Properties

Label 7600.2.a.bg
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( 3 - \beta ) q^{7} + 2 \beta q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( 3 - \beta ) q^{7} + 2 \beta q^{9} -\beta q^{11} + ( -3 + 2 \beta ) q^{13} - q^{17} + q^{19} + ( 1 + 2 \beta ) q^{21} + ( 5 + 3 \beta ) q^{23} + ( 1 - \beta ) q^{27} + ( -3 - 2 \beta ) q^{29} + ( -2 + 3 \beta ) q^{31} + ( -2 - \beta ) q^{33} -6 \beta q^{37} + ( 1 - \beta ) q^{39} + 3 \beta q^{41} + ( 6 + 3 \beta ) q^{43} + ( 4 - 6 \beta ) q^{49} + ( -1 - \beta ) q^{51} + ( 3 + 6 \beta ) q^{53} + ( 1 + \beta ) q^{57} + ( 3 + 7 \beta ) q^{59} + ( 10 - 3 \beta ) q^{61} + ( -4 + 6 \beta ) q^{63} + ( 9 + 3 \beta ) q^{67} + ( 11 + 8 \beta ) q^{69} + ( 12 - \beta ) q^{71} + ( -3 + 6 \beta ) q^{73} + ( 2 - 3 \beta ) q^{77} + ( -2 - 6 \beta ) q^{79} + ( -1 - 6 \beta ) q^{81} + ( 6 - 6 \beta ) q^{83} + ( -7 - 5 \beta ) q^{87} -5 \beta q^{89} + ( -13 + 9 \beta ) q^{91} + ( 4 + \beta ) q^{93} + ( 6 + 4 \beta ) q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 6q^{7} + O(q^{10}) \) \( 2q + 2q^{3} + 6q^{7} - 6q^{13} - 2q^{17} + 2q^{19} + 2q^{21} + 10q^{23} + 2q^{27} - 6q^{29} - 4q^{31} - 4q^{33} + 2q^{39} + 12q^{43} + 8q^{49} - 2q^{51} + 6q^{53} + 2q^{57} + 6q^{59} + 20q^{61} - 8q^{63} + 18q^{67} + 22q^{69} + 24q^{71} - 6q^{73} + 4q^{77} - 4q^{79} - 2q^{81} + 12q^{83} - 14q^{87} - 26q^{91} + 8q^{93} + 12q^{97} - 8q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −0.414214 0 0 0 4.41421 0 −2.82843 0
1.2 0 2.41421 0 0 0 1.58579 0 2.82843 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.bg 2
4.b odd 2 1 950.2.a.g 2
5.b even 2 1 7600.2.a.v 2
5.c odd 4 2 1520.2.d.e 4
12.b even 2 1 8550.2.a.bn 2
20.d odd 2 1 950.2.a.f 2
20.e even 4 2 190.2.b.a 4
60.h even 2 1 8550.2.a.cb 2
60.l odd 4 2 1710.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.a 4 20.e even 4 2
950.2.a.f 2 20.d odd 2 1
950.2.a.g 2 4.b odd 2 1
1520.2.d.e 4 5.c odd 4 2
1710.2.d.c 4 60.l odd 4 2
7600.2.a.v 2 5.b even 2 1
7600.2.a.bg 2 1.a even 1 1 trivial
8550.2.a.bn 2 12.b even 2 1
8550.2.a.cb 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -1 - 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - 6 T + T^{2} \)
$11$ \( -2 + T^{2} \)
$13$ \( 1 + 6 T + T^{2} \)
$17$ \( ( 1 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 7 - 10 T + T^{2} \)
$29$ \( 1 + 6 T + T^{2} \)
$31$ \( -14 + 4 T + T^{2} \)
$37$ \( -72 + T^{2} \)
$41$ \( -18 + T^{2} \)
$43$ \( 18 - 12 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( -63 - 6 T + T^{2} \)
$59$ \( -89 - 6 T + T^{2} \)
$61$ \( 82 - 20 T + T^{2} \)
$67$ \( 63 - 18 T + T^{2} \)
$71$ \( 142 - 24 T + T^{2} \)
$73$ \( -63 + 6 T + T^{2} \)
$79$ \( -68 + 4 T + T^{2} \)
$83$ \( -36 - 12 T + T^{2} \)
$89$ \( -50 + T^{2} \)
$97$ \( 4 - 12 T + T^{2} \)
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