Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [364,2,Mod(29,364)]
 
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([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("364.29");
 
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

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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + \zeta_{6} q^{7} + 3 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + (\zeta_{6} + 3) q^{13} + 3 \zeta_{6} q^{17} + 6 \zeta_{6} q^{19} + ( - 4 \zeta_{6} + 4) q^{23} - 4 q^{25} + ( - 7 \zeta_{6} + 7) q^{29}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + q^{7} + 3 q^{9} + 2 q^{11} + 7 q^{13} + 3 q^{17} + 6 q^{19} + 4 q^{23} - 8 q^{25} + 7 q^{29} + 8 q^{31} - q^{35} - 9 q^{37} - 9 q^{41} - 10 q^{43} - 3 q^{45} - 4 q^{47} - q^{49} + 18 q^{53}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(157\) \(183\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −1.00000 0 0.500000 0.866025i 0 1.50000 2.59808i 0
113.1 0 0 0 −1.00000 0 0.500000 + 0.866025i 0 1.50000 + 2.59808i 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.a 2
3.b odd 2 1 3276.2.z.c 2
4.b odd 2 1 1456.2.s.c 2
7.b odd 2 1 2548.2.k.c 2
7.c even 3 1 2548.2.i.e 2
7.c even 3 1 2548.2.l.e 2
7.d odd 6 1 2548.2.i.d 2
7.d odd 6 1 2548.2.l.d 2
13.c even 3 1 inner 364.2.k.a 2
13.c even 3 1 4732.2.a.c 1
13.e even 6 1 4732.2.a.d 1
13.f odd 12 2 4732.2.g.b 2
39.i odd 6 1 3276.2.z.c 2
52.j odd 6 1 1456.2.s.c 2
91.g even 3 1 2548.2.i.e 2
91.h even 3 1 2548.2.l.e 2
91.m odd 6 1 2548.2.i.d 2
91.n odd 6 1 2548.2.k.c 2
91.v odd 6 1 2548.2.l.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.k.a 2 1.a even 1 1 trivial
364.2.k.a 2 13.c even 3 1 inner
1456.2.s.c 2 4.b odd 2 1
1456.2.s.c 2 52.j odd 6 1
2548.2.i.d 2 7.d odd 6 1
2548.2.i.d 2 91.m odd 6 1
2548.2.i.e 2 7.c even 3 1
2548.2.i.e 2 91.g even 3 1
2548.2.k.c 2 7.b odd 2 1
2548.2.k.c 2 91.n odd 6 1
2548.2.l.d 2 7.d odd 6 1
2548.2.l.d 2 91.v odd 6 1
2548.2.l.e 2 7.c even 3 1
2548.2.l.e 2 91.h even 3 1
3276.2.z.c 2 3.b odd 2 1
3276.2.z.c 2 39.i odd 6 1
4732.2.a.c 1 13.c even 3 1
4732.2.a.d 1 13.e even 6 1
4732.2.g.b 2 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 \(S_{2}^{\mathrm{new}}(364, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$47$ \( (T + 2)^{2} \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$73$ \( (T + 7)^{2} \) Copy content Toggle raw display
$79$ \( (T - 2)^{2} \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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