Properties

Label 2736.2.s.t
Level $2736$
Weight $2$
Character orbit 2736.s
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.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \( 3 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( 1 - \beta_{3} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( 1 - \beta_{3} ) q^{7} + \beta_{3} q^{11} + ( 1 - \beta_{1} ) q^{13} -2 \beta_{2} q^{17} + ( -1 + 2 \beta_{2} - \beta_{3} ) q^{19} + ( 6 - 6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + ( -1 + \beta_{1} ) q^{25} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( 7 - \beta_{3} ) q^{31} + ( 6 \beta_{1} - \beta_{2} ) q^{35} + ( -1 - 4 \beta_{3} ) q^{37} + ( -\beta_{1} + \beta_{2} ) q^{43} + ( -6 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{47} -2 \beta_{3} q^{49} + ( -\beta_{2} + \beta_{3} ) q^{53} -6 \beta_{1} q^{55} -3 \beta_{2} q^{59} + ( 7 - 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{61} -\beta_{3} q^{65} + ( -7 + 7 \beta_{1} + \beta_{2} - \beta_{3} ) q^{67} -6 \beta_{1} q^{71} + ( 7 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -6 + \beta_{3} ) q^{77} + ( 5 \beta_{1} - 3 \beta_{2} ) q^{79} + ( -12 + 2 \beta_{3} ) q^{83} + ( -12 + 12 \beta_{1} ) q^{85} + ( -6 + 6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{91} + ( 12 - 6 \beta_{1} + \beta_{2} ) q^{95} + ( 4 \beta_{1} + 6 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} + 2q^{13} - 4q^{19} + 12q^{23} - 2q^{25} + 28q^{31} + 12q^{35} - 4q^{37} - 2q^{43} - 12q^{47} - 12q^{55} + 14q^{61} - 14q^{67} - 12q^{71} + 14q^{73} - 24q^{77} + 10q^{79} - 48q^{83} - 24q^{85} - 12q^{89} + 2q^{91} + 36q^{95} + 8q^{97} + O(q^{100}) \)

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

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0 0 0 −1.22474 2.12132i 0 −1.44949 0 0 0
577.2 0 0 0 1.22474 + 2.12132i 0 3.44949 0 0 0
1873.1 0 0 0 −1.22474 + 2.12132i 0 −1.44949 0 0 0
1873.2 0 0 0 1.22474 2.12132i 0 3.44949 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.t 4
3.b odd 2 1 912.2.q.i 4
4.b odd 2 1 684.2.k.f 4
12.b even 2 1 228.2.i.a 4
19.c even 3 1 inner 2736.2.s.t 4
57.h odd 6 1 912.2.q.i 4
76.g odd 6 1 684.2.k.f 4
228.m even 6 1 228.2.i.a 4
228.m even 6 1 4332.2.a.l 2
228.n odd 6 1 4332.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.i.a 4 12.b even 2 1
228.2.i.a 4 228.m even 6 1
684.2.k.f 4 4.b odd 2 1
684.2.k.f 4 76.g odd 6 1
912.2.q.i 4 3.b odd 2 1
912.2.q.i 4 57.h odd 6 1
2736.2.s.t 4 1.a even 1 1 trivial
2736.2.s.t 4 19.c even 3 1 inner
4332.2.a.g 2 228.n odd 6 1
4332.2.a.l 2 228.m even 6 1

Hecke kernels

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

\( T_{5}^{4} + 6 T_{5}^{2} + 36 \)
\( T_{7}^{2} - 2 T_{7} - 5 \)
\( T_{11}^{2} - 6 \)
\( T_{13}^{2} - T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 36 + 6 T^{2} + T^{4} \)
$7$ \( ( -5 - 2 T + T^{2} )^{2} \)
$11$ \( ( -6 + T^{2} )^{2} \)
$13$ \( ( 1 - T + T^{2} )^{2} \)
$17$ \( 576 + 24 T^{2} + T^{4} \)
$19$ \( ( 19 + 2 T + T^{2} )^{2} \)
$23$ \( 900 - 360 T + 114 T^{2} - 12 T^{3} + T^{4} \)
$29$ \( 576 + 24 T^{2} + T^{4} \)
$31$ \( ( 43 - 14 T + T^{2} )^{2} \)
$37$ \( ( -95 + 2 T + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( 25 - 10 T + 9 T^{2} + 2 T^{3} + T^{4} \)
$47$ \( 144 + 144 T + 132 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( 36 + 6 T^{2} + T^{4} \)
$59$ \( 2916 + 54 T^{2} + T^{4} \)
$61$ \( 625 - 350 T + 171 T^{2} - 14 T^{3} + T^{4} \)
$67$ \( 1849 + 602 T + 153 T^{2} + 14 T^{3} + T^{4} \)
$71$ \( ( 36 + 6 T + T^{2} )^{2} \)
$73$ \( 625 - 350 T + 171 T^{2} - 14 T^{3} + T^{4} \)
$79$ \( 841 + 290 T + 129 T^{2} - 10 T^{3} + T^{4} \)
$83$ \( ( 120 + 24 T + T^{2} )^{2} \)
$89$ \( 900 + 360 T + 114 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( 40000 + 1600 T + 264 T^{2} - 8 T^{3} + T^{4} \)
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