LMFDB - Embedded newform 39.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,2,Mod(1,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, 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("39.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	39.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:	$0.311416567883$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	39.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -2.82843 q^{5} +0.414214 q^{6} +2.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -2.82843 q^{5} +0.414214 q^{6} +2.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} -1.17157 q^{10} -2.00000 q^{11} -1.82843 q^{12} -1.00000 q^{13} +1.17157 q^{14} -2.82843 q^{15} +3.00000 q^{16} +7.65685 q^{17} +0.414214 q^{18} -2.82843 q^{19} +5.17157 q^{20} +2.82843 q^{21} -0.828427 q^{22} -4.00000 q^{23} -1.58579 q^{24} +3.00000 q^{25} -0.414214 q^{26} +1.00000 q^{27} -5.17157 q^{28} +2.00000 q^{29} -1.17157 q^{30} -1.17157 q^{31} +4.41421 q^{32} -2.00000 q^{33} +3.17157 q^{34} -8.00000 q^{35} -1.82843 q^{36} -7.65685 q^{37} -1.17157 q^{38} -1.00000 q^{39} +4.48528 q^{40} +5.17157 q^{41} +1.17157 q^{42} -1.65685 q^{43} +3.65685 q^{44} -2.82843 q^{45} -1.65685 q^{46} -11.6569 q^{47} +3.00000 q^{48} +1.00000 q^{49} +1.24264 q^{50} +7.65685 q^{51} +1.82843 q^{52} -2.00000 q^{53} +0.414214 q^{54} +5.65685 q^{55} -4.48528 q^{56} -2.82843 q^{57} +0.828427 q^{58} +7.65685 q^{59} +5.17157 q^{60} +13.3137 q^{61} -0.485281 q^{62} +2.82843 q^{63} -4.17157 q^{64} +2.82843 q^{65} -0.828427 q^{66} +6.82843 q^{67} -14.0000 q^{68} -4.00000 q^{69} -3.31371 q^{70} +2.00000 q^{71} -1.58579 q^{72} +0.343146 q^{73} -3.17157 q^{74} +3.00000 q^{75} +5.17157 q^{76} -5.65685 q^{77} -0.414214 q^{78} -11.3137 q^{79} -8.48528 q^{80} +1.00000 q^{81} +2.14214 q^{82} +3.65685 q^{83} -5.17157 q^{84} -21.6569 q^{85} -0.686292 q^{86} +2.00000 q^{87} +3.17157 q^{88} +14.8284 q^{89} -1.17157 q^{90} -2.82843 q^{91} +7.31371 q^{92} -1.17157 q^{93} -4.82843 q^{94} +8.00000 q^{95} +4.41421 q^{96} +3.65685 q^{97} +0.414214 q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9} - 8 q^{10} - 4 q^{11} + 2 q^{12} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{18} + 16 q^{20} + 4 q^{22} - 8 q^{23} - 6 q^{24} + 6 q^{25} + 2 q^{26} + 2 q^{27} - 16 q^{28} + 4 q^{29} - 8 q^{30} - 8 q^{31} + 6 q^{32} - 4 q^{33} + 12 q^{34} - 16 q^{35} + 2 q^{36} - 4 q^{37} - 8 q^{38} - 2 q^{39} - 8 q^{40} + 16 q^{41} + 8 q^{42} + 8 q^{43} - 4 q^{44} + 8 q^{46} - 12 q^{47} + 6 q^{48} + 2 q^{49} - 6 q^{50} + 4 q^{51} - 2 q^{52} - 4 q^{53} - 2 q^{54} + 8 q^{56} - 4 q^{58} + 4 q^{59} + 16 q^{60} + 4 q^{61} + 16 q^{62} - 14 q^{64} + 4 q^{66} + 8 q^{67} - 28 q^{68} - 8 q^{69} + 16 q^{70} + 4 q^{71} - 6 q^{72} + 12 q^{73} - 12 q^{74} + 6 q^{75} + 16 q^{76} + 2 q^{78} + 2 q^{81} - 24 q^{82} - 4 q^{83} - 16 q^{84} - 32 q^{85} - 24 q^{86} + 4 q^{87} + 12 q^{88} + 24 q^{89} - 8 q^{90} - 8 q^{92} - 8 q^{93} - 4 q^{94} + 16 q^{95} + 6 q^{96} - 4 q^{97} - 2 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^8 + 2 * q^9 - 8 * q^10 - 4 * q^11 + 2 * q^12 - 2 * q^13 + 8 * q^14 + 6 * q^16 + 4 * q^17 - 2 * q^18 + 16 * q^20 + 4 * q^22 - 8 * q^23 - 6 * q^24 + 6 * q^25 + 2 * q^26 + 2 * q^27 - 16 * q^28 + 4 * q^29 - 8 * q^30 - 8 * q^31 + 6 * q^32 - 4 * q^33 + 12 * q^34 - 16 * q^35 + 2 * q^36 - 4 * q^37 - 8 * q^38 - 2 * q^39 - 8 * q^40 + 16 * q^41 + 8 * q^42 + 8 * q^43 - 4 * q^44 + 8 * q^46 - 12 * q^47 + 6 * q^48 + 2 * q^49 - 6 * q^50 + 4 * q^51 - 2 * q^52 - 4 * q^53 - 2 * q^54 + 8 * q^56 - 4 * q^58 + 4 * q^59 + 16 * q^60 + 4 * q^61 + 16 * q^62 - 14 * q^64 + 4 * q^66 + 8 * q^67 - 28 * q^68 - 8 * q^69 + 16 * q^70 + 4 * q^71 - 6 * q^72 + 12 * q^73 - 12 * q^74 + 6 * q^75 + 16 * q^76 + 2 * q^78 + 2 * q^81 - 24 * q^82 - 4 * q^83 - 16 * q^84 - 32 * q^85 - 24 * q^86 + 4 * q^87 + 12 * q^88 + 24 * q^89 - 8 * q^90 - 8 * q^92 - 8 * q^93 - 4 * q^94 + 16 * q^95 + 6 * q^96 - 4 * q^97 - 2 * q^98 - 4 * q^99

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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.82843	−0.914214
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0.414214	0.169102
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	−1.58579	−0.560660
$9$	1.00000	0.333333
$10$	−1.17157	−0.370484
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.82843	−0.527821
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.17157	0.313116
$15$	−2.82843	−0.730297
$16$	3.00000	0.750000
$17$	7.65685	1.85706	0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685	1.85706	0.928530	+	0.371257i	$0.121073\pi$
$18$	0.414214	0.0976311
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	5.17157	1.15640
$21$	2.82843	0.617213
$22$	−0.828427	−0.176621
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.58579	−0.323697
$25$	3.00000	0.600000
$26$	−0.414214	−0.0812340
$27$	1.00000	0.192450
$28$	−5.17157	−0.977335
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.17157	−0.213899
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	4.41421	0.780330
$33$	−2.00000	−0.348155
$34$	3.17157	0.543920
$35$	−8.00000	−1.35225
$36$	−1.82843	−0.304738
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	−1.17157	−0.190054
$39$	−1.00000	−0.160128
$40$	4.48528	0.709185
$41$	5.17157	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$41$	5.17157	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$42$	1.17157	0.180778
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	3.65685	0.551292
$45$	−2.82843	−0.421637
$46$	−1.65685	−0.244290
$47$	−11.6569	−1.70033	−0.850163	−	0.526519i	$-0.823497\pi$
$47$	−11.6569	−1.70033	−0.850163	+	0.526519i	$0.823497\pi$
$48$	3.00000	0.433013
$49$	1.00000	0.142857
$50$	1.24264	0.175736
$51$	7.65685	1.07217
$52$	1.82843	0.253557
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0.414214	0.0563673
$55$	5.65685	0.762770
$56$	−4.48528	−0.599371
$57$	−2.82843	−0.374634
$58$	0.828427	0.108778
$59$	7.65685	0.996838	0.498419	−	0.866936i	$-0.333914\pi$
$59$	7.65685	0.996838	0.498419	+	0.866936i	$0.333914\pi$
$60$	5.17157	0.667647
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	−0.485281	−0.0616308
$63$	2.82843	0.356348
$64$	−4.17157	−0.521447
$65$	2.82843	0.350823
$66$	−0.828427	−0.101972
$67$	6.82843	0.834225	0.417113	−	0.908855i	$-0.363042\pi$
$67$	6.82843	0.834225	0.417113	+	0.908855i	$0.363042\pi$
$68$	−14.0000	−1.69775
$69$	−4.00000	−0.481543
$70$	−3.31371	−0.396064
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−1.58579	−0.186887
$73$	0.343146	0.0401622	0.0200811	−	0.999798i	$-0.493608\pi$
$73$	0.343146	0.0401622	0.0200811	+	0.999798i	$0.493608\pi$
$74$	−3.17157	−0.368688
$75$	3.00000	0.346410
$76$	5.17157	0.593220
$77$	−5.65685	−0.644658
$78$	−0.414214	−0.0469005
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	−8.48528	−0.948683
$81$	1.00000	0.111111
$82$	2.14214	0.236559
$83$	3.65685	0.401392	0.200696	−	0.979654i	$-0.435680\pi$
$83$	3.65685	0.401392	0.200696	+	0.979654i	$0.435680\pi$
$84$	−5.17157	−0.564265
$85$	−21.6569	−2.34902
$86$	−0.686292	−0.0740047
$87$	2.00000	0.214423
$88$	3.17157	0.338091
$89$	14.8284	1.57181	0.785905	−	0.618347i	$-0.212197\pi$
$89$	14.8284	1.57181	0.785905	+	0.618347i	$0.212197\pi$
$90$	−1.17157	−0.123495
$91$	−2.82843	−0.296500
$92$	7.31371	0.762507
$93$	−1.17157	−0.121486
$94$	−4.82843	−0.498014
$95$	8.00000	0.820783
$96$	4.41421	0.450524
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	0.414214	0.0418419
$99$	−2.00000	−0.201008
$100$	−5.48528	−0.548528
$101$	7.65685	0.761885	0.380943	−	0.924599i	$-0.375599\pi$
$101$	7.65685	0.761885	0.380943	+	0.924599i	$0.375599\pi$
$102$	3.17157	0.314033
$103$	2.34315	0.230877	0.115439	−	0.993315i	$-0.463173\pi$
$103$	2.34315	0.230877	0.115439	+	0.993315i	$0.463173\pi$
$104$	1.58579	0.155499
$105$	−8.00000	−0.780720
$106$	−0.828427	−0.0804640
$107$	−11.3137	−1.09374	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	−11.3137	−1.09374	−0.546869	+	0.837218i	$0.684180\pi$
$108$	−1.82843	−0.175940
$109$	5.31371	0.508961	0.254480	−	0.967078i	$-0.418096\pi$
$109$	5.31371	0.508961	0.254480	+	0.967078i	$0.418096\pi$
$110$	2.34315	0.223410
$111$	−7.65685	−0.726756
$112$	8.48528	0.801784
$113$	−5.31371	−0.499872	−0.249936	−	0.968262i	$-0.580410\pi$
$113$	−5.31371	−0.499872	−0.249936	+	0.968262i	$0.580410\pi$
$114$	−1.17157	−0.109728
$115$	11.3137	1.05501
$116$	−3.65685	−0.339530
$117$	−1.00000	−0.0924500
$118$	3.17157	0.291967
$119$	21.6569	1.98528
$120$	4.48528	0.409448
$121$	−7.00000	−0.636364
$122$	5.51472	0.499279
$123$	5.17157	0.466305
$124$	2.14214	0.192369
$125$	5.65685	0.505964
$126$	1.17157	0.104372
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	−10.5563	−0.933058
$129$	−1.65685	−0.145878
$130$	1.17157	0.102754
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	3.65685	0.318288
$133$	−8.00000	−0.693688
$134$	2.82843	0.244339
$135$	−2.82843	−0.243432
$136$	−12.1421	−1.04118
$137$	−10.8284	−0.925135	−0.462567	−	0.886584i	$-0.653072\pi$
$137$	−10.8284	−0.925135	−0.462567	+	0.886584i	$0.653072\pi$
$138$	−1.65685	−0.141041
$139$	−7.31371	−0.620341	−0.310170	−	0.950681i	$-0.600386\pi$
$139$	−7.31371	−0.620341	−0.310170	+	0.950681i	$0.600386\pi$
$140$	14.6274	1.23624
$141$	−11.6569	−0.981684
$142$	0.828427	0.0695201
$143$	2.00000	0.167248
$144$	3.00000	0.250000
$145$	−5.65685	−0.469776
$146$	0.142136	0.0117632
$147$	1.00000	0.0824786
$148$	14.0000	1.15079
$149$	−9.17157	−0.751365	−0.375682	−	0.926749i	$-0.622592\pi$
$149$	−9.17157	−0.751365	−0.375682	+	0.926749i	$0.622592\pi$
$150$	1.24264	0.101461
$151$	−3.51472	−0.286024	−0.143012	−	0.989721i	$-0.545679\pi$
$151$	−3.51472	−0.286024	−0.143012	+	0.989721i	$0.545679\pi$
$152$	4.48528	0.363804
$153$	7.65685	0.619020
$154$	−2.34315	−0.188816
$155$	3.31371	0.266163
$156$	1.82843	0.146391
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−4.68629	−0.372821
$159$	−2.00000	−0.158610
$160$	−12.4853	−0.987048
$161$	−11.3137	−0.891645
$162$	0.414214	0.0325437
$163$	18.8284	1.47476	0.737378	−	0.675480i	$-0.236064\pi$
$163$	18.8284	1.47476	0.737378	+	0.675480i	$0.236064\pi$
$164$	−9.45584	−0.738377
$165$	5.65685	0.440386
$166$	1.51472	0.117565
$167$	−3.65685	−0.282976	−0.141488	−	0.989940i	$-0.545189\pi$
$167$	−3.65685	−0.282976	−0.141488	+	0.989940i	$0.545189\pi$
$168$	−4.48528	−0.346047
$169$	1.00000	0.0769231
$170$	−8.97056	−0.688011
$171$	−2.82843	−0.216295
$172$	3.02944	0.230992
$173$	−11.6569	−0.886254	−0.443127	−	0.896459i	$-0.646131\pi$
$173$	−11.6569	−0.886254	−0.443127	+	0.896459i	$0.646131\pi$
$174$	0.828427	0.0628029
$175$	8.48528	0.641427
$176$	−6.00000	−0.452267
$177$	7.65685	0.575524
$178$	6.14214	0.460373
$179$	−23.3137	−1.74255	−0.871274	−	0.490797i	$-0.836706\pi$
$179$	−23.3137	−1.74255	−0.871274	+	0.490797i	$0.836706\pi$
$180$	5.17157	0.385466
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−1.17157	−0.0868428
$183$	13.3137	0.984178
$184$	6.34315	0.467623
$185$	21.6569	1.59224
$186$	−0.485281	−0.0355826
$187$	−15.3137	−1.11985
$188$	21.3137	1.55446
$189$	2.82843	0.205738
$190$	3.31371	0.240402
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	−4.17157	−0.301057
$193$	5.31371	0.382489	0.191245	−	0.981542i	$-0.438748\pi$
$193$	5.31371	0.382489	0.191245	+	0.981542i	$0.438748\pi$
$194$	1.51472	0.108750
$195$	2.82843	0.202548
$196$	−1.82843	−0.130602
$197$	0.485281	0.0345749	0.0172874	−	0.999851i	$-0.494497\pi$
$197$	0.485281	0.0345749	0.0172874	+	0.999851i	$0.494497\pi$
$198$	−0.828427	−0.0588738
$199$	21.6569	1.53521	0.767607	−	0.640921i	$-0.221447\pi$
$199$	21.6569	1.53521	0.767607	+	0.640921i	$0.221447\pi$
$200$	−4.75736	−0.336396
$201$	6.82843	0.481640
$202$	3.17157	0.223151
$203$	5.65685	0.397033
$204$	−14.0000	−0.980196
$205$	−14.6274	−1.02162
$206$	0.970563	0.0676223
$207$	−4.00000	−0.278019
$208$	−3.00000	−0.208013
$209$	5.65685	0.391293
$210$	−3.31371	−0.228668
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	3.65685	0.251154
$213$	2.00000	0.137038
$214$	−4.68629	−0.320348
$215$	4.68629	0.319602
$216$	−1.58579	−0.107899
$217$	−3.31371	−0.224949
$218$	2.20101	0.149071
$219$	0.343146	0.0231876
$220$	−10.3431	−0.697335
$221$	−7.65685	−0.515056
$222$	−3.17157	−0.212862
$223$	−12.4853	−0.836076	−0.418038	−	0.908429i	$-0.637282\pi$
$223$	−12.4853	−0.836076	−0.418038	+	0.908429i	$0.637282\pi$
$224$	12.4853	0.834208
$225$	3.00000	0.200000
$226$	−2.20101	−0.146409
$227$	−17.3137	−1.14915	−0.574576	−	0.818452i	$-0.694833\pi$
$227$	−17.3137	−1.14915	−0.574576	+	0.818452i	$0.694833\pi$
$228$	5.17157	0.342496
$229$	−1.31371	−0.0868123	−0.0434062	−	0.999058i	$-0.513821\pi$
$229$	−1.31371	−0.0868123	−0.0434062	+	0.999058i	$0.513821\pi$
$230$	4.68629	0.309005
$231$	−5.65685	−0.372194
$232$	−3.17157	−0.208224
$233$	6.97056	0.456657	0.228328	−	0.973584i	$-0.426674\pi$
$233$	6.97056	0.456657	0.228328	+	0.973584i	$0.426674\pi$
$234$	−0.414214	−0.0270780
$235$	32.9706	2.15076
$236$	−14.0000	−0.911322
$237$	−11.3137	−0.734904
$238$	8.97056	0.581475
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	−8.48528	−0.547723
$241$	0.343146	0.0221040	0.0110520	−	0.999939i	$-0.496482\pi$
$241$	0.343146	0.0221040	0.0110520	+	0.999939i	$0.496482\pi$
$242$	−2.89949	−0.186387
$243$	1.00000	0.0641500
$244$	−24.3431	−1.55841
$245$	−2.82843	−0.180702
$246$	2.14214	0.136578
$247$	2.82843	0.179969
$248$	1.85786	0.117975
$249$	3.65685	0.231744
$250$	2.34315	0.148194
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−5.17157	−0.325778
$253$	8.00000	0.502956
$254$	2.34315	0.147022
$255$	−21.6569	−1.35620
$256$	3.97056	0.248160
$257$	−4.34315	−0.270918	−0.135459	−	0.990783i	$-0.543251\pi$
$257$	−4.34315	−0.270918	−0.135459	+	0.990783i	$0.543251\pi$
$258$	−0.686292	−0.0427266
$259$	−21.6569	−1.34569
$260$	−5.17157	−0.320727
$261$	2.00000	0.123797
$262$	−3.31371	−0.204722
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	3.17157	0.195197
$265$	5.65685	0.347498
$266$	−3.31371	−0.203177
$267$	14.8284	0.907485
$268$	−12.4853	−0.762660
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−1.17157	−0.0712997
$271$	−27.7990	−1.68867	−0.844334	−	0.535817i	$-0.820004\pi$
$271$	−27.7990	−1.68867	−0.844334	+	0.535817i	$0.820004\pi$
$272$	22.9706	1.39279
$273$	−2.82843	−0.171184
$274$	−4.48528	−0.270966
$275$	−6.00000	−0.361814
$276$	7.31371	0.440234
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−3.02944	−0.181694
$279$	−1.17157	−0.0701402
$280$	12.6863	0.758151
$281$	21.1716	1.26299	0.631495	−	0.775380i	$-0.282442\pi$
$281$	21.1716	1.26299	0.631495	+	0.775380i	$0.282442\pi$
$282$	−4.82843	−0.287529
$283$	28.9706	1.72212	0.861061	−	0.508502i	$-0.169801\pi$
$283$	28.9706	1.72212	0.861061	+	0.508502i	$0.169801\pi$
$284$	−3.65685	−0.216994
$285$	8.00000	0.473879
$286$	0.828427	0.0489859
$287$	14.6274	0.863429
$288$	4.41421	0.260110
$289$	41.6274	2.44867
$290$	−2.34315	−0.137594
$291$	3.65685	0.214369
$292$	−0.627417	−0.0367168
$293$	2.14214	0.125145	0.0625724	−	0.998040i	$-0.480070\pi$
$293$	2.14214	0.125145	0.0625724	+	0.998040i	$0.480070\pi$
$294$	0.414214	0.0241574
$295$	−21.6569	−1.26091
$296$	12.1421	0.705747
$297$	−2.00000	−0.116052
$298$	−3.79899	−0.220070
$299$	4.00000	0.231326
$300$	−5.48528	−0.316693
$301$	−4.68629	−0.270113
$302$	−1.45584	−0.0837744
$303$	7.65685	0.439875
$304$	−8.48528	−0.486664
$305$	−37.6569	−2.15623
$306$	3.17157	0.181307
$307$	−22.8284	−1.30289	−0.651444	−	0.758697i	$-0.725836\pi$
$307$	−22.8284	−1.30289	−0.651444	+	0.758697i	$0.725836\pi$
$308$	10.3431	0.589355
$309$	2.34315	0.133297
$310$	1.37258	0.0779575
$311$	−10.6274	−0.602626	−0.301313	−	0.953525i	$-0.597425\pi$
$311$	−10.6274	−0.602626	−0.301313	+	0.953525i	$0.597425\pi$
$312$	1.58579	0.0897775
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−4.14214	−0.233754
$315$	−8.00000	−0.450749
$316$	20.6863	1.16369
$317$	8.48528	0.476581	0.238290	−	0.971194i	$-0.423413\pi$
$317$	8.48528	0.476581	0.238290	+	0.971194i	$0.423413\pi$
$318$	−0.828427	−0.0464559
$319$	−4.00000	−0.223957
$320$	11.7990	0.659584
$321$	−11.3137	−0.631470
$322$	−4.68629	−0.261157
$323$	−21.6569	−1.20502
$324$	−1.82843	−0.101579
$325$	−3.00000	−0.166410
$326$	7.79899	0.431946
$327$	5.31371	0.293849
$328$	−8.20101	−0.452825
$329$	−32.9706	−1.81773
$330$	2.34315	0.128986
$331$	26.1421	1.43690	0.718451	−	0.695578i	$-0.244852\pi$
$331$	26.1421	1.43690	0.718451	+	0.695578i	$0.244852\pi$
$332$	−6.68629	−0.366958
$333$	−7.65685	−0.419593
$334$	−1.51472	−0.0828817
$335$	−19.3137	−1.05522
$336$	8.48528	0.462910
$337$	9.31371	0.507350	0.253675	−	0.967290i	$-0.418361\pi$
$337$	9.31371	0.507350	0.253675	+	0.967290i	$0.418361\pi$
$338$	0.414214	0.0225302
$339$	−5.31371	−0.288601
$340$	39.5980	2.14750
$341$	2.34315	0.126888
$342$	−1.17157	−0.0633514
$343$	−16.9706	−0.916324
$344$	2.62742	0.141661
$345$	11.3137	0.609110
$346$	−4.82843	−0.259578
$347$	−8.68629	−0.466305	−0.233152	−	0.972440i	$-0.574904\pi$
$347$	−8.68629	−0.466305	−0.233152	+	0.972440i	$0.574904\pi$
$348$	−3.65685	−0.196028
$349$	3.65685	0.195747	0.0978735	−	0.995199i	$-0.468796\pi$
$349$	3.65685	0.195747	0.0978735	+	0.995199i	$0.468796\pi$
$350$	3.51472	0.187870
$351$	−1.00000	−0.0533761
$352$	−8.82843	−0.470557
$353$	−33.4558	−1.78067	−0.890337	−	0.455301i	$-0.849532\pi$
$353$	−33.4558	−1.78067	−0.890337	+	0.455301i	$0.849532\pi$
$354$	3.17157	0.168567
$355$	−5.65685	−0.300235
$356$	−27.1127	−1.43697
$357$	21.6569	1.14620
$358$	−9.65685	−0.510381
$359$	34.9706	1.84568	0.922838	−	0.385189i	$-0.125864\pi$
$359$	34.9706	1.84568	0.922838	+	0.385189i	$0.125864\pi$
$360$	4.48528	0.236395
$361$	−11.0000	−0.578947
$362$	5.79899	0.304788
$363$	−7.00000	−0.367405
$364$	5.17157	0.271064
$365$	−0.970563	−0.0508016
$366$	5.51472	0.288259
$367$	−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
$369$	5.17157	0.269221
$370$	8.97056	0.466357
$371$	−5.65685	−0.293689
$372$	2.14214	0.111065
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−6.34315	−0.327996
$375$	5.65685	0.292119
$376$	18.4853	0.953306
$377$	−2.00000	−0.103005
$378$	1.17157	0.0602592
$379$	0.485281	0.0249272	0.0124636	−	0.999922i	$-0.496033\pi$
$379$	0.485281	0.0249272	0.0124636	+	0.999922i	$0.496033\pi$
$380$	−14.6274	−0.750371
$381$	5.65685	0.289809
$382$	1.37258	0.0702275
$383$	30.9706	1.58252	0.791261	−	0.611479i	$-0.209425\pi$
$383$	30.9706	1.58252	0.791261	+	0.611479i	$0.209425\pi$
$384$	−10.5563	−0.538701
$385$	16.0000	0.815436
$386$	2.20101	0.112028
$387$	−1.65685	−0.0842226
$388$	−6.68629	−0.339445
$389$	−26.9706	−1.36746	−0.683731	−	0.729734i	$-0.739644\pi$
$389$	−26.9706	−1.36746	−0.683731	+	0.729734i	$0.739644\pi$
$390$	1.17157	0.0593249
$391$	−30.6274	−1.54890
$392$	−1.58579	−0.0800943
$393$	−8.00000	−0.403547
$394$	0.201010	0.0101267
$395$	32.0000	1.61009
$396$	3.65685	0.183764
$397$	30.9706	1.55437	0.777184	−	0.629273i	$-0.216647\pi$
$397$	30.9706	1.55437	0.777184	+	0.629273i	$0.216647\pi$
$398$	8.97056	0.449654
$399$	−8.00000	−0.400501
$400$	9.00000	0.450000
$401$	26.1421	1.30548	0.652738	−	0.757584i	$-0.273620\pi$
$401$	26.1421	1.30548	0.652738	+	0.757584i	$0.273620\pi$
$402$	2.82843	0.141069
$403$	1.17157	0.0583602
$404$	−14.0000	−0.696526
$405$	−2.82843	−0.140546
$406$	2.34315	0.116288
$407$	15.3137	0.759072
$408$	−12.1421	−0.601125
$409$	−34.9706	−1.72918	−0.864592	−	0.502475i	$-0.832423\pi$
$409$	−34.9706	−1.72918	−0.864592	+	0.502475i	$0.832423\pi$
$410$	−6.05887	−0.299226
$411$	−10.8284	−0.534127
$412$	−4.28427	−0.211071
$413$	21.6569	1.06566
$414$	−1.65685	−0.0814299
$415$	−10.3431	−0.507725
$416$	−4.41421	−0.216425
$417$	−7.31371	−0.358154
$418$	2.34315	0.114607
$419$	14.6274	0.714596	0.357298	−	0.933990i	$-0.383698\pi$
$419$	14.6274	0.714596	0.357298	+	0.933990i	$0.383698\pi$
$420$	14.6274	0.713745
$421$	37.3137	1.81856	0.909279	−	0.416186i	$-0.136634\pi$
$421$	37.3137	1.81856	0.909279	+	0.416186i	$0.136634\pi$
$422$	−4.97056	−0.241963
$423$	−11.6569	−0.566776
$424$	3.17157	0.154025
$425$	22.9706	1.11424
$426$	0.828427	0.0401374
$427$	37.6569	1.82234
$428$	20.6863	0.999910
$429$	2.00000	0.0965609
$430$	1.94113	0.0936094
$431$	8.34315	0.401875	0.200938	−	0.979604i	$-0.435601\pi$
$431$	8.34315	0.401875	0.200938	+	0.979604i	$0.435601\pi$
$432$	3.00000	0.144338
$433$	−21.3137	−1.02427	−0.512136	−	0.858905i	$-0.671146\pi$
$433$	−21.3137	−1.02427	−0.512136	+	0.858905i	$0.671146\pi$
$434$	−1.37258	−0.0658861
$435$	−5.65685	−0.271225
$436$	−9.71573	−0.465299
$437$	11.3137	0.541208
$438$	0.142136	0.00679150
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	−8.97056	−0.427655
$441$	1.00000	0.0476190
$442$	−3.17157	−0.150856
$443$	−25.9411	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$443$	−25.9411	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$444$	14.0000	0.664411
$445$	−41.9411	−1.98820
$446$	−5.17157	−0.244881
$447$	−9.17157	−0.433801
$448$	−11.7990	−0.557450
$449$	−31.7990	−1.50069	−0.750344	−	0.661048i	$-0.770112\pi$
$449$	−31.7990	−1.50069	−0.750344	+	0.661048i	$0.770112\pi$
$450$	1.24264	0.0585786
$451$	−10.3431	−0.487040
$452$	9.71573	0.456989
$453$	−3.51472	−0.165136
$454$	−7.17157	−0.336579
$455$	8.00000	0.375046
$456$	4.48528	0.210043
$457$	−7.65685	−0.358173	−0.179086	−	0.983833i	$-0.557314\pi$
$457$	−7.65685	−0.358173	−0.179086	+	0.983833i	$0.557314\pi$
$458$	−0.544156	−0.0254267
$459$	7.65685	0.357391
$460$	−20.6863	−0.964503
$461$	5.17157	0.240864	0.120432	−	0.992722i	$-0.461572\pi$
$461$	5.17157	0.240864	0.120432	+	0.992722i	$0.461572\pi$
$462$	−2.34315	−0.109013
$463$	−24.4853	−1.13793	−0.568964	−	0.822363i	$-0.692656\pi$
$463$	−24.4853	−1.13793	−0.568964	+	0.822363i	$0.692656\pi$
$464$	6.00000	0.278543
$465$	3.31371	0.153670
$466$	2.88730	0.133752
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	1.82843	0.0845191
$469$	19.3137	0.891824
$470$	13.6569	0.629944
$471$	−10.0000	−0.460776
$472$	−12.1421	−0.558887
$473$	3.31371	0.152364
$474$	−4.68629	−0.215248
$475$	−8.48528	−0.389331
$476$	−39.5980	−1.81497
$477$	−2.00000	−0.0915737
$478$	0.828427	0.0378914
$479$	25.3137	1.15661	0.578306	−	0.815820i	$-0.303714\pi$
$479$	25.3137	1.15661	0.578306	+	0.815820i	$0.303714\pi$
$480$	−12.4853	−0.569873
$481$	7.65685	0.349123
$482$	0.142136	0.00647410
$483$	−11.3137	−0.514792
$484$	12.7990	0.581772
$485$	−10.3431	−0.469658
$486$	0.414214	0.0187891
$487$	−7.79899	−0.353406	−0.176703	−	0.984264i	$-0.556543\pi$
$487$	−7.79899	−0.353406	−0.176703	+	0.984264i	$0.556543\pi$
$488$	−21.1127	−0.955727
$489$	18.8284	0.851451
$490$	−1.17157	−0.0529263
$491$	30.6274	1.38220	0.691098	−	0.722761i	$-0.257127\pi$
$491$	30.6274	1.38220	0.691098	+	0.722761i	$0.257127\pi$
$492$	−9.45584	−0.426302
$493$	15.3137	0.689695
$494$	1.17157	0.0527116
$495$	5.65685	0.254257
$496$	−3.51472	−0.157816
$497$	5.65685	0.253745
$498$	1.51472	0.0678762
$499$	26.1421	1.17028	0.585141	−	0.810931i	$-0.301039\pi$
$499$	26.1421	1.17028	0.585141	+	0.810931i	$0.301039\pi$
$500$	−10.3431	−0.462560
$501$	−3.65685	−0.163376
$502$	0	0
$503$	−7.31371	−0.326102	−0.163051	−	0.986618i	$-0.552134\pi$
$503$	−7.31371	−0.326102	−0.163051	+	0.986618i	$0.552134\pi$
$504$	−4.48528	−0.199790
$505$	−21.6569	−0.963717
$506$	3.31371	0.147312
$507$	1.00000	0.0444116
$508$	−10.3431	−0.458903
$509$	11.7990	0.522981	0.261491	−	0.965206i	$-0.415786\pi$
$509$	11.7990	0.522981	0.261491	+	0.965206i	$0.415786\pi$
$510$	−8.97056	−0.397223
$511$	0.970563	0.0429352
$512$	22.7574	1.00574
$513$	−2.82843	−0.124878
$514$	−1.79899	−0.0793500
$515$	−6.62742	−0.292039
$516$	3.02944	0.133364
$517$	23.3137	1.02534
$518$	−8.97056	−0.394144
$519$	−11.6569	−0.511679
$520$	−4.48528	−0.196693
$521$	25.3137	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$521$	25.3137	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$522$	0.828427	0.0362593
$523$	−15.3137	−0.669622	−0.334811	−	0.942285i	$-0.608672\pi$
$523$	−15.3137	−0.669622	−0.334811	+	0.942285i	$0.608672\pi$
$524$	14.6274	0.639002
$525$	8.48528	0.370328
$526$	4.97056	0.216727
$527$	−8.97056	−0.390764
$528$	−6.00000	−0.261116
$529$	−7.00000	−0.304348
$530$	2.34315	0.101780
$531$	7.65685	0.332279
$532$	14.6274	0.634179
$533$	−5.17157	−0.224006
$534$	6.14214	0.265796
$535$	32.0000	1.38348
$536$	−10.8284	−0.467717
$537$	−23.3137	−1.00606
$538$	7.45584	0.321444
$539$	−2.00000	−0.0861461
$540$	5.17157	0.222549
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	−11.5147	−0.494600
$543$	14.0000	0.600798
$544$	33.7990	1.44912
$545$	−15.0294	−0.643790
$546$	−1.17157	−0.0501387
$547$	23.3137	0.996822	0.498411	−	0.866941i	$-0.333917\pi$
$547$	23.3137	0.996822	0.498411	+	0.866941i	$0.333917\pi$
$548$	19.7990	0.845771
$549$	13.3137	0.568215
$550$	−2.48528	−0.105973
$551$	−5.65685	−0.240990
$552$	6.34315	0.269982
$553$	−32.0000	−1.36078
$554$	−0.828427	−0.0351965
$555$	21.6569	0.919282
$556$	13.3726	0.567124
$557$	−7.79899	−0.330454	−0.165227	−	0.986256i	$-0.552836\pi$
$557$	−7.79899	−0.330454	−0.165227	+	0.986256i	$0.552836\pi$
$558$	−0.485281	−0.0205436
$559$	1.65685	0.0700775
$560$	−24.0000	−1.01419
$561$	−15.3137	−0.646545
$562$	8.76955	0.369921
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	21.3137	0.897469
$565$	15.0294	0.632293
$566$	12.0000	0.504398
$567$	2.82843	0.118783
$568$	−3.17157	−0.133076
$569$	−42.9706	−1.80142	−0.900710	−	0.434421i	$-0.856953\pi$
$569$	−42.9706	−1.80142	−0.900710	+	0.434421i	$0.856953\pi$
$570$	3.31371	0.138796
$571$	−12.9706	−0.542801	−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706	−0.542801	−0.271401	+	0.962466i	$0.587487\pi$
$572$	−3.65685	−0.152901
$573$	3.31371	0.138432
$574$	6.05887	0.252893
$575$	−12.0000	−0.500435
$576$	−4.17157	−0.173816
$577$	−31.9411	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$577$	−31.9411	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$578$	17.2426	0.717199
$579$	5.31371	0.220830
$580$	10.3431	0.429476
$581$	10.3431	0.429106
$582$	1.51472	0.0627871
$583$	4.00000	0.165663
$584$	−0.544156	−0.0225173
$585$	2.82843	0.116941
$586$	0.887302	0.0366541
$587$	−10.9706	−0.452804	−0.226402	−	0.974034i	$-0.572696\pi$
$587$	−10.9706	−0.452804	−0.226402	+	0.974034i	$0.572696\pi$
$588$	−1.82843	−0.0754031
$589$	3.31371	0.136539
$590$	−8.97056	−0.369312
$591$	0.485281	0.0199618
$592$	−22.9706	−0.944084
$593$	−20.4853	−0.841230	−0.420615	−	0.907239i	$-0.638186\pi$
$593$	−20.4853	−0.841230	−0.420615	+	0.907239i	$0.638186\pi$
$594$	−0.828427	−0.0339908
$595$	−61.2548	−2.51120
$596$	16.7696	0.686908
$597$	21.6569	0.886356
$598$	1.65685	0.0677538
$599$	−23.3137	−0.952572	−0.476286	−	0.879290i	$-0.658017\pi$
$599$	−23.3137	−0.952572	−0.476286	+	0.879290i	$0.658017\pi$
$600$	−4.75736	−0.194218
$601$	−0.627417	−0.0255929	−0.0127964	−	0.999918i	$-0.504073\pi$
$601$	−0.627417	−0.0255929	−0.0127964	+	0.999918i	$0.504073\pi$
$602$	−1.94113	−0.0791144
$603$	6.82843	0.278075
$604$	6.42641	0.261487
$605$	19.7990	0.804943
$606$	3.17157	0.128836
$607$	41.9411	1.70234	0.851169	−	0.524892i	$-0.175894\pi$
$607$	41.9411	1.70234	0.851169	+	0.524892i	$0.175894\pi$
$608$	−12.4853	−0.506345
$609$	5.65685	0.229227
$610$	−15.5980	−0.631544
$611$	11.6569	0.471586
$612$	−14.0000	−0.565916
$613$	−47.6569	−1.92484	−0.962421	−	0.271561i	$-0.912460\pi$
$613$	−47.6569	−1.92484	−0.962421	+	0.271561i	$0.912460\pi$
$614$	−9.45584	−0.381607
$615$	−14.6274	−0.589834
$616$	8.97056	0.361434
$617$	−34.8284	−1.40214	−0.701070	−	0.713093i	$-0.747294\pi$
$617$	−34.8284	−1.40214	−0.701070	+	0.713093i	$0.747294\pi$
$618$	0.970563	0.0390418
$619$	23.7990	0.956562	0.478281	−	0.878207i	$-0.341260\pi$
$619$	23.7990	0.956562	0.478281	+	0.878207i	$0.341260\pi$
$620$	−6.05887	−0.243330
$621$	−4.00000	−0.160514
$622$	−4.40202	−0.176505
$623$	41.9411	1.68034
$624$	−3.00000	−0.120096
$625$	−31.0000	−1.24000
$626$	2.48528	0.0993318
$627$	5.65685	0.225913
$628$	18.2843	0.729622
$629$	−58.6274	−2.33763
$630$	−3.31371	−0.132021
$631$	43.1127	1.71629	0.858145	−	0.513408i	$-0.171617\pi$
$631$	43.1127	1.71629	0.858145	+	0.513408i	$0.171617\pi$
$632$	17.9411	0.713660
$633$	−12.0000	−0.476957
$634$	3.51472	0.139587
$635$	−16.0000	−0.634941
$636$	3.65685	0.145004
$637$	−1.00000	−0.0396214
$638$	−1.65685	−0.0655955
$639$	2.00000	0.0791188
$640$	29.8579	1.18024
$641$	30.2843	1.19616	0.598078	−	0.801438i	$-0.295931\pi$
$641$	30.2843	1.19616	0.598078	+	0.801438i	$0.295931\pi$
$642$	−4.68629	−0.184953
$643$	22.8284	0.900265	0.450133	−	0.892962i	$-0.351377\pi$
$643$	22.8284	0.900265	0.450133	+	0.892962i	$0.351377\pi$
$644$	20.6863	0.815154
$645$	4.68629	0.184523
$646$	−8.97056	−0.352942
$647$	11.3137	0.444788	0.222394	−	0.974957i	$-0.428613\pi$
$647$	11.3137	0.444788	0.222394	+	0.974957i	$0.428613\pi$
$648$	−1.58579	−0.0622956
$649$	−15.3137	−0.601116
$650$	−1.24264	−0.0487404
$651$	−3.31371	−0.129874
$652$	−34.4264	−1.34824
$653$	−25.3137	−0.990602	−0.495301	−	0.868721i	$-0.664942\pi$
$653$	−25.3137	−0.990602	−0.495301	+	0.868721i	$0.664942\pi$
$654$	2.20101	0.0860663
$655$	22.6274	0.884126
$656$	15.5147	0.605748
$657$	0.343146	0.0133874
$658$	−13.6569	−0.532400
$659$	−47.3137	−1.84308	−0.921540	−	0.388283i	$-0.873068\pi$
$659$	−47.3137	−1.84308	−0.921540	+	0.388283i	$0.873068\pi$
$660$	−10.3431	−0.402606
$661$	−34.9706	−1.36020	−0.680099	−	0.733121i	$-0.738063\pi$
$661$	−34.9706	−1.36020	−0.680099	+	0.733121i	$0.738063\pi$
$662$	10.8284	0.420859
$663$	−7.65685	−0.297368
$664$	−5.79899	−0.225044
$665$	22.6274	0.877454
$666$	−3.17157	−0.122896
$667$	−8.00000	−0.309761
$668$	6.68629	0.258700
$669$	−12.4853	−0.482709
$670$	−8.00000	−0.309067
$671$	−26.6274	−1.02794
$672$	12.4853	0.481630
$673$	16.6274	0.640940	0.320470	−	0.947259i	$-0.396159\pi$
$673$	16.6274	0.640940	0.320470	+	0.947259i	$0.396159\pi$
$674$	3.85786	0.148599
$675$	3.00000	0.115470
$676$	−1.82843	−0.0703241
$677$	26.6863	1.02564	0.512819	−	0.858497i	$-0.328601\pi$
$677$	26.6863	1.02564	0.512819	+	0.858497i	$0.328601\pi$
$678$	−2.20101	−0.0845293
$679$	10.3431	0.396934
$680$	34.3431	1.31700
$681$	−17.3137	−0.663463
$682$	0.970563	0.0371648
$683$	47.9411	1.83442	0.917208	−	0.398408i	$-0.130437\pi$
$683$	47.9411	1.83442	0.917208	+	0.398408i	$0.130437\pi$
$684$	5.17157	0.197740
$685$	30.6274	1.17021
$686$	−7.02944	−0.268385
$687$	−1.31371	−0.0501211
$688$	−4.97056	−0.189501
$689$	2.00000	0.0761939
$690$	4.68629	0.178404
$691$	−5.85786	−0.222844	−0.111422	−	0.993773i	$-0.535540\pi$
$691$	−5.85786	−0.222844	−0.111422	+	0.993773i	$0.535540\pi$
$692$	21.3137	0.810226
$693$	−5.65685	−0.214886
$694$	−3.59798	−0.136577
$695$	20.6863	0.784676
$696$	−3.17157	−0.120218
$697$	39.5980	1.49988
$698$	1.51472	0.0573329
$699$	6.97056	0.263651
$700$	−15.5147	−0.586401
$701$	5.02944	0.189959	0.0949796	−	0.995479i	$-0.469721\pi$
$701$	5.02944	0.189959	0.0949796	+	0.995479i	$0.469721\pi$
$702$	−0.414214	−0.0156335
$703$	21.6569	0.816804
$704$	8.34315	0.314444
$705$	32.9706	1.24174
$706$	−13.8579	−0.521548
$707$	21.6569	0.814490
$708$	−14.0000	−0.526152
$709$	−4.62742	−0.173786	−0.0868931	−	0.996218i	$-0.527694\pi$
$709$	−4.62742	−0.173786	−0.0868931	+	0.996218i	$0.527694\pi$
$710$	−2.34315	−0.0879367
$711$	−11.3137	−0.424297
$712$	−23.5147	−0.881251
$713$	4.68629	0.175503
$714$	8.97056	0.335715
$715$	−5.65685	−0.211554
$716$	42.6274	1.59306
$717$	2.00000	0.0746914
$718$	14.4853	0.540586
$719$	−29.9411	−1.11662	−0.558308	−	0.829634i	$-0.688549\pi$
$719$	−29.9411	−1.11662	−0.558308	+	0.829634i	$0.688549\pi$
$720$	−8.48528	−0.316228
$721$	6.62742	0.246818
$722$	−4.55635	−0.169570
$723$	0.343146	0.0127617
$724$	−25.5980	−0.951341
$725$	6.00000	0.222834
$726$	−2.89949	−0.107610
$727$	−10.3431	−0.383606	−0.191803	−	0.981433i	$-0.561433\pi$
$727$	−10.3431	−0.383606	−0.191803	+	0.981433i	$0.561433\pi$
$728$	4.48528	0.166236
$729$	1.00000	0.0370370
$730$	−0.402020	−0.0148794
$731$	−12.6863	−0.469219
$732$	−24.3431	−0.899749
$733$	−36.6274	−1.35286	−0.676432	−	0.736505i	$-0.736475\pi$
$733$	−36.6274	−1.35286	−0.676432	+	0.736505i	$0.736475\pi$
$734$	−9.94113	−0.366934
$735$	−2.82843	−0.104328
$736$	−17.6569	−0.650840
$737$	−13.6569	−0.503057
$738$	2.14214	0.0788531
$739$	−18.1421	−0.667369	−0.333685	−	0.942685i	$-0.608292\pi$
$739$	−18.1421	−0.667369	−0.333685	+	0.942685i	$0.608292\pi$
$740$	−39.5980	−1.45565
$741$	2.82843	0.103905
$742$	−2.34315	−0.0860196
$743$	2.00000	0.0733729	0.0366864	−	0.999327i	$-0.488320\pi$
$743$	2.00000	0.0733729	0.0366864	+	0.999327i	$0.488320\pi$
$744$	1.85786	0.0681126
$745$	25.9411	0.950409
$746$	4.14214	0.151654
$747$	3.65685	0.133797
$748$	28.0000	1.02378
$749$	−32.0000	−1.16925
$750$	2.34315	0.0855596
$751$	−0.970563	−0.0354163	−0.0177082	−	0.999843i	$-0.505637\pi$
$751$	−0.970563	−0.0354163	−0.0177082	+	0.999843i	$0.505637\pi$
$752$	−34.9706	−1.27525
$753$	0	0
$754$	−0.828427	−0.0301695
$755$	9.94113	0.361795
$756$	−5.17157	−0.188088
$757$	51.9411	1.88783	0.943916	−	0.330185i	$-0.107111\pi$
$757$	51.9411	1.88783	0.943916	+	0.330185i	$0.107111\pi$
$758$	0.201010	0.00730102
$759$	8.00000	0.290382
$760$	−12.6863	−0.460180
$761$	32.4853	1.17759	0.588795	−	0.808282i	$-0.299602\pi$
$761$	32.4853	1.17759	0.588795	+	0.808282i	$0.299602\pi$
$762$	2.34315	0.0848832
$763$	15.0294	0.544102
$764$	−6.05887	−0.219202
$765$	−21.6569	−0.783005
$766$	12.8284	0.463510
$767$	−7.65685	−0.276473
$768$	3.97056	0.143275
$769$	42.0000	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	6.62742	0.238836
$771$	−4.34315	−0.156415
$772$	−9.71573	−0.349677
$773$	34.1421	1.22801	0.614004	−	0.789303i	$-0.289558\pi$
$773$	34.1421	1.22801	0.614004	+	0.789303i	$0.289558\pi$
$774$	−0.686292	−0.0246682
$775$	−3.51472	−0.126252
$776$	−5.79899	−0.208172
$777$	−21.6569	−0.776935
$778$	−11.1716	−0.400520
$779$	−14.6274	−0.524082
$780$	−5.17157	−0.185172
$781$	−4.00000	−0.143131
$782$	−12.6863	−0.453661
$783$	2.00000	0.0714742
$784$	3.00000	0.107143
$785$	28.2843	1.00951
$786$	−3.31371	−0.118196
$787$	−40.7696	−1.45328	−0.726639	−	0.687020i	$-0.758919\pi$
$787$	−40.7696	−1.45328	−0.726639	+	0.687020i	$0.758919\pi$
$788$	−0.887302	−0.0316088
$789$	12.0000	0.427211
$790$	13.2548	0.471586
$791$	−15.0294	−0.534385
$792$	3.17157	0.112697
$793$	−13.3137	−0.472784
$794$	12.8284	0.455264
$795$	5.65685	0.200628
$796$	−39.5980	−1.40351
$797$	24.3431	0.862278	0.431139	−	0.902285i	$-0.358112\pi$
$797$	24.3431	0.862278	0.431139	+	0.902285i	$0.358112\pi$
$798$	−3.31371	−0.117304
$799$	−89.2548	−3.15761
$800$	13.2426	0.468198
$801$	14.8284	0.523937
$802$	10.8284	0.382365
$803$	−0.686292	−0.0242187
$804$	−12.4853	−0.440322
$805$	32.0000	1.12785
$806$	0.485281	0.0170933
$807$	18.0000	0.633630
$808$	−12.1421	−0.427159
$809$	18.6863	0.656975	0.328488	−	0.944508i	$-0.393461\pi$
$809$	18.6863	0.656975	0.328488	+	0.944508i	$0.393461\pi$
$810$	−1.17157	−0.0411649
$811$	−30.1421	−1.05843	−0.529217	−	0.848487i	$-0.677514\pi$
$811$	−30.1421	−1.05843	−0.529217	+	0.848487i	$0.677514\pi$
$812$	−10.3431	−0.362973
$813$	−27.7990	−0.974953
$814$	6.34315	0.222327
$815$	−53.2548	−1.86544
$816$	22.9706	0.804131
$817$	4.68629	0.163953
$818$	−14.4853	−0.506466
$819$	−2.82843	−0.0988332
$820$	26.7452	0.933982
$821$	−23.7990	−0.830590	−0.415295	−	0.909687i	$-0.636322\pi$
$821$	−23.7990	−0.830590	−0.415295	+	0.909687i	$0.636322\pi$
$822$	−4.48528	−0.156442
$823$	15.0294	0.523893	0.261947	−	0.965082i	$-0.415636\pi$
$823$	15.0294	0.523893	0.261947	+	0.965082i	$0.415636\pi$
$824$	−3.71573	−0.129444
$825$	−6.00000	−0.208893
$826$	8.97056	0.312126
$827$	−26.0000	−0.904109	−0.452054	−	0.891990i	$-0.649309\pi$
$827$	−26.0000	−0.904109	−0.452054	+	0.891990i	$0.649309\pi$
$828$	7.31371	0.254169
$829$	−17.3137	−0.601330	−0.300665	−	0.953730i	$-0.597209\pi$
$829$	−17.3137	−0.601330	−0.300665	+	0.953730i	$0.597209\pi$
$830$	−4.28427	−0.148709
$831$	−2.00000	−0.0693792
$832$	4.17157	0.144623
$833$	7.65685	0.265294
$834$	−3.02944	−0.104901
$835$	10.3431	0.357939
$836$	−10.3431	−0.357725
$837$	−1.17157	−0.0404955
$838$	6.05887	0.209300
$839$	43.2548	1.49332	0.746661	−	0.665204i	$-0.231656\pi$
$839$	43.2548	1.49332	0.746661	+	0.665204i	$0.231656\pi$
$840$	12.6863	0.437719
$841$	−25.0000	−0.862069
$842$	15.4558	0.532644
$843$	21.1716	0.729188
$844$	21.9411	0.755245
$845$	−2.82843	−0.0973009
$846$	−4.82843	−0.166005
$847$	−19.7990	−0.680301
$848$	−6.00000	−0.206041
$849$	28.9706	0.994267
$850$	9.51472	0.326352
$851$	30.6274	1.04989
$852$	−3.65685	−0.125282
$853$	3.65685	0.125208	0.0626042	−	0.998038i	$-0.480059\pi$
$853$	3.65685	0.125208	0.0626042	+	0.998038i	$0.480059\pi$
$854$	15.5980	0.533752
$855$	8.00000	0.273594
$856$	17.9411	0.613215
$857$	−49.5980	−1.69423	−0.847117	−	0.531406i	$-0.821664\pi$
$857$	−49.5980	−1.69423	−0.847117	+	0.531406i	$0.821664\pi$
$858$	0.828427	0.0282820
$859$	−0.686292	−0.0234160	−0.0117080	−	0.999931i	$-0.503727\pi$
$859$	−0.686292	−0.0234160	−0.0117080	+	0.999931i	$0.503727\pi$
$860$	−8.56854	−0.292185
$861$	14.6274	0.498501
$862$	3.45584	0.117707
$863$	−28.3431	−0.964812	−0.482406	−	0.875948i	$-0.660237\pi$
$863$	−28.3431	−0.964812	−0.482406	+	0.875948i	$0.660237\pi$
$864$	4.41421	0.150175
$865$	32.9706	1.12103
$866$	−8.82843	−0.300002
$867$	41.6274	1.41374
$868$	6.05887	0.205652
$869$	22.6274	0.767583
$870$	−2.34315	−0.0794401
$871$	−6.82843	−0.231372
$872$	−8.42641	−0.285354
$873$	3.65685	0.123766
$874$	4.68629	0.158516
$875$	16.0000	0.540899
$876$	−0.627417	−0.0211985
$877$	42.2843	1.42784	0.713919	−	0.700228i	$-0.246918\pi$
$877$	42.2843	1.42784	0.713919	+	0.700228i	$0.246918\pi$
$878$	7.02944	0.237232
$879$	2.14214	0.0722524
$880$	16.9706	0.572078
$881$	−25.5980	−0.862418	−0.431209	−	0.902252i	$-0.641913\pi$
$881$	−25.5980	−0.862418	−0.431209	+	0.902252i	$0.641913\pi$
$882$	0.414214	0.0139473
$883$	27.5980	0.928746	0.464373	−	0.885640i	$-0.346280\pi$
$883$	27.5980	0.928746	0.464373	+	0.885640i	$0.346280\pi$
$884$	14.0000	0.470871
$885$	−21.6569	−0.727987
$886$	−10.7452	−0.360991
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	12.1421	0.407463
$889$	16.0000	0.536623
$890$	−17.3726	−0.582330
$891$	−2.00000	−0.0670025
$892$	22.8284	0.764352
$893$	32.9706	1.10332
$894$	−3.79899	−0.127057
$895$	65.9411	2.20417
$896$	−29.8579	−0.997481
$897$	4.00000	0.133556
$898$	−13.1716	−0.439541
$899$	−2.34315	−0.0781483
$900$	−5.48528	−0.182843
$901$	−15.3137	−0.510174
$902$	−4.28427	−0.142651
$903$	−4.68629	−0.155950
$904$	8.42641	0.280258
$905$	−39.5980	−1.31628
$906$	−1.45584	−0.0483672
$907$	−12.9706	−0.430680	−0.215340	−	0.976539i	$-0.569086\pi$
$907$	−12.9706	−0.430680	−0.215340	+	0.976539i	$0.569086\pi$
$908$	31.6569	1.05057
$909$	7.65685	0.253962
$910$	3.31371	0.109848
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	−8.48528	−0.280976
$913$	−7.31371	−0.242048
$914$	−3.17157	−0.104906
$915$	−37.6569	−1.24490
$916$	2.40202	0.0793650
$917$	−22.6274	−0.747223
$918$	3.17157	0.104678
$919$	3.31371	0.109309	0.0546546	−	0.998505i	$-0.482594\pi$
$919$	3.31371	0.109309	0.0546546	+	0.998505i	$0.482594\pi$
$920$	−17.9411	−0.591501
$921$	−22.8284	−0.752222
$922$	2.14214	0.0705475
$923$	−2.00000	−0.0658308
$924$	10.3431	0.340265
$925$	−22.9706	−0.755267
$926$	−10.1421	−0.333291
$927$	2.34315	0.0769590
$928$	8.82843	0.289807
$929$	11.7990	0.387112	0.193556	−	0.981089i	$-0.437998\pi$
$929$	11.7990	0.387112	0.193556	+	0.981089i	$0.437998\pi$
$930$	1.37258	0.0450088
$931$	−2.82843	−0.0926980
$932$	−12.7452	−0.417482
$933$	−10.6274	−0.347926
$934$	−3.31371	−0.108428
$935$	43.3137	1.41651
$936$	1.58579	0.0518331
$937$	−21.3137	−0.696289	−0.348144	−	0.937441i	$-0.613188\pi$
$937$	−21.3137	−0.696289	−0.348144	+	0.937441i	$0.613188\pi$
$938$	8.00000	0.261209
$939$	6.00000	0.195803
$940$	−60.2843	−1.96626
$941$	34.1421	1.11300	0.556501	−	0.830847i	$-0.312144\pi$
$941$	34.1421	1.11300	0.556501	+	0.830847i	$0.312144\pi$
$942$	−4.14214	−0.134958
$943$	−20.6863	−0.673638
$944$	22.9706	0.747628
$945$	−8.00000	−0.260240
$946$	1.37258	0.0446265
$947$	−21.0294	−0.683365	−0.341682	−	0.939815i	$-0.610997\pi$
$947$	−21.0294	−0.683365	−0.341682	+	0.939815i	$0.610997\pi$
$948$	20.6863	0.671860
$949$	−0.343146	−0.0111390
$950$	−3.51472	−0.114033
$951$	8.48528	0.275154
$952$	−34.3431	−1.11307
$953$	40.3431	1.30684	0.653421	−	0.756994i	$-0.273333\pi$
$953$	40.3431	1.30684	0.653421	+	0.756994i	$0.273333\pi$
$954$	−0.828427	−0.0268213
$955$	−9.37258	−0.303290
$956$	−3.65685	−0.118271
$957$	−4.00000	−0.129302
$958$	10.4853	0.338764
$959$	−30.6274	−0.989011
$960$	11.7990	0.380811
$961$	−29.6274	−0.955723
$962$	3.17157	0.102256
$963$	−11.3137	−0.364579
$964$	−0.627417	−0.0202077
$965$	−15.0294	−0.483815
$966$	−4.68629	−0.150779
$967$	18.1421	0.583412	0.291706	−	0.956508i	$-0.405777\pi$
$967$	18.1421	0.583412	0.291706	+	0.956508i	$0.405777\pi$
$968$	11.1005	0.356784
$969$	−21.6569	−0.695718
$970$	−4.28427	−0.137560
$971$	−15.3137	−0.491440	−0.245720	−	0.969341i	$-0.579024\pi$
$971$	−15.3137	−0.491440	−0.245720	+	0.969341i	$0.579024\pi$
$972$	−1.82843	−0.0586468
$973$	−20.6863	−0.663172
$974$	−3.23045	−0.103510
$975$	−3.00000	−0.0960769
$976$	39.9411	1.27848
$977$	42.1421	1.34825	0.674123	−	0.738619i	$-0.264522\pi$
$977$	42.1421	1.34825	0.674123	+	0.738619i	$0.264522\pi$
$978$	7.79899	0.249384
$979$	−29.6569	−0.947837
$980$	5.17157	0.165200
$981$	5.31371	0.169654
$982$	12.6863	0.404836
$983$	25.3137	0.807382	0.403691	−	0.914895i	$-0.367727\pi$
$983$	25.3137	0.807382	0.403691	+	0.914895i	$0.367727\pi$
$984$	−8.20101	−0.261439
$985$	−1.37258	−0.0437341
$986$	6.34315	0.202007
$987$	−32.9706	−1.04946
$988$	−5.17157	−0.164530
$989$	6.62742	0.210740
$990$	2.34315	0.0744701
$991$	4.68629	0.148865	0.0744325	−	0.997226i	$-0.476285\pi$
$991$	4.68629	0.148865	0.0744325	+	0.997226i	$0.476285\pi$
$992$	−5.17157	−0.164198
$993$	26.1421	0.829596
$994$	2.34315	0.0743201
$995$	−61.2548	−1.94191
$996$	−6.68629	−0.211863
$997$	−39.2548	−1.24321	−0.621607	−	0.783330i	$-0.713520\pi$
$997$	−39.2548	−1.24321	−0.621607	+	0.783330i	$0.713520\pi$
$998$	10.8284	0.342768
$999$	−7.65685	−0.242252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.2.a.b.1.2	✓	2
3.2	odd	2		117.2.a.c.1.1		2
4.3	odd	2		624.2.a.k.1.1		2
5.2	odd	4		975.2.c.h.274.3		4
5.3	odd	4		975.2.c.h.274.2		4
5.4	even	2		975.2.a.l.1.1		2
7.6	odd	2		1911.2.a.h.1.2		2
8.3	odd	2		2496.2.a.bi.1.2		2
8.5	even	2		2496.2.a.bf.1.2		2
9.2	odd	6		1053.2.e.e.352.2		4
9.4	even	3		1053.2.e.m.703.1		4
9.5	odd	6		1053.2.e.e.703.2		4
9.7	even	3		1053.2.e.m.352.1		4
11.10	odd	2		4719.2.a.p.1.1		2
12.11	even	2		1872.2.a.w.1.2		2
13.2	odd	12		507.2.j.f.316.3		8
13.3	even	3		507.2.e.h.22.1		4
13.4	even	6		507.2.e.d.484.2		4
13.5	odd	4		507.2.b.e.337.2		4
13.6	odd	12		507.2.j.f.361.2		8
13.7	odd	12		507.2.j.f.361.3		8
13.8	odd	4		507.2.b.e.337.3		4
13.9	even	3		507.2.e.h.484.1		4
13.10	even	6		507.2.e.d.22.2		4
13.11	odd	12		507.2.j.f.316.2		8
13.12	even	2		507.2.a.h.1.1		2
15.2	even	4		2925.2.c.u.2224.2		4
15.8	even	4		2925.2.c.u.2224.3		4
15.14	odd	2		2925.2.a.v.1.2		2
21.20	even	2		5733.2.a.u.1.1		2
24.5	odd	2		7488.2.a.cl.1.1		2
24.11	even	2		7488.2.a.co.1.1		2
39.5	even	4		1521.2.b.j.1351.3		4
39.8	even	4		1521.2.b.j.1351.2		4
39.38	odd	2		1521.2.a.f.1.2		2
52.51	odd	2		8112.2.a.bm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.2	✓	2	1.1	even	1	trivial
117.2.a.c.1.1		2	3.2	odd	2
507.2.a.h.1.1		2	13.12	even	2
507.2.b.e.337.2		4	13.5	odd	4
507.2.b.e.337.3		4	13.8	odd	4
507.2.e.d.22.2		4	13.10	even	6
507.2.e.d.484.2		4	13.4	even	6
507.2.e.h.22.1		4	13.3	even	3
507.2.e.h.484.1		4	13.9	even	3
507.2.j.f.316.2		8	13.11	odd	12
507.2.j.f.316.3		8	13.2	odd	12
507.2.j.f.361.2		8	13.6	odd	12
507.2.j.f.361.3		8	13.7	odd	12
624.2.a.k.1.1		2	4.3	odd	2
975.2.a.l.1.1		2	5.4	even	2
975.2.c.h.274.2		4	5.3	odd	4
975.2.c.h.274.3		4	5.2	odd	4
1053.2.e.e.352.2		4	9.2	odd	6
1053.2.e.e.703.2		4	9.5	odd	6
1053.2.e.m.352.1		4	9.7	even	3
1053.2.e.m.703.1		4	9.4	even	3
1521.2.a.f.1.2		2	39.38	odd	2
1521.2.b.j.1351.2		4	39.8	even	4
1521.2.b.j.1351.3		4	39.5	even	4
1872.2.a.w.1.2		2	12.11	even	2
1911.2.a.h.1.2		2	7.6	odd	2
2496.2.a.bf.1.2		2	8.5	even	2
2496.2.a.bi.1.2		2	8.3	odd	2
2925.2.a.v.1.2		2	15.14	odd	2
2925.2.c.u.2224.2		4	15.2	even	4
2925.2.c.u.2224.3		4	15.8	even	4
4719.2.a.p.1.1		2	11.10	odd	2
5733.2.a.u.1.1		2	21.20	even	2
7488.2.a.cl.1.1		2	24.5	odd	2
7488.2.a.co.1.1		2	24.11	even	2
8112.2.a.bm.1.2		2	52.51	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	39.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -2.82843 q^{5} +0.414214 q^{6} +2.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -2.82843 q^{5} +0.414214 q^{6} +2.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} -1.17157 q^{10} -2.00000 q^{11} -1.82843 q^{12} -1.00000 q^{13} +1.17157 q^{14} -2.82843 q^{15} +3.00000 q^{16} +7.65685 q^{17} +0.414214 q^{18} -2.82843 q^{19} +5.17157 q^{20} +2.82843 q^{21} -0.828427 q^{22} -4.00000 q^{23} -1.58579 q^{24} +3.00000 q^{25} -0.414214 q^{26} +1.00000 q^{27} -5.17157 q^{28} +2.00000 q^{29} -1.17157 q^{30} -1.17157 q^{31} +4.41421 q^{32} -2.00000 q^{33} +3.17157 q^{34} -8.00000 q^{35} -1.82843 q^{36} -7.65685 q^{37} -1.17157 q^{38} -1.00000 q^{39} +4.48528 q^{40} +5.17157 q^{41} +1.17157 q^{42} -1.65685 q^{43} +3.65685 q^{44} -2.82843 q^{45} -1.65685 q^{46} -11.6569 q^{47} +3.00000 q^{48} +1.00000 q^{49} +1.24264 q^{50} +7.65685 q^{51} +1.82843 q^{52} -2.00000 q^{53} +0.414214 q^{54} +5.65685 q^{55} -4.48528 q^{56} -2.82843 q^{57} +0.828427 q^{58} +7.65685 q^{59} +5.17157 q^{60} +13.3137 q^{61} -0.485281 q^{62} +2.82843 q^{63} -4.17157 q^{64} +2.82843 q^{65} -0.828427 q^{66} +6.82843 q^{67} -14.0000 q^{68} -4.00000 q^{69} -3.31371 q^{70} +2.00000 q^{71} -1.58579 q^{72} +0.343146 q^{73} -3.17157 q^{74} +3.00000 q^{75} +5.17157 q^{76} -5.65685 q^{77} -0.414214 q^{78} -11.3137 q^{79} -8.48528 q^{80} +1.00000 q^{81} +2.14214 q^{82} +3.65685 q^{83} -5.17157 q^{84} -21.6569 q^{85} -0.686292 q^{86} +2.00000 q^{87} +3.17157 q^{88} +14.8284 q^{89} -1.17157 q^{90} -2.82843 q^{91} +7.31371 q^{92} -1.17157 q^{93} -4.82843 q^{94} +8.00000 q^{95} +4.41421 q^{96} +3.65685 q^{97} +0.414214 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9} - 8 q^{10} - 4 q^{11} + 2 q^{12} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{18} + 16 q^{20} + 4 q^{22} - 8 q^{23} - 6 q^{24} + 6 q^{25} + 2 q^{26} + 2 q^{27} - 16 q^{28} + 4 q^{29} - 8 q^{30} - 8 q^{31} + 6 q^{32} - 4 q^{33} + 12 q^{34} - 16 q^{35} + 2 q^{36} - 4 q^{37} - 8 q^{38} - 2 q^{39} - 8 q^{40} + 16 q^{41} + 8 q^{42} + 8 q^{43} - 4 q^{44} + 8 q^{46} - 12 q^{47} + 6 q^{48} + 2 q^{49} - 6 q^{50} + 4 q^{51} - 2 q^{52} - 4 q^{53} - 2 q^{54} + 8 q^{56} - 4 q^{58} + 4 q^{59} + 16 q^{60} + 4 q^{61} + 16 q^{62} - 14 q^{64} + 4 q^{66} + 8 q^{67} - 28 q^{68} - 8 q^{69} + 16 q^{70} + 4 q^{71} - 6 q^{72} + 12 q^{73} - 12 q^{74} + 6 q^{75} + 16 q^{76} + 2 q^{78} + 2 q^{81} - 24 q^{82} - 4 q^{83} - 16 q^{84} - 32 q^{85} - 24 q^{86} + 4 q^{87} + 12 q^{88} + 24 q^{89} - 8 q^{90} - 8 q^{92} - 8 q^{93} - 4 q^{94} + 16 q^{95} + 6 q^{96} - 4 q^{97} - 2 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^8 + 2 * q^9 - 8 * q^10 - 4 * q^11 + 2 * q^12 - 2 * q^13 + 8 * q^14 + 6 * q^16 + 4 * q^17 - 2 * q^18 + 16 * q^20 + 4 * q^22 - 8 * q^23 - 6 * q^24 + 6 * q^25 + 2 * q^26 + 2 * q^27 - 16 * q^28 + 4 * q^29 - 8 * q^30 - 8 * q^31 + 6 * q^32 - 4 * q^33 + 12 * q^34 - 16 * q^35 + 2 * q^36 - 4 * q^37 - 8 * q^38 - 2 * q^39 - 8 * q^40 + 16 * q^41 + 8 * q^42 + 8 * q^43 - 4 * q^44 + 8 * q^46 - 12 * q^47 + 6 * q^48 + 2 * q^49 - 6 * q^50 + 4 * q^51 - 2 * q^52 - 4 * q^53 - 2 * q^54 + 8 * q^56 - 4 * q^58 + 4 * q^59 + 16 * q^60 + 4 * q^61 + 16 * q^62 - 14 * q^64 + 4 * q^66 + 8 * q^67 - 28 * q^68 - 8 * q^69 + 16 * q^70 + 4 * q^71 - 6 * q^72 + 12 * q^73 - 12 * q^74 + 6 * q^75 + 16 * q^76 + 2 * q^78 + 2 * q^81 - 24 * q^82 - 4 * q^83 - 16 * q^84 - 32 * q^85 - 24 * q^86 + 4 * q^87 + 12 * q^88 + 24 * q^89 - 8 * q^90 - 8 * q^92 - 8 * q^93 - 4 * q^94 + 16 * q^95 + 6 * q^96 - 4 * q^97 - 2 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more