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Properties

Label 6010.2.a.d
Level $6010$
Weight $2$
Character orbit 6010.a
Self dual yes
Analytic conductor $47.990$
Analytic rank $1$
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: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 21 q^{2} - q^{3} + 21 q^{4} + 21 q^{5} + q^{6} - 21 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q - 21 q^{2} - q^{3} + 21 q^{4} + 21 q^{5} + q^{6} - 21 q^{8} + 12 q^{9} - 21 q^{10} - 22 q^{11} - q^{12} - 2 q^{13} - q^{15} + 21 q^{16} - 7 q^{17} - 12 q^{18} - 27 q^{19} + 21 q^{20} - 18 q^{21} + 22 q^{22} - 15 q^{23} + q^{24} + 21 q^{25} + 2 q^{26} - 4 q^{27} - 24 q^{29} + q^{30} - 20 q^{31} - 21 q^{32} + 21 q^{33} + 7 q^{34} + 12 q^{36} + 12 q^{37} + 27 q^{38} - 20 q^{39} - 21 q^{40} - 27 q^{41} + 18 q^{42} - 21 q^{43} - 22 q^{44} + 12 q^{45} + 15 q^{46} - 9 q^{47} - q^{48} + 17 q^{49} - 21 q^{50} - 43 q^{51} - 2 q^{52} - 8 q^{53} + 4 q^{54} - 22 q^{55} - 14 q^{57} + 24 q^{58} - 64 q^{59} - q^{60} - 26 q^{61} + 20 q^{62} + q^{63} + 21 q^{64} - 2 q^{65} - 21 q^{66} - 4 q^{67} - 7 q^{68} - 13 q^{69} - 31 q^{71} - 12 q^{72} + 23 q^{73} - 12 q^{74} - q^{75} - 27 q^{76} - 14 q^{77} + 20 q^{78} - 29 q^{79} + 21 q^{80} - 43 q^{81} + 27 q^{82} - 36 q^{83} - 18 q^{84} - 7 q^{85} + 21 q^{86} - 35 q^{87} + 22 q^{88} - 43 q^{89} - 12 q^{90} - 75 q^{91} - 15 q^{92} - 9 q^{93} + 9 q^{94} - 27 q^{95} + q^{96} + 41 q^{97} - 17 q^{98} - 107 q^{99}+O(q^{100}) \) 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 −3.16548 1.00000 1.00000 3.16548 4.36641 −1.00000 7.02023 −1.00000
1.2 −1.00000 −2.58313 1.00000 1.00000 2.58313 −4.43901 −1.00000 3.67258 −1.00000
1.3 −1.00000 −2.31293 1.00000 1.00000 2.31293 −0.119805 −1.00000 2.34966 −1.00000
1.4 −1.00000 −2.23576 1.00000 1.00000 2.23576 1.08514 −1.00000 1.99861 −1.00000
1.5 −1.00000 −2.07728 1.00000 1.00000 2.07728 −2.70771 −1.00000 1.31508 −1.00000
1.6 −1.00000 −1.89868 1.00000 1.00000 1.89868 3.14560 −1.00000 0.605001 −1.00000
1.7 −1.00000 −1.64904 1.00000 1.00000 1.64904 4.21137 −1.00000 −0.280653 −1.00000
1.8 −1.00000 −0.768651 1.00000 1.00000 0.768651 0.702391 −1.00000 −2.40918 −1.00000
1.9 −1.00000 −0.596244 1.00000 1.00000 0.596244 0.300294 −1.00000 −2.64449 −1.00000
1.10 −1.00000 −0.337730 1.00000 1.00000 0.337730 −4.14906 −1.00000 −2.88594 −1.00000
1.11 −1.00000 −0.287692 1.00000 1.00000 0.287692 −2.91926 −1.00000 −2.91723 −1.00000
1.12 −1.00000 −0.129218 1.00000 1.00000 0.129218 0.702356 −1.00000 −2.98330 −1.00000
1.13 −1.00000 0.867143 1.00000 1.00000 −0.867143 2.93618 −1.00000 −2.24806 −1.00000
1.14 −1.00000 1.33381 1.00000 1.00000 −1.33381 4.17566 −1.00000 −1.22096 −1.00000
1.15 −1.00000 1.34686 1.00000 1.00000 −1.34686 −2.49387 −1.00000 −1.18597 −1.00000
1.16 −1.00000 1.44989 1.00000 1.00000 −1.44989 −0.786145 −1.00000 −0.897809 −1.00000
1.17 −1.00000 1.95440 1.00000 1.00000 −1.95440 1.07262 −1.00000 0.819671 −1.00000
1.18 −1.00000 2.06697 1.00000 1.00000 −2.06697 −2.56157 −1.00000 1.27235 −1.00000
1.19 −1.00000 2.41398 1.00000 1.00000 −2.41398 −0.315761 −1.00000 2.82729 −1.00000
1.20 −1.00000 2.62573 1.00000 1.00000 −2.62573 1.98922 −1.00000 3.89445 −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.d ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6010.2.a.d ✓ 21 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{21} + T_{3}^{20} - 37 T_{3}^{19} - 34 T_{3}^{18} + 577 T_{3}^{17} + 484 T_{3}^{16} - 4958 T_{3}^{15} + \cdots - 273 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\). Copy content Toggle raw display