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Properties

Label 2325.1.cq.b
Level $2325$
Weight $1$
Character orbit 2325.cq
Analytic conductor $1.160$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -15
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] = [2325,1,Mod(68,2325)] 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("2325.68"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, 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: \( N \) \(=\) \( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2325.cq (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.16032615437\)
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 6.2.19324676925.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{2} - \zeta_{24}^{5} q^{3} + ( - \zeta_{24}^{10} + \cdots - \zeta_{24}^{2}) q^{4} + (\zeta_{24}^{4} + 1) q^{6} + (\zeta_{24}^{5} + \zeta_{24}) q^{8} + \zeta_{24}^{10} q^{9} + \cdots + ( - \zeta_{24}^{9} - \zeta_{24}^{5}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{6} - 8 q^{16} - 8 q^{31} + 8 q^{36} + 4 q^{51} + 8 q^{76} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2325\mathbb{Z}\right)^\times\).

\(n\) \(652\) \(776\) \(1801\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(-1\) \(\zeta_{24}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−1.22474 − 1.22474i −0.258819 + 0.965926i 2.00000i 0 1.50000 − 0.866025i 0 1.22474 − 1.22474i −0.866025 − 0.500000i 0
68.2 1.22474 + 1.22474i 0.258819 − 0.965926i 2.00000i 0 1.50000 − 0.866025i 0 −1.22474 + 1.22474i −0.866025 − 0.500000i 0
1607.1 −1.22474 + 1.22474i −0.258819 − 0.965926i − 2.00000i 0 1.50000 + 0.866025i 0 1.22474 + 1.22474i −0.866025 + 0.500000i 0
1607.2 1.22474 − 1.22474i 0.258819 + 0.965926i − 2.00000i 0 1.50000 + 0.866025i 0 −1.22474 − 1.22474i −0.866025 + 0.500000i 0
1793.1 −1.22474 − 1.22474i −0.965926 + 0.258819i 2.00000i 0 1.50000 + 0.866025i 0 1.22474 − 1.22474i 0.866025 − 0.500000i 0
1793.2 1.22474 + 1.22474i 0.965926 − 0.258819i 2.00000i 0 1.50000 + 0.866025i 0 −1.22474 + 1.22474i 0.866025 − 0.500000i 0
2207.1 −1.22474 + 1.22474i −0.965926 − 0.258819i − 2.00000i 0 1.50000 − 0.866025i 0 1.22474 + 1.22474i 0.866025 + 0.500000i 0
2207.2 1.22474 − 1.22474i 0.965926 + 0.258819i − 2.00000i 0 1.50000 − 0.866025i 0 −1.22474 − 1.22474i 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.e even 4 2 inner
31.e odd 6 1 inner
93.g even 6 1 inner
155.i odd 6 1 inner
155.p even 12 2 inner
465.t even 6 1 inner
465.bc odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2325.1.cq.b ✓ 8
3.b odd 2 1 inner 2325.1.cq.b ✓ 8
5.b even 2 1 inner 2325.1.cq.b ✓ 8
5.c odd 4 2 inner 2325.1.cq.b ✓ 8
15.d odd 2 1 CM 2325.1.cq.b ✓ 8
15.e even 4 2 inner 2325.1.cq.b ✓ 8
31.e odd 6 1 inner 2325.1.cq.b ✓ 8
93.g even 6 1 inner 2325.1.cq.b ✓ 8
155.i odd 6 1 inner 2325.1.cq.b ✓ 8
155.p even 12 2 inner 2325.1.cq.b ✓ 8
465.t even 6 1 inner 2325.1.cq.b ✓ 8
465.bc odd 12 2 inner 2325.1.cq.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2325.1.cq.b ✓ 8 1.a even 1 1 trivial
2325.1.cq.b ✓ 8 3.b odd 2 1 inner
2325.1.cq.b ✓ 8 5.b even 2 1 inner
2325.1.cq.b ✓ 8 5.c odd 4 2 inner
2325.1.cq.b ✓ 8 15.d odd 2 1 CM
2325.1.cq.b ✓ 8 15.e even 4 2 inner
2325.1.cq.b ✓ 8 31.e odd 6 1 inner
2325.1.cq.b ✓ 8 93.g even 6 1 inner
2325.1.cq.b ✓ 8 155.i odd 6 1 inner
2325.1.cq.b ✓ 8 155.p even 12 2 inner
2325.1.cq.b ✓ 8 465.t even 6 1 inner
2325.1.cq.b ✓ 8 465.bc odd 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 9 \) acting on \(S_{1}^{\mathrm{new}}(2325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T + 1)^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 16T^{4} + 256 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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