Properties

Label 7800.2.a.bc
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta q^{7} + q^{9} - \beta q^{11} - q^{13} + ( - 2 \beta - 1) q^{17} + 2 \beta q^{19} + \beta q^{21} - 2 q^{23} + q^{27} + ( - 4 \beta - 3) q^{29} + (3 \beta - 4) q^{31} - \beta q^{33} + (2 \beta + 4) q^{37} - q^{39} + ( - 2 \beta - 6) q^{41} + (4 \beta - 2) q^{43} + ( - 5 \beta - 2) q^{47} - 4 q^{49} + ( - 2 \beta - 1) q^{51} + ( - 4 \beta + 5) q^{53} + 2 \beta q^{57} - \beta q^{59} + (2 \beta - 9) q^{61} + \beta q^{63} + ( - \beta - 6) q^{67} - 2 q^{69} + (6 \beta - 4) q^{71} - 2 \beta q^{73} - 3 q^{77} + 14 q^{79} + q^{81} + (\beta - 12) q^{83} + ( - 4 \beta - 3) q^{87} + (4 \beta + 4) q^{89} - \beta q^{91} + (3 \beta - 4) q^{93} + (4 \beta + 2) q^{97} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{9} - 2 q^{13} - 2 q^{17} - 4 q^{23} + 2 q^{27} - 6 q^{29} - 8 q^{31} + 8 q^{37} - 2 q^{39} - 12 q^{41} - 4 q^{43} - 4 q^{47} - 8 q^{49} - 2 q^{51} + 10 q^{53} - 18 q^{61} - 12 q^{67} - 4 q^{69} - 8 q^{71} - 6 q^{77} + 28 q^{79} + 2 q^{81} - 24 q^{83} - 6 q^{87} + 8 q^{89} - 8 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
−1.73205
1.73205
0 1.00000 0 0 0 −1.73205 0 1.00000 0
1.2 0 1.00000 0 0 0 1.73205 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.bc yes 2
5.b even 2 1 7800.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.y 2 5.b even 2 1
7800.2.a.bc yes 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} - 11 \) Copy content Toggle raw display
\( T_{19}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3 \) Copy content Toggle raw display
$11$ \( T^{2} - 3 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 11 \) Copy content Toggle raw display
$19$ \( T^{2} - 12 \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 39 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T - 11 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 71 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T - 23 \) Copy content Toggle raw display
$59$ \( T^{2} - 3 \) Copy content Toggle raw display
$61$ \( T^{2} + 18T + 69 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 33 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
$73$ \( T^{2} - 12 \) Copy content Toggle raw display
$79$ \( (T - 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 24T + 141 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
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