Properties

Label 7623.2.a.by
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + 3 \beta q^{4} - \beta q^{5} - q^{7} + (4 \beta + 1) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + 3 \beta q^{4} - \beta q^{5} - q^{7} + (4 \beta + 1) q^{8} + ( - 2 \beta - 1) q^{10} + ( - 4 \beta + 1) q^{13} + ( - \beta - 1) q^{14} + (3 \beta + 5) q^{16} - 3 q^{17} - 4 q^{19} + ( - 3 \beta - 3) q^{20} + (2 \beta + 4) q^{23} + (\beta - 4) q^{25} + ( - 7 \beta - 3) q^{26} - 3 \beta q^{28} + 3 q^{29} + (2 \beta - 3) q^{31} + (3 \beta + 6) q^{32} + ( - 3 \beta - 3) q^{34} + \beta q^{35} + ( - 4 \beta - 2) q^{37} + ( - 4 \beta - 4) q^{38} + ( - 5 \beta - 4) q^{40} + ( - 5 \beta + 5) q^{41} + 9 q^{43} + (8 \beta + 6) q^{46} + ( - 5 \beta - 2) q^{47} + q^{49} + ( - 2 \beta - 3) q^{50} + ( - 9 \beta - 12) q^{52} + ( - 5 \beta + 4) q^{53} + ( - 4 \beta - 1) q^{56} + (3 \beta + 3) q^{58} + ( - \beta - 11) q^{59} + (6 \beta - 3) q^{61} + (\beta - 1) q^{62} + (6 \beta - 1) q^{64} + (3 \beta + 4) q^{65} + ( - 6 \beta + 6) q^{67} - 9 \beta q^{68} + (2 \beta + 1) q^{70} + ( - 2 \beta - 9) q^{71} + (3 \beta - 1) q^{73} + ( - 10 \beta - 6) q^{74} - 12 \beta q^{76} + ( - 3 \beta - 2) q^{79} + ( - 8 \beta - 3) q^{80} - 5 \beta q^{82} + (8 \beta - 10) q^{83} + 3 \beta q^{85} + (9 \beta + 9) q^{86} + (7 \beta - 14) q^{89} + (4 \beta - 1) q^{91} + (18 \beta + 6) q^{92} + ( - 12 \beta - 7) q^{94} + 4 \beta q^{95} + ( - 12 \beta + 3) q^{97} + (\beta + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{4} - q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 3 q^{4} - q^{5} - 2 q^{7} + 6 q^{8} - 4 q^{10} - 2 q^{13} - 3 q^{14} + 13 q^{16} - 6 q^{17} - 8 q^{19} - 9 q^{20} + 10 q^{23} - 7 q^{25} - 13 q^{26} - 3 q^{28} + 6 q^{29} - 4 q^{31} + 15 q^{32} - 9 q^{34} + q^{35} - 8 q^{37} - 12 q^{38} - 13 q^{40} + 5 q^{41} + 18 q^{43} + 20 q^{46} - 9 q^{47} + 2 q^{49} - 8 q^{50} - 33 q^{52} + 3 q^{53} - 6 q^{56} + 9 q^{58} - 23 q^{59} - q^{62} + 4 q^{64} + 11 q^{65} + 6 q^{67} - 9 q^{68} + 4 q^{70} - 20 q^{71} + q^{73} - 22 q^{74} - 12 q^{76} - 7 q^{79} - 14 q^{80} - 5 q^{82} - 12 q^{83} + 3 q^{85} + 27 q^{86} - 21 q^{89} + 2 q^{91} + 30 q^{92} - 26 q^{94} + 4 q^{95} - 6 q^{97} + 3 q^{98}+O(q^{100}) \) 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
−0.618034
1.61803
0.381966 0 −1.85410 0.618034 0 −1.00000 −1.47214 0 0.236068
1.2 2.61803 0 4.85410 −1.61803 0 −1.00000 7.47214 0 −4.23607
\(n\): 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.by 2
3.b odd 2 1 2541.2.a.m 2
11.b odd 2 1 7623.2.a.u 2
11.d odd 10 2 693.2.m.e 4
33.d even 2 1 2541.2.a.bf 2
33.f even 10 2 231.2.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.a 4 33.f even 10 2
693.2.m.e 4 11.d odd 10 2
2541.2.a.m 2 3.b odd 2 1
2541.2.a.bf 2 33.d even 2 1
7623.2.a.u 2 11.b odd 2 1
7623.2.a.by 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T - 25 \) Copy content Toggle raw display
$43$ \( (T - 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$59$ \( T^{2} + 23T + 131 \) Copy content Toggle raw display
$61$ \( T^{2} - 45 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$71$ \( T^{2} + 20T + 95 \) Copy content Toggle raw display
$73$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$79$ \( T^{2} + 7T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$89$ \( T^{2} + 21T + 49 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 171 \) Copy content Toggle raw display
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