Properties

Label 2736.2.f.f
Level $2736$
Weight $2$
Character orbit 2736.f
Analytic conductor $21.847$
Analytic rank $0$
Dimension $4$
CM no
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: \(\Q(\sqrt{-2}, \sqrt{5})\)
Defining polynomial: \(x^{4} + 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 171)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + 2 q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} + 2 q^{7} -\beta_{1} q^{11} + 2 \beta_{2} q^{13} -3 \beta_{1} q^{17} + ( 3 - \beta_{2} ) q^{19} + 5 \beta_{1} q^{23} + 3 q^{25} + \beta_{3} q^{29} + 2 \beta_{2} q^{31} -2 \beta_{1} q^{35} -\beta_{3} q^{41} -4 q^{43} -5 \beta_{1} q^{47} -3 q^{49} + 3 \beta_{3} q^{53} -2 q^{55} + 2 \beta_{3} q^{59} + 8 q^{61} + 2 \beta_{3} q^{65} + 2 \beta_{2} q^{67} + 6 q^{73} -2 \beta_{1} q^{77} -4 \beta_{2} q^{79} -\beta_{1} q^{83} -6 q^{85} -3 \beta_{3} q^{89} + 4 \beta_{2} q^{91} + ( -3 \beta_{1} - \beta_{3} ) q^{95} + 2 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{7} + O(q^{10}) \) \( 4q + 8q^{7} + 12q^{19} + 12q^{25} - 16q^{43} - 12q^{49} - 8q^{55} + 32q^{61} + 24q^{73} - 24q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 6 x^{2} + 4\):

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

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
2.28825i
0.874032i
0.874032i
2.28825i
0 0 0 1.41421i 0 2.00000 0 0 0
1025.2 0 0 0 1.41421i 0 2.00000 0 0 0
1025.3 0 0 0 1.41421i 0 2.00000 0 0 0
1025.4 0 0 0 1.41421i 0 2.00000 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
19.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.f 4
3.b odd 2 1 inner 2736.2.f.f 4
4.b odd 2 1 171.2.d.b 4
12.b even 2 1 171.2.d.b 4
19.b odd 2 1 inner 2736.2.f.f 4
57.d even 2 1 inner 2736.2.f.f 4
76.d even 2 1 171.2.d.b 4
228.b odd 2 1 171.2.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.d.b 4 4.b odd 2 1
171.2.d.b 4 12.b even 2 1
171.2.d.b 4 76.d even 2 1
171.2.d.b 4 228.b odd 2 1
2736.2.f.f 4 1.a even 1 1 trivial
2736.2.f.f 4 3.b odd 2 1 inner
2736.2.f.f 4 19.b odd 2 1 inner
2736.2.f.f 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 \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{7} - 2 \)
\( T_{11}^{2} + 2 \)
\( T_{29}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 2 + T^{2} )^{2} \)
$7$ \( ( -2 + T )^{4} \)
$11$ \( ( 2 + T^{2} )^{2} \)
$13$ \( ( 40 + T^{2} )^{2} \)
$17$ \( ( 18 + T^{2} )^{2} \)
$19$ \( ( 19 - 6 T + T^{2} )^{2} \)
$23$ \( ( 50 + T^{2} )^{2} \)
$29$ \( ( -20 + T^{2} )^{2} \)
$31$ \( ( 40 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( -20 + T^{2} )^{2} \)
$43$ \( ( 4 + T )^{4} \)
$47$ \( ( 50 + T^{2} )^{2} \)
$53$ \( ( -180 + T^{2} )^{2} \)
$59$ \( ( -80 + T^{2} )^{2} \)
$61$ \( ( -8 + T )^{4} \)
$67$ \( ( 40 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( -6 + T )^{4} \)
$79$ \( ( 160 + T^{2} )^{2} \)
$83$ \( ( 2 + T^{2} )^{2} \)
$89$ \( ( -180 + T^{2} )^{2} \)
$97$ \( ( 40 + T^{2} )^{2} \)
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