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 $2$

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Newspace parameters

Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 121.e (of order \(11\), degree \(10\), minimal)

Newform invariants

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.

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

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.

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 111.10
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(121, [\chi])\).