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Properties

Label 504.1.cu.a.305.2
Level $504$
Weight $1$
Character 504.305
Analytic conductor $0.252$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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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] = [504,1,Mod(233,504)] 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("504.233"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 504.cu (of order \(6\), degree \(2\), 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: \(0.251528766367\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.21168.2

Embedding invariants

Embedding label 305.2
Root \(1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 504.305
Dual form 504.1.cu.a.233.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.22474 - 0.707107i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-1.22474 - 0.707107i) q^{11} -1.00000 q^{13} +(0.500000 + 0.866025i) q^{19} +(0.500000 - 0.866025i) q^{25} +(0.500000 - 0.866025i) q^{31} +(1.22474 + 0.707107i) q^{35} +(-0.500000 - 0.866025i) q^{37} +1.41421i q^{41} -1.00000 q^{43} +(-1.22474 + 0.707107i) q^{47} +(-0.500000 + 0.866025i) q^{49} -2.00000 q^{55} +(-1.22474 + 0.707107i) q^{65} +(0.500000 - 0.866025i) q^{67} -1.41421i q^{71} +(-0.500000 + 0.866025i) q^{73} -1.41421i q^{77} +(-0.500000 - 0.866025i) q^{79} +1.41421i q^{83} +(-0.500000 - 0.866025i) q^{91} +(1.22474 + 0.707107i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{7} - 4 q^{13} + 2 q^{19} + 2 q^{25} + 2 q^{31} - 2 q^{37} - 4 q^{43} - 2 q^{49} - 8 q^{55} + 2 q^{67} - 2 q^{73} - 2 q^{79} - 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.22474 − 0.707107i 1.22474 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.22474 − 0.707107i −1.22474 − 0.707107i −0.258819 − 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(12\) 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
\(16\) 0 0
\(17\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(36\) 0 0
\(37\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 0 0
\(43\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 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
\(49\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) −2.00000 −2.00000
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(66\) 0 0
\(67\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 1.41421i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(72\) 0 0
\(73\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.41421i − 1.41421i
\(78\) 0 0
\(79\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 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 504.1.cu.a.305.2 yes 4
3.2 odd 2 inner 504.1.cu.a.305.1 yes 4
4.3 odd 2 1008.1.dc.a.305.2 4
7.2 even 3 inner 504.1.cu.a.233.1 ✓ 4
7.3 odd 6 3528.1.d.b.1961.1 2
7.4 even 3 3528.1.d.a.1961.2 2
7.5 odd 6 3528.1.cu.a.1745.2 4
7.6 odd 2 3528.1.cu.a.2321.1 4
12.11 even 2 1008.1.dc.a.305.1 4
21.2 odd 6 inner 504.1.cu.a.233.2 yes 4
21.5 even 6 3528.1.cu.a.1745.1 4
21.11 odd 6 3528.1.d.a.1961.1 2
21.17 even 6 3528.1.d.b.1961.2 2
21.20 even 2 3528.1.cu.a.2321.2 4
28.23 odd 6 1008.1.dc.a.737.1 4
84.23 even 6 1008.1.dc.a.737.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.1.cu.a.233.1 ✓ 4 7.2 even 3 inner
504.1.cu.a.233.2 yes 4 21.2 odd 6 inner
504.1.cu.a.305.1 yes 4 3.2 odd 2 inner
504.1.cu.a.305.2 yes 4 1.1 even 1 trivial
1008.1.dc.a.305.1 4 12.11 even 2
1008.1.dc.a.305.2 4 4.3 odd 2
1008.1.dc.a.737.1 4 28.23 odd 6
1008.1.dc.a.737.2 4 84.23 even 6
3528.1.d.a.1961.1 2 21.11 odd 6
3528.1.d.a.1961.2 2 7.4 even 3
3528.1.d.b.1961.1 2 7.3 odd 6
3528.1.d.b.1961.2 2 21.17 even 6
3528.1.cu.a.1745.1 4 21.5 even 6
3528.1.cu.a.1745.2 4 7.5 odd 6
3528.1.cu.a.2321.1 4 7.6 odd 2
3528.1.cu.a.2321.2 4 21.20 even 2