Properties

Label 6010.2.a.g
Level $6010$
Weight $2$
Character orbit 6010.a
Self dual yes
Analytic conductor $47.990$
Analytic rank $0$
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] = [6010,2,Mod(1,6010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6010 = 2 \cdot 5 \cdot 601 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6010.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: \(47.9900916148\)
Analytic rank: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 27 q^{2} + 6 q^{3} + 27 q^{4} + 27 q^{5} - 6 q^{6} - 27 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q - 27 q^{2} + 6 q^{3} + 27 q^{4} + 27 q^{5} - 6 q^{6} - 27 q^{8} + 37 q^{9} - 27 q^{10} + 18 q^{11} + 6 q^{12} - 6 q^{13} + 6 q^{15} + 27 q^{16} + 3 q^{17} - 37 q^{18} + 27 q^{19} + 27 q^{20} + 16 q^{21} - 18 q^{22} + 15 q^{23} - 6 q^{24} + 27 q^{25} + 6 q^{26} + 27 q^{27} + 25 q^{29} - 6 q^{30} + 9 q^{31} - 27 q^{32} + 11 q^{33} - 3 q^{34} + 37 q^{36} - 16 q^{37} - 27 q^{38} + 20 q^{39} - 27 q^{40} + 39 q^{41} - 16 q^{42} + 9 q^{43} + 18 q^{44} + 37 q^{45} - 15 q^{46} + 31 q^{47} + 6 q^{48} + 27 q^{49} - 27 q^{50} + 39 q^{51} - 6 q^{52} - 5 q^{53} - 27 q^{54} + 18 q^{55} - 10 q^{57} - 25 q^{58} + 46 q^{59} + 6 q^{60} + 18 q^{61} - 9 q^{62} + 23 q^{63} + 27 q^{64} - 6 q^{65} - 11 q^{66} + 11 q^{67} + 3 q^{68} + 17 q^{69} + 50 q^{71} - 37 q^{72} - 29 q^{73} + 16 q^{74} + 6 q^{75} + 27 q^{76} - 6 q^{77} - 20 q^{78} + 56 q^{79} + 27 q^{80} + 51 q^{81} - 39 q^{82} + 44 q^{83} + 16 q^{84} + 3 q^{85} - 9 q^{86} + 42 q^{87} - 18 q^{88} + 34 q^{89} - 37 q^{90} + 43 q^{91} + 15 q^{92} - 20 q^{93} - 31 q^{94} + 27 q^{95} - 6 q^{96} - 37 q^{97} - 27 q^{98} + 67 q^{99}+O(q^{100}) \) 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 −1.00000 −3.12246 1.00000 1.00000 3.12246 −0.227875 −1.00000 6.74974 −1.00000
1.2 −1.00000 −2.89176 1.00000 1.00000 2.89176 4.20794 −1.00000 5.36230 −1.00000
1.3 −1.00000 −2.88499 1.00000 1.00000 2.88499 0.210491 −1.00000 5.32318 −1.00000
1.4 −1.00000 −2.52918 1.00000 1.00000 2.52918 −2.70859 −1.00000 3.39676 −1.00000
1.5 −1.00000 −2.12852 1.00000 1.00000 2.12852 −4.55968 −1.00000 1.53060 −1.00000
1.6 −1.00000 −2.00989 1.00000 1.00000 2.00989 −0.566476 −1.00000 1.03966 −1.00000
1.7 −1.00000 −1.92694 1.00000 1.00000 1.92694 −3.44591 −1.00000 0.713113 −1.00000
1.8 −1.00000 −1.21224 1.00000 1.00000 1.21224 2.32820 −1.00000 −1.53047 −1.00000
1.9 −1.00000 −1.11003 1.00000 1.00000 1.11003 1.51458 −1.00000 −1.76783 −1.00000
1.10 −1.00000 −0.805731 1.00000 1.00000 0.805731 2.46129 −1.00000 −2.35080 −1.00000
1.11 −1.00000 −0.716010 1.00000 1.00000 0.716010 3.53027 −1.00000 −2.48733 −1.00000
1.12 −1.00000 −0.409284 1.00000 1.00000 0.409284 −1.22599 −1.00000 −2.83249 −1.00000
1.13 −1.00000 −0.114121 1.00000 1.00000 0.114121 −3.33992 −1.00000 −2.98698 −1.00000
1.14 −1.00000 −0.0770650 1.00000 1.00000 0.0770650 −1.82853 −1.00000 −2.99406 −1.00000
1.15 −1.00000 1.03782 1.00000 1.00000 −1.03782 −3.65688 −1.00000 −1.92294 −1.00000
1.16 −1.00000 1.08273 1.00000 1.00000 −1.08273 3.52082 −1.00000 −1.82769 −1.00000
1.17 −1.00000 1.23890 1.00000 1.00000 −1.23890 −4.00661 −1.00000 −1.46513 −1.00000
1.18 −1.00000 1.45178 1.00000 1.00000 −1.45178 3.96566 −1.00000 −0.892348 −1.00000
1.19 −1.00000 1.60467 1.00000 1.00000 −1.60467 −3.11642 −1.00000 −0.425041 −1.00000
1.20 −1.00000 1.83794 1.00000 1.00000 −1.83794 2.63564 −1.00000 0.378009 −1.00000
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
\(2\) \(1\)
\(5\) \(-1\)
\(601\) \(1\)

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 6010.2.a.g 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6010.2.a.g 27 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{27} - 6 T_{3}^{26} - 41 T_{3}^{25} + 297 T_{3}^{24} + 641 T_{3}^{23} - 6360 T_{3}^{22} - 3968 T_{3}^{21} + 77392 T_{3}^{20} - 9446 T_{3}^{19} - 591646 T_{3}^{18} + 324442 T_{3}^{17} + 2970438 T_{3}^{16} - 2400660 T_{3}^{15} + \cdots - 11264 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\). Copy content Toggle raw display