Properties

Label 7623.2.a.bb
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( -1 + \beta ) q^{4} + ( 1 + 2 \beta ) q^{5} - q^{7} + ( -1 + 2 \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( -1 + \beta ) q^{4} + ( 1 + 2 \beta ) q^{5} - q^{7} + ( -1 + 2 \beta ) q^{8} + ( -2 - 3 \beta ) q^{10} + ( 3 + 2 \beta ) q^{13} + \beta q^{14} -3 \beta q^{16} + ( -2 + 4 \beta ) q^{17} + 3 q^{19} + ( 1 + \beta ) q^{20} + ( -2 - 4 \beta ) q^{23} + 8 \beta q^{25} + ( -2 - 5 \beta ) q^{26} + ( 1 - \beta ) q^{28} + 3 q^{29} + ( 5 - \beta ) q^{32} + ( -4 - 2 \beta ) q^{34} + ( -1 - 2 \beta ) q^{35} + ( -3 + 4 \beta ) q^{37} -3 \beta q^{38} + ( 3 + 4 \beta ) q^{40} + ( -8 + 4 \beta ) q^{41} + ( -2 + 8 \beta ) q^{43} + ( 4 + 6 \beta ) q^{46} + 3 q^{47} + q^{49} + ( -8 - 8 \beta ) q^{50} + ( -1 + 3 \beta ) q^{52} + ( 4 - 8 \beta ) q^{53} + ( 1 - 2 \beta ) q^{56} -3 \beta q^{58} + ( -5 + 4 \beta ) q^{59} + ( 10 - 4 \beta ) q^{61} + ( 1 + 2 \beta ) q^{64} + ( 7 + 12 \beta ) q^{65} + ( 1 - 6 \beta ) q^{67} + ( 6 - 2 \beta ) q^{68} + ( 2 + 3 \beta ) q^{70} + ( -8 + 4 \beta ) q^{71} + ( 9 + 2 \beta ) q^{73} + ( -4 - \beta ) q^{74} + ( -3 + 3 \beta ) q^{76} + ( 6 - 12 \beta ) q^{79} + ( -6 - 9 \beta ) q^{80} + ( -4 + 4 \beta ) q^{82} -6 q^{83} + ( 6 + 8 \beta ) q^{85} + ( -8 - 6 \beta ) q^{86} + ( -2 + 4 \beta ) q^{89} + ( -3 - 2 \beta ) q^{91} + ( -2 - 2 \beta ) q^{92} -3 \beta q^{94} + ( 3 + 6 \beta ) q^{95} + 2 q^{97} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} + 4q^{5} - 2q^{7} + O(q^{10}) \) \( 2q - q^{2} - q^{4} + 4q^{5} - 2q^{7} - 7q^{10} + 8q^{13} + q^{14} - 3q^{16} + 6q^{19} + 3q^{20} - 8q^{23} + 8q^{25} - 9q^{26} + q^{28} + 6q^{29} + 9q^{32} - 10q^{34} - 4q^{35} - 2q^{37} - 3q^{38} + 10q^{40} - 12q^{41} + 4q^{43} + 14q^{46} + 6q^{47} + 2q^{49} - 24q^{50} + q^{52} - 3q^{58} - 6q^{59} + 16q^{61} + 4q^{64} + 26q^{65} - 4q^{67} + 10q^{68} + 7q^{70} - 12q^{71} + 20q^{73} - 9q^{74} - 3q^{76} - 21q^{80} - 4q^{82} - 12q^{83} + 20q^{85} - 22q^{86} - 8q^{91} - 6q^{92} - 3q^{94} + 12q^{95} + 4q^{97} - q^{98} + O(q^{100}) \)

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
−1.61803 0 0.618034 4.23607 0 −1.00000 2.23607 0 −6.85410
1.2 0.618034 0 −1.61803 −0.236068 0 −1.00000 −2.23607 0 −0.145898
\(n\): 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.bb 2
3.b odd 2 1 2541.2.a.ba yes 2
11.b odd 2 1 7623.2.a.bq 2
33.d even 2 1 2541.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.r 2 33.d even 2 1
2541.2.a.ba yes 2 3.b odd 2 1
7623.2.a.bb 2 1.a even 1 1 trivial
7623.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{5}^{2} - 4 T_{5} - 1 \)
\( T_{13}^{2} - 8 T_{13} + 11 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ \( \)
$5$ \( 1 - 4 T + 9 T^{2} - 20 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( \)
$13$ \( 1 - 8 T + 37 T^{2} - 104 T^{3} + 169 T^{4} \)
$17$ \( 1 + 14 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 3 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 8 T + 42 T^{2} + 184 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 3 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( 1 + 2 T + 55 T^{2} + 74 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 12 T + 98 T^{2} + 492 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T + 10 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 - 3 T + 47 T^{2} )^{2} \)
$53$ \( 1 + 26 T^{2} + 2809 T^{4} \)
$59$ \( 1 + 6 T + 107 T^{2} + 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 16 T + 166 T^{2} - 976 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T + 93 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 12 T + 158 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 20 T + 241 T^{2} - 1460 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 22 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 158 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 2 T + 97 T^{2} )^{2} \)
show more
show less