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Properties

Label 3024.2.h.f
Level $3024$
Weight $2$
Character orbit 3024.h
Analytic conductor $24.147$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,2,Mod(2591,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.h (of order \(2\), degree \(1\), 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: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.195105024.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (2 \beta_{7} + \beta_{3}) q^{5} - \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{7} + \beta_{3}) q^{5} - \beta_{7} q^{7} + (\beta_{2} + 2) q^{11} + (\beta_{6} + 2 \beta_{2} - \beta_1 + 1) q^{13} + (\beta_{7} + 2 \beta_{5} - 2 \beta_{4}) q^{17} + ( - 4 \beta_{7} + \beta_{4} - 2 \beta_{3}) q^{19} + ( - \beta_1 + 4) q^{23} + (2 \beta_{6} + 3 \beta_1) q^{25} + ( - 4 \beta_{7} + \beta_{5} + \cdots - 3 \beta_{3}) q^{29}+ \cdots + (4 \beta_{6} + \beta_{2} - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{11} + 4 q^{13} + 36 q^{23} - 12 q^{25} + 12 q^{35} - 20 q^{37} + 24 q^{47} - 8 q^{49} + 48 q^{59} + 36 q^{61} + 36 q^{71} - 16 q^{73} + 72 q^{83} + 28 q^{85} + 84 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} + 10\nu^{5} + 2\nu^{4} - 22\nu^{3} + 18\nu^{2} + 9\nu - 108 ) / 54 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 2\nu^{4} + 6\nu^{3} - 2\nu^{2} - 9\nu + 18 ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 4\nu^{6} + 5\nu^{5} + 4\nu^{4} - 20\nu^{3} + 12\nu^{2} + 72\nu - 108 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{7} + 7\nu^{6} - 2\nu^{5} - 16\nu^{4} + 8\nu^{3} + 60\nu^{2} - 90\nu + 81 ) / 54 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{7} - 8\nu^{6} + 4\nu^{5} + 26\nu^{4} - 52\nu^{3} - 18\nu^{2} + 153\nu - 216 ) / 54 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 8\nu^{6} - 2\nu^{5} - 20\nu^{4} + 38\nu^{3} + 32\nu^{2} - 141\nu + 144 ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\nu^{7} - 25\nu^{6} + 8\nu^{5} + 70\nu^{4} - 104\nu^{3} - 54\nu^{2} + 378\nu - 459 ) / 108 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + \beta_{4} + \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 10\beta_{7} + 6\beta_{6} + \beta_{4} - 3\beta_{3} + 4\beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8\beta_{6} + 4\beta_{5} - 8\beta_{4} + 3\beta_{3} + 5\beta_{2} + 7\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{7} + 4\beta_{6} + 12\beta_{5} + 8\beta_{2} - 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -16\beta_{7} + 24\beta_{6} + 36\beta_{5} - 16\beta_{4} + 7\beta_{3} - 9\beta_{2} - 11\beta _1 + 19 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
1.30512 − 1.13871i
1.72124 − 0.193255i
0.560908 − 1.63871i
−1.58726 + 0.693255i
−1.58726 − 0.693255i
0.560908 + 1.63871i
1.72124 + 0.193255i
1.30512 + 1.13871i
0 0 0 − 4.27743i 0 1.00000i 0 0 0
2591.2 0 0 0 − 2.38651i 0 1.00000i 0 0 0
2591.3 0 0 0 − 1.27743i 0 − 1.00000i 0 0 0
2591.4 0 0 0 − 0.613491i 0 1.00000i 0 0 0
2591.5 0 0 0 0.613491i 0 − 1.00000i 0 0 0
2591.6 0 0 0 1.27743i 0 1.00000i 0 0 0
2591.7 0 0 0 2.38651i 0 − 1.00000i 0 0 0
2591.8 0 0 0 4.27743i 0 − 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.h.f yes 8
3.b odd 2 1 3024.2.h.a ✓ 8
4.b odd 2 1 3024.2.h.a ✓ 8
12.b even 2 1 inner 3024.2.h.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.h.a ✓ 8 3.b odd 2 1
3024.2.h.a ✓ 8 4.b odd 2 1
3024.2.h.f yes 8 1.a even 1 1 trivial
3024.2.h.f yes 8 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{8} + 26T_{5}^{6} + 153T_{5}^{4} + 224T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{4} - 6T_{11}^{3} + 2T_{11}^{2} + 24T_{11} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 26 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 6 T^{3} + 2 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} - 33 T^{2} + \cdots + 46)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 92 T^{6} + \cdots + 76729 \) Copy content Toggle raw display
$19$ \( T^{8} + 88 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{4} - 18 T^{3} + \cdots + 244)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 206 T^{6} + \cdots + 114244 \) Copy content Toggle raw display
$31$ \( T^{8} + 90 T^{6} + \cdots + 54756 \) Copy content Toggle raw display
$37$ \( (T^{4} + 10 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 146 T^{6} + \cdots + 59536 \) Copy content Toggle raw display
$43$ \( T^{8} + 220 T^{6} + \cdots + 231361 \) Copy content Toggle raw display
$47$ \( (T^{4} - 12 T^{3} + \cdots - 936)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 206 T^{6} + \cdots + 394384 \) Copy content Toggle raw display
$59$ \( (T^{4} - 24 T^{3} + \cdots - 263)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 18 T^{3} + \cdots - 13248)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 126 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$71$ \( (T^{4} - 18 T^{3} + \cdots - 3632)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + \cdots + 736)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 510 T^{6} + \cdots + 15397776 \) Copy content Toggle raw display
$83$ \( (T^{4} - 36 T^{3} + \cdots + 4996)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 26 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + \cdots + 472)^{2} \) Copy content Toggle raw display
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