Properties

Label 10000.2.a.u
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.18625.1
Defining polynomial: \( x^{4} - x^{3} - 14x^{2} + 9x + 41 \) 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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{7} + ( - \beta_{3} - \beta_{2} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{7} + ( - \beta_{3} - \beta_{2} + 5) q^{9} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{3} + \beta_1 - 1) q^{13} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{17} + (\beta_{3} - \beta_{2} - 1) q^{19} + ( - \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 2) q^{21} + (2 \beta_{2} + \beta_1 + 1) q^{23} + ( - 8 \beta_{2} + 2 \beta_1 + 1) q^{27} + (\beta_{3} + \beta_1 - 3) q^{29} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 5) q^{31} + ( - \beta_{3} - 6 \beta_{2} - \beta_1 + 9) q^{33} + ( - \beta_{3} - 2 \beta_{2} - 2) q^{37} + (\beta_{3} - 6 \beta_{2} - \beta_1 + 7) q^{39} + ( - \beta_{2} + \beta_1 + 1) q^{41} + (3 \beta_{2} - 2 \beta_1 - 5) q^{43} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{47} + (2 \beta_{2} + 3 \beta_1 + 4) q^{49} + ( - 2 \beta_{3} + 6 \beta_{2} + \beta_1 - 6) q^{51} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 5) q^{53} + ( - 2 \beta_{3} + 7 \beta_{2} - \beta_1 + 1) q^{57} + ( - \beta_{3} - \beta_{2} - 8) q^{59} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{61} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 16) q^{63} + ( - 5 \beta_{2} + \beta_1 + 8) q^{67} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 5) q^{69} + ( - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 5) q^{71} + (\beta_{2} + 3 \beta_1 - 1) q^{73} + (\beta_{3} + 4 \beta_1 + 6) q^{77} + (3 \beta_{3} - \beta_1 - 3) q^{79} + ( - 5 \beta_{3} + \beta_1 + 7) q^{81} + ( - \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 4) q^{83} + ( - \beta_{3} + 8 \beta_{2} - 3 \beta_1 + 7) q^{87} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{89} + ( - \beta_{3} + 2 \beta_1 + 4) q^{91} + (3 \beta_{3} - 5 \beta_1 - 9) q^{93} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{97} + ( - 2 \beta_{3} - 8 \beta_{2} + 6 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 3 q^{7} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 3 q^{7} + 17 q^{9} - 8 q^{11} - 4 q^{13} + 2 q^{17} - 5 q^{19} - 7 q^{21} + 9 q^{23} - 10 q^{27} - 10 q^{29} - 18 q^{31} + 22 q^{33} - 13 q^{37} + 16 q^{39} + 3 q^{41} - 16 q^{43} + 8 q^{47} + 23 q^{49} - 13 q^{51} + 16 q^{53} + 15 q^{57} - 35 q^{59} + 8 q^{61} + 54 q^{63} + 23 q^{67} + 19 q^{69} + 17 q^{71} + q^{73} + 29 q^{77} - 10 q^{79} + 24 q^{81} - 11 q^{83} + 40 q^{87} - 5 q^{89} + 17 q^{91} - 38 q^{93} - 3 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 14x^{2} + 9x + 41 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 8\nu + 1 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{2} + 8\beta _1 - 1 \) 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
−1.64791
−3.08634
3.26594
2.46831
0 −3.26594 0 0 0 3.04834 0 7.66637 0
1.2 0 −2.46831 0 0 0 0.710571 0 3.09254 0
1.3 0 1.64791 0 0 0 −4.90244 0 −0.284403 0
1.4 0 3.08634 0 0 0 4.14353 0 6.52550 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.u 4
4.b odd 2 1 1250.2.a.j yes 4
5.b even 2 1 10000.2.a.v 4
20.d odd 2 1 1250.2.a.g 4
20.e even 4 2 1250.2.b.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1250.2.a.g 4 20.d odd 2 1
1250.2.a.j yes 4 4.b odd 2 1
1250.2.b.d 8 20.e even 4 2
10000.2.a.u 4 1.a even 1 1 trivial
10000.2.a.v 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3}^{4} + T_{3}^{3} - 14T_{3}^{2} - 9T_{3} + 41 \) Copy content Toggle raw display
\( T_{7}^{4} - 3T_{7}^{3} - 21T_{7}^{2} + 78T_{7} - 44 \) Copy content Toggle raw display
\( T_{11}^{4} + 8T_{11}^{3} - T_{11}^{2} - 108T_{11} - 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 14 T^{2} - 9 T + 41 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} - 21 T^{2} + 78 T - 44 \) Copy content Toggle raw display
$11$ \( T^{4} + 8 T^{3} - T^{2} - 108 T - 144 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} - 19 T^{2} - 6 T + 36 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} - 31 T^{2} - 48 T - 9 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} - 15 T^{2} - 10 T + 20 \) Copy content Toggle raw display
$23$ \( T^{4} - 9 T^{3} + T^{2} + 96 T - 144 \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} - 5 T^{2} - 150 T - 180 \) Copy content Toggle raw display
$31$ \( T^{4} + 18 T^{3} + 69 T^{2} + \cdots - 1604 \) Copy content Toggle raw display
$37$ \( T^{4} + 13 T^{3} + 39 T^{2} + 22 T - 4 \) Copy content Toggle raw display
$41$ \( T^{4} - 3 T^{3} - 11 T^{2} + 18 T + 36 \) Copy content Toggle raw display
$43$ \( T^{4} + 16 T^{3} + 31 T^{2} + \cdots - 169 \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} - T^{2} + 108 T - 144 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + 71 T^{2} - 96 T + 36 \) Copy content Toggle raw display
$59$ \( T^{4} + 35 T^{3} + 440 T^{2} + \cdots + 4545 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} - 21 T^{2} + 268 T - 484 \) Copy content Toggle raw display
$67$ \( T^{4} - 23 T^{3} + 134 T^{2} + \cdots - 599 \) Copy content Toggle raw display
$71$ \( T^{4} - 17 T^{3} - 31 T^{2} + \cdots - 2844 \) Copy content Toggle raw display
$73$ \( T^{4} - T^{3} - 139 T^{2} + 154 T + 2836 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} - 125 T^{2} + \cdots + 4220 \) Copy content Toggle raw display
$83$ \( T^{4} + 11 T^{3} - 79 T^{2} + \cdots - 2304 \) Copy content Toggle raw display
$89$ \( T^{4} + 5 T^{3} - 85 T^{2} - 300 T + 900 \) Copy content Toggle raw display
$97$ \( T^{4} + 3 T^{3} - 106 T^{2} + \cdots + 701 \) Copy content Toggle raw display
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