Properties

Label 7600.2.a.z
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
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: \(1\)
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 760)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( -2 + 2 \beta ) q^{7} - q^{9} +O(q^{10})\) \( q + \beta q^{3} + ( -2 + 2 \beta ) q^{7} - q^{9} + ( -2 + 2 \beta ) q^{11} + ( 2 + \beta ) q^{13} -2 \beta q^{17} + q^{19} + ( 4 - 2 \beta ) q^{21} + ( 2 - 4 \beta ) q^{23} -4 \beta q^{27} + ( -2 - 4 \beta ) q^{29} + ( -4 + 2 \beta ) q^{31} + ( 4 - 2 \beta ) q^{33} + ( -2 - \beta ) q^{37} + ( 2 + 2 \beta ) q^{39} + ( -2 - 2 \beta ) q^{41} + ( -6 + 2 \beta ) q^{43} + ( -2 - 2 \beta ) q^{47} + ( 5 - 8 \beta ) q^{49} -4 q^{51} + ( 2 - 7 \beta ) q^{53} + \beta q^{57} + ( 4 - 2 \beta ) q^{59} + 4 \beta q^{61} + ( 2 - 2 \beta ) q^{63} -7 \beta q^{67} + ( -8 + 2 \beta ) q^{69} -6 \beta q^{71} + ( 4 + 2 \beta ) q^{73} + ( 12 - 8 \beta ) q^{77} + 6 \beta q^{79} -5 q^{81} + ( 10 - 4 \beta ) q^{83} + ( -8 - 2 \beta ) q^{87} + ( -2 + 8 \beta ) q^{89} + 2 \beta q^{91} + ( 4 - 4 \beta ) q^{93} + ( 6 + 5 \beta ) q^{97} + ( 2 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} - 2 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{7} - 2 q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{19} + 8 q^{21} + 4 q^{23} - 4 q^{29} - 8 q^{31} + 8 q^{33} - 4 q^{37} + 4 q^{39} - 4 q^{41} - 12 q^{43} - 4 q^{47} + 10 q^{49} - 8 q^{51} + 4 q^{53} + 8 q^{59} + 4 q^{63} - 16 q^{69} + 8 q^{73} + 24 q^{77} - 10 q^{81} + 20 q^{83} - 16 q^{87} - 4 q^{89} + 8 q^{93} + 12 q^{97} + 4 q^{99} + O(q^{100}) \)

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 −1.41421 0 0 0 −4.82843 0 −1.00000 0
1.2 0 1.41421 0 0 0 0.828427 0 −1.00000 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.z 2
4.b odd 2 1 3800.2.a.o 2
5.b even 2 1 7600.2.a.bb 2
5.c odd 4 2 1520.2.d.g 4
20.d odd 2 1 3800.2.a.m 2
20.e even 4 2 760.2.d.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.d 4 20.e even 4 2
1520.2.d.g 4 5.c odd 4 2
3800.2.a.m 2 20.d odd 2 1
3800.2.a.o 2 4.b odd 2 1
7600.2.a.z 2 1.a even 1 1 trivial
7600.2.a.bb 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{2} - 2 \)
\( T_{7}^{2} + 4 T_{7} - 4 \)
\( T_{11}^{2} + 4 T_{11} - 4 \)
\( T_{13}^{2} - 4 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -4 + 4 T + T^{2} \)
$11$ \( -4 + 4 T + T^{2} \)
$13$ \( 2 - 4 T + T^{2} \)
$17$ \( -8 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -28 - 4 T + T^{2} \)
$29$ \( -28 + 4 T + T^{2} \)
$31$ \( 8 + 8 T + T^{2} \)
$37$ \( 2 + 4 T + T^{2} \)
$41$ \( -4 + 4 T + T^{2} \)
$43$ \( 28 + 12 T + T^{2} \)
$47$ \( -4 + 4 T + T^{2} \)
$53$ \( -94 - 4 T + T^{2} \)
$59$ \( 8 - 8 T + T^{2} \)
$61$ \( -32 + T^{2} \)
$67$ \( -98 + T^{2} \)
$71$ \( -72 + T^{2} \)
$73$ \( 8 - 8 T + T^{2} \)
$79$ \( -72 + T^{2} \)
$83$ \( 68 - 20 T + T^{2} \)
$89$ \( -124 + 4 T + T^{2} \)
$97$ \( -14 - 12 T + T^{2} \)
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