Properties

Label 5390.2.a.cl
Level $5390$
Weight $2$
Character orbit 5390.a
Self dual yes
Analytic conductor $43.039$
Analytic rank $1$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.27410432.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 21x^{2} - 14x - 1 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 21x^{2} - 14x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 2\nu^{3} + 10\nu^{2} - 9\nu - 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 6\nu^{3} + 14\nu^{2} + 11\nu - 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 6\nu^{3} + 18\nu^{2} + 7\nu - 17 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - \nu^{4} - 10\nu^{3} + 4\nu^{2} + 23\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 2\beta_{3} + \beta_{2} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 7\beta_{4} - 11\beta_{3} + 2\beta_{2} + 11\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{5} + 13\beta_{4} - 27\beta_{3} + 12\beta_{2} + 44\beta _1 + 29 \) 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
−2.04192
−1.77881
−0.0652853
0.632276
2.14654
3.10721
1.00000 −3.04192 1.00000 −1.00000 −3.04192 0 1.00000 6.25328 −1.00000
1.2 1.00000 −2.77881 1.00000 −1.00000 −2.77881 0 1.00000 4.72180 −1.00000
1.3 1.00000 −1.06529 1.00000 −1.00000 −1.06529 0 1.00000 −1.86517 −1.00000
1.4 1.00000 −0.367724 1.00000 −1.00000 −0.367724 0 1.00000 −2.86478 −1.00000
1.5 1.00000 1.14654 1.00000 −1.00000 1.14654 0 1.00000 −1.68545 −1.00000
1.6 1.00000 2.10721 1.00000 −1.00000 2.10721 0 1.00000 1.44032 −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.cl 6
7.b odd 2 1 5390.2.a.cm yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5390.2.a.cl 6 1.a even 1 1 trivial
5390.2.a.cm yes 6 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5390))\):

\( T_{3}^{6} + 4T_{3}^{5} - 4T_{3}^{4} - 24T_{3}^{3} - 2T_{3}^{2} + 24T_{3} + 8 \) Copy content Toggle raw display
\( T_{13}^{6} + 12T_{13}^{5} + 38T_{13}^{4} - 92T_{13}^{2} - 16T_{13} + 56 \) Copy content Toggle raw display
\( T_{17}^{6} + 8T_{17}^{5} - 4T_{17}^{4} - 144T_{17}^{3} - 284T_{17}^{2} + 224 \) Copy content Toggle raw display
\( T_{19}^{6} + 12T_{19}^{5} + 10T_{19}^{4} - 216T_{19}^{3} - 178T_{19}^{2} + 936T_{19} - 68 \) Copy content Toggle raw display
\( T_{31}^{6} + 8T_{31}^{5} - 102T_{31}^{4} - 688T_{31}^{3} + 2340T_{31}^{2} + 14144T_{31} + 5704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 4 T^{5} - 4 T^{4} - 24 T^{3} + \cdots + 8 \) 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
$13$ \( T^{6} + 12 T^{5} + 38 T^{4} - 92 T^{2} + \cdots + 56 \) Copy content Toggle raw display
$17$ \( T^{6} + 8 T^{5} - 4 T^{4} - 144 T^{3} + \cdots + 224 \) Copy content Toggle raw display
$19$ \( T^{6} + 12 T^{5} + 10 T^{4} - 216 T^{3} + \cdots - 68 \) Copy content Toggle raw display
$23$ \( T^{6} + 4 T^{5} - 72 T^{4} - 464 T^{3} + \cdots + 136 \) Copy content Toggle raw display
$29$ \( T^{6} - 8 T^{5} - 4 T^{4} + 112 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( T^{6} + 8 T^{5} - 102 T^{4} + \cdots + 5704 \) Copy content Toggle raw display
$37$ \( T^{6} + 12 T^{5} - 128 T^{3} + 162 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$41$ \( T^{6} + 8 T^{5} - 120 T^{4} + \cdots + 45808 \) Copy content Toggle raw display
$43$ \( T^{6} + 4 T^{5} - 176 T^{4} + \cdots - 26512 \) Copy content Toggle raw display
$47$ \( T^{6} + 8 T^{5} - 122 T^{4} + \cdots + 15472 \) Copy content Toggle raw display
$53$ \( T^{6} + 12 T^{5} + 8 T^{4} - 400 T^{3} + \cdots - 568 \) Copy content Toggle raw display
$59$ \( T^{6} + 4 T^{5} - 88 T^{4} + \cdots - 4976 \) Copy content Toggle raw display
$61$ \( T^{6} - 4 T^{5} - 190 T^{4} + \cdots - 56336 \) Copy content Toggle raw display
$67$ \( T^{6} - 192 T^{4} - 288 T^{3} + \cdots + 17536 \) Copy content Toggle raw display
$71$ \( T^{6} - 72 T^{4} - 96 T^{3} + \cdots - 1792 \) Copy content Toggle raw display
$73$ \( T^{6} + 8 T^{5} - 260 T^{4} + \cdots - 15904 \) Copy content Toggle raw display
$79$ \( T^{6} - 180 T^{4} - 112 T^{3} + \cdots - 35216 \) Copy content Toggle raw display
$83$ \( T^{6} + 28 T^{5} + 190 T^{4} + \cdots - 4088 \) Copy content Toggle raw display
$89$ \( T^{6} - 326 T^{4} - 864 T^{3} + \cdots - 187336 \) Copy content Toggle raw display
$97$ \( T^{6} + 24 T^{5} + 94 T^{4} + \cdots - 163996 \) Copy content Toggle raw display
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