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Properties

Label 209.3.q.a
Level $209$
Weight $3$
Character orbit 209.q
Analytic conductor $5.695$
Analytic rank $0$
Dimension $228$
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] = [209,3,Mod(43,209)]
 
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([9, 16]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 209.q (of order \(18\), 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: \(5.69483752513\)
Analytic rank: \(0\)
Dimension: \(228\)
Relative dimension: \(38\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 228 q - 12 q^{4} - 12 q^{5} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 228 q - 12 q^{4} - 12 q^{5} - 48 q^{9} + 12 q^{11} - 6 q^{12} - 132 q^{14} + 60 q^{15} - 36 q^{16} + 12 q^{20} - 150 q^{22} - 12 q^{23} + 252 q^{25} - 246 q^{26} + 54 q^{27} - 42 q^{31} - 114 q^{33} - 168 q^{34} + 318 q^{36} - 120 q^{37} + 24 q^{38} + 78 q^{42} + 207 q^{44} + 372 q^{45} + 240 q^{47} - 618 q^{48} + 246 q^{49} - 102 q^{53} + 168 q^{55} - 900 q^{56} - 192 q^{58} - 84 q^{59} + 294 q^{60} + 570 q^{64} - 282 q^{66} + 1080 q^{67} - 240 q^{69} - 102 q^{70} - 456 q^{71} - 24 q^{75} - 66 q^{77} - 204 q^{78} - 72 q^{80} + 726 q^{81} - 966 q^{82} - 258 q^{86} + 105 q^{88} - 858 q^{89} - 426 q^{91} + 1194 q^{92} - 702 q^{93} + 870 q^{97} - 1104 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.30386 − 3.58233i 0.390427 − 2.21422i −8.06887 + 6.77059i 6.17904 + 5.18483i −8.44115 + 1.48840i −7.56606 + 4.36827i 21.5692 + 12.4530i 3.70688 + 1.34920i 10.5172 − 28.8957i
43.2 −1.24995 − 3.43421i 0.955242 − 5.41745i −7.16723 + 6.01402i −4.53792 − 3.80777i −19.7986 + 3.49103i 2.84635 − 1.64334i 16.9521 + 9.78732i −19.9790 − 7.27176i −7.40449 + 20.3437i
43.3 −1.24105 − 3.40976i −0.103545 + 0.587231i −7.02210 + 5.89224i 0.554895 + 0.465612i 2.13082 − 0.375721i 8.07990 − 4.66493i 16.2361 + 9.37394i 8.12312 + 2.95657i 0.898973 − 2.46991i
43.4 −1.18587 − 3.25816i −0.469616 + 2.66333i −6.14511 + 5.15636i −7.25331 − 6.08625i 9.23444 − 1.62828i −0.294363 + 0.169951i 12.0766 + 6.97244i 1.58447 + 0.576700i −11.2285 + 30.8499i
43.5 −1.18571 − 3.25771i −0.809669 + 4.59186i −6.14260 + 5.15425i 2.12176 + 1.78037i 15.9190 − 2.80695i −1.67598 + 0.967630i 12.0651 + 6.96579i −11.9724 − 4.35759i 3.28413 − 9.02308i
43.6 −1.01387 − 2.78558i 0.526510 − 2.98599i −3.66734 + 3.07727i −0.671626 − 0.563561i −8.85152 + 1.56076i −1.60918 + 0.929062i 2.02136 + 1.16703i −0.181676 − 0.0661246i −0.888905 + 2.44225i
43.7 −0.922961 − 2.53581i 0.210178 − 1.19198i −2.51432 + 2.10977i −4.25825 − 3.57309i −3.21663 + 0.567178i −9.19804 + 5.31049i −1.67748 − 0.968494i 7.08059 + 2.57712i −5.13051 + 14.0959i
43.8 −0.853961 − 2.34624i 0.808439 − 4.58489i −1.71141 + 1.43605i 7.36947 + 6.18372i −11.4476 + 2.01852i 7.65229 − 4.41805i −3.81844 − 2.20458i −11.9104 − 4.33503i 8.21524 − 22.5712i
43.9 −0.824807 − 2.26614i −0.147389 + 0.835885i −1.39090 + 1.16711i 3.14482 + 2.63882i 2.01580 − 0.355440i −4.73738 + 2.73513i −4.56189 − 2.63381i 7.78025 + 2.83178i 3.38606 − 9.30313i
43.10 −0.775174 − 2.12977i 0.0687643 − 0.389982i −0.870856 + 0.730735i −0.730277 − 0.612775i −0.883876 + 0.155851i 10.8763 − 6.27943i −5.61987 − 3.24463i 8.30988 + 3.02455i −0.738980 + 2.03033i
43.11 −0.771794 − 2.12049i −0.625216 + 3.54577i −0.836616 + 0.702004i 3.53330 + 2.96479i 8.00130 − 1.41085i 0.511282 − 0.295189i −5.68272 − 3.28092i −3.72439 − 1.35557i 3.55981 − 9.78051i
43.12 −0.643083 − 1.76686i −1.02006 + 5.78506i 0.355949 − 0.298677i −4.22201 − 3.54268i 10.8774 − 1.91797i 2.41735 − 1.39566i −7.27000 − 4.19734i −23.9691 − 8.72406i −3.54431 + 9.73792i
43.13 −0.470507 − 1.29271i 0.960003 − 5.44445i 1.61446 − 1.35470i 1.44259 + 1.21047i −7.48976 + 1.32065i −9.06223 + 5.23208i −7.27630 − 4.20097i −20.2632 − 7.37519i 0.886041 − 2.43438i
43.14 −0.414279 − 1.13822i 0.561253 − 3.18302i 1.94025 − 1.62807i −6.91003 − 5.79820i −3.85551 + 0.679830i 6.82669 − 3.94139i −6.85288 − 3.95651i −1.35939 − 0.494779i −3.73697 + 10.2672i
43.15 −0.357612 − 0.982532i 0.654631 − 3.71260i 2.22670 − 1.86842i −0.897867 − 0.753400i −3.88185 + 0.684475i 0.248107 − 0.143245i −6.25410 − 3.61081i −4.89760 − 1.78258i −0.419151 + 1.15161i
43.16 −0.351491 − 0.965713i −0.527752 + 2.99303i 2.25512 − 1.89227i −2.85907 − 2.39905i 3.07591 − 0.542365i −8.10717 + 4.68067i −6.18007 − 3.56807i −0.222461 − 0.0809691i −1.31185 + 3.60429i
43.17 −0.232784 − 0.639569i 0.203055 − 1.15158i 2.70932 − 2.27339i 4.25811 + 3.57298i −0.783785 + 0.138202i 1.82930 − 1.05615i −4.44239 − 2.56482i 7.17232 + 2.61051i 1.29394 − 3.55508i
43.18 −0.230833 − 0.634209i −0.281936 + 1.59894i 2.71524 − 2.27836i −3.57129 − 2.99667i 1.07914 − 0.190281i 2.77404 − 1.60159i −4.40968 − 2.54593i 5.98012 + 2.17659i −1.07614 + 2.95668i
43.19 −0.120508 − 0.331094i −0.760924 + 4.31541i 2.96908 − 2.49135i 5.36799 + 4.50428i 1.52051 − 0.268106i 11.1842 − 6.45718i −2.40322 − 1.38750i −9.58656 − 3.48922i 0.844452 − 2.32011i
43.20 0.120508 + 0.331094i −0.760924 + 4.31541i 2.96908 − 2.49135i 5.36799 + 4.50428i −1.52051 + 0.268106i −11.1842 + 6.45718i 2.40322 + 1.38750i −9.58656 − 3.48922i −0.844452 + 2.32011i
See next 80 embeddings (of 228 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
19.e even 9 1 inner
209.q odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.3.q.a ✓ 228
11.b odd 2 1 inner 209.3.q.a ✓ 228
19.e even 9 1 inner 209.3.q.a ✓ 228
209.q odd 18 1 inner 209.3.q.a ✓ 228
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.3.q.a ✓ 228 1.a even 1 1 trivial
209.3.q.a ✓ 228 11.b odd 2 1 inner
209.3.q.a ✓ 228 19.e even 9 1 inner
209.3.q.a ✓ 228 209.q odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(209, [\chi])\).