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Properties

Label 3024.2.r.m
Level $3024$
Weight $2$
Character orbit 3024.r
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(1009,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.r (of order \(3\), degree \(2\), not 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.2091141441.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{6} + 3x^{5} - 15x^{4} + 9x^{3} + 9x^{2} - 27x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 504)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} + 1) q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2} + 1) q^{5} + \beta_{3} q^{7} + ( - 2 \beta_{3} - \beta_1) q^{11} + ( - \beta_{7} + \beta_{3} + 1) q^{13} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{17}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{5} - 4 q^{7} + 7 q^{11} + 3 q^{13} - 6 q^{17} + 8 q^{19} + 2 q^{23} - 5 q^{25} + 9 q^{29} - 3 q^{31} - 6 q^{35} + 6 q^{37} - 9 q^{41} - 8 q^{43} + 3 q^{47} - 4 q^{49} - 12 q^{53} + 56 q^{55} + 10 q^{59} + 20 q^{61} - q^{65} - 11 q^{67} - 6 q^{71} - 48 q^{73} + 7 q^{77} - 21 q^{79} + 8 q^{83} + 9 q^{85} - 12 q^{89} - 6 q^{91} + 36 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + x^{6} + 3x^{5} - 15x^{4} + 9x^{3} + 9x^{2} - 27x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} + 2\nu^{5} - 6\nu^{4} + 6\nu^{3} + 9\nu^{2} - 9\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - \nu^{5} + \nu^{4} + 3\nu^{3} - 6\nu^{2} + 9\nu + 9 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 2\nu^{6} - 2\nu^{5} - 3\nu^{4} + 3\nu^{3} - 18\nu^{2} + 9\nu + 54 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 2\nu^{6} - 7\nu^{5} + 15\nu^{4} - 12\nu^{3} - 9\nu^{2} + 72\nu - 54 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 5\nu^{6} - 4\nu^{5} + 12\nu^{4} - 21\nu^{3} + 36\nu^{2} + 72\nu - 81 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - \nu^{5} + 5\nu^{4} - 12\nu^{3} + 6\nu^{2} + 36\nu - 36 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 8\nu^{6} - 8\nu^{5} - 15\nu^{4} + 12\nu^{3} - 72\nu^{2} + 9\nu + 216 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} + 2\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 3\beta_{5} - 2\beta_{4} + \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} - 2\beta_{6} + 2\beta_{4} + 9\beta_{3} + 2\beta_{2} + 4\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} - \beta_{6} - 3\beta_{5} + 4\beta_{4} + \beta_{2} - 4\beta _1 + 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{7} + 10\beta_{6} - 12\beta_{5} - 7\beta_{4} + 18\beta_{3} + 8\beta_{2} + \beta _1 + 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{7} + 2\beta_{6} + 9\beta_{5} - 20\beta_{4} - 9\beta_{3} + 16\beta_{2} - 22\beta _1 - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -24\beta_{7} - 8\beta_{6} + 3\beta_{5} + 5\beta_{4} + 108\beta_{3} - 19\beta_{2} - 5\beta _1 - 18 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
1.65525 − 0.510048i
0.335492 − 1.69925i
0.199732 + 1.72050i
−1.69047 − 0.377226i
1.65525 + 0.510048i
0.335492 + 1.69925i
0.199732 − 1.72050i
−1.69047 + 0.377226i
0 0 0 −1.15525 + 2.00095i 0 −0.500000 − 0.866025i 0 0 0
1009.2 0 0 0 0.164508 − 0.284936i 0 −0.500000 − 0.866025i 0 0 0
1009.3 0 0 0 0.300268 − 0.520080i 0 −0.500000 − 0.866025i 0 0 0
1009.4 0 0 0 2.19047 − 3.79401i 0 −0.500000 − 0.866025i 0 0 0
2017.1 0 0 0 −1.15525 − 2.00095i 0 −0.500000 + 0.866025i 0 0 0
2017.2 0 0 0 0.164508 + 0.284936i 0 −0.500000 + 0.866025i 0 0 0
2017.3 0 0 0 0.300268 + 0.520080i 0 −0.500000 + 0.866025i 0 0 0
2017.4 0 0 0 2.19047 + 3.79401i 0 −0.500000 + 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1009.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.r.m 8
3.b odd 2 1 1008.2.r.l 8
4.b odd 2 1 1512.2.r.e 8
9.c even 3 1 inner 3024.2.r.m 8
9.c even 3 1 9072.2.a.cg 4
9.d odd 6 1 1008.2.r.l 8
9.d odd 6 1 9072.2.a.cj 4
12.b even 2 1 504.2.r.e ✓ 8
36.f odd 6 1 1512.2.r.e 8
36.f odd 6 1 4536.2.a.y 4
36.h even 6 1 504.2.r.e ✓ 8
36.h even 6 1 4536.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.e ✓ 8 12.b even 2 1
504.2.r.e ✓ 8 36.h even 6 1
1008.2.r.l 8 3.b odd 2 1
1008.2.r.l 8 9.d odd 6 1
1512.2.r.e 8 4.b odd 2 1
1512.2.r.e 8 36.f odd 6 1
3024.2.r.m 8 1.a even 1 1 trivial
3024.2.r.m 8 9.c even 3 1 inner
4536.2.a.y 4 36.f odd 6 1
4536.2.a.z 4 36.h even 6 1
9072.2.a.cg 4 9.c even 3 1
9072.2.a.cj 4 9.d odd 6 1

Hecke kernels

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

\( T_{5}^{8} - 3T_{5}^{7} + 17T_{5}^{6} + 6T_{5}^{5} + 93T_{5}^{4} - 84T_{5}^{3} + 65T_{5}^{2} - 18T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{8} - 7T_{11}^{7} + 42T_{11}^{6} - 73T_{11}^{5} + 148T_{11}^{4} - 126T_{11}^{3} + 249T_{11}^{2} - 180T_{11} + 225 \) 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
$5$ \( T^{8} - 3 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} - 7 T^{7} + \cdots + 225 \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} + \cdots + 100 \) Copy content Toggle raw display
$17$ \( (T^{4} + 3 T^{3} - 27 T^{2} + \cdots - 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + \cdots + 157)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( T^{8} - 9 T^{7} + \cdots + 944784 \) Copy content Toggle raw display
$31$ \( T^{8} + 3 T^{7} + \cdots + 481636 \) Copy content Toggle raw display
$37$ \( (T^{4} - 3 T^{3} - 8 T^{2} + \cdots - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 9 T^{7} + \cdots + 11881 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$47$ \( T^{8} - 3 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$53$ \( (T^{4} + 6 T^{3} + \cdots + 4068)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 10 T^{7} + \cdots + 19881 \) Copy content Toggle raw display
$61$ \( T^{8} - 20 T^{7} + \cdots + 34152336 \) Copy content Toggle raw display
$67$ \( T^{8} + 11 T^{7} + \cdots + 1265625 \) Copy content Toggle raw display
$71$ \( (T^{4} + 3 T^{3} - 8 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 24 T^{3} + \cdots - 2897)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 21 T^{7} + \cdots + 3928324 \) Copy content Toggle raw display
$83$ \( T^{8} - 8 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$89$ \( (T^{4} + 6 T^{3} + \cdots + 1674)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} + \cdots + 145161 \) Copy content Toggle raw display
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