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Properties

Label 6033.2.a.c
Level $6033$
Weight $2$
Character orbit 6033.a
Self dual yes
Analytic conductor $48.174$
Analytic rank $0$
Dimension $82$
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: \(0\)
Dimension: \(82\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 82 q + 13 q^{2} - 82 q^{3} + 87 q^{4} + 7 q^{5} - 13 q^{6} + 30 q^{7} + 39 q^{8} + 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 82 q + 13 q^{2} - 82 q^{3} + 87 q^{4} + 7 q^{5} - 13 q^{6} + 30 q^{7} + 39 q^{8} + 82 q^{9} - 9 q^{10} + 28 q^{11} - 87 q^{12} - 14 q^{13} + 21 q^{14} - 7 q^{15} + 93 q^{16} + 25 q^{17} + 13 q^{18} - 7 q^{19} + 40 q^{20} - 30 q^{21} + 31 q^{22} + 97 q^{23} - 39 q^{24} + 83 q^{25} + 22 q^{26} - 82 q^{27} + 53 q^{28} + 45 q^{29} + 9 q^{30} - 11 q^{31} + 86 q^{32} - 28 q^{33} - 30 q^{34} + 69 q^{35} + 87 q^{36} + 8 q^{37} + 33 q^{38} + 14 q^{39} - 38 q^{40} + 12 q^{41} - 21 q^{42} + 68 q^{43} + 77 q^{44} + 7 q^{45} - 14 q^{46} + 85 q^{47} - 93 q^{48} + 68 q^{49} + 56 q^{50} - 25 q^{51} - 18 q^{52} + 58 q^{53} - 13 q^{54} + 68 q^{55} + 59 q^{56} + 7 q^{57} + 27 q^{58} + 40 q^{59} - 40 q^{60} - 116 q^{61} + 79 q^{62} + 30 q^{63} + 127 q^{64} + 66 q^{65} - 31 q^{66} + 51 q^{67} + 94 q^{68} - 97 q^{69} + q^{70} + 101 q^{71} + 39 q^{72} + 12 q^{73} + 72 q^{74} - 83 q^{75} - 3 q^{76} + 101 q^{77} - 22 q^{78} + 26 q^{79} + 61 q^{80} + 82 q^{81} + 31 q^{82} + 94 q^{83} - 53 q^{84} - 8 q^{85} + 68 q^{86} - 45 q^{87} + 91 q^{88} + 40 q^{89} - 9 q^{90} - 6 q^{91} + 180 q^{92} + 11 q^{93} - 31 q^{94} + 153 q^{95} - 86 q^{96} - 39 q^{97} + 115 q^{98} + 28 q^{99}+O(q^{100}) \) 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 −2.71390 −1.00000 5.36527 0.573934 2.71390 0.517323 −9.13302 1.00000 −1.55760
1.2 −2.65350 −1.00000 5.04106 3.99915 2.65350 3.78753 −8.06945 1.00000 −10.6117
1.3 −2.60707 −1.00000 4.79682 3.79845 2.60707 0.267933 −7.29150 1.00000 −9.90282
1.4 −2.48894 −1.00000 4.19481 0.184549 2.48894 2.04165 −5.46273 1.00000 −0.459332
1.5 −2.47960 −1.00000 4.14839 −2.94703 2.47960 −3.66351 −5.32715 1.00000 7.30744
1.6 −2.47475 −1.00000 4.12439 0.484092 2.47475 −1.35174 −5.25735 1.00000 −1.19801
1.7 −2.42144 −1.00000 3.86336 1.61348 2.42144 −0.957963 −4.51200 1.00000 −3.90694
1.8 −2.34798 −1.00000 3.51301 0.242883 2.34798 −0.571157 −3.55251 1.00000 −0.570284
1.9 −2.32093 −1.00000 3.38673 −1.03535 2.32093 −3.14382 −3.21851 1.00000 2.40297
1.10 −2.18071 −1.00000 2.75549 −4.11072 2.18071 −0.856446 −1.64750 1.00000 8.96427
1.11 −2.08527 −1.00000 2.34836 −0.259814 2.08527 3.95786 −0.726422 1.00000 0.541783
1.12 −2.05866 −1.00000 2.23808 −1.30769 2.05866 4.57045 −0.490135 1.00000 2.69208
1.13 −2.01503 −1.00000 2.06036 −1.94191 2.01503 −0.407570 −0.121626 1.00000 3.91301
1.14 −1.97726 −1.00000 1.90954 3.20847 1.97726 −2.72854 0.178858 1.00000 −6.34396
1.15 −1.85935 −1.00000 1.45718 1.62853 1.85935 1.58306 1.00928 1.00000 −3.02802
1.16 −1.75798 −1.00000 1.09048 3.87577 1.75798 1.73463 1.59891 1.00000 −6.81351
1.17 −1.65413 −1.00000 0.736138 −3.06078 1.65413 3.39552 2.09059 1.00000 5.06292
1.18 −1.52431 −1.00000 0.323513 −2.18374 1.52431 −1.69385 2.55548 1.00000 3.32869
1.19 −1.49139 −1.00000 0.224256 0.671800 1.49139 1.23937 2.64833 1.00000 −1.00192
1.20 −1.48355 −1.00000 0.200920 1.86269 1.48355 −2.05902 2.66903 1.00000 −2.76340
See all 82 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.82
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.c ✓ 82
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6033.2.a.c ✓ 82 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{82} - 13 T_{2}^{81} - 41 T_{2}^{80} + 1235 T_{2}^{79} - 1463 T_{2}^{78} - 54274 T_{2}^{77} + \cdots - 140728 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6033))\). Copy content Toggle raw display