Properties

Label 2736.3.o.l
Level $2736$
Weight $3$
Character orbit 2736.o
Analytic conductor $74.551$
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: \( k \) \(=\) \( 3 \)
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: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 57)
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.

\(f(q)\) \(=\) \( q + ( 3 + \beta_{1} ) q^{5} + ( -9 - \beta_{1} ) q^{7} +O(q^{10})\) \( q + ( 3 + \beta_{1} ) q^{5} + ( -9 - \beta_{1} ) q^{7} + ( -3 + \beta_{1} ) q^{11} + ( -\beta_{2} + 2 \beta_{3} ) q^{13} + ( -1 - 5 \beta_{1} ) q^{17} + ( -1 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{19} + ( 14 + 2 \beta_{1} ) q^{23} + ( -2 + 5 \beta_{1} ) q^{25} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{29} + ( -\beta_{2} - 4 \beta_{3} ) q^{31} + ( -41 - 11 \beta_{1} ) q^{35} + ( 7 \beta_{2} - 10 \beta_{3} ) q^{37} + ( 3 \beta_{2} + \beta_{3} ) q^{41} + ( -1 + 15 \beta_{1} ) q^{43} + ( 17 - 9 \beta_{1} ) q^{47} + ( 46 + 17 \beta_{1} ) q^{49} + ( 9 \beta_{2} - 3 \beta_{3} ) q^{53} + ( 5 - \beta_{1} ) q^{55} + ( -7 \beta_{2} - \beta_{3} ) q^{59} + ( -19 - 3 \beta_{1} ) q^{61} + ( -3 \beta_{2} + 13 \beta_{3} ) q^{65} + ( -8 \beta_{2} - 18 \beta_{3} ) q^{67} + ( 5 \beta_{2} - 17 \beta_{3} ) q^{71} + ( 21 - 3 \beta_{1} ) q^{73} + ( 13 - 5 \beta_{1} ) q^{77} + ( 10 \beta_{2} - 3 \beta_{3} ) q^{79} + ( 2 - 28 \beta_{1} ) q^{83} + ( -73 - 11 \beta_{1} ) q^{85} + ( -\beta_{2} - 23 \beta_{3} ) q^{89} + ( 9 \beta_{2} - 25 \beta_{3} ) q^{91} + ( -31 - 5 \beta_{1} - 7 \beta_{2} + 17 \beta_{3} ) q^{95} + ( -13 \beta_{2} + 5 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{5} - 34q^{7} + O(q^{10}) \) \( 4q + 10q^{5} - 34q^{7} - 14q^{11} + 6q^{17} + 52q^{23} - 18q^{25} - 142q^{35} - 34q^{43} + 86q^{47} + 150q^{49} + 22q^{55} - 70q^{61} + 90q^{73} + 62q^{77} + 64q^{83} - 270q^{85} - 114q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 9 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 6 \nu^{2} + 6 \nu + 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{3} + 2 \nu^{2} - 2 \nu - 25 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 4\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{3} - 5 \beta_{2} + 4 \beta_{1} + 20\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{3} + \beta_{2} + 14\)\()/2\)

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
−1.63746 + 1.52274i
−1.63746 1.52274i
2.13746 0.656712i
2.13746 + 0.656712i
0 0 0 −1.27492 0 −4.72508 0 0 0
721.2 0 0 0 −1.27492 0 −4.72508 0 0 0
721.3 0 0 0 6.27492 0 −12.2749 0 0 0
721.4 0 0 0 6.27492 0 −12.2749 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.b odd 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.l 4
3.b odd 2 1 912.3.o.b 4
4.b odd 2 1 171.3.c.f 4
12.b even 2 1 57.3.c.b 4
19.b odd 2 1 inner 2736.3.o.l 4
57.d even 2 1 912.3.o.b 4
76.d even 2 1 171.3.c.f 4
228.b odd 2 1 57.3.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.c.b 4 12.b even 2 1
57.3.c.b 4 228.b odd 2 1
171.3.c.f 4 4.b odd 2 1
171.3.c.f 4 76.d even 2 1
912.3.o.b 4 3.b odd 2 1
912.3.o.b 4 57.d even 2 1
2736.3.o.l 4 1.a even 1 1 trivial
2736.3.o.l 4 19.b odd 2 1 inner

Hecke kernels

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

\( T_{5}^{2} - 5 T_{5} - 8 \)
\( T_{7}^{2} + 17 T_{7} + 58 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -8 - 5 T + T^{2} )^{2} \)
$7$ \( ( 58 + 17 T + T^{2} )^{2} \)
$11$ \( ( -2 + 7 T + T^{2} )^{2} \)
$13$ \( 3136 + 188 T^{2} + T^{4} \)
$17$ \( ( -354 - 3 T + T^{2} )^{2} \)
$19$ \( 130321 + 494 T^{2} + T^{4} \)
$23$ \( ( 112 - 26 T + T^{2} )^{2} \)
$29$ \( 50176 + 752 T^{2} + T^{4} \)
$31$ \( 222784 + 956 T^{2} + T^{4} \)
$37$ \( 7354944 + 6108 T^{2} + T^{4} \)
$41$ \( 12544 + 992 T^{2} + T^{4} \)
$43$ \( ( -3134 + 17 T + T^{2} )^{2} \)
$47$ \( ( -692 - 43 T + T^{2} )^{2} \)
$53$ \( 4064256 + 6768 T^{2} + T^{4} \)
$59$ \( 256 + 4832 T^{2} + T^{4} \)
$61$ \( ( 178 + 35 T + T^{2} )^{2} \)
$67$ \( 162205696 + 25904 T^{2} + T^{4} \)
$71$ \( 112896 + 11616 T^{2} + T^{4} \)
$73$ \( ( 378 - 45 T + T^{2} )^{2} \)
$79$ \( 5456896 + 8396 T^{2} + T^{4} \)
$83$ \( ( -10916 - 32 T + T^{2} )^{2} \)
$89$ \( 94945536 + 24288 T^{2} + T^{4} \)
$97$ \( 21086464 + 14048 T^{2} + T^{4} \)
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