LMFDB - Embedded newform 192.9.g.d.127.7

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:	8.0.3468738816.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 18x^{5} + 77x^{4} + 8x^{2} + 88x + 484 $ x^8 - 2x^7 + 2x^6 + 18x^5 + 77x^4 + 8x^2 + 88x + 484
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{12} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.7
Root			$2.64070 - 2.64070i$ of defining polynomial
Character	$\chi$	$=$	192.127
Dual form			192.9.g.d.127.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} +149.264 q^{5} +3207.68i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} +149.264 q^{5} +3207.68i q^{7} -2187.00 q^{9} +3235.02i q^{11} -176.686 q^{13} +6980.37i q^{15} +80443.0 q^{17} +239555. i q^{19} -150008. q^{21} +85338.9i q^{23} -368345. q^{25} -102276. i q^{27} +310329. q^{29} -494114. i q^{31} -151287. q^{33} +478790. i q^{35} +855708. q^{37} -8262.78i q^{39} -1.41237e6 q^{41} +2.15193e6i q^{43} -326439. q^{45} -1.10493e6i q^{47} -4.52442e6 q^{49} +3.76194e6i q^{51} +1.47339e7 q^{53} +482870. i q^{55} -1.12029e7 q^{57} +1.62594e7i q^{59} -9.65310e6 q^{61} -7.01520e6i q^{63} -26372.7 q^{65} +2.25759e7i q^{67} -3.99091e6 q^{69} -3.04163e7i q^{71} -4.01565e7 q^{73} -1.72258e7i q^{75} -1.03769e7 q^{77} -4.96729e7i q^{79} +4.78297e6 q^{81} -4.44132e7i q^{83} +1.20072e7 q^{85} +1.45126e7i q^{87} -1.11247e8 q^{89} -566752. i q^{91} +2.31074e7 q^{93} +3.57569e7i q^{95} +3.96005e6 q^{97} -7.07498e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 560 q^{5} - 17496 q^{9}+O(q^{10}) $ 8 * q - 560 * q^5 - 17496 * q^9	$ 8 q - 560 q^{5} - 17496 q^{9} + 54256 q^{13} + 436176 q^{17} - 54432 q^{21} + 2456472 q^{25} - 195952 q^{29} - 1065312 q^{33} + 7023408 q^{37} - 12035120 q^{41} + 1224720 q^{45} - 9602040 q^{49} - 27342192 q^{53} + 2744928 q^{57} - 50803280 q^{61} + 34049888 q^{65} + 26687232 q^{69} + 59541648 q^{73} + 178489472 q^{77} + 38263752 q^{81} - 29428704 q^{85} - 170794992 q^{89} + 121951008 q^{93} + 272647184 q^{97}+O(q^{100}) $ 8 * q - 560 * q^5 - 17496 * q^9 + 54256 * q^13 + 436176 * q^17 - 54432 * q^21 + 2456472 * q^25 - 195952 * q^29 - 1065312 * q^33 + 7023408 * q^37 - 12035120 * q^41 + 1224720 * q^45 - 9602040 * q^49 - 27342192 * q^53 + 2744928 * q^57 - 50803280 * q^61 + 34049888 * q^65 + 26687232 * q^69 + 59541648 * q^73 + 178489472 * q^77 + 38263752 * q^81 - 29428704 * q^85 - 170794992 * q^89 + 121951008 * q^93 + 272647184 * 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
$5$	149.264	0.238822	0.119411	−	0.992845i	$-0.461899\pi$
$5$	149.264	0.238822	0.119411	+	0.992845i	$0.461899\pi$
$6$	0	0
$7$	3207.68i	1.33598i	0.744171	+	0.667989i	$0.232845\pi$
$7$	3207.68i	1.33598i	−0.744171	+	0.667989i	$0.767155\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	3235.02i	0.220956i	0.993879	+	0.110478i	$0.0352382\pi$
$11$	3235.02i	0.220956i	−0.993879	+	0.110478i	$0.964762\pi$
$12$	0	0
$13$	−176.686	−0.00618626	−0.00309313	−	0.999995i	$-0.500985\pi$
$13$	−176.686	−0.00618626	−0.00309313	+	0.999995i	$0.500985\pi$
$14$	0	0
$15$	6980.37i	0.137884i
$16$	0	0
$17$	80443.0	0.963146	0.481573	−	0.876406i	$-0.340066\pi$
$17$	80443.0	0.963146	0.481573	+	0.876406i	$0.340066\pi$
$18$	0	0
$19$	239555.i	1.83819i	0.394030	+	0.919097i	$0.371080\pi$
$19$	239555.i	1.83819i	−0.394030	+	0.919097i	$0.628920\pi$
$20$	0	0
$21$	−150008.	−0.771327
$22$	0	0
$23$	85338.9i	0.304955i	0.988307	+	0.152478i	$0.0487252\pi$
$23$	85338.9i	0.304955i	−0.988307	+	0.152478i	$0.951275\pi$
$24$	0	0
$25$	−368345.	−0.942964
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	310329.	0.438763	0.219382	−	0.975639i	$-0.429596\pi$
$29$	310329.	0.438763	0.219382	+	0.975639i	$0.429596\pi$
$30$	0	0
$31$	− 494114.i	− 0.535033i	−0.963553	−	0.267516i	$-0.913797\pi$
$31$	− 494114.i	− 0.535033i	0.963553	−	0.267516i	$-0.0862029\pi$
$32$	0	0
$33$	−151287.	−0.127569
$34$	0	0
$35$	478790.i	0.319060i
$36$	0	0
$37$	855708.	0.456582	0.228291	−	0.973593i	$-0.426686\pi$
$37$	855708.	0.456582	0.228291	+	0.973593i	$0.426686\pi$
$38$	0	0
$39$	− 8262.78i	− 0.00357164i
$40$	0	0
$41$	−1.41237e6	−0.499818	−0.249909	−	0.968269i	$-0.580401\pi$
$41$	−1.41237e6	−0.499818	−0.249909	+	0.968269i	$0.580401\pi$
$42$	0	0
$43$	2.15193e6i	0.629439i	0.949185	+	0.314720i	$0.101910\pi$
$43$	2.15193e6i	0.629439i	−0.949185	+	0.314720i	$0.898090\pi$
$44$	0	0
$45$	−326439.	−0.0796072
$46$	0	0
$47$	− 1.10493e6i	− 0.226434i	−0.993570	−	0.113217i	$-0.963884\pi$
$47$	− 1.10493e6i	− 0.226434i	0.993570	−	0.113217i	$-0.0361156\pi$
$48$	0	0
$49$	−4.52442e6	−0.784835
$50$	0	0
$51$	3.76194e6i	0.556073i
$52$	0	0
$53$	1.47339e7	1.86730	0.933651	−	0.358184i	$-0.116604\pi$
$53$	1.47339e7	1.86730	0.933651	+	0.358184i	$0.116604\pi$
$54$	0	0
$55$	482870.i	0.0527691i
$56$	0	0
$57$	−1.12029e7	−1.06128
$58$	0	0
$59$	1.62594e7i	1.34183i	0.741535	+	0.670914i	$0.234098\pi$
$59$	1.62594e7i	1.34183i	−0.741535	+	0.670914i	$0.765902\pi$
$60$	0	0
$61$	−9.65310e6	−0.697184	−0.348592	−	0.937275i	$-0.613340\pi$
$61$	−9.65310e6	−0.697184	−0.348592	+	0.937275i	$0.613340\pi$
$62$	0	0
$63$	− 7.01520e6i	− 0.445326i
$64$	0	0
$65$	−26372.7	−0.00147741
$66$	0	0
$67$	2.25759e7i	1.12033i	0.828381	+	0.560164i	$0.189262\pi$
$67$	2.25759e7i	1.12033i	−0.828381	+	0.560164i	$0.810738\pi$
$68$	0	0
$69$	−3.99091e6	−0.176066
$70$	0	0
$71$	− 3.04163e7i	− 1.19694i	−0.801144	−	0.598472i	$-0.795775\pi$
$71$	− 3.04163e7i	− 1.19694i	0.801144	−	0.598472i	$-0.204225\pi$
$72$	0	0
$73$	−4.01565e7	−1.41405	−0.707024	−	0.707190i	$-0.749963\pi$
$73$	−4.01565e7	−1.41405	−0.707024	+	0.707190i	$0.749963\pi$
$74$	0	0
$75$	− 1.72258e7i	− 0.544421i
$76$	0	0
$77$	−1.03769e7	−0.295192
$78$	0	0
$79$	− 4.96729e7i	− 1.27530i	−0.770328	−	0.637648i	$-0.779908\pi$
$79$	− 4.96729e7i	− 1.27530i	0.770328	−	0.637648i	$-0.220092\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 4.44132e7i	− 0.935836i	−0.883772	−	0.467918i	$-0.845004\pi$
$83$	− 4.44132e7i	− 0.935836i	0.883772	−	0.467918i	$-0.154996\pi$
$84$	0	0
$85$	1.20072e7	0.230020
$86$	0	0
$87$	1.45126e7i	0.253320i
$88$	0	0
$89$	−1.11247e8	−1.77307	−0.886537	−	0.462659i	$-0.846896\pi$
$89$	−1.11247e8	−1.77307	−0.886537	+	0.462659i	$0.846896\pi$
$90$	0	0
$91$	− 566752.i	− 0.00826470i
$92$	0	0
$93$	2.31074e7	0.308901
$94$	0	0
$95$	3.57569e7i	0.439001i
$96$	0	0
$97$	3.96005e6	0.0447316	0.0223658	−	0.999750i	$-0.492880\pi$
$97$	3.96005e6	0.0447316	0.0223658	+	0.999750i	$0.492880\pi$
$98$	0	0
$99$	− 7.07498e6i	− 0.0736520i
$100$	0	0
$101$	−1.74661e8	−1.67846	−0.839229	−	0.543778i	$-0.816993\pi$
$101$	−1.74661e8	−1.67846	−0.839229	+	0.543778i	$0.816993\pi$
$102$	0	0
$103$	1.11312e7i	0.0988997i	0.998777	+	0.0494498i	$0.0157468\pi$
$103$	1.11312e7i	0.0988997i	−0.998777	+	0.0494498i	$0.984253\pi$
$104$	0	0
$105$	−2.23908e7	−0.184210
$106$	0	0
$107$	2.58612e7i	0.197294i	0.995122	+	0.0986471i	$0.0314515\pi$
$107$	2.58612e7i	0.197294i	−0.995122	+	0.0986471i	$0.968548\pi$
$108$	0	0
$109$	−2.23824e8	−1.58562	−0.792812	−	0.609466i	$-0.791384\pi$
$109$	−2.23824e8	−1.58562	−0.792812	+	0.609466i	$0.791384\pi$
$110$	0	0
$111$	4.00175e7i	0.263608i
$112$	0	0
$113$	−1.13591e8	−0.696675	−0.348338	−	0.937369i	$-0.613254\pi$
$113$	−1.13591e8	−0.696675	−0.348338	+	0.937369i	$0.613254\pi$
$114$	0	0
$115$	1.27380e7i	0.0728299i
$116$	0	0
$117$	386412.	0.00206209
$118$	0	0
$119$	2.58035e8i	1.28674i
$120$	0	0
$121$	2.03894e8	0.951178
$122$	0	0
$123$	− 6.60499e7i	− 0.288570i
$124$	0	0
$125$	−1.13287e8	−0.464022
$126$	0	0
$127$	1.26585e8i	0.486594i	0.969952	+	0.243297i	$0.0782290\pi$
$127$	1.26585e8i	0.486594i	−0.969952	+	0.243297i	$0.921771\pi$
$128$	0	0
$129$	−1.00636e8	−0.363407
$130$	0	0
$131$	− 2.37678e8i	− 0.807055i	−0.914967	−	0.403528i	$-0.867784\pi$
$131$	− 2.37678e8i	− 0.807055i	0.914967	−	0.403528i	$-0.132216\pi$
$132$	0	0
$133$	−7.68417e8	−2.45579
$134$	0	0
$135$	− 1.52661e7i	− 0.0459613i
$136$	0	0
$137$	−2.32073e8	−0.658783	−0.329392	−	0.944193i	$-0.606844\pi$
$137$	−2.32073e8	−0.658783	−0.329392	+	0.944193i	$0.606844\pi$
$138$	0	0
$139$	1.04348e7i	0.0279529i	0.999902	+	0.0139764i	$0.00444898\pi$
$139$	1.04348e7i	0.0279529i	−0.999902	+	0.0139764i	$0.995551\pi$
$140$	0	0
$141$	5.16724e7	0.130732
$142$	0	0
$143$	− 571581.i	− 0.00136689i
$144$	0	0
$145$	4.63208e7	0.104786
$146$	0	0
$147$	− 2.11586e8i	− 0.453125i
$148$	0	0
$149$	−4.71028e8	−0.955656	−0.477828	−	0.878453i	$-0.658576\pi$
$149$	−4.71028e8	−0.955656	−0.477828	+	0.878453i	$0.658576\pi$
$150$	0	0
$151$	− 6.01838e8i	− 1.15764i	−0.815457	−	0.578818i	$-0.803514\pi$
$151$	− 6.01838e8i	− 1.15764i	0.815457	−	0.578818i	$-0.196486\pi$
$152$	0	0
$153$	−1.75929e8	−0.321049
$154$	0	0
$155$	− 7.37532e7i	− 0.127777i
$156$	0	0
$157$	−4.87702e7	−0.0802706	−0.0401353	−	0.999194i	$-0.512779\pi$
$157$	−4.87702e7	−0.0802706	−0.0401353	+	0.999194i	$0.512779\pi$
$158$	0	0
$159$	6.89037e8i	1.07809i
$160$	0	0
$161$	−2.73740e8	−0.407413
$162$	0	0
$163$	7.37869e8i	1.04527i	0.852556	+	0.522635i	$0.175051\pi$
$163$	7.37869e8i	1.04527i	−0.852556	+	0.522635i	$0.824949\pi$
$164$	0	0
$165$	−2.25816e7	−0.0304662
$166$	0	0
$167$	1.45451e9i	1.87004i	0.354600	+	0.935018i	$0.384617\pi$
$167$	1.45451e9i	1.87004i	−0.354600	+	0.935018i	$0.615383\pi$
$168$	0	0
$169$	−8.15700e8	−0.999962
$170$	0	0
$171$	− 5.23908e8i	− 0.612732i
$172$	0	0
$173$	1.13360e9	1.26554	0.632768	−	0.774341i	$-0.281919\pi$
$173$	1.13360e9	1.26554	0.632768	+	0.774341i	$0.281919\pi$
$174$	0	0
$175$	− 1.18153e9i	− 1.25978i
$176$	0	0
$177$	−7.60378e8	−0.774705
$178$	0	0
$179$	3.48018e8i	0.338992i	0.985531	+	0.169496i	$0.0542140\pi$
$179$	3.48018e8i	0.338992i	−0.985531	+	0.169496i	$0.945786\pi$
$180$	0	0
$181$	−1.15025e9	−1.07172	−0.535858	−	0.844308i	$-0.680012\pi$
$181$	−1.15025e9	−1.07172	−0.535858	+	0.844308i	$0.680012\pi$
$182$	0	0
$183$	− 4.51431e8i	− 0.402519i
$184$	0	0
$185$	1.27726e8	0.109042
$186$	0	0
$187$	2.60234e8i	0.212813i
$188$	0	0
$189$	3.28068e8	0.257109
$190$	0	0
$191$	− 1.71289e9i	− 1.28705i	−0.765424	−	0.643527i	$-0.777471\pi$
$191$	− 1.71289e9i	− 1.28705i	0.765424	−	0.643527i	$-0.222529\pi$
$192$	0	0
$193$	1.78226e9	1.28452	0.642261	−	0.766486i	$-0.277996\pi$
$193$	1.78226e9	1.28452	0.642261	+	0.766486i	$0.277996\pi$
$194$	0	0
$195$	− 1.23333e6i	0 0.000852985i
$196$	0	0
$197$	2.49946e9	1.65951	0.829756	−	0.558126i	$-0.188479\pi$
$197$	2.49946e9	1.65951	0.829756	+	0.558126i	$0.188479\pi$
$198$	0	0
$199$	− 2.17140e9i	− 1.38461i	−0.721604	−	0.692306i	$-0.756595\pi$
$199$	− 2.17140e9i	− 1.38461i	0.721604	−	0.692306i	$-0.243405\pi$
$200$	0	0
$201$	−1.05577e9	−0.646822
$202$	0	0
$203$	9.95436e8i	0.586177i
$204$	0	0
$205$	−2.10815e8	−0.119367
$206$	0	0
$207$	− 1.86636e8i	− 0.101652i
$208$	0	0
$209$	−7.74965e8	−0.406160
$210$	0	0
$211$	1.98390e9i	1.00090i	0.865766	+	0.500449i	$0.166832\pi$
$211$	1.98390e9i	1.00090i	−0.865766	+	0.500449i	$0.833168\pi$
$212$	0	0
$213$	1.42243e9	0.691056
$214$	0	0
$215$	3.21204e8i	0.150324i
$216$	0	0
$217$	1.58496e9	0.714791
$218$	0	0
$219$	− 1.87793e9i	− 0.816401i
$220$	0	0
$221$	−1.42131e7	−0.00595827
$222$	0	0
$223$	− 9.04205e8i	− 0.365635i	−0.983147	−	0.182817i	$-0.941478\pi$
$223$	− 9.04205e8i	− 0.365635i	0.983147	−	0.182817i	$-0.0585217\pi$
$224$	0	0
$225$	8.05571e8	0.314321
$226$	0	0
$227$	− 1.18796e9i	− 0.447402i	−0.974658	−	0.223701i	$-0.928186\pi$
$227$	− 1.18796e9i	− 0.447402i	0.974658	−	0.223701i	$-0.0718138\pi$
$228$	0	0
$229$	2.73614e9	0.994940	0.497470	−	0.867481i	$-0.334262\pi$
$229$	2.73614e9	0.994940	0.497470	+	0.867481i	$0.334262\pi$
$230$	0	0
$231$	− 4.85280e8i	− 0.170429i
$232$	0	0
$233$	−3.83321e9	−1.30059	−0.650293	−	0.759683i	$-0.725354\pi$
$233$	−3.83321e9	−1.30059	−0.650293	+	0.759683i	$0.725354\pi$
$234$	0	0
$235$	− 1.64925e8i	− 0.0540775i
$236$	0	0
$237$	2.32297e9	0.736292
$238$	0	0
$239$	5.01203e9i	1.53611i	0.640385	+	0.768054i	$0.278775\pi$
$239$	5.01203e9i	1.53611i	−0.640385	+	0.768054i	$0.721225\pi$
$240$	0	0
$241$	4.52278e9	1.34072	0.670359	−	0.742037i	$-0.266140\pi$
$241$	4.52278e9	1.34072	0.670359	+	0.742037i	$0.266140\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	−6.75331e8	−0.187436
$246$	0	0
$247$	− 4.23260e7i	− 0.0113716i
$248$	0	0
$249$	2.07700e9	0.540305
$250$	0	0
$251$	4.93482e9i	1.24330i	0.783294	+	0.621651i	$0.213538\pi$
$251$	4.93482e9i	1.24330i	−0.783294	+	0.621651i	$0.786462\pi$
$252$	0	0
$253$	−2.76073e8	−0.0673816
$254$	0	0
$255$	5.61521e8i	0.132802i
$256$	0	0
$257$	5.31230e9	1.21773	0.608864	−	0.793275i	$-0.291626\pi$
$257$	5.31230e9	1.21773	0.608864	+	0.793275i	$0.291626\pi$
$258$	0	0
$259$	2.74484e9i	0.609983i
$260$	0	0
$261$	−6.78689e8	−0.146254
$262$	0	0
$263$	9.22967e9i	1.92914i	0.263830	+	0.964569i	$0.415014\pi$
$263$	9.22967e9i	1.92914i	−0.263830	+	0.964569i	$0.584986\pi$
$264$	0	0
$265$	2.19924e9	0.445952
$266$	0	0
$267$	− 5.20249e9i	− 1.02368i
$268$	0	0
$269$	2.92523e9	0.558664	0.279332	−	0.960195i	$-0.409887\pi$
$269$	2.92523e9	0.558664	0.279332	+	0.960195i	$0.409887\pi$
$270$	0	0
$271$	− 2.04474e9i	− 0.379106i	−0.981870	−	0.189553i	$-0.939296\pi$
$271$	− 2.04474e9i	− 0.379106i	0.981870	−	0.189553i	$-0.0607039\pi$
$272$	0	0
$273$	2.65044e7	0.00477163
$274$	0	0
$275$	− 1.19160e9i	− 0.208354i
$276$	0	0
$277$	3.15752e9	0.536323	0.268162	−	0.963374i	$-0.413584\pi$
$277$	3.15752e9	0.536323	0.268162	+	0.963374i	$0.413584\pi$
$278$	0	0
$279$	1.08063e9i	0.178344i
$280$	0	0
$281$	1.16146e10	1.86286	0.931431	−	0.363917i	$-0.118561\pi$
$281$	1.16146e10	1.86286	0.931431	+	0.363917i	$0.118561\pi$
$282$	0	0
$283$	− 9.59477e9i	− 1.49585i	−0.663782	−	0.747926i	$-0.731050\pi$
$283$	− 9.59477e9i	− 1.49585i	0.663782	−	0.747926i	$-0.268950\pi$
$284$	0	0
$285$	−1.67218e9	−0.253457
$286$	0	0
$287$	− 4.53042e9i	− 0.667746i
$288$	0	0
$289$	−5.04689e8	−0.0723490
$290$	0	0
$291$	1.85193e8i	0.0258258i
$292$	0	0
$293$	−1.10305e9	−0.149667	−0.0748335	−	0.997196i	$-0.523843\pi$
$293$	−1.10305e9	−0.149667	−0.0748335	+	0.997196i	$0.523843\pi$
$294$	0	0
$295$	2.42694e9i	0.320458i
$296$	0	0
$297$	3.30864e8	0.0425230
$298$	0	0
$299$	− 1.50782e7i	− 0.00188653i
$300$	0	0
$301$	−6.90270e9	−0.840916
$302$	0	0
$303$	− 8.16809e9i	− 0.969058i
$304$	0	0
$305$	−1.44086e9	−0.166503
$306$	0	0
$307$	− 9.19662e9i	− 1.03532i	−0.855586	−	0.517660i	$-0.826803\pi$
$307$	− 9.19662e9i	− 1.03532i	0.855586	−	0.517660i	$-0.173197\pi$
$308$	0	0
$309$	−5.20557e8	−0.0570997
$310$	0	0
$311$	7.75359e9i	0.828822i	0.910090	+	0.414411i	$0.136012\pi$
$311$	7.75359e9i	0.828822i	−0.910090	+	0.414411i	$0.863988\pi$
$312$	0	0
$313$	1.32776e10	1.38339	0.691693	−	0.722192i	$-0.256865\pi$
$313$	1.32776e10	1.38339	0.691693	+	0.722192i	$0.256865\pi$
$314$	0	0
$315$	− 1.04711e9i	− 0.106353i
$316$	0	0
$317$	−3.22186e8	−0.0319058	−0.0159529	−	0.999873i	$-0.505078\pi$
$317$	−3.22186e8	−0.0319058	−0.0159529	+	0.999873i	$0.505078\pi$
$318$	0	0
$319$	1.00392e9i	0.0969473i
$320$	0	0
$321$	−1.20941e9	−0.113908
$322$	0	0
$323$	1.92705e10i	1.77045i
$324$	0	0
$325$	6.50814e7	0.00583342
$326$	0	0
$327$	− 1.04672e10i	− 0.915460i
$328$	0	0
$329$	3.54426e9	0.302511
$330$	0	0
$331$	− 1.16547e10i	− 0.970931i	−0.874256	−	0.485466i	$-0.838650\pi$
$331$	− 1.16547e10i	− 0.970931i	0.874256	−	0.485466i	$-0.161350\pi$
$332$	0	0
$333$	−1.87143e9	−0.152194
$334$	0	0
$335$	3.36976e9i	0.267559i
$336$	0	0
$337$	−2.36011e10	−1.82983	−0.914917	−	0.403641i	$-0.867744\pi$
$337$	−2.36011e10	−1.82983	−0.914917	+	0.403641i	$0.867744\pi$
$338$	0	0
$339$	− 5.31213e9i	− 0.402226i
$340$	0	0
$341$	1.59847e9	0.118219
$342$	0	0
$343$	3.97875e9i	0.287455i
$344$	0	0
$345$	−5.95697e8	−0.0420484
$346$	0	0
$347$	2.40544e9i	0.165912i	0.996553	+	0.0829559i	$0.0264361\pi$
$347$	2.40544e9i	0.165912i	−0.996553	+	0.0829559i	$0.973564\pi$
$348$	0	0
$349$	2.05149e10	1.38283	0.691415	−	0.722458i	$-0.256988\pi$
$349$	2.05149e10	1.38283	0.691415	+	0.722458i	$0.256988\pi$
$350$	0	0
$351$	1.80707e7i	0.00119055i
$352$	0	0
$353$	−1.10340e10	−0.710612	−0.355306	−	0.934750i	$-0.615623\pi$
$353$	−1.10340e10	−0.710612	−0.355306	+	0.934750i	$0.615623\pi$
$354$	0	0
$355$	− 4.54005e9i	− 0.285856i
$356$	0	0
$357$	−1.20671e10	−0.742901
$358$	0	0
$359$	1.81015e10i	1.08978i	0.838509	+	0.544888i	$0.183428\pi$
$359$	1.81015e10i	1.08978i	−0.838509	+	0.544888i	$0.816572\pi$
$360$	0	0
$361$	−4.04032e10	−2.37896
$362$	0	0
$363$	9.53516e9i	0.549163i
$364$	0	0
$365$	−5.99390e9	−0.337705
$366$	0	0
$367$	− 2.78190e10i	− 1.53348i	−0.641959	−	0.766739i	$-0.721878\pi$
$367$	− 2.78190e10i	− 1.53348i	0.641959	−	0.766739i	$-0.278122\pi$
$368$	0	0
$369$	3.08885e9	0.166606
$370$	0	0
$371$	4.72617e10i	2.49467i
$372$	0	0
$373$	8.10899e8	0.0418920	0.0209460	−	0.999781i	$-0.493332\pi$
$373$	8.10899e8	0.0418920	0.0209460	+	0.999781i	$0.493332\pi$
$374$	0	0
$375$	− 5.29789e9i	− 0.267903i
$376$	0	0
$377$	−5.48307e7	−0.00271430
$378$	0	0
$379$	− 6.25965e9i	− 0.303384i	−0.988428	−	0.151692i	$-0.951528\pi$
$379$	− 6.25965e9i	− 0.303384i	0.988428	−	0.151692i	$-0.0484722\pi$
$380$	0	0
$381$	−5.91979e9	−0.280935
$382$	0	0
$383$	− 1.50020e10i	− 0.697195i	−0.937272	−	0.348598i	$-0.886658\pi$
$383$	− 1.50020e10i	− 0.697195i	0.937272	−	0.348598i	$-0.113342\pi$
$384$	0	0
$385$	−1.54889e9	−0.0704983
$386$	0	0
$387$	− 4.70627e9i	− 0.209813i
$388$	0	0
$389$	−4.77166e9	−0.208387	−0.104194	−	0.994557i	$-0.533226\pi$
$389$	−4.77166e9	−0.208387	−0.104194	+	0.994557i	$0.533226\pi$
$390$	0	0
$391$	6.86492e9i	0.293716i
$392$	0	0
$393$	1.11151e10	0.465954
$394$	0	0
$395$	− 7.41435e9i	− 0.304568i
$396$	0	0
$397$	−1.06262e10	−0.427777	−0.213889	−	0.976858i	$-0.568613\pi$
$397$	−1.06262e10	−0.427777	−0.213889	+	0.976858i	$0.568613\pi$
$398$	0	0
$399$	− 3.59353e10i	− 1.41785i
$400$	0	0
$401$	−2.34131e10	−0.905485	−0.452743	−	0.891641i	$-0.649554\pi$
$401$	−2.34131e10	−0.905485	−0.452743	+	0.891641i	$0.649554\pi$
$402$	0	0
$403$	8.73029e7i	0.00330985i
$404$	0	0
$405$	7.13923e8	0.0265357
$406$	0	0
$407$	2.76823e9i	0.100884i
$408$	0	0
$409$	−3.53006e10	−1.26150	−0.630752	−	0.775985i	$-0.717253\pi$
$409$	−3.53006e10	−1.26150	−0.630752	+	0.775985i	$0.717253\pi$
$410$	0	0
$411$	− 1.08530e10i	− 0.380349i
$412$	0	0
$413$	−5.21551e10	−1.79265
$414$	0	0
$415$	− 6.62927e9i	− 0.223498i
$416$	0	0
$417$	−4.87989e8	−0.0161386
$418$	0	0
$419$	5.73589e10i	1.86099i	0.366299	+	0.930497i	$0.380625\pi$
$419$	5.73589e10i	1.86099i	−0.366299	+	0.930497i	$0.619375\pi$
$420$	0	0
$421$	4.22283e10	1.34423	0.672117	−	0.740445i	$-0.265385\pi$
$421$	4.22283e10	1.34423	0.672117	+	0.740445i	$0.265385\pi$
$422$	0	0
$423$	2.41648e9i	0.0754782i
$424$	0	0
$425$	−2.96308e10	−0.908213
$426$	0	0
$427$	− 3.09641e10i	− 0.931422i
$428$	0	0
$429$	2.67302e7	0.000789175 0
$430$	0	0
$431$	4.42681e10i	1.28287i	0.767179	+	0.641433i	$0.221660\pi$
$431$	4.42681e10i	1.28287i	−0.767179	+	0.641433i	$0.778340\pi$
$432$	0	0
$433$	1.90037e8	0.00540614	0.00270307	−	0.999996i	$-0.499140\pi$
$433$	1.90037e8	0.00540614	0.00270307	+	0.999996i	$0.499140\pi$
$434$	0	0
$435$	2.16621e9i	0.0604983i
$436$	0	0
$437$	−2.04434e10	−0.560567
$438$	0	0
$439$	6.76040e10i	1.82018i	0.414409	+	0.910091i	$0.363988\pi$
$439$	6.76040e10i	1.82018i	−0.414409	+	0.910091i	$0.636012\pi$
$440$	0	0
$441$	9.89490e9	0.261612
$442$	0	0
$443$	1.35521e10i	0.351877i	0.984401	+	0.175939i	$0.0562960\pi$
$443$	1.35521e10i	0.351877i	−0.984401	+	0.175939i	$0.943704\pi$
$444$	0	0
$445$	−1.66051e10	−0.423448
$446$	0	0
$447$	− 2.20278e10i	− 0.551748i
$448$	0	0
$449$	1.25951e10	0.309897	0.154948	−	0.987923i	$-0.450479\pi$
$449$	1.25951e10	0.309897	0.154948	+	0.987923i	$0.450479\pi$
$450$	0	0
$451$	− 4.56903e9i	− 0.110438i
$452$	0	0
$453$	2.81452e10	0.668361
$454$	0	0
$455$	− 8.45954e7i	− 0.00197379i
$456$	0	0
$457$	3.90759e10	0.895869	0.447935	−	0.894066i	$-0.352160\pi$
$457$	3.90759e10	0.895869	0.447935	+	0.894066i	$0.352160\pi$
$458$	0	0
$459$	− 8.22737e9i	− 0.185358i
$460$	0	0
$461$	−2.22395e9	−0.0492404	−0.0246202	−	0.999697i	$-0.507838\pi$
$461$	−2.22395e9	−0.0492404	−0.0246202	+	0.999697i	$0.507838\pi$
$462$	0	0
$463$	− 1.65068e10i	− 0.359203i	−0.983739	−	0.179602i	$-0.942519\pi$
$463$	− 1.65068e10i	− 0.359203i	0.983739	−	0.179602i	$-0.0574809\pi$
$464$	0	0
$465$	3.44910e9	0.0737723
$466$	0	0
$467$	7.20146e9i	0.151409i	0.997130	+	0.0757047i	$0.0241206\pi$
$467$	7.20146e9i	0.151409i	−0.997130	+	0.0757047i	$0.975879\pi$
$468$	0	0
$469$	−7.24162e10	−1.49673
$470$	0	0
$471$	− 2.28076e9i	− 0.0463442i
$472$	0	0
$473$	−6.96152e9	−0.139078
$474$	0	0
$475$	− 8.82391e10i	− 1.73335i
$476$	0	0
$477$	−3.22231e10	−0.622434
$478$	0	0
$479$	2.84033e10i	0.539543i	0.962924	+	0.269772i	$0.0869482\pi$
$479$	2.84033e10i	0.539543i	−0.962924	+	0.269772i	$0.913052\pi$
$480$	0	0
$481$	−1.51191e8	−0.00282453
$482$	0	0
$483$	− 1.28016e10i	− 0.235220i
$484$	0	0
$485$	5.91092e8	0.0106829
$486$	0	0
$487$	6.74024e9i	0.119828i	0.998204	+	0.0599142i	$0.0190827\pi$
$487$	6.74024e9i	0.119828i	−0.998204	+	0.0599142i	$0.980917\pi$
$488$	0	0
$489$	−3.45067e10	−0.603487
$490$	0	0
$491$	− 7.72506e10i	− 1.32916i	−0.747219	−	0.664578i	$-0.768611\pi$
$491$	− 7.72506e10i	− 1.32916i	0.747219	−	0.664578i	$-0.231389\pi$
$492$	0	0
$493$	2.49638e10	0.422593
$494$	0	0
$495$	− 1.05604e9i	− 0.0175897i
$496$	0	0
$497$	9.75659e10	1.59909
$498$	0	0
$499$	2.02155e10i	0.326049i	0.986622	+	0.163024i	$0.0521249\pi$
$499$	2.02155e10i	0.326049i	−0.986622	+	0.163024i	$0.947875\pi$
$500$	0	0
$501$	−6.80206e10	−1.07967
$502$	0	0
$503$	6.27908e10i	0.980899i	0.871470	+	0.490449i	$0.163167\pi$
$503$	6.27908e10i	0.980899i	−0.871470	+	0.490449i	$0.836833\pi$
$504$	0	0
$505$	−2.60705e10	−0.400852
$506$	0	0
$507$	− 3.81465e10i	− 0.577328i
$508$	0	0
$509$	7.85179e10	1.16976	0.584881	−	0.811119i	$-0.301141\pi$
$509$	7.85179e10	1.16976	0.584881	+	0.811119i	$0.301141\pi$
$510$	0	0
$511$	− 1.28809e11i	− 1.88913i
$512$	0	0
$513$	2.45007e10	0.353761
$514$	0	0
$515$	1.66149e9i	0.0236194i
$516$	0	0
$517$	3.57446e9	0.0500320
$518$	0	0
$519$	5.30131e10i	0.730658i
$520$	0	0
$521$	1.08134e11	1.46761	0.733806	−	0.679359i	$-0.237742\pi$
$521$	1.08134e11	1.46761	0.733806	+	0.679359i	$0.237742\pi$
$522$	0	0
$523$	1.90750e8i	0.00254952i	0.999999	+	0.00127476i	$0.000405769\pi$
$523$	1.90750e8i	0.00254952i	−0.999999	+	0.00127476i	$0.999594\pi$
$524$	0	0
$525$	5.52549e10	0.727334
$526$	0	0
$527$	− 3.97480e10i	− 0.515315i
$528$	0	0
$529$	7.10282e10	0.907002
$530$	0	0
$531$	− 3.55594e10i	− 0.447276i
$532$	0	0
$533$	2.49545e8	0.00309200
$534$	0	0
$535$	3.86014e9i	0.0471181i
$536$	0	0
$537$	−1.62752e10	−0.195717
$538$	0	0
$539$	− 1.46366e10i	− 0.173414i
$540$	0	0
$541$	−8.80067e9	−0.102737	−0.0513685	−	0.998680i	$-0.516358\pi$
$541$	−8.80067e9	−0.102737	−0.0513685	+	0.998680i	$0.516358\pi$
$542$	0	0
$543$	− 5.37921e10i	− 0.618755i
$544$	0	0
$545$	−3.34087e10	−0.378681
$546$	0	0
$547$	1.38634e11i	1.54853i	0.632862	+	0.774265i	$0.281880\pi$
$547$	1.38634e11i	1.54853i	−0.632862	+	0.774265i	$0.718120\pi$
$548$	0	0
$549$	2.11113e10	0.232395
$550$	0	0
$551$	7.43409e10i	0.806532i
$552$	0	0
$553$	1.59335e11	1.70377
$554$	0	0
$555$	5.97315e9i	0.0629552i
$556$	0	0
$557$	9.91532e10	1.03012	0.515058	−	0.857156i	$-0.327771\pi$
$557$	9.91532e10	1.03012	0.515058	+	0.857156i	$0.327771\pi$
$558$	0	0
$559$	− 3.80215e8i	− 0.00389387i
$560$	0	0
$561$	−1.21699e10	−0.122868
$562$	0	0
$563$	1.30106e11i	1.29498i	0.762073	+	0.647492i	$0.224182\pi$
$563$	1.30106e11i	1.29498i	−0.762073	+	0.647492i	$0.775818\pi$
$564$	0	0
$565$	−1.69550e10	−0.166381
$566$	0	0
$567$	1.53422e10i	0.148442i
$568$	0	0
$569$	−7.48058e10	−0.713651	−0.356826	−	0.934171i	$-0.616141\pi$
$569$	−7.48058e10	−0.713651	−0.356826	+	0.934171i	$0.616141\pi$
$570$	0	0
$571$	4.89520e10i	0.460496i	0.973132	+	0.230248i	$0.0739537\pi$
$571$	4.89520e10i	0.460496i	−0.973132	+	0.230248i	$0.926046\pi$
$572$	0	0
$573$	8.01040e10	0.743081
$574$	0	0
$575$	− 3.14342e10i	− 0.287562i
$576$	0	0
$577$	−8.33708e9	−0.0752161	−0.0376080	−	0.999293i	$-0.511974\pi$
$577$	−8.33708e9	−0.0752161	−0.0376080	+	0.999293i	$0.511974\pi$
$578$	0	0
$579$	8.33480e10i	0.741619i
$580$	0	0
$581$	1.42463e11	1.25025
$582$	0	0
$583$	4.76644e10i	0.412591i
$584$	0	0
$585$	5.76772e7	0.000492471 0
$586$	0	0
$587$	− 1.05583e11i	− 0.889289i	−0.895707	−	0.444644i	$-0.853330\pi$
$587$	− 1.05583e11i	− 0.889289i	0.895707	−	0.444644i	$-0.146670\pi$
$588$	0	0
$589$	1.18368e11	0.983494
$590$	0	0
$591$	1.16888e11i	0.958120i
$592$	0	0
$593$	−1.10652e11	−0.894832	−0.447416	−	0.894326i	$-0.647656\pi$
$593$	−1.10652e11	−0.894832	−0.447416	+	0.894326i	$0.647656\pi$
$594$	0	0
$595$	3.85153e10i	0.307302i
$596$	0	0
$597$	1.01547e11	0.799407
$598$	0	0
$599$	8.69285e10i	0.675235i	0.941283	+	0.337617i	$0.109621\pi$
$599$	8.69285e10i	0.675235i	−0.941283	+	0.337617i	$0.890379\pi$
$600$	0	0
$601$	3.34335e10	0.256262	0.128131	−	0.991757i	$-0.459102\pi$
$601$	3.34335e10	0.256262	0.128131	+	0.991757i	$0.459102\pi$
$602$	0	0
$603$	− 4.93734e10i	− 0.373443i
$604$	0	0
$605$	3.04339e10	0.227162
$606$	0	0
$607$	− 1.12699e11i	− 0.830167i	−0.909783	−	0.415084i	$-0.863752\pi$
$607$	− 1.12699e11i	− 0.830167i	0.909783	−	0.415084i	$-0.136248\pi$
$608$	0	0
$609$	−4.65519e10	−0.338430
$610$	0	0
$611$	1.95225e8i	0.00140078i
$612$	0	0
$613$	−4.91478e10	−0.348067	−0.174033	−	0.984740i	$-0.555680\pi$
$613$	−4.91478e10	−0.348067	−0.174033	+	0.984740i	$0.555680\pi$
$614$	0	0
$615$	− 9.85884e9i	− 0.0689168i
$616$	0	0
$617$	2.19370e10	0.151369	0.0756845	−	0.997132i	$-0.475886\pi$
$617$	2.19370e10	0.151369	0.0756845	+	0.997132i	$0.475886\pi$
$618$	0	0
$619$	6.88531e10i	0.468987i	0.972118	+	0.234493i	$0.0753432\pi$
$619$	6.88531e10i	0.468987i	−0.972118	+	0.234493i	$0.924657\pi$
$620$	0	0
$621$	8.72811e9	0.0586886
$622$	0	0
$623$	− 3.56844e11i	− 2.36879i
$624$	0	0
$625$	1.26975e11	0.832146
$626$	0	0
$627$	− 3.62415e10i	− 0.234497i
$628$	0	0
$629$	6.88357e10	0.439755
$630$	0	0
$631$	− 2.01724e11i	− 1.27245i	−0.771505	−	0.636223i	$-0.780496\pi$
$631$	− 2.01724e11i	− 1.27245i	0.771505	−	0.636223i	$-0.219504\pi$
$632$	0	0
$633$	−9.27777e10	−0.577868
$634$	0	0
$635$	1.88945e10i	0.116209i
$636$	0	0
$637$	7.99400e8	0.00485519
$638$	0	0
$639$	6.65205e10i	0.398981i
$640$	0	0
$641$	−3.11594e10	−0.184568	−0.0922841	−	0.995733i	$-0.529417\pi$
$641$	−3.11594e10	−0.184568	−0.0922841	+	0.995733i	$0.529417\pi$
$642$	0	0
$643$	1.11577e11i	0.652728i	0.945244	+	0.326364i	$0.105823\pi$
$643$	1.11577e11i	0.652728i	−0.945244	+	0.326364i	$0.894177\pi$
$644$	0	0
$645$	−1.50212e10	−0.0867895
$646$	0	0
$647$	9.54210e10i	0.544536i	0.962221	+	0.272268i	$0.0877737\pi$
$647$	9.54210e10i	0.544536i	−0.962221	+	0.272268i	$0.912226\pi$
$648$	0	0
$649$	−5.25995e10	−0.296485
$650$	0	0
$651$	7.41212e10i	0.412685i
$652$	0	0
$653$	−1.57875e11	−0.868281	−0.434140	−	0.900845i	$-0.642948\pi$
$653$	−1.57875e11	−0.868281	−0.434140	+	0.900845i	$0.642948\pi$
$654$	0	0
$655$	− 3.54766e10i	− 0.192742i
$656$	0	0
$657$	8.78222e10	0.471349
$658$	0	0
$659$	− 1.05320e11i	− 0.558433i	−0.960228	−	0.279216i	$-0.909925\pi$
$659$	− 1.05320e11i	− 0.558433i	0.960228	−	0.279216i	$-0.0900747\pi$
$660$	0	0
$661$	−1.99905e10	−0.104717	−0.0523585	−	0.998628i	$-0.516674\pi$
$661$	−1.99905e10	−0.104717	−0.0523585	+	0.998628i	$0.516674\pi$
$662$	0	0
$663$	− 6.64682e8i	− 0.00344001i
$664$	0	0
$665$	−1.14697e11	−0.586495
$666$	0	0
$667$	2.64831e10i	0.133803i
$668$	0	0
$669$	4.22855e10	0.211099
$670$	0	0
$671$	− 3.12279e10i	− 0.154047i
$672$	0	0
$673$	1.01495e11	0.494750	0.247375	−	0.968920i	$-0.420432\pi$
$673$	1.01495e11	0.494750	0.247375	+	0.968920i	$0.420432\pi$
$674$	0	0
$675$	3.76728e10i	0.181474i
$676$	0	0
$677$	−1.34737e10	−0.0641406	−0.0320703	−	0.999486i	$-0.510210\pi$
$677$	−1.34737e10	−0.0641406	−0.0320703	+	0.999486i	$0.510210\pi$
$678$	0	0
$679$	1.27026e10i	0.0597604i
$680$	0	0
$681$	5.55553e10	0.258307
$682$	0	0
$683$	− 2.42663e11i	− 1.11512i	−0.830138	−	0.557558i	$-0.811738\pi$
$683$	− 2.42663e11i	− 1.11512i	0.830138	−	0.557558i	$-0.188262\pi$
$684$	0	0
$685$	−3.46400e10	−0.157332
$686$	0	0
$687$	1.27957e11i	0.574429i
$688$	0	0
$689$	−2.60327e9	−0.0115516
$690$	0	0
$691$	2.15881e11i	0.946896i	0.880822	+	0.473448i	$0.156991\pi$
$691$	2.15881e11i	0.946896i	−0.880822	+	0.473448i	$0.843009\pi$
$692$	0	0
$693$	2.26943e10	0.0983974
$694$	0	0
$695$	1.55754e9i	0.00667575i
$696$	0	0
$697$	−1.13615e11	−0.481398
$698$	0	0
$699$	− 1.79262e11i	− 0.750894i
$700$	0	0
$701$	−2.45336e11	−1.01599	−0.507994	−	0.861361i	$-0.669613\pi$
$701$	−2.45336e11	−1.01599	−0.507994	+	0.861361i	$0.669613\pi$
$702$	0	0
$703$	2.04989e11i	0.839286i
$704$	0	0
$705$	7.71280e9	0.0312216
$706$	0	0
$707$	− 5.60257e11i	− 2.24238i
$708$	0	0
$709$	1.40515e10	0.0556080	0.0278040	−	0.999613i	$-0.491149\pi$
$709$	1.40515e10	0.0556080	0.0278040	+	0.999613i	$0.491149\pi$
$710$	0	0
$711$	1.08635e11i	0.425099i
$712$	0	0
$713$	4.21672e10	0.163161
$714$	0	0
$715$	− 8.53162e7i	0 0.000326443i
$716$	0	0
$717$	−2.34389e11	−0.886873
$718$	0	0
$719$	− 1.91174e11i	− 0.715343i	−0.933848	−	0.357671i	$-0.883571\pi$
$719$	− 1.91174e11i	− 0.715343i	0.933848	−	0.357671i	$-0.116429\pi$
$720$	0	0
$721$	−3.57055e10	−0.132128
$722$	0	0
$723$	2.11510e11i	0.774064i
$724$	0	0
$725$	−1.14308e11	−0.413738
$726$	0	0
$727$	3.91555e11i	1.40170i	0.713308	+	0.700850i	$0.247196\pi$
$727$	3.91555e11i	1.40170i	−0.713308	+	0.700850i	$0.752804\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	1.73107e11i	0.606242i
$732$	0	0
$733$	−5.52391e11	−1.91351	−0.956754	−	0.290898i	$-0.906046\pi$
$733$	−5.52391e11	−1.91351	−0.956754	+	0.290898i	$0.906046\pi$
$734$	0	0
$735$	− 3.15821e10i	− 0.108216i
$736$	0	0
$737$	−7.30333e10	−0.247543
$738$	0	0
$739$	− 1.83932e10i	− 0.0616709i	−0.999524	−	0.0308354i	$-0.990183\pi$
$739$	− 1.83932e10i	− 0.0616709i	0.999524	−	0.0308354i	$-0.00981678\pi$
$740$	0	0
$741$	1.97939e9	0.00656537
$742$	0	0
$743$	− 1.79242e11i	− 0.588145i	−0.955783	−	0.294072i	$-0.904989\pi$
$743$	− 1.79242e11i	− 0.588145i	0.955783	−	0.294072i	$-0.0950106\pi$
$744$	0	0
$745$	−7.03073e10	−0.228231
$746$	0	0
$747$	9.71316e10i	0.311945i
$748$	0	0
$749$	−8.29546e10	−0.263581
$750$	0	0
$751$	3.38227e11i	1.06328i	0.846970	+	0.531641i	$0.178424\pi$
$751$	3.38227e11i	1.06328i	−0.846970	+	0.531641i	$0.821576\pi$
$752$	0	0
$753$	−2.30779e11	−0.717821
$754$	0	0
$755$	− 8.98325e10i	− 0.276468i
$756$	0	0
$757$	−3.36576e11	−1.02494	−0.512471	−	0.858705i	$-0.671270\pi$
$757$	−3.36576e11	−1.02494	−0.512471	+	0.858705i	$0.671270\pi$
$758$	0	0
$759$	− 1.29106e10i	− 0.0389028i
$760$	0	0
$761$	3.50280e11	1.04442	0.522212	−	0.852816i	$-0.325107\pi$
$761$	3.50280e11	1.04442	0.522212	+	0.852816i	$0.325107\pi$
$762$	0	0
$763$	− 7.17955e11i	− 2.11836i
$764$	0	0
$765$	−2.62597e10	−0.0766734
$766$	0	0
$767$	− 2.87281e9i	− 0.00830090i
$768$	0	0
$769$	−3.89703e11	−1.11437	−0.557184	−	0.830389i	$-0.688118\pi$
$769$	−3.89703e11	−1.11437	−0.557184	+	0.830389i	$0.688118\pi$
$770$	0	0
$771$	2.48432e11i	0.703055i
$772$	0	0
$773$	5.62526e11	1.57552	0.787761	−	0.615981i	$-0.211240\pi$
$773$	5.62526e11	1.57552	0.787761	+	0.615981i	$0.211240\pi$
$774$	0	0
$775$	1.82005e11i	0.504517i
$776$	0	0
$777$	−1.28363e11	−0.352174
$778$	0	0
$779$	− 3.38340e11i	− 0.918763i
$780$	0	0
$781$	9.83973e10	0.264472
$782$	0	0
$783$	− 3.17391e10i	− 0.0844400i
$784$	0	0
$785$	−7.27962e9	−0.0191704
$786$	0	0
$787$	− 7.83441e10i	− 0.204224i	−0.994773	−	0.102112i	$-0.967440\pi$
$787$	− 7.83441e10i	− 0.204224i	0.994773	−	0.102112i	$-0.0325600\pi$
$788$	0	0
$789$	−4.31629e11	−1.11379
$790$	0	0
$791$	− 3.64364e11i	− 0.930742i
$792$	0	0
$793$	1.70556e9	0.00431296
$794$	0	0
$795$	1.02848e11i	0.257471i
$796$	0	0
$797$	−4.07970e11	−1.01110	−0.505551	−	0.862797i	$-0.668711\pi$
$797$	−4.07970e11	−1.01110	−0.505551	+	0.862797i	$0.668711\pi$
$798$	0	0
$799$	− 8.88837e10i	− 0.218090i
$800$	0	0
$801$	2.43296e11	0.591024
$802$	0	0
$803$	− 1.29907e11i	− 0.312442i
$804$	0	0
$805$	−4.08594e10	−0.0972991
$806$	0	0
$807$	1.36799e11i	0.322545i
$808$	0	0
$809$	−6.48839e10	−0.151476	−0.0757378	−	0.997128i	$-0.524131\pi$
$809$	−6.48839e10	−0.151476	−0.0757378	+	0.997128i	$0.524131\pi$
$810$	0	0
$811$	5.68236e11i	1.31355i	0.754088	+	0.656773i	$0.228079\pi$
$811$	5.68236e11i	1.31355i	−0.754088	+	0.656773i	$0.771921\pi$
$812$	0	0
$813$	9.56230e10	0.218877
$814$	0	0
$815$	1.10137e11i	0.249633i
$816$	0	0
$817$	−5.15506e11	−1.15703
$818$	0	0
$819$	1.23949e9i	0.00275490i
$820$	0	0
$821$	−8.62816e11	−1.89909	−0.949545	−	0.313631i	$-0.898454\pi$
$821$	−8.62816e11	−1.89909	−0.949545	+	0.313631i	$0.898454\pi$
$822$	0	0
$823$	5.20649e10i	0.113487i	0.998389	+	0.0567435i	$0.0180717\pi$
$823$	5.20649e10i	0.113487i	−0.998389	+	0.0567435i	$0.981928\pi$
$824$	0	0
$825$	5.57258e10	0.120293
$826$	0	0
$827$	6.71579e11i	1.43574i	0.696179	+	0.717869i	$0.254882\pi$
$827$	6.71579e11i	1.43574i	−0.696179	+	0.717869i	$0.745118\pi$
$828$	0	0
$829$	7.34136e11	1.55439	0.777193	−	0.629263i	$-0.216643\pi$
$829$	7.34136e11	1.55439	0.777193	+	0.629263i	$0.216643\pi$
$830$	0	0
$831$	1.47662e11i	0.309646i
$832$	0	0
$833$	−3.63958e11	−0.755911
$834$	0	0
$835$	2.17105e11i	0.446605i
$836$	0	0
$837$	−5.05359e10	−0.102967
$838$	0	0
$839$	− 6.60308e11i	− 1.33260i	−0.745685	−	0.666299i	$-0.767878\pi$
$839$	− 6.60308e11i	− 1.33260i	0.745685	−	0.666299i	$-0.232122\pi$
$840$	0	0
$841$	−4.03942e11	−0.807487
$842$	0	0
$843$	5.43163e11i	1.07552i
$844$	0	0
$845$	−1.21754e11	−0.238813
$846$	0	0
$847$	6.54026e11i	1.27075i
$848$	0	0
$849$	4.48703e11	0.863631
$850$	0	0
$851$	7.30252e10i	0.139237i
$852$	0	0
$853$	3.67260e11	0.693709	0.346854	−	0.937919i	$-0.387250\pi$
$853$	3.67260e11	0.693709	0.346854	+	0.937919i	$0.387250\pi$
$854$	0	0
$855$	− 7.82003e10i	− 0.146334i
$856$	0	0
$857$	5.83979e11	1.08261	0.541307	−	0.840825i	$-0.317930\pi$
$857$	5.83979e11	1.08261	0.541307	+	0.840825i	$0.317930\pi$
$858$	0	0
$859$	4.73945e11i	0.870473i	0.900316	+	0.435236i	$0.143335\pi$
$859$	4.73945e11i	0.870473i	−0.900316	+	0.435236i	$0.856665\pi$
$860$	0	0
$861$	2.11867e11	0.385523
$862$	0	0
$863$	− 4.46554e11i	− 0.805065i	−0.915406	−	0.402533i	$-0.868130\pi$
$863$	− 4.46554e11i	− 0.805065i	0.915406	−	0.402533i	$-0.131870\pi$
$864$	0	0
$865$	1.69205e11	0.302238
$866$	0	0
$867$	− 2.36020e10i	− 0.0417707i
$868$	0	0
$869$	1.60693e11	0.281784
$870$	0	0
$871$	− 3.98884e9i	− 0.00693064i
$872$	0	0
$873$	−8.66064e9	−0.0149105
$874$	0	0
$875$	− 3.63387e11i	− 0.619923i
$876$	0	0
$877$	6.58064e11	1.11242	0.556211	−	0.831041i	$-0.312254\pi$
$877$	6.58064e11	1.11242	0.556211	+	0.831041i	$0.312254\pi$
$878$	0	0
$879$	− 5.15847e10i	− 0.0864103i
$880$	0	0
$881$	−1.09401e12	−1.81600	−0.908001	−	0.418968i	$-0.862392\pi$
$881$	−1.09401e12	−1.81600	−0.908001	+	0.418968i	$0.862392\pi$
$882$	0	0
$883$	− 7.41168e11i	− 1.21920i	−0.792710	−	0.609599i	$-0.791331\pi$
$883$	− 7.41168e11i	− 1.21920i	0.792710	−	0.609599i	$-0.208669\pi$
$884$	0	0
$885$	−1.13497e11	−0.185016
$886$	0	0
$887$	− 5.47892e11i	− 0.885116i	−0.896740	−	0.442558i	$-0.854071\pi$
$887$	− 5.47892e11i	− 0.885116i	0.896740	−	0.442558i	$-0.145929\pi$
$888$	0	0
$889$	−4.06044e11	−0.650079
$890$	0	0
$891$	1.54730e10i	0.0245507i
$892$	0	0
$893$	2.64691e11	0.416231
$894$	0	0
$895$	5.19464e10i	0.0809587i
$896$	0	0
$897$	7.05137e8	0.00108919
$898$	0	0
$899$	− 1.53338e11i	− 0.234753i
$900$	0	0
$901$	1.18524e12	1.79849
$902$	0	0
$903$	− 3.22807e11i	− 0.485503i
$904$	0	0
$905$	−1.71691e11	−0.255949
$906$	0	0
$907$	2.33352e11i	0.344813i	0.985026	+	0.172406i	$0.0551542\pi$
$907$	2.33352e11i	0.344813i	−0.985026	+	0.172406i	$0.944846\pi$
$908$	0	0
$909$	3.81984e11	0.559486
$910$	0	0
$911$	2.97901e11i	0.432513i	0.976337	+	0.216256i	$0.0693847\pi$
$911$	2.97901e11i	0.432513i	−0.976337	+	0.216256i	$0.930615\pi$
$912$	0	0
$913$	1.43677e11	0.206778
$914$	0	0
$915$	− 6.73821e10i	− 0.0961303i
$916$	0	0
$917$	7.62394e11	1.07821
$918$	0	0
$919$	− 8.26039e11i	− 1.15808i	−0.815299	−	0.579040i	$-0.803428\pi$
$919$	− 8.26039e11i	− 1.15808i	0.815299	−	0.579040i	$-0.196572\pi$
$920$	0	0
$921$	4.30083e11	0.597742
$922$	0	0
$923$	5.37414e9i	0.00740460i
$924$	0	0
$925$	−3.15196e11	−0.430540
$926$	0	0
$927$	− 2.43440e10i	− 0.0329666i
$928$	0	0
$929$	7.14762e11	0.959619	0.479809	−	0.877373i	$-0.340706\pi$
$929$	7.14762e11	0.959619	0.479809	+	0.877373i	$0.340706\pi$
$930$	0	0
$931$	− 1.08385e12i	− 1.44268i
$932$	0	0
$933$	−3.62600e11	−0.478521
$934$	0	0
$935$	3.88435e10i	0.0508243i
$936$	0	0
$937$	3.24075e11	0.420424	0.210212	−	0.977656i	$-0.432585\pi$
$937$	3.24075e11	0.420424	0.210212	+	0.977656i	$0.432585\pi$
$938$	0	0
$939$	6.20933e11i	0.798698i
$940$	0	0
$941$	6.78533e10	0.0865391	0.0432696	−	0.999063i	$-0.486223\pi$
$941$	6.78533e10	0.0865391	0.0432696	+	0.999063i	$0.486223\pi$
$942$	0	0
$943$	− 1.20530e11i	− 0.152422i
$944$	0	0
$945$	4.89687e10	0.0614032
$946$	0	0
$947$	8.07653e11i	1.00421i	0.864806	+	0.502105i	$0.167441\pi$
$947$	8.07653e11i	1.00421i	−0.864806	+	0.502105i	$0.832559\pi$
$948$	0	0
$949$	7.09507e9	0.00874766
$950$	0	0
$951$	− 1.50671e10i	− 0.0184208i
$952$	0	0
$953$	7.72345e11	0.936353	0.468176	−	0.883635i	$-0.344911\pi$
$953$	7.72345e11	0.936353	0.468176	+	0.883635i	$0.344911\pi$
$954$	0	0
$955$	− 2.55672e11i	− 0.307376i
$956$	0	0
$957$	−4.69486e10	−0.0559725
$958$	0	0
$959$	− 7.44416e11i	− 0.880119i
$960$	0	0
$961$	6.08743e11	0.713740
$962$	0	0
$963$	− 5.65585e10i	− 0.0657647i
$964$	0	0
$965$	2.66026e11	0.306772
$966$	0	0
$967$	5.00769e11i	0.572706i	0.958124	+	0.286353i	$0.0924430\pi$
$967$	5.00769e11i	0.572706i	−0.958124	+	0.286353i	$0.907557\pi$
$968$	0	0
$969$	−9.01194e11	−1.02217
$970$	0	0
$971$	− 8.51918e11i	− 0.958343i	−0.877721	−	0.479171i	$-0.840937\pi$
$971$	− 8.51918e11i	− 0.958343i	0.877721	−	0.479171i	$-0.159063\pi$
$972$	0	0
$973$	−3.34716e10	−0.0373444
$974$	0	0
$975$	3.04356e9i	0.00336793i
$976$	0	0
$977$	9.46264e11	1.03857	0.519283	−	0.854603i	$-0.326199\pi$
$977$	9.46264e11	1.03857	0.519283	+	0.854603i	$0.326199\pi$
$978$	0	0
$979$	− 3.59884e11i	− 0.391771i
$980$	0	0
$981$	4.89503e11	0.528541
$982$	0	0
$983$	− 9.35217e11i	− 1.00161i	−0.865560	−	0.500804i	$-0.833038\pi$
$983$	− 9.35217e11i	− 1.00161i	0.865560	−	0.500804i	$-0.166962\pi$
$984$	0	0
$985$	3.73078e11	0.396328
$986$	0	0
$987$	1.65748e11i	0.174655i
$988$	0	0
$989$	−1.83643e11	−0.191951
$990$	0	0
$991$	9.50357e11i	0.985354i	0.870212	+	0.492677i	$0.163981\pi$
$991$	9.50357e11i	0.985354i	−0.870212	+	0.492677i	$0.836019\pi$
$992$	0	0
$993$	5.45036e11	0.560568
$994$	0	0
$995$	− 3.24112e11i	− 0.330676i
$996$	0	0
$997$	−1.84651e12	−1.86884	−0.934419	−	0.356175i	$-0.884081\pi$
$997$	−1.84651e12	−1.86884	−0.934419	+	0.356175i	$0.884081\pi$
$998$	0	0
$999$	− 8.75183e10i	− 0.0878692i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.9.g.d.127.7		8
4.3	odd	2	inner	192.9.g.d.127.3		8
8.3	odd	2		96.9.g.b.31.6	yes	8
8.5	even	2		96.9.g.b.31.2	✓	8
24.5	odd	2		288.9.g.c.127.6		8
24.11	even	2		288.9.g.c.127.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.9.g.b.31.2	✓	8	8.5	even	2
96.9.g.b.31.6	yes	8	8.3	odd	2
192.9.g.d.127.3		8	4.3	odd	2	inner
192.9.g.d.127.7		8	1.1	even	1	trivial
288.9.g.c.127.5		8	24.11	even	2
288.9.g.c.127.6		8	24.5	odd	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)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} +149.264 q^{5} +3207.68i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} +149.264 q^{5} +3207.68i q^{7} -2187.00 q^{9} +3235.02i q^{11} -176.686 q^{13} +6980.37i q^{15} +80443.0 q^{17} +239555. i q^{19} -150008. q^{21} +85338.9i q^{23} -368345. q^{25} -102276. i q^{27} +310329. q^{29} -494114. i q^{31} -151287. q^{33} +478790. i q^{35} +855708. q^{37} -8262.78i q^{39} -1.41237e6 q^{41} +2.15193e6i q^{43} -326439. q^{45} -1.10493e6i q^{47} -4.52442e6 q^{49} +3.76194e6i q^{51} +1.47339e7 q^{53} +482870. i q^{55} -1.12029e7 q^{57} +1.62594e7i q^{59} -9.65310e6 q^{61} -7.01520e6i q^{63} -26372.7 q^{65} +2.25759e7i q^{67} -3.99091e6 q^{69} -3.04163e7i q^{71} -4.01565e7 q^{73} -1.72258e7i q^{75} -1.03769e7 q^{77} -4.96729e7i q^{79} +4.78297e6 q^{81} -4.44132e7i q^{83} +1.20072e7 q^{85} +1.45126e7i q^{87} -1.11247e8 q^{89} -566752. i q^{91} +2.31074e7 q^{93} +3.57569e7i q^{95} +3.96005e6 q^{97} -7.07498e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 560 q^{5} - 17496 q^{9}+O(q^{10}) \) 8 * q - 560 * q^5 - 17496 * q^9	\( 8 q - 560 q^{5} - 17496 q^{9} + 54256 q^{13} + 436176 q^{17} - 54432 q^{21} + 2456472 q^{25} - 195952 q^{29} - 1065312 q^{33} + 7023408 q^{37} - 12035120 q^{41} + 1224720 q^{45} - 9602040 q^{49} - 27342192 q^{53} + 2744928 q^{57} - 50803280 q^{61} + 34049888 q^{65} + 26687232 q^{69} + 59541648 q^{73} + 178489472 q^{77} + 38263752 q^{81} - 29428704 q^{85} - 170794992 q^{89} + 121951008 q^{93} + 272647184 q^{97}+O(q^{100}) \) 8 * q - 560 * q^5 - 17496 * q^9 + 54256 * q^13 + 436176 * q^17 - 54432 * q^21 + 2456472 * q^25 - 195952 * q^29 - 1065312 * q^33 + 7023408 * q^37 - 12035120 * q^41 + 1224720 * q^45 - 9602040 * q^49 - 27342192 * q^53 + 2744928 * q^57 - 50803280 * q^61 + 34049888 * q^65 + 26687232 * q^69 + 59541648 * q^73 + 178489472 * q^77 + 38263752 * q^81 - 29428704 * q^85 - 170794992 * q^89 + 121951008 * q^93 + 272647184 * q^97

Introduction

L-functions

Modular forms

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