Properties

Label 2916.1.d.a.1459.3
Level $2916$
Weight $1$
Character 2916.1459
Self dual yes
Analytic conductor $1.455$
Analytic rank $0$
Dimension $3$
Projective image $D_{9}$
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] = [2916,1,Mod(1459,2916)] 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("2916.1459"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2916, 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: \( N \) \(=\) \( 2916 = 2^{2} \cdot 3^{6} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2916.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3] 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: \(1.45527357684\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.8033551259904.2

Embedding invariants

Embedding label 1459.3
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 2916.1459

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.87939 q^{5} -1.00000 q^{8} -1.87939 q^{10} +0.347296 q^{13} +1.00000 q^{16} -1.53209 q^{17} +1.87939 q^{20} +2.53209 q^{25} -0.347296 q^{26} -0.347296 q^{29} -1.00000 q^{32} +1.53209 q^{34} +1.53209 q^{37} -1.87939 q^{40} +1.00000 q^{41} +1.00000 q^{49} -2.53209 q^{50} +0.347296 q^{52} +1.00000 q^{53} +0.347296 q^{58} +0.347296 q^{61} +1.00000 q^{64} +0.652704 q^{65} -1.53209 q^{68} -1.87939 q^{73} -1.53209 q^{74} +1.87939 q^{80} -1.00000 q^{82} -2.87939 q^{85} -1.53209 q^{89} -1.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 3 q^{16} + 3 q^{25} - 3 q^{32} + 3 q^{41} + 3 q^{49} - 3 q^{50} + 3 q^{53} + 3 q^{64} + 3 q^{65} - 3 q^{82} - 3 q^{85} - 3 q^{97} - 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2916\mathbb{Z}\right)^\times\).

\(n\) \(1459\) \(2189\)
\(\chi(n)\) \(-1\) \(1\)

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
\(5\) 1.87939 1.87939 0.939693 − 0.342020i \(-0.111111\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) −1.87939 −1.87939
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0.347296 0.347296 0.173648 − 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) −1.53209 −1.53209 −0.766044 − 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.87939 1.87939
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.53209 2.53209
\(26\) −0.347296 −0.347296
\(27\) 0 0
\(28\) 0 0
\(29\) −0.347296 −0.347296 −0.173648 − 0.984808i \(-0.555556\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) 1.53209 1.53209
\(35\) 0 0
\(36\) 0 0
\(37\) 1.53209 1.53209 0.766044 − 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.87939 −1.87939
\(41\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 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
\(50\) −2.53209 −2.53209
\(51\) 0 0
\(52\) 0.347296 0.347296
\(53\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0.347296 0.347296
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0.347296 0.347296 0.173648 − 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0.652704 0.652704
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −1.53209 −1.53209
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.87939 −1.87939 −0.939693 − 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(74\) −1.53209 −1.53209
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.87939 1.87939
\(81\) 0 0
\(82\) −1.00000 −1.00000
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −2.87939 −2.87939
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.53209 −1.53209 −0.766044 − 0.642788i \(-0.777778\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) −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 2916.1.d.a.1459.3 ✓ 3
3.2 odd 2 2916.1.d.b.1459.1 yes 3
4.3 odd 2 CM 2916.1.d.a.1459.3 ✓ 3
9.2 odd 6 2916.1.f.a.2431.3 6
9.4 even 3 2916.1.f.b.487.1 6
9.5 odd 6 2916.1.f.a.487.3 6
9.7 even 3 2916.1.f.b.2431.1 6
12.11 even 2 2916.1.d.b.1459.1 yes 3
27.2 odd 18 2916.1.j.b.2755.1 6
27.4 even 9 2916.1.j.a.1135.1 6
27.5 odd 18 2916.1.j.c.2107.1 6
27.7 even 9 2916.1.j.a.1783.1 6
27.11 odd 18 2916.1.j.c.811.1 6
27.13 even 9 2916.1.j.f.163.1 6
27.14 odd 18 2916.1.j.b.163.1 6
27.16 even 9 2916.1.j.g.811.1 6
27.20 odd 18 2916.1.j.h.1783.1 6
27.22 even 9 2916.1.j.g.2107.1 6
27.23 odd 18 2916.1.j.h.1135.1 6
27.25 even 9 2916.1.j.f.2755.1 6
36.7 odd 6 2916.1.f.b.2431.1 6
36.11 even 6 2916.1.f.a.2431.3 6
36.23 even 6 2916.1.f.a.487.3 6
36.31 odd 6 2916.1.f.b.487.1 6
108.7 odd 18 2916.1.j.a.1783.1 6
108.11 even 18 2916.1.j.c.811.1 6
108.23 even 18 2916.1.j.h.1135.1 6
108.31 odd 18 2916.1.j.a.1135.1 6
108.43 odd 18 2916.1.j.g.811.1 6
108.47 even 18 2916.1.j.h.1783.1 6
108.59 even 18 2916.1.j.c.2107.1 6
108.67 odd 18 2916.1.j.f.163.1 6
108.79 odd 18 2916.1.j.f.2755.1 6
108.83 even 18 2916.1.j.b.2755.1 6
108.95 even 18 2916.1.j.b.163.1 6
108.103 odd 18 2916.1.j.g.2107.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2916.1.d.a.1459.3 ✓ 3 1.1 even 1 trivial
2916.1.d.a.1459.3 ✓ 3 4.3 odd 2 CM
2916.1.d.b.1459.1 yes 3 3.2 odd 2
2916.1.d.b.1459.1 yes 3 12.11 even 2
2916.1.f.a.487.3 6 9.5 odd 6
2916.1.f.a.487.3 6 36.23 even 6
2916.1.f.a.2431.3 6 9.2 odd 6
2916.1.f.a.2431.3 6 36.11 even 6
2916.1.f.b.487.1 6 9.4 even 3
2916.1.f.b.487.1 6 36.31 odd 6
2916.1.f.b.2431.1 6 9.7 even 3
2916.1.f.b.2431.1 6 36.7 odd 6
2916.1.j.a.1135.1 6 27.4 even 9
2916.1.j.a.1135.1 6 108.31 odd 18
2916.1.j.a.1783.1 6 27.7 even 9
2916.1.j.a.1783.1 6 108.7 odd 18
2916.1.j.b.163.1 6 27.14 odd 18
2916.1.j.b.163.1 6 108.95 even 18
2916.1.j.b.2755.1 6 27.2 odd 18
2916.1.j.b.2755.1 6 108.83 even 18
2916.1.j.c.811.1 6 27.11 odd 18
2916.1.j.c.811.1 6 108.11 even 18
2916.1.j.c.2107.1 6 27.5 odd 18
2916.1.j.c.2107.1 6 108.59 even 18
2916.1.j.f.163.1 6 27.13 even 9
2916.1.j.f.163.1 6 108.67 odd 18
2916.1.j.f.2755.1 6 27.25 even 9
2916.1.j.f.2755.1 6 108.79 odd 18
2916.1.j.g.811.1 6 27.16 even 9
2916.1.j.g.811.1 6 108.43 odd 18
2916.1.j.g.2107.1 6 27.22 even 9
2916.1.j.g.2107.1 6 108.103 odd 18
2916.1.j.h.1135.1 6 27.23 odd 18
2916.1.j.h.1135.1 6 108.23 even 18
2916.1.j.h.1783.1 6 27.20 odd 18
2916.1.j.h.1783.1 6 108.47 even 18