Properties

Label 7260.2.a.w
Level $7260$
Weight $2$
Character orbit 7260.a
Self dual yes
Analytic conductor $57.971$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7260 = 2^{2} \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7260.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.9713918674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 660)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + ( -1 - \beta ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} + ( -1 - \beta ) q^{7} + q^{9} + ( 1 + \beta ) q^{13} - q^{15} + ( -3 - \beta ) q^{17} + 2 \beta q^{19} + ( 1 + \beta ) q^{21} + q^{25} - q^{27} -8 q^{29} + ( 2 + 2 \beta ) q^{31} + ( -1 - \beta ) q^{35} + ( 4 - 2 \beta ) q^{37} + ( -1 - \beta ) q^{39} -8 q^{41} + ( 7 - \beta ) q^{43} + q^{45} + ( 2 - 2 \beta ) q^{47} + ( 7 + 2 \beta ) q^{49} + ( 3 + \beta ) q^{51} + 2 q^{53} -2 \beta q^{57} + 8 q^{59} -2 \beta q^{61} + ( -1 - \beta ) q^{63} + ( 1 + \beta ) q^{65} -4 q^{67} -4 \beta q^{71} + ( 3 - \beta ) q^{73} - q^{75} + ( 4 + 2 \beta ) q^{79} + q^{81} + ( 7 + \beta ) q^{83} + ( -3 - \beta ) q^{85} + 8 q^{87} + 6 q^{89} + ( -14 - 2 \beta ) q^{91} + ( -2 - 2 \beta ) q^{93} + 2 \beta q^{95} + ( 2 - 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} - 2q^{7} + 2q^{9} + 2q^{13} - 2q^{15} - 6q^{17} + 2q^{21} + 2q^{25} - 2q^{27} - 16q^{29} + 4q^{31} - 2q^{35} + 8q^{37} - 2q^{39} - 16q^{41} + 14q^{43} + 2q^{45} + 4q^{47} + 14q^{49} + 6q^{51} + 4q^{53} + 16q^{59} - 2q^{63} + 2q^{65} - 8q^{67} + 6q^{73} - 2q^{75} + 8q^{79} + 2q^{81} + 14q^{83} - 6q^{85} + 16q^{87} + 12q^{89} - 28q^{91} - 4q^{93} + 4q^{97} + O(q^{100}) \)

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
2.30278
−1.30278
0 −1.00000 0 1.00000 0 −4.60555 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 2.60555 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)

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 7260.2.a.w 2
11.b odd 2 1 660.2.a.e 2
33.d even 2 1 1980.2.a.h 2
44.c even 2 1 2640.2.a.bc 2
55.d odd 2 1 3300.2.a.w 2
55.e even 4 2 3300.2.c.l 4
132.d odd 2 1 7920.2.a.bo 2
165.d even 2 1 9900.2.a.bl 2
165.l odd 4 2 9900.2.c.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.a.e 2 11.b odd 2 1
1980.2.a.h 2 33.d even 2 1
2640.2.a.bc 2 44.c even 2 1
3300.2.a.w 2 55.d odd 2 1
3300.2.c.l 4 55.e even 4 2
7260.2.a.w 2 1.a even 1 1 trivial
7920.2.a.bo 2 132.d odd 2 1
9900.2.a.bl 2 165.d even 2 1
9900.2.c.q 4 165.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7260))\):

\( T_{7}^{2} + 2 T_{7} - 12 \)
\( T_{13}^{2} - 2 T_{13} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -12 + 2 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -12 - 2 T + T^{2} \)
$17$ \( -4 + 6 T + T^{2} \)
$19$ \( -52 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 8 + T )^{2} \)
$31$ \( -48 - 4 T + T^{2} \)
$37$ \( -36 - 8 T + T^{2} \)
$41$ \( ( 8 + T )^{2} \)
$43$ \( 36 - 14 T + T^{2} \)
$47$ \( -48 - 4 T + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( -52 + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( -208 + T^{2} \)
$73$ \( -4 - 6 T + T^{2} \)
$79$ \( -36 - 8 T + T^{2} \)
$83$ \( 36 - 14 T + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( -204 - 4 T + T^{2} \)
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