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Properties

Label 5586.2.a.cf
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $8$
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] = [5586,2,Mod(1,5586)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, 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("5586.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.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: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 22x^{6} + 60x^{5} + 87x^{4} - 176x^{3} - 40x^{2} + 64x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + \beta_{7} q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + \beta_{7} q^{5} + q^{6} + q^{8} + q^{9} + \beta_{7} q^{10} + ( - \beta_1 + 1) q^{11} + q^{12} + (\beta_{3} - 1) q^{13} + \beta_{7} q^{15} + q^{16} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + 1) q^{17}+ \cdots + ( - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 4 q^{5} + 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 4 q^{5} + 8 q^{6} + 8 q^{8} + 8 q^{9} + 4 q^{10} + 8 q^{11} + 8 q^{12} - 4 q^{13} + 4 q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + 8 q^{19} + 4 q^{20} + 8 q^{22} + 4 q^{23} + 8 q^{24} + 24 q^{25} - 4 q^{26} + 8 q^{27} + 16 q^{29} + 4 q^{30} + 4 q^{31} + 8 q^{32} + 8 q^{33} + 8 q^{34} + 8 q^{36} + 12 q^{37} + 8 q^{38} - 4 q^{39} + 4 q^{40} + 16 q^{43} + 8 q^{44} + 4 q^{45} + 4 q^{46} + 12 q^{47} + 8 q^{48} + 24 q^{50} + 8 q^{51} - 4 q^{52} + 24 q^{53} + 8 q^{54} - 8 q^{55} + 8 q^{57} + 16 q^{58} - 8 q^{59} + 4 q^{60} + 4 q^{62} + 8 q^{64} + 8 q^{65} + 8 q^{66} + 8 q^{67} + 8 q^{68} + 4 q^{69} + 24 q^{71} + 8 q^{72} - 16 q^{73} + 12 q^{74} + 24 q^{75} + 8 q^{76} - 4 q^{78} + 20 q^{79} + 4 q^{80} + 8 q^{81} + 16 q^{83} + 48 q^{85} + 16 q^{86} + 16 q^{87} + 8 q^{88} + 4 q^{90} + 4 q^{92} + 4 q^{93} + 12 q^{94} + 4 q^{95} + 8 q^{96} - 8 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 22x^{6} + 60x^{5} + 87x^{4} - 176x^{3} - 40x^{2} + 64x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9\nu^{7} + 56\nu^{6} - 502\nu^{5} - 1524\nu^{4} + 4267\nu^{3} + 6100\nu^{2} - 6148\nu - 4644 ) / 1972 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 268\nu^{7} - 1893\nu^{6} - 3226\nu^{5} + 37114\nu^{4} - 14484\nu^{3} - 159895\nu^{2} + 87638\nu + 96380 ) / 13804 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 675 \nu^{7} + 1716 \nu^{6} + 18916 \nu^{5} - 19796 \nu^{4} - 116909 \nu^{3} + 47332 \nu^{2} + \cdots - 24408 ) / 13804 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1983 \nu^{7} + 7710 \nu^{6} + 42902 \nu^{5} - 107912 \nu^{4} - 146761 \nu^{3} + 237182 \nu^{2} + \cdots + 43144 ) / 27608 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -167\nu^{7} + 659\nu^{6} + 3618\nu^{5} - 9518\nu^{4} - 13005\nu^{3} + 25125\nu^{2} + 4524\nu - 6512 ) / 1972 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -167\nu^{7} + 659\nu^{6} + 3618\nu^{5} - 9518\nu^{4} - 13005\nu^{3} + 25125\nu^{2} + 580\nu - 4540 ) / 1972 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2859 \nu^{7} - 13434 \nu^{6} - 55610 \nu^{5} + 215936 \nu^{4} + 149277 \nu^{3} - 653770 \nu^{2} + \cdots + 139824 ) / 27608 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 4\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{7} - 21\beta_{6} + 17\beta_{5} + 12\beta_{4} + 4\beta_{3} - 8\beta_{2} + 10\beta _1 + 35 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 75\beta_{7} - 27\beta_{6} + 51\beta_{5} + 53\beta_{4} + 31\beta_{3} - 48\beta_{2} + 114\beta _1 + 302 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 298\beta_{7} - 545\beta_{6} + 445\beta_{5} + 390\beta_{4} + 170\beta_{3} - 260\beta_{2} + 448\beta _1 + 1267 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2113 \beta_{7} - 1757 \beta_{6} + 1951 \beta_{5} + 1939 \beta_{4} + 1023 \beta_{3} - 1466 \beta_{2} + \cdots + 8642 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10348 \beta_{7} - 15443 \beta_{6} + 13263 \beta_{5} + 12296 \beta_{4} + 5792 \beta_{3} - 8360 \beta_{2} + \cdots + 43669 ) / 2 \) 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
−0.514116
1.35228
0.804035
−1.97842
−0.261561
−3.71976
5.71612
2.60141
1.00000 1.00000 1.00000 −3.93854 1.00000 0 1.00000 1.00000 −3.93854
1.2 1.00000 1.00000 1.00000 −3.05440 1.00000 0 1.00000 1.00000 −3.05440
1.3 1.00000 1.00000 1.00000 −0.856159 1.00000 0 1.00000 1.00000 −0.856159
1.4 1.00000 1.00000 1.00000 −0.153106 1.00000 0 1.00000 1.00000 −0.153106
1.5 1.00000 1.00000 1.00000 2.05470 1.00000 0 1.00000 1.00000 2.05470
1.6 1.00000 1.00000 1.00000 2.63704 1.00000 0 1.00000 1.00000 2.63704
1.7 1.00000 1.00000 1.00000 3.15280 1.00000 0 1.00000 1.00000 3.15280
1.8 1.00000 1.00000 1.00000 4.15767 1.00000 0 1.00000 1.00000 4.15767
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(19\) \(-1\)

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 5586.2.a.cf yes 8
7.b odd 2 1 5586.2.a.ce ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5586.2.a.ce ✓ 8 7.b odd 2 1
5586.2.a.cf yes 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5}^{8} - 4T_{5}^{7} - 24T_{5}^{6} + 104T_{5}^{5} + 122T_{5}^{4} - 704T_{5}^{3} + 128T_{5}^{2} + 768T_{5} + 112 \) Copy content Toggle raw display
\( T_{11}^{8} - 8T_{11}^{7} - 28T_{11}^{6} + 368T_{11}^{5} - 532T_{11}^{4} - 2464T_{11}^{3} + 7584T_{11}^{2} - 5760T_{11} + 448 \) Copy content Toggle raw display
\( T_{13}^{8} + 4T_{13}^{7} - 80T_{13}^{6} - 272T_{13}^{5} + 2074T_{13}^{4} + 4928T_{13}^{3} - 20800T_{13}^{2} - 20544T_{13} + 59248 \) Copy content Toggle raw display
\( T_{17}^{8} - 8T_{17}^{7} - 48T_{17}^{6} + 480T_{17}^{5} + 24T_{17}^{4} - 5312T_{17}^{3} + 1920T_{17}^{2} + 17920T_{17} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 112 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + \cdots + 448 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 59248 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T - 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 4 T^{7} + \cdots - 8 \) Copy content Toggle raw display
$29$ \( T^{8} - 16 T^{7} + \cdots + 31744 \) Copy content Toggle raw display
$31$ \( T^{8} - 4 T^{7} + \cdots - 149536 \) Copy content Toggle raw display
$37$ \( T^{8} - 12 T^{7} + \cdots + 109856 \) Copy content Toggle raw display
$41$ \( T^{8} - 228 T^{6} + \cdots + 691712 \) Copy content Toggle raw display
$43$ \( T^{8} - 16 T^{7} + \cdots - 13555456 \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 2896352 \) Copy content Toggle raw display
$53$ \( T^{8} - 24 T^{7} + \cdots - 2142464 \) Copy content Toggle raw display
$59$ \( T^{8} + 8 T^{7} + \cdots + 260096 \) Copy content Toggle raw display
$61$ \( T^{8} - 236 T^{6} + \cdots - 2787904 \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} + \cdots - 387136 \) Copy content Toggle raw display
$71$ \( T^{8} - 24 T^{7} + \cdots + 14336 \) Copy content Toggle raw display
$73$ \( T^{8} + 16 T^{7} + \cdots + 3523072 \) Copy content Toggle raw display
$79$ \( T^{8} - 20 T^{7} + \cdots - 1016968 \) Copy content Toggle raw display
$83$ \( T^{8} - 16 T^{7} + \cdots - 974848 \) Copy content Toggle raw display
$89$ \( T^{8} - 440 T^{6} + \cdots + 185104 \) Copy content Toggle raw display
$97$ \( T^{8} + 8 T^{7} + \cdots - 44800 \) Copy content Toggle raw display
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