Properties

Label 6039.2.a.m
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.

\(\operatorname{Tr}(f)(q) = \) \( 25q - 5q^{2} + 25q^{4} - 12q^{5} - 4q^{7} - 15q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 25q - 5q^{2} + 25q^{4} - 12q^{5} - 4q^{7} - 15q^{8} - 12q^{10} + 25q^{11} - 4q^{13} - 14q^{14} + 21q^{16} - 16q^{17} - 18q^{19} - 28q^{20} - 5q^{22} - 8q^{23} + 29q^{25} - 16q^{26} + 18q^{28} - 28q^{29} - 8q^{31} - 35q^{32} + 6q^{34} - 22q^{35} + 4q^{37} + 4q^{38} - 12q^{40} - 58q^{41} - 26q^{43} + 25q^{44} + 8q^{46} - 20q^{47} + 23q^{49} - 27q^{50} - 2q^{52} - 36q^{53} - 12q^{55} - 70q^{56} + 12q^{58} - 18q^{59} + 25q^{61} - 42q^{62} + 35q^{64} - 76q^{65} - 8q^{67} - 28q^{68} + 76q^{70} - 24q^{71} + 2q^{73} - 40q^{74} - 64q^{76} - 4q^{77} - 22q^{79} - 36q^{80} + 30q^{82} - 14q^{83} - 70q^{86} - 15q^{88} - 82q^{89} - 6q^{91} - 48q^{92} - 16q^{94} - 34q^{95} + 16q^{97} - 35q^{98} + O(q^{100}) \)

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 −2.81809 0 5.94161 −0.621459 0 3.29277 −11.1078 0 1.75133
1.2 −2.53522 0 4.42736 3.72510 0 −0.951731 −6.15390 0 −9.44395
1.3 −2.52609 0 4.38115 −1.93230 0 0.113912 −6.01501 0 4.88117
1.4 −2.43689 0 3.93843 −3.08470 0 4.69684 −4.72373 0 7.51707
1.5 −2.03124 0 2.12592 −3.40634 0 −3.47017 −0.255767 0 6.91908
1.6 −1.98740 0 1.94976 1.69625 0 1.47319 0.0998540 0 −3.37113
1.7 −1.84847 0 1.41684 3.30338 0 −3.95713 1.07795 0 −6.10619
1.8 −1.38389 0 −0.0848502 −4.17750 0 3.01863 2.88520 0 5.78120
1.9 −1.18061 0 −0.606158 0.892999 0 −4.86985 3.07686 0 −1.05428
1.10 −1.12582 0 −0.732534 0.691275 0 −1.12590 3.07634 0 −0.778249
1.11 −0.800654 0 −1.35895 −0.757549 0 0.484724 2.68936 0 0.606535
1.12 −0.649968 0 −1.57754 3.31582 0 1.03899 2.32529 0 −2.15517
1.13 −0.536820 0 −1.71182 −1.82707 0 2.96010 1.99258 0 0.980806
1.14 0.228656 0 −1.94772 0.872276 0 −1.27506 −0.902667 0 0.199451
1.15 0.325798 0 −1.89386 −3.19527 0 −2.43974 −1.26861 0 −1.04101
1.16 0.608354 0 −1.62991 1.59846 0 −1.08777 −2.20827 0 0.972431
1.17 0.870092 0 −1.24294 −3.50217 0 −4.57282 −2.82166 0 −3.04721
1.18 1.04320 0 −0.911728 1.64856 0 5.01087 −3.03752 0 1.71978
1.19 1.23685 0 −0.470198 −0.403292 0 −0.303505 −3.05527 0 −0.498813
1.20 1.53628 0 0.360147 −2.16725 0 2.96367 −2.51927 0 −3.32950
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.m 25
3.b odd 2 1 6039.2.a.p yes 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6039.2.a.m 25 1.a even 1 1 trivial
6039.2.a.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).