LMFDB - Embedded newform 384.10.d.b.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,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, 10, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
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:	$197.773761087$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225 $ x^8 + 13062x^6 + 45211107x^4 + 45928424926*x^2 + 852972309225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.1
Root			$90.8862i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.10.d.b.193.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.0000i q^{3} -2178.24i q^{5} +7658.16 q^{7} -6561.00 q^{9} +O(q^{10})$	$q-81.0000i q^{3} -2178.24i q^{5} +7658.16 q^{7} -6561.00 q^{9} -61495.3i q^{11} +28283.5i q^{13} -176438. q^{15} +193168. q^{17} +366846. i q^{19} -620311. i q^{21} +820746. q^{23} -2.79162e6 q^{25} +531441. i q^{27} -4.44100e6i q^{29} -2.59982e6 q^{31} -4.98112e6 q^{33} -1.66813e7i q^{35} +2.14501e7i q^{37} +2.29096e6 q^{39} -3.15678e7 q^{41} -9.42768e6i q^{43} +1.42915e7i q^{45} -3.59591e7 q^{47} +1.82938e7 q^{49} -1.56466e7i q^{51} +3.97195e7i q^{53} -1.33952e8 q^{55} +2.97145e7 q^{57} -9.36289e7i q^{59} -3.86390e7i q^{61} -5.02452e7 q^{63} +6.16083e7 q^{65} -1.80133e8i q^{67} -6.64805e7i q^{69} +1.54189e8 q^{71} -3.79531e8 q^{73} +2.26121e8i q^{75} -4.70940e8i q^{77} -5.21268e8 q^{79} +4.30467e7 q^{81} -5.93034e8i q^{83} -4.20768e8i q^{85} -3.59721e8 q^{87} +1.05828e8 q^{89} +2.16599e8i q^{91} +2.10585e8i q^{93} +7.99080e8 q^{95} +2.91252e8 q^{97} +4.03470e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 13632 q^{7} - 52488 q^{9}+O(q^{10}) $ 8 * q + 13632 * q^7 - 52488 * q^9	$ 8 q + 13632 q^{7} - 52488 q^{9} + 90720 q^{15} + 8304 q^{17} + 4612608 q^{23} - 4754904 q^{25} + 7499328 q^{31} - 6213024 q^{33} + 23211360 q^{39} - 43518896 q^{41} - 49382016 q^{47} - 74106808 q^{49} - 19030656 q^{55} + 38141280 q^{57} - 89439552 q^{63} - 110270336 q^{65} + 741751296 q^{71} - 1507903440 q^{73} - 1008373440 q^{79} + 344373768 q^{81} - 423468000 q^{87} - 1337034448 q^{89} + 543950208 q^{95} - 904817936 q^{97}+O(q^{100}) $ 8 * q + 13632 * q^7 - 52488 * q^9 + 90720 * q^15 + 8304 * q^17 + 4612608 * q^23 - 4754904 * q^25 + 7499328 * q^31 - 6213024 * q^33 + 23211360 * q^39 - 43518896 * q^41 - 49382016 * q^47 - 74106808 * q^49 - 19030656 * q^55 + 38141280 * q^57 - 89439552 * q^63 - 110270336 * q^65 + 741751296 * q^71 - 1507903440 * q^73 - 1008373440 * q^79 + 344373768 * q^81 - 423468000 * q^87 - 1337034448 * q^89 + 543950208 * q^95 - 904817936 * 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
$3$	− 81.0000i	− 0.577350i
$3$	− 81.0000i	− 0.577350i
$4$	0	0
$5$	− 2178.24i	− 1.55862i	−0.626636	−	0.779312i	$-0.715569\pi$
$5$	− 2178.24i	− 1.55862i	0.626636	−	0.779312i	$-0.284431\pi$
$6$	0	0
$7$	7658.16	1.20554	0.602772	−	0.797913i	$-0.294063\pi$
$7$	7658.16	1.20554	0.602772	+	0.797913i	$0.294063\pi$
$8$	0	0
$9$	−6561.00	−0.333333
$10$	0	0
$11$	− 61495.3i	− 1.26641i	−0.773984	−	0.633205i	$-0.781739\pi$
$11$	− 61495.3i	− 1.26641i	0.773984	−	0.633205i	$-0.218261\pi$
$12$	0	0
$13$	28283.5i	0.274655i	0.990526	+	0.137328i	$0.0438513\pi$
$13$	28283.5i	0.274655i	−0.990526	+	0.137328i	$0.956149\pi$
$14$	0	0
$15$	−176438.	−0.899872
$16$	0	0
$17$	193168.	0.560939	0.280470	−	0.959863i	$-0.409510\pi$
$17$	193168.	0.560939	0.280470	+	0.959863i	$0.409510\pi$
$18$	0	0
$19$	366846.i	0.645792i	0.946434	+	0.322896i	$0.104656\pi$
$19$	366846.i	0.645792i	−0.946434	+	0.322896i	$0.895344\pi$
$20$	0	0
$21$	− 620311.i	− 0.696021i
$22$	0	0
$23$	820746.	0.611553	0.305776	−	0.952103i	$-0.401084\pi$
$23$	820746.	0.611553	0.305776	+	0.952103i	$0.401084\pi$
$24$	0	0
$25$	−2.79162e6	−1.42931
$26$	0	0
$27$	531441.i	0.192450i
$28$	0	0
$29$	− 4.44100e6i	− 1.16598i	−0.812481	−	0.582988i	$-0.801884\pi$
$29$	− 4.44100e6i	− 1.16598i	0.812481	−	0.582988i	$-0.198116\pi$
$30$	0	0
$31$	−2.59982e6	−0.505609	−0.252805	−	0.967517i	$-0.581353\pi$
$31$	−2.59982e6	−0.505609	−0.252805	+	0.967517i	$0.581353\pi$
$32$	0	0
$33$	−4.98112e6	−0.731163
$34$	0	0
$35$	− 1.66813e7i	− 1.87899i
$36$	0	0
$37$	2.14501e7i	1.88157i	0.338999	+	0.940787i	$0.389912\pi$
$37$	2.14501e7i	1.88157i	−0.338999	+	0.940787i	$0.610088\pi$
$38$	0	0
$39$	2.29096e6	0.158572
$40$	0	0
$41$	−3.15678e7	−1.74468	−0.872341	−	0.488898i	$-0.837399\pi$
$41$	−3.15678e7	−1.74468	−0.872341	+	0.488898i	$0.837399\pi$
$42$	0	0
$43$	− 9.42768e6i	− 0.420530i	−0.977644	−	0.210265i	$-0.932567\pi$
$43$	− 9.42768e6i	− 0.420530i	0.977644	−	0.210265i	$-0.0674327\pi$
$44$	0	0
$45$	1.42915e7i	0.519541i
$46$	0	0
$47$	−3.59591e7	−1.07490	−0.537450	−	0.843295i	$-0.680612\pi$
$47$	−3.59591e7	−1.07490	−0.537450	+	0.843295i	$0.680612\pi$
$48$	0	0
$49$	1.82938e7	0.453337
$50$	0	0
$51$	− 1.56466e7i	− 0.323859i
$52$	0	0
$53$	3.97195e7i	0.691453i	0.938335	+	0.345726i	$0.112367\pi$
$53$	3.97195e7i	0.691453i	−0.938335	+	0.345726i	$0.887633\pi$
$54$	0	0
$55$	−1.33952e8	−1.97386
$56$	0	0
$57$	2.97145e7	0.372848
$58$	0	0
$59$	− 9.36289e7i	− 1.00595i	−0.864301	−	0.502975i	$-0.832239\pi$
$59$	− 9.36289e7i	− 1.00595i	0.864301	−	0.502975i	$-0.167761\pi$
$60$	0	0
$61$	− 3.86390e7i	− 0.357308i	−0.983912	−	0.178654i	$-0.942826\pi$
$61$	− 3.86390e7i	− 0.357308i	0.983912	−	0.178654i	$-0.0571742\pi$
$62$	0	0
$63$	−5.02452e7	−0.401848
$64$	0	0
$65$	6.16083e7	0.428084
$66$	0	0
$67$	− 1.80133e8i	− 1.09209i	−0.837757	−	0.546043i	$-0.816134\pi$
$67$	− 1.80133e8i	− 1.09209i	0.837757	−	0.546043i	$-0.183866\pi$
$68$	0	0
$69$	− 6.64805e7i	− 0.353080i
$70$	0	0
$71$	1.54189e8	0.720097	0.360049	−	0.932934i	$-0.382760\pi$
$71$	1.54189e8	0.720097	0.360049	+	0.932934i	$0.382760\pi$
$72$	0	0
$73$	−3.79531e8	−1.56421	−0.782105	−	0.623147i	$-0.785854\pi$
$73$	−3.79531e8	−1.56421	−0.782105	+	0.623147i	$0.785854\pi$
$74$	0	0
$75$	2.26121e8i	0.825212i
$76$	0	0
$77$	− 4.70940e8i	− 1.52671i
$78$	0	0
$79$	−5.21268e8	−1.50570	−0.752851	−	0.658190i	$-0.771322\pi$
$79$	−5.21268e8	−1.50570	−0.752851	+	0.658190i	$0.771322\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	− 5.93034e8i	− 1.37160i	−0.727790	−	0.685801i	$-0.759452\pi$
$83$	− 5.93034e8i	− 1.37160i	0.727790	−	0.685801i	$-0.240548\pi$
$84$	0	0
$85$	− 4.20768e8i	− 0.874294i
$86$	0	0
$87$	−3.59721e8	−0.673176
$88$	0	0
$89$	1.05828e8	0.178790	0.0893952	−	0.995996i	$-0.471507\pi$
$89$	1.05828e8	0.178790	0.0893952	+	0.995996i	$0.471507\pi$
$90$	0	0
$91$	2.16599e8i	0.331109i
$92$	0	0
$93$	2.10585e8i	0.291914i
$94$	0	0
$95$	7.99080e8	1.00655
$96$	0	0
$97$	2.91252e8	0.334038	0.167019	−	0.985954i	$-0.446586\pi$
$97$	2.91252e8	0.334038	0.167019	+	0.985954i	$0.446586\pi$
$98$	0	0
$99$	4.03470e8i	0.422137i
$100$	0	0
$101$	− 1.38250e9i	− 1.32197i	−0.750401	−	0.660983i	$-0.770140\pi$
$101$	− 1.38250e9i	− 1.32197i	0.750401	−	0.660983i	$-0.229860\pi$
$102$	0	0
$103$	5.22223e8	0.457181	0.228591	−	0.973523i	$-0.426588\pi$
$103$	5.22223e8	0.457181	0.228591	+	0.973523i	$0.426588\pi$
$104$	0	0
$105$	−1.35119e9	−1.08484
$106$	0	0
$107$	2.56206e9i	1.88957i	0.327695	+	0.944784i	$0.393728\pi$
$107$	2.56206e9i	1.88957i	−0.327695	+	0.944784i	$0.606272\pi$
$108$	0	0
$109$	4.72077e8i	0.320327i	0.987090	+	0.160164i	$0.0512022\pi$
$109$	4.72077e8i	0.320327i	−0.987090	+	0.160164i	$0.948798\pi$
$110$	0	0
$111$	1.73746e9	1.08633
$112$	0	0
$113$	9.44729e8	0.545073	0.272536	−	0.962146i	$-0.412138\pi$
$113$	9.44729e8	0.545073	0.272536	+	0.962146i	$0.412138\pi$
$114$	0	0
$115$	− 1.78779e9i	− 0.953180i
$116$	0	0
$117$	− 1.85568e8i	− 0.0915517i
$118$	0	0
$119$	1.47931e9	0.676237
$120$	0	0
$121$	−1.42372e9	−0.603796
$122$	0	0
$123$	2.55699e9i	1.00729i
$124$	0	0
$125$	1.82644e9i	0.669132i
$126$	0	0
$127$	−2.56290e9	−0.874209	−0.437105	−	0.899411i	$-0.643996\pi$
$127$	−2.56290e9	−0.874209	−0.437105	+	0.899411i	$0.643996\pi$
$128$	0	0
$129$	−7.63642e8	−0.242793
$130$	0	0
$131$	1.68489e9i	0.499863i	0.968264	+	0.249931i	$0.0804081\pi$
$131$	1.68489e9i	0.499863i	−0.968264	+	0.249931i	$0.919592\pi$
$132$	0	0
$133$	2.80936e9i	0.778531i
$134$	0	0
$135$	1.15761e9	0.299957
$136$	0	0
$137$	5.71074e7	0.0138500	0.00692501	−	0.999976i	$-0.497796\pi$
$137$	5.71074e7	0.0138500	0.00692501	+	0.999976i	$0.497796\pi$
$138$	0	0
$139$	5.87092e9i	1.33395i	0.745081	+	0.666974i	$0.232411\pi$
$139$	5.87092e9i	1.33395i	−0.745081	+	0.666974i	$0.767589\pi$
$140$	0	0
$141$	2.91269e9i	0.620594i
$142$	0	0
$143$	1.73930e9	0.347826
$144$	0	0
$145$	−9.67357e9	−1.81732
$146$	0	0
$147$	− 1.48180e9i	− 0.261734i
$148$	0	0
$149$	− 1.06329e10i	− 1.76731i	−0.468136	−	0.883656i	$-0.655074\pi$
$149$	− 1.06329e10i	− 1.76731i	0.468136	−	0.883656i	$-0.344926\pi$
$150$	0	0
$151$	−4.05058e9	−0.634046	−0.317023	−	0.948418i	$-0.602683\pi$
$151$	−4.05058e9	−0.634046	−0.317023	+	0.948418i	$0.602683\pi$
$152$	0	0
$153$	−1.26738e9	−0.186980
$154$	0	0
$155$	5.66303e9i	0.788054i
$156$	0	0
$157$	− 5.91477e8i	− 0.0776944i	−0.999245	−	0.0388472i	$-0.987631\pi$
$157$	− 5.91477e8i	− 0.0776944i	0.999245	−	0.0388472i	$-0.0123686\pi$
$158$	0	0
$159$	3.21728e9	0.399210
$160$	0	0
$161$	6.28541e9	0.737254
$162$	0	0
$163$	− 1.71216e10i	− 1.89977i	−0.312598	−	0.949886i	$-0.601199\pi$
$163$	− 1.71216e10i	− 1.89977i	0.312598	−	0.949886i	$-0.398801\pi$
$164$	0	0
$165$	1.08501e10i	1.13961i
$166$	0	0
$167$	−1.79932e10	−1.79013	−0.895065	−	0.445936i	$-0.852871\pi$
$167$	−1.79932e10	−1.79013	−0.895065	+	0.445936i	$0.852871\pi$
$168$	0	0
$169$	9.80454e9	0.924565
$170$	0	0
$171$	− 2.40688e9i	− 0.215264i
$172$	0	0
$173$	− 1.65957e10i	− 1.40861i	−0.709900	−	0.704303i	$-0.751260\pi$
$173$	− 1.65957e10i	− 1.40861i	0.709900	−	0.704303i	$-0.248740\pi$
$174$	0	0
$175$	−2.13787e10	−1.72310
$176$	0	0
$177$	−7.58394e9	−0.580785
$178$	0	0
$179$	1.76363e10i	1.28401i	0.766701	+	0.642004i	$0.221897\pi$
$179$	1.76363e10i	1.28401i	−0.766701	+	0.642004i	$0.778103\pi$
$180$	0	0
$181$	− 4.78894e8i	− 0.0331654i	−0.999862	−	0.0165827i	$-0.994721\pi$
$181$	− 4.78894e8i	− 0.0331654i	0.999862	−	0.0165827i	$-0.00527868\pi$
$182$	0	0
$183$	−3.12976e9	−0.206292
$184$	0	0
$185$	4.67235e10	2.93267
$186$	0	0
$187$	− 1.18789e10i	− 0.710380i
$188$	0	0
$189$	4.06986e9i	0.232007i
$190$	0	0
$191$	−2.02911e10	−1.10320	−0.551601	−	0.834108i	$-0.685983\pi$
$191$	−2.02911e10	−1.10320	−0.551601	+	0.834108i	$0.685983\pi$
$192$	0	0
$193$	6.29052e9	0.326346	0.163173	−	0.986597i	$-0.447827\pi$
$193$	6.29052e9	0.326346	0.163173	+	0.986597i	$0.447827\pi$
$194$	0	0
$195$	− 4.99027e9i	− 0.247154i
$196$	0	0
$197$	− 6.17930e9i	− 0.292308i	−0.989262	−	0.146154i	$-0.953310\pi$
$197$	− 6.17930e9i	− 0.292308i	0.989262	−	0.146154i	$-0.0466896\pi$
$198$	0	0
$199$	1.98276e10	0.896255	0.448128	−	0.893970i	$-0.352091\pi$
$199$	1.98276e10	0.896255	0.448128	+	0.893970i	$0.352091\pi$
$200$	0	0
$201$	−1.45908e10	−0.630516
$202$	0	0
$203$	− 3.40099e10i	− 1.40564i
$204$	0	0
$205$	6.87623e10i	2.71930i
$206$	0	0
$207$	−5.38492e9	−0.203851
$208$	0	0
$209$	2.25593e10	0.817838
$210$	0	0
$211$	5.37775e10i	1.86780i	0.357539	+	0.933898i	$0.383616\pi$
$211$	5.37775e10i	1.86780i	−0.357539	+	0.933898i	$0.616384\pi$
$212$	0	0
$213$	− 1.24893e10i	− 0.415748i
$214$	0	0
$215$	−2.05358e10	−0.655448
$216$	0	0
$217$	−1.99098e10	−0.609534
$218$	0	0
$219$	3.07420e10i	0.903097i
$220$	0	0
$221$	5.46347e9i	0.154065i
$222$	0	0
$223$	−4.21444e10	−1.14122	−0.570609	−	0.821222i	$-0.693293\pi$
$223$	−4.21444e10	−1.14122	−0.570609	+	0.821222i	$0.693293\pi$
$224$	0	0
$225$	1.83158e10	0.476436
$226$	0	0
$227$	6.21393e10i	1.55328i	0.629944	+	0.776641i	$0.283078\pi$
$227$	6.21393e10i	1.55328i	−0.629944	+	0.776641i	$0.716922\pi$
$228$	0	0
$229$	− 2.28989e10i	− 0.550244i	−0.961409	−	0.275122i	$-0.911282\pi$
$229$	− 2.28989e10i	− 0.550244i	0.961409	−	0.275122i	$-0.0887182\pi$
$230$	0	0
$231$	−3.81462e10	−0.881449
$232$	0	0
$233$	−6.44391e10	−1.43234	−0.716172	−	0.697923i	$-0.754108\pi$
$233$	−6.44391e10	−1.43234	−0.716172	+	0.697923i	$0.754108\pi$
$234$	0	0
$235$	7.83277e10i	1.67537i
$236$	0	0
$237$	4.22227e10i	0.869318i
$238$	0	0
$239$	1.85246e10	0.367246	0.183623	−	0.982997i	$-0.441217\pi$
$239$	1.85246e10	0.367246	0.183623	+	0.982997i	$0.441217\pi$
$240$	0	0
$241$	−8.59459e10	−1.64115	−0.820575	−	0.571539i	$-0.806347\pi$
$241$	−8.59459e10	−1.64115	−0.820575	+	0.571539i	$0.806347\pi$
$242$	0	0
$243$	− 3.48678e9i	− 0.0641500i
$244$	0	0
$245$	− 3.98483e10i	− 0.706581i
$246$	0	0
$247$	−1.03757e10	−0.177370
$248$	0	0
$249$	−4.80357e10	−0.791894
$250$	0	0
$251$	− 3.84815e10i	− 0.611957i	−0.952038	−	0.305978i	$-0.901016\pi$
$251$	− 3.84815e10i	− 0.611957i	0.952038	−	0.305978i	$-0.0989835\pi$
$252$	0	0
$253$	− 5.04720e10i	− 0.774477i
$254$	0	0
$255$	−3.40822e10	−0.504774
$256$	0	0
$257$	1.23429e11	1.76489	0.882447	−	0.470412i	$-0.155895\pi$
$257$	1.23429e11	1.76489	0.882447	+	0.470412i	$0.155895\pi$
$258$	0	0
$259$	1.64268e11i	2.26832i
$260$	0	0
$261$	2.91374e10i	0.388659i
$262$	0	0
$263$	2.36808e10	0.305207	0.152604	−	0.988287i	$-0.451234\pi$
$263$	2.36808e10	0.305207	0.152604	+	0.988287i	$0.451234\pi$
$264$	0	0
$265$	8.65188e10	1.07771
$266$	0	0
$267$	− 8.57204e9i	− 0.103225i
$268$	0	0
$269$	5.33379e10i	0.621085i	0.950560	+	0.310542i	$0.100511\pi$
$269$	5.33379e10i	0.621085i	−0.950560	+	0.310542i	$0.899489\pi$
$270$	0	0
$271$	1.06039e11	1.19427	0.597137	−	0.802139i	$-0.296305\pi$
$271$	1.06039e11	1.19427	0.597137	+	0.802139i	$0.296305\pi$
$272$	0	0
$273$	1.75445e10	0.191166
$274$	0	0
$275$	1.71671e11i	1.81009i
$276$	0	0
$277$	1.39984e10i	0.142863i	0.997446	+	0.0714314i	$0.0227567\pi$
$277$	1.39984e10i	0.142863i	−0.997446	+	0.0714314i	$0.977243\pi$
$278$	0	0
$279$	1.70574e10	0.168536
$280$	0	0
$281$	−9.66573e10	−0.924818	−0.462409	−	0.886667i	$-0.653015\pi$
$281$	−9.66573e10	−0.924818	−0.462409	+	0.886667i	$0.653015\pi$
$282$	0	0
$283$	1.62302e11i	1.50413i	0.659087	+	0.752066i	$0.270943\pi$
$283$	1.62302e11i	1.50413i	−0.659087	+	0.752066i	$0.729057\pi$
$284$	0	0
$285$	− 6.47255e10i	− 0.581130i
$286$	0	0
$287$	−2.41751e11	−2.10329
$288$	0	0
$289$	−8.12738e10	−0.685347
$290$	0	0
$291$	− 2.35914e10i	− 0.192857i
$292$	0	0
$293$	1.22365e10i	0.0969957i	0.998823	+	0.0484979i	$0.0154434\pi$
$293$	1.22365e10i	0.0969957i	−0.998823	+	0.0484979i	$0.984557\pi$
$294$	0	0
$295$	−2.03947e11	−1.56790
$296$	0	0
$297$	3.26811e10	0.243721
$298$	0	0
$299$	2.32136e10i	0.167966i
$300$	0	0
$301$	− 7.21987e10i	− 0.506967i
$302$	0	0
$303$	−1.11983e11	−0.763237
$304$	0	0
$305$	−8.41653e10	−0.556908
$306$	0	0
$307$	− 1.03830e11i	− 0.667116i	−0.942730	−	0.333558i	$-0.891751\pi$
$307$	− 1.03830e11i	− 0.667116i	0.942730	−	0.333558i	$-0.108249\pi$
$308$	0	0
$309$	− 4.23001e10i	− 0.263954i
$310$	0	0
$311$	−1.95717e10	−0.118633	−0.0593167	−	0.998239i	$-0.518892\pi$
$311$	−1.95717e10	−0.118633	−0.0593167	+	0.998239i	$0.518892\pi$
$312$	0	0
$313$	2.36501e11	1.39278	0.696391	−	0.717662i	$-0.254788\pi$
$313$	2.36501e11	1.39278	0.696391	+	0.717662i	$0.254788\pi$
$314$	0	0
$315$	1.09446e11i	0.626330i
$316$	0	0
$317$	− 1.54506e11i	− 0.859369i	−0.902979	−	0.429684i	$-0.858625\pi$
$317$	− 1.54506e11i	− 0.859369i	0.902979	−	0.429684i	$-0.141375\pi$
$318$	0	0
$319$	−2.73100e11	−1.47660
$320$	0	0
$321$	2.07527e11	1.09094
$322$	0	0
$323$	7.08631e10i	0.362250i
$324$	0	0
$325$	− 7.89567e10i	− 0.392567i
$326$	0	0
$327$	3.82382e10	0.184941
$328$	0	0
$329$	−2.75380e11	−1.29584
$330$	0	0
$331$	4.67240e10i	0.213951i	0.994262	+	0.106975i	$0.0341166\pi$
$331$	4.67240e10i	0.213951i	−0.994262	+	0.106975i	$0.965883\pi$
$332$	0	0
$333$	− 1.40734e11i	− 0.627191i
$334$	0	0
$335$	−3.92373e11	−1.70215
$336$	0	0
$337$	−4.64738e10	−0.196279	−0.0981394	−	0.995173i	$-0.531289\pi$
$337$	−4.64738e10	−0.196279	−0.0981394	+	0.995173i	$0.531289\pi$
$338$	0	0
$339$	− 7.65231e10i	− 0.314698i
$340$	0	0
$341$	1.59876e11i	0.640309i
$342$	0	0
$343$	−1.68938e11	−0.659027
$344$	0	0
$345$	−1.44811e11	−0.550319
$346$	0	0
$347$	1.67034e11i	0.618476i	0.950985	+	0.309238i	$0.100074\pi$
$347$	1.67034e11i	0.618476i	−0.950985	+	0.309238i	$0.899926\pi$
$348$	0	0
$349$	− 4.14205e11i	− 1.49452i	−0.664534	−	0.747258i	$-0.731370\pi$
$349$	− 4.14205e11i	− 1.49452i	0.664534	−	0.747258i	$-0.268630\pi$
$350$	0	0
$351$	−1.50310e10	−0.0528574
$352$	0	0
$353$	−3.92533e11	−1.34552	−0.672759	−	0.739862i	$-0.734891\pi$
$353$	−3.92533e11	−1.34552	−0.672759	+	0.739862i	$0.734891\pi$
$354$	0	0
$355$	− 3.35861e11i	− 1.12236i
$356$	0	0
$357$	− 1.19824e11i	− 0.390426i
$358$	0	0
$359$	−5.24918e11	−1.66789	−0.833943	−	0.551850i	$-0.813922\pi$
$359$	−5.24918e11	−1.66789	−0.833943	+	0.551850i	$0.813922\pi$
$360$	0	0
$361$	1.88112e11	0.582953
$362$	0	0
$363$	1.15321e11i	0.348602i
$364$	0	0
$365$	8.26712e11i	2.43801i
$366$	0	0
$367$	−2.29807e11	−0.661250	−0.330625	−	0.943762i	$-0.607260\pi$
$367$	−2.29807e11	−0.661250	−0.330625	+	0.943762i	$0.607260\pi$
$368$	0	0
$369$	2.07116e11	0.581561
$370$	0	0
$371$	3.04178e11i	0.833577i
$372$	0	0
$373$	− 5.03584e10i	− 0.134704i	−0.997729	−	0.0673522i	$-0.978545\pi$
$373$	− 5.03584e10i	− 0.134704i	0.997729	−	0.0673522i	$-0.0214551\pi$
$374$	0	0
$375$	1.47942e11	0.386323
$376$	0	0
$377$	1.25607e11	0.320241
$378$	0	0
$379$	4.42233e11i	1.10097i	0.834846	+	0.550484i	$0.185557\pi$
$379$	4.42233e11i	1.10097i	−0.834846	+	0.550484i	$0.814443\pi$
$380$	0	0
$381$	2.07595e11i	0.504725i
$382$	0	0
$383$	−3.96927e11	−0.942576	−0.471288	−	0.881979i	$-0.656211\pi$
$383$	−3.96927e11	−0.942576	−0.471288	+	0.881979i	$0.656211\pi$
$384$	0	0
$385$	−1.02582e12	−2.37957
$386$	0	0
$387$	6.18550e10i	0.140177i
$388$	0	0
$389$	7.68232e11i	1.70106i	0.525929	+	0.850529i	$0.323718\pi$
$389$	7.68232e11i	1.70106i	−0.525929	+	0.850529i	$0.676282\pi$
$390$	0	0
$391$	1.58542e11	0.343044
$392$	0	0
$393$	1.36476e11	0.288596
$394$	0	0
$395$	1.13545e12i	2.34682i
$396$	0	0
$397$	3.41102e11i	0.689172i	0.938755	+	0.344586i	$0.111981\pi$
$397$	3.41102e11i	0.689172i	−0.938755	+	0.344586i	$0.888019\pi$
$398$	0	0
$399$	2.27559e11	0.449485
$400$	0	0
$401$	5.83691e11	1.12728	0.563642	−	0.826019i	$-0.309400\pi$
$401$	5.83691e11	1.12728	0.563642	+	0.826019i	$0.309400\pi$
$402$	0	0
$403$	− 7.35318e10i	− 0.138868i
$404$	0	0
$405$	− 9.37662e10i	− 0.173180i
$406$	0	0
$407$	1.31908e12	2.38284
$408$	0	0
$409$	7.80713e11	1.37955	0.689773	−	0.724026i	$-0.257710\pi$
$409$	7.80713e11	1.37955	0.689773	+	0.724026i	$0.257710\pi$
$410$	0	0
$411$	− 4.62570e9i	− 0.00799631i
$412$	0	0
$413$	− 7.17025e11i	− 1.21272i
$414$	0	0
$415$	−1.29177e12	−2.13781
$416$	0	0
$417$	4.75544e11	0.770156
$418$	0	0
$419$	− 5.88488e11i	− 0.932769i	−0.884582	−	0.466385i	$-0.845556\pi$
$419$	− 5.88488e11i	− 0.932769i	0.884582	−	0.466385i	$-0.154444\pi$
$420$	0	0
$421$	2.50950e11i	0.389330i	0.980870	+	0.194665i	$0.0623620\pi$
$421$	2.50950e11i	0.389330i	−0.980870	+	0.194665i	$0.937638\pi$
$422$	0	0
$423$	2.35928e11	0.358300
$424$	0	0
$425$	−5.39253e11	−0.801756
$426$	0	0
$427$	− 2.95904e11i	− 0.430750i
$428$	0	0
$429$	− 1.40883e11i	− 0.200817i
$430$	0	0
$431$	1.22079e12	1.70409	0.852046	−	0.523467i	$-0.175362\pi$
$431$	1.22079e12	1.70409	0.852046	+	0.523467i	$0.175362\pi$
$432$	0	0
$433$	2.85938e11	0.390909	0.195455	−	0.980713i	$-0.437382\pi$
$433$	2.85938e11	0.390909	0.195455	+	0.980713i	$0.437382\pi$
$434$	0	0
$435$	7.83559e11i	1.04923i
$436$	0	0
$437$	3.01088e11i	0.394936i
$438$	0	0
$439$	4.26567e11	0.548147	0.274073	−	0.961709i	$-0.411629\pi$
$439$	4.26567e11	0.548147	0.274073	+	0.961709i	$0.411629\pi$
$440$	0	0
$441$	−1.20025e11	−0.151112
$442$	0	0
$443$	− 2.78131e11i	− 0.343109i	−0.985175	−	0.171555i	$-0.945121\pi$
$443$	− 2.78131e11i	− 0.343109i	0.985175	−	0.171555i	$-0.0548790\pi$
$444$	0	0
$445$	− 2.30518e11i	− 0.278667i
$446$	0	0
$447$	−8.61265e11	−1.02036
$448$	0	0
$449$	6.47195e11	0.751496	0.375748	−	0.926722i	$-0.377386\pi$
$449$	6.47195e11	0.751496	0.375748	+	0.926722i	$0.377386\pi$
$450$	0	0
$451$	1.94127e12i	2.20948i
$452$	0	0
$453$	3.28097e11i	0.366067i
$454$	0	0
$455$	4.71806e11	0.516074
$456$	0	0
$457$	9.87836e11	1.05940	0.529702	−	0.848184i	$-0.322304\pi$
$457$	9.87836e11	1.05940	0.529702	+	0.848184i	$0.322304\pi$
$458$	0	0
$459$	1.02658e11i	0.107953i
$460$	0	0
$461$	− 5.23286e11i	− 0.539616i	−0.962914	−	0.269808i	$-0.913040\pi$
$461$	− 5.23286e11i	− 0.539616i	0.962914	−	0.269808i	$-0.0869603\pi$
$462$	0	0
$463$	9.21613e11	0.932039	0.466020	−	0.884774i	$-0.345688\pi$
$463$	9.21613e11	0.932039	0.466020	+	0.884774i	$0.345688\pi$
$464$	0	0
$465$	4.58705e11	0.454983
$466$	0	0
$467$	− 1.04570e11i	− 0.101738i	−0.998705	−	0.0508689i	$-0.983801\pi$
$467$	− 1.04570e11i	− 0.101738i	0.998705	−	0.0508689i	$-0.0161991\pi$
$468$	0	0
$469$	− 1.37949e12i	− 1.31656i
$470$	0	0
$471$	−4.79097e10	−0.0448569
$472$	0	0
$473$	−5.79758e11	−0.532563
$474$	0	0
$475$	− 1.02409e12i	− 0.923036i
$476$	0	0
$477$	− 2.60600e11i	− 0.230484i
$478$	0	0
$479$	−1.93040e12	−1.67547	−0.837737	−	0.546074i	$-0.816122\pi$
$479$	−1.93040e12	−1.67547	−0.837737	+	0.546074i	$0.816122\pi$
$480$	0	0
$481$	−6.06683e11	−0.516784
$482$	0	0
$483$	− 5.09118e11i	− 0.425654i
$484$	0	0
$485$	− 6.34417e11i	− 0.520639i
$486$	0	0
$487$	1.28669e12	1.03656	0.518281	−	0.855210i	$-0.326572\pi$
$487$	1.28669e12	1.03656	0.518281	+	0.855210i	$0.326572\pi$
$488$	0	0
$489$	−1.38685e12	−1.09683
$490$	0	0
$491$	− 1.58967e12i	− 1.23436i	−0.786823	−	0.617179i	$-0.788276\pi$
$491$	− 1.58967e12i	− 1.23436i	0.786823	−	0.617179i	$-0.211724\pi$
$492$	0	0
$493$	− 8.57860e11i	− 0.654042i
$494$	0	0
$495$	8.78857e11	0.657953
$496$	0	0
$497$	1.18080e12	0.868109
$498$	0	0
$499$	− 1.27745e12i	− 0.922344i	−0.887311	−	0.461172i	$-0.847429\pi$
$499$	− 1.27745e12i	− 0.922344i	0.887311	−	0.461172i	$-0.152571\pi$
$500$	0	0
$501$	1.45745e12i	1.03353i
$502$	0	0
$503$	2.40613e11	0.167596	0.0837979	−	0.996483i	$-0.473295\pi$
$503$	2.40613e11	0.167596	0.0837979	+	0.996483i	$0.473295\pi$
$504$	0	0
$505$	−3.01143e12	−2.06045
$506$	0	0
$507$	− 7.94168e11i	− 0.533798i
$508$	0	0
$509$	5.46994e11i	0.361204i	0.983556	+	0.180602i	$0.0578046\pi$
$509$	5.46994e11i	0.361204i	−0.983556	+	0.180602i	$0.942195\pi$
$510$	0	0
$511$	−2.90651e12	−1.88572
$512$	0	0
$513$	−1.94957e11	−0.124283
$514$	0	0
$515$	− 1.13753e12i	− 0.712574i
$516$	0	0
$517$	2.21131e12i	1.36127i
$518$	0	0
$519$	−1.34426e12	−0.813259
$520$	0	0
$521$	2.49243e12	1.48202	0.741009	−	0.671495i	$-0.234347\pi$
$521$	2.49243e12	1.48202	0.741009	+	0.671495i	$0.234347\pi$
$522$	0	0
$523$	− 1.54454e12i	− 0.902697i	−0.892348	−	0.451349i	$-0.850943\pi$
$523$	− 1.54454e12i	− 0.902697i	0.892348	−	0.451349i	$-0.149057\pi$
$524$	0	0
$525$	1.73167e12i	0.994829i
$526$	0	0
$527$	−5.02202e11	−0.283616
$528$	0	0
$529$	−1.12753e12	−0.626004
$530$	0	0
$531$	6.14299e11i	0.335316i
$532$	0	0
$533$	− 8.92846e11i	− 0.479186i
$534$	0	0
$535$	5.58079e12	2.94513
$536$	0	0
$537$	1.42854e12	0.741322
$538$	0	0
$539$	− 1.12498e12i	− 0.574110i
$540$	0	0
$541$	4.66468e11i	0.234118i	0.993125	+	0.117059i	$0.0373466\pi$
$541$	4.66468e11i	0.234118i	−0.993125	+	0.117059i	$0.962653\pi$
$542$	0	0
$543$	−3.87904e10	−0.0191481
$544$	0	0
$545$	1.02830e12	0.499269
$546$	0	0
$547$	1.63835e12i	0.782462i	0.920293	+	0.391231i	$0.127951\pi$
$547$	1.63835e12i	0.782462i	−0.920293	+	0.391231i	$0.872049\pi$
$548$	0	0
$549$	2.53511e11i	0.119103i
$550$	0	0
$551$	1.62916e12	0.752978
$552$	0	0
$553$	−3.99195e12	−1.81519
$554$	0	0
$555$	− 3.78460e12i	− 1.69318i
$556$	0	0
$557$	− 6.77404e11i	− 0.298194i	−0.988823	−	0.149097i	$-0.952363\pi$
$557$	− 6.77404e11i	− 0.298194i	0.988823	−	0.149097i	$-0.0476367\pi$
$558$	0	0
$559$	2.66647e11	0.115501
$560$	0	0
$561$	−9.62194e11	−0.410138
$562$	0	0
$563$	− 2.75360e12i	− 1.15508i	−0.816361	−	0.577542i	$-0.804012\pi$
$563$	− 2.75360e12i	− 1.15508i	0.816361	−	0.577542i	$-0.195988\pi$
$564$	0	0
$565$	− 2.05785e12i	− 0.849563i
$566$	0	0
$567$	3.29659e11	0.133949
$568$	0	0
$569$	2.14461e12	0.857716	0.428858	−	0.903372i	$-0.358916\pi$
$569$	2.14461e12	0.857716	0.428858	+	0.903372i	$0.358916\pi$
$570$	0	0
$571$	− 3.43875e12i	− 1.35375i	−0.736098	−	0.676875i	$-0.763334\pi$
$571$	− 3.43875e12i	− 1.35375i	0.736098	−	0.676875i	$-0.236666\pi$
$572$	0	0
$573$	1.64358e12i	0.636934i
$574$	0	0
$575$	−2.29121e12	−0.874098
$576$	0	0
$577$	−2.85300e12	−1.07155	−0.535773	−	0.844362i	$-0.679980\pi$
$577$	−2.85300e12	−1.07155	−0.535773	+	0.844362i	$0.679980\pi$
$578$	0	0
$579$	− 5.09532e11i	− 0.188416i
$580$	0	0
$581$	− 4.54154e12i	− 1.65353i
$582$	0	0
$583$	2.44256e12	0.875663
$584$	0	0
$585$	−4.04212e11	−0.142695
$586$	0	0
$587$	− 2.36076e10i	− 0.00820691i	−0.999992	−	0.00410346i	$-0.998694\pi$
$587$	− 2.36076e10i	− 0.00820691i	0.999992	−	0.00410346i	$-0.00130617\pi$
$588$	0	0
$589$	− 9.53732e11i	− 0.326518i
$590$	0	0
$591$	−5.00523e11	−0.168764
$592$	0	0
$593$	−2.05586e12	−0.682727	−0.341364	−	0.939931i	$-0.610889\pi$
$593$	−2.05586e12	−0.682727	−0.341364	+	0.939931i	$0.610889\pi$
$594$	0	0
$595$	− 3.22231e12i	− 1.05400i
$596$	0	0
$597$	− 1.60604e12i	− 0.517453i
$598$	0	0
$599$	5.72088e12	1.81569	0.907846	−	0.419304i	$-0.137726\pi$
$599$	5.72088e12	1.81569	0.907846	+	0.419304i	$0.137726\pi$
$600$	0	0
$601$	3.10411e12	0.970516	0.485258	−	0.874371i	$-0.338726\pi$
$601$	3.10411e12	0.970516	0.485258	+	0.874371i	$0.338726\pi$
$602$	0	0
$603$	1.18185e12i	0.364028i
$604$	0	0
$605$	3.10121e12i	0.941091i
$606$	0	0
$607$	1.37507e12	0.411127	0.205564	−	0.978644i	$-0.434097\pi$
$607$	1.37507e12	0.411127	0.205564	+	0.978644i	$0.434097\pi$
$608$	0	0
$609$	−2.75480e12	−0.811544
$610$	0	0
$611$	− 1.01705e12i	− 0.295227i
$612$	0	0
$613$	− 6.69362e12i	− 1.91465i	−0.289016	−	0.957324i	$-0.593328\pi$
$613$	− 6.69362e12i	− 1.91465i	0.289016	−	0.957324i	$-0.406672\pi$
$614$	0	0
$615$	5.56974e12	1.56999
$616$	0	0
$617$	4.17726e11	0.116040	0.0580201	−	0.998315i	$-0.481521\pi$
$617$	4.17726e11	0.116040	0.0580201	+	0.998315i	$0.481521\pi$
$618$	0	0
$619$	6.34992e12i	1.73844i	0.494423	+	0.869221i	$0.335379\pi$
$619$	6.34992e12i	1.73844i	−0.494423	+	0.869221i	$0.664621\pi$
$620$	0	0
$621$	4.36178e11i	0.117693i
$622$	0	0
$623$	8.10445e11	0.215540
$624$	0	0
$625$	−1.47394e12	−0.386385
$626$	0	0
$627$	− 1.82730e12i	− 0.472179i
$628$	0	0
$629$	4.14348e12i	1.05545i
$630$	0	0
$631$	4.73127e12	1.18808	0.594040	−	0.804435i	$-0.297532\pi$
$631$	4.73127e12	1.18808	0.594040	+	0.804435i	$0.297532\pi$
$632$	0	0
$633$	4.35598e12	1.07837
$634$	0	0
$635$	5.58263e12i	1.36256i
$636$	0	0
$637$	5.17411e11i	0.124511i
$638$	0	0
$639$	−1.01163e12	−0.240032
$640$	0	0
$641$	3.90673e12	0.914014	0.457007	−	0.889463i	$-0.348921\pi$
$641$	3.90673e12	0.914014	0.457007	+	0.889463i	$0.348921\pi$
$642$	0	0
$643$	− 2.06347e12i	− 0.476046i	−0.971260	−	0.238023i	$-0.923501\pi$
$643$	− 2.06347e12i	− 0.476046i	0.971260	−	0.238023i	$-0.0764993\pi$
$644$	0	0
$645$	1.66340e12i	0.378423i
$646$	0	0
$647$	4.14405e12	0.929728	0.464864	−	0.885382i	$-0.346103\pi$
$647$	4.14405e12	0.929728	0.464864	+	0.885382i	$0.346103\pi$
$648$	0	0
$649$	−5.75773e12	−1.27394
$650$	0	0
$651$	1.61269e12i	0.351915i
$652$	0	0
$653$	− 8.11599e12i	− 1.74676i	−0.487044	−	0.873378i	$-0.661925\pi$
$653$	− 8.11599e12i	− 1.74676i	0.487044	−	0.873378i	$-0.338075\pi$
$654$	0	0
$655$	3.67010e12	0.779098
$656$	0	0
$657$	2.49011e12	0.521403
$658$	0	0
$659$	8.57180e11i	0.177047i	0.996074	+	0.0885234i	$0.0282148\pi$
$659$	8.57180e11i	0.177047i	−0.996074	+	0.0885234i	$0.971785\pi$
$660$	0	0
$661$	1.57325e12i	0.320546i	0.987073	+	0.160273i	$0.0512375\pi$
$661$	1.57325e12i	0.320546i	−0.987073	+	0.160273i	$0.948762\pi$
$662$	0	0
$663$	4.42541e11	0.0889494
$664$	0	0
$665$	6.11948e12	1.21344
$666$	0	0
$667$	− 3.64493e12i	− 0.713055i
$668$	0	0
$669$	3.41370e12i	0.658882i
$670$	0	0
$671$	−2.37612e12	−0.452498
$672$	0	0
$673$	4.91142e12	0.922867	0.461434	−	0.887175i	$-0.347335\pi$
$673$	4.91142e12	0.922867	0.461434	+	0.887175i	$0.347335\pi$
$674$	0	0
$675$	− 1.48358e12i	− 0.275071i
$676$	0	0
$677$	4.67942e12i	0.856137i	0.903746	+	0.428069i	$0.140806\pi$
$677$	4.67942e12i	0.856137i	−0.903746	+	0.428069i	$0.859194\pi$
$678$	0	0
$679$	2.23045e12	0.402697
$680$	0	0
$681$	5.03329e12	0.896787
$682$	0	0
$683$	− 5.92535e12i	− 1.04189i	−0.853591	−	0.520944i	$-0.825580\pi$
$683$	− 5.92535e12i	− 1.04189i	0.853591	−	0.520944i	$-0.174420\pi$
$684$	0	0
$685$	− 1.24394e11i	− 0.0215870i
$686$	0	0
$687$	−1.85481e12	−0.317683
$688$	0	0
$689$	−1.12341e12	−0.189911
$690$	0	0
$691$	− 4.43776e12i	− 0.740479i	−0.928936	−	0.370239i	$-0.879276\pi$
$691$	− 4.43776e12i	− 0.740479i	0.928936	−	0.370239i	$-0.120724\pi$
$692$	0	0
$693$	3.08984e12i	0.508905i
$694$	0	0
$695$	1.27883e13	2.07912
$696$	0	0
$697$	−6.09789e12	−0.978661
$698$	0	0
$699$	5.21956e12i	0.826965i
$700$	0	0
$701$	9.55720e12i	1.49486i	0.664343	+	0.747428i	$0.268712\pi$
$701$	9.55720e12i	1.49486i	−0.664343	+	0.747428i	$0.731288\pi$
$702$	0	0
$703$	−7.86888e12	−1.21511
$704$	0	0
$705$	6.34454e12	0.967273
$706$	0	0
$707$	− 1.05874e13i	− 1.59369i
$708$	0	0
$709$	− 5.67373e12i	− 0.843258i	−0.906769	−	0.421629i	$-0.861459\pi$
$709$	− 5.67373e12i	− 0.843258i	0.906769	−	0.421629i	$-0.138541\pi$
$710$	0	0
$711$	3.42004e12	0.501901
$712$	0	0
$713$	−2.13379e12	−0.309207
$714$	0	0
$715$	− 3.78862e12i	− 0.542130i
$716$	0	0
$717$	− 1.50049e12i	− 0.212030i
$718$	0	0
$719$	−3.80229e11	−0.0530598	−0.0265299	−	0.999648i	$-0.508446\pi$
$719$	−3.80229e11	−0.0530598	−0.0265299	+	0.999648i	$0.508446\pi$
$720$	0	0
$721$	3.99927e12	0.551152
$722$	0	0
$723$	6.96161e12i	0.947518i
$724$	0	0
$725$	1.23976e13i	1.66654i
$726$	0	0
$727$	−1.32242e13	−1.75576	−0.877880	−	0.478881i	$-0.841043\pi$
$727$	−1.32242e13	−1.75576	−0.877880	+	0.478881i	$0.841043\pi$
$728$	0	0
$729$	−2.82430e11	−0.0370370
$730$	0	0
$731$	− 1.82113e12i	− 0.235892i
$732$	0	0
$733$	− 1.53563e13i	− 1.96480i	−0.186797	−	0.982399i	$-0.559811\pi$
$733$	− 1.53563e13i	− 1.96480i	0.186797	−	0.982399i	$-0.440189\pi$
$734$	0	0
$735$	−3.22771e12	−0.407945
$736$	0	0
$737$	−1.10773e13	−1.38303
$738$	0	0
$739$	− 6.82404e12i	− 0.841669i	−0.907137	−	0.420835i	$-0.861737\pi$
$739$	− 6.82404e12i	− 0.841669i	0.907137	−	0.420835i	$-0.138263\pi$
$740$	0	0
$741$	8.40430e11i	0.102405i
$742$	0	0
$743$	−6.28798e12	−0.756940	−0.378470	−	0.925614i	$-0.623550\pi$
$743$	−6.28798e12	−0.756940	−0.378470	+	0.925614i	$0.623550\pi$
$744$	0	0
$745$	−2.31610e13	−2.75458
$746$	0	0
$747$	3.89089e12i	0.457200i
$748$	0	0
$749$	1.96207e13i	2.27796i
$750$	0	0
$751$	−1.19773e13	−1.37397	−0.686986	−	0.726671i	$-0.741067\pi$
$751$	−1.19773e13	−1.37397	−0.686986	+	0.726671i	$0.741067\pi$
$752$	0	0
$753$	−3.11700e12	−0.353313
$754$	0	0
$755$	8.82315e12i	0.988240i
$756$	0	0
$757$	− 5.00435e12i	− 0.553881i	−0.960887	−	0.276940i	$-0.910680\pi$
$757$	− 5.00435e12i	− 0.553881i	0.960887	−	0.276940i	$-0.0893205\pi$
$758$	0	0
$759$	−4.08823e12	−0.447144
$760$	0	0
$761$	2.44206e12	0.263952	0.131976	−	0.991253i	$-0.457868\pi$
$761$	2.44206e12	0.263952	0.131976	+	0.991253i	$0.457868\pi$
$762$	0	0
$763$	3.61524e12i	0.386168i
$764$	0	0
$765$	2.76066e12i	0.291431i
$766$	0	0
$767$	2.64815e12	0.276289
$768$	0	0
$769$	8.52460e12	0.879034	0.439517	−	0.898234i	$-0.355150\pi$
$769$	8.52460e12	0.879034	0.439517	+	0.898234i	$0.355150\pi$
$770$	0	0
$771$	− 9.99776e12i	− 1.01896i
$772$	0	0
$773$	5.84609e12i	0.588922i	0.955664	+	0.294461i	$0.0951401\pi$
$773$	5.84609e12i	0.588922i	−0.955664	+	0.294461i	$0.904860\pi$
$774$	0	0
$775$	7.25769e12	0.722672
$776$	0	0
$777$	1.33057e13	1.30962
$778$	0	0
$779$	− 1.15805e13i	− 1.12670i
$780$	0	0
$781$	− 9.48190e12i	− 0.911939i
$782$	0	0
$783$	2.36013e12	0.224392
$784$	0	0
$785$	−1.28838e12	−0.121096
$786$	0	0
$787$	− 1.32719e13i	− 1.23324i	−0.787262	−	0.616618i	$-0.788502\pi$
$787$	− 1.32719e13i	− 1.23324i	0.787262	−	0.616618i	$-0.211498\pi$
$788$	0	0
$789$	− 1.91814e12i	− 0.176211i
$790$	0	0
$791$	7.23489e12	0.657109
$792$	0	0
$793$	1.09285e12	0.0981363
$794$	0	0
$795$	− 7.00802e12i	− 0.622219i
$796$	0	0
$797$	2.02632e12i	0.177887i	0.996037	+	0.0889436i	$0.0283491\pi$
$797$	2.02632e12i	0.177887i	−0.996037	+	0.0889436i	$0.971651\pi$
$798$	0	0
$799$	−6.94616e12	−0.602954
$800$	0	0
$801$	−6.94335e11	−0.0595968
$802$	0	0
$803$	2.33394e13i	1.98093i
$804$	0	0
$805$	− 1.36911e13i	− 1.14910i
$806$	0	0
$807$	4.32037e12	0.358584
$808$	0	0
$809$	−3.71860e12	−0.305219	−0.152609	−	0.988287i	$-0.548768\pi$
$809$	−3.71860e12	−0.305219	−0.152609	+	0.988287i	$0.548768\pi$
$810$	0	0
$811$	8.32783e12i	0.675987i	0.941149	+	0.337993i	$0.109748\pi$
$811$	8.32783e12i	0.675987i	−0.941149	+	0.337993i	$0.890252\pi$
$812$	0	0
$813$	− 8.58916e12i	− 0.689515i
$814$	0	0
$815$	−3.72951e13	−2.96103
$816$	0	0
$817$	3.45851e12	0.271575
$818$	0	0
$819$	− 1.42111e12i	− 0.110370i
$820$	0	0
$821$	− 3.72889e12i	− 0.286441i	−0.989691	−	0.143220i	$-0.954254\pi$
$821$	− 3.72889e12i	− 0.286441i	0.989691	−	0.143220i	$-0.0457458\pi$
$822$	0	0
$823$	4.95662e12	0.376605	0.188303	−	0.982111i	$-0.439701\pi$
$823$	4.95662e12	0.376605	0.188303	+	0.982111i	$0.439701\pi$
$824$	0	0
$825$	1.39054e13	1.04506
$826$	0	0
$827$	− 2.05532e12i	− 0.152794i	−0.997077	−	0.0763968i	$-0.975658\pi$
$827$	− 2.05532e12i	− 0.152794i	0.997077	−	0.0763968i	$-0.0243416\pi$
$828$	0	0
$829$	4.49719e12i	0.330709i	0.986234	+	0.165354i	$0.0528768\pi$
$829$	4.49719e12i	0.330709i	−0.986234	+	0.165354i	$0.947123\pi$
$830$	0	0
$831$	1.13387e12	0.0824818
$832$	0	0
$833$	3.53378e12	0.254294
$834$	0	0
$835$	3.91936e13i	2.79014i
$836$	0	0
$837$	− 1.38165e12i	− 0.0973045i
$838$	0	0
$839$	−9.05888e12	−0.631169	−0.315585	−	0.948897i	$-0.602201\pi$
$839$	−9.05888e12	−0.631169	−0.315585	+	0.948897i	$0.602201\pi$
$840$	0	0
$841$	−5.21531e12	−0.359499
$842$	0	0
$843$	7.82924e12i	0.533944i
$844$	0	0
$845$	− 2.13567e13i	− 1.44105i
$846$	0	0
$847$	−1.09031e13	−0.727903
$848$	0	0
$849$	1.31465e13	0.868411
$850$	0	0
$851$	1.76051e13i	1.15068i
$852$	0	0
$853$	− 2.43668e13i	− 1.57589i	−0.615743	−	0.787947i	$-0.711144\pi$
$853$	− 2.43668e13i	− 1.57589i	0.615743	−	0.787947i	$-0.288856\pi$
$854$	0	0
$855$	−5.24276e12	−0.335516
$856$	0	0
$857$	1.54772e13	0.980117	0.490059	−	0.871689i	$-0.336975\pi$
$857$	1.54772e13	0.980117	0.490059	+	0.871689i	$0.336975\pi$
$858$	0	0
$859$	1.56032e12i	0.0977790i	0.998804	+	0.0488895i	$0.0155682\pi$
$859$	1.56032e12i	0.0977790i	−0.998804	+	0.0488895i	$0.984432\pi$
$860$	0	0
$861$	1.95818e13i	1.21434i
$862$	0	0
$863$	1.09332e13	0.670962	0.335481	−	0.942047i	$-0.391101\pi$
$863$	1.09332e13	0.670962	0.335481	+	0.942047i	$0.391101\pi$
$864$	0	0
$865$	−3.61496e13	−2.19549
$866$	0	0
$867$	6.58318e12i	0.395685i
$868$	0	0
$869$	3.20555e13i	1.90684i
$870$	0	0
$871$	5.09478e12	0.299947
$872$	0	0
$873$	−1.91090e12	−0.111346
$874$	0	0
$875$	1.39872e13i	0.806668i
$876$	0	0
$877$	− 1.51832e13i	− 0.866693i	−0.901227	−	0.433347i	$-0.857333\pi$
$877$	− 1.51832e13i	− 0.866693i	0.901227	−	0.433347i	$-0.142667\pi$
$878$	0	0
$879$	9.91156e11	0.0560005
$880$	0	0
$881$	2.87715e13	1.60905	0.804527	−	0.593916i	$-0.202419\pi$
$881$	2.87715e13	1.60905	0.804527	+	0.593916i	$0.202419\pi$
$882$	0	0
$883$	2.30013e13i	1.27329i	0.771155	+	0.636647i	$0.219679\pi$
$883$	2.30013e13i	1.27329i	−0.771155	+	0.636647i	$0.780321\pi$
$884$	0	0
$885$	1.65197e13i	0.905225i
$886$	0	0
$887$	−1.27341e13	−0.690735	−0.345367	−	0.938468i	$-0.612246\pi$
$887$	−1.27341e13	−0.690735	−0.345367	+	0.938468i	$0.612246\pi$
$888$	0	0
$889$	−1.96271e13	−1.05390
$890$	0	0
$891$	− 2.64717e12i	− 0.140712i
$892$	0	0
$893$	− 1.31914e13i	− 0.694162i
$894$	0	0
$895$	3.84161e13	2.00129
$896$	0	0
$897$	1.88030e12	0.0969752
$898$	0	0
$899$	1.15458e13i	0.589528i
$900$	0	0
$901$	7.67255e12i	0.387863i
$902$	0	0
$903$	−5.84809e12	−0.292698
$904$	0	0
$905$	−1.04315e12	−0.0516924
$906$	0	0
$907$	2.44366e12i	0.119897i	0.998201	+	0.0599484i	$0.0190936\pi$
$907$	2.44366e12i	0.119897i	−0.998201	+	0.0599484i	$0.980906\pi$
$908$	0	0
$909$	9.07061e12i	0.440655i
$910$	0	0
$911$	−3.69051e12	−0.177523	−0.0887613	−	0.996053i	$-0.528291\pi$
$911$	−3.69051e12	−0.177523	−0.0887613	+	0.996053i	$0.528291\pi$
$912$	0	0
$913$	−3.64688e13	−1.73701
$914$	0	0
$915$	6.81739e12i	0.321531i
$916$	0	0
$917$	1.29032e13i	0.602607i
$918$	0	0
$919$	−1.04045e13	−0.481173	−0.240587	−	0.970628i	$-0.577340\pi$
$919$	−1.04045e13	−0.481173	−0.240587	+	0.970628i	$0.577340\pi$
$920$	0	0
$921$	−8.41025e12	−0.385160
$922$	0	0
$923$	4.36100e12i	0.197778i
$924$	0	0
$925$	− 5.98805e13i	− 2.68935i
$926$	0	0
$927$	−3.42631e12	−0.152394
$928$	0	0
$929$	−3.78338e13	−1.66652	−0.833258	−	0.552885i	$-0.813527\pi$
$929$	−3.78338e13	−1.66652	−0.833258	+	0.552885i	$0.813527\pi$
$930$	0	0
$931$	6.71100e12i	0.292761i
$932$	0	0
$933$	1.58531e12i	0.0684930i
$934$	0	0
$935$	−2.58752e13	−1.10721
$936$	0	0
$937$	−7.67680e12	−0.325351	−0.162675	−	0.986680i	$-0.552012\pi$
$937$	−7.67680e12	−0.325351	−0.162675	+	0.986680i	$0.552012\pi$
$938$	0	0
$939$	− 1.91566e13i	− 0.804123i
$940$	0	0
$941$	1.98175e12i	0.0823941i	0.999151	+	0.0411970i	$0.0131171\pi$
$941$	1.98175e12i	0.0823941i	−0.999151	+	0.0411970i	$0.986883\pi$
$942$	0	0
$943$	−2.59091e13	−1.06696
$944$	0	0
$945$	8.86514e12	0.361612
$946$	0	0
$947$	1.68460e13i	0.680647i	0.940308	+	0.340324i	$0.110537\pi$
$947$	1.68460e13i	0.680647i	−0.940308	+	0.340324i	$0.889463\pi$
$948$	0	0
$949$	− 1.07345e13i	− 0.429618i
$950$	0	0
$951$	−1.25150e13	−0.496157
$952$	0	0
$953$	2.24737e13	0.882585	0.441293	−	0.897363i	$-0.354520\pi$
$953$	2.24737e13	0.882585	0.441293	+	0.897363i	$0.354520\pi$
$954$	0	0
$955$	4.41989e13i	1.71948i
$956$	0	0
$957$	2.21211e13i	0.852518i
$958$	0	0
$959$	4.37338e11	0.0166968
$960$	0	0
$961$	−1.96806e13	−0.744359
$962$	0	0
$963$	− 1.68097e13i	− 0.629856i
$964$	0	0
$965$	− 1.37023e13i	− 0.508651i
$966$	0	0
$967$	3.69178e13	1.35774	0.678871	−	0.734258i	$-0.262470\pi$
$967$	3.69178e13	1.35774	0.678871	+	0.734258i	$0.262470\pi$
$968$	0	0
$969$	5.73991e12	0.209145
$970$	0	0
$971$	1.73104e13i	0.624913i	0.949932	+	0.312456i	$0.101152\pi$
$971$	1.73104e13i	0.624913i	−0.949932	+	0.312456i	$0.898848\pi$
$972$	0	0
$973$	4.49604e13i	1.60813i
$974$	0	0
$975$	−6.39549e12	−0.226649
$976$	0	0
$977$	2.58899e13	0.909085	0.454543	−	0.890725i	$-0.349803\pi$
$977$	2.58899e13	0.909085	0.454543	+	0.890725i	$0.349803\pi$
$978$	0	0
$979$	− 6.50790e12i	− 0.226422i
$980$	0	0
$981$	− 3.09730e12i	− 0.106776i
$982$	0	0
$983$	−2.22442e13	−0.759848	−0.379924	−	0.925018i	$-0.624050\pi$
$983$	−2.22442e13	−0.759848	−0.379924	+	0.925018i	$0.624050\pi$
$984$	0	0
$985$	−1.34600e13	−0.455599
$986$	0	0
$987$	2.23058e13i	0.748154i
$988$	0	0
$989$	− 7.73774e12i	− 0.257176i
$990$	0	0
$991$	−3.85698e13	−1.27033	−0.635165	−	0.772377i	$-0.719068\pi$
$991$	−3.85698e13	−1.27033	−0.635165	+	0.772377i	$0.719068\pi$
$992$	0	0
$993$	3.78465e12	0.123525
$994$	0	0
$995$	− 4.31894e13i	− 1.39693i
$996$	0	0
$997$	1.29804e13i	0.416062i	0.978122	+	0.208031i	$0.0667056\pi$
$997$	1.29804e13i	0.416062i	−0.978122	+	0.208031i	$0.933294\pi$
$998$	0	0
$999$	−1.13995e13	−0.362109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.d.b.193.1	yes	8
4.3	odd	2		384.10.d.a.193.5	yes	8
8.3	odd	2		384.10.d.a.193.4	✓	8
8.5	even	2	inner	384.10.d.b.193.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.d.a.193.4	✓	8	8.3	odd	2
384.10.d.a.193.5	yes	8	4.3	odd	2
384.10.d.b.193.1	yes	8	1.1	even	1	trivial
384.10.d.b.193.8	yes	8	8.5	even	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-81.0000i q^{3} -2178.24i q^{5} +7658.16 q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q-81.0000i q^{3} -2178.24i q^{5} +7658.16 q^{7} -6561.00 q^{9} -61495.3i q^{11} +28283.5i q^{13} -176438. q^{15} +193168. q^{17} +366846. i q^{19} -620311. i q^{21} +820746. q^{23} -2.79162e6 q^{25} +531441. i q^{27} -4.44100e6i q^{29} -2.59982e6 q^{31} -4.98112e6 q^{33} -1.66813e7i q^{35} +2.14501e7i q^{37} +2.29096e6 q^{39} -3.15678e7 q^{41} -9.42768e6i q^{43} +1.42915e7i q^{45} -3.59591e7 q^{47} +1.82938e7 q^{49} -1.56466e7i q^{51} +3.97195e7i q^{53} -1.33952e8 q^{55} +2.97145e7 q^{57} -9.36289e7i q^{59} -3.86390e7i q^{61} -5.02452e7 q^{63} +6.16083e7 q^{65} -1.80133e8i q^{67} -6.64805e7i q^{69} +1.54189e8 q^{71} -3.79531e8 q^{73} +2.26121e8i q^{75} -4.70940e8i q^{77} -5.21268e8 q^{79} +4.30467e7 q^{81} -5.93034e8i q^{83} -4.20768e8i q^{85} -3.59721e8 q^{87} +1.05828e8 q^{89} +2.16599e8i q^{91} +2.10585e8i q^{93} +7.99080e8 q^{95} +2.91252e8 q^{97} +4.03470e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 13632 q^{7} - 52488 q^{9}+O(q^{10}) \) 8 * q + 13632 * q^7 - 52488 * q^9	\( 8 q + 13632 q^{7} - 52488 q^{9} + 90720 q^{15} + 8304 q^{17} + 4612608 q^{23} - 4754904 q^{25} + 7499328 q^{31} - 6213024 q^{33} + 23211360 q^{39} - 43518896 q^{41} - 49382016 q^{47} - 74106808 q^{49} - 19030656 q^{55} + 38141280 q^{57} - 89439552 q^{63} - 110270336 q^{65} + 741751296 q^{71} - 1507903440 q^{73} - 1008373440 q^{79} + 344373768 q^{81} - 423468000 q^{87} - 1337034448 q^{89} + 543950208 q^{95} - 904817936 q^{97}+O(q^{100}) \) 8 * q + 13632 * q^7 - 52488 * q^9 + 90720 * q^15 + 8304 * q^17 + 4612608 * q^23 - 4754904 * q^25 + 7499328 * q^31 - 6213024 * q^33 + 23211360 * q^39 - 43518896 * q^41 - 49382016 * q^47 - 74106808 * q^49 - 19030656 * q^55 + 38141280 * q^57 - 89439552 * q^63 - 110270336 * q^65 + 741751296 * q^71 - 1507903440 * q^73 - 1008373440 * q^79 + 344373768 * q^81 - 423468000 * q^87 - 1337034448 * q^89 + 543950208 * q^95 - 904817936 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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