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Properties

Label 364.2.k.d
Level $364$
Weight $2$
Character orbit 364.k
Analytic conductor $2.907$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [364,2,Mod(29,364)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(364, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("364.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 364.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,1,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.90655463357\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{2} - 2) q^{5} + \beta_1 q^{7} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{9} + ( - \beta_{3} + 2 \beta_1 + 2) q^{11} + (3 \beta_1 + 4) q^{13} + ( - \beta_{3} + 5 \beta_1 + 5) q^{15}+ \cdots + (\beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} - 6 q^{5} - 2 q^{7} - 5 q^{9} + 3 q^{11} + 10 q^{13} + 9 q^{15} - 6 q^{17} - q^{19} - 2 q^{21} - 12 q^{23} + 10 q^{25} - 20 q^{27} + 9 q^{29} - 4 q^{31} + 9 q^{33} + 3 q^{35} + 8 q^{37} - 2 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} - \nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 3\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} + \nu^{2} + \nu - 8 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 5\beta _1 + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} - \beta_{2} - \beta _1 + 7 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−0.895644 − 1.09445i
1.39564 + 0.228425i
−0.895644 + 1.09445i
1.39564 − 0.228425i
0 −0.895644 − 1.55130i 0 −3.79129 0 −0.500000 + 0.866025i 0 −0.104356 + 0.180750i 0
29.2 0 1.39564 + 2.41733i 0 0.791288 0 −0.500000 + 0.866025i 0 −2.39564 + 4.14938i 0
113.1 0 −0.895644 + 1.55130i 0 −3.79129 0 −0.500000 − 0.866025i 0 −0.104356 − 0.180750i 0
113.2 0 1.39564 − 2.41733i 0 0.791288 0 −0.500000 − 0.866025i 0 −2.39564 − 4.14938i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 364.2.k.d ✓ 4
3.b odd 2 1 3276.2.z.e 4
4.b odd 2 1 1456.2.s.k 4
7.b odd 2 1 2548.2.k.e 4
7.c even 3 1 2548.2.i.k 4
7.c even 3 1 2548.2.l.j 4
7.d odd 6 1 2548.2.i.i 4
7.d odd 6 1 2548.2.l.l 4
13.c even 3 1 inner 364.2.k.d ✓ 4
13.c even 3 1 4732.2.a.g 2
13.e even 6 1 4732.2.a.h 2
13.f odd 12 2 4732.2.g.e 4
39.i odd 6 1 3276.2.z.e 4
52.j odd 6 1 1456.2.s.k 4
91.g even 3 1 2548.2.i.k 4
91.h even 3 1 2548.2.l.j 4
91.m odd 6 1 2548.2.i.i 4
91.n odd 6 1 2548.2.k.e 4
91.v odd 6 1 2548.2.l.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.k.d ✓ 4 1.a even 1 1 trivial
364.2.k.d ✓ 4 13.c even 3 1 inner
1456.2.s.k 4 4.b odd 2 1
1456.2.s.k 4 52.j odd 6 1
2548.2.i.i 4 7.d odd 6 1
2548.2.i.i 4 91.m odd 6 1
2548.2.i.k 4 7.c even 3 1
2548.2.i.k 4 91.g even 3 1
2548.2.k.e 4 7.b odd 2 1
2548.2.k.e 4 91.n odd 6 1
2548.2.l.j 4 7.c even 3 1
2548.2.l.j 4 91.h even 3 1
2548.2.l.l 4 7.d odd 6 1
2548.2.l.l 4 91.v odd 6 1
3276.2.z.e 4 3.b odd 2 1
3276.2.z.e 4 39.i odd 6 1
4732.2.a.g 2 13.c even 3 1
4732.2.a.h 2 13.e even 6 1
4732.2.g.e 4 13.f odd 12 2

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{3} + 6T_{3}^{2} + 5T_{3} + 25 \) Copy content Toggle raw display
\( T_{5}^{2} + 3T_{5} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T - 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 3 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( (T^{2} - 5 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 9 T^{3} + \cdots + 225 \) Copy content Toggle raw display
$31$ \( (T + 1)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$43$ \( T^{4} - 5 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T - 75)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T + 15)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$73$ \( (T - 14)^{4} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T + 9)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 3 T^{3} + \cdots + 2025 \) Copy content Toggle raw display
$97$ \( T^{4} - 23 T^{3} + \cdots + 7225 \) Copy content Toggle raw display
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