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Properties

Label 5004.2.a.o
Level $5004$
Weight $2$
Character orbit 5004.a
Self dual yes
Analytic conductor $39.957$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5004,2,Mod(1,5004)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5004.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5004, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5004 = 2^{2} \cdot 3^{2} \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5004.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.9571411714\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 49x^{12} + 914x^{10} - 8121x^{8} + 34862x^{6} - 63222x^{4} + 25939x^{2} - 124 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{5} + 1) q^{7} - \beta_{9} q^{11} + (\beta_{8} + \beta_{5}) q^{13} - \beta_{12} q^{17} + (\beta_{10} + \beta_{5} + 1) q^{19} + ( - \beta_{13} + \beta_1) q^{23} + (\beta_{11} - \beta_{8} - \beta_{7} + 2) q^{25}+ \cdots + (\beta_{11} + \beta_{10} - 2 \beta_{8} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 8 q^{7} - 4 q^{13} + 10 q^{19} + 28 q^{25} + 24 q^{31} + 6 q^{37} + 20 q^{43} + 34 q^{49} + 20 q^{55} + 10 q^{61} + 64 q^{67} + 24 q^{73} + 74 q^{79} + 18 q^{85} + 84 q^{91} + 50 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 49x^{12} + 914x^{10} - 8121x^{8} + 34862x^{6} - 63222x^{4} + 25939x^{2} - 124 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14111 \nu^{13} - 333514 \nu^{11} + 32232536 \nu^{9} - 595803969 \nu^{7} + \cdots + 3066717384 \nu ) / 287927468 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 277733 \nu^{13} - 14977789 \nu^{11} + 315543923 \nu^{9} - 3271735060 \nu^{7} + \cdots + 23815932712 \nu ) / 719818670 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1223713 \nu^{13} + 63550554 \nu^{11} - 1272195038 \nu^{9} + 12293603795 \nu^{7} + \cdots - 57725267622 \nu ) / 1439637340 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 489179 \nu^{12} + 21539387 \nu^{10} - 345351049 \nu^{8} + 2457106610 \nu^{6} + \cdots - 4324680466 ) / 719818670 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 263678 \nu^{12} + 12682904 \nu^{10} - 225068693 \nu^{8} + 1792871140 \nu^{6} + \cdots - 2086803072 ) / 359909335 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 266139 \nu^{12} - 11573908 \nu^{10} + 182295806 \nu^{8} - 1252795233 \nu^{6} + 3414316989 \nu^{4} + \cdots + 167290820 ) / 287927468 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 420725 \nu^{12} - 18428564 \nu^{10} + 285962838 \nu^{8} - 1821335587 \nu^{6} + 3717050935 \nu^{4} + \cdots + 332597260 ) / 287927468 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1884608 \nu^{13} - 92590559 \nu^{11} + 1734610748 \nu^{9} - 15519243110 \nu^{7} + \cdots + 46122963767 \nu ) / 719818670 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2686613 \nu^{12} - 118295424 \nu^{10} + 1874701098 \nu^{8} - 12786842735 \nu^{6} + \cdots + 3853486072 ) / 1439637340 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 171716 \nu^{12} - 7500618 \nu^{10} + 117064661 \nu^{8} - 768532705 \nu^{6} + 1782841981 \nu^{4} + \cdots - 378901049 ) / 71981867 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5731731 \nu^{13} - 280659078 \nu^{11} + 5221361046 \nu^{9} - 46103203845 \nu^{7} + \cdots + 127455362354 \nu ) / 1439637340 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5971287 \nu^{13} + 291810446 \nu^{11} - 5432118632 \nu^{9} + 48246303235 \nu^{7} + \cdots - 158620556088 \nu ) / 1439637340 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{8} - \beta_{7} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - \beta_{9} + 2\beta_{4} + \beta_{3} + \beta_{2} + 12\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 18\beta_{11} - 5\beta_{10} - 15\beta_{8} - 20\beta_{7} - 10\beta_{5} + 77 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{13} + 23\beta_{12} - 34\beta_{9} + 41\beta_{4} + 22\beta_{3} + 22\beta_{2} + 163\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 311\beta_{11} - 144\beta_{10} - 226\beta_{8} - 338\beta_{7} - 5\beta_{6} - 244\beta_{5} + 985 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -185\beta_{13} + 428\beta_{12} - 757\beta_{9} + 717\beta_{4} + 365\beta_{3} + 424\beta_{2} + 2387\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5285\beta_{11} - 3069\beta_{10} - 3528\beta_{8} - 5501\beta_{7} - 161\beta_{6} - 4770\beta_{5} + 13714 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -4177\beta_{13} + 7419\beta_{12} - 14665\beta_{9} + 12110\beta_{4} + 5717\beta_{3} + 7748\beta_{2} + 36519\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 88949\beta_{11} - 58371\beta_{10} - 56377\beta_{8} - 88960\beta_{7} - 3322\beta_{6} - 86688\beta_{5} + 201182 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 83344 \beta_{13} + 124854 \beta_{12} - 266569 \beta_{9} + 202893 \beta_{4} + 88971 \beta_{3} + \cdots + 574150 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1488916 \beta_{11} - 1049999 \beta_{10} - 914268 \beta_{8} - 1441799 \beta_{7} - 57725 \beta_{6} + \cdots + 3057389 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1559423 \beta_{13} + 2074172 \beta_{12} - 4683594 \beta_{9} + 3393022 \beta_{4} + \cdots + 9190106 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.07765
−3.47095
−3.21058
−2.23146
−2.11569
−0.746316
−0.0695509
0.0695509
0.746316
2.11569
2.23146
3.21058
3.47095
4.07765
0 0 0 −4.07765 0 −0.612864 0 0 0
1.2 0 0 0 −3.47095 0 4.18423 0 0 0
1.3 0 0 0 −3.21058 0 −0.0193926 0 0 0
1.4 0 0 0 −2.23146 0 4.32595 0 0 0
1.5 0 0 0 −2.11569 0 2.00731 0 0 0
1.6 0 0 0 −0.746316 0 −0.935742 0 0 0
1.7 0 0 0 −0.0695509 0 −4.94949 0 0 0
1.8 0 0 0 0.0695509 0 −4.94949 0 0 0
1.9 0 0 0 0.746316 0 −0.935742 0 0 0
1.10 0 0 0 2.11569 0 2.00731 0 0 0
1.11 0 0 0 2.23146 0 4.32595 0 0 0
1.12 0 0 0 3.21058 0 −0.0193926 0 0 0
1.13 0 0 0 3.47095 0 4.18423 0 0 0
1.14 0 0 0 4.07765 0 −0.612864 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(139\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5004.2.a.o ✓ 14
3.b odd 2 1 inner 5004.2.a.o ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5004.2.a.o ✓ 14 1.a even 1 1 trivial
5004.2.a.o ✓ 14 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5004))\):

\( T_{5}^{14} - 49T_{5}^{12} + 914T_{5}^{10} - 8121T_{5}^{8} + 34862T_{5}^{6} - 63222T_{5}^{4} + 25939T_{5}^{2} - 124 \) Copy content Toggle raw display
\( T_{7}^{7} - 4T_{7}^{6} - 25T_{7}^{5} + 108T_{7}^{4} + 26T_{7}^{3} - 199T_{7}^{2} - 107T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 49 T^{12} + \cdots - 124 \) Copy content Toggle raw display
$7$ \( (T^{7} - 4 T^{6} - 25 T^{5} + \cdots - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} - 122 T^{12} + \cdots - 85115119 \) Copy content Toggle raw display
$13$ \( (T^{7} + 2 T^{6} + \cdots - 16964)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} - 144 T^{12} + \cdots - 2864896 \) Copy content Toggle raw display
$19$ \( (T^{7} - 5 T^{6} + \cdots - 3344)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} - 152 T^{12} + \cdots - 126976 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 15245024256 \) Copy content Toggle raw display
$31$ \( (T^{7} - 12 T^{6} + \cdots + 2366)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} - 3 T^{6} + \cdots + 20258)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} - 329 T^{12} + \cdots - 17530624 \) Copy content Toggle raw display
$43$ \( (T^{7} - 10 T^{6} + \cdots - 87552)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 6196374364 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 4298899456 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 1708589056 \) Copy content Toggle raw display
$61$ \( (T^{7} - 5 T^{6} + \cdots + 2395008)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} - 32 T^{6} + \cdots + 461092)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 1425200852736 \) Copy content Toggle raw display
$73$ \( (T^{7} - 12 T^{6} + \cdots + 1491200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} - 37 T^{6} + \cdots + 1906208)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 590379376 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 684988749218844 \) Copy content Toggle raw display
$97$ \( (T^{7} - 25 T^{6} + \cdots - 7648)^{2} \) Copy content Toggle raw display
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