Properties

Label 4020.2.a.g
Level $4020$
Weight $2$
Character orbit 4020.a
Self dual yes
Analytic conductor $32.100$
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] = [4020,2,Mod(1,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, 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("4020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.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: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.195727752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 15x^{4} - 25x^{3} - 3x^{2} + 9x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + (\beta_{4} + \beta_1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + (\beta_{4} + \beta_1) q^{7} + q^{9} + (\beta_{4} - \beta_{2}) q^{11} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{13} - q^{15} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{17} + ( - \beta_{5} - \beta_1 - 1) q^{19} + (\beta_{4} + \beta_1) q^{21} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{23} + q^{25} + q^{27} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 3) q^{29} + ( - \beta_{5} - \beta_{3} + 2 \beta_1 - 1) q^{31} + (\beta_{4} - \beta_{2}) q^{33} + ( - \beta_{4} - \beta_1) q^{35} + (2 \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{37} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{39} + (\beta_{5} - \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{41} + ( - \beta_{5} + 3 \beta_{2} + 2 \beta_1 - 1) q^{43} - q^{45} + (\beta_{5} - 3 \beta_{3} - 2 \beta_{2} + \beta_1 + 4) q^{47} + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{49} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{51} + (\beta_{5} + 4 \beta_{3} + \beta_1 + 3) q^{53} + ( - \beta_{4} + \beta_{2}) q^{55} + ( - \beta_{5} - \beta_1 - 1) q^{57} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{59} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_1 + 3) q^{61} + (\beta_{4} + \beta_1) q^{63} + ( - \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{65} + q^{67} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{69} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{71} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1) q^{73} + q^{75} + ( - \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 5) q^{77} + ( - 3 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{79} + q^{81} + (\beta_{2} + \beta_1 + 4) q^{83} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{85} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 3) q^{87} + ( - 4 \beta_{4} - 4 \beta_{3} + \beta_{2} - 3 \beta_1 + 6) q^{89} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 5) q^{91} + ( - \beta_{5} - \beta_{3} + 2 \beta_1 - 1) q^{93} + (\beta_{5} + \beta_1 + 1) q^{95} + (2 \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{97} + (\beta_{4} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{15} + 6 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 6 q^{25} + 6 q^{27} + 21 q^{29} - 3 q^{31} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 9 q^{41} - 3 q^{43} - 6 q^{45} + 21 q^{47} + 3 q^{49} + 6 q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 15 q^{59} + 15 q^{61} + 3 q^{63} - 3 q^{65} + 6 q^{67} + 6 q^{69} + 18 q^{71} - 6 q^{73} + 6 q^{75} + 33 q^{77} + 9 q^{79} + 6 q^{81} + 24 q^{83} - 6 q^{85} + 21 q^{87} + 24 q^{89} - 21 q^{91} - 3 q^{93} + 3 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 15x^{4} - 25x^{3} - 3x^{2} + 9x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{5} - \nu^{4} - 14\nu^{3} - 11\nu^{2} + 8\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{5} + \nu^{4} + 30\nu^{3} + 34\nu^{2} - 16\nu - 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\nu^{5} + 2\nu^{4} + 43\nu^{3} + 48\nu^{2} - 17\nu - 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 4\nu^{5} - 2\nu^{4} - 59\nu^{3} - 70\nu^{2} + 22\nu + 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 3\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{5} + 2\beta_{4} + 4\beta_{3} + 2\beta_{2} + 16\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 18\beta_{5} + 16\beta_{4} + 19\beta_{3} + 14\beta_{2} + 68\beta _1 + 78 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 71\beta_{5} + 55\beta_{4} + 86\beta_{3} + 54\beta_{2} + 317\beta _1 + 313 \) 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
−1.75666
−0.457832
0.600664
−0.587102
−2.33297
4.53390
0 1.00000 0 −1.00000 0 −3.63834 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −2.59172 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 1.05225 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 1.68378 0 1.00000 0
1.5 0 1.00000 0 −1.00000 0 3.15496 0 1.00000 0
1.6 0 1.00000 0 −1.00000 0 3.33907 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(67\) \(-1\)

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 4020.2.a.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.g 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 3T_{7}^{5} - 18T_{7}^{4} + 60T_{7}^{3} + 51T_{7}^{2} - 264T_{7} + 176 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} - 18 T^{4} + 60 T^{3} + \cdots + 176 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} - 36 T^{4} + 90 T^{3} + \cdots + 72 \) Copy content Toggle raw display
$13$ \( T^{6} - 3 T^{5} - 39 T^{4} + 119 T^{3} + \cdots + 46 \) Copy content Toggle raw display
$17$ \( T^{6} - 6 T^{5} - 24 T^{4} + 125 T^{3} + \cdots - 30 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} - 33 T^{4} - 27 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} - 60 T^{4} + \cdots - 2640 \) Copy content Toggle raw display
$29$ \( T^{6} - 21 T^{5} + 75 T^{4} + \cdots + 21516 \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} - 72 T^{4} + \cdots + 3370 \) Copy content Toggle raw display
$37$ \( T^{6} - 102 T^{4} - 180 T^{3} + \cdots + 9476 \) Copy content Toggle raw display
$41$ \( T^{6} - 9 T^{5} - 60 T^{4} + 424 T^{3} + \cdots - 960 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} - 168 T^{4} + \cdots - 6040 \) Copy content Toggle raw display
$47$ \( T^{6} - 21 T^{5} + 21 T^{4} + \cdots + 119256 \) Copy content Toggle raw display
$53$ \( T^{6} - 15 T^{5} - 171 T^{4} + \cdots + 287100 \) Copy content Toggle raw display
$59$ \( T^{6} - 15 T^{5} - 9 T^{4} + \cdots + 18198 \) Copy content Toggle raw display
$61$ \( T^{6} - 15 T^{5} - 33 T^{4} + \cdots + 25524 \) Copy content Toggle raw display
$67$ \( (T - 1)^{6} \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} - 126 T^{4} + \cdots + 473094 \) Copy content Toggle raw display
$73$ \( T^{6} + 6 T^{5} - 210 T^{4} + \cdots - 108284 \) Copy content Toggle raw display
$79$ \( T^{6} - 9 T^{5} - 390 T^{4} + \cdots - 76530 \) Copy content Toggle raw display
$83$ \( T^{6} - 24 T^{5} + 210 T^{4} + \cdots - 360 \) Copy content Toggle raw display
$89$ \( T^{6} - 24 T^{5} - 258 T^{4} + \cdots + 1927500 \) Copy content Toggle raw display
$97$ \( T^{6} - 9 T^{5} - 129 T^{4} + \cdots - 13450 \) Copy content Toggle raw display
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