↑

Properties

Label 185.2.u.a
Level $185$
Weight $2$
Character orbit 185.u
Analytic conductor $1.477$
Analytic rank $0$
Dimension $68$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [185,2,Mod(8,185)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.u (of order \(12\), degree \(4\), 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: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 6 q^{2} - 4 q^{3} + 30 q^{4} - 8 q^{6} - 2 q^{7} - 6 q^{10} - 10 q^{12} - 6 q^{13} - 16 q^{15} - 26 q^{16} - 10 q^{17} - 8 q^{18} - 4 q^{19} - 28 q^{20} - 12 q^{21} - 14 q^{22} + 20 q^{25} - 24 q^{26}+ \cdots - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −2.38259 + 1.37559i 0.185681 + 0.692973i 2.78450 − 4.82289i −1.62458 − 1.53647i −1.39565 − 1.39565i 0.502511 + 1.87540i 9.81895i 2.15234 − 1.24266i 5.98426 + 1.42604i
8.2 −2.10315 + 1.21425i 0.180195 + 0.672498i 1.94883 − 3.37547i 2.14998 + 0.614465i −1.19556 − 1.19556i −1.19887 − 4.47423i 4.60849i 2.17829 − 1.25764i −5.26786 + 1.31832i
8.3 −1.90916 + 1.10225i −0.817378 − 3.05050i 1.42993 − 2.47671i 1.22028 − 1.87374i 4.92293 + 4.92293i 0.0394498 + 0.147229i 1.89556i −6.03934 + 3.48681i −0.264372 + 4.92234i
8.4 −1.72086 + 0.993536i 0.630160 + 2.35179i 0.974229 − 1.68741i −0.468054 + 2.18653i −3.42100 − 3.42100i 0.704548 + 2.62941i − 0.102418i −2.53574 + 1.46401i −1.36695 − 4.22774i
8.5 −1.54509 + 0.892058i −0.471934 − 1.76128i 0.591534 − 1.02457i −0.319125 + 2.21318i 2.30034 + 2.30034i −0.134169 − 0.500724i − 1.45750i −0.281310 + 0.162414i −1.48121 − 3.70424i
8.6 −1.25283 + 0.723320i −0.212077 − 0.791482i 0.0463849 − 0.0803411i −2.18370 − 0.481096i 0.838191 + 0.838191i 0.425785 + 1.58905i − 2.75908i 2.01661 − 1.16429i 3.08379 − 0.976784i
8.7 −0.713738 + 0.412077i 0.475172 + 1.77336i −0.660386 + 1.14382i 2.23300 − 0.117085i −1.06991 − 1.06991i 0.320840 + 1.19739i − 2.73682i −0.320958 + 0.185305i −1.54553 + 1.00374i
8.8 −0.614089 + 0.354544i −0.0751630 − 0.280512i −0.748597 + 1.29661i 0.0904746 − 2.23424i 0.145611 + 0.145611i −0.977086 − 3.64653i − 2.47982i 2.52504 − 1.45783i 0.736577 + 1.40410i
8.9 −0.212566 + 0.122725i −0.491666 − 1.83492i −0.969877 + 1.67988i 2.03028 + 0.937007i 0.329702 + 0.329702i 1.00868 + 3.76446i − 0.967013i −0.527123 + 0.304334i −0.546561 + 0.0499898i
8.10 −0.138202 + 0.0797909i 0.607844 + 2.26850i −0.987267 + 1.71000i −2.23170 + 0.139678i −0.265011 − 0.265011i −0.345018 − 1.28763i − 0.634263i −2.17856 + 1.25779i 0.297280 − 0.197373i
8.11 0.407096 − 0.235037i −0.838017 − 3.12752i −0.889515 + 1.54069i −2.02380 + 0.950915i −1.07624 − 1.07624i −0.968227 − 3.61347i 1.77643i −6.48105 + 3.74183i −0.600381 + 0.862782i
8.12 0.877658 − 0.506716i 0.458707 + 1.71192i −0.486478 + 0.842604i −0.241001 − 2.22304i 1.27004 + 1.27004i 1.12037 + 4.18129i 3.01289i −0.122169 + 0.0705344i −1.33797 − 1.82895i
8.13 0.971183 − 0.560713i 0.168506 + 0.628874i −0.371203 + 0.642942i 0.550691 + 2.16720i 0.516268 + 0.516268i −0.377247 − 1.40791i 3.07540i 2.23099 − 1.28806i 1.75000 + 1.79596i
8.14 1.11517 − 0.643846i −0.352022 − 1.31376i −0.170924 + 0.296050i 1.94466 − 1.10377i −1.23843 − 1.23843i −0.459253 − 1.71395i 3.01558i 0.996020 − 0.575052i 1.45798 − 2.48295i
8.15 1.80193 − 1.04034i −0.294240 − 1.09812i 1.16463 − 2.01721i −1.89677 − 1.18417i −1.67262 − 1.67262i −0.0206204 − 0.0769563i − 0.685107i 1.47879 − 0.853779i −4.64979 − 0.160500i
8.16 2.08716 − 1.20502i −0.740695 − 2.76431i 1.90415 − 3.29809i 0.749673 + 2.10665i −4.87700 − 4.87700i 0.856324 + 3.19584i − 4.35808i −4.49471 + 2.59502i 4.10325 + 3.49355i
8.17 2.10002 − 1.21245i 0.586925 + 2.19044i 1.94006 − 3.36028i −1.71236 + 1.43799i 3.88834 + 3.88834i −0.132002 − 0.492639i − 4.55908i −1.85545 + 1.07124i −1.85251 + 5.09596i
23.1 −2.34255 − 1.35247i −0.438550 − 0.117509i 2.65837 + 4.60443i 1.31482 + 1.80866i 0.868398 + 0.868398i −3.48545 − 0.933922i − 8.97159i −2.41956 − 1.39693i −0.633857 − 6.01514i
23.2 −1.89659 − 1.09500i 2.68684 + 0.719938i 1.39805 + 2.42149i −1.55121 + 1.61051i −4.30752 − 4.30752i 2.04761 + 0.548654i − 1.74345i 4.10274 + 2.36872i 4.70553 − 1.35592i
23.3 −1.85990 − 1.07382i −2.04873 − 0.548954i 1.30616 + 2.26234i −2.22787 − 0.191263i 3.22096 + 3.22096i −0.905285 − 0.242570i − 1.31504i 1.29785 + 0.749313i 3.93825 + 2.74806i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.u even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.u.a yes 68
5.b even 2 1 925.2.y.b 68
5.c odd 4 1 185.2.p.a ✓ 68
5.c odd 4 1 925.2.t.b 68
37.g odd 12 1 185.2.p.a ✓ 68
185.p even 12 1 925.2.y.b 68
185.q odd 12 1 925.2.t.b 68
185.u even 12 1 inner 185.2.u.a yes 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.p.a ✓ 68 5.c odd 4 1
185.2.p.a ✓ 68 37.g odd 12 1
185.2.u.a yes 68 1.a even 1 1 trivial
185.2.u.a yes 68 185.u even 12 1 inner
925.2.t.b 68 5.c odd 4 1
925.2.t.b 68 185.q odd 12 1
925.2.y.b 68 5.b even 2 1
925.2.y.b 68 185.p even 12 1

Hecke kernels

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