Properties

Label 7623.2.a.cd
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: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + (\beta_1 + 2) q^{4} + (\beta_{2} + \beta_1 - 1) q^{5} + q^{7} + (2 \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + (\beta_1 + 2) q^{4} + (\beta_{2} + \beta_1 - 1) q^{5} + q^{7} + (2 \beta_1 + 1) q^{8} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{10} + ( - \beta_{2} + 3 \beta_1 + 1) q^{13} + (\beta_{2} + 1) q^{14} + (\beta_{2} + 2 \beta_1 - 1) q^{16} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{17} + (\beta_{2} - \beta_1 + 3) q^{19} + (3 \beta_{2} + 2 \beta_1 + 2) q^{20} + ( - 2 \beta_{2} - 4) q^{23} + ( - 3 \beta_{2} + \beta_1 + 4) q^{25} + (2 \beta_{2} + 5 \beta_1 + 1) q^{26} + (\beta_1 + 2) q^{28} + (\beta_{2} - 3 \beta_1 - 1) q^{29} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{31} + ( - 2 \beta_{2} + \beta_1 + 2) q^{32} + (4 \beta_{2} + 6 \beta_1) q^{34} + (\beta_{2} + \beta_1 - 1) q^{35} + (3 \beta_{2} - \beta_1 + 1) q^{37} + (2 \beta_{2} - \beta_1 + 5) q^{38} + (3 \beta_{2} + \beta_1 + 7) q^{40} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{41} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{43} + ( - 2 \beta_{2} - 2 \beta_1 - 10) q^{46} + ( - 3 \beta_{2} - \beta_1 - 1) q^{47} + q^{49} + (7 \beta_{2} - \beta_1 - 4) q^{50} + (\beta_{2} + 6 \beta_1 + 10) q^{52} + 2 \beta_1 q^{53} + (2 \beta_1 + 1) q^{56} + ( - 2 \beta_{2} - 5 \beta_1 - 1) q^{58} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{59} + 2 q^{61} + (2 \beta_{2} - 12) q^{62} + (2 \beta_{2} - 4 \beta_1 - 1) q^{64} + (7 \beta_{2} - \beta_1 + 7) q^{65} + (\beta_{2} - 5 \beta_1 - 1) q^{67} + (8 \beta_1 + 14) q^{68} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{70} + ( - 4 \beta_1 + 4) q^{71} + (\beta_{2} - 3 \beta_1 + 7) q^{73} + ( - 2 \beta_{2} + \beta_1 + 9) q^{74} + (\beta_{2} + 2 \beta_1 + 4) q^{76} + (4 \beta_1 - 4) q^{79} + ( - 2 \beta_{2} + \beta_1 + 13) q^{80} + (6 \beta_{2} + 2 \beta_1) q^{82} + ( - 6 \beta_1 + 2) q^{83} + (12 \beta_{2} - 2 \beta_1 + 6) q^{85} + (6 \beta_{2} - 10 \beta_1 - 6) q^{86} + ( - 4 \beta_{2} - 10) q^{89} + ( - \beta_{2} + 3 \beta_1 + 1) q^{91} + ( - 4 \beta_{2} - 6 \beta_1 - 10) q^{92} + (2 \beta_{2} - 5 \beta_1 - 11) q^{94} + ( - \beta_{2} + 5 \beta_1 - 3) q^{95} + (4 \beta_{2} + 2 \beta_1) q^{97} + (\beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{10} + 4 q^{13} + 2 q^{14} - 4 q^{16} + 8 q^{17} + 8 q^{19} + 3 q^{20} - 10 q^{23} + 15 q^{25} + q^{26} + 6 q^{28} - 4 q^{29} - 2 q^{31} + 8 q^{32} - 4 q^{34} - 4 q^{35} + 13 q^{38} + 18 q^{40} + 14 q^{41} + 14 q^{43} - 28 q^{46} + 3 q^{49} - 19 q^{50} + 29 q^{52} + 3 q^{56} - q^{58} + 6 q^{61} - 38 q^{62} - 5 q^{64} + 14 q^{65} - 4 q^{67} + 42 q^{68} + 11 q^{70} + 12 q^{71} + 20 q^{73} + 29 q^{74} + 11 q^{76} - 12 q^{79} + 41 q^{80} - 6 q^{82} + 6 q^{83} + 6 q^{85} - 24 q^{86} - 26 q^{89} + 4 q^{91} - 26 q^{92} - 35 q^{94} - 8 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : 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
−0.254102
−1.86081
2.11491
−1.93543 0 1.74590 −4.18953 0 1.00000 0.491797 0 8.10856
1.2 1.46260 0 0.139194 −2.39821 0 1.00000 −2.72161 0 −3.50761
1.3 2.47283 0 4.11491 2.58774 0 1.00000 5.22982 0 6.39905
\(n\): 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.cd 3
3.b odd 2 1 2541.2.a.bg 3
11.b odd 2 1 693.2.a.l 3
33.d even 2 1 231.2.a.e 3
77.b even 2 1 4851.2.a.bi 3
132.d odd 2 1 3696.2.a.bo 3
165.d even 2 1 5775.2.a.bp 3
231.h odd 2 1 1617.2.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 33.d even 2 1
693.2.a.l 3 11.b odd 2 1
1617.2.a.t 3 231.h odd 2 1
2541.2.a.bg 3 3.b odd 2 1
3696.2.a.bo 3 132.d odd 2 1
4851.2.a.bi 3 77.b even 2 1
5775.2.a.bp 3 165.d even 2 1
7623.2.a.cd 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{3} - 2T_{2}^{2} - 4T_{2} + 7 \) Copy content Toggle raw display
\( T_{5}^{3} + 4T_{5}^{2} - 7T_{5} - 26 \) Copy content Toggle raw display
\( T_{13}^{3} - 4T_{13}^{2} - 27T_{13} + 94 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2 T^{2} - 4 T + 7 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 4 T^{2} - 7 T - 26 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} - 27 T + 94 \) Copy content Toggle raw display
$17$ \( T^{3} - 8 T^{2} - 40 T + 328 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} + 15 T - 4 \) Copy content Toggle raw display
$23$ \( T^{3} + 10 T^{2} + 12 T - 64 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 27 T - 94 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 76 T - 256 \) Copy content Toggle raw display
$37$ \( T^{3} - 43T + 106 \) Copy content Toggle raw display
$41$ \( T^{3} - 14 T^{2} + 40 T + 32 \) Copy content Toggle raw display
$43$ \( T^{3} - 14 T^{2} - 44 T + 848 \) Copy content Toggle raw display
$47$ \( T^{3} - 61T - 32 \) Copy content Toggle raw display
$53$ \( T^{3} - 16T - 8 \) Copy content Toggle raw display
$59$ \( T^{3} - 57T + 52 \) Copy content Toggle raw display
$61$ \( (T - 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} + 4 T^{2} - 85 T - 236 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} - 16 T + 256 \) Copy content Toggle raw display
$73$ \( T^{3} - 20 T^{2} + 101 T - 134 \) Copy content Toggle raw display
$79$ \( T^{3} + 12 T^{2} - 16 T - 256 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} - 132 T + 496 \) Copy content Toggle raw display
$89$ \( T^{3} + 26 T^{2} + 140 T - 328 \) Copy content Toggle raw display
$97$ \( T^{3} + 4 T^{2} - 120 T - 232 \) Copy content Toggle raw display
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