Properties

Label 6039.2.a.l
Level 6039
Weight 2
Character orbit 6039.a
Self dual Yes
Analytic conductor 48.222
Analytic rank 0
Dimension 21
CM No

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Newspace parameters

Level: \( N \) = \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6039.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(21\)
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.

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

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 −2.68161 0 5.19104 3.56342 0 3.18744 −8.55713 0 −9.55570
1.2 −2.67686 0 5.16560 −1.61394 0 −4.48909 −8.47390 0 4.32029
1.3 −2.54333 0 4.46852 0.268217 0 2.27426 −6.27827 0 −0.682163
1.4 −2.16302 0 2.67865 −3.42546 0 1.52570 −1.46794 0 7.40934
1.5 −2.08860 0 2.36225 0.0546248 0 −2.53368 −0.756597 0 −0.114089
1.6 −1.61525 0 0.609041 −2.50050 0 4.06392 2.24675 0 4.03894
1.7 −1.29392 0 −0.325780 3.15298 0 5.03078 3.00937 0 −4.07969
1.8 −0.942064 0 −1.11252 −4.16220 0 −0.914635 2.93219 0 3.92106
1.9 −0.731918 0 −1.46430 −2.76447 0 −1.34613 2.53558 0 2.02337
1.10 −0.543169 0 −1.70497 0.382346 0 −4.84691 2.01242 0 −0.207678
1.11 −0.472572 0 −1.77668 3.20957 0 4.06194 1.78475 0 −1.51675
1.12 0.404310 0 −1.83653 −0.0975766 0 1.38056 −1.55115 0 −0.0394512
1.13 0.634118 0 −1.59789 1.64555 0 −3.41198 −2.28149 0 1.04347
1.14 1.29453 0 −0.324186 −2.94198 0 −0.962580 −3.00873 0 −3.80849
1.15 1.45442 0 0.115325 −3.82594 0 3.26042 −2.74110 0 −5.56450
1.16 1.47389 0 0.172365 −0.593969 0 3.31761 −2.69374 0 −0.875448
1.17 2.19302 0 2.80936 3.87665 0 3.34814 1.77494 0 8.50160
1.18 2.40042 0 3.76203 3.90458 0 −3.05635 4.22962 0 9.37264
1.19 2.55546 0 4.53040 −4.10969 0 −4.64385 6.46634 0 −10.5022
1.20 2.60374 0 4.77947 −0.851432 0 3.04314 7.23702 0 −2.21691
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)
\(61\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{21} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).