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Properties

Label 5408.2.a.br
Level $5408$
Weight $2$
Character orbit 5408.a
Self dual yes
Analytic conductor $43.183$
Analytic rank $0$
Dimension $9$
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] = [5408,2,Mod(1,5408)] 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("5408.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5408 = 2^{5} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5408.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,7,0,-6,0,0,0,10,0,1,0,0,0,-18,0,-7,0,5,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(43.1830974131\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 14x^{7} + 23x^{6} + 63x^{5} - 85x^{4} - 99x^{3} + 98x^{2} + 35x - 7 \) 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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{3} - 1) q^{5} + ( - \beta_{6} + \beta_{4}) q^{7} + ( - \beta_{7} + \beta_{5} + \beta_{2} + \cdots + 2) q^{9} + ( - \beta_{7} - \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{11}+ \cdots + (4 \beta_{8} - \beta_{7} - 5 \beta_{6} + \cdots + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 7 q^{3} - 6 q^{5} + 10 q^{9} + q^{11} - 18 q^{15} - 7 q^{17} + 5 q^{19} - 12 q^{21} + 12 q^{23} + 11 q^{25} + 34 q^{27} - 8 q^{29} - 20 q^{31} + 12 q^{33} + 6 q^{35} - 6 q^{37} + 31 q^{41} + 33 q^{43}+ \cdots + 47 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 14x^{7} + 23x^{6} + 63x^{5} - 85x^{4} - 99x^{3} + 98x^{2} + 35x - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\nu^{8} + 27\nu^{7} - 420\nu^{6} - 281\nu^{5} + 1716\nu^{4} + 294\nu^{3} - 1657\nu^{2} + 891\nu - 341 ) / 547 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -78\nu^{8} - 128\nu^{7} + 1201\nu^{6} + 2102\nu^{5} - 4610\nu^{4} - 8444\nu^{3} + 4229\nu^{2} + 8904\nu + 320 ) / 547 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -94\nu^{8} - 14\nu^{7} + 1251\nu^{6} + 794\nu^{5} - 4111\nu^{4} - 3556\nu^{3} + 2095\nu^{2} + 2820\nu + 1676 ) / 547 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 99\nu^{8} - 90\nu^{7} - 1335\nu^{6} + 572\nu^{5} + 5220\nu^{4} - 980\nu^{3} - 5599\nu^{2} + 312\nu - 869 ) / 547 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -108\nu^{8} - 51\nu^{7} + 1705\nu^{6} + 1017\nu^{5} - 7435\nu^{4} - 3655\nu^{3} + 9390\nu^{2} + 2146\nu - 693 ) / 547 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 124\nu^{8} - 63\nu^{7} - 1755\nu^{6} + 291\nu^{5} + 6936\nu^{4} - 686\nu^{3} - 7803\nu^{2} + 1203\nu + 978 ) / 547 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -185\nu^{8} + 19\nu^{7} + 2561\nu^{6} + 876\nu^{5} - 9854\nu^{4} - 4473\nu^{3} + 10949\nu^{2} + 3362\nu - 868 ) / 547 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{5} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{8} - 8\beta_{7} + 2\beta_{6} + 8\beta_{5} + 3\beta_{4} - \beta_{3} + 10\beta_{2} + 2\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{8} - 5\beta_{7} + 13\beta_{6} + 14\beta_{5} + 13\beta_{4} - 10\beta_{3} + 21\beta_{2} + 32\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -26\beta_{8} - 65\beta_{7} + 34\beta_{6} + 69\beta_{5} + 40\beta_{4} - 17\beta_{3} + 101\beta_{2} + 36\beta _1 + 178 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 53 \beta_{8} - 92 \beta_{7} + 141 \beta_{6} + 162 \beta_{5} + 143 \beta_{4} - 95 \beta_{3} + \cdots + 304 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 276 \beta_{8} - 566 \beta_{7} + 416 \beta_{6} + 647 \beta_{5} + 446 \beta_{4} - 215 \beta_{3} + \cdots + 1529 \) 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
3.22467
1.96612
1.89279
1.21063
0.148538
−0.425773
−1.48637
−2.18831
−2.34228
0 −2.22467 0 1.49069 0 1.90209 0 1.94914 0
1.2 0 −0.966118 0 −1.52445 0 0.857710 0 −2.06662 0
1.3 0 −0.892791 0 2.29689 0 0.141905 0 −2.20292 0
1.4 0 −0.210627 0 −2.48480 0 −3.64637 0 −2.95564 0
1.5 0 0.851462 0 −4.11908 0 4.22319 0 −2.27501 0
1.6 0 1.42577 0 3.06984 0 1.57496 0 −0.967171 0
1.7 0 2.48637 0 −0.177816 0 −4.36509 0 3.18204 0
1.8 0 3.18831 0 −1.00589 0 1.74428 0 7.16533 0
1.9 0 3.34228 0 −3.54539 0 −2.43267 0 8.17085 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

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

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 5408.2.a.br yes 9
4.b odd 2 1 5408.2.a.bp ✓ 9
13.b even 2 1 5408.2.a.bs yes 9
52.b odd 2 1 5408.2.a.bq yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5408.2.a.bp ✓ 9 4.b odd 2 1
5408.2.a.bq yes 9 52.b odd 2 1
5408.2.a.br yes 9 1.a even 1 1 trivial
5408.2.a.bs yes 9 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5408))\):

\( T_{3}^{9} - 7T_{3}^{8} + 6T_{3}^{7} + 47T_{3}^{6} - 79T_{3}^{5} - 71T_{3}^{4} + 133T_{3}^{3} + 48T_{3}^{2} - 58T_{3} - 13 \) Copy content Toggle raw display
\( T_{5}^{9} + 6T_{5}^{8} - 10T_{5}^{7} - 102T_{5}^{6} - 31T_{5}^{5} + 492T_{5}^{4} + 425T_{5}^{3} - 610T_{5}^{2} - 704T_{5} - 104 \) Copy content Toggle raw display
\( T_{7}^{9} - 33T_{7}^{7} + 18T_{7}^{6} + 314T_{7}^{5} - 410T_{7}^{4} - 761T_{7}^{3} + 1744T_{7}^{2} - 964T_{7} + 104 \) Copy content Toggle raw display
\( T_{37}^{9} + 6 T_{37}^{8} - 178 T_{37}^{7} - 950 T_{37}^{6} + 9233 T_{37}^{5} + 38340 T_{37}^{4} + \cdots + 75608 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 7 T^{8} + \cdots - 13 \) Copy content Toggle raw display
$5$ \( T^{9} + 6 T^{8} + \cdots - 104 \) Copy content Toggle raw display
$7$ \( T^{9} - 33 T^{7} + \cdots + 104 \) Copy content Toggle raw display
$11$ \( T^{9} - T^{8} + \cdots - 71 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 7 T^{8} + \cdots + 122473 \) Copy content Toggle raw display
$19$ \( T^{9} - 5 T^{8} + \cdots + 727 \) Copy content Toggle raw display
$23$ \( T^{9} - 12 T^{8} + \cdots - 66248 \) Copy content Toggle raw display
$29$ \( T^{9} + 8 T^{8} + \cdots - 109096 \) Copy content Toggle raw display
$31$ \( T^{9} + 20 T^{8} + \cdots + 4525144 \) Copy content Toggle raw display
$37$ \( T^{9} + 6 T^{8} + \cdots + 75608 \) Copy content Toggle raw display
$41$ \( T^{9} - 31 T^{8} + \cdots + 4774393 \) Copy content Toggle raw display
$43$ \( T^{9} - 33 T^{8} + \cdots - 1403857 \) Copy content Toggle raw display
$47$ \( T^{9} - 14 T^{8} + \cdots - 729352 \) Copy content Toggle raw display
$53$ \( T^{9} - 14 T^{8} + \cdots + 59088952 \) Copy content Toggle raw display
$59$ \( T^{9} + 9 T^{8} + \cdots - 1368737 \) Copy content Toggle raw display
$61$ \( T^{9} - 8 T^{8} + \cdots - 627136 \) Copy content Toggle raw display
$67$ \( T^{9} + 3 T^{8} + \cdots - 636721 \) Copy content Toggle raw display
$71$ \( T^{9} - 26 T^{8} + \cdots - 4797304 \) Copy content Toggle raw display
$73$ \( T^{9} - 5 T^{8} + \cdots - 143437489 \) Copy content Toggle raw display
$79$ \( T^{9} - 44 T^{8} + \cdots - 19152328 \) Copy content Toggle raw display
$83$ \( T^{9} + 17 T^{8} + \cdots + 21802661 \) Copy content Toggle raw display
$89$ \( T^{9} - 41 T^{8} + \cdots + 4925843 \) Copy content Toggle raw display
$97$ \( T^{9} - 15 T^{8} + \cdots + 156600977 \) Copy content Toggle raw display
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