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Properties

Label 5415.2.a.bk
Level $5415$
Weight $2$
Character orbit 5415.a
Self dual yes
Analytic conductor $43.239$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5415,2,Mod(1,5415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5415, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(43.2389926945\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 9x^{7} + 30x^{6} + 18x^{5} - 87x^{4} + 13x^{3} + 60x^{2} - 12x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{7} + \beta_{6} + \beta_{3} + 1) q^{4} - q^{5} - \beta_1 q^{6} + ( - \beta_{8} + \beta_1 - 1) q^{7} + ( - \beta_{8} - \beta_{7} + \beta_{6} + \cdots - 1) q^{8} + q^{9}+ \cdots + ( - \beta_{7} - \beta_{6} - 2 \beta_{4} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} + 9 q^{3} + 9 q^{4} - 9 q^{5} - 3 q^{6} - 6 q^{7} - 6 q^{8} + 9 q^{9} + 3 q^{10} + 9 q^{12} + 15 q^{13} - 12 q^{14} - 9 q^{15} + 9 q^{16} - 18 q^{17} - 3 q^{18} - 9 q^{20} - 6 q^{21} + 3 q^{22}+ \cdots - 21 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 9x^{7} + 30x^{6} + 18x^{5} - 87x^{4} + 13x^{3} + 60x^{2} - 12x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} - 13\nu^{5} + 12\nu^{4} + 44\nu^{3} - 35\nu^{2} - 25\nu + 8 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + \nu^{7} + 13\nu^{6} - 12\nu^{5} - 44\nu^{4} + 35\nu^{3} + 25\nu^{2} - 8\nu - 6 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} + 5\nu^{7} + 3\nu^{6} - 40\nu^{5} + 22\nu^{4} + 79\nu^{3} - 79\nu^{2} - 6\nu + 32 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} + \nu^{7} - 21\nu^{6} - 2\nu^{5} + 110\nu^{4} - 7\nu^{3} - 167\nu^{2} - 12\nu + 52 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{8} + 5\nu^{7} + 31\nu^{6} - 38\nu^{5} - 102\nu^{4} + 73\nu^{3} + 113\nu^{2} - 20\nu - 32 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{8} - 7\nu^{7} - 57\nu^{6} + 62\nu^{5} + 190\nu^{4} - 143\nu^{3} - 151\nu^{2} + 36\nu + 8 ) / 12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{8} + 13\nu^{7} + 79\nu^{6} - 126\nu^{5} - 266\nu^{4} + 341\nu^{3} + 241\nu^{2} - 176\nu - 60 ) / 12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - 2\beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 7\beta_{7} + 4\beta_{6} + \beta_{5} + \beta_{4} + 8\beta_{3} - 3\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{8} + 11\beta_{7} - 7\beta_{6} + 7\beta_{5} + 3\beta_{4} + 12\beta_{3} - 19\beta_{2} + 18\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{8} + 49\beta_{7} + 12\beta_{6} + 9\beta_{5} + 13\beta_{4} + 57\beta_{3} - 37\beta_{2} + \beta _1 + 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 61\beta_{8} + 99\beta_{7} - 48\beta_{6} + 44\beta_{5} + 40\beta_{4} + 108\beta_{3} - 154\beta_{2} + 84\beta _1 + 127 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 125 \beta_{8} + 356 \beta_{7} + 6 \beta_{6} + 68 \beta_{5} + 129 \beta_{4} + 407 \beta_{3} - 345 \beta_{2} + \cdots + 632 \) 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.

comment: embeddings in the coefficient field
 
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
2.76659
2.22034
1.61008
1.23207
0.579596
−0.302176
−0.814097
−2.10511
−2.18729
−2.76659 1.00000 5.65402 −1.00000 −2.76659 −1.54612 −10.1092 1.00000 2.76659
1.2 −2.22034 1.00000 2.92989 −1.00000 −2.22034 1.64839 −2.06467 1.00000 2.22034
1.3 −1.61008 1.00000 0.592361 −1.00000 −1.61008 2.39880 2.26641 1.00000 1.61008
1.4 −1.23207 1.00000 −0.481999 −1.00000 −1.23207 −4.04135 3.05800 1.00000 1.23207
1.5 −0.579596 1.00000 −1.66407 −1.00000 −0.579596 3.72320 2.12368 1.00000 0.579596
1.6 0.302176 1.00000 −1.90869 −1.00000 0.302176 −1.63020 −1.18111 1.00000 −0.302176
1.7 0.814097 1.00000 −1.33725 −1.00000 0.814097 −2.29770 −2.71684 1.00000 −0.814097
1.8 2.10511 1.00000 2.43149 −1.00000 2.10511 −1.13913 0.908342 1.00000 −2.10511
1.9 2.18729 1.00000 2.78424 −1.00000 2.18729 −3.11590 1.71536 1.00000 −2.18729
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(19\) \( -1 \)

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 5415.2.a.bk 9
19.b odd 2 1 5415.2.a.bl 9
19.f odd 18 2 285.2.u.b ✓ 18
57.j even 18 2 855.2.bs.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.u.b ✓ 18 19.f odd 18 2
855.2.bs.d 18 57.j even 18 2
5415.2.a.bk 9 1.a even 1 1 trivial
5415.2.a.bl 9 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):

\( T_{2}^{9} + 3T_{2}^{8} - 9T_{2}^{7} - 30T_{2}^{6} + 18T_{2}^{5} + 87T_{2}^{4} + 13T_{2}^{3} - 60T_{2}^{2} - 12T_{2} + 8 \) Copy content Toggle raw display
\( T_{7}^{9} + 6T_{7}^{8} - 12T_{7}^{7} - 128T_{7}^{6} - 99T_{7}^{5} + 669T_{7}^{4} + 1170T_{7}^{3} - 543T_{7}^{2} - 2262T_{7} - 1223 \) Copy content Toggle raw display
\( T_{11}^{9} - 57T_{11}^{7} - 55T_{11}^{6} + 906T_{11}^{5} + 1557T_{11}^{4} - 3185T_{11}^{3} - 8574T_{11}^{2} - 6168T_{11} - 1432 \) Copy content Toggle raw display
\( T_{13}^{9} - 15 T_{13}^{8} + 60 T_{13}^{7} + 130 T_{13}^{6} - 1641 T_{13}^{5} + 4827 T_{13}^{4} + \cdots - 449 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 3 T^{8} + \cdots + 8 \) Copy content Toggle raw display
$3$ \( (T - 1)^{9} \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 6 T^{8} + \cdots - 1223 \) Copy content Toggle raw display
$11$ \( T^{9} - 57 T^{7} + \cdots - 1432 \) Copy content Toggle raw display
$13$ \( T^{9} - 15 T^{8} + \cdots - 449 \) Copy content Toggle raw display
$17$ \( T^{9} + 18 T^{8} + \cdots - 179848 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + 9 T^{8} + \cdots + 3392 \) Copy content Toggle raw display
$29$ \( T^{9} + 15 T^{8} + \cdots - 168328 \) Copy content Toggle raw display
$31$ \( T^{9} + 12 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$37$ \( T^{9} - 6 T^{8} + \cdots - 1232499 \) Copy content Toggle raw display
$41$ \( T^{9} + 18 T^{8} + \cdots - 1123176 \) Copy content Toggle raw display
$43$ \( T^{9} + 6 T^{8} + \cdots + 33119 \) Copy content Toggle raw display
$47$ \( T^{9} + 27 T^{8} + \cdots - 67416 \) Copy content Toggle raw display
$53$ \( T^{9} + 12 T^{8} + \cdots - 141065432 \) Copy content Toggle raw display
$59$ \( T^{9} + 36 T^{8} + \cdots + 218952 \) Copy content Toggle raw display
$61$ \( T^{9} + 9 T^{8} + \cdots + 2666701 \) Copy content Toggle raw display
$67$ \( T^{9} - 24 T^{8} + \cdots + 1018827 \) Copy content Toggle raw display
$71$ \( T^{9} + 18 T^{8} + \cdots - 79512 \) Copy content Toggle raw display
$73$ \( T^{9} + 69 T^{8} + \cdots - 97284133 \) Copy content Toggle raw display
$79$ \( T^{9} + 42 T^{8} + \cdots + 2924875 \) Copy content Toggle raw display
$83$ \( T^{9} + 9 T^{8} + \cdots - 11569464 \) Copy content Toggle raw display
$89$ \( T^{9} + 6 T^{8} + \cdots - 4046088 \) Copy content Toggle raw display
$97$ \( T^{9} + 21 T^{8} + \cdots - 180922239 \) Copy content Toggle raw display
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