↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [663,2,Mod(29,663)]
 
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, 16, 39]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.cr (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.

\(\operatorname{Tr}(f)(q) = \) \( 1280 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1280 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9} - 16 q^{10} - 32 q^{12} - 32 q^{13} - 8 q^{15} - 128 q^{18} - 16 q^{19} - 32 q^{21} - 16 q^{22} - 8 q^{24} - 128 q^{25} - 32 q^{27} + 32 q^{28} - 8 q^{30} - 128 q^{31} - 64 q^{34} + 8 q^{36} - 16 q^{37} - 64 q^{40} + 40 q^{42} - 16 q^{43} + 56 q^{45} - 48 q^{46} - 56 q^{48} + 16 q^{49} - 32 q^{51} - 8 q^{54} + 16 q^{55} - 192 q^{57} + 48 q^{58} - 112 q^{60} - 16 q^{61} - 8 q^{63} - 192 q^{64} - 208 q^{66} + 80 q^{69} - 128 q^{70} + 64 q^{72} - 64 q^{73} - 8 q^{75} - 16 q^{76} - 120 q^{78} - 64 q^{79} - 8 q^{81} + 112 q^{82} - 112 q^{85} - 56 q^{87} + 112 q^{88} + 128 q^{90} - 64 q^{91} - 72 q^{93} + 48 q^{94} - 128 q^{96} - 16 q^{97} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −2.21114 − 1.69667i 1.71890 + 0.213055i 1.49283 + 5.57131i −0.683069 + 3.43402i −3.43924 − 3.38750i 0.829303 − 2.44305i 4.01867 − 9.70192i 2.90921 + 0.732440i 7.33676 − 6.43416i
29.2 −2.14229 − 1.64384i −0.756536 − 1.55809i 1.36957 + 5.11131i 0.413828 − 2.08045i −0.940532 + 4.58151i −0.251003 + 0.739430i 3.40143 − 8.21177i −1.85531 + 2.35751i −4.30647 + 3.77667i
29.3 −2.10866 − 1.61803i −1.64074 + 0.554945i 1.31078 + 4.89190i −0.0874422 + 0.439602i 4.35769 + 1.48458i −1.24695 + 3.67340i 3.11699 − 7.52507i 2.38407 − 1.82104i 0.895676 − 0.785486i
29.4 −2.09390 − 1.60671i 0.133799 + 1.72688i 1.28528 + 4.79672i −0.204213 + 1.02665i 2.49442 − 3.83088i −0.800577 + 2.35842i 2.99564 − 7.23211i −2.96420 + 0.462110i 2.07713 − 1.82159i
29.5 −2.01338 − 1.54492i 0.532973 + 1.64801i 1.14928 + 4.28917i 0.500953 − 2.51846i 1.47297 − 4.14147i 0.977638 − 2.88003i 2.37013 − 5.72201i −2.43188 + 1.75669i −4.89943 + 4.29668i
29.6 −1.95542 − 1.50044i 1.10219 − 1.33611i 1.05468 + 3.93613i 0.0550568 − 0.276789i −4.15999 + 0.958870i −0.830800 + 2.44746i 1.95716 − 4.72500i −0.570358 − 2.94528i −0.522965 + 0.458628i
29.7 −1.92837 − 1.47969i −1.40330 + 1.01525i 1.01148 + 3.77491i −0.241263 + 1.21291i 4.20834 + 0.118676i 1.31765 − 3.88168i 1.77483 − 4.28482i 0.938529 − 2.84941i 2.25997 − 1.98194i
29.8 −1.92699 − 1.47863i 1.05825 − 1.37117i 1.00931 + 3.76679i 0.439736 − 2.21070i −4.06670 + 1.07746i 1.13021 − 3.32948i 1.76575 − 4.26290i −0.760194 − 2.90209i −4.11619 + 3.60980i
29.9 −1.89784 − 1.45627i −1.43564 − 0.968989i 0.963461 + 3.59568i −0.733756 + 3.68884i 1.31351 + 3.92966i −0.155810 + 0.459000i 1.57688 − 3.80693i 1.12212 + 2.78224i 6.76449 − 5.93230i
29.10 −1.79830 − 1.37988i −1.65808 + 0.500781i 0.812159 + 3.03102i 0.838173 − 4.21378i 3.67273 + 1.38740i −0.319263 + 0.940519i 0.987084 − 2.38303i 2.49844 − 1.66067i −7.32180 + 6.42104i
29.11 −1.77843 − 1.36464i −0.258311 − 1.71268i 0.782944 + 2.92199i −0.649132 + 3.26341i −1.87780 + 3.39839i 0.531881 − 1.56687i 0.879348 − 2.12293i −2.86655 + 0.884809i 5.60781 − 4.91792i
29.12 −1.71661 − 1.31720i 1.72151 − 0.190753i 0.694087 + 2.59037i −0.114054 + 0.573387i −3.20642 − 1.94013i −0.537793 + 1.58429i 0.564504 − 1.36283i 2.92723 − 0.656768i 0.951050 − 0.834049i
29.13 −1.70766 − 1.31034i −1.58433 − 0.699924i 0.681496 + 2.54338i 0.210783 − 1.05968i 1.78837 + 3.27124i 1.28430 − 3.78341i 0.521491 − 1.25899i 2.02021 + 2.21782i −1.74848 + 1.53337i
29.14 −1.63927 − 1.25786i 1.58847 + 0.690471i 0.587373 + 2.19210i 0.576008 − 2.89579i −1.73543 − 3.12995i −0.606103 + 1.78552i 0.213048 − 0.514342i 2.04650 + 2.19359i −4.58673 + 4.02245i
29.15 −1.56351 − 1.19972i −0.622150 + 1.61646i 0.487588 + 1.81970i −0.408944 + 2.05590i 2.91204 − 1.78094i 0.341405 − 1.00575i −0.0875646 + 0.211400i −2.22586 − 2.01136i 3.10590 − 2.72380i
29.16 −1.51356 − 1.16140i −0.804927 − 1.53365i 0.424387 + 1.58383i 0.0167698 − 0.0843073i −0.562872 + 3.25612i −1.18769 + 3.49881i −0.263046 + 0.635050i −1.70418 + 2.46896i −0.123296 + 0.108128i
29.17 −1.46984 − 1.12785i 0.909024 + 1.47434i 0.370746 + 1.38364i −0.615571 + 3.09469i 0.326710 − 3.19228i −1.32359 + 3.89918i −0.402387 + 0.971447i −1.34735 + 2.68042i 4.39511 − 3.85441i
29.18 −1.34566 − 1.03256i 1.31240 − 1.13032i 0.226984 + 0.847116i −0.566258 + 2.84677i −2.93317 + 0.165892i 0.818926 − 2.41248i −0.728936 + 1.75981i 0.444770 − 2.96685i 3.70146 − 3.24610i
29.19 −1.26550 − 0.971049i 1.54754 + 0.777897i 0.140905 + 0.525864i 0.369264 − 1.85641i −1.20303 − 2.48716i 1.30660 − 3.84911i −0.888528 + 2.14510i 1.78975 + 2.40765i −2.26997 + 1.99071i
29.20 −1.20122 − 0.921726i 1.08199 − 1.35252i 0.0757046 + 0.282534i 0.776638 − 3.90442i −2.54635 + 0.627372i −0.378828 + 1.11599i −0.989361 + 2.38853i −0.658609 − 2.92681i −4.53172 + 3.97421i
See next 80 embeddings (of 1280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.80
Significant digits:
Format:

Inner twists

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

Hecke kernels

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