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Properties

Label 324.1.d.b.163.1
Level $324$
Weight $1$
Character 324.163
Self dual yes
Analytic conductor $0.162$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -4
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] = [324,1,Mod(163,324)] 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("324.163"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 324.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1] 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: \(0.161697064093\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.324.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.324.1
Stark unit: Root of $x^{3} - 57x^{2} - 15x - 1$

Embedding invariants

Embedding label 163.1
Character \(\chi\) \(=\) 324.163

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{13} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{20} -1.00000 q^{26} -1.00000 q^{29} +1.00000 q^{32} -1.00000 q^{34} -1.00000 q^{37} -1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{49} -1.00000 q^{52} +2.00000 q^{53} -1.00000 q^{58} -1.00000 q^{61} +1.00000 q^{64} +1.00000 q^{65} -1.00000 q^{68} -1.00000 q^{73} -1.00000 q^{74} -1.00000 q^{80} +2.00000 q^{82} +1.00000 q^{85} -1.00000 q^{89} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 1.00000
\(3\) 0 0
\(4\) 1.00000 1.00000
\(5\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000 1.00000
\(9\) 0 0
\(10\) −1.00000 −1.00000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −1.00000 −1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −1.00000
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.00000 1.00000
\(33\) 0 0
\(34\) −1.00000 −1.00000
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −1.00000
\(41\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −1.00000 −1.00000
\(53\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −1.00000 −1.00000
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 1.00000 1.00000
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −1.00000 −1.00000
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) −1.00000 −1.00000
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −1.00000 −1.00000
\(81\) 0 0
\(82\) 2.00000 2.00000
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 1.00000 1.00000
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(98\) 1.00000 1.00000
\(99\) 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 324.1.d.b.163.1 yes 1
3.2 odd 2 324.1.d.a.163.1 ✓ 1
4.3 odd 2 CM 324.1.d.b.163.1 yes 1
9.2 odd 6 324.1.f.b.271.1 2
9.4 even 3 324.1.f.a.55.1 2
9.5 odd 6 324.1.f.b.55.1 2
9.7 even 3 324.1.f.a.271.1 2
12.11 even 2 324.1.d.a.163.1 ✓ 1
27.2 odd 18 2916.1.j.e.2755.1 6
27.4 even 9 2916.1.j.d.1135.1 6
27.5 odd 18 2916.1.j.e.2107.1 6
27.7 even 9 2916.1.j.d.1783.1 6
27.11 odd 18 2916.1.j.e.811.1 6
27.13 even 9 2916.1.j.d.163.1 6
27.14 odd 18 2916.1.j.e.163.1 6
27.16 even 9 2916.1.j.d.811.1 6
27.20 odd 18 2916.1.j.e.1783.1 6
27.22 even 9 2916.1.j.d.2107.1 6
27.23 odd 18 2916.1.j.e.1135.1 6
27.25 even 9 2916.1.j.d.2755.1 6
36.7 odd 6 324.1.f.a.271.1 2
36.11 even 6 324.1.f.b.271.1 2
36.23 even 6 324.1.f.b.55.1 2
36.31 odd 6 324.1.f.a.55.1 2
108.7 odd 18 2916.1.j.d.1783.1 6
108.11 even 18 2916.1.j.e.811.1 6
108.23 even 18 2916.1.j.e.1135.1 6
108.31 odd 18 2916.1.j.d.1135.1 6
108.43 odd 18 2916.1.j.d.811.1 6
108.47 even 18 2916.1.j.e.1783.1 6
108.59 even 18 2916.1.j.e.2107.1 6
108.67 odd 18 2916.1.j.d.163.1 6
108.79 odd 18 2916.1.j.d.2755.1 6
108.83 even 18 2916.1.j.e.2755.1 6
108.95 even 18 2916.1.j.e.163.1 6
108.103 odd 18 2916.1.j.d.2107.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.1.d.a.163.1 ✓ 1 3.2 odd 2
324.1.d.a.163.1 ✓ 1 12.11 even 2
324.1.d.b.163.1 yes 1 1.1 even 1 trivial
324.1.d.b.163.1 yes 1 4.3 odd 2 CM
324.1.f.a.55.1 2 9.4 even 3
324.1.f.a.55.1 2 36.31 odd 6
324.1.f.a.271.1 2 9.7 even 3
324.1.f.a.271.1 2 36.7 odd 6
324.1.f.b.55.1 2 9.5 odd 6
324.1.f.b.55.1 2 36.23 even 6
324.1.f.b.271.1 2 9.2 odd 6
324.1.f.b.271.1 2 36.11 even 6
2916.1.j.d.163.1 6 27.13 even 9
2916.1.j.d.163.1 6 108.67 odd 18
2916.1.j.d.811.1 6 27.16 even 9
2916.1.j.d.811.1 6 108.43 odd 18
2916.1.j.d.1135.1 6 27.4 even 9
2916.1.j.d.1135.1 6 108.31 odd 18
2916.1.j.d.1783.1 6 27.7 even 9
2916.1.j.d.1783.1 6 108.7 odd 18
2916.1.j.d.2107.1 6 27.22 even 9
2916.1.j.d.2107.1 6 108.103 odd 18
2916.1.j.d.2755.1 6 27.25 even 9
2916.1.j.d.2755.1 6 108.79 odd 18
2916.1.j.e.163.1 6 27.14 odd 18
2916.1.j.e.163.1 6 108.95 even 18
2916.1.j.e.811.1 6 27.11 odd 18
2916.1.j.e.811.1 6 108.11 even 18
2916.1.j.e.1135.1 6 27.23 odd 18
2916.1.j.e.1135.1 6 108.23 even 18
2916.1.j.e.1783.1 6 27.20 odd 18
2916.1.j.e.1783.1 6 108.47 even 18
2916.1.j.e.2107.1 6 27.5 odd 18
2916.1.j.e.2107.1 6 108.59 even 18
2916.1.j.e.2755.1 6 27.2 odd 18
2916.1.j.e.2755.1 6 108.83 even 18