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Properties

Label 201.3.h.b
Level $201$
Weight $3$
Character orbit 201.h
Analytic conductor $5.477$
Analytic rank $0$
Dimension $24$
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: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 34 q^{4} + 21 q^{7} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 34 q^{4} + 21 q^{7} - 72 q^{9} + 30 q^{10} + 12 q^{11} + 102 q^{12} + 30 q^{13} + 6 q^{14} - 6 q^{15} - 98 q^{16} + 4 q^{17} + 39 q^{19} + 108 q^{20} - 21 q^{21} - 82 q^{22} - 13 q^{23} - 36 q^{24} - 222 q^{25} + 29 q^{26} + 189 q^{28} - 90 q^{30} - 93 q^{31} - 33 q^{32} - 12 q^{33} + 72 q^{34} - 93 q^{35} - 102 q^{36} - 33 q^{37} - 210 q^{38} + 30 q^{39} + 274 q^{40} - 15 q^{41} - 9 q^{44} - 228 q^{46} - 131 q^{47} + 294 q^{48} + 295 q^{49} - 423 q^{50} - 12 q^{51} - 56 q^{55} + 349 q^{56} - 117 q^{57} - 116 q^{59} - 108 q^{60} + 120 q^{61} + 282 q^{62} - 63 q^{63} - 276 q^{64} + 136 q^{65} - 25 q^{67} + 10 q^{68} + 39 q^{69} + 169 q^{71} + 187 q^{73} + 849 q^{74} + 386 q^{76} - 348 q^{77} + 87 q^{78} + 51 q^{80} + 216 q^{81} - 14 q^{82} + 104 q^{83} + 189 q^{84} - 243 q^{85} + 641 q^{86} - 766 q^{88} + 152 q^{89} - 90 q^{90} + 32 q^{91} - 216 q^{92} + 93 q^{93} + 762 q^{95} + 33 q^{96} - 84 q^{97} - 1086 q^{98} - 36 q^{99}+O(q^{100}) \) 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 −3.23066 + 1.86522i 1.73205i 4.95809 − 8.58766i − 1.91835i −3.23066 − 5.59566i −11.0264 − 6.36611i 22.0700i −3.00000 3.57814 + 6.19753i
97.2 −3.01185 + 1.73889i 1.73205i 4.04748 − 7.01044i − 4.44467i −3.01185 − 5.21667i 8.21480 + 4.74282i 14.2414i −3.00000 7.72880 + 13.3867i
97.3 −2.78358 + 1.60710i 1.73205i 3.16555 − 5.48289i 6.80030i −2.78358 − 4.82130i 9.81487 + 5.66662i 7.49262i −3.00000 −10.9288 − 18.9292i
97.4 −1.79024 + 1.03359i 1.73205i 0.136637 − 0.236663i 0.118573i −1.79024 − 3.10078i −4.18772 − 2.41778i − 7.70385i −3.00000 −0.122556 − 0.212274i
97.5 −0.893138 + 0.515653i 1.73205i −1.46820 + 2.54300i − 8.85241i −0.893138 − 1.54696i 7.85559 + 4.53543i − 7.15356i −3.00000 4.56478 + 7.90642i
97.6 −0.415796 + 0.240060i 1.73205i −1.88474 + 3.26447i 7.03218i −0.415796 − 0.720180i 1.25782 + 0.726201i − 3.73029i −3.00000 −1.68815 − 2.92395i
97.7 0.535032 − 0.308901i 1.73205i −1.80916 + 3.13356i − 6.48720i 0.535032 + 0.926703i −10.3489 − 5.97496i 4.70662i −3.00000 −2.00390 − 3.47086i
97.8 1.15796 − 0.668551i 1.73205i −1.10608 + 1.91579i 0.749427i 1.15796 + 2.00565i 2.89468 + 1.67125i 8.30629i −3.00000 0.501030 + 0.867810i
97.9 2.00217 − 1.15595i 1.73205i 0.672461 − 1.16474i − 1.12578i 2.00217 + 3.46786i 6.42822 + 3.71133i 6.13830i −3.00000 −1.30135 − 2.25401i
97.10 2.13074 − 1.23018i 1.73205i 1.02670 − 1.77830i 7.82361i 2.13074 + 3.69055i −7.43732 − 4.29394i 4.78934i −3.00000 9.62447 + 16.6701i
97.11 2.98997 − 1.72626i 1.73205i 3.95993 − 6.85880i − 6.27203i 2.98997 + 5.17877i −1.92187 − 1.10959i − 13.5334i −3.00000 −10.8271 − 18.7532i
97.12 3.30938 − 1.91067i 1.73205i 5.30134 − 9.18219i 8.30841i 3.30938 + 5.73202i 8.95628 + 5.17091i − 25.2311i −3.00000 15.8747 + 27.4957i
172.1 −3.23066 − 1.86522i − 1.73205i 4.95809 + 8.58766i 1.91835i −3.23066 + 5.59566i −11.0264 + 6.36611i − 22.0700i −3.00000 3.57814 − 6.19753i
172.2 −3.01185 − 1.73889i − 1.73205i 4.04748 + 7.01044i 4.44467i −3.01185 + 5.21667i 8.21480 − 4.74282i − 14.2414i −3.00000 7.72880 − 13.3867i
172.3 −2.78358 − 1.60710i − 1.73205i 3.16555 + 5.48289i − 6.80030i −2.78358 + 4.82130i 9.81487 − 5.66662i − 7.49262i −3.00000 −10.9288 + 18.9292i
172.4 −1.79024 − 1.03359i − 1.73205i 0.136637 + 0.236663i − 0.118573i −1.79024 + 3.10078i −4.18772 + 2.41778i 7.70385i −3.00000 −0.122556 + 0.212274i
172.5 −0.893138 − 0.515653i − 1.73205i −1.46820 − 2.54300i 8.85241i −0.893138 + 1.54696i 7.85559 − 4.53543i 7.15356i −3.00000 4.56478 − 7.90642i
172.6 −0.415796 − 0.240060i − 1.73205i −1.88474 − 3.26447i − 7.03218i −0.415796 + 0.720180i 1.25782 − 0.726201i 3.73029i −3.00000 −1.68815 + 2.92395i
172.7 0.535032 + 0.308901i − 1.73205i −1.80916 − 3.13356i 6.48720i 0.535032 − 0.926703i −10.3489 + 5.97496i − 4.70662i −3.00000 −2.00390 + 3.47086i
172.8 1.15796 + 0.668551i − 1.73205i −1.10608 − 1.91579i − 0.749427i 1.15796 − 2.00565i 2.89468 − 1.67125i − 8.30629i −3.00000 0.501030 − 0.867810i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.12
Significant digits:
Format:

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.b ✓ 24
67.d odd 6 1 inner 201.3.h.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.3.h.b ✓ 24 1.a even 1 1 trivial
201.3.h.b ✓ 24 67.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 41 T_{2}^{22} + 1063 T_{2}^{20} - 51 T_{2}^{19} - 16998 T_{2}^{18} + 1779 T_{2}^{17} + \cdots + 7001316 \) acting on \(S_{3}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display