Properties

Label 10000.2.a.c
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.

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

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 −0.618034 0 −2.00000 0
1.2 0 −1.00000 0 0 0 1.61803 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.c 2
4.b odd 2 1 625.2.a.b 2
5.b even 2 1 10000.2.a.l 2
12.b even 2 1 5625.2.a.f 2
20.d odd 2 1 625.2.a.c 2
20.e even 4 2 625.2.b.a 4
25.d even 5 2 400.2.u.b 4
60.h even 2 1 5625.2.a.d 2
100.h odd 10 2 125.2.d.a 4
100.h odd 10 2 625.2.d.b 4
100.j odd 10 2 25.2.d.a 4
100.j odd 10 2 625.2.d.h 4
100.l even 20 4 125.2.e.a 8
100.l even 20 4 625.2.e.c 8
300.n 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.j odd 10 2
125.2.d.a 4 100.h odd 10 2
125.2.e.a 8 100.l even 20 4
225.2.h.b 4 300.n even 10 2
400.2.u.b 4 25.d even 5 2
625.2.a.b 2 4.b odd 2 1
625.2.a.c 2 20.d odd 2 1
625.2.b.a 4 20.e even 4 2
625.2.d.b 4 100.h odd 10 2
625.2.d.h 4 100.j odd 10 2
625.2.e.c 8 100.l even 20 4
5625.2.a.d 2 60.h even 2 1
5625.2.a.f 2 12.b even 2 1
10000.2.a.c 2 1.a even 1 1 trivial
10000.2.a.l 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

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

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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