Properties

Label 2790.2.a.bj
Level $2790$
Weight $2$
Character orbit 2790.a
Self dual yes
Analytic conductor $22.278$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + ( - \beta_{2} + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} + ( - \beta_{2} + 1) q^{7} - q^{8} + q^{10} + ( - \beta_{2} - 1) q^{11} + ( - \beta_1 + 2) q^{13} + (\beta_{2} - 1) q^{14} + q^{16} + ( - \beta_{3} + \beta_1 - 1) q^{17} + ( - \beta_{2} + \beta_1 + 1) q^{19} - q^{20} + (\beta_{2} + 1) q^{22} + (\beta_{3} - \beta_{2}) q^{23} + q^{25} + (\beta_1 - 2) q^{26} + ( - \beta_{2} + 1) q^{28} - 2 \beta_1 q^{29} + q^{31} - q^{32} + (\beta_{3} - \beta_1 + 1) q^{34} + (\beta_{2} - 1) q^{35} + ( - 2 \beta_{3} + \beta_1) q^{37} + (\beta_{2} - \beta_1 - 1) q^{38} + q^{40} - 2 \beta_1 q^{41} + (2 \beta_{3} + \beta_{2} - \beta_1 + 5) q^{43} + ( - \beta_{2} - 1) q^{44} + ( - \beta_{3} + \beta_{2}) q^{46} + (2 \beta_{3} + 2) q^{47} + ( - 3 \beta_{2} + \beta_1) q^{49} - q^{50} + ( - \beta_1 + 2) q^{52} + (\beta_{2} + \beta_1 + 1) q^{53} + (\beta_{2} + 1) q^{55} + (\beta_{2} - 1) q^{56} + 2 \beta_1 q^{58} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{59} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 - 1) q^{61} - q^{62} + q^{64} + (\beta_1 - 2) q^{65} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{67} + ( - \beta_{3} + \beta_1 - 1) q^{68} + ( - \beta_{2} + 1) q^{70} + ( - \beta_{2} + 2 \beta_1 - 1) q^{71} + (3 \beta_{2} - \beta_1 + 5) q^{73} + (2 \beta_{3} - \beta_1) q^{74} + ( - \beta_{2} + \beta_1 + 1) q^{76} + ( - \beta_{2} + \beta_1 + 5) q^{77} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{79} - q^{80} + 2 \beta_1 q^{82} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{83} + (\beta_{3} - \beta_1 + 1) q^{85} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 5) q^{86} + (\beta_{2} + 1) q^{88} + (2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 5) q^{89} + (2 \beta_{3} - 2 \beta_1 + 6) q^{91} + (\beta_{3} - \beta_{2}) q^{92} + ( - 2 \beta_{3} - 2) q^{94} + (\beta_{2} - \beta_1 - 1) q^{95} + (2 \beta_{2} - 2 \beta_1 + 4) q^{97} + (3 \beta_{2} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 5 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 5 q^{7} - 4 q^{8} + 4 q^{10} - 3 q^{11} + 6 q^{13} - 5 q^{14} + 4 q^{16} + 7 q^{19} - 4 q^{20} + 3 q^{22} - q^{23} + 4 q^{25} - 6 q^{26} + 5 q^{28} - 4 q^{29} + 4 q^{31} - 4 q^{32} - 5 q^{35} + 6 q^{37} - 7 q^{38} + 4 q^{40} - 4 q^{41} + 13 q^{43} - 3 q^{44} + q^{46} + 4 q^{47} + 5 q^{49} - 4 q^{50} + 6 q^{52} + 5 q^{53} + 3 q^{55} - 5 q^{56} + 4 q^{58} - 8 q^{59} - 4 q^{61} - 4 q^{62} + 4 q^{64} - 6 q^{65} + 8 q^{67} + 5 q^{70} + q^{71} + 15 q^{73} - 6 q^{74} + 7 q^{76} + 23 q^{77} + 5 q^{79} - 4 q^{80} + 4 q^{82} + 2 q^{83} - 13 q^{86} + 3 q^{88} + 9 q^{89} + 16 q^{91} - q^{92} - 4 q^{94} - 7 q^{95} + 10 q^{97} - 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.36865
−0.787711
−2.10710
1.52616
−1.00000 0 1.00000 −1.00000 0 −1.81471 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 −0.662077 −1.00000 0 1.00000
1.3 −1.00000 0 1.00000 −1.00000 0 2.92682 −1.00000 0 1.00000
1.4 −1.00000 0 1.00000 −1.00000 0 4.54997 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(1\)
\(31\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2790.2.a.bj 4
3.b odd 2 1 2790.2.a.bk yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2790.2.a.bj 4 1.a even 1 1 trivial
2790.2.a.bk yes 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2790))\):

\( T_{7}^{4} - 5T_{7}^{3} - 4T_{7}^{2} + 24T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} + 3T_{11}^{3} - 10T_{11}^{2} - 20T_{11} + 24 \) Copy content Toggle raw display
\( T_{13}^{4} - 6T_{13}^{3} - 12T_{13}^{2} + 56T_{13} + 64 \) Copy content Toggle raw display
\( T_{17}^{4} - 40T_{17}^{2} + 80T_{17} + 48 \) Copy content Toggle raw display
\( T_{19}^{4} - 7T_{19}^{3} - 12T_{19}^{2} + 48T_{19} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 5 T^{3} - 4 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 3 T^{3} - 10 T^{2} - 20 T + 24 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} - 12 T^{2} + 56 T + 64 \) Copy content Toggle raw display
$17$ \( T^{4} - 40 T^{2} + 80 T + 48 \) Copy content Toggle raw display
$19$ \( T^{4} - 7 T^{3} - 12 T^{2} + 48 T + 64 \) Copy content Toggle raw display
$23$ \( T^{4} + T^{3} - 36 T^{2} + 80 T - 48 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} - 96 T^{2} + \cdots + 1536 \) Copy content Toggle raw display
$31$ \( (T - 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} - 100 T^{2} + \cdots + 2272 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} - 96 T^{2} + \cdots + 1536 \) Copy content Toggle raw display
$43$ \( T^{4} - 13 T^{3} - 68 T^{2} + \cdots + 2624 \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} - 112 T^{2} + \cdots + 1536 \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} - 38 T^{2} + 68 T - 24 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} - 152 T^{2} + \cdots + 3072 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} - 256 T^{2} + \cdots + 15376 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} - 160 T^{2} + \cdots - 5552 \) Copy content Toggle raw display
$71$ \( T^{4} - T^{3} - 98 T^{2} + 28 T + 2136 \) Copy content Toggle raw display
$73$ \( T^{4} - 15 T^{3} - 36 T^{2} + \cdots - 3008 \) Copy content Toggle raw display
$79$ \( T^{4} - 5 T^{3} - 156 T^{2} - 208 T - 64 \) Copy content Toggle raw display
$83$ \( T^{4} - 2 T^{3} - 144 T^{2} + \cdots - 768 \) Copy content Toggle raw display
$89$ \( T^{4} - 9 T^{3} - 238 T^{2} + \cdots - 3768 \) Copy content Toggle raw display
$97$ \( T^{4} - 10 T^{3} - 84 T^{2} + \cdots - 128 \) Copy content Toggle raw display
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