Properties

Label 6015.2.a.h
Level $6015$
Weight $2$
Character orbit 6015.a
Self dual yes
Analytic conductor $48.030$
Analytic rank $0$
Dimension $39$
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: \(39\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 39 q + 39 q^{3} + 48 q^{4} - 39 q^{5} + 22 q^{7} + 3 q^{8} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 39 q + 39 q^{3} + 48 q^{4} - 39 q^{5} + 22 q^{7} + 3 q^{8} + 39 q^{9} - q^{11} + 48 q^{12} + 30 q^{13} + 8 q^{14} - 39 q^{15} + 58 q^{16} + 32 q^{17} + 27 q^{19} - 48 q^{20} + 22 q^{21} + 23 q^{22} - 8 q^{23} + 3 q^{24} + 39 q^{25} - 4 q^{26} + 39 q^{27} + 60 q^{28} - 9 q^{29} + 19 q^{31} + q^{32} - q^{33} + 26 q^{34} - 22 q^{35} + 48 q^{36} + 44 q^{37} + 14 q^{38} + 30 q^{39} - 3 q^{40} + 31 q^{41} + 8 q^{42} + 75 q^{43} + q^{44} - 39 q^{45} + 19 q^{46} - 16 q^{47} + 58 q^{48} + 91 q^{49} + 32 q^{51} + 94 q^{52} + 17 q^{53} + q^{55} + 27 q^{56} + 27 q^{57} + 26 q^{58} - q^{59} - 48 q^{60} + 55 q^{61} + 11 q^{62} + 22 q^{63} + 77 q^{64} - 30 q^{65} + 23 q^{66} + 84 q^{67} + 36 q^{68} - 8 q^{69} - 8 q^{70} - 2 q^{71} + 3 q^{72} + 79 q^{73} + 20 q^{74} + 39 q^{75} + 58 q^{76} + 32 q^{77} - 4 q^{78} + 29 q^{79} - 58 q^{80} + 39 q^{81} + 53 q^{82} + 9 q^{83} + 60 q^{84} - 32 q^{85} - 17 q^{86} - 9 q^{87} + 57 q^{88} + 37 q^{89} + 71 q^{91} + 7 q^{92} + 19 q^{93} + 32 q^{94} - 27 q^{95} + q^{96} + 91 q^{97} - 9 q^{98} - q^{99}+O(q^{100}) \) 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 −2.73278 1.00000 5.46807 −1.00000 −2.73278 −1.71200 −9.47746 1.00000 2.73278
1.2 −2.72880 1.00000 5.44634 −1.00000 −2.72880 4.68686 −9.40437 1.00000 2.72880
1.3 −2.61259 1.00000 4.82563 −1.00000 −2.61259 1.06107 −7.38220 1.00000 2.61259
1.4 −2.40618 1.00000 3.78972 −1.00000 −2.40618 −5.00600 −4.30638 1.00000 2.40618
1.5 −2.39683 1.00000 3.74481 −1.00000 −2.39683 3.91052 −4.18201 1.00000 2.39683
1.6 −2.30681 1.00000 3.32135 −1.00000 −2.30681 0.182032 −3.04810 1.00000 2.30681
1.7 −2.17202 1.00000 2.71767 −1.00000 −2.17202 −0.722026 −1.55879 1.00000 2.17202
1.8 −2.05828 1.00000 2.23653 −1.00000 −2.05828 4.53201 −0.486836 1.00000 2.05828
1.9 −1.76442 1.00000 1.11320 −1.00000 −1.76442 −1.91400 1.56470 1.00000 1.76442
1.10 −1.60401 1.00000 0.572848 −1.00000 −1.60401 0.763953 2.28917 1.00000 1.60401
1.11 −1.46868 1.00000 0.157024 −1.00000 −1.46868 −1.36125 2.70674 1.00000 1.46868
1.12 −1.45937 1.00000 0.129770 −1.00000 −1.45937 4.22534 2.72936 1.00000 1.45937
1.13 −1.39292 1.00000 −0.0597773 −1.00000 −1.39292 0.772686 2.86910 1.00000 1.39292
1.14 −1.06568 1.00000 −0.864332 −1.00000 −1.06568 −3.38476 3.05245 1.00000 1.06568
1.15 −0.832555 1.00000 −1.30685 −1.00000 −0.832555 2.56306 2.75314 1.00000 0.832555
1.16 −0.749262 1.00000 −1.43861 −1.00000 −0.749262 −2.59303 2.57642 1.00000 0.749262
1.17 −0.693654 1.00000 −1.51884 −1.00000 −0.693654 5.04357 2.44086 1.00000 0.693654
1.18 −0.407003 1.00000 −1.83435 −1.00000 −0.407003 −1.58531 1.56059 1.00000 0.407003
1.19 −0.369183 1.00000 −1.86370 −1.00000 −0.369183 −1.65507 1.42641 1.00000 0.369183
1.20 0.104331 1.00000 −1.98912 −1.00000 0.104331 −3.27326 −0.416187 1.00000 −0.104331
See all 39 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.39
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.h 39
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6015.2.a.h 39 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{39} - 63 T_{2}^{37} - T_{2}^{36} + 1820 T_{2}^{35} + 58 T_{2}^{34} - 31968 T_{2}^{33} - 1530 T_{2}^{32} + 381697 T_{2}^{31} + 24317 T_{2}^{30} - 3281393 T_{2}^{29} - 260029 T_{2}^{28} + 20989235 T_{2}^{27} + \cdots + 22239 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\). Copy content Toggle raw display