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: \( x^{2} - x - 8 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{3} - \beta q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta q^{7} + q^{9} - \beta q^{11} + q^{13} + (\beta - 2) q^{17} - 2 \beta q^{19} - \beta q^{21} + ( - \beta + 4) q^{23} + q^{27} - 2 q^{29} + 2 \beta q^{31} - \beta q^{33} + ( - \beta - 2) q^{37} + q^{39} + ( - \beta + 2) q^{41} + 4 q^{43} + ( - 2 \beta + 8) q^{47} + (\beta + 1) q^{49} + (\beta - 2) q^{51} + (\beta + 6) q^{53} - 2 \beta q^{57} - 8 q^{59} + (3 \beta - 2) q^{61} - \beta q^{63} + 4 q^{67} + ( - \beta + 4) q^{69} + ( - \beta - 8) q^{71} + (4 \beta + 2) q^{73} + (\beta + 8) q^{77} - 5 \beta q^{79} + q^{81} + 12 q^{83} - 2 q^{87} + (\beta + 10) q^{89} - \beta q^{91} + 2 \beta q^{93} + (\beta - 6) q^{97} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{17} - 2 q^{19} - q^{21} + 7 q^{23} + 2 q^{27} - 4 q^{29} + 2 q^{31} - q^{33} - 5 q^{37} + 2 q^{39} + 3 q^{41} + 8 q^{43} + 14 q^{47} + 3 q^{49} - 3 q^{51} + 13 q^{53} - 2 q^{57} - 16 q^{59} - q^{61} - q^{63} + 8 q^{67} + 7 q^{69} - 17 q^{71} + 8 q^{73} + 17 q^{77} - 5 q^{79} + 2 q^{81} + 24 q^{83} - 4 q^{87} + 21 q^{89} - q^{91} + 2 q^{93} - 11 q^{97} - q^{99}+O(q^{100}) \) 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
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

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