Properties

Label 633.2.d.a
Level $633$
Weight $2$
Character orbit 633.d
Analytic conductor $5.055$
Analytic rank $0$
Dimension $68$
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] = [633,2,Mod(632,633)]
 
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([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.632");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.d (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: \(5.05453044795\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 64 q^{4} - 14 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 64 q^{4} - 14 q^{6} - 4 q^{9} - 16 q^{13} + 56 q^{16} + 4 q^{19} + 8 q^{21} - 38 q^{24} - 60 q^{25} - 14 q^{30} - 32 q^{34} - 18 q^{36} - 28 q^{37} + 40 q^{43} - 2 q^{45} - 8 q^{46} - 80 q^{49} + 16 q^{51} - 16 q^{52} + 52 q^{54} + 16 q^{55} - 40 q^{58} + 28 q^{64} + 18 q^{66} - 10 q^{69} + 80 q^{70} - 8 q^{76} + 32 q^{78} - 40 q^{79} - 28 q^{81} - 44 q^{82} + 84 q^{84} - 44 q^{87} - 10 q^{93} - 56 q^{96} + 68 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
632.1 −2.69215 −0.336216 1.69911i 5.24765 2.13595i 0.905142 + 4.57424i 2.13995i −8.74315 −2.77392 + 1.14253i 5.75028i
632.2 −2.69215 −0.336216 + 1.69911i 5.24765 2.13595i 0.905142 4.57424i 2.13995i −8.74315 −2.77392 1.14253i 5.75028i
632.3 −2.60443 1.06993 1.36207i 4.78303 2.83866i −2.78656 + 3.54742i 2.67377i −7.24820 −0.710484 2.91465i 7.39308i
632.4 −2.60443 1.06993 + 1.36207i 4.78303 2.83866i −2.78656 3.54742i 2.67377i −7.24820 −0.710484 + 2.91465i 7.39308i
632.5 −2.47350 1.19250 1.25616i 4.11822 0.579575i −2.94966 + 3.10712i 4.56706i −5.23944 −0.155884 2.99595i 1.43358i
632.6 −2.47350 1.19250 + 1.25616i 4.11822 0.579575i −2.94966 3.10712i 4.56706i −5.23944 −0.155884 + 2.99595i 1.43358i
632.7 −2.34810 −1.26767 1.18026i 3.51358 2.68207i 2.97662 + 2.77136i 0.262668i −3.55404 0.213986 + 2.99236i 6.29778i
632.8 −2.34810 −1.26767 + 1.18026i 3.51358 2.68207i 2.97662 2.77136i 0.262668i −3.55404 0.213986 2.99236i 6.29778i
632.9 −2.29624 1.68029 0.420273i 3.27272 4.22206i −3.85835 + 0.965047i 1.75525i −2.92248 2.64674 1.41236i 9.69486i
632.10 −2.29624 1.68029 + 0.420273i 3.27272 4.22206i −3.85835 0.965047i 1.75525i −2.92248 2.64674 + 1.41236i 9.69486i
632.11 −2.01825 −1.66864 0.464362i 2.07332 1.97456i 3.36773 + 0.937196i 5.19790i −0.147976 2.56874 + 1.54971i 3.98515i
632.12 −2.01825 −1.66864 + 0.464362i 2.07332 1.97456i 3.36773 0.937196i 5.19790i −0.147976 2.56874 1.54971i 3.98515i
632.13 −1.88878 −0.343145 1.69772i 1.56748 0.692716i 0.648125 + 3.20661i 1.71180i 0.816942 −2.76450 + 1.16513i 1.30839i
632.14 −1.88878 −0.343145 + 1.69772i 1.56748 0.692716i 0.648125 3.20661i 1.71180i 0.816942 −2.76450 1.16513i 1.30839i
632.15 −1.76207 1.49484 0.874902i 1.10489 0.0681843i −2.63401 + 1.54164i 2.98322i 1.57725 1.46909 2.61568i 0.120145i
632.16 −1.76207 1.49484 + 0.874902i 1.10489 0.0681843i −2.63401 1.54164i 2.98322i 1.57725 1.46909 + 2.61568i 0.120145i
632.17 −1.55050 −1.58549 0.697289i 0.404037 3.05483i 2.45830 + 1.08114i 1.87042i 2.47453 2.02757 + 2.21109i 4.73649i
632.18 −1.55050 −1.58549 + 0.697289i 0.404037 3.05483i 2.45830 1.08114i 1.87042i 2.47453 2.02757 2.21109i 4.73649i
632.19 −1.34820 0.793870 1.53941i −0.182353 1.28346i −1.07030 + 2.07543i 2.59500i 2.94225 −1.73954 2.44418i 1.73036i
632.20 −1.34820 0.793870 + 1.53941i −0.182353 1.28346i −1.07030 2.07543i 2.59500i 2.94225 −1.73954 + 2.44418i 1.73036i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 632.68
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.d.a 68
3.b odd 2 1 inner 633.2.d.a 68
211.b odd 2 1 inner 633.2.d.a 68
633.d even 2 1 inner 633.2.d.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.d.a 68 1.a even 1 1 trivial
633.2.d.a 68 3.b odd 2 1 inner
633.2.d.a 68 211.b odd 2 1 inner
633.2.d.a 68 633.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(633, [\chi])\).