Properties

Label 7605.2.a.bt
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: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
−1.48119
2.17009
0.311108
−1.67513 0 0.806063 −1.00000 0 −3.63752 2.00000 0 1.67513
1.2 −0.539189 0 −1.70928 −1.00000 0 4.80098 2.00000 0 0.539189
1.3 2.21432 0 2.90321 −1.00000 0 3.83654 2.00000 0 −2.21432
\(n\): 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.bt 3
3.b odd 2 1 2535.2.a.z 3
13.b even 2 1 7605.2.a.bu 3
13.e even 6 2 585.2.j.g 6
39.d odd 2 1 2535.2.a.y 3
39.h odd 6 2 195.2.i.e 6
195.y odd 6 2 975.2.i.m 6
195.bf even 12 4 975.2.bb.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.e 6 39.h odd 6 2
585.2.j.g 6 13.e even 6 2
975.2.i.m 6 195.y odd 6 2
975.2.bb.j 12 195.bf even 12 4
2535.2.a.y 3 39.d odd 2 1
2535.2.a.z 3 3.b odd 2 1
7605.2.a.bt 3 1.a even 1 1 trivial
7605.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{3} - 4T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} - 5T_{7}^{2} - 13T_{7} + 67 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} - 16T_{11} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 4T - 2 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 5 T^{2} - 13 T + 67 \) Copy content Toggle raw display
$11$ \( T^{3} - 4 T^{2} - 16 T + 32 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 8 T + 34 \) Copy content Toggle raw display
$19$ \( T^{3} - 40T - 76 \) Copy content Toggle raw display
$23$ \( T^{3} + 8 T^{2} - 40 T - 304 \) Copy content Toggle raw display
$29$ \( T^{3} + 6 T^{2} - 18 T - 54 \) Copy content Toggle raw display
$31$ \( T^{3} - 9 T^{2} - T + 109 \) Copy content Toggle raw display
$37$ \( T^{3} - 14 T^{2} + 12 T + 152 \) Copy content Toggle raw display
$41$ \( T^{3} + 20 T^{2} + 118 T + 214 \) Copy content Toggle raw display
$43$ \( T^{3} + 15 T^{2} + 41 T - 107 \) Copy content Toggle raw display
$47$ \( T^{3} - 4T - 2 \) Copy content Toggle raw display
$53$ \( T^{3} - 16 T^{2} + 32 T - 16 \) Copy content Toggle raw display
$59$ \( T^{3} - 10 T^{2} - 102 T + 970 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} - T + 109 \) Copy content Toggle raw display
$67$ \( T^{3} - 11 T^{2} - 43 T + 247 \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} - 18 T + 46 \) Copy content Toggle raw display
$73$ \( T^{3} - 15 T^{2} + 71 T - 103 \) Copy content Toggle raw display
$79$ \( T^{3} - 17 T^{2} + 63 T - 67 \) Copy content Toggle raw display
$83$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$89$ \( T^{3} + 22 T^{2} + 86 T - 134 \) Copy content Toggle raw display
$97$ \( T^{3} - T^{2} - 151 T - 547 \) Copy content Toggle raw display
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