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Properties

Label 663.2.z.f
Level $663$
Weight $2$
Character orbit 663.z
Analytic conductor $5.294$
Analytic rank $0$
Dimension $22$
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] = [663,2,Mod(205,663)]
 
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, 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("663.205");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.z (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: \(22\)
Relative dimension: \(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 11 q^{3} + 16 q^{4} + 3 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 11 q^{3} + 16 q^{4} + 3 q^{7} - 11 q^{9} + 13 q^{10} + 6 q^{11} + 32 q^{12} + q^{13} + 18 q^{14} - 18 q^{16} - 11 q^{17} - 6 q^{19} - 39 q^{20} - 11 q^{22} - 10 q^{23} - 46 q^{25} + 44 q^{26} - 22 q^{27} + 39 q^{28} - 3 q^{29} - 13 q^{30} - 15 q^{32} + 6 q^{33} + 7 q^{35} + 16 q^{36} - 6 q^{37} - 38 q^{38} + 2 q^{39} + 108 q^{40} + 6 q^{41} + 9 q^{42} - 15 q^{43} - 57 q^{46} + 18 q^{48} + 34 q^{49} - 66 q^{50} - 22 q^{51} + 34 q^{52} - 10 q^{53} + 9 q^{55} - 15 q^{56} + 18 q^{58} - 51 q^{59} - 46 q^{61} + 26 q^{62} - 3 q^{63} - 96 q^{64} + 25 q^{65} - 22 q^{66} - 24 q^{67} + 16 q^{68} + 10 q^{69} + 9 q^{71} - 21 q^{74} - 23 q^{75} - 120 q^{76} - 28 q^{77} + 37 q^{78} + 68 q^{79} + 57 q^{80} - 11 q^{81} + 3 q^{82} + 39 q^{84} + 3 q^{87} + 67 q^{88} - 3 q^{89} - 26 q^{90} + 29 q^{91} + 40 q^{92} + 3 q^{93} + 47 q^{94} - 36 q^{95} + 15 q^{97} - 156 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
205.1 −2.34255 + 1.35247i 0.500000 + 0.866025i 2.65836 − 4.60441i − 1.50456i −2.34255 − 1.35247i 4.03889 + 2.33186i 8.97152i −0.500000 + 0.866025i 2.03488 + 3.52451i
205.2 −2.04171 + 1.17878i 0.500000 + 0.866025i 1.77906 − 3.08142i − 3.59766i −2.04171 − 1.17878i −3.47211 − 2.00462i 3.67335i −0.500000 + 0.866025i 4.24086 + 7.34538i
205.3 −1.56364 + 0.902766i 0.500000 + 0.866025i 0.629974 − 1.09115i − 0.537668i −1.56364 − 0.902766i −2.86997 − 1.65698i − 1.33619i −0.500000 + 0.866025i 0.485389 + 0.840718i
205.4 −1.21298 + 0.700316i 0.500000 + 0.866025i −0.0191141 + 0.0331066i − 0.691348i −1.21298 − 0.700316i 3.40132 + 1.96375i − 2.85481i −0.500000 + 0.866025i 0.484163 + 0.838594i
205.5 −0.895497 + 0.517015i 0.500000 + 0.866025i −0.465390 + 0.806080i 3.65775i −0.895497 − 0.517015i −2.72063 − 1.57076i − 3.03052i −0.500000 + 0.866025i −1.89111 − 3.27550i
205.6 0.319686 − 0.184571i 0.500000 + 0.866025i −0.931867 + 1.61404i 3.61390i 0.319686 + 0.184571i 2.12008 + 1.22403i 1.42626i −0.500000 + 0.866025i 0.667019 + 1.15531i
205.7 0.523841 − 0.302440i 0.500000 + 0.866025i −0.817060 + 1.41519i − 0.897196i 0.523841 + 0.302440i −2.51522 − 1.45216i 2.19821i −0.500000 + 0.866025i −0.271348 − 0.469988i
205.8 1.36825 − 0.789959i 0.500000 + 0.866025i 0.248069 − 0.429669i − 1.82584i 1.36825 + 0.789959i −1.03494 − 0.597523i 2.37598i −0.500000 + 0.866025i −1.44233 − 2.49820i
205.9 1.39737 − 0.806772i 0.500000 + 0.866025i 0.301763 − 0.522668i 1.43426i 1.39737 + 0.806772i 1.43206 + 0.826802i 2.25327i −0.500000 + 0.866025i 1.15712 + 2.00419i
205.10 2.09061 − 1.20701i 0.500000 + 0.866025i 1.91377 − 3.31474i − 3.65533i 2.09061 + 1.20701i 3.66530 + 2.11616i − 4.41173i −0.500000 + 0.866025i −4.41203 − 7.64187i
205.11 2.35662 − 1.36060i 0.500000 + 0.866025i 2.70244 − 4.68077i 4.00369i 2.35662 + 1.36060i −0.544792 − 0.314536i − 9.26535i −0.500000 + 0.866025i 5.44741 + 9.43518i
511.1 −2.34255 − 1.35247i 0.500000 − 0.866025i 2.65836 + 4.60441i 1.50456i −2.34255 + 1.35247i 4.03889 − 2.33186i − 8.97152i −0.500000 − 0.866025i 2.03488 − 3.52451i
511.2 −2.04171 − 1.17878i 0.500000 − 0.866025i 1.77906 + 3.08142i 3.59766i −2.04171 + 1.17878i −3.47211 + 2.00462i − 3.67335i −0.500000 − 0.866025i 4.24086 − 7.34538i
511.3 −1.56364 − 0.902766i 0.500000 − 0.866025i 0.629974 + 1.09115i 0.537668i −1.56364 + 0.902766i −2.86997 + 1.65698i 1.33619i −0.500000 − 0.866025i 0.485389 − 0.840718i
511.4 −1.21298 − 0.700316i 0.500000 − 0.866025i −0.0191141 − 0.0331066i 0.691348i −1.21298 + 0.700316i 3.40132 − 1.96375i 2.85481i −0.500000 − 0.866025i 0.484163 − 0.838594i
511.5 −0.895497 − 0.517015i 0.500000 − 0.866025i −0.465390 − 0.806080i − 3.65775i −0.895497 + 0.517015i −2.72063 + 1.57076i 3.03052i −0.500000 − 0.866025i −1.89111 + 3.27550i
511.6 0.319686 + 0.184571i 0.500000 − 0.866025i −0.931867 − 1.61404i − 3.61390i 0.319686 − 0.184571i 2.12008 − 1.22403i − 1.42626i −0.500000 − 0.866025i 0.667019 − 1.15531i
511.7 0.523841 + 0.302440i 0.500000 − 0.866025i −0.817060 − 1.41519i 0.897196i 0.523841 − 0.302440i −2.51522 + 1.45216i − 2.19821i −0.500000 − 0.866025i −0.271348 + 0.469988i
511.8 1.36825 + 0.789959i 0.500000 − 0.866025i 0.248069 + 0.429669i 1.82584i 1.36825 − 0.789959i −1.03494 + 0.597523i − 2.37598i −0.500000 − 0.866025i −1.44233 + 2.49820i
511.9 1.39737 + 0.806772i 0.500000 − 0.866025i 0.301763 + 0.522668i − 1.43426i 1.39737 − 0.806772i 1.43206 − 0.826802i − 2.25327i −0.500000 − 0.866025i 1.15712 − 2.00419i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 205.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.z.f ✓ 22
13.e even 6 1 inner 663.2.z.f ✓ 22
13.f odd 12 2 8619.2.a.br 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.z.f ✓ 22 1.a even 1 1 trivial
663.2.z.f ✓ 22 13.e even 6 1 inner
8619.2.a.br 22 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 19 T_{2}^{20} + 231 T_{2}^{18} + 3 T_{2}^{17} - 1688 T_{2}^{16} - 33 T_{2}^{15} + \cdots + 3888 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display