LMFDB - Embedded newform 192.9.g.b.127.2

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:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	192.127
Dual form			192.9.g.b.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} +90.0000 q^{5} -921.451i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} +90.0000 q^{5} -921.451i q^{7} -2187.00 q^{9} +5175.37i q^{11} +16358.0 q^{13} +4208.88i q^{15} -42678.0 q^{17} -114447. i q^{19} +43092.0 q^{21} +250662. i q^{23} -382525. q^{25} -102276. i q^{27} +1.27053e6 q^{29} +512943. i q^{31} -242028. q^{33} -82930.6i q^{35} +2.26214e6 q^{37} +764988. i q^{39} -872694. q^{41} -1.66208e6i q^{43} -196830. q^{45} -797007. i q^{47} +4.91573e6 q^{49} -1.99585e6i q^{51} -1.06169e6 q^{53} +465783. i q^{55} +5.35216e6 q^{57} +2.33723e7i q^{59} -1.53010e7 q^{61} +2.01521e6i q^{63} +1.47222e6 q^{65} +9.18022e6i q^{67} -1.17223e7 q^{69} -2.28310e7i q^{71} +1.89164e7 q^{73} -1.78889e7i q^{75} +4.76885e6 q^{77} +5.41593e7i q^{79} +4.78297e6 q^{81} +6.52089e7i q^{83} -3.84102e6 q^{85} +5.94168e7i q^{87} -8.98132e7 q^{89} -1.50731e7i q^{91} -2.39880e7 q^{93} -1.03002e7i q^{95} -7.57782e7 q^{97} -1.13185e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 180 q^{5} - 4374 q^{9}+O(q^{10}) $ 2 * q + 180 * q^5 - 4374 * q^9	$ 2 q + 180 q^{5} - 4374 q^{9} + 32716 q^{13} - 85356 q^{17} + 86184 q^{21} - 765050 q^{25} + 2541060 q^{29} - 484056 q^{33} + 4524284 q^{37} - 1745388 q^{41} - 393660 q^{45} + 9831458 q^{49} - 2123388 q^{53} + 10704312 q^{57} - 30602020 q^{61} + 2944440 q^{65} - 23444640 q^{69} + 37832708 q^{73} + 9537696 q^{77} + 9565938 q^{81} - 7682040 q^{85} - 179626428 q^{89} - 47975976 q^{93} - 151556476 q^{97}+O(q^{100}) $ 2 * q + 180 * q^5 - 4374 * q^9 + 32716 * q^13 - 85356 * q^17 + 86184 * q^21 - 765050 * q^25 + 2541060 * q^29 - 484056 * q^33 + 4524284 * q^37 - 1745388 * q^41 - 393660 * q^45 + 9831458 * q^49 - 2123388 * q^53 + 10704312 * q^57 - 30602020 * q^61 + 2944440 * q^65 - 23444640 * q^69 + 37832708 * q^73 + 9537696 * q^77 + 9565938 * q^81 - 7682040 * q^85 - 179626428 * q^89 - 47975976 * q^93 - 151556476 * 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
$5$	90.0000	0.144000	0.0720000	−	0.997405i	$-0.477062\pi$
$5$	90.0000	0.144000	0.0720000	+	0.997405i	$0.477062\pi$
$6$	0	0
$7$	− 921.451i	− 0.383778i	−0.981417	−	0.191889i	$-0.938539\pi$
$7$	− 921.451i	− 0.383778i	0.981417	−	0.191889i	$-0.0614614\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	5175.37i	0.353485i	0.984257	+	0.176742i	$0.0565559\pi$
$11$	5175.37i	0.353485i	−0.984257	+	0.176742i	$0.943444\pi$
$12$	0	0
$13$	16358.0	0.572739	0.286370	−	0.958119i	$-0.407552\pi$
$13$	16358.0	0.572739	0.286370	+	0.958119i	$0.407552\pi$
$14$	0	0
$15$	4208.88i	0.0831384i
$16$	0	0
$17$	−42678.0	−0.510985	−0.255493	−	0.966811i	$-0.582238\pi$
$17$	−42678.0	−0.510985	−0.255493	+	0.966811i	$0.582238\pi$
$18$	0	0
$19$	− 114447.i	− 0.878193i	−0.898440	−	0.439096i	$-0.855299\pi$
$19$	− 114447.i	− 0.878193i	0.898440	−	0.439096i	$-0.144701\pi$
$20$	0	0
$21$	43092.0	0.221574
$22$	0	0
$23$	250662.i	0.895731i	0.894101	+	0.447866i	$0.147816\pi$
$23$	250662.i	0.895731i	−0.894101	+	0.447866i	$0.852184\pi$
$24$	0	0
$25$	−382525.	−0.979264
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	1.27053e6	1.79636	0.898179	−	0.439630i	$-0.144890\pi$
$29$	1.27053e6	1.79636	0.898179	+	0.439630i	$0.144890\pi$
$30$	0	0
$31$	512943.i	0.555421i	0.960665	+	0.277711i	$0.0895757\pi$
$31$	512943.i	0.555421i	−0.960665	+	0.277711i	$0.910424\pi$
$32$	0	0
$33$	−242028.	−0.204084
$34$	0	0
$35$	− 82930.6i	− 0.0552640i
$36$	0	0
$37$	2.26214e6	1.20702	0.603508	−	0.797357i	$-0.293769\pi$
$37$	2.26214e6	1.20702	0.603508	+	0.797357i	$0.293769\pi$
$38$	0	0
$39$	764988.i	0.330671i
$40$	0	0
$41$	−872694.	−0.308835	−0.154418	−	0.988006i	$-0.549350\pi$
$41$	−872694.	−0.308835	−0.154418	+	0.988006i	$0.549350\pi$
$42$	0	0
$43$	− 1.66208e6i	− 0.486160i	−0.970006	−	0.243080i	$-0.921842\pi$
$43$	− 1.66208e6i	− 0.486160i	0.970006	−	0.243080i	$-0.0781577\pi$
$44$	0	0
$45$	−196830.	−0.0480000
$46$	0	0
$47$	− 797007.i	− 0.163332i	−0.996660	−	0.0816659i	$-0.973976\pi$
$47$	− 797007.i	− 0.163332i	0.996660	−	0.0816659i	$-0.0260240\pi$
$48$	0	0
$49$	4.91573e6	0.852714
$50$	0	0
$51$	− 1.99585e6i	− 0.295017i
$52$	0	0
$53$	−1.06169e6	−0.134554	−0.0672769	−	0.997734i	$-0.521431\pi$
$53$	−1.06169e6	−0.134554	−0.0672769	+	0.997734i	$0.521431\pi$
$54$	0	0
$55$	465783.i	0.0509018i
$56$	0	0
$57$	5.35216e6	0.507025
$58$	0	0
$59$	2.33723e7i	1.92883i	0.264402	+	0.964413i	$0.414826\pi$
$59$	2.33723e7i	1.92883i	−0.264402	+	0.964413i	$0.585174\pi$
$60$	0	0
$61$	−1.53010e7	−1.10510	−0.552549	−	0.833480i	$-0.686345\pi$
$61$	−1.53010e7	−1.10510	−0.552549	+	0.833480i	$0.686345\pi$
$62$	0	0
$63$	2.01521e6i	0.127926i
$64$	0	0
$65$	1.47222e6	0.0824744
$66$	0	0
$67$	9.18022e6i	0.455569i	0.973712	+	0.227784i	$0.0731481\pi$
$67$	9.18022e6i	0.455569i	−0.973712	+	0.227784i	$0.926852\pi$
$68$	0	0
$69$	−1.17223e7	−0.517151
$70$	0	0
$71$	− 2.28310e7i	− 0.898444i	−0.893420	−	0.449222i	$-0.851701\pi$
$71$	− 2.28310e7i	− 0.898444i	0.893420	−	0.449222i	$-0.148299\pi$
$72$	0	0
$73$	1.89164e7	0.666110	0.333055	−	0.942907i	$-0.391921\pi$
$73$	1.89164e7	0.666110	0.333055	+	0.942907i	$0.391921\pi$
$74$	0	0
$75$	− 1.78889e7i	− 0.565378i
$76$	0	0
$77$	4.76885e6	0.135660
$78$	0	0
$79$	5.41593e7i	1.39048i	0.718778	+	0.695240i	$0.244702\pi$
$79$	5.41593e7i	1.39048i	−0.718778	+	0.695240i	$0.755298\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	6.52089e7i	1.37403i	0.726645	+	0.687013i	$0.241078\pi$
$83$	6.52089e7i	1.37403i	−0.726645	+	0.687013i	$0.758922\pi$
$84$	0	0
$85$	−3.84102e6	−0.0735819
$86$	0	0
$87$	5.94168e7i	1.03713i
$88$	0	0
$89$	−8.98132e7	−1.43146	−0.715732	−	0.698375i	$-0.753907\pi$
$89$	−8.98132e7	−1.43146	−0.715732	+	0.698375i	$0.753907\pi$
$90$	0	0
$91$	− 1.50731e7i	− 0.219805i
$92$	0	0
$93$	−2.39880e7	−0.320673
$94$	0	0
$95$	− 1.03002e7i	− 0.126460i
$96$	0	0
$97$	−7.57782e7	−0.855968	−0.427984	−	0.903786i	$-0.640776\pi$
$97$	−7.57782e7	−0.855968	−0.427984	+	0.903786i	$0.640776\pi$
$98$	0	0
$99$	− 1.13185e7i	− 0.117828i
$100$	0	0
$101$	1.68976e8	1.62383	0.811914	−	0.583778i	$-0.198426\pi$
$101$	1.68976e8	1.62383	0.811914	+	0.583778i	$0.198426\pi$
$102$	0	0
$103$	1.59060e8i	1.41323i	0.707601	+	0.706613i	$0.249778\pi$
$103$	1.59060e8i	1.41323i	−0.707601	+	0.706613i	$0.750222\pi$
$104$	0	0
$105$	3.87828e6	0.0319067
$106$	0	0
$107$	1.75980e8i	1.34254i	0.741213	+	0.671270i	$0.234251\pi$
$107$	1.75980e8i	1.34254i	−0.741213	+	0.671270i	$0.765749\pi$
$108$	0	0
$109$	1.37980e7	0.0977483	0.0488742	−	0.998805i	$-0.484437\pi$
$109$	1.37980e7	0.0977483	0.0488742	+	0.998805i	$0.484437\pi$
$110$	0	0
$111$	1.05790e8i	0.696871i
$112$	0	0
$113$	−9.25151e7	−0.567412	−0.283706	−	0.958911i	$-0.591564\pi$
$113$	−9.25151e7	−0.567412	−0.283706	+	0.958911i	$0.591564\pi$
$114$	0	0
$115$	2.25596e7i	0.128985i
$116$	0	0
$117$	−3.57749e7	−0.190913
$118$	0	0
$119$	3.93257e7i	0.196105i
$120$	0	0
$121$	1.87574e8	0.875049
$122$	0	0
$123$	− 4.08119e7i	− 0.178306i
$124$	0	0
$125$	−6.95835e7	−0.285014
$126$	0	0
$127$	2.21744e8i	0.852389i	0.904632	+	0.426195i	$0.140146\pi$
$127$	2.21744e8i	0.852389i	−0.904632	+	0.426195i	$0.859854\pi$
$128$	0	0
$129$	7.77279e7	0.280684
$130$	0	0
$131$	3.60662e8i	1.22466i	0.790603	+	0.612329i	$0.209767\pi$
$131$	3.60662e8i	1.22466i	−0.790603	+	0.612329i	$0.790233\pi$
$132$	0	0
$133$	−1.05457e8	−0.337031
$134$	0	0
$135$	− 9.20483e6i	− 0.0277128i
$136$	0	0
$137$	4.40725e8	1.25108	0.625541	−	0.780191i	$-0.284878\pi$
$137$	4.40725e8	1.25108	0.625541	+	0.780191i	$0.284878\pi$
$138$	0	0
$139$	5.44242e8i	1.45792i	0.684558	+	0.728958i	$0.259995\pi$
$139$	5.44242e8i	1.45792i	−0.684558	+	0.728958i	$0.740005\pi$
$140$	0	0
$141$	3.72723e7	0.0942996
$142$	0	0
$143$	8.46587e7i	0.202454i
$144$	0	0
$145$	1.14348e8	0.258676
$146$	0	0
$147$	2.29886e8i	0.492315i
$148$	0	0
$149$	−4.40746e7	−0.0894217	−0.0447109	−	0.999000i	$-0.514237\pi$
$149$	−4.40746e7	−0.0894217	−0.0447109	+	0.999000i	$0.514237\pi$
$150$	0	0
$151$	− 1.11931e8i	− 0.215300i	−0.994189	−	0.107650i	$-0.965667\pi$
$151$	− 1.11931e8i	− 0.215300i	0.994189	−	0.107650i	$-0.0343326\pi$
$152$	0	0
$153$	9.33368e7	0.170328
$154$	0	0
$155$	4.61649e7i	0.0799807i
$156$	0	0
$157$	−2.89510e8	−0.476503	−0.238251	−	0.971204i	$-0.576574\pi$
$157$	−2.89510e8	−0.476503	−0.238251	+	0.971204i	$0.576574\pi$
$158$	0	0
$159$	− 4.96505e7i	− 0.0776847i
$160$	0	0
$161$	2.30973e8	0.343762
$162$	0	0
$163$	3.06936e8i	0.434808i	0.976082	+	0.217404i	$0.0697589\pi$
$163$	3.06936e8i	0.434808i	−0.976082	+	0.217404i	$0.930241\pi$
$164$	0	0
$165$	−2.17825e7	−0.0293882
$166$	0	0
$167$	1.19626e9i	1.53801i	0.639243	+	0.769004i	$0.279248\pi$
$167$	1.19626e9i	1.53801i	−0.639243	+	0.769004i	$0.720752\pi$
$168$	0	0
$169$	−5.48147e8	−0.671970
$170$	0	0
$171$	2.50296e8i	0.292731i
$172$	0	0
$173$	−8.26090e7	−0.0922237	−0.0461119	−	0.998936i	$-0.514683\pi$
$173$	−8.26090e7	−0.0922237	−0.0461119	+	0.998936i	$0.514683\pi$
$174$	0	0
$175$	3.52478e8i	0.375820i
$176$	0	0
$177$	−1.09301e9	−1.11361
$178$	0	0
$179$	− 1.13619e9i	− 1.10673i	−0.832940	−	0.553364i	$-0.813344\pi$
$179$	− 1.13619e9i	− 1.10673i	0.832940	−	0.553364i	$-0.186656\pi$
$180$	0	0
$181$	8.20024e8	0.764033	0.382017	−	0.924155i	$-0.375230\pi$
$181$	8.20024e8	0.764033	0.382017	+	0.924155i	$0.375230\pi$
$182$	0	0
$183$	− 7.15557e8i	− 0.638029i
$184$	0	0
$185$	2.03593e8	0.173810
$186$	0	0
$187$	− 2.20874e8i	− 0.180625i
$188$	0	0
$189$	−9.42422e7	−0.0738581
$190$	0	0
$191$	1.24962e9i	0.938956i	0.882944	+	0.469478i	$0.155558\pi$
$191$	1.24962e9i	0.938956i	−0.882944	+	0.469478i	$0.844442\pi$
$192$	0	0
$193$	−2.01195e9	−1.45007	−0.725033	−	0.688714i	$-0.758176\pi$
$193$	−2.01195e9	−1.45007	−0.725033	+	0.688714i	$0.758176\pi$
$194$	0	0
$195$	6.88489e7i	0.0476166i
$196$	0	0
$197$	6.32559e8	0.419988	0.209994	−	0.977703i	$-0.432656\pi$
$197$	6.32559e8	0.419988	0.209994	+	0.977703i	$0.432656\pi$
$198$	0	0
$199$	− 1.45714e9i	− 0.929156i	−0.885532	−	0.464578i	$-0.846206\pi$
$199$	− 1.45714e9i	− 0.929156i	0.885532	−	0.464578i	$-0.153794\pi$
$200$	0	0
$201$	−4.29317e8	−0.263023
$202$	0	0
$203$	− 1.17073e9i	− 0.689403i
$204$	0	0
$205$	−7.85425e7	−0.0444722
$206$	0	0
$207$	− 5.48199e8i	− 0.298577i
$208$	0	0
$209$	5.92305e8	0.310428
$210$	0	0
$211$	− 3.57401e8i	− 0.180312i	−0.995928	−	0.0901562i	$-0.971263\pi$
$211$	− 3.57401e8i	− 0.180312i	0.995928	−	0.0901562i	$-0.0287366\pi$
$212$	0	0
$213$	1.06770e9	0.518717
$214$	0	0
$215$	− 1.49587e8i	− 0.0700070i
$216$	0	0
$217$	4.72652e8	0.213159
$218$	0	0
$219$	8.84630e8i	0.384579i
$220$	0	0
$221$	−6.98127e8	−0.292661
$222$	0	0
$223$	− 1.66879e9i	− 0.674813i	−0.941359	−	0.337406i	$-0.890450\pi$
$223$	− 1.66879e9i	− 0.674813i	0.941359	−	0.337406i	$-0.109550\pi$
$224$	0	0
$225$	8.36582e8	0.326421
$226$	0	0
$227$	− 3.34498e9i	− 1.25977i	−0.776690	−	0.629883i	$-0.783103\pi$
$227$	− 3.34498e9i	− 1.25977i	0.776690	−	0.629883i	$-0.216897\pi$
$228$	0	0
$229$	−1.42427e9	−0.517906	−0.258953	−	0.965890i	$-0.583378\pi$
$229$	−1.42427e9	−0.517906	−0.258953	+	0.965890i	$0.583378\pi$
$230$	0	0
$231$	2.23017e8i	0.0783231i
$232$	0	0
$233$	3.82745e9	1.29863	0.649316	−	0.760519i	$-0.275055\pi$
$233$	3.82745e9	1.29863	0.649316	+	0.760519i	$0.275055\pi$
$234$	0	0
$235$	− 7.17306e7i	− 0.0235198i
$236$	0	0
$237$	−2.53278e9	−0.802794
$238$	0	0
$239$	− 3.55074e9i	− 1.08825i	−0.839006	−	0.544123i	$-0.816863\pi$
$239$	− 3.55074e9i	− 1.08825i	0.839006	−	0.544123i	$-0.183137\pi$
$240$	0	0
$241$	2.46612e9	0.731048	0.365524	−	0.930802i	$-0.380890\pi$
$241$	2.46612e9	0.731048	0.365524	+	0.930802i	$0.380890\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	4.42416e8	0.122791
$246$	0	0
$247$	− 1.87212e9i	− 0.502975i
$248$	0	0
$249$	−3.04952e9	−0.793294
$250$	0	0
$251$	2.51003e9i	0.632389i	0.948694	+	0.316195i	$0.102405\pi$
$251$	2.51003e9i	0.632389i	−0.948694	+	0.316195i	$0.897595\pi$
$252$	0	0
$253$	−1.29727e9	−0.316627
$254$	0	0
$255$	− 1.79627e8i	− 0.0424825i
$256$	0	0
$257$	3.24232e9	0.743231	0.371615	−	0.928387i	$-0.378804\pi$
$257$	3.24232e9	0.743231	0.371615	+	0.928387i	$0.378804\pi$
$258$	0	0
$259$	− 2.08445e9i	− 0.463226i
$260$	0	0
$261$	−2.77865e9	−0.598786
$262$	0	0
$263$	− 8.13109e9i	− 1.69952i	−0.527172	−	0.849759i	$-0.676748\pi$
$263$	− 8.13109e9i	− 1.69952i	0.527172	−	0.849759i	$-0.323252\pi$
$264$	0	0
$265$	−9.55525e7	−0.0193757
$266$	0	0
$267$	− 4.20015e9i	− 0.826456i
$268$	0	0
$269$	7.44829e9	1.42248	0.711242	−	0.702947i	$-0.248133\pi$
$269$	7.44829e9	1.42248	0.711242	+	0.702947i	$0.248133\pi$
$270$	0	0
$271$	7.41657e7i	0.0137507i	0.999976	+	0.00687537i	$0.00218851\pi$
$271$	7.41657e7i	0.0137507i	−0.999976	+	0.00687537i	$0.997811\pi$
$272$	0	0
$273$	7.04899e8	0.126904
$274$	0	0
$275$	− 1.97971e9i	− 0.346155i
$276$	0	0
$277$	−2.00834e9	−0.341128	−0.170564	−	0.985347i	$-0.554559\pi$
$277$	−2.00834e9	−0.341128	−0.170564	+	0.985347i	$0.554559\pi$
$278$	0	0
$279$	− 1.12181e9i	− 0.185140i
$280$	0	0
$281$	−1.37037e9	−0.219793	−0.109896	−	0.993943i	$-0.535052\pi$
$281$	−1.37037e9	−0.219793	−0.109896	+	0.993943i	$0.535052\pi$
$282$	0	0
$283$	− 6.48210e9i	− 1.01058i	−0.862950	−	0.505289i	$-0.831386\pi$
$283$	− 6.48210e9i	− 1.01058i	0.862950	−	0.505289i	$-0.168614\pi$
$284$	0	0
$285$	4.81694e8	0.0730116
$286$	0	0
$287$	8.04145e8i	0.118524i
$288$	0	0
$289$	−5.15435e9	−0.738894
$290$	0	0
$291$	− 3.54380e9i	− 0.494193i
$292$	0	0
$293$	1.10320e10	1.49687	0.748436	−	0.663207i	$-0.230805\pi$
$293$	1.10320e10	1.49687	0.748436	+	0.663207i	$0.230805\pi$
$294$	0	0
$295$	2.10350e9i	0.277751i
$296$	0	0
$297$	5.29315e8	0.0680281
$298$	0	0
$299$	4.10034e9i	0.513020i
$300$	0	0
$301$	−1.53153e9	−0.186577
$302$	0	0
$303$	7.90223e9i	0.937517i
$304$	0	0
$305$	−1.37709e9	−0.159134
$306$	0	0
$307$	− 9.80026e9i	− 1.10328i	−0.834084	−	0.551638i	$-0.814003\pi$
$307$	− 9.80026e9i	− 1.10328i	0.834084	−	0.551638i	$-0.185997\pi$
$308$	0	0
$309$	−7.43849e9	−0.815926
$310$	0	0
$311$	− 6.60957e9i	− 0.706531i	−0.935523	−	0.353266i	$-0.885071\pi$
$311$	− 6.60957e9i	− 0.706531i	0.935523	−	0.353266i	$-0.114929\pi$
$312$	0	0
$313$	−8.98046e9	−0.935667	−0.467834	−	0.883817i	$-0.654965\pi$
$313$	−8.98046e9	−0.935667	−0.467834	+	0.883817i	$0.654965\pi$
$314$	0	0
$315$	1.81369e8i	0.0184213i
$316$	0	0
$317$	1.29468e10	1.28211	0.641057	−	0.767493i	$-0.278496\pi$
$317$	1.29468e10	1.28211	0.641057	+	0.767493i	$0.278496\pi$
$318$	0	0
$319$	6.57546e9i	0.634985i
$320$	0	0
$321$	−8.22975e9	−0.775115
$322$	0	0
$323$	4.88437e9i	0.448744i
$324$	0	0
$325$	−6.25734e9	−0.560863
$326$	0	0
$327$	6.45267e8i	0.0564350i
$328$	0	0
$329$	−7.34403e8	−0.0626831
$330$	0	0
$331$	8.27574e9i	0.689438i	0.938706	+	0.344719i	$0.112026\pi$
$331$	8.27574e9i	0.689438i	−0.938706	+	0.344719i	$0.887974\pi$
$332$	0	0
$333$	−4.94730e9	−0.402339
$334$	0	0
$335$	8.26220e8i	0.0656019i
$336$	0	0
$337$	−1.22715e9	−0.0951436	−0.0475718	−	0.998868i	$-0.515148\pi$
$337$	−1.22715e9	−0.0951436	−0.0475718	+	0.998868i	$0.515148\pi$
$338$	0	0
$339$	− 4.32650e9i	− 0.327596i
$340$	0	0
$341$	−2.65467e9	−0.196333
$342$	0	0
$343$	− 9.84159e9i	− 0.711031i
$344$	0	0
$345$	−1.05501e9	−0.0744697
$346$	0	0
$347$	1.34897e10i	0.930429i	0.885198	+	0.465214i	$0.154023\pi$
$347$	1.34897e10i	0.930429i	−0.885198	+	0.465214i	$0.845977\pi$
$348$	0	0
$349$	−9.63945e9	−0.649756	−0.324878	−	0.945756i	$-0.605323\pi$
$349$	−9.63945e9	−0.649756	−0.324878	+	0.945756i	$0.605323\pi$
$350$	0	0
$351$	− 1.67303e9i	− 0.110224i
$352$	0	0
$353$	2.39489e10	1.54237	0.771183	−	0.636614i	$-0.219666\pi$
$353$	2.39489e10	1.54237	0.771183	+	0.636614i	$0.219666\pi$
$354$	0	0
$355$	− 2.05479e9i	− 0.129376i
$356$	0	0
$357$	−1.83908e9	−0.113221
$358$	0	0
$359$	1.22556e9i	0.0737831i	0.999319	+	0.0368915i	$0.0117456\pi$
$359$	1.22556e9i	0.0737831i	−0.999319	+	0.0368915i	$0.988254\pi$
$360$	0	0
$361$	3.88545e9	0.228777
$362$	0	0
$363$	8.77199e9i	0.505210i
$364$	0	0
$365$	1.70247e9	0.0959198
$366$	0	0
$367$	2.19414e10i	1.20948i	0.796422	+	0.604742i	$0.206724\pi$
$367$	2.19414e10i	1.20948i	−0.796422	+	0.604742i	$0.793276\pi$
$368$	0	0
$369$	1.90858e9	0.102945
$370$	0	0
$371$	9.78299e8i	0.0516388i
$372$	0	0
$373$	−2.32459e10	−1.20091	−0.600457	−	0.799657i	$-0.705014\pi$
$373$	−2.32459e10	−1.20091	−0.600457	+	0.799657i	$0.705014\pi$
$374$	0	0
$375$	− 3.25410e9i	− 0.164553i
$376$	0	0
$377$	2.07833e10	1.02884
$378$	0	0
$379$	3.77864e10i	1.83138i	0.401887	+	0.915689i	$0.368354\pi$
$379$	3.77864e10i	1.83138i	−0.401887	+	0.915689i	$0.631646\pi$
$380$	0	0
$381$	−1.03700e10	−0.492127
$382$	0	0
$383$	3.04157e10i	1.41352i	0.707452	+	0.706761i	$0.249845\pi$
$383$	3.04157e10i	1.41352i	−0.707452	+	0.706761i	$0.750155\pi$
$384$	0	0
$385$	4.29196e8	0.0195350
$386$	0	0
$387$	3.63498e9i	0.162053i
$388$	0	0
$389$	1.53539e10	0.670532	0.335266	−	0.942123i	$-0.391174\pi$
$389$	1.53539e10	0.670532	0.335266	+	0.942123i	$0.391174\pi$
$390$	0	0
$391$	− 1.06978e10i	− 0.457706i
$392$	0	0
$393$	−1.68665e10	−0.707057
$394$	0	0
$395$	4.87434e9i	0.200229i
$396$	0	0
$397$	2.10954e10	0.849231	0.424615	−	0.905374i	$-0.360409\pi$
$397$	2.10954e10	0.849231	0.424615	+	0.905374i	$0.360409\pi$
$398$	0	0
$399$	− 4.93175e9i	− 0.194585i
$400$	0	0
$401$	3.89388e10	1.50593	0.752966	−	0.658059i	$-0.228622\pi$
$401$	3.89388e10	1.50593	0.752966	+	0.658059i	$0.228622\pi$
$402$	0	0
$403$	8.39073e9i	0.318112i
$404$	0	0
$405$	4.30467e8	0.0160000
$406$	0	0
$407$	1.17074e10i	0.426661i
$408$	0	0
$409$	−1.41946e10	−0.507258	−0.253629	−	0.967302i	$-0.581624\pi$
$409$	−1.41946e10	−0.507258	−0.253629	+	0.967302i	$0.581624\pi$
$410$	0	0
$411$	2.06107e10i	0.722312i
$412$	0	0
$413$	2.15364e10	0.740241
$414$	0	0
$415$	5.86881e9i	0.197860i
$416$	0	0
$417$	−2.54517e10	−0.841729
$418$	0	0
$419$	− 3.39978e10i	− 1.10305i	−0.834159	−	0.551524i	$-0.814047\pi$
$419$	− 3.39978e10i	− 1.10305i	0.834159	−	0.551524i	$-0.185953\pi$
$420$	0	0
$421$	1.11314e10	0.354342	0.177171	−	0.984180i	$-0.443305\pi$
$421$	1.11314e10	0.354342	0.177171	+	0.984180i	$0.443305\pi$
$422$	0	0
$423$	1.74305e9i	0.0544439i
$424$	0	0
$425$	1.63254e10	0.500389
$426$	0	0
$427$	1.40991e10i	0.424112i
$428$	0	0
$429$	−3.95909e9	−0.116887
$430$	0	0
$431$	4.91968e10i	1.42570i	0.701318	+	0.712849i	$0.252595\pi$
$431$	4.91968e10i	1.42570i	−0.701318	+	0.712849i	$0.747405\pi$
$432$	0	0
$433$	−4.76011e10	−1.35414	−0.677072	−	0.735916i	$-0.736752\pi$
$433$	−4.76011e10	−1.35414	−0.677072	+	0.735916i	$0.736752\pi$
$434$	0	0
$435$	5.34751e9i	0.149346i
$436$	0	0
$437$	2.86876e10	0.786625
$438$	0	0
$439$	1.30610e10i	0.351657i	0.984421	+	0.175829i	$0.0562605\pi$
$439$	1.30610e10i	0.351657i	−0.984421	+	0.175829i	$0.943740\pi$
$440$	0	0
$441$	−1.07507e10	−0.284238
$442$	0	0
$443$	− 1.48073e10i	− 0.384468i	−0.981349	−	0.192234i	$-0.938427\pi$
$443$	− 1.48073e10i	− 0.384468i	0.981349	−	0.192234i	$-0.0615732\pi$
$444$	0	0
$445$	−8.08319e9	−0.206131
$446$	0	0
$447$	− 2.06116e9i	− 0.0516276i
$448$	0	0
$449$	2.97486e9	0.0731949	0.0365975	−	0.999330i	$-0.488348\pi$
$449$	2.97486e9	0.0731949	0.0365975	+	0.999330i	$0.488348\pi$
$450$	0	0
$451$	− 4.51651e9i	− 0.109168i
$452$	0	0
$453$	5.23451e9	0.124304
$454$	0	0
$455$	− 1.35658e9i	− 0.0316519i
$456$	0	0
$457$	−3.07215e10	−0.704333	−0.352166	−	0.935937i	$-0.614555\pi$
$457$	−3.07215e10	−0.704333	−0.352166	+	0.935937i	$0.614555\pi$
$458$	0	0
$459$	4.36493e9i	0.0983392i
$460$	0	0
$461$	−8.09097e9	−0.179142	−0.0895709	−	0.995980i	$-0.528550\pi$
$461$	−8.09097e9	−0.179142	−0.0895709	+	0.995980i	$0.528550\pi$
$462$	0	0
$463$	4.56191e10i	0.992711i	0.868119	+	0.496356i	$0.165329\pi$
$463$	4.56191e10i	0.992711i	−0.868119	+	0.496356i	$0.834671\pi$
$464$	0	0
$465$	−2.15892e9	−0.0461769
$466$	0	0
$467$	− 4.09800e10i	− 0.861598i	−0.902448	−	0.430799i	$-0.858232\pi$
$467$	− 4.09800e10i	− 0.861598i	0.902448	−	0.430799i	$-0.141768\pi$
$468$	0	0
$469$	8.45913e9	0.174837
$470$	0	0
$471$	− 1.35391e10i	− 0.275109i
$472$	0	0
$473$	8.60189e9	0.171850
$474$	0	0
$475$	4.37788e10i	0.859983i
$476$	0	0
$477$	2.32192e9	0.0448513
$478$	0	0
$479$	4.58901e10i	0.871720i	0.900014	+	0.435860i	$0.143556\pi$
$479$	4.58901e10i	0.871720i	−0.900014	+	0.435860i	$0.856444\pi$
$480$	0	0
$481$	3.70041e10	0.691305
$482$	0	0
$483$	1.08015e10i	0.198471i
$484$	0	0
$485$	−6.82004e9	−0.123259
$486$	0	0
$487$	− 1.01277e11i	− 1.80051i	−0.435360	−	0.900256i	$-0.643379\pi$
$487$	− 1.01277e11i	− 1.80051i	0.435360	−	0.900256i	$-0.356621\pi$
$488$	0	0
$489$	−1.43540e10	−0.251036
$490$	0	0
$491$	8.24892e10i	1.41929i	0.704559	+	0.709645i	$0.251145\pi$
$491$	8.24892e10i	1.41929i	−0.704559	+	0.709645i	$0.748855\pi$
$492$	0	0
$493$	−5.42237e10	−0.917913
$494$	0	0
$495$	− 1.01867e9i	− 0.0169673i
$496$	0	0
$497$	−2.10376e10	−0.344803
$498$	0	0
$499$	− 9.46533e10i	− 1.52663i	−0.646026	−	0.763315i	$-0.723570\pi$
$499$	− 9.46533e10i	− 1.52663i	0.646026	−	0.763315i	$-0.276430\pi$
$500$	0	0
$501$	−5.59434e10	−0.887970
$502$	0	0
$503$	− 9.80310e10i	− 1.53141i	−0.643192	−	0.765705i	$-0.722390\pi$
$503$	− 9.80310e10i	− 1.53141i	0.643192	−	0.765705i	$-0.277610\pi$
$504$	0	0
$505$	1.52078e10	0.233831
$506$	0	0
$507$	− 2.56343e10i	− 0.387962i
$508$	0	0
$509$	−1.42476e10	−0.212261	−0.106131	−	0.994352i	$-0.533846\pi$
$509$	−1.42476e10	−0.212261	−0.106131	+	0.994352i	$0.533846\pi$
$510$	0	0
$511$	− 1.74305e10i	− 0.255638i
$512$	0	0
$513$	−1.17052e10	−0.169008
$514$	0	0
$515$	1.43154e10i	0.203504i
$516$	0	0
$517$	4.12480e9	0.0577352
$518$	0	0
$519$	− 3.86324e9i	− 0.0532454i
$520$	0	0
$521$	8.85973e10	1.20246	0.601229	−	0.799077i	$-0.294678\pi$
$521$	8.85973e10	1.20246	0.601229	+	0.799077i	$0.294678\pi$
$522$	0	0
$523$	3.72959e10i	0.498488i	0.968441	+	0.249244i	$0.0801821\pi$
$523$	3.72959e10i	0.498488i	−0.968441	+	0.249244i	$0.919818\pi$
$524$	0	0
$525$	−1.64838e10	−0.216980
$526$	0	0
$527$	− 2.18914e10i	− 0.283812i
$528$	0	0
$529$	1.54794e10	0.197665
$530$	0	0
$531$	− 5.11152e10i	− 0.642942i
$532$	0	0
$533$	−1.42755e10	−0.176882
$534$	0	0
$535$	1.58382e10i	0.193326i
$536$	0	0
$537$	5.31346e10	0.638969
$538$	0	0
$539$	2.54407e10i	0.301421i
$540$	0	0
$541$	−8.90331e10	−1.03935	−0.519676	−	0.854364i	$-0.673947\pi$
$541$	−8.90331e10	−1.03935	−0.519676	+	0.854364i	$0.673947\pi$
$542$	0	0
$543$	3.83487e10i	0.441115i
$544$	0	0
$545$	1.24182e9	0.0140758
$546$	0	0
$547$	− 1.11679e11i	− 1.24745i	−0.781645	−	0.623723i	$-0.785619\pi$
$547$	− 1.11679e11i	− 1.24745i	0.781645	−	0.623723i	$-0.214381\pi$
$548$	0	0
$549$	3.34633e10	0.368366
$550$	0	0
$551$	− 1.45408e11i	− 1.57755i
$552$	0	0
$553$	4.99051e10	0.533636
$554$	0	0
$555$	9.52109e9i	0.100349i
$556$	0	0
$557$	1.51601e11	1.57501	0.787503	−	0.616310i	$-0.211373\pi$
$557$	1.51601e11	1.57501	0.787503	+	0.616310i	$0.211373\pi$
$558$	0	0
$559$	− 2.71884e10i	− 0.278443i
$560$	0	0
$561$	1.03293e10	0.104284
$562$	0	0
$563$	− 9.17147e10i	− 0.912863i	−0.889759	−	0.456431i	$-0.849127\pi$
$563$	− 9.17147e10i	− 0.912863i	0.889759	−	0.456431i	$-0.150873\pi$
$564$	0	0
$565$	−8.32636e9	−0.0817074
$566$	0	0
$567$	− 4.40727e9i	− 0.0426420i
$568$	0	0
$569$	−7.26500e10	−0.693085	−0.346543	−	0.938034i	$-0.612644\pi$
$569$	−7.26500e10	−0.693085	−0.346543	+	0.938034i	$0.612644\pi$
$570$	0	0
$571$	− 6.84484e10i	− 0.643901i	−0.946757	−	0.321950i	$-0.895662\pi$
$571$	− 6.84484e10i	− 0.643901i	0.946757	−	0.321950i	$-0.104338\pi$
$572$	0	0
$573$	−5.84391e10	−0.542107
$574$	0	0
$575$	− 9.58846e10i	− 0.877158i
$576$	0	0
$577$	−1.62814e11	−1.46889	−0.734445	−	0.678668i	$-0.762557\pi$
$577$	−1.62814e11	−1.46889	−0.734445	+	0.678668i	$0.762557\pi$
$578$	0	0
$579$	− 9.40896e10i	− 0.837196i
$580$	0	0
$581$	6.00869e10	0.527321
$582$	0	0
$583$	− 5.49466e9i	− 0.0475627i
$584$	0	0
$585$	−3.21975e9	−0.0274915
$586$	0	0
$587$	3.09698e10i	0.260847i	0.991458	+	0.130423i	$0.0416337\pi$
$587$	3.09698e10i	0.260847i	−0.991458	+	0.130423i	$0.958366\pi$
$588$	0	0
$589$	5.87048e10	0.487767
$590$	0	0
$591$	2.95819e10i	0.242480i
$592$	0	0
$593$	−1.25721e11	−1.01669	−0.508347	−	0.861152i	$-0.669743\pi$
$593$	−1.25721e11	−1.01669	−0.508347	+	0.861152i	$0.669743\pi$
$594$	0	0
$595$	3.53931e9i	0.0282391i
$596$	0	0
$597$	6.81437e10	0.536449
$598$	0	0
$599$	− 8.69133e10i	− 0.675117i	−0.941305	−	0.337559i	$-0.890399\pi$
$599$	− 8.69133e10i	− 0.675117i	0.941305	−	0.337559i	$-0.109601\pi$
$600$	0	0
$601$	1.19412e11	0.915269	0.457634	−	0.889140i	$-0.348697\pi$
$601$	1.19412e11	0.915269	0.457634	+	0.889140i	$0.348697\pi$
$602$	0	0
$603$	− 2.00771e10i	− 0.151856i
$604$	0	0
$605$	1.68817e10	0.126007
$606$	0	0
$607$	2.21728e11i	1.63330i	0.577136	+	0.816648i	$0.304170\pi$
$607$	2.21728e11i	1.63330i	−0.577136	+	0.816648i	$0.695830\pi$
$608$	0	0
$609$	5.47497e10	0.398027
$610$	0	0
$611$	− 1.30374e10i	− 0.0935464i
$612$	0	0
$613$	2.09579e11	1.48425	0.742123	−	0.670263i	$-0.233819\pi$
$613$	2.09579e11	1.48425	0.742123	+	0.670263i	$0.233819\pi$
$614$	0	0
$615$	− 3.67307e9i	− 0.0256761i
$616$	0	0
$617$	−1.95007e11	−1.34558	−0.672788	−	0.739835i	$-0.734904\pi$
$617$	−1.95007e11	−1.34558	−0.672788	+	0.739835i	$0.734904\pi$
$618$	0	0
$619$	− 1.27224e11i	− 0.866574i	−0.901256	−	0.433287i	$-0.857354\pi$
$619$	− 1.27224e11i	− 0.866574i	0.901256	−	0.433287i	$-0.142646\pi$
$620$	0	0
$621$	2.56367e10	0.172384
$622$	0	0
$623$	8.27585e10i	0.549364i
$624$	0	0
$625$	1.43161e11	0.938222
$626$	0	0
$627$	2.76994e10i	0.179226i
$628$	0	0
$629$	−9.65437e10	−0.616767
$630$	0	0
$631$	− 2.89620e11i	− 1.82689i	−0.406966	−	0.913443i	$-0.633413\pi$
$631$	− 2.89620e11i	− 1.82689i	0.406966	−	0.913443i	$-0.366587\pi$
$632$	0	0
$633$	1.67140e10	0.104103
$634$	0	0
$635$	1.99570e10i	0.122744i
$636$	0	0
$637$	8.04115e10	0.488383
$638$	0	0
$639$	4.99314e10i	0.299481i
$640$	0	0
$641$	1.34888e11	0.798991	0.399496	−	0.916735i	$-0.369185\pi$
$641$	1.34888e11	0.798991	0.399496	+	0.916735i	$0.369185\pi$
$642$	0	0
$643$	− 1.40627e11i	− 0.822670i	−0.911484	−	0.411335i	$-0.865063\pi$
$643$	− 1.40627e11i	− 0.822670i	0.911484	−	0.411335i	$-0.134937\pi$
$644$	0	0
$645$	6.99551e9	0.0404185
$646$	0	0
$647$	9.22875e10i	0.526655i	0.964707	+	0.263327i	$0.0848199\pi$
$647$	9.22875e10i	0.526655i	−0.964707	+	0.263327i	$0.915180\pi$
$648$	0	0
$649$	−1.20960e11	−0.681810
$650$	0	0
$651$	2.21038e10i	0.123067i
$652$	0	0
$653$	−1.79607e11	−0.987805	−0.493902	−	0.869517i	$-0.664430\pi$
$653$	−1.79607e11	−0.987805	−0.493902	+	0.869517i	$0.664430\pi$
$654$	0	0
$655$	3.24596e10i	0.176351i
$656$	0	0
$657$	−4.13701e10	−0.222037
$658$	0	0
$659$	− 5.95800e10i	− 0.315907i	−0.987447	−	0.157953i	$-0.949510\pi$
$659$	− 5.95800e10i	− 0.315907i	0.987447	−	0.157953i	$-0.0504896\pi$
$660$	0	0
$661$	−3.13906e11	−1.64435	−0.822174	−	0.569236i	$-0.807239\pi$
$661$	−3.13906e11	−1.64435	−0.822174	+	0.569236i	$0.807239\pi$
$662$	0	0
$663$	− 3.26482e10i	− 0.168968i
$664$	0	0
$665$	−9.49116e9	−0.0485325
$666$	0	0
$667$	3.18474e11i	1.60905i
$668$	0	0
$669$	7.80418e10	0.389603
$670$	0	0
$671$	− 7.91884e10i	− 0.390635i
$672$	0	0
$673$	5.89957e10	0.287581	0.143791	−	0.989608i	$-0.454071\pi$
$673$	5.89957e10	0.287581	0.143791	+	0.989608i	$0.454071\pi$
$674$	0	0
$675$	3.91231e10i	0.188459i
$676$	0	0
$677$	−8.71335e10	−0.414792	−0.207396	−	0.978257i	$-0.566499\pi$
$677$	−8.71335e10	−0.414792	−0.207396	+	0.978257i	$0.566499\pi$
$678$	0	0
$679$	6.98259e10i	0.328502i
$680$	0	0
$681$	1.56429e11	0.727326
$682$	0	0
$683$	4.38375e10i	0.201448i	0.994914	+	0.100724i	$0.0321159\pi$
$683$	4.38375e10i	0.201448i	−0.994914	+	0.100724i	$0.967884\pi$
$684$	0	0
$685$	3.96653e10	0.180156
$686$	0	0
$687$	− 6.66066e10i	− 0.299013i
$688$	0	0
$689$	−1.73672e10	−0.0770642
$690$	0	0
$691$	2.55445e11i	1.12043i	0.828347	+	0.560216i	$0.189282\pi$
$691$	2.55445e11i	1.12043i	−0.828347	+	0.560216i	$0.810718\pi$
$692$	0	0
$693$	−1.04295e10	−0.0452199
$694$	0	0
$695$	4.89818e10i	0.209940i
$696$	0	0
$697$	3.72448e10	0.157810
$698$	0	0
$699$	1.78992e11i	0.749766i
$700$	0	0
$701$	−4.08255e11	−1.69067	−0.845337	−	0.534233i	$-0.820600\pi$
$701$	−4.08255e11	−1.69067	−0.845337	+	0.534233i	$0.820600\pi$
$702$	0	0
$703$	− 2.58895e11i	− 1.05999i
$704$	0	0
$705$	3.35451e9	0.0135791
$706$	0	0
$707$	− 1.55703e11i	− 0.623189i
$708$	0	0
$709$	9.56878e10	0.378679	0.189340	−	0.981912i	$-0.439365\pi$
$709$	9.56878e10	0.378679	0.189340	+	0.981912i	$0.439365\pi$
$710$	0	0
$711$	− 1.18446e11i	− 0.463493i
$712$	0	0
$713$	−1.28576e11	−0.497508
$714$	0	0
$715$	7.61928e9i	0.0291534i
$716$	0	0
$717$	1.66051e11	0.628299
$718$	0	0
$719$	3.37132e11i	1.26149i	0.775990	+	0.630746i	$0.217251\pi$
$719$	3.37132e11i	1.26149i	−0.775990	+	0.630746i	$0.782749\pi$
$720$	0	0
$721$	1.46566e11	0.542365
$722$	0	0
$723$	1.15329e11i	0.422071i
$724$	0	0
$725$	−4.86009e11	−1.75911
$726$	0	0
$727$	2.71347e11i	0.971377i	0.874132	+	0.485688i	$0.161431\pi$
$727$	2.71347e11i	0.971377i	−0.874132	+	0.485688i	$0.838569\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	7.09344e10i	0.248420i
$732$	0	0
$733$	−3.31640e9	−0.0114882	−0.00574409	−	0.999984i	$-0.501828\pi$
$733$	−3.31640e9	−0.0114882	−0.00574409	+	0.999984i	$0.501828\pi$
$734$	0	0
$735$	2.06897e10i	0.0708933i
$736$	0	0
$737$	−4.75110e10	−0.161037
$738$	0	0
$739$	3.59220e11i	1.20443i	0.798333	+	0.602216i	$0.205716\pi$
$739$	3.59220e11i	1.20443i	−0.798333	+	0.602216i	$0.794284\pi$
$740$	0	0
$741$	8.75506e10	0.290393
$742$	0	0
$743$	4.22337e11i	1.38581i	0.721028	+	0.692906i	$0.243670\pi$
$743$	4.22337e11i	1.38581i	−0.721028	+	0.692906i	$0.756330\pi$
$744$	0	0
$745$	−3.96671e9	−0.0128767
$746$	0	0
$747$	− 1.42612e11i	− 0.458009i
$748$	0	0
$749$	1.62157e11	0.515237
$750$	0	0
$751$	2.60036e11i	0.817473i	0.912653	+	0.408736i	$0.134030\pi$
$751$	2.60036e11i	0.817473i	−0.912653	+	0.408736i	$0.865970\pi$
$752$	0	0
$753$	−1.17383e11	−0.365110
$754$	0	0
$755$	− 1.00738e10i	− 0.0310032i
$756$	0	0
$757$	6.09599e11	1.85635	0.928177	−	0.372140i	$-0.121376\pi$
$757$	6.09599e11	1.85635	0.928177	+	0.372140i	$0.121376\pi$
$758$	0	0
$759$	− 6.06673e10i	− 0.182805i
$760$	0	0
$761$	−8.45871e10	−0.252212	−0.126106	−	0.992017i	$-0.540248\pi$
$761$	−8.45871e10	−0.252212	−0.126106	+	0.992017i	$0.540248\pi$
$762$	0	0
$763$	− 1.27142e10i	− 0.0375137i
$764$	0	0
$765$	8.40031e9	0.0245273
$766$	0	0
$767$	3.82324e11i	1.10471i
$768$	0	0
$769$	6.82479e10	0.195157	0.0975784	−	0.995228i	$-0.468890\pi$
$769$	6.82479e10	0.195157	0.0975784	+	0.995228i	$0.468890\pi$
$770$	0	0
$771$	1.51628e11i	0.429104i
$772$	0	0
$773$	3.06208e10	0.0857627	0.0428813	−	0.999080i	$-0.486346\pi$
$773$	3.06208e10	0.0857627	0.0428813	+	0.999080i	$0.486346\pi$
$774$	0	0
$775$	− 1.96214e11i	− 0.543904i
$776$	0	0
$777$	9.74802e10	0.267444
$778$	0	0
$779$	9.98772e10i	0.271217i
$780$	0	0
$781$	1.18159e11	0.317586
$782$	0	0
$783$	− 1.29945e11i	− 0.345709i
$784$	0	0
$785$	−2.60559e10	−0.0686164
$786$	0	0
$787$	− 4.62193e11i	− 1.20483i	−0.798185	−	0.602413i	$-0.794206\pi$
$787$	− 4.62193e11i	− 1.20483i	0.798185	−	0.602413i	$-0.205794\pi$
$788$	0	0
$789$	3.80253e11	0.981217
$790$	0	0
$791$	8.52481e10i	0.217760i
$792$	0	0
$793$	−2.50294e11	−0.632933
$794$	0	0
$795$	− 4.46855e9i	− 0.0111866i
$796$	0	0
$797$	3.34741e11	0.829614	0.414807	−	0.909909i	$-0.363849\pi$
$797$	3.34741e11	0.829614	0.414807	+	0.909909i	$0.363849\pi$
$798$	0	0
$799$	3.40146e10i	0.0834601i
$800$	0	0
$801$	1.96421e11	0.477154
$802$	0	0
$803$	9.78991e10i	0.235460i
$804$	0	0
$805$	2.07876e10	0.0495017
$806$	0	0
$807$	3.48322e11i	0.821271i
$808$	0	0
$809$	−5.80925e11	−1.35621	−0.678104	−	0.734966i	$-0.737198\pi$
$809$	−5.80925e11	−1.35621	−0.678104	+	0.734966i	$0.737198\pi$
$810$	0	0
$811$	6.93631e11i	1.60341i	0.597719	+	0.801706i	$0.296074\pi$
$811$	6.93631e11i	1.60341i	−0.597719	+	0.801706i	$0.703926\pi$
$812$	0	0
$813$	−3.46839e9	−0.00793899
$814$	0	0
$815$	2.76242e10i	0.0626123i
$816$	0	0
$817$	−1.90220e11	−0.426942
$818$	0	0
$819$	3.29649e10i	0.0732682i
$820$	0	0
$821$	−7.20568e11	−1.58600	−0.792999	−	0.609223i	$-0.791481\pi$
$821$	−7.20568e11	−1.58600	−0.792999	+	0.609223i	$0.791481\pi$
$822$	0	0
$823$	5.20292e11i	1.13409i	0.823686	+	0.567046i	$0.191914\pi$
$823$	5.20292e11i	1.13409i	−0.823686	+	0.567046i	$0.808086\pi$
$824$	0	0
$825$	9.25818e10	0.199853
$826$	0	0
$827$	− 2.76065e11i	− 0.590187i	−0.955468	−	0.295093i	$-0.904649\pi$
$827$	− 2.76065e11i	− 0.590187i	0.955468	−	0.295093i	$-0.0953508\pi$
$828$	0	0
$829$	5.58542e11	1.18260	0.591300	−	0.806452i	$-0.298615\pi$
$829$	5.58542e11	1.18260	0.591300	+	0.806452i	$0.298615\pi$
$830$	0	0
$831$	− 9.39206e10i	− 0.196950i
$832$	0	0
$833$	−2.09793e11	−0.435725
$834$	0	0
$835$	1.07663e11i	0.221473i
$836$	0	0
$837$	5.24617e10	0.106891
$838$	0	0
$839$	− 1.37279e11i	− 0.277049i	−0.990359	−	0.138525i	$-0.955764\pi$
$839$	− 1.37279e11i	− 0.277049i	0.990359	−	0.138525i	$-0.0442360\pi$
$840$	0	0
$841$	1.11400e12	2.22690
$842$	0	0
$843$	− 6.40859e10i	− 0.126897i
$844$	0	0
$845$	−4.93332e10	−0.0967637
$846$	0	0
$847$	− 1.72841e11i	− 0.335824i
$848$	0	0
$849$	3.03138e11	0.583457
$850$	0	0
$851$	5.67034e11i	1.08116i
$852$	0	0
$853$	−5.08270e11	−0.960060	−0.480030	−	0.877252i	$-0.659374\pi$
$853$	−5.08270e11	−0.960060	−0.480030	+	0.877252i	$0.659374\pi$
$854$	0	0
$855$	2.25266e10i	0.0421533i
$856$	0	0
$857$	−5.97042e11	−1.10683	−0.553416	−	0.832905i	$-0.686676\pi$
$857$	−5.97042e11	−1.10683	−0.553416	+	0.832905i	$0.686676\pi$
$858$	0	0
$859$	− 8.74635e11i	− 1.60640i	−0.595708	−	0.803201i	$-0.703128\pi$
$859$	− 8.74635e11i	− 1.60640i	0.595708	−	0.803201i	$-0.296872\pi$
$860$	0	0
$861$	−3.76061e10	−0.0684299
$862$	0	0
$863$	− 8.44217e11i	− 1.52199i	−0.648760	−	0.760993i	$-0.724712\pi$
$863$	− 8.44217e11i	− 1.52199i	0.648760	−	0.760993i	$-0.275288\pi$
$864$	0	0
$865$	−7.43481e9	−0.0132802
$866$	0	0
$867$	− 2.41045e11i	− 0.426601i
$868$	0	0
$869$	−2.80294e11	−0.491513
$870$	0	0
$871$	1.50170e11i	0.260922i
$872$	0	0
$873$	1.65727e11	0.285323
$874$	0	0
$875$	6.41178e10i	0.109382i
$876$	0	0
$877$	−6.03933e11	−1.02092	−0.510458	−	0.859903i	$-0.670524\pi$
$877$	−6.03933e11	−1.02092	−0.510458	+	0.859903i	$0.670524\pi$
$878$	0	0
$879$	5.15917e11i	0.864220i
$880$	0	0
$881$	7.65645e11	1.27094	0.635468	−	0.772127i	$-0.280807\pi$
$881$	7.65645e11	1.27094	0.635468	+	0.772127i	$0.280807\pi$
$882$	0	0
$883$	7.59490e11i	1.24934i	0.780891	+	0.624668i	$0.214766\pi$
$883$	7.59490e11i	1.24934i	−0.780891	+	0.624668i	$0.785234\pi$
$884$	0	0
$885$	−9.83712e10	−0.160360
$886$	0	0
$887$	− 2.76654e11i	− 0.446932i	−0.974712	−	0.223466i	$-0.928263\pi$
$887$	− 2.76654e11i	− 0.446932i	0.974712	−	0.223466i	$-0.0717372\pi$
$888$	0	0
$889$	2.04327e11	0.327128
$890$	0	0
$891$	2.47536e10i	0.0392761i
$892$	0	0
$893$	−9.12150e10	−0.143437
$894$	0	0
$895$	− 1.02258e11i	− 0.159369i
$896$	0	0
$897$	−1.91754e11	−0.296192
$898$	0	0
$899$	6.51710e11i	0.997736i
$900$	0	0
$901$	4.53110e10	0.0687550
$902$	0	0
$903$	− 7.16225e10i	− 0.107720i
$904$	0	0
$905$	7.38022e10	0.110021
$906$	0	0
$907$	− 1.00823e12i	− 1.48980i	−0.667174	−	0.744902i	$-0.732496\pi$
$907$	− 1.00823e12i	− 1.48980i	0.667174	−	0.744902i	$-0.267504\pi$
$908$	0	0
$909$	−3.69551e11	−0.541276
$910$	0	0
$911$	− 1.30302e12i	− 1.89181i	−0.324437	−	0.945907i	$-0.605175\pi$
$911$	− 1.30302e12i	− 1.89181i	0.324437	−	0.945907i	$-0.394825\pi$
$912$	0	0
$913$	−3.37480e11	−0.485697
$914$	0	0
$915$	− 6.44002e10i	− 0.0918761i
$916$	0	0
$917$	3.32332e11	0.469997
$918$	0	0
$919$	− 8.83053e11i	− 1.23801i	−0.785387	−	0.619006i	$-0.787536\pi$
$919$	− 8.83053e11i	− 1.23801i	0.785387	−	0.619006i	$-0.212464\pi$
$920$	0	0
$921$	4.58313e11	0.636977
$922$	0	0
$923$	− 3.73469e11i	− 0.514574i
$924$	0	0
$925$	−8.65326e11	−1.18199
$926$	0	0
$927$	− 3.47864e11i	− 0.471075i
$928$	0	0
$929$	−6.58431e11	−0.883990	−0.441995	−	0.897017i	$-0.645729\pi$
$929$	−6.58431e11	−0.883990	−0.441995	+	0.897017i	$0.645729\pi$
$930$	0	0
$931$	− 5.62590e11i	− 0.748848i
$932$	0	0
$933$	3.09099e11	0.407916
$934$	0	0
$935$	− 1.98787e10i	− 0.0260101i
$936$	0	0
$937$	−3.25948e11	−0.422854	−0.211427	−	0.977394i	$-0.567811\pi$
$937$	−3.25948e11	−0.422854	−0.211427	+	0.977394i	$0.567811\pi$
$938$	0	0
$939$	− 4.19975e11i	− 0.540208i
$940$	0	0
$941$	5.07812e11	0.647656	0.323828	−	0.946116i	$-0.395030\pi$
$941$	5.07812e11	0.647656	0.323828	+	0.946116i	$0.395030\pi$
$942$	0	0
$943$	− 2.18752e11i	− 0.276633i
$944$	0	0
$945$	−8.48180e9	−0.0106356
$946$	0	0
$947$	1.00994e11i	0.125573i	0.998027	+	0.0627867i	$0.0199988\pi$
$947$	1.00994e11i	0.125573i	−0.998027	+	0.0627867i	$0.980001\pi$
$948$	0	0
$949$	3.09434e11	0.381507
$950$	0	0
$951$	6.05464e11i	0.740229i
$952$	0	0
$953$	−4.54997e11	−0.551616	−0.275808	−	0.961213i	$-0.588945\pi$
$953$	−4.54997e11	−0.551616	−0.275808	+	0.961213i	$0.588945\pi$
$954$	0	0
$955$	1.12466e11i	0.135210i
$956$	0	0
$957$	−3.07504e11	−0.366609
$958$	0	0
$959$	− 4.06107e11i	− 0.480138i
$960$	0	0
$961$	5.89780e11	0.691507
$962$	0	0
$963$	− 3.84867e11i	− 0.447513i
$964$	0	0
$965$	−1.81076e11	−0.208810
$966$	0	0
$967$	1.62872e12i	1.86269i	0.364134	+	0.931346i	$0.381365\pi$
$967$	1.62872e12i	1.86269i	−0.364134	+	0.931346i	$0.618635\pi$
$968$	0	0
$969$	−2.28419e11	−0.259082
$970$	0	0
$971$	− 5.60458e11i	− 0.630472i	−0.949013	−	0.315236i	$-0.897916\pi$
$971$	− 5.60458e11i	− 0.630472i	0.949013	−	0.315236i	$-0.102084\pi$
$972$	0	0
$973$	5.01492e11	0.559516
$974$	0	0
$975$	− 2.92627e11i	− 0.323814i
$976$	0	0
$977$	−5.16303e11	−0.566665	−0.283332	−	0.959022i	$-0.591440\pi$
$977$	−5.16303e11	−0.566665	−0.283332	+	0.959022i	$0.591440\pi$
$978$	0	0
$979$	− 4.64816e11i	− 0.506000i
$980$	0	0
$981$	−3.01762e10	−0.0325828
$982$	0	0
$983$	6.31070e11i	0.675870i	0.941169	+	0.337935i	$0.109728\pi$
$983$	6.31070e11i	0.675870i	−0.941169	+	0.337935i	$0.890272\pi$
$984$	0	0
$985$	5.69304e10	0.0604782
$986$	0	0
$987$	− 3.43446e10i	− 0.0361901i
$988$	0	0
$989$	4.16622e11	0.435468
$990$	0	0
$991$	− 1.28282e12i	− 1.33006i	−0.746815	−	0.665032i	$-0.768418\pi$
$991$	− 1.28282e12i	− 1.33006i	0.746815	−	0.665032i	$-0.231582\pi$
$992$	0	0
$993$	−3.87018e11	−0.398047
$994$	0	0
$995$	− 1.31143e11i	− 0.133799i
$996$	0	0
$997$	1.44928e12	1.46680	0.733402	−	0.679795i	$-0.237931\pi$
$997$	1.44928e12	1.46680	0.733402	+	0.679795i	$0.237931\pi$
$998$	0	0
$999$	− 2.31363e11i	− 0.232290i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.9.g.b.127.2		2
4.3	odd	2	inner	192.9.g.b.127.1		2
8.3	odd	2		48.9.g.a.31.2	yes	2
8.5	even	2		48.9.g.a.31.1	✓	2
24.5	odd	2		144.9.g.f.127.1		2
24.11	even	2		144.9.g.f.127.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.9.g.a.31.1	✓	2	8.5	even	2
48.9.g.a.31.2	yes	2	8.3	odd	2
144.9.g.f.127.1		2	24.5	odd	2
144.9.g.f.127.2		2	24.11	even	2
192.9.g.b.127.1		2	4.3	odd	2	inner
192.9.g.b.127.2		2	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	192.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} +90.0000 q^{5} -921.451i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} +90.0000 q^{5} -921.451i q^{7} -2187.00 q^{9} +5175.37i q^{11} +16358.0 q^{13} +4208.88i q^{15} -42678.0 q^{17} -114447. i q^{19} +43092.0 q^{21} +250662. i q^{23} -382525. q^{25} -102276. i q^{27} +1.27053e6 q^{29} +512943. i q^{31} -242028. q^{33} -82930.6i q^{35} +2.26214e6 q^{37} +764988. i q^{39} -872694. q^{41} -1.66208e6i q^{43} -196830. q^{45} -797007. i q^{47} +4.91573e6 q^{49} -1.99585e6i q^{51} -1.06169e6 q^{53} +465783. i q^{55} +5.35216e6 q^{57} +2.33723e7i q^{59} -1.53010e7 q^{61} +2.01521e6i q^{63} +1.47222e6 q^{65} +9.18022e6i q^{67} -1.17223e7 q^{69} -2.28310e7i q^{71} +1.89164e7 q^{73} -1.78889e7i q^{75} +4.76885e6 q^{77} +5.41593e7i q^{79} +4.78297e6 q^{81} +6.52089e7i q^{83} -3.84102e6 q^{85} +5.94168e7i q^{87} -8.98132e7 q^{89} -1.50731e7i q^{91} -2.39880e7 q^{93} -1.03002e7i q^{95} -7.57782e7 q^{97} -1.13185e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 180 q^{5} - 4374 q^{9}+O(q^{10}) \) 2 * q + 180 * q^5 - 4374 * q^9	\( 2 q + 180 q^{5} - 4374 q^{9} + 32716 q^{13} - 85356 q^{17} + 86184 q^{21} - 765050 q^{25} + 2541060 q^{29} - 484056 q^{33} + 4524284 q^{37} - 1745388 q^{41} - 393660 q^{45} + 9831458 q^{49} - 2123388 q^{53} + 10704312 q^{57} - 30602020 q^{61} + 2944440 q^{65} - 23444640 q^{69} + 37832708 q^{73} + 9537696 q^{77} + 9565938 q^{81} - 7682040 q^{85} - 179626428 q^{89} - 47975976 q^{93} - 151556476 q^{97}+O(q^{100}) \) 2 * q + 180 * q^5 - 4374 * q^9 + 32716 * q^13 - 85356 * q^17 + 86184 * q^21 - 765050 * q^25 + 2541060 * q^29 - 484056 * q^33 + 4524284 * q^37 - 1745388 * q^41 - 393660 * q^45 + 9831458 * q^49 - 2123388 * q^53 + 10704312 * q^57 - 30602020 * q^61 + 2944440 * q^65 - 23444640 * q^69 + 37832708 * q^73 + 9537696 * q^77 + 9565938 * q^81 - 7682040 * q^85 - 179626428 * q^89 - 47975976 * q^93 - 151556476 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more