Properties

Label 3024.2.t.h
Level $3024$
Weight $2$
Character orbit 3024.t
Analytic conductor $24.147$
Analytic rank $0$
Dimension $6$
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(289,3024)]
 
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, 4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.t (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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 126)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{7} - \beta_{3} q^{11} + (2 \beta_{5} - 3 \beta_{4} + \beta_{2} + 3) q^{13} + (2 \beta_{5} - 2 \beta_{4} + 2) q^{17} + ( - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{19} + ( - \beta_1 + 2) q^{23} + ( - 2 \beta_{3} + \beta_1 - 1) q^{25} + (3 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{29} + ( - \beta_{5} - 6 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{31} + (3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{35} + \beta_{4} q^{37} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - 1) q^{41} + (3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{43} + (3 \beta_{5} + 3 \beta_{4} + 3 \beta_{2} - 3) q^{47} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{49} + (\beta_{5} + 5 \beta_{4} + \beta_{2} - 5) q^{53} + ( - 2 \beta_{3} + \beta_1 + 4) q^{55} + ( - \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{59} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{2} + 3) q^{61} + ( - \beta_{5} - 2 \beta_{4} + 5 \beta_{2} + 2) q^{65} + (\beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - \beta_1) q^{67} + ( - 2 \beta_{3} + 7 \beta_1 + 4) q^{71} + (\beta_{5} - 8 \beta_{4} - 4 \beta_{2} + 8) q^{73} + (3 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{77} + (4 \beta_{5} - \beta_{2}) q^{79} + (2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{83} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 2) q^{85} + (4 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1) q^{89} + (3 \beta_{5} - \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - \beta_1 + 8) q^{91} + (3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{95} + (2 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{5} + 4 q^{7} + 2 q^{11} + 8 q^{13} + 4 q^{17} + 3 q^{19} + 14 q^{23} - 4 q^{25} + 5 q^{29} - 20 q^{31} - 13 q^{35} + 3 q^{37} + 6 q^{43} - 9 q^{47} - 12 q^{49} - 15 q^{53} + 26 q^{55} - 14 q^{59} + 8 q^{61} + 12 q^{65} - q^{67} + 14 q^{71} + 19 q^{73} - 13 q^{77} - 5 q^{79} + 2 q^{83} - 2 q^{85} + 9 q^{89} + 46 q^{91} - 4 q^{95} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
0 0 0 −1.58836 0 2.64400 0.0963576i 0 0 0
289.2 0 0 0 −0.593579 0 0.0665372 + 2.64491i 0 0 0
289.3 0 0 0 3.18194 0 −0.710533 2.54856i 0 0 0
1873.1 0 0 0 −1.58836 0 2.64400 + 0.0963576i 0 0 0
1873.2 0 0 0 −0.593579 0 0.0665372 2.64491i 0 0 0
1873.3 0 0 0 3.18194 0 −0.710533 + 2.54856i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g 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.t.h 6
3.b odd 2 1 1008.2.t.h 6
4.b odd 2 1 378.2.h.c 6
7.c even 3 1 3024.2.q.g 6
9.c even 3 1 3024.2.q.g 6
9.d odd 6 1 1008.2.q.g 6
12.b even 2 1 126.2.h.d yes 6
21.h odd 6 1 1008.2.q.g 6
28.d even 2 1 2646.2.h.o 6
28.f even 6 1 2646.2.e.p 6
28.f even 6 1 2646.2.f.m 6
28.g odd 6 1 378.2.e.d 6
28.g odd 6 1 2646.2.f.l 6
36.f odd 6 1 378.2.e.d 6
36.f odd 6 1 1134.2.g.l 6
36.h even 6 1 126.2.e.c 6
36.h even 6 1 1134.2.g.m 6
63.g even 3 1 inner 3024.2.t.h 6
63.n odd 6 1 1008.2.t.h 6
84.h odd 2 1 882.2.h.p 6
84.j odd 6 1 882.2.e.o 6
84.j odd 6 1 882.2.f.o 6
84.n even 6 1 126.2.e.c 6
84.n even 6 1 882.2.f.n 6
252.n even 6 1 2646.2.h.o 6
252.n even 6 1 7938.2.a.bz 3
252.o even 6 1 126.2.h.d yes 6
252.o even 6 1 7938.2.a.bv 3
252.r odd 6 1 882.2.f.o 6
252.s odd 6 1 882.2.e.o 6
252.u odd 6 1 1134.2.g.l 6
252.u odd 6 1 2646.2.f.l 6
252.bb even 6 1 882.2.f.n 6
252.bb even 6 1 1134.2.g.m 6
252.bi even 6 1 2646.2.e.p 6
252.bj even 6 1 2646.2.f.m 6
252.bl odd 6 1 378.2.h.c 6
252.bl odd 6 1 7938.2.a.ca 3
252.bn odd 6 1 882.2.h.p 6
252.bn odd 6 1 7938.2.a.bw 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 36.h even 6 1
126.2.e.c 6 84.n even 6 1
126.2.h.d yes 6 12.b even 2 1
126.2.h.d yes 6 252.o even 6 1
378.2.e.d 6 28.g odd 6 1
378.2.e.d 6 36.f odd 6 1
378.2.h.c 6 4.b odd 2 1
378.2.h.c 6 252.bl odd 6 1
882.2.e.o 6 84.j odd 6 1
882.2.e.o 6 252.s odd 6 1
882.2.f.n 6 84.n even 6 1
882.2.f.n 6 252.bb even 6 1
882.2.f.o 6 84.j odd 6 1
882.2.f.o 6 252.r odd 6 1
882.2.h.p 6 84.h odd 2 1
882.2.h.p 6 252.bn odd 6 1
1008.2.q.g 6 9.d odd 6 1
1008.2.q.g 6 21.h odd 6 1
1008.2.t.h 6 3.b odd 2 1
1008.2.t.h 6 63.n odd 6 1
1134.2.g.l 6 36.f odd 6 1
1134.2.g.l 6 252.u odd 6 1
1134.2.g.m 6 36.h even 6 1
1134.2.g.m 6 252.bb even 6 1
2646.2.e.p 6 28.f even 6 1
2646.2.e.p 6 252.bi even 6 1
2646.2.f.l 6 28.g odd 6 1
2646.2.f.l 6 252.u odd 6 1
2646.2.f.m 6 28.f even 6 1
2646.2.f.m 6 252.bj even 6 1
2646.2.h.o 6 28.d even 2 1
2646.2.h.o 6 252.n even 6 1
3024.2.q.g 6 7.c even 3 1
3024.2.q.g 6 9.c even 3 1
3024.2.t.h 6 1.a even 1 1 trivial
3024.2.t.h 6 63.g even 3 1 inner
7938.2.a.bv 3 252.o even 6 1
7938.2.a.bw 3 252.bn odd 6 1
7938.2.a.bz 3 252.n even 6 1
7938.2.a.ca 3 252.bl odd 6 1

Hecke kernels

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

\( T_{5}^{3} - T_{5}^{2} - 6T_{5} - 3 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 6T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} - T^{2} - 6 T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 4 T^{5} + 14 T^{4} - 55 T^{3} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( (T^{3} - T^{2} - 6 T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 8 T^{5} + 63 T^{4} + \cdots + 4761 \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + 28 T^{4} + 240 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( (T^{3} - 7 T^{2} + 12 T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 5 T^{5} + 91 T^{4} + \cdots + 131769 \) Copy content Toggle raw display
$31$ \( T^{6} + 20 T^{5} + 279 T^{4} + \cdots + 40401 \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{6} + 33 T^{4} - 18 T^{3} + 1089 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} - 6 T^{5} + 105 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} + 135 T^{4} + \cdots + 35721 \) Copy content Toggle raw display
$53$ \( T^{6} + 15 T^{5} + 159 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{6} + 14 T^{5} + 157 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$61$ \( T^{6} - 8 T^{5} + 69 T^{4} + \cdots + 8649 \) Copy content Toggle raw display
$67$ \( T^{6} + T^{5} + 113 T^{4} + \cdots + 44521 \) Copy content Toggle raw display
$71$ \( (T^{3} - 7 T^{2} - 198 T + 1593)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 19 T^{5} + 353 T^{4} + \cdots + 398161 \) Copy content Toggle raw display
$79$ \( T^{6} + 5 T^{5} + 99 T^{4} + \cdots + 103041 \) Copy content Toggle raw display
$83$ \( T^{6} - 2 T^{5} + 67 T^{4} + \cdots + 21609 \) Copy content Toggle raw display
$89$ \( T^{6} - 9 T^{5} + 123 T^{4} + 396 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$97$ \( T^{6} - 28 T^{5} + 572 T^{4} + \cdots + 61504 \) Copy content Toggle raw display
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