LMFDB - Embedded newform 5070.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +6.46410 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +7.46410 q^{19} +1.00000 q^{20} -2.00000 q^{21} +6.46410 q^{22} -3.73205 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} +0.267949 q^{29} -1.00000 q^{30} -1.73205 q^{31} +1.00000 q^{32} -6.46410 q^{33} +4.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +9.19615 q^{37} +7.46410 q^{38} +1.00000 q^{40} -2.00000 q^{41} -2.00000 q^{42} -11.9282 q^{43} +6.46410 q^{44} +1.00000 q^{45} -3.73205 q^{46} +3.53590 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +0.928203 q^{53} -1.00000 q^{54} +6.46410 q^{55} +2.00000 q^{56} -7.46410 q^{57} +0.267949 q^{58} -8.46410 q^{59} -1.00000 q^{60} -10.3923 q^{61} -1.73205 q^{62} +2.00000 q^{63} +1.00000 q^{64} -6.46410 q^{66} +11.4641 q^{67} +4.00000 q^{68} +3.73205 q^{69} +2.00000 q^{70} -12.3923 q^{71} +1.00000 q^{72} -2.00000 q^{73} +9.19615 q^{74} -1.00000 q^{75} +7.46410 q^{76} +12.9282 q^{77} -13.9282 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +8.92820 q^{83} -2.00000 q^{84} +4.00000 q^{85} -11.9282 q^{86} -0.267949 q^{87} +6.46410 q^{88} -0.535898 q^{89} +1.00000 q^{90} -3.73205 q^{92} +1.73205 q^{93} +3.53590 q^{94} +7.46410 q^{95} -1.00000 q^{96} +0.535898 q^{97} -3.00000 q^{98} +6.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 4 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 6 q^{11} - 2 q^{12} + 4 q^{14} - 2 q^{15} + 2 q^{16} + 8 q^{17} + 2 q^{18} + 8 q^{19} + 2 q^{20} - 4 q^{21} + 6 q^{22} - 4 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{27} + 4 q^{28} + 4 q^{29} - 2 q^{30} + 2 q^{32} - 6 q^{33} + 8 q^{34} + 4 q^{35} + 2 q^{36} + 8 q^{37} + 8 q^{38} + 2 q^{40} - 4 q^{41} - 4 q^{42} - 10 q^{43} + 6 q^{44} + 2 q^{45} - 4 q^{46} + 14 q^{47} - 2 q^{48} - 6 q^{49} + 2 q^{50} - 8 q^{51} - 12 q^{53} - 2 q^{54} + 6 q^{55} + 4 q^{56} - 8 q^{57} + 4 q^{58} - 10 q^{59} - 2 q^{60} + 4 q^{63} + 2 q^{64} - 6 q^{66} + 16 q^{67} + 8 q^{68} + 4 q^{69} + 4 q^{70} - 4 q^{71} + 2 q^{72} - 4 q^{73} + 8 q^{74} - 2 q^{75} + 8 q^{76} + 12 q^{77} - 14 q^{79} + 2 q^{80} + 2 q^{81} - 4 q^{82} + 4 q^{83} - 4 q^{84} + 8 q^{85} - 10 q^{86} - 4 q^{87} + 6 q^{88} - 8 q^{89} + 2 q^{90} - 4 q^{92} + 14 q^{94} + 8 q^{95} - 2 q^{96} + 8 q^{97} - 6 q^{98} + 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 4 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 6 * q^11 - 2 * q^12 + 4 * q^14 - 2 * q^15 + 2 * q^16 + 8 * q^17 + 2 * q^18 + 8 * q^19 + 2 * q^20 - 4 * q^21 + 6 * q^22 - 4 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^27 + 4 * q^28 + 4 * q^29 - 2 * q^30 + 2 * q^32 - 6 * q^33 + 8 * q^34 + 4 * q^35 + 2 * q^36 + 8 * q^37 + 8 * q^38 + 2 * q^40 - 4 * q^41 - 4 * q^42 - 10 * q^43 + 6 * q^44 + 2 * q^45 - 4 * q^46 + 14 * q^47 - 2 * q^48 - 6 * q^49 + 2 * q^50 - 8 * q^51 - 12 * q^53 - 2 * q^54 + 6 * q^55 + 4 * q^56 - 8 * q^57 + 4 * q^58 - 10 * q^59 - 2 * q^60 + 4 * q^63 + 2 * q^64 - 6 * q^66 + 16 * q^67 + 8 * q^68 + 4 * q^69 + 4 * q^70 - 4 * q^71 + 2 * q^72 - 4 * q^73 + 8 * q^74 - 2 * q^75 + 8 * q^76 + 12 * q^77 - 14 * q^79 + 2 * q^80 + 2 * q^81 - 4 * q^82 + 4 * q^83 - 4 * q^84 + 8 * q^85 - 10 * q^86 - 4 * q^87 + 6 * q^88 - 8 * q^89 + 2 * q^90 - 4 * q^92 + 14 * q^94 + 8 * q^95 - 2 * q^96 + 8 * q^97 - 6 * q^98 + 6 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	6.46410	1.94900	0.974500	−	0.224388i	$-0.0720382\pi$
$11$	6.46410	1.94900	0.974500	+	0.224388i	$0.0720382\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	1.00000	0.235702
$19$	7.46410	1.71238	0.856191	−	0.516659i	$-0.172825\pi$
$19$	7.46410	1.71238	0.856191	+	0.516659i	$0.172825\pi$
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	6.46410	1.37815
$23$	−3.73205	−0.778186	−0.389093	−	0.921198i	$-0.627212\pi$
$23$	−3.73205	−0.778186	−0.389093	+	0.921198i	$0.627212\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	0.267949	0.0497569	0.0248785	−	0.999690i	$-0.492080\pi$
$29$	0.267949	0.0497569	0.0248785	+	0.999690i	$0.492080\pi$
$30$	−1.00000	−0.182574
$31$	−1.73205	−0.311086	−0.155543	−	0.987829i	$-0.549713\pi$
$31$	−1.73205	−0.311086	−0.155543	+	0.987829i	$0.549713\pi$
$32$	1.00000	0.176777
$33$	−6.46410	−1.12526
$34$	4.00000	0.685994
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	9.19615	1.51184	0.755919	−	0.654665i	$-0.227190\pi$
$37$	9.19615	1.51184	0.755919	+	0.654665i	$0.227190\pi$
$38$	7.46410	1.21084
$39$	0	0
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−2.00000	−0.308607
$43$	−11.9282	−1.81903	−0.909517	−	0.415667i	$-0.863548\pi$
$43$	−11.9282	−1.81903	−0.909517	+	0.415667i	$0.863548\pi$
$44$	6.46410	0.974500
$45$	1.00000	0.149071
$46$	−3.73205	−0.550261
$47$	3.53590	0.515764	0.257882	−	0.966176i	$-0.416975\pi$
$47$	3.53590	0.515764	0.257882	+	0.966176i	$0.416975\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	−4.00000	−0.560112
$52$	0	0
$53$	0.928203	0.127499	0.0637493	−	0.997966i	$-0.479694\pi$
$53$	0.928203	0.127499	0.0637493	+	0.997966i	$0.479694\pi$
$54$	−1.00000	−0.136083
$55$	6.46410	0.871619
$56$	2.00000	0.267261
$57$	−7.46410	−0.988644
$58$	0.267949	0.0351835
$59$	−8.46410	−1.10193	−0.550966	−	0.834528i	$-0.685741\pi$
$59$	−8.46410	−1.10193	−0.550966	+	0.834528i	$0.685741\pi$
$60$	−1.00000	−0.129099
$61$	−10.3923	−1.33060	−0.665299	−	0.746577i	$-0.731696\pi$
$61$	−10.3923	−1.33060	−0.665299	+	0.746577i	$0.731696\pi$
$62$	−1.73205	−0.219971
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.46410	−0.795676
$67$	11.4641	1.40056	0.700281	−	0.713867i	$-0.253058\pi$
$67$	11.4641	1.40056	0.700281	+	0.713867i	$0.253058\pi$
$68$	4.00000	0.485071
$69$	3.73205	0.449286
$70$	2.00000	0.239046
$71$	−12.3923	−1.47070	−0.735348	−	0.677690i	$-0.762981\pi$
$71$	−12.3923	−1.47070	−0.735348	+	0.677690i	$0.762981\pi$
$72$	1.00000	0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	9.19615	1.06903
$75$	−1.00000	−0.115470
$76$	7.46410	0.856191
$77$	12.9282	1.47331
$78$	0	0
$79$	−13.9282	−1.56705	−0.783523	−	0.621363i	$-0.786579\pi$
$79$	−13.9282	−1.56705	−0.783523	+	0.621363i	$0.786579\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	8.92820	0.979998	0.489999	−	0.871723i	$-0.336997\pi$
$83$	8.92820	0.979998	0.489999	+	0.871723i	$0.336997\pi$
$84$	−2.00000	−0.218218
$85$	4.00000	0.433861
$86$	−11.9282	−1.28625
$87$	−0.267949	−0.0287272
$88$	6.46410	0.689076
$89$	−0.535898	−0.0568051	−0.0284026	−	0.999597i	$-0.509042\pi$
$89$	−0.535898	−0.0568051	−0.0284026	+	0.999597i	$0.509042\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−3.73205	−0.389093
$93$	1.73205	0.179605
$94$	3.53590	0.364700
$95$	7.46410	0.765801
$96$	−1.00000	−0.102062
$97$	0.535898	0.0544122	0.0272061	−	0.999630i	$-0.491339\pi$
$97$	0.535898	0.0544122	0.0272061	+	0.999630i	$0.491339\pi$
$98$	−3.00000	−0.303046
$99$	6.46410	0.649667
$100$	1.00000	0.100000
$101$	−2.92820	−0.291367	−0.145684	−	0.989331i	$-0.546538\pi$
$101$	−2.92820	−0.291367	−0.145684	+	0.989331i	$0.546538\pi$
$102$	−4.00000	−0.396059
$103$	11.8564	1.16825	0.584123	−	0.811665i	$-0.301438\pi$
$103$	11.8564	1.16825	0.584123	+	0.811665i	$0.301438\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0.928203	0.0901551
$107$	−7.85641	−0.759507	−0.379754	−	0.925088i	$-0.623991\pi$
$107$	−7.85641	−0.759507	−0.379754	+	0.925088i	$0.623991\pi$
$108$	−1.00000	−0.0962250
$109$	−15.8564	−1.51877	−0.759384	−	0.650643i	$-0.774500\pi$
$109$	−15.8564	−1.51877	−0.759384	+	0.650643i	$0.774500\pi$
$110$	6.46410	0.616328
$111$	−9.19615	−0.872860
$112$	2.00000	0.188982
$113$	0.803848	0.0756196	0.0378098	−	0.999285i	$-0.487962\pi$
$113$	0.803848	0.0756196	0.0378098	+	0.999285i	$0.487962\pi$
$114$	−7.46410	−0.699077
$115$	−3.73205	−0.348016
$116$	0.267949	0.0248785
$117$	0	0
$118$	−8.46410	−0.779184
$119$	8.00000	0.733359
$120$	−1.00000	−0.0912871
$121$	30.7846	2.79860
$122$	−10.3923	−0.940875
$123$	2.00000	0.180334
$124$	−1.73205	−0.155543
$125$	1.00000	0.0894427
$126$	2.00000	0.178174
$127$	4.92820	0.437307	0.218654	−	0.975803i	$-0.429834\pi$
$127$	4.92820	0.437307	0.218654	+	0.975803i	$0.429834\pi$
$128$	1.00000	0.0883883
$129$	11.9282	1.05022
$130$	0	0
$131$	−18.6603	−1.63035	−0.815177	−	0.579212i	$-0.803360\pi$
$131$	−18.6603	−1.63035	−0.815177	+	0.579212i	$0.803360\pi$
$132$	−6.46410	−0.562628
$133$	14.9282	1.29444
$134$	11.4641	0.990348
$135$	−1.00000	−0.0860663
$136$	4.00000	0.342997
$137$	2.46410	0.210522	0.105261	−	0.994445i	$-0.466432\pi$
$137$	2.46410	0.210522	0.105261	+	0.994445i	$0.466432\pi$
$138$	3.73205	0.317693
$139$	−12.9282	−1.09656	−0.548278	−	0.836296i	$-0.684716\pi$
$139$	−12.9282	−1.09656	−0.548278	+	0.836296i	$0.684716\pi$
$140$	2.00000	0.169031
$141$	−3.53590	−0.297776
$142$	−12.3923	−1.03994
$143$	0	0
$144$	1.00000	0.0833333
$145$	0.267949	0.0222520
$146$	−2.00000	−0.165521
$147$	3.00000	0.247436
$148$	9.19615	0.755919
$149$	−13.5359	−1.10890	−0.554452	−	0.832216i	$-0.687072\pi$
$149$	−13.5359	−1.10890	−0.554452	+	0.832216i	$0.687072\pi$
$150$	−1.00000	−0.0816497
$151$	10.3923	0.845714	0.422857	−	0.906196i	$-0.361027\pi$
$151$	10.3923	0.845714	0.422857	+	0.906196i	$0.361027\pi$
$152$	7.46410	0.605419
$153$	4.00000	0.323381
$154$	12.9282	1.04178
$155$	−1.73205	−0.139122
$156$	0	0
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	−13.9282	−1.10807
$159$	−0.928203	−0.0736113
$160$	1.00000	0.0790569
$161$	−7.46410	−0.588254
$162$	1.00000	0.0785674
$163$	−15.0526	−1.17901	−0.589504	−	0.807766i	$-0.700677\pi$
$163$	−15.0526	−1.17901	−0.589504	+	0.807766i	$0.700677\pi$
$164$	−2.00000	−0.156174
$165$	−6.46410	−0.503230
$166$	8.92820	0.692963
$167$	16.3205	1.26292	0.631459	−	0.775409i	$-0.282456\pi$
$167$	16.3205	1.26292	0.631459	+	0.775409i	$0.282456\pi$
$168$	−2.00000	−0.154303
$169$	0	0
$170$	4.00000	0.306786
$171$	7.46410	0.570794
$172$	−11.9282	−0.909517
$173$	−10.9282	−0.830856	−0.415428	−	0.909626i	$-0.636368\pi$
$173$	−10.9282	−0.830856	−0.415428	+	0.909626i	$0.636368\pi$
$174$	−0.267949	−0.0203132
$175$	2.00000	0.151186
$176$	6.46410	0.487250
$177$	8.46410	0.636201
$178$	−0.535898	−0.0401673
$179$	19.7321	1.47484	0.737421	−	0.675433i	$-0.236043\pi$
$179$	19.7321	1.47484	0.737421	+	0.675433i	$0.236043\pi$
$180$	1.00000	0.0745356
$181$	2.92820	0.217652	0.108826	−	0.994061i	$-0.465291\pi$
$181$	2.92820	0.217652	0.108826	+	0.994061i	$0.465291\pi$
$182$	0	0
$183$	10.3923	0.768221
$184$	−3.73205	−0.275130
$185$	9.19615	0.676115
$186$	1.73205	0.127000
$187$	25.8564	1.89081
$188$	3.53590	0.257882
$189$	−2.00000	−0.145479
$190$	7.46410	0.541503
$191$	21.4641	1.55309	0.776544	−	0.630063i	$-0.216971\pi$
$191$	21.4641	1.55309	0.776544	+	0.630063i	$0.216971\pi$
$192$	−1.00000	−0.0721688
$193$	11.3205	0.814868	0.407434	−	0.913235i	$-0.366424\pi$
$193$	11.3205	0.814868	0.407434	+	0.913235i	$0.366424\pi$
$194$	0.535898	0.0384753
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−4.39230	−0.312939	−0.156469	−	0.987683i	$-0.550011\pi$
$197$	−4.39230	−0.312939	−0.156469	+	0.987683i	$0.550011\pi$
$198$	6.46410	0.459384
$199$	5.07180	0.359530	0.179765	−	0.983710i	$-0.442466\pi$
$199$	5.07180	0.359530	0.179765	+	0.983710i	$0.442466\pi$
$200$	1.00000	0.0707107
$201$	−11.4641	−0.808615
$202$	−2.92820	−0.206028
$203$	0.535898	0.0376127
$204$	−4.00000	−0.280056
$205$	−2.00000	−0.139686
$206$	11.8564	0.826075
$207$	−3.73205	−0.259395
$208$	0	0
$209$	48.2487	3.33743
$210$	−2.00000	−0.138013
$211$	−11.3205	−0.779336	−0.389668	−	0.920955i	$-0.627410\pi$
$211$	−11.3205	−0.779336	−0.389668	+	0.920955i	$0.627410\pi$
$212$	0.928203	0.0637493
$213$	12.3923	0.849107
$214$	−7.85641	−0.537053
$215$	−11.9282	−0.813497
$216$	−1.00000	−0.0680414
$217$	−3.46410	−0.235159
$218$	−15.8564	−1.07393
$219$	2.00000	0.135147
$220$	6.46410	0.435810
$221$	0	0
$222$	−9.19615	−0.617205
$223$	20.5359	1.37519	0.687593	−	0.726097i	$-0.258667\pi$
$223$	20.5359	1.37519	0.687593	+	0.726097i	$0.258667\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	0.803848	0.0534711
$227$	−16.3923	−1.08800	−0.543998	−	0.839087i	$-0.683090\pi$
$227$	−16.3923	−1.08800	−0.543998	+	0.839087i	$0.683090\pi$
$228$	−7.46410	−0.494322
$229$	7.85641	0.519166	0.259583	−	0.965721i	$-0.416415\pi$
$229$	7.85641	0.519166	0.259583	+	0.965721i	$0.416415\pi$
$230$	−3.73205	−0.246084
$231$	−12.9282	−0.850613
$232$	0.267949	0.0175917
$233$	−6.12436	−0.401220	−0.200610	−	0.979671i	$-0.564292\pi$
$233$	−6.12436	−0.401220	−0.200610	+	0.979671i	$0.564292\pi$
$234$	0	0
$235$	3.53590	0.230657
$236$	−8.46410	−0.550966
$237$	13.9282	0.904734
$238$	8.00000	0.518563
$239$	−16.3923	−1.06033	−0.530165	−	0.847894i	$-0.677870\pi$
$239$	−16.3923	−1.06033	−0.530165	+	0.847894i	$0.677870\pi$
$240$	−1.00000	−0.0645497
$241$	−17.7321	−1.14222	−0.571111	−	0.820873i	$-0.693487\pi$
$241$	−17.7321	−1.14222	−0.571111	+	0.820873i	$0.693487\pi$
$242$	30.7846	1.97891
$243$	−1.00000	−0.0641500
$244$	−10.3923	−0.665299
$245$	−3.00000	−0.191663
$246$	2.00000	0.127515
$247$	0	0
$248$	−1.73205	−0.109985
$249$	−8.92820	−0.565802
$250$	1.00000	0.0632456
$251$	−15.7321	−0.992998	−0.496499	−	0.868037i	$-0.665381\pi$
$251$	−15.7321	−0.992998	−0.496499	+	0.868037i	$0.665381\pi$
$252$	2.00000	0.125988
$253$	−24.1244	−1.51669
$254$	4.92820	0.309223
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	5.33975	0.333084	0.166542	−	0.986034i	$-0.446740\pi$
$257$	5.33975	0.333084	0.166542	+	0.986034i	$0.446740\pi$
$258$	11.9282	0.742617
$259$	18.3923	1.14284
$260$	0	0
$261$	0.267949	0.0165856
$262$	−18.6603	−1.15283
$263$	6.12436	0.377644	0.188822	−	0.982011i	$-0.439533\pi$
$263$	6.12436	0.377644	0.188822	+	0.982011i	$0.439533\pi$
$264$	−6.46410	−0.397838
$265$	0.928203	0.0570191
$266$	14.9282	0.915307
$267$	0.535898	0.0327964
$268$	11.4641	0.700281
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	−1.00000	−0.0608581
$271$	1.19615	0.0726611	0.0363305	−	0.999340i	$-0.488433\pi$
$271$	1.19615	0.0726611	0.0363305	+	0.999340i	$0.488433\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	2.46410	0.148862
$275$	6.46410	0.389800
$276$	3.73205	0.224643
$277$	3.92820	0.236023	0.118011	−	0.993012i	$-0.462348\pi$
$277$	3.92820	0.236023	0.118011	+	0.993012i	$0.462348\pi$
$278$	−12.9282	−0.775382
$279$	−1.73205	−0.103695
$280$	2.00000	0.119523
$281$	8.92820	0.532612	0.266306	−	0.963889i	$-0.414197\pi$
$281$	8.92820	0.532612	0.266306	+	0.963889i	$0.414197\pi$
$282$	−3.53590	−0.210560
$283$	9.92820	0.590170	0.295085	−	0.955471i	$-0.404652\pi$
$283$	9.92820	0.590170	0.295085	+	0.955471i	$0.404652\pi$
$284$	−12.3923	−0.735348
$285$	−7.46410	−0.442135
$286$	0	0
$287$	−4.00000	−0.236113
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	0.267949	0.0157345
$291$	−0.535898	−0.0314149
$292$	−2.00000	−0.117041
$293$	31.8564	1.86107	0.930536	−	0.366202i	$-0.119342\pi$
$293$	31.8564	1.86107	0.930536	+	0.366202i	$0.119342\pi$
$294$	3.00000	0.174964
$295$	−8.46410	−0.492799
$296$	9.19615	0.534516
$297$	−6.46410	−0.375085
$298$	−13.5359	−0.784114
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−23.8564	−1.37506
$302$	10.3923	0.598010
$303$	2.92820	0.168221
$304$	7.46410	0.428096
$305$	−10.3923	−0.595062
$306$	4.00000	0.228665
$307$	−19.4641	−1.11087	−0.555437	−	0.831558i	$-0.687449\pi$
$307$	−19.4641	−1.11087	−0.555437	+	0.831558i	$0.687449\pi$
$308$	12.9282	0.736653
$309$	−11.8564	−0.674487
$310$	−1.73205	−0.0983739
$311$	28.3923	1.60998	0.804990	−	0.593288i	$-0.202171\pi$
$311$	28.3923	1.60998	0.804990	+	0.593288i	$0.202171\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	−5.00000	−0.282166
$315$	2.00000	0.112687
$316$	−13.9282	−0.783523
$317$	14.5359	0.816417	0.408209	−	0.912889i	$-0.366154\pi$
$317$	14.5359	0.816417	0.408209	+	0.912889i	$0.366154\pi$
$318$	−0.928203	−0.0520511
$319$	1.73205	0.0969762
$320$	1.00000	0.0559017
$321$	7.85641	0.438502
$322$	−7.46410	−0.415958
$323$	29.8564	1.66125
$324$	1.00000	0.0555556
$325$	0	0
$326$	−15.0526	−0.833684
$327$	15.8564	0.876861
$328$	−2.00000	−0.110432
$329$	7.07180	0.389881
$330$	−6.46410	−0.355837
$331$	−16.7846	−0.922566	−0.461283	−	0.887253i	$-0.652611\pi$
$331$	−16.7846	−0.922566	−0.461283	+	0.887253i	$0.652611\pi$
$332$	8.92820	0.489999
$333$	9.19615	0.503946
$334$	16.3205	0.893018
$335$	11.4641	0.626351
$336$	−2.00000	−0.109109
$337$	−9.32051	−0.507720	−0.253860	−	0.967241i	$-0.581700\pi$
$337$	−9.32051	−0.507720	−0.253860	+	0.967241i	$0.581700\pi$
$338$	0	0
$339$	−0.803848	−0.0436590
$340$	4.00000	0.216930
$341$	−11.1962	−0.606306
$342$	7.46410	0.403612
$343$	−20.0000	−1.07990
$344$	−11.9282	−0.643126
$345$	3.73205	0.200927
$346$	−10.9282	−0.587504
$347$	1.60770	0.0863056	0.0431528	−	0.999068i	$-0.486260\pi$
$347$	1.60770	0.0863056	0.0431528	+	0.999068i	$0.486260\pi$
$348$	−0.267949	−0.0143636
$349$	21.4641	1.14895	0.574474	−	0.818523i	$-0.305207\pi$
$349$	21.4641	1.14895	0.574474	+	0.818523i	$0.305207\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	6.46410	0.344538
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	8.46410	0.449862
$355$	−12.3923	−0.657715
$356$	−0.535898	−0.0284026
$357$	−8.00000	−0.423405
$358$	19.7321	1.04287
$359$	−5.07180	−0.267679	−0.133840	−	0.991003i	$-0.542731\pi$
$359$	−5.07180	−0.267679	−0.133840	+	0.991003i	$0.542731\pi$
$360$	1.00000	0.0527046
$361$	36.7128	1.93225
$362$	2.92820	0.153903
$363$	−30.7846	−1.61577
$364$	0	0
$365$	−2.00000	−0.104685
$366$	10.3923	0.543214
$367$	15.6077	0.814715	0.407358	−	0.913269i	$-0.366450\pi$
$367$	15.6077	0.814715	0.407358	+	0.913269i	$0.366450\pi$
$368$	−3.73205	−0.194547
$369$	−2.00000	−0.104116
$370$	9.19615	0.478085
$371$	1.85641	0.0963798
$372$	1.73205	0.0898027
$373$	−15.7846	−0.817296	−0.408648	−	0.912692i	$-0.634000\pi$
$373$	−15.7846	−0.817296	−0.408648	+	0.912692i	$0.634000\pi$
$374$	25.8564	1.33700
$375$	−1.00000	−0.0516398
$376$	3.53590	0.182350
$377$	0	0
$378$	−2.00000	−0.102869
$379$	−27.8564	−1.43089	−0.715444	−	0.698670i	$-0.753775\pi$
$379$	−27.8564	−1.43089	−0.715444	+	0.698670i	$0.753775\pi$
$380$	7.46410	0.382900
$381$	−4.92820	−0.252479
$382$	21.4641	1.09820
$383$	25.3923	1.29749	0.648743	−	0.761007i	$-0.275295\pi$
$383$	25.3923	1.29749	0.648743	+	0.761007i	$0.275295\pi$
$384$	−1.00000	−0.0510310
$385$	12.9282	0.658882
$386$	11.3205	0.576199
$387$	−11.9282	−0.606345
$388$	0.535898	0.0272061
$389$	23.7321	1.20326	0.601631	−	0.798774i	$-0.294518\pi$
$389$	23.7321	1.20326	0.601631	+	0.798774i	$0.294518\pi$
$390$	0	0
$391$	−14.9282	−0.754952
$392$	−3.00000	−0.151523
$393$	18.6603	0.941285
$394$	−4.39230	−0.221281
$395$	−13.9282	−0.700804
$396$	6.46410	0.324833
$397$	−12.1244	−0.608504	−0.304252	−	0.952592i	$-0.598406\pi$
$397$	−12.1244	−0.608504	−0.304252	+	0.952592i	$0.598406\pi$
$398$	5.07180	0.254226
$399$	−14.9282	−0.747345
$400$	1.00000	0.0500000
$401$	−32.0000	−1.59800	−0.799002	−	0.601329i	$-0.794638\pi$
$401$	−32.0000	−1.59800	−0.799002	+	0.601329i	$0.794638\pi$
$402$	−11.4641	−0.571777
$403$	0	0
$404$	−2.92820	−0.145684
$405$	1.00000	0.0496904
$406$	0.535898	0.0265962
$407$	59.4449	2.94657
$408$	−4.00000	−0.198030
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	−2.00000	−0.0987730
$411$	−2.46410	−0.121545
$412$	11.8564	0.584123
$413$	−16.9282	−0.832982
$414$	−3.73205	−0.183420
$415$	8.92820	0.438268
$416$	0	0
$417$	12.9282	0.633097
$418$	48.2487	2.35992
$419$	22.3923	1.09394	0.546968	−	0.837154i	$-0.315782\pi$
$419$	22.3923	1.09394	0.546968	+	0.837154i	$0.315782\pi$
$420$	−2.00000	−0.0975900
$421$	4.39230	0.214068	0.107034	−	0.994255i	$-0.465865\pi$
$421$	4.39230	0.214068	0.107034	+	0.994255i	$0.465865\pi$
$422$	−11.3205	−0.551074
$423$	3.53590	0.171921
$424$	0.928203	0.0450775
$425$	4.00000	0.194029
$426$	12.3923	0.600409
$427$	−20.7846	−1.00584
$428$	−7.85641	−0.379754
$429$	0	0
$430$	−11.9282	−0.575229
$431$	−28.3923	−1.36761	−0.683805	−	0.729665i	$-0.739676\pi$
$431$	−28.3923	−1.36761	−0.683805	+	0.729665i	$0.739676\pi$
$432$	−1.00000	−0.0481125
$433$	−19.3205	−0.928484	−0.464242	−	0.885708i	$-0.653673\pi$
$433$	−19.3205	−0.928484	−0.464242	+	0.885708i	$0.653673\pi$
$434$	−3.46410	−0.166282
$435$	−0.267949	−0.0128472
$436$	−15.8564	−0.759384
$437$	−27.8564	−1.33255
$438$	2.00000	0.0955637
$439$	17.8564	0.852240	0.426120	−	0.904667i	$-0.359880\pi$
$439$	17.8564	0.852240	0.426120	+	0.904667i	$0.359880\pi$
$440$	6.46410	0.308164
$441$	−3.00000	−0.142857
$442$	0	0
$443$	16.3923	0.778822	0.389411	−	0.921064i	$-0.372679\pi$
$443$	16.3923	0.778822	0.389411	+	0.921064i	$0.372679\pi$
$444$	−9.19615	−0.436430
$445$	−0.535898	−0.0254040
$446$	20.5359	0.972403
$447$	13.5359	0.640226
$448$	2.00000	0.0944911
$449$	15.7128	0.741533	0.370767	−	0.928726i	$-0.379095\pi$
$449$	15.7128	0.741533	0.370767	+	0.928726i	$0.379095\pi$
$450$	1.00000	0.0471405
$451$	−12.9282	−0.608765
$452$	0.803848	0.0378098
$453$	−10.3923	−0.488273
$454$	−16.3923	−0.769329
$455$	0	0
$456$	−7.46410	−0.349539
$457$	−24.5359	−1.14774	−0.573870	−	0.818946i	$-0.694559\pi$
$457$	−24.5359	−1.14774	−0.573870	+	0.818946i	$0.694559\pi$
$458$	7.85641	0.367106
$459$	−4.00000	−0.186704
$460$	−3.73205	−0.174008
$461$	0.464102	0.0216154	0.0108077	−	0.999942i	$-0.496560\pi$
$461$	0.464102	0.0216154	0.0108077	+	0.999942i	$0.496560\pi$
$462$	−12.9282	−0.601474
$463$	−7.07180	−0.328654	−0.164327	−	0.986406i	$-0.552545\pi$
$463$	−7.07180	−0.328654	−0.164327	+	0.986406i	$0.552545\pi$
$464$	0.267949	0.0124392
$465$	1.73205	0.0803219
$466$	−6.12436	−0.283705
$467$	15.8564	0.733747	0.366873	−	0.930271i	$-0.380428\pi$
$467$	15.8564	0.733747	0.366873	+	0.930271i	$0.380428\pi$
$468$	0	0
$469$	22.9282	1.05873
$470$	3.53590	0.163099
$471$	5.00000	0.230388
$472$	−8.46410	−0.389592
$473$	−77.1051	−3.54530
$474$	13.9282	0.639744
$475$	7.46410	0.342476
$476$	8.00000	0.366679
$477$	0.928203	0.0424995
$478$	−16.3923	−0.749767
$479$	5.46410	0.249661	0.124831	−	0.992178i	$-0.460161\pi$
$479$	5.46410	0.249661	0.124831	+	0.992178i	$0.460161\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−17.7321	−0.807673
$483$	7.46410	0.339628
$484$	30.7846	1.39930
$485$	0.535898	0.0243339
$486$	−1.00000	−0.0453609
$487$	23.1769	1.05025	0.525123	−	0.851026i	$-0.324019\pi$
$487$	23.1769	1.05025	0.525123	+	0.851026i	$0.324019\pi$
$488$	−10.3923	−0.470438
$489$	15.0526	0.680700
$490$	−3.00000	−0.135526
$491$	17.3205	0.781664	0.390832	−	0.920462i	$-0.372187\pi$
$491$	17.3205	0.781664	0.390832	+	0.920462i	$0.372187\pi$
$492$	2.00000	0.0901670
$493$	1.07180	0.0482713
$494$	0	0
$495$	6.46410	0.290540
$496$	−1.73205	−0.0777714
$497$	−24.7846	−1.11174
$498$	−8.92820	−0.400082
$499$	6.53590	0.292587	0.146293	−	0.989241i	$-0.453266\pi$
$499$	6.53590	0.292587	0.146293	+	0.989241i	$0.453266\pi$
$500$	1.00000	0.0447214
$501$	−16.3205	−0.729147
$502$	−15.7321	−0.702156
$503$	−31.1769	−1.39011	−0.695055	−	0.718957i	$-0.744620\pi$
$503$	−31.1769	−1.39011	−0.695055	+	0.718957i	$0.744620\pi$
$504$	2.00000	0.0890871
$505$	−2.92820	−0.130303
$506$	−24.1244	−1.07246
$507$	0	0
$508$	4.92820	0.218654
$509$	−1.39230	−0.0617128	−0.0308564	−	0.999524i	$-0.509823\pi$
$509$	−1.39230	−0.0617128	−0.0308564	+	0.999524i	$0.509823\pi$
$510$	−4.00000	−0.177123
$511$	−4.00000	−0.176950
$512$	1.00000	0.0441942
$513$	−7.46410	−0.329548
$514$	5.33975	0.235526
$515$	11.8564	0.522456
$516$	11.9282	0.525110
$517$	22.8564	1.00522
$518$	18.3923	0.808111
$519$	10.9282	0.479695
$520$	0	0
$521$	−17.3205	−0.758825	−0.379413	−	0.925228i	$-0.623874\pi$
$521$	−17.3205	−0.758825	−0.379413	+	0.925228i	$0.623874\pi$
$522$	0.267949	0.0117278
$523$	11.7846	0.515305	0.257653	−	0.966238i	$-0.417051\pi$
$523$	11.7846	0.515305	0.257653	+	0.966238i	$0.417051\pi$
$524$	−18.6603	−0.815177
$525$	−2.00000	−0.0872872
$526$	6.12436	0.267035
$527$	−6.92820	−0.301797
$528$	−6.46410	−0.281314
$529$	−9.07180	−0.394426
$530$	0.928203	0.0403186
$531$	−8.46410	−0.367311
$532$	14.9282	0.647220
$533$	0	0
$534$	0.535898	0.0231906
$535$	−7.85641	−0.339662
$536$	11.4641	0.495174
$537$	−19.7321	−0.851501
$538$	12.0000	0.517357
$539$	−19.3923	−0.835286
$540$	−1.00000	−0.0430331
$541$	−26.9282	−1.15773	−0.578867	−	0.815422i	$-0.696505\pi$
$541$	−26.9282	−1.15773	−0.578867	+	0.815422i	$0.696505\pi$
$542$	1.19615	0.0513791
$543$	−2.92820	−0.125661
$544$	4.00000	0.171499
$545$	−15.8564	−0.679214
$546$	0	0
$547$	22.9282	0.980339	0.490170	−	0.871627i	$-0.336935\pi$
$547$	22.9282	0.980339	0.490170	+	0.871627i	$0.336935\pi$
$548$	2.46410	0.105261
$549$	−10.3923	−0.443533
$550$	6.46410	0.275630
$551$	2.00000	0.0852029
$552$	3.73205	0.158847
$553$	−27.8564	−1.18457
$554$	3.92820	0.166893
$555$	−9.19615	−0.390355
$556$	−12.9282	−0.548278
$557$	17.7128	0.750516	0.375258	−	0.926920i	$-0.377554\pi$
$557$	17.7128	0.750516	0.375258	+	0.926920i	$0.377554\pi$
$558$	−1.73205	−0.0733236
$559$	0	0
$560$	2.00000	0.0845154
$561$	−25.8564	−1.09166
$562$	8.92820	0.376614
$563$	4.67949	0.197217	0.0986085	−	0.995126i	$-0.468561\pi$
$563$	4.67949	0.197217	0.0986085	+	0.995126i	$0.468561\pi$
$564$	−3.53590	−0.148888
$565$	0.803848	0.0338181
$566$	9.92820	0.417314
$567$	2.00000	0.0839921
$568$	−12.3923	−0.519970
$569$	−29.3205	−1.22918	−0.614590	−	0.788847i	$-0.710678\pi$
$569$	−29.3205	−1.22918	−0.614590	+	0.788847i	$0.710678\pi$
$570$	−7.46410	−0.312637
$571$	17.1769	0.718832	0.359416	−	0.933178i	$-0.382976\pi$
$571$	17.1769	0.718832	0.359416	+	0.933178i	$0.382976\pi$
$572$	0	0
$573$	−21.4641	−0.896676
$574$	−4.00000	−0.166957
$575$	−3.73205	−0.155637
$576$	1.00000	0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	−1.00000	−0.0415945
$579$	−11.3205	−0.470464
$580$	0.267949	0.0111260
$581$	17.8564	0.740809
$582$	−0.535898	−0.0222137
$583$	6.00000	0.248495
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	31.8564	1.31598
$587$	2.39230	0.0987410	0.0493705	−	0.998781i	$-0.484278\pi$
$587$	2.39230	0.0987410	0.0493705	+	0.998781i	$0.484278\pi$
$588$	3.00000	0.123718
$589$	−12.9282	−0.532697
$590$	−8.46410	−0.348462
$591$	4.39230	0.180675
$592$	9.19615	0.377960
$593$	−45.1051	−1.85225	−0.926123	−	0.377223i	$-0.876879\pi$
$593$	−45.1051	−1.85225	−0.926123	+	0.377223i	$0.876879\pi$
$594$	−6.46410	−0.265225
$595$	8.00000	0.327968
$596$	−13.5359	−0.554452
$597$	−5.07180	−0.207575
$598$	0	0
$599$	−10.3923	−0.424618	−0.212309	−	0.977203i	$-0.568098\pi$
$599$	−10.3923	−0.424618	−0.212309	+	0.977203i	$0.568098\pi$
$600$	−1.00000	−0.0408248
$601$	−19.7846	−0.807031	−0.403516	−	0.914973i	$-0.632212\pi$
$601$	−19.7846	−0.807031	−0.403516	+	0.914973i	$0.632212\pi$
$602$	−23.8564	−0.972315
$603$	11.4641	0.466854
$604$	10.3923	0.422857
$605$	30.7846	1.25157
$606$	2.92820	0.118950
$607$	−19.1769	−0.778367	−0.389183	−	0.921160i	$-0.627243\pi$
$607$	−19.1769	−0.778367	−0.389183	+	0.921160i	$0.627243\pi$
$608$	7.46410	0.302709
$609$	−0.535898	−0.0217157
$610$	−10.3923	−0.420772
$611$	0	0
$612$	4.00000	0.161690
$613$	39.0526	1.57732	0.788659	−	0.614831i	$-0.210776\pi$
$613$	39.0526	1.57732	0.788659	+	0.614831i	$0.210776\pi$
$614$	−19.4641	−0.785507
$615$	2.00000	0.0806478
$616$	12.9282	0.520892
$617$	−22.4641	−0.904371	−0.452185	−	0.891924i	$-0.649355\pi$
$617$	−22.4641	−0.904371	−0.452185	+	0.891924i	$0.649355\pi$
$618$	−11.8564	−0.476935
$619$	24.2487	0.974638	0.487319	−	0.873224i	$-0.337975\pi$
$619$	24.2487	0.974638	0.487319	+	0.873224i	$0.337975\pi$
$620$	−1.73205	−0.0695608
$621$	3.73205	0.149762
$622$	28.3923	1.13843
$623$	−1.07180	−0.0429406
$624$	0	0
$625$	1.00000	0.0400000
$626$	−28.0000	−1.11911
$627$	−48.2487	−1.92687
$628$	−5.00000	−0.199522
$629$	36.7846	1.46670
$630$	2.00000	0.0796819
$631$	−31.4641	−1.25257	−0.626283	−	0.779596i	$-0.715425\pi$
$631$	−31.4641	−1.25257	−0.626283	+	0.779596i	$0.715425\pi$
$632$	−13.9282	−0.554034
$633$	11.3205	0.449950
$634$	14.5359	0.577294
$635$	4.92820	0.195570
$636$	−0.928203	−0.0368057
$637$	0	0
$638$	1.73205	0.0685725
$639$	−12.3923	−0.490232
$640$	1.00000	0.0395285
$641$	−0.143594	−0.00567160	−0.00283580	−	0.999996i	$-0.500903\pi$
$641$	−0.143594	−0.00567160	−0.00283580	+	0.999996i	$0.500903\pi$
$642$	7.85641	0.310068
$643$	−20.5359	−0.809857	−0.404928	−	0.914348i	$-0.632704\pi$
$643$	−20.5359	−0.809857	−0.404928	+	0.914348i	$0.632704\pi$
$644$	−7.46410	−0.294127
$645$	11.9282	0.469673
$646$	29.8564	1.17468
$647$	13.3205	0.523683	0.261842	−	0.965111i	$-0.415670\pi$
$647$	13.3205	0.523683	0.261842	+	0.965111i	$0.415670\pi$
$648$	1.00000	0.0392837
$649$	−54.7128	−2.14767
$650$	0	0
$651$	3.46410	0.135769
$652$	−15.0526	−0.589504
$653$	4.24871	0.166265	0.0831325	−	0.996539i	$-0.473508\pi$
$653$	4.24871	0.166265	0.0831325	+	0.996539i	$0.473508\pi$
$654$	15.8564	0.620035
$655$	−18.6603	−0.729116
$656$	−2.00000	−0.0780869
$657$	−2.00000	−0.0780274
$658$	7.07180	0.275687
$659$	−0.267949	−0.0104378	−0.00521891	−	0.999986i	$-0.501661\pi$
$659$	−0.267949	−0.0104378	−0.00521891	+	0.999986i	$0.501661\pi$
$660$	−6.46410	−0.251615
$661$	−8.67949	−0.337593	−0.168797	−	0.985651i	$-0.553988\pi$
$661$	−8.67949	−0.337593	−0.168797	+	0.985651i	$0.553988\pi$
$662$	−16.7846	−0.652352
$663$	0	0
$664$	8.92820	0.346481
$665$	14.9282	0.578891
$666$	9.19615	0.356344
$667$	−1.00000	−0.0387202
$668$	16.3205	0.631459
$669$	−20.5359	−0.793964
$670$	11.4641	0.442897
$671$	−67.1769	−2.59334
$672$	−2.00000	−0.0771517
$673$	−32.0000	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$673$	−32.0000	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$674$	−9.32051	−0.359013
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	32.3923	1.24494	0.622469	−	0.782645i	$-0.286130\pi$
$677$	32.3923	1.24494	0.622469	+	0.782645i	$0.286130\pi$
$678$	−0.803848	−0.0308716
$679$	1.07180	0.0411318
$680$	4.00000	0.153393
$681$	16.3923	0.628154
$682$	−11.1962	−0.428723
$683$	40.7846	1.56058	0.780290	−	0.625418i	$-0.215072\pi$
$683$	40.7846	1.56058	0.780290	+	0.625418i	$0.215072\pi$
$684$	7.46410	0.285397
$685$	2.46410	0.0941485
$686$	−20.0000	−0.763604
$687$	−7.85641	−0.299741
$688$	−11.9282	−0.454758
$689$	0	0
$690$	3.73205	0.142077
$691$	27.1769	1.03386	0.516929	−	0.856028i	$-0.327075\pi$
$691$	27.1769	1.03386	0.516929	+	0.856028i	$0.327075\pi$
$692$	−10.9282	−0.415428
$693$	12.9282	0.491102
$694$	1.60770	0.0610273
$695$	−12.9282	−0.490395
$696$	−0.267949	−0.0101566
$697$	−8.00000	−0.303022
$698$	21.4641	0.812428
$699$	6.12436	0.231644
$700$	2.00000	0.0755929
$701$	0.267949	0.0101203	0.00506015	−	0.999987i	$-0.498389\pi$
$701$	0.267949	0.0101203	0.00506015	+	0.999987i	$0.498389\pi$
$702$	0	0
$703$	68.6410	2.58884
$704$	6.46410	0.243625
$705$	−3.53590	−0.133170
$706$	−2.00000	−0.0752710
$707$	−5.85641	−0.220253
$708$	8.46410	0.318100
$709$	22.9282	0.861087	0.430543	−	0.902570i	$-0.358322\pi$
$709$	22.9282	0.861087	0.430543	+	0.902570i	$0.358322\pi$
$710$	−12.3923	−0.465075
$711$	−13.9282	−0.522348
$712$	−0.535898	−0.0200836
$713$	6.46410	0.242083
$714$	−8.00000	−0.299392
$715$	0	0
$716$	19.7321	0.737421
$717$	16.3923	0.612182
$718$	−5.07180	−0.189278
$719$	34.6410	1.29189	0.645946	−	0.763383i	$-0.276463\pi$
$719$	34.6410	1.29189	0.645946	+	0.763383i	$0.276463\pi$
$720$	1.00000	0.0372678
$721$	23.7128	0.883111
$722$	36.7128	1.36631
$723$	17.7321	0.659462
$724$	2.92820	0.108826
$725$	0.267949	0.00995138
$726$	−30.7846	−1.14252
$727$	31.7128	1.17616	0.588082	−	0.808802i	$-0.299883\pi$
$727$	31.7128	1.17616	0.588082	+	0.808802i	$0.299883\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	−47.7128	−1.76472
$732$	10.3923	0.384111
$733$	−50.9282	−1.88108	−0.940538	−	0.339688i	$-0.889678\pi$
$733$	−50.9282	−1.88108	−0.940538	+	0.339688i	$0.889678\pi$
$734$	15.6077	0.576091
$735$	3.00000	0.110657
$736$	−3.73205	−0.137565
$737$	74.1051	2.72970
$738$	−2.00000	−0.0736210
$739$	−19.3205	−0.710716	−0.355358	−	0.934730i	$-0.615641\pi$
$739$	−19.3205	−0.710716	−0.355358	+	0.934730i	$0.615641\pi$
$740$	9.19615	0.338057
$741$	0	0
$742$	1.85641	0.0681508
$743$	40.4641	1.48448	0.742242	−	0.670132i	$-0.233763\pi$
$743$	40.4641	1.48448	0.742242	+	0.670132i	$0.233763\pi$
$744$	1.73205	0.0635001
$745$	−13.5359	−0.495917
$746$	−15.7846	−0.577916
$747$	8.92820	0.326666
$748$	25.8564	0.945404
$749$	−15.7128	−0.574134
$750$	−1.00000	−0.0365148
$751$	−14.0718	−0.513487	−0.256744	−	0.966480i	$-0.582650\pi$
$751$	−14.0718	−0.513487	−0.256744	+	0.966480i	$0.582650\pi$
$752$	3.53590	0.128941
$753$	15.7321	0.573308
$754$	0	0
$755$	10.3923	0.378215
$756$	−2.00000	−0.0727393
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−27.8564	−1.01179
$759$	24.1244	0.875659
$760$	7.46410	0.270751
$761$	−5.07180	−0.183852	−0.0919262	−	0.995766i	$-0.529302\pi$
$761$	−5.07180	−0.183852	−0.0919262	+	0.995766i	$0.529302\pi$
$762$	−4.92820	−0.178530
$763$	−31.7128	−1.14808
$764$	21.4641	0.776544
$765$	4.00000	0.144620
$766$	25.3923	0.917461
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	11.5885	0.417890	0.208945	−	0.977927i	$-0.432997\pi$
$769$	11.5885	0.417890	0.208945	+	0.977927i	$0.432997\pi$
$770$	12.9282	0.465900
$771$	−5.33975	−0.192306
$772$	11.3205	0.407434
$773$	27.7128	0.996761	0.498380	−	0.866959i	$-0.333928\pi$
$773$	27.7128	0.996761	0.498380	+	0.866959i	$0.333928\pi$
$774$	−11.9282	−0.428750
$775$	−1.73205	−0.0622171
$776$	0.535898	0.0192376
$777$	−18.3923	−0.659820
$778$	23.7321	0.850835
$779$	−14.9282	−0.534858
$780$	0	0
$781$	−80.1051	−2.86639
$782$	−14.9282	−0.533831
$783$	−0.267949	−0.00957572
$784$	−3.00000	−0.107143
$785$	−5.00000	−0.178458
$786$	18.6603	0.665589
$787$	−1.73205	−0.0617409	−0.0308705	−	0.999523i	$-0.509828\pi$
$787$	−1.73205	−0.0617409	−0.0308705	+	0.999523i	$0.509828\pi$
$788$	−4.39230	−0.156469
$789$	−6.12436	−0.218033
$790$	−13.9282	−0.495543
$791$	1.60770	0.0571631
$792$	6.46410	0.229692
$793$	0	0
$794$	−12.1244	−0.430277
$795$	−0.928203	−0.0329200
$796$	5.07180	0.179765
$797$	10.1436	0.359305	0.179652	−	0.983730i	$-0.442503\pi$
$797$	10.1436	0.359305	0.179652	+	0.983730i	$0.442503\pi$
$798$	−14.9282	−0.528453
$799$	14.1436	0.500364
$800$	1.00000	0.0353553
$801$	−0.535898	−0.0189350
$802$	−32.0000	−1.12996
$803$	−12.9282	−0.456226
$804$	−11.4641	−0.404308
$805$	−7.46410	−0.263075
$806$	0	0
$807$	−12.0000	−0.422420
$808$	−2.92820	−0.103014
$809$	−42.7846	−1.50423	−0.752113	−	0.659034i	$-0.770965\pi$
$809$	−42.7846	−1.50423	−0.752113	+	0.659034i	$0.770965\pi$
$810$	1.00000	0.0351364
$811$	40.7846	1.43214	0.716071	−	0.698028i	$-0.245939\pi$
$811$	40.7846	1.43214	0.716071	+	0.698028i	$0.245939\pi$
$812$	0.535898	0.0188063
$813$	−1.19615	−0.0419509
$814$	59.4449	2.08354
$815$	−15.0526	−0.527268
$816$	−4.00000	−0.140028
$817$	−89.0333	−3.11488
$818$	4.00000	0.139857
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	−14.6077	−0.509812	−0.254906	−	0.966966i	$-0.582045\pi$
$821$	−14.6077	−0.509812	−0.254906	+	0.966966i	$0.582045\pi$
$822$	−2.46410	−0.0859454
$823$	−39.1769	−1.36562	−0.682811	−	0.730595i	$-0.739243\pi$
$823$	−39.1769	−1.36562	−0.682811	+	0.730595i	$0.739243\pi$
$824$	11.8564	0.413037
$825$	−6.46410	−0.225051
$826$	−16.9282	−0.589008
$827$	−17.3205	−0.602293	−0.301147	−	0.953578i	$-0.597369\pi$
$827$	−17.3205	−0.602293	−0.301147	+	0.953578i	$0.597369\pi$
$828$	−3.73205	−0.129698
$829$	−16.5359	−0.574315	−0.287158	−	0.957883i	$-0.592710\pi$
$829$	−16.5359	−0.574315	−0.287158	+	0.957883i	$0.592710\pi$
$830$	8.92820	0.309902
$831$	−3.92820	−0.136268
$832$	0	0
$833$	−12.0000	−0.415775
$834$	12.9282	0.447667
$835$	16.3205	0.564794
$836$	48.2487	1.66872
$837$	1.73205	0.0598684
$838$	22.3923	0.773529
$839$	43.5692	1.50418	0.752088	−	0.659062i	$-0.229047\pi$
$839$	43.5692	1.50418	0.752088	+	0.659062i	$0.229047\pi$
$840$	−2.00000	−0.0690066
$841$	−28.9282	−0.997524
$842$	4.39230	0.151369
$843$	−8.92820	−0.307504
$844$	−11.3205	−0.389668
$845$	0	0
$846$	3.53590	0.121567
$847$	61.5692	2.11554
$848$	0.928203	0.0318746
$849$	−9.92820	−0.340735
$850$	4.00000	0.137199
$851$	−34.3205	−1.17649
$852$	12.3923	0.424553
$853$	−43.8372	−1.50096	−0.750478	−	0.660895i	$-0.770177\pi$
$853$	−43.8372	−1.50096	−0.750478	+	0.660895i	$0.770177\pi$
$854$	−20.7846	−0.711235
$855$	7.46410	0.255267
$856$	−7.85641	−0.268526
$857$	−24.5167	−0.837473	−0.418737	−	0.908108i	$-0.637527\pi$
$857$	−24.5167	−0.837473	−0.418737	+	0.908108i	$0.637527\pi$
$858$	0	0
$859$	−35.1769	−1.20022	−0.600110	−	0.799917i	$-0.704877\pi$
$859$	−35.1769	−1.20022	−0.600110	+	0.799917i	$0.704877\pi$
$860$	−11.9282	−0.406748
$861$	4.00000	0.136320
$862$	−28.3923	−0.967046
$863$	35.5359	1.20966	0.604828	−	0.796356i	$-0.293242\pi$
$863$	35.5359	1.20966	0.604828	+	0.796356i	$0.293242\pi$
$864$	−1.00000	−0.0340207
$865$	−10.9282	−0.371570
$866$	−19.3205	−0.656538
$867$	1.00000	0.0339618
$868$	−3.46410	−0.117579
$869$	−90.0333	−3.05417
$870$	−0.267949	−0.00908433
$871$	0	0
$872$	−15.8564	−0.536966
$873$	0.535898	0.0181374
$874$	−27.8564	−0.942257
$875$	2.00000	0.0676123
$876$	2.00000	0.0675737
$877$	−3.87564	−0.130871	−0.0654356	−	0.997857i	$-0.520844\pi$
$877$	−3.87564	−0.130871	−0.0654356	+	0.997857i	$0.520844\pi$
$878$	17.8564	0.602625
$879$	−31.8564	−1.07449
$880$	6.46410	0.217905
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	−3.00000	−0.101015
$883$	−29.9282	−1.00716	−0.503582	−	0.863947i	$-0.667985\pi$
$883$	−29.9282	−1.00716	−0.503582	+	0.863947i	$0.667985\pi$
$884$	0	0
$885$	8.46410	0.284518
$886$	16.3923	0.550710
$887$	14.1244	0.474249	0.237125	−	0.971479i	$-0.423795\pi$
$887$	14.1244	0.474249	0.237125	+	0.971479i	$0.423795\pi$
$888$	−9.19615	−0.308603
$889$	9.85641	0.330573
$890$	−0.535898	−0.0179634
$891$	6.46410	0.216556
$892$	20.5359	0.687593
$893$	26.3923	0.883185
$894$	13.5359	0.452708
$895$	19.7321	0.659570
$896$	2.00000	0.0668153
$897$	0	0
$898$	15.7128	0.524343
$899$	−0.464102	−0.0154787
$900$	1.00000	0.0333333
$901$	3.71281	0.123692
$902$	−12.9282	−0.430462
$903$	23.8564	0.793891
$904$	0.803848	0.0267356
$905$	2.92820	0.0973368
$906$	−10.3923	−0.345261
$907$	20.8564	0.692526	0.346263	−	0.938138i	$-0.387451\pi$
$907$	20.8564	0.692526	0.346263	+	0.938138i	$0.387451\pi$
$908$	−16.3923	−0.543998
$909$	−2.92820	−0.0971224
$910$	0	0
$911$	24.2487	0.803396	0.401698	−	0.915772i	$-0.368420\pi$
$911$	24.2487	0.803396	0.401698	+	0.915772i	$0.368420\pi$
$912$	−7.46410	−0.247161
$913$	57.7128	1.91002
$914$	−24.5359	−0.811575
$915$	10.3923	0.343559
$916$	7.85641	0.259583
$917$	−37.3205	−1.23243
$918$	−4.00000	−0.132020
$919$	14.6410	0.482963	0.241481	−	0.970405i	$-0.422367\pi$
$919$	14.6410	0.482963	0.241481	+	0.970405i	$0.422367\pi$
$920$	−3.73205	−0.123042
$921$	19.4641	0.641364
$922$	0.464102	0.0152844
$923$	0	0
$924$	−12.9282	−0.425307
$925$	9.19615	0.302368
$926$	−7.07180	−0.232394
$927$	11.8564	0.389415
$928$	0.267949	0.00879586
$929$	−6.14359	−0.201565	−0.100782	−	0.994908i	$-0.532135\pi$
$929$	−6.14359	−0.201565	−0.100782	+	0.994908i	$0.532135\pi$
$930$	1.73205	0.0567962
$931$	−22.3923	−0.733878
$932$	−6.12436	−0.200610
$933$	−28.3923	−0.929522
$934$	15.8564	0.518837
$935$	25.8564	0.845595
$936$	0	0
$937$	−24.6410	−0.804987	−0.402493	−	0.915423i	$-0.631856\pi$
$937$	−24.6410	−0.804987	−0.402493	+	0.915423i	$0.631856\pi$
$938$	22.9282	0.748632
$939$	28.0000	0.913745
$940$	3.53590	0.115328
$941$	−14.7846	−0.481965	−0.240982	−	0.970530i	$-0.577470\pi$
$941$	−14.7846	−0.481965	−0.240982	+	0.970530i	$0.577470\pi$
$942$	5.00000	0.162909
$943$	7.46410	0.243065
$944$	−8.46410	−0.275483
$945$	−2.00000	−0.0650600
$946$	−77.1051	−2.50690
$947$	9.46410	0.307542	0.153771	−	0.988107i	$-0.450858\pi$
$947$	9.46410	0.307542	0.153771	+	0.988107i	$0.450858\pi$
$948$	13.9282	0.452367
$949$	0	0
$950$	7.46410	0.242167
$951$	−14.5359	−0.471359
$952$	8.00000	0.259281
$953$	12.2679	0.397398	0.198699	−	0.980061i	$-0.436328\pi$
$953$	12.2679	0.397398	0.198699	+	0.980061i	$0.436328\pi$
$954$	0.928203	0.0300517
$955$	21.4641	0.694562
$956$	−16.3923	−0.530165
$957$	−1.73205	−0.0559893
$958$	5.46410	0.176537
$959$	4.92820	0.159140
$960$	−1.00000	−0.0322749
$961$	−28.0000	−0.903226
$962$	0	0
$963$	−7.85641	−0.253169
$964$	−17.7321	−0.571111
$965$	11.3205	0.364420
$966$	7.46410	0.240154
$967$	−41.4641	−1.33340	−0.666698	−	0.745328i	$-0.732293\pi$
$967$	−41.4641	−1.33340	−0.666698	+	0.745328i	$0.732293\pi$
$968$	30.7846	0.989455
$969$	−29.8564	−0.959126
$970$	0.535898	0.0172067
$971$	4.53590	0.145564	0.0727820	−	0.997348i	$-0.476812\pi$
$971$	4.53590	0.145564	0.0727820	+	0.997348i	$0.476812\pi$
$972$	−1.00000	−0.0320750
$973$	−25.8564	−0.828918
$974$	23.1769	0.742636
$975$	0	0
$976$	−10.3923	−0.332650
$977$	−12.6077	−0.403356	−0.201678	−	0.979452i	$-0.564639\pi$
$977$	−12.6077	−0.403356	−0.201678	+	0.979452i	$0.564639\pi$
$978$	15.0526	0.481328
$979$	−3.46410	−0.110713
$980$	−3.00000	−0.0958315
$981$	−15.8564	−0.506256
$982$	17.3205	0.552720
$983$	36.6077	1.16760	0.583802	−	0.811896i	$-0.301564\pi$
$983$	36.6077	1.16760	0.583802	+	0.811896i	$0.301564\pi$
$984$	2.00000	0.0637577
$985$	−4.39230	−0.139950
$986$	1.07180	0.0341330
$987$	−7.07180	−0.225098
$988$	0	0
$989$	44.5167	1.41555
$990$	6.46410	0.205443
$991$	−52.8564	−1.67904	−0.839520	−	0.543329i	$-0.817163\pi$
$991$	−52.8564	−1.67904	−0.839520	+	0.543329i	$0.817163\pi$
$992$	−1.73205	−0.0549927
$993$	16.7846	0.532643
$994$	−24.7846	−0.786120
$995$	5.07180	0.160787
$996$	−8.92820	−0.282901
$997$	35.5692	1.12649	0.563244	−	0.826290i	$-0.309553\pi$
$997$	35.5692	1.12649	0.563244	+	0.826290i	$0.309553\pi$
$998$	6.53590	0.206890
$999$	−9.19615	−0.290953

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bg.1.2		2
13.5	odd	4		5070.2.b.o.1351.2		4
13.6	odd	12		390.2.bb.b.361.1	yes	4
13.8	odd	4		5070.2.b.o.1351.3		4
13.11	odd	12		390.2.bb.b.121.1	✓	4
13.12	even	2		5070.2.a.y.1.1		2
39.11	even	12		1170.2.bs.e.901.2		4
39.32	even	12		1170.2.bs.e.361.2		4
65.19	odd	12		1950.2.bc.b.751.2		4
65.24	odd	12		1950.2.bc.b.901.2		4
65.32	even	12		1950.2.y.c.49.2		4
65.37	even	12		1950.2.y.f.199.1		4
65.58	even	12		1950.2.y.f.49.1		4
65.63	even	12		1950.2.y.c.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.b.121.1	✓	4	13.11	odd	12
390.2.bb.b.361.1	yes	4	13.6	odd	12
1170.2.bs.e.361.2		4	39.32	even	12
1170.2.bs.e.901.2		4	39.11	even	12
1950.2.y.c.49.2		4	65.32	even	12
1950.2.y.c.199.2		4	65.63	even	12
1950.2.y.f.49.1		4	65.58	even	12
1950.2.y.f.199.1		4	65.37	even	12
1950.2.bc.b.751.2		4	65.19	odd	12
1950.2.bc.b.901.2		4	65.24	odd	12
5070.2.a.y.1.1		2	13.12	even	2
5070.2.a.bg.1.2		2	1.1	even	1	trivial
5070.2.b.o.1351.2		4	13.5	odd	4
5070.2.b.o.1351.3		4	13.8	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +6.46410 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +7.46410 q^{19} +1.00000 q^{20} -2.00000 q^{21} +6.46410 q^{22} -3.73205 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} +0.267949 q^{29} -1.00000 q^{30} -1.73205 q^{31} +1.00000 q^{32} -6.46410 q^{33} +4.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +9.19615 q^{37} +7.46410 q^{38} +1.00000 q^{40} -2.00000 q^{41} -2.00000 q^{42} -11.9282 q^{43} +6.46410 q^{44} +1.00000 q^{45} -3.73205 q^{46} +3.53590 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +0.928203 q^{53} -1.00000 q^{54} +6.46410 q^{55} +2.00000 q^{56} -7.46410 q^{57} +0.267949 q^{58} -8.46410 q^{59} -1.00000 q^{60} -10.3923 q^{61} -1.73205 q^{62} +2.00000 q^{63} +1.00000 q^{64} -6.46410 q^{66} +11.4641 q^{67} +4.00000 q^{68} +3.73205 q^{69} +2.00000 q^{70} -12.3923 q^{71} +1.00000 q^{72} -2.00000 q^{73} +9.19615 q^{74} -1.00000 q^{75} +7.46410 q^{76} +12.9282 q^{77} -13.9282 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +8.92820 q^{83} -2.00000 q^{84} +4.00000 q^{85} -11.9282 q^{86} -0.267949 q^{87} +6.46410 q^{88} -0.535898 q^{89} +1.00000 q^{90} -3.73205 q^{92} +1.73205 q^{93} +3.53590 q^{94} +7.46410 q^{95} -1.00000 q^{96} +0.535898 q^{97} -3.00000 q^{98} +6.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 4 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 6 q^{11} - 2 q^{12} + 4 q^{14} - 2 q^{15} + 2 q^{16} + 8 q^{17} + 2 q^{18} + 8 q^{19} + 2 q^{20} - 4 q^{21} + 6 q^{22} - 4 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{27} + 4 q^{28} + 4 q^{29} - 2 q^{30} + 2 q^{32} - 6 q^{33} + 8 q^{34} + 4 q^{35} + 2 q^{36} + 8 q^{37} + 8 q^{38} + 2 q^{40} - 4 q^{41} - 4 q^{42} - 10 q^{43} + 6 q^{44} + 2 q^{45} - 4 q^{46} + 14 q^{47} - 2 q^{48} - 6 q^{49} + 2 q^{50} - 8 q^{51} - 12 q^{53} - 2 q^{54} + 6 q^{55} + 4 q^{56} - 8 q^{57} + 4 q^{58} - 10 q^{59} - 2 q^{60} + 4 q^{63} + 2 q^{64} - 6 q^{66} + 16 q^{67} + 8 q^{68} + 4 q^{69} + 4 q^{70} - 4 q^{71} + 2 q^{72} - 4 q^{73} + 8 q^{74} - 2 q^{75} + 8 q^{76} + 12 q^{77} - 14 q^{79} + 2 q^{80} + 2 q^{81} - 4 q^{82} + 4 q^{83} - 4 q^{84} + 8 q^{85} - 10 q^{86} - 4 q^{87} + 6 q^{88} - 8 q^{89} + 2 q^{90} - 4 q^{92} + 14 q^{94} + 8 q^{95} - 2 q^{96} + 8 q^{97} - 6 q^{98} + 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 4 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 6 * q^11 - 2 * q^12 + 4 * q^14 - 2 * q^15 + 2 * q^16 + 8 * q^17 + 2 * q^18 + 8 * q^19 + 2 * q^20 - 4 * q^21 + 6 * q^22 - 4 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^27 + 4 * q^28 + 4 * q^29 - 2 * q^30 + 2 * q^32 - 6 * q^33 + 8 * q^34 + 4 * q^35 + 2 * q^36 + 8 * q^37 + 8 * q^38 + 2 * q^40 - 4 * q^41 - 4 * q^42 - 10 * q^43 + 6 * q^44 + 2 * q^45 - 4 * q^46 + 14 * q^47 - 2 * q^48 - 6 * q^49 + 2 * q^50 - 8 * q^51 - 12 * q^53 - 2 * q^54 + 6 * q^55 + 4 * q^56 - 8 * q^57 + 4 * q^58 - 10 * q^59 - 2 * q^60 + 4 * q^63 + 2 * q^64 - 6 * q^66 + 16 * q^67 + 8 * q^68 + 4 * q^69 + 4 * q^70 - 4 * q^71 + 2 * q^72 - 4 * q^73 + 8 * q^74 - 2 * q^75 + 8 * q^76 + 12 * q^77 - 14 * q^79 + 2 * q^80 + 2 * q^81 - 4 * q^82 + 4 * q^83 - 4 * q^84 + 8 * q^85 - 10 * q^86 - 4 * q^87 + 6 * q^88 - 8 * q^89 + 2 * q^90 - 4 * q^92 + 14 * q^94 + 8 * q^95 - 2 * q^96 + 8 * q^97 - 6 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more