Properties

Label 209.2.j.b
Level $209$
Weight $2$
Character orbit 209.j
Analytic conductor $1.669$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [209,2,Mod(23,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.j (of order \(9\), degree \(6\), 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: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{7} - 15 q^{8} - 15 q^{10} - 21 q^{11} - 30 q^{12} - 18 q^{13} - 3 q^{14} - 9 q^{15} - 3 q^{16} + 12 q^{18} - 6 q^{19} + 30 q^{20} + 9 q^{21} + 6 q^{22} + 15 q^{23} + 3 q^{24} - 6 q^{26} - 24 q^{27} - 9 q^{28} - 15 q^{29} + 30 q^{30} - 54 q^{31} + 42 q^{32} - 45 q^{34} - 12 q^{35} + 6 q^{36} + 114 q^{37} - 60 q^{38} - 48 q^{40} + 15 q^{41} + 84 q^{42} - 6 q^{43} + 3 q^{44} + 3 q^{45} - 15 q^{46} - 42 q^{47} + 12 q^{48} - 21 q^{49} - 33 q^{50} + 93 q^{51} + 3 q^{52} + 48 q^{54} + 3 q^{55} + 114 q^{56} - 15 q^{57} + 60 q^{58} - 33 q^{59} - 81 q^{60} + 63 q^{61} - 24 q^{62} + 87 q^{63} + 15 q^{64} + 24 q^{65} - 15 q^{66} - 51 q^{67} + 48 q^{68} - 54 q^{69} - 84 q^{70} + 9 q^{71} - 51 q^{72} - 27 q^{73} - 33 q^{74} + 54 q^{75} - 60 q^{76} + 12 q^{77} - 87 q^{78} + 24 q^{79} + 120 q^{80} - 36 q^{81} + 111 q^{82} - 12 q^{83} + 48 q^{84} + 39 q^{85} - 60 q^{86} - 75 q^{87} - 15 q^{88} - 21 q^{89} - 63 q^{90} - 3 q^{91} + 6 q^{92} - 72 q^{93} + 84 q^{94} - 72 q^{95} - 6 q^{96} - 18 q^{97} - 120 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −2.38547 0.868238i 0.291201 + 1.65148i 3.40452 + 2.85673i 2.29004 1.92157i 0.739230 4.19238i −0.775424 + 1.34307i −3.10248 5.37365i 0.176489 0.0642366i −7.13118 + 2.59554i
23.2 −1.96733 0.716050i 0.502296 + 2.84866i 1.82557 + 1.53184i −2.76559 + 2.32060i 1.05160 5.96393i −0.132423 + 0.229364i −0.401042 0.694626i −5.04350 + 1.83568i 7.10249 2.58510i
23.3 −1.69844 0.618183i −0.255473 1.44886i 0.970468 + 0.814320i 1.59794 1.34083i −0.461753 + 2.61873i 1.29928 2.25041i 0.662558 + 1.14758i 0.785155 0.285773i −3.54289 + 1.28951i
23.4 −0.0793104 0.0288666i 0.115804 + 0.656759i −1.52663 1.28100i 1.81015 1.51890i 0.00977392 0.0554307i −0.266530 + 0.461644i 0.168500 + 0.291851i 2.40116 0.873949i −0.187409 + 0.0682115i
23.5 0.248801 + 0.0905562i −0.369765 2.09704i −1.47839 1.24051i −2.83976 + 2.38284i 0.0979021 0.555231i −1.58658 + 2.74803i −0.520257 0.901111i −1.44178 + 0.524764i −0.922316 + 0.335696i
23.6 1.68033 + 0.611589i 0.505415 + 2.86635i 0.917370 + 0.769764i 0.525977 0.441347i −0.903766 + 5.12551i 1.17848 2.04120i −0.717465 1.24269i −5.14143 + 1.87133i 1.15374 0.419925i
23.7 1.99568 + 0.726369i 0.0787632 + 0.446688i 1.92305 + 1.61363i −0.260317 + 0.218432i −0.167274 + 0.948659i −1.83015 + 3.16990i 0.541945 + 0.938677i 2.62575 0.955695i −0.678172 + 0.246834i
100.1 −2.38547 + 0.868238i 0.291201 1.65148i 3.40452 2.85673i 2.29004 + 1.92157i 0.739230 + 4.19238i −0.775424 1.34307i −3.10248 + 5.37365i 0.176489 + 0.0642366i −7.13118 2.59554i
100.2 −1.96733 + 0.716050i 0.502296 2.84866i 1.82557 1.53184i −2.76559 2.32060i 1.05160 + 5.96393i −0.132423 0.229364i −0.401042 + 0.694626i −5.04350 1.83568i 7.10249 + 2.58510i
100.3 −1.69844 + 0.618183i −0.255473 + 1.44886i 0.970468 0.814320i 1.59794 + 1.34083i −0.461753 2.61873i 1.29928 + 2.25041i 0.662558 1.14758i 0.785155 + 0.285773i −3.54289 1.28951i
100.4 −0.0793104 + 0.0288666i 0.115804 0.656759i −1.52663 + 1.28100i 1.81015 + 1.51890i 0.00977392 + 0.0554307i −0.266530 0.461644i 0.168500 0.291851i 2.40116 + 0.873949i −0.187409 0.0682115i
100.5 0.248801 0.0905562i −0.369765 + 2.09704i −1.47839 + 1.24051i −2.83976 2.38284i 0.0979021 + 0.555231i −1.58658 2.74803i −0.520257 + 0.901111i −1.44178 0.524764i −0.922316 0.335696i
100.6 1.68033 0.611589i 0.505415 2.86635i 0.917370 0.769764i 0.525977 + 0.441347i −0.903766 5.12551i 1.17848 + 2.04120i −0.717465 + 1.24269i −5.14143 1.87133i 1.15374 + 0.419925i
100.7 1.99568 0.726369i 0.0787632 0.446688i 1.92305 1.61363i −0.260317 0.218432i −0.167274 0.948659i −1.83015 3.16990i 0.541945 0.938677i 2.62575 + 0.955695i −0.678172 0.246834i
111.1 −1.96241 1.64666i −2.67281 + 0.972823i 0.792280 + 4.49324i 0.366121 2.07637i 6.84707 + 2.49213i 1.57957 + 2.73590i 3.28232 5.68514i 3.89939 3.27198i −4.13756 + 3.47183i
111.2 −1.28771 1.08051i −1.02863 + 0.374390i 0.143380 + 0.813151i −0.0882338 + 0.500399i 1.72910 + 0.629342i 0.0395283 + 0.0684651i −0.986992 + 1.70952i −1.38023 + 1.15815i 0.654306 0.549028i
111.3 −0.368271 0.309016i 2.42559 0.882842i −0.307164 1.74201i 0.229238 1.30007i −1.16609 0.424421i 1.88709 + 3.26854i −0.905934 + 1.56912i 2.80593 2.35446i −0.486166 + 0.407942i
111.4 −0.0229450 0.0192531i −1.38899 + 0.505551i −0.347141 1.96873i −0.508658 + 2.88474i 0.0416038 + 0.0151425i −2.21460 3.83580i −0.0598916 + 0.103735i −0.624422 + 0.523953i 0.0672114 0.0563971i
111.5 0.938345 + 0.787365i −1.27599 + 0.464422i −0.0867486 0.491976i −0.392586 + 2.22647i −1.56299 0.568881i 2.23878 + 3.87769i 1.53089 2.65157i −0.885675 + 0.743170i −2.12142 + 1.78009i
111.6 0.948012 + 0.795477i −2.09808 + 0.763637i −0.0813525 0.461373i 0.604339 3.42738i −2.59646 0.945033i −0.971037 1.68188i 1.52743 2.64559i 1.52065 1.27598i 3.29932 2.76846i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.j.b 42
19.e even 9 1 inner 209.2.j.b 42
19.e even 9 1 3971.2.a.t 21
19.f odd 18 1 3971.2.a.s 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.j.b 42 1.a even 1 1 trivial
209.2.j.b 42 19.e even 9 1 inner
3971.2.a.s 21 19.f odd 18 1
3971.2.a.t 21 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{42} + 3 T_{2}^{41} + 3 T_{2}^{40} + 13 T_{2}^{39} + 33 T_{2}^{38} - 54 T_{2}^{37} + 98 T_{2}^{36} + 960 T_{2}^{35} + 435 T_{2}^{34} - 1098 T_{2}^{33} + 4728 T_{2}^{32} - 7194 T_{2}^{31} - 11964 T_{2}^{30} + 101247 T_{2}^{29} + 174462 T_{2}^{28} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display