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Properties

Label 5096.2.a.bf
Level $5096$
Weight $2$
Character orbit 5096.a
Self dual yes
Analytic conductor $40.692$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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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] = [5096,2,Mod(1,5096)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5096, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5096.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5096 = 2^{3} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5096.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-2,0,-6,0,0,0,12,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.6917648700\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 19x^{8} + 36x^{7} + 110x^{6} - 188x^{5} - 218x^{4} + 276x^{3} + 154x^{2} - 48x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{9} - 1) q^{5} + ( - \beta_{9} + \beta_{8} + 2) q^{9} + (\beta_{6} - \beta_{2} + \beta_1) q^{11} - q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{15} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 2) q^{17}+ \cdots + (\beta_{9} - \beta_{8} + \beta_{7} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{3} - 6 q^{5} + 12 q^{9} - 2 q^{11} - 10 q^{13} + 4 q^{15} - 16 q^{17} - 14 q^{19} + 8 q^{23} + 12 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} - 10 q^{33} - 6 q^{37} + 2 q^{39} - 22 q^{41} + 2 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 19x^{8} + 36x^{7} + 110x^{6} - 188x^{5} - 218x^{4} + 276x^{3} + 154x^{2} - 48x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 549 \nu^{9} + 5443 \nu^{8} - 12499 \nu^{7} - 101067 \nu^{6} + 83097 \nu^{5} + 544211 \nu^{4} + \cdots + 63124 ) / 46445 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 577 \nu^{9} + 576 \nu^{8} + 11517 \nu^{7} - 9764 \nu^{6} - 72216 \nu^{5} + 47312 \nu^{4} + \cdots + 12068 ) / 13270 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 289 \nu^{9} + 277 \nu^{8} + 5504 \nu^{7} - 4373 \nu^{6} - 30582 \nu^{5} + 17039 \nu^{4} + \cdots - 1154 ) / 6635 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5017 \nu^{9} + 9056 \nu^{8} + 100347 \nu^{7} - 160884 \nu^{6} - 644406 \nu^{5} + 804302 \nu^{4} + \cdots - 307152 ) / 92890 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5827 \nu^{9} + 17776 \nu^{8} + 112697 \nu^{7} - 331554 \nu^{6} - 689346 \nu^{5} + 1828712 \nu^{4} + \cdots + 415928 ) / 92890 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9873 \nu^{9} - 21654 \nu^{8} - 189753 \nu^{7} + 392776 \nu^{6} + 1117494 \nu^{5} - 2067258 \nu^{4} + \cdots - 399772 ) / 92890 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10443 \nu^{9} + 15749 \nu^{8} + 191563 \nu^{7} - 270331 \nu^{6} - 1014944 \nu^{5} + 1291578 \nu^{4} + \cdots - 151028 ) / 92890 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10443 \nu^{9} + 15749 \nu^{8} + 191563 \nu^{7} - 270331 \nu^{6} - 1014944 \nu^{5} + 1291578 \nu^{4} + \cdots + 313422 ) / 92890 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{8} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{4} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{9} + 11\beta_{8} - 3\beta_{5} - \beta_{4} + 5\beta_{3} + \beta_{2} - \beta _1 + 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{9} + 2\beta_{8} + 10\beta_{7} + 12\beta_{6} - 3\beta_{5} + 17\beta_{4} - \beta_{3} + 69\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 111 \beta_{9} + 111 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 42 \beta_{5} - 11 \beta_{4} + 81 \beta_{3} + \cdots + 343 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 75 \beta_{9} + 33 \beta_{8} + 90 \beta_{7} + 124 \beta_{6} - 49 \beta_{5} + 218 \beta_{4} - 7 \beta_{3} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1121 \beta_{9} + 1115 \beta_{8} + 30 \beta_{7} - 95 \beta_{6} - 484 \beta_{5} - 95 \beta_{4} + \cdots + 3217 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1027 \beta_{9} + 433 \beta_{8} + 818 \beta_{7} + 1240 \beta_{6} - 621 \beta_{5} + 2510 \beta_{4} + \cdots + 405 \) 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
3.24723
2.53124
1.97465
1.67486
0.291109
−0.0697696
−0.690495
−1.36379
−2.43321
−3.16182
0 −3.24723 0 −2.31239 0 0 0 7.54451 0
1.2 0 −2.53124 0 0.175881 0 0 0 3.40715 0
1.3 0 −1.97465 0 −4.40355 0 0 0 0.899254 0
1.4 0 −1.67486 0 1.96101 0 0 0 −0.194860 0
1.5 0 −0.291109 0 0.189897 0 0 0 −2.91526 0
1.6 0 0.0697696 0 2.59235 0 0 0 −2.99513 0
1.7 0 0.690495 0 −3.44303 0 0 0 −2.52322 0
1.8 0 1.36379 0 2.86444 0 0 0 −1.14008 0
1.9 0 2.43321 0 −1.82666 0 0 0 2.92050 0
1.10 0 3.16182 0 −1.79794 0 0 0 6.99712 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( +1 \)
\(13\) \( +1 \)

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 5096.2.a.bf ✓ 10
7.b odd 2 1 5096.2.a.bg yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5096.2.a.bf ✓ 10 1.a even 1 1 trivial
5096.2.a.bg yes 10 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5096))\):

\( T_{3}^{10} + 2 T_{3}^{9} - 19 T_{3}^{8} - 36 T_{3}^{7} + 110 T_{3}^{6} + 188 T_{3}^{5} - 218 T_{3}^{4} + \cdots - 4 \) Copy content Toggle raw display
\( T_{5}^{10} + 6 T_{5}^{9} - 13 T_{5}^{8} - 116 T_{5}^{7} + 7 T_{5}^{6} + 730 T_{5}^{5} + 413 T_{5}^{4} + \cdots - 56 \) Copy content Toggle raw display
\( T_{11}^{10} + 2 T_{11}^{9} - 59 T_{11}^{8} - 112 T_{11}^{7} + 1204 T_{11}^{6} + 2220 T_{11}^{5} + \cdots + 21176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 2 T^{9} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{10} + 6 T^{9} + \cdots - 56 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 2 T^{9} + \cdots + 21176 \) Copy content Toggle raw display
$13$ \( (T + 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} + 16 T^{9} + \cdots + 28672 \) Copy content Toggle raw display
$19$ \( T^{10} + 14 T^{9} + \cdots + 21724 \) Copy content Toggle raw display
$23$ \( T^{10} - 8 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{10} - 2 T^{9} + \cdots - 1224 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + \cdots - 3675784 \) Copy content Toggle raw display
$37$ \( T^{10} + 6 T^{9} + \cdots + 6436024 \) Copy content Toggle raw display
$41$ \( T^{10} + 22 T^{9} + \cdots - 2873824 \) Copy content Toggle raw display
$43$ \( T^{10} - 2 T^{9} + \cdots + 3344072 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 376452664 \) Copy content Toggle raw display
$53$ \( T^{10} + 6 T^{9} + \cdots - 22806328 \) Copy content Toggle raw display
$59$ \( T^{10} + 24 T^{9} + \cdots + 3186800 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 570654688 \) Copy content Toggle raw display
$67$ \( T^{10} + 14 T^{9} + \cdots - 175616 \) Copy content Toggle raw display
$71$ \( T^{10} + 8 T^{9} + \cdots - 4235200 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 136816274 \) Copy content Toggle raw display
$79$ \( T^{10} + 16 T^{9} + \cdots + 34111368 \) Copy content Toggle raw display
$83$ \( T^{10} + 2 T^{9} + \cdots - 764316 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 4151864304 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 1459226594 \) Copy content Toggle raw display
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