Properties

Label 7623.2.a.y
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: \(\Z[a_1, a_2]\)
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.

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

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 2.23607 0 1.00000 2.23607 0 −3.61803
1.2 0.618034 0 −1.61803 −2.23607 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.y 2
3.b odd 2 1 2541.2.a.y yes 2
11.b odd 2 1 7623.2.a.bn 2
33.d even 2 1 2541.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.q 2 33.d even 2 1
2541.2.a.y yes 2 3.b odd 2 1
7623.2.a.y 2 1.a even 1 1 trivial
7623.2.a.bn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ \( \)
$5$ \( 1 + 5 T^{2} + 25 T^{4} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( \)
$13$ \( 1 + 4 T + 25 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 6 T + 27 T^{2} - 114 T^{3} + 361 T^{4} \)
$23$ \( ( 1 - 2 T + 23 T^{2} )^{2} \)
$29$ \( ( 1 + 5 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} \)
$37$ \( 1 - 6 T + 63 T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 8 T + 82 T^{2} + 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 10 T + 99 T^{2} + 470 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 4 T + 90 T^{2} + 212 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 2 T + 39 T^{2} - 118 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 16 T + 166 T^{2} - 976 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T + 13 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 4 T + 126 T^{2} + 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 8 T + 117 T^{2} + 584 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 8 T + 154 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 24 T + 290 T^{2} + 1992 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 8 T + 174 T^{2} + 712 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 8 T + 30 T^{2} + 776 T^{3} + 9409 T^{4} \)
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