Properties

Label 7605.2.a.bw
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: 3.3.756.1
Defining polynomial: \( x^{3} - 6x - 2 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + ( - \beta_1 - 1) q^{7} + (2 \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + ( - \beta_1 - 1) q^{7} + (2 \beta_1 + 2) q^{8} + \beta_1 q^{10} - 2 \beta_1 q^{11} + ( - \beta_{2} - \beta_1 - 4) q^{14} + (2 \beta_1 + 4) q^{16} + (2 \beta_{2} - \beta_1) q^{17} + ( - \beta_{2} - 4) q^{19} + (\beta_{2} + 2) q^{20} + ( - 2 \beta_{2} - 8) q^{22} - 2 \beta_{2} q^{23} + q^{25} + ( - \beta_{2} - 4 \beta_1 - 4) q^{28} + (\beta_{2} - \beta_1 - 2) q^{29} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{31} + (2 \beta_{2} + 4) q^{32} + ( - \beta_{2} + 4 \beta_1) q^{34} + ( - \beta_1 - 1) q^{35} + (2 \beta_{2} - 2) q^{37} + ( - 6 \beta_1 - 2) q^{38} + (2 \beta_1 + 2) q^{40} + ( - \beta_{2} - \beta_1) q^{41} + (\beta_{2} - \beta_1 + 3) q^{43} + ( - 8 \beta_1 - 4) q^{44} + ( - 4 \beta_1 - 4) q^{46} + (3 \beta_1 - 4) q^{47} + (\beta_{2} + 2 \beta_1 - 2) q^{49} + \beta_1 q^{50} + 2 \beta_1 q^{53} - 2 \beta_1 q^{55} + ( - 2 \beta_{2} - 4 \beta_1 - 10) q^{56} + ( - \beta_{2} - 2) q^{58} + ( - \beta_{2} - \beta_1 - 2) q^{59} + (\beta_{2} - 2 \beta_1 - 1) q^{61} + (2 \beta_{2} - 7 \beta_1 + 2) q^{62} + (4 \beta_1 - 4) q^{64} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{67} + 14 q^{68} + ( - \beta_{2} - \beta_1 - 4) q^{70} + ( - 3 \beta_{2} + \beta_1 - 4) q^{71} + (3 \beta_1 - 7) q^{73} + (2 \beta_1 + 4) q^{74} + ( - 4 \beta_{2} - 2 \beta_1 - 16) q^{76} + (2 \beta_{2} + 2 \beta_1 + 8) q^{77} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{79} + (2 \beta_1 + 4) q^{80} + ( - \beta_{2} - 2 \beta_1 - 6) q^{82} + ( - 2 \beta_{2} - 6) q^{83} + (2 \beta_{2} - \beta_1) q^{85} + ( - \beta_{2} + 5 \beta_1 - 2) q^{86} + ( - 4 \beta_{2} - 4 \beta_1 - 16) q^{88} + ( - \beta_{2} + \beta_1 + 10) q^{89} + ( - 4 \beta_1 - 16) q^{92} + (3 \beta_{2} - 4 \beta_1 + 12) q^{94} + ( - \beta_{2} - 4) q^{95} + (\beta_{2} + 3 \beta_1 - 5) q^{97} + (2 \beta_{2} + 10) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 12 q^{14} + 12 q^{16} - 12 q^{19} + 6 q^{20} - 24 q^{22} + 3 q^{25} - 12 q^{28} - 6 q^{29} - 3 q^{31} + 12 q^{32} - 3 q^{35} - 6 q^{37} - 6 q^{38} + 6 q^{40} + 9 q^{43} - 12 q^{44} - 12 q^{46} - 12 q^{47} - 6 q^{49} - 30 q^{56} - 6 q^{58} - 6 q^{59} - 3 q^{61} + 6 q^{62} - 12 q^{64} - 9 q^{67} + 42 q^{68} - 12 q^{70} - 12 q^{71} - 21 q^{73} + 12 q^{74} - 48 q^{76} + 24 q^{77} - 3 q^{79} + 12 q^{80} - 18 q^{82} - 18 q^{83} - 6 q^{86} - 48 q^{88} + 30 q^{89} - 48 q^{92} + 36 q^{94} - 12 q^{95} - 15 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 2 \) : 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
−2.26180
−0.339877
2.60168
−2.26180 0 3.11575 1.00000 0 1.26180 −2.52360 0 −2.26180
1.2 −0.339877 0 −1.88448 1.00000 0 −0.660123 1.32025 0 −0.339877
1.3 2.60168 0 4.76873 1.00000 0 −3.60168 7.20336 0 2.60168
\(n\): 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.bw 3
3.b odd 2 1 2535.2.a.ba 3
13.b even 2 1 7605.2.a.bv 3
13.e even 6 2 585.2.j.f 6
39.d odd 2 1 2535.2.a.bb 3
39.h odd 6 2 195.2.i.d 6
195.y odd 6 2 975.2.i.l 6
195.bf even 12 4 975.2.bb.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.d 6 39.h odd 6 2
585.2.j.f 6 13.e even 6 2
975.2.i.l 6 195.y odd 6 2
975.2.bb.k 12 195.bf even 12 4
2535.2.a.ba 3 3.b odd 2 1
2535.2.a.bb 3 39.d odd 2 1
7605.2.a.bv 3 13.b even 2 1
7605.2.a.bw 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{3} - 6T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 3T_{7} - 3 \) Copy content Toggle raw display
\( T_{11}^{3} - 24T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 6T - 2 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} - 3 T - 3 \) Copy content Toggle raw display
$11$ \( T^{3} - 24T + 16 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 42T + 98 \) Copy content Toggle raw display
$19$ \( T^{3} + 12 T^{2} + 36 T + 4 \) Copy content Toggle raw display
$23$ \( T^{3} - 48T - 96 \) Copy content Toggle raw display
$29$ \( T^{3} + 6T^{2} - 14 \) Copy content Toggle raw display
$31$ \( T^{3} + 3 T^{2} - 93 T - 363 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} - 36 T + 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 24T + 26 \) Copy content Toggle raw display
$43$ \( T^{3} - 9 T^{2} + 15 T + 11 \) Copy content Toggle raw display
$47$ \( T^{3} + 12 T^{2} - 6 T - 206 \) Copy content Toggle raw display
$53$ \( T^{3} - 24T - 16 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} - 12 T - 14 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} - 21 T - 67 \) Copy content Toggle raw display
$67$ \( T^{3} + 9 T^{2} - 81 T - 351 \) Copy content Toggle raw display
$71$ \( T^{3} + 12 T^{2} - 48 T - 682 \) Copy content Toggle raw display
$73$ \( T^{3} + 21 T^{2} + 93 T - 89 \) Copy content Toggle raw display
$79$ \( T^{3} + 3 T^{2} - 93 T - 363 \) Copy content Toggle raw display
$83$ \( T^{3} + 18 T^{2} + 60 T - 168 \) Copy content Toggle raw display
$89$ \( T^{3} - 30 T^{2} + 288 T - 882 \) Copy content Toggle raw display
$97$ \( T^{3} + 15 T^{2} - 9 T - 589 \) Copy content Toggle raw display
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