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Properties

Label 3204.1.bd.a
Level $3204$
Weight $1$
Character orbit 3204.bd
Analytic conductor $1.599$
Analytic rank $0$
Dimension $10$
Projective image $D_{22}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3204,1,Mod(235,3204)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3204, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3204.235");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3204 = 2^{2} \cdot 3^{2} \cdot 89 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3204.bd (of order \(22\), degree \(10\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.59900430048\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
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: no (minimal twist has level 356)
Projective image: \(D_{22}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{22} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q - \zeta_{22}^{7} q^{2} - \zeta_{22}^{3} q^{4} + (\zeta_{22}^{5} - \zeta_{22}^{4}) q^{5} + \zeta_{22}^{10} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{22}^{7} q^{2} - \zeta_{22}^{3} q^{4} + (\zeta_{22}^{5} - \zeta_{22}^{4}) q^{5} + \zeta_{22}^{10} q^{8} + (\zeta_{22} - 1) q^{10} + (\zeta_{22}^{3} + \zeta_{22}^{2}) q^{13} + \zeta_{22}^{6} q^{16} + (\zeta_{22}^{6} + \zeta_{22}^{2}) q^{17} + ( - \zeta_{22}^{8} + \zeta_{22}^{7}) q^{20} + (\zeta_{22}^{10} + \cdots + \zeta_{22}^{8}) q^{25} + \cdots + \zeta_{22}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - q^{4} + 2 q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} - q^{4} + 2 q^{5} - q^{8} - 9 q^{10} - q^{16} - 2 q^{17} + 2 q^{20} - 3 q^{25} - q^{32} - 2 q^{34} + 2 q^{40} + 11 q^{41} + q^{49} - 3 q^{50} - 2 q^{53} - q^{64} - 2 q^{68} - 2 q^{73} - 11 q^{74} - 9 q^{80} - 7 q^{85} + q^{89} + 2 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3204\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(713\) \(1603\)
\(\chi(n)\) \(\zeta_{22}^{5}\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
−0.415415 − 0.909632i
−0.415415 + 0.909632i
0.654861 − 0.755750i
0.959493 − 0.281733i
0.959493 + 0.281733i
0.142315 + 0.989821i
−0.841254 + 0.540641i
−0.841254 − 0.540641i
0.142315 − 0.989821i
0.654861 + 0.755750i
−0.142315 + 0.989821i 0 −0.959493 − 0.281733i −0.698939 + 1.53046i 0 0 0.415415 − 0.909632i 0 −1.41542 − 0.909632i
559.1 −0.142315 − 0.989821i 0 −0.959493 + 0.281733i −0.698939 − 1.53046i 0 0 0.415415 + 0.909632i 0 −1.41542 + 0.909632i
667.1 −0.959493 − 0.281733i 0 0.841254 + 0.540641i 0.544078 + 0.627899i 0 0 −0.654861 − 0.755750i 0 −0.345139 − 0.755750i
1207.1 0.415415 + 0.909632i 0 −0.654861 + 0.755750i −0.273100 − 0.0801894i 0 0 −0.959493 − 0.281733i 0 −0.0405070 − 0.281733i
1675.1 0.415415 − 0.909632i 0 −0.654861 − 0.755750i −0.273100 + 0.0801894i 0 0 −0.959493 + 0.281733i 0 −0.0405070 + 0.281733i
1891.1 0.841254 + 0.540641i 0 0.415415 + 0.909632i −0.186393 + 1.29639i 0 0 −0.142315 + 0.989821i 0 −0.857685 + 0.989821i
2395.1 −0.654861 + 0.755750i 0 −0.142315 − 0.989821i 1.61435 + 1.03748i 0 0 0.841254 + 0.540641i 0 −1.84125 + 0.540641i
2503.1 −0.654861 − 0.755750i 0 −0.142315 + 0.989821i 1.61435 − 1.03748i 0 0 0.841254 − 0.540641i 0 −1.84125 − 0.540641i
2755.1 0.841254 − 0.540641i 0 0.415415 − 0.909632i −0.186393 − 1.29639i 0 0 −0.142315 − 0.989821i 0 −0.857685 − 0.989821i
2935.1 −0.959493 + 0.281733i 0 0.841254 − 0.540641i 0.544078 − 0.627899i 0 0 −0.654861 + 0.755750i 0 −0.345139 + 0.755750i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
89.f even 22 1 inner
356.j odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3204.1.bd.a 10
3.b odd 2 1 356.1.j.a ✓ 10
4.b odd 2 1 CM 3204.1.bd.a 10
12.b even 2 1 356.1.j.a ✓ 10
89.f even 22 1 inner 3204.1.bd.a 10
267.l odd 22 1 356.1.j.a ✓ 10
356.j odd 22 1 inner 3204.1.bd.a 10
1068.u even 22 1 356.1.j.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
356.1.j.a ✓ 10 3.b odd 2 1
356.1.j.a ✓ 10 12.b even 2 1
356.1.j.a ✓ 10 267.l odd 22 1
356.1.j.a ✓ 10 1068.u even 22 1
3204.1.bd.a 10 1.a even 1 1 trivial
3204.1.bd.a 10 4.b odd 2 1 CM
3204.1.bd.a 10 89.f even 22 1 inner
3204.1.bd.a 10 356.j odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3204, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + 11 T^{6} + \cdots + 11 \) Copy content Toggle raw display
$17$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{10} \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} + 11 T^{6} + \cdots + 11 \) Copy content Toggle raw display
$31$ \( T^{10} \) Copy content Toggle raw display
$37$ \( T^{10} + 11 T^{8} + \cdots + 11 \) Copy content Toggle raw display
$41$ \( T^{10} - 11 T^{9} + \cdots + 11 \) Copy content Toggle raw display
$43$ \( T^{10} \) Copy content Toggle raw display
$47$ \( T^{10} \) Copy content Toggle raw display
$53$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{10} \) Copy content Toggle raw display
$61$ \( T^{10} - 22 T^{5} + \cdots + 11 \) Copy content Toggle raw display
$67$ \( T^{10} \) Copy content Toggle raw display
$71$ \( T^{10} \) Copy content Toggle raw display
$73$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{10} \) Copy content Toggle raw display
$83$ \( T^{10} \) Copy content Toggle raw display
$89$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
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