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Properties

Label 663.2.by.b
Level $663$
Weight $2$
Character orbit 663.by
Analytic conductor $5.294$
Analytic rank $0$
Dimension $288$
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(14,663)]
 
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([8, 0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.by (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: \(288\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q + 16 q^{7} - 24 q^{12} - 24 q^{21} - 160 q^{24} - 32 q^{25} - 80 q^{34} - 32 q^{37} - 32 q^{40} + 232 q^{42} - 32 q^{43} - 16 q^{46} + 80 q^{48} - 176 q^{49} - 80 q^{51} - 288 q^{52} + 16 q^{54} + 16 q^{55} - 56 q^{57} + 128 q^{58} + 16 q^{61} - 72 q^{63} + 32 q^{64} + 168 q^{66} + 64 q^{69} - 32 q^{70} - 64 q^{73} - 208 q^{75} - 256 q^{76} - 32 q^{79} - 56 q^{81} + 64 q^{82} + 464 q^{85} - 96 q^{88} + 200 q^{90} + 88 q^{93} + 96 q^{94} + 152 q^{96} - 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −1.03580 − 2.50063i −1.73048 + 0.0736695i −3.76607 + 3.76607i −1.31294 − 1.96495i 1.97665 + 4.25099i −0.182503 − 0.121945i 8.31716 + 3.44508i 2.98915 − 0.254968i −3.55369 + 5.31847i
14.2 −1.01653 − 2.45413i −1.07662 + 1.35679i −3.57520 + 3.57520i 2.31755 + 3.46846i 4.42417 + 1.26294i −0.517158 − 0.345554i 7.50006 + 3.10662i −0.681782 − 2.92150i 6.15618 − 9.21337i
14.3 −0.976261 − 2.35690i 1.71424 − 0.247770i −3.18769 + 3.18769i −0.817413 − 1.22335i −2.25751 − 3.79840i 3.00645 + 2.00884i 5.91131 + 2.44854i 2.87722 − 0.849473i −2.08530 + 3.12087i
14.4 −0.883667 − 2.13336i −0.724347 − 1.57332i −2.35615 + 2.35615i −1.42126 − 2.12706i −2.71637 + 2.93558i −1.61850 − 1.08145i 2.84184 + 1.17713i −1.95064 + 2.27925i −3.28187 + 4.91167i
14.5 −0.849666 − 2.05127i 1.56087 + 0.750780i −2.07158 + 2.07158i 1.07378 + 1.60702i 0.213834 − 3.83970i −0.463054 − 0.309403i 1.90699 + 0.789901i 1.87266 + 2.34375i 2.38409 − 3.56804i
14.6 −0.789516 − 1.90606i −0.122940 + 1.72768i −1.59552 + 1.59552i −2.23665 − 3.34739i 3.39013 − 1.12970i −2.07214 − 1.38456i 0.488720 + 0.202434i −2.96977 − 0.424804i −4.61445 + 6.90602i
14.7 −0.779986 − 1.88305i −0.378221 − 1.69025i −1.52329 + 1.52329i 0.850532 + 1.27291i −2.88782 + 2.03058i 3.38347 + 2.26076i 0.290481 + 0.120321i −2.71390 + 1.27858i 1.73355 − 2.59445i
14.8 −0.672812 − 1.62431i −0.943735 + 1.45237i −0.771502 + 0.771502i 0.566089 + 0.847212i 2.99405 + 0.555752i 1.38555 + 0.925793i −1.47639 − 0.611540i −1.21873 − 2.74130i 0.995265 − 1.48952i
14.9 −0.659844 − 1.59300i 1.68384 − 0.405826i −0.688052 + 0.688052i 1.75186 + 2.62185i −1.75755 − 2.41458i −3.98528 − 2.66288i −1.63593 − 0.677624i 2.67061 − 1.36669i 3.02066 − 4.52074i
14.10 −0.534354 − 1.29004i −1.72215 + 0.184951i 0.0355335 − 0.0355335i 0.491004 + 0.734839i 1.15883 + 2.12282i −1.23144 − 0.822823i −2.64492 − 1.09556i 2.93159 − 0.637024i 0.685605 − 1.02608i
14.11 −0.461591 − 1.11438i 1.21828 − 1.23117i 0.385438 − 0.385438i −1.67489 − 2.50665i −1.93434 − 0.789330i 1.40879 + 0.941324i −2.83620 − 1.17479i −0.0315773 − 2.99983i −2.02024 + 3.02350i
14.12 −0.386559 − 0.933237i 0.0564905 − 1.73113i 0.692711 − 0.692711i −1.74957e−5 0 2.61842e-5i −1.63739 + 0.616465i −2.76978 − 1.85071i −2.78071 − 1.15181i −2.99362 − 0.195585i −1.76729e−5 0 2.64494e-5i
14.13 −0.363759 − 0.878192i 0.956083 + 1.44427i 0.775313 − 0.775313i −1.33440 − 1.99707i 0.920559 − 1.36499i 4.12532 + 2.75645i −2.71929 − 1.12636i −1.17181 + 2.76168i −1.26841 + 1.89831i
14.14 −0.295800 − 0.714124i −1.39470 − 1.02704i 0.991738 − 0.991738i 1.47112 + 2.20168i −0.320883 + 1.29979i 0.659392 + 0.440592i −2.42983 − 1.00647i 0.890371 + 2.86483i 1.13712 − 1.70182i
14.15 −0.245418 − 0.592493i 1.36205 − 1.06996i 1.12340 − 1.12340i 1.83828 + 2.75118i −0.968216 − 0.544418i 0.702156 + 0.469166i −2.12629 − 0.880739i 0.710373 − 2.91468i 1.17891 − 1.76436i
14.16 −0.117632 − 0.283989i 1.72008 + 0.203316i 1.34740 − 1.34740i 0.965130 + 1.44442i −0.144597 − 0.512399i 2.65089 + 1.77127i −1.10912 − 0.459414i 2.91733 + 0.699438i 0.296669 − 0.443996i
14.17 −0.102771 − 0.248111i −1.19324 + 1.25546i 1.36322 − 1.36322i −1.74201 − 2.60710i 0.434124 + 0.167032i 0.226426 + 0.151293i −0.974551 − 0.403672i −0.152344 − 2.99613i −0.467823 + 0.700147i
14.18 −0.0562211 − 0.135730i 0.913738 + 1.47142i 1.39895 − 1.39895i −0.540942 − 0.809577i 0.148344 − 0.206746i −3.32590 − 2.22230i −0.539990 − 0.223671i −1.33017 + 2.68899i −0.0794714 + 0.118937i
14.19 0.0562211 + 0.135730i −1.40727 + 1.00974i 1.39895 − 1.39895i 0.540942 + 0.809577i −0.216171 − 0.134240i −3.32590 − 2.22230i 0.539990 + 0.223671i 0.960832 − 2.84197i −0.0794714 + 0.118937i
14.20 0.102771 + 0.248111i 0.621970 + 1.61653i 1.36322 − 1.36322i 1.74201 + 2.60710i −0.337158 + 0.320450i 0.226426 + 0.151293i 0.974551 + 0.403672i −2.22631 + 2.01086i −0.467823 + 0.700147i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.e odd 16 1 inner
51.i 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.by.b ✓ 288
3.b odd 2 1 inner 663.2.by.b ✓ 288
17.e odd 16 1 inner 663.2.by.b ✓ 288
51.i even 16 1 inner 663.2.by.b ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.by.b ✓ 288 1.a even 1 1 trivial
663.2.by.b ✓ 288 3.b odd 2 1 inner
663.2.by.b ✓ 288 17.e odd 16 1 inner
663.2.by.b ✓ 288 51.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} - 32 T_{2}^{282} + 20260 T_{2}^{280} - 1088 T_{2}^{278} + 512 T_{2}^{276} + \cdots + 67\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display