Properties

Label 6035.2.a.a
Level 6035
Weight 2
Character orbit 6035.a
Self dual Yes
Analytic conductor 48.190
Analytic rank 1
Dimension 36
CM No

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Newspace parameters

Level: \( N \) = \( 6035 = 5 \cdot 17 \cdot 71 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6035.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1897176198\)
Analytic rank: \(1\)
Dimension: \(36\)
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.

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

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 −2.62226 −1.54085 4.87626 1.00000 4.04052 0.0125838 −7.54231 −0.625776 −2.62226
1.2 −2.61440 2.49037 4.83510 1.00000 −6.51084 2.00109 −7.41211 3.20197 −2.61440
1.3 −2.37331 2.40096 3.63259 1.00000 −5.69821 −2.59145 −3.87465 2.76459 −2.37331
1.4 −2.27701 0.848689 3.18476 1.00000 −1.93247 1.85809 −2.69770 −2.27973 −2.27701
1.5 −2.15542 −0.292006 2.64584 1.00000 0.629396 −5.08250 −1.39207 −2.91473 −2.15542
1.6 −2.13074 −1.60683 2.54004 1.00000 3.42373 2.82688 −1.15069 −0.418101 −2.13074
1.7 −2.04946 −2.65729 2.20029 1.00000 5.44602 −0.551924 −0.410497 4.06121 −2.04946
1.8 −1.90835 −2.27887 1.64180 1.00000 4.34888 −0.428939 0.683573 2.19324 −1.90835
1.9 −1.70160 0.687277 0.895431 1.00000 −1.16947 −0.626542 1.87953 −2.52765 −1.70160
1.10 −1.14290 −1.19489 −0.693776 1.00000 1.36564 −1.41810 3.07872 −1.57223 −1.14290
1.11 −1.12696 2.34856 −0.729961 1.00000 −2.64673 −2.68480 3.07656 2.51573 −1.12696
1.12 −1.12140 0.383871 −0.742456 1.00000 −0.430474 1.34550 3.07540 −2.85264 −1.12140
1.13 −1.03393 2.68090 −0.930989 1.00000 −2.77186 1.41622 3.03044 4.18720 −1.03393
1.14 −0.904707 −2.66665 −1.18150 1.00000 2.41254 3.83492 2.87833 4.11101 −0.904707
1.15 −0.890323 −0.682987 −1.20732 1.00000 0.608079 2.96744 2.85556 −2.53353 −0.890323
1.16 −0.599472 −1.15414 −1.64063 1.00000 0.691877 −3.77052 2.18246 −1.66795 −0.599472
1.17 −0.256013 0.196473 −1.93446 1.00000 −0.0502997 2.99337 1.00727 −2.96140 −0.256013
1.18 −0.174528 1.77073 −1.96954 1.00000 −0.309043 −0.0296109 0.692797 0.135502 −0.174528
1.19 0.0925953 0.181897 −1.99143 1.00000 0.0168428 −0.294707 −0.369587 −2.96691 0.0925953
1.20 0.139816 −2.98888 −1.98045 1.00000 −0.417894 2.58191 −0.556532 5.93339 0.139816
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(17\) \(-1\)
\(71\) \(1\)