Properties

Label 633.2.bd.a
Level $633$
Weight $2$
Character orbit 633.bd
Analytic conductor $5.055$
Analytic rank $0$
Dimension $48$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [633,2,Mod(2,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(210))
 
chi = DirichletCharacter(H, H._module([105, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.bd (of order \(210\), degree \(48\), 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: \(5.05453044795\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{210}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 3 q^{3} + 2 q^{4} - 16 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 3 q^{3} + 2 q^{4} - 16 q^{7} + 3 q^{9} - 14 q^{13} - 4 q^{16} - 36 q^{19} + 54 q^{21} + 10 q^{25} + 10 q^{28} + 3 q^{31} - 6 q^{36} + 39 q^{37} - 63 q^{39} + 43 q^{43} + 12 q^{48} - 35 q^{49} - 14 q^{52} + 3 q^{57} + 27 q^{61} - 16 q^{64} - 77 q^{67} + 112 q^{73} - 15 q^{75} + 6 q^{76} - 8 q^{79} - 9 q^{81} - 12 q^{84} - 63 q^{91} + 87 q^{93} - 70 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 0 1.38587 + 1.03892i 1.99910 + 0.0598309i 0 0 1.24511 2.23309i 0 0.841282 + 2.87963i 0
17.1 0 1.21555 1.23387i 1.89265 0.646419i 0 0 0.415774 0.505657i 0 −0.0448782 2.99966i 0
29.1 0 −0.797856 + 1.53734i 1.19964 1.60027i 0 0 −0.152905 2.55218i 0 −1.72685 2.45316i 0
35.1 0 0.0518151 + 1.73128i −0.503174 + 1.93567i 0 0 −0.969587 2.36578i 0 −2.99463 + 0.179412i 0
41.1 0 0.103584 + 1.72895i 1.74682 + 0.973978i 0 0 3.36271 + 3.41339i 0 −2.97854 + 0.358183i 0
92.1 0 −1.05954 + 1.37018i 1.54483 + 1.27023i 0 0 −2.75838 2.33810i 0 −0.754760 2.90350i 0
116.1 0 −1.69940 0.334729i 1.29320 1.52566i 0 0 −2.09071 + 1.47171i 0 2.77591 + 1.13768i 0
131.1 0 −0.797856 1.53734i 1.19964 + 1.60027i 0 0 −0.152905 + 2.55218i 0 −1.72685 + 2.45316i 0
149.1 0 1.21555 + 1.23387i 1.89265 + 0.646419i 0 0 0.415774 + 0.505657i 0 −0.0448782 + 2.99966i 0
152.1 0 1.54911 + 0.774769i −0.0299188 1.99978i 0 0 3.49968 0.909735i 0 1.79947 + 2.40040i 0
155.1 0 0.410635 1.68267i 0.894626 + 1.78875i 0 0 0.108174 + 3.61437i 0 −2.66276 1.38193i 0
158.1 0 1.66255 + 0.485714i −1.99642 + 0.119608i 0 0 −4.07609 2.51885i 0 2.52816 + 1.61505i 0
164.1 0 −1.63053 + 0.584273i 1.68544 + 1.07670i 0 0 −0.155203 + 0.635979i 0 2.31725 1.90535i 0
167.1 0 1.67634 + 0.435761i −0.386512 + 1.96230i 0 0 2.52222 + 0.418842i 0 2.62022 + 1.46097i 0
191.1 0 −1.69940 + 0.334729i 1.29320 + 1.52566i 0 0 −2.09071 1.47171i 0 2.77591 1.13768i 0
218.1 0 −0.206797 1.71966i −1.05137 1.70136i 0 0 0.654570 + 0.00979307i 0 −2.91447 + 0.711241i 0
233.1 0 0.656838 + 1.60267i 0.327636 1.97298i 0 0 −4.51071 + 1.61634i 0 −2.13713 + 2.10539i 0
296.1 0 −1.32126 + 1.11994i −1.77517 0.921285i 0 0 −2.45417 0.716984i 0 0.491454 2.95947i 0
302.1 0 −1.70865 + 0.283741i −1.15123 + 1.63544i 0 0 2.57954 + 4.03795i 0 2.83898 0.969629i 0
317.1 0 1.38587 1.03892i 1.99910 0.0598309i 0 0 1.24511 + 2.23309i 0 0.841282 2.87963i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
211.p odd 210 1 inner
633.bd even 210 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.bd.a 48
3.b odd 2 1 CM 633.2.bd.a 48
211.p odd 210 1 inner 633.2.bd.a 48
633.bd even 210 1 inner 633.2.bd.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.bd.a 48 1.a even 1 1 trivial
633.2.bd.a 48 3.b odd 2 1 CM
633.2.bd.a 48 211.p odd 210 1 inner
633.2.bd.a 48 633.bd even 210 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display