Properties

Label 7600.2.a.bm
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: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} + \beta_{2} ) q^{3} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{7} + ( 4 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} + \beta_{2} ) q^{3} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{7} + ( 4 + \beta_{1} ) q^{9} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{11} + ( -2 \beta_{1} - \beta_{2} ) q^{13} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{17} + q^{19} + ( -2 - 5 \beta_{1} + 3 \beta_{2} ) q^{21} + ( -5 + \beta_{1} - \beta_{2} ) q^{23} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{27} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{29} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{31} + ( 4 \beta_{1} - 6 \beta_{2} ) q^{33} + ( -5 - 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -9 - 2 \beta_{1} - 3 \beta_{2} ) q^{39} + ( -4 + 4 \beta_{1} ) q^{41} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -3 - 4 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 9 - 2 \beta_{1} - 3 \beta_{2} ) q^{49} + ( 5 - 4 \beta_{1} + 3 \beta_{2} ) q^{51} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{53} + ( -1 + \beta_{1} + \beta_{2} ) q^{57} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{59} + ( 8 - 2 \beta_{1} + 4 \beta_{2} ) q^{61} + ( -1 + \beta_{1} - 9 \beta_{2} ) q^{63} + ( 1 - 3 \beta_{1} - 4 \beta_{2} ) q^{67} + ( 5 - 7 \beta_{1} - 2 \beta_{2} ) q^{69} + 2 \beta_{1} q^{71} + ( -3 - 3 \beta_{1} + 3 \beta_{2} ) q^{73} + ( -18 + 4 \beta_{1} + 6 \beta_{2} ) q^{77} + ( -8 - 4 \beta_{2} ) q^{79} + ( -2 + 5 \beta_{1} + \beta_{2} ) q^{81} + ( -2 - 6 \beta_{1} + 4 \beta_{2} ) q^{83} + ( 5 + 6 \beta_{1} + 7 \beta_{2} ) q^{87} -10 q^{89} + 7 \beta_{1} q^{91} + ( -14 - 2 \beta_{1} ) q^{93} + 6 \beta_{2} q^{97} + ( -6 - 6 \beta_{1} + 8 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{3} - 2q^{7} + 13q^{9} + O(q^{10}) \) \( 3q - 2q^{3} - 2q^{7} + 13q^{9} - 2q^{11} - 2q^{13} - 4q^{17} + 3q^{19} - 11q^{21} - 14q^{23} + 7q^{27} + 14q^{29} + 4q^{31} + 4q^{33} - 17q^{37} - 29q^{39} - 8q^{41} - 2q^{43} - 13q^{47} + 25q^{49} + 11q^{51} - 10q^{53} - 2q^{57} + 6q^{59} + 22q^{61} - 2q^{63} + 8q^{69} + 2q^{71} - 12q^{73} - 50q^{77} - 24q^{79} - q^{81} - 12q^{83} + 21q^{87} - 30q^{89} + 7q^{91} - 44q^{93} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
0.713538
−1.91223
2.19869
0 −2.77733 0 0 0 4.69527 0 4.71354 0
1.2 0 −2.25561 0 0 0 −4.22547 0 2.08777 0
1.3 0 3.03293 0 0 0 −2.46980 0 6.19869 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.bm 3
4.b odd 2 1 950.2.a.m yes 3
5.b even 2 1 7600.2.a.cb 3
12.b even 2 1 8550.2.a.cj 3
20.d odd 2 1 950.2.a.k 3
20.e even 4 2 950.2.b.g 6
60.h even 2 1 8550.2.a.co 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.k 3 20.d odd 2 1
950.2.a.m yes 3 4.b odd 2 1
950.2.b.g 6 20.e even 4 2
7600.2.a.bm 3 1.a even 1 1 trivial
7600.2.a.cb 3 5.b even 2 1
8550.2.a.cj 3 12.b even 2 1
8550.2.a.co 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{3} + 2 T_{3}^{2} - 9 T_{3} - 19 \)
\( T_{7}^{3} + 2 T_{7}^{2} - 21 T_{7} - 49 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 32 T_{11} - 24 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 23 T_{13} + 21 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -19 - 9 T + 2 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( -49 - 21 T + 2 T^{2} + T^{3} \)
$11$ \( -24 - 32 T + 2 T^{2} + T^{3} \)
$13$ \( 21 - 23 T + 2 T^{2} + T^{3} \)
$17$ \( 7 - 15 T + 4 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( 63 + 57 T + 14 T^{2} + T^{3} \)
$29$ \( -25 + 41 T - 14 T^{2} + T^{3} \)
$31$ \( 152 - 36 T - 4 T^{2} + T^{3} \)
$37$ \( -9 + 63 T + 17 T^{2} + T^{3} \)
$41$ \( -64 - 48 T + 8 T^{2} + T^{3} \)
$43$ \( 72 - 40 T + 2 T^{2} + T^{3} \)
$47$ \( -525 - 25 T + 13 T^{2} + T^{3} \)
$53$ \( -3 + 11 T + 10 T^{2} + T^{3} \)
$59$ \( 175 - 29 T - 6 T^{2} + T^{3} \)
$61$ \( 536 + 72 T - 22 T^{2} + T^{3} \)
$67$ \( 469 - 131 T + T^{3} \)
$71$ \( 24 - 16 T - 2 T^{2} + T^{3} \)
$73$ \( -243 - 27 T + 12 T^{2} + T^{3} \)
$79$ \( -320 + 112 T + 24 T^{2} + T^{3} \)
$83$ \( -1544 - 164 T + 12 T^{2} + T^{3} \)
$89$ \( ( 10 + T )^{3} \)
$97$ \( 648 - 180 T + T^{3} \)
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