Properties

Label 2736.2.s.v
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{5} + ( -1 - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{5} + ( -1 - \beta_{3} ) q^{7} + ( -2 - \beta_{3} ) q^{11} + 2 \beta_{2} q^{13} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{25} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( 3 + \beta_{3} ) q^{31} + ( 6 + 6 \beta_{2} ) q^{35} + ( 3 - \beta_{3} ) q^{37} + ( 5 + 2 \beta_{1} + 5 \beta_{2} ) q^{41} + ( 6 - 2 \beta_{1} + 6 \beta_{2} ) q^{43} + ( \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{47} + ( 1 + 2 \beta_{3} ) q^{49} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{53} + ( 5 - \beta_{1} + 5 \beta_{2} ) q^{55} -3 \beta_{1} q^{59} + ( -3 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} ) q^{61} + ( -2 + 2 \beta_{3} ) q^{65} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{67} + ( 8 + 2 \beta_{1} + 8 \beta_{2} ) q^{71} + ( -7 - 2 \beta_{1} - 7 \beta_{2} ) q^{73} + ( 9 + 3 \beta_{3} ) q^{77} + ( -4 - 4 \beta_{2} ) q^{79} + 3 \beta_{3} q^{83} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -9 - 4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{95} + ( 9 + 2 \beta_{1} + 9 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q + 2q^{5} - 4q^{7} - 8q^{11} - 4q^{13} - 12q^{19} - 2q^{23} - 6q^{25} - 2q^{29} + 12q^{31} + 12q^{35} + 12q^{37} + 10q^{41} + 12q^{43} - 14q^{47} + 4q^{49} - 4q^{53} + 10q^{55} + 14q^{61} - 8q^{65} - 4q^{67} + 16q^{71} - 14q^{73} + 36q^{77} - 8q^{79} + 4q^{91} - 28q^{95} + 18q^{97} + O(q^{100}) \)

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 −3.64575 0 0 0
577.2 0 0 0 1.82288 + 3.15731i 0 1.64575 0 0 0
1873.1 0 0 0 −0.822876 + 1.42526i 0 −3.64575 0 0 0
1873.2 0 0 0 1.82288 3.15731i 0 1.64575 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.v 4
3.b odd 2 1 304.2.i.e 4
4.b odd 2 1 342.2.g.f 4
12.b even 2 1 38.2.c.b 4
19.c even 3 1 inner 2736.2.s.v 4
24.f even 2 1 1216.2.i.l 4
24.h odd 2 1 1216.2.i.k 4
57.f even 6 1 5776.2.a.z 2
57.h odd 6 1 304.2.i.e 4
57.h odd 6 1 5776.2.a.ba 2
60.h even 2 1 950.2.e.k 4
60.l odd 4 2 950.2.j.g 8
76.f even 6 1 6498.2.a.bg 2
76.g odd 6 1 342.2.g.f 4
76.g odd 6 1 6498.2.a.ba 2
228.b odd 2 1 722.2.c.j 4
228.m even 6 1 38.2.c.b 4
228.m even 6 1 722.2.a.j 2
228.n odd 6 1 722.2.a.g 2
228.n odd 6 1 722.2.c.j 4
228.u odd 18 6 722.2.e.o 12
228.v even 18 6 722.2.e.n 12
456.u even 6 1 1216.2.i.l 4
456.x odd 6 1 1216.2.i.k 4
1140.bn even 6 1 950.2.e.k 4
1140.bu odd 12 2 950.2.j.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b 4 12.b even 2 1
38.2.c.b 4 228.m even 6 1
304.2.i.e 4 3.b odd 2 1
304.2.i.e 4 57.h odd 6 1
342.2.g.f 4 4.b odd 2 1
342.2.g.f 4 76.g odd 6 1
722.2.a.g 2 228.n odd 6 1
722.2.a.j 2 228.m even 6 1
722.2.c.j 4 228.b odd 2 1
722.2.c.j 4 228.n odd 6 1
722.2.e.n 12 228.v even 18 6
722.2.e.o 12 228.u odd 18 6
950.2.e.k 4 60.h even 2 1
950.2.e.k 4 1140.bn even 6 1
950.2.j.g 8 60.l odd 4 2
950.2.j.g 8 1140.bu odd 12 2
1216.2.i.k 4 24.h odd 2 1
1216.2.i.k 4 456.x odd 6 1
1216.2.i.l 4 24.f even 2 1
1216.2.i.l 4 456.u even 6 1
2736.2.s.v 4 1.a even 1 1 trivial
2736.2.s.v 4 19.c even 3 1 inner
5776.2.a.z 2 57.f even 6 1
5776.2.a.ba 2 57.h odd 6 1
6498.2.a.ba 2 76.g odd 6 1
6498.2.a.bg 2 76.f even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( ( -6 + 2 T + T^{2} )^{2} \)
$11$ \( ( -3 + 4 T + T^{2} )^{2} \)
$13$ \( ( 4 + 2 T + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( 361 + 228 T + 67 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
$31$ \( ( 2 - 6 T + T^{2} )^{2} \)
$37$ \( ( 2 - 6 T + T^{2} )^{2} \)
$41$ \( 9 + 30 T + 103 T^{2} - 10 T^{3} + T^{4} \)
$43$ \( 64 - 96 T + 136 T^{2} - 12 T^{3} + T^{4} \)
$47$ \( 1764 + 588 T + 154 T^{2} + 14 T^{3} + T^{4} \)
$53$ \( 11664 - 432 T + 124 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( 3969 + 63 T^{2} + T^{4} \)
$61$ \( 196 + 196 T + 210 T^{2} - 14 T^{3} + T^{4} \)
$67$ \( 9 - 12 T + 19 T^{2} + 4 T^{3} + T^{4} \)
$71$ \( 1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4} \)
$73$ \( 441 + 294 T + 175 T^{2} + 14 T^{3} + T^{4} \)
$79$ \( ( 16 + 4 T + T^{2} )^{2} \)
$83$ \( ( -63 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( 2809 - 954 T + 271 T^{2} - 18 T^{3} + T^{4} \)
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