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Properties

Label 6015.2.a.i
Level $6015$
Weight $2$
Character orbit 6015.a
Self dual yes
Analytic conductor $48.030$
Analytic rank $0$
Dimension $43$
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] = [6015,2,Mod(1,6015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6015, 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("6015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6015 = 3 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6015.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: \(48.0300168158\)
Analytic rank: \(0\)
Dimension: \(43\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 43 q + 3 q^{2} + 43 q^{3} + 61 q^{4} + 43 q^{5} + 3 q^{6} + 14 q^{7} + 18 q^{8} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 43 q + 3 q^{2} + 43 q^{3} + 61 q^{4} + 43 q^{5} + 3 q^{6} + 14 q^{7} + 18 q^{8} + 43 q^{9} + 3 q^{10} + 19 q^{11} + 61 q^{12} + 8 q^{13} + 6 q^{14} + 43 q^{15} + 85 q^{16} + 40 q^{17} + 3 q^{18} + 43 q^{19} + 61 q^{20} + 14 q^{21} + 19 q^{22} + 12 q^{23} + 18 q^{24} + 43 q^{25} + 43 q^{27} + 36 q^{28} + 41 q^{29} + 3 q^{30} + 33 q^{31} + 4 q^{32} + 19 q^{33} + 20 q^{34} + 14 q^{35} + 61 q^{36} + 12 q^{37} + 10 q^{38} + 8 q^{39} + 18 q^{40} + 47 q^{41} + 6 q^{42} + 73 q^{43} + 5 q^{44} + 43 q^{45} + 21 q^{46} + 32 q^{47} + 85 q^{48} + 87 q^{49} + 3 q^{50} + 40 q^{51} + 18 q^{52} + 17 q^{53} + 3 q^{54} + 19 q^{55} + 15 q^{56} + 43 q^{57} - 16 q^{58} + 21 q^{59} + 61 q^{60} + 77 q^{61} + 15 q^{62} + 14 q^{63} + 112 q^{64} + 8 q^{65} + 19 q^{66} + 26 q^{67} + 50 q^{68} + 12 q^{69} + 6 q^{70} + 2 q^{71} + 18 q^{72} + 49 q^{73} - 34 q^{74} + 43 q^{75} + 50 q^{76} - 2 q^{77} + 59 q^{79} + 85 q^{80} + 43 q^{81} - 45 q^{82} - 3 q^{83} + 36 q^{84} + 40 q^{85} - 35 q^{86} + 41 q^{87} - 13 q^{88} + 57 q^{89} + 3 q^{90} + 11 q^{91} - 9 q^{92} + 33 q^{93} + 52 q^{94} + 43 q^{95} + 4 q^{96} + 7 q^{97} - 32 q^{98} + 19 q^{99}+O(q^{100}) \) 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 −2.80316 1.00000 5.85773 1.00000 −2.80316 1.72166 −10.8139 1.00000 −2.80316
1.2 −2.71814 1.00000 5.38826 1.00000 −2.71814 −4.08781 −9.20975 1.00000 −2.71814
1.3 −2.63729 1.00000 4.95532 1.00000 −2.63729 3.21800 −7.79405 1.00000 −2.63729
1.4 −2.53504 1.00000 4.42643 1.00000 −2.53504 3.85984 −6.15109 1.00000 −2.53504
1.5 −2.53356 1.00000 4.41894 1.00000 −2.53356 1.97049 −6.12853 1.00000 −2.53356
1.6 −2.26589 1.00000 3.13426 1.00000 −2.26589 −0.747264 −2.57010 1.00000 −2.26589
1.7 −2.07772 1.00000 2.31692 1.00000 −2.07772 −4.42665 −0.658463 1.00000 −2.07772
1.8 −1.89266 1.00000 1.58218 1.00000 −1.89266 −3.83242 0.790801 1.00000 −1.89266
1.9 −1.87550 1.00000 1.51750 1.00000 −1.87550 3.82548 0.904923 1.00000 −1.87550
1.10 −1.83425 1.00000 1.36446 1.00000 −1.83425 0.959721 1.16574 1.00000 −1.83425
1.11 −1.80049 1.00000 1.24176 1.00000 −1.80049 −4.47883 1.36520 1.00000 −1.80049
1.12 −1.69499 1.00000 0.872981 1.00000 −1.69499 4.08936 1.91028 1.00000 −1.69499
1.13 −1.41364 1.00000 −0.00162729 1.00000 −1.41364 −1.83352 2.82958 1.00000 −1.41364
1.14 −1.30521 1.00000 −0.296439 1.00000 −1.30521 1.84814 2.99732 1.00000 −1.30521
1.15 −1.15381 1.00000 −0.668729 1.00000 −1.15381 −0.585541 3.07920 1.00000 −1.15381
1.16 −0.979152 1.00000 −1.04126 1.00000 −0.979152 4.18408 2.97786 1.00000 −0.979152
1.17 −0.762858 1.00000 −1.41805 1.00000 −0.762858 −2.97870 2.60748 1.00000 −0.762858
1.18 −0.497996 1.00000 −1.75200 1.00000 −0.497996 2.05942 1.86848 1.00000 −0.497996
1.19 −0.439798 1.00000 −1.80658 1.00000 −0.439798 −0.915251 1.67413 1.00000 −0.439798
1.20 −0.211503 1.00000 −1.95527 1.00000 −0.211503 3.03461 0.836552 1.00000 −0.211503
See all 43 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.43
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.i ✓ 43
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6015.2.a.i ✓ 43 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{43} - 3 T_{2}^{42} - 69 T_{2}^{41} + 206 T_{2}^{40} + 2205 T_{2}^{39} - 6530 T_{2}^{38} + \cdots - 11944 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\). Copy content Toggle raw display