Properties

Label 7605.2.a.co
Level $7605$
Weight $2$
Character orbit 7605.a
Self dual yes
Analytic conductor $60.726$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.3352656.1
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 6x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_{4} q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_{4} q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8} + \beta_1 q^{10} + (\beta_{4} + \beta_1 + 1) q^{11} + (\beta_{4} + \beta_{3} - 1) q^{14} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{16} + (2 \beta_{2} - \beta_1) q^{17} + (\beta_{2} - 1) q^{19} + (\beta_{2} + 1) q^{20} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{22} + ( - \beta_{3} - \beta_{2} - 1) q^{23} + q^{25} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{28} + (\beta_{4} - \beta_{3} + 3) q^{29} + (\beta_{4} - \beta_{3} - \beta_1 + 2) q^{31} + (2 \beta_{4} + 2 \beta_{2} + 2) q^{32} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{34} + \beta_{4} q^{35} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{37} + (\beta_{3} + \beta_{2} + 1) q^{38} + (\beta_{3} + \beta_{2} + 1) q^{40} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} + 1) q^{41} + ( - \beta_{2} + 3 \beta_1 + 2) q^{43} + (2 \beta_{3} + 4 \beta_{2} + 2) q^{44} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{46} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{47} + ( - 2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 1) q^{49} + \beta_1 q^{50} + (2 \beta_{4} - 2 \beta_{2} - 4) q^{53} + (\beta_{4} + \beta_1 + 1) q^{55} + ( - \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{56} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{58} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{59} + ( - \beta_{4} + \beta_{3} + \beta_1 + 2) q^{61} + (\beta_{3} - 3 \beta_{2} + 3 \beta_1 - 5) q^{62} + (2 \beta_{3} + 2 \beta_1) q^{64} + ( - \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{67} + (2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 9) q^{68} + (\beta_{4} + \beta_{3} - 1) q^{70} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{71} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 - 2) q^{73} + ( - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4) q^{74} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 4) q^{76} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 7) q^{77} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 - 2) q^{79} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{80} + ( - 2 \beta_{4} + \beta_{3} + 4 \beta_1 + 2) q^{82} + ( - 3 \beta_{3} + \beta_{2} + 3) q^{83} + (2 \beta_{2} - \beta_1) q^{85} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 8) q^{86} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1 + 2) q^{88} + ( - \beta_{3} - 3 \beta_1 + 2) q^{89} + ( - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 4) q^{92} + ( - 3 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 3) q^{94} + (\beta_{2} - 1) q^{95} + ( - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{97} + ( - 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{11} - 4 q^{14} + 4 q^{16} - 4 q^{19} + 6 q^{20} + 14 q^{22} - 6 q^{23} + 5 q^{25} - 2 q^{28} + 16 q^{29} + 9 q^{31} + 14 q^{32} + q^{35} - 4 q^{37} + 6 q^{38} + 6 q^{40} + 6 q^{41} + 15 q^{43} + 14 q^{44} - 16 q^{46} + 10 q^{47} + 10 q^{49} + 2 q^{50} - 20 q^{53} + 8 q^{55} - 2 q^{56} - 4 q^{58} + 12 q^{59} + 11 q^{61} - 22 q^{62} + 4 q^{64} + 5 q^{67} + 50 q^{68} - 4 q^{70} + 10 q^{71} + q^{73} - 26 q^{74} + 24 q^{76} + 42 q^{77} - 17 q^{79} + 4 q^{80} + 16 q^{82} + 16 q^{83} + 44 q^{86} + 20 q^{88} + 4 q^{89} - 34 q^{92} - 16 q^{94} - 4 q^{95} - 11 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 6x - 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \) 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
−2.12283
−0.946366
0.626791
1.81031
2.63209
−2.12283 0 2.50640 1.00000 0 1.46707 −1.07500 0 −2.12283
1.2 −0.946366 0 −1.10439 1.00000 0 −1.56306 2.93789 0 −0.946366
1.3 0.626791 0 −1.60713 1.00000 0 4.43127 −2.26092 0 0.626791
1.4 1.81031 0 1.27724 1.00000 0 −4.42503 −1.30843 0 1.81031
1.5 2.63209 0 4.92789 1.00000 0 1.08975 7.70645 0 2.63209
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.co 5
3.b odd 2 1 7605.2.a.cm 5
13.b even 2 1 7605.2.a.cl 5
13.c even 3 2 585.2.j.h 10
39.d odd 2 1 7605.2.a.cn 5
39.i odd 6 2 585.2.j.i yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.j.h 10 13.c even 3 2
585.2.j.i yes 10 39.i odd 6 2
7605.2.a.cl 5 13.b even 2 1
7605.2.a.cm 5 3.b odd 2 1
7605.2.a.cn 5 39.d odd 2 1
7605.2.a.co 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{5} - 2T_{2}^{4} - 6T_{2}^{3} + 10T_{2}^{2} + 6T_{2} - 6 \) Copy content Toggle raw display
\( T_{7}^{5} - T_{7}^{4} - 22T_{7}^{3} + 22T_{7}^{2} + 47T_{7} - 49 \) Copy content Toggle raw display
\( T_{11}^{5} - 8T_{11}^{4} + 64T_{11}^{2} + 48T_{11} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 2 T^{4} - 6 T^{3} + 10 T^{2} + \cdots - 6 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - T^{4} - 22 T^{3} + 22 T^{2} + \cdots - 49 \) Copy content Toggle raw display
$11$ \( T^{5} - 8 T^{4} + 64 T^{2} + 48 T - 24 \) Copy content Toggle raw display
$13$ \( T^{5} \) Copy content Toggle raw display
$17$ \( T^{5} - 50 T^{3} - 14 T^{2} + \cdots + 642 \) Copy content Toggle raw display
$19$ \( T^{5} + 4 T^{4} - 8 T^{3} - 36 T^{2} + \cdots + 12 \) Copy content Toggle raw display
$23$ \( T^{5} + 6 T^{4} - 20 T^{3} - 36 T^{2} + \cdots - 72 \) Copy content Toggle raw display
$29$ \( T^{5} - 16 T^{4} + 58 T^{3} + \cdots + 258 \) Copy content Toggle raw display
$31$ \( T^{5} - 9 T^{4} - 30 T^{3} + 178 T^{2} + \cdots + 603 \) Copy content Toggle raw display
$37$ \( T^{5} + 4 T^{4} - 64 T^{3} + 56 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$41$ \( T^{5} - 6 T^{4} - 98 T^{3} + \cdots - 11898 \) Copy content Toggle raw display
$43$ \( T^{5} - 15 T^{4} + 30 T^{3} + \cdots + 2059 \) Copy content Toggle raw display
$47$ \( T^{5} - 10 T^{4} - 90 T^{3} + \cdots - 7434 \) Copy content Toggle raw display
$53$ \( T^{5} + 20 T^{4} - 1872 T^{2} + \cdots - 15552 \) Copy content Toggle raw display
$59$ \( T^{5} - 12 T^{4} - 42 T^{3} + \cdots - 2754 \) Copy content Toggle raw display
$61$ \( T^{5} - 11 T^{4} - 14 T^{3} + \cdots - 3023 \) Copy content Toggle raw display
$67$ \( T^{5} - 5 T^{4} - 44 T^{3} + 4 T^{2} + \cdots - 17 \) Copy content Toggle raw display
$71$ \( T^{5} - 10 T^{4} - 22 T^{3} + \cdots + 162 \) Copy content Toggle raw display
$73$ \( T^{5} - T^{4} - 236 T^{3} + 144 T^{2} + \cdots + 9693 \) Copy content Toggle raw display
$79$ \( T^{5} + 17 T^{4} - 26 T^{3} + \cdots + 2349 \) Copy content Toggle raw display
$83$ \( T^{5} - 16 T^{4} - 116 T^{3} + \cdots - 6264 \) Copy content Toggle raw display
$89$ \( T^{5} - 4 T^{4} - 102 T^{3} + 78 T^{2} + \cdots - 54 \) Copy content Toggle raw display
$97$ \( T^{5} + 11 T^{4} - 130 T^{3} + \cdots - 7679 \) Copy content Toggle raw display
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