Properties

Label 7605.2.a.bq
Level $7605$
Weight $2$
Character orbit 7605.a
Self dual yes
Analytic conductor $60.726$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2535)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8} - \beta_1 q^{10} - q^{11} + ( - \beta_{2} + \beta_1 - 2) q^{14} + ( - 3 \beta_{2} + \beta_1 - 3) q^{16} + ( - 2 \beta_1 + 4) q^{17} + ( - 3 \beta_{2} + 1) q^{19} + \beta_{2} q^{20} + \beta_1 q^{22} + (\beta_{2} - \beta_1 - 1) q^{23} + q^{25} + q^{28} + (3 \beta_{2} + 2 \beta_1) q^{29} + (4 \beta_{2} - 2 \beta_1 + 1) q^{31} + (4 \beta_{2} - \beta_1 + 3) q^{32} + (2 \beta_{2} - 4 \beta_1 + 4) q^{34} + (\beta_1 - 1) q^{35} + (\beta_{2} - 4 \beta_1) q^{37} + (3 \beta_{2} - \beta_1 + 3) q^{38} + ( - \beta_{2} + 2 \beta_1 - 1) q^{40} + ( - \beta_1 - 7) q^{41} + ( - \beta_1 - 5) q^{43} - \beta_{2} q^{44} + (\beta_1 + 1) q^{46} + (4 \beta_{2} + \beta_1 + 5) q^{47} + (\beta_{2} - 2 \beta_1 - 4) q^{49} - \beta_1 q^{50} + ( - 6 \beta_{2} + 7 \beta_1 - 6) q^{53} - q^{55} + (2 \beta_{2} - 3 \beta_1 + 4) q^{56} + ( - 5 \beta_{2} - 7) q^{58} + ( - 2 \beta_{2} + 7 \beta_1 - 2) q^{59} + ( - 7 \beta_{2} + 7 \beta_1 - 8) q^{61} + ( - 2 \beta_{2} - \beta_1) q^{62} + (3 \beta_{2} - 5 \beta_1 + 4) q^{64} + (3 \beta_{2} - \beta_1 + 8) q^{67} + (2 \beta_{2} - 2) q^{68} + ( - \beta_{2} + \beta_1 - 2) q^{70} + ( - 3 \beta_{2} + 6 \beta_1 - 1) q^{71} + (7 \beta_{2} - 10 \beta_1 + 5) q^{73} + (3 \beta_{2} + 7) q^{74} + (4 \beta_{2} - 3 \beta_1 - 3) q^{76} + ( - \beta_1 + 1) q^{77} + ( - 2 \beta_{2} - 7 \beta_1 + 1) q^{79} + ( - 3 \beta_{2} + \beta_1 - 3) q^{80} + (\beta_{2} + 7 \beta_1 + 2) q^{82} + ( - 5 \beta_{2} - 2 \beta_1 + 8) q^{83} + ( - 2 \beta_1 + 4) q^{85} + (\beta_{2} + 5 \beta_1 + 2) q^{86} + (\beta_{2} - 2 \beta_1 + 1) q^{88} + (7 \beta_{2} - 4 \beta_1 + 5) q^{89} + ( - 3 \beta_{2} + \beta_1) q^{92} + ( - 5 \beta_{2} - 5 \beta_1 - 6) q^{94} + ( - 3 \beta_{2} + 1) q^{95} + ( - 12 \beta_{2} + 6 \beta_1 - 9) q^{97} + (\beta_{2} + 4 \beta_1 + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - q^{4} + 3 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - q^{4} + 3 q^{5} - 2 q^{7} - q^{10} - 3 q^{11} - 4 q^{14} - 5 q^{16} + 10 q^{17} + 6 q^{19} - q^{20} + q^{22} - 5 q^{23} + 3 q^{25} + 3 q^{28} - q^{29} - 3 q^{31} + 4 q^{32} + 6 q^{34} - 2 q^{35} - 5 q^{37} + 5 q^{38} - 22 q^{41} - 16 q^{43} + q^{44} + 4 q^{46} + 12 q^{47} - 15 q^{49} - q^{50} - 5 q^{53} - 3 q^{55} + 7 q^{56} - 16 q^{58} + 3 q^{59} - 10 q^{61} + q^{62} + 4 q^{64} + 20 q^{67} - 8 q^{68} - 4 q^{70} + 6 q^{71} - 2 q^{73} + 18 q^{74} - 16 q^{76} + 2 q^{77} - 2 q^{79} - 5 q^{80} + 12 q^{82} + 27 q^{83} + 10 q^{85} + 10 q^{86} + 4 q^{89} + 4 q^{92} - 18 q^{94} + 6 q^{95} - 9 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
0.445042
−1.24698
−1.80194 0 1.24698 1.00000 0 0.801938 1.35690 0 −1.80194
1.2 −0.445042 0 −1.80194 1.00000 0 −0.554958 1.69202 0 −0.445042
1.3 1.24698 0 −0.445042 1.00000 0 −2.24698 −3.04892 0 1.24698
\(n\): 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.bq 3
3.b odd 2 1 2535.2.a.bd yes 3
13.b even 2 1 7605.2.a.bz 3
39.d odd 2 1 2535.2.a.v 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2535.2.a.v 3 39.d odd 2 1
2535.2.a.bd yes 3 3.b odd 2 1
7605.2.a.bq 3 1.a even 1 1 trivial
7605.2.a.bz 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{3} + T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - T_{7} - 1 \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2T^{2} - T - 1 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 10 T^{2} + 24 T - 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} - 9 T + 41 \) Copy content Toggle raw display
$23$ \( T^{3} + 5 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$29$ \( T^{3} + T^{2} - 44 T - 127 \) Copy content Toggle raw display
$31$ \( T^{3} + 3 T^{2} - 25 T + 29 \) Copy content Toggle raw display
$37$ \( T^{3} + 5 T^{2} - 22 T - 97 \) Copy content Toggle raw display
$41$ \( T^{3} + 22 T^{2} + 159 T + 377 \) Copy content Toggle raw display
$43$ \( T^{3} + 16 T^{2} + 83 T + 139 \) Copy content Toggle raw display
$47$ \( T^{3} - 12 T^{2} - T + 41 \) Copy content Toggle raw display
$53$ \( T^{3} + 5 T^{2} - 92 T - 83 \) Copy content Toggle raw display
$59$ \( T^{3} - 3 T^{2} - 88 T + 377 \) Copy content Toggle raw display
$61$ \( T^{3} + 10 T^{2} - 81 T - 433 \) Copy content Toggle raw display
$67$ \( T^{3} - 20 T^{2} + 117 T - 169 \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} - 51 T + 307 \) Copy content Toggle raw display
$73$ \( T^{3} + 2 T^{2} - 183 T - 743 \) Copy content Toggle raw display
$79$ \( T^{3} + 2 T^{2} - 155 T + 223 \) Copy content Toggle raw display
$83$ \( T^{3} - 27 T^{2} + 152 T + 377 \) Copy content Toggle raw display
$89$ \( T^{3} - 4 T^{2} - 81 T + 421 \) Copy content Toggle raw display
$97$ \( T^{3} + 9 T^{2} - 225 T - 2241 \) Copy content Toggle raw display
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