Properties

Label 4015.2.a.f
Level 4015
Weight 2
Character orbit 4015.a
Self dual yes
Analytic conductor 32.060
Analytic rank 1
Dimension 31
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4015.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0599364115\)
Analytic rank: \(1\)
Dimension: \(31\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 31q - 7q^{2} - 4q^{3} + 39q^{4} - 31q^{5} - 5q^{6} - 11q^{7} - 24q^{8} + 31q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 31q - 7q^{2} - 4q^{3} + 39q^{4} - 31q^{5} - 5q^{6} - 11q^{7} - 24q^{8} + 31q^{9} + 7q^{10} + 31q^{11} - 4q^{12} - 24q^{13} - 9q^{14} + 4q^{15} + 43q^{16} - 49q^{17} - 35q^{18} - 22q^{19} - 39q^{20} - 8q^{21} - 7q^{22} - q^{23} - 13q^{24} + 31q^{25} - 9q^{26} - 22q^{27} - 34q^{28} - 12q^{29} + 5q^{30} + 4q^{31} - 45q^{32} - 4q^{33} + 2q^{34} + 11q^{35} + 34q^{36} - 18q^{37} - 7q^{38} - q^{39} + 24q^{40} - 58q^{41} - 21q^{42} - 41q^{43} + 39q^{44} - 31q^{45} + 23q^{46} - 31q^{47} - 29q^{48} + 44q^{49} - 7q^{50} + 8q^{51} - 89q^{52} - 46q^{53} - 47q^{54} - 31q^{55} + 10q^{56} - 47q^{57} - 34q^{58} - 9q^{59} + 4q^{60} - 5q^{61} - 50q^{62} - 61q^{63} + 78q^{64} + 24q^{65} - 5q^{66} + q^{67} - 115q^{68} - 19q^{69} + 9q^{70} - 8q^{71} - 93q^{72} + 31q^{73} - 19q^{74} - 4q^{75} - 7q^{76} - 11q^{77} + 57q^{78} - 43q^{80} + 43q^{81} + 20q^{82} - 29q^{83} - 32q^{84} + 49q^{85} + 25q^{86} - 62q^{87} - 24q^{88} - 77q^{89} + 35q^{90} - 11q^{91} - 25q^{92} - 38q^{94} + 22q^{95} - 23q^{96} - 39q^{97} - 65q^{98} + 31q^{99} + O(q^{100}) \)

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 −2.77028 −2.37765 5.67444 −1.00000 6.58674 −3.31727 −10.1792 2.65320 2.77028
1.2 −2.76022 1.97999 5.61884 −1.00000 −5.46521 −3.73820 −9.98882 0.920356 2.76022
1.3 −2.63633 −0.0191781 4.95023 −1.00000 0.0505599 0.546812 −7.77778 −2.99963 2.63633
1.4 −2.58131 3.02998 4.66316 −1.00000 −7.82131 4.31568 −6.87443 6.18077 2.58131
1.5 −2.42903 −2.96305 3.90019 −1.00000 7.19735 2.52750 −4.61563 5.77969 2.42903
1.6 −2.19172 −1.93564 2.80362 −1.00000 4.24237 −4.64679 −1.76130 0.746693 2.19172
1.7 −2.11520 0.824774 2.47406 −1.00000 −1.74456 0.827865 −1.00272 −2.31975 2.11520
1.8 −2.08368 −0.642257 2.34171 −1.00000 1.33826 4.42756 −0.712022 −2.58751 2.08368
1.9 −1.69949 −0.262694 0.888269 −1.00000 0.446447 0.253523 1.88938 −2.93099 1.69949
1.10 −1.69343 3.34184 0.867694 −1.00000 −5.65917 −3.70695 1.91748 8.16792 1.69343
1.11 −1.60539 2.52956 0.577267 −1.00000 −4.06093 −3.16259 2.28404 3.39869 1.60539
1.12 −1.24926 −2.63422 −0.439347 −1.00000 3.29082 1.69239 3.04738 3.93909 1.24926
1.13 −0.754723 −0.390285 −1.43039 −1.00000 0.294557 1.05438 2.58900 −2.84768 0.754723
1.14 −0.602631 −3.44000 −1.63684 −1.00000 2.07305 −0.349257 2.19167 8.83357 0.602631
1.15 −0.595851 −0.809318 −1.64496 −1.00000 0.482233 −4.26782 2.17186 −2.34500 0.595851
1.16 −0.485073 −0.214641 −1.76470 −1.00000 0.104117 3.34739 1.82616 −2.95393 0.485073
1.17 −0.403680 1.58032 −1.83704 −1.00000 −0.637944 3.59112 1.54894 −0.502579 0.403680
1.18 0.315426 2.38391 −1.90051 −1.00000 0.751948 0.00485401 −1.23032 2.68302 −0.315426
1.19 0.416759 1.40268 −1.82631 −1.00000 0.584579 −1.52897 −1.59465 −1.03249 −0.416759
1.20 0.773764 −2.02237 −1.40129 −1.00000 −1.56484 −2.27321 −2.63180 1.08999 −0.773764
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
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 4015.2.a.f 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4015.2.a.f 31 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)
\(73\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{31} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database