Properties

Label 2736.2.k.m
Level $2736$
Weight $2$
Character orbit 2736.k
Analytic conductor $21.847$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{19})\)
Defining polynomial: \(x^{4} - 9 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} -\beta_{3} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} -\beta_{3} q^{7} + 5 \beta_{1} q^{11} + \beta_{2} q^{17} + \beta_{3} q^{19} + 4 \beta_{1} q^{23} + 14 q^{25} + 19 \beta_{1} q^{35} -3 \beta_{3} q^{43} -13 \beta_{1} q^{47} -12 q^{49} -5 \beta_{3} q^{55} -15 q^{61} + 11 q^{73} + 5 \beta_{2} q^{77} -16 \beta_{1} q^{83} -19 q^{85} -19 \beta_{1} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 56q^{25} - 48q^{49} - 60q^{61} + 44q^{73} - 76q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 9 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 4 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 14 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 9\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 7 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
2.17945 + 0.500000i
2.17945 0.500000i
−2.17945 0.500000i
−2.17945 + 0.500000i
0 0 0 −4.35890 0 4.35890i 0 0 0
2431.2 0 0 0 −4.35890 0 4.35890i 0 0 0
2431.3 0 0 0 4.35890 0 4.35890i 0 0 0
2431.4 0 0 0 4.35890 0 4.35890i 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
4.b odd 2 1 inner
12.b even 2 1 inner
57.d even 2 1 inner
76.d even 2 1 inner
228.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.k.m 4
3.b odd 2 1 inner 2736.2.k.m 4
4.b odd 2 1 inner 2736.2.k.m 4
12.b even 2 1 inner 2736.2.k.m 4
19.b odd 2 1 CM 2736.2.k.m 4
57.d even 2 1 inner 2736.2.k.m 4
76.d even 2 1 inner 2736.2.k.m 4
228.b odd 2 1 inner 2736.2.k.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.k.m 4 1.a even 1 1 trivial
2736.2.k.m 4 3.b odd 2 1 inner
2736.2.k.m 4 4.b odd 2 1 inner
2736.2.k.m 4 12.b even 2 1 inner
2736.2.k.m 4 19.b odd 2 1 CM
2736.2.k.m 4 57.d even 2 1 inner
2736.2.k.m 4 76.d even 2 1 inner
2736.2.k.m 4 228.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{2} - 19 \)
\( T_{7}^{2} + 19 \)
\( T_{11}^{2} + 25 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -19 + T^{2} )^{2} \)
$7$ \( ( 19 + T^{2} )^{2} \)
$11$ \( ( 25 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( -19 + T^{2} )^{2} \)
$19$ \( ( 19 + T^{2} )^{2} \)
$23$ \( ( 16 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 171 + T^{2} )^{2} \)
$47$ \( ( 169 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 15 + T )^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -11 + T )^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 256 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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