Properties

Label 7440.2.a.bg
Level $7440$
Weight $2$
Character orbit 7440.a
Self dual yes
Analytic conductor $59.409$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7440.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4086991038\)
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.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - \beta q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - \beta q^{7} + q^{9} + (\beta - 4) q^{11} - 2 q^{13} - q^{15} + ( - 2 \beta + 2) q^{17} + ( - \beta + 4) q^{19} + \beta q^{21} - \beta q^{23} + q^{25} - q^{27} + ( - 2 \beta - 2) q^{29} + q^{31} + ( - \beta + 4) q^{33} - \beta q^{35} + ( - 2 \beta + 6) q^{37} + 2 q^{39} + (2 \beta - 6) q^{41} + (\beta - 4) q^{43} + q^{45} + 2 \beta q^{47} + (\beta + 1) q^{49} + (2 \beta - 2) q^{51} + (\beta - 2) q^{53} + (\beta - 4) q^{55} + (\beta - 4) q^{57} + ( - 2 \beta + 4) q^{59} + (4 \beta - 2) q^{61} - \beta q^{63} - 2 q^{65} + ( - 2 \beta - 4) q^{67} + \beta q^{69} - \beta q^{71} + ( - 3 \beta + 2) q^{73} - q^{75} + (3 \beta - 8) q^{77} + ( - \beta + 8) q^{79} + q^{81} + 12 q^{83} + ( - 2 \beta + 2) q^{85} + (2 \beta + 2) q^{87} + ( - \beta + 2) q^{89} + 2 \beta q^{91} - q^{93} + ( - \beta + 4) q^{95} + 2 q^{97} + (\beta - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 4 q^{13} - 2 q^{15} + 2 q^{17} + 7 q^{19} + q^{21} - q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 2 q^{31} + 7 q^{33} - q^{35} + 10 q^{37} + 4 q^{39} - 10 q^{41} - 7 q^{43} + 2 q^{45} + 2 q^{47} + 3 q^{49} - 2 q^{51} - 3 q^{53} - 7 q^{55} - 7 q^{57} + 6 q^{59} - q^{63} - 4 q^{65} - 10 q^{67} + q^{69} - q^{71} + q^{73} - 2 q^{75} - 13 q^{77} + 15 q^{79} + 2 q^{81} + 24 q^{83} + 2 q^{85} + 6 q^{87} + 3 q^{89} + 2 q^{91} - 2 q^{93} + 7 q^{95} + 4 q^{97} - 7 q^{99}+O(q^{100}) \) 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
3.37228
−2.37228
0 −1.00000 0 1.00000 0 −3.37228 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 2.37228 0 1.00000 0
\(n\): 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 7440.2.a.bg 2
4.b odd 2 1 930.2.a.r 2
12.b even 2 1 2790.2.a.bd 2
20.d odd 2 1 4650.2.a.by 2
20.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 4.b odd 2 1
2790.2.a.bd 2 12.b even 2 1
4650.2.a.by 2 20.d odd 2 1
4650.2.d.bh 4 20.e even 4 2
7440.2.a.bg 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7440))\):

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

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) 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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