Properties

Label 2790.2.a.bg
Level $2790$
Weight $2$
Character orbit 2790.a
Self dual yes
Analytic conductor $22.278$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 310)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} - 2 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - q^{5} - 2 q^{7} + q^{8} - q^{10} + (\beta - 2) q^{11} + ( - \beta + 2) q^{13} - 2 q^{14} + q^{16} - 2 \beta q^{17} + 2 \beta q^{19} - q^{20} + (\beta - 2) q^{22} - 2 q^{23} + q^{25} + ( - \beta + 2) q^{26} - 2 q^{28} + ( - \beta - 8) q^{29} - q^{31} + q^{32} - 2 \beta q^{34} + 2 q^{35} + (\beta + 2) q^{37} + 2 \beta q^{38} - q^{40} + (\beta - 8) q^{43} + (\beta - 2) q^{44} - 2 q^{46} - 6 q^{47} - 3 q^{49} + q^{50} + ( - \beta + 2) q^{52} + ( - 3 \beta + 2) q^{53} + ( - \beta + 2) q^{55} - 2 q^{56} + ( - \beta - 8) q^{58} + (2 \beta - 4) q^{59} + ( - \beta - 4) q^{61} - q^{62} + q^{64} + (\beta - 2) q^{65} + (2 \beta - 8) q^{67} - 2 \beta q^{68} + 2 q^{70} + 4 \beta q^{71} - 4 q^{73} + (\beta + 2) q^{74} + 2 \beta q^{76} + ( - 2 \beta + 4) q^{77} + 2 \beta q^{79} - q^{80} + ( - 5 \beta - 4) q^{83} + 2 \beta q^{85} + (\beta - 8) q^{86} + (\beta - 2) q^{88} + ( - 4 \beta - 6) q^{89} + (2 \beta - 4) q^{91} - 2 q^{92} - 6 q^{94} - 2 \beta q^{95} + ( - 2 \beta + 4) q^{97} - 3 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8} - 2 q^{10} - 4 q^{11} + 4 q^{13} - 4 q^{14} + 2 q^{16} - 2 q^{20} - 4 q^{22} - 4 q^{23} + 2 q^{25} + 4 q^{26} - 4 q^{28} - 16 q^{29} - 2 q^{31} + 2 q^{32} + 4 q^{35} + 4 q^{37} - 2 q^{40} - 16 q^{43} - 4 q^{44} - 4 q^{46} - 12 q^{47} - 6 q^{49} + 2 q^{50} + 4 q^{52} + 4 q^{53} + 4 q^{55} - 4 q^{56} - 16 q^{58} - 8 q^{59} - 8 q^{61} - 2 q^{62} + 2 q^{64} - 4 q^{65} - 16 q^{67} + 4 q^{70} - 8 q^{73} + 4 q^{74} + 8 q^{77} - 2 q^{80} - 8 q^{83} - 16 q^{86} - 4 q^{88} - 12 q^{89} - 8 q^{91} - 4 q^{92} - 12 q^{94} + 8 q^{97} - 6 q^{98}+O(q^{100}) \) 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
−2.44949
2.44949
1.00000 0 1.00000 −1.00000 0 −2.00000 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 −2.00000 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.bg 2
3.b odd 2 1 310.2.a.d 2
12.b even 2 1 2480.2.a.q 2
15.d odd 2 1 1550.2.a.i 2
15.e even 4 2 1550.2.b.g 4
24.f even 2 1 9920.2.a.bq 2
24.h odd 2 1 9920.2.a.bo 2
93.c even 2 1 9610.2.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.d 2 3.b odd 2 1
1550.2.a.i 2 15.d odd 2 1
1550.2.b.g 4 15.e even 4 2
2480.2.a.q 2 12.b even 2 1
2790.2.a.bg 2 1.a even 1 1 trivial
9610.2.a.f 2 93.c even 2 1
9920.2.a.bo 2 24.h odd 2 1
9920.2.a.bq 2 24.f even 2 1

Hecke kernels

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

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 24 \) Copy content Toggle raw display
\( T_{19}^{2} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 24 \) Copy content Toggle raw display
$19$ \( T^{2} - 24 \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16T + 58 \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16T + 58 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 50 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T - 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 10 \) Copy content Toggle raw display
$67$ \( T^{2} + 16T + 40 \) Copy content Toggle raw display
$71$ \( T^{2} - 96 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 24 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 134 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T - 60 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 8 \) Copy content Toggle raw display
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