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Properties

Label 2535.1.y.a
Level $2535$
Weight $1$
Character orbit 2535.y
Analytic conductor $1.265$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -39
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] = [2535,1,Mod(1499,2535)] 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("2535.1499"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2535, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2535.y (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.26512980702\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.12675.1

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( - \zeta_{24}^{7} + \zeta_{24}) q^{2} + \zeta_{24}^{10} q^{3} - \zeta_{24}^{8} q^{4} - \zeta_{24}^{9} q^{5} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{6} - \zeta_{24}^{8} q^{9} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{10} + \cdots + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{9} - 4 q^{10} + 4 q^{16} - 4 q^{30} - 4 q^{36} + 4 q^{49} + 4 q^{55} + 8 q^{64} - 16 q^{66} + 4 q^{75} - 4 q^{81} - 8 q^{90} - 8 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2535\mathbb{Z}\right)^\times\).

\(n\) \(1522\) \(1691\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{24}^{8}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1499.1
−0.965926 + 0.258819i
−0.258819 − 0.965926i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 − 0.965926i
−1.22474 − 0.707107i −0.866025 − 0.500000i 0.500000 + 0.866025i −0.707107 − 0.707107i 0.707107 + 1.22474i 0 0 0.500000 + 0.866025i 0.366025 + 1.36603i
1499.2 −1.22474 − 0.707107i 0.866025 + 0.500000i 0.500000 + 0.866025i 0.707107 − 0.707107i −0.707107 − 1.22474i 0 0 0.500000 + 0.866025i −1.36603 + 0.366025i
1499.3 1.22474 + 0.707107i −0.866025 − 0.500000i 0.500000 + 0.866025i 0.707107 + 0.707107i −0.707107 − 1.22474i 0 0 0.500000 + 0.866025i 0.366025 + 1.36603i
1499.4 1.22474 + 0.707107i 0.866025 + 0.500000i 0.500000 + 0.866025i −0.707107 + 0.707107i 0.707107 + 1.22474i 0 0 0.500000 + 0.866025i −1.36603 + 0.366025i
1544.1 −1.22474 + 0.707107i −0.866025 + 0.500000i 0.500000 − 0.866025i −0.707107 + 0.707107i 0.707107 − 1.22474i 0 0 0.500000 − 0.866025i 0.366025 − 1.36603i
1544.2 −1.22474 + 0.707107i 0.866025 − 0.500000i 0.500000 − 0.866025i 0.707107 + 0.707107i −0.707107 + 1.22474i 0 0 0.500000 − 0.866025i −1.36603 − 0.366025i
1544.3 1.22474 − 0.707107i −0.866025 + 0.500000i 0.500000 − 0.866025i 0.707107 − 0.707107i −0.707107 + 1.22474i 0 0 0.500000 − 0.866025i 0.366025 − 1.36603i
1544.4 1.22474 − 0.707107i 0.866025 − 0.500000i 0.500000 − 0.866025i −0.707107 − 0.707107i 0.707107 − 1.22474i 0 0 0.500000 − 0.866025i −1.36603 − 0.366025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1499.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner
15.d odd 2 1 inner
39.h odd 6 1 inner
39.i odd 6 1 inner
65.d even 2 1 inner
65.l even 6 1 inner
65.n even 6 1 inner
195.e odd 2 1 inner
195.x odd 6 1 inner
195.y odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2535.1.y.a 8
3.b odd 2 1 inner 2535.1.y.a 8
5.b even 2 1 inner 2535.1.y.a 8
13.b even 2 1 inner 2535.1.y.a 8
13.c even 3 1 195.1.e.a ✓ 4
13.c even 3 1 inner 2535.1.y.a 8
13.d odd 4 2 2535.1.x.e 8
13.e even 6 1 195.1.e.a ✓ 4
13.e even 6 1 inner 2535.1.y.a 8
13.f odd 12 2 2535.1.f.e 4
13.f odd 12 2 2535.1.x.e 8
15.d odd 2 1 inner 2535.1.y.a 8
39.d odd 2 1 CM 2535.1.y.a 8
39.f even 4 2 2535.1.x.e 8
39.h odd 6 1 195.1.e.a ✓ 4
39.h odd 6 1 inner 2535.1.y.a 8
39.i odd 6 1 195.1.e.a ✓ 4
39.i odd 6 1 inner 2535.1.y.a 8
39.k even 12 2 2535.1.f.e 4
39.k even 12 2 2535.1.x.e 8
52.i odd 6 1 3120.1.be.e 4
52.j odd 6 1 3120.1.be.e 4
65.d even 2 1 inner 2535.1.y.a 8
65.g odd 4 2 2535.1.x.e 8
65.l even 6 1 195.1.e.a ✓ 4
65.l even 6 1 inner 2535.1.y.a 8
65.n even 6 1 195.1.e.a ✓ 4
65.n even 6 1 inner 2535.1.y.a 8
65.q odd 12 1 975.1.g.b 2
65.q odd 12 1 975.1.g.c 2
65.r odd 12 1 975.1.g.b 2
65.r odd 12 1 975.1.g.c 2
65.s odd 12 2 2535.1.f.e 4
65.s odd 12 2 2535.1.x.e 8
156.p even 6 1 3120.1.be.e 4
156.r even 6 1 3120.1.be.e 4
195.e odd 2 1 inner 2535.1.y.a 8
195.n even 4 2 2535.1.x.e 8
195.x odd 6 1 195.1.e.a ✓ 4
195.x odd 6 1 inner 2535.1.y.a 8
195.y odd 6 1 195.1.e.a ✓ 4
195.y odd 6 1 inner 2535.1.y.a 8
195.bf even 12 1 975.1.g.b 2
195.bf even 12 1 975.1.g.c 2
195.bh even 12 2 2535.1.f.e 4
195.bh even 12 2 2535.1.x.e 8
195.bl even 12 1 975.1.g.b 2
195.bl even 12 1 975.1.g.c 2
260.v odd 6 1 3120.1.be.e 4
260.w odd 6 1 3120.1.be.e 4
780.br even 6 1 3120.1.be.e 4
780.cb even 6 1 3120.1.be.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.1.e.a ✓ 4 13.c even 3 1
195.1.e.a ✓ 4 13.e even 6 1
195.1.e.a ✓ 4 39.h odd 6 1
195.1.e.a ✓ 4 39.i odd 6 1
195.1.e.a ✓ 4 65.l even 6 1
195.1.e.a ✓ 4 65.n even 6 1
195.1.e.a ✓ 4 195.x odd 6 1
195.1.e.a ✓ 4 195.y odd 6 1
975.1.g.b 2 65.q odd 12 1
975.1.g.b 2 65.r odd 12 1
975.1.g.b 2 195.bf even 12 1
975.1.g.b 2 195.bl even 12 1
975.1.g.c 2 65.q odd 12 1
975.1.g.c 2 65.r odd 12 1
975.1.g.c 2 195.bf even 12 1
975.1.g.c 2 195.bl even 12 1
2535.1.f.e 4 13.f odd 12 2
2535.1.f.e 4 39.k even 12 2
2535.1.f.e 4 65.s odd 12 2
2535.1.f.e 4 195.bh even 12 2
2535.1.x.e 8 13.d odd 4 2
2535.1.x.e 8 13.f odd 12 2
2535.1.x.e 8 39.f even 4 2
2535.1.x.e 8 39.k even 12 2
2535.1.x.e 8 65.g odd 4 2
2535.1.x.e 8 65.s odd 12 2
2535.1.x.e 8 195.n even 4 2
2535.1.x.e 8 195.bh even 12 2
2535.1.y.a 8 1.a even 1 1 trivial
2535.1.y.a 8 3.b odd 2 1 inner
2535.1.y.a 8 5.b even 2 1 inner
2535.1.y.a 8 13.b even 2 1 inner
2535.1.y.a 8 13.c even 3 1 inner
2535.1.y.a 8 13.e even 6 1 inner
2535.1.y.a 8 15.d odd 2 1 inner
2535.1.y.a 8 39.d odd 2 1 CM
2535.1.y.a 8 39.h odd 6 1 inner
2535.1.y.a 8 39.i odd 6 1 inner
2535.1.y.a 8 65.d even 2 1 inner
2535.1.y.a 8 65.l even 6 1 inner
2535.1.y.a 8 65.n even 6 1 inner
2535.1.y.a 8 195.e odd 2 1 inner
2535.1.y.a 8 195.x odd 6 1 inner
2535.1.y.a 8 195.y odd 6 1 inner
3120.1.be.e 4 52.i odd 6 1
3120.1.be.e 4 52.j odd 6 1
3120.1.be.e 4 156.p even 6 1
3120.1.be.e 4 156.r even 6 1
3120.1.be.e 4 260.v odd 6 1
3120.1.be.e 4 260.w odd 6 1
3120.1.be.e 4 780.br even 6 1
3120.1.be.e 4 780.cb even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2535, [\chi])\):

\( T_{2}^{4} - 2T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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