LMFDB - Embedded newform 192.9.g.f.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,9,Mod(127,192)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	192.g (of order $2$, degree $1$, not 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:	$78.2166931317$
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} - 342x^{6} + 43937x^{4} - 2512824x^{2} + 53993104 $ x^8 - 342x^6 + 43937x^4 - 2512824*x^2 + 53993104
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$9.62695 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	192.127
Dual form			192.9.g.f.127.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654i q^{3} -455.790 q^{5} -1689.76i q^{7} -2187.00 q^{9} +O(q^{10})$	$q-46.7654i q^{3} -455.790 q^{5} -1689.76i q^{7} -2187.00 q^{9} +6803.31i q^{11} +15357.7 q^{13} +21315.2i q^{15} -93787.7 q^{17} +14546.9i q^{19} -79022.4 q^{21} +336401. i q^{23} -182880. q^{25} +102276. i q^{27} +10707.6 q^{29} -740903. i q^{31} +318159. q^{33} +770177. i q^{35} +1.33541e6 q^{37} -718209. i q^{39} -3.17759e6 q^{41} +4.00670e6i q^{43} +996813. q^{45} -1.57345e6i q^{47} +2.90950e6 q^{49} +4.38601e6i q^{51} +7.67006e6 q^{53} -3.10088e6i q^{55} +680289. q^{57} +1.53199e7i q^{59} +1.93832e7 q^{61} +3.69551e6i q^{63} -6.99990e6 q^{65} -1.37063e7i q^{67} +1.57319e7 q^{69} -1.95800e7i q^{71} -3.69806e7 q^{73} +8.55247e6i q^{75} +1.14960e7 q^{77} -6.14961e7i q^{79} +4.78297e6 q^{81} +1.79749e7i q^{83} +4.27475e7 q^{85} -500744. i q^{87} +7.04735e7 q^{89} -2.59509e7i q^{91} -3.46486e7 q^{93} -6.63031e6i q^{95} +1.04776e8 q^{97} -1.48788e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1232 q^{5} - 17496 q^{9}+O(q^{10}) $ 8 * q + 1232 * q^5 - 17496 * q^9	$ 8 q + 1232 q^{5} - 17496 q^{9} + 54256 q^{13} - 204336 q^{17} + 298080 q^{21} - 781416 q^{25} + 333712 q^{29} - 775008 q^{33} - 3945168 q^{37} + 11803088 q^{41} - 2694384 q^{45} - 3977208 q^{49} - 788592 q^{53} - 1194912 q^{57} + 40252336 q^{61} - 20747936 q^{65} - 26687232 q^{69} - 56712048 q^{73} + 52014208 q^{77} + 38263752 q^{81} - 52697568 q^{85} + 61660176 q^{89} - 81738720 q^{93} + 27830288 q^{97}+O(q^{100}) $ 8 * q + 1232 * q^5 - 17496 * q^9 + 54256 * q^13 - 204336 * q^17 + 298080 * q^21 - 781416 * q^25 + 333712 * q^29 - 775008 * q^33 - 3945168 * q^37 + 11803088 * q^41 - 2694384 * q^45 - 3977208 * q^49 - 788592 * q^53 - 1194912 * q^57 + 40252336 * q^61 - 20747936 * q^65 - 26687232 * q^69 - 56712048 * q^73 + 52014208 * q^77 + 38263752 * q^81 - 52697568 * q^85 + 61660176 * q^89 - 81738720 * q^93 + 27830288 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 46.7654i	− 0.577350i
$3$	− 46.7654i	− 0.577350i
$4$	0	0
$5$	−455.790	−0.729264	−0.364632	−	0.931152i	$-0.618805\pi$
$5$	−455.790	−0.729264	−0.364632	+	0.931152i	$0.618805\pi$
$6$	0	0
$7$	− 1689.76i	− 0.703775i	−0.936042	−	0.351887i	$-0.885540\pi$
$7$	− 1689.76i	− 0.703775i	0.936042	−	0.351887i	$-0.114460\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	6803.31i	0.464675i	0.972635	+	0.232338i	$0.0746374\pi$
$11$	6803.31i	0.464675i	−0.972635	+	0.232338i	$0.925363\pi$
$12$	0	0
$13$	15357.7	0.537716	0.268858	−	0.963180i	$-0.413354\pi$
$13$	15357.7	0.537716	0.268858	+	0.963180i	$0.413354\pi$
$14$	0	0
$15$	21315.2i	0.421041i
$16$	0	0
$17$	−93787.7	−1.12292	−0.561462	−	0.827503i	$-0.689761\pi$
$17$	−93787.7	−1.12292	−0.561462	+	0.827503i	$0.689761\pi$
$18$	0	0
$19$	14546.9i	0.111623i	0.998441	+	0.0558117i	$0.0177746\pi$
$19$	14546.9i	0.111623i	−0.998441	+	0.0558117i	$0.982225\pi$
$20$	0	0
$21$	−79022.4	−0.406325
$22$	0	0
$23$	336401.i	1.20212i	0.799205	+	0.601058i	$0.205254\pi$
$23$	336401.i	1.20212i	−0.799205	+	0.601058i	$0.794746\pi$
$24$	0	0
$25$	−182880.	−0.468174
$26$	0	0
$27$	102276.i	0.192450i
$28$	0	0
$29$	10707.6	0.0151391	0.00756953	−	0.999971i	$-0.497591\pi$
$29$	10707.6	0.0151391	0.00756953	+	0.999971i	$0.497591\pi$
$30$	0	0
$31$	− 740903.i	− 0.802259i	−0.916021	−	0.401130i	$-0.868618\pi$
$31$	− 740903.i	− 0.802259i	0.916021	−	0.401130i	$-0.131382\pi$
$32$	0	0
$33$	318159.	0.268280
$34$	0	0
$35$	770177.i	0.513238i
$36$	0	0
$37$	1.33541e6	0.712540	0.356270	−	0.934383i	$-0.384048\pi$
$37$	1.33541e6	0.712540	0.356270	+	0.934383i	$0.384048\pi$
$38$	0	0
$39$	− 718209.i	− 0.310451i
$40$	0	0
$41$	−3.17759e6	−1.12451	−0.562253	−	0.826965i	$-0.690065\pi$
$41$	−3.17759e6	−1.12451	−0.562253	+	0.826965i	$0.690065\pi$
$42$	0	0
$43$	4.00670e6i	1.17196i	0.810325	+	0.585981i	$0.199291\pi$
$43$	4.00670e6i	1.17196i	−0.810325	+	0.585981i	$0.800709\pi$
$44$	0	0
$45$	996813.	0.243088
$46$	0	0
$47$	− 1.57345e6i	− 0.322450i	−0.986918	−	0.161225i	$-0.948456\pi$
$47$	− 1.57345e6i	− 0.322450i	0.986918	−	0.161225i	$-0.0515445\pi$
$48$	0	0
$49$	2.90950e6	0.504701
$50$	0	0
$51$	4.38601e6i	0.648320i
$52$	0	0
$53$	7.67006e6	0.972065	0.486033	−	0.873941i	$-0.338444\pi$
$53$	7.67006e6	0.972065	0.486033	+	0.873941i	$0.338444\pi$
$54$	0	0
$55$	− 3.10088e6i	− 0.338871i
$56$	0	0
$57$	680289.	0.0644457
$58$	0	0
$59$	1.53199e7i	1.26430i	0.774848	+	0.632148i	$0.217827\pi$
$59$	1.53199e7i	1.26430i	−0.774848	+	0.632148i	$0.782173\pi$
$60$	0	0
$61$	1.93832e7	1.39993	0.699965	−	0.714178i	$-0.253199\pi$
$61$	1.93832e7	1.39993	0.699965	+	0.714178i	$0.253199\pi$
$62$	0	0
$63$	3.69551e6i	0.234592i
$64$	0	0
$65$	−6.99990e6	−0.392137
$66$	0	0
$67$	− 1.37063e7i	− 0.680176i	−0.940394	−	0.340088i	$-0.889543\pi$
$67$	− 1.37063e7i	− 0.680176i	0.940394	−	0.340088i	$-0.110457\pi$
$68$	0	0
$69$	1.57319e7	0.694042
$70$	0	0
$71$	− 1.95800e7i	− 0.770510i	−0.922810	−	0.385255i	$-0.874113\pi$
$71$	− 1.95800e7i	− 0.770510i	0.922810	−	0.385255i	$-0.125887\pi$
$72$	0	0
$73$	−3.69806e7	−1.30221	−0.651107	−	0.758986i	$-0.725695\pi$
$73$	−3.69806e7	−1.30221	−0.651107	+	0.758986i	$0.725695\pi$
$74$	0	0
$75$	8.55247e6i	0.270300i
$76$	0	0
$77$	1.14960e7	0.327027
$78$	0	0
$79$	− 6.14961e7i	− 1.57884i	−0.613851	−	0.789422i	$-0.710381\pi$
$79$	− 6.14961e7i	− 1.57884i	0.613851	−	0.789422i	$-0.289619\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	1.79749e7i	0.378751i	0.981905	+	0.189375i	$0.0606463\pi$
$83$	1.79749e7i	0.378751i	−0.981905	+	0.189375i	$0.939354\pi$
$84$	0	0
$85$	4.27475e7	0.818908
$86$	0	0
$87$	− 500744.i	− 0.00874055i
$88$	0	0
$89$	7.04735e7	1.12322	0.561611	−	0.827401i	$-0.310182\pi$
$89$	7.04735e7	1.12322	0.561611	+	0.827401i	$0.310182\pi$
$90$	0	0
$91$	− 2.59509e7i	− 0.378431i
$92$	0	0
$93$	−3.46486e7	−0.463185
$94$	0	0
$95$	− 6.63031e6i	− 0.0814029i
$96$	0	0
$97$	1.04776e8	1.18351	0.591757	−	0.806116i	$-0.298434\pi$
$97$	1.04776e8	1.18351	0.591757	+	0.806116i	$0.298434\pi$
$98$	0	0
$99$	− 1.48788e7i	− 0.154892i
$100$	0	0
$101$	5.62286e7	0.540346	0.270173	−	0.962812i	$-0.412919\pi$
$101$	5.62286e7	0.540346	0.270173	+	0.962812i	$0.412919\pi$
$102$	0	0
$103$	1.22106e8i	1.08490i	0.840090	+	0.542448i	$0.182502\pi$
$103$	1.22106e8i	1.08490i	−0.840090	+	0.542448i	$0.817498\pi$
$104$	0	0
$105$	3.60176e7	0.296318
$106$	0	0
$107$	2.28826e6i	0.0174570i	0.999962	+	0.00872850i	$0.00277840\pi$
$107$	2.28826e6i	0.0174570i	−0.999962	+	0.00872850i	$0.997222\pi$
$108$	0	0
$109$	1.50447e8	1.06580	0.532901	−	0.846177i	$-0.321102\pi$
$109$	1.50447e8	1.06580	0.532901	+	0.846177i	$0.321102\pi$
$110$	0	0
$111$	− 6.24512e7i	− 0.411385i
$112$	0	0
$113$	2.57360e8	1.57844	0.789220	−	0.614111i	$-0.210485\pi$
$113$	2.57360e8	1.57844	0.789220	+	0.614111i	$0.210485\pi$
$114$	0	0
$115$	− 1.53328e8i	− 0.876660i
$116$	0	0
$117$	−3.35873e7	−0.179239
$118$	0	0
$119$	1.58479e8i	0.790285i
$120$	0	0
$121$	1.68074e8	0.784077
$122$	0	0
$123$	1.48601e8i	0.649234i
$124$	0	0
$125$	2.61398e8	1.07069
$126$	0	0
$127$	3.20725e8i	1.23287i	0.787405	+	0.616436i	$0.211424\pi$
$127$	3.20725e8i	1.23287i	−0.787405	+	0.616436i	$0.788576\pi$
$128$	0	0
$129$	1.87375e8	0.676633
$130$	0	0
$131$	− 1.62608e8i	− 0.552149i	−0.961136	−	0.276075i	$-0.910966\pi$
$131$	− 1.62608e8i	− 0.552149i	0.961136	−	0.276075i	$-0.0890337\pi$
$132$	0	0
$133$	2.45808e7	0.0785577
$134$	0	0
$135$	− 4.66163e7i	− 0.140347i
$136$	0	0
$137$	3.73201e8	1.05940	0.529700	−	0.848185i	$-0.322304\pi$
$137$	3.73201e8	1.05940	0.529700	+	0.848185i	$0.322304\pi$
$138$	0	0
$139$	− 2.91334e8i	− 0.780428i	−0.920724	−	0.390214i	$-0.872401\pi$
$139$	− 2.91334e8i	− 0.780428i	0.920724	−	0.390214i	$-0.127599\pi$
$140$	0	0
$141$	−7.35831e7	−0.186167
$142$	0	0
$143$	1.04483e8i	0.249864i
$144$	0	0
$145$	−4.88041e6	−0.0110404
$146$	0	0
$147$	− 1.36064e8i	− 0.291389i
$148$	0	0
$149$	−6.00711e8	−1.21877	−0.609383	−	0.792876i	$-0.708583\pi$
$149$	−6.00711e8	−1.21877	−0.609383	+	0.792876i	$0.708583\pi$
$150$	0	0
$151$	5.53985e8i	1.06559i	0.846244	+	0.532795i	$0.178858\pi$
$151$	5.53985e8i	1.06559i	−0.846244	+	0.532795i	$0.821142\pi$
$152$	0	0
$153$	2.05114e8	0.374308
$154$	0	0
$155$	3.37696e8i	0.585059i
$156$	0	0
$157$	9.83548e8	1.61881	0.809407	−	0.587248i	$-0.199789\pi$
$157$	9.83548e8	1.61881	0.809407	+	0.587248i	$0.199789\pi$
$158$	0	0
$159$	− 3.58693e8i	− 0.561222i
$160$	0	0
$161$	5.68439e8	0.846019
$162$	0	0
$163$	− 7.01240e6i	− 0.00993382i	−0.999988	−	0.00496691i	$-0.998419\pi$
$163$	− 7.01240e6i	− 0.00993382i	0.999988	−	0.00496691i	$-0.00158102\pi$
$164$	0	0
$165$	−1.45014e8	−0.195647
$166$	0	0
$167$	− 1.07398e9i	− 1.38079i	−0.723431	−	0.690396i	$-0.757436\pi$
$167$	− 1.07398e9i	− 1.38079i	0.723431	−	0.690396i	$-0.242564\pi$
$168$	0	0
$169$	−5.79871e8	−0.710861
$170$	0	0
$171$	− 3.18140e7i	− 0.0372078i
$172$	0	0
$173$	−3.38472e7	−0.0377867	−0.0188933	−	0.999822i	$-0.506014\pi$
$173$	−3.38472e7	−0.0377867	−0.0188933	+	0.999822i	$0.506014\pi$
$174$	0	0
$175$	3.09025e8i	0.329489i
$176$	0	0
$177$	7.16442e8	0.729942
$178$	0	0
$179$	1.55294e9i	1.51266i	0.654188	+	0.756332i	$0.273010\pi$
$179$	1.55294e9i	1.51266i	−0.654188	+	0.756332i	$0.726990\pi$
$180$	0	0
$181$	1.27084e9	1.18407	0.592033	−	0.805913i	$-0.298325\pi$
$181$	1.27084e9	1.18407	0.592033	+	0.805913i	$0.298325\pi$
$182$	0	0
$183$	− 9.06462e8i	− 0.808249i
$184$	0	0
$185$	−6.08669e8	−0.519630
$186$	0	0
$187$	− 6.38067e8i	− 0.521795i
$188$	0	0
$189$	1.72822e8	0.135442
$190$	0	0
$191$	3.90488e8i	0.293409i	0.989180	+	0.146705i	$0.0468667\pi$
$191$	3.90488e8i	0.293409i	−0.989180	+	0.146705i	$0.953133\pi$
$192$	0	0
$193$	−1.80145e9	−1.29836	−0.649178	−	0.760637i	$-0.724887\pi$
$193$	−1.80145e9	−1.29836	−0.649178	+	0.760637i	$0.724887\pi$
$194$	0	0
$195$	3.27353e8i	0.226401i
$196$	0	0
$197$	5.96764e8	0.396221	0.198110	−	0.980180i	$-0.436520\pi$
$197$	5.96764e8	0.396221	0.198110	+	0.980180i	$0.436520\pi$
$198$	0	0
$199$	− 4.63439e8i	− 0.295516i	−0.989024	−	0.147758i	$-0.952794\pi$
$199$	− 4.63439e8i	− 0.295516i	0.989024	−	0.147758i	$-0.0472056\pi$
$200$	0	0
$201$	−6.40980e8	−0.392700
$202$	0	0
$203$	− 1.80933e7i	− 0.0106545i
$204$	0	0
$205$	1.44831e9	0.820062
$206$	0	0
$207$	− 7.35710e8i	− 0.400705i
$208$	0	0
$209$	−9.89668e7	−0.0518686
$210$	0	0
$211$	− 3.26587e9i	− 1.64767i	−0.566833	−	0.823833i	$-0.691832\pi$
$211$	− 3.26587e9i	− 1.64767i	0.566833	−	0.823833i	$-0.308168\pi$
$212$	0	0
$213$	−9.15664e8	−0.444854
$214$	0	0
$215$	− 1.82622e9i	− 0.854670i
$216$	0	0
$217$	−1.25195e9	−0.564610
$218$	0	0
$219$	1.72941e9i	0.751834i
$220$	0	0
$221$	−1.44036e9	−0.603814
$222$	0	0
$223$	− 8.00369e8i	− 0.323647i	−0.986820	−	0.161823i	$-0.948263\pi$
$223$	− 8.00369e8i	− 0.323647i	0.986820	−	0.161823i	$-0.0517375\pi$
$224$	0	0
$225$	3.99959e8	0.156058
$226$	0	0
$227$	3.56089e9i	1.34108i	0.741872	+	0.670541i	$0.233938\pi$
$227$	3.56089e9i	1.34108i	−0.741872	+	0.670541i	$0.766062\pi$
$228$	0	0
$229$	−1.50719e9	−0.548058	−0.274029	−	0.961721i	$-0.588356\pi$
$229$	−1.50719e9	−0.548058	−0.274029	+	0.961721i	$0.588356\pi$
$230$	0	0
$231$	− 5.37614e8i	− 0.188809i
$232$	0	0
$233$	−3.73819e9	−1.26835	−0.634173	−	0.773191i	$-0.718659\pi$
$233$	−3.73819e9	−1.26835	−0.634173	+	0.773191i	$0.718659\pi$
$234$	0	0
$235$	7.17164e8i	0.235151i
$236$	0	0
$237$	−2.87589e9	−0.911546
$238$	0	0
$239$	4.67896e9i	1.43403i	0.697058	+	0.717014i	$0.254492\pi$
$239$	4.67896e9i	1.43403i	−0.697058	+	0.717014i	$0.745508\pi$
$240$	0	0
$241$	−4.10707e9	−1.21748	−0.608742	−	0.793368i	$-0.708326\pi$
$241$	−4.10707e9	−1.21748	−0.608742	+	0.793368i	$0.708326\pi$
$242$	0	0
$243$	− 2.23677e8i	− 0.0641500i
$244$	0	0
$245$	−1.32612e9	−0.368060
$246$	0	0
$247$	2.23407e8i	0.0600217i
$248$	0	0
$249$	8.40602e8	0.218672
$250$	0	0
$251$	3.48115e9i	0.877057i	0.898717	+	0.438529i	$0.144500\pi$
$251$	3.48115e9i	0.877057i	−0.898717	+	0.438529i	$0.855500\pi$
$252$	0	0
$253$	−2.28864e9	−0.558594
$254$	0	0
$255$	− 1.99910e9i	− 0.472796i
$256$	0	0
$257$	3.32719e9	0.762685	0.381342	−	0.924434i	$-0.375462\pi$
$257$	3.32719e9	0.762685	0.381342	+	0.924434i	$0.375462\pi$
$258$	0	0
$259$	− 2.25653e9i	− 0.501468i
$260$	0	0
$261$	−2.34175e7	−0.00504636
$262$	0	0
$263$	4.91281e9i	1.02685i	0.858135	+	0.513425i	$0.171623\pi$
$263$	4.91281e9i	1.02685i	−0.858135	+	0.513425i	$0.828377\pi$
$264$	0	0
$265$	−3.49594e9	−0.708892
$266$	0	0
$267$	− 3.29572e9i	− 0.648493i
$268$	0	0
$269$	−7.37829e9	−1.40912	−0.704558	−	0.709646i	$-0.748855\pi$
$269$	−7.37829e9	−1.40912	−0.704558	+	0.709646i	$0.748855\pi$
$270$	0	0
$271$	3.89984e9i	0.723052i	0.932362	+	0.361526i	$0.117744\pi$
$271$	3.89984e9i	0.723052i	−0.932362	+	0.361526i	$0.882256\pi$
$272$	0	0
$273$	−1.21360e9	−0.218487
$274$	0	0
$275$	− 1.24419e9i	− 0.217549i
$276$	0	0
$277$	−9.38474e9	−1.59405	−0.797027	−	0.603943i	$-0.793595\pi$
$277$	−9.38474e9	−1.59405	−0.797027	+	0.603943i	$0.793595\pi$
$278$	0	0
$279$	1.62036e9i	0.267420i
$280$	0	0
$281$	4.48974e9	0.720106	0.360053	−	0.932932i	$-0.382759\pi$
$281$	4.48974e9	0.720106	0.360053	+	0.932932i	$0.382759\pi$
$282$	0	0
$283$	− 5.99019e9i	− 0.933888i	−0.884287	−	0.466944i	$-0.845355\pi$
$283$	− 5.99019e9i	− 0.933888i	0.884287	−	0.466944i	$-0.154645\pi$
$284$	0	0
$285$	−3.10069e8	−0.0469980
$286$	0	0
$287$	5.36937e9i	0.791399i
$288$	0	0
$289$	1.82037e9	0.260956
$290$	0	0
$291$	− 4.89987e9i	− 0.683302i
$292$	0	0
$293$	−1.09529e9	−0.148613	−0.0743066	−	0.997235i	$-0.523674\pi$
$293$	−1.09529e9	−0.148613	−0.0743066	+	0.997235i	$0.523674\pi$
$294$	0	0
$295$	− 6.98267e9i	− 0.922006i
$296$	0	0
$297$	−6.95815e8	−0.0894268
$298$	0	0
$299$	5.16636e9i	0.646398i
$300$	0	0
$301$	6.77038e9	0.824797
$302$	0	0
$303$	− 2.62955e9i	− 0.311969i
$304$	0	0
$305$	−8.83467e9	−1.02092
$306$	0	0
$307$	− 1.16757e10i	− 1.31440i	−0.753714	−	0.657202i	$-0.771740\pi$
$307$	− 1.16757e10i	− 1.31440i	0.753714	−	0.657202i	$-0.228260\pi$
$308$	0	0
$309$	5.71033e9	0.626365
$310$	0	0
$311$	5.41321e9i	0.578647i	0.957231	+	0.289323i	$0.0934303\pi$
$311$	5.41321e9i	0.578647i	−0.957231	+	0.289323i	$0.906570\pi$
$312$	0	0
$313$	1.50436e10	1.56738	0.783691	−	0.621151i	$-0.213335\pi$
$313$	1.50436e10	1.56738	0.783691	+	0.621151i	$0.213335\pi$
$314$	0	0
$315$	− 1.68438e9i	− 0.171079i
$316$	0	0
$317$	−1.36257e9	−0.134934	−0.0674672	−	0.997721i	$-0.521492\pi$
$317$	−1.36257e9	−0.134934	−0.0674672	+	0.997721i	$0.521492\pi$
$318$	0	0
$319$	7.28470e7i	0.00703475i
$320$	0	0
$321$	1.07011e8	0.0100788
$322$	0	0
$323$	− 1.36432e9i	− 0.125344i
$324$	0	0
$325$	−2.80863e9	−0.251745
$326$	0	0
$327$	− 7.03570e9i	− 0.615342i
$328$	0	0
$329$	−2.65876e9	−0.226932
$330$	0	0
$331$	1.66087e10i	1.38364i	0.722069	+	0.691821i	$0.243191\pi$
$331$	1.66087e10i	1.38364i	−0.722069	+	0.691821i	$0.756809\pi$
$332$	0	0
$333$	−2.92055e9	−0.237513
$334$	0	0
$335$	6.24720e9i	0.496028i
$336$	0	0
$337$	−1.92166e9	−0.148990	−0.0744948	−	0.997221i	$-0.523734\pi$
$337$	−1.92166e9	−0.148990	−0.0744948	+	0.997221i	$0.523734\pi$
$338$	0	0
$339$	− 1.20356e10i	− 0.911313i
$340$	0	0
$341$	5.04060e9	0.372790
$342$	0	0
$343$	− 1.46575e10i	− 1.05897i
$344$	0	0
$345$	−7.17046e9	−0.506140
$346$	0	0
$347$	1.88173e10i	1.29789i	0.760833	+	0.648947i	$0.224791\pi$
$347$	1.88173e10i	1.29789i	−0.760833	+	0.648947i	$0.775209\pi$
$348$	0	0
$349$	−1.56042e10	−1.05182	−0.525910	−	0.850540i	$-0.676275\pi$
$349$	−1.56042e10	−1.05182	−0.525910	+	0.850540i	$0.676275\pi$
$350$	0	0
$351$	1.57072e9i	0.103484i
$352$	0	0
$353$	−6.78029e9	−0.436666	−0.218333	−	0.975874i	$-0.570062\pi$
$353$	−6.78029e9	−0.436666	−0.218333	+	0.975874i	$0.570062\pi$
$354$	0	0
$355$	8.92435e9i	0.561906i
$356$	0	0
$357$	7.41133e9	0.456271
$358$	0	0
$359$	1.33198e9i	0.0801900i	0.999196	+	0.0400950i	$0.0127661\pi$
$359$	1.33198e9i	0.0801900i	−0.999196	+	0.0400950i	$0.987234\pi$
$360$	0	0
$361$	1.67720e10	0.987540
$362$	0	0
$363$	− 7.86003e9i	− 0.452687i
$364$	0	0
$365$	1.68554e10	0.949658
$366$	0	0
$367$	− 1.87117e10i	− 1.03145i	−0.856754	−	0.515725i	$-0.827523\pi$
$367$	− 1.87117e10i	− 1.03145i	0.856754	−	0.515725i	$-0.172477\pi$
$368$	0	0
$369$	6.94938e9	0.374835
$370$	0	0
$371$	− 1.29606e10i	− 0.684115i
$372$	0	0
$373$	1.69388e10	0.875081	0.437541	−	0.899199i	$-0.355850\pi$
$373$	1.69388e10	0.875081	0.437541	+	0.899199i	$0.355850\pi$
$374$	0	0
$375$	− 1.22244e10i	− 0.618161i
$376$	0	0
$377$	1.64444e8	0.00814053
$378$	0	0
$379$	3.28410e10i	1.59169i	0.605498	+	0.795847i	$0.292974\pi$
$379$	3.28410e10i	1.59169i	−0.605498	+	0.795847i	$0.707026\pi$
$380$	0	0
$381$	1.49988e10	0.711799
$382$	0	0
$383$	3.74844e10i	1.74203i	0.491255	+	0.871016i	$0.336538\pi$
$383$	3.74844e10i	1.74203i	−0.491255	+	0.871016i	$0.663462\pi$
$384$	0	0
$385$	−5.23976e9	−0.238489
$386$	0	0
$387$	− 8.76266e9i	− 0.390654i
$388$	0	0
$389$	4.42667e10	1.93321	0.966604	−	0.256274i	$-0.0824948\pi$
$389$	4.42667e10	1.93321	0.966604	+	0.256274i	$0.0824948\pi$
$390$	0	0
$391$	− 3.15503e10i	− 1.34988i
$392$	0	0
$393$	−7.60442e9	−0.318783
$394$	0	0
$395$	2.80293e10i	1.15139i
$396$	0	0
$397$	2.16236e10	0.870495	0.435248	−	0.900311i	$-0.356661\pi$
$397$	2.16236e10	0.870495	0.435248	+	0.900311i	$0.356661\pi$
$398$	0	0
$399$	− 1.14953e9i	− 0.0453553i
$400$	0	0
$401$	5.59819e9	0.216506	0.108253	−	0.994123i	$-0.465474\pi$
$401$	5.59819e9	0.216506	0.108253	+	0.994123i	$0.465474\pi$
$402$	0	0
$403$	− 1.13786e10i	− 0.431388i
$404$	0	0
$405$	−2.18003e9	−0.0810293
$406$	0	0
$407$	9.08524e9i	0.331100i
$408$	0	0
$409$	2.03559e10	0.727440	0.363720	−	0.931508i	$-0.381506\pi$
$409$	2.03559e10	0.727440	0.363720	+	0.931508i	$0.381506\pi$
$410$	0	0
$411$	− 1.74529e10i	− 0.611645i
$412$	0	0
$413$	2.58871e10	0.889780
$414$	0	0
$415$	− 8.19277e9i	− 0.276209i
$416$	0	0
$417$	−1.36244e10	−0.450580
$418$	0	0
$419$	4.45043e10i	1.44393i	0.691930	+	0.721965i	$0.256761\pi$
$419$	4.45043e10i	1.44393i	−0.691930	+	0.721965i	$0.743239\pi$
$420$	0	0
$421$	−6.02330e9	−0.191737	−0.0958685	−	0.995394i	$-0.530563\pi$
$421$	−6.02330e9	−0.191737	−0.0958685	+	0.995394i	$0.530563\pi$
$422$	0	0
$423$	3.44114e9i	0.107483i
$424$	0	0
$425$	1.71519e10	0.525723
$426$	0	0
$427$	− 3.27530e10i	− 0.985235i
$428$	0	0
$429$	4.88620e9	0.144259
$430$	0	0
$431$	3.56553e10i	1.03327i	0.856205	+	0.516637i	$0.172816\pi$
$431$	3.56553e10i	1.03327i	−0.856205	+	0.516637i	$0.827184\pi$
$432$	0	0
$433$	4.62137e10	1.31468	0.657338	−	0.753596i	$-0.271682\pi$
$433$	4.62137e10	1.31468	0.657338	+	0.753596i	$0.271682\pi$
$434$	0	0
$435$	2.28234e8i	0.00637417i
$436$	0	0
$437$	−4.89358e9	−0.134184
$438$	0	0
$439$	− 3.76751e10i	− 1.01437i	−0.861837	−	0.507185i	$-0.830686\pi$
$439$	− 3.76751e10i	− 1.01437i	0.861837	−	0.507185i	$-0.169314\pi$
$440$	0	0
$441$	−6.36308e9	−0.168234
$442$	0	0
$443$	4.34035e10i	1.12696i	0.826128	+	0.563482i	$0.190539\pi$
$443$	4.34035e10i	1.12696i	−0.826128	+	0.563482i	$0.809461\pi$
$444$	0	0
$445$	−3.21211e10	−0.819126
$446$	0	0
$447$	2.80925e10i	0.703655i
$448$	0	0
$449$	4.97918e10	1.22510	0.612551	−	0.790431i	$-0.290143\pi$
$449$	4.97918e10	1.22510	0.612551	+	0.790431i	$0.290143\pi$
$450$	0	0
$451$	− 2.16181e10i	− 0.522530i
$452$	0	0
$453$	2.59073e10	0.615219
$454$	0	0
$455$	1.18282e10i	0.275976i
$456$	0	0
$457$	−4.97522e9	−0.114064	−0.0570319	−	0.998372i	$-0.518164\pi$
$457$	−4.97522e9	−0.114064	−0.0570319	+	0.998372i	$0.518164\pi$
$458$	0	0
$459$	− 9.59221e9i	− 0.216107i
$460$	0	0
$461$	−2.43202e10	−0.538472	−0.269236	−	0.963074i	$-0.586771\pi$
$461$	−2.43202e10	−0.538472	−0.269236	+	0.963074i	$0.586771\pi$
$462$	0	0
$463$	4.95954e10i	1.07924i	0.841909	+	0.539619i	$0.181432\pi$
$463$	4.95954e10i	1.07924i	−0.841909	+	0.539619i	$0.818568\pi$
$464$	0	0
$465$	1.57925e10	0.337784
$466$	0	0
$467$	− 8.32165e10i	− 1.74961i	−0.484474	−	0.874806i	$-0.660989\pi$
$467$	− 8.32165e10i	− 1.74961i	0.484474	−	0.874806i	$-0.339011\pi$
$468$	0	0
$469$	−2.31604e10	−0.478691
$470$	0	0
$471$	− 4.59960e10i	− 0.934623i
$472$	0	0
$473$	−2.72589e10	−0.544582
$474$	0	0
$475$	− 2.66034e9i	− 0.0522591i
$476$	0	0
$477$	−1.67744e10	−0.324022
$478$	0	0
$479$	4.26328e10i	0.809844i	0.914351	+	0.404922i	$0.132701\pi$
$479$	4.26328e10i	0.809844i	−0.914351	+	0.404922i	$0.867299\pi$
$480$	0	0
$481$	2.05089e10	0.383144
$482$	0	0
$483$	− 2.65833e10i	− 0.488449i
$484$	0	0
$485$	−4.77557e10	−0.863094
$486$	0	0
$487$	− 6.66609e10i	− 1.18510i	−0.805533	−	0.592550i	$-0.798121\pi$
$487$	− 6.66609e10i	− 1.18510i	0.805533	−	0.592550i	$-0.201879\pi$
$488$	0	0
$489$	−3.27938e8	−0.00573530
$490$	0	0
$491$	4.56152e10i	0.784845i	0.919785	+	0.392422i	$0.128363\pi$
$491$	4.56152e10i	0.784845i	−0.919785	+	0.392422i	$0.871637\pi$
$492$	0	0
$493$	−1.00424e9	−0.0170000
$494$	0	0
$495$	6.78163e9i	0.112957i
$496$	0	0
$497$	−3.30855e10	−0.542266
$498$	0	0
$499$	− 4.30599e10i	− 0.694497i	−0.937773	−	0.347249i	$-0.887116\pi$
$499$	− 4.30599e10i	− 0.694497i	0.937773	−	0.347249i	$-0.112884\pi$
$500$	0	0
$501$	−5.02249e10	−0.797201
$502$	0	0
$503$	− 3.00633e10i	− 0.469639i	−0.972039	−	0.234819i	$-0.924550\pi$
$503$	− 3.00633e10i	− 0.469639i	0.972039	−	0.234819i	$-0.0754499\pi$
$504$	0	0
$505$	−2.56285e10	−0.394055
$506$	0	0
$507$	2.71179e10i	0.410416i
$508$	0	0
$509$	−4.28086e10	−0.637763	−0.318882	−	0.947795i	$-0.603307\pi$
$509$	−4.28086e10	−0.637763	−0.318882	+	0.947795i	$0.603307\pi$
$510$	0	0
$511$	6.24884e10i	0.916465i
$512$	0	0
$513$	−1.48779e9	−0.0214819
$514$	0	0
$515$	− 5.56547e10i	− 0.791175i
$516$	0	0
$517$	1.07047e10	0.149835
$518$	0	0
$519$	1.58288e9i	0.0218162i
$520$	0	0
$521$	2.00634e10	0.272304	0.136152	−	0.990688i	$-0.456526\pi$
$521$	2.00634e10	0.272304	0.136152	+	0.990688i	$0.456526\pi$
$522$	0	0
$523$	− 4.87294e10i	− 0.651304i	−0.945490	−	0.325652i	$-0.894416\pi$
$523$	− 4.87294e10i	− 0.651304i	0.945490	−	0.325652i	$-0.105584\pi$
$524$	0	0
$525$	1.44517e10	0.190231
$526$	0	0
$527$	6.94876e10i	0.900876i
$528$	0	0
$529$	−3.48549e10	−0.445083
$530$	0	0
$531$	− 3.35047e10i	− 0.421432i
$532$	0	0
$533$	−4.88005e10	−0.604666
$534$	0	0
$535$	− 1.04296e9i	− 0.0127308i
$536$	0	0
$537$	7.26238e10	0.873337
$538$	0	0
$539$	1.97942e10i	0.234522i
$540$	0	0
$541$	6.88063e10	0.803229	0.401614	−	0.915809i	$-0.368449\pi$
$541$	6.88063e10	0.803229	0.401614	+	0.915809i	$0.368449\pi$
$542$	0	0
$543$	− 5.94313e10i	− 0.683621i
$544$	0	0
$545$	−6.85722e10	−0.777252
$546$	0	0
$547$	− 8.25362e10i	− 0.921924i	−0.887420	−	0.460962i	$-0.847504\pi$
$547$	− 8.25362e10i	− 0.921924i	0.887420	−	0.460962i	$-0.152496\pi$
$548$	0	0
$549$	−4.23910e10	−0.466643
$550$	0	0
$551$	1.55762e8i	0.00168987i
$552$	0	0
$553$	−1.03914e11	−1.11115
$554$	0	0
$555$	2.84646e10i	0.300008i
$556$	0	0
$557$	−3.74890e9	−0.0389478	−0.0194739	−	0.999810i	$-0.506199\pi$
$557$	−3.74890e9	−0.0389478	−0.0194739	+	0.999810i	$0.506199\pi$
$558$	0	0
$559$	6.15338e10i	0.630183i
$560$	0	0
$561$	−2.98394e10	−0.301258
$562$	0	0
$563$	5.87145e10i	0.584402i	0.956357	+	0.292201i	$0.0943876\pi$
$563$	5.87145e10i	0.584402i	−0.956357	+	0.292201i	$0.905612\pi$
$564$	0	0
$565$	−1.17302e11	−1.15110
$566$	0	0
$567$	− 8.08209e9i	− 0.0781972i
$568$	0	0
$569$	−5.06599e10	−0.483299	−0.241649	−	0.970364i	$-0.577688\pi$
$569$	−5.06599e10	−0.483299	−0.241649	+	0.970364i	$0.577688\pi$
$570$	0	0
$571$	2.04586e11i	1.92456i	0.272067	+	0.962278i	$0.412293\pi$
$571$	2.04586e11i	1.92456i	−0.272067	+	0.962278i	$0.587707\pi$
$572$	0	0
$573$	1.82613e10	0.169400
$574$	0	0
$575$	− 6.15212e10i	− 0.562799i
$576$	0	0
$577$	−5.83766e10	−0.526667	−0.263333	−	0.964705i	$-0.584822\pi$
$577$	−5.83766e10	−0.526667	−0.263333	+	0.964705i	$0.584822\pi$
$578$	0	0
$579$	8.42456e10i	0.749606i
$580$	0	0
$581$	3.03733e10	0.266555
$582$	0	0
$583$	5.21818e10i	0.451695i
$584$	0	0
$585$	1.53088e10	0.130712
$586$	0	0
$587$	1.89365e11i	1.59495i	0.603349	+	0.797477i	$0.293833\pi$
$587$	1.89365e11i	1.59495i	−0.603349	+	0.797477i	$0.706167\pi$
$588$	0	0
$589$	1.07778e10	0.0895508
$590$	0	0
$591$	− 2.79079e10i	− 0.228758i
$592$	0	0
$593$	−1.98001e10	−0.160121	−0.0800605	−	0.996790i	$-0.525511\pi$
$593$	−1.98001e10	−0.160121	−0.0800605	+	0.996790i	$0.525511\pi$
$594$	0	0
$595$	− 7.22331e10i	− 0.576327i
$596$	0	0
$597$	−2.16729e10	−0.170616
$598$	0	0
$599$	− 2.18765e11i	− 1.69930i	−0.527343	−	0.849652i	$-0.676812\pi$
$599$	− 2.18765e11i	− 1.69930i	0.527343	−	0.849652i	$-0.323188\pi$
$600$	0	0
$601$	−2.58060e11	−1.97798	−0.988992	−	0.147972i	$-0.952725\pi$
$601$	−2.58060e11	−1.97798	−0.988992	+	0.147972i	$0.952725\pi$
$602$	0	0
$603$	2.99757e10i	0.226725i
$604$	0	0
$605$	−7.66064e10	−0.571799
$606$	0	0
$607$	8.74343e10i	0.644061i	0.946729	+	0.322030i	$0.104365\pi$
$607$	8.74343e10i	0.644061i	−0.946729	+	0.322030i	$0.895635\pi$
$608$	0	0
$609$	−8.46138e8	−0.00615138
$610$	0	0
$611$	− 2.41646e10i	− 0.173387i
$612$	0	0
$613$	1.79871e11	1.27385	0.636925	−	0.770926i	$-0.280206\pi$
$613$	1.79871e11	1.27385	0.636925	+	0.770926i	$0.280206\pi$
$614$	0	0
$615$	− 6.77309e10i	− 0.473463i
$616$	0	0
$617$	1.01331e11	0.699202	0.349601	−	0.936899i	$-0.386317\pi$
$617$	1.01331e11	0.699202	0.349601	+	0.936899i	$0.386317\pi$
$618$	0	0
$619$	1.04565e11i	0.712234i	0.934441	+	0.356117i	$0.115900\pi$
$619$	1.04565e11i	0.712234i	−0.934441	+	0.356117i	$0.884100\pi$
$620$	0	0
$621$	−3.44057e10	−0.231347
$622$	0	0
$623$	− 1.19084e11i	− 0.790496i
$624$	0	0
$625$	−4.77050e10	−0.312639
$626$	0	0
$627$	4.62822e9i	0.0299464i
$628$	0	0
$629$	−1.25245e11	−0.800128
$630$	0	0
$631$	1.38666e11i	0.874687i	0.899295	+	0.437344i	$0.144081\pi$
$631$	1.38666e11i	0.874687i	−0.899295	+	0.437344i	$0.855919\pi$
$632$	0	0
$633$	−1.52730e11	−0.951280
$634$	0	0
$635$	− 1.46183e11i	− 0.899089i
$636$	0	0
$637$	4.46833e10	0.271386
$638$	0	0
$639$	4.28214e10i	0.256837i
$640$	0	0
$641$	−6.29564e10	−0.372913	−0.186457	−	0.982463i	$-0.559700\pi$
$641$	−6.29564e10	−0.372913	−0.186457	+	0.982463i	$0.559700\pi$
$642$	0	0
$643$	− 1.36211e11i	− 0.796834i	−0.917204	−	0.398417i	$-0.869560\pi$
$643$	− 1.36211e11i	− 0.796834i	0.917204	−	0.398417i	$-0.130440\pi$
$644$	0	0
$645$	−8.54037e10	−0.493444
$646$	0	0
$647$	− 2.53047e11i	− 1.44405i	−0.691864	−	0.722027i	$-0.743210\pi$
$647$	− 2.53047e11i	− 1.44405i	0.691864	−	0.722027i	$-0.256790\pi$
$648$	0	0
$649$	−1.04226e11	−0.587487
$650$	0	0
$651$	5.85480e10i	0.325978i
$652$	0	0
$653$	8.08800e10	0.444824	0.222412	−	0.974953i	$-0.428607\pi$
$653$	8.08800e10	0.444824	0.222412	+	0.974953i	$0.428607\pi$
$654$	0	0
$655$	7.41151e10i	0.402663i
$656$	0	0
$657$	8.08765e10	0.434071
$658$	0	0
$659$	− 8.97635e10i	− 0.475946i	−0.971272	−	0.237973i	$-0.923517\pi$
$659$	− 8.97635e10i	− 0.475946i	0.971272	−	0.237973i	$-0.0764830\pi$
$660$	0	0
$661$	1.12878e11	0.591296	0.295648	−	0.955297i	$-0.404464\pi$
$661$	1.12878e11	0.591296	0.295648	+	0.955297i	$0.404464\pi$
$662$	0	0
$663$	6.73592e10i	0.348612i
$664$	0	0
$665$	−1.12037e10	−0.0572893
$666$	0	0
$667$	3.60204e9i	0.0181989i
$668$	0	0
$669$	−3.74296e10	−0.186857
$670$	0	0
$671$	1.31870e11i	0.650513i
$672$	0	0
$673$	1.26871e10	0.0618444	0.0309222	−	0.999522i	$-0.490156\pi$
$673$	1.26871e10	0.0618444	0.0309222	+	0.999522i	$0.490156\pi$
$674$	0	0
$675$	− 1.87043e10i	− 0.0901001i
$676$	0	0
$677$	1.20385e11	0.573082	0.286541	−	0.958068i	$-0.407495\pi$
$677$	1.20385e11	0.573082	0.286541	+	0.958068i	$0.407495\pi$
$678$	0	0
$679$	− 1.77046e11i	− 0.832927i
$680$	0	0
$681$	1.66526e11	0.774274
$682$	0	0
$683$	− 8.09582e10i	− 0.372030i	−0.982547	−	0.186015i	$-0.940443\pi$
$683$	− 8.09582e10i	− 0.372030i	0.982547	−	0.186015i	$-0.0595573\pi$
$684$	0	0
$685$	−1.70101e11	−0.772583
$686$	0	0
$687$	7.04844e10i	0.316422i
$688$	0	0
$689$	1.17795e11	0.522695
$690$	0	0
$691$	− 5.81335e10i	− 0.254985i	−0.991840	−	0.127492i	$-0.959307\pi$
$691$	− 5.81335e10i	− 0.254985i	0.991840	−	0.127492i	$-0.0406928\pi$
$692$	0	0
$693$	−2.51417e10	−0.109009
$694$	0	0
$695$	1.32787e11i	0.569138i
$696$	0	0
$697$	2.98018e11	1.26273
$698$	0	0
$699$	1.74818e11i	0.732280i
$700$	0	0
$701$	1.42599e10	0.0590531	0.0295266	−	0.999564i	$-0.490600\pi$
$701$	1.42599e10	0.0590531	0.0295266	+	0.999564i	$0.490600\pi$
$702$	0	0
$703$	1.94261e10i	0.0795361i
$704$	0	0
$705$	3.35384e10	0.135765
$706$	0	0
$707$	− 9.50131e10i	− 0.380282i
$708$	0	0
$709$	3.59144e11	1.42130	0.710648	−	0.703548i	$-0.248402\pi$
$709$	3.59144e11	1.42130	0.710648	+	0.703548i	$0.248402\pi$
$710$	0	0
$711$	1.34492e11i	0.526281i
$712$	0	0
$713$	2.49241e11	0.964409
$714$	0	0
$715$	− 4.76225e10i	− 0.182217i
$716$	0	0
$717$	2.18813e11	0.827937
$718$	0	0
$719$	− 1.87449e11i	− 0.701404i	−0.936487	−	0.350702i	$-0.885943\pi$
$719$	− 1.87449e11i	− 0.701404i	0.936487	−	0.350702i	$-0.114057\pi$
$720$	0	0
$721$	2.06330e11	0.763522
$722$	0	0
$723$	1.92068e11i	0.702915i
$724$	0	0
$725$	−1.95821e9	−0.00708772
$726$	0	0
$727$	1.12950e11i	0.404341i	0.979350	+	0.202171i	$0.0647996\pi$
$727$	1.12950e11i	0.404341i	−0.979350	+	0.202171i	$0.935200\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	− 3.75779e11i	− 1.31602i
$732$	0	0
$733$	−2.07098e11	−0.717397	−0.358699	−	0.933453i	$-0.616779\pi$
$733$	−2.07098e11	−0.717397	−0.358699	+	0.933453i	$0.616779\pi$
$734$	0	0
$735$	6.20166e10i	0.212500i
$736$	0	0
$737$	9.32483e10	0.316061
$738$	0	0
$739$	5.08051e10i	0.170345i	0.996366	+	0.0851726i	$0.0271442\pi$
$739$	5.08051e10i	0.170345i	−0.996366	+	0.0851726i	$0.972856\pi$
$740$	0	0
$741$	1.04477e10	0.0346535
$742$	0	0
$743$	− 5.69279e10i	− 0.186797i	−0.995629	−	0.0933985i	$-0.970227\pi$
$743$	− 5.69279e10i	− 0.186797i	0.995629	−	0.0933985i	$-0.0297731\pi$
$744$	0	0
$745$	2.73798e11	0.888803
$746$	0	0
$747$	− 3.93111e10i	− 0.126250i
$748$	0	0
$749$	3.86661e9	0.0122858
$750$	0	0
$751$	2.75630e10i	0.0866497i	0.999061	+	0.0433248i	$0.0137950\pi$
$751$	2.75630e10i	0.0866497i	−0.999061	+	0.0433248i	$0.986205\pi$
$752$	0	0
$753$	1.62797e11	0.506369
$754$	0	0
$755$	− 2.52501e11i	− 0.777097i
$756$	0	0
$757$	−2.84625e11	−0.866743	−0.433371	−	0.901215i	$-0.642676\pi$
$757$	−2.84625e11	−0.866743	−0.433371	+	0.901215i	$0.642676\pi$
$758$	0	0
$759$	1.07029e11i	0.322504i
$760$	0	0
$761$	3.24323e11	0.967028	0.483514	−	0.875337i	$-0.339360\pi$
$761$	3.24323e11	0.967028	0.483514	+	0.875337i	$0.339360\pi$
$762$	0	0
$763$	− 2.54219e11i	− 0.750085i
$764$	0	0
$765$	−9.34887e10	−0.272969
$766$	0	0
$767$	2.35279e11i	0.679833i
$768$	0	0
$769$	4.03018e11	1.15244	0.576221	−	0.817294i	$-0.304527\pi$
$769$	4.03018e11	1.15244	0.576221	+	0.817294i	$0.304527\pi$
$770$	0	0
$771$	− 1.55597e11i	− 0.440336i
$772$	0	0
$773$	3.04590e11	0.853097	0.426548	−	0.904465i	$-0.359729\pi$
$773$	3.04590e11	0.853097	0.426548	+	0.904465i	$0.359729\pi$
$774$	0	0
$775$	1.35497e11i	0.375597i
$776$	0	0
$777$	−1.05528e11	−0.289522
$778$	0	0
$779$	− 4.62239e10i	− 0.125521i
$780$	0	0
$781$	1.33209e11	0.358037
$782$	0	0
$783$	1.09513e9i	0.00291352i
$784$	0	0
$785$	−4.48292e11	−1.18054
$786$	0	0
$787$	5.85835e11i	1.52713i	0.645730	+	0.763565i	$0.276553\pi$
$787$	5.85835e11i	1.52713i	−0.645730	+	0.763565i	$0.723447\pi$
$788$	0	0
$789$	2.29749e11	0.592852
$790$	0	0
$791$	− 4.34878e11i	− 1.11087i
$792$	0	0
$793$	2.97682e11	0.752765
$794$	0	0
$795$	1.63489e11i	0.409279i
$796$	0	0
$797$	−4.31674e11	−1.06985	−0.534925	−	0.844900i	$-0.679660\pi$
$797$	−4.31674e11	−1.06985	−0.534925	+	0.844900i	$0.679660\pi$
$798$	0	0
$799$	1.47570e11i	0.362086i
$800$	0	0
$801$	−1.54126e11	−0.374408
$802$	0	0
$803$	− 2.51590e11i	− 0.605107i
$804$	0	0
$805$	−2.59089e11	−0.616971
$806$	0	0
$807$	3.45049e11i	0.813554i
$808$	0	0
$809$	−1.05205e10	−0.0245608	−0.0122804	−	0.999925i	$-0.503909\pi$
$809$	−1.05205e10	−0.0245608	−0.0122804	+	0.999925i	$0.503909\pi$
$810$	0	0
$811$	4.65876e11i	1.07693i	0.842648	+	0.538464i	$0.180995\pi$
$811$	4.65876e11i	1.07693i	−0.842648	+	0.538464i	$0.819005\pi$
$812$	0	0
$813$	1.82377e11	0.417454
$814$	0	0
$815$	3.19618e9i	0.00724438i
$816$	0	0
$817$	−5.82850e10	−0.130818
$818$	0	0
$819$	5.67546e10i	0.126144i
$820$	0	0
$821$	5.66593e11	1.24709	0.623546	−	0.781786i	$-0.285691\pi$
$821$	5.66593e11	1.24709	0.623546	+	0.781786i	$0.285691\pi$
$822$	0	0
$823$	− 4.94664e10i	− 0.107823i	−0.998546	−	0.0539114i	$-0.982831\pi$
$823$	− 4.94664e10i	− 0.107823i	0.998546	−	0.0539114i	$-0.0171689\pi$
$824$	0	0
$825$	−5.81851e10	−0.125602
$826$	0	0
$827$	− 3.83529e11i	− 0.819928i	−0.912102	−	0.409964i	$-0.865541\pi$
$827$	− 3.83529e11i	− 0.819928i	0.912102	−	0.409964i	$-0.134459\pi$
$828$	0	0
$829$	5.61549e11	1.18897	0.594483	−	0.804108i	$-0.297357\pi$
$829$	5.61549e11	1.18897	0.594483	+	0.804108i	$0.297357\pi$
$830$	0	0
$831$	4.38881e11i	0.920328i
$832$	0	0
$833$	−2.72875e11	−0.566740
$834$	0	0
$835$	4.89507e11i	1.00696i
$836$	0	0
$837$	7.57765e10	0.154395
$838$	0	0
$839$	− 2.21865e11i	− 0.447755i	−0.974617	−	0.223877i	$-0.928129\pi$
$839$	− 2.21865e11i	− 0.447755i	0.974617	−	0.223877i	$-0.0718715\pi$
$840$	0	0
$841$	−5.00132e11	−0.999771
$842$	0	0
$843$	− 2.09965e11i	− 0.415753i
$844$	0	0
$845$	2.64300e11	0.518405
$846$	0	0
$847$	− 2.84005e11i	− 0.551814i
$848$	0	0
$849$	−2.80134e11	−0.539181
$850$	0	0
$851$	4.49235e11i	0.856556i
$852$	0	0
$853$	−2.73278e11	−0.516189	−0.258095	−	0.966120i	$-0.583095\pi$
$853$	−2.73278e11	−0.516189	−0.258095	+	0.966120i	$0.583095\pi$
$854$	0	0
$855$	1.45005e10i	0.0271343i
$856$	0	0
$857$	2.78664e11	0.516605	0.258302	−	0.966064i	$-0.416837\pi$
$857$	2.78664e11	0.516605	0.258302	+	0.966064i	$0.416837\pi$
$858$	0	0
$859$	2.77264e11i	0.509239i	0.967041	+	0.254619i	$0.0819502\pi$
$859$	2.77264e11i	0.509239i	−0.967041	+	0.254619i	$0.918050\pi$
$860$	0	0
$861$	2.51101e11	0.456915
$862$	0	0
$863$	− 8.42997e11i	− 1.51979i	−0.650047	−	0.759894i	$-0.725251\pi$
$863$	− 8.42997e11i	− 1.51979i	0.650047	−	0.759894i	$-0.274749\pi$
$864$	0	0
$865$	1.54272e10	0.0275565
$866$	0	0
$867$	− 8.51302e10i	− 0.150663i
$868$	0	0
$869$	4.18377e11	0.733650
$870$	0	0
$871$	− 2.10498e11i	− 0.365742i
$872$	0	0
$873$	−2.29144e11	−0.394505
$874$	0	0
$875$	− 4.41701e11i	− 0.753522i
$876$	0	0
$877$	1.59168e11	0.269066	0.134533	−	0.990909i	$-0.457047\pi$
$877$	1.59168e11	0.269066	0.134533	+	0.990909i	$0.457047\pi$
$878$	0	0
$879$	5.12215e10i	0.0858019i
$880$	0	0
$881$	2.76086e11	0.458290	0.229145	−	0.973392i	$-0.426407\pi$
$881$	2.76086e11	0.458290	0.229145	+	0.973392i	$0.426407\pi$
$882$	0	0
$883$	7.01305e11i	1.15362i	0.816877	+	0.576811i	$0.195703\pi$
$883$	7.01305e11i	1.15362i	−0.816877	+	0.576811i	$0.804297\pi$
$884$	0	0
$885$	−3.26547e11	−0.532320
$886$	0	0
$887$	7.11458e11i	1.14936i	0.818379	+	0.574678i	$0.194873\pi$
$887$	7.11458e11i	1.14936i	−0.818379	+	0.574678i	$0.805127\pi$
$888$	0	0
$889$	5.41949e11	0.867664
$890$	0	0
$891$	3.25400e10i	0.0516306i
$892$	0	0
$893$	2.28888e10	0.0359929
$894$	0	0
$895$	− 7.07814e11i	− 1.10313i
$896$	0	0
$897$	2.41607e11	0.373198
$898$	0	0
$899$	− 7.93328e9i	− 0.0121455i
$900$	0	0
$901$	−7.19357e11	−1.09155
$902$	0	0
$903$	− 3.16619e11i	− 0.476197i
$904$	0	0
$905$	−5.79236e11	−0.863497
$906$	0	0
$907$	5.42602e10i	0.0801774i	0.999196	+	0.0400887i	$0.0127641\pi$
$907$	5.42602e10i	0.0801774i	−0.999196	+	0.0400887i	$0.987236\pi$
$908$	0	0
$909$	−1.22972e11	−0.180115
$910$	0	0
$911$	− 7.14023e11i	− 1.03667i	−0.855179	−	0.518333i	$-0.826553\pi$
$911$	− 7.14023e11i	− 1.03667i	0.855179	−	0.518333i	$-0.173447\pi$
$912$	0	0
$913$	−1.22289e11	−0.175996
$914$	0	0
$915$	4.13157e11i	0.589427i
$916$	0	0
$917$	−2.74769e11	−0.388589
$918$	0	0
$919$	1.58012e11i	0.221527i	0.993847	+	0.110764i	$0.0353296\pi$
$919$	1.58012e11i	0.221527i	−0.993847	+	0.110764i	$0.964670\pi$
$920$	0	0
$921$	−5.46018e11	−0.758872
$922$	0	0
$923$	− 3.00704e11i	− 0.414316i
$924$	0	0
$925$	−2.44221e11	−0.333593
$926$	0	0
$927$	− 2.67046e11i	− 0.361632i
$928$	0	0
$929$	−1.26528e12	−1.69873	−0.849365	−	0.527805i	$-0.823015\pi$
$929$	−1.26528e12	−1.69873	−0.849365	+	0.527805i	$0.823015\pi$
$930$	0	0
$931$	4.23241e10i	0.0563364i
$932$	0	0
$933$	2.53151e11	0.334082
$934$	0	0
$935$	2.90824e11i	0.380526i
$936$	0	0
$937$	−6.74824e11	−0.875452	−0.437726	−	0.899108i	$-0.644216\pi$
$937$	−6.74824e11	−0.875452	−0.437726	+	0.899108i	$0.644216\pi$
$938$	0	0
$939$	− 7.03520e11i	− 0.904928i
$940$	0	0
$941$	−3.19026e11	−0.406881	−0.203441	−	0.979087i	$-0.565212\pi$
$941$	−3.19026e11	−0.406881	−0.203441	+	0.979087i	$0.565212\pi$
$942$	0	0
$943$	− 1.06894e12i	− 1.35179i
$944$	0	0
$945$	−7.87706e10	−0.0987726
$946$	0	0
$947$	− 6.73069e11i	− 0.836873i	−0.908246	−	0.418437i	$-0.862578\pi$
$947$	− 6.73069e11i	− 0.836873i	0.908246	−	0.418437i	$-0.137422\pi$
$948$	0	0
$949$	−5.67937e11	−0.700222
$950$	0	0
$951$	6.37212e10i	0.0779044i
$952$	0	0
$953$	−5.41655e11	−0.656676	−0.328338	−	0.944560i	$-0.606489\pi$
$953$	−5.41655e11	−0.656676	−0.328338	+	0.944560i	$0.606489\pi$
$954$	0	0
$955$	− 1.77980e11i	− 0.213973i
$956$	0	0
$957$	3.40672e9	0.00406152
$958$	0	0
$959$	− 6.30621e11i	− 0.745580i
$960$	0	0
$961$	3.03953e11	0.356380
$962$	0	0
$963$	− 5.00441e9i	− 0.00581900i
$964$	0	0
$965$	8.21084e11	0.946844
$966$	0	0
$967$	1.61996e12i	1.85267i	0.376697	+	0.926336i	$0.377060\pi$
$967$	1.61996e12i	1.85267i	−0.376697	+	0.926336i	$0.622940\pi$
$968$	0	0
$969$	−6.38027e10	−0.0723676
$970$	0	0
$971$	5.73599e11i	0.645255i	0.946526	+	0.322627i	$0.104566\pi$
$971$	5.73599e11i	0.645255i	−0.946526	+	0.322627i	$0.895434\pi$
$972$	0	0
$973$	−4.92286e11	−0.549245
$974$	0	0
$975$	1.31346e11i	0.145345i
$976$	0	0
$977$	1.32544e11	0.145473	0.0727366	−	0.997351i	$-0.476827\pi$
$977$	1.32544e11	0.145473	0.0727366	+	0.997351i	$0.476827\pi$
$978$	0	0
$979$	4.79453e11i	0.521934i
$980$	0	0
$981$	−3.29027e11	−0.355268
$982$	0	0
$983$	1.02796e12i	1.10093i	0.834857	+	0.550466i	$0.185550\pi$
$983$	1.02796e12i	1.10093i	−0.834857	+	0.550466i	$0.814450\pi$
$984$	0	0
$985$	−2.71999e11	−0.288950
$986$	0	0
$987$	1.24338e11i	0.131019i
$988$	0	0
$989$	−1.34786e12	−1.40883
$990$	0	0
$991$	4.84738e11i	0.502588i	0.967911	+	0.251294i	$0.0808561\pi$
$991$	4.84738e11i	0.502588i	−0.967911	+	0.251294i	$0.919144\pi$
$992$	0	0
$993$	7.76712e11	0.798846
$994$	0	0
$995$	2.11231e11i	0.215509i
$996$	0	0
$997$	−9.70160e11	−0.981890	−0.490945	−	0.871191i	$-0.663348\pi$
$997$	−9.70160e11	−0.981890	−0.490945	+	0.871191i	$0.663348\pi$
$998$	0	0
$999$	1.36581e11i	0.137128i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.9.g.f.127.1		8
4.3	odd	2	inner	192.9.g.f.127.5		8
8.3	odd	2		96.9.g.a.31.4	✓	8
8.5	even	2		96.9.g.a.31.8	yes	8
24.5	odd	2		288.9.g.f.127.1		8
24.11	even	2		288.9.g.f.127.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.9.g.a.31.4	✓	8	8.3	odd	2
96.9.g.a.31.8	yes	8	8.5	even	2
192.9.g.f.127.1		8	1.1	even	1	trivial
192.9.g.f.127.5		8	4.3	odd	2	inner
288.9.g.f.127.1		8	24.5	odd	2
288.9.g.f.127.2		8	24.11	even	2

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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	192.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-46.7654i q^{3} -455.790 q^{5} -1689.76i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q-46.7654i q^{3} -455.790 q^{5} -1689.76i q^{7} -2187.00 q^{9} +6803.31i q^{11} +15357.7 q^{13} +21315.2i q^{15} -93787.7 q^{17} +14546.9i q^{19} -79022.4 q^{21} +336401. i q^{23} -182880. q^{25} +102276. i q^{27} +10707.6 q^{29} -740903. i q^{31} +318159. q^{33} +770177. i q^{35} +1.33541e6 q^{37} -718209. i q^{39} -3.17759e6 q^{41} +4.00670e6i q^{43} +996813. q^{45} -1.57345e6i q^{47} +2.90950e6 q^{49} +4.38601e6i q^{51} +7.67006e6 q^{53} -3.10088e6i q^{55} +680289. q^{57} +1.53199e7i q^{59} +1.93832e7 q^{61} +3.69551e6i q^{63} -6.99990e6 q^{65} -1.37063e7i q^{67} +1.57319e7 q^{69} -1.95800e7i q^{71} -3.69806e7 q^{73} +8.55247e6i q^{75} +1.14960e7 q^{77} -6.14961e7i q^{79} +4.78297e6 q^{81} +1.79749e7i q^{83} +4.27475e7 q^{85} -500744. i q^{87} +7.04735e7 q^{89} -2.59509e7i q^{91} -3.46486e7 q^{93} -6.63031e6i q^{95} +1.04776e8 q^{97} -1.48788e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1232 q^{5} - 17496 q^{9}+O(q^{10}) \) 8 * q + 1232 * q^5 - 17496 * q^9	\( 8 q + 1232 q^{5} - 17496 q^{9} + 54256 q^{13} - 204336 q^{17} + 298080 q^{21} - 781416 q^{25} + 333712 q^{29} - 775008 q^{33} - 3945168 q^{37} + 11803088 q^{41} - 2694384 q^{45} - 3977208 q^{49} - 788592 q^{53} - 1194912 q^{57} + 40252336 q^{61} - 20747936 q^{65} - 26687232 q^{69} - 56712048 q^{73} + 52014208 q^{77} + 38263752 q^{81} - 52697568 q^{85} + 61660176 q^{89} - 81738720 q^{93} + 27830288 q^{97}+O(q^{100}) \) 8 * q + 1232 * q^5 - 17496 * q^9 + 54256 * q^13 - 204336 * q^17 + 298080 * q^21 - 781416 * q^25 + 333712 * q^29 - 775008 * q^33 - 3945168 * q^37 + 11803088 * q^41 - 2694384 * q^45 - 3977208 * q^49 - 788592 * q^53 - 1194912 * q^57 + 40252336 * q^61 - 20747936 * q^65 - 26687232 * q^69 - 56712048 * q^73 + 52014208 * q^77 + 38263752 * q^81 - 52697568 * q^85 + 61660176 * q^89 - 81738720 * q^93 + 27830288 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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