Properties

Label 2736.2.f.e
Level $2736$
Weight $2$
Character orbit 2736.f
Analytic conductor $21.847$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{19})\)
Defining polynomial: \(x^{4} + 20 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} -\beta_{3} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} -\beta_{3} q^{7} + ( -\beta_{1} - \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{2} ) q^{17} + \beta_{3} q^{19} + ( 2 \beta_{1} + \beta_{2} ) q^{23} + ( -5 + \beta_{3} ) q^{25} + ( \beta_{1} - 3 \beta_{2} ) q^{35} + q^{43} + ( -\beta_{1} + 2 \beta_{2} ) q^{47} + 12 q^{49} + ( 7 + 2 \beta_{3} ) q^{55} + \beta_{3} q^{61} + 11 q^{73} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{77} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -13 + 4 \beta_{3} ) q^{85} + ( -\beta_{1} + 3 \beta_{2} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 20q^{25} + 4q^{43} + 48q^{49} + 28q^{55} + 44q^{73} - 52q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 20 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 10 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 10\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} - 11 \beta_{1}\)

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
3.78931i
2.37510i
2.37510i
3.78931i
0 0 0 3.78931i 0 4.35890 0 0 0
1025.2 0 0 0 2.37510i 0 −4.35890 0 0 0
1025.3 0 0 0 2.37510i 0 −4.35890 0 0 0
1025.4 0 0 0 3.78931i 0 4.35890 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
3.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.f.e 4
3.b odd 2 1 inner 2736.2.f.e 4
4.b odd 2 1 171.2.d.a 4
12.b even 2 1 171.2.d.a 4
19.b odd 2 1 CM 2736.2.f.e 4
57.d even 2 1 inner 2736.2.f.e 4
76.d even 2 1 171.2.d.a 4
228.b odd 2 1 171.2.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.d.a 4 4.b odd 2 1
171.2.d.a 4 12.b even 2 1
171.2.d.a 4 76.d even 2 1
171.2.d.a 4 228.b odd 2 1
2736.2.f.e 4 1.a even 1 1 trivial
2736.2.f.e 4 3.b odd 2 1 inner
2736.2.f.e 4 19.b odd 2 1 CM
2736.2.f.e 4 57.d 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 \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{4} + 20 T_{5}^{2} + 81 \)
\( T_{7}^{2} - 19 \)
\( T_{11}^{4} + 44 T_{11}^{2} + 9 \)
\( T_{29} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 81 + 20 T^{2} + T^{4} \)
$7$ \( ( -19 + T^{2} )^{2} \)
$11$ \( 9 + 44 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 225 + 68 T^{2} + T^{4} \)
$19$ \( ( -19 + T^{2} )^{2} \)
$23$ \( 900 + 92 T^{2} + T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( -1 + T )^{4} \)
$47$ \( 5625 + 188 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -19 + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -11 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( 8100 + 332 T^{2} + T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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