LMFDB - Embedded newform 192.9.g.c.127.3

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:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{1801})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 451x^{2} + 450x + 202500 $ x^4 - x^3 + 451x^2 + 450x + 202500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.3
Root			$10.8595 - 18.8093i$ of defining polynomial
Character	$\chi$	$=$	192.127
Dual form			192.9.g.c.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} -952.517 q^{5} +3101.27i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} -952.517 q^{5} +3101.27i q^{7} -2187.00 q^{9} -7175.83i q^{11} +26897.5 q^{13} -44544.8i q^{15} +146374. q^{17} -220215. i q^{19} -145032. q^{21} -96574.1i q^{23} +516663. q^{25} -102276. i q^{27} -169321. q^{29} +429668. i q^{31} +335580. q^{33} -2.95401e6i q^{35} -2.94894e6 q^{37} +1.25787e6i q^{39} +3.15721e6 q^{41} +4.89907e6i q^{43} +2.08315e6 q^{45} +808105. i q^{47} -3.85305e6 q^{49} +6.84524e6i q^{51} +1.21447e7 q^{53} +6.83510e6i q^{55} +1.02984e7 q^{57} +2.47774e6i q^{59} +6.21934e6 q^{61} -6.78247e6i q^{63} -2.56203e7 q^{65} +1.52139e7i q^{67} +4.51632e6 q^{69} +1.92087e7i q^{71} -3.17283e7 q^{73} +2.41619e7i q^{75} +2.22542e7 q^{77} -5.67932e7i q^{79} +4.78297e6 q^{81} -8.52789e7i q^{83} -1.39424e8 q^{85} -7.91835e6i q^{87} +3.06662e7 q^{89} +8.34163e7i q^{91} -2.00936e7 q^{93} +2.09758e8i q^{95} +5.15412e6 q^{97} +1.56935e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 264 q^{5} - 8748 q^{9}+O(q^{10}) $ 4 * q + 264 * q^5 - 8748 * q^9	$ 4 q + 264 q^{5} - 8748 q^{9} - 14632 q^{13} + 332904 q^{17} - 250128 q^{21} + 2604428 q^{25} - 2343576 q^{29} + 2002320 q^{33} - 4315784 q^{37} + 9035496 q^{41} - 577368 q^{45} + 3458884 q^{49} + 42186600 q^{53} + 2253744 q^{57} + 48148408 q^{61} - 125450832 q^{65} - 23514624 q^{69} - 21215480 q^{73} + 32354496 q^{77} + 19131876 q^{81} - 235297584 q^{85} - 12675576 q^{89} - 193564080 q^{93} + 263153800 q^{97}+O(q^{100}) $ 4 * q + 264 * q^5 - 8748 * q^9 - 14632 * q^13 + 332904 * q^17 - 250128 * q^21 + 2604428 * q^25 - 2343576 * q^29 + 2002320 * q^33 - 4315784 * q^37 + 9035496 * q^41 - 577368 * q^45 + 3458884 * q^49 + 42186600 * q^53 + 2253744 * q^57 + 48148408 * q^61 - 125450832 * q^65 - 23514624 * q^69 - 21215480 * q^73 + 32354496 * q^77 + 19131876 * q^81 - 235297584 * q^85 - 12675576 * q^89 - 193564080 * q^93 + 263153800 * 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
$5$	−952.517	−1.52403	−0.762013	−	0.647561i	$-0.775789\pi$
$5$	−952.517	−1.52403	−0.762013	+	0.647561i	$0.775789\pi$
$6$	0	0
$7$	3101.27i	1.29166i	0.763483	+	0.645828i	$0.223488\pi$
$7$	3101.27i	1.29166i	−0.763483	+	0.645828i	$0.776512\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	− 7175.83i	− 0.490119i	−0.969508	−	0.245059i	$-0.921193\pi$
$11$	− 7175.83i	− 0.490119i	0.969508	−	0.245059i	$-0.0788075\pi$
$12$	0	0
$13$	26897.5	0.941756	0.470878	−	0.882198i	$-0.343937\pi$
$13$	26897.5	0.941756	0.470878	+	0.882198i	$0.343937\pi$
$14$	0	0
$15$	− 44544.8i	− 0.879897i
$16$	0	0
$17$	146374.	1.75254	0.876271	−	0.481819i	$-0.160024\pi$
$17$	146374.	1.75254	0.876271	+	0.481819i	$0.160024\pi$
$18$	0	0
$19$	− 220215.i	− 1.68979i	−0.534935	−	0.844893i	$-0.679664\pi$
$19$	− 220215.i	− 1.68979i	0.534935	−	0.844893i	$-0.320336\pi$
$20$	0	0
$21$	−145032.	−0.745738
$22$	0	0
$23$	− 96574.1i	− 0.345104i	−0.985000	−	0.172552i	$-0.944799\pi$
$23$	− 96574.1i	− 0.345104i	0.985000	−	0.172552i	$-0.0552012\pi$
$24$	0	0
$25$	516663.	1.32266
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	−169321.	−0.239397	−0.119698	−	0.992810i	$-0.538193\pi$
$29$	−169321.	−0.239397	−0.119698	+	0.992810i	$0.538193\pi$
$30$	0	0
$31$	429668.i	0.465250i	0.972567	+	0.232625i	$0.0747314\pi$
$31$	429668.i	0.465250i	−0.972567	+	0.232625i	$0.925269\pi$
$32$	0	0
$33$	335580.	0.282970
$34$	0	0
$35$	− 2.95401e6i	− 1.96852i
$36$	0	0
$37$	−2.94894e6	−1.57347	−0.786737	−	0.617289i	$-0.788231\pi$
$37$	−2.94894e6	−1.57347	−0.786737	+	0.617289i	$0.788231\pi$
$38$	0	0
$39$	1.25787e6i	0.543723i
$40$	0	0
$41$	3.15721e6	1.11729	0.558647	−	0.829406i	$-0.311321\pi$
$41$	3.15721e6	1.11729	0.558647	+	0.829406i	$0.311321\pi$
$42$	0	0
$43$	4.89907e6i	1.43298i	0.697597	+	0.716490i	$0.254252\pi$
$43$	4.89907e6i	1.43298i	−0.697597	+	0.716490i	$0.745748\pi$
$44$	0	0
$45$	2.08315e6	0.508009
$46$	0	0
$47$	808105.i	0.165606i	0.996566	+	0.0828031i	$0.0263873\pi$
$47$	808105.i	0.165606i	−0.996566	+	0.0828031i	$0.973613\pi$
$48$	0	0
$49$	−3.85305e6	−0.668375
$50$	0	0
$51$	6.84524e6i	1.01183i
$52$	0	0
$53$	1.21447e7	1.53916	0.769579	−	0.638551i	$-0.220466\pi$
$53$	1.21447e7	1.53916	0.769579	+	0.638551i	$0.220466\pi$
$54$	0	0
$55$	6.83510e6i	0.746954i
$56$	0	0
$57$	1.02984e7	0.975598
$58$	0	0
$59$	2.47774e6i	0.204478i	0.994760	+	0.102239i	$0.0326007\pi$
$59$	2.47774e6i	0.204478i	−0.994760	+	0.102239i	$0.967399\pi$
$60$	0	0
$61$	6.21934e6	0.449184	0.224592	−	0.974453i	$-0.427895\pi$
$61$	6.21934e6	0.449184	0.224592	+	0.974453i	$0.427895\pi$
$62$	0	0
$63$	− 6.78247e6i	− 0.430552i
$64$	0	0
$65$	−2.56203e7	−1.43526
$66$	0	0
$67$	1.52139e7i	0.754989i	0.926012	+	0.377494i	$0.123214\pi$
$67$	1.52139e7i	0.754989i	−0.926012	+	0.377494i	$0.876786\pi$
$68$	0	0
$69$	4.51632e6	0.199246
$70$	0	0
$71$	1.92087e7i	0.755899i	0.925826	+	0.377949i	$0.123371\pi$
$71$	1.92087e7i	0.755899i	−0.925826	+	0.377949i	$0.876629\pi$
$72$	0	0
$73$	−3.17283e7	−1.11726	−0.558631	−	0.829416i	$-0.688673\pi$
$73$	−3.17283e7	−1.11726	−0.558631	+	0.829416i	$0.688673\pi$
$74$	0	0
$75$	2.41619e7i	0.763636i
$76$	0	0
$77$	2.22542e7	0.633065
$78$	0	0
$79$	− 5.67932e7i	− 1.45810i	−0.684460	−	0.729051i	$-0.739962\pi$
$79$	− 5.67932e7i	− 1.45810i	0.684460	−	0.729051i	$-0.260038\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	− 8.52789e7i	− 1.79692i	−0.439053	−	0.898461i	$-0.644686\pi$
$83$	− 8.52789e7i	− 1.79692i	0.439053	−	0.898461i	$-0.355314\pi$
$84$	0	0
$85$	−1.39424e8	−2.67092
$86$	0	0
$87$	− 7.91835e6i	− 0.138216i
$88$	0	0
$89$	3.06662e7	0.488765	0.244383	−	0.969679i	$-0.421415\pi$
$89$	3.06662e7	0.488765	0.244383	+	0.969679i	$0.421415\pi$
$90$	0	0
$91$	8.34163e7i	1.21642i
$92$	0	0
$93$	−2.00936e7	−0.268612
$94$	0	0
$95$	2.09758e8i	2.57528i
$96$	0	0
$97$	5.15412e6	0.0582194	0.0291097	−	0.999576i	$-0.490733\pi$
$97$	5.15412e6	0.0582194	0.0291097	+	0.999576i	$0.490733\pi$
$98$	0	0
$99$	1.56935e7i	0.163373i
$100$	0	0
$101$	1.01284e8	0.973316	0.486658	−	0.873593i	$-0.338216\pi$
$101$	1.01284e8	0.973316	0.486658	+	0.873593i	$0.338216\pi$
$102$	0	0
$103$	2.04461e8i	1.81661i	0.418310	+	0.908304i	$0.362623\pi$
$103$	2.04461e8i	1.81661i	−0.418310	+	0.908304i	$0.637377\pi$
$104$	0	0
$105$	1.38145e8	1.13652
$106$	0	0
$107$	1.61055e8i	1.22868i	0.789041	+	0.614340i	$0.210578\pi$
$107$	1.61055e8i	1.22868i	−0.789041	+	0.614340i	$0.789422\pi$
$108$	0	0
$109$	−5.73094e7	−0.405994	−0.202997	−	0.979179i	$-0.565068\pi$
$109$	−5.73094e7	−0.405994	−0.202997	+	0.979179i	$0.565068\pi$
$110$	0	0
$111$	− 1.37908e8i	− 0.908445i
$112$	0	0
$113$	1.49424e7	0.0916446	0.0458223	−	0.998950i	$-0.485409\pi$
$113$	1.49424e7	0.0916446	0.0458223	+	0.998950i	$0.485409\pi$
$114$	0	0
$115$	9.19884e7i	0.525947i
$116$	0	0
$117$	−5.88248e7	−0.313919
$118$	0	0
$119$	4.53945e8i	2.26368i
$120$	0	0
$121$	1.62866e8	0.759784
$122$	0	0
$123$	1.47648e8i	0.645070i
$124$	0	0
$125$	−1.20053e8	−0.491738
$126$	0	0
$127$	− 8.68934e7i	− 0.334019i	−0.985955	−	0.167010i	$-0.946589\pi$
$127$	− 8.68934e7i	− 0.334019i	0.985955	−	0.167010i	$-0.0534111\pi$
$128$	0	0
$129$	−2.29107e8	−0.827331
$130$	0	0
$131$	2.93118e8i	0.995308i	0.867376	+	0.497654i	$0.165805\pi$
$131$	2.93118e8i	0.995308i	−0.867376	+	0.497654i	$0.834195\pi$
$132$	0	0
$133$	6.82944e8	2.18262
$134$	0	0
$135$	9.74195e7i	0.293299i
$136$	0	0
$137$	−4.93337e8	−1.40043	−0.700215	−	0.713932i	$-0.746913\pi$
$137$	−4.93337e8	−1.40043	−0.700215	+	0.713932i	$0.746913\pi$
$138$	0	0
$139$	5.21073e8i	1.39585i	0.716170	+	0.697926i	$0.245894\pi$
$139$	5.21073e8i	1.39585i	−0.716170	+	0.697926i	$0.754106\pi$
$140$	0	0
$141$	−3.77913e7	−0.0956128
$142$	0	0
$143$	− 1.93012e8i	− 0.461572i
$144$	0	0
$145$	1.61281e8	0.364847
$146$	0	0
$147$	− 1.80189e8i	− 0.385886i
$148$	0	0
$149$	4.09228e8	0.830271	0.415135	−	0.909760i	$-0.363734\pi$
$149$	4.09228e8	0.830271	0.415135	+	0.909760i	$0.363734\pi$
$150$	0	0
$151$	8.92557e6i	0.0171683i	0.999963	+	0.00858417i	$0.00273246\pi$
$151$	8.92557e6i	0.0171683i	−0.999963	+	0.00858417i	$0.997268\pi$
$152$	0	0
$153$	−3.20120e8	−0.584181
$154$	0	0
$155$	− 4.09266e8i	− 0.709053i
$156$	0	0
$157$	−2.58929e8	−0.426169	−0.213085	−	0.977034i	$-0.568351\pi$
$157$	−2.58929e8	−0.426169	−0.213085	+	0.977034i	$0.568351\pi$
$158$	0	0
$159$	5.67952e8i	0.888634i
$160$	0	0
$161$	2.99502e8	0.445755
$162$	0	0
$163$	− 2.01640e8i	− 0.285645i	−0.989748	−	0.142823i	$-0.954382\pi$
$163$	− 2.01640e8i	− 0.285645i	0.989748	−	0.142823i	$-0.0456179\pi$
$164$	0	0
$165$	−3.19646e8	−0.431254
$166$	0	0
$167$	− 1.45117e9i	− 1.86575i	−0.360202	−	0.932874i	$-0.617292\pi$
$167$	− 1.45117e9i	− 1.86575i	0.360202	−	0.932874i	$-0.382708\pi$
$168$	0	0
$169$	−9.22554e7	−0.113095
$170$	0	0
$171$	4.81609e8i	0.563262i
$172$	0	0
$173$	−6.36420e8	−0.710492	−0.355246	−	0.934773i	$-0.615603\pi$
$173$	−6.36420e8	−0.710492	−0.355246	+	0.934773i	$0.615603\pi$
$174$	0	0
$175$	1.60231e9i	1.70842i
$176$	0	0
$177$	−1.15872e8	−0.118056
$178$	0	0
$179$	1.84104e9i	1.79329i	0.442751	+	0.896644i	$0.354002\pi$
$179$	1.84104e9i	1.79329i	−0.442751	+	0.896644i	$0.645998\pi$
$180$	0	0
$181$	7.25192e8	0.675676	0.337838	−	0.941204i	$-0.390304\pi$
$181$	7.25192e8	0.675676	0.337838	+	0.941204i	$0.390304\pi$
$182$	0	0
$183$	2.90850e8i	0.259337i
$184$	0	0
$185$	2.80892e9	2.39802
$186$	0	0
$187$	− 1.05036e9i	− 0.858954i
$188$	0	0
$189$	3.17185e8	0.248579
$190$	0	0
$191$	1.59853e9i	1.20112i	0.799579	+	0.600561i	$0.205056\pi$
$191$	1.59853e9i	1.20112i	−0.799579	+	0.600561i	$0.794944\pi$
$192$	0	0
$193$	−5.05071e8	−0.364018	−0.182009	−	0.983297i	$-0.558260\pi$
$193$	−5.05071e8	−0.364018	−0.182009	+	0.983297i	$0.558260\pi$
$194$	0	0
$195$	− 1.19814e9i	− 0.828649i
$196$	0	0
$197$	−2.23833e9	−1.48614	−0.743070	−	0.669214i	$-0.766631\pi$
$197$	−2.23833e9	−1.48614	−0.743070	+	0.669214i	$0.766631\pi$
$198$	0	0
$199$	1.11582e9i	0.711510i	0.934579	+	0.355755i	$0.115776\pi$
$199$	1.11582e9i	0.711510i	−0.934579	+	0.355755i	$0.884224\pi$
$200$	0	0
$201$	−7.11482e8	−0.435893
$202$	0	0
$203$	− 5.25109e8i	− 0.309218i
$204$	0	0
$205$	−3.00729e9	−1.70279
$206$	0	0
$207$	2.11208e8i	0.115035i
$208$	0	0
$209$	−1.58022e9	−0.828196
$210$	0	0
$211$	2.94087e7i	0.0148370i	0.999972	+	0.00741850i	$0.00236140\pi$
$211$	2.94087e7i	0.0148370i	−0.999972	+	0.00741850i	$0.997639\pi$
$212$	0	0
$213$	−8.98300e8	−0.436418
$214$	0	0
$215$	− 4.66645e9i	− 2.18390i
$216$	0	0
$217$	−1.33251e9	−0.600942
$218$	0	0
$219$	− 1.48378e9i	− 0.645051i
$220$	0	0
$221$	3.93709e9	1.65047
$222$	0	0
$223$	− 3.32442e9i	− 1.34430i	−0.740414	−	0.672151i	$-0.765371\pi$
$223$	− 3.32442e9i	− 1.34430i	0.740414	−	0.672151i	$-0.234629\pi$
$224$	0	0
$225$	−1.12994e9	−0.440886
$226$	0	0
$227$	3.68532e8i	0.138794i	0.997589	+	0.0693971i	$0.0221076\pi$
$227$	3.68532e8i	0.138794i	−0.997589	+	0.0693971i	$0.977892\pi$
$228$	0	0
$229$	1.32990e9	0.483591	0.241795	−	0.970327i	$-0.422264\pi$
$229$	1.32990e9	0.483591	0.241795	+	0.970327i	$0.422264\pi$
$230$	0	0
$231$	1.04072e9i	0.365500i
$232$	0	0
$233$	2.49591e9	0.846846	0.423423	−	0.905932i	$-0.360828\pi$
$233$	2.49591e9	0.846846	0.423423	+	0.905932i	$0.360828\pi$
$234$	0	0
$235$	− 7.69734e8i	− 0.252388i
$236$	0	0
$237$	2.65595e9	0.841835
$238$	0	0
$239$	5.77198e7i	0.0176902i	0.999961	+	0.00884511i	$0.00281552\pi$
$239$	5.77198e7i	0.0176902i	−0.999961	+	0.00884511i	$0.997184\pi$
$240$	0	0
$241$	−1.57199e9	−0.465996	−0.232998	−	0.972477i	$-0.574854\pi$
$241$	−1.57199e9	−0.465996	−0.232998	+	0.972477i	$0.574854\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	3.67009e9	1.01862
$246$	0	0
$247$	− 5.92322e9i	− 1.59137i
$248$	0	0
$249$	3.98810e9	1.03745
$250$	0	0
$251$	− 2.63479e9i	− 0.663822i	−0.943311	−	0.331911i	$-0.892307\pi$
$251$	− 2.63479e9i	− 0.663822i	0.943311	−	0.331911i	$-0.107693\pi$
$252$	0	0
$253$	−6.92999e8	−0.169142
$254$	0	0
$255$	− 6.52020e9i	− 1.54206i
$256$	0	0
$257$	3.67876e9	0.843276	0.421638	−	0.906764i	$-0.361455\pi$
$257$	3.67876e9	0.843276	0.421638	+	0.906764i	$0.361455\pi$
$258$	0	0
$259$	− 9.14545e9i	− 2.03239i
$260$	0	0
$261$	3.70304e8	0.0797989
$262$	0	0
$263$	− 1.21643e9i	− 0.254252i	−0.991887	−	0.127126i	$-0.959425\pi$
$263$	− 1.21643e9i	− 0.254252i	0.991887	−	0.127126i	$-0.0405753\pi$
$264$	0	0
$265$	−1.15680e10	−2.34572
$266$	0	0
$267$	1.43412e9i	0.282189i
$268$	0	0
$269$	9.26695e9	1.76981	0.884907	−	0.465768i	$-0.154222\pi$
$269$	9.26695e9	1.76981	0.884907	+	0.465768i	$0.154222\pi$
$270$	0	0
$271$	− 3.54288e7i	− 0.00656871i	−0.999995	−	0.00328435i	$-0.998955\pi$
$271$	− 3.54288e7i	− 0.00656871i	0.999995	−	0.00328435i	$-0.00104544\pi$
$272$	0	0
$273$	−3.90099e9	−0.702303
$274$	0	0
$275$	− 3.70748e9i	− 0.648259i
$276$	0	0
$277$	−5.73085e9	−0.973419	−0.486709	−	0.873564i	$-0.661803\pi$
$277$	−5.73085e9	−0.973419	−0.486709	+	0.873564i	$0.661803\pi$
$278$	0	0
$279$	− 9.39683e8i	− 0.155083i
$280$	0	0
$281$	6.79380e9	1.08965	0.544825	−	0.838550i	$-0.316596\pi$
$281$	6.79380e9	1.08965	0.544825	+	0.838550i	$0.316596\pi$
$282$	0	0
$283$	5.67140e9i	0.884188i	0.896969	+	0.442094i	$0.145764\pi$
$283$	5.67140e9i	0.884188i	−0.896969	+	0.442094i	$0.854236\pi$
$284$	0	0
$285$	−9.80941e9	−1.48684
$286$	0	0
$287$	9.79133e9i	1.44316i
$288$	0	0
$289$	1.44496e10	2.07140
$290$	0	0
$291$	2.41034e8i	0.0336130i
$292$	0	0
$293$	1.52415e9	0.206803	0.103402	−	0.994640i	$-0.467027\pi$
$293$	1.52415e9	0.206803	0.103402	+	0.994640i	$0.467027\pi$
$294$	0	0
$295$	− 2.36009e9i	− 0.311631i
$296$	0	0
$297$	−7.33914e8	−0.0943234
$298$	0	0
$299$	− 2.59760e9i	− 0.325003i
$300$	0	0
$301$	−1.51933e10	−1.85092
$302$	0	0
$303$	4.73657e9i	0.561944i
$304$	0	0
$305$	−5.92402e9	−0.684569
$306$	0	0
$307$	5.36906e9i	0.604428i	0.953240	+	0.302214i	$0.0977257\pi$
$307$	5.36906e9i	0.604428i	−0.953240	+	0.302214i	$0.902274\pi$
$308$	0	0
$309$	−9.56169e9	−1.04882
$310$	0	0
$311$	6.21354e9i	0.664198i	0.943245	+	0.332099i	$0.107757\pi$
$311$	6.21354e9i	0.664198i	−0.943245	+	0.332099i	$0.892243\pi$
$312$	0	0
$313$	2.95026e8	0.0307385	0.0153693	−	0.999882i	$-0.495108\pi$
$313$	2.95026e8	0.0307385	0.0153693	+	0.999882i	$0.495108\pi$
$314$	0	0
$315$	6.46041e9i	0.656173i
$316$	0	0
$317$	−1.22463e9	−0.121274	−0.0606369	−	0.998160i	$-0.519313\pi$
$317$	−1.22463e9	−0.121274	−0.0606369	+	0.998160i	$0.519313\pi$
$318$	0	0
$319$	1.21502e9i	0.117333i
$320$	0	0
$321$	−7.53180e9	−0.709379
$322$	0	0
$323$	− 3.22337e10i	− 2.96142i
$324$	0	0
$325$	1.38969e10	1.24562
$326$	0	0
$327$	− 2.68010e9i	− 0.234401i
$328$	0	0
$329$	−2.50615e9	−0.213906
$330$	0	0
$331$	1.49808e10i	1.24802i	0.781415	+	0.624012i	$0.214498\pi$
$331$	1.49808e10i	1.24802i	−0.781415	+	0.624012i	$0.785502\pi$
$332$	0	0
$333$	6.44934e9	0.524491
$334$	0	0
$335$	− 1.44915e10i	− 1.15062i
$336$	0	0
$337$	1.22500e10	0.949768	0.474884	−	0.880049i	$-0.342490\pi$
$337$	1.22500e10	0.949768	0.474884	+	0.880049i	$0.342490\pi$
$338$	0	0
$339$	6.98787e8i	0.0529110i
$340$	0	0
$341$	3.08322e9	0.228028
$342$	0	0
$343$	5.92885e9i	0.428346i
$344$	0	0
$345$	−4.30187e9	−0.303656
$346$	0	0
$347$	9.66765e9i	0.666811i	0.942783	+	0.333406i	$0.108198\pi$
$347$	9.66765e9i	0.666811i	−0.942783	+	0.333406i	$0.891802\pi$
$348$	0	0
$349$	8.41057e9	0.566923	0.283461	−	0.958984i	$-0.408517\pi$
$349$	8.41057e9	0.566923	0.283461	+	0.958984i	$0.408517\pi$
$350$	0	0
$351$	− 2.75096e9i	− 0.181241i
$352$	0	0
$353$	1.96120e10	1.26305	0.631527	−	0.775354i	$-0.282428\pi$
$353$	1.96120e10	1.26305	0.631527	+	0.775354i	$0.282428\pi$
$354$	0	0
$355$	− 1.82966e10i	− 1.15201i
$356$	0	0
$357$	−2.12289e10	−1.30694
$358$	0	0
$359$	− 3.86050e9i	− 0.232416i	−0.993225	−	0.116208i	$-0.962926\pi$
$359$	− 3.86050e9i	− 0.232416i	0.993225	−	0.116208i	$-0.0370739\pi$
$360$	0	0
$361$	−3.15109e10	−1.85538
$362$	0	0
$363$	7.61651e9i	0.438661i
$364$	0	0
$365$	3.02217e10	1.70274
$366$	0	0
$367$	4.07167e9i	0.224444i	0.993683	+	0.112222i	$0.0357968\pi$
$367$	4.07167e9i	0.224444i	−0.993683	+	0.112222i	$0.964203\pi$
$368$	0	0
$369$	−6.90481e9	−0.372431
$370$	0	0
$371$	3.76639e10i	1.98806i
$372$	0	0
$373$	−9.73315e8	−0.0502827	−0.0251413	−	0.999684i	$-0.508004\pi$
$373$	−9.73315e8	−0.0502827	−0.0251413	+	0.999684i	$0.508004\pi$
$374$	0	0
$375$	− 5.61433e9i	− 0.283905i
$376$	0	0
$377$	−4.55430e9	−0.225453
$378$	0	0
$379$	1.03474e10i	0.501506i	0.968051	+	0.250753i	$0.0806781\pi$
$379$	1.03474e10i	0.501506i	−0.968051	+	0.250753i	$0.919322\pi$
$380$	0	0
$381$	4.06360e9	0.192846
$382$	0	0
$383$	1.47254e10i	0.684342i	0.939638	+	0.342171i	$0.111162\pi$
$383$	1.47254e10i	0.684342i	−0.939638	+	0.342171i	$0.888838\pi$
$384$	0	0
$385$	−2.11974e10	−0.964808
$386$	0	0
$387$	− 1.07143e10i	− 0.477660i
$388$	0	0
$389$	2.12190e10	0.926673	0.463336	−	0.886183i	$-0.346652\pi$
$389$	2.12190e10	0.926673	0.463336	+	0.886183i	$0.346652\pi$
$390$	0	0
$391$	− 1.41359e10i	− 0.604808i
$392$	0	0
$393$	−1.37078e10	−0.574641
$394$	0	0
$395$	5.40964e10i	2.22219i
$396$	0	0
$397$	−2.05017e10	−0.825331	−0.412665	−	0.910883i	$-0.635402\pi$
$397$	−2.05017e10	−0.825331	−0.412665	+	0.910883i	$0.635402\pi$
$398$	0	0
$399$	3.19381e10i	1.26014i
$400$	0	0
$401$	3.20594e10	1.23987	0.619937	−	0.784652i	$-0.287158\pi$
$401$	3.20594e10	1.23987	0.619937	+	0.784652i	$0.287158\pi$
$402$	0	0
$403$	1.15570e10i	0.438152i
$404$	0	0
$405$	−4.55586e9	−0.169336
$406$	0	0
$407$	2.11611e10i	0.771189i
$408$	0	0
$409$	2.67550e10	0.956117	0.478058	−	0.878328i	$-0.341341\pi$
$409$	2.67550e10	0.956117	0.478058	+	0.878328i	$0.341341\pi$
$410$	0	0
$411$	− 2.30711e10i	− 0.808539i
$412$	0	0
$413$	−7.68413e9	−0.264116
$414$	0	0
$415$	8.12296e10i	2.73856i
$416$	0	0
$417$	−2.43682e10	−0.805895
$418$	0	0
$419$	− 1.08653e10i	− 0.352521i	−0.984344	−	0.176261i	$-0.943600\pi$
$419$	− 1.08653e10i	− 0.352521i	0.984344	−	0.176261i	$-0.0564002\pi$
$420$	0	0
$421$	−2.03721e9	−0.0648495	−0.0324247	−	0.999474i	$-0.510323\pi$
$421$	−2.03721e9	−0.0648495	−0.0324247	+	0.999474i	$0.510323\pi$
$422$	0	0
$423$	− 1.76733e9i	− 0.0552021i
$424$	0	0
$425$	7.56260e10	2.31801
$426$	0	0
$427$	1.92878e10i	0.580192i
$428$	0	0
$429$	9.02627e9	0.266489
$430$	0	0
$431$	3.19963e10i	0.927236i	0.886035	+	0.463618i	$0.153449\pi$
$431$	3.19963e10i	0.927236i	−0.886035	+	0.463618i	$0.846551\pi$
$432$	0	0
$433$	−2.78313e10	−0.791738	−0.395869	−	0.918307i	$-0.629557\pi$
$433$	−2.78313e10	−0.791738	−0.395869	+	0.918307i	$0.629557\pi$
$434$	0	0
$435$	7.54236e9i	0.210644i
$436$	0	0
$437$	−2.12670e10	−0.583151
$438$	0	0
$439$	− 3.88247e10i	− 1.04532i	−0.852540	−	0.522661i	$-0.824939\pi$
$439$	− 3.88247e10i	− 1.04532i	0.852540	−	0.522661i	$-0.175061\pi$
$440$	0	0
$441$	8.42662e9	0.222792
$442$	0	0
$443$	4.68689e10i	1.21694i	0.793576	+	0.608471i	$0.208217\pi$
$443$	4.68689e10i	1.21694i	−0.793576	+	0.608471i	$0.791783\pi$
$444$	0	0
$445$	−2.92101e10	−0.744891
$446$	0	0
$447$	1.91377e10i	0.479357i
$448$	0	0
$449$	3.48851e10	0.858331	0.429166	−	0.903226i	$-0.358808\pi$
$449$	3.48851e10	0.858331	0.429166	+	0.903226i	$0.358808\pi$
$450$	0	0
$451$	− 2.26556e10i	− 0.547607i
$452$	0	0
$453$	−4.17408e8	−0.00991214
$454$	0	0
$455$	− 7.94554e10i	− 1.85386i
$456$	0	0
$457$	−7.46508e9	−0.171147	−0.0855736	−	0.996332i	$-0.527272\pi$
$457$	−7.46508e9	−0.171147	−0.0855736	+	0.996332i	$0.527272\pi$
$458$	0	0
$459$	− 1.49705e10i	− 0.337277i
$460$	0	0
$461$	5.60087e10	1.24008	0.620042	−	0.784568i	$-0.287115\pi$
$461$	5.60087e10	1.24008	0.620042	+	0.784568i	$0.287115\pi$
$462$	0	0
$463$	4.51649e10i	0.982826i	0.870927	+	0.491413i	$0.163519\pi$
$463$	4.51649e10i	0.982826i	−0.870927	+	0.491413i	$0.836481\pi$
$464$	0	0
$465$	1.91395e10	0.409372
$466$	0	0
$467$	− 1.30108e10i	− 0.273550i	−0.990602	−	0.136775i	$-0.956326\pi$
$467$	− 1.30108e10i	− 0.273550i	0.990602	−	0.136775i	$-0.0436737\pi$
$468$	0	0
$469$	−4.71822e10	−0.975186
$470$	0	0
$471$	− 1.21089e10i	− 0.246049i
$472$	0	0
$473$	3.51549e10	0.702330
$474$	0	0
$475$	− 1.13777e11i	− 2.23501i
$476$	0	0
$477$	−2.65605e10	−0.513053
$478$	0	0
$479$	5.63024e9i	0.106951i	0.998569	+	0.0534755i	$0.0170299\pi$
$479$	5.63024e9i	0.106951i	−0.998569	+	0.0534755i	$0.982970\pi$
$480$	0	0
$481$	−7.93192e10	−1.48183
$482$	0	0
$483$	1.40063e10i	0.257357i
$484$	0	0
$485$	−4.90939e9	−0.0887279
$486$	0	0
$487$	2.21903e10i	0.394501i	0.980353	+	0.197250i	$0.0632012\pi$
$487$	2.21903e10i	0.394501i	−0.980353	+	0.197250i	$0.936799\pi$
$488$	0	0
$489$	9.42979e9	0.164917
$490$	0	0
$491$	2.38353e10i	0.410104i	0.978751	+	0.205052i	$0.0657364\pi$
$491$	2.38353e10i	0.410104i	−0.978751	+	0.205052i	$0.934264\pi$
$492$	0	0
$493$	−2.47842e10	−0.419553
$494$	0	0
$495$	− 1.49484e10i	− 0.248985i
$496$	0	0
$497$	−5.95712e10	−0.976361
$498$	0	0
$499$	8.49679e10i	1.37042i	0.728347	+	0.685209i	$0.240289\pi$
$499$	8.49679e10i	1.37042i	−0.728347	+	0.685209i	$0.759711\pi$
$500$	0	0
$501$	6.78646e10	1.07719
$502$	0	0
$503$	− 8.35912e10i	− 1.30584i	−0.757428	−	0.652918i	$-0.773544\pi$
$503$	− 8.35912e10i	− 1.30584i	0.757428	−	0.652918i	$-0.226456\pi$
$504$	0	0
$505$	−9.64744e10	−1.48336
$506$	0	0
$507$	− 4.31436e9i	− 0.0652956i
$508$	0	0
$509$	4.04693e10	0.602912	0.301456	−	0.953480i	$-0.402527\pi$
$509$	4.04693e10	0.602912	0.301456	+	0.953480i	$0.402527\pi$
$510$	0	0
$511$	− 9.83978e10i	− 1.44312i
$512$	0	0
$513$	−2.25226e10	−0.325199
$514$	0	0
$515$	− 1.94752e11i	− 2.76856i
$516$	0	0
$517$	5.79883e9	0.0811667
$518$	0	0
$519$	− 2.97624e10i	− 0.410203i
$520$	0	0
$521$	−8.68115e10	−1.17822	−0.589110	−	0.808053i	$-0.700522\pi$
$521$	−8.68115e10	−1.17822	−0.589110	+	0.808053i	$0.700522\pi$
$522$	0	0
$523$	1.28187e11i	1.71332i	0.515881	+	0.856660i	$0.327465\pi$
$523$	1.28187e11i	1.71332i	−0.515881	+	0.856660i	$0.672535\pi$
$524$	0	0
$525$	−7.49326e10	−0.986355
$526$	0	0
$527$	6.28922e10i	0.815369i
$528$	0	0
$529$	6.89844e10	0.880904
$530$	0	0
$531$	− 5.41882e9i	− 0.0681595i
$532$	0	0
$533$	8.49209e10	1.05222
$534$	0	0
$535$	− 1.53408e11i	− 1.87254i
$536$	0	0
$537$	−8.60968e10	−1.03536
$538$	0	0
$539$	2.76488e10i	0.327583i
$540$	0	0
$541$	9.63677e10	1.12497	0.562487	−	0.826806i	$-0.309845\pi$
$541$	9.63677e10	1.12497	0.562487	+	0.826806i	$0.309845\pi$
$542$	0	0
$543$	3.39139e10i	0.390102i
$544$	0	0
$545$	5.45882e10	0.618746
$546$	0	0
$547$	− 1.13025e11i	− 1.26248i	−0.775587	−	0.631241i	$-0.782546\pi$
$547$	− 1.13025e11i	− 1.26248i	0.775587	−	0.631241i	$-0.217454\pi$
$548$	0	0
$549$	−1.36017e10	−0.149728
$550$	0	0
$551$	3.72869e10i	0.404529i
$552$	0	0
$553$	1.76131e11	1.88337
$554$	0	0
$555$	1.31360e11i	1.38449i
$556$	0	0
$557$	−1.42898e11	−1.48458	−0.742291	−	0.670078i	$-0.766261\pi$
$557$	−1.42898e11	−1.48458	−0.742291	+	0.670078i	$0.766261\pi$
$558$	0	0
$559$	1.31773e11i	1.34952i
$560$	0	0
$561$	4.91202e10	0.495917
$562$	0	0
$563$	− 8.62444e10i	− 0.858415i	−0.903206	−	0.429207i	$-0.858793\pi$
$563$	− 8.62444e10i	− 0.858415i	0.903206	−	0.429207i	$-0.141207\pi$
$564$	0	0
$565$	−1.42329e10	−0.139669
$566$	0	0
$567$	1.48333e10i	0.143517i
$568$	0	0
$569$	−7.08653e10	−0.676059	−0.338029	−	0.941136i	$-0.609760\pi$
$569$	−7.08653e10	−0.676059	−0.338029	+	0.941136i	$0.609760\pi$
$570$	0	0
$571$	1.62595e11i	1.52955i	0.644298	+	0.764774i	$0.277150\pi$
$571$	1.62595e11i	1.52955i	−0.644298	+	0.764774i	$0.722850\pi$
$572$	0	0
$573$	−7.47558e10	−0.693468
$574$	0	0
$575$	− 4.98963e10i	− 0.456453i
$576$	0	0
$577$	7.57766e10	0.683647	0.341824	−	0.939764i	$-0.388955\pi$
$577$	7.57766e10	0.683647	0.341824	+	0.939764i	$0.388955\pi$
$578$	0	0
$579$	− 2.36198e10i	− 0.210166i
$580$	0	0
$581$	2.64473e11	2.32101
$582$	0	0
$583$	− 8.71483e10i	− 0.754371i
$584$	0	0
$585$	5.60316e10	0.478420
$586$	0	0
$587$	− 1.92106e11i	− 1.61804i	−0.587784	−	0.809018i	$-0.700000\pi$
$587$	− 1.92106e11i	− 1.61804i	0.587784	−	0.809018i	$-0.300000\pi$
$588$	0	0
$589$	9.46191e10	0.786172
$590$	0	0
$591$	− 1.04676e11i	− 0.858023i
$592$	0	0
$593$	−5.12974e10	−0.414836	−0.207418	−	0.978252i	$-0.566506\pi$
$593$	−5.12974e10	−0.414836	−0.207418	+	0.978252i	$0.566506\pi$
$594$	0	0
$595$	− 4.32390e11i	− 3.44991i
$596$	0	0
$597$	−5.21817e10	−0.410791
$598$	0	0
$599$	6.95617e10i	0.540335i	0.962813	+	0.270167i	$0.0870790\pi$
$599$	6.95617e10i	0.540335i	−0.962813	+	0.270167i	$0.912921\pi$
$600$	0	0
$601$	1.46315e11	1.12148	0.560739	−	0.827993i	$-0.310517\pi$
$601$	1.46315e11	1.12148	0.560739	+	0.827993i	$0.310517\pi$
$602$	0	0
$603$	− 3.32727e10i	− 0.251663i
$604$	0	0
$605$	−1.55133e11	−1.15793
$606$	0	0
$607$	− 6.48265e10i	− 0.477527i	−0.971078	−	0.238763i	$-0.923258\pi$
$607$	− 6.48265e10i	− 0.477527i	0.971078	−	0.238763i	$-0.0767420\pi$
$608$	0	0
$609$	2.45569e10	0.178527
$610$	0	0
$611$	2.17360e10i	0.155961i
$612$	0	0
$613$	−5.62833e9	−0.0398600	−0.0199300	−	0.999801i	$-0.506344\pi$
$613$	−5.62833e9	−0.0398600	−0.0199300	+	0.999801i	$0.506344\pi$
$614$	0	0
$615$	− 1.40637e11i	− 0.983104i
$616$	0	0
$617$	1.70885e11	1.17913	0.589567	−	0.807719i	$-0.299298\pi$
$617$	1.70885e11	1.17913	0.589567	+	0.807719i	$0.299298\pi$
$618$	0	0
$619$	2.59388e10i	0.176680i	0.996090	+	0.0883401i	$0.0281562\pi$
$619$	2.59388e10i	0.176680i	−0.996090	+	0.0883401i	$0.971844\pi$
$620$	0	0
$621$	−9.87720e9	−0.0664152
$622$	0	0
$623$	9.51041e10i	0.631316i
$624$	0	0
$625$	−8.74688e10	−0.573236
$626$	0	0
$627$	− 7.38997e10i	− 0.478159i
$628$	0	0
$629$	−4.31649e11	−2.75758
$630$	0	0
$631$	3.47218e9i	0.0219021i	0.999940	+	0.0109510i	$0.00348589\pi$
$631$	3.47218e9i	0.0219021i	−0.999940	+	0.0109510i	$0.996514\pi$
$632$	0	0
$633$	−1.37531e9	−0.00856615
$634$	0	0
$635$	8.27674e10i	0.509055i
$636$	0	0
$637$	−1.03637e11	−0.629446
$638$	0	0
$639$	− 4.20093e10i	− 0.251966i
$640$	0	0
$641$	−1.20527e11	−0.713927	−0.356963	−	0.934118i	$-0.616188\pi$
$641$	−1.20527e11	−0.713927	−0.356963	+	0.934118i	$0.616188\pi$
$642$	0	0
$643$	1.04525e11i	0.611470i	0.952117	+	0.305735i	$0.0989021\pi$
$643$	1.04525e11i	0.611470i	−0.952117	+	0.305735i	$0.901098\pi$
$644$	0	0
$645$	2.18228e11	1.26087
$646$	0	0
$647$	4.75874e10i	0.271566i	0.990739	+	0.135783i	$0.0433550\pi$
$647$	4.75874e10i	0.271566i	−0.990739	+	0.135783i	$0.956645\pi$
$648$	0	0
$649$	1.77798e10	0.100219
$650$	0	0
$651$	− 6.23155e10i	− 0.346954i
$652$	0	0
$653$	1.53529e11	0.844381	0.422190	−	0.906507i	$-0.361261\pi$
$653$	1.53529e11	0.844381	0.422190	+	0.906507i	$0.361261\pi$
$654$	0	0
$655$	− 2.79200e11i	− 1.51688i
$656$	0	0
$657$	6.93897e10	0.372421
$658$	0	0
$659$	1.08583e11i	0.575731i	0.957671	+	0.287865i	$0.0929456\pi$
$659$	1.08583e11i	0.575731i	−0.957671	+	0.287865i	$0.907054\pi$
$660$	0	0
$661$	−1.09736e11	−0.574834	−0.287417	−	0.957806i	$-0.592796\pi$
$661$	−1.09736e11	−0.574834	−0.287417	+	0.957806i	$0.592796\pi$
$662$	0	0
$663$	1.84120e11i	0.952897i
$664$	0	0
$665$	−6.50515e11	−3.32637
$666$	0	0
$667$	1.63520e10i	0.0826166i
$668$	0	0
$669$	1.55468e11	0.776133
$670$	0	0
$671$	− 4.46289e10i	− 0.220154i
$672$	0	0
$673$	−2.53333e11	−1.23490	−0.617449	−	0.786611i	$-0.711834\pi$
$673$	−2.53333e11	−1.23490	−0.617449	+	0.786611i	$0.711834\pi$
$674$	0	0
$675$	− 5.28421e10i	− 0.254545i
$676$	0	0
$677$	−8.20339e10	−0.390516	−0.195258	−	0.980752i	$-0.562554\pi$
$677$	−8.20339e10	−0.390516	−0.195258	+	0.980752i	$0.562554\pi$
$678$	0	0
$679$	1.59843e10i	0.0751994i
$680$	0	0
$681$	−1.72345e10	−0.0801329
$682$	0	0
$683$	− 3.43876e10i	− 0.158022i	−0.996874	−	0.0790112i	$-0.974824\pi$
$683$	− 3.43876e10i	− 0.158022i	0.996874	−	0.0790112i	$-0.0251763\pi$
$684$	0	0
$685$	4.69912e11	2.13429
$686$	0	0
$687$	6.21934e10i	0.279201i
$688$	0	0
$689$	3.26662e11	1.44951
$690$	0	0
$691$	1.89939e11i	0.833111i	0.909110	+	0.416556i	$0.136763\pi$
$691$	1.89939e11i	0.833111i	−0.909110	+	0.416556i	$0.863237\pi$
$692$	0	0
$693$	−4.86698e10	−0.211022
$694$	0	0
$695$	− 4.96331e11i	− 2.12732i
$696$	0	0
$697$	4.62133e11	1.95810
$698$	0	0
$699$	1.16722e11i	0.488927i
$700$	0	0
$701$	1.49808e11	0.620388	0.310194	−	0.950673i	$-0.399606\pi$
$701$	1.49808e11	0.620388	0.310194	+	0.950673i	$0.399606\pi$
$702$	0	0
$703$	6.49400e11i	2.65883i
$704$	0	0
$705$	3.59969e10	0.145716
$706$	0	0
$707$	3.14107e11i	1.25719i
$708$	0	0
$709$	1.29962e11	0.514317	0.257158	−	0.966369i	$-0.417214\pi$
$709$	1.29962e11	0.514317	0.257158	+	0.966369i	$0.417214\pi$
$710$	0	0
$711$	1.24207e11i	0.486034i
$712$	0	0
$713$	4.14948e10	0.160559
$714$	0	0
$715$	1.83847e11i	0.703449i
$716$	0	0
$717$	−2.69929e9	−0.0102135
$718$	0	0
$719$	3.22515e11i	1.20680i	0.797440	+	0.603398i	$0.206187\pi$
$719$	3.22515e11i	1.20680i	−0.797440	+	0.603398i	$0.793813\pi$
$720$	0	0
$721$	−6.34088e11	−2.34643
$722$	0	0
$723$	− 7.35148e10i	− 0.269043i
$724$	0	0
$725$	−8.74817e10	−0.316640
$726$	0	0
$727$	4.13304e11i	1.47956i	0.672850	+	0.739779i	$0.265070\pi$
$727$	4.13304e11i	1.47956i	−0.672850	+	0.739779i	$0.734930\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	7.17097e11i	2.51136i
$732$	0	0
$733$	−3.98880e11	−1.38174	−0.690871	−	0.722978i	$-0.742773\pi$
$733$	−3.98880e11	−1.38174	−0.690871	+	0.722978i	$0.742773\pi$
$734$	0	0
$735$	1.71633e11i	0.588101i
$736$	0	0
$737$	1.09172e11	0.370034
$738$	0	0
$739$	− 1.18702e11i	− 0.397998i	−0.980000	−	0.198999i	$-0.936231\pi$
$739$	− 1.18702e11i	− 0.397998i	0.980000	−	0.198999i	$-0.0637691\pi$
$740$	0	0
$741$	2.77002e11	0.918776
$742$	0	0
$743$	− 2.31061e11i	− 0.758179i	−0.925360	−	0.379089i	$-0.876237\pi$
$743$	− 2.31061e11i	− 0.758179i	0.925360	−	0.379089i	$-0.123763\pi$
$744$	0	0
$745$	−3.89796e11	−1.26535
$746$	0	0
$747$	1.86505e11i	0.598974i
$748$	0	0
$749$	−4.99474e11	−1.58703
$750$	0	0
$751$	− 1.94612e11i	− 0.611801i	−0.952064	−	0.305900i	$-0.901043\pi$
$751$	− 1.94612e11i	− 0.611801i	0.952064	−	0.305900i	$-0.0989574\pi$
$752$	0	0
$753$	1.23217e11	0.383258
$754$	0	0
$755$	− 8.50175e9i	− 0.0261650i
$756$	0	0
$757$	−3.63936e10	−0.110826	−0.0554130	−	0.998464i	$-0.517648\pi$
$757$	−3.63936e10	−0.110826	−0.0554130	+	0.998464i	$0.517648\pi$
$758$	0	0
$759$	− 3.24084e10i	− 0.0976540i
$760$	0	0
$761$	−3.62354e10	−0.108043	−0.0540213	−	0.998540i	$-0.517204\pi$
$761$	−3.62354e10	−0.108043	−0.0540213	+	0.998540i	$0.517204\pi$
$762$	0	0
$763$	− 1.77732e11i	− 0.524405i
$764$	0	0
$765$	3.04920e11	0.890307
$766$	0	0
$767$	6.66450e10i	0.192569i
$768$	0	0
$769$	4.96871e11	1.42082	0.710409	−	0.703789i	$-0.248510\pi$
$769$	4.96871e11	1.42082	0.710409	+	0.703789i	$0.248510\pi$
$770$	0	0
$771$	1.72039e11i	0.486865i
$772$	0	0
$773$	−6.63957e11	−1.85961	−0.929806	−	0.368051i	$-0.880025\pi$
$773$	−6.63957e11	−1.85961	−0.929806	+	0.368051i	$0.880025\pi$
$774$	0	0
$775$	2.21993e11i	0.615366i
$776$	0	0
$777$	4.27691e11	1.17340
$778$	0	0
$779$	− 6.95263e11i	− 1.88799i
$780$	0	0
$781$	1.37838e11	0.370480
$782$	0	0
$783$	1.73174e10i	0.0460719i
$784$	0	0
$785$	2.46634e11	0.649493
$786$	0	0
$787$	6.50846e10i	0.169660i	0.996395	+	0.0848299i	$0.0270347\pi$
$787$	6.50846e10i	0.169660i	−0.996395	+	0.0848299i	$0.972965\pi$
$788$	0	0
$789$	5.68869e10	0.146793
$790$	0	0
$791$	4.63404e10i	0.118373i
$792$	0	0
$793$	1.67285e11	0.423022
$794$	0	0
$795$	− 5.40983e11i	− 1.35430i
$796$	0	0
$797$	−5.57879e10	−0.138263	−0.0691316	−	0.997608i	$-0.522023\pi$
$797$	−5.57879e10	−0.138263	−0.0691316	+	0.997608i	$0.522023\pi$
$798$	0	0
$799$	1.18286e11i	0.290232i
$800$	0	0
$801$	−6.70670e10	−0.162922
$802$	0	0
$803$	2.27677e11i	0.547591i
$804$	0	0
$805$	−2.85281e11	−0.679342
$806$	0	0
$807$	4.33372e11i	1.02180i
$808$	0	0
$809$	−6.06402e11	−1.41568	−0.707842	−	0.706370i	$-0.750331\pi$
$809$	−6.06402e11	−1.41568	−0.707842	+	0.706370i	$0.750331\pi$
$810$	0	0
$811$	− 4.98163e11i	− 1.15156i	−0.817603	−	0.575782i	$-0.804698\pi$
$811$	− 4.98163e11i	− 1.15156i	0.817603	−	0.575782i	$-0.195302\pi$
$812$	0	0
$813$	1.65684e9	0.00379244
$814$	0	0
$815$	1.92066e11i	0.435331i
$816$	0	0
$817$	1.07885e12	2.42143
$818$	0	0
$819$	− 1.82431e11i	− 0.405475i
$820$	0	0
$821$	3.16956e11	0.697632	0.348816	−	0.937191i	$-0.386584\pi$
$821$	3.16956e11	0.697632	0.348816	+	0.937191i	$0.386584\pi$
$822$	0	0
$823$	− 6.55634e11i	− 1.42910i	−0.699585	−	0.714549i	$-0.746632\pi$
$823$	− 6.55634e11i	− 1.42910i	0.699585	−	0.714549i	$-0.253368\pi$
$824$	0	0
$825$	1.73382e11	0.374272
$826$	0	0
$827$	− 8.69774e10i	− 0.185945i	−0.995669	−	0.0929724i	$-0.970363\pi$
$827$	− 8.69774e10i	− 0.185945i	0.995669	−	0.0929724i	$-0.0296368\pi$
$828$	0	0
$829$	3.79641e11	0.803813	0.401906	−	0.915681i	$-0.368348\pi$
$829$	3.79641e11	0.803813	0.401906	+	0.915681i	$0.368348\pi$
$830$	0	0
$831$	− 2.68005e11i	− 0.562004i
$832$	0	0
$833$	−5.63986e11	−1.17135
$834$	0	0
$835$	1.38227e12i	2.84345i
$836$	0	0
$837$	4.39446e10	0.0895373
$838$	0	0
$839$	2.07825e11i	0.419421i	0.977764	+	0.209710i	$0.0672521\pi$
$839$	2.07825e11i	0.419421i	−0.977764	+	0.209710i	$0.932748\pi$
$840$	0	0
$841$	−4.71577e11	−0.942689
$842$	0	0
$843$	3.17714e11i	0.629110i
$844$	0	0
$845$	8.78748e10	0.172360
$846$	0	0
$847$	5.05092e11i	0.981379i
$848$	0	0
$849$	−2.65225e11	−0.510486
$850$	0	0
$851$	2.84791e11i	0.543011i
$852$	0	0
$853$	7.32516e11	1.38363	0.691817	−	0.722073i	$-0.256811\pi$
$853$	7.32516e11	1.38363	0.691817	+	0.722073i	$0.256811\pi$
$854$	0	0
$855$	− 4.58741e11i	− 0.858426i
$856$	0	0
$857$	2.23294e11	0.413955	0.206978	−	0.978346i	$-0.433637\pi$
$857$	2.23294e11	0.413955	0.206978	+	0.978346i	$0.433637\pi$
$858$	0	0
$859$	2.41807e11i	0.444116i	0.975034	+	0.222058i	$0.0712774\pi$
$859$	2.41807e11i	0.444116i	−0.975034	+	0.222058i	$0.928723\pi$
$860$	0	0
$861$	−4.57895e11	−0.833208
$862$	0	0
$863$	− 1.52874e11i	− 0.275607i	−0.990460	−	0.137804i	$-0.955996\pi$
$863$	− 1.52874e11i	− 0.275607i	0.990460	−	0.137804i	$-0.0440043\pi$
$864$	0	0
$865$	6.06200e11	1.08281
$866$	0	0
$867$	6.75741e11i	1.19592i
$868$	0	0
$869$	−4.07538e11	−0.714643
$870$	0	0
$871$	4.09215e11i	0.711015i
$872$	0	0
$873$	−1.12721e10	−0.0194065
$874$	0	0
$875$	− 3.72317e11i	− 0.635156i
$876$	0	0
$877$	1.94466e11	0.328735	0.164367	−	0.986399i	$-0.447442\pi$
$877$	1.94466e11	0.328735	0.164367	+	0.986399i	$0.447442\pi$
$878$	0	0
$879$	7.12774e10i	0.119398i
$880$	0	0
$881$	−9.23735e11	−1.53336	−0.766679	−	0.642030i	$-0.778092\pi$
$881$	−9.23735e11	−1.53336	−0.766679	+	0.642030i	$0.778092\pi$
$882$	0	0
$883$	− 1.15008e12i	− 1.89184i	−0.324405	−	0.945918i	$-0.605164\pi$
$883$	− 1.15008e12i	− 1.89184i	0.324405	−	0.945918i	$-0.394836\pi$
$884$	0	0
$885$	1.10370e11	0.179920
$886$	0	0
$887$	1.00865e12i	1.62947i	0.579831	+	0.814737i	$0.303119\pi$
$887$	1.00865e12i	1.62947i	−0.579831	+	0.814737i	$0.696881\pi$
$888$	0	0
$889$	2.69479e11	0.431438
$890$	0	0
$891$	− 3.43218e10i	− 0.0544576i
$892$	0	0
$893$	1.77957e11	0.279839
$894$	0	0
$895$	− 1.75362e12i	− 2.73302i
$896$	0	0
$897$	1.21478e11	0.187641
$898$	0	0
$899$	− 7.27517e10i	− 0.111379i
$900$	0	0
$901$	1.77767e12	2.69744
$902$	0	0
$903$	− 7.10522e11i	− 1.06863i
$904$	0	0
$905$	−6.90758e11	−1.02975
$906$	0	0
$907$	1.63003e11i	0.240861i	0.992722	+	0.120430i	$0.0384274\pi$
$907$	1.63003e11i	0.240861i	−0.992722	+	0.120430i	$0.961573\pi$
$908$	0	0
$909$	−2.21507e11	−0.324439
$910$	0	0
$911$	− 2.86326e11i	− 0.415707i	−0.978160	−	0.207854i	$-0.933352\pi$
$911$	− 2.86326e11i	− 0.415707i	0.978160	−	0.207854i	$-0.0666478\pi$
$912$	0	0
$913$	−6.11947e11	−0.880705
$914$	0	0
$915$	− 2.77039e11i	− 0.395236i
$916$	0	0
$917$	−9.09037e11	−1.28560
$918$	0	0
$919$	4.28536e11i	0.600793i	0.953814	+	0.300396i	$0.0971190\pi$
$919$	4.28536e11i	0.600793i	−0.953814	+	0.300396i	$0.902881\pi$
$920$	0	0
$921$	−2.51086e11	−0.348967
$922$	0	0
$923$	5.16665e11i	0.711872i
$924$	0	0
$925$	−1.52361e12	−2.08117
$926$	0	0
$927$	− 4.47156e11i	− 0.605536i
$928$	0	0
$929$	−7.91477e11	−1.06261	−0.531307	−	0.847180i	$-0.678299\pi$
$929$	−7.91477e11	−1.06261	−0.531307	+	0.847180i	$0.678299\pi$
$930$	0	0
$931$	8.48497e11i	1.12941i
$932$	0	0
$933$	−2.90578e11	−0.383475
$934$	0	0
$935$	1.00048e12i	1.30907i
$936$	0	0
$937$	−8.10675e11	−1.05169	−0.525846	−	0.850580i	$-0.676251\pi$
$937$	−8.10675e11	−1.05169	−0.525846	+	0.850580i	$0.676251\pi$
$938$	0	0
$939$	1.37970e10i	0.0177469i
$940$	0	0
$941$	1.14478e12	1.46004	0.730021	−	0.683425i	$-0.239510\pi$
$941$	1.14478e12	1.46004	0.730021	+	0.683425i	$0.239510\pi$
$942$	0	0
$943$	− 3.04904e11i	− 0.385582i
$944$	0	0
$945$	−3.02124e11	−0.378841
$946$	0	0
$947$	− 1.36874e12i	− 1.70185i	−0.525284	−	0.850927i	$-0.676041\pi$
$947$	− 1.36874e12i	− 1.70185i	0.525284	−	0.850927i	$-0.323959\pi$
$948$	0	0
$949$	−8.53411e11	−1.05219
$950$	0	0
$951$	− 5.72702e10i	− 0.0700175i
$952$	0	0
$953$	1.59249e11	0.193065	0.0965327	−	0.995330i	$-0.469225\pi$
$953$	1.59249e11	0.193065	0.0965327	+	0.995330i	$0.469225\pi$
$954$	0	0
$955$	− 1.52263e12i	− 1.83054i
$956$	0	0
$957$	−5.68207e10	−0.0677421
$958$	0	0
$959$	− 1.52997e12i	− 1.80887i
$960$	0	0
$961$	6.68277e11	0.783543
$962$	0	0
$963$	− 3.52227e11i	− 0.409560i
$964$	0	0
$965$	4.81088e11	0.554773
$966$	0	0
$967$	− 9.07631e11i	− 1.03801i	−0.854770	−	0.519007i	$-0.826302\pi$
$967$	− 9.07631e11i	− 1.03801i	0.854770	−	0.519007i	$-0.173698\pi$
$968$	0	0
$969$	1.50742e12	1.70978
$970$	0	0
$971$	− 1.07988e12i	− 1.21478i	−0.794405	−	0.607389i	$-0.792217\pi$
$971$	− 1.07988e12i	− 1.21478i	0.794405	−	0.607389i	$-0.207783\pi$
$972$	0	0
$973$	−1.61599e12	−1.80296
$974$	0	0
$975$	6.49895e11i	0.719159i
$976$	0	0
$977$	1.39047e12	1.52610	0.763049	−	0.646341i	$-0.223702\pi$
$977$	1.39047e12	1.52610	0.763049	+	0.646341i	$0.223702\pi$
$978$	0	0
$979$	− 2.20056e11i	− 0.239553i
$980$	0	0
$981$	1.25336e11	0.135331
$982$	0	0
$983$	1.23832e12i	1.32623i	0.748516	+	0.663117i	$0.230767\pi$
$983$	1.23832e12i	1.32623i	−0.748516	+	0.663117i	$0.769233\pi$
$984$	0	0
$985$	2.13205e12	2.26492
$986$	0	0
$987$	− 1.17201e11i	− 0.123499i
$988$	0	0
$989$	4.73124e11	0.494526
$990$	0	0
$991$	− 1.64879e12i	− 1.70951i	−0.519033	−	0.854754i	$-0.673708\pi$
$991$	− 1.64879e12i	− 1.70951i	0.519033	−	0.854754i	$-0.326292\pi$
$992$	0	0
$993$	−7.00582e11	−0.720547
$994$	0	0
$995$	− 1.06284e12i	− 1.08436i
$996$	0	0
$997$	−1.01638e10	−0.0102867	−0.00514335	−	0.999987i	$-0.501637\pi$
$997$	−1.01638e10	−0.0102867	−0.00514335	+	0.999987i	$0.501637\pi$
$998$	0	0
$999$	3.01606e11i	0.302815i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.9.g.c.127.3		4
4.3	odd	2	inner	192.9.g.c.127.1		4
8.3	odd	2		48.9.g.c.31.4	yes	4
8.5	even	2		48.9.g.c.31.2	✓	4
24.5	odd	2		144.9.g.i.127.2		4
24.11	even	2		144.9.g.i.127.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.9.g.c.31.2	✓	4	8.5	even	2
48.9.g.c.31.4	yes	4	8.3	odd	2
144.9.g.i.127.1		4	24.11	even	2
144.9.g.i.127.2		4	24.5	odd	2
192.9.g.c.127.1		4	4.3	odd	2	inner
192.9.g.c.127.3		4	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)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} -952.517 q^{5} +3101.27i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} -952.517 q^{5} +3101.27i q^{7} -2187.00 q^{9} -7175.83i q^{11} +26897.5 q^{13} -44544.8i q^{15} +146374. q^{17} -220215. i q^{19} -145032. q^{21} -96574.1i q^{23} +516663. q^{25} -102276. i q^{27} -169321. q^{29} +429668. i q^{31} +335580. q^{33} -2.95401e6i q^{35} -2.94894e6 q^{37} +1.25787e6i q^{39} +3.15721e6 q^{41} +4.89907e6i q^{43} +2.08315e6 q^{45} +808105. i q^{47} -3.85305e6 q^{49} +6.84524e6i q^{51} +1.21447e7 q^{53} +6.83510e6i q^{55} +1.02984e7 q^{57} +2.47774e6i q^{59} +6.21934e6 q^{61} -6.78247e6i q^{63} -2.56203e7 q^{65} +1.52139e7i q^{67} +4.51632e6 q^{69} +1.92087e7i q^{71} -3.17283e7 q^{73} +2.41619e7i q^{75} +2.22542e7 q^{77} -5.67932e7i q^{79} +4.78297e6 q^{81} -8.52789e7i q^{83} -1.39424e8 q^{85} -7.91835e6i q^{87} +3.06662e7 q^{89} +8.34163e7i q^{91} -2.00936e7 q^{93} +2.09758e8i q^{95} +5.15412e6 q^{97} +1.56935e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 264 q^{5} - 8748 q^{9}+O(q^{10}) \) 4 * q + 264 * q^5 - 8748 * q^9	\( 4 q + 264 q^{5} - 8748 q^{9} - 14632 q^{13} + 332904 q^{17} - 250128 q^{21} + 2604428 q^{25} - 2343576 q^{29} + 2002320 q^{33} - 4315784 q^{37} + 9035496 q^{41} - 577368 q^{45} + 3458884 q^{49} + 42186600 q^{53} + 2253744 q^{57} + 48148408 q^{61} - 125450832 q^{65} - 23514624 q^{69} - 21215480 q^{73} + 32354496 q^{77} + 19131876 q^{81} - 235297584 q^{85} - 12675576 q^{89} - 193564080 q^{93} + 263153800 q^{97}+O(q^{100}) \) 4 * q + 264 * q^5 - 8748 * q^9 - 14632 * q^13 + 332904 * q^17 - 250128 * q^21 + 2604428 * q^25 - 2343576 * q^29 + 2002320 * q^33 - 4315784 * q^37 + 9035496 * q^41 - 577368 * q^45 + 3458884 * q^49 + 42186600 * q^53 + 2253744 * q^57 + 48148408 * q^61 - 125450832 * q^65 - 23514624 * q^69 - 21215480 * q^73 + 32354496 * q^77 + 19131876 * q^81 - 235297584 * q^85 - 12675576 * q^89 - 193564080 * q^93 + 263153800 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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