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Properties

Label 5290.2.a.bg
Level $5290$
Weight $2$
Character orbit 5290.a
Self dual yes
Analytic conductor $42.241$
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] = [5290,2,Mod(1,5290)] 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("5290.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5290, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-10,-5,10,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(42.2408626693\)
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: \( x^{10} - x^{9} - 16x^{8} + 6x^{7} + 85x^{6} + 9x^{5} - 149x^{4} - 20x^{3} + 103x^{2} + 6x - 23 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta_{7} - 1) q^{3} + q^{4} - q^{5} + (\beta_{7} + 1) q^{6} + (\beta_{8} - \beta_{6} - \beta_{3} + 1) q^{7} - q^{8} + ( - \beta_{9} + \beta_{7} + \cdots + 2 \beta_{4}) q^{9} + q^{10}+ \cdots + ( - 4 \beta_{9} + 3 \beta_{8} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} - 5 q^{3} + 10 q^{4} - 10 q^{5} + 5 q^{6} + 5 q^{7} - 10 q^{8} + 3 q^{9} + 10 q^{10} + 4 q^{11} - 5 q^{12} + 2 q^{13} - 5 q^{14} + 5 q^{15} + 10 q^{16} - 4 q^{17} - 3 q^{18} + 13 q^{19}+ \cdots + 62 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 16x^{8} + 6x^{7} + 85x^{6} + 9x^{5} - 149x^{4} - 20x^{3} + 103x^{2} + 6x - 23 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} + 18\nu^{8} - 14\nu^{7} - 228\nu^{6} + 134\nu^{5} + 910\nu^{4} - 3\nu^{3} - 872\nu^{2} - 68\nu + 230 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{9} + 29\nu^{8} + 72\nu^{7} - 283\nu^{6} - 170\nu^{5} + 771\nu^{4} + 45\nu^{3} - 720\nu^{2} - 15\nu + 230 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10 \nu^{9} + 19 \nu^{8} + 136 \nu^{7} - 164 \nu^{6} - 615 \nu^{5} + 291 \nu^{4} + 844 \nu^{3} + \cdots + 46 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13 \nu^{9} + 27 \nu^{8} + 163 \nu^{7} - 227 \nu^{6} - 650 \nu^{5} + 399 \nu^{4} + 628 \nu^{3} + \cdots + 115 ) / 23 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7 \nu^{9} + 12 \nu^{8} - 132 \nu^{7} - 244 \nu^{6} + 695 \nu^{5} + 1427 \nu^{4} - 715 \nu^{3} + \cdots + 598 ) / 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7 \nu^{9} - 35 \nu^{8} + 178 \nu^{7} + 520 \nu^{6} - 1040 \nu^{5} - 2439 \nu^{4} + 1060 \nu^{3} + \cdots - 644 ) / 23 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31 \nu^{9} + 29 \nu^{8} + 463 \nu^{7} - 99 \nu^{6} - 2171 \nu^{5} - 839 \nu^{4} + 2529 \nu^{3} + \cdots - 322 ) / 23 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25 \nu^{9} + 13 \nu^{8} + 386 \nu^{7} + 73 \nu^{6} - 1825 \nu^{5} - 1400 \nu^{4} + 1949 \nu^{3} + \cdots - 575 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 9\beta_{8} - 8\beta_{7} - 9\beta_{5} - 8\beta_{4} + \beta_{3} - 9\beta_{2} + 4\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{9} + 12 \beta_{8} - 13 \beta_{7} + 9 \beta_{6} - 30 \beta_{5} - 11 \beta_{4} + 9 \beta_{3} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8 \beta_{9} + 74 \beta_{8} - 65 \beta_{7} + 2 \beta_{6} - 82 \beta_{5} - 60 \beta_{4} + 15 \beta_{3} + \cdots + 134 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 62 \beta_{9} + 129 \beta_{8} - 137 \beta_{7} + 68 \beta_{6} - 274 \beta_{5} - 101 \beta_{4} + 79 \beta_{3} + \cdots + 274 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 48 \beta_{9} + 614 \beta_{8} - 552 \beta_{7} + 39 \beta_{6} - 760 \beta_{5} - 449 \beta_{4} + \cdots + 1074 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 385 \beta_{9} + 1297 \beta_{8} - 1345 \beta_{7} + 497 \beta_{6} - 2477 \beta_{5} - 867 \beta_{4} + \cdots + 2550 \) 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
−0.646593
−1.72370
0.859877
−1.04270
0.686792
−2.36930
2.96169
−1.69071
3.03342
0.931223
−1.00000 −3.27170 1.00000 −1.00000 3.27170 −0.278370 −1.00000 7.70402 1.00000
1.2 −1.00000 −2.31608 1.00000 −1.00000 2.31608 1.26741 −1.00000 2.36423 1.00000
1.3 −1.00000 −2.03496 1.00000 −1.00000 2.03496 5.05411 −1.00000 1.14107 1.00000
1.4 −1.00000 −1.54336 1.00000 −1.00000 1.54336 −3.57936 −1.00000 −0.618038 1.00000
1.5 −1.00000 −1.40820 1.00000 −1.00000 1.40820 −4.28047 −1.00000 −1.01699 1.00000
1.6 −1.00000 0.408195 1.00000 −1.00000 −0.408195 2.53066 −1.00000 −2.83338 1.00000
1.7 −1.00000 0.543361 1.00000 −1.00000 −0.543361 2.61223 −1.00000 −2.70476 1.00000
1.8 −1.00000 1.03496 1.00000 −1.00000 −1.03496 −1.06188 −1.00000 −1.92886 1.00000
1.9 −1.00000 1.31608 1.00000 −1.00000 −1.31608 0.848046 −1.00000 −1.26793 1.00000
1.10 −1.00000 2.27170 1.00000 −1.00000 −2.27170 1.88763 −1.00000 2.16062 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(23\) \( +1 \)

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 5290.2.a.bg 10
23.b odd 2 1 5290.2.a.bh 10
23.c even 11 2 230.2.g.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.g.c ✓ 20 23.c even 11 2
5290.2.a.bg 10 1.a even 1 1 trivial
5290.2.a.bh 10 23.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5290))\):

\( T_{3}^{10} + 5T_{3}^{9} - 4T_{3}^{8} - 46T_{3}^{7} - 16T_{3}^{6} + 134T_{3}^{5} + 58T_{3}^{4} - 171T_{3}^{3} - 28T_{3}^{2} + 89T_{3} - 23 \) Copy content Toggle raw display
\( T_{7}^{10} - 5 T_{7}^{9} - 26 T_{7}^{8} + 156 T_{7}^{7} + 50 T_{7}^{6} - 1256 T_{7}^{5} + 1642 T_{7}^{4} + \cdots + 307 \) Copy content Toggle raw display
\( T_{11}^{10} - 4 T_{11}^{9} - 46 T_{11}^{8} + 119 T_{11}^{7} + 686 T_{11}^{6} - 269 T_{11}^{5} + \cdots + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 5 T^{9} + \cdots - 23 \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 5 T^{9} + \cdots + 307 \) Copy content Toggle raw display
$11$ \( T^{10} - 4 T^{9} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{10} - 2 T^{9} + \cdots + 736 \) Copy content Toggle raw display
$17$ \( T^{10} + 4 T^{9} + \cdots + 23264 \) Copy content Toggle raw display
$19$ \( T^{10} - 13 T^{9} + \cdots + 313312 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} + 25 T^{9} + \cdots + 648163 \) Copy content Toggle raw display
$31$ \( T^{10} + 9 T^{9} + \cdots - 389344 \) Copy content Toggle raw display
$37$ \( T^{10} + 7 T^{9} + \cdots - 9793376 \) Copy content Toggle raw display
$41$ \( T^{10} + 12 T^{9} + \cdots - 2016893 \) Copy content Toggle raw display
$43$ \( T^{10} + 2 T^{9} + \cdots - 52999 \) Copy content Toggle raw display
$47$ \( T^{10} + 13 T^{9} + \cdots + 590921 \) Copy content Toggle raw display
$53$ \( T^{10} - 5 T^{9} + \cdots - 31266784 \) Copy content Toggle raw display
$59$ \( T^{10} + 24 T^{9} + \cdots + 35409056 \) Copy content Toggle raw display
$61$ \( T^{10} - 24 T^{9} + \cdots + 10279457 \) Copy content Toggle raw display
$67$ \( T^{10} + 2 T^{9} + \cdots - 612569 \) Copy content Toggle raw display
$71$ \( T^{10} + 18 T^{9} + \cdots + 81046624 \) Copy content Toggle raw display
$73$ \( T^{10} - 12 T^{9} + \cdots - 33056 \) Copy content Toggle raw display
$79$ \( T^{10} + 8 T^{9} + \cdots - 233056 \) Copy content Toggle raw display
$83$ \( T^{10} + 5 T^{9} + \cdots + 23338811 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 147674647 \) Copy content Toggle raw display
$97$ \( T^{10} + 20 T^{9} + \cdots + 486496 \) Copy content Toggle raw display
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