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Properties

Label 4020.2.a.h
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: \( x^{7} - 3x^{6} - 28x^{5} + 90x^{4} + 143x^{3} - 418x^{2} - 256x + 160 \) 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.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} + \beta_1 q^{7} + q^{9} + (\beta_{2} - 1) q^{11} + (\beta_{6} + \beta_{3} + 1) q^{13} + q^{15} + (\beta_{6} - \beta_{4} - \beta_1 + 1) q^{17} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{19} - \beta_1 q^{21} + ( - \beta_{3} + \beta_1 - 1) q^{23} + q^{25} - q^{27} + (\beta_{3} - 1) q^{29} + ( - \beta_{6} - \beta_{3} + \cdots + \beta_1) q^{31}+ \cdots + (\beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{3} - 7 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{3} - 7 q^{5} + 3 q^{7} + 7 q^{9} - 5 q^{11} + 5 q^{13} + 7 q^{15} + 3 q^{17} + 2 q^{19} - 3 q^{21} - 3 q^{23} + 7 q^{25} - 7 q^{27} - 8 q^{29} + 7 q^{31} + 5 q^{33} - 3 q^{35} - 5 q^{37} - 5 q^{39} + 7 q^{41} + 3 q^{43} - 7 q^{45} - 6 q^{47} + 16 q^{49} - 3 q^{51} - 9 q^{53} + 5 q^{55} - 2 q^{57} - 22 q^{59} + 19 q^{61} + 3 q^{63} - 5 q^{65} - 7 q^{67} + 3 q^{69} + 23 q^{73} - 7 q^{75} + 9 q^{77} + 25 q^{79} + 7 q^{81} - 20 q^{83} - 3 q^{85} + 8 q^{87} + q^{89} + 23 q^{91} - 7 q^{93} - 2 q^{95} + 3 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 28x^{5} + 90x^{4} + 143x^{3} - 418x^{2} - 256x + 160 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + \nu^{5} - 24\nu^{4} - 10\nu^{3} + 99\nu^{2} + 86\nu + 60 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + \nu^{5} + 26\nu^{4} - 34\nu^{3} - 115\nu^{2} + 64\nu + 60 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + \nu^{5} - 28\nu^{4} - 14\nu^{3} + 187\nu^{2} + 78\nu - 180 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 27\nu^{4} - 10\nu^{3} - 141\nu^{2} - 7\nu + 30 ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{6} + 3\nu^{5} - 188\nu^{4} - 18\nu^{3} + 1017\nu^{2} + 610\nu - 360 ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - \beta_{3} + 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} - 3\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 17\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 25\beta_{5} + 16\beta_{4} - 23\beta_{3} + 4\beta_{2} - 19\beta _1 + 144 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -46\beta_{6} - 83\beta_{5} + 14\beta_{4} + 47\beta_{3} + 28\beta_{2} + 318\beta _1 - 264 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 74\beta_{6} + 554\beta_{5} + 281\beta_{4} - 490\beta_{3} + 98\beta_{2} - 690\beta _1 + 2709 \) 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
−4.78907
−2.11010
−0.941196
0.403734
3.15736
3.41543
3.86385
0 −1.00000 0 −1.00000 0 −4.78907 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −2.11010 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 −0.941196 0 1.00000 0
1.4 0 −1.00000 0 −1.00000 0 0.403734 0 1.00000 0
1.5 0 −1.00000 0 −1.00000 0 3.15736 0 1.00000 0
1.6 0 −1.00000 0 −1.00000 0 3.41543 0 1.00000 0
1.7 0 −1.00000 0 −1.00000 0 3.86385 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.h ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.h ✓ 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{7} - 3T_{7}^{6} - 28T_{7}^{5} + 90T_{7}^{4} + 143T_{7}^{3} - 418T_{7}^{2} - 256T_{7} + 160 \) 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
$7$ \( T^{7} - 3 T^{6} + \cdots + 160 \) Copy content Toggle raw display
$11$ \( T^{7} + 5 T^{6} + \cdots - 10160 \) Copy content Toggle raw display
$13$ \( T^{7} - 5 T^{6} + \cdots + 4796 \) Copy content Toggle raw display
$17$ \( T^{7} - 3 T^{6} + \cdots + 21278 \) Copy content Toggle raw display
$19$ \( T^{7} - 2 T^{6} + \cdots + 3968 \) Copy content Toggle raw display
$23$ \( T^{7} + 3 T^{6} + \cdots + 4688 \) Copy content Toggle raw display
$29$ \( T^{7} + 8 T^{6} + \cdots + 1852 \) Copy content Toggle raw display
$31$ \( T^{7} - 7 T^{6} + \cdots + 67000 \) Copy content Toggle raw display
$37$ \( T^{7} + 5 T^{6} + \cdots + 116 \) Copy content Toggle raw display
$41$ \( T^{7} - 7 T^{6} + \cdots - 2272 \) Copy content Toggle raw display
$43$ \( T^{7} - 3 T^{6} + \cdots - 496 \) Copy content Toggle raw display
$47$ \( T^{7} + 6 T^{6} + \cdots - 200 \) Copy content Toggle raw display
$53$ \( T^{7} + 9 T^{6} + \cdots + 5104 \) Copy content Toggle raw display
$59$ \( T^{7} + 22 T^{6} + \cdots + 93554 \) Copy content Toggle raw display
$61$ \( T^{7} - 19 T^{6} + \cdots - 27008 \) Copy content Toggle raw display
$67$ \( (T + 1)^{7} \) Copy content Toggle raw display
$71$ \( T^{7} - 244 T^{5} + \cdots + 302488 \) Copy content Toggle raw display
$73$ \( T^{7} - 23 T^{6} + \cdots - 12031316 \) Copy content Toggle raw display
$79$ \( T^{7} - 25 T^{6} + \cdots + 51400 \) Copy content Toggle raw display
$83$ \( T^{7} + 20 T^{6} + \cdots + 3121376 \) Copy content Toggle raw display
$89$ \( T^{7} - T^{6} + \cdots - 1620 \) Copy content Toggle raw display
$97$ \( T^{7} - 3 T^{6} + \cdots - 271748 \) Copy content Toggle raw display
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