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

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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: \(71\)
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.

\(\operatorname{Tr}(f)(q) = \) \(71q \) \(\mathstrut -\mathstrut 11q^{2} \) \(\mathstrut +\mathstrut 71q^{3} \) \(\mathstrut +\mathstrut 53q^{4} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 11q^{6} \) \(\mathstrut -\mathstrut 46q^{7} \) \(\mathstrut -\mathstrut 33q^{8} \) \(\mathstrut +\mathstrut 71q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(71q \) \(\mathstrut -\mathstrut 11q^{2} \) \(\mathstrut +\mathstrut 71q^{3} \) \(\mathstrut +\mathstrut 53q^{4} \) \(\mathstrut -\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 11q^{6} \) \(\mathstrut -\mathstrut 46q^{7} \) \(\mathstrut -\mathstrut 33q^{8} \) \(\mathstrut +\mathstrut 71q^{9} \) \(\mathstrut -\mathstrut 41q^{10} \) \(\mathstrut -\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 53q^{12} \) \(\mathstrut -\mathstrut 67q^{13} \) \(\mathstrut -\mathstrut 7q^{14} \) \(\mathstrut -\mathstrut 8q^{15} \) \(\mathstrut +\mathstrut 21q^{16} \) \(\mathstrut -\mathstrut 25q^{17} \) \(\mathstrut -\mathstrut 11q^{18} \) \(\mathstrut -\mathstrut 43q^{19} \) \(\mathstrut -\mathstrut 8q^{20} \) \(\mathstrut -\mathstrut 46q^{21} \) \(\mathstrut -\mathstrut 49q^{22} \) \(\mathstrut -\mathstrut 75q^{23} \) \(\mathstrut -\mathstrut 33q^{24} \) \(\mathstrut +\mathstrut 19q^{25} \) \(\mathstrut +\mathstrut 71q^{27} \) \(\mathstrut -\mathstrut 89q^{28} \) \(\mathstrut -\mathstrut 35q^{29} \) \(\mathstrut -\mathstrut 41q^{30} \) \(\mathstrut -\mathstrut 82q^{31} \) \(\mathstrut -\mathstrut 62q^{32} \) \(\mathstrut -\mathstrut 18q^{33} \) \(\mathstrut -\mathstrut 28q^{34} \) \(\mathstrut -\mathstrut 51q^{35} \) \(\mathstrut +\mathstrut 53q^{36} \) \(\mathstrut -\mathstrut 66q^{37} \) \(\mathstrut -\mathstrut 29q^{38} \) \(\mathstrut -\mathstrut 67q^{39} \) \(\mathstrut -\mathstrut 102q^{40} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut -\mathstrut 7q^{42} \) \(\mathstrut -\mathstrut 112q^{43} \) \(\mathstrut -\mathstrut 25q^{44} \) \(\mathstrut -\mathstrut 8q^{45} \) \(\mathstrut -\mathstrut 36q^{46} \) \(\mathstrut -\mathstrut 67q^{47} \) \(\mathstrut +\mathstrut 21q^{48} \) \(\mathstrut +\mathstrut 7q^{49} \) \(\mathstrut -\mathstrut 24q^{50} \) \(\mathstrut -\mathstrut 25q^{51} \) \(\mathstrut -\mathstrut 134q^{52} \) \(\mathstrut -\mathstrut 40q^{53} \) \(\mathstrut -\mathstrut 11q^{54} \) \(\mathstrut -\mathstrut 112q^{55} \) \(\mathstrut +\mathstrut 9q^{56} \) \(\mathstrut -\mathstrut 43q^{57} \) \(\mathstrut -\mathstrut 47q^{58} \) \(\mathstrut -\mathstrut 18q^{59} \) \(\mathstrut -\mathstrut 8q^{60} \) \(\mathstrut -\mathstrut 144q^{61} \) \(\mathstrut -\mathstrut 19q^{62} \) \(\mathstrut -\mathstrut 46q^{63} \) \(\mathstrut -\mathstrut 17q^{64} \) \(\mathstrut -\mathstrut 31q^{65} \) \(\mathstrut -\mathstrut 49q^{66} \) \(\mathstrut -\mathstrut 85q^{67} \) \(\mathstrut -\mathstrut 22q^{68} \) \(\mathstrut -\mathstrut 75q^{69} \) \(\mathstrut -\mathstrut 11q^{70} \) \(\mathstrut -\mathstrut 44q^{71} \) \(\mathstrut -\mathstrut 33q^{72} \) \(\mathstrut -\mathstrut 98q^{73} \) \(\mathstrut +\mathstrut 6q^{74} \) \(\mathstrut +\mathstrut 19q^{75} \) \(\mathstrut -\mathstrut 85q^{76} \) \(\mathstrut -\mathstrut 39q^{77} \) \(\mathstrut -\mathstrut 126q^{79} \) \(\mathstrut +\mathstrut 21q^{80} \) \(\mathstrut +\mathstrut 71q^{81} \) \(\mathstrut -\mathstrut 69q^{82} \) \(\mathstrut -\mathstrut 43q^{83} \) \(\mathstrut -\mathstrut 89q^{84} \) \(\mathstrut -\mathstrut 112q^{85} \) \(\mathstrut +\mathstrut 32q^{86} \) \(\mathstrut -\mathstrut 35q^{87} \) \(\mathstrut -\mathstrut 85q^{88} \) \(\mathstrut +\mathstrut 8q^{89} \) \(\mathstrut -\mathstrut 41q^{90} \) \(\mathstrut -\mathstrut 40q^{91} \) \(\mathstrut -\mathstrut 96q^{92} \) \(\mathstrut -\mathstrut 82q^{93} \) \(\mathstrut -\mathstrut 99q^{94} \) \(\mathstrut -\mathstrut 103q^{95} \) \(\mathstrut -\mathstrut 62q^{96} \) \(\mathstrut -\mathstrut 67q^{97} \) \(\mathstrut -\mathstrut 11q^{98} \) \(\mathstrut -\mathstrut 18q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.73636 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:

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 \(T_{2}^{71} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6033))\).