Properties

Label 7800.2.a.z
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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
−1.41421
1.41421
0 −1.00000 0 0 0 −0.414214 0 1.00000 0
1.2 0 −1.00000 0 0 0 2.41421 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.z 2
5.b even 2 1 7800.2.a.ba yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.z 2 1.a even 1 1 trivial
7800.2.a.ba yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{2} - 2 T_{7} - 1 \)
\( T_{11}^{2} + 2 T_{11} - 1 \)
\( T_{17}^{2} - 2 T_{17} - 7 \)
\( T_{19} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -1 - 2 T + T^{2} \)
$11$ \( -1 + 2 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -7 - 2 T + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( -4 - 4 T + T^{2} \)
$29$ \( -23 + 6 T + T^{2} \)
$31$ \( 7 - 6 T + T^{2} \)
$37$ \( -28 + 4 T + T^{2} \)
$41$ \( -32 + T^{2} \)
$43$ \( -68 + 4 T + T^{2} \)
$47$ \( -25 - 10 T + T^{2} \)
$53$ \( 1 + 6 T + T^{2} \)
$59$ \( 23 + 10 T + T^{2} \)
$61$ \( -7 - 2 T + T^{2} \)
$67$ \( 7 - 10 T + T^{2} \)
$71$ \( -28 + 4 T + T^{2} \)
$73$ \( -124 + 4 T + T^{2} \)
$79$ \( 28 + 12 T + T^{2} \)
$83$ \( -161 - 2 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( -284 + 4 T + T^{2} \)
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