# Properties

 Label 201.2.m.a Level 201 Weight 2 Character orbit 201.m Analytic conductor 1.605 Analytic rank 0 Dimension 100 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 201.m (of order $$33$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.60499308063$$ Analytic rank: $$0$$ Dimension: $$100$$ Relative dimension: $$5$$ over $$\Q(\zeta_{33})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$100q + 2q^{2} + 10q^{3} + 6q^{4} - 2q^{5} + 9q^{6} + q^{7} - 45q^{8} - 10q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$100q + 2q^{2} + 10q^{3} + 6q^{4} - 2q^{5} + 9q^{6} + q^{7} - 45q^{8} - 10q^{9} - 25q^{10} + 9q^{11} + 5q^{12} - 3q^{13} + 22q^{14} - 9q^{15} - 46q^{17} + 2q^{18} - 14q^{19} - 16q^{20} - q^{21} - 17q^{22} + 12q^{23} + 12q^{24} - 44q^{25} - 7q^{26} + 10q^{27} - 90q^{28} + 43q^{29} - 30q^{30} - 10q^{31} + 20q^{32} - 42q^{33} + 20q^{34} + 3q^{35} - 5q^{36} + 50q^{37} + 16q^{38} + 3q^{39} + 55q^{40} - 48q^{41} - 45q^{43} - 147q^{44} - 2q^{45} + 47q^{46} + 44q^{47} - 64q^{49} - 54q^{50} + 24q^{51} - 34q^{52} + 68q^{53} - 2q^{54} - 17q^{55} + 111q^{56} + 3q^{57} + 88q^{58} - 2q^{59} - 6q^{60} + 21q^{61} + 86q^{62} + 23q^{63} + 19q^{64} + 6q^{65} + 94q^{66} + 20q^{67} - 202q^{68} + 21q^{69} - 20q^{70} + q^{71} + 32q^{72} + 10q^{73} + 67q^{74} + 44q^{75} - 90q^{76} - 62q^{77} + 51q^{78} - 29q^{79} + 199q^{80} - 10q^{81} - 36q^{82} + 43q^{83} - 75q^{84} + 93q^{85} - 83q^{86} + 12q^{87} + 54q^{88} + 21q^{89} - 25q^{90} + 58q^{91} - 192q^{92} - q^{93} + 14q^{94} - 109q^{95} - 9q^{96} - 35q^{97} - 15q^{98} - 2q^{99} + O(q^{100})$$

## 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 −2.41489 0.230594i −0.841254 0.540641i 3.81465 + 0.735214i −0.381162 2.65104i 1.90686 + 1.49958i −0.198983 + 0.279433i −4.38721 1.28820i 0.415415 + 0.909632i 0.309150 + 6.48987i
4.2 −0.579656 0.0553505i −0.841254 0.540641i −1.63092 0.314334i 0.376717 + 2.62012i 0.457713 + 0.359950i 1.92893 2.70880i 2.04539 + 0.600580i 0.415415 + 0.909632i −0.0733413 1.53962i
4.3 0.0298103 + 0.00284654i −0.841254 0.540641i −1.96298 0.378333i 0.0906218 + 0.630289i −0.0235391 0.0185113i −1.85236 + 2.60128i −0.114906 0.0337394i 0.415415 + 0.909632i 0.000907323 0.0190471i
4.4 1.11009 + 0.106001i −0.841254 0.540641i −0.742792 0.143161i −0.436900 3.03870i −0.876559 0.689334i 0.656215 0.921525i −2.94933 0.866001i 0.415415 + 0.909632i −0.162893 3.41955i
4.5 2.13798 + 0.204153i −0.841254 0.540641i 2.56544 + 0.494448i 0.515825 + 3.58764i −1.68821 1.32763i 1.76546 2.47924i 1.26251 + 0.370705i 0.415415 + 0.909632i 0.370399 + 7.77563i
10.1 −1.45951 1.39164i 0.142315 + 0.989821i 0.0983437 + 2.06449i 0.189954 0.415942i 1.16976 1.64270i −0.918966 + 3.78803i 0.0882556 0.101852i −0.959493 + 0.281733i −0.856081 + 0.342723i
10.2 −0.928460 0.885285i 0.142315 + 0.989821i −0.0168551 0.353832i 0.471271 1.03194i 0.744140 1.04500i 0.634488 2.61540i −1.97780 + 2.28250i −0.959493 + 0.281733i −1.35112 + 0.540906i
10.3 0.576780 + 0.549958i 0.142315 + 0.989821i −0.0649432 1.36333i 1.13405 2.48322i −0.462276 + 0.649176i 0.467919 1.92879i 1.75610 2.02664i −0.959493 + 0.281733i 2.01976 0.808591i
10.4 0.752307 + 0.717323i 0.142315 + 0.989821i −0.0437507 0.918439i −1.22616 + 2.68491i −0.602958 + 0.846736i −0.560955 + 2.31229i 1.98733 2.29351i −0.959493 + 0.281733i −2.84839 + 1.14032i
10.5 1.66018 + 1.58298i 0.142315 + 0.989821i 0.155214 + 3.25834i −0.764993 + 1.67510i −1.33060 + 1.86857i 0.871467 3.59223i −1.89582 + 2.18789i −0.959493 + 0.281733i −3.92168 + 1.57000i
16.1 −2.51507 0.484739i −0.415415 0.909632i 4.23385 + 1.69498i −1.33736 + 0.392683i 0.603862 + 2.48915i −0.572823 1.65506i −5.51729 3.54575i −0.654861 + 0.755750i 3.55389 0.339355i
16.2 −1.41154 0.272052i −0.415415 0.909632i 0.0616901 + 0.0246970i 2.51664 0.738953i 0.338907 + 1.39699i 0.418058 + 1.20790i 2.33827 + 1.50272i −0.654861 + 0.755750i −3.75337 + 0.358403i
16.3 −0.932247 0.179676i −0.415415 0.909632i −1.01993 0.408320i −1.95677 + 0.574560i 0.223831 + 0.922642i 0.380730 + 1.10005i 2.47484 + 1.59049i −0.654861 + 0.755750i 1.92743 0.184047i
16.4 0.618425 + 0.119192i −0.415415 0.909632i −1.48849 0.595903i 0.388724 0.114140i −0.148482 0.612053i −1.43299 4.14037i −1.90915 1.22694i −0.654861 + 0.755750i 0.254001 0.0242541i
16.5 2.35612 + 0.454105i −0.415415 0.909632i 3.48835 + 1.39652i −0.238878 + 0.0701410i −0.565699 2.33184i −0.0543848 0.157135i 3.54765 + 2.27993i −0.654861 + 0.755750i −0.594677 + 0.0567848i
19.1 −1.46772 0.756664i 0.959493 0.281733i 0.421557 + 0.591994i 1.22831 + 1.41754i −1.62145 0.312508i −0.160533 3.37000i 0.299217 + 2.08110i 0.841254 0.540641i −0.730211 3.00997i
19.2 −1.29041 0.665251i 0.959493 0.281733i 0.0624755 + 0.0877346i 1.62071 + 1.87040i −1.42556 0.274754i 0.228443 + 4.79562i 0.390970 + 2.71926i 0.841254 0.540641i −0.847092 3.49176i
19.3 0.309019 + 0.159310i 0.959493 0.281733i −1.09000 1.53069i −0.990324 1.14290i 0.341385 + 0.0657965i −0.0746351 1.56678i −0.191932 1.33492i 0.841254 0.540641i −0.123954 0.510945i
19.4 1.36337 + 0.702865i 0.959493 0.281733i 0.204640 + 0.287377i 0.953826 + 1.10077i 1.50616 + 0.290289i 0.0382883 + 0.803771i −0.359576 2.50090i 0.841254 0.540641i 0.526720 + 2.17117i
19.5 2.24987 + 1.15989i 0.959493 0.281733i 2.55645 + 3.59004i −2.75020 3.17390i 2.48551 + 0.479043i 0.118800 + 2.49393i 0.867175 + 6.03133i 0.841254 0.540641i −2.50622 10.3308i
See all 100 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.g even 33 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.m.a 100
3.b odd 2 1 603.2.z.b 100
67.g even 33 1 inner 201.2.m.a 100
201.o odd 66 1 603.2.z.b 100

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.m.a 100 1.a even 1 1 trivial
201.2.m.a 100 67.g even 33 1 inner
603.2.z.b 100 3.b odd 2 1
603.2.z.b 100 201.o odd 66 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{100} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database