Properties

Label 2736.2.k.b
Level $2736$
Weight $2$
Character orbit 2736.k
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.k (of order \(2\), degree \(1\), 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -3 q^{5} + ( 1 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -3 q^{5} + ( 1 - 2 \zeta_{6} ) q^{7} + ( 3 - 6 \zeta_{6} ) q^{11} + ( 4 - 8 \zeta_{6} ) q^{13} -3 q^{17} + ( 3 + 2 \zeta_{6} ) q^{19} + ( -2 + 4 \zeta_{6} ) q^{23} + 4 q^{25} -4 q^{31} + ( -3 + 6 \zeta_{6} ) q^{35} + ( -4 + 8 \zeta_{6} ) q^{37} + ( 4 - 8 \zeta_{6} ) q^{41} + ( 5 - 10 \zeta_{6} ) q^{43} + ( -5 + 10 \zeta_{6} ) q^{47} + 4 q^{49} + ( -4 + 8 \zeta_{6} ) q^{53} + ( -9 + 18 \zeta_{6} ) q^{55} -12 q^{59} + 7 q^{61} + ( -12 + 24 \zeta_{6} ) q^{65} + 8 q^{67} -12 q^{71} -5 q^{73} -9 q^{77} -8 q^{79} + ( 2 - 4 \zeta_{6} ) q^{83} + 9 q^{85} + ( 4 - 8 \zeta_{6} ) q^{89} -12 q^{91} + ( -9 - 6 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{5} + O(q^{10}) \) \( 2q - 6q^{5} - 6q^{17} + 8q^{19} + 8q^{25} - 8q^{31} + 8q^{49} - 24q^{59} + 14q^{61} + 16q^{67} - 24q^{71} - 10q^{73} - 18q^{77} - 16q^{79} + 18q^{85} - 24q^{91} - 24q^{95} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −3.00000 0 1.73205i 0 0 0
2431.2 0 0 0 −3.00000 0 1.73205i 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
76.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.k.b 2
3.b odd 2 1 912.2.k.c 2
4.b odd 2 1 2736.2.k.a 2
12.b even 2 1 912.2.k.f yes 2
19.b odd 2 1 2736.2.k.a 2
24.f even 2 1 3648.2.k.a 2
24.h odd 2 1 3648.2.k.d 2
57.d even 2 1 912.2.k.f yes 2
76.d even 2 1 inner 2736.2.k.b 2
228.b odd 2 1 912.2.k.c 2
456.l odd 2 1 3648.2.k.d 2
456.p even 2 1 3648.2.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.c 2 3.b odd 2 1
912.2.k.c 2 228.b odd 2 1
912.2.k.f yes 2 12.b even 2 1
912.2.k.f yes 2 57.d even 2 1
2736.2.k.a 2 4.b odd 2 1
2736.2.k.a 2 19.b odd 2 1
2736.2.k.b 2 1.a even 1 1 trivial
2736.2.k.b 2 76.d even 2 1 inner
3648.2.k.a 2 24.f even 2 1
3648.2.k.a 2 456.p even 2 1
3648.2.k.d 2 24.h odd 2 1
3648.2.k.d 2 456.l odd 2 1

Hecke kernels

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

\( T_{5} + 3 \)
\( T_{7}^{2} + 3 \)
\( T_{11}^{2} + 27 \)
\( T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 3 + T )^{2} \)
$7$ \( 3 + T^{2} \)
$11$ \( 27 + T^{2} \)
$13$ \( 48 + T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( 19 - 8 T + T^{2} \)
$23$ \( 12 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 48 + T^{2} \)
$41$ \( 48 + T^{2} \)
$43$ \( 75 + T^{2} \)
$47$ \( 75 + T^{2} \)
$53$ \( 48 + T^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( ( -7 + T )^{2} \)
$67$ \( ( -8 + T )^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( ( 5 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 12 + T^{2} \)
$89$ \( 48 + T^{2} \)
$97$ \( 48 + T^{2} \)
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