Properties

Label 10000.2.a.l
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 + \beta ) q^{7} -2 q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 + \beta ) q^{7} -2 q^{9} + ( 2 + 2 \beta ) q^{11} + ( 3 - 3 \beta ) q^{13} + ( 2 + 2 \beta ) q^{17} + ( 4 - 3 \beta ) q^{19} + ( -1 + \beta ) q^{21} + ( 7 - 2 \beta ) q^{23} -5 q^{27} + ( -2 - \beta ) q^{29} + 3 q^{31} + ( 2 + 2 \beta ) q^{33} + ( -3 + 2 \beta ) q^{37} + ( 3 - 3 \beta ) q^{39} + ( -4 + 2 \beta ) q^{41} -3 \beta q^{43} + ( -1 + \beta ) q^{47} + ( -5 - \beta ) q^{49} + ( 2 + 2 \beta ) q^{51} + ( -3 + 4 \beta ) q^{53} + ( 4 - 3 \beta ) q^{57} + ( 6 + 3 \beta ) q^{59} + ( -1 + 6 \beta ) q^{61} + ( 2 - 2 \beta ) q^{63} + ( 8 - 2 \beta ) q^{67} + ( 7 - 2 \beta ) q^{69} + ( 5 + \beta ) q^{71} + 9 q^{73} + 2 \beta q^{77} + 5 \beta q^{79} + q^{81} + ( -3 - 2 \beta ) q^{83} + ( -2 - \beta ) q^{87} + ( 4 - 8 \beta ) q^{89} + ( -6 + 3 \beta ) q^{91} + 3 q^{93} + ( -1 + 3 \beta ) q^{97} + ( -4 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - q^{7} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - q^{7} - 4q^{9} + 6q^{11} + 3q^{13} + 6q^{17} + 5q^{19} - q^{21} + 12q^{23} - 10q^{27} - 5q^{29} + 6q^{31} + 6q^{33} - 4q^{37} + 3q^{39} - 6q^{41} - 3q^{43} - q^{47} - 11q^{49} + 6q^{51} - 2q^{53} + 5q^{57} + 15q^{59} + 4q^{61} + 2q^{63} + 14q^{67} + 12q^{69} + 11q^{71} + 18q^{73} + 2q^{77} + 5q^{79} + 2q^{81} - 8q^{83} - 5q^{87} - 9q^{91} + 6q^{93} + q^{97} - 12q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
0 1.00000 0 0 0 −1.61803 0 −2.00000 0
1.2 0 1.00000 0 0 0 0.618034 0 −2.00000 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.l 2
4.b odd 2 1 625.2.a.c 2
5.b even 2 1 10000.2.a.c 2
12.b even 2 1 5625.2.a.d 2
20.d odd 2 1 625.2.a.b 2
20.e even 4 2 625.2.b.a 4
25.e even 10 2 400.2.u.b 4
60.h even 2 1 5625.2.a.f 2
100.h odd 10 2 25.2.d.a 4
100.h odd 10 2 625.2.d.h 4
100.j odd 10 2 125.2.d.a 4
100.j odd 10 2 625.2.d.b 4
100.l even 20 4 125.2.e.a 8
100.l even 20 4 625.2.e.c 8
300.r even 10 2 225.2.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 100.h odd 10 2
125.2.d.a 4 100.j odd 10 2
125.2.e.a 8 100.l even 20 4
225.2.h.b 4 300.r even 10 2
400.2.u.b 4 25.e even 10 2
625.2.a.b 2 20.d odd 2 1
625.2.a.c 2 4.b odd 2 1
625.2.b.a 4 20.e even 4 2
625.2.d.b 4 100.j odd 10 2
625.2.d.h 4 100.h odd 10 2
625.2.e.c 8 100.l even 20 4
5625.2.a.d 2 12.b even 2 1
5625.2.a.f 2 60.h even 2 1
10000.2.a.c 2 5.b even 2 1
10000.2.a.l 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3} - 1 \)
\( T_{7}^{2} + T_{7} - 1 \)
\( T_{11}^{2} - 6 T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -1 + T + T^{2} \)
$11$ \( 4 - 6 T + T^{2} \)
$13$ \( -9 - 3 T + T^{2} \)
$17$ \( 4 - 6 T + T^{2} \)
$19$ \( -5 - 5 T + T^{2} \)
$23$ \( 31 - 12 T + T^{2} \)
$29$ \( 5 + 5 T + T^{2} \)
$31$ \( ( -3 + T )^{2} \)
$37$ \( -1 + 4 T + T^{2} \)
$41$ \( 4 + 6 T + T^{2} \)
$43$ \( -9 + 3 T + T^{2} \)
$47$ \( -1 + T + T^{2} \)
$53$ \( -19 + 2 T + T^{2} \)
$59$ \( 45 - 15 T + T^{2} \)
$61$ \( -41 - 4 T + T^{2} \)
$67$ \( 44 - 14 T + T^{2} \)
$71$ \( 29 - 11 T + T^{2} \)
$73$ \( ( -9 + T )^{2} \)
$79$ \( -25 - 5 T + T^{2} \)
$83$ \( 11 + 8 T + T^{2} \)
$89$ \( -80 + T^{2} \)
$97$ \( -11 - T + T^{2} \)
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