LMFDB - Embedded newform 384.9.g.a.127.20

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			127.20
Character	$\chi$	$=$	384.127
Dual form			384.9.g.a.127.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} +1083.37 q^{5} +399.103i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} +1083.37 q^{5} +399.103i q^{7} -2187.00 q^{9} -4273.65i q^{11} +38046.6 q^{13} +50664.4i q^{15} -125236. q^{17} -216366. i q^{19} -18664.2 q^{21} +300640. i q^{23} +783073. q^{25} -102276. i q^{27} +434034. q^{29} -787745. i q^{31} +199859. q^{33} +432378. i q^{35} +1.35223e6 q^{37} +1.77927e6i q^{39} -623263. q^{41} -4.76099e6i q^{43} -2.36934e6 q^{45} +7.12911e6i q^{47} +5.60552e6 q^{49} -5.85671e6i q^{51} +7.12847e6 q^{53} -4.62995e6i q^{55} +1.01185e7 q^{57} -1.33263e7i q^{59} +2.19853e7 q^{61} -872839. i q^{63} +4.12187e7 q^{65} -1.18413e7i q^{67} -1.40595e7 q^{69} -1.35279e7i q^{71} -2.79556e7 q^{73} +3.66207e7i q^{75} +1.70563e6 q^{77} +3.52247e7i q^{79} +4.78297e6 q^{81} +2.93657e6i q^{83} -1.35677e8 q^{85} +2.02978e7i q^{87} -9.53819e7 q^{89} +1.51845e7i q^{91} +3.68392e7 q^{93} -2.34406e8i q^{95} +7.13673e7 q^{97} +9.34646e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1083.37	1.73340	0.866699	−	0.498832i	$-0.166238\pi$
$5$	1083.37	1.73340	0.866699	+	0.498832i	$0.166238\pi$
$6$	0	0
$7$	399.103i	0.166224i	0.996540	+	0.0831119i	$0.0264859\pi$
$7$	399.103i	0.166224i	−0.996540	+	0.0831119i	$0.973514\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	− 4273.65i	− 0.291896i	−0.989292	−	0.145948i	$-0.953377\pi$
$11$	− 4273.65i	− 0.291896i	0.989292	−	0.145948i	$-0.0466232\pi$
$12$	0	0
$13$	38046.6	1.33212	0.666059	−	0.745899i	$-0.267980\pi$
$13$	38046.6	1.33212	0.666059	+	0.745899i	$0.267980\pi$
$14$	0	0
$15$	50664.4i	1.00078i
$16$	0	0
$17$	−125236.	−1.49946	−0.749728	−	0.661746i	$-0.769816\pi$
$17$	−125236.	−1.49946	−0.749728	+	0.661746i	$0.769816\pi$
$18$	0	0
$19$	− 216366.i	− 1.66026i	−0.557571	−	0.830129i	$-0.688267\pi$
$19$	− 216366.i	− 1.66026i	0.557571	−	0.830129i	$-0.311733\pi$
$20$	0	0
$21$	−18664.2	−0.0959694
$22$	0	0
$23$	300640.i	1.07432i	0.843479	+	0.537162i	$0.180504\pi$
$23$	300640.i	1.07432i	−0.843479	+	0.537162i	$0.819496\pi$
$24$	0	0
$25$	783073.	2.00467
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	434034.	0.613666	0.306833	−	0.951763i	$-0.400731\pi$
$29$	434034.	0.613666	0.306833	+	0.951763i	$0.400731\pi$
$30$	0	0
$31$	− 787745.i	− 0.852980i	−0.904492	−	0.426490i	$-0.859750\pi$
$31$	− 787745.i	− 0.852980i	0.904492	−	0.426490i	$-0.140250\pi$
$32$	0	0
$33$	199859.	0.168526
$34$	0	0
$35$	432378.i	0.288132i
$36$	0	0
$37$	1.35223e6	0.721511	0.360755	−	0.932660i	$-0.382519\pi$
$37$	1.35223e6	0.721511	0.360755	+	0.932660i	$0.382519\pi$
$38$	0	0
$39$	1.77927e6i	0.769099i
$40$	0	0
$41$	−623263.	−0.220565	−0.110282	−	0.993900i	$-0.535176\pi$
$41$	−623263.	−0.220565	−0.110282	+	0.993900i	$0.535176\pi$
$42$	0	0
$43$	− 4.76099e6i	− 1.39259i	−0.717756	−	0.696295i	$-0.754831\pi$
$43$	− 4.76099e6i	− 1.39259i	0.717756	−	0.696295i	$-0.245169\pi$
$44$	0	0
$45$	−2.36934e6	−0.577799
$46$	0	0
$47$	7.12911e6i	1.46098i	0.682925	+	0.730489i	$0.260708\pi$
$47$	7.12911e6i	1.46098i	−0.682925	+	0.730489i	$0.739292\pi$
$48$	0	0
$49$	5.60552e6	0.972370
$50$	0	0
$51$	− 5.85671e6i	− 0.865712i
$52$	0	0
$53$	7.12847e6	0.903426	0.451713	−	0.892163i	$-0.350813\pi$
$53$	7.12847e6	0.903426	0.451713	+	0.892163i	$0.350813\pi$
$54$	0	0
$55$	− 4.62995e6i	− 0.505971i
$56$	0	0
$57$	1.01185e7	0.958550
$58$	0	0
$59$	− 1.33263e7i	− 1.09977i	−0.835240	−	0.549885i	$-0.814672\pi$
$59$	− 1.33263e7i	− 1.09977i	0.835240	−	0.549885i	$-0.185328\pi$
$60$	0	0
$61$	2.19853e7	1.58786	0.793932	−	0.608007i	$-0.208031\pi$
$61$	2.19853e7	1.58786	0.793932	+	0.608007i	$0.208031\pi$
$62$	0	0
$63$	− 872839.i	− 0.0554079i
$64$	0	0
$65$	4.12187e7	2.30909
$66$	0	0
$67$	− 1.18413e7i	− 0.587627i	−0.955863	−	0.293814i	$-0.905076\pi$
$67$	− 1.18413e7i	− 0.587627i	0.955863	−	0.293814i	$-0.0949245\pi$
$68$	0	0
$69$	−1.40595e7	−0.620261
$70$	0	0
$71$	− 1.35279e7i	− 0.532350i	−0.963925	−	0.266175i	$-0.914240\pi$
$71$	− 1.35279e7i	− 0.532350i	0.963925	−	0.266175i	$-0.0857600\pi$
$72$	0	0
$73$	−2.79556e7	−0.984414	−0.492207	−	0.870478i	$-0.663810\pi$
$73$	−2.79556e7	−0.984414	−0.492207	+	0.870478i	$0.663810\pi$
$74$	0	0
$75$	3.66207e7i	1.15739i
$76$	0	0
$77$	1.70563e6	0.0485200
$78$	0	0
$79$	3.52247e7i	0.904355i	0.891928	+	0.452178i	$0.149353\pi$
$79$	3.52247e7i	0.904355i	−0.891928	+	0.452178i	$0.850647\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	2.93657e6i	0.0618768i	0.999521	+	0.0309384i	$0.00984957\pi$
$83$	2.93657e6i	0.0618768i	−0.999521	+	0.0309384i	$0.990150\pi$
$84$	0	0
$85$	−1.35677e8	−2.59915
$86$	0	0
$87$	2.02978e7i	0.354300i
$88$	0	0
$89$	−9.53819e7	−1.52022	−0.760109	−	0.649795i	$-0.774855\pi$
$89$	−9.53819e7	−1.52022	−0.760109	+	0.649795i	$0.774855\pi$
$90$	0	0
$91$	1.51845e7i	0.221430i
$92$	0	0
$93$	3.68392e7	0.492468
$94$	0	0
$95$	− 2.34406e8i	− 2.87789i
$96$	0	0
$97$	7.13673e7	0.806143	0.403072	−	0.915168i	$-0.367943\pi$
$97$	7.13673e7	0.806143	0.403072	+	0.915168i	$0.367943\pi$
$98$	0	0
$99$	9.34646e6i	0.0972986i
$100$	0	0
$101$	−5.24923e7	−0.504441	−0.252220	−	0.967670i	$-0.581161\pi$
$101$	−5.24923e7	−0.504441	−0.252220	+	0.967670i	$0.581161\pi$
$102$	0	0
$103$	− 1.89309e8i	− 1.68199i	−0.541043	−	0.840995i	$-0.681970\pi$
$103$	− 1.89309e8i	− 1.68199i	0.541043	−	0.840995i	$-0.318030\pi$
$104$	0	0
$105$	−2.02203e7	−0.166353
$106$	0	0
$107$	− 1.16562e8i	− 0.889244i	−0.895718	−	0.444622i	$-0.853338\pi$
$107$	− 1.16562e8i	− 0.889244i	0.895718	−	0.444622i	$-0.146662\pi$
$108$	0	0
$109$	−3.10038e7	−0.219638	−0.109819	−	0.993952i	$-0.535027\pi$
$109$	−3.10038e7	−0.219638	−0.109819	+	0.993952i	$0.535027\pi$
$110$	0	0
$111$	6.32374e7i	0.416564i
$112$	0	0
$113$	−7.19661e7	−0.441382	−0.220691	−	0.975344i	$-0.570831\pi$
$113$	−7.19661e7	−0.441382	−0.220691	+	0.975344i	$0.570831\pi$
$114$	0	0
$115$	3.25705e8i	1.86223i
$116$	0	0
$117$	−8.32080e7	−0.444040
$118$	0	0
$119$	− 4.99822e7i	− 0.249245i
$120$	0	0
$121$	1.96095e8	0.914797
$122$	0	0
$123$	− 2.91471e7i	− 0.127343i
$124$	0	0
$125$	4.25168e8	1.74149
$126$	0	0
$127$	2.95872e8i	1.13734i	0.822567	+	0.568668i	$0.192541\pi$
$127$	2.95872e8i	1.13734i	−0.822567	+	0.568668i	$0.807459\pi$
$128$	0	0
$129$	2.22649e8	0.804012
$130$	0	0
$131$	− 7.09864e7i	− 0.241040i	−0.992711	−	0.120520i	$-0.961544\pi$
$131$	− 7.09864e7i	− 0.241040i	0.992711	−	0.120520i	$-0.0384562\pi$
$132$	0	0
$133$	8.63526e7	0.275974
$134$	0	0
$135$	− 1.10803e8i	− 0.333592i
$136$	0	0
$137$	−4.24637e8	−1.20541	−0.602706	−	0.797963i	$-0.705911\pi$
$137$	−4.24637e8	−1.20541	−0.602706	+	0.797963i	$0.705911\pi$
$138$	0	0
$139$	4.94227e8i	1.32394i	0.749532	+	0.661968i	$0.230279\pi$
$139$	4.94227e8i	1.32394i	−0.749532	+	0.661968i	$0.769721\pi$
$140$	0	0
$141$	−3.33395e8	−0.843496
$142$	0	0
$143$	− 1.62598e8i	− 0.388840i
$144$	0	0
$145$	4.70221e8	1.06373
$146$	0	0
$147$	2.62144e8i	0.561398i
$148$	0	0
$149$	8.27614e8	1.67912	0.839562	−	0.543265i	$-0.182812\pi$
$149$	8.27614e8	1.67912	0.839562	+	0.543265i	$0.182812\pi$
$150$	0	0
$151$	6.87696e7i	0.132278i	0.997810	+	0.0661392i	$0.0210681\pi$
$151$	6.87696e7i	0.132278i	−0.997810	+	0.0661392i	$0.978932\pi$
$152$	0	0
$153$	2.73891e8	0.499819
$154$	0	0
$155$	− 8.53422e8i	− 1.47855i
$156$	0	0
$157$	1.22869e8	0.202229	0.101114	−	0.994875i	$-0.467759\pi$
$157$	1.22869e8	0.202229	0.101114	+	0.994875i	$0.467759\pi$
$158$	0	0
$159$	3.33366e8i	0.521594i
$160$	0	0
$161$	−1.19986e8	−0.178578
$162$	0	0
$163$	− 5.41586e8i	− 0.767215i	−0.923496	−	0.383607i	$-0.874682\pi$
$163$	− 5.41586e8i	− 0.767215i	0.923496	−	0.383607i	$-0.125318\pi$
$164$	0	0
$165$	2.16522e8	0.292123
$166$	0	0
$167$	− 4.70718e8i	− 0.605194i	−0.953119	−	0.302597i	$-0.902146\pi$
$167$	− 4.70718e8i	− 0.605194i	0.953119	−	0.302597i	$-0.0978537\pi$
$168$	0	0
$169$	6.31816e8	0.774540
$170$	0	0
$171$	4.73193e8i	0.553419i
$172$	0	0
$173$	1.65123e9	1.84342	0.921709	−	0.387882i	$-0.126793\pi$
$173$	1.65123e9	1.84342	0.921709	+	0.387882i	$0.126793\pi$
$174$	0	0
$175$	3.12527e8i	0.333223i
$176$	0	0
$177$	6.23210e8	0.634953
$178$	0	0
$179$	1.30626e9i	1.27238i	0.771532	+	0.636191i	$0.219491\pi$
$179$	1.30626e9i	1.27238i	−0.771532	+	0.636191i	$0.780509\pi$
$180$	0	0
$181$	1.14074e9	1.06285	0.531424	−	0.847106i	$-0.321657\pi$
$181$	1.14074e9	1.06285	0.531424	+	0.847106i	$0.321657\pi$
$182$	0	0
$183$	1.02815e9i	0.916753i
$184$	0	0
$185$	1.46497e9	1.25066
$186$	0	0
$187$	5.35215e8i	0.437685i
$188$	0	0
$189$	4.08186e7	0.0319898
$190$	0	0
$191$	− 3.43690e8i	− 0.258246i	−0.991629	−	0.129123i	$-0.958784\pi$
$191$	− 3.43690e8i	− 0.258246i	0.991629	−	0.129123i	$-0.0412162\pi$
$192$	0	0
$193$	−2.90950e8	−0.209695	−0.104848	−	0.994488i	$-0.533435\pi$
$193$	−2.90950e8	−0.209695	−0.104848	+	0.994488i	$0.533435\pi$
$194$	0	0
$195$	1.92761e9i	1.33315i
$196$	0	0
$197$	1.12669e9	0.748063	0.374032	−	0.927416i	$-0.377975\pi$
$197$	1.12669e9	0.748063	0.374032	+	0.927416i	$0.377975\pi$
$198$	0	0
$199$	− 1.01757e9i	− 0.648865i	−0.945909	−	0.324432i	$-0.894827\pi$
$199$	− 1.01757e9i	− 0.648865i	0.945909	−	0.324432i	$-0.105173\pi$
$200$	0	0
$201$	5.53765e8	0.339267
$202$	0	0
$203$	1.73225e8i	0.102006i
$204$	0	0
$205$	−6.75227e8	−0.382326
$206$	0	0
$207$	− 6.57499e8i	− 0.358108i
$208$	0	0
$209$	−9.24674e8	−0.484622
$210$	0	0
$211$	7.36560e8i	0.371602i	0.982587	+	0.185801i	$0.0594880\pi$
$211$	7.36560e8i	0.371602i	−0.982587	+	0.185801i	$0.940512\pi$
$212$	0	0
$213$	6.32638e8	0.307353
$214$	0	0
$215$	− 5.15793e9i	− 2.41391i
$216$	0	0
$217$	3.14392e8	0.141786
$218$	0	0
$219$	− 1.30736e9i	− 0.568352i
$220$	0	0
$221$	−4.76481e9	−1.99745
$222$	0	0
$223$	− 1.76140e9i	− 0.712259i	−0.934437	−	0.356130i	$-0.884096\pi$
$223$	− 1.76140e9i	− 0.712259i	0.934437	−	0.356130i	$-0.115904\pi$
$224$	0	0
$225$	−1.71258e9	−0.668222
$226$	0	0
$227$	2.73722e9i	1.03087i	0.856927	+	0.515437i	$0.172371\pi$
$227$	2.73722e9i	1.03087i	−0.856927	+	0.515437i	$0.827629\pi$
$228$	0	0
$229$	3.45080e9	1.25481	0.627404	−	0.778694i	$-0.284117\pi$
$229$	3.45080e9	1.25481	0.627404	+	0.778694i	$0.284117\pi$
$230$	0	0
$231$	7.97643e7i	0.0280130i
$232$	0	0
$233$	2.22088e8	0.0753533	0.0376767	−	0.999290i	$-0.488004\pi$
$233$	2.22088e8	0.0753533	0.0376767	+	0.999290i	$0.488004\pi$
$234$	0	0
$235$	7.72348e9i	2.53246i
$236$	0	0
$237$	−1.64730e9	−0.522130
$238$	0	0
$239$	− 3.65262e9i	− 1.11947i	−0.828672	−	0.559735i	$-0.810903\pi$
$239$	− 3.65262e9i	− 1.11947i	0.828672	−	0.559735i	$-0.189097\pi$
$240$	0	0
$241$	2.46594e9	0.730996	0.365498	−	0.930812i	$-0.380899\pi$
$241$	2.46594e9	0.730996	0.365498	+	0.930812i	$0.380899\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	6.07287e9	1.68550
$246$	0	0
$247$	− 8.23202e9i	− 2.21166i
$248$	0	0
$249$	−1.37330e8	−0.0357246
$250$	0	0
$251$	4.70566e9i	1.18557i	0.805362	+	0.592783i	$0.201971\pi$
$251$	4.70566e9i	1.18557i	−0.805362	+	0.592783i	$0.798029\pi$
$252$	0	0
$253$	1.28483e9	0.313591
$254$	0	0
$255$	− 6.34501e9i	− 1.50062i
$256$	0	0
$257$	6.61097e9	1.51542	0.757709	−	0.652592i	$-0.226319\pi$
$257$	6.61097e9	1.51542	0.757709	+	0.652592i	$0.226319\pi$
$258$	0	0
$259$	5.39678e8i	0.119932i
$260$	0	0
$261$	−9.49233e8	−0.204555
$262$	0	0
$263$	8.75549e9i	1.83003i	0.403422	+	0.915014i	$0.367821\pi$
$263$	8.75549e9i	1.83003i	−0.403422	+	0.915014i	$0.632179\pi$
$264$	0	0
$265$	7.72279e9	1.56600
$266$	0	0
$267$	− 4.46057e9i	− 0.877698i
$268$	0	0
$269$	4.61409e9	0.881204	0.440602	−	0.897702i	$-0.354765\pi$
$269$	4.61409e9	0.881204	0.440602	+	0.897702i	$0.354765\pi$
$270$	0	0
$271$	5.85054e9i	1.08472i	0.840146	+	0.542361i	$0.182469\pi$
$271$	5.85054e9i	1.08472i	−0.840146	+	0.542361i	$0.817531\pi$
$272$	0	0
$273$	−7.10111e8	−0.127843
$274$	0	0
$275$	− 3.34658e9i	− 0.585154i
$276$	0	0
$277$	−5.53729e9	−0.940543	−0.470271	−	0.882522i	$-0.655844\pi$
$277$	−5.53729e9	−0.940543	−0.470271	+	0.882522i	$0.655844\pi$
$278$	0	0
$279$	1.72280e9i	0.284327i
$280$	0	0
$281$	7.98151e8	0.128015	0.0640073	−	0.997949i	$-0.479612\pi$
$281$	7.98151e8	0.128015	0.0640073	+	0.997949i	$0.479612\pi$
$282$	0	0
$283$	− 1.02273e10i	− 1.59446i	−0.603673	−	0.797232i	$-0.706297\pi$
$283$	− 1.02273e10i	− 1.59446i	0.603673	−	0.797232i	$-0.293703\pi$
$284$	0	0
$285$	1.09621e10	1.66155
$286$	0	0
$287$	− 2.48747e8i	− 0.0366631i
$288$	0	0
$289$	8.70833e9	1.24837
$290$	0	0
$291$	3.33752e9i	0.465427i
$292$	0	0
$293$	−8.23842e9	−1.11782	−0.558912	−	0.829227i	$-0.688781\pi$
$293$	−8.23842e9	−1.11782	−0.558912	+	0.829227i	$0.688781\pi$
$294$	0	0
$295$	− 1.44374e10i	− 1.90634i
$296$	0	0
$297$	−4.37091e8	−0.0561754
$298$	0	0
$299$	1.14383e10i	1.43113i
$300$	0	0
$301$	1.90013e9	0.231481
$302$	0	0
$303$	− 2.45482e9i	− 0.291239i
$304$	0	0
$305$	2.38183e10	2.75240
$306$	0	0
$307$	− 1.41125e10i	− 1.58873i	−0.607441	−	0.794365i	$-0.707804\pi$
$307$	− 1.41125e10i	− 1.58873i	0.607441	−	0.794365i	$-0.292196\pi$
$308$	0	0
$309$	8.85313e9	0.971097
$310$	0	0
$311$	7.79656e9i	0.833415i	0.909041	+	0.416708i	$0.136816\pi$
$311$	7.79656e9i	0.833415i	−0.909041	+	0.416708i	$0.863184\pi$
$312$	0	0
$313$	−1.30513e10	−1.35981	−0.679903	−	0.733302i	$-0.737978\pi$
$313$	−1.30513e10	−1.35981	−0.679903	+	0.733302i	$0.737978\pi$
$314$	0	0
$315$	− 9.45611e8i	− 0.0960440i
$316$	0	0
$317$	7.25259e9	0.718218	0.359109	−	0.933296i	$-0.383081\pi$
$317$	7.25259e9	0.718218	0.359109	+	0.933296i	$0.383081\pi$
$318$	0	0
$319$	− 1.85491e9i	− 0.179127i
$320$	0	0
$321$	5.45106e9	0.513406
$322$	0	0
$323$	2.70969e10i	2.48948i
$324$	0	0
$325$	2.97933e10	2.67045
$326$	0	0
$327$	− 1.44990e9i	− 0.126808i
$328$	0	0
$329$	−2.84525e9	−0.242849
$330$	0	0
$331$	2.16575e10i	1.80425i	0.431472	+	0.902126i	$0.357994\pi$
$331$	2.16575e10i	1.80425i	−0.431472	+	0.902126i	$0.642006\pi$
$332$	0	0
$333$	−2.95732e9	−0.240504
$334$	0	0
$335$	− 1.28286e10i	− 1.01859i
$336$	0	0
$337$	1.26612e10	0.981648	0.490824	−	0.871259i	$-0.336696\pi$
$337$	1.26612e10	0.981648	0.490824	+	0.871259i	$0.336696\pi$
$338$	0	0
$339$	− 3.36552e9i	− 0.254832i
$340$	0	0
$341$	−3.36654e9	−0.248981
$342$	0	0
$343$	4.53793e9i	0.327855i
$344$	0	0
$345$	−1.52317e10	−1.07516
$346$	0	0
$347$	− 2.78815e10i	− 1.92309i	−0.274654	−	0.961543i	$-0.588563\pi$
$347$	− 2.78815e10i	− 1.92309i	0.274654	−	0.961543i	$-0.411437\pi$
$348$	0	0
$349$	4.51164e9	0.304111	0.152056	−	0.988372i	$-0.451411\pi$
$349$	4.51164e9	0.304111	0.152056	+	0.988372i	$0.451411\pi$
$350$	0	0
$351$	− 3.89125e9i	− 0.256366i
$352$	0	0
$353$	1.69463e9	0.109138	0.0545690	−	0.998510i	$-0.482622\pi$
$353$	1.69463e9	0.109138	0.0545690	+	0.998510i	$0.482622\pi$
$354$	0	0
$355$	− 1.46558e10i	− 0.922775i
$356$	0	0
$357$	2.33743e9	0.143902
$358$	0	0
$359$	1.64958e10i	0.993105i	0.868007	+	0.496553i	$0.165401\pi$
$359$	1.64958e10i	0.993105i	−0.868007	+	0.496553i	$0.834599\pi$
$360$	0	0
$361$	−2.98309e10	−1.75646
$362$	0	0
$363$	9.17045e9i	0.528158i
$364$	0	0
$365$	−3.02864e10	−1.70638
$366$	0	0
$367$	− 1.07251e10i	− 0.591204i	−0.955311	−	0.295602i	$-0.904480\pi$
$367$	− 1.07251e10i	− 0.591204i	0.955311	−	0.295602i	$-0.0955202\pi$
$368$	0	0
$369$	1.36308e9	0.0735216
$370$	0	0
$371$	2.84500e9i	0.150171i
$372$	0	0
$373$	−1.53529e10	−0.793150	−0.396575	−	0.918002i	$-0.629801\pi$
$373$	−1.53529e10	−0.793150	−0.396575	+	0.918002i	$0.629801\pi$
$374$	0	0
$375$	1.98831e10i	1.00545i
$376$	0	0
$377$	1.65136e10	0.817476
$378$	0	0
$379$	2.78963e10i	1.35204i	0.736884	+	0.676019i	$0.236296\pi$
$379$	2.78963e10i	1.35204i	−0.736884	+	0.676019i	$0.763704\pi$
$380$	0	0
$381$	−1.38366e10	−0.656641
$382$	0	0
$383$	1.07563e10i	0.499882i	0.968261	+	0.249941i	$0.0804112\pi$
$383$	1.07563e10i	0.499882i	−0.968261	+	0.249941i	$0.919589\pi$
$384$	0	0
$385$	1.84783e9	0.0841045
$386$	0	0
$387$	1.04123e10i	0.464196i
$388$	0	0
$389$	2.34980e10	1.02620	0.513101	−	0.858328i	$-0.328497\pi$
$389$	2.34980e10	1.02620	0.513101	+	0.858328i	$0.328497\pi$
$390$	0	0
$391$	− 3.76510e10i	− 1.61090i
$392$	0	0
$393$	3.31970e9	0.139165
$394$	0	0
$395$	3.81615e10i	1.56761i
$396$	0	0
$397$	−2.52199e10	−1.01527	−0.507635	−	0.861572i	$-0.669480\pi$
$397$	−2.52199e10	−1.01527	−0.507635	+	0.861572i	$0.669480\pi$
$398$	0	0
$399$	4.03831e9i	0.159334i
$400$	0	0
$401$	−4.63475e10	−1.79246	−0.896230	−	0.443590i	$-0.853704\pi$
$401$	−4.63475e10	−1.79246	−0.896230	+	0.443590i	$0.853704\pi$
$402$	0	0
$403$	− 2.99710e10i	− 1.13627i
$404$	0	0
$405$	5.18174e9	0.192600
$406$	0	0
$407$	− 5.77894e9i	− 0.210606i
$408$	0	0
$409$	1.08244e10	0.386822	0.193411	−	0.981118i	$-0.438045\pi$
$409$	1.08244e10	0.386822	0.193411	+	0.981118i	$0.438045\pi$
$410$	0	0
$411$	− 1.98583e10i	− 0.695945i
$412$	0	0
$413$	5.31857e9	0.182808
$414$	0	0
$415$	3.18140e9i	0.107257i
$416$	0	0
$417$	−2.31127e10	−0.764375
$418$	0	0
$419$	1.78708e10i	0.579813i	0.957055	+	0.289907i	$0.0936243\pi$
$419$	1.78708e10i	0.579813i	−0.957055	+	0.289907i	$0.906376\pi$
$420$	0	0
$421$	−1.67824e10	−0.534225	−0.267113	−	0.963665i	$-0.586070\pi$
$421$	−1.67824e10	−0.534225	−0.267113	+	0.963665i	$0.586070\pi$
$422$	0	0
$423$	− 1.55914e10i	− 0.486993i
$424$	0	0
$425$	−9.80690e10	−3.00591
$426$	0	0
$427$	8.77441e9i	0.263941i
$428$	0	0
$429$	7.60395e9	0.224497
$430$	0	0
$431$	3.93772e8i	0.0114113i	0.999984	+	0.00570565i	$0.00181618\pi$
$431$	3.93772e8i	0.0114113i	−0.999984	+	0.00570565i	$0.998184\pi$
$432$	0	0
$433$	1.54627e10	0.439879	0.219940	−	0.975514i	$-0.429414\pi$
$433$	1.54627e10	0.439879	0.219940	+	0.975514i	$0.429414\pi$
$434$	0	0
$435$	2.19901e10i	0.614143i
$436$	0	0
$437$	6.50484e10	1.78365
$438$	0	0
$439$	2.63295e10i	0.708898i	0.935075	+	0.354449i	$0.115332\pi$
$439$	2.63295e10i	0.708898i	−0.935075	+	0.354449i	$0.884668\pi$
$440$	0	0
$441$	−1.22593e10	−0.324123
$442$	0	0
$443$	− 6.73590e10i	− 1.74896i	−0.485059	−	0.874481i	$-0.661202\pi$
$443$	− 6.73590e10i	− 1.74896i	0.485059	−	0.874481i	$-0.338798\pi$
$444$	0	0
$445$	−1.03334e11	−2.63514
$446$	0	0
$447$	3.87037e10i	0.969442i
$448$	0	0
$449$	2.15836e10	0.531054	0.265527	−	0.964103i	$-0.414454\pi$
$449$	2.15836e10	0.531054	0.265527	+	0.964103i	$0.414454\pi$
$450$	0	0
$451$	2.66361e9i	0.0643819i
$452$	0	0
$453$	−3.21604e9	−0.0763709
$454$	0	0
$455$	1.64505e10i	0.383826i
$456$	0	0
$457$	5.99321e10	1.37403	0.687013	−	0.726645i	$-0.258922\pi$
$457$	5.99321e10	1.37403	0.687013	+	0.726645i	$0.258922\pi$
$458$	0	0
$459$	1.28086e10i	0.288571i
$460$	0	0
$461$	−4.65071e10	−1.02971	−0.514856	−	0.857277i	$-0.672154\pi$
$461$	−4.65071e10	−1.02971	−0.514856	+	0.857277i	$0.672154\pi$
$462$	0	0
$463$	6.97762e10i	1.51839i	0.650863	+	0.759195i	$0.274407\pi$
$463$	6.97762e10i	1.51839i	−0.650863	+	0.759195i	$0.725593\pi$
$464$	0	0
$465$	3.99106e10	0.853643
$466$	0	0
$467$	− 6.74656e10i	− 1.41845i	−0.704981	−	0.709226i	$-0.749044\pi$
$467$	− 6.74656e10i	− 1.41845i	0.704981	−	0.709226i	$-0.250956\pi$
$468$	0	0
$469$	4.72592e9	0.0976776
$470$	0	0
$471$	5.74600e9i	0.116757i
$472$	0	0
$473$	−2.03468e10	−0.406491
$474$	0	0
$475$	− 1.69431e11i	− 3.32826i
$476$	0	0
$477$	−1.55900e10	−0.301142
$478$	0	0
$479$	− 3.60583e10i	− 0.684956i	−0.939526	−	0.342478i	$-0.888734\pi$
$479$	− 3.60583e10i	− 0.684956i	0.939526	−	0.342478i	$-0.111266\pi$
$480$	0	0
$481$	5.14477e10	0.961138
$482$	0	0
$483$	− 5.61121e9i	− 0.103102i
$484$	0	0
$485$	7.73174e10	1.39737
$486$	0	0
$487$	1.05290e11i	1.87185i	0.352203	+	0.935924i	$0.385433\pi$
$487$	1.05290e11i	1.87185i	−0.352203	+	0.935924i	$0.614567\pi$
$488$	0	0
$489$	2.53275e10	0.442952
$490$	0	0
$491$	8.18993e9i	0.140914i	0.997515	+	0.0704570i	$0.0224458\pi$
$491$	8.18993e9i	0.140914i	−0.997515	+	0.0704570i	$0.977554\pi$
$492$	0	0
$493$	−5.43568e10	−0.920166
$494$	0	0
$495$	1.01257e10i	0.168657i
$496$	0	0
$497$	5.39904e9	0.0884893
$498$	0	0
$499$	3.95174e9i	0.0637361i	0.999492	+	0.0318681i	$0.0101456\pi$
$499$	3.95174e9i	0.0637361i	−0.999492	+	0.0318681i	$0.989854\pi$
$500$	0	0
$501$	2.20133e10	0.349409
$502$	0	0
$503$	− 9.67274e9i	− 0.151105i	−0.997142	−	0.0755523i	$-0.975928\pi$
$503$	− 9.67274e9i	− 0.151105i	0.997142	−	0.0755523i	$-0.0240720\pi$
$504$	0	0
$505$	−5.68687e10	−0.874396
$506$	0	0
$507$	2.95471e10i	0.447181i
$508$	0	0
$509$	−1.62059e10	−0.241436	−0.120718	−	0.992687i	$-0.538520\pi$
$509$	−1.62059e10	−0.241436	−0.120718	+	0.992687i	$0.538520\pi$
$510$	0	0
$511$	− 1.11572e10i	− 0.163633i
$512$	0	0
$513$	−2.21291e10	−0.319517
$514$	0	0
$515$	− 2.05093e11i	− 2.91556i
$516$	0	0
$517$	3.04673e10	0.426453
$518$	0	0
$519$	7.72205e10i	1.06430i
$520$	0	0
$521$	−8.32977e10	−1.13053	−0.565265	−	0.824909i	$-0.691226\pi$
$521$	−8.32977e10	−1.13053	−0.565265	+	0.824909i	$0.691226\pi$
$522$	0	0
$523$	− 3.54782e10i	− 0.474192i	−0.971486	−	0.237096i	$-0.923804\pi$
$523$	− 3.54782e10i	− 0.474192i	0.971486	−	0.237096i	$-0.0761956\pi$
$524$	0	0
$525$	−1.46154e10	−0.192387
$526$	0	0
$527$	9.86541e10i	1.27901i
$528$	0	0
$529$	−1.20734e10	−0.154172
$530$	0	0
$531$	2.91446e10i	0.366590i
$532$	0	0
$533$	−2.37131e10	−0.293819
$534$	0	0
$535$	− 1.26280e11i	− 1.54141i
$536$	0	0
$537$	−6.10877e10	−0.734610
$538$	0	0
$539$	− 2.39560e10i	− 0.283831i
$540$	0	0
$541$	−9.54293e10	−1.11402	−0.557009	−	0.830506i	$-0.688051\pi$
$541$	−9.54293e10	−1.11402	−0.557009	+	0.830506i	$0.688051\pi$
$542$	0	0
$543$	5.33470e10i	0.613636i
$544$	0	0
$545$	−3.35886e10	−0.380721
$546$	0	0
$547$	7.43876e10i	0.830904i	0.909615	+	0.415452i	$0.136377\pi$
$547$	7.43876e10i	0.830904i	−0.909615	+	0.415452i	$0.863623\pi$
$548$	0	0
$549$	−4.80818e10	−0.529288
$550$	0	0
$551$	− 9.39105e10i	− 1.01884i
$552$	0	0
$553$	−1.40583e10	−0.150325
$554$	0	0
$555$	6.85097e10i	0.722072i
$556$	0	0
$557$	−1.83358e11	−1.90493	−0.952463	−	0.304653i	$-0.901459\pi$
$557$	−1.83358e11	−1.90493	−0.952463	+	0.304653i	$0.901459\pi$
$558$	0	0
$559$	− 1.81140e11i	− 1.85509i
$560$	0	0
$561$	−2.50295e10	−0.252698
$562$	0	0
$563$	8.27020e10i	0.823156i	0.911374	+	0.411578i	$0.135022\pi$
$563$	8.27020e10i	0.823156i	−0.911374	+	0.411578i	$0.864978\pi$
$564$	0	0
$565$	−7.79662e10	−0.765090
$566$	0	0
$567$	1.90890e9i	0.0184693i
$568$	0	0
$569$	−1.33574e11	−1.27430	−0.637150	−	0.770740i	$-0.719887\pi$
$569$	−1.33574e11	−1.27430	−0.637150	+	0.770740i	$0.719887\pi$
$570$	0	0
$571$	− 1.68600e11i	− 1.58603i	−0.609200	−	0.793016i	$-0.708510\pi$
$571$	− 1.68600e11i	− 1.58603i	0.609200	−	0.793016i	$-0.291490\pi$
$572$	0	0
$573$	1.60728e10	0.149098
$574$	0	0
$575$	2.35423e11i	2.15366i
$576$	0	0
$577$	1.81240e11	1.63513	0.817563	−	0.575839i	$-0.195325\pi$
$577$	1.81240e11	1.63513	0.817563	+	0.575839i	$0.195325\pi$
$578$	0	0
$579$	− 1.36064e10i	− 0.121068i
$580$	0	0
$581$	−1.17199e9	−0.0102854
$582$	0	0
$583$	− 3.04646e10i	− 0.263706i
$584$	0	0
$585$	−9.01453e10	−0.769697
$586$	0	0
$587$	1.05306e11i	0.886953i	0.896286	+	0.443477i	$0.146255\pi$
$587$	1.05306e11i	0.886953i	−0.896286	+	0.443477i	$0.853745\pi$
$588$	0	0
$589$	−1.70442e11	−1.41617
$590$	0	0
$591$	5.26899e10i	0.431895i
$592$	0	0
$593$	−1.07051e10	−0.0865706	−0.0432853	−	0.999063i	$-0.513782\pi$
$593$	−1.07051e10	−0.0865706	−0.0432853	+	0.999063i	$0.513782\pi$
$594$	0	0
$595$	− 5.41493e10i	− 0.432041i
$596$	0	0
$597$	4.75873e10	0.374622
$598$	0	0
$599$	1.44979e11i	1.12615i	0.826406	+	0.563075i	$0.190382\pi$
$599$	1.44979e11i	1.12615i	−0.826406	+	0.563075i	$0.809618\pi$
$600$	0	0
$601$	−2.11181e11	−1.61866	−0.809331	−	0.587353i	$-0.800170\pi$
$601$	−2.11181e11	−1.61866	−0.809331	+	0.587353i	$0.800170\pi$
$602$	0	0
$603$	2.58970e10i	0.195876i
$604$	0	0
$605$	2.12444e11	1.58571
$606$	0	0
$607$	− 1.11663e11i	− 0.822533i	−0.911515	−	0.411267i	$-0.865086\pi$
$607$	− 1.11663e11i	− 0.822533i	0.911515	−	0.411267i	$-0.134914\pi$
$608$	0	0
$609$	−8.10091e9	−0.0588932
$610$	0	0
$611$	2.71239e11i	1.94620i
$612$	0	0
$613$	−1.70169e11	−1.20514	−0.602572	−	0.798065i	$-0.705857\pi$
$613$	−1.70169e11	−1.20514	−0.602572	+	0.798065i	$0.705857\pi$
$614$	0	0
$615$	− 3.15772e10i	− 0.220736i
$616$	0	0
$617$	−1.53977e11	−1.06247	−0.531234	−	0.847225i	$-0.678271\pi$
$617$	−1.53977e11	−1.06247	−0.531234	+	0.847225i	$0.678271\pi$
$618$	0	0
$619$	− 2.66594e11i	− 1.81588i	−0.419100	−	0.907940i	$-0.637654\pi$
$619$	− 2.66594e11i	− 1.81588i	0.419100	−	0.907940i	$-0.362346\pi$
$620$	0	0
$621$	3.07482e10	0.206754
$622$	0	0
$623$	− 3.80672e10i	− 0.252696i
$624$	0	0
$625$	1.54727e11	1.01402
$626$	0	0
$627$	− 4.32427e10i	− 0.279797i
$628$	0	0
$629$	−1.69348e11	−1.08187
$630$	0	0
$631$	1.20683e11i	0.761252i	0.924729	+	0.380626i	$0.124291\pi$
$631$	1.20683e11i	0.761252i	−0.924729	+	0.380626i	$0.875709\pi$
$632$	0	0
$633$	−3.44455e10	−0.214545
$634$	0	0
$635$	3.20540e11i	1.97146i
$636$	0	0
$637$	2.13271e11	1.29531
$638$	0	0
$639$	2.95856e10i	0.177450i
$640$	0	0
$641$	7.89134e10	0.467432	0.233716	−	0.972305i	$-0.424911\pi$
$641$	7.89134e10	0.467432	0.233716	+	0.972305i	$0.424911\pi$
$642$	0	0
$643$	7.30055e9i	0.0427083i	0.999772	+	0.0213541i	$0.00679775\pi$
$643$	7.30055e9i	0.0427083i	−0.999772	+	0.0213541i	$0.993202\pi$
$644$	0	0
$645$	2.41212e11	1.39367
$646$	0	0
$647$	− 1.49365e10i	− 0.0852378i	−0.999091	−	0.0426189i	$-0.986430\pi$
$647$	− 1.49365e10i	− 0.0852378i	0.999091	−	0.0426189i	$-0.0135701\pi$
$648$	0	0
$649$	−5.69519e10	−0.321018
$650$	0	0
$651$	1.47026e10i	0.0818599i
$652$	0	0
$653$	1.40142e11	0.770754	0.385377	−	0.922759i	$-0.374071\pi$
$653$	1.40142e11	0.770754	0.385377	+	0.922759i	$0.374071\pi$
$654$	0	0
$655$	− 7.69047e10i	− 0.417819i
$656$	0	0
$657$	6.11390e10	0.328138
$658$	0	0
$659$	− 8.16253e10i	− 0.432796i	−0.976305	−	0.216398i	$-0.930569\pi$
$659$	− 8.16253e10i	− 0.432796i	0.976305	−	0.216398i	$-0.0694309\pi$
$660$	0	0
$661$	1.24459e11	0.651958	0.325979	−	0.945377i	$-0.394306\pi$
$661$	1.24459e11	0.651958	0.325979	+	0.945377i	$0.394306\pi$
$662$	0	0
$663$	− 2.22828e11i	− 1.15323i
$664$	0	0
$665$	9.35521e10	0.478373
$666$	0	0
$667$	1.30488e11i	0.659276i
$668$	0	0
$669$	8.23724e10	0.411223
$670$	0	0
$671$	− 9.39574e10i	− 0.463491i
$672$	0	0
$673$	1.14330e11	0.557315	0.278657	−	0.960391i	$-0.410111\pi$
$673$	1.14330e11	0.557315	0.278657	+	0.960391i	$0.410111\pi$
$674$	0	0
$675$	− 8.00895e10i	− 0.385798i
$676$	0	0
$677$	1.84315e11	0.877418	0.438709	−	0.898629i	$-0.355436\pi$
$677$	1.84315e11	0.877418	0.438709	+	0.898629i	$0.355436\pi$
$678$	0	0
$679$	2.84829e10i	0.134000i
$680$	0	0
$681$	−1.28007e11	−0.595176
$682$	0	0
$683$	− 1.75497e11i	− 0.806469i	−0.915097	−	0.403234i	$-0.867886\pi$
$683$	− 1.75497e11i	− 0.806469i	0.915097	−	0.403234i	$-0.132114\pi$
$684$	0	0
$685$	−4.60041e11	−2.08946
$686$	0	0
$687$	1.61378e11i	0.724464i
$688$	0	0
$689$	2.71214e11	1.20347
$690$	0	0
$691$	− 1.35838e11i	− 0.595813i	−0.954595	−	0.297907i	$-0.903712\pi$
$691$	− 1.35838e11i	− 0.595813i	0.954595	−	0.297907i	$-0.0962884\pi$
$692$	0	0
$693$	−3.73020e9	−0.0161733
$694$	0	0
$695$	5.35432e11i	2.29491i
$696$	0	0
$697$	7.80551e10	0.330727
$698$	0	0
$699$	1.03861e10i	0.0435053i
$700$	0	0
$701$	−1.54296e11	−0.638973	−0.319487	−	0.947591i	$-0.603510\pi$
$701$	−1.54296e11	−0.638973	−0.319487	+	0.947591i	$0.603510\pi$
$702$	0	0
$703$	− 2.92577e11i	− 1.19789i
$704$	0	0
$705$	−3.61192e11	−1.46211
$706$	0	0
$707$	− 2.09498e10i	− 0.0838500i
$708$	0	0
$709$	3.80878e10	0.150731	0.0753653	−	0.997156i	$-0.475988\pi$
$709$	3.80878e10	0.150731	0.0753653	+	0.997156i	$0.475988\pi$
$710$	0	0
$711$	− 7.70364e10i	− 0.301452i
$712$	0	0
$713$	2.36828e11	0.916377
$714$	0	0
$715$	− 1.76154e11i	− 0.674014i
$716$	0	0
$717$	1.70816e11	0.646326
$718$	0	0
$719$	3.67632e11i	1.37562i	0.725893	+	0.687808i	$0.241427\pi$
$719$	3.67632e11i	1.37562i	−0.725893	+	0.687808i	$0.758573\pi$
$720$	0	0
$721$	7.55540e10	0.279587
$722$	0	0
$723$	1.15321e11i	0.422041i
$724$	0	0
$725$	3.39881e11	1.23020
$726$	0	0
$727$	− 2.37076e11i	− 0.848691i	−0.905500	−	0.424345i	$-0.860504\pi$
$727$	− 2.37076e11i	− 0.848691i	0.905500	−	0.424345i	$-0.139496\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	5.96247e11i	2.08813i
$732$	0	0
$733$	2.55021e11	0.883405	0.441702	−	0.897162i	$-0.354375\pi$
$733$	2.55021e11	0.883405	0.441702	+	0.897162i	$0.354375\pi$
$734$	0	0
$735$	2.84000e11i	0.973126i
$736$	0	0
$737$	−5.06057e10	−0.171526
$738$	0	0
$739$	− 2.95124e11i	− 0.989524i	−0.869029	−	0.494762i	$-0.835255\pi$
$739$	− 2.95124e11i	− 0.989524i	0.869029	−	0.494762i	$-0.164745\pi$
$740$	0	0
$741$	3.84973e11	1.27690
$742$	0	0
$743$	− 6.00101e10i	− 0.196911i	−0.995141	−	0.0984553i	$-0.968610\pi$
$743$	− 6.00101e10i	− 0.196911i	0.995141	−	0.0984553i	$-0.0313901\pi$
$744$	0	0
$745$	8.96615e11	2.91059
$746$	0	0
$747$	− 6.42228e9i	− 0.0206256i
$748$	0	0
$749$	4.65202e10	0.147814
$750$	0	0
$751$	− 1.34479e11i	− 0.422761i	−0.977404	−	0.211381i	$-0.932204\pi$
$751$	− 1.34479e11i	− 0.422761i	0.977404	−	0.211381i	$-0.0677960\pi$
$752$	0	0
$753$	−2.20062e11	−0.684487
$754$	0	0
$755$	7.45032e10i	0.229291i
$756$	0	0
$757$	2.37299e11	0.722625	0.361313	−	0.932445i	$-0.382329\pi$
$757$	2.37299e11	0.722625	0.361313	+	0.932445i	$0.382329\pi$
$758$	0	0
$759$	6.00855e10i	0.181052i
$760$	0	0
$761$	2.94377e11	0.877739	0.438870	−	0.898551i	$-0.355379\pi$
$761$	2.94377e11	0.877739	0.438870	+	0.898551i	$0.355379\pi$
$762$	0	0
$763$	− 1.23737e10i	− 0.0365091i
$764$	0	0
$765$	2.96727e11	0.866385
$766$	0	0
$767$	− 5.07021e11i	− 1.46502i
$768$	0	0
$769$	2.86371e11	0.818886	0.409443	−	0.912336i	$-0.365723\pi$
$769$	2.86371e11	0.818886	0.409443	+	0.912336i	$0.365723\pi$
$770$	0	0
$771$	3.09164e11i	0.874927i
$772$	0	0
$773$	2.16776e11	0.607146	0.303573	−	0.952808i	$-0.401820\pi$
$773$	2.16776e11	0.607146	0.303573	+	0.952808i	$0.401820\pi$
$774$	0	0
$775$	− 6.16862e11i	− 1.70994i
$776$	0	0
$777$	−2.52383e10	−0.0692429
$778$	0	0
$779$	1.34853e11i	0.366194i
$780$	0	0
$781$	−5.78135e10	−0.155391
$782$	0	0
$783$	− 4.43913e10i	− 0.118100i
$784$	0	0
$785$	1.33113e11	0.350543
$786$	0	0
$787$	3.63459e11i	0.947451i	0.880673	+	0.473725i	$0.157091\pi$
$787$	3.63459e11i	0.947451i	−0.880673	+	0.473725i	$0.842909\pi$
$788$	0	0
$789$	−4.09454e11	−1.05657
$790$	0	0
$791$	− 2.87219e10i	− 0.0733681i
$792$	0	0
$793$	8.36467e11	2.11522
$794$	0	0
$795$	3.61159e11i	0.904129i
$796$	0	0
$797$	−5.38732e11	−1.33518	−0.667590	−	0.744529i	$-0.732674\pi$
$797$	−5.38732e11	−1.33518	−0.667590	+	0.744529i	$0.732674\pi$
$798$	0	0
$799$	− 8.92822e11i	− 2.19067i
$800$	0	0
$801$	2.08600e11	0.506739
$802$	0	0
$803$	1.19472e11i	0.287346i
$804$	0	0
$805$	−1.29990e11	−0.309547
$806$	0	0
$807$	2.15779e11i	0.508764i
$808$	0	0
$809$	−5.20326e11	−1.21473	−0.607367	−	0.794422i	$-0.707774\pi$
$809$	−5.20326e11	−1.21473	−0.607367	+	0.794422i	$0.707774\pi$
$810$	0	0
$811$	− 4.90908e11i	− 1.13479i	−0.823445	−	0.567397i	$-0.807951\pi$
$811$	− 4.90908e11i	− 1.13479i	0.823445	−	0.567397i	$-0.192049\pi$
$812$	0	0
$813$	−2.73602e11	−0.626265
$814$	0	0
$815$	− 5.86740e11i	− 1.32989i
$816$	0	0
$817$	−1.03012e12	−2.31206
$818$	0	0
$819$	− 3.32086e10i	− 0.0738099i
$820$	0	0
$821$	1.07712e11	0.237079	0.118539	−	0.992949i	$-0.462179\pi$
$821$	1.07712e11	0.237079	0.118539	+	0.992949i	$0.462179\pi$
$822$	0	0
$823$	− 6.38714e11i	− 1.39222i	−0.717936	−	0.696109i	$-0.754913\pi$
$823$	− 6.38714e11i	− 1.39222i	0.717936	−	0.696109i	$-0.245087\pi$
$824$	0	0
$825$	1.56504e11	0.337839
$826$	0	0
$827$	4.81782e11i	1.02998i	0.857196	+	0.514990i	$0.172204\pi$
$827$	4.81782e11i	1.02998i	−0.857196	+	0.514990i	$0.827796\pi$
$828$	0	0
$829$	−5.77091e11	−1.22187	−0.610937	−	0.791679i	$-0.709207\pi$
$829$	−5.77091e11	−1.22187	−0.610937	+	0.791679i	$0.709207\pi$
$830$	0	0
$831$	− 2.58954e11i	− 0.543023i
$832$	0	0
$833$	−7.02013e11	−1.45803
$834$	0	0
$835$	− 5.09963e11i	− 1.04904i
$836$	0	0
$837$	−8.05673e10	−0.164156
$838$	0	0
$839$	1.97382e11i	0.398346i	0.979964	+	0.199173i	$0.0638255\pi$
$839$	1.97382e11i	0.398346i	−0.979964	+	0.199173i	$0.936174\pi$
$840$	0	0
$841$	−3.11860e11	−0.623414
$842$	0	0
$843$	3.73258e10i	0.0739093i
$844$	0	0
$845$	6.84493e11	1.34259
$846$	0	0
$847$	7.82621e10i	0.152061i
$848$	0	0
$849$	4.78283e11	0.920564
$850$	0	0
$851$	4.06533e11i	0.775136i
$852$	0	0
$853$	−6.55773e11	−1.23867	−0.619337	−	0.785125i	$-0.712599\pi$
$853$	−6.55773e11	−1.23867	−0.619337	+	0.785125i	$0.712599\pi$
$854$	0	0
$855$	5.12645e11i	0.959296i
$856$	0	0
$857$	−2.54255e11	−0.471353	−0.235676	−	0.971832i	$-0.575731\pi$
$857$	−2.54255e11	−0.471353	−0.235676	+	0.971832i	$0.575731\pi$
$858$	0	0
$859$	9.47136e11i	1.73956i	0.493439	+	0.869781i	$0.335740\pi$
$859$	9.47136e11i	1.73956i	−0.493439	+	0.869781i	$0.664260\pi$
$860$	0	0
$861$	1.16327e10	0.0211675
$862$	0	0
$863$	− 1.75920e11i	− 0.317155i	−0.987347	−	0.158577i	$-0.949309\pi$
$863$	− 1.75920e11i	− 0.317155i	0.987347	−	0.158577i	$-0.0506907\pi$
$864$	0	0
$865$	1.78890e12	3.19538
$866$	0	0
$867$	4.07248e11i	0.720747i
$868$	0	0
$869$	1.50538e11	0.263977
$870$	0	0
$871$	− 4.50523e11i	− 0.782789i
$872$	0	0
$873$	−1.56080e11	−0.268714
$874$	0	0
$875$	1.69686e11i	0.289476i
$876$	0	0
$877$	1.49628e11	0.252938	0.126469	−	0.991971i	$-0.459636\pi$
$877$	1.49628e11	0.252938	0.126469	+	0.991971i	$0.459636\pi$
$878$	0	0
$879$	− 3.85273e11i	− 0.645376i
$880$	0	0
$881$	−1.06276e11	−0.176413	−0.0882067	−	0.996102i	$-0.528114\pi$
$881$	−1.06276e11	−0.176413	−0.0882067	+	0.996102i	$0.528114\pi$
$882$	0	0
$883$	5.79271e11i	0.952881i	0.879207	+	0.476441i	$0.158073\pi$
$883$	5.79271e11i	0.952881i	−0.879207	+	0.476441i	$0.841927\pi$
$884$	0	0
$885$	6.75169e11	1.10063
$886$	0	0
$887$	− 8.55030e11i	− 1.38130i	−0.723191	−	0.690648i	$-0.757325\pi$
$887$	− 8.55030e11i	− 1.38130i	0.723191	−	0.690648i	$-0.242675\pi$
$888$	0	0
$889$	−1.18083e11	−0.189052
$890$	0	0
$891$	− 2.04407e10i	− 0.0324329i
$892$	0	0
$893$	1.54250e12	2.42560
$894$	0	0
$895$	1.41517e12i	2.20554i
$896$	0	0
$897$	−5.34918e11	−0.826262
$898$	0	0
$899$	− 3.41908e11i	− 0.523445i
$900$	0	0
$901$	−8.92742e11	−1.35465
$902$	0	0
$903$	8.88601e10i	0.133646i
$904$	0	0
$905$	1.23584e12	1.84234
$906$	0	0
$907$	− 7.33664e11i	− 1.08410i	−0.840347	−	0.542049i	$-0.817649\pi$
$907$	− 7.33664e11i	− 1.08410i	0.840347	−	0.542049i	$-0.182351\pi$
$908$	0	0
$909$	1.14801e11	0.168147
$910$	0	0
$911$	2.49857e9i	0.00362760i	0.999998	+	0.00181380i	$0.000577350\pi$
$911$	2.49857e9i	0.00362760i	−0.999998	+	0.00181380i	$0.999423\pi$
$912$	0	0
$913$	1.25499e10	0.0180616
$914$	0	0
$915$	1.11387e12i	1.58910i
$916$	0	0
$917$	2.83309e10	0.0400666
$918$	0	0
$919$	1.45830e11i	0.204449i	0.994761	+	0.102224i	$0.0325959\pi$
$919$	1.45830e11i	0.204449i	−0.994761	+	0.102224i	$0.967404\pi$
$920$	0	0
$921$	6.59975e11	0.917253
$922$	0	0
$923$	− 5.14692e11i	− 0.709154i
$924$	0	0
$925$	1.05889e12	1.44639
$926$	0	0
$927$	4.14020e11i	0.560663i
$928$	0	0
$929$	−1.37384e12	−1.84448	−0.922239	−	0.386621i	$-0.873642\pi$
$929$	−1.37384e12	−1.84448	−0.922239	+	0.386621i	$0.873642\pi$
$930$	0	0
$931$	− 1.21285e12i	− 1.61438i
$932$	0	0
$933$	−3.64609e11	−0.481173
$934$	0	0
$935$	5.79838e11i	0.758682i
$936$	0	0
$937$	1.31669e12	1.70814	0.854072	−	0.520156i	$-0.174126\pi$
$937$	1.31669e12	1.70814	0.854072	+	0.520156i	$0.174126\pi$
$938$	0	0
$939$	− 6.10350e11i	− 0.785085i
$940$	0	0
$941$	1.36533e12	1.74132	0.870661	−	0.491883i	$-0.163691\pi$
$941$	1.36533e12	1.74132	0.870661	+	0.491883i	$0.163691\pi$
$942$	0	0
$943$	− 1.87378e11i	− 0.236958i
$944$	0	0
$945$	4.42218e10	0.0554510
$946$	0	0
$947$	− 8.58935e11i	− 1.06797i	−0.845493	−	0.533987i	$-0.820693\pi$
$947$	− 8.58935e11i	− 1.06797i	0.845493	−	0.533987i	$-0.179307\pi$
$948$	0	0
$949$	−1.06362e12	−1.31136
$950$	0	0
$951$	3.39170e11i	0.414663i
$952$	0	0
$953$	−5.21038e11	−0.631681	−0.315841	−	0.948812i	$-0.602286\pi$
$953$	−5.21038e11	−0.631681	−0.315841	+	0.948812i	$0.602286\pi$
$954$	0	0
$955$	− 3.72344e11i	− 0.447642i
$956$	0	0
$957$	8.67455e10	0.103419
$958$	0	0
$959$	− 1.69474e11i	− 0.200368i
$960$	0	0
$961$	2.32349e11	0.272426
$962$	0	0
$963$	2.54921e11i	0.296415i
$964$	0	0
$965$	−3.15207e11	−0.363485
$966$	0	0
$967$	− 1.44761e12i	− 1.65557i	−0.561048	−	0.827783i	$-0.689602\pi$
$967$	− 1.44761e12i	− 1.65557i	0.561048	−	0.827783i	$-0.310398\pi$
$968$	0	0
$969$	−1.26720e12	−1.43730
$970$	0	0
$971$	− 5.90410e11i	− 0.664166i	−0.943250	−	0.332083i	$-0.892249\pi$
$971$	− 5.90410e11i	− 0.664166i	0.943250	−	0.332083i	$-0.107751\pi$
$972$	0	0
$973$	−1.97248e11	−0.220070
$974$	0	0
$975$	1.39329e12i	1.54179i
$976$	0	0
$977$	1.79731e11	0.197262	0.0986310	−	0.995124i	$-0.468554\pi$
$977$	1.79731e11	0.197262	0.0986310	+	0.995124i	$0.468554\pi$
$978$	0	0
$979$	4.07628e11i	0.443745i
$980$	0	0
$981$	6.78052e10	0.0732128
$982$	0	0
$983$	1.47368e12i	1.57830i	0.614201	+	0.789150i	$0.289479\pi$
$983$	1.47368e12i	1.57830i	−0.614201	+	0.789150i	$0.710521\pi$
$984$	0	0
$985$	1.22062e12	1.29669
$986$	0	0
$987$	− 1.33059e11i	− 0.140209i
$988$	0	0
$989$	1.43134e12	1.49609
$990$	0	0
$991$	1.58331e12i	1.64162i	0.571204	+	0.820808i	$0.306476\pi$
$991$	1.58331e12i	1.64162i	−0.571204	+	0.820808i	$0.693524\pi$
$992$	0	0
$993$	−1.01282e12	−1.04169
$994$	0	0
$995$	− 1.10241e12i	− 1.12474i
$996$	0	0
$997$	5.14404e11	0.520624	0.260312	−	0.965525i	$-0.416175\pi$
$997$	5.14404e11	0.520624	0.260312	+	0.965525i	$0.416175\pi$
$998$	0	0
$999$	− 1.38300e11i	− 0.138855i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.a.127.20	yes	32
4.3	odd	2	inner	384.9.g.a.127.19	✓	32
8.3	odd	2		384.9.g.b.127.14	yes	32
8.5	even	2		384.9.g.b.127.13	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.19	✓	32	4.3	odd	2	inner
384.9.g.a.127.20	yes	32	1.1	even	1	trivial
384.9.g.b.127.13	yes	32	8.5	even	2
384.9.g.b.127.14	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)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} +1083.37 q^{5} +399.103i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} +1083.37 q^{5} +399.103i q^{7} -2187.00 q^{9} -4273.65i q^{11} +38046.6 q^{13} +50664.4i q^{15} -125236. q^{17} -216366. i q^{19} -18664.2 q^{21} +300640. i q^{23} +783073. q^{25} -102276. i q^{27} +434034. q^{29} -787745. i q^{31} +199859. q^{33} +432378. i q^{35} +1.35223e6 q^{37} +1.77927e6i q^{39} -623263. q^{41} -4.76099e6i q^{43} -2.36934e6 q^{45} +7.12911e6i q^{47} +5.60552e6 q^{49} -5.85671e6i q^{51} +7.12847e6 q^{53} -4.62995e6i q^{55} +1.01185e7 q^{57} -1.33263e7i q^{59} +2.19853e7 q^{61} -872839. i q^{63} +4.12187e7 q^{65} -1.18413e7i q^{67} -1.40595e7 q^{69} -1.35279e7i q^{71} -2.79556e7 q^{73} +3.66207e7i q^{75} +1.70563e6 q^{77} +3.52247e7i q^{79} +4.78297e6 q^{81} +2.93657e6i q^{83} -1.35677e8 q^{85} +2.02978e7i q^{87} -9.53819e7 q^{89} +1.51845e7i q^{91} +3.68392e7 q^{93} -2.34406e8i q^{95} +7.13673e7 q^{97} +9.34646e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 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

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