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Properties

Label 185.2.w.a
Level $185$
Weight $2$
Character orbit 185.w
Analytic conductor $1.477$
Analytic rank $0$
Dimension $72$
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(21,185)]
 
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, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.w (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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{2} - 6 q^{4} - 6 q^{7} - 18 q^{8} + 6 q^{10} + 30 q^{12} - 30 q^{13} - 18 q^{14} - 18 q^{16} - 30 q^{18} - 18 q^{19} + 30 q^{22} - 30 q^{24} - 24 q^{27} + 30 q^{28} + 18 q^{29} + 24 q^{30} - 36 q^{32}+ \cdots - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −0.912722 + 2.50768i −2.54963 + 0.927988i −3.92332 − 3.29206i −0.984808 − 0.173648i − 7.24065i −0.681286 + 3.86376i 7.21416 − 4.16510i 3.34130 − 2.80368i 1.33431 − 2.31109i
21.2 −0.707909 + 1.94497i −0.121836 + 0.0443448i −1.74967 − 1.46814i 0.984808 + 0.173648i − 0.268359i −0.426398 + 2.41822i 0.509113 − 0.293937i −2.28526 + 1.91756i −1.03489 + 1.79249i
21.3 −0.350159 + 0.962054i 2.03239 − 0.739729i 0.729153 + 0.611832i 0.984808 + 0.173648i 2.21429i 0.426620 − 2.41948i −2.61720 + 1.51104i 1.28527 − 1.07847i −0.511898 + 0.886634i
21.4 −0.302982 + 0.832436i 2.30523 − 0.839035i 0.930937 + 0.781149i −0.984808 − 0.173648i 2.17317i −0.623850 + 3.53803i −2.46667 + 1.42413i 2.31198 − 1.93998i 0.442930 − 0.767177i
21.5 −0.106080 + 0.291452i −1.76040 + 0.640734i 1.45840 + 1.22374i −0.984808 − 0.173648i − 0.581042i −0.295101 + 1.67360i −1.04858 + 0.605396i 0.390342 − 0.327536i 0.155078 − 0.268604i
21.6 −0.0584921 + 0.160706i −0.624082 + 0.227147i 1.50968 + 1.26678i 0.984808 + 0.173648i − 0.113580i −0.0729992 + 0.413999i −0.588097 + 0.339538i −1.96025 + 1.64485i −0.0855097 + 0.148107i
21.7 0.0874964 − 0.240394i 1.14636 − 0.417243i 1.48196 + 1.24351i −0.984808 − 0.173648i − 0.312087i 0.829933 − 4.70679i 0.871695 − 0.503274i −1.15807 + 0.971738i −0.127911 + 0.221549i
21.8 0.544164 − 1.49508i −3.11875 + 1.13513i −0.407055 − 0.341560i 0.984808 + 0.173648i 5.28047i −0.619542 + 3.51360i 2.02358 − 1.16831i 6.13994 − 5.15202i 0.795514 − 1.37787i
21.9 0.643934 − 1.76920i 1.84845 − 0.672780i −1.18331 − 0.992917i −0.984808 − 0.173648i − 3.70349i −0.303772 + 1.72278i 0.742360 − 0.428602i 0.665996 − 0.558837i −0.941369 + 1.63050i
21.10 0.684008 − 1.87930i −0.691854 + 0.251814i −1.53180 − 1.28533i 0.984808 + 0.173648i 1.47244i 0.818589 − 4.64245i 0.000650079 0 0.000375323i −1.88288 + 1.57993i 0.999953 − 1.73197i
21.11 0.904057 − 2.48388i 1.58444 − 0.576689i −3.82023 − 3.20555i 0.984808 + 0.173648i − 4.45691i −0.275913 + 1.56478i −6.83760 + 3.94769i −0.120255 + 0.100906i 1.32164 − 2.28915i
21.12 0.921981 − 2.53312i −1.92971 + 0.702356i −4.03457 − 3.38541i −0.984808 − 0.173648i 5.53575i −0.123578 + 0.700845i −7.62637 + 4.40308i 0.932337 − 0.782323i −1.34785 + 2.33454i
41.1 −2.37605 − 0.418962i −0.261904 − 1.48533i 3.59070 + 1.30691i −0.642788 − 0.766044i 3.63895i 2.36937 − 1.98814i −3.80522 − 2.19695i 0.681459 − 0.248031i 1.20635 + 2.08946i
41.2 −1.82036 − 0.320979i −0.118478 − 0.671921i 1.33130 + 0.484554i 0.642788 + 0.766044i 1.26117i −1.02434 + 0.859519i 0.933680 + 0.539060i 2.38164 − 0.866845i −0.924221 − 1.60080i
41.3 −1.49156 − 0.263002i 0.406811 + 2.30714i 0.276189 + 0.100525i 0.642788 + 0.766044i − 3.54822i 2.32391 − 1.94999i 2.23779 + 1.29199i −2.33831 + 0.851075i −0.757284 − 1.31165i
41.4 −1.43188 − 0.252480i −0.0435088 − 0.246751i 0.107162 + 0.0390040i −0.642788 − 0.766044i 0.364304i −3.65531 + 3.06717i 2.37476 + 1.37107i 2.76008 − 1.00459i 0.726987 + 1.25918i
41.5 −0.540933 − 0.0953810i 0.128393 + 0.728150i −1.59587 − 0.580851i −0.642788 − 0.766044i − 0.406126i 1.81619 − 1.52397i 1.75924 + 1.01569i 2.30536 − 0.839082i 0.274639 + 0.475688i
41.6 0.105693 + 0.0186364i 0.492384 + 2.79245i −1.86856 − 0.680101i 0.642788 + 0.766044i 0.304317i −3.01176 + 2.52717i −0.370707 − 0.214028i −4.73626 + 1.72386i 0.0536615 + 0.0929444i
41.7 0.772863 + 0.136277i −0.381820 − 2.16541i −1.30064 − 0.473394i −0.642788 − 0.766044i − 1.72560i −0.668410 + 0.560863i −2.29999 − 1.32790i −1.72413 + 0.627533i −0.392393 − 0.679644i
41.8 1.24400 + 0.219350i −0.234364 − 1.32915i −0.379974 − 0.138299i 0.642788 + 0.766044i − 1.70486i 2.83337 − 2.37748i −2.63025 − 1.51858i 1.10737 − 0.403051i 0.631593 + 1.09395i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.w.a ✓ 72
5.b even 2 1 925.2.bb.b 72
5.c odd 4 1 925.2.ba.b 72
5.c odd 4 1 925.2.ba.c 72
37.h even 18 1 inner 185.2.w.a ✓ 72
37.i odd 36 1 6845.2.a.t 36
37.i odd 36 1 6845.2.a.u 36
185.v even 18 1 925.2.bb.b 72
185.y odd 36 1 925.2.ba.b 72
185.y odd 36 1 925.2.ba.c 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.w.a ✓ 72 1.a even 1 1 trivial
185.2.w.a ✓ 72 37.h even 18 1 inner
925.2.ba.b 72 5.c odd 4 1
925.2.ba.b 72 185.y odd 36 1
925.2.ba.c 72 5.c odd 4 1
925.2.ba.c 72 185.y odd 36 1
925.2.bb.b 72 5.b even 2 1
925.2.bb.b 72 185.v even 18 1
6845.2.a.t 36 37.i odd 36 1
6845.2.a.u 36 37.i odd 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(185, [\chi])\).