Properties

Label 7623.2.a.bj
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: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \beta ) q^{2} + 3 q^{4} -\beta q^{5} - q^{7} + ( 1 - 2 \beta ) q^{8} +O(q^{10})\) \( q + ( 1 - 2 \beta ) q^{2} + 3 q^{4} -\beta q^{5} - q^{7} + ( 1 - 2 \beta ) q^{8} + ( 2 + \beta ) q^{10} + ( -2 + 2 \beta ) q^{13} + ( -1 + 2 \beta ) q^{14} - q^{16} + ( 3 - 3 \beta ) q^{17} + ( -1 + 3 \beta ) q^{19} -3 \beta q^{20} + ( -5 - \beta ) q^{23} + ( -4 + \beta ) q^{25} + ( -6 + 2 \beta ) q^{26} -3 q^{28} -6 q^{29} + 5 \beta q^{31} + ( -3 + 6 \beta ) q^{32} + ( 9 - 3 \beta ) q^{34} + \beta q^{35} + ( 4 - \beta ) q^{37} + ( -7 - \beta ) q^{38} + ( 2 + \beta ) q^{40} + ( 8 + \beta ) q^{41} + ( -6 + 6 \beta ) q^{43} + ( -3 + 11 \beta ) q^{46} + ( -2 + 4 \beta ) q^{47} + q^{49} + ( -6 + 7 \beta ) q^{50} + ( -6 + 6 \beta ) q^{52} + ( -8 - 2 \beta ) q^{53} + ( -1 + 2 \beta ) q^{56} + ( -6 + 12 \beta ) q^{58} + ( -2 + 2 \beta ) q^{59} + ( -10 - 5 \beta ) q^{62} -13 q^{64} -2 q^{65} + ( 9 - 9 \beta ) q^{68} + ( -2 - \beta ) q^{70} -8 \beta q^{71} + ( -10 + 6 \beta ) q^{73} + ( 6 - 7 \beta ) q^{74} + ( -3 + 9 \beta ) q^{76} + 10 q^{79} + \beta q^{80} + ( 6 - 17 \beta ) q^{82} + ( 14 - 4 \beta ) q^{83} + 3 q^{85} + ( -18 + 6 \beta ) q^{86} + ( -5 + \beta ) q^{89} + ( 2 - 2 \beta ) q^{91} + ( -15 - 3 \beta ) q^{92} -10 q^{94} + ( -3 - 2 \beta ) q^{95} -6 q^{97} + ( 1 - 2 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{4} - q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + 6q^{4} - q^{5} - 2q^{7} + 5q^{10} - 2q^{13} - 2q^{16} + 3q^{17} + q^{19} - 3q^{20} - 11q^{23} - 7q^{25} - 10q^{26} - 6q^{28} - 12q^{29} + 5q^{31} + 15q^{34} + q^{35} + 7q^{37} - 15q^{38} + 5q^{40} + 17q^{41} - 6q^{43} + 5q^{46} + 2q^{49} - 5q^{50} - 6q^{52} - 18q^{53} - 2q^{59} - 25q^{62} - 26q^{64} - 4q^{65} + 9q^{68} - 5q^{70} - 8q^{71} - 14q^{73} + 5q^{74} + 3q^{76} + 20q^{79} + q^{80} - 5q^{82} + 24q^{83} + 6q^{85} - 30q^{86} - 9q^{89} + 2q^{91} - 33q^{92} - 20q^{94} - 8q^{95} - 12q^{97} + O(q^{100}) \)

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
1.61803
−0.618034
−2.23607 0 3.00000 −1.61803 0 −1.00000 −2.23607 0 3.61803
1.2 2.23607 0 3.00000 0.618034 0 −1.00000 2.23607 0 1.38197
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.bj 2
3.b odd 2 1 2541.2.a.v 2
11.b odd 2 1 7623.2.a.bk 2
11.d odd 10 2 693.2.m.a 4
33.d even 2 1 2541.2.a.w 2
33.f even 10 2 231.2.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.e 4 33.f even 10 2
693.2.m.a 4 11.d odd 10 2
2541.2.a.v 2 3.b odd 2 1
2541.2.a.w 2 33.d even 2 1
7623.2.a.bj 2 1.a even 1 1 trivial
7623.2.a.bk 2 11.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} - 5 \)
\( T_{5}^{2} + T_{5} - 1 \)
\( T_{13}^{2} + 2 T_{13} - 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + 4 T^{4} \)
$3$ \( \)
$5$ \( 1 + T + 9 T^{2} + 5 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( \)
$13$ \( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 3 T + 25 T^{2} - 51 T^{3} + 289 T^{4} \)
$19$ \( 1 - T + 27 T^{2} - 19 T^{3} + 361 T^{4} \)
$23$ \( 1 + 11 T + 75 T^{2} + 253 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 5 T + 37 T^{2} - 155 T^{3} + 961 T^{4} \)
$37$ \( 1 - 7 T + 85 T^{2} - 259 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 17 T + 153 T^{2} - 697 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 6 T + 50 T^{2} + 258 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 74 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 18 T + 182 T^{2} + 954 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 67 T^{2} )^{2} \)
$71$ \( 1 + 8 T + 78 T^{2} + 568 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 14 T + 150 T^{2} + 1022 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 24 T + 290 T^{2} - 1992 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 9 T + 197 T^{2} + 801 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 6 T + 97 T^{2} )^{2} \)
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