Properties

Label 10000.2.a.a
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $2$
Dimension $2$
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: \(2\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
0 −2.61803 0 0 0 −3.00000 0 3.85410 0
1.2 0 −0.381966 0 0 0 −3.00000 0 −2.85410 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.a 2
4.b odd 2 1 1250.2.a.d 2
5.b even 2 1 10000.2.a.n 2
20.d odd 2 1 1250.2.a.a 2
20.e even 4 2 1250.2.b.b 4
25.e even 10 2 400.2.u.c 4
100.h odd 10 2 50.2.d.a 4
100.j odd 10 2 250.2.d.a 4
100.l even 20 4 250.2.e.b 8
300.r even 10 2 450.2.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 100.h odd 10 2
250.2.d.a 4 100.j odd 10 2
250.2.e.b 8 100.l even 20 4
400.2.u.c 4 25.e even 10 2
450.2.h.a 4 300.r even 10 2
1250.2.a.a 2 20.d odd 2 1
1250.2.a.d 2 4.b odd 2 1
1250.2.b.b 4 20.e even 4 2
10000.2.a.a 2 1.a even 1 1 trivial
10000.2.a.n 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3}^{2} + 3 T_{3} + 1 \)
\( T_{7} + 3 \)
\( T_{11}^{2} + 4 T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 3 + T )^{2} \)
$11$ \( -1 + 4 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 9 + 9 T + T^{2} \)
$19$ \( -5 + 5 T + T^{2} \)
$23$ \( 11 + 8 T + T^{2} \)
$29$ \( 5 + 10 T + T^{2} \)
$31$ \( -1 + 4 T + T^{2} \)
$37$ \( -61 - T + T^{2} \)
$41$ \( -11 + 6 T + T^{2} \)
$43$ \( 11 + 8 T + T^{2} \)
$47$ \( -71 + 6 T + T^{2} \)
$53$ \( 16 + 12 T + T^{2} \)
$59$ \( -20 + T^{2} \)
$61$ \( 19 + 11 T + T^{2} \)
$67$ \( 59 + 16 T + T^{2} \)
$71$ \( ( -3 + T )^{2} \)
$73$ \( -44 + 2 T + T^{2} \)
$79$ \( 20 - 10 T + T^{2} \)
$83$ \( 61 + 18 T + T^{2} \)
$89$ \( -20 + T^{2} \)
$97$ \( -101 - T + T^{2} \)
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