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Properties

Label 6020.2.a.f
Level $6020$
Weight $2$
Character orbit 6020.a
Self dual yes
Analytic conductor $48.070$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 12x^{6} + 26x^{5} + 55x^{4} - 52x^{3} - 82x^{2} + 22x + 27 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - q^{5} + q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} - q^{5} + q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + (\beta_{6} + 1) q^{11} + ( - \beta_{5} + \beta_{4} - 1) q^{13} + ( - \beta_1 + 1) q^{15} + (\beta_{5} - \beta_1) q^{17} + ( - \beta_{2} - 2) q^{19} + (\beta_1 - 1) q^{21} + ( - \beta_{7} - \beta_{4} + \beta_{3}) q^{23} + q^{25} + (\beta_{7} - \beta_{6} - \beta_{3} + \cdots - 2) q^{27}+ \cdots + (\beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9} + 6 q^{11} - 13 q^{13} + 5 q^{15} - 2 q^{17} - 14 q^{19} - 5 q^{21} + 6 q^{23} + 8 q^{25} - 11 q^{27} - 7 q^{29} - 18 q^{31} - q^{33} - 8 q^{35} + 2 q^{37} + 9 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - q^{47} + 8 q^{49} - 19 q^{51} + 5 q^{53} - 6 q^{55} - 4 q^{57} - 12 q^{59} - 23 q^{61} + 11 q^{63} + 13 q^{65} + 8 q^{67} - 18 q^{69} + 20 q^{71} - 4 q^{73} - 5 q^{75} + 6 q^{77} + 24 q^{79} - 8 q^{81} - 14 q^{83} + 2 q^{85} + 10 q^{87} - 21 q^{89} - 13 q^{91} + q^{93} + 14 q^{95} + 7 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 12x^{6} + 26x^{5} + 55x^{4} - 52x^{3} - 82x^{2} + 22x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{7} + 23\nu^{6} + 65\nu^{5} - 126\nu^{4} - 349\nu^{3} + 66\nu^{2} + 580\nu - 9 ) / 87 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\nu^{7} - 46\nu^{6} - 130\nu^{5} + 339\nu^{4} + 437\nu^{3} - 480\nu^{2} - 290\nu + 18 ) / 87 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{7} - 9\nu^{6} - 62\nu^{5} + 101\nu^{4} + 274\nu^{3} - 220\nu^{2} - 261\nu + 59 ) / 29 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -25\nu^{7} + 107\nu^{6} + 170\nu^{5} - 885\nu^{4} - 364\nu^{3} + 1926\nu^{2} + 232\nu - 927 ) / 87 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -32\nu^{7} + 130\nu^{6} + 235\nu^{5} - 1011\nu^{4} - 626\nu^{3} + 1818\nu^{2} + 377\nu - 675 ) / 87 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{3} + 2\beta_{2} + 7\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} - 3\beta_{6} + \beta_{4} - \beta_{3} + 10\beta_{2} + 15\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16\beta_{7} - 15\beta_{6} - 2\beta_{5} + 7\beta_{4} - 9\beta_{3} + 28\beta_{2} + 64\beta _1 + 78 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 57\beta_{7} - 51\beta_{6} - 5\beta_{5} + 35\beta_{4} - 17\beta_{3} + 106\beta_{2} + 180\beta _1 + 324 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 232\beta_{7} - 203\beta_{6} - 35\beta_{5} + 162\beta_{4} - 84\beta_{3} + 338\beta_{2} + 659\beta _1 + 1018 \) 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
−2.11466
−1.73402
−1.16692
−0.604219
0.704685
1.41294
2.96620
3.53600
0 −3.11466 0 −1.00000 0 1.00000 0 6.70113 0
1.2 0 −2.73402 0 −1.00000 0 1.00000 0 4.47486 0
1.3 0 −2.16692 0 −1.00000 0 1.00000 0 1.69555 0
1.4 0 −1.60422 0 −1.00000 0 1.00000 0 −0.426481 0
1.5 0 −0.295315 0 −1.00000 0 1.00000 0 −2.91279 0
1.6 0 0.412945 0 −1.00000 0 1.00000 0 −2.82948 0
1.7 0 1.96620 0 −1.00000 0 1.00000 0 0.865935 0
1.8 0 2.53600 0 −1.00000 0 1.00000 0 3.43127 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.f ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6020.2.a.f ✓ 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6020))\):

\( T_{3}^{8} + 5T_{3}^{7} - 5T_{3}^{6} - 53T_{3}^{5} - 30T_{3}^{4} + 139T_{3}^{3} + 137T_{3}^{2} - 33T_{3} - 18 \) Copy content Toggle raw display
\( T_{11}^{8} - 6T_{11}^{7} - 12T_{11}^{6} + 104T_{11}^{5} - 33T_{11}^{4} - 320T_{11}^{3} + 104T_{11}^{2} + 259T_{11} + 59 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{7} + \cdots - 18 \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 59 \) Copy content Toggle raw display
$13$ \( T^{8} + 13 T^{7} + \cdots - 13 \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots - 361 \) Copy content Toggle raw display
$19$ \( T^{8} + 14 T^{7} + \cdots - 572 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 603 \) Copy content Toggle raw display
$29$ \( T^{8} + 7 T^{7} + \cdots - 40688 \) Copy content Toggle raw display
$31$ \( T^{8} + 18 T^{7} + \cdots - 49933 \) Copy content Toggle raw display
$37$ \( T^{8} - 2 T^{7} + \cdots + 42418 \) Copy content Toggle raw display
$41$ \( T^{8} + 18 T^{7} + \cdots - 456431 \) Copy content Toggle raw display
$43$ \( (T + 1)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + T^{7} + \cdots + 1662112 \) Copy content Toggle raw display
$53$ \( T^{8} - 5 T^{7} + \cdots - 780417 \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + \cdots + 12116 \) Copy content Toggle raw display
$61$ \( T^{8} + 23 T^{7} + \cdots - 435898 \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} + \cdots - 13257 \) Copy content Toggle raw display
$71$ \( T^{8} - 20 T^{7} + \cdots + 105516 \) Copy content Toggle raw display
$73$ \( T^{8} + 4 T^{7} + \cdots + 1369008 \) Copy content Toggle raw display
$79$ \( T^{8} - 24 T^{7} + \cdots - 11738728 \) Copy content Toggle raw display
$83$ \( T^{8} + 14 T^{7} + \cdots + 3566611 \) Copy content Toggle raw display
$89$ \( T^{8} + 21 T^{7} + \cdots + 12928166 \) Copy content Toggle raw display
$97$ \( T^{8} - 7 T^{7} + \cdots + 5598801 \) Copy content Toggle raw display
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