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Properties

Label 322.2.o.a
Level $322$
Weight $2$
Character orbit 322.o
Analytic conductor $2.571$
Analytic rank $0$
Dimension $160$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,2,Mod(5,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([55, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.o (of order \(66\), degree \(20\), 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: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(8\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9} + 27 q^{12} + 11 q^{14} + 8 q^{16} - 66 q^{17} + 32 q^{18} - 66 q^{21} - 30 q^{23} - 6 q^{24} + 42 q^{25} + 6 q^{26} - 22 q^{28} + 16 q^{29}+ \cdots + 132 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.723734 + 0.690079i −0.577099 + 2.99428i 0.0475819 − 0.998867i 1.83118 + 1.44006i −1.64862 − 2.56530i −2.56146 + 0.662521i 0.654861 + 0.755750i −5.84754 − 2.34100i −2.31904 + 0.221442i
5.2 −0.723734 + 0.690079i −0.543074 + 2.81773i 0.0475819 − 0.998867i −2.82379 − 2.22065i −1.55142 − 2.41405i 1.58807 − 2.11613i 0.654861 + 0.755750i −4.85959 − 1.94549i 3.57610 − 0.341476i
5.3 −0.723734 + 0.690079i −0.347280 + 1.80186i 0.0475819 − 0.998867i −0.315754 − 0.248312i −0.992088 − 1.54372i 0.597169 + 2.57748i 0.654861 + 0.755750i −0.340999 − 0.136515i 0.399877 − 0.0381836i
5.4 −0.723734 + 0.690079i −0.0911992 + 0.473186i 0.0475819 − 0.998867i 3.41260 + 2.68370i −0.260532 − 0.405396i 0.336071 − 2.62432i 0.654861 + 0.755750i 2.56952 + 1.02868i −4.32178 + 0.412680i
5.5 −0.723734 + 0.690079i −0.0174618 + 0.0906004i 0.0475819 − 0.998867i −0.471311 − 0.370643i −0.0498837 − 0.0776206i −2.54557 − 0.721146i 0.654861 + 0.755750i 2.77720 + 1.11182i 0.596877 − 0.0569948i
5.6 −0.723734 + 0.690079i 0.101468 − 0.526466i 0.0475819 − 0.998867i −1.55500 − 1.22286i 0.289867 + 0.451043i −1.61196 − 2.09799i 0.654861 + 0.755750i 2.51823 + 1.00815i 1.96928 − 0.188043i
5.7 −0.723734 + 0.690079i 0.359896 − 1.86732i 0.0475819 − 0.998867i 2.44923 + 1.92610i 1.02813 + 1.59980i −0.00653026 + 2.64574i 0.654861 + 0.755750i −0.572239 − 0.229090i −3.10175 + 0.296181i
5.8 −0.723734 + 0.690079i 0.485721 − 2.52016i 0.0475819 − 0.998867i −0.469279 − 0.369045i 1.38758 + 2.15911i 2.39792 − 1.11803i 0.654861 + 0.755750i −3.33018 − 1.33320i 0.594303 − 0.0567491i
17.1 0.995472 − 0.0950560i −1.61555 − 1.69434i 0.981929 − 0.189251i −1.08486 − 0.559284i −1.76929 − 1.53310i −0.926600 − 2.47819i 0.959493 − 0.281733i −0.118039 + 2.47795i −1.13311 − 0.453629i
17.2 0.995472 − 0.0950560i −0.939050 − 0.984847i 0.981929 − 0.189251i 3.18388 + 1.64140i −1.02841 − 0.891125i −0.445049 + 2.60805i 0.959493 − 0.281733i 0.0546365 − 1.14696i 3.32548 + 1.33132i
17.3 0.995472 − 0.0950560i −0.539036 − 0.565325i 0.981929 − 0.189251i −2.85594 − 1.47234i −0.590333 − 0.511527i −2.40085 + 1.11171i 0.959493 − 0.281733i 0.113714 − 2.38714i −2.98297 − 1.19420i
17.4 0.995472 − 0.0950560i −0.522185 − 0.547652i 0.981929 − 0.189251i −0.212151 − 0.109371i −0.571878 − 0.495536i 2.45288 + 0.991667i 0.959493 − 0.281733i 0.115500 − 2.42465i −0.221587 − 0.0887099i
17.5 0.995472 − 0.0950560i 0.0400565 + 0.0420101i 0.981929 − 0.189251i 0.891984 + 0.459850i 0.0438685 + 0.0380122i −0.448926 − 2.60739i 0.959493 − 0.281733i 0.142585 − 2.99324i 0.931657 + 0.372979i
17.6 0.995472 − 0.0950560i 1.19209 + 1.25023i 0.981929 − 0.189251i 2.98731 + 1.54006i 1.30553 + 1.13125i −2.61962 − 0.370945i 0.959493 − 0.281733i 0.000754041 − 0.0158293i 3.12017 + 1.24913i
17.7 0.995472 − 0.0950560i 1.63351 + 1.71318i 0.981929 − 0.189251i −0.844196 − 0.435213i 1.78896 + 1.55014i 2.35126 − 1.21309i 0.959493 − 0.281733i −0.123871 + 2.60037i −0.881743 − 0.352997i
17.8 0.995472 − 0.0950560i 1.74321 + 1.82823i 0.981929 − 0.189251i −1.19856 − 0.617900i 1.90910 + 1.65425i −1.38944 + 2.25155i 0.959493 − 0.281733i −0.160884 + 3.37737i −1.25187 − 0.501171i
19.1 0.995472 + 0.0950560i −1.61555 + 1.69434i 0.981929 + 0.189251i −1.08486 + 0.559284i −1.76929 + 1.53310i −0.926600 + 2.47819i 0.959493 + 0.281733i −0.118039 − 2.47795i −1.13311 + 0.453629i
19.2 0.995472 + 0.0950560i −0.939050 + 0.984847i 0.981929 + 0.189251i 3.18388 − 1.64140i −1.02841 + 0.891125i −0.445049 − 2.60805i 0.959493 + 0.281733i 0.0546365 + 1.14696i 3.32548 − 1.33132i
19.3 0.995472 + 0.0950560i −0.539036 + 0.565325i 0.981929 + 0.189251i −2.85594 + 1.47234i −0.590333 + 0.511527i −2.40085 − 1.11171i 0.959493 + 0.281733i 0.113714 + 2.38714i −2.98297 + 1.19420i
19.4 0.995472 + 0.0950560i −0.522185 + 0.547652i 0.981929 + 0.189251i −0.212151 + 0.109371i −0.571878 + 0.495536i 2.45288 − 0.991667i 0.959493 + 0.281733i 0.115500 + 2.42465i −0.221587 + 0.0887099i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.d odd 22 1 inner
161.o even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.o.a ✓ 160
7.d odd 6 1 inner 322.2.o.a ✓ 160
23.d odd 22 1 inner 322.2.o.a ✓ 160
161.o even 66 1 inner 322.2.o.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.o.a ✓ 160 1.a even 1 1 trivial
322.2.o.a ✓ 160 7.d odd 6 1 inner
322.2.o.a ✓ 160 23.d odd 22 1 inner
322.2.o.a ✓ 160 161.o even 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{160} - 6 T_{3}^{159} + 46 T_{3}^{158} - 204 T_{3}^{157} + 892 T_{3}^{156} - 3249 T_{3}^{155} + \cdots + 43\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display