Properties

Label 7623.2.a.ba
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.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 1) q^{4} + (\beta + 1) q^{5} + q^{7} + (2 \beta - 1) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 1) q^{4} + (\beta + 1) q^{5} + q^{7} + (2 \beta - 1) q^{8} + ( - 2 \beta - 1) q^{10} - q^{13} - \beta q^{14} - 3 \beta q^{16} + ( - 2 \beta + 3) q^{17} + \beta q^{20} + ( - 2 \beta + 2) q^{23} + (3 \beta - 3) q^{25} + \beta q^{26} + (\beta - 1) q^{28} + (6 \beta - 3) q^{29} + (2 \beta - 9) q^{31} + ( - \beta + 5) q^{32} + ( - \beta + 2) q^{34} + (\beta + 1) q^{35} + ( - 8 \beta + 2) q^{37} + (3 \beta + 1) q^{40} + (5 \beta - 2) q^{41} - q^{43} + 2 q^{46} + ( - 7 \beta + 3) q^{47} + q^{49} - 3 q^{50} + ( - \beta + 1) q^{52} + (\beta - 7) q^{53} + (2 \beta - 1) q^{56} + ( - 3 \beta - 6) q^{58} + (3 \beta + 6) q^{59} + ( - 2 \beta - 7) q^{61} + (7 \beta - 2) q^{62} + (2 \beta + 1) q^{64} + ( - \beta - 1) q^{65} + (2 \beta - 8) q^{67} + (3 \beta - 5) q^{68} + ( - 2 \beta - 1) q^{70} + ( - 4 \beta + 5) q^{71} + ( - 9 \beta + 6) q^{73} + (6 \beta + 8) q^{74} + ( - 3 \beta + 9) q^{79} + ( - 6 \beta - 3) q^{80} + ( - 3 \beta - 5) q^{82} + 6 q^{83} + ( - \beta + 1) q^{85} + \beta q^{86} + (\beta + 7) q^{89} - q^{91} + (2 \beta - 4) q^{92} + (4 \beta + 7) q^{94} - 17 q^{97} - \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 3 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 3 q^{5} + 2 q^{7} - 4 q^{10} - 2 q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + q^{20} + 2 q^{23} - 3 q^{25} + q^{26} - q^{28} - 16 q^{31} + 9 q^{32} + 3 q^{34} + 3 q^{35} - 4 q^{37} + 5 q^{40} + q^{41} - 2 q^{43} + 4 q^{46} - q^{47} + 2 q^{49} - 6 q^{50} + q^{52} - 13 q^{53} - 15 q^{58} + 15 q^{59} - 16 q^{61} + 3 q^{62} + 4 q^{64} - 3 q^{65} - 14 q^{67} - 7 q^{68} - 4 q^{70} + 6 q^{71} + 3 q^{73} + 22 q^{74} + 15 q^{79} - 12 q^{80} - 13 q^{82} + 12 q^{83} + q^{85} + q^{86} + 15 q^{89} - 2 q^{91} - 6 q^{92} + 18 q^{94} - 34 q^{97} - q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
−1.61803 0 0.618034 2.61803 0 1.00000 2.23607 0 −4.23607
1.2 0.618034 0 −1.61803 0.381966 0 1.00000 −2.23607 0 0.236068
\(n\): 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.ba 2
3.b odd 2 1 2541.2.a.bb 2
11.b odd 2 1 7623.2.a.bp 2
11.c even 5 2 693.2.m.b 4
33.d even 2 1 2541.2.a.s 2
33.h odd 10 2 231.2.j.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.d 4 33.h odd 10 2
693.2.m.b 4 11.c even 5 2
2541.2.a.s 2 33.d even 2 1
2541.2.a.bb 2 3.b odd 2 1
7623.2.a.ba 2 1.a even 1 1 trivial
7623.2.a.bp 2 11.b odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 45 \) Copy content Toggle raw display
$31$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$41$ \( T^{2} - T - 31 \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + T - 61 \) Copy content Toggle raw display
$53$ \( T^{2} + 13T + 41 \) Copy content Toggle raw display
$59$ \( T^{2} - 15T + 45 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$73$ \( T^{2} - 3T - 99 \) Copy content Toggle raw display
$79$ \( T^{2} - 15T + 45 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 15T + 55 \) Copy content Toggle raw display
$97$ \( (T + 17)^{2} \) Copy content Toggle raw display
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