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Properties

Label 945.2.l.b
Level $945$
Weight $2$
Character orbit 945.l
Analytic conductor $7.546$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(46,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.l (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 14 q^{4} - 12 q^{5} - 11 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 14 q^{4} - 12 q^{5} - 11 q^{7} - 12 q^{8} + q^{10} - q^{11} - 4 q^{13} - 8 q^{14} + 10 q^{16} + 7 q^{17} - 2 q^{19} - 7 q^{20} + 19 q^{22} - q^{23} - 12 q^{25} - 11 q^{26} - 28 q^{28} - 16 q^{31} + 34 q^{32} + q^{34} + 7 q^{35} + 17 q^{37} + 35 q^{38} + 6 q^{40} - 20 q^{41} + 31 q^{43} + 7 q^{44} - 10 q^{46} - 62 q^{47} - 11 q^{49} + q^{50} - 4 q^{52} - 8 q^{53} + 2 q^{55} + 45 q^{58} - 42 q^{59} - 10 q^{61} - 14 q^{62} - 56 q^{64} + 8 q^{65} - 86 q^{67} + 48 q^{68} - 5 q^{70} - 24 q^{71} - 18 q^{73} - 9 q^{74} - 13 q^{76} - 35 q^{77} - 80 q^{79} - 5 q^{80} + 5 q^{82} + 60 q^{83} + 7 q^{85} - 12 q^{86} + 50 q^{88} + 4 q^{89} - 33 q^{91} + 18 q^{92} + 22 q^{94} + 4 q^{95} + 6 q^{97} + 15 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −2.34008 0 3.47596 −0.500000 + 0.866025i 0 −1.61838 + 2.09305i −3.45385 0 1.17004 − 2.02657i
46.2 −2.32555 0 3.40817 −0.500000 + 0.866025i 0 2.02670 − 1.70073i −3.27475 0 1.16277 − 2.01398i
46.3 −2.16352 0 2.68083 −0.500000 + 0.866025i 0 −2.41085 − 1.08987i −1.47299 0 1.08176 − 1.87367i
46.4 −1.19807 0 −0.564635 −0.500000 + 0.866025i 0 0.433740 − 2.60996i 3.07261 0 0.599034 − 1.03756i
46.5 −0.518491 0 −1.73117 −0.500000 + 0.866025i 0 −0.619045 + 2.57231i 1.93458 0 0.259245 − 0.449026i
46.6 −0.297462 0 −1.91152 −0.500000 + 0.866025i 0 1.24794 + 2.33295i 1.16353 0 0.148731 − 0.257610i
46.7 0.308078 0 −1.90509 −0.500000 + 0.866025i 0 −2.36933 − 1.17741i −1.20307 0 −0.154039 + 0.266804i
46.8 0.609814 0 −1.62813 −0.500000 + 0.866025i 0 0.731085 − 2.54274i −2.21248 0 −0.304907 + 0.528114i
46.9 1.03554 0 −0.927661 −0.500000 + 0.866025i 0 2.63139 − 0.275284i −3.03170 0 −0.517769 + 0.896802i
46.10 1.61038 0 0.593327 −0.500000 + 0.866025i 0 −2.03107 − 1.69552i −2.26528 0 −0.805191 + 1.39463i
46.11 1.71927 0 0.955889 −0.500000 + 0.866025i 0 −2.52983 + 0.774581i −1.79511 0 −0.859635 + 1.48893i
46.12 2.56008 0 4.55403 −0.500000 + 0.866025i 0 −0.992363 + 2.45259i 6.53853 0 −1.28004 + 2.21710i
226.1 −2.34008 0 3.47596 −0.500000 − 0.866025i 0 −1.61838 − 2.09305i −3.45385 0 1.17004 + 2.02657i
226.2 −2.32555 0 3.40817 −0.500000 − 0.866025i 0 2.02670 + 1.70073i −3.27475 0 1.16277 + 2.01398i
226.3 −2.16352 0 2.68083 −0.500000 − 0.866025i 0 −2.41085 + 1.08987i −1.47299 0 1.08176 + 1.87367i
226.4 −1.19807 0 −0.564635 −0.500000 − 0.866025i 0 0.433740 + 2.60996i 3.07261 0 0.599034 + 1.03756i
226.5 −0.518491 0 −1.73117 −0.500000 − 0.866025i 0 −0.619045 − 2.57231i 1.93458 0 0.259245 + 0.449026i
226.6 −0.297462 0 −1.91152 −0.500000 − 0.866025i 0 1.24794 − 2.33295i 1.16353 0 0.148731 + 0.257610i
226.7 0.308078 0 −1.90509 −0.500000 − 0.866025i 0 −2.36933 + 1.17741i −1.20307 0 −0.154039 − 0.266804i
226.8 0.609814 0 −1.62813 −0.500000 − 0.866025i 0 0.731085 + 2.54274i −2.21248 0 −0.304907 − 0.528114i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.l.b 24
3.b odd 2 1 315.2.l.b yes 24
7.c even 3 1 945.2.k.b 24
9.c even 3 1 945.2.k.b 24
9.d odd 6 1 315.2.k.b ✓ 24
21.h odd 6 1 315.2.k.b ✓ 24
63.h even 3 1 inner 945.2.l.b 24
63.j odd 6 1 315.2.l.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.b ✓ 24 9.d odd 6 1
315.2.k.b ✓ 24 21.h odd 6 1
315.2.l.b yes 24 3.b odd 2 1
315.2.l.b yes 24 63.j odd 6 1
945.2.k.b 24 7.c even 3 1
945.2.k.b 24 9.c even 3 1
945.2.l.b 24 1.a even 1 1 trivial
945.2.l.b 24 63.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + T_{2}^{11} - 15 T_{2}^{10} - 12 T_{2}^{9} + 80 T_{2}^{8} + 42 T_{2}^{7} - 186 T_{2}^{6} + \cdots + 3 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display