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Properties

Label 211.2.i.a
Level $211$
Weight $2$
Character orbit 211.i
Analytic conductor $1.685$
Analytic rank $0$
Dimension $136$
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(19,211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(211, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 211.i (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: \(1.68484348265\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 12 q^{2} - 7 q^{3} + 8 q^{4} - 6 q^{5} - 19 q^{6} - 2 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q - 12 q^{2} - 7 q^{3} + 8 q^{4} - 6 q^{5} - 19 q^{6} - 2 q^{7} + 9 q^{8} - 2 q^{9} - 20 q^{10} + 18 q^{11} - 14 q^{12} - 14 q^{13} - 5 q^{14} + 17 q^{15} + 36 q^{16} - 12 q^{17} - 35 q^{18} - 61 q^{19} - 6 q^{20} - 16 q^{21} - 5 q^{22} - 2 q^{23} - 40 q^{24} - 28 q^{25} + 8 q^{27} - 15 q^{28} - 26 q^{29} + 8 q^{30} + 25 q^{31} + 79 q^{32} - 32 q^{33} - 14 q^{34} + 52 q^{35} - 39 q^{36} - 52 q^{37} + 35 q^{38} - 50 q^{39} - 56 q^{40} - 25 q^{41} - 7 q^{42} + 22 q^{43} + 48 q^{44} + 4 q^{45} - 30 q^{46} + 2 q^{47} - 146 q^{48} + 49 q^{49} + 97 q^{50} + 43 q^{51} + 69 q^{52} + 4 q^{53} + 38 q^{54} - 8 q^{55} - 37 q^{56} + 63 q^{57} + 66 q^{58} - 101 q^{59} + 54 q^{60} + 18 q^{61} - 98 q^{62} - 78 q^{63} + 73 q^{64} - 13 q^{65} - 209 q^{66} + 40 q^{67} - 9 q^{68} + 83 q^{69} - 175 q^{70} + 21 q^{71} + 72 q^{72} + 15 q^{73} + 39 q^{74} + 61 q^{75} + 81 q^{76} - 25 q^{77} - 65 q^{78} - 28 q^{79} - 87 q^{80} - 174 q^{81} + 80 q^{82} + 23 q^{83} - 29 q^{84} + 76 q^{85} - 92 q^{86} + 42 q^{87} + 100 q^{88} + 24 q^{89} + 159 q^{90} - 97 q^{91} + 125 q^{92} + 112 q^{93} + 62 q^{94} - 33 q^{95} - 184 q^{96} - 20 q^{97} + 41 q^{98} - 38 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.84178 + 2.04550i −0.172147 + 1.63787i −0.582875 − 5.54568i 2.56095 − 1.86064i −3.03321 − 3.36872i 1.35582 − 0.288189i 7.96361 + 5.78590i 0.281469 + 0.0598281i −0.910763 + 8.66533i
19.2 −1.79784 + 1.99671i 0.215682 − 2.05208i −0.545542 − 5.19049i −2.46098 + 1.78801i 3.70964 + 4.11997i 0.899023 − 0.191093i 6.99730 + 5.08384i −1.23006 − 0.261458i 0.854332 − 8.12843i
19.3 −1.34892 + 1.49813i −0.225758 + 2.14794i −0.215744 − 2.05266i −1.54971 + 1.12593i −2.91336 − 3.23561i −0.366585 + 0.0779201i 0.104330 + 0.0758005i −1.62824 − 0.346092i 0.403648 − 3.84045i
19.4 −1.32901 + 1.47602i 0.131321 − 1.24943i −0.203299 − 1.93426i 0.493615 − 0.358632i 1.66966 + 1.85435i −1.35967 + 0.289008i −0.0885193 − 0.0643131i 1.39060 + 0.295582i −0.126673 + 1.20521i
19.5 −0.918826 + 1.02046i 0.313789 − 2.98550i 0.0119602 + 0.113793i 0.738313 − 0.536416i 2.75827 + 3.06337i 3.22920 − 0.686387i −2.34894 − 1.70660i −5.88032 − 1.24990i −0.130991 + 1.24629i
19.6 −0.597204 + 0.663262i −0.0926254 + 0.881271i 0.125793 + 1.19684i 2.48141 − 1.80285i −0.529198 − 0.587734i 2.32909 − 0.495063i −2.31305 − 1.68053i 2.16638 + 0.460479i −0.286145 + 2.72249i
19.7 −0.525679 + 0.583826i −0.0456553 + 0.434381i 0.144543 + 1.37523i −0.505018 + 0.366917i −0.229603 − 0.255000i −4.15084 + 0.882287i −2.15003 − 1.56209i 2.74784 + 0.584072i 0.0512618 − 0.487723i
19.8 −0.143512 + 0.159386i −0.320528 + 3.04962i 0.204249 + 1.94330i −0.805696 + 0.585372i −0.440067 − 0.488744i 1.51434 − 0.321883i −0.686074 − 0.498462i −6.26301 − 1.33124i 0.0223267 − 0.212424i
19.9 0.0472204 − 0.0524436i 0.0880027 − 0.837290i 0.208536 + 1.98409i −2.04254 + 1.48399i −0.0397550 − 0.0441523i 2.78721 − 0.592441i 0.228084 + 0.165713i 2.24113 + 0.476367i −0.0186237 + 0.177192i
19.10 0.340177 − 0.377805i 0.270501 − 2.57365i 0.182041 + 1.73200i 3.23322 − 2.34908i −0.880319 − 0.977694i −2.76439 + 0.587590i 1.53887 + 1.11806i −3.61606 − 0.768616i 0.212377 − 2.02063i
19.11 0.400336 − 0.444619i 0.124444 − 1.18401i 0.171640 + 1.63305i 0.161871 − 0.117606i −0.476613 − 0.529332i 1.04467 − 0.222053i 1.76286 + 1.28079i 1.54805 + 0.329048i 0.0125130 − 0.119053i
19.12 0.658763 − 0.731630i −0.260628 + 2.47971i 0.107743 + 1.02510i 2.58190 − 1.87586i 1.64254 + 1.82422i −2.33028 + 0.495316i 2.41394 + 1.75383i −3.14658 − 0.668827i 0.328423 − 3.12474i
19.13 0.870563 − 0.966858i −0.126543 + 1.20398i 0.0321223 + 0.305623i −3.17963 + 2.31014i 1.05391 + 1.17049i −4.11763 + 0.875230i 2.42858 + 1.76447i 1.50089 + 0.319025i −0.534494 + 5.08537i
19.14 1.21205 − 1.34612i 0.348500 − 3.31576i −0.133914 − 1.27410i −2.56581 + 1.86417i −4.04101 − 4.48800i 1.30676 − 0.277761i 1.05347 + 0.765394i −7.93835 − 1.68735i −0.600500 + 5.71337i
19.15 1.30504 − 1.44940i −0.212667 + 2.02339i −0.188558 − 1.79401i −2.10277 + 1.52775i 2.65516 + 2.94885i 4.74579 − 1.00875i 0.309435 + 0.224818i −1.11443 − 0.236880i −0.529885 + 5.04152i
19.16 1.48117 − 1.64501i 0.119496 − 1.13693i −0.303125 − 2.88404i 0.325917 − 0.236793i −1.69327 − 1.88057i −1.56735 + 0.333150i −1.61161 − 1.17091i 1.65611 + 0.352016i 0.0932136 − 0.886868i
19.17 1.70552 − 1.89417i −0.241641 + 2.29906i −0.470032 − 4.47206i 1.32594 − 0.963349i 3.94269 + 4.37880i −0.598863 + 0.127292i −5.14835 − 3.74050i −2.29284 − 0.487358i 0.436662 − 4.15456i
21.1 −2.34938 + 1.04601i 0.210068 + 0.233304i 3.08718 − 3.42866i −0.00881170 − 0.0271196i −0.737568 − 0.328387i 0.136275 + 1.29657i −2.07713 + 6.39275i 0.303283 − 2.88555i 0.0490695 + 0.0544971i
21.2 −2.21582 + 0.986547i −2.06679 − 2.29540i 2.59833 − 2.88573i 1.07622 + 3.31225i 6.84417 + 3.04722i −0.224489 − 2.13587i −1.41146 + 4.34404i −0.683669 + 6.50468i −5.65239 − 6.27762i
21.3 −2.02039 + 0.899535i 2.08754 + 2.31845i 1.93455 − 2.14853i −0.853121 − 2.62564i −6.30318 − 2.80635i 0.298629 + 2.84127i −0.609020 + 1.87437i −0.703795 + 6.69616i 4.08549 + 4.53740i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.i even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.2.i.a ✓ 136
211.i even 15 1 inner 211.2.i.a ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.2.i.a ✓ 136 1.a even 1 1 trivial
211.2.i.a ✓ 136 211.i even 15 1 inner

Hecke kernels

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