Properties

Label 15.11.d.a.14.1
Level $15$
Weight $11$
Character 15.14
Self dual yes
Analytic conductor $9.530$
Analytic rank $0$
Dimension $1$
CM discriminant -15
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [15,11,Mod(14,15)] 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("15.14"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 15.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-61] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.53035879011\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 14.1
Character \(\chi\) \(=\) 15.14

$q$-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)
 
\(f(q)\) \(=\) \(q-61.0000 q^{2} +243.000 q^{3} +2697.00 q^{4} -3125.00 q^{5} -14823.0 q^{6} -102053. q^{8} +59049.0 q^{9} +190625. q^{10} +655371. q^{12} -759375. q^{15} +3.46350e6 q^{16} +2.41921e6 q^{17} -3.60199e6 q^{18} -269302. q^{19} -8.42812e6 q^{20} +1.09507e7 q^{23} -2.47989e7 q^{24} +9.76562e6 q^{25} +1.43489e7 q^{27} +4.63219e7 q^{30} +9.19680e6 q^{31} -1.06772e8 q^{32} -1.47572e8 q^{34} +1.59255e8 q^{36} +1.64274e7 q^{38} +3.18916e8 q^{40} -1.84528e8 q^{45} -6.67992e8 q^{46} +3.11808e8 q^{47} +8.41632e8 q^{48} +2.82475e8 q^{49} -5.95703e8 q^{50} +5.87869e8 q^{51} -8.36230e8 q^{53} -8.75283e8 q^{54} -6.54404e7 q^{57} -2.04803e9 q^{60} -4.78013e8 q^{61} -5.61005e8 q^{62} +2.96643e9 q^{64} +6.52462e9 q^{68} +2.66102e9 q^{69} -6.02613e9 q^{72} +2.37305e9 q^{75} -7.26307e8 q^{76} -1.24515e9 q^{79} -1.08235e10 q^{80} +3.48678e9 q^{81} +2.64223e9 q^{83} -7.56004e9 q^{85} +1.12562e10 q^{90} +2.95340e10 q^{92} +2.23482e9 q^{93} -1.90203e10 q^{94} +8.41569e8 q^{95} -2.59455e10 q^{96} -1.72310e10 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −61.0000 −1.90625 −0.953125 − 0.302577i \(-0.902153\pi\)
−0.953125 + 0.302577i \(0.902153\pi\)
\(3\) 243.000 1.00000
\(4\) 2697.00 2.63379
\(5\) −3125.00 −1.00000
\(6\) −14823.0 −1.90625
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −102053. −3.11441
\(9\) 59049.0 1.00000
\(10\) 190625. 1.90625
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 655371. 2.63379
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −759375. −1.00000
\(16\) 3.46350e6 3.30306
\(17\) 2.41921e6 1.70384 0.851922 − 0.523669i \(-0.175437\pi\)
0.851922 + 0.523669i \(0.175437\pi\)
\(18\) −3.60199e6 −1.90625
\(19\) −269302. −0.108761 −0.0543803 − 0.998520i \(-0.517318\pi\)
−0.0543803 + 0.998520i \(0.517318\pi\)
\(20\) −8.42812e6 −2.63379
\(21\) 0 0
\(22\) 0 0
\(23\) 1.09507e7 1.70138 0.850692 − 0.525665i \(-0.176183\pi\)
0.850692 + 0.525665i \(0.176183\pi\)
\(24\) −2.47989e7 −3.11441
\(25\) 9.76562e6 1.00000
\(26\) 0 0
\(27\) 1.43489e7 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 4.63219e7 1.90625
\(31\) 9.19680e6 0.321239 0.160620 − 0.987016i \(-0.448651\pi\)
0.160620 + 0.987016i \(0.448651\pi\)
\(32\) −1.06772e8 −3.18204
\(33\) 0 0
\(34\) −1.47572e8 −3.24795
\(35\) 0 0
\(36\) 1.59255e8 2.63379
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.64274e7 0.207325
\(39\) 0 0
\(40\) 3.18916e8 3.11441
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −1.84528e8 −1.00000
\(46\) −6.67992e8 −3.24326
\(47\) 3.11808e8 1.35956 0.679779 − 0.733417i \(-0.262076\pi\)
0.679779 + 0.733417i \(0.262076\pi\)
\(48\) 8.41632e8 3.30306
\(49\) 2.82475e8 1.00000
\(50\) −5.95703e8 −1.90625
\(51\) 5.87869e8 1.70384
\(52\) 0 0
\(53\) −8.36230e8 −1.99961 −0.999807 − 0.0196489i \(-0.993745\pi\)
−0.999807 + 0.0196489i \(0.993745\pi\)
\(54\) −8.75283e8 −1.90625
\(55\) 0 0
\(56\) 0 0
\(57\) −6.54404e7 −0.108761
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −2.04803e9 −2.63379
\(61\) −4.78013e8 −0.565967 −0.282983 − 0.959125i \(-0.591324\pi\)
−0.282983 + 0.959125i \(0.591324\pi\)
\(62\) −5.61005e8 −0.612362
\(63\) 0 0
\(64\) 2.96643e9 2.76271
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 6.52462e9 4.48756
\(69\) 2.66102e9 1.70138
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −6.02613e9 −3.11441
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 2.37305e9 1.00000
\(76\) −7.26307e8 −0.286452
\(77\) 0 0
\(78\) 0 0
\(79\) −1.24515e9 −0.404656 −0.202328 − 0.979318i \(-0.564851\pi\)
−0.202328 + 0.979318i \(0.564851\pi\)
\(80\) −1.08235e10 −3.30306
\(81\) 3.48678e9 1.00000
\(82\) 0 0
\(83\) 2.64223e9 0.670781 0.335390 − 0.942079i \(-0.391132\pi\)
0.335390 + 0.942079i \(0.391132\pi\)
\(84\) 0 0
\(85\) −7.56004e9 −1.70384
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 1.12562e10 1.90625
\(91\) 0 0
\(92\) 2.95340e10 4.48108
\(93\) 2.23482e9 0.321239
\(94\) −1.90203e10 −2.59166
\(95\) 8.41569e8 0.108761
\(96\) −2.59455e10 −3.18204
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −1.72310e10 −1.90625
\(99\) 0 0
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 15.11.d.a.14.1 ✓ 1
3.2 odd 2 15.11.d.b.14.1 yes 1
5.2 odd 4 75.11.c.c.26.1 2
5.3 odd 4 75.11.c.c.26.2 2
5.4 even 2 15.11.d.b.14.1 yes 1
15.2 even 4 75.11.c.c.26.2 2
15.8 even 4 75.11.c.c.26.1 2
15.14 odd 2 CM 15.11.d.a.14.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.11.d.a.14.1 ✓ 1 1.1 even 1 trivial
15.11.d.a.14.1 ✓ 1 15.14 odd 2 CM
15.11.d.b.14.1 yes 1 3.2 odd 2
15.11.d.b.14.1 yes 1 5.4 even 2
75.11.c.c.26.1 2 5.2 odd 4
75.11.c.c.26.1 2 15.8 even 4
75.11.c.c.26.2 2 5.3 odd 4
75.11.c.c.26.2 2 15.2 even 4