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Properties

Label 211.2.o.a
Level $211$
Weight $2$
Character orbit 211.o
Analytic conductor $1.685$
Analytic rank $0$
Dimension $816$
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(4,211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(211, base_ring=CyclotomicField(210))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 211.o (of order \(105\), degree \(48\), 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: \(816\)
Relative dimension: \(17\) over \(\Q(\zeta_{105})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{105}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 816 q - 51 q^{2} - 49 q^{3} - 71 q^{4} - 50 q^{5} - 51 q^{6} - 54 q^{7} - 51 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 816 q - 51 q^{2} - 49 q^{3} - 71 q^{4} - 50 q^{5} - 51 q^{6} - 54 q^{7} - 51 q^{8} - 54 q^{9} - 57 q^{10} - 74 q^{11} - 42 q^{13} - 23 q^{14} + 67 q^{15} - 43 q^{16} - 51 q^{17} - 21 q^{18} + 12 q^{19} - 64 q^{20} - 12 q^{21} - 79 q^{22} + 16 q^{23} - 37 q^{24} - 56 q^{26} - 22 q^{27} - 41 q^{28} + 26 q^{29} + 55 q^{30} - 25 q^{31} - 149 q^{32} - 24 q^{33} - 14 q^{34} - 38 q^{35} - 129 q^{36} + 10 q^{37} - 105 q^{38} + 50 q^{39} + 28 q^{40} - 45 q^{41} - 56 q^{42} + 13 q^{43} - 216 q^{44} - 11 q^{45} + 212 q^{46} - 2 q^{47} + 202 q^{48} - 63 q^{49} - 258 q^{50} - 190 q^{51} + q^{52} - 88 q^{53} - 3 q^{54} + 8 q^{55} + 44 q^{56} + 70 q^{57} + 186 q^{58} - 11 q^{59} - 96 q^{60} - 11 q^{61} + 329 q^{62} - 34 q^{63} - 59 q^{64} - 99 q^{65} + 181 q^{66} - 96 q^{67} - 47 q^{68} - 97 q^{69} + 322 q^{70} + 21 q^{71} + 12 q^{72} - 197 q^{73} - 172 q^{74} + 23 q^{75} + 122 q^{76} + 158 q^{77} - 19 q^{78} - 42 q^{79} - 95 q^{80} + 76 q^{81} - 213 q^{82} - 107 q^{83} + 15 q^{84} + 218 q^{85} + 232 q^{86} - 168 q^{87} - 135 q^{88} - 80 q^{89} + 86 q^{90} + 125 q^{91} + 239 q^{92} - 252 q^{93} + 113 q^{94} - 226 q^{95} + 226 q^{96} - 64 q^{97} + 162 q^{98} + 178 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.75180 − 0.0823583i −0.711965 − 2.43698i 5.56921 + 0.333659i 1.32956 + 1.39061i 1.75848 + 6.76473i 1.03940 + 1.68200i −9.81398 − 0.883274i −2.90384 + 1.85504i −3.54415 − 3.93618i
4.2 −2.44264 − 0.0731055i 0.367420 + 1.25764i 3.96474 + 0.237533i −0.122699 − 0.128333i −0.805536 − 3.09884i −2.68128 − 4.33895i −4.79929 − 0.431944i 1.08149 − 0.690884i 0.290328 + 0.322442i
4.3 −2.37743 − 0.0711538i 0.558747 + 1.91254i 3.65069 + 0.218718i 1.56281 + 1.63457i −1.19230 − 4.58668i 1.81509 + 2.93725i −3.92586 − 0.353334i −0.817437 + 0.522199i −3.59917 − 3.99728i
4.4 −1.86118 − 0.0557030i 0.358819 + 1.22820i 1.46447 + 0.0877384i −2.40280 − 2.51313i −0.599412 − 2.30589i 1.22165 + 1.97692i 0.988282 + 0.0889470i 1.14843 − 0.733648i 4.33206 + 4.81124i
4.5 −1.78281 − 0.0533574i −0.778252 − 2.66388i 1.17913 + 0.0706433i −2.40023 − 2.51044i 1.24534 + 4.79071i −1.09626 − 1.77401i 1.45446 + 0.130904i −3.96242 + 2.53129i 4.14519 + 4.60370i
4.6 −1.62581 − 0.0486586i −0.394618 − 1.35074i 0.644463 + 0.0386107i 0.529965 + 0.554300i 0.575849 + 2.21524i 0.782375 + 1.26607i 2.19408 + 0.197471i 0.859389 − 0.548999i −0.834650 − 0.926973i
4.7 −0.592280 − 0.0177263i −0.0673593 − 0.230564i −1.64594 − 0.0986105i 1.10051 + 1.15105i 0.0358085 + 0.137753i 0.220418 + 0.356688i 2.15343 + 0.193812i 2.47954 − 1.58399i −0.631409 − 0.701250i
4.8 −0.386109 − 0.0115558i 0.286528 + 0.980756i −1.84747 − 0.110685i −1.22504 − 1.28129i −0.0992974 − 0.381989i −1.53445 − 2.48310i 1.48150 + 0.133337i 1.64838 − 1.05302i 0.458193 + 0.508874i
4.9 0.119006 + 0.00356172i −0.863037 − 2.95409i −1.98227 − 0.118761i 1.23005 + 1.28653i −0.0921851 − 0.354629i −1.96250 − 3.17578i −0.472640 − 0.0425383i −5.45366 + 3.48393i 0.141802 + 0.157487i
4.10 0.532583 + 0.0159396i 0.730732 + 2.50122i −1.71303 − 0.102630i −0.133530 − 0.139662i 0.349307 + 1.34376i 1.36955 + 2.21625i −1.97205 − 0.177487i −3.19399 + 2.04040i −0.0688898 − 0.0765098i
4.11 1.04242 + 0.0311983i −0.229276 − 0.784789i −0.910762 − 0.0545650i −2.92192 − 3.05608i −0.214517 − 0.825230i −0.244360 − 0.395431i −3.02506 − 0.272260i 1.96484 − 1.25518i −2.95051 − 3.27687i
4.12 1.11108 + 0.0332533i −0.279454 − 0.956543i −0.763035 − 0.0457145i 2.95464 + 3.09031i −0.278687 − 1.07209i 1.50112 + 2.42917i −3.06047 − 0.275447i 1.69128 − 1.08043i 3.18007 + 3.53183i
4.13 1.49732 + 0.0448130i 0.621605 + 2.12769i 0.243527 + 0.0145901i 2.35550 + 2.46366i 0.835391 + 3.21369i −2.42275 − 3.92059i −2.61993 − 0.235798i −1.61252 + 1.03012i 3.41653 + 3.79444i
4.14 1.91403 + 0.0572847i 0.214879 + 0.735510i 1.66380 + 0.0996806i −0.566829 − 0.592857i 0.369151 + 1.42010i 1.73371 + 2.80555i −0.635503 − 0.0571963i 2.03336 − 1.29896i −1.05097 − 1.16722i
4.15 2.21450 + 0.0662775i −0.871634 − 2.98352i 2.90320 + 0.173935i −0.188658 − 0.197320i −1.73249 − 6.66477i 1.18016 + 1.90978i 2.00446 + 0.180404i −5.61346 + 3.58602i −0.404704 − 0.449470i
4.16 2.21594 + 0.0663206i −0.193810 − 0.663393i 2.90958 + 0.174317i 0.252650 + 0.264251i −0.385476 − 1.48289i −1.49391 − 2.41749i 2.01989 + 0.181793i 2.12563 − 1.35791i 0.542332 + 0.602320i
4.17 2.44288 + 0.0731126i 0.815227 + 2.79044i 3.96589 + 0.237602i −2.30913 − 2.41516i 1.78748 + 6.87631i −1.14192 − 1.84790i 4.80255 + 0.432237i −4.59380 + 2.93463i −5.46434 − 6.06877i
6.1 −2.07602 − 1.60536i −3.11215 − 0.0931432i 1.22952 + 4.72985i −1.25557 + 2.33324i 6.31137 + 5.18949i −1.92448 − 0.788726i 2.97776 − 6.96683i 6.68218 + 0.400339i 6.35228 − 2.82822i
6.2 −2.03140 − 1.57085i 2.23998 + 0.0670399i 1.15583 + 4.44640i −0.696817 + 1.29490i −4.44498 − 3.65486i −0.692818 − 0.283944i 2.61817 − 6.12552i 2.01837 + 0.120923i 3.44962 − 1.53587i
6.3 −1.63809 − 1.26671i 0.0493203 + 0.00147610i 0.575604 + 2.21430i −0.0818133 + 0.152035i −0.0789213 − 0.0648926i −1.88456 − 0.772366i 0.234301 − 0.548175i −2.99220 − 0.179267i 0.326601 − 0.145412i
See next 80 embeddings (of 816 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.o even 105 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.2.o.a ✓ 816
211.o even 105 1 inner 211.2.o.a ✓ 816
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.2.o.a ✓ 816 1.a even 1 1 trivial
211.2.o.a ✓ 816 211.o even 105 1 inner

Hecke kernels

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