LMFDB - Embedded newform 192.5.g.a.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,5,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, 5, names="a")

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
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:	$19.8470329121$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	192.127
Dual form			192.5.g.a.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} -6.00000 q^{5} +62.3538i q^{7} -27.0000 q^{9} +O(q^{10})$	$q-5.19615i q^{3} -6.00000 q^{5} +62.3538i q^{7} -27.0000 q^{9} -187.061i q^{11} +86.0000 q^{13} +31.1769i q^{15} +426.000 q^{17} -62.3538i q^{19} +324.000 q^{21} -748.246i q^{23} -589.000 q^{25} +140.296i q^{27} -1182.00 q^{29} -1558.85i q^{31} -972.000 q^{33} -374.123i q^{35} +430.000 q^{37} -446.869i q^{39} +2250.00 q^{41} -2681.21i q^{43} +162.000 q^{45} +374.123i q^{47} -1487.00 q^{49} -2213.56i q^{51} +1602.00 q^{53} +1122.37i q^{55} -324.000 q^{57} -3554.17i q^{59} -2114.00 q^{61} -1683.55i q^{63} -516.000 q^{65} +1309.43i q^{67} -3888.00 q^{69} -2992.98i q^{71} +4066.00 q^{73} +3060.53i q^{75} +11664.0 q^{77} -5549.49i q^{79} +729.000 q^{81} +9166.01i q^{83} -2556.00 q^{85} +6141.85i q^{87} -2046.00 q^{89} +5362.43i q^{91} -8100.00 q^{93} +374.123i q^{95} -2942.00 q^{97} +5050.66i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{5} - 54 q^{9}+O(q^{10}) $ 2 * q - 12 * q^5 - 54 * q^9	$ 2 q - 12 q^{5} - 54 q^{9} + 172 q^{13} + 852 q^{17} + 648 q^{21} - 1178 q^{25} - 2364 q^{29} - 1944 q^{33} + 860 q^{37} + 4500 q^{41} + 324 q^{45} - 2974 q^{49} + 3204 q^{53} - 648 q^{57} - 4228 q^{61} - 1032 q^{65} - 7776 q^{69} + 8132 q^{73} + 23328 q^{77} + 1458 q^{81} - 5112 q^{85} - 4092 q^{89} - 16200 q^{93} - 5884 q^{97}+O(q^{100}) $ 2 * q - 12 * q^5 - 54 * q^9 + 172 * q^13 + 852 * q^17 + 648 * q^21 - 1178 * q^25 - 2364 * q^29 - 1944 * q^33 + 860 * q^37 + 4500 * q^41 + 324 * q^45 - 2974 * q^49 + 3204 * q^53 - 648 * q^57 - 4228 * q^61 - 1032 * q^65 - 7776 * q^69 + 8132 * q^73 + 23328 * q^77 + 1458 * q^81 - 5112 * q^85 - 4092 * q^89 - 16200 * q^93 - 5884 * 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
$3$	− 5.19615i	− 0.577350i
$3$	− 5.19615i	− 0.577350i
$4$	0	0
$5$	−6.00000	−0.240000	−0.120000	−	0.992774i	$-0.538289\pi$
$5$	−6.00000	−0.240000	−0.120000	+	0.992774i	$0.538289\pi$
$6$	0	0
$7$	62.3538i	1.27253i	0.771472	+	0.636264i	$0.219521\pi$
$7$	62.3538i	1.27253i	−0.771472	+	0.636264i	$0.780479\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	− 187.061i	− 1.54596i	−0.634429	−	0.772981i	$-0.718765\pi$
$11$	− 187.061i	− 1.54596i	0.634429	−	0.772981i	$-0.281235\pi$
$12$	0	0
$13$	86.0000	0.508876	0.254438	−	0.967089i	$-0.418110\pi$
$13$	86.0000	0.508876	0.254438	+	0.967089i	$0.418110\pi$
$14$	0	0
$15$	31.1769i	0.138564i
$16$	0	0
$17$	426.000	1.47405	0.737024	−	0.675866i	$-0.236230\pi$
$17$	426.000	1.47405	0.737024	+	0.675866i	$0.236230\pi$
$18$	0	0
$19$	− 62.3538i	− 0.172725i	−0.996264	−	0.0863626i	$-0.972476\pi$
$19$	− 62.3538i	− 0.172725i	0.996264	−	0.0863626i	$-0.0275244\pi$
$20$	0	0
$21$	324.000	0.734694
$22$	0	0
$23$	− 748.246i	− 1.41445i	−0.706987	−	0.707227i	$-0.749946\pi$
$23$	− 748.246i	− 1.41445i	0.706987	−	0.707227i	$-0.250054\pi$
$24$	0	0
$25$	−589.000	−0.942400
$26$	0	0
$27$	140.296i	0.192450i
$28$	0	0
$29$	−1182.00	−1.40547	−0.702735	−	0.711452i	$-0.748038\pi$
$29$	−1182.00	−1.40547	−0.702735	+	0.711452i	$0.748038\pi$
$30$	0	0
$31$	− 1558.85i	− 1.62211i	−0.584971	−	0.811054i	$-0.698894\pi$
$31$	− 1558.85i	− 1.62211i	0.584971	−	0.811054i	$-0.301106\pi$
$32$	0	0
$33$	−972.000	−0.892562
$34$	0	0
$35$	− 374.123i	− 0.305407i
$36$	0	0
$37$	430.000	0.314098	0.157049	−	0.987591i	$-0.449802\pi$
$37$	430.000	0.314098	0.157049	+	0.987591i	$0.449802\pi$
$38$	0	0
$39$	− 446.869i	− 0.293800i
$40$	0	0
$41$	2250.00	1.33849	0.669244	−	0.743042i	$-0.266618\pi$
$41$	2250.00	1.33849	0.669244	+	0.743042i	$0.266618\pi$
$42$	0	0
$43$	− 2681.21i	− 1.45009i	−0.688702	−	0.725045i	$-0.741819\pi$
$43$	− 2681.21i	− 1.45009i	0.688702	−	0.725045i	$-0.258181\pi$
$44$	0	0
$45$	162.000	0.0800000
$46$	0	0
$47$	374.123i	0.169363i	0.996408	+	0.0846815i	$0.0269873\pi$
$47$	374.123i	0.169363i	−0.996408	+	0.0846815i	$0.973013\pi$
$48$	0	0
$49$	−1487.00	−0.619325
$50$	0	0
$51$	− 2213.56i	− 0.851042i
$52$	0	0
$53$	1602.00	0.570310	0.285155	−	0.958481i	$-0.407955\pi$
$53$	1602.00	0.570310	0.285155	+	0.958481i	$0.407955\pi$
$54$	0	0
$55$	1122.37i	0.371031i
$56$	0	0
$57$	−324.000	−0.0997230
$58$	0	0
$59$	− 3554.17i	− 1.02102i	−0.859872	−	0.510510i	$-0.829457\pi$
$59$	− 3554.17i	− 1.02102i	0.859872	−	0.510510i	$-0.170543\pi$
$60$	0	0
$61$	−2114.00	−0.568127	−0.284063	−	0.958805i	$-0.591683\pi$
$61$	−2114.00	−0.568127	−0.284063	+	0.958805i	$0.591683\pi$
$62$	0	0
$63$	− 1683.55i	− 0.424176i
$64$	0	0
$65$	−516.000	−0.122130
$66$	0	0
$67$	1309.43i	0.291698i	0.989307	+	0.145849i	$0.0465913\pi$
$67$	1309.43i	0.291698i	−0.989307	+	0.145849i	$0.953409\pi$
$68$	0	0
$69$	−3888.00	−0.816635
$70$	0	0
$71$	− 2992.98i	− 0.593728i	−0.954920	−	0.296864i	$-0.904059\pi$
$71$	− 2992.98i	− 0.593728i	0.954920	−	0.296864i	$-0.0959408\pi$
$72$	0	0
$73$	4066.00	0.762995	0.381497	−	0.924370i	$-0.375408\pi$
$73$	4066.00	0.762995	0.381497	+	0.924370i	$0.375408\pi$
$74$	0	0
$75$	3060.53i	0.544095i
$76$	0	0
$77$	11664.0	1.96728
$78$	0	0
$79$	− 5549.49i	− 0.889199i	−0.895729	−	0.444599i	$-0.853346\pi$
$79$	− 5549.49i	− 0.889199i	0.895729	−	0.444599i	$-0.146654\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	9166.01i	1.33053i	0.746608	+	0.665264i	$0.231681\pi$
$83$	9166.01i	1.33053i	−0.746608	+	0.665264i	$0.768319\pi$
$84$	0	0
$85$	−2556.00	−0.353772
$86$	0	0
$87$	6141.85i	0.811448i
$88$	0	0
$89$	−2046.00	−0.258301	−0.129150	−	0.991625i	$-0.541225\pi$
$89$	−2046.00	−0.258301	−0.129150	+	0.991625i	$0.541225\pi$
$90$	0	0
$91$	5362.43i	0.647558i
$92$	0	0
$93$	−8100.00	−0.936524
$94$	0	0
$95$	374.123i	0.0414541i
$96$	0	0
$97$	−2942.00	−0.312679	−0.156340	−	0.987703i	$-0.549969\pi$
$97$	−2942.00	−0.312679	−0.156340	+	0.987703i	$0.549969\pi$
$98$	0	0
$99$	5050.66i	0.515321i
$100$	0	0
$101$	11682.0	1.14518	0.572591	−	0.819841i	$-0.305938\pi$
$101$	11682.0	1.14518	0.572591	+	0.819841i	$0.305938\pi$
$102$	0	0
$103$	16648.5i	1.56928i	0.619953	+	0.784639i	$0.287152\pi$
$103$	16648.5i	1.56928i	−0.619953	+	0.784639i	$0.712848\pi$
$104$	0	0
$105$	−1944.00	−0.176327
$106$	0	0
$107$	− 9540.14i	− 0.833272i	−0.909073	−	0.416636i	$-0.863209\pi$
$107$	− 9540.14i	− 0.833272i	0.909073	−	0.416636i	$-0.136791\pi$
$108$	0	0
$109$	−17530.0	−1.47547	−0.737733	−	0.675093i	$-0.764103\pi$
$109$	−17530.0	−1.47547	−0.737733	+	0.675093i	$0.764103\pi$
$110$	0	0
$111$	− 2234.35i	− 0.181344i
$112$	0	0
$113$	12018.0	0.941186	0.470593	−	0.882350i	$-0.344040\pi$
$113$	12018.0	0.941186	0.470593	+	0.882350i	$0.344040\pi$
$114$	0	0
$115$	4489.48i	0.339469i
$116$	0	0
$117$	−2322.00	−0.169625
$118$	0	0
$119$	26562.7i	1.87577i
$120$	0	0
$121$	−20351.0	−1.39000
$122$	0	0
$123$	− 11691.3i	− 0.772777i
$124$	0	0
$125$	7284.00	0.466176
$126$	0	0
$127$	10787.2i	0.668809i	0.942430	+	0.334404i	$0.108535\pi$
$127$	10787.2i	0.668809i	−0.942430	+	0.334404i	$0.891465\pi$
$128$	0	0
$129$	−13932.0	−0.837209
$130$	0	0
$131$	22260.3i	1.29715i	0.761153	+	0.648573i	$0.224634\pi$
$131$	22260.3i	1.29715i	−0.761153	+	0.648573i	$0.775366\pi$
$132$	0	0
$133$	3888.00	0.219798
$134$	0	0
$135$	− 841.777i	− 0.0461880i
$136$	0	0
$137$	26970.0	1.43694	0.718472	−	0.695556i	$-0.244842\pi$
$137$	26970.0	1.43694	0.718472	+	0.695556i	$0.244842\pi$
$138$	0	0
$139$	− 2307.09i	− 0.119409i	−0.998216	−	0.0597043i	$-0.980984\pi$
$139$	− 2307.09i	− 0.119409i	0.998216	−	0.0597043i	$-0.0190158\pi$
$140$	0	0
$141$	1944.00	0.0977818
$142$	0	0
$143$	− 16087.3i	− 0.786703i
$144$	0	0
$145$	7092.00	0.337313
$146$	0	0
$147$	7726.68i	0.357568i
$148$	0	0
$149$	−13926.0	−0.627269	−0.313635	−	0.949544i	$-0.601547\pi$
$149$	−13926.0	−0.627269	−0.313635	+	0.949544i	$0.601547\pi$
$150$	0	0
$151$	− 18145.0i	− 0.795797i	−0.917429	−	0.397898i	$-0.869740\pi$
$151$	− 18145.0i	− 0.795797i	0.917429	−	0.397898i	$-0.130260\pi$
$152$	0	0
$153$	−11502.0	−0.491349
$154$	0	0
$155$	9353.07i	0.389306i
$156$	0	0
$157$	30766.0	1.24816	0.624082	−	0.781359i	$-0.285473\pi$
$157$	30766.0	1.24816	0.624082	+	0.781359i	$0.285473\pi$
$158$	0	0
$159$	− 8324.24i	− 0.329268i
$160$	0	0
$161$	46656.0	1.79993
$162$	0	0
$163$	26874.5i	1.01150i	0.862681	+	0.505749i	$0.168784\pi$
$163$	26874.5i	1.01150i	−0.862681	+	0.505749i	$0.831216\pi$
$164$	0	0
$165$	5832.00	0.214215
$166$	0	0
$167$	10101.3i	0.362197i	0.983465	+	0.181099i	$0.0579653\pi$
$167$	10101.3i	0.362197i	−0.983465	+	0.181099i	$0.942035\pi$
$168$	0	0
$169$	−21165.0	−0.741045
$170$	0	0
$171$	1683.55i	0.0575751i
$172$	0	0
$173$	14010.0	0.468108	0.234054	−	0.972224i	$-0.424801\pi$
$173$	14010.0	0.468108	0.234054	+	0.972224i	$0.424801\pi$
$174$	0	0
$175$	− 36726.4i	− 1.19923i
$176$	0	0
$177$	−18468.0	−0.589486
$178$	0	0
$179$	− 14403.7i	− 0.449541i	−0.974412	−	0.224770i	$-0.927837\pi$
$179$	− 14403.7i	− 0.449541i	0.974412	−	0.224770i	$-0.0721632\pi$
$180$	0	0
$181$	−41722.0	−1.27353	−0.636763	−	0.771059i	$-0.719727\pi$
$181$	−41722.0	−1.27353	−0.636763	+	0.771059i	$0.719727\pi$
$182$	0	0
$183$	10984.7i	0.328008i
$184$	0	0
$185$	−2580.00	−0.0753835
$186$	0	0
$187$	− 79688.2i	− 2.27882i
$188$	0	0
$189$	−8748.00	−0.244898
$190$	0	0
$191$	− 53873.7i	− 1.47676i	−0.674384	−	0.738380i	$-0.735591\pi$
$191$	− 53873.7i	− 1.47676i	0.674384	−	0.738380i	$-0.264409\pi$
$192$	0	0
$193$	−40990.0	−1.10043	−0.550216	−	0.835022i	$-0.685455\pi$
$193$	−40990.0	−1.10043	−0.550216	+	0.835022i	$0.685455\pi$
$194$	0	0
$195$	2681.21i	0.0705119i
$196$	0	0
$197$	−5646.00	−0.145482	−0.0727409	−	0.997351i	$-0.523175\pi$
$197$	−5646.00	−0.145482	−0.0727409	+	0.997351i	$0.523175\pi$
$198$	0	0
$199$	46328.9i	1.16989i	0.811072	+	0.584946i	$0.198884\pi$
$199$	46328.9i	1.16989i	−0.811072	+	0.584946i	$0.801116\pi$
$200$	0	0
$201$	6804.00	0.168412
$202$	0	0
$203$	− 73702.2i	− 1.78850i
$204$	0	0
$205$	−13500.0	−0.321237
$206$	0	0
$207$	20202.6i	0.471485i
$208$	0	0
$209$	−11664.0	−0.267027
$210$	0	0
$211$	21013.2i	0.471985i	0.971755	+	0.235992i	$0.0758341\pi$
$211$	21013.2i	0.471985i	−0.971755	+	0.235992i	$0.924166\pi$
$212$	0	0
$213$	−15552.0	−0.342789
$214$	0	0
$215$	16087.3i	0.348021i
$216$	0	0
$217$	97200.0	2.06418
$218$	0	0
$219$	− 21127.6i	− 0.440515i
$220$	0	0
$221$	36636.0	0.750107
$222$	0	0
$223$	− 27373.3i	− 0.550450i	−0.961380	−	0.275225i	$-0.911248\pi$
$223$	− 27373.3i	− 0.550450i	0.961380	−	0.275225i	$-0.0887524\pi$
$224$	0	0
$225$	15903.0	0.314133
$226$	0	0
$227$	− 50693.7i	− 0.983789i	−0.870655	−	0.491894i	$-0.836305\pi$
$227$	− 50693.7i	− 0.983789i	0.870655	−	0.491894i	$-0.163695\pi$
$228$	0	0
$229$	75254.0	1.43502	0.717511	−	0.696547i	$-0.245281\pi$
$229$	75254.0	1.43502	0.717511	+	0.696547i	$0.245281\pi$
$230$	0	0
$231$	− 60607.9i	− 1.13581i
$232$	0	0
$233$	−100830.	−1.85728	−0.928641	−	0.370979i	$-0.879022\pi$
$233$	−100830.	−1.85728	−0.928641	+	0.370979i	$0.879022\pi$
$234$	0	0
$235$	− 2244.74i	− 0.0406471i
$236$	0	0
$237$	−28836.0	−0.513379
$238$	0	0
$239$	− 49384.2i	− 0.864555i	−0.901741	−	0.432277i	$-0.857710\pi$
$239$	− 49384.2i	− 0.864555i	0.901741	−	0.432277i	$-0.142290\pi$
$240$	0	0
$241$	−18062.0	−0.310979	−0.155490	−	0.987838i	$-0.549696\pi$
$241$	−18062.0	−0.310979	−0.155490	+	0.987838i	$0.549696\pi$
$242$	0	0
$243$	− 3788.00i	− 0.0641500i
$244$	0	0
$245$	8922.00	0.148638
$246$	0	0
$247$	− 5362.43i	− 0.0878957i
$248$	0	0
$249$	47628.0	0.768181
$250$	0	0
$251$	100826.i	1.60039i	0.599740	+	0.800195i	$0.295271\pi$
$251$	100826.i	1.60039i	−0.599740	+	0.800195i	$0.704729\pi$
$252$	0	0
$253$	−139968.	−2.18669
$254$	0	0
$255$	13281.4i	0.204250i
$256$	0	0
$257$	−18174.0	−0.275159	−0.137580	−	0.990491i	$-0.543932\pi$
$257$	−18174.0	−0.275159	−0.137580	+	0.990491i	$0.543932\pi$
$258$	0	0
$259$	26812.1i	0.399698i
$260$	0	0
$261$	31914.0	0.468490
$262$	0	0
$263$	− 6360.09i	− 0.0919500i	−0.998943	−	0.0459750i	$-0.985361\pi$
$263$	− 6360.09i	− 0.0919500i	0.998943	−	0.0459750i	$-0.0146395\pi$
$264$	0	0
$265$	−9612.00	−0.136874
$266$	0	0
$267$	10631.3i	0.149130i
$268$	0	0
$269$	30882.0	0.426777	0.213388	−	0.976967i	$-0.431550\pi$
$269$	30882.0	0.426777	0.213388	+	0.976967i	$0.431550\pi$
$270$	0	0
$271$	− 85487.1i	− 1.16402i	−0.813180	−	0.582012i	$-0.802266\pi$
$271$	− 85487.1i	− 1.16402i	0.813180	−	0.582012i	$-0.197734\pi$
$272$	0	0
$273$	27864.0	0.373868
$274$	0	0
$275$	110179.i	1.45692i
$276$	0	0
$277$	38438.0	0.500958	0.250479	−	0.968122i	$-0.419412\pi$
$277$	38438.0	0.500958	0.250479	+	0.968122i	$0.419412\pi$
$278$	0	0
$279$	42088.8i	0.540703i
$280$	0	0
$281$	−38382.0	−0.486088	−0.243044	−	0.970015i	$-0.578146\pi$
$281$	−38382.0	−0.486088	−0.243044	+	0.970015i	$0.578146\pi$
$282$	0	0
$283$	− 24006.2i	− 0.299744i	−0.988705	−	0.149872i	$-0.952114\pi$
$283$	− 24006.2i	− 0.299744i	0.988705	−	0.149872i	$-0.0478862\pi$
$284$	0	0
$285$	1944.00	0.0239335
$286$	0	0
$287$	140296.i	1.70326i
$288$	0	0
$289$	97955.0	1.17282
$290$	0	0
$291$	15287.1i	0.180526i
$292$	0	0
$293$	−43710.0	−0.509150	−0.254575	−	0.967053i	$-0.581936\pi$
$293$	−43710.0	−0.509150	−0.254575	+	0.967053i	$0.581936\pi$
$294$	0	0
$295$	21325.0i	0.245045i
$296$	0	0
$297$	26244.0	0.297521
$298$	0	0
$299$	− 64349.2i	− 0.719781i
$300$	0	0
$301$	167184.	1.84528
$302$	0	0
$303$	− 60701.5i	− 0.661171i
$304$	0	0
$305$	12684.0	0.136350
$306$	0	0
$307$	− 59298.5i	− 0.629168i	−0.949230	−	0.314584i	$-0.898135\pi$
$307$	− 59298.5i	− 0.629168i	0.949230	−	0.314584i	$-0.101865\pi$
$308$	0	0
$309$	86508.0	0.906023
$310$	0	0
$311$	60982.0i	0.630494i	0.949010	+	0.315247i	$0.102087\pi$
$311$	60982.0i	0.630494i	−0.949010	+	0.315247i	$0.897913\pi$
$312$	0	0
$313$	2258.00	0.0230481	0.0115241	−	0.999934i	$-0.496332\pi$
$313$	2258.00	0.0230481	0.0115241	+	0.999934i	$0.496332\pi$
$314$	0	0
$315$	10101.3i	0.101802i
$316$	0	0
$317$	−121614.	−1.21022	−0.605111	−	0.796141i	$-0.706871\pi$
$317$	−121614.	−1.21022	−0.605111	+	0.796141i	$0.706871\pi$
$318$	0	0
$319$	221107.i	2.17280i
$320$	0	0
$321$	−49572.0	−0.481090
$322$	0	0
$323$	− 26562.7i	− 0.254605i
$324$	0	0
$325$	−50654.0	−0.479564
$326$	0	0
$327$	91088.6i	0.851860i
$328$	0	0
$329$	−23328.0	−0.215519
$330$	0	0
$331$	193484.i	1.76599i	0.469380	+	0.882996i	$0.344478\pi$
$331$	193484.i	1.76599i	−0.469380	+	0.882996i	$0.655522\pi$
$332$	0	0
$333$	−11610.0	−0.104699
$334$	0	0
$335$	− 7856.58i	− 0.0700074i
$336$	0	0
$337$	−145166.	−1.27822	−0.639109	−	0.769116i	$-0.720697\pi$
$337$	−145166.	−1.27822	−0.639109	+	0.769116i	$0.720697\pi$
$338$	0	0
$339$	− 62447.4i	− 0.543394i
$340$	0	0
$341$	−291600.	−2.50772
$342$	0	0
$343$	56991.4i	0.484419i
$344$	0	0
$345$	23328.0	0.195992
$346$	0	0
$347$	− 14403.7i	− 0.119623i	−0.998210	−	0.0598117i	$-0.980950\pi$
$347$	− 14403.7i	− 0.119623i	0.998210	−	0.0598117i	$-0.0190500\pi$
$348$	0	0
$349$	29854.0	0.245105	0.122552	−	0.992462i	$-0.460892\pi$
$349$	29854.0	0.245105	0.122552	+	0.992462i	$0.460892\pi$
$350$	0	0
$351$	12065.5i	0.0979332i
$352$	0	0
$353$	−4878.00	−0.0391465	−0.0195732	−	0.999808i	$-0.506231\pi$
$353$	−4878.00	−0.0391465	−0.0195732	+	0.999808i	$0.506231\pi$
$354$	0	0
$355$	17957.9i	0.142495i
$356$	0	0
$357$	138024.	1.08297
$358$	0	0
$359$	− 160125.i	− 1.24242i	−0.783643	−	0.621211i	$-0.786641\pi$
$359$	− 160125.i	− 1.24242i	0.783643	−	0.621211i	$-0.213359\pi$
$360$	0	0
$361$	126433.	0.970166
$362$	0	0
$363$	105747.i	0.802517i
$364$	0	0
$365$	−24396.0	−0.183119
$366$	0	0
$367$	61543.2i	0.456928i	0.973552	+	0.228464i	$0.0733704\pi$
$367$	61543.2i	0.456928i	−0.973552	+	0.228464i	$0.926630\pi$
$368$	0	0
$369$	−60750.0	−0.446163
$370$	0	0
$371$	99890.8i	0.725735i
$372$	0	0
$373$	189598.	1.36275	0.681375	−	0.731935i	$-0.261382\pi$
$373$	189598.	1.36275	0.681375	+	0.731935i	$0.261382\pi$
$374$	0	0
$375$	− 37848.8i	− 0.269147i
$376$	0	0
$377$	−101652.	−0.715209
$378$	0	0
$379$	144474.i	1.00580i	0.864345	+	0.502899i	$0.167733\pi$
$379$	144474.i	1.00580i	−0.864345	+	0.502899i	$0.832267\pi$
$380$	0	0
$381$	56052.0	0.386137
$382$	0	0
$383$	97272.0i	0.663117i	0.943435	+	0.331559i	$0.107574\pi$
$383$	97272.0i	0.663117i	−0.943435	+	0.331559i	$0.892426\pi$
$384$	0	0
$385$	−69984.0	−0.472147
$386$	0	0
$387$	72392.8i	0.483363i
$388$	0	0
$389$	−160326.	−1.05951	−0.529755	−	0.848151i	$-0.677716\pi$
$389$	−160326.	−1.05951	−0.529755	+	0.848151i	$0.677716\pi$
$390$	0	0
$391$	− 318753.i	− 2.08497i
$392$	0	0
$393$	115668.	0.748907
$394$	0	0
$395$	33296.9i	0.213408i
$396$	0	0
$397$	−93026.0	−0.590233	−0.295116	−	0.955461i	$-0.595358\pi$
$397$	−93026.0	−0.590233	−0.295116	+	0.955461i	$0.595358\pi$
$398$	0	0
$399$	− 20202.6i	− 0.126900i
$400$	0	0
$401$	185370.	1.15279	0.576396	−	0.817171i	$-0.304459\pi$
$401$	185370.	1.15279	0.576396	+	0.817171i	$0.304459\pi$
$402$	0	0
$403$	− 134061.i	− 0.825451i
$404$	0	0
$405$	−4374.00	−0.0266667
$406$	0	0
$407$	− 80436.4i	− 0.485584i
$408$	0	0
$409$	90130.0	0.538794	0.269397	−	0.963029i	$-0.413176\pi$
$409$	90130.0	0.538794	0.269397	+	0.963029i	$0.413176\pi$
$410$	0	0
$411$	− 140140.i	− 0.829620i
$412$	0	0
$413$	221616.	1.29927
$414$	0	0
$415$	− 54996.1i	− 0.319327i
$416$	0	0
$417$	−11988.0	−0.0689405
$418$	0	0
$419$	− 73141.0i	− 0.416613i	−0.978064	−	0.208307i	$-0.933205\pi$
$419$	− 73141.0i	− 0.416613i	0.978064	−	0.208307i	$-0.0667952\pi$
$420$	0	0
$421$	80774.0	0.455730	0.227865	−	0.973693i	$-0.426826\pi$
$421$	80774.0	0.455730	0.227865	+	0.973693i	$0.426826\pi$
$422$	0	0
$423$	− 10101.3i	− 0.0564543i
$424$	0	0
$425$	−250914.	−1.38914
$426$	0	0
$427$	− 131816.i	− 0.722957i
$428$	0	0
$429$	−83592.0	−0.454203
$430$	0	0
$431$	34045.2i	0.183274i	0.995792	+	0.0916371i	$0.0292100\pi$
$431$	34045.2i	0.183274i	−0.995792	+	0.0916371i	$0.970790\pi$
$432$	0	0
$433$	176978.	0.943938	0.471969	−	0.881615i	$-0.343543\pi$
$433$	176978.	0.943938	0.471969	+	0.881615i	$0.343543\pi$
$434$	0	0
$435$	− 36851.1i	− 0.194748i
$436$	0	0
$437$	−46656.0	−0.244312
$438$	0	0
$439$	141606.i	0.734770i	0.930069	+	0.367385i	$0.119747\pi$
$439$	141606.i	0.734770i	−0.930069	+	0.367385i	$0.880253\pi$
$440$	0	0
$441$	40149.0	0.206442
$442$	0	0
$443$	70896.3i	0.361257i	0.983551	+	0.180628i	$0.0578132\pi$
$443$	70896.3i	0.361257i	−0.983551	+	0.180628i	$0.942187\pi$
$444$	0	0
$445$	12276.0	0.0619922
$446$	0	0
$447$	72361.6i	0.362154i
$448$	0	0
$449$	53514.0	0.265445	0.132723	−	0.991153i	$-0.457628\pi$
$449$	53514.0	0.265445	0.132723	+	0.991153i	$0.457628\pi$
$450$	0	0
$451$	− 420888.i	− 2.06925i
$452$	0	0
$453$	−94284.0	−0.459454
$454$	0	0
$455$	− 32174.6i	− 0.155414i
$456$	0	0
$457$	−38798.0	−0.185771	−0.0928853	−	0.995677i	$-0.529609\pi$
$457$	−38798.0	−0.185771	−0.0928853	+	0.995677i	$0.529609\pi$
$458$	0	0
$459$	59766.1i	0.283681i
$460$	0	0
$461$	280890.	1.32170	0.660852	−	0.750516i	$-0.270195\pi$
$461$	280890.	1.32170	0.660852	+	0.750516i	$0.270195\pi$
$462$	0	0
$463$	− 114918.i	− 0.536076i	−0.963408	−	0.268038i	$-0.913625\pi$
$463$	− 114918.i	− 0.536076i	0.963408	−	0.268038i	$-0.0863753\pi$
$464$	0	0
$465$	48600.0	0.224766
$466$	0	0
$467$	− 409103.i	− 1.87586i	−0.346831	−	0.937928i	$-0.612742\pi$
$467$	− 409103.i	− 1.87586i	0.346831	−	0.937928i	$-0.387258\pi$
$468$	0	0
$469$	−81648.0	−0.371193
$470$	0	0
$471$	− 159865.i	− 0.720628i
$472$	0	0
$473$	−501552.	−2.24178
$474$	0	0
$475$	36726.4i	0.162776i
$476$	0	0
$477$	−43254.0	−0.190103
$478$	0	0
$479$	317630.i	1.38437i	0.721722	+	0.692183i	$0.243351\pi$
$479$	317630.i	1.38437i	−0.721722	+	0.692183i	$0.756649\pi$
$480$	0	0
$481$	36980.0	0.159837
$482$	0	0
$483$	− 242432.i	− 1.03919i
$484$	0	0
$485$	17652.0	0.0750430
$486$	0	0
$487$	− 238379.i	− 1.00510i	−0.864548	−	0.502550i	$-0.832395\pi$
$487$	− 238379.i	− 1.00510i	0.864548	−	0.502550i	$-0.167605\pi$
$488$	0	0
$489$	139644.	0.583989
$490$	0	0
$491$	75759.9i	0.314251i	0.987579	+	0.157125i	$0.0502227\pi$
$491$	75759.9i	0.314251i	−0.987579	+	0.157125i	$0.949777\pi$
$492$	0	0
$493$	−503532.	−2.07173
$494$	0	0
$495$	− 30304.0i	− 0.123677i
$496$	0	0
$497$	186624.	0.755535
$498$	0	0
$499$	245986.i	0.987891i	0.869493	+	0.493946i	$0.164446\pi$
$499$	245986.i	0.987891i	−0.869493	+	0.493946i	$0.835554\pi$
$500$	0	0
$501$	52488.0	0.209115
$502$	0	0
$503$	− 374871.i	− 1.48165i	−0.671697	−	0.740826i	$-0.734434\pi$
$503$	− 374871.i	− 1.48165i	0.671697	−	0.740826i	$-0.265566\pi$
$504$	0	0
$505$	−70092.0	−0.274844
$506$	0	0
$507$	109977.i	0.427843i
$508$	0	0
$509$	168210.	0.649256	0.324628	−	0.945842i	$-0.394761\pi$
$509$	168210.	0.649256	0.324628	+	0.945842i	$0.394761\pi$
$510$	0	0
$511$	253531.i	0.970932i
$512$	0	0
$513$	8748.00	0.0332410
$514$	0	0
$515$	− 99890.8i	− 0.376627i
$516$	0	0
$517$	69984.0	0.261829
$518$	0	0
$519$	− 72798.1i	− 0.270262i
$520$	0	0
$521$	154890.	0.570621	0.285311	−	0.958435i	$-0.407903\pi$
$521$	154890.	0.570621	0.285311	+	0.958435i	$0.407903\pi$
$522$	0	0
$523$	39470.0i	0.144299i	0.997394	+	0.0721495i	$0.0229859\pi$
$523$	39470.0i	0.144299i	−0.997394	+	0.0721495i	$0.977014\pi$
$524$	0	0
$525$	−190836.	−0.692376
$526$	0	0
$527$	− 664068.i	− 2.39107i
$528$	0	0
$529$	−280031.	−1.00068
$530$	0	0
$531$	95962.5i	0.340340i
$532$	0	0
$533$	193500.	0.681125
$534$	0	0
$535$	57240.8i	0.199985i
$536$	0	0
$537$	−74844.0	−0.259542
$538$	0	0
$539$	278160.i	0.957454i
$540$	0	0
$541$	218342.	0.746007	0.373003	−	0.927830i	$-0.378328\pi$
$541$	218342.	0.746007	0.373003	+	0.927830i	$0.378328\pi$
$542$	0	0
$543$	216794.i	0.735271i
$544$	0	0
$545$	105180.	0.354112
$546$	0	0
$547$	351863.i	1.17598i	0.808870	+	0.587988i	$0.200080\pi$
$547$	351863.i	1.17598i	−0.808870	+	0.587988i	$0.799920\pi$
$548$	0	0
$549$	57078.0	0.189376
$550$	0	0
$551$	73702.2i	0.242760i
$552$	0	0
$553$	346032.	1.13153
$554$	0	0
$555$	13406.1i	0.0435227i
$556$	0	0
$557$	185802.	0.598880	0.299440	−	0.954115i	$-0.403200\pi$
$557$	185802.	0.598880	0.299440	+	0.954115i	$0.403200\pi$
$558$	0	0
$559$	− 230584.i	− 0.737915i
$560$	0	0
$561$	−414072.	−1.31568
$562$	0	0
$563$	− 248231.i	− 0.783138i	−0.920149	−	0.391569i	$-0.871932\pi$
$563$	− 248231.i	− 0.783138i	0.920149	−	0.391569i	$-0.128068\pi$
$564$	0	0
$565$	−72108.0	−0.225885
$566$	0	0
$567$	45455.9i	0.141392i
$568$	0	0
$569$	361866.	1.11769	0.558847	−	0.829270i	$-0.311244\pi$
$569$	361866.	1.11769	0.558847	+	0.829270i	$0.311244\pi$
$570$	0	0
$571$	− 275666.i	− 0.845496i	−0.906247	−	0.422748i	$-0.861066\pi$
$571$	− 275666.i	− 0.845496i	0.906247	−	0.422748i	$-0.138934\pi$
$572$	0	0
$573$	−279936.	−0.852608
$574$	0	0
$575$	440717.i	1.33298i
$576$	0	0
$577$	−477358.	−1.43381	−0.716907	−	0.697169i	$-0.754443\pi$
$577$	−477358.	−1.43381	−0.716907	+	0.697169i	$0.754443\pi$
$578$	0	0
$579$	212990.i	0.635335i
$580$	0	0
$581$	−571536.	−1.69313
$582$	0	0
$583$	− 299673.i	− 0.881678i
$584$	0	0
$585$	13932.0	0.0407101
$586$	0	0
$587$	255339.i	0.741039i	0.928825	+	0.370519i	$0.120820\pi$
$587$	255339.i	0.741039i	−0.928825	+	0.370519i	$0.879180\pi$
$588$	0	0
$589$	−97200.0	−0.280179
$590$	0	0
$591$	29337.5i	0.0839939i
$592$	0	0
$593$	232818.	0.662075	0.331037	−	0.943618i	$-0.392601\pi$
$593$	232818.	0.662075	0.331037	+	0.943618i	$0.392601\pi$
$594$	0	0
$595$	− 159376.i	− 0.450184i
$596$	0	0
$597$	240732.	0.675437
$598$	0	0
$599$	692876.i	1.93109i	0.260243	+	0.965543i	$0.416197\pi$
$599$	692876.i	1.93109i	−0.260243	+	0.965543i	$0.583803\pi$
$600$	0	0
$601$	−171550.	−0.474943	−0.237472	−	0.971394i	$-0.576319\pi$
$601$	−171550.	−0.474943	−0.237472	+	0.971394i	$0.576319\pi$
$602$	0	0
$603$	− 35354.6i	− 0.0972325i
$604$	0	0
$605$	122106.	0.333600
$606$	0	0
$607$	689197.i	1.87054i	0.353941	+	0.935268i	$0.384841\pi$
$607$	689197.i	1.87054i	−0.353941	+	0.935268i	$0.615159\pi$
$608$	0	0
$609$	−382968.	−1.03259
$610$	0	0
$611$	32174.6i	0.0861847i
$612$	0	0
$613$	245086.	0.652225	0.326113	−	0.945331i	$-0.394261\pi$
$613$	245086.	0.652225	0.326113	+	0.945331i	$0.394261\pi$
$614$	0	0
$615$	70148.1i	0.185466i
$616$	0	0
$617$	511602.	1.34388	0.671942	−	0.740604i	$-0.265460\pi$
$617$	511602.	1.34388	0.671942	+	0.740604i	$0.265460\pi$
$618$	0	0
$619$	15152.0i	0.0395447i	0.999805	+	0.0197723i	$0.00629414\pi$
$619$	15152.0i	0.0395447i	−0.999805	+	0.0197723i	$0.993706\pi$
$620$	0	0
$621$	104976.	0.272212
$622$	0	0
$623$	− 127576.i	− 0.328695i
$624$	0	0
$625$	324421.	0.830518
$626$	0	0
$627$	60607.9i	0.154168i
$628$	0	0
$629$	183180.	0.462995
$630$	0	0
$631$	610880.i	1.53425i	0.641495	+	0.767127i	$0.278315\pi$
$631$	610880.i	1.53425i	−0.641495	+	0.767127i	$0.721685\pi$
$632$	0	0
$633$	109188.	0.272501
$634$	0	0
$635$	− 64723.3i	− 0.160514i
$636$	0	0
$637$	−127882.	−0.315160
$638$	0	0
$639$	80810.6i	0.197909i
$640$	0	0
$641$	730122.	1.77697	0.888484	−	0.458909i	$-0.151759\pi$
$641$	730122.	1.77697	0.888484	+	0.458909i	$0.151759\pi$
$642$	0	0
$643$	40592.3i	0.0981798i	0.998794	+	0.0490899i	$0.0156321\pi$
$643$	40592.3i	0.0981798i	−0.998794	+	0.0490899i	$0.984368\pi$
$644$	0	0
$645$	83592.0	0.200930
$646$	0	0
$647$	47887.7i	0.114397i	0.998363	+	0.0571987i	$0.0182168\pi$
$647$	47887.7i	0.114397i	−0.998363	+	0.0571987i	$0.981783\pi$
$648$	0	0
$649$	−664848.	−1.57846
$650$	0	0
$651$	− 505066.i	− 1.19175i
$652$	0	0
$653$	587754.	1.37838	0.689190	−	0.724580i	$-0.257966\pi$
$653$	587754.	1.37838	0.689190	+	0.724580i	$0.257966\pi$
$654$	0	0
$655$	− 133562.i	− 0.311315i
$656$	0	0
$657$	−109782.	−0.254332
$658$	0	0
$659$	583445.i	1.34347i	0.740790	+	0.671736i	$0.234451\pi$
$659$	583445.i	1.34347i	−0.740790	+	0.671736i	$0.765549\pi$
$660$	0	0
$661$	250798.	0.574012	0.287006	−	0.957929i	$-0.407340\pi$
$661$	250798.	0.574012	0.287006	+	0.957929i	$0.407340\pi$
$662$	0	0
$663$	− 190366.i	− 0.433075i
$664$	0	0
$665$	−23328.0	−0.0527514
$666$	0	0
$667$	884427.i	1.98797i
$668$	0	0
$669$	−142236.	−0.317802
$670$	0	0
$671$	395448.i	0.878303i
$672$	0	0
$673$	−257230.	−0.567926	−0.283963	−	0.958835i	$-0.591649\pi$
$673$	−257230.	−0.567926	−0.283963	+	0.958835i	$0.591649\pi$
$674$	0	0
$675$	− 82634.4i	− 0.181365i
$676$	0	0
$677$	−513270.	−1.11987	−0.559936	−	0.828536i	$-0.689174\pi$
$677$	−513270.	−1.11987	−0.559936	+	0.828536i	$0.689174\pi$
$678$	0	0
$679$	− 183445.i	− 0.397893i
$680$	0	0
$681$	−263412.	−0.567991
$682$	0	0
$683$	64910.3i	0.139147i	0.997577	+	0.0695733i	$0.0221638\pi$
$683$	64910.3i	0.139147i	−0.997577	+	0.0695733i	$0.977836\pi$
$684$	0	0
$685$	−161820.	−0.344867
$686$	0	0
$687$	− 391031.i	− 0.828510i
$688$	0	0
$689$	137772.	0.290217
$690$	0	0
$691$	− 191863.i	− 0.401823i	−0.979609	−	0.200911i	$-0.935610\pi$
$691$	− 191863.i	− 0.401823i	0.979609	−	0.200911i	$-0.0643903\pi$
$692$	0	0
$693$	−314928.	−0.655760
$694$	0	0
$695$	13842.6i	0.0286580i
$696$	0	0
$697$	958500.	1.97300
$698$	0	0
$699$	523928.i	1.07230i
$700$	0	0
$701$	−392622.	−0.798985	−0.399492	−	0.916736i	$-0.630814\pi$
$701$	−392622.	−0.798985	−0.399492	+	0.916736i	$0.630814\pi$
$702$	0	0
$703$	− 26812.1i	− 0.0542526i
$704$	0	0
$705$	−11664.0	−0.0234676
$706$	0	0
$707$	728417.i	1.45727i
$708$	0	0
$709$	−480106.	−0.955091	−0.477545	−	0.878607i	$-0.658473\pi$
$709$	−480106.	−0.955091	−0.477545	+	0.878607i	$0.658473\pi$
$710$	0	0
$711$	149836.i	0.296400i
$712$	0	0
$713$	−1.16640e6	−2.29440
$714$	0	0
$715$	96523.7i	0.188809i
$716$	0	0
$717$	−256608.	−0.499151
$718$	0	0
$719$	− 511426.i	− 0.989293i	−0.869094	−	0.494647i	$-0.835297\pi$
$719$	− 511426.i	− 0.989293i	0.869094	−	0.494647i	$-0.164703\pi$
$720$	0	0
$721$	−1.03810e6	−1.99695
$722$	0	0
$723$	93852.9i	0.179544i
$724$	0	0
$725$	696198.	1.32451
$726$	0	0
$727$	− 380296.i	− 0.719537i	−0.933042	−	0.359768i	$-0.882856\pi$
$727$	− 380296.i	− 0.719537i	0.933042	−	0.359768i	$-0.117144\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	− 1.14220e6i	− 2.13750i
$732$	0	0
$733$	−477706.	−0.889104	−0.444552	−	0.895753i	$-0.646637\pi$
$733$	−477706.	−0.889104	−0.444552	+	0.895753i	$0.646637\pi$
$734$	0	0
$735$	− 46360.1i	− 0.0858162i
$736$	0	0
$737$	244944.	0.450954
$738$	0	0
$739$	143726.i	0.263175i	0.991305	+	0.131588i	$0.0420075\pi$
$739$	143726.i	0.263175i	−0.991305	+	0.131588i	$0.957993\pi$
$740$	0	0
$741$	−27864.0	−0.0507466
$742$	0	0
$743$	− 533873.i	− 0.967076i	−0.875323	−	0.483538i	$-0.839351\pi$
$743$	− 533873.i	− 0.967076i	0.875323	−	0.483538i	$-0.160649\pi$
$744$	0	0
$745$	83556.0	0.150545
$746$	0	0
$747$	− 247482.i	− 0.443510i
$748$	0	0
$749$	594864.	1.06036
$750$	0	0
$751$	− 746812.i	− 1.32413i	−0.749446	−	0.662066i	$-0.769680\pi$
$751$	− 746812.i	− 1.32413i	0.749446	−	0.662066i	$-0.230320\pi$
$752$	0	0
$753$	523908.	0.923985
$754$	0	0
$755$	108870.i	0.190991i
$756$	0	0
$757$	482870.	0.842633	0.421317	−	0.906914i	$-0.361568\pi$
$757$	482870.	0.842633	0.421317	+	0.906914i	$0.361568\pi$
$758$	0	0
$759$	727295.i	1.26249i
$760$	0	0
$761$	96954.0	0.167416	0.0837079	−	0.996490i	$-0.473324\pi$
$761$	96954.0	0.167416	0.0837079	+	0.996490i	$0.473324\pi$
$762$	0	0
$763$	− 1.09306e6i	− 1.87757i
$764$	0	0
$765$	69012.0	0.117924
$766$	0	0
$767$	− 305658.i	− 0.519572i
$768$	0	0
$769$	1.05427e6	1.78279	0.891396	−	0.453225i	$-0.149726\pi$
$769$	1.05427e6	1.78279	0.891396	+	0.453225i	$0.149726\pi$
$770$	0	0
$771$	94434.9i	0.158863i
$772$	0	0
$773$	921426.	1.54206	0.771030	−	0.636798i	$-0.219742\pi$
$773$	921426.	1.54206	0.771030	+	0.636798i	$0.219742\pi$
$774$	0	0
$775$	918160.i	1.52867i
$776$	0	0
$777$	139320.	0.230766
$778$	0	0
$779$	− 140296.i	− 0.231191i
$780$	0	0
$781$	−559872.	−0.917882
$782$	0	0
$783$	− 165830.i	− 0.270483i
$784$	0	0
$785$	−184596.	−0.299559
$786$	0	0
$787$	199220.i	0.321651i	0.986983	+	0.160825i	$0.0514156\pi$
$787$	199220.i	0.321651i	−0.986983	+	0.160825i	$0.948584\pi$
$788$	0	0
$789$	−33048.0	−0.0530874
$790$	0	0
$791$	749368.i	1.19768i
$792$	0	0
$793$	−181804.	−0.289106
$794$	0	0
$795$	49945.4i	0.0790244i
$796$	0	0
$797$	96522.0	0.151953	0.0759766	−	0.997110i	$-0.475793\pi$
$797$	96522.0	0.151953	0.0759766	+	0.997110i	$0.475793\pi$
$798$	0	0
$799$	159376.i	0.249649i
$800$	0	0
$801$	55242.0	0.0861002
$802$	0	0
$803$	− 760592.i	− 1.17956i
$804$	0	0
$805$	−279936.	−0.431983
$806$	0	0
$807$	− 160468.i	− 0.246400i
$808$	0	0
$809$	−336486.	−0.514126	−0.257063	−	0.966395i	$-0.582755\pi$
$809$	−336486.	−0.514126	−0.257063	+	0.966395i	$0.582755\pi$
$810$	0	0
$811$	− 842213.i	− 1.28050i	−0.768166	−	0.640251i	$-0.778830\pi$
$811$	− 842213.i	− 1.28050i	0.768166	−	0.640251i	$-0.221170\pi$
$812$	0	0
$813$	−444204.	−0.672050
$814$	0	0
$815$	− 161247.i	− 0.242760i
$816$	0	0
$817$	−167184.	−0.250467
$818$	0	0
$819$	− 144786.i	− 0.215853i
$820$	0	0
$821$	−696462.	−1.03326	−0.516632	−	0.856208i	$-0.672814\pi$
$821$	−696462.	−1.03326	−0.516632	+	0.856208i	$0.672814\pi$
$822$	0	0
$823$	15027.3i	0.0221861i	0.999938	+	0.0110930i	$0.00353110\pi$
$823$	15027.3i	0.0221861i	−0.999938	+	0.0110930i	$0.996469\pi$
$824$	0	0
$825$	572508.	0.841150
$826$	0	0
$827$	− 842338.i	− 1.23162i	−0.787896	−	0.615808i	$-0.788830\pi$
$827$	− 842338.i	− 1.23162i	0.787896	−	0.615808i	$-0.211170\pi$
$828$	0	0
$829$	1.07025e6	1.55731	0.778654	−	0.627453i	$-0.215903\pi$
$829$	1.07025e6	1.55731	0.778654	+	0.627453i	$0.215903\pi$
$830$	0	0
$831$	− 199730.i	− 0.289228i
$832$	0	0
$833$	−633462.	−0.912915
$834$	0	0
$835$	− 60607.9i	− 0.0869274i
$836$	0	0
$837$	218700.	0.312175
$838$	0	0
$839$	− 479626.i	− 0.681363i	−0.940179	−	0.340681i	$-0.889342\pi$
$839$	− 479626.i	− 0.681363i	0.940179	−	0.340681i	$-0.110658\pi$
$840$	0	0
$841$	689843.	0.975345
$842$	0	0
$843$	199439.i	0.280643i
$844$	0	0
$845$	126990.	0.177851
$846$	0	0
$847$	− 1.26896e6i	− 1.76881i
$848$	0	0
$849$	−124740.	−0.173057
$850$	0	0
$851$	− 321746.i	− 0.444277i
$852$	0	0
$853$	−929042.	−1.27684	−0.638421	−	0.769687i	$-0.720412\pi$
$853$	−929042.	−1.27684	−0.638421	+	0.769687i	$0.720412\pi$
$854$	0	0
$855$	− 10101.3i	− 0.0138180i
$856$	0	0
$857$	−365430.	−0.497557	−0.248778	−	0.968560i	$-0.580029\pi$
$857$	−365430.	−0.497557	−0.248778	+	0.968560i	$0.580029\pi$
$858$	0	0
$859$	212814.i	0.288412i	0.989548	+	0.144206i	$0.0460628\pi$
$859$	212814.i	0.288412i	−0.989548	+	0.144206i	$0.953937\pi$
$860$	0	0
$861$	729000.	0.983380
$862$	0	0
$863$	− 240935.i	− 0.323503i	−0.986832	−	0.161752i	$-0.948286\pi$
$863$	− 240935.i	− 0.323503i	0.986832	−	0.161752i	$-0.0517143\pi$
$864$	0	0
$865$	−84060.0	−0.112346
$866$	0	0
$867$	− 508989.i	− 0.677127i
$868$	0	0
$869$	−1.03810e6	−1.37467
$870$	0	0
$871$	112611.i	0.148438i
$872$	0	0
$873$	79434.0	0.104226
$874$	0	0
$875$	454185.i	0.593222i
$876$	0	0
$877$	1.11437e6	1.44887	0.724434	−	0.689344i	$-0.242101\pi$
$877$	1.11437e6	1.44887	0.724434	+	0.689344i	$0.242101\pi$
$878$	0	0
$879$	227124.i	0.293958i
$880$	0	0
$881$	−76446.0	−0.0984925	−0.0492462	−	0.998787i	$-0.515682\pi$
$881$	−76446.0	−0.0984925	−0.0492462	+	0.998787i	$0.515682\pi$
$882$	0	0
$883$	644676.i	0.826838i	0.910541	+	0.413419i	$0.135665\pi$
$883$	644676.i	0.826838i	−0.910541	+	0.413419i	$0.864335\pi$
$884$	0	0
$885$	110808.	0.141477
$886$	0	0
$887$	− 403679.i	− 0.513084i	−0.966533	−	0.256542i	$-0.917417\pi$
$887$	− 403679.i	− 0.513084i	0.966533	−	0.256542i	$-0.0825832\pi$
$888$	0	0
$889$	−672624.	−0.851077
$890$	0	0
$891$	− 136368.i	− 0.171774i
$892$	0	0
$893$	23328.0	0.0292533
$894$	0	0
$895$	86422.4i	0.107890i
$896$	0	0
$897$	−334368.	−0.415566
$898$	0	0
$899$	1.84256e6i	2.27982i
$900$	0	0
$901$	682452.	0.840664
$902$	0	0
$903$	− 868714.i	− 1.06537i
$904$	0	0
$905$	250332.	0.305646
$906$	0	0
$907$	− 256337.i	− 0.311599i	−0.987789	−	0.155799i	$-0.950205\pi$
$907$	− 256337.i	− 0.311599i	0.987789	−	0.155799i	$-0.0497954\pi$
$908$	0	0
$909$	−315414.	−0.381727
$910$	0	0
$911$	− 716820.i	− 0.863720i	−0.901941	−	0.431860i	$-0.857857\pi$
$911$	− 716820.i	− 0.863720i	0.901941	−	0.431860i	$-0.142143\pi$
$912$	0	0
$913$	1.71461e6	2.05695
$914$	0	0
$915$	− 65908.0i	− 0.0787220i
$916$	0	0
$917$	−1.38802e6	−1.65065
$918$	0	0
$919$	542291.i	0.642098i	0.947063	+	0.321049i	$0.104035\pi$
$919$	542291.i	0.642098i	−0.947063	+	0.321049i	$0.895965\pi$
$920$	0	0
$921$	−308124.	−0.363251
$922$	0	0
$923$	− 257397.i	− 0.302134i
$924$	0	0
$925$	−253270.	−0.296006
$926$	0	0
$927$	− 449509.i	− 0.523093i
$928$	0	0
$929$	470010.	0.544598	0.272299	−	0.962213i	$-0.412216\pi$
$929$	470010.	0.544598	0.272299	+	0.962213i	$0.412216\pi$
$930$	0	0
$931$	92720.1i	0.106973i
$932$	0	0
$933$	316872.	0.364016
$934$	0	0
$935$	478129.i	0.546918i
$936$	0	0
$937$	−621070.	−0.707394	−0.353697	−	0.935360i	$-0.615076\pi$
$937$	−621070.	−0.707394	−0.353697	+	0.935360i	$0.615076\pi$
$938$	0	0
$939$	− 11732.9i	− 0.0133068i
$940$	0	0
$941$	−1.07204e6	−1.21068	−0.605342	−	0.795965i	$-0.706964\pi$
$941$	−1.07204e6	−1.21068	−0.605342	+	0.795965i	$0.706964\pi$
$942$	0	0
$943$	− 1.68355e6i	− 1.89323i
$944$	0	0
$945$	52488.0	0.0587755
$946$	0	0
$947$	138238.i	0.154145i	0.997025	+	0.0770724i	$0.0245573\pi$
$947$	138238.i	0.154145i	−0.997025	+	0.0770724i	$0.975443\pi$
$948$	0	0
$949$	349676.	0.388270
$950$	0	0
$951$	631925.i	0.698722i
$952$	0	0
$953$	89610.0	0.0986667	0.0493334	−	0.998782i	$-0.484290\pi$
$953$	89610.0	0.0986667	0.0493334	+	0.998782i	$0.484290\pi$
$954$	0	0
$955$	323242.i	0.354423i
$956$	0	0
$957$	1.14890e6	1.25447
$958$	0	0
$959$	1.68168e6i	1.82855i
$960$	0	0
$961$	−1.50648e6	−1.63123
$962$	0	0
$963$	257584.i	0.277757i
$964$	0	0
$965$	245940.	0.264104
$966$	0	0
$967$	684084.i	0.731571i	0.930699	+	0.365785i	$0.119200\pi$
$967$	684084.i	0.731571i	−0.930699	+	0.365785i	$0.880800\pi$
$968$	0	0
$969$	−138024.	−0.146997
$970$	0	0
$971$	− 201465.i	− 0.213679i	−0.994276	−	0.106839i	$-0.965927\pi$
$971$	− 201465.i	− 0.213679i	0.994276	−	0.106839i	$-0.0340731\pi$
$972$	0	0
$973$	143856.	0.151951
$974$	0	0
$975$	263206.i	0.276877i
$976$	0	0
$977$	−735750.	−0.770799	−0.385400	−	0.922750i	$-0.625936\pi$
$977$	−735750.	−0.770799	−0.385400	+	0.922750i	$0.625936\pi$
$978$	0	0
$979$	382728.i	0.399323i
$980$	0	0
$981$	473310.	0.491822
$982$	0	0
$983$	1.20954e6i	1.25174i	0.779929	+	0.625868i	$0.215255\pi$
$983$	1.20954e6i	1.25174i	−0.779929	+	0.625868i	$0.784745\pi$
$984$	0	0
$985$	33876.0	0.0349156
$986$	0	0
$987$	121216.i	0.124430i
$988$	0	0
$989$	−2.00621e6	−2.05108
$990$	0	0
$991$	869649.i	0.885516i	0.896641	+	0.442758i	$0.146000\pi$
$991$	869649.i	0.885516i	−0.896641	+	0.442758i	$0.854000\pi$
$992$	0	0
$993$	1.00537e6	1.01960
$994$	0	0
$995$	− 277973.i	− 0.280774i
$996$	0	0
$997$	352702.	0.354828	0.177414	−	0.984136i	$-0.443227\pi$
$997$	352702.	0.354828	0.177414	+	0.984136i	$0.443227\pi$
$998$	0	0
$999$	60327.3i	0.0604482i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.5.g.a.127.1		2
3.2	odd	2		576.5.g.g.127.2		2
4.3	odd	2	inner	192.5.g.a.127.2		2
8.3	odd	2		48.5.g.b.31.1	✓	2
8.5	even	2		48.5.g.b.31.2	yes	2
12.11	even	2		576.5.g.g.127.1		2
16.3	odd	4		768.5.b.d.127.2		4
16.5	even	4		768.5.b.d.127.1		4
16.11	odd	4		768.5.b.d.127.3		4
16.13	even	4		768.5.b.d.127.4		4
24.5	odd	2		144.5.g.d.127.2		2
24.11	even	2		144.5.g.d.127.1		2
40.3	even	4		1200.5.j.a.799.3		4
40.13	odd	4		1200.5.j.a.799.2		4
40.19	odd	2		1200.5.e.a.751.2		2
40.27	even	4		1200.5.j.a.799.1		4
40.29	even	2		1200.5.e.a.751.1		2
40.37	odd	4		1200.5.j.a.799.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.g.b.31.1	✓	2	8.3	odd	2
48.5.g.b.31.2	yes	2	8.5	even	2
144.5.g.d.127.1		2	24.11	even	2
144.5.g.d.127.2		2	24.5	odd	2
192.5.g.a.127.1		2	1.1	even	1	trivial
192.5.g.a.127.2		2	4.3	odd	2	inner
576.5.g.g.127.1		2	12.11	even	2
576.5.g.g.127.2		2	3.2	odd	2
768.5.b.d.127.1		4	16.5	even	4
768.5.b.d.127.2		4	16.3	odd	4
768.5.b.d.127.3		4	16.11	odd	4
768.5.b.d.127.4		4	16.13	even	4
1200.5.e.a.751.1		2	40.29	even	2
1200.5.e.a.751.2		2	40.19	odd	2
1200.5.j.a.799.1		4	40.27	even	4
1200.5.j.a.799.2		4	40.13	odd	4
1200.5.j.a.799.3		4	40.3	even	4
1200.5.j.a.799.4		4	40.37	odd	4

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	192.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} -6.00000 q^{5} +62.3538i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q-5.19615i q^{3} -6.00000 q^{5} +62.3538i q^{7} -27.0000 q^{9} -187.061i q^{11} +86.0000 q^{13} +31.1769i q^{15} +426.000 q^{17} -62.3538i q^{19} +324.000 q^{21} -748.246i q^{23} -589.000 q^{25} +140.296i q^{27} -1182.00 q^{29} -1558.85i q^{31} -972.000 q^{33} -374.123i q^{35} +430.000 q^{37} -446.869i q^{39} +2250.00 q^{41} -2681.21i q^{43} +162.000 q^{45} +374.123i q^{47} -1487.00 q^{49} -2213.56i q^{51} +1602.00 q^{53} +1122.37i q^{55} -324.000 q^{57} -3554.17i q^{59} -2114.00 q^{61} -1683.55i q^{63} -516.000 q^{65} +1309.43i q^{67} -3888.00 q^{69} -2992.98i q^{71} +4066.00 q^{73} +3060.53i q^{75} +11664.0 q^{77} -5549.49i q^{79} +729.000 q^{81} +9166.01i q^{83} -2556.00 q^{85} +6141.85i q^{87} -2046.00 q^{89} +5362.43i q^{91} -8100.00 q^{93} +374.123i q^{95} -2942.00 q^{97} +5050.66i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{5} - 54 q^{9}+O(q^{10}) \) 2 * q - 12 * q^5 - 54 * q^9	\( 2 q - 12 q^{5} - 54 q^{9} + 172 q^{13} + 852 q^{17} + 648 q^{21} - 1178 q^{25} - 2364 q^{29} - 1944 q^{33} + 860 q^{37} + 4500 q^{41} + 324 q^{45} - 2974 q^{49} + 3204 q^{53} - 648 q^{57} - 4228 q^{61} - 1032 q^{65} - 7776 q^{69} + 8132 q^{73} + 23328 q^{77} + 1458 q^{81} - 5112 q^{85} - 4092 q^{89} - 16200 q^{93} - 5884 q^{97}+O(q^{100}) \) 2 * q - 12 * q^5 - 54 * q^9 + 172 * q^13 + 852 * q^17 + 648 * q^21 - 1178 * q^25 - 2364 * q^29 - 1944 * q^33 + 860 * q^37 + 4500 * q^41 + 324 * q^45 - 2974 * q^49 + 3204 * q^53 - 648 * q^57 - 4228 * q^61 - 1032 * q^65 - 7776 * q^69 + 8132 * q^73 + 23328 * q^77 + 1458 * q^81 - 5112 * q^85 - 4092 * q^89 - 16200 * q^93 - 5884 * q^97

Introduction

L-functions

Modular forms

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Database

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