Properties

Label 7600.2.a.bl
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: 3.3.321.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1900)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + ( 2 - \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + ( 2 - \beta_{1} ) q^{9} -\beta_{2} q^{11} + ( 1 + \beta_{1} + \beta_{2} ) q^{13} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{17} - q^{19} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{21} + ( -5 - \beta_{1} ) q^{23} + ( 2 \beta_{1} + \beta_{2} ) q^{27} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{29} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{31} + ( 4 - \beta_{1} - \beta_{2} ) q^{33} + ( -4 + \beta_{2} ) q^{37} + ( -4 - \beta_{1} ) q^{39} -3 \beta_{1} q^{41} + ( -5 - 2 \beta_{2} ) q^{43} + ( -1 - 5 \beta_{1} + \beta_{2} ) q^{47} + ( 6 + 3 \beta_{1} ) q^{49} + ( 5 - 5 \beta_{1} - 2 \beta_{2} ) q^{51} + ( 1 - 4 \beta_{1} ) q^{53} + ( 1 + \beta_{2} ) q^{57} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -3 + 5 \beta_{1} + \beta_{2} ) q^{61} + ( -7 + 2 \beta_{1} ) q^{63} + ( -4 - 4 \beta_{1} + 3 \beta_{2} ) q^{67} + ( 4 + 2 \beta_{1} + 5 \beta_{2} ) q^{69} + ( 1 + 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 4 + \beta_{1} - 2 \beta_{2} ) q^{73} + ( -2 - \beta_{1} + 3 \beta_{2} ) q^{77} + ( 1 + 5 \beta_{2} ) q^{79} + ( -8 + \beta_{2} ) q^{81} + ( -10 + \beta_{1} ) q^{83} + ( -8 + 5 \beta_{1} - \beta_{2} ) q^{87} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 6 + 5 \beta_{1} ) q^{91} + ( 5 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 1 + \beta_{1} + 5 \beta_{2} ) q^{97} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{3} - 2q^{7} + 5q^{9} + O(q^{10}) \) \( 3q - 2q^{3} - 2q^{7} + 5q^{9} + q^{11} + 3q^{13} + 6q^{17} - 3q^{19} - 8q^{21} - 16q^{23} + q^{27} + 3q^{29} + q^{31} + 12q^{33} - 13q^{37} - 13q^{39} - 3q^{41} - 13q^{43} - 9q^{47} + 21q^{49} + 12q^{51} - q^{53} + 2q^{57} + 6q^{59} - 5q^{61} - 19q^{63} - 19q^{67} + 9q^{69} + 3q^{71} + 15q^{73} - 10q^{77} - 2q^{79} - 25q^{81} - 29q^{83} - 18q^{87} + 14q^{89} + 23q^{91} + 7q^{93} - q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 1\):

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

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
−1.69963
2.46050
0.239123
0 −2.58836 0 0 0 −2.81089 0 3.69963 0
1.2 0 −1.59358 0 0 0 4.51459 0 −0.460505 0
1.3 0 2.18194 0 0 0 −3.70370 0 1.76088 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.bl 3
4.b odd 2 1 1900.2.a.i yes 3
5.b even 2 1 7600.2.a.ca 3
20.d odd 2 1 1900.2.a.g 3
20.e even 4 2 1900.2.c.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.g 3 20.d odd 2 1
1900.2.a.i yes 3 4.b odd 2 1
1900.2.c.f 6 20.e even 4 2
7600.2.a.bl 3 1.a even 1 1 trivial
7600.2.a.ca 3 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{3} + 2 T_{3}^{2} - 5 T_{3} - 9 \)
\( T_{7}^{3} + 2 T_{7}^{2} - 19 T_{7} - 47 \)
\( T_{11}^{3} - T_{11}^{2} - 6 T_{11} - 3 \)
\( T_{13}^{3} - 3 T_{13}^{2} - 6 T_{13} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -9 - 5 T + 2 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( -47 - 19 T + 2 T^{2} + T^{3} \)
$11$ \( -3 - 6 T - T^{2} + T^{3} \)
$13$ \( 7 - 6 T - 3 T^{2} + T^{3} \)
$17$ \( 99 - 15 T - 6 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( 129 + 81 T + 16 T^{2} + T^{3} \)
$29$ \( -27 - 24 T - 3 T^{2} + T^{3} \)
$31$ \( 109 - 40 T - T^{2} + T^{3} \)
$37$ \( 59 + 50 T + 13 T^{2} + T^{3} \)
$41$ \( -27 - 36 T + 3 T^{2} + T^{3} \)
$43$ \( -69 + 31 T + 13 T^{2} + T^{3} \)
$47$ \( -621 - 96 T + 9 T^{2} + T^{3} \)
$53$ \( 3 - 69 T + T^{2} + T^{3} \)
$59$ \( 72 - 24 T - 6 T^{2} + T^{3} \)
$61$ \( -489 - 98 T + 5 T^{2} + T^{3} \)
$67$ \( -1323 - 26 T + 19 T^{2} + T^{3} \)
$71$ \( 27 - 33 T - 3 T^{2} + T^{3} \)
$73$ \( 49 + 42 T - 15 T^{2} + T^{3} \)
$79$ \( 529 - 157 T + 2 T^{2} + T^{3} \)
$83$ \( 861 + 276 T + 29 T^{2} + T^{3} \)
$89$ \( 489 - 9 T - 14 T^{2} + T^{3} \)
$97$ \( 683 - 154 T + T^{2} + T^{3} \)
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