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Properties

Label 185.2.f.d
Level $185$
Weight $2$
Character orbit 185.f
Analytic conductor $1.477$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [185,2,Mod(43,185)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.f (of order \(4\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 32 q^{4} - 14 q^{6} + 6 q^{7} - 6 q^{10} - 36 q^{12} + 28 q^{14} - 2 q^{15} + 24 q^{16} - 16 q^{17} - 4 q^{19} + 10 q^{20} + 56 q^{22} + 18 q^{24} - 40 q^{26} - 26 q^{27} - 2 q^{28} + 4 q^{29}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 − 2.64177i 2.13485 − 2.13485i −4.97897 1.88841 + 1.19746i −5.63980 − 5.63980i −2.38316 + 2.38316i 7.86977i − 6.11519i 3.16342 − 4.98875i
43.2 − 2.38796i 0.646256 − 0.646256i −3.70235 −1.50931 − 1.64985i −1.54323 − 1.54323i 2.20744 − 2.20744i 4.06513i 2.16471i −3.93976 + 3.60417i
43.3 − 1.72182i −0.229804 + 0.229804i −0.964650 2.09318 + 0.786516i 0.395681 + 0.395681i 1.16936 − 1.16936i − 1.78268i 2.89438i 1.35424 − 3.60407i
43.4 − 1.64338i −1.31039 + 1.31039i −0.700698 −1.24291 + 1.85881i 2.15347 + 2.15347i −2.64033 + 2.64033i − 2.13525i − 0.434259i 3.05473 + 2.04257i
43.5 − 1.40625i 1.57253 − 1.57253i 0.0224648 −2.07072 + 0.843869i −2.21137 − 2.21137i −0.0416138 + 0.0416138i − 2.84409i − 1.94570i 1.18669 + 2.91195i
43.6 − 0.917493i 1.17965 − 1.17965i 1.15821 0.796235 − 2.08950i −1.08232 − 1.08232i −2.45118 + 2.45118i − 2.89763i 0.216841i −1.91710 − 0.730540i
43.7 0.0562904i −0.608697 + 0.608697i 1.99683 −1.63670 + 1.52355i −0.0342638 − 0.0342638i 3.37670 − 3.37670i 0.224983i 2.25898i −0.0857611 − 0.0921308i
43.8 0.165127i −1.77117 + 1.77117i 1.97273 2.13211 − 0.673884i −0.292468 − 0.292468i −0.477215 + 0.477215i 0.656006i − 3.27410i 0.111276 + 0.352069i
43.9 1.55778i 0.374562 − 0.374562i −0.426681 0.557533 − 2.16545i 0.583485 + 0.583485i 2.24838 − 2.24838i 2.45089i 2.71941i 3.37329 + 0.868515i
43.10 1.89107i −1.70537 + 1.70537i −1.57614 0.501674 + 2.17906i −3.22498 − 3.22498i 0.431915 − 0.431915i 0.801541i − 2.81659i −4.12076 + 0.948700i
43.11 2.35455i 2.15478 − 2.15478i −3.54389 0.725340 + 2.11516i 5.07354 + 5.07354i 0.853954 − 0.853954i − 3.63515i − 6.28619i −4.98023 + 1.70785i
43.12 2.69386i −0.437197 + 0.437197i −5.25686 −2.23483 + 0.0742518i −1.17775 − 1.17775i 0.705745 − 0.705745i − 8.77351i 2.61772i −0.200024 − 6.02032i
142.1 − 2.69386i −0.437197 − 0.437197i −5.25686 −2.23483 − 0.0742518i −1.17775 + 1.17775i 0.705745 + 0.705745i 8.77351i − 2.61772i −0.200024 + 6.02032i
142.2 − 2.35455i 2.15478 + 2.15478i −3.54389 0.725340 − 2.11516i 5.07354 − 5.07354i 0.853954 + 0.853954i 3.63515i 6.28619i −4.98023 − 1.70785i
142.3 − 1.89107i −1.70537 − 1.70537i −1.57614 0.501674 − 2.17906i −3.22498 + 3.22498i 0.431915 + 0.431915i − 0.801541i 2.81659i −4.12076 − 0.948700i
142.4 − 1.55778i 0.374562 + 0.374562i −0.426681 0.557533 + 2.16545i 0.583485 − 0.583485i 2.24838 + 2.24838i − 2.45089i − 2.71941i 3.37329 − 0.868515i
142.5 − 0.165127i −1.77117 − 1.77117i 1.97273 2.13211 + 0.673884i −0.292468 + 0.292468i −0.477215 − 0.477215i − 0.656006i 3.27410i 0.111276 − 0.352069i
142.6 − 0.0562904i −0.608697 − 0.608697i 1.99683 −1.63670 − 1.52355i −0.0342638 + 0.0342638i 3.37670 + 3.37670i − 0.224983i − 2.25898i −0.0857611 + 0.0921308i
142.7 0.917493i 1.17965 + 1.17965i 1.15821 0.796235 + 2.08950i −1.08232 + 1.08232i −2.45118 − 2.45118i 2.89763i − 0.216841i −1.91710 + 0.730540i
142.8 1.40625i 1.57253 + 1.57253i 0.0224648 −2.07072 − 0.843869i −2.21137 + 2.21137i −0.0416138 − 0.0416138i 2.84409i 1.94570i 1.18669 − 2.91195i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.f.d ✓ 24
5.b even 2 1 925.2.f.d 24
5.c odd 4 1 185.2.k.d yes 24
5.c odd 4 1 925.2.k.d 24
37.d odd 4 1 185.2.k.d yes 24
185.f even 4 1 inner 185.2.f.d ✓ 24
185.j odd 4 1 925.2.k.d 24
185.k even 4 1 925.2.f.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.f.d ✓ 24 1.a even 1 1 trivial
185.2.f.d ✓ 24 185.f even 4 1 inner
185.2.k.d yes 24 5.c odd 4 1
185.2.k.d yes 24 37.d odd 4 1
925.2.f.d 24 5.b even 2 1
925.2.f.d 24 185.k even 4 1
925.2.k.d 24 5.c odd 4 1
925.2.k.d 24 185.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(185, [\chi])\):

\( T_{2}^{24} + 40 T_{2}^{22} + 698 T_{2}^{20} + 6984 T_{2}^{18} + 44289 T_{2}^{16} + 185636 T_{2}^{14} + \cdots + 16 \) Copy content Toggle raw display
\( T_{3}^{24} - 4 T_{3}^{23} + 8 T_{3}^{22} + 6 T_{3}^{21} + 88 T_{3}^{20} - 344 T_{3}^{19} + 690 T_{3}^{18} + \cdots + 1024 \) Copy content Toggle raw display