Properties

Label 165.3.q.a
Level $165$
Weight $3$
Character orbit 165.q
Analytic conductor $4.496$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,3,Mod(26,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 165.q (of order \(10\), 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: \(4.49592436194\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 8 q^{3} + 64 q^{4} + 24 q^{6} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 8 q^{3} + 64 q^{4} + 24 q^{6} + 30 q^{9} - 48 q^{12} - 30 q^{15} - 96 q^{16} - 16 q^{18} + 72 q^{19} - 104 q^{21} - 80 q^{22} - 160 q^{24} + 160 q^{25} - 142 q^{27} - 456 q^{28} + 80 q^{30} - 28 q^{31} + 320 q^{33} + 72 q^{34} + 348 q^{36} + 92 q^{37} + 266 q^{39} + 406 q^{42} + 296 q^{43} - 224 q^{46} - 246 q^{48} - 280 q^{49} - 188 q^{51} - 424 q^{52} - 1056 q^{54} - 180 q^{55} - 420 q^{57} + 128 q^{58} - 370 q^{60} - 268 q^{61} + 238 q^{63} + 404 q^{64} - 50 q^{66} - 80 q^{67} + 246 q^{69} + 180 q^{70} + 840 q^{72} + 152 q^{73} + 60 q^{75} + 1904 q^{76} + 804 q^{78} + 872 q^{79} + 738 q^{81} - 180 q^{82} - 1080 q^{84} - 120 q^{85} - 1412 q^{87} - 1660 q^{88} - 160 q^{90} - 484 q^{91} - 808 q^{93} - 1240 q^{94} + 846 q^{96} - 1080 q^{97} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −2.30025 + 3.16603i −2.99037 + 0.240127i −3.49650 10.7611i 1.31433 + 1.80902i 6.11838 10.0200i −1.29634 3.98973i 27.2253 + 8.84603i 8.88468 1.43614i −8.75069
26.2 −2.20174 + 3.03043i 0.615997 2.93608i −3.09980 9.54021i −1.31433 1.80902i 7.54132 + 8.33121i 2.97551 + 9.15766i 21.4859 + 6.98121i −8.24109 3.61723i 8.37591
26.3 −2.04878 + 2.81990i 2.94142 + 0.589968i −2.51827 7.75045i −1.31433 1.80902i −7.68995 + 7.08579i −3.41857 10.5213i 13.7549 + 4.46924i 8.30388 + 3.47068i 7.79401
26.4 −1.84427 + 2.53842i 2.61041 1.47844i −1.80619 5.55887i 1.31433 + 1.80902i −1.06139 + 9.35296i −0.0257605 0.0792825i 5.50549 + 1.78884i 4.62843 7.71865i −7.01603
26.5 −1.73462 + 2.38749i −2.66585 1.37596i −1.45517 4.47855i −1.31433 1.80902i 7.90931 3.97794i −0.926259 2.85073i 1.98999 + 0.646587i 5.21349 + 7.33618i 6.59887
26.6 −1.61829 + 2.22739i 0.208595 + 2.99274i −1.10632 3.40491i 1.31433 + 1.80902i −7.00356 4.37850i −3.69283 11.3654i −1.09940 0.357218i −8.91298 + 1.24854i −6.15635
26.7 −1.59966 + 2.20175i −0.632879 2.93248i −1.05270 3.23988i 1.31433 + 1.80902i 7.46898 + 3.29755i −0.283495 0.872508i −1.53586 0.499032i −8.19893 + 3.71182i −6.08548
26.8 −1.46951 + 2.02260i −1.97655 + 2.25682i −0.695407 2.14024i −1.31433 1.80902i −1.66009 7.31420i 0.101982 + 0.313867i −4.16008 1.35169i −1.18647 8.92145i 5.59034
26.9 −1.30528 + 1.79656i −2.99034 + 0.240553i −0.287817 0.885809i 1.31433 + 1.80902i 3.47106 5.68633i 4.07959 + 12.5557i −6.48085 2.10576i 8.88427 1.43867i −4.96558
26.10 −1.14753 + 1.57945i 2.62795 + 1.44702i 0.0582553 + 0.179291i −1.31433 1.80902i −5.30115 + 2.49021i 1.64114 + 5.05091i −7.77703 2.52691i 4.81228 + 7.60539i 4.36548
26.11 −0.971166 + 1.33669i 2.00130 + 2.23491i 0.392477 + 1.20792i 1.31433 + 1.80902i −4.93098 + 0.504659i 1.32808 + 4.08740i −8.28130 2.69076i −0.989618 + 8.94543i −3.69453
26.12 −0.881600 + 1.21342i 1.06116 2.80605i 0.540903 + 1.66473i −1.31433 1.80902i 2.46939 + 3.76145i −3.68024 11.3266i −8.20270 2.66522i −6.74786 5.95536i 3.35381
26.13 −0.462120 + 0.636053i −0.392907 2.97416i 1.04506 + 3.21636i −1.31433 1.80902i 2.07329 + 1.12451i 3.54884 + 10.9222i −5.51962 1.79343i −8.69125 + 2.33714i 1.75801
26.14 −0.356298 + 0.490402i −1.91621 2.30827i 1.12252 + 3.45477i 1.31433 + 1.80902i 1.81473 0.117281i −0.651691 2.00570i −4.40019 1.42971i −1.65626 + 8.84629i −1.35544
26.15 −0.0434364 + 0.0597850i −0.712635 + 2.91413i 1.23438 + 3.79903i −1.31433 1.80902i −0.143267 0.169184i 1.56117 + 4.80478i −0.561868 0.182562i −7.98430 4.15342i 0.165242
26.16 −0.00589937 + 0.00811978i −2.77949 1.12890i 1.23604 + 3.80413i −1.31433 1.80902i 0.0255637 0.0159090i −1.26110 3.88128i −0.0763620 0.0248115i 6.45115 + 6.27556i 0.0224425
26.17 0.00589937 0.00811978i 2.91221 + 0.720441i 1.23604 + 3.80413i 1.31433 + 1.80902i 0.0230300 0.0193964i −1.26110 3.88128i 0.0763620 + 0.0248115i 7.96193 + 4.19615i 0.0224425
26.18 0.0434364 0.0597850i −1.13635 + 2.77646i 1.23438 + 3.79903i 1.31433 + 1.80902i 0.116632 + 0.188536i 1.56117 + 4.80478i 0.561868 + 0.182562i −6.41742 6.31005i 0.165242
26.19 0.356298 0.490402i 2.90702 0.741111i 1.12252 + 3.45477i −1.31433 1.80902i 0.672322 1.68966i −0.651691 2.00570i 4.40019 + 1.42971i 7.90151 4.30885i −1.35544
26.20 0.462120 0.636053i 2.06604 2.17520i 1.04506 + 3.21636i 1.31433 + 1.80902i −0.428787 2.31931i 3.54884 + 10.9222i 5.51962 + 1.79343i −0.462994 8.98808i 1.75801
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.3.q.a 128
3.b odd 2 1 inner 165.3.q.a 128
11.c even 5 1 inner 165.3.q.a 128
33.h odd 10 1 inner 165.3.q.a 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.3.q.a 128 1.a even 1 1 trivial
165.3.q.a 128 3.b odd 2 1 inner
165.3.q.a 128 11.c even 5 1 inner
165.3.q.a 128 33.h odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(165, [\chi])\).