Properties

Label 211.2.d.b
Level $211$
Weight $2$
Character orbit 211.d
Analytic conductor $1.685$
Analytic rank $0$
Dimension $60$
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(55,211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(211, base_ring=CyclotomicField(10))
 
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("211.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 211.d (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: \(1.68484348265\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 2 q^{2} - 5 q^{3} - 8 q^{4} - 2 q^{5} + 15 q^{6} - 12 q^{7} - 8 q^{8} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 2 q^{2} - 5 q^{3} - 8 q^{4} - 2 q^{5} + 15 q^{6} - 12 q^{7} - 8 q^{8} - 32 q^{9} - 6 q^{10} - 7 q^{11} - 34 q^{12} + 6 q^{13} + 12 q^{14} - 20 q^{15} + 4 q^{16} + 12 q^{17} - 10 q^{18} - 43 q^{19} + 22 q^{20} - q^{21} + 30 q^{22} + 16 q^{23} + 20 q^{24} - q^{25} - 34 q^{26} + 13 q^{27} - 69 q^{28} + 10 q^{29} + 52 q^{30} - 18 q^{31} - 28 q^{32} - 30 q^{33} + 60 q^{34} + 4 q^{35} + 57 q^{36} - 23 q^{37} - 64 q^{38} + 56 q^{39} + 96 q^{40} + 50 q^{41} + 52 q^{42} - 130 q^{43} + 58 q^{44} - 18 q^{45} - 116 q^{46} - q^{47} + 42 q^{48} - 39 q^{49} - 22 q^{50} + 55 q^{51} + 41 q^{52} + 23 q^{53} - 60 q^{54} + 9 q^{55} + 29 q^{56} - 14 q^{57} + 24 q^{58} + 8 q^{59} + 9 q^{60} + 20 q^{61} - 102 q^{62} - 56 q^{63} + 18 q^{64} - 25 q^{65} - 26 q^{66} - 14 q^{67} + 56 q^{68} - 13 q^{69} - 61 q^{70} - 6 q^{71} + 29 q^{72} + 12 q^{73} + 24 q^{74} - 9 q^{75} - 72 q^{76} + 64 q^{77} + 35 q^{78} + 60 q^{79} + 51 q^{80} - 37 q^{81} - 111 q^{82} + 52 q^{83} - 91 q^{84} - 26 q^{85} + 22 q^{86} + 3 q^{87} + 78 q^{88} + 8 q^{89} + 138 q^{90} - 2 q^{91} - 56 q^{92} - 18 q^{93} + 76 q^{94} + 17 q^{95} - 42 q^{96} - 19 q^{97} + 34 q^{98} - 77 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −0.823713 2.53513i 1.09276 + 0.793934i −4.13033 + 3.00086i −2.20904 + 1.60496i 1.11261 3.42425i −1.40617 + 4.32774i 6.69675 + 4.86547i −0.363265 1.11802i 5.88840 + 4.27817i
55.2 −0.783034 2.40993i 0.126121 + 0.0916323i −3.57660 + 2.59855i 1.92447 1.39821i 0.122071 0.375694i 1.02238 3.14656i 4.96291 + 3.60576i −0.919541 2.83006i −4.87651 3.54299i
55.3 −0.601014 1.84973i 2.58570 + 1.87862i −1.44225 + 1.04785i 1.58388 1.15076i 1.92090 5.91192i −0.332824 + 1.02433i −0.341891 0.248398i 2.22958 + 6.86194i −3.08052 2.23813i
55.4 −0.494908 1.52317i 0.161827 + 0.117574i −0.457083 + 0.332090i −0.346918 + 0.252050i 0.0989960 0.304678i 0.262972 0.809344i −1.85933 1.35088i −0.914687 2.81512i 0.555609 + 0.403673i
55.5 −0.369787 1.13809i −1.30350 0.947050i 0.459534 0.333871i −2.74223 + 1.99234i −0.595807 + 1.83371i −0.492385 + 1.51540i −2.48613 1.80628i −0.124836 0.384205i 3.28150 + 2.38415i
55.6 −0.306400 0.943001i −2.37872 1.72824i 0.822664 0.597700i 1.68704 1.22570i −0.900893 + 2.77266i 0.670719 2.06426i −2.42002 1.75825i 1.74444 + 5.36882i −1.67275 1.21532i
55.7 −0.174343 0.536572i 0.321270 + 0.233417i 1.36052 0.988476i 3.24511 2.35771i 0.0692335 0.213079i −1.30076 + 4.00333i −1.68045 1.22092i −0.878320 2.70319i −1.83084 1.33018i
55.8 0.00729930 + 0.0224649i 2.04906 + 1.48873i 1.61758 1.17524i −2.91308 + 2.11648i −0.0184875 + 0.0568986i −0.724895 + 2.23100i 0.0764286 + 0.0555286i 1.05528 + 3.24781i −0.0688100 0.0499934i
55.9 0.0769537 + 0.236839i 1.22311 + 0.888642i 1.56786 1.13912i 0.142587 0.103595i −0.116342 + 0.358065i 0.992095 3.05335i 0.793375 + 0.576421i −0.220736 0.679355i 0.0355080 + 0.0257981i
55.10 0.157742 + 0.485481i −1.10206 0.800692i 1.40723 1.02241i 0.529046 0.384375i 0.214879 0.661331i −0.670812 + 2.06455i 1.54429 + 1.12199i −0.353628 1.08835i 0.270059 + 0.196210i
55.11 0.207152 + 0.637549i −1.13296 0.823147i 1.25448 0.911431i −2.19752 + 1.59659i 0.290101 0.892838i 1.58364 4.87395i 1.92561 + 1.39904i −0.321013 0.987977i −1.47312 1.07029i
55.12 0.429487 + 1.32183i −2.68143 1.94817i 0.0552709 0.0401567i −1.41131 + 1.02537i 1.42350 4.38110i −0.906281 + 2.78925i 2.32564 + 1.68968i 2.46764 + 7.59460i −1.96150 1.42512i
55.13 0.600382 + 1.84779i −1.20922 0.878551i −1.43582 + 1.04318i 3.00251 2.18145i 0.897379 2.76185i 0.713624 2.19631i 0.354019 + 0.257210i −0.236686 0.728445i 5.83351 + 4.23829i
55.14 0.723652 + 2.22717i 1.60963 + 1.16946i −2.81859 + 2.04783i −1.52835 + 1.11041i −1.43979 + 4.43120i 0.795165 2.44727i −2.81145 2.04264i 0.296207 + 0.911631i −3.57907 2.60034i
55.15 0.850529 + 2.61766i −1.17060 0.850488i −4.51071 + 3.27722i −0.384227 + 0.279157i 1.23066 3.78759i −0.970402 + 2.98659i −7.96171 5.78452i −0.280085 0.862012i −1.05754 0.768345i
71.1 −2.00568 + 1.45721i −0.880080 2.70861i 1.28126 3.94330i 1.09752 3.37781i 5.71218 + 4.15014i −2.94866 2.14233i 1.64423 + 5.06042i −4.13497 + 3.00423i 2.72092 + 8.37413i
71.2 −1.84928 + 1.34358i 0.783219 + 2.41050i 0.996596 3.06721i −0.917830 + 2.82479i −4.68709 3.40537i −1.47046 1.06835i 0.865332 + 2.66322i −2.77002 + 2.01254i −2.09801 6.45701i
71.3 −1.84532 + 1.34070i 0.186810 + 0.574942i 0.989681 3.04592i 0.379148 1.16690i −1.11555 0.810494i −0.287656 0.208994i 0.847703 + 2.60896i 2.13139 1.54855i 0.864813 + 2.66162i
71.4 −1.36230 + 0.989770i −0.936978 2.88372i 0.258187 0.794617i −0.871071 + 2.68088i 4.13067 + 3.00111i 1.63788 + 1.18999i −0.605946 1.86491i −5.01087 + 3.64061i −1.46679 4.51433i
71.5 −1.15099 + 0.836242i −0.102365 0.315048i 0.00743956 0.0228966i 0.347159 1.06844i 0.381278 + 0.277015i 2.21212 + 1.60720i −0.868693 2.67356i 2.33827 1.69886i 0.493903 + 1.52008i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.2.d.b 60
211.d even 5 1 inner 211.2.d.b 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.2.d.b 60 1.a even 1 1 trivial
211.2.d.b 60 211.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 2 T_{2}^{59} + 21 T_{2}^{58} + 38 T_{2}^{57} + 280 T_{2}^{56} + 528 T_{2}^{55} + 3112 T_{2}^{54} + 6056 T_{2}^{53} + 30782 T_{2}^{52} + 60044 T_{2}^{51} + 251223 T_{2}^{50} + 478633 T_{2}^{49} + 1794064 T_{2}^{48} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(211, [\chi])\). Copy content Toggle raw display