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Properties

Label 2904.1.bh.b
Level $2904$
Weight $1$
Character orbit 2904.bh
Analytic conductor $1.449$
Analytic rank $0$
Dimension $10$
Projective image $D_{22}$
CM discriminant -8
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] = [2904,1,Mod(131,2904)] 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("2904.131"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2904, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 11, 11, 15])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2904 = 2^{3} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2904.bh (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.44928479669\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{22}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{22} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{22}^{10} q^{2} - \zeta_{22}^{2} q^{3} - \zeta_{22}^{9} q^{4} - \zeta_{22} q^{6} - \zeta_{22}^{8} q^{8} + \zeta_{22}^{4} q^{9} - \zeta_{22}^{3} q^{11} - q^{12} - \zeta_{22}^{7} q^{16} + ( - \zeta_{22}^{9} - \zeta_{22}^{3}) q^{17} + \cdots - \zeta_{22}^{7} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} + q^{3} - q^{4} - q^{6} + q^{8} - q^{9} - q^{11} - 10 q^{12} - q^{16} - 2 q^{17} + q^{18} + q^{22} - q^{24} + q^{25} + q^{27} + q^{32} + q^{33} + 2 q^{34} - q^{36} + 2 q^{41}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2904\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1453\) \(1937\) \(2785\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{22}^{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1
−0.841254 + 0.540641i
0.142315 + 0.989821i
0.959493 + 0.281733i
0.654861 − 0.755750i
−0.415415 − 0.909632i
−0.415415 + 0.909632i
0.654861 + 0.755750i
0.959493 − 0.281733i
0.142315 − 0.989821i
−0.841254 − 0.540641i
−0.841254 − 0.540641i −0.415415 + 0.909632i 0.415415 + 0.909632i 0 0.841254 − 0.540641i 0 0.142315 − 0.989821i −0.654861 − 0.755750i 0
395.1 0.142315 − 0.989821i 0.959493 − 0.281733i −0.959493 − 0.281733i 0 −0.142315 − 0.989821i 0 −0.415415 + 0.909632i 0.841254 − 0.540641i 0
659.1 0.959493 − 0.281733i −0.841254 − 0.540641i 0.841254 − 0.540641i 0 −0.959493 − 0.281733i 0 0.654861 − 0.755750i 0.415415 + 0.909632i 0
923.1 0.654861 + 0.755750i 0.142315 + 0.989821i −0.142315 + 0.989821i 0 −0.654861 + 0.755750i 0 −0.841254 + 0.540641i −0.959493 + 0.281733i 0
1187.1 −0.415415 + 0.909632i 0.654861 − 0.755750i −0.654861 − 0.755750i 0 0.415415 + 0.909632i 0 0.959493 − 0.281733i −0.142315 − 0.989821i 0
1715.1 −0.415415 − 0.909632i 0.654861 + 0.755750i −0.654861 + 0.755750i 0 0.415415 − 0.909632i 0 0.959493 + 0.281733i −0.142315 + 0.989821i 0
1979.1 0.654861 − 0.755750i 0.142315 − 0.989821i −0.142315 − 0.989821i 0 −0.654861 − 0.755750i 0 −0.841254 − 0.540641i −0.959493 − 0.281733i 0
2243.1 0.959493 + 0.281733i −0.841254 + 0.540641i 0.841254 + 0.540641i 0 −0.959493 + 0.281733i 0 0.654861 + 0.755750i 0.415415 − 0.909632i 0
2507.1 0.142315 + 0.989821i 0.959493 + 0.281733i −0.959493 + 0.281733i 0 −0.142315 + 0.989821i 0 −0.415415 − 0.909632i 0.841254 + 0.540641i 0
2771.1 −0.841254 + 0.540641i −0.415415 − 0.909632i 0.415415 − 0.909632i 0 0.841254 + 0.540641i 0 0.142315 + 0.989821i −0.654861 + 0.755750i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 131.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
363.j even 22 1 inner
2904.bh odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2904.1.bh.b yes 10
3.b odd 2 1 2904.1.bh.a ✓ 10
8.d odd 2 1 CM 2904.1.bh.b yes 10
24.f even 2 1 2904.1.bh.a ✓ 10
121.f odd 22 1 2904.1.bh.a ✓ 10
363.j even 22 1 inner 2904.1.bh.b yes 10
968.w even 22 1 2904.1.bh.a ✓ 10
2904.bh odd 22 1 inner 2904.1.bh.b yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2904.1.bh.a ✓ 10 3.b odd 2 1
2904.1.bh.a ✓ 10 24.f even 2 1
2904.1.bh.a ✓ 10 121.f odd 22 1
2904.1.bh.a ✓ 10 968.w even 22 1
2904.1.bh.b yes 10 1.a even 1 1 trivial
2904.1.bh.b yes 10 8.d odd 2 1 CM
2904.1.bh.b yes 10 363.j even 22 1 inner
2904.1.bh.b yes 10 2904.bh odd 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{10} + 2 T_{17}^{9} + 4 T_{17}^{8} + 8 T_{17}^{7} + 16 T_{17}^{6} + 10 T_{17}^{5} + 20 T_{17}^{4} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(2904, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{10} + 22 T^{5} + \cdots + 11 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} \) Copy content Toggle raw display
$31$ \( T^{10} \) Copy content Toggle raw display
$37$ \( T^{10} \) Copy content Toggle raw display
$41$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{10} + 11 T^{6} + \cdots + 11 \) Copy content Toggle raw display
$47$ \( T^{10} \) Copy content Toggle raw display
$53$ \( T^{10} \) Copy content Toggle raw display
$59$ \( T^{10} + 11 T^{6} + \cdots + 11 \) Copy content Toggle raw display
$61$ \( T^{10} \) Copy content Toggle raw display
$67$ \( T^{10} + 2 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{10} \) Copy content Toggle raw display
$73$ \( T^{10} + 22 T^{5} + \cdots + 11 \) Copy content Toggle raw display
$79$ \( T^{10} \) Copy content Toggle raw display
$83$ \( T^{10} - 2 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{10} + 11 T^{7} + \cdots + 11 \) Copy content Toggle raw display
$97$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
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