Properties

Label 6015.2.a.g
Level 6015
Weight 2
Character orbit 6015.a
Self dual yes
Analytic conductor 48.030
Analytic rank 0
Dimension 36
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6015 = 3 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6015.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0300168158\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

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

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 −2.70933 −1.00000 5.34047 1.00000 2.70933 0.462460 −9.05044 1.00000 −2.70933
1.2 −2.61390 −1.00000 4.83247 1.00000 2.61390 −4.22901 −7.40380 1.00000 −2.61390
1.3 −2.55061 −1.00000 4.50559 1.00000 2.55061 −0.898526 −6.39078 1.00000 −2.55061
1.4 −2.43402 −1.00000 3.92446 1.00000 2.43402 −1.65420 −4.68418 1.00000 −2.43402
1.5 −2.31869 −1.00000 3.37634 1.00000 2.31869 3.37785 −3.19130 1.00000 −2.31869
1.6 −2.12459 −1.00000 2.51388 1.00000 2.12459 3.70349 −1.09177 1.00000 −2.12459
1.7 −1.92662 −1.00000 1.71187 1.00000 1.92662 2.81069 0.555119 1.00000 −1.92662
1.8 −1.76666 −1.00000 1.12107 1.00000 1.76666 −4.11244 1.55276 1.00000 −1.76666
1.9 −1.73556 −1.00000 1.01218 1.00000 1.73556 −3.79849 1.71442 1.00000 −1.73556
1.10 −1.65221 −1.00000 0.729796 1.00000 1.65221 −0.892368 2.09864 1.00000 −1.65221
1.11 −1.50420 −1.00000 0.262613 1.00000 1.50420 1.83015 2.61337 1.00000 −1.50420
1.12 −0.967785 −1.00000 −1.06339 1.00000 0.967785 0.0385606 2.96471 1.00000 −0.967785
1.13 −0.942366 −1.00000 −1.11195 1.00000 0.942366 5.06770 2.93259 1.00000 −0.942366
1.14 −0.713173 −1.00000 −1.49138 1.00000 0.713173 −4.34956 2.48996 1.00000 −0.713173
1.15 −0.567973 −1.00000 −1.67741 1.00000 0.567973 3.22749 2.08867 1.00000 −0.567973
1.16 −0.558858 −1.00000 −1.68768 1.00000 0.558858 −1.43354 2.06089 1.00000 −0.558858
1.17 −0.347552 −1.00000 −1.87921 1.00000 0.347552 −4.57897 1.34823 1.00000 −0.347552
1.18 0.108652 −1.00000 −1.98819 1.00000 −0.108652 0.790651 −0.433326 1.00000 0.108652
1.19 0.274127 −1.00000 −1.92485 1.00000 −0.274127 0.857432 −1.07591 1.00000 0.274127
1.20 0.288786 −1.00000 −1.91660 1.00000 −0.288786 −0.447752 −1.13106 1.00000 0.288786
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 admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6015.2.a.g 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6015.2.a.g 36 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(401\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{36} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database