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Properties

Label 445.2.o.b
Level $445$
Weight $2$
Character orbit 445.o
Analytic conductor $3.553$
Analytic rank $0$
Dimension $160$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [445,2,Mod(16,445)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(445, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("445.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 445 = 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 445.o (of order \(11\), degree \(10\), 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: \(3.55334288995\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(16\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 2 q^{2} - 2 q^{3} - 22 q^{4} - 16 q^{5} - 6 q^{6} - 2 q^{7} - 6 q^{8} - 32 q^{9} - 2 q^{10} - q^{11} - 16 q^{12} - 8 q^{13} + 27 q^{14} - 2 q^{15} - 34 q^{16} + 19 q^{17} + 53 q^{18} - 14 q^{19}+ \cdots + 50 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.07319 + 2.34995i −1.90713 + 2.20094i −3.06082 − 3.53237i −0.959493 + 0.281733i −3.12540 − 6.84368i 3.43982 − 1.01002i 6.62820 − 1.94621i −0.780069 − 5.42550i 0.367657 − 2.55711i
16.2 −0.888306 + 1.94512i 2.03962 − 2.35385i −1.68468 − 1.94422i −0.959493 + 0.281733i 2.76671 + 6.05825i −4.62862 + 1.35908i 1.17477 − 0.344945i −0.953608 − 6.63249i 0.304320 − 2.11659i
16.3 −0.866073 + 1.89644i 0.739954 − 0.853952i −1.53666 − 1.77341i −0.959493 + 0.281733i 0.978611 + 2.14286i 2.06090 − 0.605134i 0.693238 − 0.203553i 0.245242 + 1.70569i 0.296703 − 2.06362i
16.4 −0.717449 + 1.57099i −0.959245 + 1.10703i −0.643567 − 0.742716i −0.959493 + 0.281733i −1.05092 − 2.30120i −1.20449 + 0.353670i −1.68569 + 0.494962i 0.121585 + 0.845641i 0.245787 − 1.70949i
16.5 −0.454095 + 0.994329i −0.941402 + 1.08644i 0.527233 + 0.608459i −0.959493 + 0.281733i −0.652789 − 1.42941i 2.30535 − 0.676911i −2.94209 + 0.863875i 0.132840 + 0.923919i 0.155566 − 1.08199i
16.6 −0.277760 + 0.608209i 0.985626 − 1.13747i 1.01695 + 1.17363i −0.959493 + 0.281733i 0.418054 + 0.915411i −1.44253 + 0.423565i −2.27937 + 0.669284i 0.104558 + 0.727217i 0.0951562 − 0.661826i
16.7 −0.253332 + 0.554720i −2.20686 + 2.54685i 1.06618 + 1.23044i −0.959493 + 0.281733i −0.853719 − 1.86938i −3.40176 + 0.998847i −2.12290 + 0.623340i −1.18928 − 8.27159i 0.0867877 − 0.603621i
16.8 0.0146750 − 0.0321338i 1.73061 − 1.99723i 1.30890 + 1.51056i −0.959493 + 0.281733i −0.0387817 − 0.0849202i 2.84179 − 0.834426i 0.135538 − 0.0397976i −0.566969 − 3.94336i −0.00502743 + 0.0349665i
16.9 0.218657 − 0.478792i −0.200028 + 0.230845i 1.12829 + 1.30212i −0.959493 + 0.281733i 0.0667890 + 0.146248i −3.86392 + 1.13455i 1.88023 − 0.552084i 0.413666 + 2.87711i −0.0749086 + 0.521000i
16.10 0.344093 − 0.753459i −1.84517 + 2.12944i 0.860421 + 0.992978i −0.959493 + 0.281733i 0.969536 + 2.12299i 4.86694 − 1.42906i 2.63375 − 0.773340i −0.702916 − 4.88889i −0.117881 + 0.819881i
16.11 0.344743 − 0.754882i 0.778249 − 0.898148i 0.858722 + 0.991019i −0.959493 + 0.281733i −0.409699 − 0.897117i 2.17157 − 0.637629i 2.63666 − 0.774194i 0.225947 + 1.57150i −0.118104 + 0.821429i
16.12 0.621107 − 1.36003i 2.10489 − 2.42918i −0.154199 − 0.177955i −0.959493 + 0.281733i −1.99640 − 4.37151i −1.64289 + 0.482395i 2.53137 − 0.743277i −1.04338 − 7.25686i −0.212782 + 1.47993i
16.13 0.747311 − 1.63638i −0.340472 + 0.392925i −0.809555 − 0.934276i −0.959493 + 0.281733i 0.388538 + 0.850780i 1.89875 − 0.557524i 1.31834 − 0.387099i 0.388475 + 2.70190i −0.256017 + 1.78064i
16.14 0.877930 − 1.92240i −1.26146 + 1.45581i −1.61513 − 1.86396i −0.959493 + 0.281733i 1.69116 + 3.70313i −2.66220 + 0.781693i −0.945703 + 0.277683i −0.101139 − 0.703437i −0.300765 + 2.09187i
16.15 1.02859 − 2.25229i 1.32871 − 1.53341i −2.70509 − 3.12184i −0.959493 + 0.281733i −2.08699 − 4.56987i −2.41307 + 0.708542i −5.06220 + 1.48640i −0.158939 − 1.10544i −0.352378 + 2.45084i
16.16 1.16393 − 2.54865i −1.14770 + 1.32452i −3.83117 − 4.42140i −0.959493 + 0.281733i 2.03989 + 4.46674i 0.877188 − 0.257566i −10.3511 + 3.03936i −0.0101867 − 0.0708500i −0.398745 + 2.77333i
91.1 −2.61220 + 0.767013i −0.500240 − 0.321485i 4.55280 − 2.92591i −0.654861 + 0.755750i 1.55331 + 0.456094i 0.997025 − 1.15063i −6.08294 + 7.02009i −1.09936 − 2.40726i 1.13096 − 2.47646i
91.2 −2.31795 + 0.680611i −2.67313 − 1.71792i 3.22714 − 2.07396i −0.654861 + 0.755750i 7.36541 + 2.16268i −0.0792012 + 0.0914030i −2.90476 + 3.35227i 2.94814 + 6.45553i 1.00356 − 2.19749i
91.3 −2.00203 + 0.587849i 1.98305 + 1.27443i 1.98005 − 1.27250i −0.654861 + 0.755750i −4.71930 − 1.38571i 0.642378 − 0.741344i −0.483284 + 0.557740i 1.06207 + 2.32561i 0.866784 − 1.89799i
91.4 −1.96084 + 0.575756i 0.169433 + 0.108888i 1.83091 − 1.17665i −0.654861 + 0.755750i −0.394924 − 0.115960i −3.08094 + 3.55559i −0.236079 + 0.272450i −1.22939 − 2.69200i 0.848953 − 1.85895i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 445.2.o.b ✓ 160
89.e even 11 1 inner 445.2.o.b ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
445.2.o.b ✓ 160 1.a even 1 1 trivial
445.2.o.b ✓ 160 89.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} + 2 T_{2}^{159} + 29 T_{2}^{158} + 60 T_{2}^{157} + 502 T_{2}^{156} + 1063 T_{2}^{155} + \cdots + 6285049 \) acting on \(S_{2}^{\mathrm{new}}(445, [\chi])\). Copy content Toggle raw display