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Properties

Label 6010.2.a.f
Level $6010$
Weight $2$
Character orbit 6010.a
Self dual yes
Analytic conductor $47.990$
Analytic rank $1$
Dimension $22$
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: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 22 q^{2} - 6 q^{3} + 22 q^{4} - 22 q^{5} - 6 q^{6} - 12 q^{7} + 22 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 22 q^{2} - 6 q^{3} + 22 q^{4} - 22 q^{5} - 6 q^{6} - 12 q^{7} + 22 q^{8} + 12 q^{9} - 22 q^{10} - 4 q^{11} - 6 q^{12} - 20 q^{13} - 12 q^{14} + 6 q^{15} + 22 q^{16} - 23 q^{17} + 12 q^{18} + q^{19} - 22 q^{20} - 8 q^{21} - 4 q^{22} - 17 q^{23} - 6 q^{24} + 22 q^{25} - 20 q^{26} - 21 q^{27} - 12 q^{28} - 13 q^{29} + 6 q^{30} - 13 q^{31} + 22 q^{32} - 21 q^{33} - 23 q^{34} + 12 q^{35} + 12 q^{36} - 16 q^{37} + q^{38} - 4 q^{39} - 22 q^{40} - 31 q^{41} - 8 q^{42} - 9 q^{43} - 4 q^{44} - 12 q^{45} - 17 q^{46} - 41 q^{47} - 6 q^{48} - 6 q^{49} + 22 q^{50} - 7 q^{51} - 20 q^{52} - 15 q^{53} - 21 q^{54} + 4 q^{55} - 12 q^{56} - 26 q^{57} - 13 q^{58} - 32 q^{59} + 6 q^{60} - 22 q^{61} - 13 q^{62} - 55 q^{63} + 22 q^{64} + 20 q^{65} - 21 q^{66} - 19 q^{67} - 23 q^{68} - 37 q^{69} + 12 q^{70} - 36 q^{71} + 12 q^{72} - 47 q^{73} - 16 q^{74} - 6 q^{75} + q^{76} - 26 q^{77} - 4 q^{78} - 10 q^{79} - 22 q^{80} - 18 q^{81} - 31 q^{82} - 48 q^{83} - 8 q^{84} + 23 q^{85} - 9 q^{86} - 50 q^{87} - 4 q^{88} - 42 q^{89} - 12 q^{90} + 25 q^{91} - 17 q^{92} - 48 q^{93} - 41 q^{94} - q^{95} - 6 q^{96} - 67 q^{97} - 6 q^{98} - 3 q^{99}+O(q^{100}) \) 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 −3.26920 1.00000 −1.00000 −3.26920 0.898513 1.00000 7.68765 −1.00000
1.2 1.00000 −3.13304 1.00000 −1.00000 −3.13304 −3.32735 1.00000 6.81597 −1.00000
1.3 1.00000 −2.63651 1.00000 −1.00000 −2.63651 2.17132 1.00000 3.95120 −1.00000
1.4 1.00000 −2.29642 1.00000 −1.00000 −2.29642 −4.86493 1.00000 2.27356 −1.00000
1.5 1.00000 −1.98091 1.00000 −1.00000 −1.98091 −1.31383 1.00000 0.924017 −1.00000
1.6 1.00000 −1.88167 1.00000 −1.00000 −1.88167 −3.66727 1.00000 0.540685 −1.00000
1.7 1.00000 −1.55148 1.00000 −1.00000 −1.55148 2.51624 1.00000 −0.592901 −1.00000
1.8 1.00000 −1.51693 1.00000 −1.00000 −1.51693 −1.44279 1.00000 −0.698933 −1.00000
1.9 1.00000 −1.30771 1.00000 −1.00000 −1.30771 4.54753 1.00000 −1.28989 −1.00000
1.10 1.00000 −1.30184 1.00000 −1.00000 −1.30184 0.511613 1.00000 −1.30521 −1.00000
1.11 1.00000 −0.310973 1.00000 −1.00000 −0.310973 1.47909 1.00000 −2.90330 −1.00000
1.12 1.00000 0.0740024 1.00000 −1.00000 0.0740024 2.79514 1.00000 −2.99452 −1.00000
1.13 1.00000 0.214164 1.00000 −1.00000 0.214164 −0.767476 1.00000 −2.95413 −1.00000
1.14 1.00000 0.530792 1.00000 −1.00000 0.530792 −3.74336 1.00000 −2.71826 −1.00000
1.15 1.00000 0.724060 1.00000 −1.00000 0.724060 −1.42945 1.00000 −2.47574 −1.00000
1.16 1.00000 1.17759 1.00000 −1.00000 1.17759 1.86080 1.00000 −1.61328 −1.00000
1.17 1.00000 1.34202 1.00000 −1.00000 1.34202 2.38251 1.00000 −1.19898 −1.00000
1.18 1.00000 1.73340 1.00000 −1.00000 1.73340 −0.557678 1.00000 0.00467827 −1.00000
1.19 1.00000 2.04538 1.00000 −1.00000 2.04538 −3.30792 1.00000 1.18360 −1.00000
1.20 1.00000 2.14756 1.00000 −1.00000 2.14756 −1.75294 1.00000 1.61200 −1.00000
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
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.f ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6010.2.a.f ✓ 22 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{22} + 6 T_{3}^{21} - 21 T_{3}^{20} - 179 T_{3}^{19} + 105 T_{3}^{18} + 2208 T_{3}^{17} + \cdots - 140 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\). Copy content Toggle raw display