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Properties

Label 209.2.j.c
Level $209$
Weight $2$
Character orbit 209.j
Analytic conductor $1.669$
Analytic rank $0$
Dimension $48$
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: \(48\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{4} - 36 q^{6} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{4} - 36 q^{6} + 9 q^{8} - 15 q^{10} + 24 q^{11} + 12 q^{12} - 3 q^{14} - 21 q^{15} + 30 q^{16} - 3 q^{17} - 54 q^{18} + 12 q^{19} + 3 q^{21} - 48 q^{23} + 33 q^{24} - 30 q^{25} + 12 q^{27} + 15 q^{28} + 9 q^{29} - 6 q^{30} + 27 q^{31} - 6 q^{34} + 9 q^{35} + 6 q^{36} - 96 q^{37} + 21 q^{38} - 30 q^{40} + 18 q^{41} - 60 q^{42} + 24 q^{43} - 12 q^{44} + 9 q^{45} + 54 q^{46} - 15 q^{47} + 90 q^{48} - 18 q^{49} + 30 q^{50} - 69 q^{51} + 69 q^{52} + 30 q^{53} + 3 q^{57} - 132 q^{58} - 15 q^{59} + 75 q^{60} - 42 q^{61} + 39 q^{62} - 75 q^{63} + 3 q^{64} + 3 q^{65} - 9 q^{66} + 33 q^{67} - 48 q^{68} + 72 q^{69} + 30 q^{70} + 87 q^{71} + 48 q^{72} - 36 q^{73} - 9 q^{74} - 126 q^{75} + 21 q^{76} - 57 q^{78} + 51 q^{79} - 117 q^{80} + 60 q^{81} - 24 q^{82} + 6 q^{83} + 54 q^{84} - 9 q^{85} + 66 q^{86} + 39 q^{87} - 9 q^{88} - 24 q^{89} + 99 q^{90} + 60 q^{91} - 75 q^{92} - 6 q^{93} - 108 q^{94} + 42 q^{95} - 246 q^{96} + 33 q^{97} - 39 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −2.48446 − 0.904268i −0.198191 − 1.12400i 3.82273 + 3.20765i −1.47046 + 1.23386i −0.523997 + 2.97173i 0.903436 − 1.56480i −3.95293 − 6.84667i 1.59499 − 0.580530i 4.76904 − 1.73579i
23.2 −1.53574 − 0.558963i −0.0856969 − 0.486011i 0.513960 + 0.431264i −0.273084 + 0.229145i −0.140054 + 0.794287i −2.27250 + 3.93608i 1.08605 + 1.88109i 2.59021 − 0.942761i 0.547469 − 0.199262i
23.3 −0.902342 − 0.328426i 0.466687 + 2.64671i −0.825731 − 0.692870i 1.99678 − 1.67550i 0.448137 − 2.54151i 2.18176 − 3.77893i 1.47779 + 2.55960i −3.96821 + 1.44431i −2.35206 + 0.856080i
23.4 −0.268629 − 0.0977731i −0.489784 − 2.77770i −1.46949 − 1.23305i 1.80064 − 1.51092i −0.140014 + 0.794061i −0.216960 + 0.375785i 0.560058 + 0.970049i −4.65667 + 1.69489i −0.631433 + 0.229823i
23.5 0.456501 + 0.166153i 0.423835 + 2.40369i −1.35130 − 1.13388i −1.55871 + 1.30791i −0.205898 + 1.16771i −1.39807 + 2.42152i −0.914272 − 1.58357i −2.77900 + 1.01147i −0.928864 + 0.338079i
23.6 1.28226 + 0.466703i 0.0764514 + 0.433577i −0.105719 − 0.0887087i 2.67296 − 2.24288i −0.104322 + 0.591638i −0.946420 + 1.63925i −1.45871 − 2.52655i 2.63693 − 0.959765i 4.47417 − 1.62847i
23.7 2.13434 + 0.776837i 0.114559 + 0.649697i 2.41985 + 2.03050i −3.35396 + 2.81431i −0.260200 + 1.47567i 1.56295 − 2.70710i 1.31611 + 2.27956i 2.41010 − 0.877203i −9.34475 + 3.40121i
23.8 2.25776 + 0.821757i −0.481509 − 2.73077i 2.89010 + 2.42508i 0.185826 − 0.155927i 1.15690 − 6.56111i −0.753892 + 1.30578i 2.12966 + 3.68868i −4.40619 + 1.60372i 0.547685 − 0.199341i
100.1 −2.48446 + 0.904268i −0.198191 + 1.12400i 3.82273 − 3.20765i −1.47046 − 1.23386i −0.523997 − 2.97173i 0.903436 + 1.56480i −3.95293 + 6.84667i 1.59499 + 0.580530i 4.76904 + 1.73579i
100.2 −1.53574 + 0.558963i −0.0856969 + 0.486011i 0.513960 − 0.431264i −0.273084 − 0.229145i −0.140054 − 0.794287i −2.27250 − 3.93608i 1.08605 − 1.88109i 2.59021 + 0.942761i 0.547469 + 0.199262i
100.3 −0.902342 + 0.328426i 0.466687 − 2.64671i −0.825731 + 0.692870i 1.99678 + 1.67550i 0.448137 + 2.54151i 2.18176 + 3.77893i 1.47779 − 2.55960i −3.96821 − 1.44431i −2.35206 − 0.856080i
100.4 −0.268629 + 0.0977731i −0.489784 + 2.77770i −1.46949 + 1.23305i 1.80064 + 1.51092i −0.140014 − 0.794061i −0.216960 − 0.375785i 0.560058 − 0.970049i −4.65667 − 1.69489i −0.631433 − 0.229823i
100.5 0.456501 − 0.166153i 0.423835 − 2.40369i −1.35130 + 1.13388i −1.55871 − 1.30791i −0.205898 − 1.16771i −1.39807 − 2.42152i −0.914272 + 1.58357i −2.77900 − 1.01147i −0.928864 − 0.338079i
100.6 1.28226 − 0.466703i 0.0764514 − 0.433577i −0.105719 + 0.0887087i 2.67296 + 2.24288i −0.104322 − 0.591638i −0.946420 − 1.63925i −1.45871 + 2.52655i 2.63693 + 0.959765i 4.47417 + 1.62847i
100.7 2.13434 − 0.776837i 0.114559 − 0.649697i 2.41985 − 2.03050i −3.35396 − 2.81431i −0.260200 − 1.47567i 1.56295 + 2.70710i 1.31611 − 2.27956i 2.41010 + 0.877203i −9.34475 − 3.40121i
100.8 2.25776 − 0.821757i −0.481509 + 2.73077i 2.89010 − 2.42508i 0.185826 + 0.155927i 1.15690 + 6.56111i −0.753892 − 1.30578i 2.12966 − 3.68868i −4.40619 − 1.60372i 0.547685 + 0.199341i
111.1 −1.72255 − 1.44539i 2.01001 − 0.731585i 0.530725 + 3.00989i 0.706666 − 4.00770i −4.51977 − 1.64506i 0.415979 + 0.720498i 1.18764 − 2.05705i 1.20680 − 1.01263i −7.00996 + 5.88206i
111.2 −0.978552 − 0.821102i 0.838053 − 0.305026i −0.0639420 − 0.362633i −0.608001 + 3.44815i −1.07054 − 0.389643i 2.22321 + 3.85071i −1.51260 + 2.61989i −1.68884 + 1.41711i 3.42624 − 2.87496i
111.3 −0.925337 − 0.776450i 3.04091 − 1.10680i −0.0939224 − 0.532660i −0.404489 + 2.29397i −3.67324 − 1.33695i −2.44265 − 4.23079i −1.53462 + 2.65803i 5.72397 − 4.80298i 2.15544 − 1.80863i
111.4 −0.728147 − 0.610988i −1.24034 + 0.451446i −0.190404 − 1.07984i 0.399689 − 2.26675i 1.17898 + 0.429112i 0.119508 + 0.206994i −1.47165 + 2.54898i −0.963502 + 0.808474i −1.67599 + 1.40632i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.8
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.c ✓ 48
19.e even 9 1 inner 209.2.j.c ✓ 48
19.e even 9 1 3971.2.a.u 24
19.f odd 18 1 3971.2.a.w 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.j.c ✓ 48 1.a even 1 1 trivial
209.2.j.c ✓ 48 19.e even 9 1 inner
3971.2.a.u 24 19.e even 9 1
3971.2.a.w 24 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 3 T_{2}^{46} - 7 T_{2}^{45} - 21 T_{2}^{44} + 51 T_{2}^{43} + 263 T_{2}^{42} - 78 T_{2}^{41} + \cdots + 3249 \) acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display