Properties

Label 6015.2.a.b
Level $6015$
Weight $2$
Character orbit 6015.a
Self dual yes
Analytic conductor $48.030$
Analytic rank $1$
Dimension $23$
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: \(1\)
Dimension: \(23\)
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.

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

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 −2.62638 1.00000 4.89789 1.00000 −2.62638 −0.417980 −7.61099 1.00000 −2.62638
1.2 −2.46180 1.00000 4.06045 1.00000 −2.46180 −1.67554 −5.07240 1.00000 −2.46180
1.3 −2.22931 1.00000 2.96981 1.00000 −2.22931 −0.230360 −2.16201 1.00000 −2.22931
1.4 −2.14529 1.00000 2.60225 1.00000 −2.14529 −2.89107 −1.29200 1.00000 −2.14529
1.5 −1.79507 1.00000 1.22228 1.00000 −1.79507 3.49730 1.39607 1.00000 −1.79507
1.6 −1.63766 1.00000 0.681937 1.00000 −1.63766 −0.197583 2.15854 1.00000 −1.63766
1.7 −1.27466 1.00000 −0.375250 1.00000 −1.27466 −0.831191 3.02763 1.00000 −1.27466
1.8 −1.13202 1.00000 −0.718528 1.00000 −1.13202 −1.31252 3.07743 1.00000 −1.13202
1.9 −1.12824 1.00000 −0.727075 1.00000 −1.12824 −4.89351 3.07679 1.00000 −1.12824
1.10 −0.791085 1.00000 −1.37418 1.00000 −0.791085 2.45382 2.66927 1.00000 −0.791085
1.11 −0.762017 1.00000 −1.41933 1.00000 −0.762017 2.32113 2.60559 1.00000 −0.762017
1.12 −0.108293 1.00000 −1.98827 1.00000 −0.108293 −2.69972 0.431901 1.00000 −0.108293
1.13 0.0493063 1.00000 −1.99757 1.00000 0.0493063 2.05010 −0.197105 1.00000 0.0493063
1.14 0.186838 1.00000 −1.96509 1.00000 0.186838 −4.07485 −0.740831 1.00000 0.186838
1.15 0.592282 1.00000 −1.64920 1.00000 0.592282 −0.542800 −2.16136 1.00000 0.592282
1.16 0.783438 1.00000 −1.38623 1.00000 0.783438 −0.845679 −2.65290 1.00000 0.783438
1.17 1.02584 1.00000 −0.947645 1.00000 1.02584 0.748026 −3.02382 1.00000 1.02584
1.18 1.03713 1.00000 −0.924357 1.00000 1.03713 2.93887 −3.03294 1.00000 1.03713
1.19 1.60756 1.00000 0.584262 1.00000 1.60756 0.859599 −2.27589 1.00000 1.60756
1.20 1.75959 1.00000 1.09616 1.00000 1.75959 −3.35674 −1.59039 1.00000 1.75959
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
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.b 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6015.2.a.b 23 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{23} + 5 T_{2}^{22} - 15 T_{2}^{21} - 106 T_{2}^{20} + 57 T_{2}^{19} + 942 T_{2}^{18} + 252 T_{2}^{17} - 4580 T_{2}^{16} - 3018 T_{2}^{15} + 13334 T_{2}^{14} + 11792 T_{2}^{13} - 23949 T_{2}^{12} - 24531 T_{2}^{11} + 26491 T_{2}^{10} + \cdots - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\). Copy content Toggle raw display