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Properties

Label 966.2.l.d
Level $966$
Weight $2$
Character orbit 966.l
Analytic conductor $7.714$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(47,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.l (of order \(6\), degree \(2\), 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: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 28 q^{4} + 8 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 28 q^{4} + 8 q^{7} + 10 q^{9} + 12 q^{10} + 8 q^{15} - 28 q^{16} - 8 q^{18} - 24 q^{19} - 4 q^{21} - 12 q^{22} + 6 q^{24} - 8 q^{25} + 10 q^{28} + 6 q^{30} - 6 q^{31} + 30 q^{33} + 20 q^{36} + 4 q^{37} - 10 q^{39} + 12 q^{40} + 8 q^{42} - 104 q^{43} - 6 q^{45} + 28 q^{46} + 40 q^{49} - 22 q^{51} - 6 q^{52} + 18 q^{54} + 68 q^{57} - 30 q^{58} + 4 q^{60} - 84 q^{61} - 6 q^{63} - 56 q^{64} - 30 q^{66} + 2 q^{67} + 30 q^{70} + 8 q^{72} + 54 q^{73} - 24 q^{75} + 68 q^{78} - 6 q^{79} + 22 q^{81} + 4 q^{84} - 108 q^{85} - 126 q^{87} - 6 q^{88} - 42 q^{91} + 36 q^{93} - 18 q^{94} + 6 q^{96} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −0.866025 − 0.500000i −1.72705 + 0.131543i 0.500000 + 0.866025i 1.22807 − 2.12708i 1.56144 + 0.749605i −0.955407 − 2.46722i − 1.00000i 2.96539 − 0.454363i −2.12708 + 1.22807i
47.2 −0.866025 − 0.500000i −1.71118 − 0.268099i 0.500000 + 0.866025i −0.519082 + 0.899076i 1.34787 + 1.08777i 2.02474 + 1.70307i − 1.00000i 2.85625 + 0.917529i 0.899076 − 0.519082i
47.3 −0.866025 − 0.500000i −1.67384 + 0.445275i 0.500000 + 0.866025i −1.04467 + 1.80941i 1.67222 + 0.451299i −2.62205 − 0.353363i − 1.00000i 2.60346 − 1.49064i 1.80941 − 1.04467i
47.4 −0.866025 − 0.500000i −1.20460 + 1.24456i 0.500000 + 0.866025i 0.662008 − 1.14663i 1.66550 − 0.475517i 2.07946 + 1.63580i − 1.00000i −0.0978548 − 2.99840i −1.14663 + 0.662008i
47.5 −0.866025 − 0.500000i −0.896677 + 1.48188i 0.500000 + 0.866025i 1.18968 − 2.06059i 1.51749 − 0.835007i 1.24146 − 2.33640i − 1.00000i −1.39194 − 2.65754i −2.06059 + 1.18968i
47.6 −0.866025 − 0.500000i −0.778308 − 1.54733i 0.500000 + 0.866025i −1.34998 + 2.33823i −0.0996314 + 1.72918i −2.25095 + 1.39041i − 1.00000i −1.78847 + 2.40860i 2.33823 − 1.34998i
47.7 −0.866025 − 0.500000i 0.191760 − 1.72140i 0.500000 + 0.866025i −0.750837 + 1.30049i −1.02677 + 1.39490i −0.436788 − 2.60945i − 1.00000i −2.92646 − 0.660191i 1.30049 − 0.750837i
47.8 −0.866025 − 0.500000i 0.219734 + 1.71806i 0.500000 + 0.866025i −1.55436 + 2.69223i 0.668732 − 1.59775i 1.70233 + 2.02535i − 1.00000i −2.90343 + 0.755032i 2.69223 − 1.55436i
47.9 −0.866025 − 0.500000i 0.845576 − 1.51162i 0.500000 + 0.866025i 1.37030 − 2.37343i −1.48810 + 0.886316i −2.52742 + 0.782396i − 1.00000i −1.57000 − 2.55638i −2.37343 + 1.37030i
47.10 −0.866025 − 0.500000i 1.43371 + 0.971841i 0.500000 + 0.866025i −0.427501 + 0.740454i −0.755709 − 1.55849i −2.36450 − 1.18708i − 1.00000i 1.11105 + 2.78668i 0.740454 − 0.427501i
47.11 −0.866025 − 0.500000i 1.45188 + 0.944479i 0.500000 + 0.866025i −1.52157 + 2.63544i −0.785127 − 1.54388i −0.624219 + 2.57106i − 1.00000i 1.21592 + 2.74254i 2.63544 − 1.52157i
47.12 −0.866025 − 0.500000i 1.45696 − 0.936628i 0.500000 + 0.866025i −1.06670 + 1.84758i −1.73008 + 0.0826645i 2.47771 − 0.927868i − 1.00000i 1.24546 − 2.72926i 1.84758 − 1.06670i
47.13 −0.866025 − 0.500000i 1.54271 + 0.787427i 0.500000 + 0.866025i 1.84945 − 3.20334i −0.942314 − 1.45329i 1.61347 − 2.09684i − 1.00000i 1.75992 + 2.42955i −3.20334 + 1.84945i
47.14 −0.866025 − 0.500000i 1.71535 − 0.239977i 0.500000 + 0.866025i 0.203133 − 0.351837i −1.60552 − 0.649847i 2.64215 + 0.138089i − 1.00000i 2.88482 − 0.823286i −0.351837 + 0.203133i
47.15 0.866025 + 0.500000i −1.73168 + 0.0356051i 0.500000 + 0.866025i −1.18968 + 2.06059i −1.51749 − 0.835007i 1.24146 − 2.33640i 1.00000i 2.99746 − 0.123314i −2.06059 + 1.18968i
47.16 0.866025 + 0.500000i −1.68012 + 0.420939i 0.500000 + 0.866025i −0.662008 + 1.14663i −1.66550 − 0.475517i 2.07946 + 1.63580i 1.00000i 2.64562 − 1.41446i −1.14663 + 0.662008i
47.17 0.866025 + 0.500000i −1.37801 − 1.04932i 0.500000 + 0.866025i 1.55436 − 2.69223i −0.668732 − 1.59775i 1.70233 + 2.02535i 1.00000i 0.797840 + 2.89196i 2.69223 − 1.55436i
47.18 0.866025 + 0.500000i −1.22254 + 1.22695i 0.500000 + 0.866025i 1.04467 − 1.80941i −1.67222 + 0.451299i −2.62205 − 0.353363i 1.00000i −0.0108008 − 2.99998i 1.80941 − 1.04467i
47.19 0.866025 + 0.500000i −0.977444 + 1.42990i 0.500000 + 0.866025i −1.22807 + 2.12708i −1.56144 + 0.749605i −0.955407 − 2.46722i 1.00000i −1.08921 − 2.79529i −2.12708 + 1.22807i
47.20 0.866025 + 0.500000i −0.623408 + 1.61597i 0.500000 + 0.866025i 0.519082 − 0.899076i −1.34787 + 1.08777i 2.02474 + 1.70307i 1.00000i −2.22273 − 2.01482i 0.899076 − 0.519082i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.l.d ✓ 56
3.b odd 2 1 inner 966.2.l.d ✓ 56
7.d odd 6 1 inner 966.2.l.d ✓ 56
21.g even 6 1 inner 966.2.l.d ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.l.d ✓ 56 1.a even 1 1 trivial
966.2.l.d ✓ 56 3.b odd 2 1 inner
966.2.l.d ✓ 56 7.d odd 6 1 inner
966.2.l.d ✓ 56 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{56} + 74 T_{5}^{54} + 3035 T_{5}^{52} + 85738 T_{5}^{50} + 1842285 T_{5}^{48} + \cdots + 546946904514816 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display