Properties

Label 7800.2.a.bz
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.504568.1
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta_1 q^{7} + q^{9} + ( - \beta_{3} + 1) q^{11} - q^{13} + (\beta_{4} - \beta_{2} - 1) q^{17} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{19} + \beta_1 q^{21} + ( - \beta_{4} + \beta_{2} - 1) q^{23} - q^{27} + ( - \beta_{3} + \beta_{2}) q^{29} - 2 \beta_{4} q^{31} + (\beta_{3} - 1) q^{33} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{37} + q^{39} + ( - \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1) q^{41} + (\beta_{4} - \beta_{2} + 3 \beta_1 - 1) q^{43} + (\beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2) q^{47} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{49} + ( - \beta_{4} + \beta_{2} + 1) q^{51} + ( - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{53} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{57} + (2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 3) q^{59} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{61} - \beta_1 q^{63} + (2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 2) q^{67} + (\beta_{4} - \beta_{2} + 1) q^{69} + ( - \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{71} + (2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{73} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{77} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{79} + q^{81} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{83} + (\beta_{3} - \beta_{2}) q^{87} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{89} + \beta_1 q^{91} + 2 \beta_{4} q^{93} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{97} + ( - \beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} - q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} - q^{7} + 5 q^{9} + 5 q^{11} - 5 q^{13} - 7 q^{17} + 6 q^{19} + q^{21} - 3 q^{23} - 5 q^{27} + 2 q^{29} - 5 q^{33} - 9 q^{37} + 5 q^{39} - q^{41} - 4 q^{43} - 14 q^{47} + 2 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{57} + 20 q^{59} - 19 q^{61} - q^{63} - 12 q^{67} + 3 q^{69} + 13 q^{71} - 16 q^{73} - 11 q^{77} + 7 q^{79} + 5 q^{81} + 2 q^{83} - 2 q^{87} + 9 q^{89} + q^{91} - 13 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} - \nu^{3} - 4\nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 2\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 4\nu + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{4} + 2\nu^{3} + 4\nu^{2} - 5\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 2\beta_{3} + 2\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{4} - 7\beta_{3} + 3\beta_{2} + 8\beta _1 + 20 ) / 2 \) 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
−1.84576
2.52064
0.271831
−1.23118
1.28447
0 −1.00000 0 0 0 −3.42163 0 1.00000 0
1.2 0 −1.00000 0 0 0 −2.45939 0 1.00000 0
1.3 0 −1.00000 0 0 0 −0.961637 0 1.00000 0
1.4 0 −1.00000 0 0 0 2.13051 0 1.00000 0
1.5 0 −1.00000 0 0 0 3.71215 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bz 5
5.b even 2 1 7800.2.a.ca 5
5.c odd 4 2 1560.2.l.f 10
15.e even 4 2 4680.2.l.h 10
20.e even 4 2 3120.2.l.q 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.f 10 5.c odd 4 2
3120.2.l.q 10 20.e even 4 2
4680.2.l.h 10 15.e even 4 2
7800.2.a.bz 5 1.a even 1 1 trivial
7800.2.a.ca 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{5} + T_{7}^{4} - 18T_{7}^{3} - 20T_{7}^{2} + 64T_{7} + 64 \) Copy content Toggle raw display
\( T_{11}^{5} - 5T_{11}^{4} - 10T_{11}^{3} + 66T_{11}^{2} + 8T_{11} - 184 \) Copy content Toggle raw display
\( T_{17}^{5} + 7T_{17}^{4} - 6T_{17}^{3} - 24T_{17}^{2} + 8T_{17} + 16 \) Copy content Toggle raw display
\( T_{19}^{5} - 6T_{19}^{4} - 28T_{19}^{3} + 160T_{19}^{2} - 96T_{19} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + T^{4} - 18 T^{3} - 20 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{5} - 5 T^{4} - 10 T^{3} + 66 T^{2} + \cdots - 184 \) Copy content Toggle raw display
$13$ \( (T + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} + 7 T^{4} - 6 T^{3} - 24 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{5} - 6 T^{4} - 28 T^{3} + 160 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$23$ \( T^{5} + 3 T^{4} - 22 T^{3} - 100 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$29$ \( T^{5} - 2 T^{4} - 24 T^{3} + 24 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$31$ \( T^{5} - 88 T^{3} + 64 T^{2} + \cdots - 1024 \) Copy content Toggle raw display
$37$ \( T^{5} + 9 T^{4} - 72 T^{3} - 752 T^{2} + \cdots + 368 \) Copy content Toggle raw display
$41$ \( T^{5} + T^{4} - 186 T^{3} + \cdots - 29816 \) Copy content Toggle raw display
$43$ \( T^{5} + 4 T^{4} - 180 T^{3} + \cdots + 20224 \) Copy content Toggle raw display
$47$ \( T^{5} + 14 T^{4} - 22 T^{3} + \cdots + 6400 \) Copy content Toggle raw display
$53$ \( T^{5} + 5 T^{4} - 172 T^{3} + \cdots - 944 \) Copy content Toggle raw display
$59$ \( T^{5} - 20 T^{4} - 106 T^{3} + \cdots - 45088 \) Copy content Toggle raw display
$61$ \( T^{5} + 19 T^{4} - 104 T^{3} + \cdots - 62912 \) Copy content Toggle raw display
$67$ \( T^{5} + 12 T^{4} - 112 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( T^{5} - 13 T^{4} - 210 T^{3} + \cdots + 3104 \) Copy content Toggle raw display
$73$ \( T^{5} + 16 T^{4} - 36 T^{3} + \cdots - 13792 \) Copy content Toggle raw display
$79$ \( T^{5} - 7 T^{4} - 192 T^{3} + \cdots - 2752 \) Copy content Toggle raw display
$83$ \( T^{5} - 2 T^{4} - 150 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{5} - 9 T^{4} - 10 T^{3} + 206 T^{2} + \cdots - 200 \) Copy content Toggle raw display
$97$ \( T^{5} + 13 T^{4} - 62 T^{3} + \cdots + 1552 \) Copy content Toggle raw display
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