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Properties

Label 2304.1.z.a
Level $2304$
Weight $1$
Character orbit 2304.z
Analytic conductor $1.150$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,1,Mod(319,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.319");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2304.z (of order \(12\), degree \(4\), 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: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.165888.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{5} q^{3} + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{5} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{7} + \zeta_{24}^{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{5} q^{3} + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{5} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{7} + \zeta_{24}^{10} q^{9} + \zeta_{24}^{11} q^{11} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{15} + q^{17} - \zeta_{24}^{9} q^{19} + ( - \zeta_{24}^{10} + \zeta_{24}^{4}) q^{21} + \zeta_{24}^{2} q^{25} - \zeta_{24}^{3} q^{27} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{29} - \zeta_{24}^{4} q^{33} + \zeta_{24}^{9} q^{35} + ( - \zeta_{24}^{6} + 1) q^{37} - \zeta_{24}^{10} q^{41} - \zeta_{24}^{11} q^{43} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{45} - \zeta_{24}^{4} q^{49} + \zeta_{24}^{5} q^{51} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{55} + \zeta_{24}^{2} q^{57} - \zeta_{24}^{7} q^{59} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{63} - \zeta_{24} q^{67} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{71} + \zeta_{24}^{6} q^{73} + \zeta_{24}^{7} q^{75} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{77} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{79} - \zeta_{24}^{8} q^{81} + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{85} + (\zeta_{24}^{7} + \zeta_{24}) q^{87} - \zeta_{24}^{6} q^{89} + (\zeta_{24}^{7} - \zeta_{24}) q^{95} + \zeta_{24}^{8} q^{97} - \zeta_{24}^{9} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} + 8 q^{17} + 4 q^{21} + 4 q^{29} - 4 q^{33} + 8 q^{37} + 4 q^{45} - 4 q^{49} + 4 q^{77} + 4 q^{81} - 4 q^{85} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-\zeta_{24}^{4}\) \(-\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−0.965926 + 0.258819i
0.965926 − 0.258819i
−0.258819 + 0.965926i
0.258819 − 0.965926i
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
0 −0.258819 + 0.965926i 0 −1.36603 + 0.366025i 0 −0.707107 − 1.22474i 0 −0.866025 − 0.500000i 0
319.2 0 0.258819 − 0.965926i 0 −1.36603 + 0.366025i 0 0.707107 + 1.22474i 0 −0.866025 − 0.500000i 0
1087.1 0 −0.965926 + 0.258819i 0 0.366025 − 1.36603i 0 0.707107 − 1.22474i 0 0.866025 − 0.500000i 0
1087.2 0 0.965926 − 0.258819i 0 0.366025 − 1.36603i 0 −0.707107 + 1.22474i 0 0.866025 − 0.500000i 0
1471.1 0 −0.965926 − 0.258819i 0 0.366025 + 1.36603i 0 0.707107 + 1.22474i 0 0.866025 + 0.500000i 0
1471.2 0 0.965926 + 0.258819i 0 0.366025 + 1.36603i 0 −0.707107 − 1.22474i 0 0.866025 + 0.500000i 0
2239.1 0 −0.258819 − 0.965926i 0 −1.36603 − 0.366025i 0 −0.707107 + 1.22474i 0 −0.866025 + 0.500000i 0
2239.2 0 0.258819 + 0.965926i 0 −1.36603 − 0.366025i 0 0.707107 − 1.22474i 0 −0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner
36.f odd 6 1 inner
144.v odd 12 1 inner
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.1.z.a ✓ 8
4.b odd 2 1 inner 2304.1.z.a ✓ 8
8.b even 2 1 2304.1.z.b yes 8
8.d odd 2 1 2304.1.z.b yes 8
9.c even 3 1 inner 2304.1.z.a ✓ 8
16.e even 4 1 inner 2304.1.z.a ✓ 8
16.e even 4 1 2304.1.z.b yes 8
16.f odd 4 1 inner 2304.1.z.a ✓ 8
16.f odd 4 1 2304.1.z.b yes 8
36.f odd 6 1 inner 2304.1.z.a ✓ 8
72.n even 6 1 2304.1.z.b yes 8
72.p odd 6 1 2304.1.z.b yes 8
144.v odd 12 1 inner 2304.1.z.a ✓ 8
144.v odd 12 1 2304.1.z.b yes 8
144.x even 12 1 inner 2304.1.z.a ✓ 8
144.x even 12 1 2304.1.z.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2304.1.z.a ✓ 8 1.a even 1 1 trivial
2304.1.z.a ✓ 8 4.b odd 2 1 inner
2304.1.z.a ✓ 8 9.c even 3 1 inner
2304.1.z.a ✓ 8 16.e even 4 1 inner
2304.1.z.a ✓ 8 16.f odd 4 1 inner
2304.1.z.a ✓ 8 36.f odd 6 1 inner
2304.1.z.a ✓ 8 144.v odd 12 1 inner
2304.1.z.a ✓ 8 144.x even 12 1 inner
2304.1.z.b yes 8 8.b even 2 1
2304.1.z.b yes 8 8.d odd 2 1
2304.1.z.b yes 8 16.e even 4 1
2304.1.z.b yes 8 16.f odd 4 1
2304.1.z.b yes 8 72.n even 6 1
2304.1.z.b yes 8 72.p odd 6 1
2304.1.z.b yes 8 144.v odd 12 1
2304.1.z.b yes 8 144.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2T_{5}^{3} + 2T_{5}^{2} + 4T_{5} + 4 \) acting on \(S_{1}^{\mathrm{new}}(2304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T - 1)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
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