Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1666,2,Mod(1,1666)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1666.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1666, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1666 = 2 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1666.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,-2,1,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.3030769767\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 238)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1666.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -4.00000 q^{11} -2.00000 q^{12} +4.00000 q^{13} +8.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -4.00000 q^{20} +4.00000 q^{22} +2.00000 q^{24} +11.0000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +6.00000 q^{29} -8.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +8.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} -6.00000 q^{38} -8.00000 q^{39} +4.00000 q^{40} -6.00000 q^{41} -4.00000 q^{44} -4.00000 q^{45} -4.00000 q^{47} -2.00000 q^{48} -11.0000 q^{50} -2.00000 q^{51} +4.00000 q^{52} +14.0000 q^{53} -4.00000 q^{54} +16.0000 q^{55} -12.0000 q^{57} -6.00000 q^{58} +6.00000 q^{59} +8.00000 q^{60} +12.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -16.0000 q^{65} -8.00000 q^{66} +4.00000 q^{67} +1.00000 q^{68} -8.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +10.0000 q^{74} -22.0000 q^{75} +6.00000 q^{76} +8.00000 q^{78} -4.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} -10.0000 q^{83} -4.00000 q^{85} -12.0000 q^{87} +4.00000 q^{88} -10.0000 q^{89} +4.00000 q^{90} +8.00000 q^{93} +4.00000 q^{94} -24.0000 q^{95} +2.00000 q^{96} -6.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.00000 −1.78885 −0.894427 − 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −2.00000 −0.577350
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 8.00000 2.06559
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −4.00000 −0.894427
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 2.00000 0.408248
\(25\) 11.0000 2.20000
\(26\) −4.00000 −0.784465
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −8.00000 −1.46059
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 8.00000 1.39262
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −6.00000 −0.973329
\(39\) −8.00000 −1.28103
\(40\) 4.00000 0.632456
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −4.00000 −0.603023
\(45\) −4.00000 −0.596285
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −2.00000 −0.288675
\(49\) 0 0
\(50\) −11.0000 −1.55563
\(51\) −2.00000 −0.280056
\(52\) 4.00000 0.554700
\(53\) 14.0000 1.92305 0.961524 − 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) −4.00000 −0.544331
\(55\) 16.0000 2.15744
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) −6.00000 −0.787839
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 8.00000 1.03280
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.0000 −1.98456
\(66\) −8.00000 −0.984732
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 10.0000 1.16248
\(75\) −22.0000 −2.54034
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 8.00000 0.905822
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −4.00000 −0.447214
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) −10.0000 −1.09764 −0.548821 − 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 4.00000 0.426401
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 4.00000 0.421637
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 4.00000 0.412568
\(95\) −24.0000 −2.46235
\(96\) 2.00000 0.204124
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1666.2.a.b.1.1 1
7.6 odd 2 238.2.a.b.1.1 ✓ 1
21.20 even 2 2142.2.a.l.1.1 1
28.27 even 2 1904.2.a.b.1.1 1
35.34 odd 2 5950.2.a.k.1.1 1
56.13 odd 2 7616.2.a.a.1.1 1
56.27 even 2 7616.2.a.i.1.1 1
119.118 odd 2 4046.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
238.2.a.b.1.1 ✓ 1 7.6 odd 2
1666.2.a.b.1.1 1 1.1 even 1 trivial
1904.2.a.b.1.1 1 28.27 even 2
2142.2.a.l.1.1 1 21.20 even 2
4046.2.a.b.1.1 1 119.118 odd 2
5950.2.a.k.1.1 1 35.34 odd 2
7616.2.a.a.1.1 1 56.13 odd 2
7616.2.a.i.1.1 1 56.27 even 2