# Properties

 Label 121.2.e.a Level $121$ Weight $2$ Character orbit 121.e Analytic conductor $0.966$ Analytic rank $0$ Dimension $100$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,2,Mod(12,121)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([18]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("121.12");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 121.e (of order $$11$$, degree $$10$$, 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: $$0.966189864457$$ Analytic rank: $$0$$ Dimension: $$100$$ Relative dimension: $$10$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$100 q - 6 q^{2} - 18 q^{3} - 16 q^{4} - 7 q^{5} - 23 q^{6} - q^{7} + 4 q^{8} + 70 q^{9}+O(q^{10})$$ 100 * q - 6 * q^2 - 18 * q^3 - 16 * q^4 - 7 * q^5 - 23 * q^6 - q^7 + 4 * q^8 + 70 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$100 q - 6 q^{2} - 18 q^{3} - 16 q^{4} - 7 q^{5} - 23 q^{6} - q^{7} + 4 q^{8} + 70 q^{9} - 13 q^{10} - 12 q^{11} - 51 q^{12} - 34 q^{13} - 17 q^{14} - 46 q^{15} + 10 q^{16} + 9 q^{17} - 31 q^{18} + 9 q^{19} + 21 q^{20} - 14 q^{21} - 20 q^{22} - 11 q^{23} - 72 q^{24} + 11 q^{25} + 33 q^{26} - 60 q^{27} + 49 q^{28} + 19 q^{29} + 26 q^{30} - 13 q^{31} + 44 q^{32} + q^{33} + 31 q^{34} + 39 q^{35} - 17 q^{36} - 16 q^{37} - 29 q^{38} + 16 q^{39} + 2 q^{40} + 39 q^{41} + 42 q^{42} + 39 q^{43} + 53 q^{44} - 33 q^{45} + 59 q^{46} + 21 q^{47} + 56 q^{48} - 11 q^{49} - 58 q^{50} - 139 q^{51} - 75 q^{52} - 73 q^{53} - 156 q^{54} - 34 q^{55} + 10 q^{56} - 41 q^{57} - 38 q^{58} + 33 q^{59} + 100 q^{60} + 39 q^{61} + 44 q^{62} - 76 q^{63} - 16 q^{64} + 36 q^{65} + 75 q^{66} - 4 q^{67} + 119 q^{68} + 32 q^{69} + 61 q^{70} + 5 q^{71} + 63 q^{72} + 37 q^{73} + 109 q^{74} + 58 q^{75} - 91 q^{76} - 53 q^{77} - 24 q^{78} - 9 q^{79} - 36 q^{80} + 28 q^{81} + 33 q^{82} + 79 q^{83} + 176 q^{84} - 11 q^{85} + 85 q^{86} + 76 q^{87} + 33 q^{88} - 48 q^{89} - 89 q^{90} - 14 q^{91} - 113 q^{92} + 31 q^{93} - 38 q^{94} + 21 q^{95} + 84 q^{96} + 40 q^{97} - 22 q^{98} - 53 q^{99}+O(q^{100})$$ 100 * q - 6 * q^2 - 18 * q^3 - 16 * q^4 - 7 * q^5 - 23 * q^6 - q^7 + 4 * q^8 + 70 * q^9 - 13 * q^10 - 12 * q^11 - 51 * q^12 - 34 * q^13 - 17 * q^14 - 46 * q^15 + 10 * q^16 + 9 * q^17 - 31 * q^18 + 9 * q^19 + 21 * q^20 - 14 * q^21 - 20 * q^22 - 11 * q^23 - 72 * q^24 + 11 * q^25 + 33 * q^26 - 60 * q^27 + 49 * q^28 + 19 * q^29 + 26 * q^30 - 13 * q^31 + 44 * q^32 + q^33 + 31 * q^34 + 39 * q^35 - 17 * q^36 - 16 * q^37 - 29 * q^38 + 16 * q^39 + 2 * q^40 + 39 * q^41 + 42 * q^42 + 39 * q^43 + 53 * q^44 - 33 * q^45 + 59 * q^46 + 21 * q^47 + 56 * q^48 - 11 * q^49 - 58 * q^50 - 139 * q^51 - 75 * q^52 - 73 * q^53 - 156 * q^54 - 34 * q^55 + 10 * q^56 - 41 * q^57 - 38 * q^58 + 33 * q^59 + 100 * q^60 + 39 * q^61 + 44 * q^62 - 76 * q^63 - 16 * q^64 + 36 * q^65 + 75 * q^66 - 4 * q^67 + 119 * q^68 + 32 * q^69 + 61 * q^70 + 5 * q^71 + 63 * q^72 + 37 * q^73 + 109 * q^74 + 58 * q^75 - 91 * q^76 - 53 * q^77 - 24 * q^78 - 9 * q^79 - 36 * q^80 + 28 * q^81 + 33 * q^82 + 79 * q^83 + 176 * q^84 - 11 * q^85 + 85 * q^86 + 76 * q^87 + 33 * q^88 - 48 * q^89 - 89 * q^90 - 14 * q^91 - 113 * q^92 + 31 * q^93 - 38 * q^94 + 21 * q^95 + 84 * q^96 + 40 * q^97 - 22 * q^98 - 53 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
12.1 −2.28929 + 1.47124i −1.98337 2.24548 4.91691i 0.00109106 0.00758846i 4.54050 2.91800i 1.07261 1.23786i 1.31883 + 9.17268i 0.933746 0.00866667 + 0.0189774i
12.2 −1.90982 + 1.22737i 2.89591 1.31016 2.86885i −0.301049 + 2.09384i −5.53069 + 3.55436i 2.32393 2.68196i 0.372795 + 2.59284i 5.38632 −1.99496 4.36836i
12.3 −1.29895 + 0.834784i −0.564640 0.159574 0.349419i −0.144022 + 1.00170i 0.733439 0.471353i −1.26378 + 1.45848i −0.355076 2.46961i −2.68118 −0.649122 1.42138i
12.4 −0.800834 + 0.514665i −2.23328 −0.454375 + 0.994941i 0.373665 2.59890i 1.78848 1.14939i 1.22249 1.41083i −0.419137 2.91516i 1.98753 1.03832 + 2.27360i
12.5 −0.322674 + 0.207370i 1.45814 −0.769714 + 1.68544i −0.307743 + 2.14040i −0.470506 + 0.302376i −0.298898 + 0.344946i −0.210316 1.46278i −0.873817 −0.344555 0.754469i
12.6 −0.0579015 + 0.0372110i 2.65925 −0.828862 + 1.81495i 0.499158 3.47172i −0.153975 + 0.0989535i −1.42926 + 1.64945i −0.0391344 0.272185i 4.07163 0.100284 + 0.219592i
12.7 0.668610 0.429690i −2.29704 −0.568424 + 1.24467i −0.598868 + 4.16522i −1.53583 + 0.987017i 2.14854 2.47954i 0.380987 + 2.64983i 2.27641 1.38934 + 3.04223i
12.8 1.05761 0.679684i 0.322850 −0.174263 + 0.381583i 0.250017 1.73891i 0.341450 0.219436i 2.04681 2.36215i 0.432885 + 3.01078i −2.89577 −0.917487 2.00902i
12.9 1.77043 1.13779i 0.298538 1.00904 2.20948i −0.0612837 + 0.426238i 0.528540 0.339672i −1.45441 + 1.67848i −0.128482 0.893611i −2.91088 0.376469 + 0.824352i
12.10 2.02796 1.30329i −3.38720 1.58323 3.46678i 0.366767 2.55092i −6.86911 + 4.41451i −0.194745 + 0.224747i −0.621365 4.32169i 8.47314 −2.58080 5.65116i
23.1 −1.14719 + 2.51200i −1.28359 −3.68436 4.25198i −0.739579 0.217160i 1.47253 3.22438i −0.216919 1.50870i 9.60824 2.82123i −1.35239 1.39394 1.60870i
23.2 −0.917263 + 2.00852i 3.12379 −1.88308 2.17319i −1.00380 0.294742i −2.86534 + 6.27421i 0.184381 + 1.28240i 1.85494 0.544660i 6.75808 1.51274 1.74580i
23.3 −0.671026 + 1.46934i −1.09234 −0.398970 0.460436i 3.18729 + 0.935873i 0.732990 1.60502i 0.233950 + 1.62716i −2.15551 + 0.632915i −1.80679 −3.51387 + 4.05523i
23.4 −0.597130 + 1.30753i −0.640035 −0.0433545 0.0500337i −3.97756 1.16792i 0.382184 0.836867i 0.623172 + 4.33426i −2.66710 + 0.783131i −2.59035 3.90221 4.50339i
23.5 −0.389908 + 0.853778i 1.43258 0.732812 + 0.845710i 0.815711 + 0.239514i −0.558572 + 1.22310i −0.497109 3.45747i −2.80893 + 0.824777i −0.947723 −0.522544 + 0.603048i
23.6 0.194417 0.425713i −3.09292 1.16629 + 1.34597i 1.44135 + 0.423218i −0.601314 + 1.31669i 0.384441 + 2.67384i 1.69784 0.498530i 6.56613 0.460391 0.531320i
23.7 0.333827 0.730979i 0.0294246 0.886832 + 1.02346i 0.779896 + 0.228998i 0.00982271 0.0215087i 0.0964221 + 0.670630i 2.58627 0.759397i −2.99913 0.427743 0.493642i
23.8 0.569265 1.24652i 2.07869 0.0799801 + 0.0923019i −3.60635 1.05892i 1.18333 2.59113i −0.0268425 0.186694i 2.79027 0.819298i 1.32097 −3.37293 + 3.89257i
23.9 0.934014 2.04521i −1.99480 −2.00076 2.30900i −1.29355 0.379821i −1.86317 + 4.07977i −0.411373 2.86116i −2.27650 + 0.668441i 0.979213 −1.98501 + 2.29082i
23.10 1.04868 2.29629i 0.748916 −2.86349 3.30464i 1.29867 + 0.381324i 0.785372 1.71973i 0.561499 + 3.90531i −5.74697 + 1.68746i −2.43913 2.23752 2.58224i
See all 100 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 12.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.2.e.a 100
121.e even 11 1 inner 121.2.e.a 100

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.2.e.a 100 1.a even 1 1 trivial
121.2.e.a 100 121.e even 11 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(121, [\chi])$$.