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Properties

Label 185.2.o.b
Level $185$
Weight $2$
Character orbit 185.o
Analytic conductor $1.477$
Analytic rank $0$
Dimension $36$
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(16,185)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.o (of order \(9\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} - 3 q^{4} + 12 q^{6} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} - 3 q^{4} + 12 q^{6} - 6 q^{8} - 3 q^{10} + 6 q^{11} - 36 q^{12} + 6 q^{13} + 15 q^{14} - 9 q^{16} + 6 q^{17} - 27 q^{18} - 3 q^{19} - 3 q^{20} - 24 q^{21} - 45 q^{22} - 30 q^{23} - 15 q^{24} - 12 q^{27} + 9 q^{28} - 15 q^{29} - 12 q^{30} + 30 q^{31} + 36 q^{32} + 42 q^{33} + 27 q^{34} - 9 q^{35} - 42 q^{36} + 24 q^{37} + 48 q^{38} - 6 q^{40} - 21 q^{41} + 84 q^{42} + 72 q^{43} + 6 q^{44} - 9 q^{45} + 15 q^{46} - 51 q^{47} + 30 q^{48} - 18 q^{49} - 3 q^{50} + 18 q^{51} + 18 q^{52} - 9 q^{53} + 9 q^{54} - 15 q^{55} - 18 q^{56} - 21 q^{57} - 69 q^{58} - 39 q^{59} - 6 q^{60} - 6 q^{61} + 90 q^{62} + 27 q^{63} + 6 q^{64} + 6 q^{65} - 69 q^{66} - 15 q^{67} + 6 q^{68} + 3 q^{69} + 27 q^{70} - 6 q^{71} - 3 q^{72} + 15 q^{74} + 6 q^{75} - 54 q^{76} - 30 q^{78} + 36 q^{79} + 18 q^{80} + 36 q^{81} + 30 q^{82} - 57 q^{83} - 60 q^{84} - 3 q^{85} - 114 q^{86} - 27 q^{87} + 15 q^{88} + 12 q^{89} + 18 q^{90} + 30 q^{91} - 30 q^{92} + 39 q^{93} - 21 q^{94} + 6 q^{95} - 39 q^{96} + 24 q^{97} - 75 q^{98} + 66 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.48953 − 0.906116i −1.62029 + 0.589737i 3.84464 + 3.22604i 0.173648 − 0.984808i 4.56813 0.167591 − 0.950454i −3.99889 − 6.92629i −0.0205879 + 0.0172753i −1.32465 + 2.29437i
16.2 −1.13382 − 0.412676i 0.157473 − 0.0573154i −0.416848 − 0.349777i 0.173648 − 0.984808i −0.202198 0.426285 − 2.41758i 1.53487 + 2.65847i −2.27662 + 1.91031i −0.603292 + 1.04493i
16.3 −0.783255 − 0.285081i −2.83432 + 1.03161i −0.999872 − 0.838992i 0.173648 − 0.984808i 2.51409 −0.135029 + 0.765789i 1.37750 + 2.38589i 4.67101 − 3.91944i −0.416761 + 0.721852i
16.4 0.348967 + 0.127014i 2.66743 − 0.970864i −1.42644 − 1.19693i 0.173648 − 0.984808i 1.05416 −0.536342 + 3.04175i −0.717119 − 1.24209i 3.87446 − 3.25106i 0.185682 − 0.321610i
16.5 0.609428 + 0.221814i 0.848369 − 0.308781i −1.20989 − 1.01522i 0.173648 − 0.984808i 0.585512 0.858109 − 4.86658i −1.16069 − 2.01037i −1.67375 + 1.40444i 0.324270 − 0.561652i
16.6 2.18217 + 0.794244i −0.158355 + 0.0576365i 2.59894 + 2.18077i 0.173648 − 0.984808i −0.391334 −0.0145685 + 0.0826222i 1.61704 + 2.80080i −2.27638 + 1.91011i 1.16111 − 2.01110i
46.1 −0.310905 − 1.76323i 0.454814 − 2.57938i −1.13294 + 0.412356i 0.766044 + 0.642788i −4.68945 −0.610054 − 0.511896i −0.711117 − 1.23169i −3.62727 − 1.32022i 0.895216 − 1.55056i
46.2 −0.278096 − 1.57716i −0.0897561 + 0.509032i −0.530708 + 0.193162i 0.766044 + 0.642788i 0.827785 3.59511 + 3.01666i −1.14925 − 1.99057i 2.56802 + 0.934683i 0.800745 − 1.38693i
46.3 0.0656119 + 0.372104i 0.213620 − 1.21150i 1.74523 − 0.635211i 0.766044 + 0.642788i 0.464821 −2.22367 − 1.86588i 0.728717 + 1.26217i 1.39697 + 0.508457i −0.188922 + 0.327223i
46.4 0.167121 + 0.947790i −0.437162 + 2.47927i 1.00901 − 0.367249i 0.766044 + 0.642788i −2.42289 0.186820 + 0.156761i 1.47911 + 2.56190i −3.13658 − 1.14162i −0.481206 + 0.833473i
46.5 0.391872 + 2.22242i 0.316944 − 1.79748i −2.90619 + 1.05777i 0.766044 + 0.642788i 4.11894 1.93485 + 1.62353i −1.23295 − 2.13553i −0.311389 − 0.113336i −1.12835 + 1.95436i
46.6 0.404089 + 2.29170i −0.284812 + 1.61525i −3.20922 + 1.16806i 0.766044 + 0.642788i −3.81676 −3.82275 − 3.20767i −1.64660 − 2.85199i 0.291164 + 0.105975i −1.16353 + 2.01529i
71.1 −1.96919 − 1.65234i −1.28474 + 1.07802i 0.800158 + 4.53792i −0.939693 − 0.342020i 4.31115 0.159622 + 0.0580977i 3.35195 − 5.80576i −0.0325280 + 0.184476i 1.28529 + 2.22620i
71.2 −1.23720 − 1.03813i 1.98766 − 1.66785i 0.105645 + 0.599145i −0.939693 − 0.342020i −4.19058 −3.65454 − 1.33014i −1.12376 + 1.94641i 0.648143 − 3.67580i 0.807525 + 1.39867i
71.3 −0.805595 − 0.675975i −1.52876 + 1.28278i −0.155255 − 0.880492i −0.939693 − 0.342020i 2.09868 2.07226 + 0.754242i −1.52175 + 2.63574i 0.170629 − 0.967684i 0.525815 + 0.910738i
71.4 0.332294 + 0.278827i 0.766533 − 0.643198i −0.314622 − 1.78431i −0.939693 − 0.342020i 0.434055 1.57472 + 0.573152i 0.826746 − 1.43197i −0.347075 + 1.96836i −0.216889 − 0.375663i
71.5 1.19546 + 1.00311i 1.72734 − 1.44941i 0.0756004 + 0.428751i −0.939693 − 0.342020i 3.51889 −0.978838 − 0.356268i 1.22086 − 2.11459i 0.361967 − 2.05282i −0.780283 − 1.35149i
71.6 1.81058 + 1.51925i −0.901996 + 0.756865i 0.622758 + 3.53183i −0.939693 − 0.342020i −2.78300 1.00042 + 0.364122i −1.87466 + 3.24701i −0.280191 + 1.58904i −1.18177 − 2.04689i
81.1 −2.48953 + 0.906116i −1.62029 − 0.589737i 3.84464 − 3.22604i 0.173648 + 0.984808i 4.56813 0.167591 + 0.950454i −3.99889 + 6.92629i −0.0205879 − 0.0172753i −1.32465 − 2.29437i
81.2 −1.13382 + 0.412676i 0.157473 + 0.0573154i −0.416848 + 0.349777i 0.173648 + 0.984808i −0.202198 0.426285 + 2.41758i 1.53487 − 2.65847i −2.27662 − 1.91031i −0.603292 − 1.04493i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.o.b ✓ 36
5.b even 2 1 925.2.p.d 36
5.c odd 4 2 925.2.bc.d 72
37.f even 9 1 inner 185.2.o.b ✓ 36
37.f even 9 1 6845.2.a.r 18
37.h even 18 1 6845.2.a.p 18
185.x even 18 1 925.2.p.d 36
185.bd odd 36 2 925.2.bc.d 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.o.b ✓ 36 1.a even 1 1 trivial
185.2.o.b ✓ 36 37.f even 9 1 inner
925.2.p.d 36 5.b even 2 1
925.2.p.d 36 185.x even 18 1
925.2.bc.d 72 5.c odd 4 2
925.2.bc.d 72 185.bd odd 36 2
6845.2.a.p 18 37.h even 18 1
6845.2.a.r 18 37.f even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 3 T_{2}^{35} + 6 T_{2}^{34} + 15 T_{2}^{33} + 36 T_{2}^{32} + 18 T_{2}^{31} + 197 T_{2}^{30} + \cdots + 3249 \) acting on \(S_{2}^{\mathrm{new}}(185, [\chi])\). Copy content Toggle raw display