Properties

Label 10000.2.a.w
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) 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 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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2}) q^{3} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{7} + (\beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2}) q^{3} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{7} + (\beta_1 - 1) q^{9} + ( - 2 \beta_{2} + 1) q^{11} + (4 \beta_{3} + 2 \beta_{2}) q^{13} + ( - 3 \beta_{3} - \beta_1 - 3) q^{17} + ( - 4 \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{19} + (2 \beta_1 + 4) q^{21} + (4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{23} + (2 \beta_{3} + 3 \beta_{2} - 1) q^{27} + (6 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 6) q^{29} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{31} + (\beta_{3} - \beta_{2} + 4) q^{33} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{37} + (2 \beta_{3} - 4 \beta_1 - 4) q^{39} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 - 1) q^{41} + ( - 5 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{43} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{47} + (4 \beta_1 + 1) q^{49} + (2 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 1) q^{51} + (4 \beta_{3} - 4 \beta_1 + 4) q^{53} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{57} + ( - 3 \beta_{3} + 2 \beta_{2} + 3) q^{59} + ( - 6 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{61} + ( - 2 \beta_{3} - 2) q^{63} + (3 \beta_{2} - 2 \beta_1 - 6) q^{67} + (6 \beta_{3} - 4 \beta_1) q^{69} + (6 \beta_{3} - 4 \beta_1 + 2) q^{71} + ( - 4 \beta_{2} + 3 \beta_1) q^{73} + (2 \beta_{3} - 2 \beta_{2} + 8) q^{77} + (4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{79} + (\beta_{2} - 5 \beta_1 - 3) q^{81} + (4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 3) q^{83} + (6 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{87} + (8 \beta_{3} + 5 \beta_{2} + 4) q^{89} + (4 \beta_{3} - 8 \beta_1 - 8) q^{91} + ( - 2 \beta_{3} - 2 \beta_1 - 6) q^{93} + ( - 9 \beta_{3} - 5) q^{97} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + 2 q^{7} - 3 q^{9} + 2 q^{11} - 6 q^{13} - 7 q^{17} - 15 q^{19} + 18 q^{21} + 6 q^{23} - 5 q^{27} + 10 q^{29} - 8 q^{31} + 13 q^{33} - 12 q^{37} - 24 q^{39} + 3 q^{41} - 14 q^{43} + 2 q^{47} + 8 q^{49} + 7 q^{51} + 4 q^{53} + 10 q^{57} + 20 q^{59} + 18 q^{61} - 4 q^{63} - 23 q^{67} - 16 q^{69} - 8 q^{71} - q^{73} + 26 q^{77} - 10 q^{79} - 16 q^{81} - 14 q^{83} - 20 q^{87} + 5 q^{89} - 48 q^{91} - 22 q^{93} - 2 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{15} + \zeta_{15}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
1.82709
−1.95630
−0.209057
1.33826
0 −1.95630 0 0 0 −3.91259 0 0.827091 0
1.2 0 −0.209057 0 0 0 −0.418114 0 −2.95630 0
1.3 0 1.33826 0 0 0 2.67652 0 −1.20906 0
1.4 0 1.82709 0 0 0 3.65418 0 0.338261 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.w 4
4.b odd 2 1 1250.2.a.e 4
5.b even 2 1 10000.2.a.s 4
20.d odd 2 1 1250.2.a.k yes 4
20.e even 4 2 1250.2.b.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1250.2.a.e 4 4.b odd 2 1
1250.2.a.k yes 4 20.d odd 2 1
1250.2.b.f 8 20.e even 4 2
10000.2.a.s 4 5.b even 2 1
10000.2.a.w 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3}^{4} - T_{3}^{3} - 4T_{3}^{2} + 4T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} - 16T_{7}^{2} + 32T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 16T_{11}^{2} + 2T_{11} + 31 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 4 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} - 16 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} - 16 T^{2} + 2 T + 31 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} - 24 T^{2} - 144 T - 144 \) Copy content Toggle raw display
$17$ \( T^{4} + 7 T^{3} - 16 T^{2} - 112 T + 61 \) Copy content Toggle raw display
$19$ \( T^{4} + 15 T^{3} + 50 T^{2} + \cdots - 755 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} - 64 T^{2} + 384 T - 464 \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} - 60 T^{2} + \cdots - 2480 \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} - 16 T^{2} - 248 T - 464 \) Copy content Toggle raw display
$37$ \( T^{4} + 12 T^{3} - 16 T^{2} + \cdots - 1424 \) Copy content Toggle raw display
$41$ \( T^{4} - 3 T^{3} - 76 T^{2} + 408 T - 359 \) Copy content Toggle raw display
$43$ \( T^{4} + 14 T^{3} + 21 T^{2} + \cdots - 1289 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} - 76 T^{2} - 88 T + 16 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} - 64 T^{2} + 256 T + 256 \) Copy content Toggle raw display
$59$ \( T^{4} - 20 T^{3} + 95 T^{2} + \cdots - 905 \) Copy content Toggle raw display
$61$ \( T^{4} - 18 T^{3} - 76 T^{2} + \cdots - 9584 \) Copy content Toggle raw display
$67$ \( T^{4} + 23 T^{3} + 134 T^{2} + \cdots - 2039 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} - 76 T^{2} - 128 T + 16 \) Copy content Toggle raw display
$73$ \( T^{4} + T^{3} - 124 T^{2} + 476 T - 59 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} - 60 T^{2} + \cdots - 2480 \) Copy content Toggle raw display
$83$ \( T^{4} + 14 T^{3} - 64 T^{2} + \cdots - 3089 \) Copy content Toggle raw display
$89$ \( T^{4} - 5 T^{3} - 160 T^{2} - 100 T + 25 \) Copy content Toggle raw display
$97$ \( (T^{2} + T - 101)^{2} \) Copy content Toggle raw display
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