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Properties

Label 6015.2.a.c
Level $6015$
Weight $2$
Character orbit 6015.a
Self dual yes
Analytic conductor $48.030$
Analytic rank $1$
Dimension $28$
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: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{2} + 28 q^{3} + 21 q^{4} - 28 q^{5} - q^{6} - 20 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{2} + 28 q^{3} + 21 q^{4} - 28 q^{5} - q^{6} - 20 q^{7} + 28 q^{9} + q^{10} - q^{11} + 21 q^{12} - 18 q^{13} - 4 q^{14} - 28 q^{15} - q^{16} - 28 q^{17} - q^{18} - 19 q^{19} - 21 q^{20} - 20 q^{21} - 35 q^{22} + 2 q^{23} + 28 q^{25} - 20 q^{26} + 28 q^{27} - 54 q^{28} + 9 q^{29} + q^{30} - 19 q^{31} - 6 q^{32} - q^{33} - 16 q^{34} + 20 q^{35} + 21 q^{36} - 32 q^{37} - 2 q^{38} - 18 q^{39} - 27 q^{41} - 4 q^{42} - 77 q^{43} + q^{44} - 28 q^{45} - 19 q^{46} + 10 q^{47} - q^{48} - 4 q^{49} - q^{50} - 28 q^{51} - 34 q^{52} - 21 q^{53} - q^{54} + q^{55} - 9 q^{56} - 19 q^{57} - 46 q^{58} - 7 q^{59} - 21 q^{60} - 31 q^{61} - 7 q^{62} - 20 q^{63} - 46 q^{64} + 18 q^{65} - 35 q^{66} - 50 q^{67} - 68 q^{68} + 2 q^{69} + 4 q^{70} - 4 q^{71} - 87 q^{73} + 4 q^{74} + 28 q^{75} - 40 q^{76} - 8 q^{77} - 20 q^{78} - 65 q^{79} + q^{80} + 28 q^{81} - 41 q^{82} + 13 q^{83} - 54 q^{84} + 28 q^{85} - 17 q^{86} + 9 q^{87} - 117 q^{88} - 33 q^{89} + q^{90} - 33 q^{91} + 3 q^{92} - 19 q^{93} - 60 q^{94} + 19 q^{95} - 6 q^{96} - 75 q^{97} - 18 q^{98} - q^{99}+O(q^{100}) \) 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 −2.60633 1.00000 4.79297 −1.00000 −2.60633 −3.84306 −7.27940 1.00000 2.60633
1.2 −2.51673 1.00000 4.33393 −1.00000 −2.51673 −0.0213252 −5.87387 1.00000 2.51673
1.3 −2.31182 1.00000 3.34451 −1.00000 −2.31182 −2.77511 −3.10826 1.00000 2.31182
1.4 −2.11452 1.00000 2.47118 −1.00000 −2.11452 1.95178 −0.996316 1.00000 2.11452
1.5 −1.92652 1.00000 1.71147 −1.00000 −1.92652 1.15268 0.555850 1.00000 1.92652
1.6 −1.85217 1.00000 1.43054 −1.00000 −1.85217 −4.50663 1.05474 1.00000 1.85217
1.7 −1.62087 1.00000 0.627218 −1.00000 −1.62087 2.78454 2.22510 1.00000 1.62087
1.8 −1.59553 1.00000 0.545711 −1.00000 −1.59553 −0.379773 2.32036 1.00000 1.59553
1.9 −1.45436 1.00000 0.115161 −1.00000 −1.45436 −2.79967 2.74123 1.00000 1.45436
1.10 −0.932405 1.00000 −1.13062 −1.00000 −0.932405 3.48092 2.91901 1.00000 0.932405
1.11 −0.711184 1.00000 −1.49422 −1.00000 −0.711184 −4.85706 2.48503 1.00000 0.711184
1.12 −0.549158 1.00000 −1.69842 −1.00000 −0.549158 0.129960 2.03102 1.00000 0.549158
1.13 −0.517037 1.00000 −1.73267 −1.00000 −0.517037 1.87988 1.92993 1.00000 0.517037
1.14 −0.327847 1.00000 −1.89252 −1.00000 −0.327847 −1.80659 1.27615 1.00000 0.327847
1.15 0.113161 1.00000 −1.98719 −1.00000 0.113161 2.66791 −0.451194 1.00000 −0.113161
1.16 0.344505 1.00000 −1.88132 −1.00000 0.344505 −3.88123 −1.33713 1.00000 −0.344505
1.17 0.403903 1.00000 −1.83686 −1.00000 0.403903 0.330671 −1.54972 1.00000 −0.403903
1.18 0.886913 1.00000 −1.21339 −1.00000 0.886913 2.28245 −2.84999 1.00000 −0.886913
1.19 0.894188 1.00000 −1.20043 −1.00000 0.894188 −1.23837 −2.86178 1.00000 −0.894188
1.20 0.902444 1.00000 −1.18560 −1.00000 0.902444 −2.02428 −2.87482 1.00000 −0.902444
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
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.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6015.2.a.c ✓ 28 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + T_{2}^{27} - 38 T_{2}^{26} - 37 T_{2}^{25} + 636 T_{2}^{24} + 602 T_{2}^{23} - 6175 T_{2}^{22} + \cdots + 88 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\). Copy content Toggle raw display