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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4015,2,Mod(1,4015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4015.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0599364115\)
Analytic rank: \(1\)
Dimension: \(31\)
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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:

Atkin-Lehner signs

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

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 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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{31} + 7 T_{2}^{30} - 26 T_{2}^{29} - 279 T_{2}^{28} + 128 T_{2}^{27} + 4885 T_{2}^{26} + \cdots + 18000 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\). Copy content Toggle raw display