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Properties

Label 364.2.v.e
Level $364$
Weight $2$
Character orbit 364.v
Analytic conductor $2.907$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [364,2,Mod(55,364)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(364, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("364.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 364.v (of order \(6\), degree \(2\), 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: \(2.90655463357\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{4} - \beta_{2} + 1) q^{2} + ( - 2 \beta_{7} + \beta_{3}) q^{3} + (\beta_{4} - \beta_{2}) q^{4} - \beta_{5} q^{5} + ( - 4 \beta_{5} - 4 \beta_{3} - \beta_1) q^{6} + (\beta_{4} + 2 \beta_{2}) q^{7}+ \cdots + ( - 7 \beta_{4} - 7 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 6 q^{4} + 20 q^{8} - 16 q^{9} - 28 q^{14} - 2 q^{16} - 16 q^{18} + 32 q^{25} - 14 q^{28} - 32 q^{29} - 14 q^{30} + 22 q^{32} + 24 q^{36} - 16 q^{37} - 14 q^{46} + 28 q^{49} + 8 q^{50} + 96 q^{53}+ \cdots - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 5\nu^{4} - 5\nu^{2} - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 5\nu^{5} - 5\nu^{3} - 12\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 36 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 7 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + 5\nu^{3} + 12\nu ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{5} - \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{6} - 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10\beta_{5} - 10\beta_{3} - 7\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/364\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(183\) \(197\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{4}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−0.228425 + 1.39564i
0.228425 − 1.39564i
−1.09445 + 0.895644i
1.09445 − 0.895644i
−0.228425 − 1.39564i
0.228425 + 1.39564i
−1.09445 − 0.895644i
1.09445 + 0.895644i
−0.895644 + 1.09445i −1.32288 − 2.29129i −0.395644 − 1.96048i 1.00000i 3.69253 + 0.604356i 2.29129 + 1.32288i 2.50000 + 1.32288i −2.00000 + 3.46410i −1.09445 − 0.895644i
55.2 −0.895644 + 1.09445i 1.32288 + 2.29129i −0.395644 − 1.96048i − 1.00000i −3.69253 − 0.604356i 2.29129 + 1.32288i 2.50000 + 1.32288i −2.00000 + 3.46410i 1.09445 + 0.895644i
55.3 1.39564 − 0.228425i −1.32288 − 2.29129i 1.89564 − 0.637600i − 1.00000i −2.36965 − 2.89564i −2.29129 − 1.32288i 2.50000 − 1.32288i −2.00000 + 3.46410i −0.228425 − 1.39564i
55.4 1.39564 − 0.228425i 1.32288 + 2.29129i 1.89564 − 0.637600i 1.00000i 2.36965 + 2.89564i −2.29129 − 1.32288i 2.50000 − 1.32288i −2.00000 + 3.46410i 0.228425 + 1.39564i
139.1 −0.895644 − 1.09445i −1.32288 + 2.29129i −0.395644 + 1.96048i − 1.00000i 3.69253 − 0.604356i 2.29129 − 1.32288i 2.50000 − 1.32288i −2.00000 − 3.46410i −1.09445 + 0.895644i
139.2 −0.895644 − 1.09445i 1.32288 − 2.29129i −0.395644 + 1.96048i 1.00000i −3.69253 + 0.604356i 2.29129 − 1.32288i 2.50000 − 1.32288i −2.00000 − 3.46410i 1.09445 − 0.895644i
139.3 1.39564 + 0.228425i −1.32288 + 2.29129i 1.89564 + 0.637600i 1.00000i −2.36965 + 2.89564i −2.29129 + 1.32288i 2.50000 + 1.32288i −2.00000 − 3.46410i −0.228425 + 1.39564i
139.4 1.39564 + 0.228425i 1.32288 − 2.29129i 1.89564 + 0.637600i − 1.00000i 2.36965 − 2.89564i −2.29129 + 1.32288i 2.50000 + 1.32288i −2.00000 − 3.46410i 0.228425 − 1.39564i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
13.c even 3 1 inner
28.d even 2 1 inner
52.j odd 6 1 inner
91.n odd 6 1 inner
364.v even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 364.2.v.e ✓ 8
4.b odd 2 1 inner 364.2.v.e ✓ 8
7.b odd 2 1 inner 364.2.v.e ✓ 8
13.c even 3 1 inner 364.2.v.e ✓ 8
28.d even 2 1 inner 364.2.v.e ✓ 8
52.j odd 6 1 inner 364.2.v.e ✓ 8
91.n odd 6 1 inner 364.2.v.e ✓ 8
364.v even 6 1 inner 364.2.v.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.v.e ✓ 8 1.a even 1 1 trivial
364.2.v.e ✓ 8 4.b odd 2 1 inner
364.2.v.e ✓ 8 7.b odd 2 1 inner
364.2.v.e ✓ 8 13.c even 3 1 inner
364.2.v.e ✓ 8 28.d even 2 1 inner
364.2.v.e ✓ 8 52.j odd 6 1 inner
364.2.v.e ✓ 8 91.n odd 6 1 inner
364.2.v.e ✓ 8 364.v even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(364, [\chi])\):

\( T_{3}^{4} + 7T_{3}^{2} + 49 \) Copy content Toggle raw display
\( T_{17}^{4} - 16T_{17}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} - T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 16 T^{2} + 256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 28 T^{2} + 784)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 28 T^{2} + 784)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T - 12)^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 63 T^{2} + 3969)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 28 T^{2} + 784)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 252)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 196 T^{2} + 38416)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 196 T^{2} + 38416)^{2} \) Copy content Toggle raw display
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