Properties

Label 7605.2.a.bs
Level $7605$
Weight $2$
Character orbit 7605.a
Self dual yes
Analytic conductor $60.726$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + (\beta_{2} + 1) q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + (\beta_{2} + 1) q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8} - \beta_1 q^{10} + (\beta_{2} + \beta_1 - 2) q^{11} + ( - \beta_{2} - 2 \beta_1 - 1) q^{14} + (2 \beta_1 + 1) q^{16} + (2 \beta_{2} + 2) q^{17} + ( - \beta_{2} + \beta_1 + 2) q^{19} + (\beta_{2} + 2) q^{20} + ( - 2 \beta_{2} + \beta_1 - 5) q^{22} + ( - \beta_1 + 3) q^{23} + q^{25} + (\beta_{2} + 2 \beta_1 + 7) q^{28} + (\beta_{2} + 1) q^{29} + (\beta_{2} - \beta_1 + 4) q^{31} + (\beta_1 - 6) q^{32} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{34} + (\beta_{2} + 1) q^{35} + (\beta_{2} + 7) q^{37} + ( - \beta_1 - 3) q^{38} + ( - \beta_{2} - \beta_1 - 1) q^{40} + (2 \beta_{2} - 2 \beta_1 + 2) q^{41} + (2 \beta_{2} - \beta_1 - 3) q^{43} + ( - \beta_{2} + 5 \beta_1 + 2) q^{44} + (\beta_{2} - 3 \beta_1 + 4) q^{46} + (\beta_{2} + 1) q^{47} + (2 \beta_1 - 1) q^{49} - \beta_1 q^{50} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{53} + (\beta_{2} + \beta_1 - 2) q^{55} + ( - \beta_{2} - 4 \beta_1 - 7) q^{56} + ( - \beta_{2} - 2 \beta_1 - 1) q^{58} + (\beta_{2} + \beta_1 + 4) q^{59} + (\beta_{2} - 2 \beta_1 - 1) q^{61} + ( - 5 \beta_1 + 3) q^{62} + ( - \beta_{2} + 2 \beta_1 - 6) q^{64} + (\beta_{2} - 2 \beta_1 + 7) q^{67} + (2 \beta_{2} + 4 \beta_1 + 14) q^{68} + ( - \beta_{2} - 2 \beta_1 - 1) q^{70} + (\beta_{2} - \beta_1 - 2) q^{71} + ( - \beta_{2} + 4 \beta_1 + 5) q^{73} + ( - \beta_{2} - 8 \beta_1 - 1) q^{74} + (3 \beta_{2} + \beta_1) q^{76} + ( - 2 \beta_{2} + 4 \beta_1 + 4) q^{77} + (2 \beta_{2} - 4 \beta_1 + 6) q^{79} + (2 \beta_1 + 1) q^{80} + ( - 4 \beta_1 + 6) q^{82} + (\beta_{2} - 2 \beta_1 - 5) q^{83} + (2 \beta_{2} + 2) q^{85} + ( - \beta_{2} + \beta_1 + 2) q^{86} + ( - 3 \beta_1 - 9) q^{88} + (2 \beta_{2} - 4 \beta_1 + 2) q^{89} + (2 \beta_{2} - 3 \beta_1 + 5) q^{92} + ( - \beta_{2} - 2 \beta_1 - 1) q^{94} + ( - \beta_{2} + \beta_1 + 2) q^{95} + (2 \beta_{2} - 4) q^{97} + ( - 2 \beta_{2} + \beta_1 - 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} + 3 q^{5} + 2 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 5 q^{4} + 3 q^{5} + 2 q^{7} - 3 q^{8} - q^{10} - 6 q^{11} - 4 q^{14} + 5 q^{16} + 4 q^{17} + 8 q^{19} + 5 q^{20} - 12 q^{22} + 8 q^{23} + 3 q^{25} + 22 q^{28} + 2 q^{29} + 10 q^{31} - 17 q^{32} - 8 q^{34} + 2 q^{35} + 20 q^{37} - 10 q^{38} - 3 q^{40} + 2 q^{41} - 12 q^{43} + 12 q^{44} + 8 q^{46} + 2 q^{47} - q^{49} - q^{50} - 6 q^{53} - 6 q^{55} - 24 q^{56} - 4 q^{58} + 12 q^{59} - 6 q^{61} + 4 q^{62} - 15 q^{64} + 18 q^{67} + 44 q^{68} - 4 q^{70} - 8 q^{71} + 20 q^{73} - 10 q^{74} - 2 q^{76} + 18 q^{77} + 12 q^{79} + 5 q^{80} + 14 q^{82} - 18 q^{83} + 4 q^{85} + 8 q^{86} - 30 q^{88} + 10 q^{92} - 4 q^{94} + 8 q^{95} - 14 q^{97} - 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
2.51414
0.571993
−2.08613
−2.51414 0 4.32088 1.00000 0 3.32088 −5.83502 0 −2.51414
1.2 −0.571993 0 −1.67282 1.00000 0 −2.67282 2.10083 0 −0.571993
1.3 2.08613 0 2.35194 1.00000 0 1.35194 0.734191 0 2.08613
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 7605.2.a.bs 3
3.b odd 2 1 845.2.a.k 3
13.b even 2 1 7605.2.a.cc 3
13.d odd 4 2 585.2.b.g 6
15.d odd 2 1 4225.2.a.bc 3
39.d odd 2 1 845.2.a.i 3
39.f even 4 2 65.2.c.a 6
39.h odd 6 2 845.2.e.k 6
39.i odd 6 2 845.2.e.i 6
39.k even 12 4 845.2.m.h 12
156.l odd 4 2 1040.2.k.d 6
195.e odd 2 1 4225.2.a.be 3
195.j odd 4 2 325.2.d.f 6
195.n even 4 2 325.2.c.g 6
195.u odd 4 2 325.2.d.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.c.a 6 39.f even 4 2
325.2.c.g 6 195.n even 4 2
325.2.d.e 6 195.u odd 4 2
325.2.d.f 6 195.j odd 4 2
585.2.b.g 6 13.d odd 4 2
845.2.a.i 3 39.d odd 2 1
845.2.a.k 3 3.b odd 2 1
845.2.e.i 6 39.i odd 6 2
845.2.e.k 6 39.h odd 6 2
845.2.m.h 12 39.k even 12 4
1040.2.k.d 6 156.l odd 4 2
4225.2.a.bc 3 15.d odd 2 1
4225.2.a.be 3 195.e odd 2 1
7605.2.a.bs 3 1.a even 1 1 trivial
7605.2.a.cc 3 13.b even 2 1

Hecke kernels

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

\( T_{2}^{3} + T_{2}^{2} - 5T_{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 8T_{7} + 12 \) Copy content Toggle raw display
\( T_{11}^{3} + 6T_{11}^{2} - 6T_{11} - 54 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 5T - 3 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 8 T + 12 \) Copy content Toggle raw display
$11$ \( T^{3} + 6 T^{2} - 6 T - 54 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 32 T + 96 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} + 10 T + 6 \) Copy content Toggle raw display
$23$ \( T^{3} - 8 T^{2} + 16 T - 6 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 8 T + 12 \) Copy content Toggle raw display
$31$ \( T^{3} - 10 T^{2} + 22 T + 6 \) Copy content Toggle raw display
$37$ \( T^{3} - 20 T^{2} + 124 T - 228 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 44 T + 72 \) Copy content Toggle raw display
$43$ \( T^{3} + 12 T^{2} + 12 T + 2 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} - 8 T + 12 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 60 T + 72 \) Copy content Toggle raw display
$59$ \( T^{3} - 12 T^{2} + 30 T - 18 \) Copy content Toggle raw display
$61$ \( T^{3} + 6 T^{2} - 12 T - 76 \) Copy content Toggle raw display
$67$ \( T^{3} - 18 T^{2} + 84 T - 108 \) Copy content Toggle raw display
$71$ \( T^{3} + 8 T^{2} + 10 T - 6 \) Copy content Toggle raw display
$73$ \( T^{3} - 20 T^{2} + 52 T + 516 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} - 48 T + 32 \) Copy content Toggle raw display
$83$ \( T^{3} + 18 T^{2} + 84 T + 36 \) Copy content Toggle raw display
$89$ \( T^{3} - 96T - 288 \) Copy content Toggle raw display
$97$ \( T^{3} + 14 T^{2} + 28 T - 24 \) Copy content Toggle raw display
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