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

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Newspace parameters

Level: \( N \) = \( 6033 = 3 \cdot 2011 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6033.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1737475394\)
Analytic rank: \(0\)
Dimension: \(82\)
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.

\(\operatorname{Tr}(f)(q) = \) \(82q \) \(\mathstrut +\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 82q^{3} \) \(\mathstrut +\mathstrut 87q^{4} \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut -\mathstrut 13q^{6} \) \(\mathstrut +\mathstrut 30q^{7} \) \(\mathstrut +\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 82q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(82q \) \(\mathstrut +\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 82q^{3} \) \(\mathstrut +\mathstrut 87q^{4} \) \(\mathstrut +\mathstrut 7q^{5} \) \(\mathstrut -\mathstrut 13q^{6} \) \(\mathstrut +\mathstrut 30q^{7} \) \(\mathstrut +\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 82q^{9} \) \(\mathstrut -\mathstrut 9q^{10} \) \(\mathstrut +\mathstrut 28q^{11} \) \(\mathstrut -\mathstrut 87q^{12} \) \(\mathstrut -\mathstrut 14q^{13} \) \(\mathstrut +\mathstrut 21q^{14} \) \(\mathstrut -\mathstrut 7q^{15} \) \(\mathstrut +\mathstrut 93q^{16} \) \(\mathstrut +\mathstrut 25q^{17} \) \(\mathstrut +\mathstrut 13q^{18} \) \(\mathstrut -\mathstrut 7q^{19} \) \(\mathstrut +\mathstrut 40q^{20} \) \(\mathstrut -\mathstrut 30q^{21} \) \(\mathstrut +\mathstrut 31q^{22} \) \(\mathstrut +\mathstrut 97q^{23} \) \(\mathstrut -\mathstrut 39q^{24} \) \(\mathstrut +\mathstrut 83q^{25} \) \(\mathstrut +\mathstrut 22q^{26} \) \(\mathstrut -\mathstrut 82q^{27} \) \(\mathstrut +\mathstrut 53q^{28} \) \(\mathstrut +\mathstrut 45q^{29} \) \(\mathstrut +\mathstrut 9q^{30} \) \(\mathstrut -\mathstrut 11q^{31} \) \(\mathstrut +\mathstrut 86q^{32} \) \(\mathstrut -\mathstrut 28q^{33} \) \(\mathstrut -\mathstrut 30q^{34} \) \(\mathstrut +\mathstrut 69q^{35} \) \(\mathstrut +\mathstrut 87q^{36} \) \(\mathstrut +\mathstrut 8q^{37} \) \(\mathstrut +\mathstrut 33q^{38} \) \(\mathstrut +\mathstrut 14q^{39} \) \(\mathstrut -\mathstrut 38q^{40} \) \(\mathstrut +\mathstrut 12q^{41} \) \(\mathstrut -\mathstrut 21q^{42} \) \(\mathstrut +\mathstrut 68q^{43} \) \(\mathstrut +\mathstrut 77q^{44} \) \(\mathstrut +\mathstrut 7q^{45} \) \(\mathstrut -\mathstrut 14q^{46} \) \(\mathstrut +\mathstrut 85q^{47} \) \(\mathstrut -\mathstrut 93q^{48} \) \(\mathstrut +\mathstrut 68q^{49} \) \(\mathstrut +\mathstrut 56q^{50} \) \(\mathstrut -\mathstrut 25q^{51} \) \(\mathstrut -\mathstrut 18q^{52} \) \(\mathstrut +\mathstrut 58q^{53} \) \(\mathstrut -\mathstrut 13q^{54} \) \(\mathstrut +\mathstrut 68q^{55} \) \(\mathstrut +\mathstrut 59q^{56} \) \(\mathstrut +\mathstrut 7q^{57} \) \(\mathstrut +\mathstrut 27q^{58} \) \(\mathstrut +\mathstrut 40q^{59} \) \(\mathstrut -\mathstrut 40q^{60} \) \(\mathstrut -\mathstrut 116q^{61} \) \(\mathstrut +\mathstrut 79q^{62} \) \(\mathstrut +\mathstrut 30q^{63} \) \(\mathstrut +\mathstrut 127q^{64} \) \(\mathstrut +\mathstrut 66q^{65} \) \(\mathstrut -\mathstrut 31q^{66} \) \(\mathstrut +\mathstrut 51q^{67} \) \(\mathstrut +\mathstrut 94q^{68} \) \(\mathstrut -\mathstrut 97q^{69} \) \(\mathstrut +\mathstrut q^{70} \) \(\mathstrut +\mathstrut 101q^{71} \) \(\mathstrut +\mathstrut 39q^{72} \) \(\mathstrut +\mathstrut 12q^{73} \) \(\mathstrut +\mathstrut 72q^{74} \) \(\mathstrut -\mathstrut 83q^{75} \) \(\mathstrut -\mathstrut 3q^{76} \) \(\mathstrut +\mathstrut 101q^{77} \) \(\mathstrut -\mathstrut 22q^{78} \) \(\mathstrut +\mathstrut 26q^{79} \) \(\mathstrut +\mathstrut 61q^{80} \) \(\mathstrut +\mathstrut 82q^{81} \) \(\mathstrut +\mathstrut 31q^{82} \) \(\mathstrut +\mathstrut 94q^{83} \) \(\mathstrut -\mathstrut 53q^{84} \) \(\mathstrut -\mathstrut 8q^{85} \) \(\mathstrut +\mathstrut 68q^{86} \) \(\mathstrut -\mathstrut 45q^{87} \) \(\mathstrut +\mathstrut 91q^{88} \) \(\mathstrut +\mathstrut 40q^{89} \) \(\mathstrut -\mathstrut 9q^{90} \) \(\mathstrut -\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 180q^{92} \) \(\mathstrut +\mathstrut 11q^{93} \) \(\mathstrut -\mathstrut 31q^{94} \) \(\mathstrut +\mathstrut 153q^{95} \) \(\mathstrut -\mathstrut 86q^{96} \) \(\mathstrut -\mathstrut 39q^{97} \) \(\mathstrut +\mathstrut 115q^{98} \) \(\mathstrut +\mathstrut 28q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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.

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:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{82} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6033))\).