LMFDB - Embedded newform 384.9.g.b.127.26

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			127.26
Character	$\chi$	$=$	384.127
Dual form			384.9.g.b.127.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} -704.478 q^{5} -2008.10i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} -704.478 q^{5} -2008.10i q^{7} -2187.00 q^{9} +4084.20i q^{11} +27820.2 q^{13} -32945.2i q^{15} +4582.51 q^{17} -132014. i q^{19} +93909.3 q^{21} +497285. i q^{23} +105664. q^{25} -102276. i q^{27} -89980.3 q^{29} +73293.8i q^{31} -190999. q^{33} +1.41466e6i q^{35} -1.83442e6 q^{37} +1.30102e6i q^{39} +4.67667e6 q^{41} +430415. i q^{43} +1.54069e6 q^{45} +8.22030e6i q^{47} +1.73235e6 q^{49} +214303. i q^{51} -3.66285e6 q^{53} -2.87723e6i q^{55} +6.17367e6 q^{57} -7.42003e6i q^{59} -1.55610e7 q^{61} +4.39170e6i q^{63} -1.95988e7 q^{65} +1.00468e7i q^{67} -2.32557e7 q^{69} -5.70613e6i q^{71} -3.27931e7 q^{73} +4.94144e6i q^{75} +8.20146e6 q^{77} -4.74652e6i q^{79} +4.78297e6 q^{81} -1.31403e7i q^{83} -3.22828e6 q^{85} -4.20796e6i q^{87} +6.90797e7 q^{89} -5.58657e7i q^{91} -3.42761e6 q^{93} +9.30008e7i q^{95} -1.35124e8 q^{97} -8.93215e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−704.478	−1.12717	−0.563583	−	0.826060i	$-0.690577\pi$
$5$	−704.478	−1.12717	−0.563583	+	0.826060i	$0.690577\pi$
$6$	0	0
$7$	− 2008.10i	− 0.836358i	−0.908365	−	0.418179i	$-0.862669\pi$
$7$	− 2008.10i	− 0.836358i	0.908365	−	0.418179i	$-0.137331\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	4084.20i	0.278956i	0.990225	+	0.139478i	$0.0445425\pi$
$11$	4084.20i	0.278956i	−0.990225	+	0.139478i	$0.955457\pi$
$12$	0	0
$13$	27820.2	0.974064	0.487032	−	0.873384i	$-0.338080\pi$
$13$	27820.2	0.974064	0.487032	+	0.873384i	$0.338080\pi$
$14$	0	0
$15$	− 32945.2i	− 0.650769i
$16$	0	0
$17$	4582.51	0.0548665	0.0274333	−	0.999624i	$-0.491267\pi$
$17$	4582.51	0.0548665	0.0274333	+	0.999624i	$0.491267\pi$
$18$	0	0
$19$	− 132014.i	− 1.01299i	−0.862243	−	0.506495i	$-0.830941\pi$
$19$	− 132014.i	− 1.01299i	0.862243	−	0.506495i	$-0.169059\pi$
$20$	0	0
$21$	93909.3	0.482871
$22$	0	0
$23$	497285.i	1.77703i	0.458850	+	0.888514i	$0.348261\pi$
$23$	497285.i	1.77703i	−0.458850	+	0.888514i	$0.651739\pi$
$24$	0	0
$25$	105664.	0.270501
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	−89980.3	−0.127220	−0.0636100	−	0.997975i	$-0.520261\pi$
$29$	−89980.3	−0.127220	−0.0636100	+	0.997975i	$0.520261\pi$
$30$	0	0
$31$	73293.8i	0.0793634i	0.999212	+	0.0396817i	$0.0126344\pi$
$31$	73293.8i	0.0793634i	−0.999212	+	0.0396817i	$0.987366\pi$
$32$	0	0
$33$	−190999.	−0.161056
$34$	0	0
$35$	1.41466e6i	0.942713i
$36$	0	0
$37$	−1.83442e6	−0.978797	−0.489399	−	0.872060i	$-0.662784\pi$
$37$	−1.83442e6	−0.978797	−0.489399	+	0.872060i	$0.662784\pi$
$38$	0	0
$39$	1.30102e6i	0.562376i
$40$	0	0
$41$	4.67667e6	1.65501	0.827507	−	0.561455i	$-0.189758\pi$
$41$	4.67667e6	1.65501	0.827507	+	0.561455i	$0.189758\pi$
$42$	0	0
$43$	430415.i	0.125897i	0.998017	+	0.0629483i	$0.0200503\pi$
$43$	430415.i	0.125897i	−0.998017	+	0.0629483i	$0.979950\pi$
$44$	0	0
$45$	1.54069e6	0.375722
$46$	0	0
$47$	8.22030e6i	1.68460i	0.539011	+	0.842299i	$0.318798\pi$
$47$	8.22030e6i	1.68460i	−0.539011	+	0.842299i	$0.681202\pi$
$48$	0	0
$49$	1.73235e6	0.300505
$50$	0	0
$51$	214303.i	0.0316772i
$52$	0	0
$53$	−3.66285e6	−0.464212	−0.232106	−	0.972691i	$-0.574562\pi$
$53$	−3.66285e6	−0.464212	−0.232106	+	0.972691i	$0.574562\pi$
$54$	0	0
$55$	− 2.87723e6i	− 0.314430i
$56$	0	0
$57$	6.17367e6	0.584849
$58$	0	0
$59$	− 7.42003e6i	− 0.612347i	−0.951976	−	0.306174i	$-0.900951\pi$
$59$	− 7.42003e6i	− 0.612347i	0.951976	−	0.306174i	$-0.0990488\pi$
$60$	0	0
$61$	−1.55610e7	−1.12387	−0.561937	−	0.827180i	$-0.689944\pi$
$61$	−1.55610e7	−1.12387	−0.561937	+	0.827180i	$0.689944\pi$
$62$	0	0
$63$	4.39170e6i	0.278786i
$64$	0	0
$65$	−1.95988e7	−1.09793
$66$	0	0
$67$	1.00468e7i	0.498571i	0.968430	+	0.249286i	$0.0801958\pi$
$67$	1.00468e7i	0.498571i	−0.968430	+	0.249286i	$0.919804\pi$
$68$	0	0
$69$	−2.32557e7	−1.02597
$70$	0	0
$71$	− 5.70613e6i	− 0.224548i	−0.993677	−	0.112274i	$-0.964187\pi$
$71$	− 5.70613e6i	− 0.224548i	0.993677	−	0.112274i	$-0.0358134\pi$
$72$	0	0
$73$	−3.27931e7	−1.15476	−0.577379	−	0.816476i	$-0.695924\pi$
$73$	−3.27931e7	−1.15476	−0.577379	+	0.816476i	$0.695924\pi$
$74$	0	0
$75$	4.94144e6i	0.156174i
$76$	0	0
$77$	8.20146e6	0.233307
$78$	0	0
$79$	− 4.74652e6i	− 0.121861i	−0.998142	−	0.0609307i	$-0.980593\pi$
$79$	− 4.74652e6i	− 0.121861i	0.998142	−	0.0609307i	$-0.0194069\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 1.31403e7i	− 0.276882i	−0.990371	−	0.138441i	$-0.955791\pi$
$83$	− 1.31403e7i	− 0.276882i	0.990371	−	0.138441i	$-0.0442091\pi$
$84$	0	0
$85$	−3.22828e6	−0.0618436
$86$	0	0
$87$	− 4.20796e6i	− 0.0734505i
$88$	0	0
$89$	6.90797e7	1.10101	0.550504	−	0.834832i	$-0.314435\pi$
$89$	6.90797e7	1.10101	0.550504	+	0.834832i	$0.314435\pi$
$90$	0	0
$91$	− 5.58657e7i	− 0.814666i
$92$	0	0
$93$	−3.42761e6	−0.0458205
$94$	0	0
$95$	9.30008e7i	1.14181i
$96$	0	0
$97$	−1.35124e8	−1.52633	−0.763163	−	0.646207i	$-0.776354\pi$
$97$	−1.35124e8	−1.52633	−0.763163	+	0.646207i	$0.776354\pi$
$98$	0	0
$99$	− 8.93215e6i	− 0.0929855i
$100$	0	0
$101$	−5.67145e7	−0.545015	−0.272508	−	0.962154i	$-0.587853\pi$
$101$	−5.67145e7	−0.545015	−0.272508	+	0.962154i	$0.587853\pi$
$102$	0	0
$103$	− 1.03382e8i	− 0.918534i	−0.888298	−	0.459267i	$-0.848112\pi$
$103$	− 1.03382e8i	− 0.918534i	0.888298	−	0.459267i	$-0.151888\pi$
$104$	0	0
$105$	−6.61571e7	−0.544276
$106$	0	0
$107$	− 1.17697e8i	− 0.897904i	−0.893556	−	0.448952i	$-0.851797\pi$
$107$	− 1.17697e8i	− 0.897904i	0.893556	−	0.448952i	$-0.148203\pi$
$108$	0	0
$109$	1.90586e8	1.35016	0.675080	−	0.737744i	$-0.264109\pi$
$109$	1.90586e8	1.35016	0.675080	+	0.737744i	$0.264109\pi$
$110$	0	0
$111$	− 8.57875e7i	− 0.565109i
$112$	0	0
$113$	1.18873e7	0.0729071	0.0364536	−	0.999335i	$-0.488394\pi$
$113$	1.18873e7	0.0729071	0.0364536	+	0.999335i	$0.488394\pi$
$114$	0	0
$115$	− 3.50326e8i	− 2.00300i
$116$	0	0
$117$	−6.08429e7	−0.324688
$118$	0	0
$119$	− 9.20211e6i	− 0.0458880i
$120$	0	0
$121$	1.97678e8	0.922183
$122$	0	0
$123$	2.18706e8i	0.955523i
$124$	0	0
$125$	2.00748e8	0.822266
$126$	0	0
$127$	− 1.56645e8i	− 0.602146i	−0.953601	−	0.301073i	$-0.902655\pi$
$127$	− 1.56645e8i	− 0.602146i	0.953601	−	0.301073i	$-0.0973448\pi$
$128$	0	0
$129$	−2.01285e7	−0.0726864
$130$	0	0
$131$	− 1.87108e8i	− 0.635340i	−0.948201	−	0.317670i	$-0.897100\pi$
$131$	− 1.87108e8i	− 0.635340i	0.948201	−	0.317670i	$-0.102900\pi$
$132$	0	0
$133$	−2.65096e8	−0.847221
$134$	0	0
$135$	7.20511e7i	0.216923i
$136$	0	0
$137$	4.00003e8	1.13549	0.567743	−	0.823206i	$-0.307817\pi$
$137$	4.00003e8	1.13549	0.567743	+	0.823206i	$0.307817\pi$
$138$	0	0
$139$	2.22885e8i	0.597064i	0.954400	+	0.298532i	$0.0964969\pi$
$139$	2.22885e8i	0.597064i	−0.954400	+	0.298532i	$0.903503\pi$
$140$	0	0
$141$	−3.84425e8	−0.972603
$142$	0	0
$143$	1.13623e8i	0.271721i
$144$	0	0
$145$	6.33891e7	0.143398
$146$	0	0
$147$	8.10142e7i	0.173497i
$148$	0	0
$149$	−1.59610e8	−0.323828	−0.161914	−	0.986805i	$-0.551767\pi$
$149$	−1.59610e8	−0.323828	−0.161914	+	0.986805i	$0.551767\pi$
$150$	0	0
$151$	− 3.13589e8i	− 0.603188i	−0.953436	−	0.301594i	$-0.902481\pi$
$151$	− 3.13589e8i	− 0.603188i	0.953436	−	0.301594i	$-0.0975187\pi$
$152$	0	0
$153$	−1.00219e7	−0.0182888
$154$	0	0
$155$	− 5.16339e7i	− 0.0894557i
$156$	0	0
$157$	−1.01224e9	−1.66603	−0.833016	−	0.553249i	$-0.813388\pi$
$157$	−1.01224e9	−1.66603	−0.833016	+	0.553249i	$0.813388\pi$
$158$	0	0
$159$	− 1.71295e8i	− 0.268013i
$160$	0	0
$161$	9.98596e8	1.48623
$162$	0	0
$163$	9.30515e8i	1.31817i	0.752066	+	0.659087i	$0.229057\pi$
$163$	9.30515e8i	1.31817i	−0.752066	+	0.659087i	$0.770943\pi$
$164$	0	0
$165$	1.34555e8	0.181536
$166$	0	0
$167$	− 2.16948e8i	− 0.278927i	−0.990227	−	0.139463i	$-0.955462\pi$
$167$	− 2.16948e8i	− 0.278927i	0.990227	−	0.139463i	$-0.0445378\pi$
$168$	0	0
$169$	−4.17650e7	−0.0511995
$170$	0	0
$171$	2.88714e8i	0.337663i
$172$	0	0
$173$	1.67601e7	0.0187108	0.00935540	−	0.999956i	$-0.497022\pi$
$173$	1.67601e7	0.0187108	0.00935540	+	0.999956i	$0.497022\pi$
$174$	0	0
$175$	− 2.12184e8i	− 0.226236i
$176$	0	0
$177$	3.47000e8	0.353539
$178$	0	0
$179$	− 1.47474e9i	− 1.43649i	−0.695788	−	0.718247i	$-0.744945\pi$
$179$	− 1.47474e9i	− 1.43649i	0.695788	−	0.718247i	$-0.255055\pi$
$180$	0	0
$181$	6.48041e8	0.603793	0.301897	−	0.953341i	$-0.402380\pi$
$181$	6.48041e8	0.603793	0.301897	+	0.953341i	$0.402380\pi$
$182$	0	0
$183$	− 7.27715e8i	− 0.648869i
$184$	0	0
$185$	1.29231e9	1.10327
$186$	0	0
$187$	1.87159e7i	0.0153054i
$188$	0	0
$189$	−2.05380e8	−0.160957
$190$	0	0
$191$	− 4.74674e8i	− 0.356666i	−0.983970	−	0.178333i	$-0.942930\pi$
$191$	− 4.74674e8i	− 0.356666i	0.983970	−	0.178333i	$-0.0570705\pi$
$192$	0	0
$193$	1.01783e9	0.733577	0.366788	−	0.930304i	$-0.380457\pi$
$193$	1.01783e9	0.733577	0.366788	+	0.930304i	$0.380457\pi$
$194$	0	0
$195$	− 9.16543e8i	− 0.633891i
$196$	0	0
$197$	6.53891e7	0.0434151	0.0217075	−	0.999764i	$-0.493090\pi$
$197$	6.53891e7	0.0434151	0.0217075	+	0.999764i	$0.493090\pi$
$198$	0	0
$199$	− 2.74390e9i	− 1.74967i	−0.484420	−	0.874835i	$-0.660969\pi$
$199$	− 2.74390e9i	− 1.74967i	0.484420	−	0.874835i	$-0.339031\pi$
$200$	0	0
$201$	−4.69841e8	−0.287850
$202$	0	0
$203$	1.80689e8i	0.106401i
$204$	0	0
$205$	−3.29461e9	−1.86547
$206$	0	0
$207$	− 1.08756e9i	− 0.592342i
$208$	0	0
$209$	5.39171e8	0.282580
$210$	0	0
$211$	− 2.01269e9i	− 1.01542i	−0.861528	−	0.507711i	$-0.830492\pi$
$211$	− 2.01269e9i	− 1.01542i	0.861528	−	0.507711i	$-0.169508\pi$
$212$	0	0
$213$	2.66849e8	0.129643
$214$	0	0
$215$	− 3.03218e8i	− 0.141906i
$216$	0	0
$217$	1.47181e8	0.0663762
$218$	0	0
$219$	− 1.53358e9i	− 0.666700i
$220$	0	0
$221$	1.27486e8	0.0534435
$222$	0	0
$223$	2.23882e9i	0.905315i	0.891684	+	0.452658i	$0.149524\pi$
$223$	2.23882e9i	0.905315i	−0.891684	+	0.452658i	$0.850476\pi$
$224$	0	0
$225$	−2.31088e8	−0.0901670
$226$	0	0
$227$	− 3.39289e9i	− 1.27781i	−0.769286	−	0.638905i	$-0.779388\pi$
$227$	− 3.39289e9i	− 1.27781i	0.769286	−	0.638905i	$-0.220612\pi$
$228$	0	0
$229$	−5.86309e8	−0.213199	−0.106599	−	0.994302i	$-0.533996\pi$
$229$	−5.86309e8	−0.213199	−0.106599	+	0.994302i	$0.533996\pi$
$230$	0	0
$231$	3.83544e8i	0.134700i
$232$	0	0
$233$	2.11212e9	0.716628	0.358314	−	0.933601i	$-0.383352\pi$
$233$	2.11212e9	0.716628	0.358314	+	0.933601i	$0.383352\pi$
$234$	0	0
$235$	− 5.79102e9i	− 1.89882i
$236$	0	0
$237$	2.21973e8	0.0703568
$238$	0	0
$239$	− 4.02306e9i	− 1.23300i	−0.787353	−	0.616502i	$-0.788549\pi$
$239$	− 4.02306e9i	− 1.23300i	0.787353	−	0.616502i	$-0.211451\pi$
$240$	0	0
$241$	6.07357e9	1.80043	0.900214	−	0.435447i	$-0.143410\pi$
$241$	6.07357e9	1.80043	0.900214	+	0.435447i	$0.143410\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	−1.22041e9	−0.338719
$246$	0	0
$247$	− 3.67265e9i	− 0.986716i
$248$	0	0
$249$	6.14513e8	0.159858
$250$	0	0
$251$	− 4.47559e9i	− 1.12760i	−0.825911	−	0.563800i	$-0.809339\pi$
$251$	− 4.47559e9i	− 1.12760i	0.825911	−	0.563800i	$-0.190661\pi$
$252$	0	0
$253$	−2.03101e9	−0.495713
$254$	0	0
$255$	− 1.50972e8i	− 0.0357054i
$256$	0	0
$257$	−8.61526e9	−1.97486	−0.987429	−	0.158062i	$-0.949475\pi$
$257$	−8.61526e9	−1.97486	−0.987429	+	0.158062i	$0.949475\pi$
$258$	0	0
$259$	3.68370e9i	0.818625i
$260$	0	0
$261$	1.96787e8	0.0424067
$262$	0	0
$263$	− 5.12216e9i	− 1.07061i	−0.844660	−	0.535303i	$-0.820197\pi$
$263$	− 5.12216e9i	− 1.07061i	0.844660	−	0.535303i	$-0.179803\pi$
$264$	0	0
$265$	2.58040e9	0.523243
$266$	0	0
$267$	3.23054e9i	0.635668i
$268$	0	0
$269$	−7.74667e9	−1.47947	−0.739735	−	0.672899i	$-0.765049\pi$
$269$	−7.74667e9	−1.47947	−0.739735	+	0.672899i	$0.765049\pi$
$270$	0	0
$271$	2.04096e9i	0.378405i	0.981938	+	0.189202i	$0.0605902\pi$
$271$	2.04096e9i	0.378405i	−0.981938	+	0.189202i	$0.939410\pi$
$272$	0	0
$273$	2.61258e9	0.470348
$274$	0	0
$275$	4.31555e8i	0.0754580i
$276$	0	0
$277$	−6.44922e9	−1.09544	−0.547720	−	0.836662i	$-0.684504\pi$
$277$	−6.44922e9	−1.09544	−0.547720	+	0.836662i	$0.684504\pi$
$278$	0	0
$279$	− 1.60294e8i	− 0.0264545i
$280$	0	0
$281$	3.08366e9	0.494586	0.247293	−	0.968941i	$-0.420459\pi$
$281$	3.08366e9	0.494586	0.247293	+	0.968941i	$0.420459\pi$
$282$	0	0
$283$	− 1.11274e10i	− 1.73480i	−0.497610	−	0.867401i	$-0.665789\pi$
$283$	− 1.11274e10i	− 1.73480i	0.497610	−	0.867401i	$-0.334211\pi$
$284$	0	0
$285$	−4.34922e9	−0.659222
$286$	0	0
$287$	− 9.39121e9i	− 1.38418i
$288$	0	0
$289$	−6.95476e9	−0.996990
$290$	0	0
$291$	− 6.31915e9i	− 0.881224i
$292$	0	0
$293$	−9.04033e9	−1.22663	−0.613316	−	0.789838i	$-0.710165\pi$
$293$	−9.04033e9	−1.22663	−0.613316	+	0.789838i	$0.710165\pi$
$294$	0	0
$295$	5.22725e9i	0.690216i
$296$	0	0
$297$	4.17715e8	0.0536852
$298$	0	0
$299$	1.38346e10i	1.73094i
$300$	0	0
$301$	8.64315e8	0.105295
$302$	0	0
$303$	− 2.65228e9i	− 0.314665i
$304$	0	0
$305$	1.09624e10	1.26679
$306$	0	0
$307$	− 7.01670e9i	− 0.789913i	−0.918700	−	0.394957i	$-0.870760\pi$
$307$	− 7.01670e9i	− 0.789913i	0.918700	−	0.394957i	$-0.129240\pi$
$308$	0	0
$309$	4.83469e9	0.530316
$310$	0	0
$311$	1.07801e10i	1.15234i	0.817329	+	0.576172i	$0.195454\pi$
$311$	1.07801e10i	1.15234i	−0.817329	+	0.576172i	$0.804546\pi$
$312$	0	0
$313$	1.61356e10	1.68115	0.840577	−	0.541692i	$-0.182216\pi$
$313$	1.61356e10	1.68115	0.840577	+	0.541692i	$0.182216\pi$
$314$	0	0
$315$	− 3.09386e9i	− 0.314238i
$316$	0	0
$317$	−1.61832e10	−1.60261	−0.801306	−	0.598255i	$-0.795861\pi$
$317$	−1.61832e10	−1.60261	−0.801306	+	0.598255i	$0.795861\pi$
$318$	0	0
$319$	− 3.67497e8i	− 0.0354888i
$320$	0	0
$321$	5.50414e9	0.518405
$322$	0	0
$323$	− 6.04954e8i	− 0.0555792i
$324$	0	0
$325$	2.93961e9	0.263485
$326$	0	0
$327$	8.91284e9i	0.779516i
$328$	0	0
$329$	1.65071e10	1.40893
$330$	0	0
$331$	1.14677e9i	0.0955357i	0.998858	+	0.0477679i	$0.0152108\pi$
$331$	1.14677e9i	0.0955357i	−0.998858	+	0.0477679i	$0.984789\pi$
$332$	0	0
$333$	4.01188e9	0.326266
$334$	0	0
$335$	− 7.07773e9i	− 0.561972i
$336$	0	0
$337$	−1.50865e10	−1.16968	−0.584842	−	0.811147i	$-0.698844\pi$
$337$	−1.50865e10	−1.16968	−0.584842	+	0.811147i	$0.698844\pi$
$338$	0	0
$339$	5.55915e8i	0.0420930i
$340$	0	0
$341$	−2.99347e8	−0.0221389
$342$	0	0
$343$	− 1.50550e10i	− 1.08769i
$344$	0	0
$345$	1.63831e10	1.15643
$346$	0	0
$347$	− 1.31995e10i	− 0.910415i	−0.890385	−	0.455208i	$-0.849565\pi$
$347$	− 1.31995e10i	− 0.910415i	0.890385	−	0.455208i	$-0.150435\pi$
$348$	0	0
$349$	−8.79573e9	−0.592885	−0.296442	−	0.955051i	$-0.595800\pi$
$349$	−8.79573e9	−0.592885	−0.296442	+	0.955051i	$0.595800\pi$
$350$	0	0
$351$	− 2.84534e9i	− 0.187459i
$352$	0	0
$353$	−1.48698e10	−0.957648	−0.478824	−	0.877911i	$-0.658937\pi$
$353$	−1.48698e10	−0.957648	−0.478824	+	0.877911i	$0.658937\pi$
$354$	0	0
$355$	4.01985e9i	0.253102i
$356$	0	0
$357$	4.30340e8	0.0264935
$358$	0	0
$359$	− 1.39585e10i	− 0.840349i	−0.907443	−	0.420174i	$-0.861969\pi$
$359$	− 1.39585e10i	− 0.840349i	0.907443	−	0.420174i	$-0.138031\pi$
$360$	0	0
$361$	−4.44065e8	−0.0261467
$362$	0	0
$363$	9.24449e9i	0.532423i
$364$	0	0
$365$	2.31020e10	1.30160
$366$	0	0
$367$	− 5.47490e9i	− 0.301795i	−0.988549	−	0.150897i	$-0.951784\pi$
$367$	− 5.47490e9i	− 0.301795i	0.988549	−	0.150897i	$-0.0482163\pi$
$368$	0	0
$369$	−1.02279e10	−0.551671
$370$	0	0
$371$	7.35536e9i	0.388247i
$372$	0	0
$373$	2.02612e10	1.04672	0.523359	−	0.852112i	$-0.324679\pi$
$373$	2.02612e10	1.04672	0.523359	+	0.852112i	$0.324679\pi$
$374$	0	0
$375$	9.38808e9i	0.474735i
$376$	0	0
$377$	−2.50327e9	−0.123920
$378$	0	0
$379$	− 3.43125e10i	− 1.66301i	−0.555515	−	0.831507i	$-0.687479\pi$
$379$	− 3.43125e10i	− 1.66301i	0.555515	−	0.831507i	$-0.312521\pi$
$380$	0	0
$381$	7.32557e9	0.347649
$382$	0	0
$383$	− 7.05178e9i	− 0.327721i	−0.986484	−	0.163860i	$-0.947605\pi$
$383$	− 7.05178e9i	− 0.327721i	0.986484	−	0.163860i	$-0.0523946\pi$
$384$	0	0
$385$	−5.77775e9	−0.262976
$386$	0	0
$387$	− 9.41318e8i	− 0.0419655i
$388$	0	0
$389$	3.34155e10	1.45932	0.729658	−	0.683813i	$-0.239679\pi$
$389$	3.34155e10	1.45932	0.729658	+	0.683813i	$0.239679\pi$
$390$	0	0
$391$	2.27881e9i	0.0974993i
$392$	0	0
$393$	8.75015e9	0.366814
$394$	0	0
$395$	3.34382e9i	0.137358i
$396$	0	0
$397$	−1.65036e10	−0.664381	−0.332190	−	0.943212i	$-0.607788\pi$
$397$	−1.65036e10	−0.664381	−0.332190	+	0.943212i	$0.607788\pi$
$398$	0	0
$399$	− 1.23973e10i	− 0.489143i
$400$	0	0
$401$	−1.13883e10	−0.440436	−0.220218	−	0.975451i	$-0.570677\pi$
$401$	−1.13883e10	−0.440436	−0.220218	+	0.975451i	$0.570677\pi$
$402$	0	0
$403$	2.03905e9i	0.0773051i
$404$	0	0
$405$	−3.36950e9	−0.125241
$406$	0	0
$407$	− 7.49215e9i	− 0.273042i
$408$	0	0
$409$	−5.32308e9	−0.190226	−0.0951130	−	0.995466i	$-0.530321\pi$
$409$	−5.32308e9	−0.190226	−0.0951130	+	0.995466i	$0.530321\pi$
$410$	0	0
$411$	1.87063e10i	0.655573i
$412$	0	0
$413$	−1.49001e10	−0.512141
$414$	0	0
$415$	9.25708e9i	0.312091i
$416$	0	0
$417$	−1.04233e10	−0.344715
$418$	0	0
$419$	4.15290e10i	1.34740i	0.739006	+	0.673699i	$0.235295\pi$
$419$	4.15290e10i	1.34740i	−0.739006	+	0.673699i	$0.764705\pi$
$420$	0	0
$421$	1.31138e10	0.417446	0.208723	−	0.977975i	$-0.433069\pi$
$421$	1.31138e10	0.417446	0.208723	+	0.977975i	$0.433069\pi$
$422$	0	0
$423$	− 1.79778e10i	− 0.561533i
$424$	0	0
$425$	4.84208e8	0.0148414
$426$	0	0
$427$	3.12479e10i	0.939961i
$428$	0	0
$429$	−5.31364e9	−0.156878
$430$	0	0
$431$	2.64000e10i	0.765060i	0.923943	+	0.382530i	$0.124947\pi$
$431$	2.64000e10i	0.765060i	−0.923943	+	0.382530i	$0.875053\pi$
$432$	0	0
$433$	−3.95779e10	−1.12590	−0.562952	−	0.826490i	$-0.690334\pi$
$433$	−3.95779e10	−1.12590	−0.562952	+	0.826490i	$0.690334\pi$
$434$	0	0
$435$	2.96442e9i	0.0827908i
$436$	0	0
$437$	6.56485e10	1.80011
$438$	0	0
$439$	2.12540e10i	0.572246i	0.958193	+	0.286123i	$0.0923666\pi$
$439$	2.12540e10i	0.572246i	−0.958193	+	0.286123i	$0.907633\pi$
$440$	0	0
$441$	−3.78866e9	−0.100168
$442$	0	0
$443$	1.20023e10i	0.311637i	0.987786	+	0.155818i	$0.0498015\pi$
$443$	1.20023e10i	0.311637i	−0.987786	+	0.155818i	$0.950199\pi$
$444$	0	0
$445$	−4.86652e10	−1.24102
$446$	0	0
$447$	− 7.46422e9i	− 0.186962i
$448$	0	0
$449$	2.52290e10	0.620747	0.310374	−	0.950615i	$-0.399546\pi$
$449$	2.52290e10	0.620747	0.310374	+	0.950615i	$0.399546\pi$
$450$	0	0
$451$	1.91005e10i	0.461677i
$452$	0	0
$453$	1.46651e10	0.348251
$454$	0	0
$455$	3.93562e10i	0.918263i
$456$	0	0
$457$	−2.14985e10	−0.492883	−0.246442	−	0.969158i	$-0.579261\pi$
$457$	−2.14985e10	−0.492883	−0.246442	+	0.969158i	$0.579261\pi$
$458$	0	0
$459$	− 4.68680e8i	− 0.0105591i
$460$	0	0
$461$	−7.38060e10	−1.63413	−0.817067	−	0.576542i	$-0.804402\pi$
$461$	−7.38060e10	−1.63413	−0.817067	+	0.576542i	$0.804402\pi$
$462$	0	0
$463$	8.39463e10i	1.82674i	0.407126	+	0.913372i	$0.366531\pi$
$463$	8.39463e10i	1.82674i	−0.407126	+	0.913372i	$0.633469\pi$
$464$	0	0
$465$	2.41468e9	0.0516473
$466$	0	0
$467$	− 5.50574e10i	− 1.15757i	−0.815479	−	0.578786i	$-0.803527\pi$
$467$	− 5.50574e10i	− 1.15757i	0.815479	−	0.578786i	$-0.196473\pi$
$468$	0	0
$469$	2.01749e10	0.416984
$470$	0	0
$471$	− 4.73376e10i	− 0.961884i
$472$	0	0
$473$	−1.75790e9	−0.0351196
$474$	0	0
$475$	− 1.39492e10i	− 0.274014i
$476$	0	0
$477$	8.01066e9	0.154737
$478$	0	0
$479$	2.77607e10i	0.527337i	0.964613	+	0.263668i	$0.0849324\pi$
$479$	2.77607e10i	0.527337i	−0.964613	+	0.263668i	$0.915068\pi$
$480$	0	0
$481$	−5.10341e10	−0.953411
$482$	0	0
$483$	4.66997e10i	0.858076i
$484$	0	0
$485$	9.51922e10	1.72042
$486$	0	0
$487$	6.64050e10i	1.18055i	0.807201	+	0.590276i	$0.200981\pi$
$487$	6.64050e10i	1.18055i	−0.807201	+	0.590276i	$0.799019\pi$
$488$	0	0
$489$	−4.35159e10	−0.761048
$490$	0	0
$491$	2.56637e10i	0.441564i	0.975323	+	0.220782i	$0.0708609\pi$
$491$	2.56637e10i	0.441564i	−0.975323	+	0.220782i	$0.929139\pi$
$492$	0	0
$493$	−4.12335e8	−0.00698012
$494$	0	0
$495$	6.29250e9i	0.104810i
$496$	0	0
$497$	−1.14585e10	−0.187802
$498$	0	0
$499$	1.21348e10i	0.195717i	0.995200	+	0.0978587i	$0.0311993\pi$
$499$	1.21348e10i	0.195717i	−0.995200	+	0.0978587i	$0.968801\pi$
$500$	0	0
$501$	1.01457e10	0.161039
$502$	0	0
$503$	5.51921e10i	0.862193i	0.902306	+	0.431097i	$0.141873\pi$
$503$	5.51921e10i	0.862193i	−0.902306	+	0.431097i	$0.858127\pi$
$504$	0	0
$505$	3.99541e10	0.614322
$506$	0	0
$507$	− 1.95316e9i	− 0.0295600i
$508$	0	0
$509$	2.36773e10	0.352746	0.176373	−	0.984323i	$-0.443564\pi$
$509$	2.36773e10	0.352746	0.176373	+	0.984323i	$0.443564\pi$
$510$	0	0
$511$	6.58516e10i	0.965791i
$512$	0	0
$513$	−1.35018e10	−0.194950
$514$	0	0
$515$	7.28303e10i	1.03534i
$516$	0	0
$517$	−3.35733e10	−0.469929
$518$	0	0
$519$	7.83793e8i	0.0108027i
$520$	0	0
$521$	9.94644e10	1.34995	0.674974	−	0.737842i	$-0.264155\pi$
$521$	9.94644e10	1.34995	0.674974	+	0.737842i	$0.264155\pi$
$522$	0	0
$523$	1.15694e11i	1.54634i	0.634201	+	0.773168i	$0.281329\pi$
$523$	1.15694e11i	1.54634i	−0.634201	+	0.773168i	$0.718671\pi$
$524$	0	0
$525$	9.92288e9	0.130617
$526$	0	0
$527$	3.35869e8i	0.00435440i
$528$	0	0
$529$	−1.68981e11	−2.15783
$530$	0	0
$531$	1.62276e10i	0.204116i
$532$	0	0
$533$	1.30106e11	1.61209
$534$	0	0
$535$	8.29149e10i	1.01209i
$536$	0	0
$537$	6.89669e10	0.829360
$538$	0	0
$539$	7.07528e9i	0.0838279i
$540$	0	0
$541$	−1.38315e11	−1.61466	−0.807329	−	0.590102i	$-0.799088\pi$
$541$	−1.38315e11	−1.61466	−0.807329	+	0.590102i	$0.799088\pi$
$542$	0	0
$543$	3.03059e10i	0.348600i
$544$	0	0
$545$	−1.34264e11	−1.52185
$546$	0	0
$547$	− 1.57661e11i	− 1.76106i	−0.473988	−	0.880531i	$-0.657186\pi$
$547$	− 1.57661e11i	− 1.76106i	0.473988	−	0.880531i	$-0.342814\pi$
$548$	0	0
$549$	3.40319e10	0.374625
$550$	0	0
$551$	1.18786e10i	0.128872i
$552$	0	0
$553$	−9.53145e9	−0.101920
$554$	0	0
$555$	6.04354e10i	0.636971i
$556$	0	0
$557$	1.48689e11	1.54475	0.772374	−	0.635168i	$-0.219069\pi$
$557$	1.48689e11	1.54475	0.772374	+	0.635168i	$0.219069\pi$
$558$	0	0
$559$	1.19743e10i	0.122631i
$560$	0	0
$561$	−8.75255e8	−0.00883656
$562$	0	0
$563$	− 1.04219e11i	− 1.03732i	−0.854980	−	0.518661i	$-0.826431\pi$
$563$	− 1.04219e11i	− 1.03732i	0.854980	−	0.518661i	$-0.173569\pi$
$564$	0	0
$565$	−8.37435e9	−0.0821784
$566$	0	0
$567$	− 9.60466e9i	− 0.0929287i
$568$	0	0
$569$	−6.10741e10	−0.582651	−0.291325	−	0.956624i	$-0.594096\pi$
$569$	−6.10741e10	−0.582651	−0.291325	+	0.956624i	$0.594096\pi$
$570$	0	0
$571$	3.70892e10i	0.348902i	0.984666	+	0.174451i	$0.0558150\pi$
$571$	3.70892e10i	0.348902i	−0.984666	+	0.174451i	$0.944185\pi$
$572$	0	0
$573$	2.21983e10	0.205921
$574$	0	0
$575$	5.25453e10i	0.480688i
$576$	0	0
$577$	−1.30323e11	−1.17576	−0.587880	−	0.808948i	$-0.700037\pi$
$577$	−1.30323e11	−1.17576	−0.587880	+	0.808948i	$0.700037\pi$
$578$	0	0
$579$	4.75991e10i	0.423531i
$580$	0	0
$581$	−2.63871e10	−0.231572
$582$	0	0
$583$	− 1.49598e10i	− 0.129495i
$584$	0	0
$585$	4.28625e10	0.365977
$586$	0	0
$587$	7.19194e10i	0.605751i	0.953030	+	0.302875i	$0.0979466\pi$
$587$	7.19194e10i	0.605751i	−0.953030	+	0.302875i	$0.902053\pi$
$588$	0	0
$589$	9.67579e9	0.0803943
$590$	0	0
$591$	3.05795e9i	0.0250657i
$592$	0	0
$593$	−8.93031e10	−0.722184	−0.361092	−	0.932530i	$-0.617596\pi$
$593$	−8.93031e10	−0.722184	−0.361092	+	0.932530i	$0.617596\pi$
$594$	0	0
$595$	6.48268e9i	0.0517234i
$596$	0	0
$597$	1.28320e11	1.01017
$598$	0	0
$599$	− 2.10614e11i	− 1.63599i	−0.575227	−	0.817994i	$-0.695086\pi$
$599$	− 2.10614e11i	− 1.63599i	0.575227	−	0.817994i	$-0.304914\pi$
$600$	0	0
$601$	−1.66081e11	−1.27298	−0.636491	−	0.771284i	$-0.719615\pi$
$601$	−1.66081e11	−1.27298	−0.636491	+	0.771284i	$0.719615\pi$
$602$	0	0
$603$	− 2.19723e10i	− 0.166190i
$604$	0	0
$605$	−1.39260e11	−1.03945
$606$	0	0
$607$	− 1.34364e11i	− 0.989758i	−0.868962	−	0.494879i	$-0.835212\pi$
$607$	− 1.34364e11i	− 0.989758i	0.868962	−	0.494879i	$-0.164788\pi$
$608$	0	0
$609$	−8.44999e9	−0.0614309
$610$	0	0
$611$	2.28691e11i	1.64091i
$612$	0	0
$613$	7.64794e10	0.541630	0.270815	−	0.962631i	$-0.412707\pi$
$613$	7.64794e10	0.541630	0.270815	+	0.962631i	$0.412707\pi$
$614$	0	0
$615$	− 1.54074e11i	− 1.07703i
$616$	0	0
$617$	2.24130e11	1.54653	0.773267	−	0.634081i	$-0.218621\pi$
$617$	2.24130e11	1.54653	0.773267	+	0.634081i	$0.218621\pi$
$618$	0	0
$619$	− 1.69554e11i	− 1.15490i	−0.816425	−	0.577451i	$-0.804047\pi$
$619$	− 1.69554e11i	− 1.15490i	0.816425	−	0.577451i	$-0.195953\pi$
$620$	0	0
$621$	5.08603e10	0.341989
$622$	0	0
$623$	− 1.38719e11i	− 0.920837i
$624$	0	0
$625$	−1.82698e11	−1.19733
$626$	0	0
$627$	2.52145e10i	0.163147i
$628$	0	0
$629$	−8.40626e9	−0.0537032
$630$	0	0
$631$	1.63018e11i	1.02830i	0.857702	+	0.514148i	$0.171892\pi$
$631$	1.63018e11i	1.02830i	−0.857702	+	0.514148i	$0.828108\pi$
$632$	0	0
$633$	9.41240e10	0.586254
$634$	0	0
$635$	1.10353e11i	0.678718i
$636$	0	0
$637$	4.81945e10	0.292712
$638$	0	0
$639$	1.24793e10i	0.0748492i
$640$	0	0
$641$	6.93255e10	0.410640	0.205320	−	0.978695i	$-0.434177\pi$
$641$	6.93255e10	0.410640	0.205320	+	0.978695i	$0.434177\pi$
$642$	0	0
$643$	− 3.10184e11i	− 1.81458i	−0.420508	−	0.907289i	$-0.638148\pi$
$643$	− 3.10184e11i	− 1.81458i	0.420508	−	0.907289i	$-0.361852\pi$
$644$	0	0
$645$	1.41801e10	0.0819295
$646$	0	0
$647$	6.87948e10i	0.392590i	0.980545	+	0.196295i	$0.0628909\pi$
$647$	6.87948e10i	0.392590i	−0.980545	+	0.196295i	$0.937109\pi$
$648$	0	0
$649$	3.03049e10	0.170818
$650$	0	0
$651$	6.88297e9i	0.0383223i
$652$	0	0
$653$	1.90064e11	1.04531	0.522657	−	0.852543i	$-0.324941\pi$
$653$	1.90064e11	1.04531	0.522657	+	0.852543i	$0.324941\pi$
$654$	0	0
$655$	1.31813e11i	0.716133i
$656$	0	0
$657$	7.17185e10	0.384919
$658$	0	0
$659$	3.58821e11i	1.90255i	0.308345	+	0.951275i	$0.400225\pi$
$659$	3.58821e11i	1.90255i	−0.308345	+	0.951275i	$0.599775\pi$
$660$	0	0
$661$	1.70755e11	0.894474	0.447237	−	0.894416i	$-0.352408\pi$
$661$	1.70755e11	0.894474	0.447237	+	0.894416i	$0.352408\pi$
$662$	0	0
$663$	5.96195e9i	0.0308556i
$664$	0	0
$665$	1.86754e11	0.954958
$666$	0	0
$667$	− 4.47458e10i	− 0.226073i
$668$	0	0
$669$	−1.04699e11	−0.522684
$670$	0	0
$671$	− 6.35542e10i	− 0.313512i
$672$	0	0
$673$	3.39762e11	1.65621	0.828104	−	0.560574i	$-0.189419\pi$
$673$	3.39762e11	1.65621	0.828104	+	0.560574i	$0.189419\pi$
$674$	0	0
$675$	− 1.08069e10i	− 0.0520579i
$676$	0	0
$677$	−1.84992e11	−0.880641	−0.440321	−	0.897841i	$-0.645135\pi$
$677$	−1.84992e11	−0.880641	−0.440321	+	0.897841i	$0.645135\pi$
$678$	0	0
$679$	2.71343e11i	1.27655i
$680$	0	0
$681$	1.58670e11	0.737744
$682$	0	0
$683$	− 2.03092e11i	− 0.933274i	−0.884449	−	0.466637i	$-0.845466\pi$
$683$	− 2.03092e11i	− 0.933274i	0.884449	−	0.466637i	$-0.154534\pi$
$684$	0	0
$685$	−2.81794e11	−1.27988
$686$	0	0
$687$	− 2.74189e10i	− 0.123090i
$688$	0	0
$689$	−1.01901e11	−0.452172
$690$	0	0
$691$	− 1.54339e11i	− 0.676960i	−0.940974	−	0.338480i	$-0.890087\pi$
$691$	− 1.54339e11i	− 0.676960i	0.940974	−	0.338480i	$-0.109913\pi$
$692$	0	0
$693$	−1.79366e10	−0.0777691
$694$	0	0
$695$	− 1.57017e11i	− 0.672990i
$696$	0	0
$697$	2.14309e10	0.0908049
$698$	0	0
$699$	9.87739e10i	0.413746i
$700$	0	0
$701$	−1.08729e11	−0.450271	−0.225136	−	0.974327i	$-0.572283\pi$
$701$	−1.08729e11	−0.450271	−0.225136	+	0.974327i	$0.572283\pi$
$702$	0	0
$703$	2.42169e11i	0.991511i
$704$	0	0
$705$	2.70819e11	1.09628
$706$	0	0
$707$	1.13888e11i	0.455828i
$708$	0	0
$709$	−1.88943e11	−0.747733	−0.373866	−	0.927483i	$-0.621968\pi$
$709$	−1.88943e11	−0.747733	−0.373866	+	0.927483i	$0.621968\pi$
$710$	0	0
$711$	1.03806e10i	0.0406205i
$712$	0	0
$713$	−3.64479e10	−0.141031
$714$	0	0
$715$	− 8.00452e10i	− 0.306275i
$716$	0	0
$717$	1.88140e11	0.711875
$718$	0	0
$719$	− 1.73917e11i	− 0.650767i	−0.945582	−	0.325384i	$-0.894507\pi$
$719$	− 1.73917e11i	− 0.650767i	0.945582	−	0.325384i	$-0.105493\pi$
$720$	0	0
$721$	−2.07601e11	−0.768224
$722$	0	0
$723$	2.84033e11i	1.03948i
$724$	0	0
$725$	−9.50771e9	−0.0344131
$726$	0	0
$727$	− 6.03007e10i	− 0.215866i	−0.994158	−	0.107933i	$-0.965577\pi$
$727$	− 6.03007e10i	− 0.215866i	0.994158	−	0.107933i	$-0.0344233\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	1.97238e9i	0.00690750i
$732$	0	0
$733$	−4.37645e11	−1.51602	−0.758012	−	0.652241i	$-0.773829\pi$
$733$	−4.37645e11	−1.51602	−0.758012	+	0.652241i	$0.773829\pi$
$734$	0	0
$735$	− 5.70727e10i	− 0.195560i
$736$	0	0
$737$	−4.10330e10	−0.139080
$738$	0	0
$739$	− 2.77150e11i	− 0.929260i	−0.885505	−	0.464630i	$-0.846187\pi$
$739$	− 2.77150e11i	− 0.929260i	0.885505	−	0.464630i	$-0.153813\pi$
$740$	0	0
$741$	1.71753e11	0.569681
$742$	0	0
$743$	− 3.58138e11i	− 1.17515i	−0.809168	−	0.587577i	$-0.800082\pi$
$743$	− 3.58138e11i	− 1.17515i	0.809168	−	0.587577i	$-0.199918\pi$
$744$	0	0
$745$	1.12442e11	0.365008
$746$	0	0
$747$	2.87379e10i	0.0922939i
$748$	0	0
$749$	−2.36347e11	−0.750969
$750$	0	0
$751$	− 6.75329e10i	− 0.212303i	−0.994350	−	0.106151i	$-0.966147\pi$
$751$	− 6.75329e10i	− 0.212303i	0.994350	−	0.106151i	$-0.0338528\pi$
$752$	0	0
$753$	2.09302e11	0.651020
$754$	0	0
$755$	2.20916e11i	0.679892i
$756$	0	0
$757$	−4.67685e11	−1.42420	−0.712098	−	0.702081i	$-0.752255\pi$
$757$	−4.67685e11	−1.42420	−0.712098	+	0.702081i	$0.752255\pi$
$758$	0	0
$759$	− 9.49810e10i	− 0.286200i
$760$	0	0
$761$	3.21194e11	0.957700	0.478850	−	0.877897i	$-0.341054\pi$
$761$	3.21194e11	0.957700	0.478850	+	0.877897i	$0.341054\pi$
$762$	0	0
$763$	− 3.82715e11i	− 1.12922i
$764$	0	0
$765$	7.06024e9	0.0206145
$766$	0	0
$767$	− 2.06427e11i	− 0.596465i
$768$	0	0
$769$	−1.13289e11	−0.323954	−0.161977	−	0.986795i	$-0.551787\pi$
$769$	−1.13289e11	−0.323954	−0.161977	+	0.986795i	$0.551787\pi$
$770$	0	0
$771$	− 4.02896e11i	− 1.14018i
$772$	0	0
$773$	2.66938e11	0.747641	0.373820	−	0.927501i	$-0.378048\pi$
$773$	2.66938e11	0.747641	0.373820	+	0.927501i	$0.378048\pi$
$774$	0	0
$775$	7.74455e9i	0.0214679i
$776$	0	0
$777$	−1.72269e11	−0.472633
$778$	0	0
$779$	− 6.17385e11i	− 1.67651i
$780$	0	0
$781$	2.33050e10	0.0626390
$782$	0	0
$783$	9.20281e9i	0.0244835i
$784$	0	0
$785$	7.13098e11	1.87789
$786$	0	0
$787$	− 3.74375e11i	− 0.975907i	−0.872870	−	0.487953i	$-0.837744\pi$
$787$	− 3.74375e11i	− 0.975907i	0.872870	−	0.487953i	$-0.162256\pi$
$788$	0	0
$789$	2.39540e11	0.618115
$790$	0	0
$791$	− 2.38709e10i	− 0.0609765i
$792$	0	0
$793$	−4.32910e11	−1.09473
$794$	0	0
$795$	1.20673e11i	0.302094i
$796$	0	0
$797$	−1.56260e11	−0.387270	−0.193635	−	0.981074i	$-0.562028\pi$
$797$	−1.56260e11	−0.387270	−0.193635	+	0.981074i	$0.562028\pi$
$798$	0	0
$799$	3.76696e10i	0.0924280i
$800$	0	0
$801$	−1.51077e11	−0.367003
$802$	0	0
$803$	− 1.33934e11i	− 0.322127i
$804$	0	0
$805$	−7.03489e11	−1.67523
$806$	0	0
$807$	− 3.62276e11i	− 0.854172i
$808$	0	0
$809$	−4.22500e11	−0.986355	−0.493177	−	0.869929i	$-0.664165\pi$
$809$	−4.22500e11	−0.986355	−0.493177	+	0.869929i	$0.664165\pi$
$810$	0	0
$811$	− 5.61012e11i	− 1.29685i	−0.761279	−	0.648424i	$-0.775428\pi$
$811$	− 5.61012e11i	− 1.29685i	0.761279	−	0.648424i	$-0.224572\pi$
$812$	0	0
$813$	−9.54461e10	−0.218472
$814$	0	0
$815$	− 6.55527e11i	− 1.48580i
$816$	0	0
$817$	5.68207e10	0.127532
$818$	0	0
$819$	1.22178e11i	0.271555i
$820$	0	0
$821$	−2.46357e10	−0.0542242	−0.0271121	−	0.999632i	$-0.508631\pi$
$821$	−2.46357e10	−0.0542242	−0.0271121	+	0.999632i	$0.508631\pi$
$822$	0	0
$823$	1.51809e11i	0.330902i	0.986218	+	0.165451i	$0.0529079\pi$
$823$	1.51809e11i	0.330902i	−0.986218	+	0.165451i	$0.947092\pi$
$824$	0	0
$825$	−2.01818e10	−0.0435657
$826$	0	0
$827$	3.48232e11i	0.744468i	0.928139	+	0.372234i	$0.121408\pi$
$827$	3.48232e11i	0.744468i	−0.928139	+	0.372234i	$0.878592\pi$
$828$	0	0
$829$	−7.04874e11	−1.49243	−0.746214	−	0.665706i	$-0.768131\pi$
$829$	−7.04874e11	−1.49243	−0.746214	+	0.665706i	$0.768131\pi$
$830$	0	0
$831$	− 3.01600e11i	− 0.632452i
$832$	0	0
$833$	7.93852e9	0.0164877
$834$	0	0
$835$	1.52835e11i	0.314397i
$836$	0	0
$837$	7.49619e9	0.0152735
$838$	0	0
$839$	6.18908e11i	1.24905i	0.781006	+	0.624523i	$0.214707\pi$
$839$	6.18908e11i	1.24905i	−0.781006	+	0.624523i	$0.785293\pi$
$840$	0	0
$841$	−4.92150e11	−0.983815
$842$	0	0
$843$	1.44209e11i	0.285549i
$844$	0	0
$845$	2.94225e10	0.0577103
$846$	0	0
$847$	− 3.96957e11i	− 0.771275i
$848$	0	0
$849$	5.20379e11	1.00159
$850$	0	0
$851$	− 9.12232e11i	− 1.73935i
$852$	0	0
$853$	−2.49864e11	−0.471963	−0.235981	−	0.971758i	$-0.575830\pi$
$853$	−2.49864e11	−0.471963	−0.235981	+	0.971758i	$0.575830\pi$
$854$	0	0
$855$	− 2.03393e11i	− 0.380602i
$856$	0	0
$857$	5.89098e11	1.09210	0.546052	−	0.837751i	$-0.316130\pi$
$857$	5.89098e11	1.09210	0.546052	+	0.837751i	$0.316130\pi$
$858$	0	0
$859$	4.08489e11i	0.750253i	0.926974	+	0.375127i	$0.122401\pi$
$859$	4.08489e11i	0.750253i	−0.926974	+	0.375127i	$0.877599\pi$
$860$	0	0
$861$	4.39183e11	0.799159
$862$	0	0
$863$	− 5.90212e11i	− 1.06406i	−0.846726	−	0.532029i	$-0.821430\pi$
$863$	− 5.90212e11i	− 1.06406i	0.846726	−	0.532029i	$-0.178570\pi$
$864$	0	0
$865$	−1.18071e10	−0.0210902
$866$	0	0
$867$	− 3.25242e11i	− 0.575612i
$868$	0	0
$869$	1.93857e10	0.0339940
$870$	0	0
$871$	2.79504e11i	0.485640i
$872$	0	0
$873$	2.95517e11	0.508775
$874$	0	0
$875$	− 4.03122e11i	− 0.687708i
$876$	0	0
$877$	−1.32377e10	−0.0223776	−0.0111888	−	0.999937i	$-0.503562\pi$
$877$	−1.32377e10	−0.0223776	−0.0111888	+	0.999937i	$0.503562\pi$
$878$	0	0
$879$	− 4.22775e11i	− 0.708196i
$880$	0	0
$881$	9.45239e11	1.56905	0.784527	−	0.620094i	$-0.212906\pi$
$881$	9.45239e11	1.56905	0.784527	+	0.620094i	$0.212906\pi$
$882$	0	0
$883$	− 1.89878e11i	− 0.312344i	−0.987730	−	0.156172i	$-0.950085\pi$
$883$	− 1.89878e11i	− 0.312344i	0.987730	−	0.156172i	$-0.0499153\pi$
$884$	0	0
$885$	−2.44454e11	−0.398497
$886$	0	0
$887$	3.49636e11i	0.564835i	0.959292	+	0.282417i	$0.0911363\pi$
$887$	3.49636e11i	0.564835i	−0.959292	+	0.282417i	$0.908864\pi$
$888$	0	0
$889$	−3.14558e11	−0.503610
$890$	0	0
$891$	1.95346e10i	0.0309952i
$892$	0	0
$893$	1.08519e12	1.70648
$894$	0	0
$895$	1.03892e12i	1.61917i
$896$	0	0
$897$	−6.46980e11	−0.999358
$898$	0	0
$899$	− 6.59500e9i	− 0.0100966i
$900$	0	0
$901$	−1.67850e10	−0.0254697
$902$	0	0
$903$	4.04200e10i	0.0607918i
$904$	0	0
$905$	−4.56531e11	−0.680575
$906$	0	0
$907$	− 9.74600e11i	− 1.44011i	−0.693914	−	0.720057i	$-0.744115\pi$
$907$	− 9.74600e11i	− 1.44011i	0.693914	−	0.720057i	$-0.255885\pi$
$908$	0	0
$909$	1.24035e11	0.181672
$910$	0	0
$911$	− 3.42444e10i	− 0.0497183i	−0.999691	−	0.0248592i	$-0.992086\pi$
$911$	− 3.42444e10i	− 0.0497183i	0.999691	−	0.0248592i	$-0.00791374\pi$
$912$	0	0
$913$	5.36678e10	0.0772379
$914$	0	0
$915$	5.12659e11i	0.731382i
$916$	0	0
$917$	−3.75730e11	−0.531371
$918$	0	0
$919$	− 1.34762e12i	− 1.88932i	−0.328055	−	0.944659i	$-0.606393\pi$
$919$	− 1.34762e12i	− 1.88932i	0.328055	−	0.944659i	$-0.393607\pi$
$920$	0	0
$921$	3.28139e11	0.456057
$922$	0	0
$923$	− 1.58746e11i	− 0.218724i
$924$	0	0
$925$	−1.93833e11	−0.264766
$926$	0	0
$927$	2.26096e11i	0.306178i
$928$	0	0
$929$	6.19743e11	0.832049	0.416024	−	0.909353i	$-0.363423\pi$
$929$	6.19743e11	0.832049	0.416024	+	0.909353i	$0.363423\pi$
$930$	0	0
$931$	− 2.28695e11i	− 0.304409i
$932$	0	0
$933$	−5.04136e11	−0.665306
$934$	0	0
$935$	− 1.31849e10i	− 0.0172517i
$936$	0	0
$937$	1.15526e11	0.149872	0.0749361	−	0.997188i	$-0.476125\pi$
$937$	1.15526e11	0.149872	0.0749361	+	0.997188i	$0.476125\pi$
$938$	0	0
$939$	7.54587e11i	0.970614i
$940$	0	0
$941$	1.40100e12	1.78681	0.893405	−	0.449252i	$-0.148310\pi$
$941$	1.40100e12	1.78681	0.893405	+	0.449252i	$0.148310\pi$
$942$	0	0
$943$	2.32564e12i	2.94101i
$944$	0	0
$945$	1.44685e11	0.181425
$946$	0	0
$947$	2.70007e11i	0.335718i	0.985811	+	0.167859i	$0.0536854\pi$
$947$	2.70007e11i	0.335718i	−0.985811	+	0.167859i	$0.946315\pi$
$948$	0	0
$949$	−9.12312e11	−1.12481
$950$	0	0
$951$	− 7.56815e11i	− 0.925268i
$952$	0	0
$953$	1.49871e11	0.181696	0.0908479	−	0.995865i	$-0.471042\pi$
$953$	1.49871e11	0.181696	0.0908479	+	0.995865i	$0.471042\pi$
$954$	0	0
$955$	3.34398e11i	0.402022i
$956$	0	0
$957$	1.71862e10	0.0204895
$958$	0	0
$959$	− 8.03245e11i	− 0.949672i
$960$	0	0
$961$	8.47519e11	0.993701
$962$	0	0
$963$	2.57403e11i	0.299301i
$964$	0	0
$965$	−7.17038e11	−0.826862
$966$	0	0
$967$	− 2.79335e11i	− 0.319462i	−0.987161	−	0.159731i	$-0.948937\pi$
$967$	− 2.79335e11i	− 0.319462i	0.987161	−	0.159731i	$-0.0510627\pi$
$968$	0	0
$969$	2.82909e10	0.0320887
$970$	0	0
$971$	4.97536e11i	0.559690i	0.960045	+	0.279845i	$0.0902831\pi$
$971$	4.97536e11i	0.559690i	−0.960045	+	0.279845i	$0.909717\pi$
$972$	0	0
$973$	4.47573e11	0.499359
$974$	0	0
$975$	1.37472e11i	0.152123i
$976$	0	0
$977$	3.48161e11	0.382121	0.191061	−	0.981578i	$-0.438807\pi$
$977$	3.48161e11	0.382121	0.191061	+	0.981578i	$0.438807\pi$
$978$	0	0
$979$	2.82135e11i	0.307133i
$980$	0	0
$981$	−4.16812e11	−0.450054
$982$	0	0
$983$	3.74180e11i	0.400744i	0.979720	+	0.200372i	$0.0642151\pi$
$983$	3.74180e11i	0.400744i	−0.979720	+	0.200372i	$0.935785\pi$
$984$	0	0
$985$	−4.60652e10	−0.0489359
$986$	0	0
$987$	7.71963e11i	0.813444i
$988$	0	0
$989$	−2.14039e11	−0.223722
$990$	0	0
$991$	− 3.20951e11i	− 0.332770i	−0.986061	−	0.166385i	$-0.946790\pi$
$991$	− 3.20951e11i	− 0.332770i	0.986061	−	0.166385i	$-0.0532095\pi$
$992$	0	0
$993$	−5.36293e10	−0.0551576
$994$	0	0
$995$	1.93302e12i	1.97217i
$996$	0	0
$997$	−1.34045e12	−1.35666	−0.678330	−	0.734757i	$-0.737296\pi$
$997$	−1.34045e12	−1.35666	−0.678330	+	0.734757i	$0.737296\pi$
$998$	0	0
$999$	1.87617e11i	0.188370i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.g.b.127.26	yes	32
4.3	odd	2	inner	384.9.g.b.127.25	yes	32
8.3	odd	2		384.9.g.a.127.8	yes	32
8.5	even	2		384.9.g.a.127.7	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.g.a.127.7	✓	32	8.5	even	2
384.9.g.a.127.8	yes	32	8.3	odd	2
384.9.g.b.127.25	yes	32	4.3	odd	2	inner
384.9.g.b.127.26	yes	32	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} -704.478 q^{5} -2008.10i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} -704.478 q^{5} -2008.10i q^{7} -2187.00 q^{9} +4084.20i q^{11} +27820.2 q^{13} -32945.2i q^{15} +4582.51 q^{17} -132014. i q^{19} +93909.3 q^{21} +497285. i q^{23} +105664. q^{25} -102276. i q^{27} -89980.3 q^{29} +73293.8i q^{31} -190999. q^{33} +1.41466e6i q^{35} -1.83442e6 q^{37} +1.30102e6i q^{39} +4.67667e6 q^{41} +430415. i q^{43} +1.54069e6 q^{45} +8.22030e6i q^{47} +1.73235e6 q^{49} +214303. i q^{51} -3.66285e6 q^{53} -2.87723e6i q^{55} +6.17367e6 q^{57} -7.42003e6i q^{59} -1.55610e7 q^{61} +4.39170e6i q^{63} -1.95988e7 q^{65} +1.00468e7i q^{67} -2.32557e7 q^{69} -5.70613e6i q^{71} -3.27931e7 q^{73} +4.94144e6i q^{75} +8.20146e6 q^{77} -4.74652e6i q^{79} +4.78297e6 q^{81} -1.31403e7i q^{83} -3.22828e6 q^{85} -4.20796e6i q^{87} +6.90797e7 q^{89} -5.58657e7i q^{91} -3.42761e6 q^{93} +9.30008e7i q^{95} -1.35124e8 q^{97} -8.93215e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 1344 q^{5} - 69984 q^{9}+O(q^{10}) \) 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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