Properties

Label 10000.2.a.t
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7625.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 4 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( -1 - \beta_{3} ) q^{7} + ( 2 + \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( -1 - \beta_{3} ) q^{7} + ( 2 + \beta_{1} + \beta_{3} ) q^{9} + ( 1 + \beta_{3} ) q^{11} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( 3 + \beta_{2} - \beta_{3} ) q^{17} + ( 3 - \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{19} + ( \beta_{1} + 4 \beta_{2} ) q^{21} + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{23} + ( -5 - 4 \beta_{2} - \beta_{3} ) q^{27} + ( 4 + \beta_{1} + \beta_{2} ) q^{29} + ( 1 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{31} + ( -\beta_{1} - 4 \beta_{2} ) q^{33} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{37} + ( 6 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{39} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{41} + ( 1 + 4 \beta_{2} + \beta_{3} ) q^{43} + ( -1 - \beta_{3} ) q^{47} + ( -2 - 4 \beta_{2} + \beta_{3} ) q^{49} + ( -1 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{51} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{53} + ( 1 - 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{57} + ( -2 + 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 5 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -6 - \beta_{1} - 2 \beta_{3} ) q^{63} + ( -7 - 3 \beta_{3} ) q^{67} + ( -10 - 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{69} + ( 3 - 4 \beta_{2} - \beta_{3} ) q^{71} + ( 5 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{73} + ( -5 + 4 \beta_{2} - \beta_{3} ) q^{77} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{79} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{81} + ( 9 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{83} + ( -6 - 5 \beta_{1} - 2 \beta_{3} ) q^{87} + ( -1 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{89} + ( 1 + 2 \beta_{1} - 3 \beta_{3} ) q^{91} + ( -6 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{93} + ( 8 + 5 \beta_{2} ) q^{97} + ( 6 + \beta_{1} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{3} - 2q^{7} + 7q^{9} + O(q^{10}) \) \( 4q - q^{3} - 2q^{7} + 7q^{9} + 2q^{11} + 11q^{13} + 12q^{17} + 5q^{19} - 7q^{21} + 4q^{23} - 10q^{27} + 15q^{29} + 12q^{31} + 7q^{33} + 12q^{37} + 11q^{39} + 13q^{41} - 6q^{43} - 2q^{47} - 2q^{49} - 13q^{51} + 11q^{53} + 8q^{61} - 21q^{63} - 22q^{67} - 31q^{69} + 22q^{71} + 21q^{73} - 26q^{77} + 10q^{79} - 16q^{81} + 24q^{83} - 25q^{87} - 5q^{89} + 12q^{91} - 23q^{93} + 22q^{97} + 21q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 9 x^{2} + 4 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 5 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} - \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{2} + 6 \beta_{1} + 5\)

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
2.96645
1.71472
−1.34841
−2.33275
0 −2.96645 0 0 0 −1.83337 0 5.79981 0
1.2 0 −1.71472 0 0 0 2.77447 0 −0.0597522 0
1.3 0 1.34841 0 0 0 0.833366 0 −1.18178 0
1.4 0 2.33275 0 0 0 −3.77447 0 2.44172 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.t 4
4.b odd 2 1 1250.2.a.l 4
5.b even 2 1 10000.2.a.x 4
20.d odd 2 1 1250.2.a.f 4
20.e even 4 2 1250.2.b.e 8
25.d even 5 2 400.2.u.d 8
100.h odd 10 2 250.2.d.d 8
100.j odd 10 2 50.2.d.b 8
100.l even 20 4 250.2.e.c 16
300.n even 10 2 450.2.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.b 8 100.j odd 10 2
250.2.d.d 8 100.h odd 10 2
250.2.e.c 16 100.l even 20 4
400.2.u.d 8 25.d even 5 2
450.2.h.e 8 300.n even 10 2
1250.2.a.f 4 20.d odd 2 1
1250.2.a.l 4 4.b odd 2 1
1250.2.b.e 8 20.e even 4 2
10000.2.a.t 4 1.a even 1 1 trivial
10000.2.a.x 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3}^{4} + T_{3}^{3} - 9 T_{3}^{2} - 4 T_{3} + 16 \)
\( T_{7}^{4} + 2 T_{7}^{3} - 11 T_{7}^{2} - 12 T_{7} + 16 \)
\( T_{11}^{4} - 2 T_{11}^{3} - 11 T_{11}^{2} + 12 T_{11} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 - 4 T - 9 T^{2} + T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 - 12 T - 11 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( 16 + 12 T - 11 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( -199 + 59 T + 26 T^{2} - 11 T^{3} + T^{4} \)
$17$ \( -109 + 2 T + 39 T^{2} - 12 T^{3} + T^{4} \)
$19$ \( 80 + 20 T - 35 T^{2} - 5 T^{3} + T^{4} \)
$23$ \( 16 - 4 T - 29 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( 5 - 105 T + 70 T^{2} - 15 T^{3} + T^{4} \)
$31$ \( -1264 + 432 T - T^{2} - 12 T^{3} + T^{4} \)
$37$ \( 71 + 102 T + 19 T^{2} - 12 T^{3} + T^{4} \)
$41$ \( -89 + 23 T + 34 T^{2} - 13 T^{3} + T^{4} \)
$43$ \( 176 - 64 T - 39 T^{2} + 6 T^{3} + T^{4} \)
$47$ \( 16 - 12 T - 11 T^{2} + 2 T^{3} + T^{4} \)
$53$ \( 256 + 464 T - 49 T^{2} - 11 T^{3} + T^{4} \)
$59$ \( -320 + 560 T - 140 T^{2} + T^{4} \)
$61$ \( -1709 + 958 T - 101 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( -944 - 572 T + 69 T^{2} + 22 T^{3} + T^{4} \)
$71$ \( -64 - 168 T + 129 T^{2} - 22 T^{3} + T^{4} \)
$73$ \( -1084 + 214 T + 81 T^{2} - 21 T^{3} + T^{4} \)
$79$ \( -320 + 240 T - 20 T^{2} - 10 T^{3} + T^{4} \)
$83$ \( -3664 + 456 T + 131 T^{2} - 24 T^{3} + T^{4} \)
$89$ \( -3100 - 1500 T - 165 T^{2} + 5 T^{3} + T^{4} \)
$97$ \( ( -1 - 11 T + T^{2} )^{2} \)
show more
show less