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Properties

Label 663.2.w.c
Level $663$
Weight $2$
Character orbit 663.w
Analytic conductor $5.294$
Analytic rank $0$
Dimension $76$
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] = [663,2,Mod(16,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.w (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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 4 q^{2} - 40 q^{4} - 48 q^{8} + 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q + 4 q^{2} - 40 q^{4} - 48 q^{8} + 38 q^{9} + 26 q^{13} - 2 q^{15} - 68 q^{16} + 10 q^{17} + 8 q^{18} - 10 q^{19} + 16 q^{21} - 168 q^{25} - 68 q^{26} - 20 q^{30} + 12 q^{32} - 8 q^{33} + 92 q^{34} - 8 q^{35} + 40 q^{36} - 136 q^{38} + 28 q^{42} + 44 q^{43} - 88 q^{47} + 38 q^{49} - 28 q^{50} + 20 q^{51} - 60 q^{52} - 96 q^{53} - 36 q^{55} - 12 q^{59} + 8 q^{60} + 296 q^{64} - 64 q^{66} + 22 q^{67} + 38 q^{68} - 2 q^{69} + 16 q^{70} - 24 q^{72} + 68 q^{76} + 80 q^{77} - 38 q^{81} + 8 q^{83} + 24 q^{84} + 16 q^{85} + 112 q^{86} + 2 q^{87} + 8 q^{89} - 12 q^{93} + 100 q^{94} - 88 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.40328 − 2.43056i −0.866025 + 0.500000i −2.93840 + 5.08946i 2.84895i 2.43056 + 1.40328i 0.309512 + 0.178697i 10.8805 0.500000 − 0.866025i 6.92452 − 3.99788i
16.2 −1.40328 − 2.43056i 0.866025 − 0.500000i −2.93840 + 5.08946i − 2.84895i −2.43056 − 1.40328i −0.309512 − 0.178697i 10.8805 0.500000 − 0.866025i −6.92452 + 3.99788i
16.3 −1.16436 − 2.01673i −0.866025 + 0.500000i −1.71148 + 2.96436i − 2.24889i 2.01673 + 1.16436i 2.19661 + 1.26822i 3.31366 0.500000 − 0.866025i −4.53542 + 2.61852i
16.4 −1.16436 − 2.01673i 0.866025 − 0.500000i −1.71148 + 2.96436i 2.24889i −2.01673 − 1.16436i −2.19661 − 1.26822i 3.31366 0.500000 − 0.866025i 4.53542 − 2.61852i
16.5 −0.999773 − 1.73166i −0.866025 + 0.500000i −0.999092 + 1.73048i 3.59710i 1.73166 + 0.999773i 2.44417 + 1.41114i −0.00363194 0.500000 − 0.866025i 6.22895 − 3.59628i
16.6 −0.999773 − 1.73166i 0.866025 − 0.500000i −0.999092 + 1.73048i − 3.59710i −1.73166 − 0.999773i −2.44417 − 1.41114i −0.00363194 0.500000 − 0.866025i −6.22895 + 3.59628i
16.7 −0.939722 − 1.62765i −0.866025 + 0.500000i −0.766154 + 1.32702i − 1.13551i 1.62765 + 0.939722i −2.78507 − 1.60796i −0.879000 0.500000 − 0.866025i −1.84822 + 1.06707i
16.8 −0.939722 − 1.62765i 0.866025 − 0.500000i −0.766154 + 1.32702i 1.13551i −1.62765 − 0.939722i 2.78507 + 1.60796i −0.879000 0.500000 − 0.866025i 1.84822 − 1.06707i
16.9 −0.650786 − 1.12719i −0.866025 + 0.500000i 0.152955 − 0.264925i − 4.13346i 1.12719 + 0.650786i 2.42988 + 1.40289i −3.00131 0.500000 − 0.866025i −4.65922 + 2.69000i
16.10 −0.650786 − 1.12719i 0.866025 − 0.500000i 0.152955 − 0.264925i 4.13346i −1.12719 − 0.650786i −2.42988 − 1.40289i −3.00131 0.500000 − 0.866025i 4.65922 − 2.69000i
16.11 −0.601601 − 1.04200i −0.866025 + 0.500000i 0.276152 − 0.478309i − 0.944477i 1.04200 + 0.601601i −3.27490 − 1.89076i −3.07094 0.500000 − 0.866025i −0.984148 + 0.568198i
16.12 −0.601601 − 1.04200i 0.866025 − 0.500000i 0.276152 − 0.478309i 0.944477i −1.04200 − 0.601601i 3.27490 + 1.89076i −3.07094 0.500000 − 0.866025i 0.984148 − 0.568198i
16.13 −0.440674 − 0.763269i −0.866025 + 0.500000i 0.611613 − 1.05935i 1.55595i 0.763269 + 0.440674i 3.44007 + 1.98613i −2.84078 0.500000 − 0.866025i 1.18761 − 0.685665i
16.14 −0.440674 − 0.763269i 0.866025 − 0.500000i 0.611613 − 1.05935i − 1.55595i −0.763269 − 0.440674i −3.44007 − 1.98613i −2.84078 0.500000 − 0.866025i −1.18761 + 0.685665i
16.15 −0.329294 − 0.570354i −0.866025 + 0.500000i 0.783131 − 1.35642i 1.30766i 0.570354 + 0.329294i 0.0703517 + 0.0406176i −2.34870 0.500000 − 0.866025i 0.745830 − 0.430605i
16.16 −0.329294 − 0.570354i 0.866025 − 0.500000i 0.783131 − 1.35642i − 1.30766i −0.570354 − 0.329294i −0.0703517 − 0.0406176i −2.34870 0.500000 − 0.866025i −0.745830 + 0.430605i
16.17 −0.325794 − 0.564291i −0.866025 + 0.500000i 0.787717 − 1.36437i 2.89244i 0.564291 + 0.325794i −2.25963 − 1.30460i −2.32971 0.500000 − 0.866025i 1.63218 − 0.942337i
16.18 −0.325794 − 0.564291i 0.866025 − 0.500000i 0.787717 − 1.36437i − 2.89244i −0.564291 − 0.325794i 2.25963 + 1.30460i −2.32971 0.500000 − 0.866025i −1.63218 + 0.942337i
16.19 0.177404 + 0.307272i −0.866025 + 0.500000i 0.937056 − 1.62303i − 4.22036i −0.307272 − 0.177404i −2.31589 − 1.33708i 1.37456 0.500000 − 0.866025i 1.29680 − 0.748707i
16.20 0.177404 + 0.307272i 0.866025 − 0.500000i 0.937056 − 1.62303i 4.22036i 0.307272 + 0.177404i 2.31589 + 1.33708i 1.37456 0.500000 − 0.866025i −1.29680 + 0.748707i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
17.b even 2 1 inner
221.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.w.c ✓ 76
13.c even 3 1 inner 663.2.w.c ✓ 76
17.b even 2 1 inner 663.2.w.c ✓ 76
221.l even 6 1 inner 663.2.w.c ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.w.c ✓ 76 1.a even 1 1 trivial
663.2.w.c ✓ 76 13.c even 3 1 inner
663.2.w.c ✓ 76 17.b even 2 1 inner
663.2.w.c ✓ 76 221.l even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{38} - 2 T_{2}^{37} + 31 T_{2}^{36} - 46 T_{2}^{35} + 529 T_{2}^{34} - 648 T_{2}^{33} + \cdots + 21904 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display