Properties

Label 633.3.c.a
Level $633$
Weight $3$
Character orbit 633.c
Analytic conductor $17.248$
Analytic rank $0$
Dimension $70$
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] = [633,3,Mod(421,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.421");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 633.c (of order \(2\), degree \(1\), 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: \(17.2480007340\)
Analytic rank: \(0\)
Dimension: \(70\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 70 q - 144 q^{4} + 12 q^{6} - 210 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 70 q - 144 q^{4} + 12 q^{6} - 210 q^{9} + 16 q^{11} + 12 q^{13} + 304 q^{16} - 16 q^{19} + 84 q^{20} - 48 q^{21} - 12 q^{24} + 354 q^{25} + 24 q^{30} + 76 q^{34} + 432 q^{36} - 228 q^{37} + 68 q^{43} - 196 q^{44} - 140 q^{46} + 308 q^{47} - 414 q^{49} + 168 q^{51} - 192 q^{52} + 106 q^{53} - 36 q^{54} + 114 q^{55} - 288 q^{56} - 268 q^{58} + 62 q^{59} + 408 q^{62} - 612 q^{64} - 14 q^{65} + 72 q^{66} - 72 q^{69} + 428 q^{70} + 92 q^{71} + 46 q^{73} + 284 q^{76} + 156 q^{78} - 348 q^{79} - 192 q^{80} + 630 q^{81} + 268 q^{82} - 22 q^{83} + 72 q^{84} - 36 q^{87} + 72 q^{93} - 330 q^{95} + 96 q^{96} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
421.1 3.89386i 1.73205i −11.1622 −8.01349 6.74437 2.08342i 27.8885i −3.00000 31.2034i
421.2 3.86519i 1.73205i −10.9397 4.52341 6.69470 6.15822i 26.8232i −3.00000 17.4838i
421.3 3.85406i 1.73205i −10.8538 3.36541 −6.67544 6.97346i 26.4150i −3.00000 12.9705i
421.4 3.66343i 1.73205i −9.42072 −6.96010 −6.34525 4.69902i 19.8584i −3.00000 25.4978i
421.5 3.59880i 1.73205i −8.95137 0.151845 −6.23331 11.0053i 17.8190i −3.00000 0.546461i
421.6 3.48277i 1.73205i −8.12969 9.79345 6.03233 12.7499i 14.3828i −3.00000 34.1083i
421.7 3.41191i 1.73205i −7.64115 0.482983 5.90961 1.24753i 12.4233i −3.00000 1.64790i
421.8 3.20877i 1.73205i −6.29622 5.84037 −5.55776 1.24166i 7.36803i −3.00000 18.7404i
421.9 3.20778i 1.73205i −6.28988 −7.87944 −5.55605 10.1392i 7.34545i −3.00000 25.2756i
421.10 3.14566i 1.73205i −5.89516 2.80095 −5.44844 0.445584i 5.96154i −3.00000 8.81083i
421.11 3.12474i 1.73205i −5.76401 −2.63058 5.41221 11.5486i 5.51207i −3.00000 8.21987i
421.12 3.10766i 1.73205i −5.65757 −8.33156 5.38263 5.74125i 5.15117i −3.00000 25.8917i
421.13 2.58659i 1.73205i −2.69047 −3.76006 4.48011 13.5456i 3.38722i −3.00000 9.72574i
421.14 2.57146i 1.73205i −2.61241 0.177066 −4.45390 9.58955i 3.56814i −3.00000 0.455318i
421.15 2.52983i 1.73205i −2.40004 2.80670 4.38179 2.89007i 4.04762i −3.00000 7.10047i
421.16 2.52154i 1.73205i −2.35816 7.67033 4.36743 9.91383i 4.13996i −3.00000 19.3410i
421.17 2.49430i 1.73205i −2.22151 −0.0577969 −4.32025 9.93281i 4.43607i −3.00000 0.144163i
421.18 2.32666i 1.73205i −1.41336 −6.54120 −4.02990 1.49967i 6.01823i −3.00000 15.2192i
421.19 2.21937i 1.73205i −0.925614 −3.95657 3.84407 6.99995i 6.82321i −3.00000 8.78110i
421.20 2.15041i 1.73205i −0.624268 5.51087 3.72462 1.27747i 7.25921i −3.00000 11.8506i
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.3.c.a 70
211.b odd 2 1 inner 633.3.c.a 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.3.c.a 70 1.a even 1 1 trivial
633.3.c.a 70 211.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(633, [\chi])\).