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Properties

Label 3024.2.b.u
Level $3024$
Weight $2$
Character orbit 3024.b
Analytic conductor $24.147$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,2,Mod(1567,3024)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.b (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: 8.0.4161798144.13
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{6} + 29x^{4} + 42x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{5} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{5} - 1) q^{7} + \beta_{2} q^{11} + (\beta_{5} + \beta_{3}) q^{13} + (\beta_{2} - \beta_1) q^{17} + (\beta_{4} + 1) q^{19} + (\beta_{2} - 2 \beta_1) q^{23} + ( - \beta_{4} - 4) q^{25} + \beta_{6} q^{29} + (\beta_{4} - 1) q^{31} + (\beta_{6} - \beta_1) q^{35} + q^{37} + ( - 2 \beta_{2} + \beta_1) q^{41} + (\beta_{5} + 2 \beta_{3}) q^{43} + \beta_{7} q^{47} + (2 \beta_{5} - 5) q^{49} + (\beta_{7} + \beta_{6}) q^{53} + (2 \beta_{4} - 3) q^{55} - \beta_{6} q^{59} - \beta_{3} q^{61} + ( - \beta_{7} - 2 \beta_{6}) q^{65} + (3 \beta_{5} + \beta_{3}) q^{67} + \beta_1 q^{71} + (3 \beta_{5} - 2 \beta_{3}) q^{73} + ( - \beta_{7} - \beta_{2}) q^{77} + ( - \beta_{5} + 2 \beta_{3}) q^{79} - \beta_{7} q^{83} + (3 \beta_{4} + 6) q^{85} + (2 \beta_{2} - 3 \beta_1) q^{89} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 6) q^{91} + ( - 3 \beta_{2} + 4 \beta_1) q^{95} + (4 \beta_{5} + \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 8 q^{19} - 32 q^{25} - 8 q^{31} + 8 q^{37} - 40 q^{49} - 24 q^{55} + 48 q^{85} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 29x^{4} + 42x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -6\nu^{7} - 29\nu^{5} - 174\nu^{3} - 455\nu ) / 203 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -10\nu^{7} - 116\nu^{5} - 493\nu^{3} - 1232\nu ) / 203 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24\nu^{6} + 116\nu^{4} + 696\nu^{2} + 602 ) / 203 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} - 135 ) / 29 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 6\nu^{4} - 22\nu^{2} - 21 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -54\nu^{7} - 261\nu^{5} - 957\nu^{3} - 441\nu ) / 203 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 90\nu^{7} + 435\nu^{5} + 2001\nu^{3} + 735\nu ) / 203 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} - 6\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{5} - 2\beta_{4} + 9\beta_{3} - 18 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 5\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12\beta_{5} - 4\beta_{4} - 11\beta_{3} - 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{7} - 23\beta_{6} - 36\beta_{2} + 102\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 29\beta_{4} + 135 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -45\beta_{7} - 103\beta_{6} + 174\beta_{2} - 444\beta_1 ) / 12 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
1.05050 + 1.81952i
−1.05050 + 1.81952i
−0.629640 + 1.09057i
0.629640 + 1.09057i
0.629640 − 1.09057i
−0.629640 − 1.09057i
−1.05050 − 1.81952i
1.05050 − 1.81952i
0 0 0 − 3.63904i 0 −1.00000 − 2.44949i 0 0 0
1567.2 0 0 0 − 3.63904i 0 −1.00000 + 2.44949i 0 0 0
1567.3 0 0 0 − 2.18114i 0 −1.00000 − 2.44949i 0 0 0
1567.4 0 0 0 − 2.18114i 0 −1.00000 + 2.44949i 0 0 0
1567.5 0 0 0 2.18114i 0 −1.00000 − 2.44949i 0 0 0
1567.6 0 0 0 2.18114i 0 −1.00000 + 2.44949i 0 0 0
1567.7 0 0 0 3.63904i 0 −1.00000 − 2.44949i 0 0 0
1567.8 0 0 0 3.63904i 0 −1.00000 + 2.44949i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.b.u ✓ 8
3.b odd 2 1 inner 3024.2.b.u ✓ 8
4.b odd 2 1 3024.2.b.v yes 8
7.b odd 2 1 3024.2.b.v yes 8
12.b even 2 1 3024.2.b.v yes 8
21.c even 2 1 3024.2.b.v yes 8
28.d even 2 1 inner 3024.2.b.u ✓ 8
84.h odd 2 1 inner 3024.2.b.u ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.b.u ✓ 8 1.a even 1 1 trivial
3024.2.b.u ✓ 8 3.b odd 2 1 inner
3024.2.b.u ✓ 8 28.d even 2 1 inner
3024.2.b.u ✓ 8 84.h odd 2 1 inner
3024.2.b.v yes 8 4.b odd 2 1
3024.2.b.v yes 8 7.b odd 2 1
3024.2.b.v yes 8 12.b even 2 1
3024.2.b.v yes 8 21.c even 2 1

Hecke kernels

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

\( T_{5}^{4} + 18T_{5}^{2} + 63 \) Copy content Toggle raw display
\( T_{11}^{4} + 30T_{11}^{2} + 63 \) Copy content Toggle raw display
\( T_{13}^{4} + 36T_{13}^{2} + 36 \) Copy content Toggle raw display
\( T_{17}^{4} + 36T_{17}^{2} + 252 \) Copy content Toggle raw display
\( T_{19}^{2} - 2T_{19} - 17 \) Copy content Toggle raw display
\( T_{29}^{4} - 108T_{29}^{2} + 2268 \) Copy content Toggle raw display
\( T_{47}^{4} - 180T_{47}^{2} + 2268 \) 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
$5$ \( (T^{4} + 18 T^{2} + 63)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 30 T^{2} + 63)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 36 T^{2} + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 36 T^{2} + 252)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 17)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 78 T^{2} + 63)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 108 T^{2} + 2268)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 17)^{4} \) Copy content Toggle raw display
$37$ \( (T - 1)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 114 T^{2} + 3087)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 108 T^{2} + 1764)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 180 T^{2} + 2268)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 216 T^{2} + 9072)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 108 T^{2} + 2268)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 132 T^{2} + 1764)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 18 T^{2} + 63)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 204 T^{2} + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 108 T^{2} + 1764)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 180 T^{2} + 2268)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 210 T^{2} + 3087)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 216 T^{2} + 7056)^{2} \) Copy content Toggle raw display
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