Properties

Label 5390.2.a.ck
Level $5390$
Weight $2$
Character orbit 5390.a
Self dual yes
Analytic conductor $43.039$
Analytic rank $0$
Dimension $6$
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] = [5390,2,Mod(1,5390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5390, 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("5390.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.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: \(43.0393666895\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.30728192.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 12x^{4} + 34x^{2} - 8x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} - q^{8} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} - q^{8} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{9} - q^{10} - q^{11} + \beta_1 q^{12} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{13} + \beta_1 q^{15} + q^{16} + (\beta_{5} + \beta_{3} + \beta_{2}) q^{17} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{18} + ( - 2 \beta_{5} + \beta_{4} + 1) q^{19} + q^{20} + q^{22} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{23} - \beta_1 q^{24} + q^{25} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{26} + (\beta_{5} - 2 \beta_{2}) q^{27} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{29} - \beta_1 q^{30} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{31} - q^{32} - \beta_1 q^{33} + ( - \beta_{5} - \beta_{3} - \beta_{2}) q^{34} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{36} + ( - \beta_{5} - \beta_{4} - \beta_{2} + 2 \beta_1 - 1) q^{37} + (2 \beta_{5} - \beta_{4} - 1) q^{38} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 3) q^{39} - q^{40} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - 1) q^{41} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 2) q^{43} - q^{44} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{45} + (\beta_{3} - \beta_{2} - \beta_1) q^{46} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{47} + \beta_1 q^{48} - q^{50} + ( - \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{51} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{52} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{53} + ( - \beta_{5} + 2 \beta_{2}) q^{54} - q^{55} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + 7 \beta_{2} - \beta_1 - 1) q^{57} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 3) q^{58} + (\beta_{5} - \beta_{4} - \beta_{2} - \beta_1 - 3) q^{59} + \beta_1 q^{60} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{61} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{62} + q^{64} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{65} + \beta_1 q^{66} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{67} + (\beta_{5} + \beta_{3} + \beta_{2}) q^{68} + (\beta_{4} + 2 \beta_{3} - 3 \beta_{2} - \beta_1 + 5) q^{69} + ( - \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - \beta_1 + 3) q^{71} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{72} + (\beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{73} + (\beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{74} + \beta_1 q^{75} + ( - 2 \beta_{5} + \beta_{4} + 1) q^{76} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 3) q^{78} + (\beta_{5} + 2 \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{79} + q^{80} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{81} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + 1) q^{82} + (3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{83} + (\beta_{5} + \beta_{3} + \beta_{2}) q^{85} + (\beta_{5} - \beta_{3} - \beta_{2} + 2) q^{86} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 1) q^{87} + q^{88} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{2} + 6) q^{89} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{90} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{92} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} - 6 \beta_{2} + \beta_1 - 1) q^{93} + (2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{94} + ( - 2 \beta_{5} + \beta_{4} + 1) q^{95} - \beta_1 q^{96} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1 + 8) q^{97} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 6 q^{8} + 6 q^{9} - 6 q^{10} - 6 q^{11} + 4 q^{13} + 6 q^{16} - 6 q^{18} + 8 q^{19} + 6 q^{20} + 6 q^{22} + 6 q^{25} - 4 q^{26} + 20 q^{29} + 8 q^{31} - 6 q^{32} + 6 q^{36} - 8 q^{37} - 8 q^{38} + 16 q^{39} - 6 q^{40} - 4 q^{41} - 12 q^{43} - 6 q^{44} + 6 q^{45} + 8 q^{47} - 6 q^{50} + 4 q^{52} + 8 q^{53} - 6 q^{55} - 8 q^{57} - 20 q^{58} - 20 q^{59} + 20 q^{61} - 8 q^{62} + 6 q^{64} + 4 q^{65} - 8 q^{67} + 32 q^{69} + 16 q^{71} - 6 q^{72} + 16 q^{73} + 8 q^{74} + 8 q^{76} - 16 q^{78} + 28 q^{79} + 6 q^{80} - 10 q^{81} + 4 q^{82} - 12 q^{83} + 12 q^{86} - 8 q^{87} + 6 q^{88} + 40 q^{89} - 6 q^{90} - 8 q^{94} + 8 q^{95} + 44 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 12x^{4} + 34x^{2} - 8x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 12\nu^{2} + 10\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} + 4\nu^{3} - 8\nu^{2} + 14\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 2\nu^{4} - 12\nu^{3} - 16\nu^{2} + 30\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 12\nu^{2} - 2\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} - \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{5} + \beta_{4} + 7\beta_{3} - 10\beta_{2} + \beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{5} + 2\beta_{4} + 2\beta_{3} - 20\beta_{2} + 40\beta _1 + 6 \) Copy content Toggle raw display

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
−2.50721
−2.40081
−0.389454
0.687705
1.71311
2.89666
−1.00000 −2.50721 1.00000 1.00000 2.50721 0 −1.00000 3.28610 −1.00000
1.2 −1.00000 −2.40081 1.00000 1.00000 2.40081 0 −1.00000 2.76390 −1.00000
1.3 −1.00000 −0.389454 1.00000 1.00000 0.389454 0 −1.00000 −2.84833 −1.00000
1.4 −1.00000 0.687705 1.00000 1.00000 −0.687705 0 −1.00000 −2.52706 −1.00000
1.5 −1.00000 1.71311 1.00000 1.00000 −1.71311 0 −1.00000 −0.0652647 −1.00000
1.6 −1.00000 2.89666 1.00000 1.00000 −2.89666 0 −1.00000 5.39066 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

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 5390.2.a.ck yes 6
7.b odd 2 1 5390.2.a.cj 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5390.2.a.cj 6 7.b odd 2 1
5390.2.a.ck yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5390))\):

\( T_{3}^{6} - 12T_{3}^{4} + 34T_{3}^{2} - 8T_{3} - 8 \) Copy content Toggle raw display
\( T_{13}^{6} - 4T_{13}^{5} - 42T_{13}^{4} + 176T_{13}^{3} + 132T_{13}^{2} - 496T_{13} + 184 \) Copy content Toggle raw display
\( T_{17}^{6} - 36T_{17}^{4} + 64T_{17}^{3} + 36T_{17}^{2} - 64T_{17} - 32 \) Copy content Toggle raw display
\( T_{19}^{6} - 8T_{19}^{5} - 62T_{19}^{4} + 448T_{19}^{3} + 786T_{19}^{2} - 3736T_{19} + 2212 \) Copy content Toggle raw display
\( T_{31}^{6} - 8T_{31}^{5} - 86T_{31}^{4} + 832T_{31}^{3} + 612T_{31}^{2} - 18016T_{31} + 33544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 12 T^{4} + 34 T^{2} - 8 T - 8 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 4 T^{5} - 42 T^{4} + 176 T^{3} + \cdots + 184 \) Copy content Toggle raw display
$17$ \( T^{6} - 36 T^{4} + 64 T^{3} + 36 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$19$ \( T^{6} - 8 T^{5} - 62 T^{4} + \cdots + 2212 \) Copy content Toggle raw display
$23$ \( T^{6} - 48 T^{4} + 168 T^{3} + \cdots + 56 \) Copy content Toggle raw display
$29$ \( T^{6} - 20 T^{5} + 52 T^{4} + \cdots - 33424 \) Copy content Toggle raw display
$31$ \( T^{6} - 8 T^{5} - 86 T^{4} + \cdots + 33544 \) Copy content Toggle raw display
$37$ \( T^{6} + 8 T^{5} - 56 T^{4} - 416 T^{3} + \cdots + 904 \) Copy content Toggle raw display
$41$ \( T^{6} + 4 T^{5} - 64 T^{4} - 208 T^{3} + \cdots - 112 \) Copy content Toggle raw display
$43$ \( T^{6} + 12 T^{5} - 24 T^{4} + \cdots + 4336 \) Copy content Toggle raw display
$47$ \( T^{6} - 8 T^{5} - 122 T^{4} + \cdots + 9872 \) Copy content Toggle raw display
$53$ \( T^{6} - 8 T^{5} - 128 T^{4} + \cdots - 2824 \) Copy content Toggle raw display
$59$ \( T^{6} + 20 T^{5} + 120 T^{4} + \cdots - 496 \) Copy content Toggle raw display
$61$ \( T^{6} - 20 T^{5} - 78 T^{4} + \cdots + 393872 \) Copy content Toggle raw display
$67$ \( T^{6} + 8 T^{5} - 320 T^{4} + \cdots - 1171712 \) Copy content Toggle raw display
$71$ \( T^{6} - 16 T^{5} - 280 T^{4} + \cdots + 1190144 \) Copy content Toggle raw display
$73$ \( T^{6} - 16 T^{5} - 52 T^{4} + \cdots + 118496 \) Copy content Toggle raw display
$79$ \( T^{6} - 28 T^{5} + 132 T^{4} + \cdots - 25616 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} - 146 T^{4} + \cdots - 20792 \) Copy content Toggle raw display
$89$ \( T^{6} - 40 T^{5} + 538 T^{4} + \cdots - 10696 \) Copy content Toggle raw display
$97$ \( T^{6} - 44 T^{5} + 678 T^{4} + \cdots + 9052 \) Copy content Toggle raw display
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