Properties

Label 2736.3.o.j
Level $2736$
Weight $3$
Character orbit 2736.o
Analytic conductor $74.551$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-7}, \sqrt{10})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 15 x^{2} + 16 x + 134\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + 2 q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} + 2 q^{7} + 2 \beta_{1} q^{11} + \beta_{3} q^{13} + ( 9 + \beta_{3} ) q^{19} -\beta_{1} q^{23} + 15 q^{25} -\beta_{2} q^{29} + \beta_{3} q^{31} -2 \beta_{1} q^{35} -3 \beta_{3} q^{37} + \beta_{2} q^{41} -34 q^{43} + 13 \beta_{1} q^{47} -45 q^{49} -3 \beta_{2} q^{53} -80 q^{55} + \beta_{2} q^{59} + 86 q^{61} -5 \beta_{2} q^{65} + 4 \beta_{3} q^{67} -6 \beta_{2} q^{71} -102 q^{73} + 4 \beta_{1} q^{77} + 7 \beta_{3} q^{79} -4 \beta_{1} q^{83} + 6 \beta_{2} q^{89} + 2 \beta_{3} q^{91} + ( -9 \beta_{1} - 5 \beta_{2} ) q^{95} -2 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{7} + O(q^{10}) \) \( 4q + 8q^{7} + 36q^{19} + 60q^{25} - 136q^{43} - 180q^{49} - 320q^{55} + 344q^{61} - 408q^{73} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 15 x^{2} + 16 x + 134\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -4 \nu^{3} + 6 \nu^{2} + 110 \nu - 56 \)\()/47\)
\(\beta_{2}\)\(=\)\((\)\( 32 \nu^{3} - 48 \nu^{2} - 128 \nu + 72 \)\()/47\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 2 \nu - 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 8 \beta_{1} + 8\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(8 \beta_{3} + \beta_{2} + 8 \beta_{1} + 136\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(12 \beta_{3} + 29 \beta_{2} + 44 \beta_{1} + 200\)\()/16\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
3.66228 1.32288i
3.66228 + 1.32288i
−2.66228 + 1.32288i
−2.66228 1.32288i
0 0 0 −6.32456 0 2.00000 0 0 0
721.2 0 0 0 −6.32456 0 2.00000 0 0 0
721.3 0 0 0 6.32456 0 2.00000 0 0 0
721.4 0 0 0 6.32456 0 2.00000 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
3.b odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.j 4
3.b odd 2 1 inner 2736.3.o.j 4
4.b odd 2 1 171.3.c.e 4
12.b even 2 1 171.3.c.e 4
19.b odd 2 1 inner 2736.3.o.j 4
57.d even 2 1 inner 2736.3.o.j 4
76.d even 2 1 171.3.c.e 4
228.b odd 2 1 171.3.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.c.e 4 4.b odd 2 1
171.3.c.e 4 12.b even 2 1
171.3.c.e 4 76.d even 2 1
171.3.c.e 4 228.b odd 2 1
2736.3.o.j 4 1.a even 1 1 trivial
2736.3.o.j 4 3.b odd 2 1 inner
2736.3.o.j 4 19.b odd 2 1 inner
2736.3.o.j 4 57.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{2} - 40 \)
\( T_{7} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -40 + T^{2} )^{2} \)
$7$ \( ( -2 + T )^{4} \)
$11$ \( ( -160 + T^{2} )^{2} \)
$13$ \( ( 280 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( 361 - 18 T + T^{2} )^{2} \)
$23$ \( ( -40 + T^{2} )^{2} \)
$29$ \( ( 448 + T^{2} )^{2} \)
$31$ \( ( 280 + T^{2} )^{2} \)
$37$ \( ( 2520 + T^{2} )^{2} \)
$41$ \( ( 448 + T^{2} )^{2} \)
$43$ \( ( 34 + T )^{4} \)
$47$ \( ( -6760 + T^{2} )^{2} \)
$53$ \( ( 4032 + T^{2} )^{2} \)
$59$ \( ( 448 + T^{2} )^{2} \)
$61$ \( ( -86 + T )^{4} \)
$67$ \( ( 4480 + T^{2} )^{2} \)
$71$ \( ( 16128 + T^{2} )^{2} \)
$73$ \( ( 102 + T )^{4} \)
$79$ \( ( 13720 + T^{2} )^{2} \)
$83$ \( ( -640 + T^{2} )^{2} \)
$89$ \( ( 16128 + T^{2} )^{2} \)
$97$ \( ( 1120 + T^{2} )^{2} \)
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