Properties

Label 121.2.g.a
Level $121$
Weight $2$
Character orbit 121.g
Analytic conductor $0.966$
Analytic rank $0$
Dimension $400$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 121.g (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.966189864457\)
Analytic rank: \(0\)
Dimension: \(400\)
Relative dimension: \(10\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 400q - 44q^{2} - 32q^{3} - 34q^{4} - 43q^{5} - 22q^{6} - 44q^{7} - 44q^{8} - 110q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 400q - 44q^{2} - 32q^{3} - 34q^{4} - 43q^{5} - 22q^{6} - 44q^{7} - 44q^{8} - 110q^{9} - 22q^{10} - 33q^{11} + 6q^{12} - 11q^{13} - 18q^{14} + 16q^{15} - 30q^{16} - 44q^{17} + 11q^{18} - 44q^{19} - 36q^{20} - 11q^{21} - 34q^{23} + 77q^{24} - 31q^{25} - 38q^{26} + 40q^{27} - 44q^{28} - 44q^{29} - 11q^{30} - 17q^{31} - 44q^{32} - 11q^{33} - 76q^{34} - 44q^{35} + 57q^{36} - 4q^{37} + 34q^{38} - 11q^{39} + 33q^{40} - 44q^{41} + 3q^{42} - 44q^{43} - 33q^{44} - 12q^{45} - 44q^{46} - 36q^{47} + 9q^{48} + 16q^{49} + 88q^{50} + 154q^{51} + 110q^{52} + 78q^{53} + 231q^{54} + 44q^{55} - 55q^{56} + 66q^{57} + 73q^{58} - 33q^{59} + 15q^{60} - 44q^{61} + 11q^{62} + 121q^{63} + 96q^{64} - 11q^{65} - 41q^{67} - 44q^{68} + 8q^{69} + 24q^{70} + 15q^{71} + 77q^{72} - 22q^{73} - 44q^{74} + 7q^{75} + 176q^{76} + 88q^{77} - 21q^{78} + 44q^{79} + 171q^{80} + 37q^{81} + 22q^{82} - 44q^{83} - 11q^{84} + 66q^{85} - 20q^{86} - 11q^{87} + 77q^{88} + 3q^{89} + 264q^{90} + 79q^{91} + 228q^{92} + 49q^{93} + 143q^{94} + 44q^{95} + 121q^{96} + 10q^{97} + 132q^{98} + 143q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.59904 0.148619i 0.517548 1.59285i 4.74598 + 0.544550i 0.803024 + 1.52190i −1.58186 + 4.06297i −0.0112634 0.00475995i −7.12374 1.23281i 0.157740 + 0.114605i −1.86091 4.07483i
4.2 −2.08449 0.119196i −0.374981 + 1.15407i 2.34393 + 0.268941i −1.51227 2.86606i 0.919206 2.36096i 2.34211 + 0.989784i −0.739221 0.127927i 1.23578 + 0.897843i 2.81069 + 6.15454i
4.3 −1.94119 0.111001i −0.618438 + 1.90336i 1.76894 + 0.202967i 0.816229 + 1.54693i 1.41178 3.62613i −4.27807 1.80793i 0.420440 + 0.0727601i −0.813251 0.590861i −1.41275 3.09349i
4.4 −0.828838 0.0473947i 1.01247 3.11606i −1.30224 0.149418i 0.0511865 + 0.0970092i −0.986857 + 2.53472i −0.0339504 0.0143476i 2.70833 + 0.468694i −6.25768 4.54647i −0.0378276 0.0828308i
4.5 −0.790346 0.0451936i −0.0254779 + 0.0784129i −1.36436 0.156546i 0.532875 + 1.00991i 0.0236801 0.0608219i 4.07073 + 1.72030i 2.63133 + 0.455369i 2.42155 + 1.75936i −0.375515 0.822262i
4.6 −0.282035 0.0161273i 0.138004 0.424731i −1.90768 0.218886i −1.62748 3.08441i −0.0457716 + 0.117563i −3.79767 1.60491i 1.09122 + 0.188843i 2.26570 + 1.64613i 0.409262 + 0.896158i
4.7 0.609785 + 0.0348688i −0.704377 + 2.16785i −1.61634 0.185458i 0.903893 + 1.71307i −0.505109 + 1.29736i −0.168606 0.0712535i −2.18283 0.377753i −1.77637 1.29061i 0.491448 + 1.07612i
4.8 1.43269 + 0.0819240i 0.403457 1.24171i 0.0589155 + 0.00675993i −0.621601 1.17807i 0.679753 1.74593i 2.51104 + 1.06117i −2.74416 0.474896i 1.04798 + 0.761401i −0.794048 1.73872i
4.9 1.51337 + 0.0865377i 0.615691 1.89490i 0.295842 + 0.0339447i 1.83167 + 3.47141i 1.09575 2.81441i −3.18667 1.34670i −2.54250 0.439998i −0.784523 0.569989i 2.47160 + 5.41204i
4.10 2.13575 + 0.122126i −0.616726 + 1.89809i 2.55954 + 0.293680i −0.685845 1.29982i −1.54898 + 3.97852i −0.751489 0.317582i 1.21486 + 0.210239i −0.795333 0.577843i −1.30605 2.85985i
5.1 −1.31555 + 2.18160i 1.84029 + 1.33705i −2.09536 3.97115i 0.0304467 0.0248961i −5.33791 + 2.25582i −2.96018 + 3.83884i 6.33320 + 0.362145i 0.671922 + 2.06796i 0.0142592 + 0.0991747i
5.2 −1.26554 + 2.09866i −1.30803 0.950337i −1.86945 3.54301i −2.56800 + 2.09984i 3.64979 1.54242i 2.42078 3.13933i 4.90801 + 0.280650i −0.119257 0.367035i −1.15695 8.04678i
5.3 −1.05343 + 1.74691i −1.35869 0.987147i −1.00866 1.91163i 3.20775 2.62296i 3.15574 1.33363i 0.212285 0.275296i 0.328744 + 0.0187983i −0.0554687 0.170715i 1.20296 + 8.36677i
5.4 −0.631909 + 1.04790i 1.62812 + 1.18290i 0.234543 + 0.444508i 0.863348 0.705956i −2.26838 + 0.958627i 1.34281 1.74139i −3.05739 0.174828i 0.324472 + 0.998622i 0.194216 + 1.35080i
5.5 −0.339119 + 0.562365i −2.60062 1.88946i 0.732082 + 1.38745i −0.891773 + 0.729199i 1.94449 0.821746i −1.70268 + 2.20808i −2.33977 0.133793i 2.26611 + 6.97437i −0.107659 0.748787i
5.6 −0.215633 + 0.357587i 0.525350 + 0.381689i 0.851964 + 1.61465i −2.63241 + 2.15251i −0.249770 + 0.105554i −0.169441 + 0.219735i −1.59487 0.0911981i −0.796745 2.45213i −0.202076 1.40547i
5.7 0.364682 0.604757i −0.929691 0.675460i 0.700597 + 1.32778i 0.792518 0.648038i −0.747530 + 0.315909i 1.96527 2.54861i 2.46858 + 0.141159i −0.518972 1.59723i −0.102889 0.715608i
5.8 0.407235 0.675323i 0.0991267 + 0.0720198i 0.643113 + 1.21884i 2.11727 1.73128i 0.0890045 0.0376136i −3.04355 + 3.94694i 2.65965 + 0.152084i −0.922412 2.83889i −0.306949 2.13488i
5.9 1.11220 1.84437i −1.96002 1.42404i −1.23139 2.33374i −0.598559 + 0.489439i −4.80638 + 2.03119i 0.115990 0.150419i −1.37332 0.0785293i 0.886746 + 2.72912i 0.236992 + 1.64832i
5.10 1.25666 2.08394i 1.16081 + 0.843378i −1.83027 3.46875i −0.796654 + 0.651421i 3.21630 1.35922i −1.00768 + 1.30679i −4.66961 0.267018i −0.290857 0.895166i 0.356397 + 2.47880i
See next 80 embeddings (of 400 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 119.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.g even 55 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.2.g.a 400
121.g even 55 1 inner 121.2.g.a 400
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.2.g.a 400 1.a even 1 1 trivial
121.2.g.a 400 121.g even 55 1 inner

Hecke kernels

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