Properties

Label 3024.2.k.g
Level 3024
Weight 2
Character orbit 3024.k
Analytic conductor 24.147
Analytic rank 0
Dimension 4
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( -1 - \beta_{3} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( -1 - \beta_{3} ) q^{7} -\beta_{1} q^{11} + \beta_{3} q^{13} + 3 \beta_{2} q^{17} + 3 \beta_{3} q^{19} + 2 \beta_{1} q^{23} -2 q^{25} -5 \beta_{1} q^{29} -\beta_{3} q^{31} + ( 3 \beta_{1} + \beta_{2} ) q^{35} + 5 q^{37} + 5 \beta_{2} q^{41} -5 q^{43} + 5 \beta_{2} q^{47} + ( -5 + 2 \beta_{3} ) q^{49} -8 \beta_{1} q^{53} + \beta_{3} q^{55} -5 \beta_{2} q^{59} -\beta_{3} q^{61} -3 \beta_{1} q^{65} -2 q^{67} -10 \beta_{1} q^{71} + ( \beta_{1} - 2 \beta_{2} ) q^{77} + 13 q^{79} -\beta_{2} q^{83} -9 q^{85} -6 \beta_{2} q^{89} + ( 6 - \beta_{3} ) q^{91} -9 \beta_{1} q^{95} -7 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{7} - 8q^{25} + 20q^{37} - 20q^{43} - 20q^{49} - 8q^{67} + 52q^{79} - 36q^{85} + 24q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 5 \beta_{1}\)\()/2\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
0.517638i
0.517638i
1.93185i
1.93185i
0 0 0 −1.73205 0 −1.00000 2.44949i 0 0 0
1889.2 0 0 0 −1.73205 0 −1.00000 + 2.44949i 0 0 0
1889.3 0 0 0 1.73205 0 −1.00000 2.44949i 0 0 0
1889.4 0 0 0 1.73205 0 −1.00000 + 2.44949i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c 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.k.g 4
3.b odd 2 1 inner 3024.2.k.g 4
4.b odd 2 1 189.2.c.c 4
7.b odd 2 1 inner 3024.2.k.g 4
12.b even 2 1 189.2.c.c 4
21.c even 2 1 inner 3024.2.k.g 4
28.d even 2 1 189.2.c.c 4
36.f odd 6 2 567.2.o.e 8
36.h even 6 2 567.2.o.e 8
84.h odd 2 1 189.2.c.c 4
252.s odd 6 2 567.2.o.e 8
252.bi even 6 2 567.2.o.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.c 4 4.b odd 2 1
189.2.c.c 4 12.b even 2 1
189.2.c.c 4 28.d even 2 1
189.2.c.c 4 84.h odd 2 1
567.2.o.e 8 36.f odd 6 2
567.2.o.e 8 36.h even 6 2
567.2.o.e 8 252.s odd 6 2
567.2.o.e 8 252.bi even 6 2
3024.2.k.g 4 1.a even 1 1 trivial
3024.2.k.g 4 3.b odd 2 1 inner
3024.2.k.g 4 7.b odd 2 1 inner
3024.2.k.g 4 21.c 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 \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{2} - 3 \)
\( T_{11}^{2} + 2 \)
\( T_{13}^{2} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 20 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 7 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 16 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 38 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 8 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 56 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 5 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 7 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 5 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 19 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 22 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 43 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 116 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 2 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 58 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 13 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 163 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 100 T^{2} + 9409 T^{4} )^{2} \)
show more
show less