LMFDB - Embedded newform 384.8.d.b.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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 277x^{4} + 19236x^{2} + 9216 $ x^6 + 277x^4 + 19236x^2 + 9216
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.4
Root			$-0.694584i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.b.193.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000i q^{3} -334.405i q^{5} +359.725 q^{7} -729.000 q^{9} +O(q^{10})$	$q+27.0000i q^{3} -334.405i q^{5} +359.725 q^{7} -729.000 q^{9} +7.65868i q^{11} -4424.87i q^{13} +9028.94 q^{15} -4955.30 q^{17} -18369.0i q^{19} +9712.58i q^{21} +36433.8 q^{23} -33701.9 q^{25} -19683.0i q^{27} -2855.15i q^{29} +90194.3 q^{31} -206.784 q^{33} -120294. i q^{35} -38156.3i q^{37} +119472. q^{39} -127357. q^{41} -644347. i q^{43} +243781. i q^{45} +603462. q^{47} -694141. q^{49} -133793. i q^{51} +291256. i q^{53} +2561.10 q^{55} +495962. q^{57} -1.72805e6i q^{59} +1.64795e6i q^{61} -262240. q^{63} -1.47970e6 q^{65} +3.21269e6i q^{67} +983714. i q^{69} -2.03751e6 q^{71} +1.45445e6 q^{73} -909950. i q^{75} +2755.02i q^{77} -4.18669e6 q^{79} +531441. q^{81} +5.85510e6i q^{83} +1.65708e6i q^{85} +77089.0 q^{87} -1.09233e7 q^{89} -1.59174e6i q^{91} +2.43524e6i q^{93} -6.14268e6 q^{95} +969168. q^{97} -5583.18i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 136 q^{7} - 4374 q^{9}+O(q^{10}) $ 6 * q + 136 * q^7 - 4374 * q^9	$ 6 q + 136 q^{7} - 4374 q^{9} - 6048 q^{15} - 25804 q^{17} + 12688 q^{23} - 18418 q^{25} + 220472 q^{31} - 264600 q^{33} - 591192 q^{39} - 619652 q^{41} - 326032 q^{47} - 581482 q^{49} - 4724352 q^{55} - 2429784 q^{57} - 99144 q^{63} - 9518720 q^{65} + 5841776 q^{71} - 11075196 q^{73} + 495800 q^{79} + 3188646 q^{81} + 965520 q^{87} + 9660740 q^{89} - 32750208 q^{95} - 9511564 q^{97}+O(q^{100}) $ 6 * q + 136 * q^7 - 4374 * q^9 - 6048 * q^15 - 25804 * q^17 + 12688 * q^23 - 18418 * q^25 + 220472 * q^31 - 264600 * q^33 - 591192 * q^39 - 619652 * q^41 - 326032 * q^47 - 581482 * q^49 - 4724352 * q^55 - 2429784 * q^57 - 99144 * q^63 - 9518720 * q^65 + 5841776 * q^71 - 11075196 * q^73 + 495800 * q^79 + 3188646 * q^81 + 965520 * q^87 + 9660740 * q^89 - 32750208 * q^95 - 9511564 * 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
$5$	− 334.405i	− 1.19640i	−0.801345	−	0.598202i	$-0.795882\pi$
$5$	− 334.405i	− 1.19640i	0.801345	−	0.598202i	$-0.204118\pi$
$6$	0	0
$7$	359.725	0.396395	0.198197	−	0.980162i	$-0.436491\pi$
$7$	359.725	0.396395	0.198197	+	0.980162i	$0.436491\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	7.65868i	0.00173492i	1.00000		0.000867460i	$0.000276121\pi$
$11$	7.65868i	0.00173492i	−1.00000		0.000867460i	$0.999724\pi$
$12$	0	0
$13$	− 4424.87i	− 0.558598i	−0.960204	−	0.279299i	$-0.909898\pi$
$13$	− 4424.87i	− 0.558598i	0.960204	−	0.279299i	$-0.0901020\pi$
$14$	0	0
$15$	9028.94	0.690744
$16$	0	0
$17$	−4955.30	−0.244624	−0.122312	−	0.992492i	$-0.539031\pi$
$17$	−4955.30	−0.244624	−0.122312	+	0.992492i	$0.539031\pi$
$18$	0	0
$19$	− 18369.0i	− 0.614394i	−0.951646	−	0.307197i	$-0.900609\pi$
$19$	− 18369.0i	− 0.614394i	0.951646	−	0.307197i	$-0.0993910\pi$
$20$	0	0
$21$	9712.58i	0.228858i
$22$	0	0
$23$	36433.8	0.624392	0.312196	−	0.950018i	$-0.398935\pi$
$23$	36433.8	0.624392	0.312196	+	0.950018i	$0.398935\pi$
$24$	0	0
$25$	−33701.9	−0.431384
$26$	0	0
$27$	− 19683.0i	− 0.192450i
$28$	0	0
$29$	− 2855.15i	− 0.0217388i	−0.999941	−	0.0108694i	$-0.996540\pi$
$29$	− 2855.15i	− 0.0217388i	0.999941	−	0.0108694i	$-0.00345991\pi$
$30$	0	0
$31$	90194.3	0.543767	0.271884	−	0.962330i	$-0.412353\pi$
$31$	90194.3	0.543767	0.271884	+	0.962330i	$0.412353\pi$
$32$	0	0
$33$	−206.784	−0.00100166
$34$	0	0
$35$	− 120294.i	− 0.474248i
$36$	0	0
$37$	− 38156.3i	− 0.123840i	−0.998081	−	0.0619199i	$-0.980278\pi$
$37$	− 38156.3i	− 0.123840i	0.998081	−	0.0619199i	$-0.0197223\pi$
$38$	0	0
$39$	119472.	0.322507
$40$	0	0
$41$	−127357.	−0.288588	−0.144294	−	0.989535i	$-0.546091\pi$
$41$	−127357.	−0.288588	−0.144294	+	0.989535i	$0.546091\pi$
$42$	0	0
$43$	− 644347.i	− 1.23589i	−0.786221	−	0.617946i	$-0.787965\pi$
$43$	− 644347.i	− 1.23589i	0.786221	−	0.617946i	$-0.212035\pi$
$44$	0	0
$45$	243781.i	0.398801i
$46$	0	0
$47$	603462.	0.847827	0.423913	−	0.905703i	$-0.360656\pi$
$47$	603462.	0.847827	0.423913	+	0.905703i	$0.360656\pi$
$48$	0	0
$49$	−694141.	−0.842871
$50$	0	0
$51$	− 133793.i	− 0.141234i
$52$	0	0
$53$	291256.i	0.268726i	0.990932	+	0.134363i	$0.0428989\pi$
$53$	291256.i	0.268726i	−0.990932	+	0.134363i	$0.957101\pi$
$54$	0	0
$55$	2561.10	0.00207567
$56$	0	0
$57$	495962.	0.354721
$58$	0	0
$59$	− 1.72805e6i	− 1.09540i	−0.836673	−	0.547702i	$-0.815503\pi$
$59$	− 1.72805e6i	− 1.09540i	0.836673	−	0.547702i	$-0.184497\pi$
$60$	0	0
$61$	1.64795e6i	0.929586i	0.885419	+	0.464793i	$0.153871\pi$
$61$	1.64795e6i	0.929586i	−0.885419	+	0.464793i	$0.846129\pi$
$62$	0	0
$63$	−262240.	−0.132132
$64$	0	0
$65$	−1.47970e6	−0.668309
$66$	0	0
$67$	3.21269e6i	1.30499i	0.757793	+	0.652495i	$0.226278\pi$
$67$	3.21269e6i	1.30499i	−0.757793	+	0.652495i	$0.773722\pi$
$68$	0	0
$69$	983714.i	0.360493i
$70$	0	0
$71$	−2.03751e6	−0.675609	−0.337805	−	0.941216i	$-0.609684\pi$
$71$	−2.03751e6	−0.675609	−0.337805	+	0.941216i	$0.609684\pi$
$72$	0	0
$73$	1.45445e6	0.437591	0.218795	−	0.975771i	$-0.429787\pi$
$73$	1.45445e6	0.437591	0.218795	+	0.975771i	$0.429787\pi$
$74$	0	0
$75$	− 909950.i	− 0.249060i
$76$	0	0
$77$	2755.02i	0 0.000687713i
$78$	0	0
$79$	−4.18669e6	−0.955380	−0.477690	−	0.878529i	$-0.658526\pi$
$79$	−4.18669e6	−0.955380	−0.477690	+	0.878529i	$0.658526\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	5.85510e6i	1.12399i	0.827142	+	0.561993i	$0.189965\pi$
$83$	5.85510e6i	1.12399i	−0.827142	+	0.561993i	$0.810035\pi$
$84$	0	0
$85$	1.65708e6i	0.292669i
$86$	0	0
$87$	77089.0	0.0125509
$88$	0	0
$89$	−1.09233e7	−1.64244	−0.821220	−	0.570612i	$-0.806706\pi$
$89$	−1.09233e7	−1.64244	−0.821220	+	0.570612i	$0.806706\pi$
$90$	0	0
$91$	− 1.59174e6i	− 0.221425i
$92$	0	0
$93$	2.43524e6i	0.313944i
$94$	0	0
$95$	−6.14268e6	−0.735064
$96$	0	0
$97$	969168.	0.107820	0.0539098	−	0.998546i	$-0.482832\pi$
$97$	969168.	0.107820	0.0539098	+	0.998546i	$0.482832\pi$
$98$	0	0
$99$	− 5583.18i	0 0.000578307i
$100$	0	0
$101$	− 1.61610e7i	− 1.56078i	−0.625292	−	0.780391i	$-0.715020\pi$
$101$	− 1.61610e7i	− 1.56078i	0.625292	−	0.780391i	$-0.284980\pi$
$102$	0	0
$103$	−1.15969e7	−1.04571	−0.522854	−	0.852422i	$-0.675133\pi$
$103$	−1.15969e7	−1.04571	−0.522854	+	0.852422i	$0.675133\pi$
$104$	0	0
$105$	3.24794e6	0.273807
$106$	0	0
$107$	− 1.36133e7i	− 1.07429i	−0.843490	−	0.537145i	$-0.819503\pi$
$107$	− 1.36133e7i	− 1.07429i	0.843490	−	0.537145i	$-0.180497\pi$
$108$	0	0
$109$	1.25378e7i	0.927320i	0.886013	+	0.463660i	$0.153464\pi$
$109$	1.25378e7i	0.927320i	−0.886013	+	0.463660i	$0.846536\pi$
$110$	0	0
$111$	1.03022e6	0.0714990
$112$	0	0
$113$	−2.14797e6	−0.140040	−0.0700201	−	0.997546i	$-0.522306\pi$
$113$	−2.14797e6	−0.140040	−0.0700201	+	0.997546i	$0.522306\pi$
$114$	0	0
$115$	− 1.21837e7i	− 0.747026i
$116$	0	0
$117$	3.22573e6i	0.186199i
$118$	0	0
$119$	−1.78255e6	−0.0969675
$120$	0	0
$121$	1.94871e7	0.999997
$122$	0	0
$123$	− 3.43863e6i	− 0.166616i
$124$	0	0
$125$	− 1.48553e7i	− 0.680295i
$126$	0	0
$127$	−2.51639e6	−0.109010	−0.0545049	−	0.998514i	$-0.517358\pi$
$127$	−2.51639e6	−0.109010	−0.0545049	+	0.998514i	$0.517358\pi$
$128$	0	0
$129$	1.73974e7	0.713542
$130$	0	0
$131$	− 3.07026e7i	− 1.19323i	−0.802527	−	0.596616i	$-0.796511\pi$
$131$	− 3.07026e7i	− 1.19323i	0.802527	−	0.596616i	$-0.203489\pi$
$132$	0	0
$133$	− 6.60778e6i	− 0.243542i
$134$	0	0
$135$	−6.58210e6	−0.230248
$136$	0	0
$137$	−1.47771e7	−0.490986	−0.245493	−	0.969398i	$-0.578950\pi$
$137$	−1.47771e7	−0.490986	−0.245493	+	0.969398i	$0.578950\pi$
$138$	0	0
$139$	− 2.19753e7i	− 0.694036i	−0.937858	−	0.347018i	$-0.887194\pi$
$139$	− 2.19753e7i	− 0.694036i	0.937858	−	0.347018i	$-0.112806\pi$
$140$	0	0
$141$	1.62935e7i	0.489493i
$142$	0	0
$143$	33888.7	0.000969123 0
$144$	0	0
$145$	−954776.	−0.0260084
$146$	0	0
$147$	− 1.87418e7i	− 0.486632i
$148$	0	0
$149$	− 3.96614e7i	− 0.982236i	−0.871093	−	0.491118i	$-0.836588\pi$
$149$	− 3.96614e7i	− 0.982236i	0.871093	−	0.491118i	$-0.163412\pi$
$150$	0	0
$151$	−1.56982e7	−0.371048	−0.185524	−	0.982640i	$-0.559398\pi$
$151$	−1.56982e7	−0.371048	−0.185524	+	0.982640i	$0.559398\pi$
$152$	0	0
$153$	3.61241e6	0.0815412
$154$	0	0
$155$	− 3.01614e7i	− 0.650565i
$156$	0	0
$157$	3.13506e6i	0.0646542i	0.999477	+	0.0323271i	$0.0102918\pi$
$157$	3.13506e6i	0.0646542i	−0.999477	+	0.0323271i	$0.989708\pi$
$158$	0	0
$159$	−7.86393e6	−0.155149
$160$	0	0
$161$	1.31062e7	0.247506
$162$	0	0
$163$	− 9.03813e7i	− 1.63464i	−0.576184	−	0.817320i	$-0.695459\pi$
$163$	− 9.03813e7i	− 1.63464i	0.576184	−	0.817320i	$-0.304541\pi$
$164$	0	0
$165$	69149.8i	0.00119839i
$166$	0	0
$167$	−8.31945e7	−1.38225	−0.691126	−	0.722735i	$-0.742885\pi$
$167$	−8.31945e7	−1.38225	−0.691126	+	0.722735i	$0.742885\pi$
$168$	0	0
$169$	4.31690e7	0.687968
$170$	0	0
$171$	1.33910e7i	0.204798i
$172$	0	0
$173$	1.29413e7i	0.190027i	0.995476	+	0.0950137i	$0.0302895\pi$
$173$	1.29413e7i	0.190027i	−0.995476	+	0.0950137i	$0.969711\pi$
$174$	0	0
$175$	−1.21234e7	−0.170998
$176$	0	0
$177$	4.66574e7	0.632432
$178$	0	0
$179$	− 5.07072e7i	− 0.660821i	−0.943837	−	0.330411i	$-0.892813\pi$
$179$	− 5.07072e7i	− 0.660821i	0.943837	−	0.330411i	$-0.107187\pi$
$180$	0	0
$181$	− 2.65693e7i	− 0.333047i	−0.986037	−	0.166523i	$-0.946746\pi$
$181$	− 2.65693e7i	− 0.333047i	0.986037	−	0.166523i	$-0.0532541\pi$
$182$	0	0
$183$	−4.44946e7	−0.536697
$184$	0	0
$185$	−1.27597e7	−0.148163
$186$	0	0
$187$	− 37951.1i	0 0.000424403i
$188$	0	0
$189$	− 7.08047e6i	− 0.0762862i
$190$	0	0
$191$	−1.30093e8	−1.35094	−0.675471	−	0.737387i	$-0.736060\pi$
$191$	−1.30093e8	−1.35094	−0.675471	+	0.737387i	$0.736060\pi$
$192$	0	0
$193$	−1.31497e8	−1.31664	−0.658318	−	0.752740i	$-0.728732\pi$
$193$	−1.31497e8	−1.31664	−0.658318	+	0.752740i	$0.728732\pi$
$194$	0	0
$195$	− 3.99519e7i	− 0.385848i
$196$	0	0
$197$	3.03533e7i	0.282862i	0.989948	+	0.141431i	$0.0451703\pi$
$197$	3.03533e7i	0.282862i	−0.989948	+	0.141431i	$0.954830\pi$
$198$	0	0
$199$	−1.96465e8	−1.76726	−0.883628	−	0.468190i	$-0.844906\pi$
$199$	−1.96465e8	−1.76726	−0.883628	+	0.468190i	$0.844906\pi$
$200$	0	0
$201$	−8.67426e7	−0.753436
$202$	0	0
$203$	− 1.02707e6i	− 0.00861714i
$204$	0	0
$205$	4.25887e7i	0.345268i
$206$	0	0
$207$	−2.65603e7	−0.208131
$208$	0	0
$209$	140682.	0.00106592
$210$	0	0
$211$	− 1.04364e8i	− 0.764826i	−0.923991	−	0.382413i	$-0.875093\pi$
$211$	− 1.04364e8i	− 0.764826i	0.923991	−	0.382413i	$-0.124907\pi$
$212$	0	0
$213$	− 5.50128e7i	− 0.390063i
$214$	0	0
$215$	−2.15473e8	−1.47863
$216$	0	0
$217$	3.24451e7	0.215546
$218$	0	0
$219$	3.92701e7i	0.252643i
$220$	0	0
$221$	2.19266e7i	0.136646i
$222$	0	0
$223$	−1.79138e8	−1.08174	−0.540868	−	0.841107i	$-0.681904\pi$
$223$	−1.79138e8	−1.08174	−0.540868	+	0.841107i	$0.681904\pi$
$224$	0	0
$225$	2.45687e7	0.143795
$226$	0	0
$227$	− 6.69053e7i	− 0.379638i	−0.981819	−	0.189819i	$-0.939210\pi$
$227$	− 6.69053e7i	− 0.379638i	0.981819	−	0.189819i	$-0.0607902\pi$
$228$	0	0
$229$	1.46993e8i	0.808856i	0.914570	+	0.404428i	$0.132529\pi$
$229$	1.46993e8i	0.808856i	−0.914570	+	0.404428i	$0.867471\pi$
$230$	0	0
$231$	−74385.5	−0.000397051 0
$232$	0	0
$233$	8.64957e7	0.447970	0.223985	−	0.974593i	$-0.428093\pi$
$233$	8.64957e7	0.447970	0.223985	+	0.974593i	$0.428093\pi$
$234$	0	0
$235$	− 2.01801e8i	− 1.01434i
$236$	0	0
$237$	− 1.13041e8i	− 0.551589i
$238$	0	0
$239$	2.31996e8	1.09923	0.549614	−	0.835419i	$-0.314775\pi$
$239$	2.31996e8	1.09923	0.549614	+	0.835419i	$0.314775\pi$
$240$	0	0
$241$	−2.86516e8	−1.31853	−0.659264	−	0.751911i	$-0.729132\pi$
$241$	−2.86516e8	−1.31853	−0.659264	+	0.751911i	$0.729132\pi$
$242$	0	0
$243$	1.43489e7i	0.0641500i
$244$	0	0
$245$	2.32124e8i	1.00842i
$246$	0	0
$247$	−8.12803e7	−0.343199
$248$	0	0
$249$	−1.58088e8	−0.648933
$250$	0	0
$251$	− 4.23350e8i	− 1.68983i	−0.534904	−	0.844913i	$-0.679652\pi$
$251$	− 4.23350e8i	− 1.68983i	0.534904	−	0.844913i	$-0.320348\pi$
$252$	0	0
$253$	279035.i	0.00108327i
$254$	0	0
$255$	−4.47411e7	−0.168972
$256$	0	0
$257$	3.70769e8	1.36250	0.681252	−	0.732049i	$-0.261436\pi$
$257$	3.70769e8	1.36250	0.681252	+	0.732049i	$0.261436\pi$
$258$	0	0
$259$	− 1.37258e7i	− 0.0490894i
$260$	0	0
$261$	2.08140e6i	0.00724627i
$262$	0	0
$263$	−2.09518e8	−0.710192	−0.355096	−	0.934830i	$-0.615552\pi$
$263$	−2.09518e8	−0.710192	−0.355096	+	0.934830i	$0.615552\pi$
$264$	0	0
$265$	9.73977e7	0.321505
$266$	0	0
$267$	− 2.94929e8i	− 0.948263i
$268$	0	0
$269$	1.69420e8i	0.530678i	0.964155	+	0.265339i	$0.0854838\pi$
$269$	1.69420e8i	0.530678i	−0.964155	+	0.265339i	$0.914516\pi$
$270$	0	0
$271$	5.20327e8	1.58812	0.794061	−	0.607839i	$-0.207963\pi$
$271$	5.20327e8	1.58812	0.794061	+	0.607839i	$0.207963\pi$
$272$	0	0
$273$	4.29769e7	0.127840
$274$	0	0
$275$	− 258112.i	0 0.000748417i
$276$	0	0
$277$	755773.i	0.00213655i	0.999999	+	0.00106827i	$0.000340042\pi$
$277$	755773.i	0.00213655i	−0.999999	+	0.00106827i	$0.999660\pi$
$278$	0	0
$279$	−6.57516e7	−0.181256
$280$	0	0
$281$	−1.42973e8	−0.384400	−0.192200	−	0.981356i	$-0.561562\pi$
$281$	−1.42973e8	−0.384400	−0.192200	+	0.981356i	$0.561562\pi$
$282$	0	0
$283$	− 2.50627e8i	− 0.657318i	−0.944449	−	0.328659i	$-0.893403\pi$
$283$	− 2.50627e8i	− 0.657318i	0.944449	−	0.328659i	$-0.106597\pi$
$284$	0	0
$285$	− 1.65852e8i	− 0.424389i
$286$	0	0
$287$	−4.58134e7	−0.114395
$288$	0	0
$289$	−3.85784e8	−0.940159
$290$	0	0
$291$	2.61675e7i	0.0622497i
$292$	0	0
$293$	1.05491e8i	0.245008i	0.992468	+	0.122504i	$0.0390925\pi$
$293$	1.05491e8i	0.245008i	−0.992468	+	0.122504i	$0.960908\pi$
$294$	0	0
$295$	−5.77869e8	−1.31055
$296$	0	0
$297$	150746.	0.000333886 0
$298$	0	0
$299$	− 1.61215e8i	− 0.348784i
$300$	0	0
$301$	− 2.31788e8i	− 0.489901i
$302$	0	0
$303$	4.36346e8	0.901118
$304$	0	0
$305$	5.51083e8	1.11216
$306$	0	0
$307$	2.31792e8i	0.457209i	0.973519	+	0.228604i	$0.0734162\pi$
$307$	2.31792e8i	0.457209i	−0.973519	+	0.228604i	$0.926584\pi$
$308$	0	0
$309$	− 3.13115e8i	− 0.603740i
$310$	0	0
$311$	−1.84093e8	−0.347037	−0.173519	−	0.984831i	$-0.555514\pi$
$311$	−1.84093e8	−0.347037	−0.173519	+	0.984831i	$0.555514\pi$
$312$	0	0
$313$	−2.52736e8	−0.465867	−0.232933	−	0.972493i	$-0.574832\pi$
$313$	−2.52736e8	−0.465867	−0.232933	+	0.972493i	$0.574832\pi$
$314$	0	0
$315$	8.76943e7i	0.158083i
$316$	0	0
$317$	− 6.94680e8i	− 1.22483i	−0.790535	−	0.612417i	$-0.790197\pi$
$317$	− 6.94680e8i	− 1.22483i	0.790535	−	0.612417i	$-0.209803\pi$
$318$	0	0
$319$	21866.7	3.77151e−5 0
$320$	0	0
$321$	3.67560e8	0.620242
$322$	0	0
$323$	9.10237e7i	0.150295i
$324$	0	0
$325$	1.49126e8i	0.240970i
$326$	0	0
$327$	−3.38521e8	−0.535388
$328$	0	0
$329$	2.17080e8	0.336074
$330$	0	0
$331$	3.17114e8i	0.480638i	0.970694	+	0.240319i	$0.0772520\pi$
$331$	3.17114e8i	0.480638i	−0.970694	+	0.240319i	$0.922748\pi$
$332$	0	0
$333$	2.78160e7i	0.0412799i
$334$	0	0
$335$	1.07434e9	1.56129
$336$	0	0
$337$	−4.82687e8	−0.687007	−0.343504	−	0.939151i	$-0.611614\pi$
$337$	−4.82687e8	−0.687007	−0.343504	+	0.939151i	$0.611614\pi$
$338$	0	0
$339$	− 5.79951e7i	− 0.0808523i
$340$	0	0
$341$	690769.i	0 0.000943393i
$342$	0	0
$343$	−5.45949e8	−0.730504
$344$	0	0
$345$	3.28959e8	0.431296
$346$	0	0
$347$	1.13188e9i	1.45428i	0.686491	+	0.727139i	$0.259150\pi$
$347$	1.13188e9i	1.45428i	−0.686491	+	0.727139i	$0.740850\pi$
$348$	0	0
$349$	3.69067e8i	0.464746i	0.972627	+	0.232373i	$0.0746490\pi$
$349$	3.69067e8i	0.464746i	−0.972627	+	0.232373i	$0.925351\pi$
$350$	0	0
$351$	−8.70948e7	−0.107502
$352$	0	0
$353$	5.51944e7	0.0667857	0.0333928	−	0.999442i	$-0.489369\pi$
$353$	5.51944e7	0.0667857	0.0333928	+	0.999442i	$0.489369\pi$
$354$	0	0
$355$	6.81354e8i	0.808302i
$356$	0	0
$357$	− 4.81287e7i	− 0.0559842i
$358$	0	0
$359$	−7.29593e8	−0.832243	−0.416121	−	0.909309i	$-0.636611\pi$
$359$	−7.29593e8	−0.832243	−0.416121	+	0.909309i	$0.636611\pi$
$360$	0	0
$361$	5.56453e8	0.622520
$362$	0	0
$363$	5.26152e8i	0.577349i
$364$	0	0
$365$	− 4.86375e8i	− 0.523535i
$366$	0	0
$367$	1.46451e9	1.54654	0.773269	−	0.634078i	$-0.218620\pi$
$367$	1.46451e9	1.54654	0.773269	+	0.634078i	$0.218620\pi$
$368$	0	0
$369$	9.28429e7	0.0961959
$370$	0	0
$371$	1.04772e8i	0.106522i
$372$	0	0
$373$	8.86627e8i	0.884626i	0.896861	+	0.442313i	$0.145842\pi$
$373$	8.86627e8i	0.884626i	−0.896861	+	0.442313i	$0.854158\pi$
$374$	0	0
$375$	4.01094e8	0.392769
$376$	0	0
$377$	−1.26337e7	−0.0121432
$378$	0	0
$379$	9.94620e8i	0.938469i	0.883074	+	0.469234i	$0.155470\pi$
$379$	9.94620e8i	0.938469i	−0.883074	+	0.469234i	$0.844530\pi$
$380$	0	0
$381$	− 6.79426e7i	− 0.0629368i
$382$	0	0
$383$	7.03399e8	0.639743	0.319872	−	0.947461i	$-0.396360\pi$
$383$	7.03399e8	0.639743	0.319872	+	0.947461i	$0.396360\pi$
$384$	0	0
$385$	921293.	0.000822783 0
$386$	0	0
$387$	4.69729e8i	0.411964i
$388$	0	0
$389$	− 1.63673e9i	− 1.40979i	−0.709313	−	0.704894i	$-0.750994\pi$
$389$	− 1.63673e9i	− 1.40979i	0.709313	−	0.704894i	$-0.249006\pi$
$390$	0	0
$391$	−1.80541e8	−0.152741
$392$	0	0
$393$	8.28969e8	0.688913
$394$	0	0
$395$	1.40005e9i	1.14302i
$396$	0	0
$397$	1.45839e9i	1.16979i	0.811110	+	0.584893i	$0.198864\pi$
$397$	1.45839e9i	1.16979i	−0.811110	+	0.584893i	$0.801136\pi$
$398$	0	0
$399$	1.78410e8	0.140609
$400$	0	0
$401$	−1.03636e9	−0.802609	−0.401305	−	0.915945i	$-0.631443\pi$
$401$	−1.03636e9	−0.802609	−0.401305	+	0.915945i	$0.631443\pi$
$402$	0	0
$403$	− 3.99098e8i	− 0.303747i
$404$	0	0
$405$	− 1.77717e8i	− 0.132934i
$406$	0	0
$407$	292227.	0.000214852 0
$408$	0	0
$409$	−1.40788e9	−1.01750	−0.508748	−	0.860916i	$-0.669891\pi$
$409$	−1.40788e9	−1.01750	−0.508748	+	0.860916i	$0.669891\pi$
$410$	0	0
$411$	− 3.98983e8i	− 0.283471i
$412$	0	0
$413$	− 6.21623e8i	− 0.434212i
$414$	0	0
$415$	1.95797e9	1.34474
$416$	0	0
$417$	5.93332e8	0.400702
$418$	0	0
$419$	− 1.18000e8i	− 0.0783669i	−0.999232	−	0.0391835i	$-0.987524\pi$
$419$	− 1.18000e8i	− 0.0783669i	0.999232	−	0.0391835i	$-0.0124757\pi$
$420$	0	0
$421$	1.02083e9i	0.666754i	0.942794	+	0.333377i	$0.108188\pi$
$421$	1.02083e9i	0.666754i	−0.942794	+	0.333377i	$0.891812\pi$
$422$	0	0
$423$	−4.39924e8	−0.282609
$424$	0	0
$425$	1.67003e8	0.105527
$426$	0	0
$427$	5.92809e8i	0.368483i
$428$	0	0
$429$	914995.i	0 0.000559523i
$430$	0	0
$431$	3.24478e9	1.95216	0.976080	−	0.217412i	$-0.0697616\pi$
$431$	3.24478e9	1.95216	0.976080	+	0.217412i	$0.0697616\pi$
$432$	0	0
$433$	7.46824e7	0.0442090	0.0221045	−	0.999756i	$-0.492963\pi$
$433$	7.46824e7	0.0442090	0.0221045	+	0.999756i	$0.492963\pi$
$434$	0	0
$435$	− 2.57790e7i	− 0.0150160i
$436$	0	0
$437$	− 6.69252e8i	− 0.383623i
$438$	0	0
$439$	1.44402e9	0.814605	0.407303	−	0.913293i	$-0.366469\pi$
$439$	1.44402e9	0.814605	0.407303	+	0.913293i	$0.366469\pi$
$440$	0	0
$441$	5.06029e8	0.280957
$442$	0	0
$443$	5.61269e8i	0.306731i	0.988170	+	0.153366i	$0.0490112\pi$
$443$	5.61269e8i	0.306731i	−0.988170	+	0.153366i	$0.950989\pi$
$444$	0	0
$445$	3.65281e9i	1.96502i
$446$	0	0
$447$	1.07086e9	0.567094
$448$	0	0
$449$	−8.13423e8	−0.424086	−0.212043	−	0.977260i	$-0.568012\pi$
$449$	−8.13423e8	−0.424086	−0.212043	+	0.977260i	$0.568012\pi$
$450$	0	0
$451$	− 975383.i	0 0.000500677i
$452$	0	0
$453$	− 4.23851e8i	− 0.214225i
$454$	0	0
$455$	−5.32286e8	−0.264914
$456$	0	0
$457$	−3.12047e9	−1.52937	−0.764687	−	0.644401i	$-0.777107\pi$
$457$	−3.12047e9	−1.52937	−0.764687	+	0.644401i	$0.777107\pi$
$458$	0	0
$459$	9.75351e7i	0.0470779i
$460$	0	0
$461$	− 1.11223e9i	− 0.528738i	−0.964422	−	0.264369i	$-0.914836\pi$
$461$	− 1.11223e9i	− 0.528738i	0.964422	−	0.264369i	$-0.0851637\pi$
$462$	0	0
$463$	3.53099e9	1.65334	0.826672	−	0.562684i	$-0.190231\pi$
$463$	3.53099e9	1.65334	0.826672	+	0.562684i	$0.190231\pi$
$464$	0	0
$465$	8.14359e8	0.375604
$466$	0	0
$467$	5.21138e8i	0.236779i	0.992967	+	0.118390i	$0.0377732\pi$
$467$	5.21138e8i	0.236779i	−0.992967	+	0.118390i	$0.962227\pi$
$468$	0	0
$469$	1.15569e9i	0.517291i
$470$	0	0
$471$	−8.46465e7	−0.0373281
$472$	0	0
$473$	4.93485e6	0.00214417
$474$	0	0
$475$	6.19068e8i	0.265040i
$476$	0	0
$477$	− 2.12326e8i	− 0.0895754i
$478$	0	0
$479$	2.62098e9	1.08966	0.544828	−	0.838548i	$-0.316595\pi$
$479$	2.62098e9	1.08966	0.544828	+	0.838548i	$0.316595\pi$
$480$	0	0
$481$	−1.68837e8	−0.0691767
$482$	0	0
$483$	3.53867e8i	0.142897i
$484$	0	0
$485$	− 3.24095e8i	− 0.128996i
$486$	0	0
$487$	−1.32738e9	−0.520769	−0.260384	−	0.965505i	$-0.583849\pi$
$487$	−1.32738e9	−0.520769	−0.260384	+	0.965505i	$0.583849\pi$
$488$	0	0
$489$	2.44030e9	0.943760
$490$	0	0
$491$	− 1.99052e8i	− 0.0758895i	−0.999280	−	0.0379447i	$-0.987919\pi$
$491$	− 1.99052e8i	− 0.0758895i	0.999280	−	0.0379447i	$-0.0120811\pi$
$492$	0	0
$493$	1.41481e7i	0.00531783i
$494$	0	0
$495$	−1.86704e6	−0.000691889 0
$496$	0	0
$497$	−7.32944e8	−0.267808
$498$	0	0
$499$	1.85791e9i	0.669379i	0.942329	+	0.334689i	$0.108631\pi$
$499$	1.85791e9i	0.669379i	−0.942329	+	0.334689i	$0.891369\pi$
$500$	0	0
$501$	− 2.24625e9i	− 0.798043i
$502$	0	0
$503$	−1.01227e9	−0.354657	−0.177328	−	0.984152i	$-0.556745\pi$
$503$	−1.01227e9	−0.354657	−0.177328	+	0.984152i	$0.556745\pi$
$504$	0	0
$505$	−5.40431e9	−1.86733
$506$	0	0
$507$	1.16556e9i	0.397199i
$508$	0	0
$509$	− 9.32060e8i	− 0.313279i	−0.987656	−	0.156640i	$-0.949934\pi$
$509$	− 9.32060e8i	− 0.313279i	0.987656	−	0.156640i	$-0.0500661\pi$
$510$	0	0
$511$	5.23201e8	0.173459
$512$	0	0
$513$	−3.61556e8	−0.118240
$514$	0	0
$515$	3.87805e9i	1.25109i
$516$	0	0
$517$	4.62172e6i	0.00147091i
$518$	0	0
$519$	−3.49415e8	−0.109712
$520$	0	0
$521$	5.23990e9	1.62327	0.811636	−	0.584164i	$-0.198577\pi$
$521$	5.23990e9	1.62327	0.811636	+	0.584164i	$0.198577\pi$
$522$	0	0
$523$	4.67012e7i	0.0142749i	0.999975	+	0.00713744i	$0.00227194\pi$
$523$	4.67012e7i	0.0142749i	−0.999975	+	0.00713744i	$0.997728\pi$
$524$	0	0
$525$	− 3.27332e8i	− 0.0987258i
$526$	0	0
$527$	−4.46939e8	−0.133018
$528$	0	0
$529$	−2.07740e9	−0.610134
$530$	0	0
$531$	1.25975e9i	0.365135i
$532$	0	0
$533$	5.63537e8i	0.161204i
$534$	0	0
$535$	−4.55237e9	−1.28529
$536$	0	0
$537$	1.36909e9	0.381525
$538$	0	0
$539$	− 5.31620e6i	− 0.00146232i
$540$	0	0
$541$	5.87911e9i	1.59632i	0.602442	+	0.798162i	$0.294194\pi$
$541$	5.87911e9i	1.59632i	−0.602442	+	0.798162i	$0.705806\pi$
$542$	0	0
$543$	7.17371e8	0.192285
$544$	0	0
$545$	4.19272e9	1.10945
$546$	0	0
$547$	2.61327e9i	0.682699i	0.939937	+	0.341349i	$0.110884\pi$
$547$	2.61327e9i	0.682699i	−0.939937	+	0.341349i	$0.889116\pi$
$548$	0	0
$549$	− 1.20135e9i	− 0.309862i
$550$	0	0
$551$	−5.24461e7	−0.0133562
$552$	0	0
$553$	−1.50606e9	−0.378707
$554$	0	0
$555$	− 3.44511e8i	− 0.0855417i
$556$	0	0
$557$	− 6.01973e8i	− 0.147599i	−0.997273	−	0.0737996i	$-0.976487\pi$
$557$	− 6.01973e8i	− 0.147599i	0.997273	−	0.0737996i	$-0.0235125\pi$
$558$	0	0
$559$	−2.85116e9	−0.690366
$560$	0	0
$561$	1.02468e6	0.000245029 0
$562$	0	0
$563$	1.61410e9i	0.381199i	0.981668	+	0.190600i	$0.0610432\pi$
$563$	1.61410e9i	0.381199i	−0.981668	+	0.190600i	$0.938957\pi$
$564$	0	0
$565$	7.18291e8i	0.167545i
$566$	0	0
$567$	1.91173e8	0.0440438
$568$	0	0
$569$	2.86279e9	0.651472	0.325736	−	0.945461i	$-0.394388\pi$
$569$	2.86279e9	0.651472	0.325736	+	0.945461i	$0.394388\pi$
$570$	0	0
$571$	− 8.50676e8i	− 0.191222i	−0.995419	−	0.0956110i	$-0.969520\pi$
$571$	− 8.50676e8i	− 0.191222i	0.995419	−	0.0956110i	$-0.0304805\pi$
$572$	0	0
$573$	− 3.51251e9i	− 0.779967i
$574$	0	0
$575$	−1.22789e9	−0.269353
$576$	0	0
$577$	1.38221e9	0.299542	0.149771	−	0.988721i	$-0.452146\pi$
$577$	1.38221e9	0.299542	0.149771	+	0.988721i	$0.452146\pi$
$578$	0	0
$579$	− 3.55042e9i	− 0.760160i
$580$	0	0
$581$	2.10622e9i	0.445542i
$582$	0	0
$583$	−2.23064e6	−0.000466219 0
$584$	0	0
$585$	1.07870e9	0.222770
$586$	0	0
$587$	− 6.95938e9i	− 1.42016i	−0.704121	−	0.710080i	$-0.748659\pi$
$587$	− 6.95938e9i	− 1.42016i	0.704121	−	0.710080i	$-0.251341\pi$
$588$	0	0
$589$	− 1.65677e9i	− 0.334087i
$590$	0	0
$591$	−8.19539e8	−0.163310
$592$	0	0
$593$	4.92113e9	0.969110	0.484555	−	0.874761i	$-0.338982\pi$
$593$	4.92113e9	0.969110	0.484555	+	0.874761i	$0.338982\pi$
$594$	0	0
$595$	5.96092e8i	0.116012i
$596$	0	0
$597$	− 5.30456e9i	− 1.02033i
$598$	0	0
$599$	−5.11728e9	−0.972849	−0.486424	−	0.873723i	$-0.661699\pi$
$599$	−5.11728e9	−0.972849	−0.486424	+	0.873723i	$0.661699\pi$
$600$	0	0
$601$	−8.63140e9	−1.62189	−0.810944	−	0.585124i	$-0.801046\pi$
$601$	−8.63140e9	−1.62189	−0.810944	+	0.585124i	$0.801046\pi$
$602$	0	0
$603$	− 2.34205e9i	− 0.434996i
$604$	0	0
$605$	− 6.51659e9i	− 1.19640i
$606$	0	0
$607$	3.06149e9	0.555613	0.277806	−	0.960637i	$-0.410393\pi$
$607$	3.06149e9	0.555613	0.277806	+	0.960637i	$0.410393\pi$
$608$	0	0
$609$	2.77308e7	0.00497511
$610$	0	0
$611$	− 2.67024e9i	− 0.473594i
$612$	0	0
$613$	4.18732e9i	0.734217i	0.930178	+	0.367109i	$0.119652\pi$
$613$	4.18732e9i	0.734217i	−0.930178	+	0.367109i	$0.880348\pi$
$614$	0	0
$615$	−1.14989e9	−0.199340
$616$	0	0
$617$	6.07057e9	1.04047	0.520237	−	0.854022i	$-0.325843\pi$
$617$	6.07057e9	1.04047	0.520237	+	0.854022i	$0.325843\pi$
$618$	0	0
$619$	− 4.67631e9i	− 0.792476i	−0.918148	−	0.396238i	$-0.870315\pi$
$619$	− 4.67631e9i	− 0.792476i	0.918148	−	0.396238i	$-0.129685\pi$
$620$	0	0
$621$	− 7.17127e8i	− 0.120164i
$622$	0	0
$623$	−3.92939e9	−0.651054
$624$	0	0
$625$	−7.60066e9	−1.24529
$626$	0	0
$627$	3.79841e6i	0 0.000615412i
$628$	0	0
$629$	1.89076e8i	0.0302942i
$630$	0	0
$631$	−2.05258e8	−0.0325235	−0.0162618	−	0.999868i	$-0.505177\pi$
$631$	−2.05258e8	−0.0325235	−0.0162618	+	0.999868i	$0.505177\pi$
$632$	0	0
$633$	2.81783e9	0.441573
$634$	0	0
$635$	8.41495e8i	0.130420i
$636$	0	0
$637$	3.07149e9i	0.470826i
$638$	0	0
$639$	1.48534e9	0.225203
$640$	0	0
$641$	2.89259e9	0.433794	0.216897	−	0.976195i	$-0.430406\pi$
$641$	2.89259e9	0.433794	0.216897	+	0.976195i	$0.430406\pi$
$642$	0	0
$643$	− 7.83603e9i	− 1.16241i	−0.813759	−	0.581203i	$-0.802582\pi$
$643$	− 7.83603e9i	− 1.16241i	0.813759	−	0.581203i	$-0.197418\pi$
$644$	0	0
$645$	− 5.81777e9i	− 0.853685i
$646$	0	0
$647$	7.62856e9	1.10733	0.553666	−	0.832739i	$-0.313229\pi$
$647$	7.62856e9	1.10733	0.553666	+	0.832739i	$0.313229\pi$
$648$	0	0
$649$	1.32346e7	0.00190044
$650$	0	0
$651$	8.76019e8i	0.124446i
$652$	0	0
$653$	− 3.00438e9i	− 0.422240i	−0.977460	−	0.211120i	$-0.932289\pi$
$653$	− 3.00438e9i	− 0.422240i	0.977460	−	0.211120i	$-0.0677111\pi$
$654$	0	0
$655$	−1.02671e10	−1.42759
$656$	0	0
$657$	−1.06029e9	−0.145864
$658$	0	0
$659$	− 1.12961e10i	− 1.53755i	−0.639518	−	0.768776i	$-0.720866\pi$
$659$	− 1.12961e10i	− 1.53755i	0.639518	−	0.768776i	$-0.279134\pi$
$660$	0	0
$661$	8.93069e9i	1.20276i	0.798962	+	0.601382i	$0.205383\pi$
$661$	8.93069e9i	1.20276i	−0.798962	+	0.601382i	$0.794617\pi$
$662$	0	0
$663$	−5.92017e8	−0.0788928
$664$	0	0
$665$	−2.20967e9	−0.291375
$666$	0	0
$667$	− 1.04024e8i	− 0.0135735i
$668$	0	0
$669$	− 4.83673e9i	− 0.624541i
$670$	0	0
$671$	−1.26211e7	−0.00161276
$672$	0	0
$673$	−9.26440e9	−1.17156	−0.585780	−	0.810470i	$-0.699212\pi$
$673$	−9.26440e9	−1.17156	−0.585780	+	0.810470i	$0.699212\pi$
$674$	0	0
$675$	6.63354e8i	0.0830198i
$676$	0	0
$677$	8.10605e9i	1.00404i	0.864857	+	0.502018i	$0.167409\pi$
$677$	8.10605e9i	1.00404i	−0.864857	+	0.502018i	$0.832591\pi$
$678$	0	0
$679$	3.48634e8	0.0427391
$680$	0	0
$681$	1.80644e9	0.219184
$682$	0	0
$683$	− 4.87846e9i	− 0.585882i	−0.956130	−	0.292941i	$-0.905366\pi$
$683$	− 4.87846e9i	− 0.585882i	0.956130	−	0.292941i	$-0.0946340\pi$
$684$	0	0
$685$	4.94156e9i	0.587417i
$686$	0	0
$687$	−3.96880e9	−0.466993
$688$	0	0
$689$	1.28877e9	0.150110
$690$	0	0
$691$	− 1.21837e10i	− 1.40478i	−0.711793	−	0.702389i	$-0.752117\pi$
$691$	− 1.21837e10i	− 1.40478i	0.711793	−	0.702389i	$-0.247883\pi$
$692$	0	0
$693$	− 2.00841e6i	0 0.000229238i
$694$	0	0
$695$	−7.34864e9	−0.830348
$696$	0	0
$697$	6.31090e8	0.0705954
$698$	0	0
$699$	2.33538e9i	0.258636i
$700$	0	0
$701$	− 7.77418e9i	− 0.852396i	−0.904630	−	0.426198i	$-0.859853\pi$
$701$	− 7.77418e9i	− 0.852396i	0.904630	−	0.426198i	$-0.140147\pi$
$702$	0	0
$703$	−7.00892e8	−0.0760864
$704$	0	0
$705$	5.44862e9	0.585632
$706$	0	0
$707$	− 5.81350e9i	− 0.618685i
$708$	0	0
$709$	− 6.03884e9i	− 0.636344i	−0.948033	−	0.318172i	$-0.896931\pi$
$709$	− 6.03884e9i	− 0.636344i	0.948033	−	0.318172i	$-0.103069\pi$
$710$	0	0
$711$	3.05210e9	0.318460
$712$	0	0
$713$	3.28612e9	0.339524
$714$	0	0
$715$	− 1.13326e7i	− 0.00115946i
$716$	0	0
$717$	6.26389e9i	0.634639i
$718$	0	0
$719$	−9.96626e9	−0.999957	−0.499978	−	0.866038i	$-0.666659\pi$
$719$	−9.96626e9	−0.999957	−0.499978	+	0.866038i	$0.666659\pi$
$720$	0	0
$721$	−4.17168e9	−0.414513
$722$	0	0
$723$	− 7.73594e9i	− 0.761253i
$724$	0	0
$725$	9.62238e7i	0.00937776i
$726$	0	0
$727$	7.90039e8	0.0762567	0.0381284	−	0.999273i	$-0.487860\pi$
$727$	7.90039e8	0.0762567	0.0381284	+	0.999273i	$0.487860\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	3.19293e9i	0.302328i
$732$	0	0
$733$	1.55021e10i	1.45387i	0.686705	+	0.726937i	$0.259057\pi$
$733$	1.55021e10i	1.45387i	−0.686705	+	0.726937i	$0.740943\pi$
$734$	0	0
$735$	−6.26736e9	−0.582209
$736$	0	0
$737$	−2.46050e7	−0.00226405
$738$	0	0
$739$	− 7.87958e9i	− 0.718203i	−0.933298	−	0.359102i	$-0.883083\pi$
$739$	− 7.87958e9i	− 0.718203i	0.933298	−	0.359102i	$-0.116917\pi$
$740$	0	0
$741$	− 2.19457e9i	− 0.198146i
$742$	0	0
$743$	−4.99093e9	−0.446397	−0.223198	−	0.974773i	$-0.571650\pi$
$743$	−4.99093e9	−0.446397	−0.223198	+	0.974773i	$0.571650\pi$
$744$	0	0
$745$	−1.32630e10	−1.17515
$746$	0	0
$747$	− 4.26836e9i	− 0.374662i
$748$	0	0
$749$	− 4.89706e9i	− 0.425843i
$750$	0	0
$751$	−2.15895e10	−1.85996	−0.929978	−	0.367615i	$-0.880174\pi$
$751$	−2.15895e10	−1.85996	−0.929978	+	0.367615i	$0.880174\pi$
$752$	0	0
$753$	1.14305e10	0.975621
$754$	0	0
$755$	5.24956e9i	0.443924i
$756$	0	0
$757$	1.68472e10i	1.41153i	0.708445	+	0.705766i	$0.249397\pi$
$757$	1.68472e10i	1.41153i	−0.708445	+	0.705766i	$0.750603\pi$
$758$	0	0
$759$	−7.53395e6	−0.000625427 0
$760$	0	0
$761$	−8.36510e9	−0.688058	−0.344029	−	0.938959i	$-0.611792\pi$
$761$	−8.36510e9	−0.688058	−0.344029	+	0.938959i	$0.611792\pi$
$762$	0	0
$763$	4.51017e9i	0.367584i
$764$	0	0
$765$	− 1.20801e9i	− 0.0975563i
$766$	0	0
$767$	−7.64641e9	−0.611890
$768$	0	0
$769$	−9.29539e9	−0.737098	−0.368549	−	0.929608i	$-0.620145\pi$
$769$	−9.29539e9	−0.737098	−0.368549	+	0.929608i	$0.620145\pi$
$770$	0	0
$771$	1.00108e10i	0.786642i
$772$	0	0
$773$	6.64734e9i	0.517631i	0.965927	+	0.258815i	$0.0833321\pi$
$773$	6.64734e9i	0.517631i	−0.965927	+	0.258815i	$0.916668\pi$
$774$	0	0
$775$	−3.03971e9	−0.234572
$776$	0	0
$777$	3.70596e8	0.0283418
$778$	0	0
$779$	2.33941e9i	0.177307i
$780$	0	0
$781$	− 1.56046e7i	− 0.00117213i
$782$	0	0
$783$	−5.61979e7	−0.00418363
$784$	0	0
$785$	1.04838e9	0.0773525
$786$	0	0
$787$	1.71417e9i	0.125355i	0.998034	+	0.0626775i	$0.0199639\pi$
$787$	1.71417e9i	0.125355i	−0.998034	+	0.0626775i	$0.980036\pi$
$788$	0	0
$789$	− 5.65698e9i	− 0.410030i
$790$	0	0
$791$	−7.72678e8	−0.0555112
$792$	0	0
$793$	7.29197e9	0.519264
$794$	0	0
$795$	2.62974e9i	0.185621i
$796$	0	0
$797$	− 2.78086e10i	− 1.94570i	−0.231438	−	0.972850i	$-0.574343\pi$
$797$	− 2.78086e10i	− 1.94570i	0.231438	−	0.972850i	$-0.425657\pi$
$798$	0	0
$799$	−2.99033e9	−0.207399
$800$	0	0
$801$	7.96309e9	0.547480
$802$	0	0
$803$	1.11391e7i	0 0.000759185i
$804$	0	0
$805$	− 4.38277e9i	− 0.296117i
$806$	0	0
$807$	−4.57433e9	−0.306387
$808$	0	0
$809$	−1.09333e10	−0.725992	−0.362996	−	0.931791i	$-0.618246\pi$
$809$	−1.09333e10	−0.725992	−0.362996	+	0.931791i	$0.618246\pi$
$810$	0	0
$811$	− 1.39081e10i	− 0.915574i	−0.889062	−	0.457787i	$-0.848642\pi$
$811$	− 1.39081e10i	− 0.915574i	0.889062	−	0.457787i	$-0.151358\pi$
$812$	0	0
$813$	1.40488e10i	0.916902i
$814$	0	0
$815$	−3.02240e10	−1.95569
$816$	0	0
$817$	−1.18360e10	−0.759324
$818$	0	0
$819$	1.16038e9i	0.0738084i
$820$	0	0
$821$	− 3.18331e9i	− 0.200760i	−0.994949	−	0.100380i	$-0.967994\pi$
$821$	− 3.18331e9i	− 0.200760i	0.994949	−	0.100380i	$-0.0320059\pi$
$822$	0	0
$823$	1.89329e10	1.18391	0.591954	−	0.805971i	$-0.298357\pi$
$823$	1.89329e10	1.18391	0.591954	+	0.805971i	$0.298357\pi$
$824$	0	0
$825$	6.96902e6	0.000432098 0
$826$	0	0
$827$	− 1.91324e9i	− 0.117625i	−0.998269	−	0.0588127i	$-0.981269\pi$
$827$	− 1.91324e9i	− 0.117625i	0.998269	−	0.0588127i	$-0.0187315\pi$
$828$	0	0
$829$	3.26316e10i	1.98929i	0.103367	+	0.994643i	$0.467038\pi$
$829$	3.26316e10i	1.98929i	−0.103367	+	0.994643i	$0.532962\pi$
$830$	0	0
$831$	−2.04059e7	−0.00123354
$832$	0	0
$833$	3.43967e9	0.206186
$834$	0	0
$835$	2.78207e10i	1.65373i
$836$	0	0
$837$	− 1.77529e9i	− 0.104648i
$838$	0	0
$839$	3.41092e10	1.99390	0.996952	−	0.0780163i	$-0.0248586\pi$
$839$	3.41092e10	1.99390	0.996952	+	0.0780163i	$0.0248586\pi$
$840$	0	0
$841$	1.72417e10	0.999527
$842$	0	0
$843$	− 3.86028e9i	− 0.221933i
$844$	0	0
$845$	− 1.44359e10i	− 0.823089i
$846$	0	0
$847$	7.01000e9	0.396393
$848$	0	0
$849$	6.76693e9	0.379502
$850$	0	0
$851$	− 1.39018e9i	− 0.0773247i
$852$	0	0
$853$	− 1.76028e10i	− 0.971094i	−0.874211	−	0.485547i	$-0.838620\pi$
$853$	− 1.76028e10i	− 0.971094i	0.874211	−	0.485547i	$-0.161380\pi$
$854$	0	0
$855$	4.47801e9	0.245021
$856$	0	0
$857$	5.03778e9	0.273405	0.136703	−	0.990612i	$-0.456350\pi$
$857$	5.03778e9	0.273405	0.136703	+	0.990612i	$0.456350\pi$
$858$	0	0
$859$	− 5.09167e9i	− 0.274084i	−0.990565	−	0.137042i	$-0.956240\pi$
$859$	− 5.09167e9i	− 0.274084i	0.990565	−	0.137042i	$-0.0437596\pi$
$860$	0	0
$861$	− 1.23696e9i	− 0.0660457i
$862$	0	0
$863$	−2.08585e9	−0.110471	−0.0552353	−	0.998473i	$-0.517591\pi$
$863$	−2.08585e9	−0.110471	−0.0552353	+	0.998473i	$0.517591\pi$
$864$	0	0
$865$	4.32764e9	0.227350
$866$	0	0
$867$	− 1.04162e10i	− 0.542801i
$868$	0	0
$869$	− 3.20645e7i	− 0.00165751i
$870$	0	0
$871$	1.42157e10	0.728964
$872$	0	0
$873$	−7.06523e8	−0.0359399
$874$	0	0
$875$	− 5.34384e9i	− 0.269665i
$876$	0	0
$877$	− 2.92237e8i	− 0.0146297i	−0.999973	−	0.00731486i	$-0.997672\pi$
$877$	− 2.92237e8i	− 0.0146297i	0.999973	−	0.00731486i	$-0.00232841\pi$
$878$	0	0
$879$	−2.84827e9	−0.141456
$880$	0	0
$881$	−1.00200e10	−0.493688	−0.246844	−	0.969055i	$-0.579393\pi$
$881$	−1.00200e10	−0.493688	−0.246844	+	0.969055i	$0.579393\pi$
$882$	0	0
$883$	− 5.93877e8i	− 0.0290291i	−0.999895	−	0.0145146i	$-0.995380\pi$
$883$	− 5.93877e8i	− 0.0290291i	0.999895	−	0.0145146i	$-0.00462029\pi$
$884$	0	0
$885$	− 1.56025e10i	− 0.756644i
$886$	0	0
$887$	3.05244e10	1.46864	0.734318	−	0.678806i	$-0.237502\pi$
$887$	3.05244e10	1.46864	0.734318	+	0.678806i	$0.237502\pi$
$888$	0	0
$889$	−9.05210e8	−0.0432109
$890$	0	0
$891$	4.07014e6i	0 0.000192769i
$892$	0	0
$893$	− 1.10850e10i	− 0.520900i
$894$	0	0
$895$	−1.69568e10	−0.790609
$896$	0	0
$897$	4.35281e9	0.201371
$898$	0	0
$899$	− 2.57518e8i	− 0.0118208i
$900$	0	0
$901$	− 1.44326e9i	− 0.0657368i
$902$	0	0
$903$	6.25827e9	0.282844
$904$	0	0
$905$	−8.88492e9	−0.398459
$906$	0	0
$907$	− 1.78107e10i	− 0.792604i	−0.918120	−	0.396302i	$-0.870293\pi$
$907$	− 1.78107e10i	− 0.792604i	0.918120	−	0.396302i	$-0.129707\pi$
$908$	0	0
$909$	1.17813e10i	0.520261i
$910$	0	0
$911$	2.81560e10	1.23384	0.616918	−	0.787028i	$-0.288381\pi$
$911$	2.81560e10	1.23384	0.616918	+	0.787028i	$0.288381\pi$
$912$	0	0
$913$	−4.48423e7	−0.00195003
$914$	0	0
$915$	1.48792e10i	0.642106i
$916$	0	0
$917$	− 1.10445e10i	− 0.472991i
$918$	0	0
$919$	2.14055e10	0.909746	0.454873	−	0.890556i	$-0.349685\pi$
$919$	2.14055e10	0.909746	0.454873	+	0.890556i	$0.349685\pi$
$920$	0	0
$921$	−6.25839e9	−0.263970
$922$	0	0
$923$	9.01573e9i	0.377394i
$924$	0	0
$925$	1.28594e9i	0.0534225i
$926$	0	0
$927$	8.45412e9	0.348569
$928$	0	0
$929$	3.97757e8	0.0162766	0.00813829	−	0.999967i	$-0.497409\pi$
$929$	3.97757e8	0.0162766	0.00813829	+	0.999967i	$0.497409\pi$
$930$	0	0
$931$	1.27506e10i	0.517855i
$932$	0	0
$933$	− 4.97052e9i	− 0.200362i
$934$	0	0
$935$	−1.26910e7	−0.000507757 0
$936$	0	0
$937$	3.39372e10	1.34768	0.673841	−	0.738876i	$-0.264643\pi$
$937$	3.39372e10	1.34768	0.673841	+	0.738876i	$0.264643\pi$
$938$	0	0
$939$	− 6.82387e9i	− 0.268968i
$940$	0	0
$941$	1.03612e10i	0.405364i	0.979245	+	0.202682i	$0.0649657\pi$
$941$	1.03612e10i	0.405364i	−0.979245	+	0.202682i	$0.935034\pi$
$942$	0	0
$943$	−4.64009e9	−0.180192
$944$	0	0
$945$	−2.36775e9	−0.0912691
$946$	0	0
$947$	1.35401e8i	0.00518080i	0.999997	+	0.00259040i	$0.000824551\pi$
$947$	1.35401e8i	0.00518080i	−0.999997	+	0.00259040i	$0.999175\pi$
$948$	0	0
$949$	− 6.43575e9i	− 0.244437i
$950$	0	0
$951$	1.87564e10	0.707158
$952$	0	0
$953$	1.54850e10	0.579544	0.289772	−	0.957096i	$-0.406421\pi$
$953$	1.54850e10	0.579544	0.289772	+	0.957096i	$0.406421\pi$
$954$	0	0
$955$	4.35037e10i	1.61627i
$956$	0	0
$957$	590400.i	0 2.17748e-5i
$958$	0	0
$959$	−5.31571e9	−0.194624
$960$	0	0
$961$	−1.93776e10	−0.704317
$962$	0	0
$963$	9.92413e9i	0.358097i
$964$	0	0
$965$	4.39733e10i	1.57523i
$966$	0	0
$967$	2.38415e10	0.847891	0.423946	−	0.905688i	$-0.360645\pi$
$967$	2.38415e10	0.847891	0.423946	+	0.905688i	$0.360645\pi$
$968$	0	0
$969$	−2.45764e9	−0.0867731
$970$	0	0
$971$	− 2.10220e8i	− 0.00736895i	−0.999993	−	0.00368448i	$-0.998827\pi$
$971$	− 2.10220e8i	− 0.00736895i	0.999993	−	0.00368448i	$-0.00117281\pi$
$972$	0	0
$973$	− 7.90505e9i	− 0.275112i
$974$	0	0
$975$	−4.02641e9	−0.139124
$976$	0	0
$977$	3.15038e10	1.08077	0.540384	−	0.841418i	$-0.318279\pi$
$977$	3.15038e10	1.08077	0.540384	+	0.841418i	$0.318279\pi$
$978$	0	0
$979$	− 8.36582e7i	− 0.00284950i
$980$	0	0
$981$	− 9.14008e9i	− 0.309107i
$982$	0	0
$983$	1.34320e10	0.451029	0.225514	−	0.974240i	$-0.427594\pi$
$983$	1.34320e10	0.451029	0.225514	+	0.974240i	$0.427594\pi$
$984$	0	0
$985$	1.01503e10	0.338417
$986$	0	0
$987$	5.86117e9i	0.194032i
$988$	0	0
$989$	− 2.34760e10i	− 0.771681i
$990$	0	0
$991$	−8.96665e9	−0.292666	−0.146333	−	0.989235i	$-0.546747\pi$
$991$	−8.96665e9	−0.292666	−0.146333	+	0.989235i	$0.546747\pi$
$992$	0	0
$993$	−8.56209e9	−0.277496
$994$	0	0
$995$	6.56989e10i	2.11435i
$996$	0	0
$997$	− 5.41294e10i	− 1.72982i	−0.501931	−	0.864908i	$-0.667377\pi$
$997$	− 5.41294e10i	− 1.72982i	0.501931	−	0.864908i	$-0.332623\pi$
$998$	0	0
$999$	−7.51031e8	−0.0238330

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.b.193.4	yes	6
4.3	odd	2		384.8.d.a.193.1	✓	6
8.3	odd	2		384.8.d.a.193.6	yes	6
8.5	even	2	inner	384.8.d.b.193.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.a.193.1	✓	6	4.3	odd	2
384.8.d.a.193.6	yes	6	8.3	odd	2
384.8.d.b.193.3	yes	6	8.5	even	2	inner
384.8.d.b.193.4	yes	6	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+27.0000i q^{3} -334.405i q^{5} +359.725 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q+27.0000i q^{3} -334.405i q^{5} +359.725 q^{7} -729.000 q^{9} +7.65868i q^{11} -4424.87i q^{13} +9028.94 q^{15} -4955.30 q^{17} -18369.0i q^{19} +9712.58i q^{21} +36433.8 q^{23} -33701.9 q^{25} -19683.0i q^{27} -2855.15i q^{29} +90194.3 q^{31} -206.784 q^{33} -120294. i q^{35} -38156.3i q^{37} +119472. q^{39} -127357. q^{41} -644347. i q^{43} +243781. i q^{45} +603462. q^{47} -694141. q^{49} -133793. i q^{51} +291256. i q^{53} +2561.10 q^{55} +495962. q^{57} -1.72805e6i q^{59} +1.64795e6i q^{61} -262240. q^{63} -1.47970e6 q^{65} +3.21269e6i q^{67} +983714. i q^{69} -2.03751e6 q^{71} +1.45445e6 q^{73} -909950. i q^{75} +2755.02i q^{77} -4.18669e6 q^{79} +531441. q^{81} +5.85510e6i q^{83} +1.65708e6i q^{85} +77089.0 q^{87} -1.09233e7 q^{89} -1.59174e6i q^{91} +2.43524e6i q^{93} -6.14268e6 q^{95} +969168. q^{97} -5583.18i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 136 q^{7} - 4374 q^{9}+O(q^{10}) \) 6 * q + 136 * q^7 - 4374 * q^9	\( 6 q + 136 q^{7} - 4374 q^{9} - 6048 q^{15} - 25804 q^{17} + 12688 q^{23} - 18418 q^{25} + 220472 q^{31} - 264600 q^{33} - 591192 q^{39} - 619652 q^{41} - 326032 q^{47} - 581482 q^{49} - 4724352 q^{55} - 2429784 q^{57} - 99144 q^{63} - 9518720 q^{65} + 5841776 q^{71} - 11075196 q^{73} + 495800 q^{79} + 3188646 q^{81} + 965520 q^{87} + 9660740 q^{89} - 32750208 q^{95} - 9511564 q^{97}+O(q^{100}) \) 6 * q + 136 * q^7 - 4374 * q^9 - 6048 * q^15 - 25804 * q^17 + 12688 * q^23 - 18418 * q^25 + 220472 * q^31 - 264600 * q^33 - 591192 * q^39 - 619652 * q^41 - 326032 * q^47 - 581482 * q^49 - 4724352 * q^55 - 2429784 * q^57 - 99144 * q^63 - 9518720 * q^65 + 5841776 * q^71 - 11075196 * q^73 + 495800 * q^79 + 3188646 * q^81 + 965520 * q^87 + 9660740 * q^89 - 32750208 * q^95 - 9511564 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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