Properties

Label 6004.2.a.g
Level $6004$
Weight $2$
Character orbit 6004.a
Self dual yes
Analytic conductor $47.942$
Analytic rank $1$
Dimension $27$
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: \(27\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −3.35817 0 −0.581328 0 3.25261 0 8.27730 0
1.2 0 −2.81775 0 −0.243389 0 0.929492 0 4.93969 0
1.3 0 −2.74782 0 −0.322226 0 −1.10048 0 4.55049 0
1.4 0 −2.67388 0 3.55353 0 1.65788 0 4.14966 0
1.5 0 −2.14389 0 2.12568 0 −4.35707 0 1.59626 0
1.6 0 −2.00223 0 −2.46425 0 4.06602 0 1.00893 0
1.7 0 −1.94547 0 −1.68966 0 −1.54452 0 0.784840 0
1.8 0 −1.68125 0 −3.32036 0 3.49635 0 −0.173410 0
1.9 0 −1.59467 0 2.11777 0 0.323115 0 −0.457036 0
1.10 0 −1.57318 0 −4.10446 0 −3.99696 0 −0.525101 0
1.11 0 −0.907672 0 1.31635 0 −2.29510 0 −2.17613 0
1.12 0 −0.604994 0 2.44978 0 2.35442 0 −2.63398 0
1.13 0 −0.338800 0 −3.08469 0 −4.48412 0 −2.88521 0
1.14 0 −0.338332 0 0.413944 0 −1.61145 0 −2.88553 0
1.15 0 0.267543 0 3.98483 0 −3.08625 0 −2.92842 0
1.16 0 0.314419 0 −4.24837 0 1.07745 0 −2.90114 0
1.17 0 0.551494 0 −2.33245 0 1.71388 0 −2.69585 0
1.18 0 0.739341 0 2.08289 0 4.12124 0 −2.45337 0
1.19 0 1.21945 0 1.36200 0 −2.61900 0 −1.51294 0
1.20 0 1.41351 0 −0.774385 0 −1.47209 0 −1.00198 0
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
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.g 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6004.2.a.g 27 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):

\( T_{3}^{27} + 4 T_{3}^{26} - 42 T_{3}^{25} - 179 T_{3}^{24} + 739 T_{3}^{23} + 3445 T_{3}^{22} - 7046 T_{3}^{21} - 37466 T_{3}^{20} + 38976 T_{3}^{19} + 254518 T_{3}^{18} - 120597 T_{3}^{17} - 1128252 T_{3}^{16} + 151681 T_{3}^{15} + \cdots - 2560 \) Copy content Toggle raw display
\( T_{5}^{27} + 10 T_{5}^{26} - 24 T_{5}^{25} - 523 T_{5}^{24} - 440 T_{5}^{23} + 11116 T_{5}^{22} + 22580 T_{5}^{21} - 126256 T_{5}^{20} - 351707 T_{5}^{19} + 851553 T_{5}^{18} + 2946572 T_{5}^{17} - 3585634 T_{5}^{16} + \cdots + 108441 \) Copy content Toggle raw display