Properties

Label 7600.2.a.bk
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.6863055362\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.993.1
Defining polynomial: \( x^{3} - x^{2} - 6x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 950)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + (\beta_{2} - 1) q^{7} + (\beta_{2} - 2 \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + (\beta_{2} - 1) q^{7} + (\beta_{2} - 2 \beta_1 + 2) q^{9} - 2 \beta_1 q^{11} + ( - \beta_{2} + \beta_1 - 2) q^{13} + (\beta_{2} - \beta_1 + 4) q^{17} - q^{19} + (\beta_1 + 2) q^{21} + ( - 2 \beta_{2} + \beta_1 + 1) q^{23} + ( - 2 \beta_{2} + 3 \beta_1 - 6) q^{27} + ( - \beta_{2} + \beta_1 + 2) q^{29} + ( - 2 \beta_1 - 2) q^{31} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{33} + (2 \beta_1 - 1) q^{37} + (\beta_{2} - 5 \beta_1 + 5) q^{39} + (2 \beta_1 + 4) q^{43} + (2 \beta_{2} - 2 \beta_1 - 1) q^{47} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{49} + ( - \beta_{2} + 7 \beta_1 - 7) q^{51} + (\beta_{2} + 4 \beta_1 - 5) q^{53} + ( - \beta_1 + 1) q^{57} + ( - 4 \beta_{2} + \beta_1 - 1) q^{59} - 2 \beta_{2} q^{61} + ( - 2 \beta_{2} + \beta_1 + 5) q^{63} + (\beta_{2} - 2 \beta_1 - 1) q^{67} + (\beta_{2} - 4 \beta_1 + 1) q^{69} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{71} + (2 \beta_{2} + \beta_1 - 5) q^{73} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{77} + (4 \beta_1 - 8) q^{79} + ( - 7 \beta_1 + 10) q^{81} + (2 \beta_{2} - 4 \beta_1 + 2) q^{83} + (\beta_{2} - \beta_1 + 1) q^{87} + ( - 4 \beta_1 + 6) q^{89} + (\beta_{2} - 2 \beta_1 - 6) q^{91} + ( - 2 \beta_{2} - 6) q^{93} + (2 \beta_{2} - 6 \beta_1 - 8) q^{97} + (2 \beta_{2} - 8 \beta_1 + 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9} - 2 q^{11} - 6 q^{13} + 12 q^{17} - 3 q^{19} + 7 q^{21} + 2 q^{23} - 17 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - q^{37} + 11 q^{39} + 14 q^{43} - 3 q^{47} + 9 q^{49} - 15 q^{51} - 10 q^{53} + 2 q^{57} - 6 q^{59} - 2 q^{61} + 14 q^{63} - 4 q^{67} + 6 q^{71} - 12 q^{73} - 10 q^{77} - 20 q^{79} + 23 q^{81} + 4 q^{83} + 3 q^{87} + 14 q^{89} - 19 q^{91} - 20 q^{93} - 28 q^{97} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x + 3 \) : 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
−2.25342
0.480031
2.77339
0 −3.25342 0 0 0 0.0778929 0 7.58473 0
1.2 0 −0.519969 0 0 0 −4.76957 0 −2.72963 0
1.3 0 1.77339 0 0 0 2.69168 0 0.144903 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(19\) \(1\)

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 7600.2.a.bk 3
4.b odd 2 1 950.2.a.l yes 3
5.b even 2 1 7600.2.a.bz 3
12.b even 2 1 8550.2.a.ci 3
20.d odd 2 1 950.2.a.j 3
20.e even 4 2 950.2.b.h 6
60.h even 2 1 8550.2.a.cp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.j 3 20.d odd 2 1
950.2.a.l yes 3 4.b odd 2 1
950.2.b.h 6 20.e even 4 2
7600.2.a.bk 3 1.a even 1 1 trivial
7600.2.a.bz 3 5.b even 2 1
8550.2.a.ci 3 12.b even 2 1
8550.2.a.cp 3 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{3} + 2T_{3}^{2} - 5T_{3} - 3 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 13T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 24T_{11} - 24 \) Copy content Toggle raw display
\( T_{13}^{3} + 6T_{13}^{2} - 3T_{13} - 35 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 5 T - 3 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 13 T + 1 \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 24 T - 24 \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} - 3 T - 35 \) Copy content Toggle raw display
$17$ \( T^{3} - 12 T^{2} + 33 T + 9 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 2 T^{2} - 51 T - 111 \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} - 3 T + 9 \) Copy content Toggle raw display
$31$ \( T^{3} + 8 T^{2} - 4 T - 56 \) Copy content Toggle raw display
$37$ \( T^{3} + T^{2} - 25T - 1 \) Copy content Toggle raw display
$41$ \( T^{3} \) Copy content Toggle raw display
$43$ \( T^{3} - 14 T^{2} + 40 T + 24 \) Copy content Toggle raw display
$47$ \( T^{3} + 3 T^{2} - 57 T + 45 \) Copy content Toggle raw display
$53$ \( T^{3} + 10 T^{2} - 105 T - 867 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} - 201 T - 1431 \) Copy content Toggle raw display
$61$ \( T^{3} + 2 T^{2} - 56 T - 120 \) Copy content Toggle raw display
$67$ \( T^{3} + 4 T^{2} - 23 T - 75 \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} - 192 T + 1512 \) Copy content Toggle raw display
$73$ \( T^{3} + 12 T^{2} - 27 T - 317 \) Copy content Toggle raw display
$79$ \( T^{3} + 20 T^{2} + 32 T - 320 \) Copy content Toggle raw display
$83$ \( T^{3} - 4 T^{2} - 108 T - 168 \) Copy content Toggle raw display
$89$ \( T^{3} - 14 T^{2} - 36 T + 312 \) Copy content Toggle raw display
$97$ \( T^{3} + 28 T^{2} + 44 T - 2440 \) Copy content Toggle raw display
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