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Properties

Label 6015.2.a.g
Level $6015$
Weight $2$
Character orbit 6015.a
Self dual yes
Analytic conductor $48.030$
Analytic rank $0$
Dimension $36$
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: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{3} + 36 q^{4} + 36 q^{5} - 4 q^{7} - 3 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{3} + 36 q^{4} + 36 q^{5} - 4 q^{7} - 3 q^{8} + 36 q^{9} + 15 q^{11} - 36 q^{12} + 8 q^{13} + 10 q^{14} - 36 q^{15} + 36 q^{16} + 32 q^{17} + 5 q^{19} + 36 q^{20} + 4 q^{21} + q^{22} - 10 q^{23} + 3 q^{24} + 36 q^{25} + 22 q^{26} - 36 q^{27} + 61 q^{29} + 15 q^{31} - q^{32} - 15 q^{33} + 26 q^{34} - 4 q^{35} + 36 q^{36} + 16 q^{37} + 20 q^{38} - 8 q^{39} - 3 q^{40} + 61 q^{41} - 10 q^{42} - 35 q^{43} + 39 q^{44} + 36 q^{45} + 11 q^{46} - 28 q^{47} - 36 q^{48} + 68 q^{49} - 32 q^{51} + 4 q^{52} + 33 q^{53} + 15 q^{55} + 23 q^{56} - 5 q^{57} + 6 q^{58} + 35 q^{59} - 36 q^{60} + 55 q^{61} + q^{62} - 4 q^{63} + 15 q^{64} + 8 q^{65} - q^{66} - 34 q^{67} + 60 q^{68} + 10 q^{69} + 10 q^{70} + 42 q^{71} - 3 q^{72} + 53 q^{73} + 54 q^{74} - 36 q^{75} + 20 q^{76} + 20 q^{77} - 22 q^{78} + 25 q^{79} + 36 q^{80} + 36 q^{81} - 15 q^{82} - 11 q^{83} + 32 q^{85} + 53 q^{86} - 61 q^{87} + 27 q^{88} + 81 q^{89} + 15 q^{91} - 7 q^{92} - 15 q^{93} + 56 q^{94} + 5 q^{95} + q^{96} + 47 q^{97} - 3 q^{98} + 15 q^{99}+O(q^{100}) \) 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 −2.70933 −1.00000 5.34047 1.00000 2.70933 0.462460 −9.05044 1.00000 −2.70933
1.2 −2.61390 −1.00000 4.83247 1.00000 2.61390 −4.22901 −7.40380 1.00000 −2.61390
1.3 −2.55061 −1.00000 4.50559 1.00000 2.55061 −0.898526 −6.39078 1.00000 −2.55061
1.4 −2.43402 −1.00000 3.92446 1.00000 2.43402 −1.65420 −4.68418 1.00000 −2.43402
1.5 −2.31869 −1.00000 3.37634 1.00000 2.31869 3.37785 −3.19130 1.00000 −2.31869
1.6 −2.12459 −1.00000 2.51388 1.00000 2.12459 3.70349 −1.09177 1.00000 −2.12459
1.7 −1.92662 −1.00000 1.71187 1.00000 1.92662 2.81069 0.555119 1.00000 −1.92662
1.8 −1.76666 −1.00000 1.12107 1.00000 1.76666 −4.11244 1.55276 1.00000 −1.76666
1.9 −1.73556 −1.00000 1.01218 1.00000 1.73556 −3.79849 1.71442 1.00000 −1.73556
1.10 −1.65221 −1.00000 0.729796 1.00000 1.65221 −0.892368 2.09864 1.00000 −1.65221
1.11 −1.50420 −1.00000 0.262613 1.00000 1.50420 1.83015 2.61337 1.00000 −1.50420
1.12 −0.967785 −1.00000 −1.06339 1.00000 0.967785 0.0385606 2.96471 1.00000 −0.967785
1.13 −0.942366 −1.00000 −1.11195 1.00000 0.942366 5.06770 2.93259 1.00000 −0.942366
1.14 −0.713173 −1.00000 −1.49138 1.00000 0.713173 −4.34956 2.48996 1.00000 −0.713173
1.15 −0.567973 −1.00000 −1.67741 1.00000 0.567973 3.22749 2.08867 1.00000 −0.567973
1.16 −0.558858 −1.00000 −1.68768 1.00000 0.558858 −1.43354 2.06089 1.00000 −0.558858
1.17 −0.347552 −1.00000 −1.87921 1.00000 0.347552 −4.57897 1.34823 1.00000 −0.347552
1.18 0.108652 −1.00000 −1.98819 1.00000 −0.108652 0.790651 −0.433326 1.00000 0.108652
1.19 0.274127 −1.00000 −1.92485 1.00000 −0.274127 0.857432 −1.07591 1.00000 0.274127
1.20 0.288786 −1.00000 −1.91660 1.00000 −0.288786 −0.447752 −1.13106 1.00000 0.288786
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
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.g ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6015.2.a.g ✓ 36 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 54 T_{2}^{34} + T_{2}^{33} + 1323 T_{2}^{32} - 49 T_{2}^{31} - 19472 T_{2}^{30} + \cdots - 712 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\). Copy content Toggle raw display