Properties

Label 7800.2.a.be
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta q^{7} + q^{9} + ( -2 - \beta ) q^{11} + q^{13} + ( -2 - \beta ) q^{17} -2 q^{19} + \beta q^{21} + ( -4 + \beta ) q^{23} + q^{27} + ( 4 - 2 \beta ) q^{29} + ( -4 + 2 \beta ) q^{31} + ( -2 - \beta ) q^{33} + ( 4 - \beta ) q^{37} + q^{39} + ( 8 - \beta ) q^{41} -4 q^{43} + ( -4 - 2 \beta ) q^{47} + ( 3 + \beta ) q^{49} + ( -2 - \beta ) q^{51} + ( -8 + \beta ) q^{53} -2 q^{57} + ( -4 + 4 \beta ) q^{59} -\beta q^{61} + \beta q^{63} + 4 q^{67} + ( -4 + \beta ) q^{69} + ( 2 - 5 \beta ) q^{71} + ( -8 + 2 \beta ) q^{73} + ( -10 - 3 \beta ) q^{77} + ( 2 - \beta ) q^{79} + q^{81} -4 q^{83} + ( 4 - 2 \beta ) q^{87} -3 \beta q^{89} + \beta q^{91} + ( -4 + 2 \beta ) q^{93} + ( -10 + 3 \beta ) q^{97} + ( -2 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + q^{7} + 2q^{9} - 5q^{11} + 2q^{13} - 5q^{17} - 4q^{19} + q^{21} - 7q^{23} + 2q^{27} + 6q^{29} - 6q^{31} - 5q^{33} + 7q^{37} + 2q^{39} + 15q^{41} - 8q^{43} - 10q^{47} + 7q^{49} - 5q^{51} - 15q^{53} - 4q^{57} - 4q^{59} - q^{61} + q^{63} + 8q^{67} - 7q^{69} - q^{71} - 14q^{73} - 23q^{77} + 3q^{79} + 2q^{81} - 8q^{83} + 6q^{87} - 3q^{89} + q^{91} - 6q^{93} - 17q^{97} - 5q^{99} + O(q^{100}) \)

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
−2.70156
3.70156
0 1.00000 0 0 0 −2.70156 0 1.00000 0
1.2 0 1.00000 0 0 0 3.70156 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 7800.2.a.be 2
5.b even 2 1 1560.2.a.m 2
15.d odd 2 1 4680.2.a.bb 2
20.d odd 2 1 3120.2.a.bf 2
60.h even 2 1 9360.2.a.ct 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.m 2 5.b even 2 1
3120.2.a.bf 2 20.d odd 2 1
4680.2.a.bb 2 15.d odd 2 1
7800.2.a.be 2 1.a even 1 1 trivial
9360.2.a.ct 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{2} - T_{7} - 10 \)
\( T_{11}^{2} + 5 T_{11} - 4 \)
\( T_{17}^{2} + 5 T_{17} - 4 \)
\( T_{19} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -10 - T + T^{2} \)
$11$ \( -4 + 5 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -4 + 5 T + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 2 + 7 T + T^{2} \)
$29$ \( -32 - 6 T + T^{2} \)
$31$ \( -32 + 6 T + T^{2} \)
$37$ \( 2 - 7 T + T^{2} \)
$41$ \( 46 - 15 T + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( -16 + 10 T + T^{2} \)
$53$ \( 46 + 15 T + T^{2} \)
$59$ \( -160 + 4 T + T^{2} \)
$61$ \( -10 + T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -256 + T + T^{2} \)
$73$ \( 8 + 14 T + T^{2} \)
$79$ \( -8 - 3 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( -90 + 3 T + T^{2} \)
$97$ \( -20 + 17 T + T^{2} \)
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