Properties

Label 7600.2.a.y
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.56155
−1.56155
0 −2.56155 0 0 0 −2.56155 0 3.56155 0
1.2 0 1.56155 0 0 0 1.56155 0 −0.561553 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.y 2
4.b odd 2 1 950.2.a.h 2
5.b even 2 1 1520.2.a.n 2
12.b even 2 1 8550.2.a.br 2
20.d odd 2 1 190.2.a.d 2
20.e even 4 2 950.2.b.f 4
40.e odd 2 1 6080.2.a.bh 2
40.f even 2 1 6080.2.a.bb 2
60.h even 2 1 1710.2.a.w 2
140.c even 2 1 9310.2.a.bc 2
380.d even 2 1 3610.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 20.d odd 2 1
950.2.a.h 2 4.b odd 2 1
950.2.b.f 4 20.e even 4 2
1520.2.a.n 2 5.b even 2 1
1710.2.a.w 2 60.h even 2 1
3610.2.a.t 2 380.d even 2 1
6080.2.a.bb 2 40.f even 2 1
6080.2.a.bh 2 40.e odd 2 1
7600.2.a.y 2 1.a even 1 1 trivial
8550.2.a.br 2 12.b even 2 1
9310.2.a.bc 2 140.c even 2 1

Hecke kernels

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

\( T_{3}^{2} + T_{3} - 4 \)
\( T_{7}^{2} + T_{7} - 4 \)
\( T_{11} + 4 \)
\( T_{13}^{2} - T_{13} - 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -4 + T + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -38 - T + T^{2} \)
$17$ \( 26 + 11 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -36 - 3 T + T^{2} \)
$29$ \( -38 - T + T^{2} \)
$31$ \( -16 - 2 T + T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( -52 - 8 T + T^{2} \)
$43$ \( 32 + 14 T + T^{2} \)
$47$ \( -64 + 4 T + T^{2} \)
$53$ \( 2 - 5 T + T^{2} \)
$59$ \( -4 + T + T^{2} \)
$61$ \( 32 - 14 T + T^{2} \)
$67$ \( -4 + T + T^{2} \)
$71$ \( -64 + 4 T + T^{2} \)
$73$ \( -18 + 9 T + T^{2} \)
$79$ \( -16 - 2 T + T^{2} \)
$83$ \( 32 - 14 T + T^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( ( 6 + T )^{2} \)
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