Properties

Label 6039.2.a.o
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
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: \(0\)
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 −2.47686 0 4.13484 0.821809 0 −3.73620 −5.28771 0 −2.03551
1.2 −2.44801 0 3.99278 1.89686 0 −0.969789 −4.87834 0 −4.64354
1.3 −2.23548 0 2.99739 −2.10645 0 −1.54970 −2.22964 0 4.70893
1.4 −1.97018 0 1.88162 1.48404 0 2.73292 0.233227 0 −2.92383
1.5 −1.83299 0 1.35986 −0.470771 0 0.751426 1.17337 0 0.862920
1.6 −1.76243 0 1.10617 −2.60834 0 0.445474 1.57532 0 4.59703
1.7 −1.07776 0 −0.838436 3.04682 0 3.44893 3.05915 0 −3.28374
1.8 −1.06662 0 −0.862332 3.84706 0 −1.30714 3.05301 0 −4.10333
1.9 −1.04470 0 −0.908610 −2.79878 0 −4.32870 3.03861 0 2.92388
1.10 −0.363962 0 −1.86753 0.137887 0 3.90557 1.40764 0 −0.0501855
1.11 0.0238689 0 −1.99943 −2.56388 0 0.894211 −0.0954618 0 −0.0611970
1.12 0.0892703 0 −1.99203 0.250715 0 −4.16429 −0.356370 0 0.0223814
1.13 0.110609 0 −1.98777 −0.0761036 0 3.79648 −0.441081 0 −0.00841771
1.14 0.226426 0 −1.94873 3.14756 0 −2.84661 −0.894095 0 0.712690
1.15 0.702784 0 −1.50609 −2.41068 0 0.380608 −2.46403 0 −1.69419
1.16 0.926278 0 −1.14201 3.01280 0 2.23888 −2.91037 0 2.79069
1.17 1.56609 0 0.452630 1.36136 0 3.35990 −2.42332 0 2.13201
1.18 1.65004 0 0.722641 −2.19434 0 −0.319690 −2.10770 0 −3.62075
1.19 1.77462 0 1.14926 −1.38341 0 −3.00047 −1.50973 0 −2.45503
1.20 1.79702 0 1.22928 −4.13243 0 4.22858 −1.38500 0 −7.42606
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.o yes 25
3.b odd 2 1 6039.2.a.n 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6039.2.a.n 25 3.b odd 2 1
6039.2.a.o yes 25 1.a even 1 1 trivial

Hecke kernels

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