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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 $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,2,Mod(4,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 201.m (of order \(33\), degree \(20\), 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: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(5\) over \(\Q(\zeta_{33})\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 100 q + 2 q^{2} + 10 q^{3} + 6 q^{4} - 2 q^{5} + 9 q^{6} + q^{7} - 45 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q + 2 q^{2} + 10 q^{3} + 6 q^{4} - 2 q^{5} + 9 q^{6} + q^{7} - 45 q^{8} - 10 q^{9} - 25 q^{10} + 9 q^{11} + 5 q^{12} - 3 q^{13} + 22 q^{14} - 9 q^{15} - 46 q^{17} + 2 q^{18} - 14 q^{19} - 16 q^{20} - q^{21} - 17 q^{22} + 12 q^{23} + 12 q^{24} - 44 q^{25} - 7 q^{26} + 10 q^{27} - 90 q^{28} + 43 q^{29} - 30 q^{30} - 10 q^{31} + 20 q^{32} - 42 q^{33} + 20 q^{34} + 3 q^{35} - 5 q^{36} + 50 q^{37} + 16 q^{38} + 3 q^{39} + 55 q^{40} - 48 q^{41} - 45 q^{43} - 147 q^{44} - 2 q^{45} + 47 q^{46} + 44 q^{47} - 64 q^{49} - 54 q^{50} + 24 q^{51} - 34 q^{52} + 68 q^{53} - 2 q^{54} - 17 q^{55} + 111 q^{56} + 3 q^{57} + 88 q^{58} - 2 q^{59} - 6 q^{60} + 21 q^{61} + 86 q^{62} + 23 q^{63} + 19 q^{64} + 6 q^{65} + 94 q^{66} + 20 q^{67} - 202 q^{68} + 21 q^{69} - 20 q^{70} + q^{71} + 32 q^{72} + 10 q^{73} + 67 q^{74} + 44 q^{75} - 90 q^{76} - 62 q^{77} + 51 q^{78} - 29 q^{79} + 199 q^{80} - 10 q^{81} - 36 q^{82} + 43 q^{83} - 75 q^{84} + 93 q^{85} - 83 q^{86} + 12 q^{87} + 54 q^{88} + 21 q^{89} - 25 q^{90} + 58 q^{91} - 192 q^{92} - q^{93} + 14 q^{94} - 109 q^{95} - 9 q^{96} - 35 q^{97} - 15 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 4.5
Significant digits:
Format:

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 \( T_{2}^{100} - 2 T_{2}^{99} - 6 T_{2}^{98} + 35 T_{2}^{97} - 43 T_{2}^{96} - 212 T_{2}^{95} + \cdots + 17161 \) acting on \(S_{2}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display