Properties

Label 7623.2.a.bt
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 2) q^{4} - q^{5} - q^{7} + (\beta + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 2) q^{4} - q^{5} - q^{7} + (\beta + 4) q^{8} - \beta q^{10} + (\beta - 4) q^{13} - \beta q^{14} + 3 \beta q^{16} + ( - 2 \beta + 1) q^{17} - 6 q^{19} + ( - \beta - 2) q^{20} + 4 q^{23} - 4 q^{25} + ( - 3 \beta + 4) q^{26} + ( - \beta - 2) q^{28} + ( - 3 \beta + 2) q^{29} + ( - 4 \beta + 2) q^{31} + (\beta + 4) q^{32} + ( - \beta - 8) q^{34} + q^{35} + (3 \beta + 2) q^{37} - 6 \beta q^{38} + ( - \beta - 4) q^{40} + (\beta - 6) q^{41} + ( - 3 \beta + 1) q^{43} + 4 \beta q^{46} + ( - \beta - 5) q^{47} + q^{49} - 4 \beta q^{50} + ( - \beta - 4) q^{52} + 5 \beta q^{53} + ( - \beta - 4) q^{56} + ( - \beta - 12) q^{58} + ( - \beta + 11) q^{59} + ( - 6 \beta + 4) q^{61} + ( - 2 \beta - 16) q^{62} + ( - \beta + 4) q^{64} + ( - \beta + 4) q^{65} + ( - 5 \beta + 1) q^{67} + ( - 5 \beta - 6) q^{68} + \beta q^{70} + 2 \beta q^{71} + (2 \beta - 4) q^{73} + (5 \beta + 12) q^{74} + ( - 6 \beta - 12) q^{76} + 2 \beta q^{79} - 3 \beta q^{80} + ( - 5 \beta + 4) q^{82} + (5 \beta - 1) q^{83} + (2 \beta - 1) q^{85} + ( - 2 \beta - 12) q^{86} + ( - 4 \beta + 5) q^{89} + ( - \beta + 4) q^{91} + (4 \beta + 8) q^{92} + ( - 6 \beta - 4) q^{94} + 6 q^{95} + (3 \beta + 6) q^{97} + \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} - 2 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} - 2 q^{5} - 2 q^{7} + 9 q^{8} - q^{10} - 7 q^{13} - q^{14} + 3 q^{16} - 12 q^{19} - 5 q^{20} + 8 q^{23} - 8 q^{25} + 5 q^{26} - 5 q^{28} + q^{29} + 9 q^{32} - 17 q^{34} + 2 q^{35} + 7 q^{37} - 6 q^{38} - 9 q^{40} - 11 q^{41} - q^{43} + 4 q^{46} - 11 q^{47} + 2 q^{49} - 4 q^{50} - 9 q^{52} + 5 q^{53} - 9 q^{56} - 25 q^{58} + 21 q^{59} + 2 q^{61} - 34 q^{62} + 7 q^{64} + 7 q^{65} - 3 q^{67} - 17 q^{68} + q^{70} + 2 q^{71} - 6 q^{73} + 29 q^{74} - 30 q^{76} + 2 q^{79} - 3 q^{80} + 3 q^{82} + 3 q^{83} - 26 q^{86} + 6 q^{89} + 7 q^{91} + 20 q^{92} - 14 q^{94} + 12 q^{95} + 15 q^{97} + q^{98}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 0 0.438447 −1.00000 0 −1.00000 2.43845 0 1.56155
1.2 2.56155 0 4.56155 −1.00000 0 −1.00000 6.56155 0 −2.56155
\(n\): 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.bt 2
3.b odd 2 1 2541.2.a.u 2
11.b odd 2 1 7623.2.a.be 2
33.d even 2 1 2541.2.a.bc yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.u 2 3.b odd 2 1
2541.2.a.bc yes 2 33.d even 2 1
7623.2.a.be 2 11.b odd 2 1
7623.2.a.bt 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 7T_{13} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$17$ \( T^{2} - 17 \) Copy content Toggle raw display
$19$ \( (T + 6)^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} - 68 \) Copy content Toggle raw display
$37$ \( T^{2} - 7T - 26 \) Copy content Toggle raw display
$41$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$47$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$53$ \( T^{2} - 5T - 100 \) Copy content Toggle raw display
$59$ \( T^{2} - 21T + 106 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T - 152 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T - 104 \) Copy content Toggle raw display
$71$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$83$ \( T^{2} - 3T - 104 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 59 \) Copy content Toggle raw display
$97$ \( T^{2} - 15T + 18 \) Copy content Toggle raw display
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