Properties

Label 3240.2.q.ba
Level $3240$
Weight $2$
Character orbit 3240.q
Analytic conductor $25.872$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.8715302549\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + \beta_{3} q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + 2 \beta_{2} q^{17} + q^{19} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} -\beta_{1} q^{25} + \beta_{3} q^{29} + ( -1 + 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{31} -\beta_{2} q^{35} + 6 q^{37} + ( -7 + 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} -2 \beta_{3} q^{43} + ( 7 \beta_{1} - \beta_{3} ) q^{47} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{49} -3 \beta_{2} q^{53} + ( -1 - \beta_{2} ) q^{55} + ( -5 + 5 \beta_{1} ) q^{59} + ( -8 \beta_{1} + 2 \beta_{3} ) q^{61} + ( \beta_{1} - \beta_{3} ) q^{65} + ( -6 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -1 + \beta_{2} ) q^{71} + ( 8 - 2 \beta_{2} ) q^{73} + ( -8 + 8 \beta_{1} ) q^{77} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{79} + 10 \beta_{1} q^{83} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -1 + \beta_{2} ) q^{89} + ( -8 + \beta_{2} ) q^{91} + ( -1 + \beta_{1} ) q^{95} -2 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - q^{7} + O(q^{10}) \) \( 4q - 2q^{5} - q^{7} + q^{11} + q^{13} - 4q^{17} + 4q^{19} - 5q^{23} - 2q^{25} + q^{29} - 5q^{31} + 2q^{35} + 24q^{37} - 12q^{41} - 2q^{43} + 13q^{47} - 3q^{49} + 6q^{53} - 2q^{55} - 10q^{59} - 14q^{61} + q^{65} - 10q^{67} - 6q^{71} + 36q^{73} - 16q^{77} - 6q^{79} + 20q^{83} + 2q^{85} - 6q^{89} - 34q^{91} - 2q^{95} - 2q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1081.1
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
0 0 0 −0.500000 + 0.866025i 0 −1.68614 2.92048i 0 0 0
1081.2 0 0 0 −0.500000 + 0.866025i 0 1.18614 + 2.05446i 0 0 0
2161.1 0 0 0 −0.500000 0.866025i 0 −1.68614 + 2.92048i 0 0 0
2161.2 0 0 0 −0.500000 0.866025i 0 1.18614 2.05446i 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
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 3240.2.q.ba 4
3.b odd 2 1 3240.2.q.bd 4
9.c even 3 1 3240.2.a.m yes 2
9.c even 3 1 inner 3240.2.q.ba 4
9.d odd 6 1 3240.2.a.i 2
9.d odd 6 1 3240.2.q.bd 4
36.f odd 6 1 6480.2.a.bo 2
36.h even 6 1 6480.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.i 2 9.d odd 6 1
3240.2.a.m yes 2 9.c even 3 1
3240.2.q.ba 4 1.a even 1 1 trivial
3240.2.q.ba 4 9.c even 3 1 inner
3240.2.q.bd 4 3.b odd 2 1
3240.2.q.bd 4 9.d odd 6 1
6480.2.a.bd 2 36.h even 6 1
6480.2.a.bo 2 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3240, [\chi])\):

\( T_{7}^{4} + T_{7}^{3} + 9 T_{7}^{2} - 8 T_{7} + 64 \)
\( T_{11}^{4} - T_{11}^{3} + 9 T_{11}^{2} + 8 T_{11} + 64 \)
\( T_{17}^{2} + 2 T_{17} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( 64 - 8 T + 9 T^{2} + T^{3} + T^{4} \)
$11$ \( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} \)
$13$ \( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} \)
$17$ \( ( -32 + 2 T + T^{2} )^{2} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 4 - 10 T + 27 T^{2} + 5 T^{3} + T^{4} \)
$29$ \( 64 + 8 T + 9 T^{2} - T^{3} + T^{4} \)
$31$ \( 4624 - 340 T + 93 T^{2} + 5 T^{3} + T^{4} \)
$37$ \( ( -6 + T )^{4} \)
$41$ \( 9 + 36 T + 141 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( 1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4} \)
$47$ \( 1156 - 442 T + 135 T^{2} - 13 T^{3} + T^{4} \)
$53$ \( ( -72 - 3 T + T^{2} )^{2} \)
$59$ \( ( 25 + 5 T + T^{2} )^{2} \)
$61$ \( 256 + 224 T + 180 T^{2} + 14 T^{3} + T^{4} \)
$67$ \( 64 - 80 T + 108 T^{2} + 10 T^{3} + T^{4} \)
$71$ \( ( -6 + 3 T + T^{2} )^{2} \)
$73$ \( ( 48 - 18 T + T^{2} )^{2} \)
$79$ \( 576 - 144 T + 60 T^{2} + 6 T^{3} + T^{4} \)
$83$ \( ( 100 - 10 T + T^{2} )^{2} \)
$89$ \( ( -6 + 3 T + T^{2} )^{2} \)
$97$ \( 1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4} \)
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