Properties

Label 211.3.e.a
Level $211$
Weight $3$
Character orbit 211.e
Analytic conductor $5.749$
Analytic rank $0$
Dimension $68$
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] = [211,3,Mod(15,211)]
 
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([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.15");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 211.e (of order \(6\), 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: \(5.74933357800\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 3 q^{2} + 55 q^{4} - 4 q^{5} - q^{6} - 24 q^{7} + 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 3 q^{2} + 55 q^{4} - 4 q^{5} - q^{6} - 24 q^{7} + 82 q^{9} + 4 q^{11} + 32 q^{13} + 12 q^{14} - 73 q^{16} + 51 q^{17} - 8 q^{19} - 46 q^{20} - 32 q^{21} - 33 q^{22} + 32 q^{24} + 252 q^{25} + 42 q^{26} - 60 q^{29} + 32 q^{30} + 126 q^{31} - 78 q^{32} - 117 q^{33} - 50 q^{34} - 45 q^{35} - 12 q^{36} + 113 q^{37} + 33 q^{38} - 162 q^{39} - 75 q^{41} + 13 q^{43} + 15 q^{44} + 83 q^{45} + 79 q^{46} - 117 q^{47} + 3 q^{48} + 130 q^{49} - 90 q^{50} - 181 q^{51} - 22 q^{52} - 140 q^{53} + 125 q^{54} - 50 q^{55} - 75 q^{56} - 234 q^{57} - 70 q^{58} - 84 q^{59} - 366 q^{61} - 75 q^{62} + 142 q^{64} + 72 q^{65} + 144 q^{66} + 130 q^{69} - 466 q^{70} + 402 q^{71} + 450 q^{72} - 54 q^{73} - 330 q^{74} - 402 q^{75} + 10 q^{76} + 69 q^{77} + 316 q^{78} - 370 q^{79} - 13 q^{80} - 206 q^{81} + 88 q^{82} - 105 q^{83} + 216 q^{84} - 168 q^{85} - 346 q^{87} + 264 q^{91} + 534 q^{92} + 310 q^{93} - 537 q^{94} + 265 q^{95} + 80 q^{96} - 50 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −3.28947 + 1.89917i 2.53131 1.46145i 5.21373 9.03044i 3.33934 −5.55111 + 9.61480i 2.80731 1.62080i 24.4137i −0.228311 + 0.395446i −10.9847 + 6.34200i
15.2 −3.06100 + 1.76727i −3.63546 + 2.09894i 4.24649 7.35513i −3.78107 7.41877 12.8497i 3.69317 2.13225i 15.8806i 4.31106 7.46697i 11.5739 6.68217i
15.3 −2.96716 + 1.71309i −2.59992 + 1.50106i 3.86935 6.70190i 5.80352 5.14291 8.90778i −3.19116 + 1.84241i 12.8094i 0.00637874 0.0110483i −17.2200 + 9.94195i
15.4 −2.77079 + 1.59971i 1.49990 0.865968i 3.11817 5.40083i −5.11467 −2.77060 + 4.79883i 6.27090 3.62051i 7.15502i −3.00020 + 5.19650i 14.1717 8.18201i
15.5 −2.72956 + 1.57591i 5.03303 2.90582i 2.96701 5.13902i −3.99896 −9.15865 + 15.8632i −9.94373 + 5.74102i 6.09572i 12.3876 21.4559i 10.9154 6.30202i
15.6 −2.45402 + 1.41683i 0.156697 0.0904691i 2.01482 3.48977i −8.94790 −0.256359 + 0.444027i −7.84079 + 4.52688i 0.0840007i −4.48363 + 7.76588i 21.9584 12.6777i
15.7 −2.36498 + 1.36542i 1.22716 0.708503i 1.72876 2.99430i 6.16488 −1.93481 + 3.35119i −7.88662 + 4.55334i 1.48144i −3.49605 + 6.05533i −14.5798 + 8.41767i
15.8 −2.00984 + 1.16038i −3.64827 + 2.10633i 0.692961 1.20024i −2.91522 4.88828 8.46676i −4.31280 + 2.49000i 6.06665i 4.37325 7.57470i 5.85911 3.38276i
15.9 −1.83582 + 1.05991i 4.02185 2.32201i 0.246821 0.427506i 6.05791 −4.92225 + 8.52560i 6.83430 3.94578i 7.43285i 6.28350 10.8833i −11.1212 + 6.42085i
15.10 −1.79689 + 1.03744i −0.231192 + 0.133478i 0.152554 0.264232i 0.253622 0.276951 0.479694i 1.69721 0.979882i 7.66644i −4.46437 + 7.73251i −0.455732 + 0.263117i
15.11 −1.45509 + 0.840099i −1.70196 + 0.982625i −0.588469 + 1.01926i 0.968548 1.65100 2.85962i 9.80511 5.66098i 8.69828i −2.56890 + 4.44946i −1.40933 + 0.813675i
15.12 −1.39681 + 0.806447i −5.02816 + 2.90301i −0.699286 + 1.21120i 8.50901 4.68224 8.10988i 2.45340 1.41647i 8.70733i 12.3549 21.3993i −11.8855 + 6.86207i
15.13 −1.10842 + 0.639949i 3.92665 2.26705i −1.18093 + 2.04543i −4.08531 −2.90160 + 5.02571i 1.75991 1.01608i 8.14253i 5.77906 10.0096i 4.52826 2.61439i
15.14 −0.668192 + 0.385781i −3.23147 + 1.86569i −1.70235 + 2.94855i −5.92170 1.43949 2.49328i −8.58189 + 4.95476i 5.71318i 2.46158 4.26359i 3.95684 2.28448i
15.15 −0.642412 + 0.370896i −1.57332 + 0.908357i −1.72487 + 2.98757i 3.64465 0.673813 1.16708i −5.13285 + 2.96345i 5.52617i −2.84977 + 4.93595i −2.34136 + 1.35179i
15.16 −0.383536 + 0.221434i 2.44507 1.41166i −1.90193 + 3.29425i −6.28491 −0.625182 + 1.08285i 1.84824 1.06708i 3.45609i −0.514408 + 0.890980i 2.41049 1.39170i
15.17 −0.269102 + 0.155366i 2.24759 1.29764i −1.95172 + 3.38048i 0.332053 −0.403219 + 0.698396i −9.33157 + 5.38758i 2.45585i −1.13224 + 1.96110i −0.0893560 + 0.0515897i
15.18 −0.0594322 + 0.0343132i 0.887522 0.512411i −1.99765 + 3.46002i 9.58936 −0.0351649 + 0.0609074i 5.09594 2.94214i 0.548688i −3.97487 + 6.88468i −0.569917 + 0.329042i
15.19 −0.0226402 + 0.0130713i −2.34985 + 1.35669i −1.99966 + 3.46351i −8.84227 0.0354674 0.0614314i 9.90132 5.71653i 0.209123i −0.818790 + 1.41819i 0.200191 0.115580i
15.20 0.717030 0.413978i 4.24124 2.44868i −1.65725 + 2.87043i 7.07258 2.02740 3.51156i −6.82168 + 3.93850i 6.05607i 7.49209 12.9767i 5.07125 2.92789i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.e odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.3.e.a 68
211.e odd 6 1 inner 211.3.e.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.3.e.a 68 1.a even 1 1 trivial
211.3.e.a 68 211.e odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(211, [\chi])\).