Properties

Label 7623.2.a.cl
Level $7623$
Weight $2$
Character orbit 7623.a
Self dual yes
Analytic conductor $60.870$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} + (\beta_{3} + \beta_1) q^{4} + 2 \beta_{2} q^{5} + q^{7} + ( - \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} + (\beta_{3} + \beta_1) q^{4} + 2 \beta_{2} q^{5} + q^{7} + ( - \beta_1 - 2) q^{8} + ( - 2 \beta_1 - 2) q^{10} + ( - 2 \beta_{2} - 2 \beta_1) q^{13} + ( - \beta_{2} - \beta_1 + 1) q^{14} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{16} + (2 \beta_{3} - 4 \beta_1 + 2) q^{17} + (2 \beta_{3} - 2 \beta_1 + 2) q^{19} + (2 \beta_{3} + 4 \beta_1 - 2) q^{20} + ( - 3 \beta_{3} + 4 \beta_1 + 3) q^{23} + ( - 4 \beta_{3} + 4 \beta_{2} + 3) q^{25} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{26} + (\beta_{3} + \beta_1) q^{28} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{31} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{32} + (6 \beta_{3} - 2 \beta_1 + 2) q^{34} + 2 \beta_{2} q^{35} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 3) q^{37} + (4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{38} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{40} + (6 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{41} + (5 \beta_{3} - 6 \beta_1) q^{43} + ( - 7 \beta_{3} - 4 \beta_{2} - \beta_1 + 3) q^{46} + (6 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{47} + q^{49} + ( - 4 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{50} + ( - 2 \beta_{3} - 4 \beta_{2} - 8 \beta_1) q^{52} + ( - 5 \beta_{3} + 2 \beta_1 + 2) q^{53} + ( - \beta_1 - 2) q^{56} + ( - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 6) q^{58} + ( - 2 \beta_{3} - 2 \beta_{2} + 6) q^{59} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 2) q^{61} + (8 \beta_{2} + 8 \beta_1 - 6) q^{62} + ( - 2 \beta_{3} - 5 \beta_{2} - \beta_1 + 1) q^{64} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{65} + (3 \beta_{3} - 2 \beta_1 - 8) q^{67} + (4 \beta_{3} - 6 \beta_{2} - 4 \beta_1 - 2) q^{68} + ( - 2 \beta_1 - 2) q^{70} + (3 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 6) q^{71} + ( - 2 \beta_{2} + 2) q^{73} + ( - 5 \beta_{3} - 3 \beta_1 - 7) q^{74} + (4 \beta_{3} - 2 \beta_{2}) q^{76} + (\beta_{3} + 2 \beta_1 + 4) q^{79} + ( - 4 \beta_{3} + 2 \beta_{2} + 10) q^{80} + (8 \beta_{3} - 2 \beta_{2} - 12 \beta_1 - 6) q^{82} + ( - 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 2) q^{83} + ( - 8 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 8) q^{85} + (11 \beta_{3} + \beta_{2} - 4 \beta_1) q^{86} + ( - 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 2) q^{89} + ( - 2 \beta_{2} - 2 \beta_1) q^{91} + (5 \beta_{2} + 8 \beta_1 + 1) q^{92} + (10 \beta_{3} - 4 \beta_1) q^{94} + ( - 4 \beta_{3} + 4 \beta_{2} + 4) q^{95} + ( - 6 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 8) q^{97} + ( - \beta_{2} - \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 3 q^{4} + 4 q^{5} + 4 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 3 q^{4} + 4 q^{5} + 4 q^{7} - 9 q^{8} - 10 q^{10} - 6 q^{13} + q^{14} - 3 q^{16} + 8 q^{17} + 10 q^{19} + 10 q^{23} + 12 q^{25} + 20 q^{26} + 3 q^{28} - 18 q^{31} - 2 q^{32} + 18 q^{34} + 4 q^{35} - 2 q^{37} + 8 q^{38} - 6 q^{40} + 10 q^{41} + 4 q^{43} - 11 q^{46} - 4 q^{47} + 4 q^{49} - 9 q^{50} - 20 q^{52} - 9 q^{56} + 14 q^{58} + 16 q^{59} - 14 q^{61} - 11 q^{64} - 28 q^{65} - 28 q^{67} - 16 q^{68} - 10 q^{70} + 18 q^{71} + 4 q^{73} - 41 q^{74} + 4 q^{76} + 20 q^{79} + 36 q^{80} - 24 q^{82} + 6 q^{83} + 20 q^{85} + 20 q^{86} + 16 q^{89} - 6 q^{91} + 22 q^{92} + 16 q^{94} + 16 q^{95} + 32 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 3x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 2\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 3\beta_1 \) 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
2.09529
−1.35567
0.737640
−0.477260
−2.39026 0 3.71333 2.58993 0 1.00000 −4.09529 0 −6.19059
1.2 0.162147 0 −1.97371 4.38705 0 1.00000 −0.644326 0 0.711349
1.3 1.45589 0 0.119606 −2.38705 0 1.00000 −2.73764 0 −3.47528
1.4 1.77222 0 1.14077 −0.589926 0 1.00000 −1.52274 0 −1.04548
\(n\): 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.cl 4
3.b odd 2 1 2541.2.a.bm 4
11.b odd 2 1 7623.2.a.ci 4
11.d odd 10 2 693.2.m.f 8
33.d even 2 1 2541.2.a.bn 4
33.f even 10 2 231.2.j.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.f 8 33.f even 10 2
693.2.m.f 8 11.d odd 10 2
2541.2.a.bm 4 3.b odd 2 1
2541.2.a.bn 4 33.d even 2 1
7623.2.a.ci 4 11.b odd 2 1
7623.2.a.cl 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + 7T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{3} - 8T_{5}^{2} + 24T_{5} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} + 6T_{13}^{3} - 8T_{13}^{2} - 16T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 5 T^{2} + 7 T - 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} - 8 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} - 8 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} - 20 T^{2} + 144 T + 304 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + 24 T^{2} - 16 \) Copy content Toggle raw display
$23$ \( T^{4} - 10 T^{3} - 9 T^{2} + 190 T + 109 \) Copy content Toggle raw display
$29$ \( T^{4} - 79T^{2} + 29 \) Copy content Toggle raw display
$31$ \( T^{4} + 18 T^{3} + 68 T^{2} + \cdots - 1744 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 77 T^{2} - 218 T + 281 \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} - 104 T^{2} + \cdots - 4496 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} - 103 T^{2} + \cdots + 1861 \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} - 80 T^{2} - 568 T - 976 \) Copy content Toggle raw display
$53$ \( T^{4} - 51 T^{2} + 20 T + 209 \) Copy content Toggle raw display
$59$ \( T^{4} - 16 T^{3} + 72 T^{2} - 104 T + 16 \) Copy content Toggle raw display
$61$ \( T^{4} + 14 T^{3} + 16 T^{2} - 256 T + 16 \) Copy content Toggle raw display
$67$ \( T^{4} + 28 T^{3} + 273 T^{2} + \cdots + 1301 \) Copy content Toggle raw display
$71$ \( T^{4} - 18 T^{3} + 43 T^{2} + \cdots - 179 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} - 8 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$79$ \( T^{4} - 20 T^{3} + 129 T^{2} + \cdots + 149 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} - 120 T^{2} + \cdots + 2864 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} - 48 T^{2} + \cdots - 304 \) Copy content Toggle raw display
$97$ \( T^{4} - 32 T^{3} + 268 T^{2} + \cdots - 8464 \) Copy content Toggle raw display
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