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Properties

Label 211.2.c.a
Level $211$
Weight $2$
Character orbit 211.c
Analytic conductor $1.685$
Analytic rank $0$
Dimension $34$
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] = [211,2,Mod(14,211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(211, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 211.c (of order \(3\), degree \(2\), 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: \(1.68484348265\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 2 q^{2} + 2 q^{3} - 18 q^{4} - 4 q^{5} + 9 q^{6} + 2 q^{7} - 24 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 2 q^{2} + 2 q^{3} - 18 q^{4} - 4 q^{5} + 9 q^{6} + 2 q^{7} - 24 q^{8} - 13 q^{9} - 10 q^{10} - 8 q^{11} + 4 q^{12} + 4 q^{13} - 2 q^{15} - 16 q^{16} + 2 q^{17} - 10 q^{18} - 9 q^{19} - 4 q^{20} + 6 q^{21} - 5 q^{22} - 8 q^{23} + 20 q^{24} + 18 q^{25} + 10 q^{26} + 2 q^{27} - 10 q^{28} + 6 q^{29} - 8 q^{30} - 10 q^{31} + 21 q^{32} + 17 q^{33} + 9 q^{34} + 3 q^{35} - q^{36} - 13 q^{37} + 40 q^{38} - 20 q^{39} - 24 q^{40} + 15 q^{41} + 72 q^{42} + 18 q^{43} - 13 q^{44} + q^{45} - 5 q^{46} + 3 q^{47} + q^{48} - 39 q^{49} + 13 q^{50} - 23 q^{51} - 24 q^{52} + 6 q^{53} + 57 q^{54} - 22 q^{55} - 3 q^{56} + 2 q^{57} - 26 q^{58} - 19 q^{59} - 4 q^{60} + 12 q^{61} - 17 q^{62} + 28 q^{63} - 28 q^{64} - 12 q^{65} - 16 q^{66} - 40 q^{67} + 44 q^{68} + 2 q^{69} - 20 q^{70} + 34 q^{71} + 13 q^{72} - 5 q^{73} - 4 q^{74} - 36 q^{75} + 24 q^{76} + 25 q^{77} + 60 q^{78} - 42 q^{79} - 53 q^{80} - 21 q^{81} + 120 q^{82} - 18 q^{83} + 74 q^{84} + 4 q^{85} - 58 q^{86} - 62 q^{87} - 60 q^{88} + 26 q^{89} + 6 q^{90} - 8 q^{91} + 30 q^{92} + 28 q^{93} - 27 q^{94} + 53 q^{95} + 24 q^{96} + 10 q^{97} - 126 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −1.27990 + 2.21685i 0.211900 − 0.367021i −2.27627 − 3.94262i 1.01023 0.542419 + 0.939498i 1.74307 − 3.01908i 6.53399 1.41020 + 2.44253i −1.29299 + 2.23953i
14.2 −1.05805 + 1.83259i 1.61684 − 2.80045i −1.23893 − 2.14588i 0.744514 3.42138 + 5.92601i −0.585760 + 1.01457i 1.01118 −3.72834 − 6.45767i −0.787731 + 1.36439i
14.3 −1.02311 + 1.77208i 0.624654 − 1.08193i −1.09352 − 1.89404i −2.94163 1.27818 + 2.21388i −0.492095 + 0.852334i 0.382740 0.719615 + 1.24641i 3.00962 − 5.21281i
14.4 −1.00528 + 1.74120i −0.762327 + 1.32039i −1.02118 − 1.76874i 3.16675 −1.53271 − 2.65473i −1.87007 + 3.23906i 0.0851778 0.337714 + 0.584938i −3.18347 + 5.51394i
14.5 −0.701959 + 1.21583i −1.12731 + 1.95256i 0.0145062 + 0.0251254i −2.46004 −1.58266 − 2.74124i 0.304653 − 0.527674i −2.84857 −1.04167 − 1.80423i 1.72685 − 2.99099i
14.6 −0.509458 + 0.882408i −0.157156 + 0.272202i 0.480904 + 0.832950i 2.15395 −0.160129 − 0.277351i 2.11719 − 3.66708i −3.01784 1.45060 + 2.51252i −1.09735 + 1.90067i
14.7 −0.233638 + 0.404672i 1.09862 − 1.90286i 0.890827 + 1.54296i 3.06202 0.513357 + 0.889161i −0.447268 + 0.774692i −1.76707 −0.913926 − 1.58297i −0.715402 + 1.23911i
14.8 −0.100504 + 0.174079i 0.223992 − 0.387965i 0.979798 + 1.69706i −2.01524 0.0450243 + 0.0779845i −1.95308 + 3.38283i −0.795914 1.39966 + 2.42427i 0.202541 − 0.350811i
14.9 −0.0213305 + 0.0369455i 1.25009 − 2.16522i 0.999090 + 1.73047i −2.73223 0.0533302 + 0.0923706i 2.39263 − 4.14416i −0.170566 −1.62545 − 2.81537i 0.0582798 − 0.100944i
14.10 0.0494392 − 0.0856313i −1.40765 + 2.43813i 0.995112 + 1.72358i 1.90276 0.139187 + 0.241078i −0.201922 + 0.349739i 0.394547 −2.46298 − 4.26601i 0.0940708 − 0.162935i
14.11 0.483234 − 0.836986i −0.380426 + 0.658917i 0.532970 + 0.923131i −0.766374 0.367670 + 0.636822i 0.625230 − 1.08293i 2.96313 1.21055 + 2.09674i −0.370338 + 0.641444i
14.12 0.737651 − 1.27765i 1.43922 − 2.49280i −0.0882584 − 0.152868i −0.420293 −2.12328 − 3.67764i −1.39047 + 2.40837i 2.69019 −2.64271 − 4.57731i −0.310029 + 0.536987i
14.13 0.863508 − 1.49564i 0.147292 − 0.255117i −0.491291 − 0.850941i 1.52306 −0.254375 − 0.440591i 0.760604 − 1.31740i 1.75710 1.45661 + 2.52292i 1.31517 − 2.27794i
14.14 0.971948 − 1.68346i −1.28135 + 2.21936i −0.889365 − 1.54042i −4.17254 2.49081 + 4.31421i −1.88554 + 3.26586i 0.430127 −1.78372 − 3.08949i −4.05549 + 7.02431i
14.15 1.18465 − 2.05188i 0.365771 − 0.633535i −1.80680 − 3.12947i −3.48547 −0.866624 − 1.50104i 1.34480 − 2.32927i −3.82313 1.23242 + 2.13462i −4.12907 + 7.15176i
14.16 1.29307 − 2.23967i −1.40258 + 2.42934i −2.34407 − 4.06005i 1.67583 3.62727 + 6.28262i 2.46134 − 4.26317i −6.95193 −2.43445 − 4.21659i 2.16697 − 3.75330i
14.17 1.34972 − 2.33779i 0.540428 − 0.936049i −2.64351 − 4.57870i 1.75471 −1.45886 − 2.52682i −1.92331 + 3.33127i −8.87316 0.915875 + 1.58634i 2.36837 − 4.10214i
196.1 −1.27990 − 2.21685i 0.211900 + 0.367021i −2.27627 + 3.94262i 1.01023 0.542419 − 0.939498i 1.74307 + 3.01908i 6.53399 1.41020 − 2.44253i −1.29299 − 2.23953i
196.2 −1.05805 − 1.83259i 1.61684 + 2.80045i −1.23893 + 2.14588i 0.744514 3.42138 − 5.92601i −0.585760 − 1.01457i 1.01118 −3.72834 + 6.45767i −0.787731 − 1.36439i
196.3 −1.02311 − 1.77208i 0.624654 + 1.08193i −1.09352 + 1.89404i −2.94163 1.27818 − 2.21388i −0.492095 − 0.852334i 0.382740 0.719615 − 1.24641i 3.00962 + 5.21281i
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.2.c.a ✓ 34
211.c even 3 1 inner 211.2.c.a ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.2.c.a ✓ 34 1.a even 1 1 trivial
211.2.c.a ✓ 34 211.c even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(211, [\chi])\).