Properties

Label 2736.2.f.a
Level $2736$
Weight $2$
Character orbit 2736.f
Analytic conductor $21.847$
Analytic rank $1$
Dimension $2$
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 342)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.41421i
1.41421i
0 0 0 1.41421i 0 −2.00000 0 0 0
1025.2 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
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.a 2
3.b odd 2 1 2736.2.f.b 2
4.b odd 2 1 342.2.b.b yes 2
12.b even 2 1 342.2.b.a 2
19.b odd 2 1 2736.2.f.b 2
57.d even 2 1 inner 2736.2.f.a 2
76.d even 2 1 342.2.b.a 2
228.b odd 2 1 342.2.b.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.b.a 2 12.b even 2 1
342.2.b.a 2 76.d even 2 1
342.2.b.b yes 2 4.b odd 2 1
342.2.b.b yes 2 228.b odd 2 1
2736.2.f.a 2 1.a even 1 1 trivial
2736.2.f.a 2 57.d even 2 1 inner
2736.2.f.b 2 3.b odd 2 1
2736.2.f.b 2 19.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 2 \)
\( T_{7} + 2 \)
\( T_{11}^{2} + 2 \)
\( T_{29} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 2 + T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( 2 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 2 + T^{2} \)
$19$ \( 19 + 2 T + T^{2} \)
$23$ \( 2 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 72 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 50 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 4 + T )^{2} \)
$67$ \( 72 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( 72 + T^{2} \)
$83$ \( 242 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 72 + T^{2} \)
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