Properties

Label 6033.2.a.d
Level 6033
Weight 2
Character orbit 6033.a
Self dual Yes
Analytic conductor 48.174
Analytic rank 1
Dimension 84
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: \(1\)
Dimension: \(84\)
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.

\(\operatorname{Tr}(f)(q) = \) \(84q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 84q^{3} \) \(\mathstrut +\mathstrut 81q^{4} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 13q^{6} \) \(\mathstrut -\mathstrut 32q^{7} \) \(\mathstrut -\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 84q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(84q \) \(\mathstrut -\mathstrut 13q^{2} \) \(\mathstrut -\mathstrut 84q^{3} \) \(\mathstrut +\mathstrut 81q^{4} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 13q^{6} \) \(\mathstrut -\mathstrut 32q^{7} \) \(\mathstrut -\mathstrut 39q^{8} \) \(\mathstrut +\mathstrut 84q^{9} \) \(\mathstrut +\mathstrut 13q^{10} \) \(\mathstrut -\mathstrut 20q^{11} \) \(\mathstrut -\mathstrut 81q^{12} \) \(\mathstrut +\mathstrut 7q^{13} \) \(\mathstrut -\mathstrut 9q^{14} \) \(\mathstrut +\mathstrut 10q^{15} \) \(\mathstrut +\mathstrut 83q^{16} \) \(\mathstrut -\mathstrut 39q^{17} \) \(\mathstrut -\mathstrut 13q^{18} \) \(\mathstrut +\mathstrut 13q^{19} \) \(\mathstrut -\mathstrut 26q^{20} \) \(\mathstrut +\mathstrut 32q^{21} \) \(\mathstrut -\mathstrut 21q^{22} \) \(\mathstrut -\mathstrut 93q^{23} \) \(\mathstrut +\mathstrut 39q^{24} \) \(\mathstrut +\mathstrut 66q^{25} \) \(\mathstrut -\mathstrut 34q^{26} \) \(\mathstrut -\mathstrut 84q^{27} \) \(\mathstrut -\mathstrut 59q^{28} \) \(\mathstrut -\mathstrut 39q^{29} \) \(\mathstrut -\mathstrut 13q^{30} \) \(\mathstrut +\mathstrut 8q^{31} \) \(\mathstrut -\mathstrut 96q^{32} \) \(\mathstrut +\mathstrut 20q^{33} \) \(\mathstrut -\mathstrut 69q^{35} \) \(\mathstrut +\mathstrut 81q^{36} \) \(\mathstrut +\mathstrut 6q^{37} \) \(\mathstrut -\mathstrut 59q^{38} \) \(\mathstrut -\mathstrut 7q^{39} \) \(\mathstrut +\mathstrut 28q^{40} \) \(\mathstrut -\mathstrut 23q^{41} \) \(\mathstrut +\mathstrut 9q^{42} \) \(\mathstrut -\mathstrut 74q^{43} \) \(\mathstrut -\mathstrut 43q^{44} \) \(\mathstrut -\mathstrut 10q^{45} \) \(\mathstrut -\mathstrut 6q^{46} \) \(\mathstrut -\mathstrut 77q^{47} \) \(\mathstrut -\mathstrut 83q^{48} \) \(\mathstrut +\mathstrut 100q^{49} \) \(\mathstrut -\mathstrut 74q^{50} \) \(\mathstrut +\mathstrut 39q^{51} \) \(\mathstrut -\mathstrut 44q^{52} \) \(\mathstrut -\mathstrut 66q^{53} \) \(\mathstrut +\mathstrut 13q^{54} \) \(\mathstrut -\mathstrut 60q^{55} \) \(\mathstrut -\mathstrut 31q^{56} \) \(\mathstrut -\mathstrut 13q^{57} \) \(\mathstrut -\mathstrut 39q^{58} \) \(\mathstrut -\mathstrut 36q^{59} \) \(\mathstrut +\mathstrut 26q^{60} \) \(\mathstrut +\mathstrut 104q^{61} \) \(\mathstrut -\mathstrut 53q^{62} \) \(\mathstrut -\mathstrut 32q^{63} \) \(\mathstrut +\mathstrut 85q^{64} \) \(\mathstrut -\mathstrut 47q^{65} \) \(\mathstrut +\mathstrut 21q^{66} \) \(\mathstrut -\mathstrut 65q^{67} \) \(\mathstrut -\mathstrut 118q^{68} \) \(\mathstrut +\mathstrut 93q^{69} \) \(\mathstrut -\mathstrut 3q^{70} \) \(\mathstrut -\mathstrut 68q^{71} \) \(\mathstrut -\mathstrut 39q^{72} \) \(\mathstrut +\mathstrut 8q^{73} \) \(\mathstrut -\mathstrut 30q^{74} \) \(\mathstrut -\mathstrut 66q^{75} \) \(\mathstrut +\mathstrut 71q^{76} \) \(\mathstrut -\mathstrut 83q^{77} \) \(\mathstrut +\mathstrut 34q^{78} \) \(\mathstrut -\mathstrut 24q^{79} \) \(\mathstrut -\mathstrut 67q^{80} \) \(\mathstrut +\mathstrut 84q^{81} \) \(\mathstrut -\mathstrut 9q^{82} \) \(\mathstrut -\mathstrut 95q^{83} \) \(\mathstrut +\mathstrut 59q^{84} \) \(\mathstrut +\mathstrut 24q^{85} \) \(\mathstrut -\mathstrut 32q^{86} \) \(\mathstrut +\mathstrut 39q^{87} \) \(\mathstrut -\mathstrut 65q^{88} \) \(\mathstrut -\mathstrut 44q^{89} \) \(\mathstrut +\mathstrut 13q^{90} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut -\mathstrut 184q^{92} \) \(\mathstrut -\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut 61q^{94} \) \(\mathstrut -\mathstrut 153q^{95} \) \(\mathstrut +\mathstrut 96q^{96} \) \(\mathstrut +\mathstrut 19q^{97} \) \(\mathstrut -\mathstrut 67q^{98} \) \(\mathstrut -\mathstrut 20q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.82117 −1.00000 5.95898 −2.46990 2.82117 0.575292 −11.1689 1.00000 6.96801
1.2 −2.80639 −1.00000 5.87581 1.59554 2.80639 −1.53809 −10.8770 1.00000 −4.47769
1.3 −2.73519 −1.00000 5.48127 −2.75382 2.73519 −2.09506 −9.52192 1.00000 7.53221
1.4 −2.66273 −1.00000 5.09011 2.84840 2.66273 −4.34717 −8.22811 1.00000 −7.58450
1.5 −2.60952 −1.00000 4.80961 2.15432 2.60952 −3.44517 −7.33175 1.00000 −5.62175
1.6 −2.60855 −1.00000 4.80454 −1.00742 2.60855 3.22569 −7.31578 1.00000 2.62791
1.7 −2.59128 −1.00000 4.71472 −3.30142 2.59128 3.98473 −7.03459 1.00000 8.55491
1.8 −2.57191 −1.00000 4.61473 2.62462 2.57191 3.44877 −6.72487 1.00000 −6.75028
1.9 −2.45280 −1.00000 4.01624 −4.16660 2.45280 −2.71400 −4.94544 1.00000 10.2199
1.10 −2.42494 −1.00000 3.88033 −3.57030 2.42494 1.93625 −4.55968 1.00000 8.65775
1.11 −2.41532 −1.00000 3.83378 −1.96861 2.41532 −4.53404 −4.42916 1.00000 4.75483
1.12 −2.31178 −1.00000 3.34433 1.56630 2.31178 1.68307 −3.10780 1.00000 −3.62093
1.13 −2.22564 −1.00000 2.95347 −0.168445 2.22564 −0.580440 −2.12207 1.00000 0.374898
1.14 −2.16814 −1.00000 2.70083 3.39119 2.16814 2.22873 −1.51950 1.00000 −7.35257
1.15 −2.10715 −1.00000 2.44009 0.912218 2.10715 4.59060 −0.927338 1.00000 −1.92218
1.16 −2.06135 −1.00000 2.24917 −2.50323 2.06135 1.09494 −0.513635 1.00000 5.16004
1.17 −2.02957 −1.00000 2.11915 3.74775 2.02957 −1.06025 −0.241816 1.00000 −7.60631
1.18 −2.00535 −1.00000 2.02142 2.82902 2.00535 −3.36485 −0.0429452 1.00000 −5.67317
1.19 −1.86532 −1.00000 1.47943 −0.830382 1.86532 −1.56075 0.971032 1.00000 1.54893
1.20 −1.84665 −1.00000 1.41011 −1.36043 1.84665 −4.64728 1.08932 1.00000 2.51223
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.84
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 \(T_{2}^{84} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6033))\).