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Properties

Label 891.2.r.a
Level $891$
Weight $2$
Character orbit 891.r
Analytic conductor $7.115$
Analytic rank $0$
Dimension $774$
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] = [891,2,Mod(34,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(54))
 
chi = DirichletCharacter(H, H._module([34, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.r (of order \(27\), degree \(18\), 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: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(774\)
Relative dimension: \(43\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 774 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 774 q + 36 q^{20} - 27 q^{21} + 72 q^{24} - 54 q^{26} + 108 q^{28} - 54 q^{30} - 324 q^{32} - 27 q^{35} + 72 q^{36} + 135 q^{40} - 18 q^{41} - 45 q^{42} + 45 q^{43} - 54 q^{47} - 117 q^{50} - 126 q^{51} - 54 q^{53} + 18 q^{55} - 54 q^{57} - 117 q^{60} - 414 q^{63} + 54 q^{67} - 45 q^{68} + 54 q^{69} - 558 q^{70} + 72 q^{71} + 216 q^{72} + 396 q^{78} + 252 q^{80} + 72 q^{81} - 252 q^{82} + 72 q^{83} + 252 q^{84} + 72 q^{86} - 36 q^{89} + 81 q^{90} + 171 q^{92} + 189 q^{93} - 396 q^{94} - 36 q^{95} + 72 q^{97} - 171 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −1.11952 + 2.59534i 0.882371 − 1.49044i −4.10996 − 4.35630i −0.125103 − 2.14794i 2.88037 + 3.95863i −1.99160 − 0.472017i 10.5952 − 3.85633i −1.44284 − 2.63025i 5.71467 + 2.07997i
34.2 −0.997848 + 2.31327i −1.52247 + 0.825893i −2.98304 − 3.16184i 0.0426934 + 0.733018i −0.391325 − 4.34599i −0.586594 − 0.139025i 5.55606 − 2.02224i 1.63580 − 2.51479i −1.73827 − 0.632679i
34.3 −0.986666 + 2.28735i −0.710469 − 1.57963i −2.88597 − 3.05895i 0.212823 + 3.65404i 4.31416 − 0.0665229i 2.86864 + 0.679881i 5.16267 − 1.87906i −1.99047 + 2.24456i −8.56804 − 3.11851i
34.4 −0.969278 + 2.24704i 1.55892 + 0.754821i −2.73720 − 2.90126i 0.144277 + 2.47715i −3.20714 + 2.77133i −3.60908 − 0.855367i 4.57316 − 1.66450i 1.86049 + 2.35342i −5.70609 − 2.07685i
34.5 −0.946807 + 2.19495i 1.68013 − 0.420905i −2.54886 − 2.70163i −0.0307944 − 0.528719i −0.666896 + 4.08631i 2.53591 + 0.601021i 3.85065 − 1.40152i 2.64568 − 1.41435i 1.18967 + 0.433003i
34.6 −0.900949 + 2.08863i 0.417463 + 1.68099i −2.17820 − 2.30876i −0.178809 − 3.07003i −3.88708 − 0.642557i −5.11927 − 1.21329i 2.50962 − 0.913429i −2.65145 + 1.40350i 6.57327 + 2.39248i
34.7 −0.800134 + 1.85492i −0.705279 − 1.58195i −1.42802 − 1.51362i −0.136461 − 2.34295i 3.49871 − 0.0424598i −2.98813 − 0.708200i 0.153640 − 0.0559205i −2.00516 + 2.23144i 4.45516 + 1.62155i
34.8 −0.768189 + 1.78086i −1.69638 − 0.349705i −1.20887 − 1.28133i −0.0988232 − 1.69673i 1.92592 − 2.75238i 2.39446 + 0.567497i −0.434519 + 0.158152i 2.75541 + 1.18646i 3.09755 + 1.12742i
34.9 −0.720605 + 1.67055i −0.531728 + 1.64841i −0.898983 − 0.952866i 0.117046 + 2.00960i −2.37059 − 2.07613i −1.44997 − 0.343649i −1.17962 + 0.429348i −2.43453 − 1.75302i −3.44148 − 1.25259i
34.10 −0.680874 + 1.57844i 0.991374 + 1.42027i −0.655408 − 0.694692i −0.100531 − 1.72604i −2.91682 + 0.597800i 3.09181 + 0.732772i −1.68794 + 0.614360i −1.03436 + 2.81605i 2.79291 + 1.01654i
34.11 −0.667589 + 1.54765i 0.434401 − 1.67669i −0.577049 − 0.611636i −0.221005 − 3.79451i 2.30492 + 1.79164i 3.60269 + 0.853852i −1.83586 + 0.668199i −2.62259 − 1.45671i 6.02009 + 2.19113i
34.12 −0.650197 + 1.50733i 1.50911 − 0.850054i −0.476793 − 0.505371i 0.223999 + 3.84591i 0.300091 + 2.82742i 0.191107 + 0.0452933i −2.01340 + 0.732816i 1.55482 − 2.56565i −5.94268 − 2.16296i
34.13 −0.542120 + 1.25678i −1.30094 − 1.14349i 0.0868914 + 0.0920995i 0.176266 + 3.02637i 2.14237 − 1.01508i 1.14645 + 0.271714i −2.73520 + 0.995530i 0.384870 + 2.97521i −3.89903 − 1.41913i
34.14 −0.372420 + 0.863366i −0.558489 + 1.63954i 0.765779 + 0.811679i −0.203743 − 3.49814i −1.20753 − 1.09278i −0.234983 − 0.0556920i −2.75309 + 1.00204i −2.37618 − 1.83133i 3.09605 + 1.12687i
34.15 −0.369728 + 0.857125i 1.13833 − 1.30546i 0.774518 + 0.820942i −0.0929874 − 1.59653i 0.698067 + 1.45835i −4.05291 − 0.960556i −2.74436 + 0.998864i −0.408426 − 2.97207i 1.40281 + 0.510580i
34.16 −0.363056 + 0.841657i 0.953223 + 1.44616i 0.795906 + 0.843611i 0.103668 + 1.77991i −1.56324 + 0.277253i 1.11218 + 0.263591i −2.72168 + 0.990609i −1.18273 + 2.75702i −1.53571 − 0.558953i
34.17 −0.328794 + 0.762231i −1.63035 + 0.584784i 0.899593 + 0.953513i 0.0329244 + 0.565291i 0.0903080 − 1.43497i 3.73742 + 0.885785i −2.58270 + 0.940025i 2.31606 − 1.90680i −0.441707 − 0.160768i
34.18 −0.239667 + 0.555610i −0.0566492 − 1.73112i 1.12122 + 1.18842i 0.0221575 + 0.380429i 0.975407 + 0.383418i −1.51911 − 0.360037i −2.06623 + 0.752047i −2.99358 + 0.196134i −0.216681 − 0.0788653i
34.19 −0.230068 + 0.533357i −1.45736 + 0.936008i 1.14094 + 1.20933i −0.138473 − 2.37749i −0.163936 − 0.992637i −1.26529 − 0.299880i −1.99916 + 0.727636i 1.24778 − 2.72820i 1.29991 + 0.473127i
34.20 −0.0737872 + 0.171058i 1.72565 + 0.148757i 1.34867 + 1.42950i −0.181173 − 3.11062i −0.152777 + 0.284210i 3.80559 + 0.901940i −0.694161 + 0.252654i 2.95574 + 0.513406i 0.545465 + 0.198533i
See next 80 embeddings (of 774 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.43
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
81.g even 27 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.r.a ✓ 774
81.g even 27 1 inner 891.2.r.a ✓ 774
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
891.2.r.a ✓ 774 1.a even 1 1 trivial
891.2.r.a ✓ 774 81.g even 27 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{774} + 108 T_{2}^{769} + 189 T_{2}^{767} - 198 T_{2}^{766} + 6255 T_{2}^{764} + \cdots + 91\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display