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Properties

Label 370.2.v.a
Level $370$
Weight $2$
Character orbit 370.v
Analytic conductor $2.954$
Analytic rank $0$
Dimension $60$
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] = [370,2,Mod(99,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.v (of order \(18\), degree \(6\), 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: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{5} - 30 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{5} - 30 q^{8} - 6 q^{9} + 9 q^{10} + 6 q^{11} - 3 q^{13} + 9 q^{14} + 18 q^{15} + 3 q^{17} + 12 q^{18} + 3 q^{19} - 6 q^{20} + 12 q^{21} - 3 q^{22} - 18 q^{23} - 15 q^{25} - 27 q^{26} + 6 q^{30} - 66 q^{33} + 21 q^{34} - 3 q^{35} - 72 q^{36} + 6 q^{37} + 12 q^{39} + 6 q^{40} - 57 q^{41} + 12 q^{42} + 60 q^{43} - 3 q^{44} + 45 q^{45} - 21 q^{46} - 18 q^{47} - 6 q^{49} + 36 q^{50} - 3 q^{52} - 6 q^{53} + 3 q^{55} + 30 q^{57} + 15 q^{58} - 12 q^{59} + 9 q^{60} + 42 q^{61} - 12 q^{62} - 9 q^{63} - 30 q^{64} - 30 q^{65} - 18 q^{67} - 24 q^{68} + 36 q^{69} + 18 q^{71} + 12 q^{72} + 3 q^{74} - 78 q^{75} - 3 q^{76} + 21 q^{77} + 6 q^{78} - 24 q^{79} - 36 q^{81} + 27 q^{82} + 30 q^{83} + 12 q^{84} + 33 q^{85} - 18 q^{86} - 30 q^{87} + 6 q^{88} + 3 q^{89} - 3 q^{90} + 57 q^{91} + 15 q^{92} + 60 q^{93} + 3 q^{94} - 150 q^{95} + 84 q^{97} - 6 q^{98} - 45 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 0.766044 − 0.642788i −2.08950 + 2.49017i 0.173648 − 0.984808i −0.926410 − 2.03513i 3.25068i −0.0369625 − 0.101554i −0.500000 − 0.866025i −1.31399 − 7.45198i −2.01783 − 0.963518i
99.2 0.766044 − 0.642788i −1.54829 + 1.84518i 0.173648 − 0.984808i 1.37327 + 1.76469i 2.40871i −0.660116 − 1.81365i −0.500000 − 0.866025i −0.486541 − 2.75931i 2.18631 + 0.469106i
99.3 0.766044 − 0.642788i −0.778276 + 0.927514i 0.173648 − 0.984808i −2.23589 − 0.0280424i 1.21078i 1.22420 + 3.36347i −0.500000 − 0.866025i 0.266377 + 1.51070i −1.73082 + 1.41572i
99.4 0.766044 − 0.642788i −0.744057 + 0.886733i 0.173648 − 0.984808i −2.02782 + 0.942308i 1.15755i −1.68887 − 4.64014i −0.500000 − 0.866025i 0.288270 + 1.63486i −0.947697 + 2.02531i
99.5 0.766044 − 0.642788i −0.173834 + 0.207168i 0.173648 − 0.984808i 1.80772 − 1.31612i 0.270438i 1.26091 + 3.46433i −0.500000 − 0.866025i 0.508244 + 2.88240i 0.538808 − 2.17018i
99.6 0.766044 − 0.642788i 0.0128020 − 0.0152568i 0.173648 − 0.984808i −0.0951277 + 2.23404i − 0.0199163i 0.768093 + 2.11032i −0.500000 − 0.866025i 0.520876 + 2.95403i 1.36314 + 1.77252i
99.7 0.766044 − 0.642788i 0.702134 − 0.836770i 0.173648 − 0.984808i 0.315694 − 2.21367i − 1.09233i −0.224189 − 0.615955i −0.500000 − 0.866025i 0.313752 + 1.77937i −1.18108 − 1.89869i
99.8 0.766044 − 0.642788i 1.01319 − 1.20747i 0.173648 − 0.984808i 2.01994 + 0.959087i − 1.57624i −1.06132 − 2.91595i −0.500000 − 0.866025i 0.0895092 + 0.507632i 2.16385 − 0.563688i
99.9 0.766044 − 0.642788i 1.56430 − 1.86427i 0.173648 − 0.984808i −2.11298 − 0.731650i − 2.43363i −0.424051 − 1.16507i −0.500000 − 0.866025i −0.507493 − 2.87813i −2.08893 + 0.797722i
99.10 0.766044 − 0.642788i 2.04153 − 2.43300i 0.173648 − 0.984808i 0.0946205 + 2.23407i − 3.17605i 1.43470 + 3.94181i −0.500000 − 0.866025i −1.23070 − 6.97964i 1.50851 + 1.65057i
139.1 0.173648 + 0.984808i −3.12092 − 0.550303i −0.939693 + 0.342020i 2.06113 + 0.867024i − 3.16907i 0.240083 − 0.286120i −0.500000 − 0.866025i 6.61825 + 2.40884i −0.495939 + 2.18038i
139.2 0.173648 + 0.984808i −2.25124 − 0.396954i −0.939693 + 0.342020i −0.146136 − 2.23129i − 2.28597i −1.61129 + 1.92026i −0.500000 − 0.866025i 2.09141 + 0.761213i 2.17201 − 0.531375i
139.3 0.173648 + 0.984808i −2.03788 − 0.359333i −0.939693 + 0.342020i −1.88707 + 1.19956i − 2.06931i 3.28437 − 3.91416i −0.500000 − 0.866025i 1.20474 + 0.438490i −1.50902 − 1.65010i
139.4 0.173648 + 0.984808i −1.09803 − 0.193612i −0.939693 + 0.342020i 1.25109 − 1.85331i − 1.11497i 0.258400 − 0.307949i −0.500000 − 0.866025i −1.65090 − 0.600879i 2.04241 + 0.910262i
139.5 0.173648 + 0.984808i 0.200382 + 0.0353327i −0.939693 + 0.342020i −2.08551 − 0.806626i 0.203473i −1.04065 + 1.24019i −0.500000 − 0.866025i −2.78017 − 1.01190i 0.432227 − 2.19390i
139.6 0.173648 + 0.984808i 0.213457 + 0.0376382i −0.939693 + 0.342020i −1.42274 + 1.72506i 0.216750i −1.62043 + 1.93115i −0.500000 − 0.866025i −2.77493 − 1.00999i −1.94591 − 1.10157i
139.7 0.173648 + 0.984808i 0.529665 + 0.0933942i −0.939693 + 0.342020i 2.21470 − 0.308411i 0.537836i 2.48431 − 2.96068i −0.500000 − 0.866025i −2.54726 − 0.927125i 0.688304 + 2.12750i
139.8 0.173648 + 0.984808i 1.98897 + 0.350709i −0.939693 + 0.342020i 2.20934 − 0.344670i 2.01965i −2.80309 + 3.34059i −0.500000 − 0.866025i 1.01393 + 0.369039i 0.723082 + 2.11593i
139.9 0.173648 + 0.984808i 2.67113 + 0.470993i −0.939693 + 0.342020i 0.257678 + 2.22117i 2.71234i 0.343566 − 0.409446i −0.500000 − 0.866025i 4.09404 + 1.49011i −2.14268 + 0.639466i
139.10 0.173648 + 0.984808i 2.90445 + 0.512134i −0.939693 + 0.342020i −0.307059 − 2.21488i 2.94926i 1.57806 − 1.88066i −0.500000 − 0.866025i 5.35450 + 1.94888i 2.12792 − 0.687005i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.v.a ✓ 60
5.b even 2 1 370.2.v.b yes 60
37.h even 18 1 370.2.v.b yes 60
185.v even 18 1 inner 370.2.v.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.v.a ✓ 60 1.a even 1 1 trivial
370.2.v.a ✓ 60 185.v even 18 1 inner
370.2.v.b yes 60 5.b even 2 1
370.2.v.b yes 60 37.h even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + 3 T_{3}^{58} - 24 T_{3}^{55} - 1324 T_{3}^{54} + 324 T_{3}^{53} - 3666 T_{3}^{52} + \cdots + 262144 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display