Properties

Label 6039.2.a.n
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $25$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(25\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 25q - 5q^{2} + 25q^{4} - 4q^{5} + 4q^{7} - 15q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 25q - 5q^{2} + 25q^{4} - 4q^{5} + 4q^{7} - 15q^{8} - 25q^{11} + 4q^{13} - 18q^{14} + 21q^{16} - 20q^{17} + 14q^{19} - 12q^{20} + 5q^{22} - 20q^{23} + 13q^{25} - 16q^{26} - 14q^{28} - 28q^{29} - 12q^{31} - 35q^{32} + 6q^{34} - 10q^{35} - 8q^{37} - 32q^{38} + 24q^{40} - 26q^{41} + 18q^{43} - 25q^{44} + 4q^{46} - 12q^{47} + 23q^{49} - 43q^{50} + 22q^{52} - 36q^{53} + 4q^{55} - 26q^{56} - 20q^{58} - 46q^{59} - 25q^{61} + 14q^{62} - 13q^{64} - 60q^{65} - 20q^{67} - 44q^{68} - 20q^{70} - 52q^{71} + 6q^{73} - 32q^{74} - 4q^{77} + 26q^{79} - 52q^{80} + 6q^{82} - 38q^{83} - 4q^{85} - 34q^{86} + 15q^{88} - 82q^{89} - 58q^{91} - 36q^{92} + 16q^{94} - 30q^{95} - 35q^{98} + O(q^{100}) \)

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 −2.77875 0 5.72143 2.25657 0 0.879084 −10.3409 0 −6.27043
1.2 −2.62068 0 4.86797 −4.12800 0 −0.639358 −7.51602 0 10.8182
1.3 −2.50682 0 4.28416 −0.826559 0 1.21549 −5.72598 0 2.07204
1.4 −2.28499 0 3.22117 −2.86039 0 3.60040 −2.79036 0 6.53597
1.5 −2.22076 0 2.93178 −0.179885 0 −5.01600 −2.06925 0 0.399482
1.6 −1.79702 0 1.22928 4.13243 0 4.22858 1.38500 0 −7.42606
1.7 −1.77462 0 1.14926 1.38341 0 −3.00047 1.50973 0 −2.45503
1.8 −1.65004 0 0.722641 2.19434 0 −0.319690 2.10770 0 −3.62075
1.9 −1.56609 0 0.452630 −1.36136 0 3.35990 2.42332 0 2.13201
1.10 −0.926278 0 −1.14201 −3.01280 0 2.23888 2.91037 0 2.79069
1.11 −0.702784 0 −1.50609 2.41068 0 0.380608 2.46403 0 −1.69419
1.12 −0.226426 0 −1.94873 −3.14756 0 −2.84661 0.894095 0 0.712690
1.13 −0.110609 0 −1.98777 0.0761036 0 3.79648 0.441081 0 −0.00841771
1.14 −0.0892703 0 −1.99203 −0.250715 0 −4.16429 0.356370 0 0.0223814
1.15 −0.0238689 0 −1.99943 2.56388 0 0.894211 0.0954618 0 −0.0611970
1.16 0.363962 0 −1.86753 −0.137887 0 3.90557 −1.40764 0 −0.0501855
1.17 1.04470 0 −0.908610 2.79878 0 −4.32870 −3.03861 0 2.92388
1.18 1.06662 0 −0.862332 −3.84706 0 −1.30714 −3.05301 0 −4.10333
1.19 1.07776 0 −0.838436 −3.04682 0 3.44893 −3.05915 0 −3.28374
1.20 1.76243 0 1.10617 2.60834 0 0.445474 −1.57532 0 4.59703
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6039.2.a.n 25
3.b odd 2 1 6039.2.a.o yes 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6039.2.a.n 25 1.a even 1 1 trivial
6039.2.a.o yes 25 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{25} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).