Properties

Label 7600.2.a.x
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: \( x^{2} - 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{3} + ( - \beta + 1) q^{7} - 2 \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{3} + ( - \beta + 1) q^{7} - 2 \beta q^{9} + \beta q^{11} + ( - 2 \beta - 1) q^{13} + q^{17} + q^{19} + (2 \beta - 3) q^{21} + (3 \beta - 1) q^{23} + ( - \beta - 1) q^{27} + ( - 2 \beta + 1) q^{29} + (\beta - 2) q^{31} + ( - \beta + 2) q^{33} + (2 \beta + 8) q^{37} + (\beta - 3) q^{39} + 5 \beta q^{41} + (3 \beta + 2) q^{43} - 8 q^{47} + ( - 2 \beta - 4) q^{49} + (\beta - 1) q^{51} + (2 \beta + 1) q^{53} + (\beta - 1) q^{57} + (\beta - 13) q^{59} + ( - \beta + 2) q^{61} + ( - 2 \beta + 4) q^{63} + ( - 5 \beta - 1) q^{67} + ( - 4 \beta + 7) q^{69} + (5 \beta + 4) q^{71} + ( - 2 \beta + 11) q^{73} + (\beta - 2) q^{77} + ( - 2 \beta - 2) q^{79} + (6 \beta - 1) q^{81} + ( - 6 \beta - 6) q^{83} + (3 \beta - 5) q^{87} + ( - 3 \beta - 8) q^{89} + ( - \beta + 3) q^{91} + ( - 3 \beta + 4) q^{93} + ( - 4 \beta + 2) q^{97} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{7} - 2 q^{13} + 2 q^{17} + 2 q^{19} - 6 q^{21} - 2 q^{23} - 2 q^{27} + 2 q^{29} - 4 q^{31} + 4 q^{33} + 16 q^{37} - 6 q^{39} + 4 q^{43} - 16 q^{47} - 8 q^{49} - 2 q^{51} + 2 q^{53} - 2 q^{57} - 26 q^{59} + 4 q^{61} + 8 q^{63} - 2 q^{67} + 14 q^{69} + 8 q^{71} + 22 q^{73} - 4 q^{77} - 4 q^{79} - 2 q^{81} - 12 q^{83} - 10 q^{87} - 16 q^{89} + 6 q^{91} + 8 q^{93} + 4 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.41421 0 0 0 2.41421 0 2.82843 0
1.2 0 0.414214 0 0 0 −0.414214 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.x 2
4.b odd 2 1 3800.2.a.p 2
5.b even 2 1 7600.2.a.bc 2
5.c odd 4 2 1520.2.d.d 4
20.d odd 2 1 3800.2.a.l 2
20.e even 4 2 760.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.c 4 20.e even 4 2
1520.2.d.d 4 5.c odd 4 2
3800.2.a.l 2 20.d odd 2 1
3800.2.a.p 2 4.b odd 2 1
7600.2.a.x 2 1.a even 1 1 trivial
7600.2.a.bc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{2} + 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 2 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 17 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$37$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$41$ \( T^{2} - 50 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$59$ \( T^{2} + 26T + 167 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 49 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T - 34 \) Copy content Toggle raw display
$73$ \( T^{2} - 22T + 113 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 16T + 46 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
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