Properties

Label 2736.2.s.u
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: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{5} + ( -2 + \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{5} + ( -2 + \beta_{3} ) q^{7} + ( 3 + \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{13} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{17} + ( -2 - 4 \beta_{2} - \beta_{3} ) q^{19} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{23} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{25} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -2 - \beta_{3} ) q^{31} + ( -9 - 3 \beta_{1} - 9 \beta_{2} ) q^{35} -5 q^{37} + ( 4 + 4 \beta_{2} ) q^{41} + ( 8 - \beta_{1} + 8 \beta_{2} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{47} + ( 4 - 4 \beta_{3} ) q^{49} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{53} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{55} + ( -5 - \beta_{1} - 5 \beta_{2} ) q^{59} + 5 \beta_{2} q^{61} + ( 13 - \beta_{3} ) q^{65} + ( -3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{67} + ( 10 + 10 \beta_{2} ) q^{71} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{73} + ( 1 + \beta_{3} ) q^{77} + ( 6 - \beta_{1} + 6 \beta_{2} ) q^{79} + ( 10 - 2 \beta_{3} ) q^{83} -12 \beta_{2} q^{85} + ( -\beta_{1} - 11 \beta_{2} - \beta_{3} ) q^{89} + ( 3 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} ) q^{91} + ( 9 - \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{95} -4 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 8q^{7} + O(q^{10}) \) \( 4q + 2q^{5} - 8q^{7} + 12q^{11} - 2q^{13} + 4q^{17} + 6q^{23} - 6q^{25} + 4q^{29} - 8q^{31} - 18q^{35} - 20q^{37} + 8q^{41} + 16q^{43} + 16q^{49} - 2q^{53} - 8q^{55} - 10q^{59} - 10q^{61} + 52q^{65} + 12q^{67} + 20q^{71} + 2q^{73} + 4q^{77} + 12q^{79} + 40q^{83} + 24q^{85} + 22q^{89} - 24q^{91} + 26q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\) \(1\)

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
−1.32288 2.29129i
1.32288 + 2.29129i
−1.32288 + 2.29129i
1.32288 2.29129i
0 0 0 −0.822876 1.42526i 0 0.645751 0 0 0
577.2 0 0 0 1.82288 + 3.15731i 0 −4.64575 0 0 0
1873.1 0 0 0 −0.822876 + 1.42526i 0 0.645751 0 0 0
1873.2 0 0 0 1.82288 3.15731i 0 −4.64575 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.u 4
3.b odd 2 1 912.2.q.g 4
4.b odd 2 1 684.2.k.g 4
12.b even 2 1 228.2.i.b 4
19.c even 3 1 inner 2736.2.s.u 4
57.h odd 6 1 912.2.q.g 4
76.g odd 6 1 684.2.k.g 4
228.m even 6 1 228.2.i.b 4
228.m even 6 1 4332.2.a.h 2
228.n odd 6 1 4332.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.i.b 4 12.b even 2 1
228.2.i.b 4 228.m even 6 1
684.2.k.g 4 4.b odd 2 1
684.2.k.g 4 76.g odd 6 1
912.2.q.g 4 3.b odd 2 1
912.2.q.g 4 57.h odd 6 1
2736.2.s.u 4 1.a even 1 1 trivial
2736.2.s.u 4 19.c even 3 1 inner
4332.2.a.h 2 228.m even 6 1
4332.2.a.m 2 228.n odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 2 T_{5}^{3} + 10 T_{5}^{2} + 12 T_{5} + 36 \)
\( T_{7}^{2} + 4 T_{7} - 3 \)
\( T_{11}^{2} - 6 T_{11} + 2 \)
\( T_{13}^{4} + 2 T_{13}^{3} + 31 T_{13}^{2} - 54 T_{13} + 729 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( ( -3 + 4 T + T^{2} )^{2} \)
$11$ \( ( 2 - 6 T + T^{2} )^{2} \)
$13$ \( 729 - 54 T + 31 T^{2} + 2 T^{3} + T^{4} \)
$17$ \( 576 + 96 T + 40 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( 361 + 10 T^{2} + T^{4} \)
$23$ \( 4 - 12 T + 34 T^{2} - 6 T^{3} + T^{4} \)
$29$ \( 576 + 96 T + 40 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( ( -3 + 4 T + T^{2} )^{2} \)
$37$ \( ( 5 + T )^{4} \)
$41$ \( ( 16 - 4 T + T^{2} )^{2} \)
$43$ \( 3249 - 912 T + 199 T^{2} - 16 T^{3} + T^{4} \)
$47$ \( 784 + 28 T^{2} + T^{4} \)
$53$ \( 3844 - 124 T + 66 T^{2} + 2 T^{3} + T^{4} \)
$59$ \( 324 + 180 T + 82 T^{2} + 10 T^{3} + T^{4} \)
$61$ \( ( 25 + 5 T + T^{2} )^{2} \)
$67$ \( 729 + 324 T + 171 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( ( 100 - 10 T + T^{2} )^{2} \)
$73$ \( 729 + 54 T + 31 T^{2} - 2 T^{3} + T^{4} \)
$79$ \( 841 - 348 T + 115 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( ( 72 - 20 T + T^{2} )^{2} \)
$89$ \( 12996 - 2508 T + 370 T^{2} - 22 T^{3} + T^{4} \)
$97$ \( 12544 + 112 T^{2} + T^{4} \)
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