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Properties

Label 1296.1.x.a.701.2
Level $1296$
Weight $1$
Character 1296.701
Analytic conductor $0.647$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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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] = [1296,1,Mod(53,1296)] 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("1296.53"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 10])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1296.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.646788256372\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 432)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.55296.2

Embedding invariants

Embedding label 701.2
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1296.701
Dual form 1296.1.x.a.917.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.258819 - 0.965926i) q^{2} +(-0.866025 - 0.500000i) q^{4} +(-0.965926 + 0.258819i) q^{5} +(-0.866025 + 0.500000i) q^{7} +(-0.707107 + 0.707107i) q^{8} +1.00000i q^{10} +(-0.258819 + 0.965926i) q^{11} +(1.36603 - 0.366025i) q^{13} +(0.258819 + 0.965926i) q^{14} +(0.500000 + 0.866025i) q^{16} +1.41421i q^{17} +(0.965926 + 0.258819i) q^{20} +(0.866025 + 0.500000i) q^{22} -1.41421i q^{26} +1.00000 q^{28} +(-0.500000 + 0.866025i) q^{31} +(0.965926 - 0.258819i) q^{32} +(1.36603 + 0.366025i) q^{34} +(0.707107 - 0.707107i) q^{35} +(0.500000 - 0.866025i) q^{40} +(1.36603 + 0.366025i) q^{43} +(0.707107 - 0.707107i) q^{44} +(-1.22474 + 0.707107i) q^{47} +(-1.36603 - 0.366025i) q^{52} +(-0.707107 - 0.707107i) q^{53} -1.00000i q^{55} +(0.258819 - 0.965926i) q^{56} +(0.707107 + 0.707107i) q^{62} -1.00000i q^{64} +(-1.22474 + 0.707107i) q^{65} +(-1.36603 + 0.366025i) q^{67} +(0.707107 - 1.22474i) q^{68} +(-0.500000 - 0.866025i) q^{70} -1.41421 q^{71} +1.00000i q^{73} +(-0.258819 - 0.965926i) q^{77} +(-0.707107 - 0.707107i) q^{80} +(0.965926 + 0.258819i) q^{83} +(-0.366025 - 1.36603i) q^{85} +(0.707107 - 1.22474i) q^{86} +(-0.500000 - 0.866025i) q^{88} +1.41421 q^{89} +(-1.00000 + 1.00000i) q^{91} +(0.366025 + 1.36603i) q^{94} +(-0.500000 - 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{13} + 4 q^{16} + 8 q^{28} - 4 q^{31} + 4 q^{34} + 4 q^{40} + 4 q^{43} - 4 q^{52} - 4 q^{67} - 4 q^{70} + 4 q^{85} - 4 q^{88} - 8 q^{91} - 4 q^{94} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.258819 − 0.965926i 0.258819 − 0.965926i
\(3\) 0 0
\(4\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(5\) −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 − 0.707107i \(-0.750000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(6\) 0 0
\(7\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(8\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(9\) 0 0
\(10\) 1.00000i 1.00000i
\(11\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(12\) 0 0
\(13\) 1.36603 − 0.366025i 1.36603 − 0.366025i 0.500000 − 0.866025i \(-0.333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(14\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(15\) 0 0
\(16\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(17\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(21\) 0 0
\(22\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 1.41421i − 1.41421i
\(27\) 0 0
\(28\) 1.00000 1.00000
\(29\) 0 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(30\) 0 0
\(31\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(32\) 0.965926 − 0.258819i 0.965926 − 0.258819i
\(33\) 0 0
\(34\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(35\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(36\) 0 0
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 1.36603 + 0.366025i 1.36603 + 0.366025i 0.866025 − 0.500000i \(-0.166667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(45\) 0 0
\(46\) 0 0
\(47\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 − 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −1.36603 − 0.366025i −1.36603 − 0.366025i
\(53\) −0.707107 − 0.707107i −0.707107 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(54\) 0 0
\(55\) − 1.00000i − 1.00000i
\(56\) 0.258819 − 0.965926i 0.258819 − 0.965926i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.258819 − 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(62\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(63\) 0 0
\(64\) − 1.00000i − 1.00000i
\(65\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(66\) 0 0
\(67\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 − 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0.707107 − 1.22474i 0.707107 − 1.22474i
\(69\) 0 0
\(70\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(71\) −1.41421 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(72\) 0 0
\(73\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.258819 − 0.965926i −0.258819 − 0.965926i
\(78\) 0 0
\(79\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(81\) 0 0
\(82\) 0 0
\(83\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 − 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(84\) 0 0
\(85\) −0.366025 − 1.36603i −0.366025 − 1.36603i
\(86\) 0.707107 − 1.22474i 0.707107 − 1.22474i
\(87\) 0 0
\(88\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(89\) 1.41421 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) 0 0
\(91\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(92\) 0 0
\(93\) 0 0
\(94\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(95\) 0 0
\(96\) 0 0
\(97\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(98\) 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 1296.1.x.a.701.2 8
3.2 odd 2 inner 1296.1.x.a.701.1 8
9.2 odd 6 inner 1296.1.x.a.269.2 8
9.4 even 3 432.1.j.a.269.2 yes 4
9.5 odd 6 432.1.j.a.269.1 yes 4
9.7 even 3 inner 1296.1.x.a.269.1 8
16.5 even 4 inner 1296.1.x.a.53.2 8
36.23 even 6 1728.1.j.a.593.1 4
36.31 odd 6 1728.1.j.a.593.2 4
48.5 odd 4 inner 1296.1.x.a.53.1 8
72.5 odd 6 3456.1.j.b.161.2 4
72.13 even 6 3456.1.j.b.161.1 4
72.59 even 6 3456.1.j.a.161.2 4
72.67 odd 6 3456.1.j.a.161.1 4
144.5 odd 12 432.1.j.a.53.2 yes 4
144.13 even 12 3456.1.j.b.1889.2 4
144.59 even 12 1728.1.j.a.1457.2 4
144.67 odd 12 3456.1.j.a.1889.2 4
144.77 odd 12 3456.1.j.b.1889.1 4
144.85 even 12 432.1.j.a.53.1 ✓ 4
144.101 odd 12 inner 1296.1.x.a.917.2 8
144.131 even 12 3456.1.j.a.1889.1 4
144.133 even 12 inner 1296.1.x.a.917.1 8
144.139 odd 12 1728.1.j.a.1457.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
432.1.j.a.53.1 ✓ 4 144.85 even 12
432.1.j.a.53.2 yes 4 144.5 odd 12
432.1.j.a.269.1 yes 4 9.5 odd 6
432.1.j.a.269.2 yes 4 9.4 even 3
1296.1.x.a.53.1 8 48.5 odd 4 inner
1296.1.x.a.53.2 8 16.5 even 4 inner
1296.1.x.a.269.1 8 9.7 even 3 inner
1296.1.x.a.269.2 8 9.2 odd 6 inner
1296.1.x.a.701.1 8 3.2 odd 2 inner
1296.1.x.a.701.2 8 1.1 even 1 trivial
1296.1.x.a.917.1 8 144.133 even 12 inner
1296.1.x.a.917.2 8 144.101 odd 12 inner
1728.1.j.a.593.1 4 36.23 even 6
1728.1.j.a.593.2 4 36.31 odd 6
1728.1.j.a.1457.1 4 144.139 odd 12
1728.1.j.a.1457.2 4 144.59 even 12
3456.1.j.a.161.1 4 72.67 odd 6
3456.1.j.a.161.2 4 72.59 even 6
3456.1.j.a.1889.1 4 144.131 even 12
3456.1.j.a.1889.2 4 144.67 odd 12
3456.1.j.b.161.1 4 72.13 even 6
3456.1.j.b.161.2 4 72.5 odd 6
3456.1.j.b.1889.1 4 144.77 odd 12
3456.1.j.b.1889.2 4 144.13 even 12