LMFDB - Embedded newform 384.8.d.e.193.4

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			193.4
Root			$49.2636 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.e.193.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000i q^{3} +8.99895i q^{5} +616.197 q^{7} -729.000 q^{9} +O(q^{10})$	$q-27.0000i q^{3} +8.99895i q^{5} +616.197 q^{7} -729.000 q^{9} +3035.60i q^{11} +12272.8i q^{13} +242.972 q^{15} -35454.2 q^{17} -46835.6i q^{19} -16637.3i q^{21} -89833.4 q^{23} +78044.0 q^{25} +19683.0i q^{27} -113126. i q^{29} +246616. q^{31} +81961.3 q^{33} +5545.13i q^{35} +383887. i q^{37} +331365. q^{39} +63822.4 q^{41} -697052. i q^{43} -6560.24i q^{45} +66539.5 q^{47} -443844. q^{49} +957264. i q^{51} -1.02190e6i q^{53} -27317.3 q^{55} -1.26456e6 q^{57} +1.66944e6i q^{59} -1.94570e6i q^{61} -449208. q^{63} -110442. q^{65} -50973.1i q^{67} +2.42550e6i q^{69} +5.04161e6 q^{71} +5.82121e6 q^{73} -2.10719e6i q^{75} +1.87053e6i q^{77} +2.30931e6 q^{79} +531441. q^{81} -3.20768e6i q^{83} -319051. i q^{85} -3.05441e6 q^{87} -5.21504e6 q^{89} +7.56246e6i q^{91} -6.65862e6i q^{93} +421471. q^{95} -3.08632e6 q^{97} -2.21296e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	8.99895i	0.0321956i	0.999870	+	0.0160978i	$0.00512432\pi$
$5$	8.99895i	0.0321956i	−0.999870	+	0.0160978i	$0.994876\pi$
$6$	0	0
$7$	616.197	0.679011	0.339505	−	0.940604i	$-0.389740\pi$
$7$	616.197	0.679011	0.339505	+	0.940604i	$0.389740\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	3035.60i	0.687655i	0.939033	+	0.343828i	$0.111724\pi$
$11$	3035.60i	0.687655i	−0.939033	+	0.343828i	$0.888276\pi$
$12$	0	0
$13$	12272.8i	1.54932i	0.632377	+	0.774660i	$0.282079\pi$
$13$	12272.8i	1.54932i	−0.632377	+	0.774660i	$0.717921\pi$
$14$	0	0
$15$	242.972	0.0185882
$16$	0	0
$17$	−35454.2	−1.75024	−0.875118	−	0.483909i	$-0.839217\pi$
$17$	−35454.2	−1.75024	−0.875118	+	0.483909i	$0.839217\pi$
$18$	0	0
$19$	− 46835.6i	− 1.56653i	−0.621689	−	0.783264i	$-0.713553\pi$
$19$	− 46835.6i	− 1.56653i	0.621689	−	0.783264i	$-0.286447\pi$
$20$	0	0
$21$	− 16637.3i	− 0.392027i
$22$	0	0
$23$	−89833.4	−1.53954	−0.769769	−	0.638323i	$-0.779629\pi$
$23$	−89833.4	−1.53954	−0.769769	+	0.638323i	$0.779629\pi$
$24$	0	0
$25$	78044.0	0.998963
$26$	0	0
$27$	19683.0i	0.192450i
$28$	0	0
$29$	− 113126.i	− 0.861333i	−0.902511	−	0.430666i	$-0.858279\pi$
$29$	− 113126.i	− 0.861333i	0.902511	−	0.430666i	$-0.141721\pi$
$30$	0	0
$31$	246616.	1.48681	0.743403	−	0.668843i	$-0.233210\pi$
$31$	246616.	1.48681	0.743403	+	0.668843i	$0.233210\pi$
$32$	0	0
$33$	81961.3	0.397018
$34$	0	0
$35$	5545.13i	0.0218612i
$36$	0	0
$37$	383887.i	1.24594i	0.782245	+	0.622970i	$0.214074\pi$
$37$	383887.i	1.24594i	−0.782245	+	0.622970i	$0.785926\pi$
$38$	0	0
$39$	331365.	0.894501
$40$	0	0
$41$	63822.4	0.144620	0.0723102	−	0.997382i	$-0.476963\pi$
$41$	63822.4	0.144620	0.0723102	+	0.997382i	$0.476963\pi$
$42$	0	0
$43$	− 697052.i	− 1.33698i	−0.743720	−	0.668491i	$-0.766940\pi$
$43$	− 697052.i	− 1.33698i	0.743720	−	0.668491i	$-0.233060\pi$
$44$	0	0
$45$	− 6560.24i	− 0.0107319i
$46$	0	0
$47$	66539.5	0.0934840	0.0467420	−	0.998907i	$-0.485116\pi$
$47$	66539.5	0.0934840	0.0467420	+	0.998907i	$0.485116\pi$
$48$	0	0
$49$	−443844.	−0.538944
$50$	0	0
$51$	957264.i	1.01050i
$52$	0	0
$53$	− 1.02190e6i	− 0.942851i	−0.881906	−	0.471426i	$-0.843740\pi$
$53$	− 1.02190e6i	− 0.942851i	0.881906	−	0.471426i	$-0.156260\pi$
$54$	0	0
$55$	−27317.3	−0.0221395
$56$	0	0
$57$	−1.26456e6	−0.904435
$58$	0	0
$59$	1.66944e6i	1.05825i	0.848544	+	0.529125i	$0.177480\pi$
$59$	1.66944e6i	1.05825i	−0.848544	+	0.529125i	$0.822520\pi$
$60$	0	0
$61$	− 1.94570e6i	− 1.09754i	−0.835972	−	0.548772i	$-0.815095\pi$
$61$	− 1.94570e6i	− 1.09754i	0.835972	−	0.548772i	$-0.184905\pi$
$62$	0	0
$63$	−449208.	−0.226337
$64$	0	0
$65$	−110442.	−0.0498814
$66$	0	0
$67$	− 50973.1i	− 0.0207052i	−0.999946	−	0.0103526i	$-0.996705\pi$
$67$	− 50973.1i	− 0.0207052i	0.999946	−	0.0103526i	$-0.00329539\pi$
$68$	0	0
$69$	2.42550e6i	0.888852i
$70$	0	0
$71$	5.04161e6	1.67173	0.835863	−	0.548938i	$-0.184968\pi$
$71$	5.04161e6	1.67173	0.835863	+	0.548938i	$0.184968\pi$
$72$	0	0
$73$	5.82121e6	1.75139	0.875696	−	0.482863i	$-0.160403\pi$
$73$	5.82121e6	1.75139	0.875696	+	0.482863i	$0.160403\pi$
$74$	0	0
$75$	− 2.10719e6i	− 0.576752i
$76$	0	0
$77$	1.87053e6i	0.466925i
$78$	0	0
$79$	2.30931e6	0.526972	0.263486	−	0.964663i	$-0.415128\pi$
$79$	2.30931e6	0.526972	0.263486	+	0.964663i	$0.415128\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	− 3.20768e6i	− 0.615768i	−0.951424	−	0.307884i	$-0.900379\pi$
$83$	− 3.20768e6i	− 0.615768i	0.951424	−	0.307884i	$-0.0996209\pi$
$84$	0	0
$85$	− 319051.i	− 0.0563500i
$86$	0	0
$87$	−3.05441e6	−0.497291
$88$	0	0
$89$	−5.21504e6	−0.784138	−0.392069	−	0.919936i	$-0.628240\pi$
$89$	−5.21504e6	−0.784138	−0.392069	+	0.919936i	$0.628240\pi$
$90$	0	0
$91$	7.56246e6i	1.05201i
$92$	0	0
$93$	− 6.65862e6i	− 0.858408i
$94$	0	0
$95$	421471.	0.0504354
$96$	0	0
$97$	−3.08632e6	−0.343352	−0.171676	−	0.985153i	$-0.554918\pi$
$97$	−3.08632e6	−0.343352	−0.171676	+	0.985153i	$0.554918\pi$
$98$	0	0
$99$	− 2.21296e6i	− 0.229218i
$100$	0	0
$101$	− 1.44652e7i	− 1.39701i	−0.715604	−	0.698507i	$-0.753848\pi$
$101$	− 1.44652e7i	− 1.39701i	0.715604	−	0.698507i	$-0.246152\pi$
$102$	0	0
$103$	1.89906e6	0.171241	0.0856205	−	0.996328i	$-0.472713\pi$
$103$	1.89906e6	0.171241	0.0856205	+	0.996328i	$0.472713\pi$
$104$	0	0
$105$	149719.	0.0126216
$106$	0	0
$107$	− 1.54233e7i	− 1.21712i	−0.793506	−	0.608562i	$-0.791747\pi$
$107$	− 1.54233e7i	− 1.21712i	0.793506	−	0.608562i	$-0.208253\pi$
$108$	0	0
$109$	− 6.59792e6i	− 0.487994i	−0.969776	−	0.243997i	$-0.921541\pi$
$109$	− 6.59792e6i	− 0.487994i	0.969776	−	0.243997i	$-0.0784587\pi$
$110$	0	0
$111$	1.03650e7	0.719344
$112$	0	0
$113$	−5.66038e6	−0.369038	−0.184519	−	0.982829i	$-0.559073\pi$
$113$	−5.66038e6	−0.369038	−0.184519	+	0.982829i	$0.559073\pi$
$114$	0	0
$115$	− 808407.i	− 0.0495664i
$116$	0	0
$117$	− 8.94686e6i	− 0.516440i
$118$	0	0
$119$	−2.18468e7	−1.18843
$120$	0	0
$121$	1.02723e7	0.527130
$122$	0	0
$123$	− 1.72321e6i	− 0.0834967i
$124$	0	0
$125$	1.40536e6i	0.0643579i
$126$	0	0
$127$	−1.21577e7	−0.526669	−0.263334	−	0.964705i	$-0.584822\pi$
$127$	−1.21577e7	−0.526669	−0.263334	+	0.964705i	$0.584822\pi$
$128$	0	0
$129$	−1.88204e7	−0.771907
$130$	0	0
$131$	3.25216e7i	1.26393i	0.774997	+	0.631965i	$0.217751\pi$
$131$	3.25216e7i	1.26393i	−0.774997	+	0.631965i	$0.782249\pi$
$132$	0	0
$133$	− 2.88599e7i	− 1.06369i
$134$	0	0
$135$	−177126.	−0.00619605
$136$	0	0
$137$	4.34472e7	1.44358	0.721789	−	0.692114i	$-0.243320\pi$
$137$	4.34472e7	1.44358	0.721789	+	0.692114i	$0.243320\pi$
$138$	0	0
$139$	2.11016e6i	0.0666444i	0.999445	+	0.0333222i	$0.0106088\pi$
$139$	2.11016e6i	0.0666444i	−0.999445	+	0.0333222i	$0.989391\pi$
$140$	0	0
$141$	− 1.79657e6i	− 0.0539730i
$142$	0	0
$143$	−3.72553e7	−1.06540
$144$	0	0
$145$	1.01802e6	0.0277312
$146$	0	0
$147$	1.19838e7i	0.311160i
$148$	0	0
$149$	− 3.58625e7i	− 0.888156i	−0.895988	−	0.444078i	$-0.853531\pi$
$149$	− 3.58625e7i	− 0.888156i	0.895988	−	0.444078i	$-0.146469\pi$
$150$	0	0
$151$	−7.80703e6	−0.184530	−0.0922649	−	0.995734i	$-0.529411\pi$
$151$	−7.80703e6	−0.184530	−0.0922649	+	0.995734i	$0.529411\pi$
$152$	0	0
$153$	2.58461e7	0.583412
$154$	0	0
$155$	2.21928e6i	0.0478687i
$156$	0	0
$157$	− 7.98173e7i	− 1.64607i	−0.567991	−	0.823035i	$-0.692279\pi$
$157$	− 7.98173e7i	− 1.64607i	0.567991	−	0.823035i	$-0.307721\pi$
$158$	0	0
$159$	−2.75913e7	−0.544355
$160$	0	0
$161$	−5.53551e7	−1.04536
$162$	0	0
$163$	− 4.10447e7i	− 0.742336i	−0.928566	−	0.371168i	$-0.878957\pi$
$163$	− 4.10447e7i	− 0.742336i	0.928566	−	0.371168i	$-0.121043\pi$
$164$	0	0
$165$	737566.i	0.0127822i
$166$	0	0
$167$	8.79088e7	1.46058	0.730289	−	0.683139i	$-0.239386\pi$
$167$	8.79088e7	1.46058	0.730289	+	0.683139i	$0.239386\pi$
$168$	0	0
$169$	−8.78727e7	−1.40040
$170$	0	0
$171$	3.41431e7i	0.522176i
$172$	0	0
$173$	− 1.29158e8i	− 1.89653i	−0.317478	−	0.948266i	$-0.602836\pi$
$173$	− 1.29158e8i	− 1.89653i	0.317478	−	0.948266i	$-0.397164\pi$
$174$	0	0
$175$	4.80905e7	0.678307
$176$	0	0
$177$	4.50749e7	0.610981
$178$	0	0
$179$	1.13096e7i	0.147388i	0.997281	+	0.0736941i	$0.0234789\pi$
$179$	1.13096e7i	0.147388i	−0.997281	+	0.0736941i	$0.976521\pi$
$180$	0	0
$181$	− 1.01595e8i	− 1.27349i	−0.771074	−	0.636746i	$-0.780280\pi$
$181$	− 1.01595e8i	− 1.27349i	0.771074	−	0.636746i	$-0.219720\pi$
$182$	0	0
$183$	−5.25340e7	−0.633668
$184$	0	0
$185$	−3.45458e6	−0.0401139
$186$	0	0
$187$	− 1.07625e8i	− 1.20356i
$188$	0	0
$189$	1.21286e7i	0.130676i
$190$	0	0
$191$	−2.54258e7	−0.264032	−0.132016	−	0.991248i	$-0.542145\pi$
$191$	−2.54258e7	−0.264032	−0.132016	+	0.991248i	$0.542145\pi$
$192$	0	0
$193$	1.01605e7	0.101734	0.0508668	−	0.998705i	$-0.483802\pi$
$193$	1.01605e7	0.101734	0.0508668	+	0.998705i	$0.483802\pi$
$194$	0	0
$195$	2.98194e6i	0.0287990i
$196$	0	0
$197$	− 5.94284e7i	− 0.553812i	−0.960897	−	0.276906i	$-0.910691\pi$
$197$	− 5.94284e7i	− 0.553812i	0.960897	−	0.276906i	$-0.0893091\pi$
$198$	0	0
$199$	−1.32802e8	−1.19459	−0.597293	−	0.802023i	$-0.703757\pi$
$199$	−1.32802e8	−1.19459	−0.597293	+	0.802023i	$0.703757\pi$
$200$	0	0
$201$	−1.37627e6	−0.0119541
$202$	0	0
$203$	− 6.97082e7i	− 0.584854i
$204$	0	0
$205$	574335.i	0.00465615i
$206$	0	0
$207$	6.54885e7	0.513179
$208$	0	0
$209$	1.42174e8	1.07723
$210$	0	0
$211$	2.57624e8i	1.88798i	0.329971	+	0.943991i	$0.392961\pi$
$211$	2.57624e8i	1.88798i	−0.329971	+	0.943991i	$0.607039\pi$
$212$	0	0
$213$	− 1.36123e8i	− 0.965171i
$214$	0	0
$215$	6.27274e6	0.0430450
$216$	0	0
$217$	1.51964e8	1.00956
$218$	0	0
$219$	− 1.57173e8i	− 1.01117i
$220$	0	0
$221$	− 4.35122e8i	− 2.71168i
$222$	0	0
$223$	1.96405e8	1.18600	0.593002	−	0.805201i	$-0.297943\pi$
$223$	1.96405e8	1.18600	0.593002	+	0.805201i	$0.297943\pi$
$224$	0	0
$225$	−5.68941e7	−0.332988
$226$	0	0
$227$	− 8.67364e7i	− 0.492165i	−0.969249	−	0.246082i	$-0.920857\pi$
$227$	− 8.67364e7i	− 0.492165i	0.969249	−	0.246082i	$-0.0791434\pi$
$228$	0	0
$229$	− 1.22165e8i	− 0.672237i	−0.941820	−	0.336118i	$-0.890886\pi$
$229$	− 1.22165e8i	− 0.672237i	0.941820	−	0.336118i	$-0.109114\pi$
$230$	0	0
$231$	5.05044e7	0.269580
$232$	0	0
$233$	−5.39072e7	−0.279191	−0.139595	−	0.990209i	$-0.544580\pi$
$233$	−5.39072e7	−0.279191	−0.139595	+	0.990209i	$0.544580\pi$
$234$	0	0
$235$	598786.i	0.00300978i
$236$	0	0
$237$	− 6.23514e7i	− 0.304247i
$238$	0	0
$239$	−2.90175e7	−0.137489	−0.0687443	−	0.997634i	$-0.521899\pi$
$239$	−2.90175e7	−0.137489	−0.0687443	+	0.997634i	$0.521899\pi$
$240$	0	0
$241$	−3.43173e8	−1.57926	−0.789630	−	0.613583i	$-0.789728\pi$
$241$	−3.43173e8	−1.57926	−0.789630	+	0.613583i	$0.789728\pi$
$242$	0	0
$243$	− 1.43489e7i	− 0.0641500i
$244$	0	0
$245$	− 3.99413e6i	− 0.0173517i
$246$	0	0
$247$	5.74803e8	2.42705
$248$	0	0
$249$	−8.66072e7	−0.355514
$250$	0	0
$251$	1.87903e8i	0.750025i	0.927020	+	0.375012i	$0.122362\pi$
$251$	1.87903e8i	0.750025i	−0.927020	+	0.375012i	$0.877638\pi$
$252$	0	0
$253$	− 2.72699e8i	− 1.05867i
$254$	0	0
$255$	−8.61438e6	−0.0325337
$256$	0	0
$257$	5.70210e7	0.209541	0.104770	−	0.994496i	$-0.466589\pi$
$257$	5.70210e7	0.209541	0.104770	+	0.994496i	$0.466589\pi$
$258$	0	0
$259$	2.36550e8i	0.846007i
$260$	0	0
$261$	8.24692e7i	0.287111i
$262$	0	0
$263$	−2.54738e8	−0.863474	−0.431737	−	0.902000i	$-0.642099\pi$
$263$	−2.54738e8	−0.863474	−0.431737	+	0.902000i	$0.642099\pi$
$264$	0	0
$265$	9.19604e6	0.0303557
$266$	0	0
$267$	1.40806e8i	0.452722i
$268$	0	0
$269$	6.31643e8i	1.97851i	0.146193	+	0.989256i	$0.453298\pi$
$269$	6.31643e8i	1.97851i	−0.146193	+	0.989256i	$0.546702\pi$
$270$	0	0
$271$	−2.74328e8	−0.837293	−0.418646	−	0.908149i	$-0.637495\pi$
$271$	−2.74328e8	−0.837293	−0.418646	+	0.908149i	$0.637495\pi$
$272$	0	0
$273$	2.04186e8	0.607376
$274$	0	0
$275$	2.36911e8i	0.686943i
$276$	0	0
$277$	− 3.83552e8i	− 1.08429i	−0.840285	−	0.542145i	$-0.817612\pi$
$277$	− 3.83552e8i	− 1.08429i	0.840285	−	0.542145i	$-0.182388\pi$
$278$	0	0
$279$	−1.79783e8	−0.495602
$280$	0	0
$281$	−4.03791e8	−1.08564	−0.542819	−	0.839850i	$-0.682643\pi$
$281$	−4.03791e8	−1.08564	−0.542819	+	0.839850i	$0.682643\pi$
$282$	0	0
$283$	− 4.57245e8i	− 1.19921i	−0.800295	−	0.599607i	$-0.795324\pi$
$283$	− 4.57245e8i	− 1.19921i	0.800295	−	0.599607i	$-0.204676\pi$
$284$	0	0
$285$	− 1.13797e7i	− 0.0291189i
$286$	0	0
$287$	3.93272e7	0.0981989
$288$	0	0
$289$	8.46664e8	2.06333
$290$	0	0
$291$	8.33306e7i	0.198234i
$292$	0	0
$293$	− 7.48249e8i	− 1.73784i	−0.494953	−	0.868920i	$-0.664815\pi$
$293$	− 7.48249e8i	− 1.73784i	0.494953	−	0.868920i	$-0.335185\pi$
$294$	0	0
$295$	−1.50232e7	−0.0340711
$296$	0	0
$297$	−5.97498e7	−0.132339
$298$	0	0
$299$	− 1.10251e9i	− 2.38524i
$300$	0	0
$301$	− 4.29522e8i	− 0.907826i
$302$	0	0
$303$	−3.90561e8	−0.806566
$304$	0	0
$305$	1.75093e7	0.0353362
$306$	0	0
$307$	− 4.34755e7i	− 0.0857551i	−0.999080	−	0.0428775i	$-0.986347\pi$
$307$	− 4.34755e7i	− 0.0857551i	0.999080	−	0.0428775i	$-0.0136525\pi$
$308$	0	0
$309$	− 5.12746e7i	− 0.0988661i
$310$	0	0
$311$	1.82116e8	0.343310	0.171655	−	0.985157i	$-0.445089\pi$
$311$	1.82116e8	0.343310	0.171655	+	0.985157i	$0.445089\pi$
$312$	0	0
$313$	5.21850e7	0.0961923	0.0480961	−	0.998843i	$-0.484685\pi$
$313$	5.21850e7	0.0961923	0.0480961	+	0.998843i	$0.484685\pi$
$314$	0	0
$315$	− 4.04240e6i	− 0.00728706i
$316$	0	0
$317$	6.76984e7i	0.119363i	0.998217	+	0.0596816i	$0.0190086\pi$
$317$	6.76984e7i	0.119363i	−0.998217	+	0.0596816i	$0.980991\pi$
$318$	0	0
$319$	3.43407e8	0.592300
$320$	0	0
$321$	−4.16430e8	−0.702707
$322$	0	0
$323$	1.66052e9i	2.74179i
$324$	0	0
$325$	9.57817e8i	1.54771i
$326$	0	0
$327$	−1.78144e8	−0.281743
$328$	0	0
$329$	4.10015e7	0.0634767
$330$	0	0
$331$	− 7.97772e8i	− 1.20915i	−0.796548	−	0.604576i	$-0.793343\pi$
$331$	− 7.97772e8i	− 1.20915i	0.796548	−	0.604576i	$-0.206657\pi$
$332$	0	0
$333$	− 2.79854e8i	− 0.415314i
$334$	0	0
$335$	458704.	0.000666617 0
$336$	0	0
$337$	1.52453e7	0.0216986	0.0108493	−	0.999941i	$-0.496546\pi$
$337$	1.52453e7	0.0216986	0.0108493	+	0.999941i	$0.496546\pi$
$338$	0	0
$339$	1.52830e8i	0.213064i
$340$	0	0
$341$	7.48627e8i	1.02241i
$342$	0	0
$343$	−7.80960e8	−1.04496
$344$	0	0
$345$	−2.18270e7	−0.0286172
$346$	0	0
$347$	− 4.45017e8i	− 0.571773i	−0.958263	−	0.285887i	$-0.907712\pi$
$347$	− 4.45017e8i	− 0.571773i	0.958263	−	0.285887i	$-0.0922881\pi$
$348$	0	0
$349$	1.38196e9i	1.74023i	0.492846	+	0.870117i	$0.335957\pi$
$349$	1.38196e9i	1.74023i	−0.492846	+	0.870117i	$0.664043\pi$
$350$	0	0
$351$	−2.41565e8	−0.298167
$352$	0	0
$353$	−3.14032e8	−0.379981	−0.189991	−	0.981786i	$-0.560846\pi$
$353$	−3.14032e8	−0.379981	−0.189991	+	0.981786i	$0.560846\pi$
$354$	0	0
$355$	4.53692e7i	0.0538223i
$356$	0	0
$357$	5.89864e8i	0.686140i
$358$	0	0
$359$	−1.14356e9	−1.30445	−0.652224	−	0.758027i	$-0.726164\pi$
$359$	−1.14356e9	−1.30445	−0.652224	+	0.758027i	$0.726164\pi$
$360$	0	0
$361$	−1.29970e9	−1.45401
$362$	0	0
$363$	− 2.77351e8i	− 0.304339i
$364$	0	0
$365$	5.23848e7i	0.0563872i
$366$	0	0
$367$	1.40602e9	1.48478	0.742388	−	0.669970i	$-0.233693\pi$
$367$	1.40602e9	1.48478	0.742388	+	0.669970i	$0.233693\pi$
$368$	0	0
$369$	−4.65266e7	−0.0482068
$370$	0	0
$371$	− 6.29693e8i	− 0.640206i
$372$	0	0
$373$	3.92421e8i	0.391536i	0.980650	+	0.195768i	$0.0627199\pi$
$373$	3.92421e8i	0.391536i	−0.980650	+	0.195768i	$0.937280\pi$
$374$	0	0
$375$	3.79447e7	0.0371571
$376$	0	0
$377$	1.38838e9	1.33448
$378$	0	0
$379$	6.58729e8i	0.621541i	0.950485	+	0.310770i	$0.100587\pi$
$379$	6.58729e8i	0.621541i	−0.950485	+	0.310770i	$0.899413\pi$
$380$	0	0
$381$	3.28257e8i	0.304072i
$382$	0	0
$383$	−1.37352e9	−1.24923	−0.624613	−	0.780935i	$-0.714743\pi$
$383$	−1.37352e9	−1.24923	−0.624613	+	0.780935i	$0.714743\pi$
$384$	0	0
$385$	−1.68328e7	−0.0150330
$386$	0	0
$387$	5.08151e8i	0.445661i
$388$	0	0
$389$	5.75082e7i	0.0495343i	0.999693	+	0.0247671i	$0.00788443\pi$
$389$	5.75082e7i	0.0495343i	−0.999693	+	0.0247671i	$0.992116\pi$
$390$	0	0
$391$	3.18497e9	2.69456
$392$	0	0
$393$	8.78085e8	0.729730
$394$	0	0
$395$	2.07814e7i	0.0169662i
$396$	0	0
$397$	4.45178e8i	0.357082i	0.983933	+	0.178541i	$0.0571376\pi$
$397$	4.45178e8i	0.357082i	−0.983933	+	0.178541i	$0.942862\pi$
$398$	0	0
$399$	−7.79219e8	−0.614121
$400$	0	0
$401$	7.79495e8	0.603682	0.301841	−	0.953358i	$-0.402399\pi$
$401$	7.79495e8	0.603682	0.301841	+	0.953358i	$0.402399\pi$
$402$	0	0
$403$	3.02666e9i	2.30354i
$404$	0	0
$405$	4.78241e6i	0.00357729i
$406$	0	0
$407$	−1.16533e9	−0.856778
$408$	0	0
$409$	1.35057e9	0.976077	0.488039	−	0.872822i	$-0.337712\pi$
$409$	1.35057e9	0.976077	0.488039	+	0.872822i	$0.337712\pi$
$410$	0	0
$411$	− 1.17307e9i	− 0.833450i
$412$	0	0
$413$	1.02870e9i	0.718564i
$414$	0	0
$415$	2.88657e7	0.0198250
$416$	0	0
$417$	5.69743e7	0.0384772
$418$	0	0
$419$	5.86568e8i	0.389556i	0.980847	+	0.194778i	$0.0623986\pi$
$419$	5.86568e8i	0.389556i	−0.980847	+	0.194778i	$0.937601\pi$
$420$	0	0
$421$	1.31673e9i	0.860021i	0.902824	+	0.430011i	$0.141490\pi$
$421$	1.31673e9i	0.860021i	−0.902824	+	0.430011i	$0.858510\pi$
$422$	0	0
$423$	−4.85073e7	−0.0311613
$424$	0	0
$425$	−2.76699e9	−1.74842
$426$	0	0
$427$	− 1.19894e9i	− 0.745245i
$428$	0	0
$429$	1.00589e9i	0.615108i
$430$	0	0
$431$	5.97794e8	0.359651	0.179825	−	0.983699i	$-0.442447\pi$
$431$	5.97794e8	0.359651	0.179825	+	0.983699i	$0.442447\pi$
$432$	0	0
$433$	−1.45917e9	−0.863769	−0.431884	−	0.901929i	$-0.642151\pi$
$433$	−1.45917e9	−0.863769	−0.431884	+	0.901929i	$0.642151\pi$
$434$	0	0
$435$	− 2.74865e7i	− 0.0160106i
$436$	0	0
$437$	4.20740e9i	2.41173i
$438$	0	0
$439$	1.42889e9	0.806069	0.403035	−	0.915185i	$-0.367955\pi$
$439$	1.42889e9	0.806069	0.403035	+	0.915185i	$0.367955\pi$
$440$	0	0
$441$	3.23562e8	0.179648
$442$	0	0
$443$	− 3.28479e9i	− 1.79512i	−0.440887	−	0.897562i	$-0.645336\pi$
$443$	− 3.28479e9i	− 1.79512i	0.440887	−	0.897562i	$-0.354664\pi$
$444$	0	0
$445$	− 4.69299e7i	− 0.0252458i
$446$	0	0
$447$	−9.68289e8	−0.512777
$448$	0	0
$449$	−8.38143e8	−0.436974	−0.218487	−	0.975840i	$-0.570112\pi$
$449$	−8.38143e8	−0.436974	−0.218487	+	0.975840i	$0.570112\pi$
$450$	0	0
$451$	1.93740e8i	0.0994491i
$452$	0	0
$453$	2.10790e8i	0.106538i
$454$	0	0
$455$	−6.80542e7	−0.0338700
$456$	0	0
$457$	1.45042e9	0.710865	0.355432	−	0.934702i	$-0.384334\pi$
$457$	1.45042e9	0.710865	0.355432	+	0.934702i	$0.384334\pi$
$458$	0	0
$459$	− 6.97846e8i	− 0.336833i
$460$	0	0
$461$	1.19272e9i	0.567001i	0.958972	+	0.283500i	$0.0914957\pi$
$461$	1.19272e9i	0.567001i	−0.958972	+	0.283500i	$0.908504\pi$
$462$	0	0
$463$	1.74225e9	0.815786	0.407893	−	0.913030i	$-0.366264\pi$
$463$	1.74225e9	0.815786	0.407893	+	0.913030i	$0.366264\pi$
$464$	0	0
$465$	5.99206e7	0.0276370
$466$	0	0
$467$	2.77787e9i	1.26213i	0.775732	+	0.631063i	$0.217381\pi$
$467$	2.77787e9i	1.26213i	−0.775732	+	0.631063i	$0.782619\pi$
$468$	0	0
$469$	− 3.14095e7i	− 0.0140590i
$470$	0	0
$471$	−2.15507e9	−0.950359
$472$	0	0
$473$	2.11598e9	0.919383
$474$	0	0
$475$	− 3.65524e9i	− 1.56490i
$476$	0	0
$477$	7.44966e8i	0.314284i
$478$	0	0
$479$	3.35409e9	1.39444	0.697221	−	0.716857i	$-0.254420\pi$
$479$	3.35409e9	1.39444	0.697221	+	0.716857i	$0.254420\pi$
$480$	0	0
$481$	−4.71136e9	−1.93036
$482$	0	0
$483$	1.49459e9i	0.603540i
$484$	0	0
$485$	− 2.77736e7i	− 0.0110544i
$486$	0	0
$487$	3.25553e9	1.27723	0.638617	−	0.769525i	$-0.279507\pi$
$487$	3.25553e9	1.27723	0.638617	+	0.769525i	$0.279507\pi$
$488$	0	0
$489$	−1.10821e9	−0.428588
$490$	0	0
$491$	1.37036e9i	0.522457i	0.965277	+	0.261228i	$0.0841276\pi$
$491$	1.37036e9i	0.522457i	−0.965277	+	0.261228i	$0.915872\pi$
$492$	0	0
$493$	4.01081e9i	1.50754i
$494$	0	0
$495$	1.99143e7	0.00737984
$496$	0	0
$497$	3.10663e9	1.13512
$498$	0	0
$499$	− 6.61371e7i	− 0.0238283i	−0.999929	−	0.0119142i	$-0.996208\pi$
$499$	− 6.61371e7i	− 0.0238283i	0.999929	−	0.0119142i	$-0.00379249\pi$
$500$	0	0
$501$	− 2.37354e9i	− 0.843265i
$502$	0	0
$503$	8.57092e8	0.300289	0.150145	−	0.988664i	$-0.452026\pi$
$503$	8.57092e8	0.300289	0.150145	+	0.988664i	$0.452026\pi$
$504$	0	0
$505$	1.30172e8	0.0449777
$506$	0	0
$507$	2.37256e9i	0.808519i
$508$	0	0
$509$	9.83151e8i	0.330452i	0.986256	+	0.165226i	$0.0528353\pi$
$509$	9.83151e8i	0.330452i	−0.986256	+	0.165226i	$0.947165\pi$
$510$	0	0
$511$	3.58702e9	1.18921
$512$	0	0
$513$	9.21864e8	0.301478
$514$	0	0
$515$	1.70895e7i	0.00551322i
$516$	0	0
$517$	2.01988e8i	0.0642848i
$518$	0	0
$519$	−3.48727e9	−1.09496
$520$	0	0
$521$	−3.95745e7	−0.0122598	−0.00612990	−	0.999981i	$-0.501951\pi$
$521$	−3.95745e7	−0.0122598	−0.00612990	+	0.999981i	$0.501951\pi$
$522$	0	0
$523$	− 2.92926e9i	− 0.895370i	−0.894191	−	0.447685i	$-0.852249\pi$
$523$	− 2.92926e9i	− 0.895370i	0.894191	−	0.447685i	$-0.147751\pi$
$524$	0	0
$525$	− 1.29844e9i	− 0.391621i
$526$	0	0
$527$	−8.74356e9	−2.60226
$528$	0	0
$529$	4.66521e9	1.37018
$530$	0	0
$531$	− 1.21702e9i	− 0.352750i
$532$	0	0
$533$	7.83279e8i	0.224064i
$534$	0	0
$535$	1.38794e8	0.0391861
$536$	0	0
$537$	3.05360e8	0.0850947
$538$	0	0
$539$	− 1.34733e9i	− 0.370608i
$540$	0	0
$541$	3.59458e9i	0.976018i	0.872839	+	0.488009i	$0.162277\pi$
$541$	3.59458e9i	0.976018i	−0.872839	+	0.488009i	$0.837723\pi$
$542$	0	0
$543$	−2.74306e9	−0.735250
$544$	0	0
$545$	5.93744e7	0.0157113
$546$	0	0
$547$	2.96194e9i	0.773786i	0.922125	+	0.386893i	$0.126452\pi$
$547$	2.96194e9i	0.773786i	−0.922125	+	0.386893i	$0.873548\pi$
$548$	0	0
$549$	1.41842e9i	0.365848i
$550$	0	0
$551$	−5.29834e9	−1.34930
$552$	0	0
$553$	1.42299e9	0.357820
$554$	0	0
$555$	9.32737e7i	0.0231598i
$556$	0	0
$557$	− 3.16328e9i	− 0.775611i	−0.921741	−	0.387805i	$-0.873233\pi$
$557$	− 3.16328e9i	− 0.775611i	0.921741	−	0.387805i	$-0.126767\pi$
$558$	0	0
$559$	8.55477e9	2.07142
$560$	0	0
$561$	−2.90588e9	−0.694876
$562$	0	0
$563$	3.08736e9i	0.729135i	0.931177	+	0.364567i	$0.118783\pi$
$563$	3.08736e9i	0.729135i	−0.931177	+	0.364567i	$0.881217\pi$
$564$	0	0
$565$	− 5.09375e7i	− 0.0118814i
$566$	0	0
$567$	3.27473e8	0.0754457
$568$	0	0
$569$	−5.45313e9	−1.24095	−0.620473	−	0.784228i	$-0.713059\pi$
$569$	−5.45313e9	−1.24095	−0.620473	+	0.784228i	$0.713059\pi$
$570$	0	0
$571$	4.86211e9i	1.09294i	0.837477	+	0.546472i	$0.184030\pi$
$571$	4.86211e9i	1.09294i	−0.837477	+	0.546472i	$0.815970\pi$
$572$	0	0
$573$	6.86495e8i	0.152439i
$574$	0	0
$575$	−7.01096e9	−1.53794
$576$	0	0
$577$	−1.41178e8	−0.0305950	−0.0152975	−	0.999883i	$-0.504870\pi$
$577$	−1.41178e8	−0.0305950	−0.0152975	+	0.999883i	$0.504870\pi$
$578$	0	0
$579$	− 2.74333e8i	− 0.0587359i
$580$	0	0
$581$	− 1.97656e9i	− 0.418113i
$582$	0	0
$583$	3.10209e9	0.648357
$584$	0	0
$585$	8.05124e7	0.0166271
$586$	0	0
$587$	− 2.79450e9i	− 0.570257i	−0.958489	−	0.285128i	$-0.907964\pi$
$587$	− 2.79450e9i	− 0.570257i	0.958489	−	0.285128i	$-0.0920363\pi$
$588$	0	0
$589$	− 1.15504e10i	− 2.32912i
$590$	0	0
$591$	−1.60457e9	−0.319743
$592$	0	0
$593$	−7.71016e9	−1.51835	−0.759175	−	0.650887i	$-0.774397\pi$
$593$	−7.71016e9	−1.51835	−0.759175	+	0.650887i	$0.774397\pi$
$594$	0	0
$595$	− 1.96598e8i	− 0.0382623i
$596$	0	0
$597$	3.58565e9i	0.689695i
$598$	0	0
$599$	2.24833e9	0.427432	0.213716	−	0.976896i	$-0.431443\pi$
$599$	2.24833e9	0.427432	0.213716	+	0.976896i	$0.431443\pi$
$600$	0	0
$601$	1.75844e9	0.330421	0.165210	−	0.986258i	$-0.447170\pi$
$601$	1.75844e9	0.330421	0.165210	+	0.986258i	$0.447170\pi$
$602$	0	0
$603$	3.71594e7i	0.00690173i
$604$	0	0
$605$	9.24397e7i	0.0169713i
$606$	0	0
$607$	1.78653e9	0.324227	0.162114	−	0.986772i	$-0.448169\pi$
$607$	1.78653e9	0.324227	0.162114	+	0.986772i	$0.448169\pi$
$608$	0	0
$609$	−1.88212e9	−0.337666
$610$	0	0
$611$	8.16625e8i	0.144837i
$612$	0	0
$613$	− 5.12242e9i	− 0.898180i	−0.893487	−	0.449090i	$-0.851748\pi$
$613$	− 5.12242e9i	− 0.898180i	0.893487	−	0.449090i	$-0.148252\pi$
$614$	0	0
$615$	1.55070e7	0.00268823
$616$	0	0
$617$	−3.14822e8	−0.0539594	−0.0269797	−	0.999636i	$-0.508589\pi$
$617$	−3.14822e8	−0.0539594	−0.0269797	+	0.999636i	$0.508589\pi$
$618$	0	0
$619$	− 4.60016e9i	− 0.779572i	−0.920906	−	0.389786i	$-0.872549\pi$
$619$	− 4.60016e9i	− 0.779572i	0.920906	−	0.389786i	$-0.127451\pi$
$620$	0	0
$621$	− 1.76819e9i	− 0.296284i
$622$	0	0
$623$	−3.21349e9	−0.532438
$624$	0	0
$625$	6.08454e9	0.996891
$626$	0	0
$627$	− 3.83870e9i	− 0.621940i
$628$	0	0
$629$	− 1.36104e10i	− 2.18069i
$630$	0	0
$631$	9.57151e9	1.51662	0.758312	−	0.651892i	$-0.226024\pi$
$631$	9.57151e9	1.51662	0.758312	+	0.651892i	$0.226024\pi$
$632$	0	0
$633$	6.95585e9	1.09003
$634$	0	0
$635$	− 1.09406e8i	− 0.0169564i
$636$	0	0
$637$	− 5.44720e9i	− 0.834998i
$638$	0	0
$639$	−3.67533e9	−0.557242
$640$	0	0
$641$	1.32428e10	1.98598	0.992992	−	0.118182i	$-0.0377066\pi$
$641$	1.32428e10	1.98598	0.992992	+	0.118182i	$0.0377066\pi$
$642$	0	0
$643$	6.68707e8i	0.0991967i	0.998769	+	0.0495984i	$0.0157941\pi$
$643$	6.68707e8i	0.0991967i	−0.998769	+	0.0495984i	$0.984206\pi$
$644$	0	0
$645$	− 1.69364e8i	− 0.0248520i
$646$	0	0
$647$	−6.36051e9	−0.923266	−0.461633	−	0.887071i	$-0.652736\pi$
$647$	−6.36051e9	−0.923266	−0.461633	+	0.887071i	$0.652736\pi$
$648$	0	0
$649$	−5.06776e9	−0.727712
$650$	0	0
$651$	− 4.10302e9i	− 0.582868i
$652$	0	0
$653$	2.19579e9i	0.308599i	0.988024	+	0.154299i	$0.0493121\pi$
$653$	2.19579e9i	0.308599i	−0.988024	+	0.154299i	$0.950688\pi$
$654$	0	0
$655$	−2.92661e8	−0.0406930
$656$	0	0
$657$	−4.24366e9	−0.583797
$658$	0	0
$659$	− 8.29618e9i	− 1.12922i	−0.825357	−	0.564611i	$-0.809026\pi$
$659$	− 8.29618e9i	− 1.12922i	0.825357	−	0.564611i	$-0.190974\pi$
$660$	0	0
$661$	6.60145e9i	0.889067i	0.895762	+	0.444533i	$0.146630\pi$
$661$	6.60145e9i	0.889067i	−0.895762	+	0.444533i	$0.853370\pi$
$662$	0	0
$663$	−1.17483e10	−1.56559
$664$	0	0
$665$	2.59709e8	0.0342462
$666$	0	0
$667$	1.01625e10i	1.32605i
$668$	0	0
$669$	− 5.30294e9i	− 0.684740i
$670$	0	0
$671$	5.90639e9	0.754733
$672$	0	0
$673$	−6.92354e9	−0.875539	−0.437769	−	0.899087i	$-0.644231\pi$
$673$	−6.92354e9	−0.875539	−0.437769	+	0.899087i	$0.644231\pi$
$674$	0	0
$675$	1.53614e9i	0.192251i
$676$	0	0
$677$	2.27425e8i	0.0281694i	0.999901	+	0.0140847i	$0.00448344\pi$
$677$	2.27425e8i	0.0281694i	−0.999901	+	0.0140847i	$0.995517\pi$
$678$	0	0
$679$	−1.90178e9	−0.233140
$680$	0	0
$681$	−2.34188e9	−0.284152
$682$	0	0
$683$	2.09687e9i	0.251825i	0.992041	+	0.125913i	$0.0401859\pi$
$683$	2.09687e9i	0.251825i	−0.992041	+	0.125913i	$0.959814\pi$
$684$	0	0
$685$	3.90980e8i	0.0464769i
$686$	0	0
$687$	−3.29845e9	−0.388116
$688$	0	0
$689$	1.25416e10	1.46078
$690$	0	0
$691$	− 6.49885e8i	− 0.0749313i	−0.999298	−	0.0374656i	$-0.988072\pi$
$691$	− 6.49885e8i	− 0.0749313i	0.999298	−	0.0374656i	$-0.0119285\pi$
$692$	0	0
$693$	− 1.36362e9i	− 0.155642i
$694$	0	0
$695$	−1.89892e7	−0.00214566
$696$	0	0
$697$	−2.26277e9	−0.253120
$698$	0	0
$699$	1.45549e9i	0.161191i
$700$	0	0
$701$	− 1.22103e10i	− 1.33879i	−0.742907	−	0.669394i	$-0.766554\pi$
$701$	− 1.22103e10i	− 1.33879i	0.742907	−	0.669394i	$-0.233446\pi$
$702$	0	0
$703$	1.79796e10	1.95180
$704$	0	0
$705$	1.61672e7	0.00173770
$706$	0	0
$707$	− 8.91344e9i	− 0.948587i
$708$	0	0
$709$	− 1.06693e10i	− 1.12428i	−0.827041	−	0.562141i	$-0.809978\pi$
$709$	− 1.06693e10i	− 1.12428i	0.827041	−	0.562141i	$-0.190022\pi$
$710$	0	0
$711$	−1.68349e9	−0.175657
$712$	0	0
$713$	−2.21543e10	−2.28899
$714$	0	0
$715$	− 3.35259e8i	− 0.0343012i
$716$	0	0
$717$	7.83471e8i	0.0793791i
$718$	0	0
$719$	4.40260e9	0.441731	0.220866	−	0.975304i	$-0.429112\pi$
$719$	4.40260e9	0.441731	0.220866	+	0.975304i	$0.429112\pi$
$720$	0	0
$721$	1.17019e9	0.116275
$722$	0	0
$723$	9.26568e9i	0.911786i
$724$	0	0
$725$	− 8.82884e9i	− 0.860440i
$726$	0	0
$727$	1.63117e10	1.57445	0.787225	−	0.616666i	$-0.211517\pi$
$727$	1.63117e10	1.57445	0.787225	+	0.616666i	$0.211517\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	2.47134e10i	2.34004i
$732$	0	0
$733$	− 1.33690e10i	− 1.25382i	−0.779091	−	0.626911i	$-0.784319\pi$
$733$	− 1.33690e10i	− 1.25382i	0.779091	−	0.626911i	$-0.215681\pi$
$734$	0	0
$735$	−1.07842e8	−0.0100180
$736$	0	0
$737$	1.54734e8	0.0142380
$738$	0	0
$739$	− 1.52697e10i	− 1.39180i	−0.718141	−	0.695898i	$-0.755007\pi$
$739$	− 1.52697e10i	− 1.39180i	0.718141	−	0.695898i	$-0.244993\pi$
$740$	0	0
$741$	− 1.55197e10i	− 1.40126i
$742$	0	0
$743$	−1.40139e9	−0.125342	−0.0626712	−	0.998034i	$-0.519962\pi$
$743$	−1.40139e9	−0.125342	−0.0626712	+	0.998034i	$0.519962\pi$
$744$	0	0
$745$	3.22725e8	0.0285947
$746$	0	0
$747$	2.33840e9i	0.205256i
$748$	0	0
$749$	− 9.50382e9i	− 0.826441i
$750$	0	0
$751$	6.12628e9	0.527785	0.263893	−	0.964552i	$-0.414994\pi$
$751$	6.12628e9	0.527785	0.263893	+	0.964552i	$0.414994\pi$
$752$	0	0
$753$	5.07338e9	0.433027
$754$	0	0
$755$	− 7.02551e7i	− 0.00594106i
$756$	0	0
$757$	− 1.54406e10i	− 1.29369i	−0.762623	−	0.646844i	$-0.776089\pi$
$757$	− 1.54406e10i	− 1.29369i	0.762623	−	0.646844i	$-0.223911\pi$
$758$	0	0
$759$	−7.36286e9	−0.611224
$760$	0	0
$761$	5.10270e9	0.419714	0.209857	−	0.977732i	$-0.432700\pi$
$761$	5.10270e9	0.419714	0.209857	+	0.977732i	$0.432700\pi$
$762$	0	0
$763$	− 4.06562e9i	− 0.331353i
$764$	0	0
$765$	2.32588e8i	0.0187833i
$766$	0	0
$767$	−2.04887e10	−1.63957
$768$	0	0
$769$	−7.19721e9	−0.570718	−0.285359	−	0.958421i	$-0.592113\pi$
$769$	−7.19721e9	−0.570718	−0.285359	+	0.958421i	$0.592113\pi$
$770$	0	0
$771$	− 1.53957e9i	− 0.120978i
$772$	0	0
$773$	1.39957e10i	1.08985i	0.838486	+	0.544924i	$0.183441\pi$
$773$	1.39957e10i	1.08985i	−0.838486	+	0.544924i	$0.816559\pi$
$774$	0	0
$775$	1.92469e10	1.48527
$776$	0	0
$777$	6.38686e9	0.488443
$778$	0	0
$779$	− 2.98916e9i	− 0.226552i
$780$	0	0
$781$	1.53043e10i	1.14957i
$782$	0	0
$783$	2.22667e9	0.165764
$784$	0	0
$785$	7.18272e8	0.0529963
$786$	0	0
$787$	− 1.50580e10i	− 1.10117i	−0.834778	−	0.550587i	$-0.814404\pi$
$787$	− 1.50580e10i	− 1.10117i	0.834778	−	0.550587i	$-0.185596\pi$
$788$	0	0
$789$	6.87794e9i	0.498527i
$790$	0	0
$791$	−3.48791e9	−0.250581
$792$	0	0
$793$	2.38792e10	1.70045
$794$	0	0
$795$	− 2.48293e8i	− 0.0175259i
$796$	0	0
$797$	5.06926e9i	0.354683i	0.984149	+	0.177342i	$0.0567498\pi$
$797$	5.06926e9i	0.354683i	−0.984149	+	0.177342i	$0.943250\pi$
$798$	0	0
$799$	−2.35911e9	−0.163619
$800$	0	0
$801$	3.80176e9	0.261379
$802$	0	0
$803$	1.76709e10i	1.20435i
$804$	0	0
$805$	− 4.98138e8i	− 0.0336561i
$806$	0	0
$807$	1.70544e10	1.14229
$808$	0	0
$809$	5.19545e9	0.344988	0.172494	−	0.985011i	$-0.444818\pi$
$809$	5.19545e9	0.344988	0.172494	+	0.985011i	$0.444818\pi$
$810$	0	0
$811$	2.10691e10i	1.38699i	0.720461	+	0.693496i	$0.243930\pi$
$811$	2.10691e10i	1.38699i	−0.720461	+	0.693496i	$0.756070\pi$
$812$	0	0
$813$	7.40685e9i	0.483411i
$814$	0	0
$815$	3.69360e8	0.0239000
$816$	0	0
$817$	−3.26468e10	−2.09442
$818$	0	0
$819$	− 5.51303e9i	− 0.350669i
$820$	0	0
$821$	− 5.98065e9i	− 0.377179i	−0.982056	−	0.188590i	$-0.939608\pi$
$821$	− 5.98065e9i	− 0.377179i	0.982056	−	0.188590i	$-0.0603916\pi$
$822$	0	0
$823$	−7.62100e9	−0.476554	−0.238277	−	0.971197i	$-0.576583\pi$
$823$	−7.62100e9	−0.476554	−0.238277	+	0.971197i	$0.576583\pi$
$824$	0	0
$825$	6.39659e9	0.396606
$826$	0	0
$827$	− 1.30836e10i	− 0.804373i	−0.915558	−	0.402187i	$-0.868250\pi$
$827$	− 1.30836e10i	− 0.804373i	0.915558	−	0.402187i	$-0.131750\pi$
$828$	0	0
$829$	− 5.21207e9i	− 0.317738i	−0.987300	−	0.158869i	$-0.949215\pi$
$829$	− 5.21207e9i	− 0.317738i	0.987300	−	0.158869i	$-0.0507848\pi$
$830$	0	0
$831$	−1.03559e10	−0.626015
$832$	0	0
$833$	1.57361e10	0.943280
$834$	0	0
$835$	7.91087e8i	0.0470242i
$836$	0	0
$837$	4.85413e9i	0.286136i
$838$	0	0
$839$	−9.15426e9	−0.535126	−0.267563	−	0.963540i	$-0.586218\pi$
$839$	−9.15426e9	−0.535126	−0.267563	+	0.963540i	$0.586218\pi$
$840$	0	0
$841$	4.45229e9	0.258106
$842$	0	0
$843$	1.09024e10i	0.626793i
$844$	0	0
$845$	− 7.90763e8i	− 0.0450866i
$846$	0	0
$847$	6.32975e9	0.357927
$848$	0	0
$849$	−1.23456e10	−0.692367
$850$	0	0
$851$	− 3.44859e10i	− 1.91817i
$852$	0	0
$853$	− 1.58918e10i	− 0.876699i	−0.898805	−	0.438349i	$-0.855563\pi$
$853$	− 1.58918e10i	− 0.876699i	0.898805	−	0.438349i	$-0.144437\pi$
$854$	0	0
$855$	−3.07252e8	−0.0168118
$856$	0	0
$857$	−3.04443e10	−1.65224	−0.826120	−	0.563495i	$-0.809457\pi$
$857$	−3.04443e10	−1.65224	−0.826120	+	0.563495i	$0.809457\pi$
$858$	0	0
$859$	1.79263e10i	0.964971i	0.875904	+	0.482486i	$0.160266\pi$
$859$	1.79263e10i	0.964971i	−0.875904	+	0.482486i	$0.839734\pi$
$860$	0	0
$861$	− 1.06183e9i	− 0.0566952i
$862$	0	0
$863$	7.12123e9	0.377153	0.188576	−	0.982059i	$-0.439613\pi$
$863$	7.12123e9	0.377153	0.188576	+	0.982059i	$0.439613\pi$
$864$	0	0
$865$	1.16229e9	0.0610600
$866$	0	0
$867$	− 2.28599e10i	− 1.19126i
$868$	0	0
$869$	7.01015e9i	0.362375i
$870$	0	0
$871$	6.25582e8	0.0320790
$872$	0	0
$873$	2.24993e9	0.114451
$874$	0	0
$875$	8.65978e8i	0.0436997i
$876$	0	0
$877$	5.49211e8i	0.0274942i	0.999906	+	0.0137471i	$0.00437597\pi$
$877$	5.49211e8i	0.0274942i	−0.999906	+	0.0137471i	$0.995624\pi$
$878$	0	0
$879$	−2.02027e10	−1.00334
$880$	0	0
$881$	2.12382e10	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$881$	2.12382e10	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$882$	0	0
$883$	− 2.26798e10i	− 1.10861i	−0.832315	−	0.554303i	$-0.812985\pi$
$883$	− 2.26798e10i	− 1.10861i	0.832315	−	0.554303i	$-0.187015\pi$
$884$	0	0
$885$	4.05627e8i	0.0196709i
$886$	0	0
$887$	−2.09480e10	−1.00788	−0.503942	−	0.863737i	$-0.668117\pi$
$887$	−2.09480e10	−1.00788	−0.503942	+	0.863737i	$0.668117\pi$
$888$	0	0
$889$	−7.49153e9	−0.357614
$890$	0	0
$891$	1.61324e9i	0.0764062i
$892$	0	0
$893$	− 3.11642e9i	− 0.146445i
$894$	0	0
$895$	−1.01775e8	−0.00474526
$896$	0	0
$897$	−2.97677e10	−1.37712
$898$	0	0
$899$	− 2.78987e10i	− 1.28064i
$900$	0	0
$901$	3.62307e10i	1.65021i
$902$	0	0
$903$	−1.15971e10	−0.524133
$904$	0	0
$905$	9.14246e8	0.0410009
$906$	0	0
$907$	2.65970e10i	1.18361i	0.806083	+	0.591803i	$0.201584\pi$
$907$	2.65970e10i	1.18361i	−0.806083	+	0.591803i	$0.798416\pi$
$908$	0	0
$909$	1.05452e10i	0.465671i
$910$	0	0
$911$	1.28193e10	0.561760	0.280880	−	0.959743i	$-0.409374\pi$
$911$	1.28193e10	0.561760	0.280880	+	0.959743i	$0.409374\pi$
$912$	0	0
$913$	9.73723e9	0.423436
$914$	0	0
$915$	− 4.72751e8i	− 0.0204013i
$916$	0	0
$917$	2.00398e10i	0.858222i
$918$	0	0
$919$	−3.23675e10	−1.37564	−0.687820	−	0.725881i	$-0.741432\pi$
$919$	−3.23675e10	−1.37564	−0.687820	+	0.725881i	$0.741432\pi$
$920$	0	0
$921$	−1.17384e9	−0.0495107
$922$	0	0
$923$	6.18746e10i	2.59004i
$924$	0	0
$925$	2.99601e10i	1.24465i
$926$	0	0
$927$	−1.38441e9	−0.0570804
$928$	0	0
$929$	−1.31144e10	−0.536653	−0.268327	−	0.963328i	$-0.586471\pi$
$929$	−1.31144e10	−0.536653	−0.268327	+	0.963328i	$0.586471\pi$
$930$	0	0
$931$	2.07877e10i	0.844271i
$932$	0	0
$933$	− 4.91713e9i	− 0.198210i
$934$	0	0
$935$	9.68513e8	0.0387494
$936$	0	0
$937$	3.21245e10	1.27570	0.637849	−	0.770161i	$-0.279824\pi$
$937$	3.21245e10	1.27570	0.637849	+	0.770161i	$0.279824\pi$
$938$	0	0
$939$	− 1.40899e9i	− 0.0555366i
$940$	0	0
$941$	− 1.15677e10i	− 0.452566i	−0.974062	−	0.226283i	$-0.927343\pi$
$941$	− 1.15677e10i	− 0.452566i	0.974062	−	0.226283i	$-0.0726574\pi$
$942$	0	0
$943$	−5.73338e9	−0.222649
$944$	0	0
$945$	−1.09145e8	−0.00420719
$946$	0	0
$947$	1.21746e10i	0.465833i	0.972497	+	0.232916i	$0.0748269\pi$
$947$	1.21746e10i	0.465833i	−0.972497	+	0.232916i	$0.925173\pi$
$948$	0	0
$949$	7.14425e10i	2.71347i
$950$	0	0
$951$	1.82786e9	0.0689144
$952$	0	0
$953$	−2.00590e10	−0.750730	−0.375365	−	0.926877i	$-0.622483\pi$
$953$	−2.00590e10	−0.750730	−0.375365	+	0.926877i	$0.622483\pi$
$954$	0	0
$955$	− 2.28805e8i	− 0.00850069i
$956$	0	0
$957$	− 9.27199e9i	− 0.341965i
$958$	0	0
$959$	2.67721e10	0.980205
$960$	0	0
$961$	3.33066e10	1.21059
$962$	0	0
$963$	1.12436e10i	0.405708i
$964$	0	0
$965$	9.14338e7i	0.00327538i
$966$	0	0
$967$	3.05520e10	1.08654	0.543271	−	0.839558i	$-0.317186\pi$
$967$	3.05520e10	1.08654	0.543271	+	0.839558i	$0.317186\pi$
$968$	0	0
$969$	4.48340e10	1.58298
$970$	0	0
$971$	− 4.05142e10i	− 1.42017i	−0.704117	−	0.710084i	$-0.748657\pi$
$971$	− 4.05142e10i	− 1.42017i	0.704117	−	0.710084i	$-0.251343\pi$
$972$	0	0
$973$	1.30028e9i	0.0452523i
$974$	0	0
$975$	2.58611e10	0.893574
$976$	0	0
$977$	2.00689e10	0.688483	0.344242	−	0.938881i	$-0.388136\pi$
$977$	2.00689e10	0.688483	0.344242	+	0.938881i	$0.388136\pi$
$978$	0	0
$979$	− 1.58308e10i	− 0.539216i
$980$	0	0
$981$	4.80988e9i	0.162665i
$982$	0	0
$983$	3.88669e10	1.30510	0.652548	−	0.757748i	$-0.273700\pi$
$983$	3.88669e10	1.30510	0.652548	+	0.757748i	$0.273700\pi$
$984$	0	0
$985$	5.34794e8	0.0178303
$986$	0	0
$987$	− 1.10704e9i	− 0.0366483i
$988$	0	0
$989$	6.26186e10i	2.05834i
$990$	0	0
$991$	−4.43097e10	−1.44624	−0.723122	−	0.690721i	$-0.757293\pi$
$991$	−4.43097e10	−1.44624	−0.723122	+	0.690721i	$0.757293\pi$
$992$	0	0
$993$	−2.15398e10	−0.698104
$994$	0	0
$995$	− 1.19508e9i	− 0.0384605i
$996$	0	0
$997$	− 3.67500e10i	− 1.17442i	−0.809433	−	0.587212i	$-0.800226\pi$
$997$	− 3.67500e10i	− 1.17442i	0.809433	−	0.587212i	$-0.199774\pi$
$998$	0	0
$999$	−7.55605e9	−0.239781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.e.193.4	yes	12
4.3	odd	2	inner	384.8.d.e.193.10	yes	12
8.3	odd	2	inner	384.8.d.e.193.3	✓	12
8.5	even	2	inner	384.8.d.e.193.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.e.193.3	✓	12	8.3	odd	2	inner
384.8.d.e.193.4	yes	12	1.1	even	1	trivial
384.8.d.e.193.9	yes	12	8.5	even	2	inner
384.8.d.e.193.10	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)

\(f(q)\)	\(=\)	\(q-27.0000i q^{3} +8.99895i q^{5} +616.197 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q-27.0000i q^{3} +8.99895i q^{5} +616.197 q^{7} -729.000 q^{9} +3035.60i q^{11} +12272.8i q^{13} +242.972 q^{15} -35454.2 q^{17} -46835.6i q^{19} -16637.3i q^{21} -89833.4 q^{23} +78044.0 q^{25} +19683.0i q^{27} -113126. i q^{29} +246616. q^{31} +81961.3 q^{33} +5545.13i q^{35} +383887. i q^{37} +331365. q^{39} +63822.4 q^{41} -697052. i q^{43} -6560.24i q^{45} +66539.5 q^{47} -443844. q^{49} +957264. i q^{51} -1.02190e6i q^{53} -27317.3 q^{55} -1.26456e6 q^{57} +1.66944e6i q^{59} -1.94570e6i q^{61} -449208. q^{63} -110442. q^{65} -50973.1i q^{67} +2.42550e6i q^{69} +5.04161e6 q^{71} +5.82121e6 q^{73} -2.10719e6i q^{75} +1.87053e6i q^{77} +2.30931e6 q^{79} +531441. q^{81} -3.20768e6i q^{83} -319051. i q^{85} -3.05441e6 q^{87} -5.21504e6 q^{89} +7.56246e6i q^{91} -6.65862e6i q^{93} +421471. q^{95} -3.08632e6 q^{97} -2.21296e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8748 q^{9}+O(q^{10}) \) 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

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