LMFDB - Embedded newform 384.8.d.e.193.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,Mod(193,384)]

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([0, 1, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	384.d (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:	$119.955849786$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 7628 x^{10} + 22070097 x^{8} - 30593373916 x^{6} + 21405948373596 x^{4} + \cdots + 90\!\cdots\!00 $ x^12 - 7628x^10 + 22070097x^8 - 30593373916x^6 + 21405948373596x^4 - 7160683139999288*x^2 + 907532972347562500
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{57}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.5
Root			$31.5402 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.e.193.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000i q^{3} +218.879i q^{5} +904.541 q^{7} -729.000 q^{9} +O(q^{10})$	$q-27.0000i q^{3} +218.879i q^{5} +904.541 q^{7} -729.000 q^{9} +1956.99i q^{11} +1492.92i q^{13} +5909.75 q^{15} +14661.8 q^{17} +32534.6i q^{19} -24422.6i q^{21} -24263.9 q^{23} +30216.8 q^{25} +19683.0i q^{27} -47760.3i q^{29} -38469.7 q^{31} +52838.8 q^{33} +197986. i q^{35} +230575. i q^{37} +40308.9 q^{39} +503187. q^{41} +74103.4i q^{43} -159563. i q^{45} -887487. q^{47} -5347.93 q^{49} -395869. i q^{51} +1.01950e6i q^{53} -428346. q^{55} +878434. q^{57} -956905. i q^{59} -1.51951e6i q^{61} -659411. q^{63} -326770. q^{65} -2.12707e6i q^{67} +655124. i q^{69} +1.47131e6 q^{71} -4.76425e6 q^{73} -815853. i q^{75} +1.77018e6i q^{77} -4.66203e6 q^{79} +531441. q^{81} +6.45883e6i q^{83} +3.20917e6i q^{85} -1.28953e6 q^{87} +3.18504e6 q^{89} +1.35041e6i q^{91} +1.03868e6i q^{93} -7.12115e6 q^{95} -1.36353e7 q^{97} -1.42665e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8748 q^{9}+O(q^{10}) $ 12 * q - 8748 * q^9	$ 12 q - 8748 q^{9} - 83096 q^{17} - 127812 q^{25} - 30672 q^{33} - 969032 q^{41} - 5087028 q^{49} - 69552 q^{57} - 240832 q^{65} + 18079656 q^{73} + 6377292 q^{81} + 43563144 q^{89} - 48458328 q^{97}+O(q^{100}) $ 12 * q - 8748 * q^9 - 83096 * q^17 - 127812 * q^25 - 30672 * q^33 - 969032 * q^41 - 5087028 * q^49 - 69552 * q^57 - 240832 * q^65 + 18079656 * q^73 + 6377292 * q^81 + 43563144 * q^89 - 48458328 * 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
$3$	− 27.0000i	− 0.577350i
$3$	− 27.0000i	− 0.577350i
$4$	0	0
$5$	218.879i	0.783087i	0.920160	+	0.391543i	$0.128059\pi$
$5$	218.879i	0.783087i	−0.920160	+	0.391543i	$0.871941\pi$
$6$	0	0
$7$	904.541	0.996748	0.498374	−	0.866962i	$-0.333931\pi$
$7$	904.541	0.996748	0.498374	+	0.866962i	$0.333931\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	1956.99i	0.443318i	0.975124	+	0.221659i	$0.0711471\pi$
$11$	1956.99i	0.443318i	−0.975124	+	0.221659i	$0.928853\pi$
$12$	0	0
$13$	1492.92i	0.188467i	0.995550	+	0.0942335i	$0.0300400\pi$
$13$	1492.92i	0.188467i	−0.995550	+	0.0942335i	$0.969960\pi$
$14$	0	0
$15$	5909.75	0.452115
$16$	0	0
$17$	14661.8	0.723797	0.361898	−	0.932218i	$-0.382129\pi$
$17$	14661.8	0.723797	0.361898	+	0.932218i	$0.382129\pi$
$18$	0	0
$19$	32534.6i	1.08820i	0.839021	+	0.544099i	$0.183128\pi$
$19$	32534.6i	1.08820i	−0.839021	+	0.544099i	$0.816872\pi$
$20$	0	0
$21$	− 24422.6i	− 0.575473i
$22$	0	0
$23$	−24263.9	−0.415827	−0.207913	−	0.978147i	$-0.566667\pi$
$23$	−24263.9	−0.415827	−0.207913	+	0.978147i	$0.566667\pi$
$24$	0	0
$25$	30216.8	0.386775
$26$	0	0
$27$	19683.0i	0.192450i
$28$	0	0
$29$	− 47760.3i	− 0.363642i	−0.983332	−	0.181821i	$-0.941801\pi$
$29$	− 47760.3i	− 0.363642i	0.983332	−	0.181821i	$-0.0581991\pi$
$30$	0	0
$31$	−38469.7	−0.231928	−0.115964	−	0.993253i	$-0.536996\pi$
$31$	−38469.7	−0.231928	−0.115964	+	0.993253i	$0.536996\pi$
$32$	0	0
$33$	52838.8	0.255950
$34$	0	0
$35$	197986.i	0.780540i
$36$	0	0
$37$	230575.i	0.748352i	0.927358	+	0.374176i	$0.122074\pi$
$37$	230575.i	0.748352i	−0.927358	+	0.374176i	$0.877926\pi$
$38$	0	0
$39$	40308.9	0.108812
$40$	0	0
$41$	503187.	1.14021	0.570106	−	0.821571i	$-0.306902\pi$
$41$	503187.	1.14021	0.570106	+	0.821571i	$0.306902\pi$
$42$	0	0
$43$	74103.4i	0.142134i	0.997472	+	0.0710671i	$0.0226404\pi$
$43$	74103.4i	0.142134i	−0.997472	+	0.0710671i	$0.977360\pi$
$44$	0	0
$45$	− 159563.i	− 0.261029i
$46$	0	0
$47$	−887487.	−1.24687	−0.623433	−	0.781877i	$-0.714263\pi$
$47$	−887487.	−1.24687	−0.623433	+	0.781877i	$0.714263\pi$
$48$	0	0
$49$	−5347.93	−0.00649381
$50$	0	0
$51$	− 395869.i	− 0.417884i
$52$	0	0
$53$	1.01950e6i	0.940633i	0.882498	+	0.470317i	$0.155860\pi$
$53$	1.01950e6i	0.940633i	−0.882498	+	0.470317i	$0.844140\pi$
$54$	0	0
$55$	−428346.	−0.347156
$56$	0	0
$57$	878434.	0.628271
$58$	0	0
$59$	− 956905.i	− 0.606578i	−0.952899	−	0.303289i	$-0.901915\pi$
$59$	− 956905.i	− 0.606578i	0.952899	−	0.303289i	$-0.0980847\pi$
$60$	0	0
$61$	− 1.51951e6i	− 0.857133i	−0.903510	−	0.428567i	$-0.859019\pi$
$61$	− 1.51951e6i	− 0.857133i	0.903510	−	0.428567i	$-0.140981\pi$
$62$	0	0
$63$	−659411.	−0.332249
$64$	0	0
$65$	−326770.	−0.147586
$66$	0	0
$67$	− 2.12707e6i	− 0.864011i	−0.901871	−	0.432006i	$-0.857806\pi$
$67$	− 2.12707e6i	− 0.864011i	0.901871	−	0.432006i	$-0.142194\pi$
$68$	0	0
$69$	655124.i	0.240078i
$70$	0	0
$71$	1.47131e6	0.487866	0.243933	−	0.969792i	$-0.421562\pi$
$71$	1.47131e6	0.487866	0.243933	+	0.969792i	$0.421562\pi$
$72$	0	0
$73$	−4.76425e6	−1.43339	−0.716695	−	0.697387i	$-0.754346\pi$
$73$	−4.76425e6	−1.43339	−0.716695	+	0.697387i	$0.754346\pi$
$74$	0	0
$75$	− 815853.i	− 0.223305i
$76$	0	0
$77$	1.77018e6i	0.441876i
$78$	0	0
$79$	−4.66203e6	−1.06385	−0.531925	−	0.846791i	$-0.678531\pi$
$79$	−4.66203e6	−1.06385	−0.531925	+	0.846791i	$0.678531\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	6.45883e6i	1.23988i	0.784648	+	0.619941i	$0.212844\pi$
$83$	6.45883e6i	1.23988i	−0.784648	+	0.619941i	$0.787156\pi$
$84$	0	0
$85$	3.20917e6i	0.566796i
$86$	0	0
$87$	−1.28953e6	−0.209949
$88$	0	0
$89$	3.18504e6	0.478906	0.239453	−	0.970908i	$-0.423032\pi$
$89$	3.18504e6	0.478906	0.239453	+	0.970908i	$0.423032\pi$
$90$	0	0
$91$	1.35041e6i	0.187854i
$92$	0	0
$93$	1.03868e6i	0.133904i
$94$	0	0
$95$	−7.12115e6	−0.852153
$96$	0	0
$97$	−1.36353e7	−1.51692	−0.758462	−	0.651717i	$-0.774049\pi$
$97$	−1.36353e7	−1.51692	−0.758462	+	0.651717i	$0.774049\pi$
$98$	0	0
$99$	− 1.42665e6i	− 0.147773i
$100$	0	0
$101$	− 695473.i	− 0.0671670i	−0.999436	−	0.0335835i	$-0.989308\pi$
$101$	− 695473.i	− 0.0671670i	0.999436	−	0.0335835i	$-0.0106920\pi$
$102$	0	0
$103$	2.88699e6	0.260324	0.130162	−	0.991493i	$-0.458450\pi$
$103$	2.88699e6	0.260324	0.130162	+	0.991493i	$0.458450\pi$
$104$	0	0
$105$	5.34561e6	0.450645
$106$	0	0
$107$	1.47604e7i	1.16481i	0.812898	+	0.582406i	$0.197888\pi$
$107$	1.47604e7i	1.16481i	−0.812898	+	0.582406i	$0.802112\pi$
$108$	0	0
$109$	1.87852e7i	1.38938i	0.719307	+	0.694692i	$0.244459\pi$
$109$	1.87852e7i	1.38938i	−0.719307	+	0.694692i	$0.755541\pi$
$110$	0	0
$111$	6.22552e6	0.432061
$112$	0	0
$113$	−1.76048e6	−0.114778	−0.0573888	−	0.998352i	$-0.518277\pi$
$113$	−1.76048e6	−0.114778	−0.0573888	+	0.998352i	$0.518277\pi$
$114$	0	0
$115$	− 5.31086e6i	− 0.325629i
$116$	0	0
$117$	− 1.08834e6i	− 0.0628224i
$118$	0	0
$119$	1.32622e7	0.721443
$120$	0	0
$121$	1.56573e7	0.803469
$122$	0	0
$123$	− 1.35860e7i	− 0.658302i
$124$	0	0
$125$	2.37138e7i	1.08597i
$126$	0	0
$127$	2.27489e7	0.985477	0.492738	−	0.870177i	$-0.335996\pi$
$127$	2.27489e7	0.985477	0.492738	+	0.870177i	$0.335996\pi$
$128$	0	0
$129$	2.00079e6	0.0820612
$130$	0	0
$131$	5.07701e6i	0.197314i	0.995121	+	0.0986572i	$0.0314547\pi$
$131$	5.07701e6i	0.197314i	−0.995121	+	0.0986572i	$0.968545\pi$
$132$	0	0
$133$	2.94289e7i	1.08466i
$134$	0	0
$135$	−4.30820e6	−0.150705
$136$	0	0
$137$	2.02216e7	0.671882	0.335941	−	0.941883i	$-0.390946\pi$
$137$	2.02216e7	0.671882	0.335941	+	0.941883i	$0.390946\pi$
$138$	0	0
$139$	− 3.10736e7i	− 0.981384i	−0.871333	−	0.490692i	$-0.836744\pi$
$139$	− 3.10736e7i	− 0.981384i	0.871333	−	0.490692i	$-0.163256\pi$
$140$	0	0
$141$	2.39622e7i	0.719878i
$142$	0	0
$143$	−2.92164e6	−0.0835508
$144$	0	0
$145$	1.04537e7	0.284763
$146$	0	0
$147$	144394.i	0.00374921i
$148$	0	0
$149$	2.49409e7i	0.617676i	0.951115	+	0.308838i	$0.0999401\pi$
$149$	2.49409e7i	0.617676i	−0.951115	+	0.308838i	$0.900060\pi$
$150$	0	0
$151$	−2.24230e7	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$151$	−2.24230e7	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$152$	0	0
$153$	−1.06885e7	−0.241266
$154$	0	0
$155$	− 8.42023e6i	− 0.181620i
$156$	0	0
$157$	− 2.43020e6i	− 0.0501180i	−0.999686	−	0.0250590i	$-0.992023\pi$
$157$	− 2.43020e6i	− 0.0501180i	0.999686	−	0.0250590i	$-0.00797736\pi$
$158$	0	0
$159$	2.75264e7	0.543075
$160$	0	0
$161$	−2.19477e7	−0.414474
$162$	0	0
$163$	1.21622e7i	0.219966i	0.993933	+	0.109983i	$0.0350797\pi$
$163$	1.21622e7i	0.219966i	−0.993933	+	0.109983i	$0.964920\pi$
$164$	0	0
$165$	1.15653e7i	0.200431i
$166$	0	0
$167$	4.96319e7	0.824619	0.412310	−	0.911044i	$-0.364722\pi$
$167$	4.96319e7	0.824619	0.412310	+	0.911044i	$0.364722\pi$
$168$	0	0
$169$	6.05197e7	0.964480
$170$	0	0
$171$	− 2.37177e7i	− 0.362732i
$172$	0	0
$173$	1.16889e8i	1.71638i	0.513334	+	0.858189i	$0.328410\pi$
$173$	1.16889e8i	1.71638i	−0.513334	+	0.858189i	$0.671590\pi$
$174$	0	0
$175$	2.73323e7	0.385517
$176$	0	0
$177$	−2.58364e7	−0.350208
$178$	0	0
$179$	6.32640e7i	0.824462i	0.911079	+	0.412231i	$0.135250\pi$
$179$	6.32640e7i	0.824462i	−0.911079	+	0.412231i	$0.864750\pi$
$180$	0	0
$181$	1.36460e8i	1.71053i	0.518191	+	0.855265i	$0.326605\pi$
$181$	1.36460e8i	1.71053i	−0.518191	+	0.855265i	$0.673395\pi$
$182$	0	0
$183$	−4.10267e7	−0.494866
$184$	0	0
$185$	−5.04681e7	−0.586025
$186$	0	0
$187$	2.86931e7i	0.320872i
$188$	0	0
$189$	1.78041e7i	0.191824i
$190$	0	0
$191$	4.55629e6	0.0473146	0.0236573	−	0.999720i	$-0.492469\pi$
$191$	4.55629e6	0.0473146	0.0236573	+	0.999720i	$0.492469\pi$
$192$	0	0
$193$	−1.40685e7	−0.140864	−0.0704318	−	0.997517i	$-0.522438\pi$
$193$	−1.40685e7	−0.140864	−0.0704318	+	0.997517i	$0.522438\pi$
$194$	0	0
$195$	8.82279e6i	0.0852089i
$196$	0	0
$197$	1.95523e8i	1.82208i	0.412321	+	0.911039i	$0.364718\pi$
$197$	1.95523e8i	1.82208i	−0.412321	+	0.911039i	$0.635282\pi$
$198$	0	0
$199$	−1.17120e8	−1.05352	−0.526761	−	0.850013i	$-0.676594\pi$
$199$	−1.17120e8	−1.05352	−0.526761	+	0.850013i	$0.676594\pi$
$200$	0	0
$201$	−5.74308e7	−0.498837
$202$	0	0
$203$	− 4.32011e7i	− 0.362459i
$204$	0	0
$205$	1.10137e8i	0.892885i
$206$	0	0
$207$	1.76884e7	0.138609
$208$	0	0
$209$	−6.36700e7	−0.482417
$210$	0	0
$211$	9.76752e7i	0.715807i	0.933759	+	0.357903i	$0.116508\pi$
$211$	9.76752e7i	0.715807i	−0.933759	+	0.357903i	$0.883492\pi$
$212$	0	0
$213$	− 3.97254e7i	− 0.281669i
$214$	0	0
$215$	−1.62197e7	−0.111303
$216$	0	0
$217$	−3.47974e7	−0.231174
$218$	0	0
$219$	1.28635e8i	0.827568i
$220$	0	0
$221$	2.18890e7i	0.136412i
$222$	0	0
$223$	−9.49268e6	−0.0573221	−0.0286610	−	0.999589i	$-0.509124\pi$
$223$	−9.49268e6	−0.0573221	−0.0286610	+	0.999589i	$0.509124\pi$
$224$	0	0
$225$	−2.20280e7	−0.128925
$226$	0	0
$227$	− 2.45827e7i	− 0.139489i	−0.997565	−	0.0697443i	$-0.977782\pi$
$227$	− 2.45827e7i	− 0.139489i	0.997565	−	0.0697443i	$-0.0222183\pi$
$228$	0	0
$229$	3.38153e8i	1.86076i	0.366602	+	0.930378i	$0.380521\pi$
$229$	3.38153e8i	1.86076i	−0.366602	+	0.930378i	$0.619479\pi$
$230$	0	0
$231$	4.77949e7	0.255117
$232$	0	0
$233$	−6.00882e7	−0.311203	−0.155602	−	0.987820i	$-0.549732\pi$
$233$	−6.00882e7	−0.311203	−0.155602	+	0.987820i	$0.549732\pi$
$234$	0	0
$235$	− 1.94253e8i	− 0.976404i
$236$	0	0
$237$	1.25875e8i	0.614215i
$238$	0	0
$239$	−2.80834e8	−1.33063	−0.665315	−	0.746563i	$-0.731703\pi$
$239$	−2.80834e8	−1.33063	−0.665315	+	0.746563i	$0.731703\pi$
$240$	0	0
$241$	1.86107e8	0.856451	0.428225	−	0.903672i	$-0.359139\pi$
$241$	1.86107e8	0.856451	0.428225	+	0.903672i	$0.359139\pi$
$242$	0	0
$243$	− 1.43489e7i	− 0.0641500i
$244$	0	0
$245$	− 1.17055e6i	− 0.00508522i
$246$	0	0
$247$	−4.85716e7	−0.205089
$248$	0	0
$249$	1.74388e8	0.715847
$250$	0	0
$251$	− 4.69957e8i	− 1.87586i	−0.346828	−	0.937929i	$-0.612741\pi$
$251$	− 4.69957e8i	− 1.87586i	0.346828	−	0.937929i	$-0.387259\pi$
$252$	0	0
$253$	− 4.74842e7i	− 0.184343i
$254$	0	0
$255$	8.66476e7	0.327240
$256$	0	0
$257$	2.21615e8	0.814392	0.407196	−	0.913341i	$-0.366507\pi$
$257$	2.21615e8	0.814392	0.407196	+	0.913341i	$0.366507\pi$
$258$	0	0
$259$	2.08565e8i	0.745919i
$260$	0	0
$261$	3.48172e7i	0.121214i
$262$	0	0
$263$	1.13618e8	0.385125	0.192562	−	0.981285i	$-0.438320\pi$
$263$	1.13618e8	0.385125	0.192562	+	0.981285i	$0.438320\pi$
$264$	0	0
$265$	−2.23147e8	−0.736598
$266$	0	0
$267$	− 8.59961e7i	− 0.276496i
$268$	0	0
$269$	5.81217e8i	1.82056i	0.413991	+	0.910281i	$0.364134\pi$
$269$	5.81217e8i	1.82056i	−0.413991	+	0.910281i	$0.635866\pi$
$270$	0	0
$271$	−5.49084e8	−1.67589	−0.837946	−	0.545753i	$-0.816244\pi$
$271$	−5.49084e8	−1.67589	−0.837946	+	0.545753i	$0.816244\pi$
$272$	0	0
$273$	3.64611e7	0.108458
$274$	0	0
$275$	5.91341e7i	0.171464i
$276$	0	0
$277$	8.37027e6i	0.0236625i	0.999930	+	0.0118312i	$0.00376609\pi$
$277$	8.37027e6i	0.0236625i	−0.999930	+	0.0118312i	$0.996234\pi$
$278$	0	0
$279$	2.80444e7	0.0773093
$280$	0	0
$281$	3.13256e8	0.842222	0.421111	−	0.907009i	$-0.361640\pi$
$281$	3.13256e8	0.842222	0.421111	+	0.907009i	$0.361640\pi$
$282$	0	0
$283$	− 1.60420e8i	− 0.420733i	−0.977623	−	0.210366i	$-0.932534\pi$
$283$	− 1.60420e8i	− 0.420733i	0.977623	−	0.210366i	$-0.0674657\pi$
$284$	0	0
$285$	1.92271e8i	0.491991i
$286$	0	0
$287$	4.55153e8	1.13650
$288$	0	0
$289$	−1.95370e8	−0.476118
$290$	0	0
$291$	3.68153e8i	0.875797i
$292$	0	0
$293$	− 3.18079e8i	− 0.738751i	−0.929280	−	0.369376i	$-0.879572\pi$
$293$	− 3.18079e8i	− 0.738751i	0.929280	−	0.369376i	$-0.120428\pi$
$294$	0	0
$295$	2.09447e8	0.475003
$296$	0	0
$297$	−3.85195e7	−0.0853165
$298$	0	0
$299$	− 3.62241e7i	− 0.0783697i
$300$	0	0
$301$	6.70296e7i	0.141672i
$302$	0	0
$303$	−1.87778e7	−0.0387789
$304$	0	0
$305$	3.32589e8	0.671210
$306$	0	0
$307$	7.12299e8i	1.40501i	0.711681	+	0.702503i	$0.247934\pi$
$307$	7.12299e8i	1.40501i	−0.711681	+	0.702503i	$0.752066\pi$
$308$	0	0
$309$	− 7.79487e7i	− 0.150298i
$310$	0	0
$311$	3.23509e8	0.609853	0.304927	−	0.952376i	$-0.401368\pi$
$311$	3.23509e8	0.609853	0.304927	+	0.952376i	$0.401368\pi$
$312$	0	0
$313$	−2.33838e8	−0.431033	−0.215517	−	0.976500i	$-0.569144\pi$
$313$	−2.33838e8	−0.431033	−0.215517	+	0.976500i	$0.569144\pi$
$314$	0	0
$315$	− 1.44331e8i	− 0.260180i
$316$	0	0
$317$	− 2.11496e8i	− 0.372902i	−0.982464	−	0.186451i	$-0.940301\pi$
$317$	− 2.11496e8i	− 0.372902i	0.982464	−	0.186451i	$-0.0596985\pi$
$318$	0	0
$319$	9.34665e7	0.161209
$320$	0	0
$321$	3.98531e8	0.672504
$322$	0	0
$323$	4.77016e8i	0.787634i
$324$	0	0
$325$	4.51113e7i	0.0728943i
$326$	0	0
$327$	5.07199e8	0.802161
$328$	0	0
$329$	−8.02769e8	−1.24281
$330$	0	0
$331$	− 3.86427e8i	− 0.585693i	−0.956160	−	0.292846i	$-0.905398\pi$
$331$	− 3.86427e8i	− 0.585693i	0.956160	−	0.292846i	$-0.0946025\pi$
$332$	0	0
$333$	− 1.68089e8i	− 0.249451i
$334$	0	0
$335$	4.65571e8	0.676596
$336$	0	0
$337$	−2.02746e8	−0.288568	−0.144284	−	0.989536i	$-0.546088\pi$
$337$	−2.02746e8	−0.288568	−0.144284	+	0.989536i	$0.546088\pi$
$338$	0	0
$339$	4.75331e7i	0.0662669i
$340$	0	0
$341$	− 7.52850e7i	− 0.102818i
$342$	0	0
$343$	−7.49766e8	−1.00322
$344$	0	0
$345$	−1.43393e8	−0.188002
$346$	0	0
$347$	− 5.75722e8i	− 0.739706i	−0.929090	−	0.369853i	$-0.879408\pi$
$347$	− 5.75722e8i	− 0.739706i	0.929090	−	0.369853i	$-0.120592\pi$
$348$	0	0
$349$	− 4.84660e8i	− 0.610307i	−0.952303	−	0.305153i	$-0.901292\pi$
$349$	− 4.84660e8i	− 0.610307i	0.952303	−	0.305153i	$-0.0987077\pi$
$350$	0	0
$351$	−2.93852e7	−0.0362705
$352$	0	0
$353$	−6.21228e7	−0.0751691	−0.0375846	−	0.999293i	$-0.511966\pi$
$353$	−6.21228e7	−0.0751691	−0.0375846	+	0.999293i	$0.511966\pi$
$354$	0	0
$355$	3.22040e8i	0.382041i
$356$	0	0
$357$	− 3.58080e8i	− 0.416525i
$358$	0	0
$359$	−4.14354e8	−0.472651	−0.236326	−	0.971674i	$-0.575943\pi$
$359$	−4.14354e8	−0.472651	−0.236326	+	0.971674i	$0.575943\pi$
$360$	0	0
$361$	−1.64627e8	−0.184173
$362$	0	0
$363$	− 4.22748e8i	− 0.463883i
$364$	0	0
$365$	− 1.04280e9i	− 1.12247i
$366$	0	0
$367$	−7.04021e8	−0.743454	−0.371727	−	0.928342i	$-0.621234\pi$
$367$	−7.04021e8	−0.743454	−0.371727	+	0.928342i	$0.621234\pi$
$368$	0	0
$369$	−3.66823e8	−0.380071
$370$	0	0
$371$	9.22177e8i	0.937574i
$372$	0	0
$373$	− 3.29622e8i	− 0.328878i	−0.986387	−	0.164439i	$-0.947419\pi$
$373$	− 3.29622e8i	− 0.328878i	0.986387	−	0.164439i	$-0.0525814\pi$
$374$	0	0
$375$	6.40272e8	0.626982
$376$	0	0
$377$	7.13024e7	0.0685345
$378$	0	0
$379$	− 1.30902e9i	− 1.23512i	−0.786525	−	0.617559i	$-0.788122\pi$
$379$	− 1.30902e9i	− 1.23512i	0.786525	−	0.617559i	$-0.211878\pi$
$380$	0	0
$381$	− 6.14219e8i	− 0.568965i
$382$	0	0
$383$	1.58491e9	1.44148	0.720739	−	0.693207i	$-0.243803\pi$
$383$	1.58491e9	1.44148	0.720739	+	0.693207i	$0.243803\pi$
$384$	0	0
$385$	−3.87456e8	−0.346027
$386$	0	0
$387$	− 5.40214e7i	− 0.0473781i
$388$	0	0
$389$	− 3.62442e8i	− 0.312187i	−0.987742	−	0.156094i	$-0.950110\pi$
$389$	− 3.62442e8i	− 0.312187i	0.987742	−	0.156094i	$-0.0498902\pi$
$390$	0	0
$391$	−3.55752e8	−0.300974
$392$	0	0
$393$	1.37079e8	0.113919
$394$	0	0
$395$	− 1.02042e9i	− 0.833088i
$396$	0	0
$397$	− 1.96171e9i	− 1.57351i	−0.617268	−	0.786753i	$-0.711761\pi$
$397$	− 1.96171e9i	− 1.57351i	0.617268	−	0.786753i	$-0.288239\pi$
$398$	0	0
$399$	7.94580e8	0.626228
$400$	0	0
$401$	−5.52413e8	−0.427818	−0.213909	−	0.976854i	$-0.568620\pi$
$401$	−5.52413e8	−0.427818	−0.213909	+	0.976854i	$0.568620\pi$
$402$	0	0
$403$	− 5.74323e7i	− 0.0437108i
$404$	0	0
$405$	1.16322e8i	0.0870097i
$406$	0	0
$407$	−4.51234e8	−0.331758
$408$	0	0
$409$	2.13185e9	1.54072	0.770362	−	0.637607i	$-0.220076\pi$
$409$	2.13185e9	1.54072	0.770362	+	0.637607i	$0.220076\pi$
$410$	0	0
$411$	− 5.45982e8i	− 0.387911i
$412$	0	0
$413$	− 8.65560e8i	− 0.604605i
$414$	0	0
$415$	−1.41371e9	−0.970936
$416$	0	0
$417$	−8.38986e8	−0.566603
$418$	0	0
$419$	2.74843e9i	1.82530i	0.408737	+	0.912652i	$0.365969\pi$
$419$	2.74843e9i	1.82530i	−0.408737	+	0.912652i	$0.634031\pi$
$420$	0	0
$421$	− 2.34645e9i	− 1.53258i	−0.642495	−	0.766290i	$-0.722101\pi$
$421$	− 2.34645e9i	− 1.53258i	0.642495	−	0.766290i	$-0.277899\pi$
$422$	0	0
$423$	6.46978e8	0.415622
$424$	0	0
$425$	4.43033e8	0.279946
$426$	0	0
$427$	− 1.37446e9i	− 0.854345i
$428$	0	0
$429$	7.88843e7i	0.0482381i
$430$	0	0
$431$	2.81903e7	0.0169601	0.00848006	−	0.999964i	$-0.497301\pi$
$431$	2.81903e7	0.0169601	0.00848006	+	0.999964i	$0.497301\pi$
$432$	0	0
$433$	1.49955e9	0.887677	0.443838	−	0.896107i	$-0.353616\pi$
$433$	1.49955e9	0.887677	0.443838	+	0.896107i	$0.353616\pi$
$434$	0	0
$435$	− 2.82251e8i	− 0.164408i
$436$	0	0
$437$	− 7.89414e8i	− 0.452502i
$438$	0	0
$439$	−2.88664e9	−1.62842	−0.814211	−	0.580568i	$-0.802830\pi$
$439$	−2.88664e9	−1.62842	−0.814211	+	0.580568i	$0.802830\pi$
$440$	0	0
$441$	3.89864e6	0.00216460
$442$	0	0
$443$	− 2.64909e8i	− 0.144772i	−0.997377	−	0.0723860i	$-0.976939\pi$
$443$	− 2.64909e8i	− 0.144772i	0.997377	−	0.0723860i	$-0.0230613\pi$
$444$	0	0
$445$	6.97140e8i	0.375025i
$446$	0	0
$447$	6.73405e8	0.356615
$448$	0	0
$449$	1.26551e8	0.0659787	0.0329893	−	0.999456i	$-0.489497\pi$
$449$	1.26551e8	0.0659787	0.0329893	+	0.999456i	$0.489497\pi$
$450$	0	0
$451$	9.84733e8i	0.505476i
$452$	0	0
$453$	6.05422e8i	0.305995i
$454$	0	0
$455$	−2.95577e8	−0.147106
$456$	0	0
$457$	6.87578e8	0.336989	0.168494	−	0.985703i	$-0.446109\pi$
$457$	6.87578e8	0.336989	0.168494	+	0.985703i	$0.446109\pi$
$458$	0	0
$459$	2.88589e8i	0.139295i
$460$	0	0
$461$	− 2.34169e9i	− 1.11321i	−0.830779	−	0.556603i	$-0.812105\pi$
$461$	− 2.34169e9i	− 1.11321i	0.830779	−	0.556603i	$-0.187895\pi$
$462$	0	0
$463$	−2.70455e9	−1.26637	−0.633187	−	0.773999i	$-0.718253\pi$
$463$	−2.70455e9	−1.26637	−0.633187	+	0.773999i	$0.718253\pi$
$464$	0	0
$465$	−2.27346e8	−0.104858
$466$	0	0
$467$	1.92358e9i	0.873979i	0.899467	+	0.436990i	$0.143955\pi$
$467$	1.92358e9i	0.873979i	−0.899467	+	0.436990i	$0.856045\pi$
$468$	0	0
$469$	− 1.92402e9i	− 0.861201i
$470$	0	0
$471$	−6.56154e7	−0.0289356
$472$	0	0
$473$	−1.45020e8	−0.0630106
$474$	0	0
$475$	9.83090e8i	0.420887i
$476$	0	0
$477$	− 7.43213e8i	− 0.313544i
$478$	0	0
$479$	1.50399e9	0.625273	0.312637	−	0.949873i	$-0.398788\pi$
$479$	1.50399e9	0.625273	0.312637	+	0.949873i	$0.398788\pi$
$480$	0	0
$481$	−3.44231e8	−0.141040
$482$	0	0
$483$	5.92587e8i	0.239297i
$484$	0	0
$485$	− 2.98449e9i	− 1.18788i
$486$	0	0
$487$	−2.17828e9	−0.854598	−0.427299	−	0.904110i	$-0.640535\pi$
$487$	−2.17828e9	−0.854598	−0.427299	+	0.904110i	$0.640535\pi$
$488$	0	0
$489$	3.28380e8	0.126998
$490$	0	0
$491$	− 2.21725e9i	− 0.845337i	−0.906284	−	0.422669i	$-0.861093\pi$
$491$	− 2.21725e9i	− 0.845337i	0.906284	−	0.422669i	$-0.138907\pi$
$492$	0	0
$493$	− 7.00252e8i	− 0.263203i
$494$	0	0
$495$	3.12264e8	0.115719
$496$	0	0
$497$	1.33086e9	0.486279
$498$	0	0
$499$	3.34308e9i	1.20447i	0.798320	+	0.602234i	$0.205723\pi$
$499$	3.34308e9i	1.20447i	−0.798320	+	0.602234i	$0.794277\pi$
$500$	0	0
$501$	− 1.34006e9i	− 0.476094i
$502$	0	0
$503$	5.18036e9	1.81498	0.907490	−	0.420073i	$-0.137996\pi$
$503$	5.18036e9	1.81498	0.907490	+	0.420073i	$0.137996\pi$
$504$	0	0
$505$	1.52225e8	0.0525976
$506$	0	0
$507$	− 1.63403e9i	− 0.556843i
$508$	0	0
$509$	− 3.12633e8i	− 0.105081i	−0.998619	−	0.0525403i	$-0.983268\pi$
$509$	− 3.12633e8i	− 0.105081i	0.998619	−	0.0525403i	$-0.0167318\pi$
$510$	0	0
$511$	−4.30946e9	−1.42873
$512$	0	0
$513$	−6.40378e8	−0.209424
$514$	0	0
$515$	6.31902e8i	0.203857i
$516$	0	0
$517$	− 1.73681e9i	− 0.552758i
$518$	0	0
$519$	3.15601e9	0.990951
$520$	0	0
$521$	−2.44796e9	−0.758355	−0.379177	−	0.925324i	$-0.623793\pi$
$521$	−2.44796e9	−0.758355	−0.379177	+	0.925324i	$0.623793\pi$
$522$	0	0
$523$	− 5.27503e9i	− 1.61239i	−0.591652	−	0.806194i	$-0.701524\pi$
$523$	− 5.27503e9i	− 1.61239i	0.591652	−	0.806194i	$-0.298476\pi$
$524$	0	0
$525$	− 7.37973e8i	− 0.222578i
$526$	0	0
$527$	−5.64036e8	−0.167869
$528$	0	0
$529$	−2.81609e9	−0.827088
$530$	0	0
$531$	6.97583e8i	0.202193i
$532$	0	0
$533$	7.51219e8i	0.214892i
$534$	0	0
$535$	−3.23075e9	−0.912149
$536$	0	0
$537$	1.70813e9	0.476003
$538$	0	0
$539$	− 1.04659e7i	− 0.00287882i
$540$	0	0
$541$	5.50854e9i	1.49571i	0.663864	+	0.747853i	$0.268915\pi$
$541$	5.50854e9i	1.49571i	−0.663864	+	0.747853i	$0.731085\pi$
$542$	0	0
$543$	3.68442e9	0.987575
$544$	0	0
$545$	−4.11169e9	−1.08801
$546$	0	0
$547$	9.96556e8i	0.260343i	0.991491	+	0.130172i	$0.0415528\pi$
$547$	9.96556e8i	0.260343i	−0.991491	+	0.130172i	$0.958447\pi$
$548$	0	0
$549$	1.10772e9i	0.285711i
$550$	0	0
$551$	1.55386e9	0.395714
$552$	0	0
$553$	−4.21700e9	−1.06039
$554$	0	0
$555$	1.36264e9i	0.338342i
$556$	0	0
$557$	− 9.27851e8i	− 0.227502i	−0.993509	−	0.113751i	$-0.963713\pi$
$557$	− 9.27851e8i	− 0.227502i	0.993509	−	0.113751i	$-0.0362866\pi$
$558$	0	0
$559$	−1.10631e8	−0.0267876
$560$	0	0
$561$	7.74713e8	0.185255
$562$	0	0
$563$	− 6.01459e9i	− 1.42045i	−0.703974	−	0.710226i	$-0.748593\pi$
$563$	− 6.01459e9i	− 1.42045i	0.703974	−	0.710226i	$-0.251407\pi$
$564$	0	0
$565$	− 3.85334e8i	− 0.0898809i
$566$	0	0
$567$	4.80710e8	0.110750
$568$	0	0
$569$	4.33955e9	0.987533	0.493766	−	0.869595i	$-0.335620\pi$
$569$	4.33955e9	0.987533	0.493766	+	0.869595i	$0.335620\pi$
$570$	0	0
$571$	− 2.36325e9i	− 0.531232i	−0.964079	−	0.265616i	$-0.914425\pi$
$571$	− 2.36325e9i	− 0.531232i	0.964079	−	0.265616i	$-0.0855753\pi$
$572$	0	0
$573$	− 1.23020e8i	− 0.0273171i
$574$	0	0
$575$	−7.33176e8	−0.160831
$576$	0	0
$577$	−2.66330e9	−0.577171	−0.288586	−	0.957454i	$-0.593185\pi$
$577$	−2.66330e9	−0.577171	−0.288586	+	0.957454i	$0.593185\pi$
$578$	0	0
$579$	3.79851e8i	0.0813276i
$580$	0	0
$581$	5.84228e9i	1.23585i
$582$	0	0
$583$	−1.99515e9	−0.416999
$584$	0	0
$585$	2.38215e8	0.0491954
$586$	0	0
$587$	− 3.21780e9i	− 0.656638i	−0.944567	−	0.328319i	$-0.893518\pi$
$587$	− 3.21780e9i	− 0.656638i	0.944567	−	0.328319i	$-0.106482\pi$
$588$	0	0
$589$	− 1.25160e9i	− 0.252383i
$590$	0	0
$591$	5.27913e9	1.05198
$592$	0	0
$593$	8.62179e9	1.69788	0.848938	−	0.528492i	$-0.177242\pi$
$593$	8.62179e9	1.69788	0.848938	+	0.528492i	$0.177242\pi$
$594$	0	0
$595$	2.90283e9i	0.564952i
$596$	0	0
$597$	3.16223e9i	0.608251i
$598$	0	0
$599$	1.04802e10	1.99240	0.996201	−	0.0870868i	$-0.0277558\pi$
$599$	1.04802e10	1.99240	0.996201	+	0.0870868i	$0.0277558\pi$
$600$	0	0
$601$	1.98735e9	0.373433	0.186717	−	0.982414i	$-0.440215\pi$
$601$	1.98735e9	0.373433	0.186717	+	0.982414i	$0.440215\pi$
$602$	0	0
$603$	1.55063e9i	0.288004i
$604$	0	0
$605$	3.42707e9i	0.629186i
$606$	0	0
$607$	−6.29420e9	−1.14230	−0.571150	−	0.820846i	$-0.693503\pi$
$607$	−6.29420e9	−1.14230	−0.571150	+	0.820846i	$0.693503\pi$
$608$	0	0
$609$	−1.16643e9	−0.209266
$610$	0	0
$611$	− 1.32495e9i	− 0.234993i
$612$	0	0
$613$	1.40995e8i	0.0247225i	0.999924	+	0.0123612i	$0.00393481\pi$
$613$	1.40995e8i	0.0247225i	−0.999924	+	0.0123612i	$0.996065\pi$
$614$	0	0
$615$	2.97370e9	0.515507
$616$	0	0
$617$	−8.20227e9	−1.40584	−0.702920	−	0.711269i	$-0.748121\pi$
$617$	−8.20227e9	−1.40584	−0.702920	+	0.711269i	$0.748121\pi$
$618$	0	0
$619$	9.32340e9i	1.58000i	0.613107	+	0.790000i	$0.289919\pi$
$619$	9.32340e9i	1.58000i	−0.613107	+	0.790000i	$0.710081\pi$
$620$	0	0
$621$	− 4.77586e8i	− 0.0800259i
$622$	0	0
$623$	2.88100e9	0.477348
$624$	0	0
$625$	−2.82978e9	−0.463630
$626$	0	0
$627$	1.71909e9i	0.278524i
$628$	0	0
$629$	3.38065e9i	0.541655i
$630$	0	0
$631$	−8.15522e9	−1.29221	−0.646104	−	0.763249i	$-0.723603\pi$
$631$	−8.15522e9	−1.29221	−0.646104	+	0.763249i	$0.723603\pi$
$632$	0	0
$633$	2.63723e9	0.413271
$634$	0	0
$635$	4.97926e9i	0.771714i
$636$	0	0
$637$	− 7.98405e6i	− 0.00122387i
$638$	0	0
$639$	−1.07259e9	−0.162622
$640$	0	0
$641$	1.02076e10	1.53081	0.765404	−	0.643550i	$-0.222539\pi$
$641$	1.02076e10	1.53081	0.765404	+	0.643550i	$0.222539\pi$
$642$	0	0
$643$	− 2.61713e9i	− 0.388229i	−0.980979	−	0.194114i	$-0.937817\pi$
$643$	− 2.61713e9i	− 0.388229i	0.980979	−	0.194114i	$-0.0621833\pi$
$644$	0	0
$645$	4.37932e8i	0.0642611i
$646$	0	0
$647$	−1.19935e9	−0.174093	−0.0870464	−	0.996204i	$-0.527743\pi$
$647$	−1.19935e9	−0.174093	−0.0870464	+	0.996204i	$0.527743\pi$
$648$	0	0
$649$	1.87266e9	0.268907
$650$	0	0
$651$	9.39531e8i	0.133468i
$652$	0	0
$653$	9.90994e9i	1.39276i	0.717675	+	0.696378i	$0.245206\pi$
$653$	9.90994e9i	1.39276i	−0.717675	+	0.696378i	$0.754794\pi$
$654$	0	0
$655$	−1.11125e9	−0.154514
$656$	0	0
$657$	3.47314e9	0.477796
$658$	0	0
$659$	3.18930e9i	0.434107i	0.976160	+	0.217054i	$0.0696446\pi$
$659$	3.18930e9i	0.434107i	−0.976160	+	0.217054i	$0.930355\pi$
$660$	0	0
$661$	1.35933e10i	1.83071i	0.402648	+	0.915355i	$0.368090\pi$
$661$	1.35933e10i	1.83071i	−0.402648	+	0.915355i	$0.631910\pi$
$662$	0	0
$663$	5.91002e8	0.0787574
$664$	0	0
$665$	−6.44138e9	−0.849382
$666$	0	0
$667$	1.15885e9i	0.151212i
$668$	0	0
$669$	2.56302e8i	0.0330949i
$670$	0	0
$671$	2.97367e9	0.379982
$672$	0	0
$673$	4.29616e9	0.543285	0.271643	−	0.962398i	$-0.412433\pi$
$673$	4.29616e9	0.543285	0.271643	+	0.962398i	$0.412433\pi$
$674$	0	0
$675$	5.94757e8i	0.0744348i
$676$	0	0
$677$	5.90012e9i	0.730803i	0.930850	+	0.365402i	$0.119068\pi$
$677$	5.90012e9i	0.730803i	−0.930850	+	0.365402i	$0.880932\pi$
$678$	0	0
$679$	−1.23337e10	−1.51199
$680$	0	0
$681$	−6.63732e8	−0.0805337
$682$	0	0
$683$	− 6.16844e9i	− 0.740803i	−0.928872	−	0.370401i	$-0.879220\pi$
$683$	− 6.16844e9i	− 0.740803i	0.928872	−	0.370401i	$-0.120780\pi$
$684$	0	0
$685$	4.42608e9i	0.526142i
$686$	0	0
$687$	9.13014e9	1.07431
$688$	0	0
$689$	−1.52203e9	−0.177278
$690$	0	0
$691$	− 5.57866e9i	− 0.643216i	−0.946873	−	0.321608i	$-0.895777\pi$
$691$	− 5.57866e9i	− 0.643216i	0.946873	−	0.321608i	$-0.104223\pi$
$692$	0	0
$693$	− 1.29046e9i	− 0.147292i
$694$	0	0
$695$	6.80136e9	0.768509
$696$	0	0
$697$	7.37763e9	0.825282
$698$	0	0
$699$	1.62238e9i	0.179673i
$700$	0	0
$701$	− 6.02296e9i	− 0.660385i	−0.943914	−	0.330192i	$-0.892886\pi$
$701$	− 6.02296e9i	− 0.660385i	0.943914	−	0.330192i	$-0.107114\pi$
$702$	0	0
$703$	−7.50166e9	−0.814355
$704$	0	0
$705$	−5.24483e9	−0.563727
$706$	0	0
$707$	− 6.29084e8i	− 0.0669485i
$708$	0	0
$709$	9.08400e9i	0.957228i	0.878025	+	0.478614i	$0.158861\pi$
$709$	9.08400e9i	0.957228i	−0.878025	+	0.478614i	$0.841139\pi$
$710$	0	0
$711$	3.39862e9	0.354617
$712$	0	0
$713$	9.33423e8	0.0964418
$714$	0	0
$715$	− 6.39487e8i	− 0.0654275i
$716$	0	0
$717$	7.58252e9i	0.768239i
$718$	0	0
$719$	9.64006e9	0.967227	0.483614	−	0.875282i	$-0.339324\pi$
$719$	9.64006e9	0.967227	0.483614	+	0.875282i	$0.339324\pi$
$720$	0	0
$721$	2.61140e9	0.259478
$722$	0	0
$723$	− 5.02488e9i	− 0.494472i
$724$	0	0
$725$	− 1.44316e9i	− 0.140647i
$726$	0	0
$727$	−1.07015e10	−1.03293	−0.516467	−	0.856307i	$-0.672753\pi$
$727$	−1.07015e10	−1.03293	−0.516467	+	0.856307i	$0.672753\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	1.08649e9i	0.102876i
$732$	0	0
$733$	1.40489e10i	1.31758i	0.752326	+	0.658791i	$0.228932\pi$
$733$	1.40489e10i	1.31758i	−0.752326	+	0.658791i	$0.771068\pi$
$734$	0	0
$735$	−3.16049e7	−0.00293595
$736$	0	0
$737$	4.16266e9	0.383031
$738$	0	0
$739$	− 8.03116e9i	− 0.732020i	−0.930611	−	0.366010i	$-0.880724\pi$
$739$	− 8.03116e9i	− 0.732020i	0.930611	−	0.366010i	$-0.119276\pi$
$740$	0	0
$741$	1.31143e9i	0.118408i
$742$	0	0
$743$	−1.30525e10	−1.16744	−0.583718	−	0.811957i	$-0.698403\pi$
$743$	−1.30525e10	−1.16744	−0.583718	+	0.811957i	$0.698403\pi$
$744$	0	0
$745$	−5.45906e9	−0.483694
$746$	0	0
$747$	− 4.70849e9i	− 0.413294i
$748$	0	0
$749$	1.33514e10i	1.16102i
$750$	0	0
$751$	−1.40425e10	−1.20977	−0.604887	−	0.796311i	$-0.706782\pi$
$751$	−1.40425e10	−1.20977	−0.604887	+	0.796311i	$0.706782\pi$
$752$	0	0
$753$	−1.26888e10	−1.08303
$754$	0	0
$755$	− 4.90794e9i	− 0.415035i
$756$	0	0
$757$	9.68111e8i	0.0811129i	0.999177	+	0.0405564i	$0.0129131\pi$
$757$	9.68111e8i	0.0811129i	−0.999177	+	0.0405564i	$0.987087\pi$
$758$	0	0
$759$	−1.28207e9	−0.106431
$760$	0	0
$761$	−2.11474e10	−1.73944	−0.869721	−	0.493544i	$-0.835701\pi$
$761$	−2.11474e10	−1.73944	−0.869721	+	0.493544i	$0.835701\pi$
$762$	0	0
$763$	1.69920e10i	1.38487i
$764$	0	0
$765$	− 2.33949e9i	− 0.188932i
$766$	0	0
$767$	1.42858e9	0.114320
$768$	0	0
$769$	2.98489e9	0.236694	0.118347	−	0.992972i	$-0.462241\pi$
$769$	2.98489e9	0.236694	0.118347	+	0.992972i	$0.462241\pi$
$770$	0	0
$771$	− 5.98361e9i	− 0.470189i
$772$	0	0
$773$	4.76718e9i	0.371222i	0.982623	+	0.185611i	$0.0594264\pi$
$773$	4.76718e9i	0.371222i	−0.982623	+	0.185611i	$0.940574\pi$
$774$	0	0
$775$	−1.16243e9	−0.0897038
$776$	0	0
$777$	5.63124e9	0.430656
$778$	0	0
$779$	1.63710e10i	1.24078i
$780$	0	0
$781$	2.87935e9i	0.216279i
$782$	0	0
$783$	9.40065e8	0.0699829
$784$	0	0
$785$	5.31921e8	0.0392467
$786$	0	0
$787$	− 1.70072e10i	− 1.24371i	−0.783131	−	0.621857i	$-0.786378\pi$
$787$	− 1.70072e10i	− 1.24371i	0.783131	−	0.621857i	$-0.213622\pi$
$788$	0	0
$789$	− 3.06768e9i	− 0.222352i
$790$	0	0
$791$	−1.59243e9	−0.114404
$792$	0	0
$793$	2.26851e9	0.161541
$794$	0	0
$795$	6.02497e9i	0.425275i
$796$	0	0
$797$	6.66070e9i	0.466032i	0.972473	+	0.233016i	$0.0748594\pi$
$797$	6.66070e9i	0.466032i	−0.972473	+	0.233016i	$0.925141\pi$
$798$	0	0
$799$	−1.30122e10	−0.902477
$800$	0	0
$801$	−2.32190e9	−0.159635
$802$	0	0
$803$	− 9.32360e9i	− 0.635447i
$804$	0	0
$805$	− 4.80389e9i	− 0.324570i
$806$	0	0
$807$	1.56929e10	1.05110
$808$	0	0
$809$	−4.80483e9	−0.319050	−0.159525	−	0.987194i	$-0.550996\pi$
$809$	−4.80483e9	−0.319050	−0.159525	+	0.987194i	$0.550996\pi$
$810$	0	0
$811$	− 1.11907e10i	− 0.736687i	−0.929690	−	0.368344i	$-0.879925\pi$
$811$	− 1.11907e10i	− 0.736687i	0.929690	−	0.368344i	$-0.120075\pi$
$812$	0	0
$813$	1.48253e10i	0.967577i
$814$	0	0
$815$	−2.66206e9	−0.172253
$816$	0	0
$817$	−2.41092e9	−0.154670
$818$	0	0
$819$	− 9.84449e8i	− 0.0626181i
$820$	0	0
$821$	− 8.88169e9i	− 0.560137i	−0.959980	−	0.280069i	$-0.909643\pi$
$821$	− 8.88169e9i	− 0.560137i	0.959980	−	0.280069i	$-0.0903572\pi$
$822$	0	0
$823$	5.97962e9	0.373916	0.186958	−	0.982368i	$-0.440137\pi$
$823$	5.97962e9	0.373916	0.186958	+	0.982368i	$0.440137\pi$
$824$	0	0
$825$	1.59662e9	0.0989948
$826$	0	0
$827$	2.78565e10i	1.71260i	0.516477	+	0.856301i	$0.327243\pi$
$827$	2.78565e10i	1.71260i	−0.516477	+	0.856301i	$0.672757\pi$
$828$	0	0
$829$	− 1.53264e10i	− 0.934325i	−0.884171	−	0.467163i	$-0.845276\pi$
$829$	− 1.53264e10i	− 0.934325i	0.884171	−	0.467163i	$-0.154724\pi$
$830$	0	0
$831$	2.25997e8	0.0136615
$832$	0	0
$833$	−7.84105e7	−0.00470020
$834$	0	0
$835$	1.08634e10i	0.645748i
$836$	0	0
$837$	− 7.57199e8i	− 0.0446345i
$838$	0	0
$839$	3.85999e9	0.225641	0.112821	−	0.993615i	$-0.464011\pi$
$839$	3.85999e9	0.225641	0.112821	+	0.993615i	$0.464011\pi$
$840$	0	0
$841$	1.49688e10	0.867765
$842$	0	0
$843$	− 8.45790e9i	− 0.486257i
$844$	0	0
$845$	1.32465e10i	0.755272i
$846$	0	0
$847$	1.41627e10	0.800856
$848$	0	0
$849$	−4.33134e9	−0.242910
$850$	0	0
$851$	− 5.59464e9i	− 0.311185i
$852$	0	0
$853$	2.15207e9i	0.118723i	0.998237	+	0.0593614i	$0.0189064\pi$
$853$	2.15207e9i	0.118723i	−0.998237	+	0.0593614i	$0.981094\pi$
$854$	0	0
$855$	5.19132e9	0.284051
$856$	0	0
$857$	1.52606e10	0.828206	0.414103	−	0.910230i	$-0.364095\pi$
$857$	1.52606e10	0.828206	0.414103	+	0.910230i	$0.364095\pi$
$858$	0	0
$859$	2.51735e10i	1.35509i	0.735481	+	0.677545i	$0.236956\pi$
$859$	2.51735e10i	1.35509i	−0.735481	+	0.677545i	$0.763044\pi$
$860$	0	0
$861$	− 1.22891e10i	− 0.656161i
$862$	0	0
$863$	1.50441e10	0.796760	0.398380	−	0.917220i	$-0.369572\pi$
$863$	1.50441e10	0.796760	0.398380	+	0.917220i	$0.369572\pi$
$864$	0	0
$865$	−2.55846e10	−1.34407
$866$	0	0
$867$	5.27498e9i	0.274887i
$868$	0	0
$869$	− 9.12357e9i	− 0.471624i
$870$	0	0
$871$	3.17555e9	0.162838
$872$	0	0
$873$	9.94014e9	0.505642
$874$	0	0
$875$	2.14501e10i	1.08243i
$876$	0	0
$877$	− 3.24650e10i	− 1.62524i	−0.582797	−	0.812618i	$-0.698042\pi$
$877$	− 3.24650e10i	− 1.62524i	0.582797	−	0.812618i	$-0.301958\pi$
$878$	0	0
$879$	−8.58813e9	−0.426518
$880$	0	0
$881$	2.20762e10	1.08770	0.543849	−	0.839183i	$-0.316967\pi$
$881$	2.20762e10	1.08770	0.543849	+	0.839183i	$0.316967\pi$
$882$	0	0
$883$	− 3.48823e10i	− 1.70507i	−0.522667	−	0.852537i	$-0.675063\pi$
$883$	− 3.48823e10i	− 1.70507i	0.522667	−	0.852537i	$-0.324937\pi$
$884$	0	0
$885$	− 5.65506e9i	− 0.274243i
$886$	0	0
$887$	1.90322e10	0.915705	0.457852	−	0.889028i	$-0.348619\pi$
$887$	1.90322e10	0.915705	0.457852	+	0.889028i	$0.348619\pi$
$888$	0	0
$889$	2.05773e10	0.982272
$890$	0	0
$891$	1.04003e9i	0.0492575i
$892$	0	0
$893$	− 2.88740e10i	− 1.35684i
$894$	0	0
$895$	−1.38472e10	−0.645626
$896$	0	0
$897$	−9.78050e8	−0.0452467
$898$	0	0
$899$	1.83732e9i	0.0843386i
$900$	0	0
$901$	1.49477e10i	0.680827i
$902$	0	0
$903$	1.80980e9	0.0817943
$904$	0	0
$905$	−2.98683e10	−1.33949
$906$	0	0
$907$	3.59008e9i	0.159764i	0.996804	+	0.0798818i	$0.0254543\pi$
$907$	3.59008e9i	0.159764i	−0.996804	+	0.0798818i	$0.974546\pi$
$908$	0	0
$909$	5.07000e8i	0.0223890i
$910$	0	0
$911$	5.18479e9	0.227205	0.113602	−	0.993526i	$-0.463761\pi$
$911$	5.18479e9	0.227205	0.113602	+	0.993526i	$0.463761\pi$
$912$	0	0
$913$	−1.26399e10	−0.549662
$914$	0	0
$915$	− 8.97990e9i	− 0.387523i
$916$	0	0
$917$	4.59237e9i	0.196673i
$918$	0	0
$919$	−8.72664e9	−0.370888	−0.185444	−	0.982655i	$-0.559372\pi$
$919$	−8.72664e9	−0.370888	−0.185444	+	0.982655i	$0.559372\pi$
$920$	0	0
$921$	1.92321e10	0.811180
$922$	0	0
$923$	2.19655e9i	0.0919466i
$924$	0	0
$925$	6.96723e9i	0.289444i
$926$	0	0
$927$	−2.10461e9	−0.0867748
$928$	0	0
$929$	3.57501e10	1.46293	0.731463	−	0.681882i	$-0.238838\pi$
$929$	3.57501e10	1.46293	0.731463	+	0.681882i	$0.238838\pi$
$930$	0	0
$931$	− 1.73993e8i	− 0.00706655i
$932$	0	0
$933$	− 8.73475e9i	− 0.352099i
$934$	0	0
$935$	−6.28033e9	−0.251271
$936$	0	0
$937$	1.61665e10	0.641989	0.320995	−	0.947081i	$-0.395983\pi$
$937$	1.61665e10	0.641989	0.320995	+	0.947081i	$0.395983\pi$
$938$	0	0
$939$	6.31364e9i	0.248857i
$940$	0	0
$941$	− 2.67594e10i	− 1.04692i	−0.852051	−	0.523459i	$-0.824641\pi$
$941$	− 2.67594e10i	− 1.04692i	0.852051	−	0.523459i	$-0.175359\pi$
$942$	0	0
$943$	−1.22092e10	−0.474131
$944$	0	0
$945$	−3.89695e9	−0.150215
$946$	0	0
$947$	− 3.63328e10i	− 1.39019i	−0.718918	−	0.695095i	$-0.755363\pi$
$947$	− 3.63328e10i	− 1.39019i	0.718918	−	0.695095i	$-0.244637\pi$
$948$	0	0
$949$	− 7.11265e9i	− 0.270147i
$950$	0	0
$951$	−5.71039e9	−0.215295
$952$	0	0
$953$	−1.03702e10	−0.388115	−0.194057	−	0.980990i	$-0.562165\pi$
$953$	−1.03702e10	−0.388115	−0.194057	+	0.980990i	$0.562165\pi$
$954$	0	0
$955$	9.97279e8i	0.0370514i
$956$	0	0
$957$	− 2.52360e9i	− 0.0930740i
$958$	0	0
$959$	1.82912e10	0.669697
$960$	0	0
$961$	−2.60327e10	−0.946209
$962$	0	0
$963$	− 1.07603e10i	− 0.388270i
$964$	0	0
$965$	− 3.07932e9i	− 0.110308i
$966$	0	0
$967$	4.82811e10	1.71706	0.858528	−	0.512766i	$-0.171379\pi$
$967$	4.82811e10	1.71706	0.858528	+	0.512766i	$0.171379\pi$
$968$	0	0
$969$	1.28794e10	0.454740
$970$	0	0
$971$	− 2.54306e10i	− 0.891433i	−0.895174	−	0.445716i	$-0.852949\pi$
$971$	− 2.54306e10i	− 0.891433i	0.895174	−	0.445716i	$-0.147051\pi$
$972$	0	0
$973$	− 2.81073e10i	− 0.978193i
$974$	0	0
$975$	1.21801e9	0.0420856
$976$	0	0
$977$	1.43404e10	0.491959	0.245980	−	0.969275i	$-0.420890\pi$
$977$	1.43404e10	0.491959	0.245980	+	0.969275i	$0.420890\pi$
$978$	0	0
$979$	6.23311e9i	0.212307i
$980$	0	0
$981$	− 1.36944e10i	− 0.463128i
$982$	0	0
$983$	−1.57764e10	−0.529750	−0.264875	−	0.964283i	$-0.585331\pi$
$983$	−1.57764e10	−0.529750	−0.264875	+	0.964283i	$0.585331\pi$
$984$	0	0
$985$	−4.27961e10	−1.42685
$986$	0	0
$987$	2.16748e10i	0.717537i
$988$	0	0
$989$	− 1.79803e9i	− 0.0591032i
$990$	0	0
$991$	2.15481e10	0.703318	0.351659	−	0.936128i	$-0.385618\pi$
$991$	2.15481e10	0.703318	0.351659	+	0.936128i	$0.385618\pi$
$992$	0	0
$993$	−1.04335e10	−0.338150
$994$	0	0
$995$	− 2.56351e10i	− 0.824999i
$996$	0	0
$997$	− 4.08176e10i	− 1.30441i	−0.758042	−	0.652205i	$-0.773844\pi$
$997$	− 4.08176e10i	− 1.30441i	0.758042	−	0.652205i	$-0.226156\pi$
$998$	0	0
$999$	−4.53841e9	−0.144020

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.e.193.5	yes	12
4.3	odd	2	inner	384.8.d.e.193.11	yes	12
8.3	odd	2	inner	384.8.d.e.193.2	✓	12
8.5	even	2	inner	384.8.d.e.193.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.e.193.2	✓	12	8.3	odd	2	inner
384.8.d.e.193.5	yes	12	1.1	even	1	trivial
384.8.d.e.193.8	yes	12	8.5	even	2	inner
384.8.d.e.193.11	yes	12	4.3	odd	2	inner

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-27.0000i q^{3} +218.879i q^{5} +904.541 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q-27.0000i q^{3} +218.879i q^{5} +904.541 q^{7} -729.000 q^{9} +1956.99i q^{11} +1492.92i q^{13} +5909.75 q^{15} +14661.8 q^{17} +32534.6i q^{19} -24422.6i q^{21} -24263.9 q^{23} +30216.8 q^{25} +19683.0i q^{27} -47760.3i q^{29} -38469.7 q^{31} +52838.8 q^{33} +197986. i q^{35} +230575. i q^{37} +40308.9 q^{39} +503187. q^{41} +74103.4i q^{43} -159563. i q^{45} -887487. q^{47} -5347.93 q^{49} -395869. i q^{51} +1.01950e6i q^{53} -428346. q^{55} +878434. q^{57} -956905. i q^{59} -1.51951e6i q^{61} -659411. q^{63} -326770. q^{65} -2.12707e6i q^{67} +655124. i q^{69} +1.47131e6 q^{71} -4.76425e6 q^{73} -815853. i q^{75} +1.77018e6i q^{77} -4.66203e6 q^{79} +531441. q^{81} +6.45883e6i q^{83} +3.20917e6i q^{85} -1.28953e6 q^{87} +3.18504e6 q^{89} +1.35041e6i q^{91} +1.03868e6i q^{93} -7.12115e6 q^{95} -1.36353e7 q^{97} -1.42665e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8748 q^{9}+O(q^{10}) \) 12 * q - 8748 * q^9	\( 12 q - 8748 q^{9} - 83096 q^{17} - 127812 q^{25} - 30672 q^{33} - 969032 q^{41} - 5087028 q^{49} - 69552 q^{57} - 240832 q^{65} + 18079656 q^{73} + 6377292 q^{81} + 43563144 q^{89} - 48458328 q^{97}+O(q^{100}) \) 12 * q - 8748 * q^9 - 83096 * q^17 - 127812 * q^25 - 30672 * q^33 - 969032 * q^41 - 5087028 * q^49 - 69552 * q^57 - 240832 * q^65 + 18079656 * q^73 + 6377292 * q^81 + 43563144 * q^89 - 48458328 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more