Properties

Label 7440.2.a.bl
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). 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 \( q + q^{3} + q^{5} - \beta q^{7} + q^{9} - \beta q^{11} + 2 q^{13} + q^{15} + ( - 2 \beta + 2) q^{17} + 3 \beta q^{19} - \beta q^{21} + (\beta - 4) q^{23} + q^{25} + q^{27} + (2 \beta + 2) q^{29} - q^{31} - \beta q^{33} - \beta q^{35} + ( - 2 \beta + 2) q^{37} + 2 q^{39} + (2 \beta + 2) q^{41} - 5 \beta q^{43} + q^{45} - 2 \beta q^{47} + (\beta - 3) q^{49} + ( - 2 \beta + 2) q^{51} + ( - \beta + 10) q^{53} - \beta q^{55} + 3 \beta q^{57} + (2 \beta + 8) q^{59} + 6 q^{61} - \beta q^{63} + 2 q^{65} + 6 \beta q^{67} + (\beta - 4) q^{69} + 3 \beta q^{71} + ( - 5 \beta + 2) q^{73} + q^{75} + (\beta + 4) q^{77} + ( - 3 \beta + 12) q^{79} + q^{81} + ( - 4 \beta - 4) q^{83} + ( - 2 \beta + 2) q^{85} + (2 \beta + 2) q^{87} + ( - 3 \beta - 6) q^{89} - 2 \beta q^{91} - q^{93} + 3 \beta q^{95} - 6 q^{97} - \beta q^{99} +O(q^{100}) \) 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 \( 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} + 3 q^{19} - q^{21} - 7 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 2 q^{31} - q^{33} - q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} - 5 q^{43} + 2 q^{45} - 2 q^{47} - 5 q^{49} + 2 q^{51} + 19 q^{53} - q^{55} + 3 q^{57} + 18 q^{59} + 12 q^{61} - q^{63} + 4 q^{65} + 6 q^{67} - 7 q^{69} + 3 q^{71} - q^{73} + 2 q^{75} + 9 q^{77} + 21 q^{79} + 2 q^{81} - 12 q^{83} + 2 q^{85} + 6 q^{87} - 15 q^{89} - 2 q^{91} - 2 q^{93} + 3 q^{95} - 12 q^{97} - q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
0 1.00000 0 1.00000 0 −2.56155 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 1.56155 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.bl 2
4.b odd 2 1 930.2.a.p 2
12.b even 2 1 2790.2.a.be 2
20.d odd 2 1 4650.2.a.ce 2
20.e even 4 2 4650.2.d.bd 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.p 2 4.b odd 2 1
2790.2.a.be 2 12.b even 2 1
4650.2.a.ce 2 20.d odd 2 1
4650.2.d.bd 4 20.e even 4 2
7440.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7440))\):

\( T_{7}^{2} + T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 16 \) Copy content Toggle raw display
\( T_{19}^{2} - 3T_{19} - 36 \) 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
$7$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$19$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$43$ \( T^{2} + 5T - 100 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} - 19T + 86 \) Copy content Toggle raw display
$59$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 6T - 144 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$73$ \( T^{2} + T - 106 \) Copy content Toggle raw display
$79$ \( T^{2} - 21T + 72 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T - 32 \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 18 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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