Properties

Label 7605.2.a.bl
Level $7605$
Weight $2$
Character orbit 7605.a
Self dual yes
Analytic conductor $60.726$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.7262307372\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 585)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 2 q^{2} + 2 q^{4} - q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 2 q^{4} - q^{5} + \beta q^{7} - 2 q^{10} + 3 q^{11} + 2 \beta q^{14} - 4 q^{16} + \beta q^{17} - 2 \beta q^{19} - 2 q^{20} + 6 q^{22} - \beta q^{23} + q^{25} + 2 \beta q^{28} + 2 \beta q^{29} + 2 \beta q^{31} - 8 q^{32} + 2 \beta q^{34} - \beta q^{35} + \beta q^{37} - 4 \beta q^{38} + 11 q^{41} + 4 q^{43} + 6 q^{44} - 2 \beta q^{46} - 4 q^{47} + 6 q^{49} + 2 q^{50} - 3 \beta q^{53} - 3 q^{55} + 4 \beta q^{58} + 12 q^{59} + 13 q^{61} + 4 \beta q^{62} - 8 q^{64} + 2 \beta q^{68} - 2 \beta q^{70} - 5 q^{71} - 2 \beta q^{73} + 2 \beta q^{74} - 4 \beta q^{76} + 3 \beta q^{77} + 13 q^{79} + 4 q^{80} + 22 q^{82} + 6 q^{83} - \beta q^{85} + 8 q^{86} - 3 q^{89} - 2 \beta q^{92} - 8 q^{94} + 2 \beta q^{95} - \beta q^{97} + 12 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 4 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{10} + 6 q^{11} - 8 q^{16} - 4 q^{20} + 12 q^{22} + 2 q^{25} - 16 q^{32} + 22 q^{41} + 8 q^{43} + 12 q^{44} - 8 q^{47} + 12 q^{49} + 4 q^{50} - 6 q^{55} + 24 q^{59} + 26 q^{61} - 16 q^{64} - 10 q^{71} + 26 q^{79} + 8 q^{80} + 44 q^{82} + 12 q^{83} + 16 q^{86} - 6 q^{89} - 16 q^{94} + 24 q^{98}+O(q^{100}) \) 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
−1.30278
2.30278
2.00000 0 2.00000 −1.00000 0 −3.60555 0 0 −2.00000
1.2 2.00000 0 2.00000 −1.00000 0 3.60555 0 0 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(13\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7605.2.a.bl 2
3.b odd 2 1 7605.2.a.w 2
13.b even 2 1 7605.2.a.w 2
13.d odd 4 2 585.2.b.f 4
39.d odd 2 1 inner 7605.2.a.bl 2
39.f even 4 2 585.2.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.b.f 4 13.d odd 4 2
585.2.b.f 4 39.f even 4 2
7605.2.a.w 2 3.b odd 2 1
7605.2.a.w 2 13.b even 2 1
7605.2.a.bl 2 1.a even 1 1 trivial
7605.2.a.bl 2 39.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 13 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 13 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 13 \) Copy content Toggle raw display
$19$ \( T^{2} - 52 \) Copy content Toggle raw display
$23$ \( T^{2} - 13 \) Copy content Toggle raw display
$29$ \( T^{2} - 52 \) Copy content Toggle raw display
$31$ \( T^{2} - 52 \) Copy content Toggle raw display
$37$ \( T^{2} - 13 \) Copy content Toggle raw display
$41$ \( (T - 11)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( (T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 117 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 13)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T + 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 52 \) Copy content Toggle raw display
$79$ \( (T - 13)^{2} \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( (T + 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 13 \) Copy content Toggle raw display
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