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Properties

Label 209.2.f.c
Level $209$
Weight $2$
Character orbit 209.f
Analytic conductor $1.669$
Analytic rank $0$
Dimension $40$
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(20,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.20");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.f (of order \(5\), degree \(4\), 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: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 3 q^{2} + 3 q^{3} - 13 q^{4} - 9 q^{5} + 10 q^{6} + 4 q^{7} - q^{8} - 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 3 q^{2} + 3 q^{3} - 13 q^{4} - 9 q^{5} + 10 q^{6} + 4 q^{7} - q^{8} - 21 q^{9} - 22 q^{10} + 8 q^{11} + 20 q^{12} + 13 q^{13} - 20 q^{14} - 9 q^{15} - 41 q^{16} + 21 q^{17} + 27 q^{18} - 10 q^{19} - 26 q^{20} - 22 q^{21} + q^{22} + 16 q^{23} + 12 q^{24} - 53 q^{25} - 14 q^{26} - 24 q^{27} + 29 q^{28} + 11 q^{29} + 10 q^{30} - 13 q^{31} - 4 q^{32} + 34 q^{33} - 38 q^{34} + 25 q^{35} - 61 q^{36} - 7 q^{37} + 3 q^{38} - 16 q^{39} + 77 q^{40} + 17 q^{41} - 12 q^{42} - 24 q^{43} + 29 q^{44} + 130 q^{45} + 32 q^{46} - 5 q^{47} - 6 q^{48} - 42 q^{49} - 5 q^{50} + 40 q^{51} - 19 q^{52} - 30 q^{53} + 52 q^{54} + 72 q^{55} + 74 q^{56} - 7 q^{57} - 67 q^{58} - 23 q^{59} - 94 q^{60} + 36 q^{61} + 35 q^{62} - 7 q^{63} - 111 q^{64} - 26 q^{65} - 9 q^{66} + 52 q^{67} + 70 q^{68} - 95 q^{69} - 38 q^{70} - 34 q^{71} + 112 q^{72} + 31 q^{73} + 80 q^{74} - 84 q^{75} + 42 q^{76} - 25 q^{77} - 90 q^{78} + 32 q^{79} - 58 q^{80} - 28 q^{81} - 50 q^{82} - 83 q^{83} + 135 q^{84} - 23 q^{85} - 45 q^{86} - 42 q^{87} + 101 q^{88} + 96 q^{89} - 4 q^{90} + 33 q^{91} - 48 q^{92} - 91 q^{93} + 67 q^{94} - 9 q^{95} - 18 q^{96} - 10 q^{97} - 90 q^{98} + 110 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −0.786175 + 2.41960i 2.60226 − 1.89065i −3.61834 − 2.62888i 0.911935 + 2.80665i 2.52879 + 7.78281i −0.317594 − 0.230746i 5.08902 − 3.69739i 2.27014 − 6.98678i −7.50789
20.2 −0.698081 + 2.14847i −0.994134 + 0.722280i −2.51059 − 1.82405i 1.16743 + 3.59299i −0.857814 − 2.64008i 2.55703 + 1.85779i 2.01632 − 1.46494i −0.460438 + 1.41708i −8.53441
20.3 −0.426067 + 1.31130i 2.21002 − 1.60567i 0.0800598 + 0.0581668i −0.963581 − 2.96560i 1.16390 + 3.58212i 0.825452 + 0.599726i −2.34130 + 1.70106i 1.37895 − 4.24397i 4.29934
20.4 −0.396170 + 1.21929i −0.404723 + 0.294048i 0.288325 + 0.209481i −0.584556 − 1.79908i −0.198190 − 0.609966i 2.75941 + 2.00483i −2.44402 + 1.77568i −0.849715 + 2.61515i 2.42517
20.5 −0.0742224 + 0.228433i −2.40577 + 1.74790i 1.57136 + 1.14166i 1.12179 + 3.45250i −0.220715 − 0.679291i −2.77639 − 2.01716i −0.766056 + 0.556573i 1.80555 − 5.55692i −0.871926
20.6 0.132152 − 0.406723i −0.197749 + 0.143673i 1.47007 + 1.06807i 0.377090 + 1.16056i 0.0323022 + 0.0994159i −0.680611 − 0.494493i 1.32064 − 0.959502i −0.908588 + 2.79635i 0.521860
20.7 0.444533 − 1.36813i 0.733866 − 0.533185i −0.0561406 − 0.0407885i −0.895501 − 2.75607i −0.403240 − 1.24104i 1.78436 + 1.29641i 2.24684 − 1.63243i −0.672778 + 2.07060i −4.16874
20.8 0.538217 − 1.65646i 1.84021 − 1.33699i −0.836151 − 0.607499i 1.31640 + 4.05146i −1.22424 − 3.76782i −3.29292 − 2.39245i 1.36181 − 0.989414i 0.671776 − 2.06751i 7.41960
20.9 0.683719 − 2.10427i −2.29879 + 1.67017i −2.34245 − 1.70189i 0.681530 + 2.09753i 1.94276 + 5.97919i 2.56872 + 1.86628i −1.60280 + 1.16450i 1.56792 − 4.82555i 4.87975
20.10 0.773078 − 2.37929i −0.894206 + 0.649679i −3.44534 − 2.50319i −0.351383 − 1.08144i 0.854484 + 2.62983i −2.42745 − 1.76364i −4.57143 + 3.32134i −0.549529 + 1.69128i −2.84472
58.1 −2.24608 + 1.63187i 0.358792 + 1.10425i 1.76382 − 5.42849i −1.10965 − 0.806208i −2.60786 − 1.89472i 0.316566 − 0.974291i 3.18106 + 9.79028i 1.33642 − 0.970964i 3.80798
58.2 −1.28891 + 0.936447i 0.935127 + 2.87802i 0.166319 − 0.511876i −2.91599 − 2.11859i −3.90041 − 2.83381i 0.0705232 − 0.217048i −0.719663 − 2.21490i −4.98151 + 3.61928i 5.74239
58.3 −1.18972 + 0.864385i −0.639384 − 1.96782i 0.0502482 − 0.154648i 0.989602 + 0.718988i 2.46165 + 1.78849i −0.728381 + 2.24173i −0.834975 − 2.56979i −1.03646 + 0.753035i −1.79884
58.4 −0.419632 + 0.304881i −1.04377 − 3.21239i −0.534895 + 1.64624i 0.137996 + 0.100260i 1.41740 + 1.02980i 1.19515 − 3.67829i −0.598018 − 1.84051i −6.80296 + 4.94264i −0.0884751
58.5 −0.207740 + 0.150932i 0.859341 + 2.64478i −0.597658 + 1.83940i 1.97532 + 1.43516i −0.577702 − 0.419725i 0.905304 − 2.78624i −0.312167 − 0.960751i −3.82934 + 2.78218i −0.626965
58.6 0.569018 − 0.413416i −0.191933 − 0.590709i −0.465165 + 1.43163i 1.10498 + 0.802813i −0.353422 − 0.256776i 0.741499 − 2.28210i 0.761863 + 2.34477i 2.11495 − 1.53660i 0.960649
58.7 0.760694 − 0.552677i 0.608863 + 1.87389i −0.344830 + 1.06128i −2.22112 − 1.61374i 1.49881 + 1.08895i −1.54336 + 4.74997i 0.905352 + 2.78639i −0.713688 + 0.518524i −2.58147
58.8 1.19498 − 0.868205i −0.280785 − 0.864168i 0.0561669 − 0.172864i −3.34220 − 2.42825i −1.08581 − 0.788886i 1.03294 − 3.17906i 0.829922 + 2.55424i 1.75910 − 1.27806i −6.10209
58.9 2.02036 − 1.46787i 0.913910 + 2.81273i 1.30915 − 4.02914i −1.44878 − 1.05260i 5.97515 + 4.34120i 0.505551 − 1.55593i −1.72592 − 5.31183i −4.64915 + 3.37780i −4.47214
58.10 2.11605 − 1.53740i −0.211143 − 0.649832i 1.49603 − 4.60431i −0.451304 − 0.327891i −1.44584 − 1.05047i −1.49579 + 4.60357i −2.29647 − 7.06780i 2.04935 − 1.48894i −1.45908
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.f.c ✓ 40
11.c even 5 1 inner 209.2.f.c ✓ 40
11.c even 5 1 2299.2.a.x 20
11.d odd 10 1 2299.2.a.y 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.f.c ✓ 40 1.a even 1 1 trivial
209.2.f.c ✓ 40 11.c even 5 1 inner
2299.2.a.x 20 11.c even 5 1
2299.2.a.y 20 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 3 T_{2}^{39} + 21 T_{2}^{38} - 51 T_{2}^{37} + 247 T_{2}^{36} - 530 T_{2}^{35} + \cdots + 6400 \) acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display