Properties

Label 663.2.cf.a
Level $663$
Weight $2$
Character orbit 663.cf
Analytic conductor $5.294$
Analytic rank $0$
Dimension $640$
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] = [663,2,Mod(110,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([12, 10, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.110");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.cf (of order \(24\), 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: \(640\)
Relative dimension: \(80\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 4 q^{3} - 304 q^{4} - 16 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q - 4 q^{3} - 304 q^{4} - 16 q^{7} - 4 q^{9} - 88 q^{10} + 16 q^{12} - 36 q^{15} - 288 q^{16} - 24 q^{19} + 32 q^{21} - 8 q^{22} - 8 q^{24} + 32 q^{25} - 64 q^{27} + 8 q^{28} + 32 q^{30} - 32 q^{31} - 48 q^{33} + 32 q^{34} - 20 q^{36} - 112 q^{37} - 12 q^{39} + 32 q^{40} + 44 q^{42} - 40 q^{43} + 56 q^{45} + 32 q^{46} - 68 q^{48} - 40 q^{49} - 64 q^{51} + 32 q^{52} + 104 q^{54} - 4 q^{57} - 48 q^{58} - 8 q^{61} + 16 q^{63} + 480 q^{64} - 120 q^{66} - 64 q^{67} - 24 q^{69} - 208 q^{72} - 96 q^{73} - 56 q^{75} - 72 q^{76} + 164 q^{78} - 64 q^{79} - 48 q^{81} + 8 q^{82} - 32 q^{84} - 8 q^{85} + 68 q^{87} - 104 q^{88} + 16 q^{90} + 96 q^{91} - 28 q^{93} - 56 q^{94} + 40 q^{96} + 64 q^{97} + 52 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
110.1 −1.36681 + 2.36739i 0.0409062 + 1.73157i −2.73634 4.73949i 1.62588 + 3.92522i −4.15520 2.26988i 0.172306 + 1.30879i 9.49301 −2.99665 + 0.141664i −11.5148 1.51595i
110.2 −1.35757 + 2.35138i 0.549761 1.64249i −2.68600 4.65229i 0.528427 + 1.27574i 3.11578 + 3.52249i 0.526117 + 3.99626i 9.15548 −2.39553 1.80595i −3.71712 0.489368i
110.3 −1.32818 + 2.30048i −1.30504 1.13880i −2.52813 4.37886i 0.291822 + 0.704520i 4.35312 1.48968i −0.218505 1.65971i 8.11856 0.406266 + 2.97236i −2.00833 0.264401i
110.4 −1.31688 + 2.28090i 1.56064 + 0.751258i −2.46834 4.27530i 0.0697335 + 0.168352i −3.76873 + 2.57036i −0.518512 3.93849i 7.73453 1.87122 + 2.34489i −0.475824 0.0626434i
110.5 −1.28851 + 2.23177i 1.10924 1.33026i −2.32052 4.01926i −1.20903 2.91885i 1.53955 + 4.18962i −0.227238 1.72605i 6.80601 −0.539168 2.95115i 8.07205 + 1.06271i
110.6 −1.27834 + 2.21416i −1.40104 + 1.01838i −2.26833 3.92886i −0.553348 1.33590i −0.463851 4.40395i −0.307457 2.33537i 6.48543 0.925799 2.85358i 3.66526 + 0.482540i
110.7 −1.20385 + 2.08512i 0.662912 + 1.60017i −1.89850 3.28829i −1.11848 2.70024i −4.13460 0.544107i 0.0587604 + 0.446330i 4.32661 −2.12110 + 2.12155i 6.97681 + 0.918515i
110.8 −1.17604 + 2.03696i −1.67248 + 0.450352i −1.76613 3.05902i 0.521848 + 1.25985i 1.04955 3.93640i 0.232315 + 1.76460i 3.60397 2.59437 1.50641i −3.17998 0.418652i
110.9 −1.17216 + 2.03024i −0.556238 + 1.64030i −1.74791 3.02746i −0.821377 1.98298i −2.67821 3.05199i 0.524203 + 3.98172i 3.50665 −2.38120 1.82480i 4.98870 + 0.656775i
110.10 −1.14218 + 1.97832i −1.42818 0.979946i −1.60917 2.78716i 0.593397 + 1.43259i 3.56990 1.70612i 0.0392968 + 0.298488i 2.78313 1.07941 + 2.79908i −3.51188 0.462348i
110.11 −1.10600 + 1.91565i 1.69205 0.370072i −1.44647 2.50536i 1.00569 + 2.42794i −1.16248 + 3.65068i −0.315622 2.39739i 1.97518 2.72609 1.25236i −5.76337 0.758762i
110.12 −1.09736 + 1.90069i 1.73138 + 0.0481161i −1.40841 2.43944i 0.840948 + 2.03023i −1.99141 + 3.23802i 0.524066 + 3.98068i 1.79270 2.99537 + 0.166615i −4.78166 0.629517i
110.13 −1.03251 + 1.78836i −0.447337 1.67329i −1.13216 1.96096i −0.690499 1.66701i 3.45433 + 0.927688i 0.340463 + 2.58607i 0.545839 −2.59978 + 1.49705i 3.69417 + 0.486347i
110.14 −0.997743 + 1.72814i 1.20097 1.24806i −0.990983 1.71643i −0.412947 0.996942i 0.958568 + 3.32070i −0.0921696 0.700097i −0.0359885 −0.115326 2.99778i 2.13487 + 0.281061i
110.15 −0.977494 + 1.69307i 0.390475 1.68746i −0.910988 1.57788i 1.61273 + 3.89347i 2.47530 + 2.31058i −0.259830 1.97361i −0.348037 −2.69506 1.31782i −8.16835 1.07538i
110.16 −0.938225 + 1.62505i −0.0494020 + 1.73135i −0.760531 1.31728i 0.367649 + 0.887583i −2.76718 1.70467i −0.497341 3.77768i −0.898702 −2.99512 0.171064i −1.78731 0.235303i
110.17 −0.861486 + 1.49214i 1.33159 + 1.10764i −0.484315 0.838859i 0.264202 + 0.637841i −2.79990 + 1.03270i 0.127074 + 0.965223i −1.77702 0.546262 + 2.94985i −1.17935 0.155265i
110.18 −0.842846 + 1.45985i 1.71216 0.261710i −0.420780 0.728812i −1.12550 2.71721i −1.06103 + 2.72009i 0.277857 + 2.11053i −1.95277 2.86302 0.896182i 4.91535 + 0.647118i
110.19 −0.836096 + 1.44816i −1.72422 0.164558i −0.398114 0.689553i −1.20374 2.90608i 1.67992 2.35936i 0.598949 + 4.54947i −2.01294 2.94584 + 0.567468i 5.21492 + 0.686557i
110.20 −0.789656 + 1.36772i −1.07437 + 1.35858i −0.247113 0.428012i 1.33211 + 3.21599i −1.00978 2.54225i 0.167968 + 1.27585i −2.37809 −0.691459 2.91923i −5.45050 0.717571i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 110.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
221.bf odd 24 1 inner
663.cf even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.cf.a 640
3.b odd 2 1 inner 663.2.cf.a 640
13.f odd 12 1 663.2.ck.a yes 640
17.d even 8 1 663.2.ck.a yes 640
39.k even 12 1 663.2.ck.a yes 640
51.g odd 8 1 663.2.ck.a yes 640
221.bf odd 24 1 inner 663.2.cf.a 640
663.cf even 24 1 inner 663.2.cf.a 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.cf.a 640 1.a even 1 1 trivial
663.2.cf.a 640 3.b odd 2 1 inner
663.2.cf.a 640 221.bf odd 24 1 inner
663.2.cf.a 640 663.cf even 24 1 inner
663.2.ck.a yes 640 13.f odd 12 1
663.2.ck.a yes 640 17.d even 8 1
663.2.ck.a yes 640 39.k even 12 1
663.2.ck.a yes 640 51.g odd 8 1

Hecke kernels

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