Properties

Label 7600.2.a.bw
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: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 475)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{7} + ( -2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{7} + ( -2 \beta_{1} + \beta_{2} ) q^{9} + ( -2 + 3 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -1 - \beta_{1} + \beta_{2} ) q^{13} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{17} + q^{19} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{21} + ( 3 + \beta_{2} ) q^{23} + ( \beta_{1} + 2 \beta_{2} ) q^{27} + ( -4 + 5 \beta_{1} ) q^{29} + ( -1 + 3 \beta_{1} - 5 \beta_{2} ) q^{31} + ( -6 + 5 \beta_{1} - 3 \beta_{2} ) q^{33} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{37} + \beta_{2} q^{39} + ( -4 + 8 \beta_{1} - 5 \beta_{2} ) q^{41} + ( -1 + 6 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} ) q^{47} + ( -4 + 2 \beta_{1} - 3 \beta_{2} ) q^{49} + ( 3 + \beta_{2} ) q^{51} + ( -7 + 4 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 1 - \beta_{1} ) q^{57} + ( -6 + 2 \beta_{1} - 6 \beta_{2} ) q^{59} + ( -5 + \beta_{1} + 3 \beta_{2} ) q^{61} + ( 1 - 4 \beta_{2} ) q^{63} + ( -5 \beta_{1} - 4 \beta_{2} ) q^{67} + ( 2 - 3 \beta_{1} ) q^{69} + ( 9 - 2 \beta_{1} + 6 \beta_{2} ) q^{71} + ( 6 - 8 \beta_{1} + 9 \beta_{2} ) q^{73} + ( -4 + \beta_{1} + 3 \beta_{2} ) q^{77} + ( -7 + 3 \beta_{1} ) q^{79} + ( -4 + 7 \beta_{1} - 4 \beta_{2} ) q^{81} + ( 6 + 5 \beta_{2} ) q^{83} + ( -14 + 9 \beta_{1} - 5 \beta_{2} ) q^{87} + ( 5 - 6 \beta_{1} + 7 \beta_{2} ) q^{89} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{91} + ( -2 + 4 \beta_{1} - 3 \beta_{2} ) q^{93} + ( -1 - \beta_{1} + \beta_{2} ) q^{97} + ( -7 + 2 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{3} - 3q^{9} + O(q^{10}) \) \( 3q + 2q^{3} - 3q^{9} - q^{11} - 5q^{13} - 2q^{17} + 3q^{19} + 8q^{23} - q^{27} - 7q^{29} + 5q^{31} - 10q^{33} - 5q^{37} - q^{39} + q^{41} + 5q^{43} + 3q^{47} - 7q^{49} + 8q^{51} - 19q^{53} + 2q^{57} - 10q^{59} - 17q^{61} + 7q^{63} - q^{67} + 3q^{69} + 19q^{71} + q^{73} - 14q^{77} - 18q^{79} - q^{81} + 13q^{83} - 28q^{87} + 2q^{89} + 7q^{91} + q^{93} - 5q^{97} - 20q^{99} + O(q^{100}) \)

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
1.80194
0.445042
−1.24698
0 −0.801938 0 0 0 1.69202 0 −2.35690 0
1.2 0 0.554958 0 0 0 −3.04892 0 −2.69202 0
1.3 0 2.24698 0 0 0 1.35690 0 2.04892 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.bw 3
4.b odd 2 1 475.2.a.d 3
5.b even 2 1 7600.2.a.bn 3
12.b even 2 1 4275.2.a.bn 3
20.d odd 2 1 475.2.a.h yes 3
20.e even 4 2 475.2.b.c 6
60.h even 2 1 4275.2.a.z 3
76.d even 2 1 9025.2.a.be 3
380.d even 2 1 9025.2.a.w 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.a.d 3 4.b odd 2 1
475.2.a.h yes 3 20.d odd 2 1
475.2.b.c 6 20.e even 4 2
4275.2.a.z 3 60.h even 2 1
4275.2.a.bn 3 12.b even 2 1
7600.2.a.bn 3 5.b even 2 1
7600.2.a.bw 3 1.a even 1 1 trivial
9025.2.a.w 3 380.d even 2 1
9025.2.a.be 3 76.d even 2 1

Hecke kernels

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

\( T_{3}^{3} - 2 T_{3}^{2} - T_{3} + 1 \)
\( T_{7}^{3} - 7 T_{7} + 7 \)
\( T_{11}^{3} + T_{11}^{2} - 16 T_{11} + 13 \)
\( T_{13}^{3} + 5 T_{13}^{2} + 6 T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 1 - T - 2 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 7 - 7 T + T^{3} \)
$11$ \( 13 - 16 T + T^{2} + T^{3} \)
$13$ \( 1 + 6 T + 5 T^{2} + T^{3} \)
$17$ \( 13 - 15 T + 2 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -13 + 19 T - 8 T^{2} + T^{3} \)
$29$ \( -91 - 42 T + 7 T^{2} + T^{3} \)
$31$ \( -43 - 36 T - 5 T^{2} + T^{3} \)
$37$ \( 1 - 8 T + 5 T^{2} + T^{3} \)
$41$ \( 421 - 114 T - T^{2} + T^{3} \)
$43$ \( 293 - 57 T - 5 T^{2} + T^{3} \)
$47$ \( 13 - 4 T - 3 T^{2} + T^{3} \)
$53$ \( -307 + 55 T + 19 T^{2} + T^{3} \)
$59$ \( -328 - 32 T + 10 T^{2} + T^{3} \)
$61$ \( -41 + 66 T + 17 T^{2} + T^{3} \)
$67$ \( 559 - 142 T + T^{2} + T^{3} \)
$71$ \( 307 + 55 T - 19 T^{2} + T^{3} \)
$73$ \( 463 - 170 T - T^{2} + T^{3} \)
$79$ \( 97 + 87 T + 18 T^{2} + T^{3} \)
$83$ \( 139 - 2 T - 13 T^{2} + T^{3} \)
$89$ \( 281 - 99 T - 2 T^{2} + T^{3} \)
$97$ \( 1 + 6 T + 5 T^{2} + T^{3} \)
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