255974i
$928$	0	0
$929$	9.82448e11	1.31901	0.659503	−	0.751702i	$-0.270767\pi$
$929$	9.82448e11	1.31901	0.659503	+	0.751702i	$0.270767\pi$
$930$	0	0
$931$	− 2.78282e11i	− 0.370413i
$932$	0	0
$933$	−3.08795e11	−0.407515
$934$	0	0
$935$	1.43186e11i	0.187350i
$936$	0	0
$937$	5.55638e11	0.720831	0.360415	−	0.932792i	$-0.382635\pi$
$937$	5.55638e11	0.720831	0.360415	+	0.932792i	$0.382635\pi$
$938$	0	0
$939$	6.93402e10i	0.0891914i
$940$	0	0
$941$	−2.71844e11	−0.346706	−0.173353	−	0.984860i	$-0.555460\pi$
$941$	−2.71844e11	−0.346706	−0.173353	+	0.984860i	$0.555460\pi$
$942$	0	0
$943$	− 1.00732e11i	− 0.127386i
$944$	0	0
$945$	8.78959e10	0.110215
$946$	0	0
$947$	− 4.13641e11i	− 0.514308i	−0.966370	−	0.257154i	$-0.917215\pi$
$947$	− 4.13641e11i	− 0.514308i	0.966370	−	0.257154i	$-0.0827848\pi$
$948$	0	0
$949$	−9.25152e11	−1.14064
$950$	0	0
$951$	5.74437e11i	0.702296i
$952$	0	0
$953$	−6.98695e11	−0.847063	−0.423532	−	0.905881i	$-0.639210\pi$
$953$	−6.98695e11	−0.847063	−0.423532	+	0.905881i	$0.639210\pi$
$954$	0	0
$955$	− 9.13333e10i	− 0.109803i
$956$	0	0
$957$	5.63126e10	0.0671364
$958$	0	0
$959$	− 5.29527e11i	− 0.626057i
$960$	0	0
$961$	4.86828e11	0.570797
$962$	0	0
$963$	2.73085e11i	0.317535i
$964$	0	0
$965$	1.21646e11	0.140278
$966$	0	0
$967$	− 1.30721e12i	− 1.49499i	−0.664268	−	0.747495i	$-0.731257\pi$
$967$	− 1.30721e12i	− 1.49499i	0.664268	−	0.747495i	$-0.268743\pi$
$968$	0	0
$969$	2.35851e11	0.267512
$970$	0	0
$971$	8.74761e11i	0.984039i	0.870584	+	0.492020i	$0.163741\pi$
$971$	8.74761e11i	0.984039i	−0.870584	+	0.492020i	$0.836259\pi$
$972$	0	0
$973$	−9.29205e11	−1.03672
$974$	0	0
$975$	6.15032e11i	0.680581i
$976$	0	0
$977$	−7.48121e11	−0.821095	−0.410548	−	0.911839i	$-0.634662\pi$
$977$	−7.48121e11	−0.821095	−0.410548	+	0.911839i	$0.634662\pi$
$978$	0	0
$979$	− 4.63150e11i	− 0.504186i
$980$	0	0
$981$	−3.36633e11	−0.363480
$982$	0	0
$983$	5.32393e10i	0.0570188i	0.999594	+	0.0285094i	$0.00907605\pi$
$983$	5.32393e10i	0.0570188i	−0.999594	+	0.0285094i	$0.990924\pi$
$984$	0	0
$985$	3.63898e10	0.0386576
$986$	0	0
$987$	8.74260e11i	0.921238i
$988$	0	0
$989$	−1.89536e12	−1.98110
$990$	0	0
$991$	− 8.33806e11i	− 0.864511i	−0.901751	−	0.432255i	$-0.857718\pi$
$991$	− 8.33806e11i	− 0.864511i	0.901751	−	0.432255i	$-0.142282\pi$
$992$	0	0
$993$	−2.25821e11	−0.232257
$994$	0	0
$995$	− 5.49803e11i	− 0.560938i
$996$	0	0
$997$	3.50909e10	0.0355151	0.0177576	−	0.999842i	$-0.494347\pi$
$997$	3.50909e10	0.0355151	0.0177576	+	0.999842i	$0.494347\pi$
$998$	0	0
$999$	− 3.56092e11i	− 0.357520i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.a.127.4	yes	32
4.3	odd	2	inner	384.9.g.a.127.3	✓	32
8.3	odd	2		384.9.g.b.127.30	yes	32
8.5	even	2		384.9.g.b.127.29	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.3	✓	32	4.3	odd	2	inner
384.9.g.a.127.4	yes	32	1.1	even	1	trivial
384.9.g.b.127.29	yes	32	8.5	even	2
384.9.g.b.127.30	yes	32	8.3	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	384.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} -230.157 q^{5} -3733.97i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} -230.157 q^{5} -3733.97i q^{7} -2187.00 q^{9} +4197.71i q^{11} -38949.6 q^{13} -10763.4i q^{15} -148205. q^{17} +34029.1i q^{19} +174621. q^{21} +388149. i q^{23} -337653. q^{25} -102276. i q^{27} -286859. q^{29} +605032. i q^{31} -196308. q^{33} +859400. i q^{35} +3.48168e6 q^{37} -1.82149e6i q^{39} -259520. q^{41} +4.88306e6i q^{43} +503353. q^{45} +5.00662e6i q^{47} -8.17776e6 q^{49} -6.93086e6i q^{51} -9.11327e6 q^{53} -966133. i q^{55} -1.59139e6 q^{57} -8.08424e6i q^{59} +1.17585e7 q^{61} +8.16620e6i q^{63} +8.96452e6 q^{65} -3.08031e7i q^{67} -1.81519e7 q^{69} -3.64486e7i q^{71} +2.37525e7 q^{73} -1.57905e7i q^{75} +1.56742e7 q^{77} -3.38140e7i q^{79} +4.78297e6 q^{81} -4.02794e7i q^{83} +3.41104e7 q^{85} -1.34151e7i q^{87} -1.10334e8 q^{89} +1.45437e8i q^{91} -2.82945e7 q^{93} -7.83204e6i q^{95} +1.13290e8 q^{97} -9.18040e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 32 * q - 1344 * q^5 - 69984 * q^9	\( 32 q - 1344 q^{5} - 69984 q^{9} - 114240 q^{13} - 154560 q^{17} + 1791712 q^{25} - 275520 q^{29} + 2421440 q^{37} - 4374720 q^{41} + 2939328 q^{45} - 14219104 q^{49} - 6224448 q^{53} + 3100032 q^{57} - 13005632 q^{61} + 75175296 q^{65} - 85710400 q^{73} + 154517760 q^{77} + 153055008 q^{81} - 384830848 q^{85} - 182669760 q^{89} + 149817600 q^{93} - 149408192 q^{97}+O(q^{100}) \) 32 * q - 1344 * q^5 - 69984 * q^9 - 114240 * q^13 - 154560 * q^17 + 1791712 * q^25 - 275520 * q^29 + 2421440 * q^37 - 4374720 * q^41 + 2939328 * q^45 - 14219104 * q^49 - 6224448 * q^53 + 3100032 * q^57 - 13005632 * q^61 + 75175296 * q^65 - 85710400 * q^73 + 154517760 * q^77 + 153055008 * q^81 - 384830848 * q^85 - 182669760 * q^89 + 149817600 * q^93 - 149408192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more