Properties

Label 7605.2.a.bm
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_1 + 4) q^{4} + q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (3 \beta_1 - 5) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_1 + 4) q^{4} + q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (3 \beta_1 - 5) q^{8} + (\beta_{2} + \beta_1 - 1) q^{10} + (2 \beta_{2} - 2 \beta_1 + 3) q^{11} + ( - \beta_{2} - 5) q^{14} + (\beta_{2} - 6 \beta_1 + 6) q^{16} + (2 \beta_{2} + 2 \beta_1 - 2) q^{17} + ( - 4 \beta_{2} + 3 \beta_1 - 3) q^{19} + ( - \beta_1 + 4) q^{20} + ( - 3 \beta_{2} + 7 \beta_1 - 5) q^{22} + ( - 5 \beta_{2} + 4 \beta_1 - 4) q^{23} + q^{25} + ( - 2 \beta_{2} - 4 \beta_1 + 3) q^{28} + (2 \beta_{2} - \beta_1 - 4) q^{29} + (2 \beta_{2} - 6 \beta_1 - 1) q^{31} + ( - 7 \beta_{2} + 7 \beta_1 - 12) q^{32} + ( - 2 \beta_1 + 12) q^{34} + ( - \beta_{2} - \beta_1) q^{35} + (2 \beta_{2} + \beta_1 + 6) q^{37} + (7 \beta_{2} - 10 \beta_1 + 4) q^{38} + (3 \beta_1 - 5) q^{40} + (3 \beta_{2} - 5 \beta_1 + 6) q^{41} + ( - 3 \beta_{2} + \beta_1 - 10) q^{43} + (8 \beta_{2} - 11 \beta_1 + 14) q^{44} + (9 \beta_{2} - 13 \beta_1 + 6) q^{46} + ( - \beta_{2} + \beta_1 - 4) q^{47} + (2 \beta_{2} + \beta_1 - 2) q^{49} + (\beta_{2} + \beta_1 - 1) q^{50} + (5 \beta_{2} - 9 \beta_1 + 1) q^{53} + (2 \beta_{2} - 2 \beta_1 + 3) q^{55} + ( - \beta_{2} + 5 \beta_1 - 9) q^{56} + ( - 8 \beta_{2} - \beta_1 + 5) q^{58} + ( - 7 \beta_{2} + 7 \beta_1 - 5) q^{59} + ( - \beta_{2} + 7) q^{61} + ( - 15 \beta_{2} + 7 \beta_1 - 13) q^{62} + (7 \beta_{2} - 14 \beta_1 + 7) q^{64} + ( - 7 \beta_{2} + 4 \beta_1 - 9) q^{67} + (4 \beta_{2} + 10 \beta_1 - 14) q^{68} + ( - \beta_{2} - 5) q^{70} + ( - 4 \beta_{2} + 3 \beta_1 + 7) q^{71} + (6 \beta_{2} + \beta_1 + 3) q^{73} + (6 \beta_{2} + 7 \beta_1 + 1) q^{74} + ( - 15 \beta_{2} + 15 \beta_1 - 14) q^{76} + (\beta_{2} - 5 \beta_1 + 2) q^{77} + ( - 7 \beta_{2} + 5 \beta_1 - 2) q^{79} + (\beta_{2} - 6 \beta_1 + 6) q^{80} + ( - 7 \beta_{2} + 14 \beta_1 - 15) q^{82} + ( - 11 \beta_1 + 2) q^{83} + (2 \beta_{2} + 2 \beta_1 - 2) q^{85} + ( - 5 \beta_{2} - 14 \beta_1 + 7) q^{86} + ( - 10 \beta_{2} + 19 \beta_1 - 21) q^{88} + ( - 2 \beta_{2} + 3 \beta_1 - 1) q^{89} + ( - 19 \beta_{2} + 20 \beta_1 - 19) q^{92} + ( - \beta_{2} - 6 \beta_1 + 5) q^{94} + ( - 4 \beta_{2} + 3 \beta_1 - 3) q^{95} + (4 \beta_{2} + 2 \beta_1 - 1) q^{97} + ( - 2 \beta_{2} - \beta_1 + 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 11 q^{4} + 3 q^{5} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 11 q^{4} + 3 q^{5} - 12 q^{8} - 3 q^{10} + 5 q^{11} - 14 q^{14} + 11 q^{16} - 6 q^{17} - 2 q^{19} + 11 q^{20} - 5 q^{22} - 3 q^{23} + 3 q^{25} + 7 q^{28} - 15 q^{29} - 11 q^{31} - 22 q^{32} + 34 q^{34} + 17 q^{37} - 5 q^{38} - 12 q^{40} + 10 q^{41} - 26 q^{43} + 23 q^{44} - 4 q^{46} - 10 q^{47} - 7 q^{49} - 3 q^{50} - 11 q^{53} + 5 q^{55} - 21 q^{56} + 22 q^{58} - q^{59} + 22 q^{61} - 17 q^{62} - 16 q^{67} - 36 q^{68} - 14 q^{70} + 28 q^{71} + 4 q^{73} + 4 q^{74} - 12 q^{76} + 6 q^{79} + 11 q^{80} - 24 q^{82} - 5 q^{83} - 6 q^{85} + 12 q^{86} - 34 q^{88} + 2 q^{89} - 18 q^{92} + 10 q^{94} - 2 q^{95} - 5 q^{97} + 28 q^{98}+O(q^{100}) \) 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
−1.24698
0.445042
1.80194
−2.69202 0 5.24698 1.00000 0 1.69202 −8.74094 0 −2.69202
1.2 −2.35690 0 3.55496 1.00000 0 1.35690 −3.66487 0 −2.35690
1.3 2.04892 0 2.19806 1.00000 0 −3.04892 0.405813 0 2.04892
\(n\): 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.bm 3
3.b odd 2 1 2535.2.a.bh yes 3
13.b even 2 1 7605.2.a.ce 3
39.d odd 2 1 2535.2.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2535.2.a.t 3 39.d odd 2 1
2535.2.a.bh yes 3 3.b odd 2 1
7605.2.a.bm 3 1.a even 1 1 trivial
7605.2.a.ce 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3 T^{2} - 4 T - 13 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 7T + 7 \) Copy content Toggle raw display
$11$ \( T^{3} - 5T^{2} - T + 13 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 6 T^{2} - 16 T - 104 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 29 T - 71 \) Copy content Toggle raw display
$23$ \( T^{3} + 3 T^{2} - 46 T - 139 \) Copy content Toggle raw display
$29$ \( T^{3} + 15 T^{2} + 68 T + 97 \) Copy content Toggle raw display
$31$ \( T^{3} + 11 T^{2} - 25 T - 379 \) Copy content Toggle raw display
$37$ \( T^{3} - 17 T^{2} + 80 T - 113 \) Copy content Toggle raw display
$41$ \( T^{3} - 10 T^{2} - 11 T + 13 \) Copy content Toggle raw display
$43$ \( T^{3} + 26 T^{2} + 209 T + 491 \) Copy content Toggle raw display
$47$ \( T^{3} + 10 T^{2} + 31 T + 29 \) Copy content Toggle raw display
$53$ \( T^{3} + 11 T^{2} - 102 T - 1079 \) Copy content Toggle raw display
$59$ \( T^{3} + T^{2} - 114 T - 127 \) Copy content Toggle raw display
$61$ \( T^{3} - 22 T^{2} + 159 T - 377 \) Copy content Toggle raw display
$67$ \( T^{3} + 16 T^{2} - T - 617 \) Copy content Toggle raw display
$71$ \( T^{3} - 28 T^{2} + 231 T - 581 \) Copy content Toggle raw display
$73$ \( T^{3} - 4 T^{2} - 95 T - 83 \) Copy content Toggle raw display
$79$ \( T^{3} - 6 T^{2} - 79 T - 113 \) Copy content Toggle raw display
$83$ \( T^{3} + 5 T^{2} - 274 T - 811 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 15 T + 29 \) Copy content Toggle raw display
$97$ \( T^{3} + 5 T^{2} - 57 T - 293 \) Copy content Toggle raw display
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