Properties

Label 209.2.n.a
Level $209$
Weight $2$
Character orbit 209.n
Analytic conductor $1.669$
Analytic rank $0$
Dimension $144$
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] = [209,2,Mod(26,209)]
 
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([6, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.n (of order \(15\), 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 \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 5 q^{2} - q^{3} + 13 q^{4} - 5 q^{5} - 14 q^{6} - 18 q^{7} - 8 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 5 q^{2} - q^{3} + 13 q^{4} - 5 q^{5} - 14 q^{6} - 18 q^{7} - 8 q^{8} + 17 q^{9} - 20 q^{11} - 64 q^{12} - 15 q^{13} - 13 q^{14} - 11 q^{15} + 13 q^{16} - 18 q^{17} - 14 q^{18} - 5 q^{19} - 60 q^{20} + 30 q^{21} - 3 q^{22} - 44 q^{24} - 13 q^{25} + 54 q^{26} - 10 q^{27} - 11 q^{28} - 30 q^{29} + 30 q^{30} - 4 q^{31} + 38 q^{32} - 16 q^{33} - 34 q^{34} + 21 q^{35} - 3 q^{36} - 12 q^{37} + 45 q^{38} - 6 q^{39} + 12 q^{40} - 23 q^{41} - 42 q^{42} - 46 q^{43} - 28 q^{45} + 8 q^{46} + 11 q^{47} - 46 q^{48} - 10 q^{49} + 148 q^{50} - 20 q^{51} + 22 q^{52} + 54 q^{54} - 5 q^{55} + 104 q^{56} - 5 q^{57} + 52 q^{58} - 4 q^{59} - 36 q^{60} + 32 q^{61} - 41 q^{62} + 24 q^{63} + 24 q^{64} - 184 q^{65} - 55 q^{66} + 6 q^{67} + 24 q^{68} + 112 q^{69} - 68 q^{70} - 5 q^{71} - 73 q^{72} + 23 q^{73} + 3 q^{74} + 30 q^{75} - 88 q^{76} + 60 q^{77} - 114 q^{78} - 5 q^{79} - 9 q^{80} - 16 q^{81} + 42 q^{82} + 100 q^{83} - 16 q^{84} - 80 q^{85} + 63 q^{86} + 164 q^{87} + 52 q^{88} - 26 q^{89} + 9 q^{90} + 33 q^{91} - 32 q^{92} + 73 q^{93} - 228 q^{94} - 62 q^{95} - 170 q^{96} - 22 q^{97} + 120 q^{98} - 45 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −0.271157 + 2.57989i 2.55309 0.542675i −4.62600 0.983287i −0.178862 0.0796343i 0.707754 + 6.73383i 0.955739 + 2.94146i 2.18790 6.73367i 3.48311 1.55078i 0.253947 0.439850i
26.2 −0.231976 + 2.20710i −0.373871 + 0.0794688i −2.86120 0.608166i −2.71951 1.21080i −0.0886667 0.843607i −0.383306 1.17970i 0.634435 1.95259i −2.60717 + 1.16079i 3.30323 5.72135i
26.3 −0.202973 + 1.93115i −0.843003 + 0.179186i −1.73186 0.368119i 2.72243 + 1.21210i −0.174929 1.66434i −0.0464276 0.142890i −0.137677 + 0.423727i −2.06209 + 0.918101i −2.89333 + 5.01140i
26.4 −0.186518 + 1.77460i 1.74061 0.369977i −1.15814 0.246170i 1.27150 + 0.566107i 0.331908 + 3.15790i −0.603916 1.85866i −0.449939 + 1.38477i 0.152194 0.0677611i −1.24177 + 2.15081i
26.5 −0.125995 + 1.19876i −2.49184 + 0.529658i 0.535145 + 0.113749i 1.68116 + 0.748502i −0.320973 3.05386i 0.653590 + 2.01154i −0.948738 + 2.91991i 3.18811 1.41944i −1.10909 + 1.92100i
26.6 −0.0795584 + 0.756948i −2.48657 + 0.528537i 1.38965 + 0.295380i −2.02530 0.901724i −0.202247 1.92425i −1.58873 4.88961i −0.804543 + 2.47613i 3.16305 1.40828i 0.843688 1.46131i
26.7 −0.0646268 + 0.614883i 1.13244 0.240707i 1.58239 + 0.336348i −1.68382 0.749683i 0.0748209 + 0.711874i 1.36446 + 4.19938i −0.691191 + 2.12727i −1.51616 + 0.675037i 0.569787 0.986900i
26.8 −0.0551358 + 0.524582i 3.09393 0.657636i 1.68415 + 0.357977i −3.18113 1.41633i 0.174397 + 1.65928i −0.546190 1.68100i −0.606640 + 1.86705i 6.39930 2.84915i 0.918374 1.59067i
26.9 −0.00441207 + 0.0419780i 0.931314 0.197957i 1.95455 + 0.415453i 1.90793 + 0.849463i 0.00420082 + 0.0399681i −0.431752 1.32880i −0.0521503 + 0.160502i −1.91248 + 0.851490i −0.0440767 + 0.0763431i
26.10 0.0181387 0.172578i −1.78742 + 0.379928i 1.92684 + 0.409563i 0.351649 + 0.156564i 0.0331458 + 0.315361i 0.447117 + 1.37609i 0.212878 0.655172i 0.309894 0.137973i 0.0333980 0.0578471i
26.11 0.0990379 0.942283i 1.14568 0.243521i 1.07821 + 0.229180i −0.566827 0.252368i −0.116000 1.10367i −0.602230 1.85347i 0.908306 2.79548i −1.48736 + 0.662217i −0.293939 + 0.509117i
26.12 0.128376 1.22142i −1.54616 + 0.328647i 0.480911 + 0.102221i −3.68103 1.63890i 0.202925 + 1.93071i 0.194103 + 0.597387i 0.945629 2.91035i −0.458021 + 0.203924i −2.47434 + 4.28568i
26.13 0.176740 1.68157i −0.842424 + 0.179063i −0.840131 0.178575i 3.33203 + 1.48352i 0.152216 + 1.44824i 1.34243 + 4.13157i 0.596219 1.83497i −2.06302 + 0.918516i 3.08353 5.34083i
26.14 0.186739 1.77670i −1.49480 + 0.317729i −1.16551 0.247736i 1.77405 + 0.789857i 0.285373 + 2.71515i −1.40791 4.33309i 0.446312 1.37361i −0.607163 + 0.270327i 1.73463 3.00446i
26.15 0.200810 1.91058i 2.50157 0.531725i −1.65369 0.351503i −1.25111 0.557030i −0.513562 4.88622i 1.08189 + 3.32973i 0.183655 0.565233i 3.23447 1.44008i −1.31548 + 2.27849i
26.16 0.244737 2.32852i 0.813081 0.172826i −3.40581 0.723927i −2.63499 1.17317i −0.203437 1.93557i −0.397027 1.22192i −1.07217 + 3.29981i −2.10940 + 0.939167i −3.37663 + 5.84850i
26.17 0.263895 2.51079i −3.12628 + 0.664512i −4.27814 0.909348i −0.536860 0.239025i 0.843441 + 8.02480i 0.0749086 + 0.230545i −1.85186 + 5.69944i 6.59142 2.93469i −0.741818 + 1.28487i
26.18 0.282066 2.68368i 2.05882 0.437617i −5.16626 1.09812i 3.37594 + 1.50307i −0.593697 5.64865i −0.679707 2.09192i −2.73649 + 8.42205i 1.30661 0.581741i 4.98598 8.63598i
49.1 −2.54591 0.541151i −0.0636153 + 0.0283233i 4.36175 + 1.94197i 1.21638 + 1.35093i 0.177286 0.0376833i 1.30543 0.948451i −5.84232 4.24470i −2.00415 + 2.22583i −2.36575 4.09760i
49.2 −2.34811 0.499107i 1.50416 0.669694i 3.43743 + 1.53044i −2.36262 2.62396i −3.86618 + 0.821782i −3.20224 + 2.32656i −3.42342 2.48726i −0.193390 + 0.214781i 4.23807 + 7.34055i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
19.c even 3 1 inner
209.n even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.n.a 144
11.c even 5 1 inner 209.2.n.a 144
19.c even 3 1 inner 209.2.n.a 144
209.n even 15 1 inner 209.2.n.a 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.n.a 144 1.a even 1 1 trivial
209.2.n.a 144 11.c even 5 1 inner
209.2.n.a 144 19.c even 3 1 inner
209.2.n.a 144 209.n even 15 1 inner

Hecke kernels

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