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Properties

Label 525.1.be.a.332.2
Level $525$
Weight $1$
Character 525.332
Analytic conductor $0.262$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -3
Inner twists $16$

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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] = [525,1,Mod(68,525)] 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("525.68"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 10])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 525.be (of order \(12\), degree \(4\), 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: \(0.262009131632\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.472696875.1

Embedding invariants

Embedding label 332.2
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 525.332
Dual form 525.1.be.a.68.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.965926 - 0.258819i) q^{3} +(-0.866025 - 0.500000i) q^{4} +(-0.258819 - 0.965926i) q^{7} +(0.866025 - 0.500000i) q^{9} +(-0.965926 - 0.258819i) q^{12} +(-0.707107 - 0.707107i) q^{13} +(0.500000 + 0.866025i) q^{16} +(0.866025 + 1.50000i) q^{19} +(-0.500000 - 0.866025i) q^{21} +(0.707107 - 0.707107i) q^{27} +(-0.258819 + 0.965926i) q^{28} -1.00000 q^{36} +(-0.448288 + 1.67303i) q^{37} +(-0.866025 - 0.500000i) q^{39} +(0.707107 + 0.707107i) q^{48} +(-0.866025 + 0.500000i) q^{49} +(0.258819 + 0.965926i) q^{52} +(1.22474 + 1.22474i) q^{57} +(-1.50000 + 0.866025i) q^{61} +(-0.707107 - 0.707107i) q^{63} -1.00000i q^{64} +(1.67303 - 0.448288i) q^{67} +(-0.965926 + 0.258819i) q^{73} -1.73205i q^{76} +(0.866025 - 0.500000i) q^{79} +(0.500000 - 0.866025i) q^{81} +1.00000i q^{84} +(-0.500000 + 0.866025i) q^{91} +(-0.707107 + 0.707107i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{16} - 4 q^{21} - 8 q^{36} - 12 q^{61} + 4 q^{81} - 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) 0.965926 − 0.258819i 0.965926 − 0.258819i
\(4\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(5\) 0 0
\(6\) 0 0
\(7\) −0.258819 − 0.965926i −0.258819 − 0.965926i
\(8\) 0 0
\(9\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −0.965926 − 0.258819i −0.965926 − 0.258819i
\(13\) −0.707107 − 0.707107i −0.707107 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(17\) 0 0 −0.258819 − 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(18\) 0 0
\(19\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(22\) 0 0
\(23\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(28\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1.00000 −1.00000
\(37\) −0.448288 + 1.67303i −0.448288 + 1.67303i 0.258819 + 0.965926i \(0.416667\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(48\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(49\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(53\) 0 0 −0.258819 − 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(58\) 0 0
\(59\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(64\) − 1.00000i − 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.67303 − 0.448288i 1.67303 − 0.448288i 0.707107 − 0.707107i \(-0.250000\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 − 0.707107i \(-0.750000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) − 1.73205i − 1.73205i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.866025 − 0.500000i 0.866025 − 0.500000i − 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 0 0
\(81\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(82\) 0 0
\(83\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 1.00000i 1.00000i
\(85\) 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
\(95\) 0 0
\(96\) 0 0
\(97\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 − 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(98\) 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 525.1.be.a.332.2 yes 8
3.2 odd 2 CM 525.1.be.a.332.2 yes 8
5.2 odd 4 inner 525.1.be.a.143.1 yes 8
5.3 odd 4 inner 525.1.be.a.143.2 yes 8
5.4 even 2 inner 525.1.be.a.332.1 yes 8
7.2 even 3 3675.1.bf.b.1832.1 8
7.3 odd 6 3675.1.k.b.2057.4 8
7.4 even 3 3675.1.k.b.2057.1 8
7.5 odd 6 inner 525.1.be.a.257.2 yes 8
7.6 odd 2 3675.1.bf.b.2432.1 8
15.2 even 4 inner 525.1.be.a.143.1 yes 8
15.8 even 4 inner 525.1.be.a.143.2 yes 8
15.14 odd 2 inner 525.1.be.a.332.1 yes 8
21.2 odd 6 3675.1.bf.b.1832.1 8
21.5 even 6 inner 525.1.be.a.257.2 yes 8
21.11 odd 6 3675.1.k.b.2057.1 8
21.17 even 6 3675.1.k.b.2057.4 8
21.20 even 2 3675.1.bf.b.2432.1 8
35.2 odd 12 3675.1.bf.b.68.2 8
35.3 even 12 3675.1.k.b.293.1 8
35.4 even 6 3675.1.k.b.2057.3 8
35.9 even 6 3675.1.bf.b.1832.2 8
35.12 even 12 inner 525.1.be.a.68.1 ✓ 8
35.13 even 4 3675.1.bf.b.668.1 8
35.17 even 12 3675.1.k.b.293.3 8
35.18 odd 12 3675.1.k.b.293.4 8
35.19 odd 6 inner 525.1.be.a.257.1 yes 8
35.23 odd 12 3675.1.bf.b.68.1 8
35.24 odd 6 3675.1.k.b.2057.2 8
35.27 even 4 3675.1.bf.b.668.2 8
35.32 odd 12 3675.1.k.b.293.2 8
35.33 even 12 inner 525.1.be.a.68.2 yes 8
35.34 odd 2 3675.1.bf.b.2432.2 8
105.2 even 12 3675.1.bf.b.68.2 8
105.17 odd 12 3675.1.k.b.293.3 8
105.23 even 12 3675.1.bf.b.68.1 8
105.32 even 12 3675.1.k.b.293.2 8
105.38 odd 12 3675.1.k.b.293.1 8
105.44 odd 6 3675.1.bf.b.1832.2 8
105.47 odd 12 inner 525.1.be.a.68.1 ✓ 8
105.53 even 12 3675.1.k.b.293.4 8
105.59 even 6 3675.1.k.b.2057.2 8
105.62 odd 4 3675.1.bf.b.668.2 8
105.68 odd 12 inner 525.1.be.a.68.2 yes 8
105.74 odd 6 3675.1.k.b.2057.3 8
105.83 odd 4 3675.1.bf.b.668.1 8
105.89 even 6 inner 525.1.be.a.257.1 yes 8
105.104 even 2 3675.1.bf.b.2432.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.1.be.a.68.1 ✓ 8 35.12 even 12 inner
525.1.be.a.68.1 ✓ 8 105.47 odd 12 inner
525.1.be.a.68.2 yes 8 35.33 even 12 inner
525.1.be.a.68.2 yes 8 105.68 odd 12 inner
525.1.be.a.143.1 yes 8 5.2 odd 4 inner
525.1.be.a.143.1 yes 8 15.2 even 4 inner
525.1.be.a.143.2 yes 8 5.3 odd 4 inner
525.1.be.a.143.2 yes 8 15.8 even 4 inner
525.1.be.a.257.1 yes 8 35.19 odd 6 inner
525.1.be.a.257.1 yes 8 105.89 even 6 inner
525.1.be.a.257.2 yes 8 7.5 odd 6 inner
525.1.be.a.257.2 yes 8 21.5 even 6 inner
525.1.be.a.332.1 yes 8 5.4 even 2 inner
525.1.be.a.332.1 yes 8 15.14 odd 2 inner
525.1.be.a.332.2 yes 8 1.1 even 1 trivial
525.1.be.a.332.2 yes 8 3.2 odd 2 CM
3675.1.k.b.293.1 8 35.3 even 12
3675.1.k.b.293.1 8 105.38 odd 12
3675.1.k.b.293.2 8 35.32 odd 12
3675.1.k.b.293.2 8 105.32 even 12
3675.1.k.b.293.3 8 35.17 even 12
3675.1.k.b.293.3 8 105.17 odd 12
3675.1.k.b.293.4 8 35.18 odd 12
3675.1.k.b.293.4 8 105.53 even 12
3675.1.k.b.2057.1 8 7.4 even 3
3675.1.k.b.2057.1 8 21.11 odd 6
3675.1.k.b.2057.2 8 35.24 odd 6
3675.1.k.b.2057.2 8 105.59 even 6
3675.1.k.b.2057.3 8 35.4 even 6
3675.1.k.b.2057.3 8 105.74 odd 6
3675.1.k.b.2057.4 8 7.3 odd 6
3675.1.k.b.2057.4 8 21.17 even 6
3675.1.bf.b.68.1 8 35.23 odd 12
3675.1.bf.b.68.1 8 105.23 even 12
3675.1.bf.b.68.2 8 35.2 odd 12
3675.1.bf.b.68.2 8 105.2 even 12
3675.1.bf.b.668.1 8 35.13 even 4
3675.1.bf.b.668.1 8 105.83 odd 4
3675.1.bf.b.668.2 8 35.27 even 4
3675.1.bf.b.668.2 8 105.62 odd 4
3675.1.bf.b.1832.1 8 7.2 even 3
3675.1.bf.b.1832.1 8 21.2 odd 6
3675.1.bf.b.1832.2 8 35.9 even 6
3675.1.bf.b.1832.2 8 105.44 odd 6
3675.1.bf.b.2432.1 8 7.6 odd 2
3675.1.bf.b.2432.1 8 21.20 even 2
3675.1.bf.b.2432.2 8 35.34 odd 2
3675.1.bf.b.2432.2 8 105.104 even 2