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Properties

Label 6015.2.a.d
Level $6015$
Weight $2$
Character orbit 6015.a
Self dual yes
Analytic conductor $48.030$
Analytic rank $1$
Dimension $29$
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: \(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q - q^{2} - 29 q^{3} + 27 q^{4} + 29 q^{5} + q^{6} + 2 q^{7} - 6 q^{8} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q - q^{2} - 29 q^{3} + 27 q^{4} + 29 q^{5} + q^{6} + 2 q^{7} - 6 q^{8} + 29 q^{9} - q^{10} - 21 q^{11} - 27 q^{12} - 8 q^{13} - 30 q^{14} - 29 q^{15} + 23 q^{16} - 28 q^{17} - q^{18} - 9 q^{19} + 27 q^{20} - 2 q^{21} - 9 q^{22} + 6 q^{24} + 29 q^{25} - 34 q^{26} - 29 q^{27} + 6 q^{28} - 61 q^{29} + q^{30} - 19 q^{31} - 8 q^{32} + 21 q^{33} - 16 q^{34} + 2 q^{35} + 27 q^{36} - 4 q^{37} + 4 q^{38} + 8 q^{39} - 6 q^{40} - 85 q^{41} + 30 q^{42} + 29 q^{43} - 69 q^{44} + 29 q^{45} - 35 q^{46} - 2 q^{47} - 23 q^{48} + q^{49} - q^{50} + 28 q^{51} - 28 q^{52} - 5 q^{53} + q^{54} - 21 q^{55} - 97 q^{56} + 9 q^{57} + 6 q^{58} - 43 q^{59} - 27 q^{60} - 59 q^{61} - 17 q^{62} + 2 q^{63} - 6 q^{64} - 8 q^{65} + 9 q^{66} + 28 q^{67} - 44 q^{68} - 30 q^{70} - 44 q^{71} - 6 q^{72} - 41 q^{73} - 50 q^{74} - 29 q^{75} - 62 q^{76} - 20 q^{77} + 34 q^{78} - 25 q^{79} + 23 q^{80} + 29 q^{81} - 29 q^{82} - 7 q^{83} - 6 q^{84} - 28 q^{85} - 43 q^{86} + 61 q^{87} - 3 q^{88} - 109 q^{89} - q^{90} - q^{91} - 11 q^{92} + 19 q^{93} - 20 q^{94} - 9 q^{95} + 8 q^{96} - 51 q^{97} - 12 q^{98} - 21 q^{99}+O(q^{100}) \) 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 −2.79568 −1.00000 5.81581 1.00000 2.79568 1.96975 −10.6678 1.00000 −2.79568
1.2 −2.50063 −1.00000 4.25317 1.00000 2.50063 4.30119 −5.63435 1.00000 −2.50063
1.3 −2.37855 −1.00000 3.65751 1.00000 2.37855 2.87479 −3.94247 1.00000 −2.37855
1.4 −2.21779 −1.00000 2.91860 1.00000 2.21779 3.34170 −2.03727 1.00000 −2.21779
1.5 −2.17430 −1.00000 2.72756 1.00000 2.17430 −3.13141 −1.58193 1.00000 −2.17430
1.6 −2.12396 −1.00000 2.51122 1.00000 2.12396 −2.76942 −1.08582 1.00000 −2.12396
1.7 −1.69527 −1.00000 0.873943 1.00000 1.69527 −0.214929 1.90897 1.00000 −1.69527
1.8 −1.64211 −1.00000 0.696514 1.00000 1.64211 −2.05425 2.14046 1.00000 −1.64211
1.9 −1.55350 −1.00000 0.413352 1.00000 1.55350 0.802812 2.46485 1.00000 −1.55350
1.10 −1.15935 −1.00000 −0.655911 1.00000 1.15935 2.44720 3.07913 1.00000 −1.15935
1.11 −0.705071 −1.00000 −1.50287 1.00000 0.705071 −2.52019 2.46978 1.00000 −0.705071
1.12 −0.462418 −1.00000 −1.78617 1.00000 0.462418 −1.50774 1.75079 1.00000 −0.462418
1.13 −0.319268 −1.00000 −1.89807 1.00000 0.319268 0.830562 1.24453 1.00000 −0.319268
1.14 −0.256270 −1.00000 −1.93433 1.00000 0.256270 0.958240 1.00825 1.00000 −0.256270
1.15 −0.0716495 −1.00000 −1.99487 1.00000 0.0716495 4.62140 0.286230 1.00000 −0.0716495
1.16 −0.0247562 −1.00000 −1.99939 1.00000 0.0247562 −4.38402 0.0990096 1.00000 −0.0247562
1.17 0.242491 −1.00000 −1.94120 1.00000 −0.242491 2.88846 −0.955706 1.00000 0.242491
1.18 0.806575 −1.00000 −1.34944 1.00000 −0.806575 2.45232 −2.70157 1.00000 0.806575
1.19 0.843993 −1.00000 −1.28768 1.00000 −0.843993 −1.98856 −2.77478 1.00000 0.843993
1.20 1.09145 −1.00000 −0.808735 1.00000 −1.09145 0.212319 −3.06560 1.00000 1.09145
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
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.d ✓ 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6015.2.a.d ✓ 29 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{29} + T_{2}^{28} - 42 T_{2}^{27} - 39 T_{2}^{26} + 781 T_{2}^{25} + 667 T_{2}^{24} - 8479 T_{2}^{23} + \cdots - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\). Copy content Toggle raw display