Properties

Label 209.2.t.a
Level $209$
Weight $2$
Character orbit 209.t
Analytic conductor $1.669$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [209,2,Mod(8,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([9, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.t (of order \(30\), 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: \(1.66887340224\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 15 q^{2} - 3 q^{3} + 13 q^{4} - q^{5} - 10 q^{6} - 10 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 15 q^{2} - 3 q^{3} + 13 q^{4} - q^{5} - 10 q^{6} - 10 q^{7} - 11 q^{9} - 12 q^{11} - 15 q^{13} - 3 q^{14} + 3 q^{15} + 13 q^{16} + 5 q^{19} - 92 q^{20} - 15 q^{22} + 20 q^{24} + 23 q^{25} - 62 q^{26} - 5 q^{28} + 60 q^{29} - 50 q^{30} - 60 q^{33} - 102 q^{34} - 25 q^{35} - q^{36} - q^{38} + 10 q^{39} + 90 q^{40} + 75 q^{41} + 60 q^{42} - 8 q^{44} - 36 q^{45} - 5 q^{47} - 126 q^{48} + 30 q^{49} - 90 q^{52} - 18 q^{53} + 29 q^{55} - 125 q^{57} + 28 q^{58} + 24 q^{59} + 114 q^{60} + 10 q^{61} - 35 q^{62} - 120 q^{64} + 13 q^{66} + 30 q^{67} - 20 q^{68} + 174 q^{70} - 27 q^{71} + 75 q^{72} + 15 q^{73} - 55 q^{74} - 44 q^{77} + 18 q^{78} - 15 q^{79} - 89 q^{80} + 26 q^{81} + 76 q^{82} - 100 q^{83} - 70 q^{85} - 33 q^{86} + 102 q^{89} - 165 q^{90} - 33 q^{91} + 18 q^{92} - 79 q^{93} + 150 q^{95} + 430 q^{96} - 54 q^{97} + 11 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −0.294203 2.79916i −0.462350 + 2.17518i −5.79244 + 1.23122i −0.435401 + 0.193853i 6.22471 + 0.654243i 2.45552 + 0.797848i 3.41103 + 10.4981i −1.77702 0.791182i 0.670722 + 1.16172i
8.2 −0.245338 2.33423i −0.0272523 + 0.128212i −3.43217 + 0.729530i 3.51782 1.56623i 0.305963 + 0.0321580i −4.02135 1.30662i 1.09435 + 3.36807i 2.72494 + 1.21322i −4.51901 7.82716i
8.3 −0.243021 2.31219i 0.586000 2.75691i −3.33085 + 0.707994i 1.05809 0.471092i −6.51691 0.684955i 1.61851 + 0.525884i 1.00960 + 3.10723i −4.51654 2.01089i −1.34639 2.33202i
8.4 −0.178386 1.69723i −0.334130 + 1.57196i −0.892457 + 0.189698i −0.0430015 + 0.0191455i 2.72757 + 0.286679i 1.13464 + 0.368667i −0.573560 1.76524i 0.381225 + 0.169732i 0.0401650 + 0.0695679i
8.5 −0.175935 1.67391i −0.355086 + 1.67055i −0.814734 + 0.173177i −3.37205 + 1.50133i 2.85883 + 0.300475i −3.69486 1.20053i −0.607010 1.86818i 0.0759839 + 0.0338302i 3.10636 + 5.38037i
8.6 −0.147836 1.40656i 0.0962713 0.452921i −0.000272568 0 5.79361e-5i −0.206887 + 0.0921118i −0.651295 0.0684538i 4.81246 + 1.56366i −0.873971 2.68981i 2.54477 + 1.13300i 0.160146 + 0.277382i
8.7 −0.106719 1.01536i 0.290180 1.36519i 0.936721 0.199106i 1.09870 0.489171i −1.41713 0.148947i −2.19688 0.713808i −0.933117 2.87184i 0.961096 + 0.427908i −0.613939 1.06337i
8.8 −0.0933513 0.888178i 0.578920 2.72360i 1.17615 0.249998i −3.51240 + 1.56382i −2.47309 0.259932i −0.765265 0.248650i −0.883786 2.72001i −4.34223 1.93328i 1.71684 + 2.97365i
8.9 −0.0555175 0.528214i −0.555682 + 2.61428i 1.68037 0.357173i 2.67341 1.19028i 1.41175 + 0.148381i −0.703891 0.228708i −0.610206 1.87802i −3.78502 1.68520i −0.777142 1.34605i
8.10 0.0208920 + 0.198774i −0.647125 + 3.04448i 1.91722 0.407518i −2.79053 + 1.24242i −0.618682 0.0650261i 3.11019 + 1.01056i 0.244584 + 0.752752i −6.10946 2.72011i −0.305261 0.528727i
8.11 0.0303126 + 0.288405i 0.0560461 0.263676i 1.87404 0.398339i −0.419601 + 0.186819i 0.0777445 + 0.00817127i −0.640936 0.208253i 0.350916 + 1.08001i 2.67425 + 1.19065i −0.0665986 0.115352i
8.12 0.0929138 + 0.884016i 0.0122736 0.0577427i 1.18344 0.251549i −2.53808 + 1.13002i 0.0521859 + 0.00548496i 0.760554 + 0.247119i 0.881693 + 2.71357i 2.73745 + 1.21879i −1.23478 2.13870i
8.13 0.105737 + 1.00602i 0.521935 2.45551i 0.955395 0.203075i 1.56886 0.698500i 2.52549 + 0.265440i −2.67054 0.867712i 0.930500 + 2.86378i −3.01649 1.34303i 0.868593 + 1.50445i
8.14 0.164885 + 1.56878i −0.307100 + 1.44479i −0.477587 + 0.101514i 2.63545 1.17338i −2.31719 0.243547i 0.651771 + 0.211773i 0.736899 + 2.26794i 0.747527 + 0.332820i 2.27531 + 3.94096i
8.15 0.180409 + 1.71647i −0.535718 + 2.52035i −0.957438 + 0.203510i −0.0762414 + 0.0339449i −4.42277 0.464852i −3.77824 1.22762i 0.544633 + 1.67621i −3.32456 1.48019i −0.0720200 0.124742i
8.16 0.212230 + 2.01923i 0.464015 2.18302i −2.07595 + 0.441257i −1.43231 + 0.637705i 4.50649 + 0.473651i 4.01889 + 1.30582i −0.0767508 0.236215i −1.80962 0.805694i −1.59165 2.75682i
8.17 0.252914 + 2.40631i 0.318632 1.49905i −3.77008 + 0.801356i 2.87604 1.28050i 3.68776 + 0.387599i 0.551658 + 0.179245i −1.38644 4.26704i 0.595024 + 0.264922i 3.80866 + 6.59680i
8.18 0.267178 + 2.54203i −0.0599442 + 0.282015i −4.43425 + 0.942528i −1.51541 + 0.674705i −0.732907 0.0770316i −2.45125 0.796461i −2.00095 6.15830i 2.66470 + 1.18640i −2.12001 3.67196i
46.1 −1.80191 2.00123i 2.39485 0.251709i −0.548964 + 5.22304i 2.42856 + 0.516207i −4.81905 4.33909i −2.04537 + 2.81521i 7.08445 5.14716i 2.73753 0.581880i −3.34301 5.79027i
46.2 −1.69399 1.88137i −1.14305 + 0.120139i −0.460883 + 4.38501i 1.39034 + 0.295526i 2.16235 + 1.94699i 2.22157 3.05773i 4.93430 3.58498i −1.64231 + 0.349084i −1.79924 3.11637i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
19.d odd 6 1 inner
209.t even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.t.a 144
11.d odd 10 1 inner 209.2.t.a 144
19.d odd 6 1 inner 209.2.t.a 144
209.t even 30 1 inner 209.2.t.a 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.t.a 144 1.a even 1 1 trivial
209.2.t.a 144 11.d odd 10 1 inner
209.2.t.a 144 19.d odd 6 1 inner
209.2.t.a 144 209.t even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(209, [\chi])\).