Properties

Label 6004.2.a.d
Level $6004$
Weight $2$
Character orbit 6004.a
Self dual yes
Analytic conductor $47.942$
Analytic rank $0$
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] = [6004,2,Mod(1,6004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6004.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: \(47.9421813736\)
Analytic rank: \(0\)
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: \( x^{8} - 4x^{7} - 15x^{6} + 56x^{5} + 87x^{4} - 248x^{3} - 241x^{2} + 340x + 248 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - \beta_{4} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} - \beta_{4} q^{7} - 3 q^{9} + ( - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{6} - \beta_{4} - \beta_{2} + \beta_1 - 2) q^{13} + (\beta_{6} - \beta_{5} - 1) q^{17} + q^{19} + (\beta_{7} - \beta_{3} + \beta_{2} + 2) q^{23} + (\beta_{6} - \beta_{5} + \beta_{2} + \beta_1) q^{25} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{29} + ( - 2 \beta_{7} + \beta_{5} - \beta_{4} - 3 \beta_{2} + 2 \beta_1 - 1) q^{31} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} + 2 \beta_1 - 2) q^{35} + (2 \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{37} + ( - \beta_{6} - 2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{41} + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{43} - 3 \beta_1 q^{45} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{47} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{49} + (\beta_{5} + 3 \beta_{3} + 2 \beta_{2} - \beta_1) q^{53} + (\beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 1) q^{55} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{59} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{61} + 3 \beta_{4} q^{63} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{65} + (\beta_{6} - \beta_{5} - \beta_1 + 3) q^{67} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 6) q^{71} + ( - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{73} + (\beta_{7} - 3 \beta_{6} - \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + 6 \beta_1 + 1) q^{77} + q^{79} + 9 q^{81} + (\beta_{7} - 2 \beta_{5} - \beta_{3} + \beta_{2}) q^{83} + ( - 2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{3} - 3) q^{85} + (\beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{89} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 6) q^{91} + \beta_1 q^{95} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 4) q^{97} + (3 \beta_{7} - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} + q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} + q^{7} - 24 q^{9} - 7 q^{11} - 12 q^{13} - 7 q^{17} + 8 q^{19} + 10 q^{23} + 6 q^{25} + 5 q^{29} + 3 q^{31} - 11 q^{35} - 15 q^{37} - 5 q^{41} + 10 q^{43} - 12 q^{45} + 18 q^{47} + 31 q^{49} + 9 q^{53} - 17 q^{55} + 8 q^{59} + 13 q^{61} - 3 q^{63} + 4 q^{65} + 21 q^{67} + 44 q^{71} - 20 q^{73} + 15 q^{77} + 8 q^{79} + 72 q^{81} - 4 q^{83} - 9 q^{85} - 10 q^{89} + 48 q^{91} + 4 q^{95} - 22 q^{97} + 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 15x^{6} + 56x^{5} + 87x^{4} - 248x^{3} - 241x^{2} + 340x + 248 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 4\nu^{5} - 10\nu^{4} + 36\nu^{3} + 37\nu^{2} - 68\nu - 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} + 14\nu^{5} - 26\nu^{4} - 73\nu^{3} + 31\nu^{2} + 140\nu + 72 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 4\nu^{6} - 10\nu^{5} + 40\nu^{4} + 25\nu^{3} - 104\nu^{2} + 4\nu + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 5\nu^{6} - 8\nu^{5} + 56\nu^{4} + 7\nu^{3} - 171\nu^{2} + 32\nu + 112 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 6\nu^{6} - 4\nu^{5} + 66\nu^{4} - 29\nu^{3} - 200\nu^{2} + 92\nu + 128 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 7\nu^{6} + 76\nu^{4} - 69\nu^{3} - 229\nu^{2} + 188\nu + 160 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} + 4\beta_{6} - 2\beta_{5} + 9\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{7} + 21\beta_{6} - 15\beta_{5} + 2\beta_{4} + 4\beta_{3} + 11\beta_{2} + 17\beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -36\beta_{7} + 84\beta_{6} - 52\beta_{5} + 10\beta_{4} + 12\beta_{3} + 14\beta_{2} + 97\beta _1 + 68 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -132\beta_{7} + 365\beta_{6} - 249\beta_{5} + 60\beta_{4} + 88\beta_{3} + 137\beta_{2} + 265\beta _1 + 369 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -598\beta_{7} + 1464\beta_{6} - 970\beta_{5} + 268\beta_{4} + 312\beta_{3} + 352\beta_{2} + 1225\beta _1 + 1032 \) 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
−2.49694
−1.94482
−1.76571
−0.654712
1.60188
2.45095
2.82169
3.98766
0 0 0 −2.49694 0 4.08498 0 −3.00000 0
1.2 0 0 0 −1.94482 0 −0.969069 0 −3.00000 0
1.3 0 0 0 −1.76571 0 2.40269 0 −3.00000 0
1.4 0 0 0 −0.654712 0 −2.24646 0 −3.00000 0
1.5 0 0 0 1.60188 0 −2.96099 0 −3.00000 0
1.6 0 0 0 2.45095 0 −5.04197 0 −3.00000 0
1.7 0 0 0 2.82169 0 4.85844 0 −3.00000 0
1.8 0 0 0 3.98766 0 0.872374 0 −3.00000 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
\(2\) \(-1\)
\(19\) \(-1\)
\(79\) \(-1\)

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 6004.2.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6004.2.a.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{8} - 4T_{5}^{7} - 15T_{5}^{6} + 56T_{5}^{5} + 87T_{5}^{4} - 248T_{5}^{3} - 241T_{5}^{2} + 340T_{5} + 248 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} - 15 T^{6} + 56 T^{5} + \cdots + 248 \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} - 43 T^{6} + 33 T^{5} + \cdots + 1352 \) Copy content Toggle raw display
$11$ \( T^{8} + 7 T^{7} - 43 T^{6} + \cdots + 1792 \) Copy content Toggle raw display
$13$ \( T^{8} + 12 T^{7} - 422 T^{5} + \cdots - 50176 \) Copy content Toggle raw display
$17$ \( T^{8} + 7 T^{7} - 33 T^{6} + \cdots - 1664 \) Copy content Toggle raw display
$19$ \( (T - 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 10 T^{7} - 36 T^{6} + \cdots + 50944 \) Copy content Toggle raw display
$29$ \( T^{8} - 5 T^{7} - 109 T^{6} + \cdots + 15472 \) Copy content Toggle raw display
$31$ \( T^{8} - 3 T^{7} - 163 T^{6} + \cdots - 283552 \) Copy content Toggle raw display
$37$ \( T^{8} + 15 T^{7} - 59 T^{6} + \cdots - 151388 \) Copy content Toggle raw display
$41$ \( T^{8} + 5 T^{7} - 137 T^{6} + \cdots + 85172 \) Copy content Toggle raw display
$43$ \( T^{8} - 10 T^{7} - 49 T^{6} + \cdots + 12334 \) Copy content Toggle raw display
$47$ \( T^{8} - 18 T^{7} - 113 T^{6} + \cdots - 634922 \) Copy content Toggle raw display
$53$ \( T^{8} - 9 T^{7} - 243 T^{6} + \cdots - 2381252 \) Copy content Toggle raw display
$59$ \( T^{8} - 8 T^{7} - 220 T^{6} + \cdots + 143872 \) Copy content Toggle raw display
$61$ \( T^{8} - 13 T^{7} - 75 T^{6} + \cdots - 479744 \) Copy content Toggle raw display
$67$ \( T^{8} - 21 T^{7} + 111 T^{6} + \cdots - 6184 \) Copy content Toggle raw display
$71$ \( T^{8} - 44 T^{7} + 714 T^{6} + \cdots + 307328 \) Copy content Toggle raw display
$73$ \( T^{8} + 20 T^{7} - 7 T^{6} + \cdots - 4804 \) Copy content Toggle raw display
$79$ \( (T - 1)^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 4 T^{7} - 88 T^{6} + \cdots - 1024 \) Copy content Toggle raw display
$89$ \( T^{8} + 10 T^{7} - 280 T^{6} + \cdots + 2323328 \) Copy content Toggle raw display
$97$ \( T^{8} + 22 T^{7} - 16 T^{6} + \cdots - 861184 \) Copy content Toggle raw display
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