↑

Properties

Label 208.3.r.a
Level $208$
Weight $3$
Character orbit 208.r
Analytic conductor $5.668$
Analytic rank $0$
Dimension $108$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,3,Mod(5,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 208.r (of order \(4\), degree \(2\), 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: \(5.66758949869\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 2 q^{2} - 4 q^{3} - 4 q^{5} + 12 q^{6} - 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 2 q^{2} - 4 q^{3} - 4 q^{5} + 12 q^{6} - 20 q^{8} - 24 q^{10} - 2 q^{13} + 12 q^{14} - 4 q^{15} - 4 q^{16} - 18 q^{18} - 4 q^{20} + 52 q^{22} + 460 q^{25} + 110 q^{26} + 32 q^{27} - 124 q^{28} - 4 q^{29} - 100 q^{30} - 4 q^{31} + 8 q^{32} - 4 q^{33} - 120 q^{34} - 4 q^{35} - 244 q^{36} - 24 q^{38} - 44 q^{40} - 104 q^{42} + 64 q^{43} + 192 q^{44} - 216 q^{46} - 4 q^{47} - 144 q^{48} - 46 q^{50} + 192 q^{52} - 4 q^{53} - 280 q^{54} + 220 q^{56} + 36 q^{57} - 172 q^{58} - 328 q^{60} - 4 q^{61} + 192 q^{63} + 204 q^{64} - 4 q^{65} - 224 q^{66} - 36 q^{70} + 428 q^{72} - 48 q^{73} - 172 q^{74} - 292 q^{75} - 12 q^{76} + 196 q^{77} + 496 q^{78} - 8 q^{79} + 100 q^{80} - 692 q^{81} + 56 q^{82} - 400 q^{84} - 600 q^{86} + 256 q^{88} + 80 q^{89} - 84 q^{90} - 196 q^{91} - 200 q^{92} - 700 q^{94} + 344 q^{96} - 4 q^{97} + 554 q^{98} + 540 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.99985 + 0.0244654i −4.06912 + 4.06912i 3.99880 − 0.0978542i 7.29978 8.03807 − 8.23717i 0.179641 − 0.179641i −7.99461 + 0.293526i − 24.1154i −14.5985 + 0.178592i
5.2 −1.99862 + 0.0742630i −1.93766 + 1.93766i 3.98897 − 0.296847i −2.13480 3.72874 − 4.01654i 2.78735 − 2.78735i −7.95039 + 0.889518i 1.49098i 4.26666 − 0.158537i
5.3 −1.97875 − 0.290744i 0.141120 − 0.141120i 3.83094 + 1.15062i 2.98975 −0.320271 + 0.238212i −8.76769 + 8.76769i −7.24594 − 3.39062i 8.96017i −5.91598 − 0.869253i
5.4 −1.94593 − 0.461910i −0.749702 + 0.749702i 3.57328 + 1.79769i −8.35280 1.80516 − 1.11257i 0.753054 − 0.753054i −6.12298 − 5.14871i 7.87589i 16.2539 + 3.85824i
5.5 −1.93458 − 0.507341i 3.14808 − 3.14808i 3.48521 + 1.96299i 1.95776 −7.68736 + 4.49306i 4.28519 − 4.28519i −5.74652 − 5.56575i − 10.8208i −3.78745 − 0.993254i
5.6 −1.88798 + 0.659937i 1.80242 − 1.80242i 3.12897 − 2.49190i 2.31890 −2.21345 + 4.59242i −7.12022 + 7.12022i −4.26294 + 6.76959i 2.50258i −4.37804 + 1.53033i
5.7 −1.85588 + 0.745467i 3.59733 − 3.59733i 2.88856 − 2.76699i −7.08449 −3.99451 + 9.35789i −0.884621 + 0.884621i −3.29811 + 7.28851i − 16.8816i 13.1479 − 5.28125i
5.8 −1.82131 + 0.826346i −1.57328 + 1.57328i 2.63431 − 3.01005i −0.859098 1.56535 − 4.16549i 3.15707 − 3.15707i −2.31053 + 7.65908i 4.04960i 1.56468 − 0.709912i
5.9 −1.81855 − 0.832401i −0.130352 + 0.130352i 2.61422 + 3.02752i 9.64222 0.345557 − 0.128546i 2.66992 − 2.66992i −2.23397 − 7.68176i 8.96602i −17.5348 − 8.02619i
5.10 −1.81382 + 0.842639i 1.28054 − 1.28054i 2.57992 − 3.05680i 4.70447 −1.24364 + 3.40170i 6.39926 − 6.39926i −2.10375 + 7.71844i 5.72045i −8.53308 + 3.96417i
5.11 −1.67505 + 1.09280i −3.62511 + 3.62511i 1.61157 − 3.66099i −7.56418 2.11071 − 10.0338i −8.60019 + 8.60019i 1.30127 + 7.89346i − 17.2828i 12.6704 − 8.26614i
5.12 −1.62271 − 1.16910i 2.22257 − 2.22257i 1.26641 + 3.79423i −5.66290 −6.20500 + 1.00819i −3.14585 + 3.14585i 2.38082 − 7.63752i − 0.879603i 9.18927 + 6.62050i
5.13 −1.51544 − 1.30515i −2.47783 + 2.47783i 0.593143 + 3.95578i 0.784813 6.98897 − 0.521063i 4.57453 − 4.57453i 4.26403 − 6.76890i − 3.27932i −1.18934 − 1.02430i
5.14 −1.40145 − 1.42686i −2.87089 + 2.87089i −0.0718681 + 3.99935i −1.99784 8.11977 + 0.0729500i −7.19604 + 7.19604i 5.80724 − 5.50235i − 7.48402i 2.79988 + 2.85065i
5.15 −1.18151 + 1.61370i 1.00109 − 1.00109i −1.20807 − 3.81321i −7.45765 0.432662 + 2.79825i 5.34436 − 5.34436i 7.58073 + 2.55588i 6.99565i 8.81129 − 12.0344i
5.16 −1.17344 + 1.61958i 3.34060 − 3.34060i −1.24609 − 3.80096i 7.72510 1.49039 + 9.33035i 0.880944 − 0.880944i 7.61816 + 2.44204i − 13.3192i −9.06492 + 12.5114i
5.17 −1.15104 − 1.63558i 3.47874 − 3.47874i −1.35022 + 3.76522i 4.45957 −9.69390 − 1.68558i −4.26667 + 4.26667i 7.71247 − 2.12553i − 15.2032i −5.13313 − 7.29396i
5.18 −1.14903 + 1.63699i −0.0442643 + 0.0442643i −1.35944 − 3.76190i −2.90131 −0.0215989 − 0.123321i −3.74480 + 3.74480i 7.72023 + 2.09716i 8.99608i 3.33370 − 4.74940i
5.19 −1.10912 + 1.66429i −1.99044 + 1.99044i −1.53971 − 3.69179i 6.70140 −1.10503 − 5.52030i −2.98180 + 2.98180i 7.85192 + 1.53211i 1.07631i −7.43265 + 11.1531i
5.20 −1.02862 − 1.71521i 0.830821 − 0.830821i −1.88390 + 3.52859i −2.29930 −2.27963 − 0.570439i 9.22401 − 9.22401i 7.99008 − 0.398264i 7.61947i 2.36510 + 3.94379i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
208.r odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.3.r.a yes 108
13.d odd 4 1 208.3.m.a ✓ 108
16.e even 4 1 208.3.m.a ✓ 108
208.r odd 4 1 inner 208.3.r.a yes 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
208.3.m.a ✓ 108 13.d odd 4 1
208.3.m.a ✓ 108 16.e even 4 1
208.3.r.a yes 108 1.a even 1 1 trivial
208.3.r.a yes 108 208.r odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(208, [\chi])\).