Properties

Label 6010.2.a.h
Level $6010$
Weight $2$
Character orbit 6010.a
Self dual yes
Analytic conductor $47.990$
Analytic rank $0$
Dimension $28$
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: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{2} + 4 q^{3} + 28 q^{4} - 28 q^{5} + 4 q^{6} + 10 q^{7} + 28 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{2} + 4 q^{3} + 28 q^{4} - 28 q^{5} + 4 q^{6} + 10 q^{7} + 28 q^{8} + 40 q^{9} - 28 q^{10} + 4 q^{11} + 4 q^{12} + 22 q^{13} + 10 q^{14} - 4 q^{15} + 28 q^{16} + 15 q^{17} + 40 q^{18} - 11 q^{19} - 28 q^{20} + 18 q^{21} + 4 q^{22} + 23 q^{23} + 4 q^{24} + 28 q^{25} + 22 q^{26} + 19 q^{27} + 10 q^{28} + 19 q^{29} - 4 q^{30} + 7 q^{31} + 28 q^{32} + 33 q^{33} + 15 q^{34} - 10 q^{35} + 40 q^{36} + 22 q^{37} - 11 q^{38} + 8 q^{39} - 28 q^{40} + 41 q^{41} + 18 q^{42} + 7 q^{43} + 4 q^{44} - 40 q^{45} + 23 q^{46} + 51 q^{47} + 4 q^{48} + 60 q^{49} + 28 q^{50} - 5 q^{51} + 22 q^{52} + 25 q^{53} + 19 q^{54} - 4 q^{55} + 10 q^{56} + 8 q^{57} + 19 q^{58} + 32 q^{59} - 4 q^{60} + 24 q^{61} + 7 q^{62} + 33 q^{63} + 28 q^{64} - 22 q^{65} + 33 q^{66} + 3 q^{67} + 15 q^{68} + 43 q^{69} - 10 q^{70} + 8 q^{71} + 40 q^{72} + 47 q^{73} + 22 q^{74} + 4 q^{75} - 11 q^{76} + 46 q^{77} + 8 q^{78} - 22 q^{79} - 28 q^{80} + 76 q^{81} + 41 q^{82} + 36 q^{83} + 18 q^{84} - 15 q^{85} + 7 q^{86} + 72 q^{87} + 4 q^{88} + 70 q^{89} - 40 q^{90} - 21 q^{91} + 23 q^{92} + 24 q^{93} + 51 q^{94} + 11 q^{95} + 4 q^{96} + 43 q^{97} + 60 q^{98} - 9 q^{99}+O(q^{100}) \) 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 −3.20632 1.00000 −1.00000 −3.20632 −2.43054 1.00000 7.28047 −1.00000
1.2 1.00000 −2.95613 1.00000 −1.00000 −2.95613 −3.28957 1.00000 5.73868 −1.00000
1.3 1.00000 −2.81740 1.00000 −1.00000 −2.81740 −2.15395 1.00000 4.93776 −1.00000
1.4 1.00000 −2.80565 1.00000 −1.00000 −2.80565 2.92877 1.00000 4.87166 −1.00000
1.5 1.00000 −2.45664 1.00000 −1.00000 −2.45664 3.60299 1.00000 3.03507 −1.00000
1.6 1.00000 −2.33584 1.00000 −1.00000 −2.33584 4.65545 1.00000 2.45616 −1.00000
1.7 1.00000 −1.85629 1.00000 −1.00000 −1.85629 2.08107 1.00000 0.445825 −1.00000
1.8 1.00000 −1.73502 1.00000 −1.00000 −1.73502 −0.456244 1.00000 0.0103032 −1.00000
1.9 1.00000 −1.14577 1.00000 −1.00000 −1.14577 0.345550 1.00000 −1.68722 −1.00000
1.10 1.00000 −0.869813 1.00000 −1.00000 −0.869813 −1.24307 1.00000 −2.24343 −1.00000
1.11 1.00000 −0.734514 1.00000 −1.00000 −0.734514 0.223072 1.00000 −2.46049 −1.00000
1.12 1.00000 −0.503837 1.00000 −1.00000 −0.503837 −4.37336 1.00000 −2.74615 −1.00000
1.13 1.00000 −0.134464 1.00000 −1.00000 −0.134464 4.82249 1.00000 −2.98192 −1.00000
1.14 1.00000 0.184156 1.00000 −1.00000 0.184156 −4.92010 1.00000 −2.96609 −1.00000
1.15 1.00000 0.321033 1.00000 −1.00000 0.321033 −2.34471 1.00000 −2.89694 −1.00000
1.16 1.00000 0.519061 1.00000 −1.00000 0.519061 −3.03480 1.00000 −2.73058 −1.00000
1.17 1.00000 0.657748 1.00000 −1.00000 0.657748 4.85947 1.00000 −2.56737 −1.00000
1.18 1.00000 0.983549 1.00000 −1.00000 0.983549 0.242326 1.00000 −2.03263 −1.00000
1.19 1.00000 1.25977 1.00000 −1.00000 1.25977 3.22153 1.00000 −1.41297 −1.00000
1.20 1.00000 1.41587 1.00000 −1.00000 1.41587 2.06613 1.00000 −0.995326 −1.00000
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
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.h 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6010.2.a.h 28 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} - 4 T_{3}^{27} - 54 T_{3}^{26} + 223 T_{3}^{25} + 1259 T_{3}^{24} - 5421 T_{3}^{23} - 16560 T_{3}^{22} + 75453 T_{3}^{21} + 134490 T_{3}^{20} - 663898 T_{3}^{19} - 690565 T_{3}^{18} + 3848189 T_{3}^{17} + \cdots - 7936 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\). Copy content Toggle raw display