Properties

Label 7623.2.a.w
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: \(\Z[a_1, \ldots, a_{13}]\)
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.

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

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
−0.618034
1.61803
−1.00000 0 −1.00000 −3.85410 0 −1.00000 3.00000 0 3.85410
1.2 −1.00000 0 −1.00000 2.85410 0 −1.00000 3.00000 0 −2.85410
\(n\): 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.w 2
3.b odd 2 1 2541.2.a.bd 2
11.b odd 2 1 7623.2.a.bu 2
11.d odd 10 2 693.2.m.c 4
33.d even 2 1 2541.2.a.n 2
33.f even 10 2 231.2.j.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.c 4 33.f even 10 2
693.2.m.c 4 11.d odd 10 2
2541.2.a.n 2 33.d even 2 1
2541.2.a.bd 2 3.b odd 2 1
7623.2.a.w 2 1.a even 1 1 trivial
7623.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + 2 T^{2} )^{2} \)
$3$ \( \)
$5$ \( 1 + T - T^{2} + 5 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( \)
$13$ \( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 9 T + 43 T^{2} - 153 T^{3} + 289 T^{4} \)
$19$ \( 1 + 3 T + 39 T^{2} + 57 T^{3} + 361 T^{4} \)
$23$ \( 1 + 5 T + 21 T^{2} + 115 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 2 T + 29 T^{2} )^{2} \)
$31$ \( 1 - T + T^{2} - 31 T^{3} + 961 T^{4} \)
$37$ \( 1 + 13 T + 85 T^{2} + 481 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 15 T + 127 T^{2} + 615 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 2 T + 82 T^{2} + 86 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 2 T + 47 T^{2} )^{2} \)
$53$ \( 1 + 2 T - 18 T^{2} + 106 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 18 T + 194 T^{2} + 1062 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 42 T^{2} + 3721 T^{4} \)
$67$ \( ( 1 - 8 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 62 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 6 T + 150 T^{2} - 438 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 14 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 12 T + 122 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 9 T + 167 T^{2} - 801 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 16 T + 238 T^{2} + 1552 T^{3} + 9409 T^{4} \)
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