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Properties

Label 6020.2.a.g
Level $6020$
Weight $2$
Character orbit 6020.a
Self dual yes
Analytic conductor $48.070$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 16x^{7} + 83x^{5} - 9x^{4} - 160x^{3} + 32x^{2} + 77x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + q^{5} - q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + q^{5} - q^{7} + (\beta_{2} + 1) q^{9} + ( - \beta_{4} - \beta_{3}) q^{11} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots - 1) q^{13}+ \cdots + ( - 2 \beta_{8} + 3 \beta_{7} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9} + q^{11} - 14 q^{13} - 11 q^{17} - 2 q^{19} - 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 14 q^{33} - 9 q^{35} - 20 q^{37} - 6 q^{39} - 6 q^{41} - 9 q^{43} + 5 q^{45} + 9 q^{49} - 8 q^{51} - 31 q^{53} + q^{55} - 16 q^{57} + 2 q^{59} - 13 q^{61} - 5 q^{63} - 14 q^{65} - 10 q^{67} - 18 q^{69} + 12 q^{71} - 32 q^{73} - q^{77} + q^{79} - 27 q^{81} - 10 q^{83} - 11 q^{85} - 5 q^{87} - q^{89} + 14 q^{91} - 49 q^{93} - 2 q^{95} - 28 q^{97} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 16x^{7} + 83x^{5} - 9x^{4} - 160x^{3} + 32x^{2} + 77x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{8} - 12\nu^{7} - 7\nu^{6} + 136\nu^{5} - 86\nu^{4} - 442\nu^{3} + 358\nu^{2} + 360\nu - 173 ) / 47 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{8} - 18\nu^{7} - 34\nu^{6} + 204\nu^{5} + 153\nu^{4} - 616\nu^{3} - 309\nu^{2} + 446\nu + 234 ) / 47 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6\nu^{8} + 11\nu^{7} - 68\nu^{6} - 156\nu^{5} + 165\nu^{4} + 507\nu^{3} - 7\nu^{2} - 330\nu - 190 ) / 47 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -12\nu^{8} + 25\nu^{7} + 136\nu^{6} - 252\nu^{5} - 424\nu^{4} + 678\nu^{3} + 296\nu^{2} - 327\nu - 43 ) / 47 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\nu^{8} + 22\nu^{7} - 183\nu^{6} - 265\nu^{5} + 800\nu^{4} + 732\nu^{3} - 1142\nu^{2} - 331\nu + 184 ) / 47 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\nu^{8} + 21\nu^{7} - 258\nu^{6} - 285\nu^{5} + 879\nu^{4} + 844\nu^{3} - 838\nu^{2} - 583\nu + 9 ) / 47 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{6} + 2\beta_{4} - \beta_{3} + 8\beta_{2} + \beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{8} - 9\beta_{7} + 2\beta_{6} - 10\beta_{5} + 2\beta_{4} - 7\beta_{3} + 12\beta_{2} + 39\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{8} - 14\beta_{7} + 12\beta_{6} - 2\beta_{5} + 22\beta_{4} - 11\beta_{3} + 62\beta_{2} + 20\beta _1 + 153 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 86\beta_{8} - 74\beta_{7} + 27\beta_{6} - 82\beta_{5} + 28\beta_{4} - 50\beta_{3} + 118\beta_{2} + 275\beta _1 + 115 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 149 \beta_{8} - 145 \beta_{7} + 111 \beta_{6} - 40 \beta_{5} + 195 \beta_{4} - 103 \beta_{3} + \cdots + 1109 \) 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
−2.49127
−2.42779
−1.89794
−0.701670
0.0257343
1.10254
1.66233
1.79476
2.93329
0 −2.49127 0 1.00000 0 −1.00000 0 3.20641 0
1.2 0 −2.42779 0 1.00000 0 −1.00000 0 2.89414 0
1.3 0 −1.89794 0 1.00000 0 −1.00000 0 0.602159 0
1.4 0 −0.701670 0 1.00000 0 −1.00000 0 −2.50766 0
1.5 0 0.0257343 0 1.00000 0 −1.00000 0 −2.99934 0
1.6 0 1.10254 0 1.00000 0 −1.00000 0 −1.78440 0
1.7 0 1.66233 0 1.00000 0 −1.00000 0 −0.236670 0
1.8 0 1.79476 0 1.00000 0 −1.00000 0 0.221173 0
1.9 0 2.93329 0 1.00000 0 −1.00000 0 5.60419 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.g ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6020.2.a.g ✓ 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6020))\):

\( T_{3}^{9} - 16T_{3}^{7} + 83T_{3}^{5} - 9T_{3}^{4} - 160T_{3}^{3} + 32T_{3}^{2} + 77T_{3} - 2 \) Copy content Toggle raw display
\( T_{11}^{9} - T_{11}^{8} - 42T_{11}^{7} + 547T_{11}^{5} + 275T_{11}^{4} - 2516T_{11}^{3} - 1789T_{11}^{2} + 2898T_{11} + 1611 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 16 T^{7} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( (T + 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} - T^{8} + \cdots + 1611 \) Copy content Toggle raw display
$13$ \( T^{9} + 14 T^{8} + \cdots + 179 \) Copy content Toggle raw display
$17$ \( T^{9} + 11 T^{8} + \cdots + 4013 \) Copy content Toggle raw display
$19$ \( T^{9} + 2 T^{8} + \cdots - 10368 \) Copy content Toggle raw display
$23$ \( T^{9} + 6 T^{8} + \cdots + 11538 \) Copy content Toggle raw display
$29$ \( T^{9} + 6 T^{8} + \cdots - 726052 \) Copy content Toggle raw display
$31$ \( T^{9} - 6 T^{8} + \cdots - 124368 \) Copy content Toggle raw display
$37$ \( T^{9} + 20 T^{8} + \cdots + 307264 \) Copy content Toggle raw display
$41$ \( T^{9} + 6 T^{8} + \cdots + 1086 \) Copy content Toggle raw display
$43$ \( (T + 1)^{9} \) Copy content Toggle raw display
$47$ \( T^{9} - 231 T^{7} + \cdots + 374496 \) Copy content Toggle raw display
$53$ \( T^{9} + 31 T^{8} + \cdots + 2779818 \) Copy content Toggle raw display
$59$ \( T^{9} - 2 T^{8} + \cdots + 16752 \) Copy content Toggle raw display
$61$ \( T^{9} + 13 T^{8} + \cdots - 424968 \) Copy content Toggle raw display
$67$ \( T^{9} + 10 T^{8} + \cdots + 29538 \) Copy content Toggle raw display
$71$ \( T^{9} - 12 T^{8} + \cdots + 30755456 \) Copy content Toggle raw display
$73$ \( T^{9} + 32 T^{8} + \cdots + 8979592 \) Copy content Toggle raw display
$79$ \( T^{9} - T^{8} + \cdots - 157361944 \) Copy content Toggle raw display
$83$ \( T^{9} + 10 T^{8} + \cdots - 58385532 \) Copy content Toggle raw display
$89$ \( T^{9} + T^{8} + \cdots - 8704212 \) Copy content Toggle raw display
$97$ \( T^{9} + 28 T^{8} + \cdots + 480655647 \) Copy content Toggle raw display
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