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Properties

Label 185.2.k.d
Level $185$
Weight $2$
Character orbit 185.k
Analytic conductor $1.477$
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] = [185,2,Mod(68,185)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.k (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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} - 4 q^{3} + 32 q^{4} + 8 q^{5} - 14 q^{6} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} - 4 q^{3} + 32 q^{4} + 8 q^{5} - 14 q^{6} + 6 q^{7} - 12 q^{8} - 6 q^{10} - 36 q^{12} + 20 q^{13} - 28 q^{14} + 6 q^{15} + 24 q^{16} + 4 q^{19} + 26 q^{20} + 20 q^{23} - 18 q^{24} - 40 q^{26} + 26 q^{27} + 2 q^{28} - 4 q^{29} + 10 q^{30} + 6 q^{31} - 32 q^{33} + 16 q^{35} + 2 q^{37} - 50 q^{38} - 46 q^{39} - 22 q^{40} - 12 q^{43} - 50 q^{45} - 8 q^{46} - 6 q^{47} - 40 q^{48} - 40 q^{50} + 2 q^{51} - 44 q^{52} + 28 q^{53} - 18 q^{54} + 30 q^{55} - 4 q^{56} - 32 q^{57} + 16 q^{58} + 10 q^{59} + 24 q^{60} - 40 q^{61} + 16 q^{62} + 68 q^{63} + 32 q^{64} - 12 q^{65} + 4 q^{66} + 20 q^{67} + 60 q^{69} - 12 q^{70} - 4 q^{71} - 48 q^{73} - 26 q^{74} + 28 q^{75} + 12 q^{76} + 40 q^{77} + 50 q^{78} + 22 q^{79} + 2 q^{80} - 28 q^{83} + 32 q^{85} + 148 q^{86} + 20 q^{89} + 14 q^{90} + 26 q^{91} + 100 q^{92} - 8 q^{93} - 32 q^{94} - 16 q^{95} - 94 q^{96} + 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −2.69386 0.437197 − 0.437197i 5.25686 0.0742518 + 2.23483i −1.17775 + 1.17775i 0.705745 − 0.705745i −8.77351 2.61772i −0.200024 − 6.02032i
68.2 −2.35455 −2.15478 + 2.15478i 3.54389 2.11516 − 0.725340i 5.07354 − 5.07354i 0.853954 − 0.853954i −3.63515 − 6.28619i −4.98023 + 1.70785i
68.3 −1.89107 1.70537 − 1.70537i 1.57614 2.17906 − 0.501674i −3.22498 + 3.22498i 0.431915 − 0.431915i 0.801541 − 2.81659i −4.12076 + 0.948700i
68.4 −1.55778 −0.374562 + 0.374562i 0.426681 −2.16545 − 0.557533i 0.583485 − 0.583485i 2.24838 − 2.24838i 2.45089 2.71941i 3.37329 + 0.868515i
68.5 −0.165127 1.77117 − 1.77117i −1.97273 −0.673884 − 2.13211i −0.292468 + 0.292468i −0.477215 + 0.477215i 0.656006 − 3.27410i 0.111276 + 0.352069i
68.6 −0.0562904 0.608697 − 0.608697i −1.99683 1.52355 + 1.63670i −0.0342638 + 0.0342638i 3.37670 − 3.37670i 0.224983 2.25898i −0.0857611 − 0.0921308i
68.7 0.917493 −1.17965 + 1.17965i −1.15821 −2.08950 − 0.796235i −1.08232 + 1.08232i −2.45118 + 2.45118i −2.89763 0.216841i −1.91710 − 0.730540i
68.8 1.40625 −1.57253 + 1.57253i −0.0224648 0.843869 + 2.07072i −2.21137 + 2.21137i −0.0416138 + 0.0416138i −2.84409 − 1.94570i 1.18669 + 2.91195i
68.9 1.64338 1.31039 − 1.31039i 0.700698 1.85881 + 1.24291i 2.15347 − 2.15347i −2.64033 + 2.64033i −2.13525 − 0.434259i 3.05473 + 2.04257i
68.10 1.72182 0.229804 − 0.229804i 0.964650 0.786516 − 2.09318i 0.395681 − 0.395681i 1.16936 − 1.16936i −1.78268 2.89438i 1.35424 − 3.60407i
68.11 2.38796 −0.646256 + 0.646256i 3.70235 −1.64985 + 1.50931i −1.54323 + 1.54323i 2.20744 − 2.20744i 4.06513 2.16471i −3.93976 + 3.60417i
68.12 2.64177 −2.13485 + 2.13485i 4.97897 1.19746 − 1.88841i −5.63980 + 5.63980i −2.38316 + 2.38316i 7.86977 − 6.11519i 3.16342 − 4.98875i
117.1 −2.69386 0.437197 + 0.437197i 5.25686 0.0742518 − 2.23483i −1.17775 − 1.17775i 0.705745 + 0.705745i −8.77351 − 2.61772i −0.200024 + 6.02032i
117.2 −2.35455 −2.15478 − 2.15478i 3.54389 2.11516 + 0.725340i 5.07354 + 5.07354i 0.853954 + 0.853954i −3.63515 6.28619i −4.98023 − 1.70785i
117.3 −1.89107 1.70537 + 1.70537i 1.57614 2.17906 + 0.501674i −3.22498 − 3.22498i 0.431915 + 0.431915i 0.801541 2.81659i −4.12076 − 0.948700i
117.4 −1.55778 −0.374562 − 0.374562i 0.426681 −2.16545 + 0.557533i 0.583485 + 0.583485i 2.24838 + 2.24838i 2.45089 − 2.71941i 3.37329 − 0.868515i
117.5 −0.165127 1.77117 + 1.77117i −1.97273 −0.673884 + 2.13211i −0.292468 − 0.292468i −0.477215 − 0.477215i 0.656006 3.27410i 0.111276 − 0.352069i
117.6 −0.0562904 0.608697 + 0.608697i −1.99683 1.52355 − 1.63670i −0.0342638 − 0.0342638i 3.37670 + 3.37670i 0.224983 − 2.25898i −0.0857611 + 0.0921308i
117.7 0.917493 −1.17965 − 1.17965i −1.15821 −2.08950 + 0.796235i −1.08232 − 1.08232i −2.45118 − 2.45118i −2.89763 − 0.216841i −1.91710 + 0.730540i
117.8 1.40625 −1.57253 − 1.57253i −0.0224648 0.843869 − 2.07072i −2.21137 − 2.21137i −0.0416138 − 0.0416138i −2.84409 1.94570i 1.18669 − 2.91195i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2}^{12} - 2 T_{2}^{11} - 18 T_{2}^{10} + 38 T_{2}^{9} + 111 T_{2}^{8} - 258 T_{2}^{7} - 256 T_{2}^{6} + \cdots + 4 \) 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