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Properties

Label 6010.2.a.e
Level $6010$
Weight $2$
Character orbit 6010.a
Self dual yes
Analytic conductor $47.990$
Analytic rank $0$
Dimension $21$
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: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 21 q^{2} + 8 q^{3} + 21 q^{4} - 21 q^{5} - 8 q^{6} - 21 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q - 21 q^{2} + 8 q^{3} + 21 q^{4} - 21 q^{5} - 8 q^{6} - 21 q^{8} + 15 q^{9} + 21 q^{10} + 8 q^{11} + 8 q^{12} + 2 q^{13} - 8 q^{15} + 21 q^{16} + 25 q^{17} - 15 q^{18} - 11 q^{19} - 21 q^{20} - 8 q^{21} - 8 q^{22} + 15 q^{23} - 8 q^{24} + 21 q^{25} - 2 q^{26} + 29 q^{27} + 3 q^{29} + 8 q^{30} - 19 q^{31} - 21 q^{32} + 11 q^{33} - 25 q^{34} + 15 q^{36} - 8 q^{37} + 11 q^{38} - 2 q^{39} + 21 q^{40} + 15 q^{41} + 8 q^{42} + 19 q^{43} + 8 q^{44} - 15 q^{45} - 15 q^{46} + 19 q^{47} + 8 q^{48} - 15 q^{49} - 21 q^{50} + 13 q^{51} + 2 q^{52} + 45 q^{53} - 29 q^{54} - 8 q^{55} + 22 q^{57} - 3 q^{58} + 34 q^{59} - 8 q^{60} - 26 q^{61} + 19 q^{62} + 5 q^{63} + 21 q^{64} - 2 q^{65} - 11 q^{66} + 19 q^{67} + 25 q^{68} - 3 q^{69} + 10 q^{71} - 15 q^{72} - 17 q^{73} + 8 q^{74} + 8 q^{75} - 11 q^{76} + 42 q^{77} + 2 q^{78} - 42 q^{79} - 21 q^{80} + q^{81} - 15 q^{82} + 76 q^{83} - 8 q^{84} - 25 q^{85} - 19 q^{86} + 6 q^{87} - 8 q^{88} + 14 q^{89} + 15 q^{90} - 19 q^{91} + 15 q^{92} + 2 q^{93} - 19 q^{94} + 11 q^{95} - 8 q^{96} - 9 q^{97} + 15 q^{98} + 21 q^{99}+O(q^{100}) \) 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 −1.00000 −2.89060 1.00000 −1.00000 2.89060 1.07596 −1.00000 5.35558 1.00000
1.2 −1.00000 −2.14296 1.00000 −1.00000 2.14296 0.736597 −1.00000 1.59228 1.00000
1.3 −1.00000 −1.93787 1.00000 −1.00000 1.93787 −0.661252 −1.00000 0.755329 1.00000
1.4 −1.00000 −1.82478 1.00000 −1.00000 1.82478 −1.84575 −1.00000 0.329836 1.00000
1.5 −1.00000 −1.73819 1.00000 −1.00000 1.73819 4.80844 −1.00000 0.0213193 1.00000
1.6 −1.00000 −1.41773 1.00000 −1.00000 1.41773 −1.60089 −1.00000 −0.990039 1.00000
1.7 −1.00000 −1.41757 1.00000 −1.00000 1.41757 −2.14903 −1.00000 −0.990504 1.00000
1.8 −1.00000 −0.220793 1.00000 −1.00000 0.220793 1.03342 −1.00000 −2.95125 1.00000
1.9 −1.00000 0.0461180 1.00000 −1.00000 −0.0461180 2.74163 −1.00000 −2.99787 1.00000
1.10 −1.00000 0.464276 1.00000 −1.00000 −0.464276 −1.37823 −1.00000 −2.78445 1.00000
1.11 −1.00000 0.474795 1.00000 −1.00000 −0.474795 0.427918 −1.00000 −2.77457 1.00000
1.12 −1.00000 0.478752 1.00000 −1.00000 −0.478752 0.579883 −1.00000 −2.77080 1.00000
1.13 −1.00000 1.19881 1.00000 −1.00000 −1.19881 2.82809 −1.00000 −1.56285 1.00000
1.14 −1.00000 1.26768 1.00000 −1.00000 −1.26768 −4.76657 −1.00000 −1.39299 1.00000
1.15 −1.00000 1.46860 1.00000 −1.00000 −1.46860 −3.37454 −1.00000 −0.843201 1.00000
1.16 −1.00000 1.81728 1.00000 −1.00000 −1.81728 0.464151 −1.00000 0.302521 1.00000
1.17 −1.00000 2.50288 1.00000 −1.00000 −2.50288 1.45988 −1.00000 3.26441 1.00000
1.18 −1.00000 2.66896 1.00000 −1.00000 −2.66896 3.89028 −1.00000 4.12335 1.00000
1.19 −1.00000 2.82986 1.00000 −1.00000 −2.82986 −4.43096 −1.00000 5.00809 1.00000
1.20 −1.00000 3.15846 1.00000 −1.00000 −3.15846 −1.63952 −1.00000 6.97584 1.00000
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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.e ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6010.2.a.e ✓ 21 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{21} - 8 T_{3}^{20} - 7 T_{3}^{19} + 201 T_{3}^{18} - 240 T_{3}^{17} - 1978 T_{3}^{16} + 4141 T_{3}^{15} + \cdots - 64 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\). Copy content Toggle raw display