Properties

Label 2790.2.a.bd
Level $2790$
Weight $2$
Character orbit 2790.a
Self dual yes
Analytic conductor $22.278$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + \beta q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} + \beta q^{7} - q^{8} + q^{10} + (\beta - 4) q^{11} - 2 q^{13} - \beta q^{14} + q^{16} + (2 \beta - 2) q^{17} + (\beta - 4) q^{19} - q^{20} + ( - \beta + 4) q^{22} - \beta q^{23} + q^{25} + 2 q^{26} + \beta q^{28} + (2 \beta + 2) q^{29} - q^{31} - q^{32} + ( - 2 \beta + 2) q^{34} - \beta q^{35} + ( - 2 \beta + 6) q^{37} + ( - \beta + 4) q^{38} + q^{40} + ( - 2 \beta + 6) q^{41} + ( - \beta + 4) q^{43} + (\beta - 4) q^{44} + \beta q^{46} + 2 \beta q^{47} + (\beta + 1) q^{49} - q^{50} - 2 q^{52} + ( - \beta + 2) q^{53} + ( - \beta + 4) q^{55} - \beta q^{56} + ( - 2 \beta - 2) q^{58} + ( - 2 \beta + 4) q^{59} + (4 \beta - 2) q^{61} + q^{62} + q^{64} + 2 q^{65} + (2 \beta + 4) q^{67} + (2 \beta - 2) q^{68} + \beta q^{70} - \beta q^{71} + ( - 3 \beta + 2) q^{73} + (2 \beta - 6) q^{74} + (\beta - 4) q^{76} + ( - 3 \beta + 8) q^{77} + (\beta - 8) q^{79} - q^{80} + (2 \beta - 6) q^{82} + 12 q^{83} + ( - 2 \beta + 2) q^{85} + (\beta - 4) q^{86} + ( - \beta + 4) q^{88} + (\beta - 2) q^{89} - 2 \beta q^{91} - \beta q^{92} - 2 \beta q^{94} + ( - \beta + 4) q^{95} + 2 q^{97} + ( - \beta - 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - 7 q^{11} - 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 7 q^{19} - 2 q^{20} + 7 q^{22} - q^{23} + 2 q^{25} + 4 q^{26} + q^{28} + 6 q^{29} - 2 q^{31} - 2 q^{32} + 2 q^{34} - q^{35} + 10 q^{37} + 7 q^{38} + 2 q^{40} + 10 q^{41} + 7 q^{43} - 7 q^{44} + q^{46} + 2 q^{47} + 3 q^{49} - 2 q^{50} - 4 q^{52} + 3 q^{53} + 7 q^{55} - q^{56} - 6 q^{58} + 6 q^{59} + 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 2 q^{68} + q^{70} - q^{71} + q^{73} - 10 q^{74} - 7 q^{76} + 13 q^{77} - 15 q^{79} - 2 q^{80} - 10 q^{82} + 24 q^{83} + 2 q^{85} - 7 q^{86} + 7 q^{88} - 3 q^{89} - 2 q^{91} - q^{92} - 2 q^{94} + 7 q^{95} + 4 q^{97} - 3 q^{98}+O(q^{100}) \) 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
−2.37228
3.37228
−1.00000 0 1.00000 −1.00000 0 −2.37228 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 3.37228 −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.bd 2
3.b odd 2 1 930.2.a.r 2
12.b even 2 1 7440.2.a.bg 2
15.d odd 2 1 4650.2.a.by 2
15.e even 4 2 4650.2.d.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.r 2 3.b odd 2 1
2790.2.a.bd 2 1.a even 1 1 trivial
4650.2.a.by 2 15.d odd 2 1
4650.2.d.bh 4 15.e even 4 2
7440.2.a.bg 2 12.b even 2 1

Hecke kernels

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

\( T_{7}^{2} - T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 7T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} - 32 \) Copy content Toggle raw display
\( T_{19}^{2} + 7T_{19} + 4 \) 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
$7$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$11$ \( T^{2} + 7T + 4 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 10T - 8 \) Copy content Toggle raw display
$41$ \( T^{2} - 10T - 8 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$61$ \( T^{2} - 132 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T - 8 \) Copy content Toggle raw display
$71$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$73$ \( T^{2} - T - 74 \) Copy content Toggle raw display
$79$ \( T^{2} + 15T + 48 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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