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: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{3} - 3 q^{7} + (3 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{3} - 3 q^{7} + (3 \beta - 1) q^{9} + (2 \beta - 3) q^{11} - q^{13} + ( - 3 \beta - 3) q^{17} + (3 \beta - 4) q^{19} + (3 \beta + 3) q^{21} + ( - 2 \beta - 3) q^{23} + ( - 2 \beta + 1) q^{27} + (4 \beta - 7) q^{29} + ( - 2 \beta - 1) q^{31} + ( - \beta + 1) q^{33} + ( - 7 \beta + 4) q^{37} + (\beta + 1) q^{39} + ( - 4 \beta - 1) q^{41} + (2 \beta - 5) q^{43} + (8 \beta - 7) q^{47} + 2 q^{49} + (9 \beta + 6) q^{51} + (4 \beta - 8) q^{53} + ( - 2 \beta + 1) q^{57} + ( - 4 \beta + 2) q^{59} + (3 \beta - 7) q^{61} + ( - 9 \beta + 3) q^{63} + (2 \beta - 9) q^{67} + (7 \beta + 5) q^{69} + 3 q^{71} + (6 \beta - 4) q^{73} + ( - 6 \beta + 9) q^{77} + ( - 2 \beta + 6) q^{79} + ( - 6 \beta + 4) q^{81} + ( - 4 \beta - 7) q^{83} + ( - \beta + 3) q^{87} + ( - 4 \beta + 2) q^{89} + 3 q^{91} + (5 \beta + 3) q^{93} + (9 \beta - 4) q^{97} + ( - 5 \beta + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 6 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 6 q^{7} + q^{9} - 4 q^{11} - 2 q^{13} - 9 q^{17} - 5 q^{19} + 9 q^{21} - 8 q^{23} - 10 q^{29} - 4 q^{31} + q^{33} + q^{37} + 3 q^{39} - 6 q^{41} - 8 q^{43} - 6 q^{47} + 4 q^{49} + 21 q^{51} - 12 q^{53} - 11 q^{61} - 3 q^{63} - 16 q^{67} + 17 q^{69} + 6 q^{71} - 2 q^{73} + 12 q^{77} + 10 q^{79} + 2 q^{81} - 18 q^{83} + 5 q^{87} + 6 q^{91} + 11 q^{93} + q^{97} + 13 q^{99}+O(q^{100}) \) 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
1.61803
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{7} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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