Properties

Label 7623.2.a.ca
Level $7623$
Weight $2$
Character orbit 7623.a
Self dual yes
Analytic conductor $60.870$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{2} - \beta_1) q^{5} + q^{7} + ( - \beta_{2} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{2} - \beta_1) q^{5} + q^{7} + ( - \beta_{2} - 1) q^{8} + ( - \beta_1 + 2) q^{10} + (\beta_1 + 2) q^{13} - \beta_1 q^{14} + ( - \beta_{2} + 2 \beta_1 - 1) q^{16} + (\beta_{2} - \beta_1) q^{17} + ( - \beta_{2} - \beta_1 + 3) q^{19} + ( - \beta_{2} + 3) q^{20} + ( - 3 \beta_{2} - \beta_1 + 1) q^{23} - 2 \beta_{2} q^{25} + ( - \beta_{2} - 2 \beta_1 - 3) q^{26} + (\beta_{2} + 1) q^{28} + ( - \beta_1 - 2) q^{29} + (2 \beta_{2} - 2 \beta_1) q^{31} + (\beta_{2} + 2 \beta_1 - 3) q^{32} + ( - \beta_1 + 2) q^{34} + (\beta_{2} - \beta_1) q^{35} + ( - \beta_{2} - 1) q^{37} + (2 \beta_{2} - 2 \beta_1 + 4) q^{38} + (\beta_{2} - 3) q^{40} + ( - \beta_{2} + 2 \beta_1 - 5) q^{41} + ( - \beta_{2} - 4 \beta_1 + 4) q^{43} + (4 \beta_{2} + 2 \beta_1 + 6) q^{46} + (2 \beta_{2} - 3 \beta_1 + 3) q^{47} + q^{49} + (2 \beta_{2} + 2 \beta_1 + 2) q^{50} + (3 \beta_{2} + 2 \beta_1 + 3) q^{52} + ( - \beta_{2} + 7) q^{53} + ( - \beta_{2} - 1) q^{56} + (\beta_{2} + 2 \beta_1 + 3) q^{58} + ( - 3 \beta_1 - 1) q^{59} + (3 \beta_{2} - 3 \beta_1 + 5) q^{61} + ( - 2 \beta_1 + 4) q^{62} + ( - \beta_{2} - 2 \beta_1 - 5) q^{64} + (2 \beta_{2} - \beta_1 - 2) q^{65} + ( - 5 \beta_{2} + 2 \beta_1) q^{67} + ( - \beta_{2} + 3) q^{68} + ( - \beta_1 + 2) q^{70} + ( - \beta_{2} + 3 \beta_1 - 1) q^{71} + (3 \beta_{2} + 5 \beta_1 - 3) q^{73} + (\beta_{2} + 2 \beta_1 + 1) q^{74} + (2 \beta_{2} - 4 \beta_1 - 2) q^{76} + ( - 2 \beta_{2} + 6 \beta_1 + 4) q^{79} + (\beta_{2} + 2 \beta_1 - 7) q^{80} + ( - \beta_{2} + 6 \beta_1 - 5) q^{82} + (6 \beta_{2} - 3 \beta_1 + 5) q^{83} + ( - 2 \beta_{2} + 5) q^{85} + (5 \beta_{2} - 3 \beta_1 + 13) q^{86} + (3 \beta_{2} + \beta_1) q^{89} + (\beta_1 + 2) q^{91} + ( - 8 \beta_1 - 12) q^{92} + (\beta_{2} - 5 \beta_1 + 7) q^{94} + (5 \beta_{2} - 5 \beta_1 - 1) q^{95} + (2 \beta_{2} - 7 \beta_1 - 6) q^{97} - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{4} - q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{4} - q^{5} + 3 q^{7} - 3 q^{8} + 5 q^{10} + 7 q^{13} - q^{14} - q^{16} - q^{17} + 8 q^{19} + 9 q^{20} + 2 q^{23} - 11 q^{26} + 3 q^{28} - 7 q^{29} - 2 q^{31} - 7 q^{32} + 5 q^{34} - q^{35} - 3 q^{37} + 10 q^{38} - 9 q^{40} - 13 q^{41} + 8 q^{43} + 20 q^{46} + 6 q^{47} + 3 q^{49} + 8 q^{50} + 11 q^{52} + 21 q^{53} - 3 q^{56} + 11 q^{58} - 6 q^{59} + 12 q^{61} + 10 q^{62} - 17 q^{64} - 7 q^{65} + 2 q^{67} + 9 q^{68} + 5 q^{70} - 4 q^{73} + 5 q^{74} - 10 q^{76} + 18 q^{79} - 19 q^{80} - 9 q^{82} + 12 q^{83} + 15 q^{85} + 36 q^{86} + q^{89} + 7 q^{91} - 44 q^{92} + 16 q^{94} - 8 q^{95} - 25 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.34292
0.470683
−1.81361
−2.34292 0 3.48929 0.146365 0 1.00000 −3.48929 0 −0.342923
1.2 −0.470683 0 −1.77846 −3.24914 0 1.00000 1.77846 0 1.52932
1.3 1.81361 0 1.28917 2.10278 0 1.00000 −1.28917 0 3.81361
\(n\): 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.ca 3
3.b odd 2 1 2541.2.a.bj yes 3
11.b odd 2 1 7623.2.a.cc 3
33.d even 2 1 2541.2.a.bh 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bh 3 33.d even 2 1
2541.2.a.bj yes 3 3.b odd 2 1
7623.2.a.ca 3 1.a even 1 1 trivial
7623.2.a.cc 3 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{3} + T_{2}^{2} - 4T_{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{3} + T_{5}^{2} - 7T_{5} + 1 \) Copy content Toggle raw display
\( T_{13}^{3} - 7T_{13}^{2} + 12T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 4T - 2 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + T^{2} - 7T + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 7 T^{2} + 12 T - 2 \) Copy content Toggle raw display
$17$ \( T^{3} + T^{2} - 7T + 1 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} + 6 T + 44 \) Copy content Toggle raw display
$23$ \( T^{3} - 2 T^{2} - 78 T + 152 \) Copy content Toggle raw display
$29$ \( T^{3} + 7 T^{2} + 12 T + 2 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 28 T + 8 \) Copy content Toggle raw display
$37$ \( T^{3} + 3 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$41$ \( T^{3} + 13 T^{2} + 40 T + 32 \) Copy content Toggle raw display
$43$ \( T^{3} - 8 T^{2} - 71 T + 422 \) Copy content Toggle raw display
$47$ \( T^{3} - 6 T^{2} - 31 T + 34 \) Copy content Toggle raw display
$53$ \( T^{3} - 21 T^{2} + 140 T - 296 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} - 27 T - 86 \) Copy content Toggle raw display
$61$ \( T^{3} - 12 T^{2} - 18 T + 292 \) Copy content Toggle raw display
$67$ \( T^{3} - 2 T^{2} - 151 T - 584 \) Copy content Toggle raw display
$71$ \( T^{3} - 34T + 76 \) Copy content Toggle raw display
$73$ \( T^{3} + 4 T^{2} - 226 T - 1628 \) Copy content Toggle raw display
$79$ \( T^{3} - 18 T^{2} - 28 T + 1208 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} - 171 T + 2056 \) Copy content Toggle raw display
$89$ \( T^{3} - T^{2} - 79 T - 73 \) Copy content Toggle raw display
$97$ \( T^{3} + 25 T^{2} + 24 T - 1882 \) Copy content Toggle raw display
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