Properties

Label 6020.2.a.e
Level $6020$
Weight $2$
Character orbit 6020.a
Self dual yes
Analytic conductor $48.070$
Analytic rank $1$
Dimension $7$
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] = [6020,2,Mod(1,6020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6020.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: \(48.0699420168\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.187391161.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 4x^{5} + 16x^{4} - x^{3} - 20x^{2} + 11x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - q^{5} - q^{7} + (\beta_{6} - \beta_{3} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} - q^{5} - q^{7} + (\beta_{6} - \beta_{3} + \beta_1 - 1) q^{9} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{5} + 2 \beta_{3} - \beta_1) q^{13} + \beta_{5} q^{15} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2}) q^{17} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1) q^{19} + \beta_{5} q^{21} + ( - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 1) q^{23} + q^{25} + (\beta_{6} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{27} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{29} + (2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{31} + (2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_1 + 1) q^{33} + q^{35} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{37} + ( - \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{39} + ( - \beta_{6} + \beta_{5} + 4 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{41} + q^{43} + ( - \beta_{6} + \beta_{3} - \beta_1 + 1) q^{45} + (\beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{47} + q^{49} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 2) q^{51} + (3 \beta_{6} - 2 \beta_{5} + \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{53} + (\beta_{5} - \beta_{3} + \beta_1) q^{55} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{57} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{59} + ( - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 3) q^{61} + ( - \beta_{6} + \beta_{3} - \beta_1 + 1) q^{63} + (\beta_{5} - 2 \beta_{3} + \beta_1) q^{65} + ( - \beta_{6} + \beta_{5} - 4 \beta_{4} + \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{67} + ( - \beta_{6} + 2 \beta_{3} - 2 \beta_1 - 1) q^{69} + (\beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 4) q^{71} + (\beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 5 \beta_1 - 1) q^{73} - \beta_{5} q^{75} + (\beta_{5} - \beta_{3} + \beta_1) q^{77} + ( - \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 2) q^{79} + ( - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 2) q^{81} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{83} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{85} + (2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{87} + (\beta_{6} + 5 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 - 5) q^{89} + (\beta_{5} - 2 \beta_{3} + \beta_1) q^{91} + (\beta_{6} - 5 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{93} + (\beta_{6} - \beta_{5} + \beta_{3} - \beta_1) q^{95} + (3 \beta_{6} - \beta_{5} + \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - \beta_1) q^{97} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{3} - 7 q^{5} - 7 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{3} - 7 q^{5} - 7 q^{7} - 2 q^{9} - 3 q^{11} - 2 q^{13} + q^{15} + q^{17} + q^{21} + 3 q^{23} + 7 q^{25} + 5 q^{27} + 3 q^{29} + 17 q^{31} + 13 q^{33} + 7 q^{35} - 4 q^{37} + 7 q^{39} - 3 q^{41} + 7 q^{43} + 2 q^{45} + 19 q^{47} + 7 q^{49} - 13 q^{51} + 10 q^{53} + 3 q^{55} - 8 q^{57} - 10 q^{59} - 15 q^{61} + 2 q^{63} + 2 q^{65} - 21 q^{67} - 14 q^{69} - 14 q^{71} - 16 q^{73} - q^{75} + 3 q^{77} - 20 q^{79} - 9 q^{81} - q^{83} - q^{85} - 2 q^{87} - 19 q^{89} + 2 q^{91} - q^{93} - 6 q^{97} - 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 4x^{5} + 16x^{4} - x^{3} - 20x^{2} + 11x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - 2\nu^{5} - 5\nu^{4} + 9\nu^{3} + 4\nu^{2} - 9\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 10\nu^{3} + 9\nu^{2} - 11\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} + 2\nu^{5} + 6\nu^{4} - 11\nu^{3} - 7\nu^{2} + 14\nu - 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 11\nu^{3} + 8\nu^{2} - 15\nu + 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{6} + 3\nu^{5} + 4\nu^{4} - 15\nu^{3} - \nu^{2} + 17\nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + \beta_{4} - \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 6\beta_{4} - 2\beta_{3} + \beta_{2} + 8\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + 16\beta_{5} + 9\beta_{4} - 8\beta_{3} + 2\beta_{2} + 27\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} + 45\beta_{5} + 35\beta_{4} - 17\beta_{3} + 10\beta_{2} + 54\beta _1 + 30 \) 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
2.52647
−1.47376
1.31298
0.114738
−1.80731
1.76939
0.557489
0 −2.29273 0 −1.00000 0 −1.00000 0 2.25660 0
1.2 0 −2.11789 0 −1.00000 0 −1.00000 0 1.48544 0
1.3 0 −0.482685 0 −1.00000 0 −1.00000 0 −2.76701 0
1.4 0 −0.399783 0 −1.00000 0 −1.00000 0 −2.84017 0
1.5 0 0.296702 0 −1.00000 0 −1.00000 0 −2.91197 0
1.6 0 1.36907 0 −1.00000 0 −1.00000 0 −1.12565 0
1.7 0 2.62731 0 −1.00000 0 −1.00000 0 3.90277 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)
\(43\) \(-1\)

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 6020.2.a.e 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6020.2.a.e 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6020))\):

\( T_{3}^{7} + T_{3}^{6} - 9T_{3}^{5} - 9T_{3}^{4} + 16T_{3}^{3} + 11T_{3}^{2} - T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{7} + 3T_{11}^{6} - 9T_{11}^{5} - 13T_{11}^{4} + 22T_{11}^{3} + 14T_{11}^{2} - 14T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + T^{6} - 9 T^{5} - 9 T^{4} + 16 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{7} \) Copy content Toggle raw display
$7$ \( (T + 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + 3 T^{6} - 9 T^{5} - 13 T^{4} + \cdots - 3 \) Copy content Toggle raw display
$13$ \( T^{7} + 2 T^{6} - 26 T^{5} - 64 T^{4} + \cdots - 487 \) Copy content Toggle raw display
$17$ \( T^{7} - T^{6} - 25 T^{5} - 11 T^{4} + \cdots - 81 \) Copy content Toggle raw display
$19$ \( T^{7} - 22 T^{5} - 33 T^{4} + 87 T^{3} + \cdots + 37 \) Copy content Toggle raw display
$23$ \( T^{7} - 3 T^{6} - 53 T^{5} + 60 T^{4} + \cdots + 439 \) Copy content Toggle raw display
$29$ \( T^{7} - 3 T^{6} - 36 T^{5} + 81 T^{4} + \cdots - 375 \) Copy content Toggle raw display
$31$ \( T^{7} - 17 T^{6} + 82 T^{5} + 47 T^{4} + \cdots - 47 \) Copy content Toggle raw display
$37$ \( T^{7} + 4 T^{6} - 99 T^{5} + \cdots + 4783 \) Copy content Toggle raw display
$41$ \( T^{7} + 3 T^{6} - 157 T^{5} + \cdots - 3149 \) Copy content Toggle raw display
$43$ \( (T - 1)^{7} \) Copy content Toggle raw display
$47$ \( T^{7} - 19 T^{6} + 84 T^{5} + \cdots - 6429 \) Copy content Toggle raw display
$53$ \( T^{7} - 10 T^{6} - 79 T^{5} + \cdots + 1257 \) Copy content Toggle raw display
$59$ \( T^{7} + 10 T^{6} - 142 T^{5} + \cdots - 675 \) Copy content Toggle raw display
$61$ \( T^{7} + 15 T^{6} - 123 T^{5} + \cdots - 505983 \) Copy content Toggle raw display
$67$ \( T^{7} + 21 T^{6} - 27 T^{5} + \cdots - 596595 \) Copy content Toggle raw display
$71$ \( T^{7} + 14 T^{6} + 4 T^{5} + \cdots - 1917 \) Copy content Toggle raw display
$73$ \( T^{7} + 16 T^{6} - 330 T^{5} + \cdots - 121743 \) Copy content Toggle raw display
$79$ \( T^{7} + 20 T^{6} + 28 T^{5} + \cdots - 4023 \) Copy content Toggle raw display
$83$ \( T^{7} + T^{6} - 215 T^{5} - 417 T^{4} + \cdots - 4317 \) Copy content Toggle raw display
$89$ \( T^{7} + 19 T^{6} - 221 T^{5} + \cdots + 76519 \) Copy content Toggle raw display
$97$ \( T^{7} + 6 T^{6} - 373 T^{5} + \cdots - 223701 \) Copy content Toggle raw display
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