Properties

Label 2736.2.bm.i
Level $2736$
Weight $2$
Character orbit 2736.bm
Analytic conductor $21.847$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.bm (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \beta_{2} ) q^{5} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( -2 + 2 \beta_{2} ) q^{5} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{7} + 2 \beta_{3} q^{11} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -2 + 2 \beta_{2} ) q^{17} + ( -3 - 2 \beta_{2} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{23} + \beta_{2} q^{25} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( -3 + 2 \beta_{1} - \beta_{3} ) q^{31} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{35} + ( 1 - 2 \beta_{2} - \beta_{3} ) q^{37} + ( -3 - 3 \beta_{2} ) q^{43} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -4 + 4 \beta_{1} - 2 \beta_{3} ) q^{49} + ( -8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{53} -4 \beta_{1} q^{55} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{59} + ( -\beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{61} + ( -2 + 4 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{67} + ( 4 - 4 \beta_{2} ) q^{71} + ( -3 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{73} + ( -16 + 4 \beta_{1} - 2 \beta_{3} ) q^{77} + ( 5 + \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{79} + ( -2 + 4 \beta_{2} ) q^{83} -4 \beta_{2} q^{85} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 2 \beta_{1} - 11 \beta_{2} + 2 \beta_{3} ) q^{91} + ( 10 - 6 \beta_{2} ) q^{95} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + O(q^{10}) \) \( 4q - 4q^{5} + 6q^{13} - 4q^{17} - 16q^{19} + 2q^{25} - 12q^{29} - 12q^{31} + 12q^{35} - 18q^{43} + 12q^{47} - 16q^{49} - 24q^{53} - 4q^{59} + 10q^{61} - 2q^{67} + 8q^{71} - 6q^{73} - 64q^{77} + 10q^{79} - 8q^{85} - 12q^{89} - 22q^{91} + 28q^{95} - 24q^{97} + O(q^{100}) \)

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−1.22474 0.707107i
1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0 0 0 −1.00000 + 1.73205i 0 4.56048i 0 0 0
559.2 0 0 0 −1.00000 + 1.73205i 0 1.09638i 0 0 0
1855.1 0 0 0 −1.00000 1.73205i 0 1.09638i 0 0 0
1855.2 0 0 0 −1.00000 1.73205i 0 4.56048i 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
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.bm.i 4
3.b odd 2 1 912.2.bb.d yes 4
4.b odd 2 1 2736.2.bm.j 4
12.b even 2 1 912.2.bb.c 4
19.d odd 6 1 2736.2.bm.j 4
57.f even 6 1 912.2.bb.c 4
76.f even 6 1 inner 2736.2.bm.i 4
228.n odd 6 1 912.2.bb.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bb.c 4 12.b even 2 1
912.2.bb.c 4 57.f even 6 1
912.2.bb.d yes 4 3.b odd 2 1
912.2.bb.d yes 4 228.n odd 6 1
2736.2.bm.i 4 1.a even 1 1 trivial
2736.2.bm.i 4 76.f even 6 1 inner
2736.2.bm.j 4 4.b odd 2 1
2736.2.bm.j 4 19.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 \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{7}^{4} + 22 T_{7}^{2} + 25 \)
\( T_{11}^{2} + 32 \)
\( T_{23}^{4} - 32 T_{23}^{2} + 1024 \)
\( T_{31}^{2} + 6 T_{31} - 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 4 + 2 T + T^{2} )^{2} \)
$7$ \( 25 + 22 T^{2} + T^{4} \)
$11$ \( ( 32 + T^{2} )^{2} \)
$13$ \( 25 + 30 T + 7 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( ( 4 + 2 T + T^{2} )^{2} \)
$19$ \( ( 19 + 8 T + T^{2} )^{2} \)
$23$ \( 1024 - 32 T^{2} + T^{4} \)
$29$ \( 400 - 240 T + 28 T^{2} + 12 T^{3} + T^{4} \)
$31$ \( ( -15 + 6 T + T^{2} )^{2} \)
$37$ \( 25 + 22 T^{2} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 27 + 9 T + T^{2} )^{2} \)
$47$ \( 400 + 240 T + 28 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( 256 + 384 T + 208 T^{2} + 24 T^{3} + T^{4} \)
$59$ \( 8464 - 368 T + 108 T^{2} + 4 T^{3} + T^{4} \)
$61$ \( 1 - 10 T + 99 T^{2} - 10 T^{3} + T^{4} \)
$67$ \( 9025 - 190 T + 99 T^{2} + 2 T^{3} + T^{4} \)
$71$ \( ( 16 - 4 T + T^{2} )^{2} \)
$73$ \( 7569 - 522 T + 123 T^{2} + 6 T^{3} + T^{4} \)
$79$ \( 1 - 10 T + 99 T^{2} - 10 T^{3} + T^{4} \)
$83$ \( ( 12 + T^{2} )^{2} \)
$89$ \( 400 - 240 T + 28 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( 6400 - 1920 T + 112 T^{2} + 24 T^{3} + T^{4} \)
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