Properties

Label 7800.2.a.bb
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -\beta q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -\beta q^{7} + q^{9} -\beta q^{11} + q^{13} + ( -2 + \beta ) q^{17} -2 \beta q^{19} -\beta q^{21} + ( 4 - \beta ) q^{23} + q^{27} -2 q^{29} + 2 \beta q^{31} -\beta q^{33} + ( -2 - \beta ) q^{37} + q^{39} + ( 2 - \beta ) q^{41} + 4 q^{43} + ( 8 - 2 \beta ) q^{47} + ( 1 + \beta ) q^{49} + ( -2 + \beta ) q^{51} + ( 6 + \beta ) q^{53} -2 \beta q^{57} -8 q^{59} + ( -2 + 3 \beta ) q^{61} -\beta q^{63} + 4 q^{67} + ( 4 - \beta ) q^{69} + ( -8 - \beta ) q^{71} + ( 2 + 4 \beta ) q^{73} + ( 8 + \beta ) q^{77} -5 \beta q^{79} + q^{81} + 12 q^{83} -2 q^{87} + ( 10 + \beta ) q^{89} -\beta q^{91} + 2 \beta q^{93} + ( -6 + \beta ) q^{97} -\beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - q^{7} + 2q^{9} - q^{11} + 2q^{13} - 3q^{17} - 2q^{19} - q^{21} + 7q^{23} + 2q^{27} - 4q^{29} + 2q^{31} - q^{33} - 5q^{37} + 2q^{39} + 3q^{41} + 8q^{43} + 14q^{47} + 3q^{49} - 3q^{51} + 13q^{53} - 2q^{57} - 16q^{59} - q^{61} - q^{63} + 8q^{67} + 7q^{69} - 17q^{71} + 8q^{73} + 17q^{77} - 5q^{79} + 2q^{81} + 24q^{83} - 4q^{87} + 21q^{89} - q^{91} + 2q^{93} - 11q^{97} - q^{99} + O(q^{100}) \)

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
3.37228
−2.37228
0 1.00000 0 0 0 −3.37228 0 1.00000 0
1.2 0 1.00000 0 0 0 2.37228 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.bb 2
5.b even 2 1 1560.2.a.n 2
15.d odd 2 1 4680.2.a.bf 2
20.d odd 2 1 3120.2.a.bd 2
60.h even 2 1 9360.2.a.co 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.n 2 5.b even 2 1
3120.2.a.bd 2 20.d odd 2 1
4680.2.a.bf 2 15.d odd 2 1
7800.2.a.bb 2 1.a even 1 1 trivial
9360.2.a.co 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{2} + T_{7} - 8 \)
\( T_{11}^{2} + T_{11} - 8 \)
\( T_{17}^{2} + 3 T_{17} - 6 \)
\( T_{19}^{2} + 2 T_{19} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -8 + T + T^{2} \)
$11$ \( -8 + T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -6 + 3 T + T^{2} \)
$19$ \( -32 + 2 T + T^{2} \)
$23$ \( 4 - 7 T + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( -32 - 2 T + T^{2} \)
$37$ \( -2 + 5 T + T^{2} \)
$41$ \( -6 - 3 T + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 16 - 14 T + T^{2} \)
$53$ \( 34 - 13 T + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( -74 + T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( 64 + 17 T + T^{2} \)
$73$ \( -116 - 8 T + T^{2} \)
$79$ \( -200 + 5 T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( 102 - 21 T + T^{2} \)
$97$ \( 22 + 11 T + T^{2} \)
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