Properties

Label 6033.2.a.b
Level $6033$
Weight $2$
Character orbit 6033.a
Self dual yes
Analytic conductor $48.174$
Analytic rank $1$
Dimension $71$
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] = [6033,2,Mod(1,6033)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6033, 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("6033.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6033 = 3 \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6033.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: \(48.1737475394\)
Analytic rank: \(1\)
Dimension: \(71\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 71 q - 11 q^{2} + 71 q^{3} + 53 q^{4} - 8 q^{5} - 11 q^{6} - 46 q^{7} - 33 q^{8} + 71 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 71 q - 11 q^{2} + 71 q^{3} + 53 q^{4} - 8 q^{5} - 11 q^{6} - 46 q^{7} - 33 q^{8} + 71 q^{9} - 41 q^{10} - 18 q^{11} + 53 q^{12} - 67 q^{13} - 7 q^{14} - 8 q^{15} + 21 q^{16} - 25 q^{17} - 11 q^{18} - 43 q^{19} - 8 q^{20} - 46 q^{21} - 49 q^{22} - 75 q^{23} - 33 q^{24} + 19 q^{25} + 71 q^{27} - 89 q^{28} - 35 q^{29} - 41 q^{30} - 82 q^{31} - 62 q^{32} - 18 q^{33} - 28 q^{34} - 51 q^{35} + 53 q^{36} - 66 q^{37} - 29 q^{38} - 67 q^{39} - 102 q^{40} + q^{41} - 7 q^{42} - 112 q^{43} - 25 q^{44} - 8 q^{45} - 36 q^{46} - 67 q^{47} + 21 q^{48} + 7 q^{49} - 24 q^{50} - 25 q^{51} - 134 q^{52} - 40 q^{53} - 11 q^{54} - 112 q^{55} + 9 q^{56} - 43 q^{57} - 47 q^{58} - 18 q^{59} - 8 q^{60} - 144 q^{61} - 19 q^{62} - 46 q^{63} - 17 q^{64} - 31 q^{65} - 49 q^{66} - 85 q^{67} - 22 q^{68} - 75 q^{69} - 11 q^{70} - 44 q^{71} - 33 q^{72} - 98 q^{73} + 6 q^{74} + 19 q^{75} - 85 q^{76} - 39 q^{77} - 126 q^{79} + 21 q^{80} + 71 q^{81} - 69 q^{82} - 43 q^{83} - 89 q^{84} - 112 q^{85} + 32 q^{86} - 35 q^{87} - 85 q^{88} + 8 q^{89} - 41 q^{90} - 40 q^{91} - 96 q^{92} - 82 q^{93} - 99 q^{94} - 103 q^{95} - 62 q^{96} - 67 q^{97} - 11 q^{98} - 18 q^{99}+O(q^{100}) \) 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 −2.73636 1.00000 5.48768 4.41934 −2.73636 −3.72834 −9.54357 1.00000 −12.0929
1.2 −2.72830 1.00000 5.44362 1.48062 −2.72830 2.42771 −9.39523 1.00000 −4.03959
1.3 −2.67601 1.00000 5.16104 0.373464 −2.67601 −4.38013 −8.45898 1.00000 −0.999393
1.4 −2.64643 1.00000 5.00359 1.18774 −2.64643 −0.455161 −7.94880 1.00000 −3.14327
1.5 −2.52118 1.00000 4.35637 −1.70056 −2.52118 −0.985883 −5.94083 1.00000 4.28742
1.6 −2.47301 1.00000 4.11576 −3.79557 −2.47301 −3.86713 −5.23229 1.00000 9.38648
1.7 −2.36106 1.00000 3.57460 2.24080 −2.36106 1.23097 −3.71772 1.00000 −5.29065
1.8 −2.34993 1.00000 3.52217 1.65833 −2.34993 1.22844 −3.57699 1.00000 −3.89696
1.9 −2.31705 1.00000 3.36871 −0.440585 −2.31705 −0.330548 −3.17136 1.00000 1.02086
1.10 −2.28528 1.00000 3.22252 −1.27062 −2.28528 −1.88714 −2.79381 1.00000 2.90374
1.11 −2.15116 1.00000 2.62747 2.30807 −2.15116 −1.03213 −1.34980 1.00000 −4.96503
1.12 −2.11027 1.00000 2.45325 −0.410067 −2.11027 2.75939 −0.956486 1.00000 0.865353
1.13 −2.08609 1.00000 2.35176 3.34556 −2.08609 −3.89610 −0.733796 1.00000 −6.97912
1.14 −2.08008 1.00000 2.32673 −1.31066 −2.08008 −0.473283 −0.679634 1.00000 2.72629
1.15 −1.93019 1.00000 1.72562 −2.39497 −1.93019 3.37261 0.529608 1.00000 4.62273
1.16 −1.79990 1.00000 1.23964 −2.94789 −1.79990 0.174421 1.36857 1.00000 5.30591
1.17 −1.66194 1.00000 0.762029 0.623860 −1.66194 3.99540 2.05743 1.00000 −1.03682
1.18 −1.62301 1.00000 0.634165 −2.89779 −1.62301 −2.91247 2.21677 1.00000 4.70315
1.19 −1.54830 1.00000 0.397234 −0.789007 −1.54830 −3.51923 2.48156 1.00000 1.22162
1.20 −1.51299 1.00000 0.289142 −0.0448522 −1.51299 0.362271 2.58851 1.00000 0.0678609
See all 71 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.71
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(2011\) \(-1\)

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 6033.2.a.b 71
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6033.2.a.b 71 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{71} + 11 T_{2}^{70} - 37 T_{2}^{69} - 825 T_{2}^{68} - 480 T_{2}^{67} + 28663 T_{2}^{66} + 63332 T_{2}^{65} - 607422 T_{2}^{64} - 2080335 T_{2}^{63} + 8646624 T_{2}^{62} + 41256389 T_{2}^{61} - 84474552 T_{2}^{60} + \cdots + 476 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6033))\). Copy content Toggle raw display