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Properties

Label 6004.2.a.f
Level $6004$
Weight $2$
Character orbit 6004.a
Self dual yes
Analytic conductor $47.942$
Analytic rank $1$
Dimension $25$
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: \(1\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −3.10476 0 −3.52338 0 0.220122 0 6.63952 0
1.2 0 −2.61174 0 0.965145 0 0.0126812 0 3.82116 0
1.3 0 −2.45149 0 1.53010 0 1.05276 0 3.00979 0
1.4 0 −2.38552 0 −2.90737 0 −2.69828 0 2.69068 0
1.5 0 −1.61532 0 1.18505 0 3.44544 0 −0.390754 0
1.6 0 −1.50951 0 −2.01073 0 3.94863 0 −0.721387 0
1.7 0 −1.45593 0 2.84743 0 −2.57267 0 −0.880256 0
1.8 0 −1.19435 0 3.68128 0 2.98419 0 −1.57353 0
1.9 0 −0.904784 0 −1.47158 0 0.136041 0 −2.18137 0
1.10 0 −0.891665 0 −1.39999 0 3.43195 0 −2.20493 0
1.11 0 −0.642577 0 −3.12522 0 −2.66487 0 −2.58709 0
1.12 0 −0.0249148 0 1.33267 0 −1.80071 0 −2.99938 0
1.13 0 0.565175 0 2.63421 0 0.231397 0 −2.68058 0
1.14 0 0.681472 0 −3.07187 0 −2.21919 0 −2.53560 0
1.15 0 0.749226 0 −1.84486 0 4.01817 0 −2.43866 0
1.16 0 1.01059 0 3.45227 0 −1.97653 0 −1.97870 0
1.17 0 1.32467 0 −0.559509 0 4.61440 0 −1.24526 0
1.18 0 1.53952 0 0.492772 0 −1.61008 0 −0.629866 0
1.19 0 1.71727 0 1.91187 0 −0.564608 0 −0.0509783 0
1.20 0 1.83994 0 −0.734016 0 1.69773 0 0.385379 0
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
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.f ✓ 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6004.2.a.f ✓ 25 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):

\( T_{3}^{25} - 4 T_{3}^{24} - 36 T_{3}^{23} + 155 T_{3}^{22} + 531 T_{3}^{21} - 2547 T_{3}^{20} + \cdots - 626 \) Copy content Toggle raw display
\( T_{5}^{25} + 8 T_{5}^{24} - 41 T_{5}^{23} - 453 T_{5}^{22} + 479 T_{5}^{21} + 10863 T_{5}^{20} + \cdots + 1440253 \) Copy content Toggle raw display