Properties

Label 10000.2.a.bb
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 10000 = 2^{4} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 10000.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.8504020213\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_1 + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_1 + 2) q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{3} - \beta_{2} - 4) q^{13} + (\beta_{3} - 2) q^{17} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{19} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{21} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{23} + (\beta_{3} + 3 \beta_{2} + 4) q^{27} + (3 \beta_{2} - \beta_1 - 1) q^{29} + ( - 2 \beta_{3} + 2 \beta_{2} - 1) q^{31} + (3 \beta_{3} + 4 \beta_{2}) q^{33} + ( - \beta_{3} - \beta_{2} - \beta_1 - 5) q^{37} + ( - 2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 5) q^{39} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 5) q^{41} + (3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 2) q^{43} + ( - 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 4) q^{47} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 4) q^{49} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{51} + (2 \beta_{2} - 2 \beta_1) q^{53} + ( - 3 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{57} - 4 \beta_{3} q^{59} + ( - 3 \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{61} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 6) q^{63} + ( - \beta_{3} + \beta_{2} - 2 \beta_1) q^{67} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{69} + ( - 6 \beta_{2} + 2 \beta_1 - 5) q^{71} + (4 \beta_{3} - 6) q^{73} + (3 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{77} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{79} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{81} + (5 \beta_{3} + 5 \beta_{2} + 6) q^{83} + (3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{87} + ( - 6 \beta_{2} - 8) q^{89} + (2 \beta_{3} - \beta_{2} + 5 \beta_1 - 5) q^{91} + ( - 2 \beta_{2} - \beta_1 - 3) q^{93} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 6) q^{97} + (\beta_{3} + 7 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 8 q^{7} + 2 q^{9} + 2 q^{11} - 14 q^{13} - 8 q^{17} - 2 q^{21} + 4 q^{23} + 10 q^{27} - 10 q^{29} - 8 q^{31} - 8 q^{33} - 18 q^{37} - 14 q^{39} - 12 q^{41} + 14 q^{43} + 8 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 2 q^{61} - 16 q^{63} - 2 q^{67} - 6 q^{69} - 8 q^{71} - 24 q^{73} - 6 q^{77} - 20 q^{79} + 4 q^{81} + 14 q^{83} - 20 q^{87} - 20 q^{89} - 18 q^{91} - 8 q^{93} - 28 q^{97} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{20} + \zeta_{20}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
−1.90211
−1.17557
1.17557
1.90211
0 −0.902113 0 0 0 5.07768 0 −2.18619 0
1.2 0 −0.175571 0 0 0 1.27346 0 −2.96917 0
1.3 0 2.17557 0 0 0 2.72654 0 1.73311 0
1.4 0 2.90211 0 0 0 −1.07768 0 5.42226 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

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 10000.2.a.bb 4
4.b odd 2 1 1250.2.a.i 4
5.b even 2 1 10000.2.a.o 4
20.d odd 2 1 1250.2.a.h 4
20.e even 4 2 1250.2.b.c 8
25.f odd 20 2 400.2.y.a 8
100.h odd 10 2 250.2.d.c 8
100.j odd 10 2 250.2.d.b 8
100.l even 20 2 50.2.e.a 8
100.l even 20 2 250.2.e.a 8
300.u odd 20 2 450.2.l.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.e.a 8 100.l even 20 2
250.2.d.b 8 100.j odd 10 2
250.2.d.c 8 100.h odd 10 2
250.2.e.a 8 100.l even 20 2
400.2.y.a 8 25.f odd 20 2
450.2.l.b 8 300.u odd 20 2
1250.2.a.h 4 20.d odd 2 1
1250.2.a.i 4 4.b odd 2 1
1250.2.b.c 8 20.e even 4 2
10000.2.a.o 4 5.b even 2 1
10000.2.a.bb 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3}^{4} - 4T_{3}^{3} + T_{3}^{2} + 6T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 8T_{7}^{3} + 14T_{7}^{2} + 8T_{7} - 19 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 26T_{11}^{2} + 82T_{11} - 59 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + T^{2} + 6 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} + 14 T^{2} + 8 T - 19 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} - 26 T^{2} + 82 T - 59 \) Copy content Toggle raw display
$13$ \( T^{4} + 14 T^{3} + 66 T^{2} + 114 T + 41 \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + 19 T^{2} + 12 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} - 15 T^{2} - 10 T + 5 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} - 14 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + 10 T^{2} - 90 T - 95 \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} - 6 T^{2} - 48 T - 19 \) Copy content Toggle raw display
$37$ \( T^{4} + 18 T^{3} + 109 T^{2} + \cdots + 241 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} - 86 T^{2} + \cdots - 5339 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} - 14 T^{2} + \cdots - 1919 \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} - 106 T^{2} + \cdots - 2939 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} - 24 T^{2} - 96 T - 64 \) Copy content Toggle raw display
$59$ \( T^{4} - 80T^{2} + 1280 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} - 71 T^{2} - 12 T + 181 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} - 26 T^{2} - 82 T - 59 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} - 86 T^{2} + \cdots + 1021 \) Copy content Toggle raw display
$73$ \( T^{4} + 24 T^{3} + 136 T^{2} + \cdots - 304 \) Copy content Toggle raw display
$79$ \( T^{4} + 20 T^{3} + 40 T^{2} + \cdots - 4720 \) Copy content Toggle raw display
$83$ \( T^{4} - 14 T^{3} - 114 T^{2} + \cdots - 4139 \) Copy content Toggle raw display
$89$ \( (T^{2} + 10 T - 20)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 28 T^{3} + 259 T^{2} + \cdots + 361 \) Copy content Toggle raw display
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