Properties

Label 2736.2.s.r
Level $2736$
Weight $2$
Character orbit 2736.s
Analytic conductor $21.847$
Analytic rank $0$
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.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 684)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - 4 \zeta_{6} ) q^{5} + 3 q^{7} +O(q^{10})\) \( q + ( 4 - 4 \zeta_{6} ) q^{5} + 3 q^{7} + 4 q^{11} -5 \zeta_{6} q^{13} + ( 5 - 2 \zeta_{6} ) q^{19} -4 \zeta_{6} q^{23} -11 \zeta_{6} q^{25} + 8 \zeta_{6} q^{29} - q^{31} + ( 12 - 12 \zeta_{6} ) q^{35} -5 q^{37} + ( 8 - 8 \zeta_{6} ) q^{41} + ( -5 + 5 \zeta_{6} ) q^{43} + 8 \zeta_{6} q^{47} + 2 q^{49} + 4 \zeta_{6} q^{53} + ( 16 - 16 \zeta_{6} ) q^{55} + ( -12 + 12 \zeta_{6} ) q^{59} + \zeta_{6} q^{61} -20 q^{65} + 3 \zeta_{6} q^{67} + ( -16 + 16 \zeta_{6} ) q^{71} + ( 15 - 15 \zeta_{6} ) q^{73} + 12 q^{77} + ( -7 + 7 \zeta_{6} ) q^{79} -12 \zeta_{6} q^{89} -15 \zeta_{6} q^{91} + ( 12 - 20 \zeta_{6} ) q^{95} + ( 2 - 2 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + 6q^{7} + O(q^{10}) \) \( 2q + 4q^{5} + 6q^{7} + 8q^{11} - 5q^{13} + 8q^{19} - 4q^{23} - 11q^{25} + 8q^{29} - 2q^{31} + 12q^{35} - 10q^{37} + 8q^{41} - 5q^{43} + 8q^{47} + 4q^{49} + 4q^{53} + 16q^{55} - 12q^{59} + q^{61} - 40q^{65} + 3q^{67} - 16q^{71} + 15q^{73} + 24q^{77} - 7q^{79} - 12q^{89} - 15q^{91} + 4q^{95} + 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 2.00000 + 3.46410i 0 3.00000 0 0 0
1873.1 0 0 0 2.00000 3.46410i 0 3.00000 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
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.r 2
3.b odd 2 1 2736.2.s.b 2
4.b odd 2 1 684.2.k.e yes 2
12.b even 2 1 684.2.k.a 2
19.c even 3 1 inner 2736.2.s.r 2
57.h odd 6 1 2736.2.s.b 2
76.g odd 6 1 684.2.k.e yes 2
228.m even 6 1 684.2.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.k.a 2 12.b even 2 1
684.2.k.a 2 228.m even 6 1
684.2.k.e yes 2 4.b odd 2 1
684.2.k.e yes 2 76.g odd 6 1
2736.2.s.b 2 3.b odd 2 1
2736.2.s.b 2 57.h odd 6 1
2736.2.s.r 2 1.a even 1 1 trivial
2736.2.s.r 2 19.c even 3 1 inner

Hecke kernels

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

\( T_{5}^{2} - 4 T_{5} + 16 \)
\( T_{7} - 3 \)
\( T_{11} - 4 \)
\( T_{13}^{2} + 5 T_{13} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 - 4 T + T^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( 25 + 5 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 19 - 8 T + T^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( 64 - 8 T + T^{2} \)
$31$ \( ( 1 + T )^{2} \)
$37$ \( ( 5 + T )^{2} \)
$41$ \( 64 - 8 T + T^{2} \)
$43$ \( 25 + 5 T + T^{2} \)
$47$ \( 64 - 8 T + T^{2} \)
$53$ \( 16 - 4 T + T^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( 9 - 3 T + T^{2} \)
$71$ \( 256 + 16 T + T^{2} \)
$73$ \( 225 - 15 T + T^{2} \)
$79$ \( 49 + 7 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( 144 + 12 T + T^{2} \)
$97$ \( 4 - 2 T + T^{2} \)
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