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Properties

Label 663.2.j.a
Level $663$
Weight $2$
Character orbit 663.j
Analytic conductor $5.294$
Analytic rank $0$
Dimension $32$
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(157,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.j (of order \(4\), 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: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{4} - 4 q^{5} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{4} - 4 q^{5} - 4 q^{6} + 4 q^{10} - 4 q^{11} - 32 q^{13} + 24 q^{14} - 12 q^{17} - 28 q^{20} + 16 q^{21} + 20 q^{23} + 4 q^{24} + 16 q^{28} + 52 q^{29} - 16 q^{30} + 20 q^{31} - 16 q^{33} + 32 q^{34} - 8 q^{35} - 20 q^{37} + 16 q^{38} - 80 q^{40} - 20 q^{41} + 44 q^{44} + 4 q^{45} - 32 q^{46} + 24 q^{47} + 16 q^{48} - 16 q^{50} + 4 q^{51} + 24 q^{52} - 4 q^{54} - 8 q^{55} - 68 q^{56} - 8 q^{57} - 64 q^{58} - 12 q^{61} + 16 q^{62} + 40 q^{64} + 4 q^{65} - 80 q^{67} + 20 q^{68} + 16 q^{69} + 4 q^{71} - 36 q^{73} + 48 q^{74} - 16 q^{75} + 4 q^{78} + 8 q^{79} + 76 q^{80} - 32 q^{81} + 4 q^{82} - 8 q^{84} + 52 q^{85} + 48 q^{86} - 60 q^{88} + 8 q^{89} + 4 q^{90} - 124 q^{92} - 48 q^{95} + 16 q^{96} + 40 q^{97} + 48 q^{98} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 − 2.28916i 0.707107 − 0.707107i −3.24026 −1.12612 + 1.12612i −1.61868 − 1.61868i −2.07325 − 2.07325i 2.83915i − 1.00000i 2.57786 + 2.57786i
157.2 − 2.27457i 0.707107 − 0.707107i −3.17368 2.96612 − 2.96612i −1.60837 − 1.60837i 2.96539 + 2.96539i 2.66962i − 1.00000i −6.74666 − 6.74666i
157.3 − 2.18004i −0.707107 + 0.707107i −2.75257 −2.15243 + 2.15243i 1.54152 + 1.54152i 2.93384 + 2.93384i 1.64064i − 1.00000i 4.69239 + 4.69239i
157.4 − 1.91545i −0.707107 + 0.707107i −1.66893 −1.35359 + 1.35359i 1.35442 + 1.35442i −0.811625 − 0.811625i − 0.634143i − 1.00000i 2.59273 + 2.59273i
157.5 − 1.44794i 0.707107 − 0.707107i −0.0965417 0.705234 − 0.705234i −1.02385 − 1.02385i −1.70047 − 1.70047i − 2.75610i − 1.00000i −1.02114 − 1.02114i
157.6 − 1.09385i −0.707107 + 0.707107i 0.803491 1.54655 − 1.54655i 0.773469 + 0.773469i −1.02590 − 1.02590i − 3.06660i − 1.00000i −1.69169 − 1.69169i
157.7 − 0.518644i 0.707107 − 0.707107i 1.73101 −0.144430 + 0.144430i −0.366737 − 0.366737i 2.76683 + 2.76683i − 1.93506i − 1.00000i 0.0749079 + 0.0749079i
157.8 0.0854616i −0.707107 + 0.707107i 1.99270 −0.427664 + 0.427664i −0.0604305 − 0.0604305i 2.11132 + 2.11132i 0.341222i − 1.00000i −0.0365488 − 0.0365488i
157.9 0.303671i 0.707107 − 0.707107i 1.90778 1.45526 − 1.45526i 0.214728 + 0.214728i −1.92916 − 1.92916i 1.18668i − 1.00000i 0.441920 + 0.441920i
157.10 0.685099i −0.707107 + 0.707107i 1.53064 −2.82114 + 2.82114i −0.484438 − 0.484438i −3.41418 − 3.41418i 2.41884i − 1.00000i −1.93276 − 1.93276i
157.11 1.08778i 0.707107 − 0.707107i 0.816737 −0.495535 + 0.495535i 0.769176 + 0.769176i 1.02138 + 1.02138i 3.06399i − 1.00000i −0.539032 − 0.539032i
157.12 1.15222i 0.707107 − 0.707107i 0.672395 −2.59606 + 2.59606i 0.814741 + 0.814741i 2.15133 + 2.15133i 3.07918i − 1.00000i −2.99123 − 2.99123i
157.13 1.49170i −0.707107 + 0.707107i −0.225164 0.731803 − 0.731803i −1.05479 − 1.05479i −0.0312384 − 0.0312384i 2.64752i − 1.00000i 1.09163 + 1.09163i
157.14 1.81235i −0.707107 + 0.707107i −1.28460 −1.46563 + 1.46563i −1.28152 − 1.28152i 0.830570 + 0.830570i 1.29656i − 1.00000i −2.65623 − 2.65623i
157.15 2.52895i −0.707107 + 0.707107i −4.39556 0.699465 − 0.699465i −1.78823 − 1.78823i −3.42121 − 3.42121i − 6.05825i − 1.00000i 1.76891 + 1.76891i
157.16 2.57244i 0.707107 − 0.707107i −4.61745 2.47817 − 2.47817i 1.81899 + 1.81899i −0.373629 − 0.373629i − 6.73323i − 1.00000i 6.37494 + 6.37494i
625.1 − 2.57244i 0.707107 + 0.707107i −4.61745 2.47817 + 2.47817i 1.81899 − 1.81899i −0.373629 + 0.373629i 6.73323i 1.00000i 6.37494 − 6.37494i
625.2 − 2.52895i −0.707107 − 0.707107i −4.39556 0.699465 + 0.699465i −1.78823 + 1.78823i −3.42121 + 3.42121i 6.05825i 1.00000i 1.76891 − 1.76891i
625.3 − 1.81235i −0.707107 − 0.707107i −1.28460 −1.46563 − 1.46563i −1.28152 + 1.28152i 0.830570 − 0.830570i − 1.29656i 1.00000i −2.65623 + 2.65623i
625.4 − 1.49170i −0.707107 − 0.707107i −0.225164 0.731803 + 0.731803i −1.05479 + 1.05479i −0.0312384 + 0.0312384i − 2.64752i 1.00000i 1.09163 − 1.09163i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.j.a ✓ 32
17.c even 4 1 inner 663.2.j.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.j.a ✓ 32 1.a even 1 1 trivial
663.2.j.a ✓ 32 17.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 44 T_{2}^{30} + 868 T_{2}^{28} + 10148 T_{2}^{26} + 78320 T_{2}^{24} + 420752 T_{2}^{22} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display