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Properties

Label 4020.2.a.i
Level $4020$
Weight $2$
Character orbit 4020.a
Self dual yes
Analytic conductor $32.100$
Analytic rank $0$
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] = [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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 21x^{5} - 12x^{4} + 93x^{3} + 18x^{2} - 120x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} - \beta_{4} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} - \beta_{4} q^{7} + q^{9} + (\beta_{6} + \beta_{3}) q^{11} + (\beta_{6} - \beta_{2} + 1) q^{13} + q^{15} + ( - \beta_{3} + 2) q^{17} + ( - \beta_{6} + \beta_{5} - \beta_{4}) q^{19} - \beta_{4} q^{21} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_1) q^{23} + q^{25} + q^{27} + ( - \beta_{5} - \beta_{3} + \beta_1 + 2) q^{29} + (\beta_{4} + \beta_{3} + \beta_{2} + \cdots + 1) q^{31}+ \cdots + (\beta_{6} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 21x^{5} - 12x^{4} + 93x^{3} + 18x^{2} - 120x + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 19\nu^{4} - 16\nu^{3} + 65\nu^{2} + 44\nu - 60 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 20\nu^{4} + 15\nu^{3} - 78\nu^{2} - 46\nu + 72 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + \nu^{5} - 21\nu^{4} - 30\nu^{3} + 74\nu^{2} + 72\nu - 64 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + \nu^{5} - 22\nu^{4} - 29\nu^{3} + 91\nu^{2} + 74\nu - 96 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + \nu^{5} - 21\nu^{4} - 32\nu^{3} + 80\nu^{2} + 90\nu - 88 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} + 3\beta_{5} - \beta_{4} + 3\beta_{3} + 3\beta_{2} + 9\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 16\beta_{5} - 14\beta_{4} + 20\beta_{3} + 20\beta_{2} + 11\beta _1 + 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -32\beta_{6} + 65\beta_{5} - 29\beta_{4} + 73\beta_{3} + 69\beta_{2} + 120\beta _1 + 113 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -70\beta_{6} + 287\beta_{5} - 217\beta_{4} + 363\beta_{3} + 367\beta_{2} + 309\beta _1 + 847 \) 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
0.819259
−1.72664
−2.67981
4.35097
0.575750
1.67614
−3.01567
0 1.00000 0 1.00000 0 −4.84231 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −2.80679 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 −0.239317 0 1.00000 0
1.4 0 1.00000 0 1.00000 0 0.772018 0 1.00000 0
1.5 0 1.00000 0 1.00000 0 1.48734 0 1.00000 0
1.6 0 1.00000 0 1.00000 0 1.75949 0 1.00000 0
1.7 0 1.00000 0 1.00000 0 4.86957 0 1.00000 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
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(67\) \(1\)

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.i ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.i ✓ 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{7} - T_{7}^{6} - 30T_{7}^{5} + 34T_{7}^{4} + 149T_{7}^{3} - 258T_{7}^{2} + 64T_{7} + 32 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( (T - 1)^{7} \) Copy content Toggle raw display
$5$ \( (T - 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - T^{6} + \cdots + 32 \) Copy content Toggle raw display
$11$ \( T^{7} - 5 T^{6} + \cdots - 1136 \) Copy content Toggle raw display
$13$ \( T^{7} - 9 T^{6} + \cdots + 588 \) Copy content Toggle raw display
$17$ \( T^{7} - 11 T^{6} + \cdots - 158 \) Copy content Toggle raw display
$19$ \( T^{7} - 2 T^{6} + \cdots + 928 \) Copy content Toggle raw display
$23$ \( T^{7} - 7 T^{6} + \cdots + 224 \) Copy content Toggle raw display
$29$ \( T^{7} - 8 T^{6} + \cdots + 4132 \) Copy content Toggle raw display
$31$ \( T^{7} - 9 T^{6} + \cdots + 24 \) Copy content Toggle raw display
$37$ \( T^{7} - 7 T^{6} + \cdots + 210548 \) Copy content Toggle raw display
$41$ \( T^{7} - 9 T^{6} + \cdots + 93792 \) Copy content Toggle raw display
$43$ \( T^{7} - 5 T^{6} + \cdots + 32368 \) Copy content Toggle raw display
$47$ \( T^{7} - 2 T^{6} + \cdots - 1736 \) Copy content Toggle raw display
$53$ \( T^{7} - 15 T^{6} + \cdots - 49824 \) Copy content Toggle raw display
$59$ \( T^{7} - 216 T^{5} + \cdots + 179214 \) Copy content Toggle raw display
$61$ \( T^{7} - 11 T^{6} + \cdots - 38096 \) Copy content Toggle raw display
$67$ \( (T + 1)^{7} \) Copy content Toggle raw display
$71$ \( T^{7} - 18 T^{6} + \cdots + 18264 \) Copy content Toggle raw display
$73$ \( T^{7} - 21 T^{6} + \cdots - 274452 \) Copy content Toggle raw display
$79$ \( T^{7} - 5 T^{6} + \cdots + 24808 \) Copy content Toggle raw display
$83$ \( T^{7} - 6 T^{6} + \cdots - 96 \) Copy content Toggle raw display
$89$ \( T^{7} - 7 T^{6} + \cdots + 7748 \) Copy content Toggle raw display
$97$ \( T^{7} + 9 T^{6} + \cdots + 822828 \) Copy content Toggle raw display
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