LMFDB - Embedded newform 117.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("117.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	117.a (trivial)

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:	yes
Analytic conductor:	$0.934249703649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	117.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} -2.82843 q^{5} -2.82843 q^{7} +4.41421 q^{8} +O(q^{10})$	\(q+2.41421 q^{2} +3.82843 q^{4} -2.82843 q^{5} -2.82843 q^{7} +4.41421 q^{8} -6.82843 q^{10} +2.00000 q^{11} -1.00000 q^{13} -6.82843 q^{14} +3.00000 q^{16} +3.65685 q^{17} +2.82843 q^{19} -10.8284 q^{20} +4.82843 q^{22} +4.00000 q^{23} +3.00000 q^{25} -2.41421 q^{26} -10.8284 q^{28} -2.00000 q^{29} -6.82843 q^{31} -1.58579 q^{32} +8.82843 q^{34} +8.00000 q^{35} +3.65685 q^{37} +6.82843 q^{38} -12.4853 q^{40} -10.8284 q^{41} +9.65685 q^{43} +7.65685 q^{44} +9.65685 q^{46} +0.343146 q^{47} +1.00000 q^{49} +7.24264 q^{50} -3.82843 q^{52} +2.00000 q^{53} -5.65685 q^{55} -12.4853 q^{56} -4.82843 q^{58} +3.65685 q^{59} -9.31371 q^{61} -16.4853 q^{62} -9.82843 q^{64} +2.82843 q^{65} +1.17157 q^{67} +14.0000 q^{68} +19.3137 q^{70} -2.00000 q^{71} +11.6569 q^{73} +8.82843 q^{74} +10.8284 q^{76} -5.65685 q^{77} +11.3137 q^{79} -8.48528 q^{80} -26.1421 q^{82} +7.65685 q^{83} -10.3431 q^{85} +23.3137 q^{86} +8.82843 q^{88} -9.17157 q^{89} +2.82843 q^{91} +15.3137 q^{92} +0.828427 q^{94} -8.00000 q^{95} -7.65685 q^{97} +2.41421 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 6 q^{8} - 8 q^{10} + 4 q^{11} - 2 q^{13} - 8 q^{14} + 6 q^{16} - 4 q^{17} - 16 q^{20} + 4 q^{22} + 8 q^{23} + 6 q^{25} - 2 q^{26} - 16 q^{28} - 4 q^{29} - 8 q^{31} - 6 q^{32} + 12 q^{34} + 16 q^{35} - 4 q^{37} + 8 q^{38} - 8 q^{40} - 16 q^{41} + 8 q^{43} + 4 q^{44} + 8 q^{46} + 12 q^{47} + 2 q^{49} + 6 q^{50} - 2 q^{52} + 4 q^{53} - 8 q^{56} - 4 q^{58} - 4 q^{59} + 4 q^{61} - 16 q^{62} - 14 q^{64} + 8 q^{67} + 28 q^{68} + 16 q^{70} - 4 q^{71} + 12 q^{73} + 12 q^{74} + 16 q^{76} - 24 q^{82} + 4 q^{83} - 32 q^{85} + 24 q^{86} + 12 q^{88} - 24 q^{89} + 8 q^{92} - 4 q^{94} - 16 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 - 8 * q^10 + 4 * q^11 - 2 * q^13 - 8 * q^14 + 6 * q^16 - 4 * q^17 - 16 * q^20 + 4 * q^22 + 8 * q^23 + 6 * q^25 - 2 * q^26 - 16 * q^28 - 4 * q^29 - 8 * q^31 - 6 * q^32 + 12 * q^34 + 16 * q^35 - 4 * q^37 + 8 * q^38 - 8 * q^40 - 16 * q^41 + 8 * q^43 + 4 * q^44 + 8 * q^46 + 12 * q^47 + 2 * q^49 + 6 * q^50 - 2 * q^52 + 4 * q^53 - 8 * q^56 - 4 * q^58 - 4 * q^59 + 4 * q^61 - 16 * q^62 - 14 * q^64 + 8 * q^67 + 28 * q^68 + 16 * q^70 - 4 * q^71 + 12 * q^73 + 12 * q^74 + 16 * q^76 - 24 * q^82 + 4 * q^83 - 32 * q^85 + 24 * q^86 + 12 * q^88 - 24 * q^89 + 8 * q^92 - 4 * q^94 - 16 * q^95 - 4 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	4.41421	1.56066
$9$	0	0
$10$	−6.82843	−2.15934
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−6.82843	−1.82497
$15$	0	0
$16$	3.00000	0.750000
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	−10.8284	−2.42131
$21$	0	0
$22$	4.82843	1.02942
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	−2.41421	−0.473466
$27$	0	0
$28$	−10.8284	−2.04638
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	8.82843	1.51406
$35$	8.00000	1.35225
$36$	0	0
$37$	3.65685	0.601183	0.300592	−	0.953753i	$-0.402816\pi$
$37$	3.65685	0.601183	0.300592	+	0.953753i	$0.402816\pi$
$38$	6.82843	1.10772
$39$	0	0
$40$	−12.4853	−1.97410
$41$	−10.8284	−1.69112	−0.845558	−	0.533883i	$-0.820732\pi$
$41$	−10.8284	−1.69112	−0.845558	+	0.533883i	$0.820732\pi$
$42$	0	0
$43$	9.65685	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	7.65685	1.15431
$45$	0	0
$46$	9.65685	1.42383
$47$	0.343146	0.0500530	0.0250265	−	0.999687i	$-0.492033\pi$
$47$	0.343146	0.0500530	0.0250265	+	0.999687i	$0.492033\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.24264	1.02426
$51$	0	0
$52$	−3.82843	−0.530907
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	−12.4853	−1.66842
$57$	0	0
$58$	−4.82843	−0.634004
$59$	3.65685	0.476082	0.238041	−	0.971255i	$-0.423495\pi$
$59$	3.65685	0.476082	0.238041	+	0.971255i	$0.423495\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	−16.4853	−2.09363
$63$	0	0
$64$	−9.82843	−1.22855
$65$	2.82843	0.350823
$66$	0	0
$67$	1.17157	0.143130	0.0715652	−	0.997436i	$-0.477201\pi$
$67$	1.17157	0.143130	0.0715652	+	0.997436i	$0.477201\pi$
$68$	14.0000	1.69775
$69$	0	0
$70$	19.3137	2.30843
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	11.6569	1.36433	0.682166	−	0.731198i	$-0.261038\pi$
$73$	11.6569	1.36433	0.682166	+	0.731198i	$0.261038\pi$
$74$	8.82843	1.02628
$75$	0	0
$76$	10.8284	1.24211
$77$	−5.65685	−0.644658
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	−8.48528	−0.948683
$81$	0	0
$82$	−26.1421	−2.88692
$83$	7.65685	0.840449	0.420224	−	0.907420i	$-0.361951\pi$
$83$	7.65685	0.840449	0.420224	+	0.907420i	$0.361951\pi$
$84$	0	0
$85$	−10.3431	−1.12187
$86$	23.3137	2.51398
$87$	0	0
$88$	8.82843	0.941113
$89$	−9.17157	−0.972185	−0.486092	−	0.873907i	$-0.661578\pi$
$89$	−9.17157	−0.972185	−0.486092	+	0.873907i	$0.661578\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	15.3137	1.59656
$93$	0	0
$94$	0.828427	0.0854457
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	2.41421	0.243872
$99$	0	0
$100$	11.4853	1.14853
$101$	3.65685	0.363871	0.181935	−	0.983311i	$-0.441764\pi$
$101$	3.65685	0.363871	0.181935	+	0.983311i	$0.441764\pi$
$102$	0	0
$103$	13.6569	1.34565	0.672825	−	0.739802i	$-0.265081\pi$
$103$	13.6569	1.34565	0.672825	+	0.739802i	$0.265081\pi$
$104$	−4.41421	−0.432849
$105$	0	0
$106$	4.82843	0.468978
$107$	−11.3137	−1.09374	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	−11.3137	−1.09374	−0.546869	+	0.837218i	$0.684180\pi$
$108$	0	0
$109$	−17.3137	−1.65835	−0.829176	−	0.558987i	$-0.811190\pi$
$109$	−17.3137	−1.65835	−0.829176	+	0.558987i	$0.811190\pi$
$110$	−13.6569	−1.30213
$111$	0	0
$112$	−8.48528	−0.801784
$113$	−17.3137	−1.62874	−0.814368	−	0.580348i	$-0.802916\pi$
$113$	−17.3137	−1.62874	−0.814368	+	0.580348i	$0.802916\pi$
$114$	0	0
$115$	−11.3137	−1.05501
$116$	−7.65685	−0.710921
$117$	0	0
$118$	8.82843	0.812723
$119$	−10.3431	−0.948155
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−22.4853	−2.03572
$123$	0	0
$124$	−26.1421	−2.34763
$125$	5.65685	0.505964
$126$	0	0
$127$	−5.65685	−0.501965	−0.250982	−	0.967992i	$-0.580754\pi$
$127$	−5.65685	−0.501965	−0.250982	+	0.967992i	$0.580754\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	6.82843	0.598893
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	2.82843	0.244339
$135$	0	0
$136$	16.1421	1.38418
$137$	5.17157	0.441837	0.220919	−	0.975292i	$-0.429094\pi$
$137$	5.17157	0.441837	0.220919	+	0.975292i	$0.429094\pi$
$138$	0	0
$139$	15.3137	1.29889	0.649446	−	0.760408i	$-0.275001\pi$
$139$	15.3137	1.29889	0.649446	+	0.760408i	$0.275001\pi$
$140$	30.6274	2.58849
$141$	0	0
$142$	−4.82843	−0.405193
$143$	−2.00000	−0.167248
$144$	0	0
$145$	5.65685	0.469776
$146$	28.1421	2.32906
$147$	0	0
$148$	14.0000	1.15079
$149$	14.8284	1.21479	0.607396	−	0.794399i	$-0.292214\pi$
$149$	14.8284	1.21479	0.607396	+	0.794399i	$0.292214\pi$
$150$	0	0
$151$	−20.4853	−1.66707	−0.833534	−	0.552468i	$-0.813686\pi$
$151$	−20.4853	−1.66707	−0.833534	+	0.552468i	$0.813686\pi$
$152$	12.4853	1.01269
$153$	0	0
$154$	−13.6569	−1.10050
$155$	19.3137	1.55131
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	27.3137	2.17296
$159$	0	0
$160$	4.48528	0.354593
$161$	−11.3137	−0.891645
$162$	0	0
$163$	13.1716	1.03168	0.515839	−	0.856686i	$-0.327480\pi$
$163$	13.1716	1.03168	0.515839	+	0.856686i	$0.327480\pi$
$164$	−41.4558	−3.23716
$165$	0	0
$166$	18.4853	1.43474
$167$	−7.65685	−0.592505	−0.296253	−	0.955110i	$-0.595737\pi$
$167$	−7.65685	−0.592505	−0.296253	+	0.955110i	$0.595737\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−24.9706	−1.91515
$171$	0	0
$172$	36.9706	2.81898
$173$	0.343146	0.0260889	0.0130444	−	0.999915i	$-0.495848\pi$
$173$	0.343146	0.0260889	0.0130444	+	0.999915i	$0.495848\pi$
$174$	0	0
$175$	−8.48528	−0.641427
$176$	6.00000	0.452267
$177$	0	0
$178$	−22.1421	−1.65962
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	6.82843	0.506157
$183$	0	0
$184$	17.6569	1.30168
$185$	−10.3431	−0.760443
$186$	0	0
$187$	7.31371	0.534831
$188$	1.31371	0.0958120
$189$	0	0
$190$	−19.3137	−1.40116
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	−17.3137	−1.24627	−0.623134	−	0.782115i	$-0.714141\pi$
$193$	−17.3137	−1.24627	−0.623134	+	0.782115i	$0.714141\pi$
$194$	−18.4853	−1.32717
$195$	0	0
$196$	3.82843	0.273459
$197$	16.4853	1.17453	0.587264	−	0.809396i	$-0.300205\pi$
$197$	16.4853	1.17453	0.587264	+	0.809396i	$0.300205\pi$
$198$	0	0
$199$	10.3431	0.733206	0.366603	−	0.930377i	$-0.380521\pi$
$199$	10.3431	0.733206	0.366603	+	0.930377i	$0.380521\pi$
$200$	13.2426	0.936396
$201$	0	0
$202$	8.82843	0.621166
$203$	5.65685	0.397033
$204$	0	0
$205$	30.6274	2.13911
$206$	32.9706	2.29717
$207$	0	0
$208$	−3.00000	−0.208013
$209$	5.65685	0.391293
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	7.65685	0.525875
$213$	0	0
$214$	−27.3137	−1.86713
$215$	−27.3137	−1.86278
$216$	0	0
$217$	19.3137	1.31110
$218$	−41.7990	−2.83098
$219$	0	0
$220$	−21.6569	−1.46010
$221$	−3.65685	−0.245987
$222$	0	0
$223$	4.48528	0.300357	0.150178	−	0.988659i	$-0.452015\pi$
$223$	4.48528	0.300357	0.150178	+	0.988659i	$0.452015\pi$
$224$	4.48528	0.299685
$225$	0	0
$226$	−41.7990	−2.78043
$227$	−5.31371	−0.352683	−0.176342	−	0.984329i	$-0.556426\pi$
$227$	−5.31371	−0.352683	−0.176342	+	0.984329i	$0.556426\pi$
$228$	0	0
$229$	21.3137	1.40845	0.704225	−	0.709977i	$-0.251295\pi$
$229$	21.3137	1.40845	0.704225	+	0.709977i	$0.251295\pi$
$230$	−27.3137	−1.80101
$231$	0	0
$232$	−8.82843	−0.579615
$233$	26.9706	1.76690	0.883450	−	0.468525i	$-0.155214\pi$
$233$	26.9706	1.76690	0.883450	+	0.468525i	$0.155214\pi$
$234$	0	0
$235$	−0.970563	−0.0633125
$236$	14.0000	0.911322
$237$	0	0
$238$	−24.9706	−1.61860
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	0	0
$241$	11.6569	0.750884	0.375442	−	0.926846i	$-0.377491\pi$
$241$	11.6569	0.750884	0.375442	+	0.926846i	$0.377491\pi$
$242$	−16.8995	−1.08634
$243$	0	0
$244$	−35.6569	−2.28270
$245$	−2.82843	−0.180702
$246$	0	0
$247$	−2.82843	−0.179969
$248$	−30.1421	−1.91403
$249$	0	0
$250$	13.6569	0.863735
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	−13.6569	−0.856907
$255$	0	0
$256$	−29.9706	−1.87316
$257$	15.6569	0.976648	0.488324	−	0.872662i	$-0.337608\pi$
$257$	15.6569	0.976648	0.488324	+	0.872662i	$0.337608\pi$
$258$	0	0
$259$	−10.3431	−0.642692
$260$	10.8284	0.671551
$261$	0	0
$262$	19.3137	1.19320
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−5.65685	−0.347498
$266$	−19.3137	−1.18420
$267$	0	0
$268$	4.48528	0.273982
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	11.7990	0.716738	0.358369	−	0.933580i	$-0.383333\pi$
$271$	11.7990	0.716738	0.358369	+	0.933580i	$0.383333\pi$
$272$	10.9706	0.665188
$273$	0	0
$274$	12.4853	0.754263
$275$	6.00000	0.361814
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	36.9706	2.21735
$279$	0	0
$280$	35.3137	2.11040
$281$	−26.8284	−1.60045	−0.800225	−	0.599700i	$-0.795287\pi$
$281$	−26.8284	−1.60045	−0.800225	+	0.599700i	$0.795287\pi$
$282$	0	0
$283$	−4.97056	−0.295469	−0.147735	−	0.989027i	$-0.547198\pi$
$283$	−4.97056	−0.295469	−0.147735	+	0.989027i	$0.547198\pi$
$284$	−7.65685	−0.454351
$285$	0	0
$286$	−4.82843	−0.285511
$287$	30.6274	1.80788
$288$	0	0
$289$	−3.62742	−0.213377
$290$	13.6569	0.801958
$291$	0	0
$292$	44.6274	2.61162
$293$	26.1421	1.52724	0.763620	−	0.645666i	$-0.223420\pi$
$293$	26.1421	1.52724	0.763620	+	0.645666i	$0.223420\pi$
$294$	0	0
$295$	−10.3431	−0.602201
$296$	16.1421	0.938243
$297$	0	0
$298$	35.7990	2.07378
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−27.3137	−1.57434
$302$	−49.4558	−2.84586
$303$	0	0
$304$	8.48528	0.486664
$305$	26.3431	1.50840
$306$	0	0
$307$	−17.1716	−0.980033	−0.490017	−	0.871713i	$-0.663009\pi$
$307$	−17.1716	−0.980033	−0.490017	+	0.871713i	$0.663009\pi$
$308$	−21.6569	−1.23401
$309$	0	0
$310$	46.6274	2.64826
$311$	−34.6274	−1.96354	−0.981770	−	0.190071i	$-0.939128\pi$
$311$	−34.6274	−1.96354	−0.981770	+	0.190071i	$0.939128\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−24.1421	−1.36242
$315$	0	0
$316$	43.3137	2.43659
$317$	8.48528	0.476581	0.238290	−	0.971194i	$-0.423413\pi$
$317$	8.48528	0.476581	0.238290	+	0.971194i	$0.423413\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	27.7990	1.55401
$321$	0	0
$322$	−27.3137	−1.52213
$323$	10.3431	0.575508
$324$	0	0
$325$	−3.00000	−0.166410
$326$	31.7990	1.76118
$327$	0	0
$328$	−47.7990	−2.63926
$329$	−0.970563	−0.0535089
$330$	0	0
$331$	−2.14214	−0.117742	−0.0588712	−	0.998266i	$-0.518750\pi$
$331$	−2.14214	−0.117742	−0.0588712	+	0.998266i	$0.518750\pi$
$332$	29.3137	1.60880
$333$	0	0
$334$	−18.4853	−1.01147
$335$	−3.31371	−0.181047
$336$	0	0
$337$	−13.3137	−0.725244	−0.362622	−	0.931936i	$-0.618118\pi$
$337$	−13.3137	−0.725244	−0.362622	+	0.931936i	$0.618118\pi$
$338$	2.41421	0.131316
$339$	0	0
$340$	−39.5980	−2.14750
$341$	−13.6569	−0.739560
$342$	0	0
$343$	16.9706	0.916324
$344$	42.6274	2.29832
$345$	0	0
$346$	0.828427	0.0445365
$347$	31.3137	1.68101	0.840504	−	0.541805i	$-0.182259\pi$
$347$	31.3137	1.68101	0.840504	+	0.541805i	$0.182259\pi$
$348$	0	0
$349$	−7.65685	−0.409862	−0.204931	−	0.978776i	$-0.565697\pi$
$349$	−7.65685	−0.409862	−0.204931	+	0.978776i	$0.565697\pi$
$350$	−20.4853	−1.09498
$351$	0	0
$352$	−3.17157	−0.169045
$353$	−17.4558	−0.929081	−0.464540	−	0.885552i	$-0.653780\pi$
$353$	−17.4558	−0.929081	−0.464540	+	0.885552i	$0.653780\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	−35.1127	−1.86097
$357$	0	0
$358$	1.65685	0.0875675
$359$	−1.02944	−0.0543316	−0.0271658	−	0.999631i	$-0.508648\pi$
$359$	−1.02944	−0.0543316	−0.0271658	+	0.999631i	$0.508648\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	33.7990	1.77644
$363$	0	0
$364$	10.8284	0.567564
$365$	−32.9706	−1.72576
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	12.0000	0.625543
$369$	0	0
$370$	−24.9706	−1.29816
$371$	−5.65685	−0.293689
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	17.6569	0.913014
$375$	0	0
$376$	1.51472	0.0781156
$377$	2.00000	0.103005
$378$	0	0
$379$	−16.4853	−0.846792	−0.423396	−	0.905945i	$-0.639162\pi$
$379$	−16.4853	−0.846792	−0.423396	+	0.905945i	$0.639162\pi$
$380$	−30.6274	−1.57115
$381$	0	0
$382$	46.6274	2.38567
$383$	2.97056	0.151789	0.0758943	−	0.997116i	$-0.475819\pi$
$383$	2.97056	0.151789	0.0758943	+	0.997116i	$0.475819\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	−41.7990	−2.12751
$387$	0	0
$388$	−29.3137	−1.48818
$389$	−6.97056	−0.353422	−0.176711	−	0.984263i	$-0.556546\pi$
$389$	−6.97056	−0.353422	−0.176711	+	0.984263i	$0.556546\pi$
$390$	0	0
$391$	14.6274	0.739740
$392$	4.41421	0.222951
$393$	0	0
$394$	39.7990	2.00504
$395$	−32.0000	−1.61009
$396$	0	0
$397$	−2.97056	−0.149088	−0.0745441	−	0.997218i	$-0.523750\pi$
$397$	−2.97056	−0.149088	−0.0745441	+	0.997218i	$0.523750\pi$
$398$	24.9706	1.25166
$399$	0	0
$400$	9.00000	0.450000
$401$	2.14214	0.106973	0.0534866	−	0.998569i	$-0.482967\pi$
$401$	2.14214	0.106973	0.0534866	+	0.998569i	$0.482967\pi$
$402$	0	0
$403$	6.82843	0.340148
$404$	14.0000	0.696526
$405$	0	0
$406$	13.6569	0.677778
$407$	7.31371	0.362527
$408$	0	0
$409$	−1.02944	−0.0509024	−0.0254512	−	0.999676i	$-0.508102\pi$
$409$	−1.02944	−0.0509024	−0.0254512	+	0.999676i	$0.508102\pi$
$410$	73.9411	3.65169
$411$	0	0
$412$	52.2843	2.57586
$413$	−10.3431	−0.508953
$414$	0	0
$415$	−21.6569	−1.06309
$416$	1.58579	0.0777496
$417$	0	0
$418$	13.6569	0.667979
$419$	30.6274	1.49625	0.748124	−	0.663559i	$-0.230955\pi$
$419$	30.6274	1.49625	0.748124	+	0.663559i	$0.230955\pi$
$420$	0	0
$421$	14.6863	0.715766	0.357883	−	0.933766i	$-0.383499\pi$
$421$	14.6863	0.715766	0.357883	+	0.933766i	$0.383499\pi$
$422$	−28.9706	−1.41026
$423$	0	0
$424$	8.82843	0.428746
$425$	10.9706	0.532150
$426$	0	0
$427$	26.3431	1.27483
$428$	−43.3137	−2.09365
$429$	0	0
$430$	−65.9411	−3.17996
$431$	−19.6569	−0.946837	−0.473419	−	0.880838i	$-0.656980\pi$
$431$	−19.6569	−0.946837	−0.473419	+	0.880838i	$0.656980\pi$
$432$	0	0
$433$	1.31371	0.0631328	0.0315664	−	0.999502i	$-0.489950\pi$
$433$	1.31371	0.0631328	0.0315664	+	0.999502i	$0.489950\pi$
$434$	46.6274	2.23819
$435$	0	0
$436$	−66.2843	−3.17444
$437$	11.3137	0.541208
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	−24.9706	−1.19042
$441$	0	0
$442$	−8.82843	−0.419925
$443$	−41.9411	−1.99268	−0.996342	−	0.0854611i	$-0.972764\pi$
$443$	−41.9411	−1.99268	−0.996342	+	0.0854611i	$0.972764\pi$
$444$	0	0
$445$	25.9411	1.22973
$446$	10.8284	0.512741
$447$	0	0
$448$	27.7990	1.31338
$449$	−7.79899	−0.368057	−0.184029	−	0.982921i	$-0.558914\pi$
$449$	−7.79899	−0.368057	−0.184029	+	0.982921i	$0.558914\pi$
$450$	0	0
$451$	−21.6569	−1.01978
$452$	−66.2843	−3.11775
$453$	0	0
$454$	−12.8284	−0.602068
$455$	−8.00000	−0.375046
$456$	0	0
$457$	3.65685	0.171060	0.0855302	−	0.996336i	$-0.472742\pi$
$457$	3.65685	0.171060	0.0855302	+	0.996336i	$0.472742\pi$
$458$	51.4558	2.40437
$459$	0	0
$460$	−43.3137	−2.01951
$461$	−10.8284	−0.504330	−0.252165	−	0.967684i	$-0.581143\pi$
$461$	−10.8284	−0.504330	−0.252165	+	0.967684i	$0.581143\pi$
$462$	0	0
$463$	−7.51472	−0.349239	−0.174619	−	0.984636i	$-0.555869\pi$
$463$	−7.51472	−0.349239	−0.174619	+	0.984636i	$0.555869\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	65.1127	3.01629
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−3.31371	−0.153013
$470$	−2.34315	−0.108081
$471$	0	0
$472$	16.1421	0.743002
$473$	19.3137	0.888045
$474$	0	0
$475$	8.48528	0.389331
$476$	−39.5980	−1.81497
$477$	0	0
$478$	−4.82843	−0.220847
$479$	−2.68629	−0.122740	−0.0613699	−	0.998115i	$-0.519547\pi$
$479$	−2.68629	−0.122740	−0.0613699	+	0.998115i	$0.519547\pi$
$480$	0	0
$481$	−3.65685	−0.166738
$482$	28.1421	1.28184
$483$	0	0
$484$	−26.7990	−1.21814
$485$	21.6569	0.983387
$486$	0	0
$487$	31.7990	1.44095	0.720475	−	0.693481i	$-0.243924\pi$
$487$	31.7990	1.44095	0.720475	+	0.693481i	$0.243924\pi$
$488$	−41.1127	−1.86108
$489$	0	0
$490$	−6.82843	−0.308477
$491$	14.6274	0.660126	0.330063	−	0.943959i	$-0.392930\pi$
$491$	14.6274	0.660126	0.330063	+	0.943959i	$0.392930\pi$
$492$	0	0
$493$	−7.31371	−0.329393
$494$	−6.82843	−0.307225
$495$	0	0
$496$	−20.4853	−0.919816
$497$	5.65685	0.253745
$498$	0	0
$499$	−2.14214	−0.0958952	−0.0479476	−	0.998850i	$-0.515268\pi$
$499$	−2.14214	−0.0958952	−0.0479476	+	0.998850i	$0.515268\pi$
$500$	21.6569	0.968524
$501$	0	0
$502$	0	0
$503$	−15.3137	−0.682805	−0.341402	−	0.939917i	$-0.610902\pi$
$503$	−15.3137	−0.682805	−0.341402	+	0.939917i	$0.610902\pi$
$504$	0	0
$505$	−10.3431	−0.460264
$506$	19.3137	0.858599
$507$	0	0
$508$	−21.6569	−0.960868
$509$	27.7990	1.23217	0.616084	−	0.787680i	$-0.288718\pi$
$509$	27.7990	1.23217	0.616084	+	0.787680i	$0.288718\pi$
$510$	0	0
$511$	−32.9706	−1.45853
$512$	−31.2426	−1.38074
$513$	0	0
$514$	37.7990	1.66724
$515$	−38.6274	−1.70213
$516$	0	0
$517$	0.686292	0.0301831
$518$	−24.9706	−1.09714
$519$	0	0
$520$	12.4853	0.547516
$521$	−2.68629	−0.117689	−0.0588443	−	0.998267i	$-0.518742\pi$
$521$	−2.68629	−0.117689	−0.0588443	+	0.998267i	$0.518742\pi$
$522$	0	0
$523$	7.31371	0.319806	0.159903	−	0.987133i	$-0.448882\pi$
$523$	7.31371	0.319806	0.159903	+	0.987133i	$0.448882\pi$
$524$	30.6274	1.33796
$525$	0	0
$526$	−28.9706	−1.26318
$527$	−24.9706	−1.08773
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−13.6569	−0.593216
$531$	0	0
$532$	−30.6274	−1.32787
$533$	10.8284	0.469031
$534$	0	0
$535$	32.0000	1.38348
$536$	5.17157	0.223378
$537$	0	0
$538$	−43.4558	−1.87351
$539$	2.00000	0.0861461
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	28.4853	1.22355
$543$	0	0
$544$	−5.79899	−0.248630
$545$	48.9706	2.09767
$546$	0	0
$547$	0.686292	0.0293437	0.0146719	−	0.999892i	$-0.495330\pi$
$547$	0.686292	0.0293437	0.0146719	+	0.999892i	$0.495330\pi$
$548$	19.7990	0.845771
$549$	0	0
$550$	14.4853	0.617654
$551$	−5.65685	−0.240990
$552$	0	0
$553$	−32.0000	−1.36078
$554$	−4.82843	−0.205140
$555$	0	0
$556$	58.6274	2.48636
$557$	−31.7990	−1.34737	−0.673683	−	0.739020i	$-0.735289\pi$
$557$	−31.7990	−1.34737	−0.673683	+	0.739020i	$0.735289\pi$
$558$	0	0
$559$	−9.65685	−0.408441
$560$	24.0000	1.01419
$561$	0	0
$562$	−64.7696	−2.73214
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	48.9706	2.06021
$566$	−12.0000	−0.504398
$567$	0	0
$568$	−8.82843	−0.370433
$569$	9.02944	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$569$	9.02944	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$570$	0	0
$571$	20.9706	0.877591	0.438795	−	0.898587i	$-0.355405\pi$
$571$	20.9706	0.877591	0.438795	+	0.898587i	$0.355405\pi$
$572$	−7.65685	−0.320149
$573$	0	0
$574$	73.9411	3.08624
$575$	12.0000	0.500435
$576$	0	0
$577$	35.9411	1.49625	0.748124	−	0.663559i	$-0.230955\pi$
$577$	35.9411	1.49625	0.748124	+	0.663559i	$0.230955\pi$
$578$	−8.75736	−0.364258
$579$	0	0
$580$	21.6569	0.899252
$581$	−21.6569	−0.898478
$582$	0	0
$583$	4.00000	0.165663
$584$	51.4558	2.12926
$585$	0	0
$586$	63.1127	2.60716
$587$	−22.9706	−0.948097	−0.474048	−	0.880499i	$-0.657208\pi$
$587$	−22.9706	−0.948097	−0.474048	+	0.880499i	$0.657208\pi$
$588$	0	0
$589$	−19.3137	−0.795807
$590$	−24.9706	−1.02802
$591$	0	0
$592$	10.9706	0.450887
$593$	3.51472	0.144332	0.0721661	−	0.997393i	$-0.477009\pi$
$593$	3.51472	0.144332	0.0721661	+	0.997393i	$0.477009\pi$
$594$	0	0
$595$	29.2548	1.19933
$596$	56.7696	2.32537
$597$	0	0
$598$	−9.65685	−0.394898
$599$	0.686292	0.0280411	0.0140206	−	0.999902i	$-0.495537\pi$
$599$	0.686292	0.0280411	0.0140206	+	0.999902i	$0.495537\pi$
$600$	0	0
$601$	44.6274	1.82039	0.910195	−	0.414180i	$-0.135931\pi$
$601$	44.6274	1.82039	0.910195	+	0.414180i	$0.135931\pi$
$602$	−65.9411	−2.68756
$603$	0	0
$604$	−78.4264	−3.19113
$605$	19.7990	0.804943
$606$	0	0
$607$	−25.9411	−1.05292	−0.526459	−	0.850201i	$-0.676481\pi$
$607$	−25.9411	−1.05292	−0.526459	+	0.850201i	$0.676481\pi$
$608$	−4.48528	−0.181902
$609$	0	0
$610$	63.5980	2.57501
$611$	−0.343146	−0.0138822
$612$	0	0
$613$	−36.3431	−1.46789	−0.733943	−	0.679211i	$-0.762322\pi$
$613$	−36.3431	−1.46789	−0.733943	+	0.679211i	$0.762322\pi$
$614$	−41.4558	−1.67302
$615$	0	0
$616$	−24.9706	−1.00609
$617$	29.1716	1.17440	0.587202	−	0.809441i	$-0.300230\pi$
$617$	29.1716	1.17440	0.587202	+	0.809441i	$0.300230\pi$
$618$	0	0
$619$	−15.7990	−0.635015	−0.317508	−	0.948256i	$-0.602846\pi$
$619$	−15.7990	−0.635015	−0.317508	+	0.948256i	$0.602846\pi$
$620$	73.9411	2.96955
$621$	0	0
$622$	−83.5980	−3.35197
$623$	25.9411	1.03931
$624$	0	0
$625$	−31.0000	−1.24000
$626$	14.4853	0.578948
$627$	0	0
$628$	−38.2843	−1.52771
$629$	13.3726	0.533200
$630$	0	0
$631$	−19.1127	−0.760865	−0.380432	−	0.924809i	$-0.624225\pi$
$631$	−19.1127	−0.760865	−0.380432	+	0.924809i	$0.624225\pi$
$632$	49.9411	1.98655
$633$	0	0
$634$	20.4853	0.813574
$635$	16.0000	0.634941
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	−9.65685	−0.382319
$639$	0	0
$640$	58.1421	2.29827
$641$	26.2843	1.03817	0.519083	−	0.854724i	$-0.326273\pi$
$641$	26.2843	1.03817	0.519083	+	0.854724i	$0.326273\pi$
$642$	0	0
$643$	17.1716	0.677181	0.338590	−	0.940934i	$-0.390050\pi$
$643$	17.1716	0.677181	0.338590	+	0.940934i	$0.390050\pi$
$644$	−43.3137	−1.70680
$645$	0	0
$646$	24.9706	0.982454
$647$	11.3137	0.444788	0.222394	−	0.974957i	$-0.428613\pi$
$647$	11.3137	0.444788	0.222394	+	0.974957i	$0.428613\pi$
$648$	0	0
$649$	7.31371	0.287088
$650$	−7.24264	−0.284080
$651$	0	0
$652$	50.4264	1.97485
$653$	2.68629	0.105123	0.0525614	−	0.998618i	$-0.483261\pi$
$653$	2.68629	0.105123	0.0525614	+	0.998618i	$0.483261\pi$
$654$	0	0
$655$	−22.6274	−0.884126
$656$	−32.4853	−1.26834
$657$	0	0
$658$	−2.34315	−0.0913453
$659$	24.6863	0.961641	0.480821	−	0.876819i	$-0.340339\pi$
$659$	24.6863	0.961641	0.480821	+	0.876819i	$0.340339\pi$
$660$	0	0
$661$	−1.02944	−0.0400405	−0.0200202	−	0.999800i	$-0.506373\pi$
$661$	−1.02944	−0.0400405	−0.0200202	+	0.999800i	$0.506373\pi$
$662$	−5.17157	−0.200999
$663$	0	0
$664$	33.7990	1.31166
$665$	22.6274	0.877454
$666$	0	0
$667$	−8.00000	−0.309761
$668$	−29.3137	−1.13418
$669$	0	0
$670$	−8.00000	−0.309067
$671$	−18.6274	−0.719103
$672$	0	0
$673$	−28.6274	−1.10351	−0.551753	−	0.834008i	$-0.686041\pi$
$673$	−28.6274	−1.10351	−0.551753	+	0.834008i	$0.686041\pi$
$674$	−32.1421	−1.23807
$675$	0	0
$676$	3.82843	0.147247
$677$	−49.3137	−1.89528	−0.947640	−	0.319341i	$-0.896538\pi$
$677$	−49.3137	−1.89528	−0.947640	+	0.319341i	$0.896538\pi$
$678$	0	0
$679$	21.6569	0.831114
$680$	−45.6569	−1.75086
$681$	0	0
$682$	−32.9706	−1.26251
$683$	19.9411	0.763026	0.381513	−	0.924363i	$-0.375403\pi$
$683$	19.9411	0.763026	0.381513	+	0.924363i	$0.375403\pi$
$684$	0	0
$685$	−14.6274	−0.558885
$686$	40.9706	1.56426
$687$	0	0
$688$	28.9706	1.10449
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−34.1421	−1.29883	−0.649414	−	0.760435i	$-0.724986\pi$
$691$	−34.1421	−1.29883	−0.649414	+	0.760435i	$0.724986\pi$
$692$	1.31371	0.0499397
$693$	0	0
$694$	75.5980	2.86966
$695$	−43.3137	−1.64298
$696$	0	0
$697$	−39.5980	−1.49988
$698$	−18.4853	−0.699678
$699$	0	0
$700$	−32.4853	−1.22783
$701$	−38.9706	−1.47190	−0.735949	−	0.677037i	$-0.763264\pi$
$701$	−38.9706	−1.47190	−0.735949	+	0.677037i	$0.763264\pi$
$702$	0	0
$703$	10.3431	0.390099
$704$	−19.6569	−0.740846
$705$	0	0
$706$	−42.1421	−1.58604
$707$	−10.3431	−0.388994
$708$	0	0
$709$	40.6274	1.52579	0.762897	−	0.646520i	$-0.223776\pi$
$709$	40.6274	1.52579	0.762897	+	0.646520i	$0.223776\pi$
$710$	13.6569	0.512533
$711$	0	0
$712$	−40.4853	−1.51725
$713$	−27.3137	−1.02291
$714$	0	0
$715$	5.65685	0.211554
$716$	2.62742	0.0981912
$717$	0	0
$718$	−2.48528	−0.0927499
$719$	−37.9411	−1.41497	−0.707483	−	0.706731i	$-0.750169\pi$
$719$	−37.9411	−1.41497	−0.707483	+	0.706731i	$0.750169\pi$
$720$	0	0
$721$	−38.6274	−1.43856
$722$	−26.5563	−0.988325
$723$	0	0
$724$	53.5980	1.99195
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−21.6569	−0.803208	−0.401604	−	0.915813i	$-0.631547\pi$
$727$	−21.6569	−0.803208	−0.401604	+	0.915813i	$0.631547\pi$
$728$	12.4853	0.462735
$729$	0	0
$730$	−79.5980	−2.94605
$731$	35.3137	1.30612
$732$	0	0
$733$	8.62742	0.318661	0.159330	−	0.987225i	$-0.449066\pi$
$733$	8.62742	0.318661	0.159330	+	0.987225i	$0.449066\pi$
$734$	−57.9411	−2.13865
$735$	0	0
$736$	−6.34315	−0.233811
$737$	2.34315	0.0863109
$738$	0	0
$739$	10.1421	0.373084	0.186542	−	0.982447i	$-0.440272\pi$
$739$	10.1421	0.373084	0.186542	+	0.982447i	$0.440272\pi$
$740$	−39.5980	−1.45565
$741$	0	0
$742$	−13.6569	−0.501359
$743$	−2.00000	−0.0733729	−0.0366864	−	0.999327i	$-0.511680\pi$
$743$	−2.00000	−0.0733729	−0.0366864	+	0.999327i	$0.511680\pi$
$744$	0	0
$745$	−41.9411	−1.53660
$746$	24.1421	0.883906
$747$	0	0
$748$	28.0000	1.02378
$749$	32.0000	1.16925
$750$	0	0
$751$	32.9706	1.20311	0.601556	−	0.798830i	$-0.294547\pi$
$751$	32.9706	1.20311	0.601556	+	0.798830i	$0.294547\pi$
$752$	1.02944	0.0375397
$753$	0	0
$754$	4.82843	0.175841
$755$	57.9411	2.10869
$756$	0	0
$757$	−15.9411	−0.579390	−0.289695	−	0.957119i	$-0.593554\pi$
$757$	−15.9411	−0.579390	−0.289695	+	0.957119i	$0.593554\pi$
$758$	−39.7990	−1.44556
$759$	0	0
$760$	−35.3137	−1.28096
$761$	−15.5147	−0.562408	−0.281204	−	0.959648i	$-0.590734\pi$
$761$	−15.5147	−0.562408	−0.281204	+	0.959648i	$0.590734\pi$
$762$	0	0
$763$	48.9706	1.77285
$764$	73.9411	2.67510
$765$	0	0
$766$	7.17157	0.259119
$767$	−3.65685	−0.132041
$768$	0	0
$769$	42.0000	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	38.6274	1.39204
$771$	0	0
$772$	−66.2843	−2.38562
$773$	−5.85786	−0.210693	−0.105346	−	0.994436i	$-0.533595\pi$
$773$	−5.85786	−0.210693	−0.105346	+	0.994436i	$0.533595\pi$
$774$	0	0
$775$	−20.4853	−0.735853
$776$	−33.7990	−1.21331
$777$	0	0
$778$	−16.8284	−0.603328
$779$	−30.6274	−1.09734
$780$	0	0
$781$	−4.00000	−0.143131
$782$	35.3137	1.26282
$783$	0	0
$784$	3.00000	0.107143
$785$	28.2843	1.00951
$786$	0	0
$787$	32.7696	1.16811	0.584054	−	0.811715i	$-0.301466\pi$
$787$	32.7696	1.16811	0.584054	+	0.811715i	$0.301466\pi$
$788$	63.1127	2.24830
$789$	0	0
$790$	−77.2548	−2.74860
$791$	48.9706	1.74119
$792$	0	0
$793$	9.31371	0.330739
$794$	−7.17157	−0.254510
$795$	0	0
$796$	39.5980	1.40351
$797$	−35.6569	−1.26303	−0.631515	−	0.775363i	$-0.717567\pi$
$797$	−35.6569	−1.26303	−0.631515	+	0.775363i	$0.717567\pi$
$798$	0	0
$799$	1.25483	0.0443928
$800$	−4.75736	−0.168198
$801$	0	0
$802$	5.17157	0.182615
$803$	23.3137	0.822723
$804$	0	0
$805$	32.0000	1.12785
$806$	16.4853	0.580669
$807$	0	0
$808$	16.1421	0.567878
$809$	−41.3137	−1.45251	−0.726256	−	0.687424i	$-0.758741\pi$
$809$	−41.3137	−1.45251	−0.726256	+	0.687424i	$0.758741\pi$
$810$	0	0
$811$	−1.85786	−0.0652384	−0.0326192	−	0.999468i	$-0.510385\pi$
$811$	−1.85786	−0.0652384	−0.0326192	+	0.999468i	$0.510385\pi$
$812$	21.6569	0.760007
$813$	0	0
$814$	17.6569	0.618872
$815$	−37.2548	−1.30498
$816$	0	0
$817$	27.3137	0.955586
$818$	−2.48528	−0.0868958
$819$	0	0
$820$	117.255	4.09472
$821$	−15.7990	−0.551389	−0.275694	−	0.961245i	$-0.588908\pi$
$821$	−15.7990	−0.551389	−0.275694	+	0.961245i	$0.588908\pi$
$822$	0	0
$823$	48.9706	1.70701	0.853503	−	0.521088i	$-0.174473\pi$
$823$	48.9706	1.70701	0.853503	+	0.521088i	$0.174473\pi$
$824$	60.2843	2.10010
$825$	0	0
$826$	−24.9706	−0.868837
$827$	26.0000	0.904109	0.452054	−	0.891990i	$-0.350691\pi$
$827$	26.0000	0.904109	0.452054	+	0.891990i	$0.350691\pi$
$828$	0	0
$829$	5.31371	0.184553	0.0922764	−	0.995733i	$-0.470586\pi$
$829$	5.31371	0.184553	0.0922764	+	0.995733i	$0.470586\pi$
$830$	−52.2843	−1.81481
$831$	0	0
$832$	9.82843	0.340739
$833$	3.65685	0.126702
$834$	0	0
$835$	21.6569	0.749466
$836$	21.6569	0.749018
$837$	0	0
$838$	73.9411	2.55425
$839$	47.2548	1.63142	0.815709	−	0.578462i	$-0.196347\pi$
$839$	47.2548	1.63142	0.815709	+	0.578462i	$0.196347\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	35.4558	1.22189
$843$	0	0
$844$	−45.9411	−1.58136
$845$	−2.82843	−0.0973009
$846$	0	0
$847$	19.7990	0.680301
$848$	6.00000	0.206041
$849$	0	0
$850$	26.4853	0.908438
$851$	14.6274	0.501421
$852$	0	0
$853$	−7.65685	−0.262166	−0.131083	−	0.991371i	$-0.541845\pi$
$853$	−7.65685	−0.262166	−0.131083	+	0.991371i	$0.541845\pi$
$854$	63.5980	2.17628
$855$	0	0
$856$	−49.9411	−1.70695
$857$	−29.5980	−1.01105	−0.505524	−	0.862813i	$-0.668701\pi$
$857$	−29.5980	−1.01105	−0.505524	+	0.862813i	$0.668701\pi$
$858$	0	0
$859$	−23.3137	−0.795453	−0.397727	−	0.917504i	$-0.630201\pi$
$859$	−23.3137	−0.795453	−0.397727	+	0.917504i	$0.630201\pi$
$860$	−104.569	−3.56576
$861$	0	0
$862$	−47.4558	−1.61635
$863$	39.6569	1.34994	0.674968	−	0.737847i	$-0.264158\pi$
$863$	39.6569	1.34994	0.674968	+	0.737847i	$0.264158\pi$
$864$	0	0
$865$	−0.970563	−0.0330001
$866$	3.17157	0.107774
$867$	0	0
$868$	73.9411	2.50973
$869$	22.6274	0.767583
$870$	0	0
$871$	−1.17157	−0.0396972
$872$	−76.4264	−2.58812
$873$	0	0
$874$	27.3137	0.923900
$875$	−16.0000	−0.540899
$876$	0	0
$877$	−14.2843	−0.482346	−0.241173	−	0.970482i	$-0.577532\pi$
$877$	−14.2843	−0.482346	−0.241173	+	0.970482i	$0.577532\pi$
$878$	−40.9706	−1.38269
$879$	0	0
$880$	−16.9706	−0.572078
$881$	−53.5980	−1.80576	−0.902881	−	0.429891i	$-0.858552\pi$
$881$	−53.5980	−1.80576	−0.902881	+	0.429891i	$0.858552\pi$
$882$	0	0
$883$	−51.5980	−1.73641	−0.868205	−	0.496205i	$-0.834726\pi$
$883$	−51.5980	−1.73641	−0.868205	+	0.496205i	$0.834726\pi$
$884$	−14.0000	−0.470871
$885$	0	0
$886$	−101.255	−3.40172
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	62.6274	2.09928
$891$	0	0
$892$	17.1716	0.574947
$893$	0.970563	0.0324786
$894$	0	0
$895$	−1.94113	−0.0648847
$896$	58.1421	1.94239
$897$	0	0
$898$	−18.8284	−0.628313
$899$	13.6569	0.455482
$900$	0	0
$901$	7.31371	0.243655
$902$	−52.2843	−1.74088
$903$	0	0
$904$	−76.4264	−2.54190
$905$	−39.5980	−1.31628
$906$	0	0
$907$	20.9706	0.696316	0.348158	−	0.937436i	$-0.386807\pi$
$907$	20.9706	0.696316	0.348158	+	0.937436i	$0.386807\pi$
$908$	−20.3431	−0.675111
$909$	0	0
$910$	−19.3137	−0.640243
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	15.3137	0.506810
$914$	8.82843	0.292018
$915$	0	0
$916$	81.5980	2.69607
$917$	−22.6274	−0.747223
$918$	0	0
$919$	−19.3137	−0.637100	−0.318550	−	0.947906i	$-0.603196\pi$
$919$	−19.3137	−0.637100	−0.318550	+	0.947906i	$0.603196\pi$
$920$	−49.9411	−1.64651
$921$	0	0
$922$	−26.1421	−0.860945
$923$	2.00000	0.0658308
$924$	0	0
$925$	10.9706	0.360710
$926$	−18.1421	−0.596188
$927$	0	0
$928$	3.17157	0.104112
$929$	27.7990	0.912055	0.456028	−	0.889966i	$-0.349272\pi$
$929$	27.7990	0.912055	0.456028	+	0.889966i	$0.349272\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	103.255	3.38222
$933$	0	0
$934$	19.3137	0.631964
$935$	−20.6863	−0.676514
$936$	0	0
$937$	1.31371	0.0429170	0.0214585	−	0.999770i	$-0.493169\pi$
$937$	1.31371	0.0429170	0.0214585	+	0.999770i	$0.493169\pi$
$938$	−8.00000	−0.261209
$939$	0	0
$940$	−3.71573	−0.121194
$941$	−5.85786	−0.190961	−0.0954805	−	0.995431i	$-0.530439\pi$
$941$	−5.85786	−0.190961	−0.0954805	+	0.995431i	$0.530439\pi$
$942$	0	0
$943$	−43.3137	−1.41049
$944$	10.9706	0.357061
$945$	0	0
$946$	46.6274	1.51599
$947$	54.9706	1.78630	0.893152	−	0.449756i	$-0.148489\pi$
$947$	54.9706	1.78630	0.893152	+	0.449756i	$0.148489\pi$
$948$	0	0
$949$	−11.6569	−0.378398
$950$	20.4853	0.664630
$951$	0	0
$952$	−45.6569	−1.47975
$953$	−51.6569	−1.67333	−0.836665	−	0.547715i	$-0.815498\pi$
$953$	−51.6569	−1.67333	−0.836665	+	0.547715i	$0.815498\pi$
$954$	0	0
$955$	−54.6274	−1.76770
$956$	−7.65685	−0.247640
$957$	0	0
$958$	−6.48528	−0.209530
$959$	−14.6274	−0.472344
$960$	0	0
$961$	15.6274	0.504110
$962$	−8.82843	−0.284640
$963$	0	0
$964$	44.6274	1.43735
$965$	48.9706	1.57642
$966$	0	0
$967$	−10.1421	−0.326149	−0.163075	−	0.986614i	$-0.552141\pi$
$967$	−10.1421	−0.326149	−0.163075	+	0.986614i	$0.552141\pi$
$968$	−30.8995	−0.993147
$969$	0	0
$970$	52.2843	1.67875
$971$	−7.31371	−0.234708	−0.117354	−	0.993090i	$-0.537441\pi$
$971$	−7.31371	−0.234708	−0.117354	+	0.993090i	$0.537441\pi$
$972$	0	0
$973$	−43.3137	−1.38857
$974$	76.7696	2.45986
$975$	0	0
$976$	−27.9411	−0.894374
$977$	−13.8579	−0.443352	−0.221676	−	0.975120i	$-0.571153\pi$
$977$	−13.8579	−0.443352	−0.221676	+	0.975120i	$0.571153\pi$
$978$	0	0
$979$	−18.3431	−0.586249
$980$	−10.8284	−0.345901
$981$	0	0
$982$	35.3137	1.12691
$983$	−2.68629	−0.0856794	−0.0428397	−	0.999082i	$-0.513640\pi$
$983$	−2.68629	−0.0856794	−0.0428397	+	0.999082i	$0.513640\pi$
$984$	0	0
$985$	−46.6274	−1.48567
$986$	−17.6569	−0.562309
$987$	0	0
$988$	−10.8284	−0.344498
$989$	38.6274	1.22828
$990$	0	0
$991$	27.3137	0.867649	0.433824	−	0.900998i	$-0.357164\pi$
$991$	27.3137	0.867649	0.433824	+	0.900998i	$0.357164\pi$
$992$	10.8284	0.343803
$993$	0	0
$994$	13.6569	0.433169
$995$	−29.2548	−0.927441
$996$	0	0
$997$	51.2548	1.62326	0.811628	−	0.584174i	$-0.198581\pi$
$997$	51.2548	1.62326	0.811628	+	0.584174i	$0.198581\pi$
$998$	−5.17157	−0.163703
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.a.c.1.2		2
3.2	odd	2		39.2.a.b.1.1	✓	2
4.3	odd	2		1872.2.a.w.1.1		2
5.2	odd	4		2925.2.c.u.2224.4		4
5.3	odd	4		2925.2.c.u.2224.1		4
5.4	even	2		2925.2.a.v.1.1		2
7.6	odd	2		5733.2.a.u.1.2		2
8.3	odd	2		7488.2.a.co.1.2		2
8.5	even	2		7488.2.a.cl.1.2		2
9.2	odd	6		1053.2.e.m.352.2		4
9.4	even	3		1053.2.e.e.703.1		4
9.5	odd	6		1053.2.e.m.703.2		4
9.7	even	3		1053.2.e.e.352.1		4
12.11	even	2		624.2.a.k.1.2		2
13.5	odd	4		1521.2.b.j.1351.1		4
13.8	odd	4		1521.2.b.j.1351.4		4
13.12	even	2		1521.2.a.f.1.1		2
15.2	even	4		975.2.c.h.274.1		4
15.8	even	4		975.2.c.h.274.4		4
15.14	odd	2		975.2.a.l.1.2		2
21.20	even	2		1911.2.a.h.1.1		2
24.5	odd	2		2496.2.a.bf.1.1		2
24.11	even	2		2496.2.a.bi.1.1		2
33.32	even	2		4719.2.a.p.1.2		2
39.2	even	12		507.2.j.f.316.1		8
39.5	even	4		507.2.b.e.337.4		4
39.8	even	4		507.2.b.e.337.1		4
39.11	even	12		507.2.j.f.316.4		8
39.17	odd	6		507.2.e.d.484.1		4
39.20	even	12		507.2.j.f.361.1		8
39.23	odd	6		507.2.e.d.22.1		4
39.29	odd	6		507.2.e.h.22.2		4
39.32	even	12		507.2.j.f.361.4		8
39.35	odd	6		507.2.e.h.484.2		4
39.38	odd	2		507.2.a.h.1.2		2
156.155	even	2		8112.2.a.bm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.1	✓	2	3.2	odd	2
117.2.a.c.1.2		2	1.1	even	1	trivial
507.2.a.h.1.2		2	39.38	odd	2
507.2.b.e.337.1		4	39.8	even	4
507.2.b.e.337.4		4	39.5	even	4
507.2.e.d.22.1		4	39.23	odd	6
507.2.e.d.484.1		4	39.17	odd	6
507.2.e.h.22.2		4	39.29	odd	6
507.2.e.h.484.2		4	39.35	odd	6
507.2.j.f.316.1		8	39.2	even	12
507.2.j.f.316.4		8	39.11	even	12
507.2.j.f.361.1		8	39.20	even	12
507.2.j.f.361.4		8	39.32	even	12
624.2.a.k.1.2		2	12.11	even	2
975.2.a.l.1.2		2	15.14	odd	2
975.2.c.h.274.1		4	15.2	even	4
975.2.c.h.274.4		4	15.8	even	4
1053.2.e.e.352.1		4	9.7	even	3
1053.2.e.e.703.1		4	9.4	even	3
1053.2.e.m.352.2		4	9.2	odd	6
1053.2.e.m.703.2		4	9.5	odd	6
1521.2.a.f.1.1		2	13.12	even	2
1521.2.b.j.1351.1		4	13.5	odd	4
1521.2.b.j.1351.4		4	13.8	odd	4
1872.2.a.w.1.1		2	4.3	odd	2
1911.2.a.h.1.1		2	21.20	even	2
2496.2.a.bf.1.1		2	24.5	odd	2
2496.2.a.bi.1.1		2	24.11	even	2
2925.2.a.v.1.1		2	5.4	even	2
2925.2.c.u.2224.1		4	5.3	odd	4
2925.2.c.u.2224.4		4	5.2	odd	4
4719.2.a.p.1.2		2	33.32	even	2
5733.2.a.u.1.2		2	7.6	odd	2
7488.2.a.cl.1.2		2	8.5	even	2
7488.2.a.co.1.2		2	8.3	odd	2
8112.2.a.bm.1.1		2	156.155	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} -2.82843 q^{5} -2.82843 q^{7} +4.41421 q^{8} +O(q^{10})\)	\(q+2.41421 q^{2} +3.82843 q^{4} -2.82843 q^{5} -2.82843 q^{7} +4.41421 q^{8} -6.82843 q^{10} +2.00000 q^{11} -1.00000 q^{13} -6.82843 q^{14} +3.00000 q^{16} +3.65685 q^{17} +2.82843 q^{19} -10.8284 q^{20} +4.82843 q^{22} +4.00000 q^{23} +3.00000 q^{25} -2.41421 q^{26} -10.8284 q^{28} -2.00000 q^{29} -6.82843 q^{31} -1.58579 q^{32} +8.82843 q^{34} +8.00000 q^{35} +3.65685 q^{37} +6.82843 q^{38} -12.4853 q^{40} -10.8284 q^{41} +9.65685 q^{43} +7.65685 q^{44} +9.65685 q^{46} +0.343146 q^{47} +1.00000 q^{49} +7.24264 q^{50} -3.82843 q^{52} +2.00000 q^{53} -5.65685 q^{55} -12.4853 q^{56} -4.82843 q^{58} +3.65685 q^{59} -9.31371 q^{61} -16.4853 q^{62} -9.82843 q^{64} +2.82843 q^{65} +1.17157 q^{67} +14.0000 q^{68} +19.3137 q^{70} -2.00000 q^{71} +11.6569 q^{73} +8.82843 q^{74} +10.8284 q^{76} -5.65685 q^{77} +11.3137 q^{79} -8.48528 q^{80} -26.1421 q^{82} +7.65685 q^{83} -10.3431 q^{85} +23.3137 q^{86} +8.82843 q^{88} -9.17157 q^{89} +2.82843 q^{91} +15.3137 q^{92} +0.828427 q^{94} -8.00000 q^{95} -7.65685 q^{97} +2.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 6 q^{8} - 8 q^{10} + 4 q^{11} - 2 q^{13} - 8 q^{14} + 6 q^{16} - 4 q^{17} - 16 q^{20} + 4 q^{22} + 8 q^{23} + 6 q^{25} - 2 q^{26} - 16 q^{28} - 4 q^{29} - 8 q^{31} - 6 q^{32} + 12 q^{34} + 16 q^{35} - 4 q^{37} + 8 q^{38} - 8 q^{40} - 16 q^{41} + 8 q^{43} + 4 q^{44} + 8 q^{46} + 12 q^{47} + 2 q^{49} + 6 q^{50} - 2 q^{52} + 4 q^{53} - 8 q^{56} - 4 q^{58} - 4 q^{59} + 4 q^{61} - 16 q^{62} - 14 q^{64} + 8 q^{67} + 28 q^{68} + 16 q^{70} - 4 q^{71} + 12 q^{73} + 12 q^{74} + 16 q^{76} - 24 q^{82} + 4 q^{83} - 32 q^{85} + 24 q^{86} + 12 q^{88} - 24 q^{89} + 8 q^{92} - 4 q^{94} - 16 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 - 8 * q^10 + 4 * q^11 - 2 * q^13 - 8 * q^14 + 6 * q^16 - 4 * q^17 - 16 * q^20 + 4 * q^22 + 8 * q^23 + 6 * q^25 - 2 * q^26 - 16 * q^28 - 4 * q^29 - 8 * q^31 - 6 * q^32 + 12 * q^34 + 16 * q^35 - 4 * q^37 + 8 * q^38 - 8 * q^40 - 16 * q^41 + 8 * q^43 + 4 * q^44 + 8 * q^46 + 12 * q^47 + 2 * q^49 + 6 * q^50 - 2 * q^52 + 4 * q^53 - 8 * q^56 - 4 * q^58 - 4 * q^59 + 4 * q^61 - 16 * q^62 - 14 * q^64 + 8 * q^67 + 28 * q^68 + 16 * q^70 - 4 * q^71 + 12 * q^73 + 12 * q^74 + 16 * q^76 - 24 * q^82 + 4 * q^83 - 32 * q^85 + 24 * q^86 + 12 * q^88 - 24 * q^89 + 8 * q^92 - 4 * q^94 - 16 * q^95 - 4 * q^97 + 2 * q^98

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