Properties

Label 3.14.a.a.1.1
Level $3$
Weight $14$
Character 3.1
Self dual yes
Analytic conductor $3.217$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3,14,Mod(1,3)] 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("3.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.21692786856\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3.1

$q$-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)
 
\(f(q)\) \(=\) \(q-12.0000 q^{2} -729.000 q^{3} -8048.00 q^{4} -30210.0 q^{5} +8748.00 q^{6} +235088. q^{7} +194880. q^{8} +531441. q^{9} +362520. q^{10} -1.11829e7 q^{11} +5.86699e6 q^{12} +8.04961e6 q^{13} -2.82106e6 q^{14} +2.20231e7 q^{15} +6.35907e7 q^{16} -1.17495e8 q^{17} -6.37729e6 q^{18} -2.14061e8 q^{19} +2.43130e8 q^{20} -1.71379e8 q^{21} +1.34195e8 q^{22} +8.30556e8 q^{23} -1.42068e8 q^{24} -3.08059e8 q^{25} -9.65954e7 q^{26} -3.87420e8 q^{27} -1.89199e9 q^{28} -1.25240e9 q^{29} -2.64277e8 q^{30} +6.15935e9 q^{31} -2.35954e9 q^{32} +8.15234e9 q^{33} +1.40994e9 q^{34} -7.10201e9 q^{35} -4.27704e9 q^{36} -5.49819e9 q^{37} +2.56874e9 q^{38} -5.86817e9 q^{39} -5.88732e9 q^{40} -4.67869e9 q^{41} +2.05655e9 q^{42} +7.11501e9 q^{43} +9.00000e10 q^{44} -1.60548e10 q^{45} -9.96667e9 q^{46} -2.95288e10 q^{47} -4.63576e10 q^{48} -4.16226e10 q^{49} +3.69671e9 q^{50} +8.56536e10 q^{51} -6.47833e10 q^{52} -2.04125e11 q^{53} +4.64905e9 q^{54} +3.37836e11 q^{55} +4.58139e10 q^{56} +1.56051e11 q^{57} +1.50288e10 q^{58} -2.99098e10 q^{59} -1.77242e11 q^{60} -1.34392e11 q^{61} -7.39122e10 q^{62} +1.24935e11 q^{63} -4.92620e11 q^{64} -2.43179e11 q^{65} -9.78281e10 q^{66} +3.48519e11 q^{67} +9.45597e11 q^{68} -6.05475e11 q^{69} +8.52241e10 q^{70} +1.31434e12 q^{71} +1.03567e11 q^{72} -1.17888e12 q^{73} +6.59783e10 q^{74} +2.24575e11 q^{75} +1.72277e12 q^{76} -2.62897e12 q^{77} +7.04180e10 q^{78} -1.07242e12 q^{79} -1.92107e12 q^{80} +2.82430e11 q^{81} +5.61443e10 q^{82} +1.12403e12 q^{83} +1.37926e12 q^{84} +3.54951e12 q^{85} -8.53802e10 q^{86} +9.13000e11 q^{87} -2.17933e12 q^{88} +2.23561e12 q^{89} +1.92658e11 q^{90} +1.89237e12 q^{91} -6.68431e12 q^{92} -4.49017e12 q^{93} +3.54345e11 q^{94} +6.46679e12 q^{95} +1.72011e12 q^{96} -1.42153e13 q^{97} +4.99472e11 q^{98} -5.94306e12 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −12.0000 −0.132583 −0.0662913 − 0.997800i \(-0.521117\pi\)
−0.0662913 + 0.997800i \(0.521117\pi\)
\(3\) −729.000 −0.577350
\(4\) −8048.00 −0.982422
\(5\) −30210.0 −0.864661 −0.432330 − 0.901715i \(-0.642309\pi\)
−0.432330 + 0.901715i \(0.642309\pi\)
\(6\) 8748.00 0.0765466
\(7\) 235088. 0.755254 0.377627 − 0.925958i \(-0.376740\pi\)
0.377627 + 0.925958i \(0.376740\pi\)
\(8\) 194880. 0.262834
\(9\) 531441. 0.333333
\(10\) 362520. 0.114639
\(11\) −1.11829e7 −1.90328 −0.951639 − 0.307218i \(-0.900602\pi\)
−0.951639 + 0.307218i \(0.900602\pi\)
\(12\) 5.86699e6 0.567202
\(13\) 8.04961e6 0.462534 0.231267 − 0.972890i \(-0.425713\pi\)
0.231267 + 0.972890i \(0.425713\pi\)
\(14\) −2.82106e6 −0.100134
\(15\) 2.20231e7 0.499212
\(16\) 6.35907e7 0.947575
\(17\) −1.17495e8 −1.18059 −0.590296 − 0.807187i \(-0.700989\pi\)
−0.590296 + 0.807187i \(0.700989\pi\)
\(18\) −6.37729e6 −0.0441942
\(19\) −2.14061e8 −1.04385 −0.521927 − 0.852990i \(-0.674787\pi\)
−0.521927 + 0.852990i \(0.674787\pi\)
\(20\) 2.43130e8 0.849462
\(21\) −1.71379e8 −0.436046
\(22\) 1.34195e8 0.252341
\(23\) 8.30556e8 1.16987 0.584935 − 0.811080i \(-0.301120\pi\)
0.584935 + 0.811080i \(0.301120\pi\)
\(24\) −1.42068e8 −0.151748
\(25\) −3.08059e8 −0.252362
\(26\) −9.65954e7 −0.0613239
\(27\) −3.87420e8 −0.192450
\(28\) −1.89199e9 −0.741978
\(29\) −1.25240e9 −0.390981 −0.195491 − 0.980706i \(-0.562630\pi\)
−0.195491 + 0.980706i \(0.562630\pi\)
\(30\) −2.64277e8 −0.0661868
\(31\) 6.15935e9 1.24648 0.623238 − 0.782032i \(-0.285817\pi\)
0.623238 + 0.782032i \(0.285817\pi\)
\(32\) −2.35954e9 −0.388466
\(33\) 8.15234e9 1.09886
\(34\) 1.40994e9 0.156526
\(35\) −7.10201e9 −0.653039
\(36\) −4.27704e9 −0.327474
\(37\) −5.49819e9 −0.352297 −0.176148 − 0.984364i \(-0.556364\pi\)
−0.176148 + 0.984364i \(0.556364\pi\)
\(38\) 2.56874e9 0.138397
\(39\) −5.86817e9 −0.267044
\(40\) −5.88732e9 −0.227263
\(41\) −4.67869e9 −0.153826 −0.0769129 − 0.997038i \(-0.524506\pi\)
−0.0769129 + 0.997038i \(0.524506\pi\)
\(42\) 2.05655e9 0.0578121
\(43\) 7.11501e9 0.171645 0.0858224 − 0.996310i \(-0.472648\pi\)
0.0858224 + 0.996310i \(0.472648\pi\)
\(44\) 9.00000e10 1.86982
\(45\) −1.60548e10 −0.288220
\(46\) −9.96667e9 −0.155104
\(47\) −2.95288e10 −0.399585 −0.199793 − 0.979838i \(-0.564027\pi\)
−0.199793 + 0.979838i \(0.564027\pi\)
\(48\) −4.63576e10 −0.547082
\(49\) −4.16226e10 −0.429591
\(50\) 3.69671e9 0.0334588
\(51\) 8.56536e10 0.681615
\(52\) −6.47833e10 −0.454403
\(53\) −2.04125e11 −1.26504 −0.632518 − 0.774545i \(-0.717979\pi\)
−0.632518 + 0.774545i \(0.717979\pi\)
\(54\) 4.64905e9 0.0255155
\(55\) 3.37836e11 1.64569
\(56\) 4.58139e10 0.198507
\(57\) 1.56051e11 0.602670
\(58\) 1.50288e10 0.0518373
\(59\) −2.99098e10 −0.0923157 −0.0461579 − 0.998934i \(-0.514698\pi\)
−0.0461579 + 0.998934i \(0.514698\pi\)
\(60\) −1.77242e11 −0.490437
\(61\) −1.34392e11 −0.333987 −0.166993 − 0.985958i \(-0.553406\pi\)
−0.166993 + 0.985958i \(0.553406\pi\)
\(62\) −7.39122e10 −0.165261
\(63\) 1.24935e11 0.251751
\(64\) −4.92620e11 −0.896071
\(65\) −2.43179e11 −0.399935
\(66\) −9.78281e10 −0.145689
\(67\) 3.48519e11 0.470695 0.235348 − 0.971911i \(-0.424377\pi\)
0.235348 + 0.971911i \(0.424377\pi\)
\(68\) 9.45597e11 1.15984
\(69\) −6.05475e11 −0.675425
\(70\) 8.52241e10 0.0865815
\(71\) 1.31434e12 1.21766 0.608831 − 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(72\) 1.03567e11 0.0876115
\(73\) −1.17888e12 −0.911737 −0.455868 − 0.890047i \(-0.650671\pi\)
−0.455868 + 0.890047i \(0.650671\pi\)
\(74\) 6.59783e10 0.0467084
\(75\) 2.24575e11 0.145701
\(76\) 1.72277e12 1.02551
\(77\) −2.62897e12 −1.43746
\(78\) 7.04180e10 0.0354054
\(79\) −1.07242e12 −0.496351 −0.248176 − 0.968715i \(-0.579831\pi\)
−0.248176 + 0.968715i \(0.579831\pi\)
\(80\) −1.92107e12 −0.819330
\(81\) 2.82430e11 0.111111
\(82\) 5.61443e10 0.0203946
\(83\) 1.12403e12 0.377371 0.188685 − 0.982038i \(-0.439577\pi\)
0.188685 + 0.982038i \(0.439577\pi\)
\(84\) 1.37926e12 0.428381
\(85\) 3.54951e12 1.02081
\(86\) −8.53802e10 −0.0227571
\(87\) 9.13000e11 0.225733
\(88\) −2.17933e12 −0.500247
\(89\) 2.23561e12 0.476827 0.238414 − 0.971164i \(-0.423373\pi\)
0.238414 + 0.971164i \(0.423373\pi\)
\(90\) 1.92658e11 0.0382130
\(91\) 1.89237e12 0.349330
\(92\) −6.68431e12 −1.14931
\(93\) −4.49017e12 −0.719653
\(94\) 3.54345e11 0.0529780
\(95\) 6.46679e12 0.902580
\(96\) 1.72011e12 0.224281
\(97\) −1.42153e13 −1.73276 −0.866380 − 0.499385i \(-0.833559\pi\)
−0.866380 + 0.499385i \(0.833559\pi\)
\(98\) 4.99472e11 0.0569563
\(99\) −5.94306e12 −0.634426
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 3.14.a.a.1.1 ✓ 1
3.2 odd 2 9.14.a.a.1.1 1
4.3 odd 2 48.14.a.c.1.1 1
5.2 odd 4 75.14.b.b.49.1 2
5.3 odd 4 75.14.b.b.49.2 2
5.4 even 2 75.14.a.a.1.1 1
7.6 odd 2 147.14.a.a.1.1 1
8.3 odd 2 192.14.a.e.1.1 1
8.5 even 2 192.14.a.j.1.1 1
12.11 even 2 144.14.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.14.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.14.a.a.1.1 1 3.2 odd 2
48.14.a.c.1.1 1 4.3 odd 2
75.14.a.a.1.1 1 5.4 even 2
75.14.b.b.49.1 2 5.2 odd 4
75.14.b.b.49.2 2 5.3 odd 4
144.14.a.k.1.1 1 12.11 even 2
147.14.a.a.1.1 1 7.6 odd 2
192.14.a.e.1.1 1 8.3 odd 2
192.14.a.j.1.1 1 8.5 even 2