Properties

Label 3024.2.h.b
Level $3024$
Weight $2$
Character orbit 3024.h
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(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: 8.0.19752615936.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{6} + 195x^{4} - 976x^{2} + 3721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} - \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} - \beta_{6} q^{7} + \beta_1 q^{11} + ( - \beta_{5} - 1) q^{13} + ( - \beta_{7} + \beta_{2}) q^{17} + (5 \beta_{6} - 2 \beta_{4}) q^{19} + \beta_1 q^{23} + ( - \beta_{5} - 3) q^{25} + (\beta_{7} - \beta_{2}) q^{29} + (3 \beta_{6} + 2 \beta_{4}) q^{31} + \beta_1 q^{35} - q^{37} + ( - 2 \beta_{7} + \beta_{2}) q^{41} + (5 \beta_{6} - \beta_{4}) q^{43} + (\beta_{3} + 3 \beta_1) q^{47} - q^{49} + 2 \beta_{2} q^{53} + (8 \beta_{6} + \beta_{4}) q^{55} + ( - \beta_{3} - \beta_1) q^{59} - 6 \beta_{5} q^{61} + (\beta_{7} + \beta_{2}) q^{65} + (3 \beta_{6} + \beta_{4}) q^{67} + (\beta_{3} - 2 \beta_1) q^{71} + (3 \beta_{5} + 7) q^{73} + \beta_{2} q^{77} + (3 \beta_{6} + \beta_{4}) q^{79} + ( - \beta_{3} - \beta_1) q^{83} + ( - 7 \beta_{5} + 5) q^{85} - 5 \beta_{2} q^{89} + (\beta_{6} + \beta_{4}) q^{91} + ( - 2 \beta_{3} - 5 \beta_1) q^{95} + ( - 6 \beta_{5} + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{13} - 24 q^{25} - 8 q^{37} - 8 q^{49} + 56 q^{73} + 40 q^{85} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{6} + 195x^{4} - 976x^{2} + 3721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{7} + 520\nu^{5} - 4355\nu^{3} + 33672\nu ) / 11895 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\nu^{7} - 195\nu^{5} + 3120\nu^{3} - 3721\nu ) / 11895 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -16\nu^{7} + 195\nu^{5} - 3120\nu^{3} + 27511\nu ) / 11895 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32\nu^{6} - 390\nu^{4} + 6240\nu^{2} - 19337 ) / 11895 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 584 ) / 195 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 16\nu^{4} - 134\nu^{2} + 488 ) / 183 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -128\nu^{7} + 1560\nu^{5} - 13065\nu^{3} + 29768\nu ) / 11895 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} + \beta_{5} + 8\beta_{4} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 8\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 48\beta_{6} - 16\beta_{5} + 67\beta_{4} - 67 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16\beta_{7} - 67\beta_{3} + 67\beta_{2} + 48\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -195\beta_{5} - 584 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -195\beta_{7} - 584\beta_{3} - 584\beta_{2} + 585\beta_1 ) / 2 \) Copy content Toggle raw display

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
2.70167 + 1.55981i
−2.70167 + 1.55981i
−2.16817 + 1.25179i
2.16817 + 1.25179i
2.16817 1.25179i
−2.16817 1.25179i
−2.70167 1.55981i
2.70167 1.55981i
0 0 0 3.11962i 0 1.00000i 0 0 0
2591.2 0 0 0 3.11962i 0 1.00000i 0 0 0
2591.3 0 0 0 2.50359i 0 1.00000i 0 0 0
2591.4 0 0 0 2.50359i 0 1.00000i 0 0 0
2591.5 0 0 0 2.50359i 0 1.00000i 0 0 0
2591.6 0 0 0 2.50359i 0 1.00000i 0 0 0
2591.7 0 0 0 3.11962i 0 1.00000i 0 0 0
2591.8 0 0 0 3.11962i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
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.b 8
3.b odd 2 1 inner 3024.2.h.b 8
4.b odd 2 1 inner 3024.2.h.b 8
12.b even 2 1 inner 3024.2.h.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.h.b 8 1.a even 1 1 trivial
3024.2.h.b 8 3.b odd 2 1 inner
3024.2.h.b 8 4.b odd 2 1 inner
3024.2.h.b 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 \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{4} + 16T_{5}^{2} + 61 \) Copy content Toggle raw display
\( T_{11}^{4} - 16T_{11}^{2} + 61 \) 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
$5$ \( (T^{4} + 16 T^{2} + 61)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{2} + 61)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 52 T^{2} + 244)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 74 T^{2} + 169)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 16 T^{2} + 61)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 52 T^{2} + 244)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 42 T^{2} + 9)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 184 T^{2} + 7381)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 56 T^{2} + 484)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 156 T^{2} + 2196)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 64 T^{2} + 976)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 52 T^{2} + 244)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 24 T^{2} + 36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 136 T^{2} + 61)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 14 T + 22)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 24 T^{2} + 36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 52 T^{2} + 244)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 400 T^{2} + 38125)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T - 44)^{4} \) Copy content Toggle raw display
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