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Properties

Label 201.3.h.a
Level $201$
Weight $3$
Character orbit 201.h
Analytic conductor $5.477$
Analytic rank $0$
Dimension $22$
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] = [201,3,Mod(97,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 201.h (of order \(6\), degree \(2\), 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: \(5.47685331364\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 26 q^{4} - 15 q^{7} - 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 26 q^{4} - 15 q^{7} - 66 q^{9} + 6 q^{10} + 30 q^{11} - 78 q^{12} - 27 q^{13} + 6 q^{14} - 6 q^{15} - 58 q^{16} - 8 q^{17} + q^{19} - 12 q^{20} - 15 q^{21} + 14 q^{22} - q^{23} - 56 q^{25} + 71 q^{26} - 75 q^{28} + 42 q^{29} + 18 q^{30} + 120 q^{31} - 105 q^{32} + 30 q^{33} - 24 q^{34} + 3 q^{35} - 78 q^{36} + 13 q^{37} + 108 q^{38} + 27 q^{39} + 82 q^{40} - 159 q^{41} + 189 q^{44} + 372 q^{46} - 35 q^{47} - 174 q^{48} - 40 q^{49} + 285 q^{50} - 24 q^{51} - 212 q^{55} - 113 q^{56} + 3 q^{57} + 136 q^{59} - 12 q^{60} + 63 q^{61} - 150 q^{62} + 45 q^{63} - 284 q^{64} - 20 q^{65} + 94 q^{67} + 154 q^{68} - 3 q^{69} - 41 q^{71} - 16 q^{73} + 441 q^{74} - 390 q^{76} - 84 q^{77} - 213 q^{78} - 615 q^{79} + 267 q^{80} + 198 q^{81} - 302 q^{82} - 154 q^{83} + 75 q^{84} + 633 q^{85} + 221 q^{86} - 126 q^{87} + 410 q^{88} + 56 q^{89} - 18 q^{90} + 272 q^{91} + 636 q^{92} + 120 q^{93} + 588 q^{95} - 105 q^{96} - 441 q^{97} + 780 q^{98} - 90 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 −3.30238 + 1.90663i − 1.73205i 5.27046 − 9.12871i 0.881143i 3.30238 + 5.71988i 1.91999 + 1.10851i 24.9422i −3.00000 −1.68001 − 2.90987i
97.2 −2.55387 + 1.47448i − 1.73205i 2.34817 − 4.06715i − 9.46283i 2.55387 + 4.42343i −6.24108 − 3.60329i 2.05347i −3.00000 13.9527 + 24.1668i
97.3 −2.11074 + 1.21863i − 1.73205i 0.970137 − 1.68033i 1.83558i 2.11074 + 3.65590i 4.69421 + 2.71020i − 5.02011i −3.00000 −2.23690 − 3.87442i
97.4 −2.09110 + 1.20729i − 1.73205i 0.915119 − 1.58503i 7.66078i 2.09110 + 3.62188i −6.04816 − 3.49191i − 5.23908i −3.00000 −9.24881 − 16.0194i
97.5 −0.443170 + 0.255864i − 1.73205i −1.86907 + 3.23732i − 5.57502i 0.443170 + 0.767593i 3.11708 + 1.79965i − 3.95982i −3.00000 1.42645 + 2.47068i
97.6 0.0720773 − 0.0416138i − 1.73205i −1.99654 + 3.45810i − 1.32016i −0.0720773 − 0.124841i 1.88003 + 1.08543i 0.665245i −3.00000 −0.0549368 − 0.0951533i
97.7 0.658194 − 0.380008i − 1.73205i −1.71119 + 2.96386i 4.20783i −0.658194 − 1.14002i −7.58591 − 4.37973i 5.64113i −3.00000 1.59901 + 2.76957i
97.8 1.57301 − 0.908176i − 1.73205i −0.350432 + 0.606966i 5.97048i −1.57301 − 2.72453i 6.73988 + 3.89127i 8.53843i −3.00000 5.42225 + 9.39162i
97.9 2.08753 − 1.20524i − 1.73205i 0.905201 − 1.56785i − 5.75683i −2.08753 − 3.61572i −7.36527 − 4.25234i 5.27798i −3.00000 −6.93836 − 12.0176i
97.10 2.82230 − 1.62946i − 1.73205i 3.31026 − 5.73353i − 4.04184i −2.82230 − 4.88837i 7.82069 + 4.51528i − 8.54003i −3.00000 −6.58601 − 11.4073i
97.11 3.28813 − 1.89840i − 1.73205i 5.20788 − 9.02031i 3.86882i −3.28813 − 5.69521i −6.43146 − 3.71321i − 24.3594i −3.00000 7.34458 + 12.7212i
172.1 −3.30238 − 1.90663i 1.73205i 5.27046 + 9.12871i − 0.881143i 3.30238 − 5.71988i 1.91999 − 1.10851i − 24.9422i −3.00000 −1.68001 + 2.90987i
172.2 −2.55387 − 1.47448i 1.73205i 2.34817 + 4.06715i 9.46283i 2.55387 − 4.42343i −6.24108 + 3.60329i − 2.05347i −3.00000 13.9527 − 24.1668i
172.3 −2.11074 − 1.21863i 1.73205i 0.970137 + 1.68033i − 1.83558i 2.11074 − 3.65590i 4.69421 − 2.71020i 5.02011i −3.00000 −2.23690 + 3.87442i
172.4 −2.09110 − 1.20729i 1.73205i 0.915119 + 1.58503i − 7.66078i 2.09110 − 3.62188i −6.04816 + 3.49191i 5.23908i −3.00000 −9.24881 + 16.0194i
172.5 −0.443170 − 0.255864i 1.73205i −1.86907 − 3.23732i 5.57502i 0.443170 − 0.767593i 3.11708 − 1.79965i 3.95982i −3.00000 1.42645 − 2.47068i
172.6 0.0720773 + 0.0416138i 1.73205i −1.99654 − 3.45810i 1.32016i −0.0720773 + 0.124841i 1.88003 − 1.08543i − 0.665245i −3.00000 −0.0549368 + 0.0951533i
172.7 0.658194 + 0.380008i 1.73205i −1.71119 − 2.96386i − 4.20783i −0.658194 + 1.14002i −7.58591 + 4.37973i − 5.64113i −3.00000 1.59901 − 2.76957i
172.8 1.57301 + 0.908176i 1.73205i −0.350432 − 0.606966i − 5.97048i −1.57301 + 2.72453i 6.73988 − 3.89127i − 8.53843i −3.00000 5.42225 − 9.39162i
172.9 2.08753 + 1.20524i 1.73205i 0.905201 + 1.56785i 5.75683i −2.08753 + 3.61572i −7.36527 + 4.25234i − 5.27798i −3.00000 −6.93836 + 12.0176i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.3.h.a ✓ 22
67.d odd 6 1 inner 201.3.h.a ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.3.h.a ✓ 22 1.a even 1 1 trivial
201.3.h.a ✓ 22 67.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 35 T_{2}^{20} + 793 T_{2}^{18} + 21 T_{2}^{17} - 10608 T_{2}^{16} - 447 T_{2}^{15} + \cdots + 13467 \) acting on \(S_{3}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display