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Properties

Label 12.9.c.a.5.1
Level $12$
Weight $9$
Character 12.5
Self dual yes
Analytic conductor $4.889$
Analytic rank $0$
Dimension $1$
CM discriminant -3
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] = [12,9,Mod(5,12)] 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("12.5"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 12.c (of order \(2\), degree \(1\), minimal)

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: \(4.88854332073\)
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 5.1
Character \(\chi\) \(=\) 12.5

$q$-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)
 
\(f(q)\) \(=\) \(q+81.0000 q^{3} +4034.00 q^{7} +6561.00 q^{9} -35806.0 q^{13} -258526. q^{19} +326754. q^{21} +390625. q^{25} +531441. q^{27} -1.80941e6 q^{31} +503522. q^{37} -2.90029e6 q^{39} +3.49219e6 q^{43} +1.05084e7 q^{49} -2.09406e7 q^{57} -2.38265e7 q^{61} +2.64671e7 q^{63} -5.42141e6 q^{67} +1.61693e7 q^{73} +3.16406e7 q^{75} -1.88870e7 q^{79} +4.30467e7 q^{81} -1.44441e8 q^{91} -1.46562e8 q^{93} +1.76908e8 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 81.0000 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 4034.00 1.68013 0.840067 − 0.542483i \(-0.182516\pi\)
0.840067 + 0.542483i \(0.182516\pi\)
\(8\) 0 0
\(9\) 6561.00 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −35806.0 −1.25367 −0.626834 − 0.779153i \(-0.715650\pi\)
−0.626834 + 0.779153i \(0.715650\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −258526. −1.98376 −0.991882 − 0.127165i \(-0.959412\pi\)
−0.991882 + 0.127165i \(0.959412\pi\)
\(20\) 0 0
\(21\) 326754. 1.68013
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 390625. 1.00000
\(26\) 0 0
\(27\) 531441. 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.80941e6 −1.95925 −0.979624 − 0.200842i \(-0.935632\pi\)
−0.979624 + 0.200842i \(0.935632\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 503522. 0.268665 0.134333 − 0.990936i \(-0.457111\pi\)
0.134333 + 0.990936i \(0.457111\pi\)
\(38\) 0 0
\(39\) −2.90029e6 −1.25367
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 3.49219e6 1.02147 0.510734 − 0.859739i \(-0.329374\pi\)
0.510734 + 0.859739i \(0.329374\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.05084e7 1.82285
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.09406e7 −1.98376
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −2.38265e7 −1.72084 −0.860422 − 0.509583i \(-0.829800\pi\)
−0.860422 + 0.509583i \(0.829800\pi\)
\(62\) 0 0
\(63\) 2.64671e7 1.68013
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.42141e6 −0.269037 −0.134519 − 0.990911i \(-0.542949\pi\)
−0.134519 + 0.990911i \(0.542949\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.61693e7 0.569376 0.284688 − 0.958620i \(-0.408110\pi\)
0.284688 + 0.958620i \(0.408110\pi\)
\(74\) 0 0
\(75\) 3.16406e7 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.88870e7 −0.484904 −0.242452 − 0.970163i \(-0.577952\pi\)
−0.242452 + 0.970163i \(0.577952\pi\)
\(80\) 0 0
\(81\) 4.30467e7 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.44441e8 −2.10633
\(92\) 0 0
\(93\) −1.46562e8 −1.95925
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.76908e8 1.99830 0.999150 − 0.0412262i \(-0.0131264\pi\)
0.999150 + 0.0412262i \(0.0131264\pi\)
\(98\) 0 0
\(99\) 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 12.9.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.9.c.a.5.1 ✓ 1
4.3 odd 2 48.9.e.a.17.1 1
5.2 odd 4 300.9.b.b.149.1 2
5.3 odd 4 300.9.b.b.149.2 2
5.4 even 2 300.9.g.a.101.1 1
8.3 odd 2 192.9.e.b.65.1 1
8.5 even 2 192.9.e.a.65.1 1
9.2 odd 6 324.9.g.a.53.1 2
9.4 even 3 324.9.g.a.269.1 2
9.5 odd 6 324.9.g.a.269.1 2
9.7 even 3 324.9.g.a.53.1 2
12.11 even 2 48.9.e.a.17.1 1
15.2 even 4 300.9.b.b.149.1 2
15.8 even 4 300.9.b.b.149.2 2
15.14 odd 2 300.9.g.a.101.1 1
24.5 odd 2 192.9.e.a.65.1 1
24.11 even 2 192.9.e.b.65.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.9.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.9.c.a.5.1 ✓ 1 3.2 odd 2 CM
48.9.e.a.17.1 1 4.3 odd 2
48.9.e.a.17.1 1 12.11 even 2
192.9.e.a.65.1 1 8.5 even 2
192.9.e.a.65.1 1 24.5 odd 2
192.9.e.b.65.1 1 8.3 odd 2
192.9.e.b.65.1 1 24.11 even 2
300.9.b.b.149.1 2 5.2 odd 4
300.9.b.b.149.1 2 15.2 even 4
300.9.b.b.149.2 2 5.3 odd 4
300.9.b.b.149.2 2 15.8 even 4
300.9.g.a.101.1 1 5.4 even 2
300.9.g.a.101.1 1 15.14 odd 2
324.9.g.a.53.1 2 9.2 odd 6
324.9.g.a.53.1 2 9.7 even 3
324.9.g.a.269.1 2 9.4 even 3
324.9.g.a.269.1 2 9.5 odd 6