Properties

Label 8049.2.a.d
Level 8049
Weight 2
Character orbit 8049.a
Self dual Yes
Analytic conductor 64.272
Analytic rank 0
Dimension 129
CM No

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Newspace parameters

Level: \( N \) = \( 8049 = 3 \cdot 2683 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8049.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2715885869\)
Analytic rank: \(0\)
Dimension: \(129\)
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.

\(\operatorname{Tr}(f)(q) = \) \(129q \) \(\mathstrut +\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 129q^{3} \) \(\mathstrut +\mathstrut 158q^{4} \) \(\mathstrut +\mathstrut 11q^{5} \) \(\mathstrut +\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 40q^{7} \) \(\mathstrut +\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut 129q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(129q \) \(\mathstrut +\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 129q^{3} \) \(\mathstrut +\mathstrut 158q^{4} \) \(\mathstrut +\mathstrut 11q^{5} \) \(\mathstrut +\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 40q^{7} \) \(\mathstrut +\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut 129q^{9} \) \(\mathstrut +\mathstrut 20q^{10} \) \(\mathstrut +\mathstrut 48q^{11} \) \(\mathstrut +\mathstrut 158q^{12} \) \(\mathstrut +\mathstrut 77q^{13} \) \(\mathstrut +\mathstrut 13q^{14} \) \(\mathstrut +\mathstrut 11q^{15} \) \(\mathstrut +\mathstrut 212q^{16} \) \(\mathstrut +\mathstrut 9q^{17} \) \(\mathstrut +\mathstrut 8q^{18} \) \(\mathstrut +\mathstrut 68q^{19} \) \(\mathstrut +\mathstrut 19q^{20} \) \(\mathstrut +\mathstrut 40q^{21} \) \(\mathstrut +\mathstrut 45q^{22} \) \(\mathstrut +\mathstrut 64q^{23} \) \(\mathstrut +\mathstrut 18q^{24} \) \(\mathstrut +\mathstrut 188q^{25} \) \(\mathstrut +\mathstrut 19q^{26} \) \(\mathstrut +\mathstrut 129q^{27} \) \(\mathstrut +\mathstrut 69q^{28} \) \(\mathstrut +\mathstrut 23q^{29} \) \(\mathstrut +\mathstrut 20q^{30} \) \(\mathstrut +\mathstrut 133q^{31} \) \(\mathstrut +\mathstrut 24q^{32} \) \(\mathstrut +\mathstrut 48q^{33} \) \(\mathstrut +\mathstrut 63q^{34} \) \(\mathstrut +\mathstrut 26q^{35} \) \(\mathstrut +\mathstrut 158q^{36} \) \(\mathstrut +\mathstrut 147q^{37} \) \(\mathstrut +\mathstrut 9q^{38} \) \(\mathstrut +\mathstrut 77q^{39} \) \(\mathstrut +\mathstrut 58q^{40} \) \(\mathstrut +\mathstrut 21q^{41} \) \(\mathstrut +\mathstrut 13q^{42} \) \(\mathstrut +\mathstrut 76q^{43} \) \(\mathstrut +\mathstrut 110q^{44} \) \(\mathstrut +\mathstrut 11q^{45} \) \(\mathstrut +\mathstrut 48q^{46} \) \(\mathstrut +\mathstrut 85q^{47} \) \(\mathstrut +\mathstrut 212q^{48} \) \(\mathstrut +\mathstrut 213q^{49} \) \(\mathstrut +\mathstrut 17q^{50} \) \(\mathstrut +\mathstrut 9q^{51} \) \(\mathstrut +\mathstrut 139q^{52} \) \(\mathstrut +\mathstrut 30q^{53} \) \(\mathstrut +\mathstrut 8q^{54} \) \(\mathstrut +\mathstrut 103q^{55} \) \(\mathstrut +\mathstrut 19q^{56} \) \(\mathstrut +\mathstrut 68q^{57} \) \(\mathstrut +\mathstrut 94q^{58} \) \(\mathstrut +\mathstrut 64q^{59} \) \(\mathstrut +\mathstrut 19q^{60} \) \(\mathstrut +\mathstrut 110q^{61} \) \(\mathstrut -\mathstrut 10q^{62} \) \(\mathstrut +\mathstrut 40q^{63} \) \(\mathstrut +\mathstrut 288q^{64} \) \(\mathstrut -\mathstrut 8q^{65} \) \(\mathstrut +\mathstrut 45q^{66} \) \(\mathstrut +\mathstrut 118q^{67} \) \(\mathstrut -\mathstrut 15q^{68} \) \(\mathstrut +\mathstrut 64q^{69} \) \(\mathstrut +\mathstrut 75q^{70} \) \(\mathstrut +\mathstrut 154q^{71} \) \(\mathstrut +\mathstrut 18q^{72} \) \(\mathstrut +\mathstrut 137q^{73} \) \(\mathstrut +\mathstrut 28q^{74} \) \(\mathstrut +\mathstrut 188q^{75} \) \(\mathstrut +\mathstrut 156q^{76} \) \(\mathstrut +\mathstrut 17q^{77} \) \(\mathstrut +\mathstrut 19q^{78} \) \(\mathstrut +\mathstrut 157q^{79} \) \(\mathstrut +\mathstrut 2q^{80} \) \(\mathstrut +\mathstrut 129q^{81} \) \(\mathstrut +\mathstrut 72q^{82} \) \(\mathstrut +\mathstrut 39q^{83} \) \(\mathstrut +\mathstrut 69q^{84} \) \(\mathstrut +\mathstrut 127q^{85} \) \(\mathstrut +\mathstrut 54q^{86} \) \(\mathstrut +\mathstrut 23q^{87} \) \(\mathstrut +\mathstrut 97q^{88} \) \(\mathstrut +\mathstrut 31q^{89} \) \(\mathstrut +\mathstrut 20q^{90} \) \(\mathstrut +\mathstrut 137q^{91} \) \(\mathstrut +\mathstrut 82q^{92} \) \(\mathstrut +\mathstrut 133q^{93} \) \(\mathstrut +\mathstrut 40q^{94} \) \(\mathstrut +\mathstrut 68q^{95} \) \(\mathstrut +\mathstrut 24q^{96} \) \(\mathstrut +\mathstrut 170q^{97} \) \(\mathstrut -\mathstrut 21q^{98} \) \(\mathstrut +\mathstrut 48q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.82087 1.00000 5.95730 1.28710 −2.82087 2.58338 −11.1630 1.00000 −3.63075
1.2 −2.77720 1.00000 5.71284 2.12496 −2.77720 −3.21138 −10.3113 1.00000 −5.90145
1.3 −2.72806 1.00000 5.44232 −3.98613 −2.72806 −2.69921 −9.39088 1.00000 10.8744
1.4 −2.70395 1.00000 5.31133 1.76368 −2.70395 −4.33108 −8.95365 1.00000 −4.76891
1.5 −2.70347 1.00000 5.30878 −2.52602 −2.70347 3.06715 −8.94519 1.00000 6.82903
1.6 −2.67215 1.00000 5.14039 −2.79285 −2.67215 3.90889 −8.39161 1.00000 7.46293
1.7 −2.65892 1.00000 5.06986 3.58637 −2.65892 4.24083 −8.16251 1.00000 −9.53588
1.8 −2.65190 1.00000 5.03256 −2.74220 −2.65190 0.184989 −8.04204 1.00000 7.27204
1.9 −2.62578 1.00000 4.89474 −0.512843 −2.62578 1.48852 −7.60096 1.00000 1.34662
1.10 −2.58791 1.00000 4.69728 1.64365 −2.58791 −4.45906 −6.98031 1.00000 −4.25362
1.11 −2.57732 1.00000 4.64257 −1.35723 −2.57732 3.16714 −6.81075 1.00000 3.49801
1.12 −2.46621 1.00000 4.08220 1.94142 −2.46621 −0.674366 −5.13515 1.00000 −4.78795
1.13 −2.42396 1.00000 3.87556 −3.49875 −2.42396 3.65352 −4.54627 1.00000 8.48081
1.14 −2.31100 1.00000 3.34070 3.34170 −2.31100 4.45674 −3.09836 1.00000 −7.72265
1.15 −2.31049 1.00000 3.33834 3.79120 −2.31049 −2.69218 −3.09222 1.00000 −8.75950
1.16 −2.30919 1.00000 3.33237 −0.223778 −2.30919 3.94364 −3.07670 1.00000 0.516746
1.17 −2.21866 1.00000 2.92244 −3.24553 −2.21866 −2.52068 −2.04658 1.00000 7.20072
1.18 −2.20805 1.00000 2.87549 −1.04969 −2.20805 −2.20196 −1.93312 1.00000 2.31777
1.19 −2.16763 1.00000 2.69860 2.25786 −2.16763 0.531131 −1.51431 1.00000 −4.89419
1.20 −2.16635 1.00000 2.69306 3.84293 −2.16635 1.73846 −1.50141 1.00000 −8.32512
See next 80 embeddings (of 129 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.129
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(2683\) \(1\)