Properties

Label 7623.2.a.bc
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: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 1) q^{4} + ( - 2 \beta + 1) q^{5} - q^{7} - 3 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta + 1) q^{4} + ( - 2 \beta + 1) q^{5} - q^{7} - 3 q^{8} + (\beta + 6) q^{10} + (2 \beta + 2) q^{13} + \beta q^{14} + (\beta - 2) q^{16} + (\beta - 5) q^{17} - 3 q^{19} + ( - 3 \beta - 5) q^{20} + ( - \beta + 5) q^{23} + 8 q^{25} + ( - 4 \beta - 6) q^{26} + ( - \beta - 1) q^{28} + (\beta - 7) q^{29} + q^{31} + (\beta + 3) q^{32} + (4 \beta - 3) q^{34} + (2 \beta - 1) q^{35} + ( - 4 \beta + 4) q^{37} + 3 \beta q^{38} + (6 \beta - 3) q^{40} - 7 q^{41} + ( - \beta + 4) q^{43} + ( - 4 \beta + 3) q^{46} + (3 \beta - 5) q^{47} + q^{49} - 8 \beta q^{50} + (6 \beta + 8) q^{52} + (3 \beta + 6) q^{53} + 3 q^{56} + (6 \beta - 3) q^{58} + (\beta - 9) q^{59} + ( - \beta - 2) q^{61} - \beta q^{62} + ( - 6 \beta + 1) q^{64} + ( - 6 \beta - 10) q^{65} + (5 \beta - 3) q^{67} + ( - 3 \beta - 2) q^{68} + ( - \beta - 6) q^{70} + (\beta + 2) q^{71} - 5 q^{73} + 12 q^{74} + ( - 3 \beta - 3) q^{76} + ( - \beta - 6) q^{79} + (3 \beta - 8) q^{80} + 7 \beta q^{82} - 3 q^{83} + (9 \beta - 11) q^{85} + ( - 3 \beta + 3) q^{86} + (9 \beta - 6) q^{89} + ( - 2 \beta - 2) q^{91} + (3 \beta + 2) q^{92} + (2 \beta - 9) q^{94} + (6 \beta - 3) q^{95} + ( - 2 \beta + 1) q^{97} - \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8} + 13 q^{10} + 6 q^{13} + q^{14} - 3 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 9 q^{23} + 16 q^{25} - 16 q^{26} - 3 q^{28} - 13 q^{29} + 2 q^{31} + 7 q^{32} - 2 q^{34} + 4 q^{37} + 3 q^{38} - 14 q^{41} + 7 q^{43} + 2 q^{46} - 7 q^{47} + 2 q^{49} - 8 q^{50} + 22 q^{52} + 15 q^{53} + 6 q^{56} - 17 q^{59} - 5 q^{61} - q^{62} - 4 q^{64} - 26 q^{65} - q^{67} - 7 q^{68} - 13 q^{70} + 5 q^{71} - 10 q^{73} + 24 q^{74} - 9 q^{76} - 13 q^{79} - 13 q^{80} + 7 q^{82} - 6 q^{83} - 13 q^{85} + 3 q^{86} - 3 q^{89} - 6 q^{91} + 7 q^{92} - 16 q^{94} - q^{98}+O(q^{100}) \) 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
2.30278
−1.30278
−2.30278 0 3.30278 −3.60555 0 −1.00000 −3.00000 0 8.30278
1.2 1.30278 0 −0.302776 3.60555 0 −1.00000 −3.00000 0 4.69722
\(n\): 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.bc 2
3.b odd 2 1 847.2.a.g yes 2
11.b odd 2 1 7623.2.a.bs 2
21.c even 2 1 5929.2.a.p 2
33.d even 2 1 847.2.a.e 2
33.f even 10 4 847.2.f.r 8
33.h odd 10 4 847.2.f.o 8
231.h odd 2 1 5929.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.e 2 33.d even 2 1
847.2.a.g yes 2 3.b odd 2 1
847.2.f.o 8 33.h odd 10 4
847.2.f.r 8 33.f even 10 4
5929.2.a.k 2 231.h odd 2 1
5929.2.a.p 2 21.c even 2 1
7623.2.a.bc 2 1.a even 1 1 trivial
7623.2.a.bs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} + T_{2} - 3 \) Copy content Toggle raw display
\( T_{5}^{2} - 13 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 13 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 9T + 17 \) Copy content Toggle raw display
$19$ \( (T + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 17 \) Copy content Toggle raw display
$29$ \( T^{2} + 13T + 39 \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$41$ \( (T + 7)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$47$ \( T^{2} + 7T - 17 \) Copy content Toggle raw display
$53$ \( T^{2} - 15T + 27 \) Copy content Toggle raw display
$59$ \( T^{2} + 17T + 69 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
$67$ \( T^{2} + T - 81 \) Copy content Toggle raw display
$71$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$73$ \( (T + 5)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 13T + 39 \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 3T - 261 \) Copy content Toggle raw display
$97$ \( T^{2} - 13 \) Copy content Toggle raw display
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