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Properties

Label 663.2.ca.a
Level $663$
Weight $2$
Character orbit 663.ca
Analytic conductor $5.294$
Analytic rank $0$
Dimension $336$
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(31,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 12, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.ca (of order \(16\), degree \(8\), 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: \(336\)
Relative dimension: \(42\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 16 q^{13} - 32 q^{14} - 32 q^{17} + 32 q^{20} + 32 q^{22} - 48 q^{26} + 48 q^{28} - 80 q^{29} + 32 q^{31} + 16 q^{33} - 32 q^{34} + 160 q^{38} + 96 q^{41} - 96 q^{42} + 64 q^{43} - 32 q^{45} - 32 q^{46} - 32 q^{49} + 48 q^{53} + 32 q^{55} + 32 q^{58} - 64 q^{59} - 96 q^{61} + 96 q^{64} - 64 q^{67} - 64 q^{70} - 128 q^{71} + 48 q^{72} - 112 q^{73} - 240 q^{74} - 64 q^{76} - 48 q^{78} + 128 q^{79} - 128 q^{80} + 32 q^{85} - 96 q^{87} + 16 q^{90} + 64 q^{91} - 256 q^{95} + 96 q^{96} - 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.06196 + 2.56379i −0.195090 + 0.980785i −4.03106 − 4.03106i −0.0112976 + 0.0567968i −2.30735 − 1.54172i −0.0710121 − 0.357002i 9.48803 − 3.93007i −0.923880 − 0.382683i −0.133618 − 0.0892804i
31.2 −0.984342 + 2.37641i 0.195090 − 0.980785i −3.26419 − 3.26419i −0.816103 + 4.10283i 2.13871 + 1.42904i −0.292042 − 1.46820i 6.21733 − 2.57530i −0.923880 − 0.382683i −8.94668 − 5.97798i
31.3 −0.932323 + 2.25083i 0.195090 − 0.980785i −2.78278 − 2.78278i −0.0223820 + 0.112522i 2.02569 + 1.35352i 0.115400 + 0.580156i 4.35636 − 1.80446i −0.923880 − 0.382683i −0.232400 − 0.155285i
31.4 −0.909257 + 2.19514i 0.195090 − 0.980785i −2.57768 − 2.57768i 0.841351 − 4.22976i 1.97557 + 1.32004i 0.906286 + 4.55621i 3.61186 − 1.49608i −0.923880 − 0.382683i 8.51991 + 5.69282i
31.5 −0.882143 + 2.12968i −0.195090 + 0.980785i −2.34316 − 2.34316i −0.559920 + 2.81491i −1.91666 − 1.28067i 0.711176 + 3.57532i 2.79782 − 1.15889i −0.923880 − 0.382683i −5.50093 − 3.67560i
31.6 −0.879851 + 2.12415i −0.195090 + 0.980785i −2.32366 − 2.32366i 0.141352 − 0.710626i −1.91168 − 1.27735i −0.0305824 − 0.153748i 2.73196 − 1.13162i −0.923880 − 0.382683i 1.38511 + 0.925499i
31.7 −0.832525 + 2.00989i −0.195090 + 0.980785i −1.93236 − 1.93236i 0.570327 − 2.86723i −1.80886 − 1.20864i −0.173257 − 0.871023i 1.47279 − 0.610049i −0.923880 − 0.382683i 5.28801 + 3.53333i
31.8 −0.727202 + 1.75562i 0.195090 − 0.980785i −1.13917 − 1.13917i −0.0286684 + 0.144126i 1.58002 + 1.05573i −0.353773 − 1.77854i −0.682884 + 0.282860i −0.923880 − 0.382683i −0.232183 − 0.155140i
31.9 −0.639591 + 1.54411i 0.195090 − 0.980785i −0.560981 − 0.560981i 0.0703333 − 0.353589i 1.38966 + 0.928542i 0.273361 + 1.37428i −1.86320 + 0.771764i −0.923880 − 0.382683i 0.500996 + 0.334755i
31.10 −0.634908 + 1.53280i 0.195090 − 0.980785i −0.532162 − 0.532162i −0.409419 + 2.05829i 1.37949 + 0.921743i 0.700065 + 3.51946i −1.91203 + 0.791989i −0.923880 − 0.382683i −2.89501 − 1.93438i
31.11 −0.580760 + 1.40208i −0.195090 + 0.980785i −0.214329 − 0.214329i −0.713546 + 3.58724i −1.26184 − 0.843133i 0.238015 + 1.19658i −2.37918 + 0.985487i −0.923880 − 0.382683i −4.61519 − 3.08377i
31.12 −0.565509 + 1.36526i −0.195090 + 0.980785i −0.129919 − 0.129919i −0.396222 + 1.99194i −1.22870 − 0.820992i −0.635870 − 3.19673i −2.47968 + 1.02712i −0.923880 − 0.382683i −2.49545 − 1.66741i
31.13 −0.524425 + 1.26607i 0.195090 − 0.980785i 0.0862911 + 0.0862911i 0.777885 − 3.91069i 1.13944 + 0.761347i −0.642023 − 3.22767i −2.68665 + 1.11285i −0.923880 − 0.382683i 4.54329 + 3.03573i
31.14 −0.514366 + 1.24179i −0.195090 + 0.980785i 0.136747 + 0.136747i 0.530594 − 2.66748i −1.11758 − 0.746743i −0.433702 − 2.18037i −2.72373 + 1.12820i −0.923880 − 0.382683i 3.03952 + 2.03094i
31.15 −0.334342 + 0.807173i 0.195090 − 0.980785i 0.874470 + 0.874470i −0.629694 + 3.16569i 0.726436 + 0.485389i −0.680255 − 3.41987i −2.61257 + 1.08216i −0.923880 − 0.382683i −2.34472 − 1.56669i
31.16 −0.235284 + 0.568026i 0.195090 − 0.980785i 1.14692 + 1.14692i 0.342829 − 1.72352i 0.511210 + 0.341579i −0.163933 − 0.824147i −2.05738 + 0.852196i −0.923880 − 0.382683i 0.898339 + 0.600251i
31.17 −0.218012 + 0.526328i −0.195090 + 0.980785i 1.18472 + 1.18472i 0.227431 − 1.14337i −0.473683 − 0.316505i 0.960346 + 4.82799i −1.93449 + 0.801293i −0.923880 − 0.382683i 0.552207 + 0.368973i
31.18 −0.175793 + 0.424402i −0.195090 + 0.980785i 1.26500 + 1.26500i 0.0429138 − 0.215742i −0.381952 − 0.255212i −0.230690 − 1.15976i −1.60805 + 0.666076i −0.923880 − 0.382683i 0.0840175 + 0.0561387i
31.19 −0.107240 + 0.258901i −0.195090 + 0.980785i 1.35868 + 1.35868i −0.174667 + 0.878109i −0.233005 − 0.155689i 0.406475 + 2.04349i −1.01527 + 0.420540i −0.923880 − 0.382683i −0.208612 − 0.139390i
31.20 −0.0761027 + 0.183728i 0.195090 − 0.980785i 1.38625 + 1.38625i −0.00685005 + 0.0344375i 0.165351 + 0.110484i 0.0792455 + 0.398394i −0.727647 + 0.301401i −0.923880 − 0.382683i −0.00580583 − 0.00387933i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
221.ba even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.ca.a ✓ 336
13.d odd 4 1 663.2.cd.a yes 336
17.e odd 16 1 663.2.cd.a yes 336
221.ba even 16 1 inner 663.2.ca.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.ca.a ✓ 336 1.a even 1 1 trivial
663.2.ca.a ✓ 336 221.ba even 16 1 inner
663.2.cd.a yes 336 13.d odd 4 1
663.2.cd.a yes 336 17.e odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(663, [\chi])\).