Properties

Label 2736.2.s.f
Level $2736$
Weight $2$
Character orbit 2736.s
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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 456)
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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{5} + 5 q^{7} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{5} + 5 q^{7} -4 q^{11} -5 \zeta_{6} q^{13} + ( -5 + 2 \zeta_{6} ) q^{19} + 6 \zeta_{6} q^{23} + \zeta_{6} q^{25} + 8 \zeta_{6} q^{29} + q^{31} + ( -10 + 10 \zeta_{6} ) q^{35} + 7 q^{37} + ( -11 + 11 \zeta_{6} ) q^{43} -10 \zeta_{6} q^{47} + 18 q^{49} + 6 \zeta_{6} q^{53} + ( 8 - 8 \zeta_{6} ) q^{55} + ( -8 + 8 \zeta_{6} ) q^{59} + \zeta_{6} q^{61} + 10 q^{65} + 5 \zeta_{6} q^{67} + ( -6 + 6 \zeta_{6} ) q^{71} + ( -1 + \zeta_{6} ) q^{73} -20 q^{77} + ( -13 + 13 \zeta_{6} ) q^{79} -4 q^{83} -12 \zeta_{6} q^{89} -25 \zeta_{6} q^{91} + ( 6 - 10 \zeta_{6} ) q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 10q^{7} + O(q^{10}) \) \( 2q - 2q^{5} + 10q^{7} - 8q^{11} - 5q^{13} - 8q^{19} + 6q^{23} + q^{25} + 8q^{29} + 2q^{31} - 10q^{35} + 14q^{37} - 11q^{43} - 10q^{47} + 36q^{49} + 6q^{53} + 8q^{55} - 8q^{59} + q^{61} + 20q^{65} + 5q^{67} - 6q^{71} - q^{73} - 40q^{77} - 13q^{79} - 8q^{83} - 12q^{89} - 25q^{91} + 2q^{95} - 2q^{97} + O(q^{100}) \)

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 −1.00000 1.73205i 0 5.00000 0 0 0
1873.1 0 0 0 −1.00000 + 1.73205i 0 5.00000 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.f 2
3.b odd 2 1 912.2.q.c 2
4.b odd 2 1 1368.2.s.b 2
12.b even 2 1 456.2.q.c 2
19.c even 3 1 inner 2736.2.s.f 2
57.h odd 6 1 912.2.q.c 2
76.g odd 6 1 1368.2.s.b 2
228.m even 6 1 456.2.q.c 2
228.m even 6 1 8664.2.a.b 1
228.n odd 6 1 8664.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.c 2 12.b even 2 1
456.2.q.c 2 228.m even 6 1
912.2.q.c 2 3.b odd 2 1
912.2.q.c 2 57.h odd 6 1
1368.2.s.b 2 4.b odd 2 1
1368.2.s.b 2 76.g odd 6 1
2736.2.s.f 2 1.a even 1 1 trivial
2736.2.s.f 2 19.c even 3 1 inner
8664.2.a.b 1 228.m even 6 1
8664.2.a.h 1 228.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 \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 + 2 T + T^{2} \)
$7$ \( ( -5 + T )^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 25 + 5 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 19 + 8 T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 64 - 8 T + T^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( ( -7 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 121 + 11 T + T^{2} \)
$47$ \( 100 + 10 T + T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( 64 + 8 T + T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( 25 - 5 T + T^{2} \)
$71$ \( 36 + 6 T + T^{2} \)
$73$ \( 1 + T + T^{2} \)
$79$ \( 169 + 13 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( 144 + 12 T + T^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
show more
show less