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Properties

Label 663.2.cs.a
Level $663$
Weight $2$
Character orbit 663.cs
Analytic conductor $5.294$
Analytic rank $0$
Dimension $1280$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [663,2,Mod(23,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([24, 40, 45]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.cs (of order \(48\), degree \(16\), 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: \(1280\)
Relative dimension: \(80\) over \(\Q(\zeta_{48})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1280 q - 8 q^{3} - 16 q^{4} - 24 q^{6} - 48 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1280 q - 8 q^{3} - 16 q^{4} - 24 q^{6} - 48 q^{7} - 8 q^{9} - 16 q^{10} - 32 q^{12} - 32 q^{13} - 24 q^{15} - 48 q^{19} - 16 q^{22} - 24 q^{24} - 32 q^{27} - 192 q^{28} - 8 q^{30} - 24 q^{36} - 48 q^{37} - 32 q^{39} - 64 q^{40} + 40 q^{42} - 16 q^{43} - 24 q^{45} + 48 q^{46} - 56 q^{48} - 48 q^{49} - 32 q^{51} - 128 q^{52} - 24 q^{54} + 16 q^{55} - 48 q^{58} - 16 q^{61} - 24 q^{63} - 192 q^{64} - 272 q^{66} - 112 q^{69} + 384 q^{72} - 8 q^{75} - 48 q^{76} + 88 q^{78} - 64 q^{79} - 8 q^{81} - 144 q^{82} - 48 q^{85} + 40 q^{87} + 112 q^{88} - 192 q^{90} + 128 q^{91} + 168 q^{93} - 80 q^{94} - 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.364017 − 2.76499i −1.14432 − 1.30020i −5.58079 + 1.49537i −0.720916 + 1.07893i −3.17849 + 3.63734i −0.146361 − 2.23304i 4.03169 + 9.73337i −0.381047 + 2.97570i 3.24564 + 1.60057i
23.2 −0.359033 − 2.72712i −1.64945 + 0.528504i −5.37645 + 1.44061i 2.04235 − 3.05659i 2.03350 + 4.30850i 0.231215 + 3.52765i 3.75379 + 9.06246i 2.44137 − 1.74348i −9.06897 − 4.47232i
23.3 −0.351364 − 2.66887i 1.71419 + 0.248102i −5.06757 + 1.35785i −0.812959 + 1.21668i 0.0598486 − 4.66213i −0.0405232 − 0.618265i 3.34420 + 8.07360i 2.87689 + 0.850588i 3.53280 + 1.74219i
23.4 −0.328525 − 2.49539i −1.21825 + 1.23120i −4.18721 + 1.12196i −0.466433 + 0.698066i 3.47256 + 2.63554i −0.271912 − 4.14857i 2.24897 + 5.42948i −0.0317173 − 2.99983i 1.89519 + 0.934602i
23.5 −0.327739 − 2.48942i 1.05807 + 1.37131i −4.15796 + 1.11412i 0.544050 − 0.814229i 3.06699 − 3.08342i 0.215950 + 3.29475i 2.21448 + 5.34622i −0.760958 + 2.90189i −2.20527 − 1.08752i
23.6 −0.324743 − 2.46667i 0.768468 − 1.55224i −4.04716 + 1.08443i 0.291908 − 0.436871i −4.07843 − 1.39148i −0.172682 − 2.63461i 2.08503 + 5.03370i −1.81891 − 2.38570i −1.17241 − 0.578169i
23.7 −0.316506 − 2.40410i 0.0169580 + 1.73197i −3.74767 + 1.00419i 0.207196 − 0.310091i 4.15846 − 0.588947i −0.0154733 − 0.236077i 1.74443 + 4.21142i −2.99942 + 0.0587415i −0.811070 − 0.399975i
23.8 −0.312837 − 2.37623i 1.65533 + 0.509779i −3.61677 + 0.969110i 2.16197 − 3.23562i 0.693504 − 4.09293i −0.310820 − 4.74220i 1.59991 + 3.86252i 2.48025 + 1.68771i −8.36493 − 4.12512i
23.9 −0.306290 − 2.32651i 0.675305 − 1.59498i −3.38697 + 0.907536i −1.45254 + 2.17389i −3.91757 − 1.08257i 0.317878 + 4.84988i 1.35279 + 3.26592i −2.08793 − 2.15420i 5.50246 + 2.71351i
23.10 −0.299619 − 2.27583i −0.739913 − 1.56605i −3.15779 + 0.846126i −0.460869 + 0.689739i −3.34239 + 2.15314i 0.241484 + 3.68434i 1.11490 + 2.69160i −1.90506 + 2.31749i 1.70781 + 0.842201i
23.11 −0.291146 − 2.21147i 1.47462 − 0.908564i −2.87399 + 0.770082i −2.07450 + 3.10471i −2.43859 − 2.99656i −0.105804 − 1.61426i 0.832572 + 2.01001i 1.34902 − 2.67958i 7.46995 + 3.68377i
23.12 −0.289134 − 2.19619i −0.885627 − 1.48851i −2.80781 + 0.752349i 2.01757 − 3.01951i −3.01299 + 2.37539i −0.00426394 − 0.0650551i 0.768740 + 1.85590i −1.43133 + 2.63653i −7.21477 − 3.55793i
23.13 −0.268336 − 2.03822i −1.69460 − 0.358233i −2.15047 + 0.576216i 0.807556 − 1.20859i −0.275434 + 3.55009i −0.151365 − 2.30939i 0.178058 + 0.429871i 2.74334 + 1.21412i −2.68007 − 1.32166i
23.14 −0.267434 − 2.03136i −1.68884 − 0.384484i −2.12306 + 0.568871i −2.31947 + 3.47133i −0.329374 + 3.53346i −0.0353567 − 0.539439i 0.155208 + 0.374705i 2.70434 + 1.29866i 7.67182 + 3.78332i
23.15 −0.258823 − 1.96596i 1.64226 − 0.550436i −1.86616 + 0.500035i 1.57674 − 2.35976i −1.50719 − 3.08615i 0.249137 + 3.80110i −0.0516106 − 0.124599i 2.39404 − 1.80792i −5.04729 − 2.48905i
23.16 −0.258286 − 1.96187i −1.18392 + 1.26425i −1.85038 + 0.495809i −0.211287 + 0.316213i 2.78608 + 1.99617i 0.0343097 + 0.523466i −0.0638686 − 0.154192i −0.196647 − 2.99355i 0.674943 + 0.332845i
23.17 −0.251352 − 1.90921i 1.45034 + 0.946839i −1.65005 + 0.442130i −1.40766 + 2.10671i 1.44317 − 3.00700i 0.0275598 + 0.420481i −0.214991 − 0.519033i 1.20699 + 2.74648i 4.37598 + 2.15799i
23.18 −0.218881 − 1.66257i −1.68364 − 0.406633i −0.784370 + 0.210171i 0.194076 − 0.290456i −0.307537 + 2.88817i 0.162913 + 2.48557i −0.762346 − 1.84047i 2.66930 + 1.36925i −0.525382 − 0.259090i
23.19 −0.215254 − 1.63502i −0.745158 + 1.56357i −0.695097 + 0.186251i 2.34044 − 3.50272i 2.71686 + 0.881782i 0.0216080 + 0.329675i −0.808041 − 1.95078i −1.88948 − 2.33021i −6.23080 − 3.07269i
23.20 −0.215032 − 1.63333i 0.681587 + 1.59231i −0.689682 + 0.184800i −0.625159 + 0.935616i 2.45420 − 1.45566i −0.319744 − 4.87836i −0.810742 − 1.95730i −2.07088 + 2.17059i 1.66260 + 0.819904i
See next 80 embeddings (of 1280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.e even 6 1 inner
17.e odd 16 1 inner
39.h odd 6 1 inner
51.i even 16 1 inner
221.bg odd 48 1 inner
663.cs even 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.cs.a ✓ 1280
3.b odd 2 1 inner 663.2.cs.a ✓ 1280
13.e even 6 1 inner 663.2.cs.a ✓ 1280
17.e odd 16 1 inner 663.2.cs.a ✓ 1280
39.h odd 6 1 inner 663.2.cs.a ✓ 1280
51.i even 16 1 inner 663.2.cs.a ✓ 1280
221.bg odd 48 1 inner 663.2.cs.a ✓ 1280
663.cs even 48 1 inner 663.2.cs.a ✓ 1280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.cs.a ✓ 1280 1.a even 1 1 trivial
663.2.cs.a ✓ 1280 3.b odd 2 1 inner
663.2.cs.a ✓ 1280 13.e even 6 1 inner
663.2.cs.a ✓ 1280 17.e odd 16 1 inner
663.2.cs.a ✓ 1280 39.h odd 6 1 inner
663.2.cs.a ✓ 1280 51.i even 16 1 inner
663.2.cs.a ✓ 1280 221.bg odd 48 1 inner
663.2.cs.a ✓ 1280 663.cs even 48 1 inner

Hecke kernels

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