Properties

Label 2736.2.s.s
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: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 456)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} - \beta_{2} ) q^{5} + ( -2 - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( -1 - \beta_{1} - \beta_{2} ) q^{5} + ( -2 - \beta_{3} ) q^{7} + ( -1 - \beta_{3} ) q^{11} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -4 - 4 \beta_{1} ) q^{17} + ( 2 - 2 \beta_{2} - \beta_{3} ) q^{19} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{25} + ( 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( 2 + 3 \beta_{3} ) q^{31} + ( -3 - 3 \beta_{1} + \beta_{2} ) q^{35} + 7 q^{37} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -4 - 4 \beta_{1} - \beta_{2} ) q^{43} -2 \beta_{1} q^{47} + ( 2 + 4 \beta_{3} ) q^{49} + ( \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{53} + ( -4 - 4 \beta_{1} ) q^{55} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{59} + 11 \beta_{1} q^{61} + ( 9 - \beta_{3} ) q^{65} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{67} -6 \beta_{2} q^{71} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{73} + ( 7 + 3 \beta_{3} ) q^{77} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{79} -8 q^{83} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{85} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{89} + ( 12 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{91} + ( -7 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{95} + ( -2 - 2 \beta_{1} - 4 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - 8q^{7} + O(q^{10}) \) \( 4q - 2q^{5} - 8q^{7} - 4q^{11} + 2q^{13} - 8q^{17} + 8q^{19} + 2q^{23} - 2q^{25} - 12q^{29} + 8q^{31} - 6q^{35} + 28q^{37} + 4q^{41} - 8q^{43} + 4q^{47} + 8q^{49} - 2q^{53} - 8q^{55} + 6q^{59} - 22q^{61} + 36q^{65} - 4q^{67} + 2q^{73} + 28q^{77} + 4q^{79} - 32q^{83} - 8q^{85} - 6q^{89} - 24q^{91} - 34q^{95} - 4q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

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

Character values

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

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

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
0.809017 + 1.40126i
−0.309017 0.535233i
0.809017 1.40126i
−0.309017 + 0.535233i
0 0 0 −1.61803 2.80252i 0 0.236068 0 0 0
577.2 0 0 0 0.618034 + 1.07047i 0 −4.23607 0 0 0
1873.1 0 0 0 −1.61803 + 2.80252i 0 0.236068 0 0 0
1873.2 0 0 0 0.618034 1.07047i 0 −4.23607 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.s 4
3.b odd 2 1 912.2.q.j 4
4.b odd 2 1 1368.2.s.h 4
12.b even 2 1 456.2.q.d 4
19.c even 3 1 inner 2736.2.s.s 4
57.h odd 6 1 912.2.q.j 4
76.g odd 6 1 1368.2.s.h 4
228.m even 6 1 456.2.q.d 4
228.m even 6 1 8664.2.a.u 2
228.n odd 6 1 8664.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.d 4 12.b even 2 1
456.2.q.d 4 228.m even 6 1
912.2.q.j 4 3.b odd 2 1
912.2.q.j 4 57.h odd 6 1
1368.2.s.h 4 4.b odd 2 1
1368.2.s.h 4 76.g odd 6 1
2736.2.s.s 4 1.a even 1 1 trivial
2736.2.s.s 4 19.c even 3 1 inner
8664.2.a.q 2 228.n odd 6 1
8664.2.a.u 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 \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{4} + 2 T_{5}^{3} + 8 T_{5}^{2} - 8 T_{5} + 16 \)
\( T_{7}^{2} + 4 T_{7} - 1 \)
\( T_{11}^{2} + 2 T_{11} - 4 \)
\( T_{13}^{4} - 2 T_{13}^{3} + 23 T_{13}^{2} + 38 T_{13} + 361 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 16 - 8 T + 8 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( ( -1 + 4 T + T^{2} )^{2} \)
$11$ \( ( -4 + 2 T + T^{2} )^{2} \)
$13$ \( 361 + 38 T + 23 T^{2} - 2 T^{3} + T^{4} \)
$17$ \( ( 16 + 4 T + T^{2} )^{2} \)
$19$ \( ( 19 - 4 T + T^{2} )^{2} \)
$23$ \( 16 + 8 T + 8 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( 256 + 192 T + 128 T^{2} + 12 T^{3} + T^{4} \)
$31$ \( ( -41 - 4 T + T^{2} )^{2} \)
$37$ \( ( -7 + T )^{4} \)
$41$ \( 256 + 64 T + 32 T^{2} - 4 T^{3} + T^{4} \)
$43$ \( 121 + 88 T + 53 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( ( 4 - 2 T + T^{2} )^{2} \)
$53$ \( 15376 - 248 T + 128 T^{2} + 2 T^{3} + T^{4} \)
$59$ \( 1296 + 216 T + 72 T^{2} - 6 T^{3} + T^{4} \)
$61$ \( ( 121 + 11 T + T^{2} )^{2} \)
$67$ \( 1 - 4 T + 17 T^{2} + 4 T^{3} + T^{4} \)
$71$ \( 32400 + 180 T^{2} + T^{4} \)
$73$ \( 361 + 38 T + 23 T^{2} - 2 T^{3} + T^{4} \)
$79$ \( 1681 + 164 T + 57 T^{2} - 4 T^{3} + T^{4} \)
$83$ \( ( 8 + T )^{4} \)
$89$ \( 1296 - 216 T + 72 T^{2} + 6 T^{3} + T^{4} \)
$97$ \( 5776 - 304 T + 92 T^{2} + 4 T^{3} + T^{4} \)
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