Properties

Label 10000.2.a.k
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1250)
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.

\(f(q)\) \(=\) \( q + q^{3} - \beta q^{7} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta q^{7} - 2 q^{9} + 3 q^{11} + ( - 4 \beta + 1) q^{13} + (3 \beta - 6) q^{17} + (2 \beta + 4) q^{19} - \beta q^{21} + (3 \beta - 3) q^{23} - 5 q^{27} + (6 \beta - 3) q^{29} + ( - 2 \beta + 9) q^{31} + 3 q^{33} + (5 \beta - 2) q^{37} + ( - 4 \beta + 1) q^{39} - 6 \beta q^{41} + ( - 5 \beta - 4) q^{43} - 3 q^{47} + (\beta - 6) q^{49} + (3 \beta - 6) q^{51} + ( - 6 \beta - 3) q^{53} + (2 \beta + 4) q^{57} + (6 \beta - 3) q^{59} + (2 \beta - 9) q^{61} + 2 \beta q^{63} + ( - 4 \beta + 9) q^{67} + (3 \beta - 3) q^{69} + (9 \beta - 9) q^{71} - 2 \beta q^{73} - 3 \beta q^{77} + ( - 8 \beta + 9) q^{79} + q^{81} - 3 \beta q^{83} + (6 \beta - 3) q^{87} + (12 \beta - 6) q^{89} + (3 \beta + 4) q^{91} + ( - 2 \beta + 9) q^{93} + ( - 2 \beta + 9) q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{7} - 4 q^{9} + 6 q^{11} - 2 q^{13} - 9 q^{17} + 10 q^{19} - q^{21} - 3 q^{23} - 10 q^{27} + 16 q^{31} + 6 q^{33} + q^{37} - 2 q^{39} - 6 q^{41} - 13 q^{43} - 6 q^{47} - 11 q^{49} - 9 q^{51} - 12 q^{53} + 10 q^{57} - 16 q^{61} + 2 q^{63} + 14 q^{67} - 3 q^{69} - 9 q^{71} - 2 q^{73} - 3 q^{77} + 10 q^{79} + 2 q^{81} - 3 q^{83} + 11 q^{91} + 16 q^{93} + 16 q^{97} - 12 q^{99}+O(q^{100}) \) 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 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.k 2
4.b odd 2 1 1250.2.a.c yes 2
5.b even 2 1 10000.2.a.d 2
20.d odd 2 1 1250.2.a.b 2
20.e even 4 2 1250.2.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1250.2.a.b 2 20.d odd 2 1
1250.2.a.c yes 2 4.b odd 2 1
1250.2.b.a 4 20.e even 4 2
10000.2.a.d 2 5.b even 2 1
10000.2.a.k 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$17$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 45 \) Copy content Toggle raw display
$31$ \( T^{2} - 16T + 59 \) Copy content Toggle raw display
$37$ \( T^{2} - T - 31 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 13T + 11 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 12T - 9 \) Copy content Toggle raw display
$59$ \( T^{2} - 45 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 29 \) Copy content Toggle raw display
$71$ \( T^{2} + 9T - 81 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T - 55 \) Copy content Toggle raw display
$83$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$89$ \( T^{2} - 180 \) Copy content Toggle raw display
$97$ \( T^{2} - 16T + 59 \) Copy content Toggle raw display
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