Properties

Label 7623.2.a.bd
Level $7623$
Weight $2$
Character orbit 7623.a
Self dual yes
Analytic conductor $60.870$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 1) q^{4} + q^{5} + q^{7} - 3 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta + 1) q^{4} + q^{5} + q^{7} - 3 q^{8} - \beta q^{10} + ( - 2 \beta + 1) q^{13} - \beta q^{14} + (\beta - 2) q^{16} - 4 q^{17} - 3 q^{19} + (\beta + 1) q^{20} + 2 q^{23} - 4 q^{25} + (\beta + 6) q^{26} + (\beta + 1) q^{28} + (2 \beta + 1) q^{29} - 2 q^{31} + (\beta + 3) q^{32} + 4 \beta q^{34} + q^{35} + ( - 4 \beta + 1) q^{37} + 3 \beta q^{38} - 3 q^{40} + (4 \beta - 2) q^{41} + (4 \beta - 4) q^{43} - 2 \beta q^{46} + ( - 2 \beta + 7) q^{47} + q^{49} + 4 \beta q^{50} + ( - 3 \beta - 5) q^{52} - 3 q^{56} + ( - 3 \beta - 6) q^{58} + (2 \beta + 3) q^{59} + (4 \beta + 2) q^{61} + 2 \beta q^{62} + ( - 6 \beta + 1) q^{64} + ( - 2 \beta + 1) q^{65} + (2 \beta - 3) q^{67} + ( - 4 \beta - 4) q^{68} - \beta q^{70} + (4 \beta + 2) q^{71} + (6 \beta - 1) q^{73} + (3 \beta + 12) q^{74} + ( - 3 \beta - 3) q^{76} + (4 \beta - 6) q^{79} + (\beta - 2) q^{80} + ( - 2 \beta - 12) q^{82} - 12 q^{83} - 4 q^{85} - 12 q^{86} + 6 q^{89} + ( - 2 \beta + 1) q^{91} + (2 \beta + 2) q^{92} + ( - 5 \beta + 6) q^{94} - 3 q^{95} + (4 \beta - 8) q^{97} - \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8} - q^{10} - q^{14} - 3 q^{16} - 8 q^{17} - 6 q^{19} + 3 q^{20} + 4 q^{23} - 8 q^{25} + 13 q^{26} + 3 q^{28} + 4 q^{29} - 4 q^{31} + 7 q^{32} + 4 q^{34} + 2 q^{35} - 2 q^{37} + 3 q^{38} - 6 q^{40} - 4 q^{43} - 2 q^{46} + 12 q^{47} + 2 q^{49} + 4 q^{50} - 13 q^{52} - 6 q^{56} - 15 q^{58} + 8 q^{59} + 8 q^{61} + 2 q^{62} - 4 q^{64} - 4 q^{67} - 12 q^{68} - q^{70} + 8 q^{71} + 4 q^{73} + 27 q^{74} - 9 q^{76} - 8 q^{79} - 3 q^{80} - 26 q^{82} - 24 q^{83} - 8 q^{85} - 24 q^{86} + 12 q^{89} + 6 q^{92} + 7 q^{94} - 6 q^{95} - 12 q^{97} - q^{98}+O(q^{100}) \) 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
2.30278
−1.30278
−2.30278 0 3.30278 1.00000 0 1.00000 −3.00000 0 −2.30278
1.2 1.30278 0 −0.302776 1.00000 0 1.00000 −3.00000 0 1.30278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 7623.2.a.bd 2
3.b odd 2 1 7623.2.a.br 2
11.b odd 2 1 693.2.a.i yes 2
33.d even 2 1 693.2.a.g 2
77.b even 2 1 4851.2.a.z 2
231.h odd 2 1 4851.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.a.g 2 33.d even 2 1
693.2.a.i yes 2 11.b odd 2 1
4851.2.a.x 2 231.h odd 2 1
4851.2.a.z 2 77.b even 2 1
7623.2.a.bd 2 1.a even 1 1 trivial
7623.2.a.br 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} + T_{2} - 3 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 13 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T + 3)^{2} \) Copy content Toggle raw display
$23$ \( (T - 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 51 \) Copy content Toggle raw display
$41$ \( T^{2} - 52 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 23 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 113 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 36 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 12T - 16 \) Copy content Toggle raw display
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