Properties

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

Related objects

Downloads

Learn more about

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: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36q - 7q^{2} - 36q^{3} + 45q^{4} - 36q^{5} + 7q^{6} + 2q^{7} - 24q^{8} + 36q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q - 7q^{2} - 36q^{3} + 45q^{4} - 36q^{5} + 7q^{6} + 2q^{7} - 24q^{8} + 36q^{9} + 7q^{10} - q^{11} - 45q^{12} - 8q^{13} - 4q^{14} + 36q^{15} + 63q^{16} - 50q^{17} - 7q^{18} + 15q^{19} - 45q^{20} - 2q^{21} + 7q^{22} - 32q^{23} + 24q^{24} + 36q^{25} - 12q^{26} - 36q^{27} - 10q^{28} + 7q^{29} - 7q^{30} + 7q^{31} - 50q^{32} + q^{33} + 8q^{34} - 2q^{35} + 45q^{36} - 10q^{37} - 48q^{38} + 8q^{39} + 24q^{40} - 31q^{41} + 4q^{42} + 27q^{43} - 9q^{44} - 36q^{45} + 23q^{46} - 46q^{47} - 63q^{48} + 48q^{49} - 7q^{50} + 50q^{51} - 14q^{52} - 39q^{53} + 7q^{54} + q^{55} - 29q^{56} - 15q^{57} - 26q^{58} - 9q^{59} + 45q^{60} + 5q^{61} - 65q^{62} + 2q^{63} + 90q^{64} + 8q^{65} - 7q^{66} + 18q^{67} - 128q^{68} + 32q^{69} + 4q^{70} + 4q^{71} - 24q^{72} - 45q^{73} - 22q^{74} - 36q^{75} + 26q^{76} - 38q^{77} + 12q^{78} + 25q^{79} - 63q^{80} + 36q^{81} - 5q^{82} - 71q^{83} + 10q^{84} + 50q^{85} + 3q^{86} - 7q^{87} - 9q^{88} - 39q^{89} + 7q^{90} + 19q^{91} - 95q^{92} - 7q^{93} + 16q^{94} - 15q^{95} + 50q^{96} - 61q^{97} - 76q^{98} - q^{99} + O(q^{100}) \)

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 −2.82805 −1.00000 5.99789 −1.00000 2.82805 0.186142 −11.3062 1.00000 2.82805
1.2 −2.70250 −1.00000 5.30351 −1.00000 2.70250 −3.42053 −8.92774 1.00000 2.70250
1.3 −2.65304 −1.00000 5.03861 −1.00000 2.65304 −2.89459 −8.06155 1.00000 2.65304
1.4 −2.61575 −1.00000 4.84213 −1.00000 2.61575 3.80178 −7.43431 1.00000 2.61575
1.5 −2.58666 −1.00000 4.69083 −1.00000 2.58666 2.48059 −6.96028 1.00000 2.58666
1.6 −2.45396 −1.00000 4.02194 −1.00000 2.45396 3.31635 −4.96176 1.00000 2.45396
1.7 −2.15201 −1.00000 2.63116 −1.00000 2.15201 −1.35819 −1.35826 1.00000 2.15201
1.8 −2.03595 −1.00000 2.14509 −1.00000 2.03595 −4.31879 −0.295397 1.00000 2.03595
1.9 −1.98597 −1.00000 1.94407 −1.00000 1.98597 −2.98220 0.111071 1.00000 1.98597
1.10 −1.98559 −1.00000 1.94257 −1.00000 1.98559 4.82942 0.114041 1.00000 1.98559
1.11 −1.50384 −1.00000 0.261530 −1.00000 1.50384 2.03402 2.61438 1.00000 1.50384
1.12 −1.46470 −1.00000 0.145344 −1.00000 1.46470 1.42914 2.71651 1.00000 1.46470
1.13 −1.30040 −1.00000 −0.308957 −1.00000 1.30040 −4.16384 3.00257 1.00000 1.30040
1.14 −1.26310 −1.00000 −0.404578 −1.00000 1.26310 −1.43647 3.03722 1.00000 1.26310
1.15 −0.886874 −1.00000 −1.21345 −1.00000 0.886874 0.962079 2.84993 1.00000 0.886874
1.16 −0.823088 −1.00000 −1.32253 −1.00000 0.823088 2.12642 2.73473 1.00000 0.823088
1.17 −0.518532 −1.00000 −1.73112 −1.00000 0.518532 3.44981 1.93471 1.00000 0.518532
1.18 −0.409079 −1.00000 −1.83265 −1.00000 0.409079 2.40403 1.56786 1.00000 0.409079
1.19 −0.139206 −1.00000 −1.98062 −1.00000 0.139206 −3.46594 0.554125 1.00000 0.139206
1.20 0.278732 −1.00000 −1.92231 −1.00000 −0.278732 −3.61104 −1.09327 1.00000 −0.278732
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

Atkin-Lehner signs

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

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

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