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Properties

Label 405.2.x.a
Level $405$
Weight $2$
Character orbit 405.x
Analytic conductor $3.234$
Analytic rank $0$
Dimension $1872$
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] = [405,2,Mod(2,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(108))
 
chi = DirichletCharacter(H, H._module([2, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.x (of order \(108\), degree \(36\), 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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(1872\)
Relative dimension: \(52\) over \(\Q(\zeta_{108})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{108}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1872 q - 36 q^{2} - 36 q^{3} - 36 q^{5} - 72 q^{6} - 36 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1872 q - 36 q^{2} - 36 q^{3} - 36 q^{5} - 72 q^{6} - 36 q^{7} - 36 q^{8} - 36 q^{10} - 72 q^{11} - 36 q^{12} - 36 q^{13} - 36 q^{15} - 72 q^{16} - 36 q^{17} - 36 q^{18} - 108 q^{20} - 72 q^{21} - 36 q^{22} - 36 q^{23} - 36 q^{25} - 108 q^{26} - 36 q^{27} - 18 q^{28} - 144 q^{30} - 72 q^{31} - 36 q^{32} - 36 q^{33} - 90 q^{35} - 72 q^{36} - 36 q^{37} - 36 q^{38} - 36 q^{40} - 108 q^{41} - 36 q^{42} - 36 q^{43} - 36 q^{45} - 72 q^{46} - 36 q^{47} - 36 q^{48} - 36 q^{50} - 288 q^{51} - 72 q^{52} - 54 q^{53} - 18 q^{55} - 324 q^{56} - 36 q^{57} - 36 q^{58} - 36 q^{60} - 72 q^{61} - 36 q^{62} - 36 q^{63} - 36 q^{65} - 216 q^{66} - 90 q^{67} - 36 q^{68} - 36 q^{70} - 72 q^{71} + 396 q^{72} - 36 q^{73} - 36 q^{75} - 72 q^{76} + 252 q^{77} - 90 q^{78} - 72 q^{81} - 72 q^{82} - 36 q^{83} - 36 q^{85} - 72 q^{86} + 252 q^{87} - 36 q^{88} - 36 q^{90} - 72 q^{91} + 396 q^{92} - 216 q^{93} - 36 q^{95} - 72 q^{96} - 90 q^{97} - 378 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.83612 − 2.06323i −0.260873 + 1.71229i −0.653398 + 5.59018i −1.50364 − 1.65501i 4.01184 − 2.60573i 0.250802 − 1.71221i 8.20864 − 5.74775i −2.86389 − 0.893381i −0.653810 + 6.14115i
2.2 −1.71334 − 1.92527i 1.31566 − 1.12652i −0.538926 + 4.61081i 0.439117 − 2.19253i −4.42302 − 0.602885i −0.344262 + 2.35026i 5.57809 − 3.90582i 0.461917 − 2.96423i −4.97356 + 2.91113i
2.3 −1.71051 − 1.92209i 1.43328 + 0.972474i −0.536380 + 4.58902i 0.751369 + 2.10605i −0.582465 − 4.41832i −0.306482 + 2.09233i 5.52264 − 3.86699i 1.10859 + 2.78766i 2.76278 − 5.04662i
2.4 −1.70633 − 1.91739i −1.72257 − 0.181001i −0.532629 + 4.55694i −0.378159 + 2.20386i 2.59222 + 3.61168i 0.513061 − 3.50264i 5.44123 − 3.80999i 2.93448 + 0.623574i 4.87092 − 3.03544i
2.5 −1.66005 − 1.86538i −0.213949 − 1.71879i −0.491700 + 4.20676i 1.87887 + 1.21238i −2.85103 + 3.25236i −0.102901 + 0.702499i 4.57249 − 3.20169i −2.90845 + 0.735464i −0.857465 − 5.51741i
2.6 −1.56301 − 1.75634i −1.64959 − 0.528067i −0.409543 + 3.50386i 0.754059 − 2.10509i 1.65086 + 3.72261i −0.343032 + 2.34186i 2.94226 − 2.06019i 2.44229 + 1.74219i −4.87585 + 1.96589i
2.7 −1.49118 − 1.67563i 1.71080 + 0.270456i −0.351919 + 3.01086i 2.08294 − 0.813251i −2.09794 − 3.26997i 0.753657 − 5.14518i 1.89504 − 1.32692i 2.85371 + 0.925395i −4.46875 − 2.27752i
2.8 −1.43127 − 1.60831i −0.264734 − 1.71170i −0.305927 + 2.61737i −1.74533 − 1.39779i −2.37403 + 2.87568i 0.657979 − 4.49199i 1.12022 − 0.784386i −2.85983 + 0.906291i 0.249961 + 4.80765i
2.9 −1.36631 − 1.53531i 1.55757 + 0.757619i −0.258190 + 2.20895i −2.19767 − 0.412602i −0.964939 − 3.42649i −0.210071 + 1.43415i 0.377101 − 0.264049i 1.85203 + 2.36008i 2.36923 + 3.93785i
2.10 −1.36218 − 1.53066i −1.44361 + 0.957072i −0.255225 + 2.18359i −2.12396 + 0.699146i 3.43141 + 0.905983i −0.534030 + 3.64580i 0.333100 − 0.233239i 1.16802 − 2.76328i 3.96336 + 2.29871i
2.11 −1.31385 − 1.47636i −0.00262291 + 1.73205i −0.221252 + 1.89293i 2.19146 − 0.444417i 2.56057 − 2.27177i 0.206840 − 1.41209i −0.152476 + 0.106765i −2.99999 − 0.00908601i −3.53536 − 2.65148i
2.12 −1.30887 − 1.47076i 0.665113 − 1.59926i −0.217821 + 1.86358i −1.65069 + 1.50839i −3.22267 + 1.11499i −0.199961 + 1.36512i −0.199550 + 0.139727i −2.11525 − 2.12737i 4.37901 + 0.453490i
2.13 −1.03315 − 1.16094i −1.35175 − 1.08294i −0.0481988 + 0.412367i 2.17432 + 0.521848i 0.139330 + 2.68814i −0.0305300 + 0.208427i −2.01753 + 1.41269i 0.654469 + 2.92774i −1.64056 − 3.06340i
2.14 −0.966586 − 1.08614i 0.658686 + 1.60192i −0.0132344 + 0.113227i 1.31714 − 1.80697i 1.10323 − 2.26382i −0.713868 + 4.87354i −2.24625 + 1.57284i −2.13226 + 2.11032i −3.23576 + 0.315991i
2.15 −0.922978 − 1.03714i −1.39946 + 1.02054i 0.00841068 − 0.0719580i −0.263454 − 2.22049i 2.35012 + 0.509501i 0.297166 − 2.02873i −2.35695 + 1.65036i 0.916983 − 2.85642i −2.05981 + 2.32271i
2.16 −0.911348 − 1.02407i 1.61255 − 0.632213i 0.0140147 − 0.119903i 1.79534 + 1.33294i −2.11702 − 1.07520i −0.604110 + 4.12423i −2.38146 + 1.66752i 2.20061 − 2.03894i −0.271150 − 3.05334i
2.17 −0.889503 − 0.999527i 1.66061 − 0.492310i 0.0243482 − 0.208312i −0.688030 + 2.12758i −1.96920 − 1.22191i 0.530260 − 3.62006i −2.42194 + 1.69586i 2.51526 − 1.63507i 2.73858 − 1.20479i
2.18 −0.867561 − 0.974870i −1.35021 + 1.08487i 0.0344757 − 0.294958i 1.36273 + 1.77284i 2.22899 + 0.375085i 0.246343 − 1.68177i −2.45545 + 1.71932i 0.646113 − 2.92960i 0.546034 − 2.86653i
2.19 −0.675160 − 0.758671i −1.38521 − 1.03981i 0.112445 − 0.962028i −1.44290 + 1.70823i 0.146362 + 1.75295i −0.111106 + 0.758516i −2.46963 + 1.72925i 0.837590 + 2.88070i 2.27017 − 0.0586441i
2.20 −0.661211 − 0.742997i 0.615793 + 1.61889i 0.117341 − 1.00392i −2.23600 + 0.0180386i 0.795660 − 1.52796i 0.368479 − 2.51558i −2.45297 + 1.71759i −2.24160 + 1.99380i 1.49187 + 1.64941i
See next 80 embeddings (of 1872 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
81.h odd 54 1 inner
405.x even 108 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.x.a ✓ 1872
5.c odd 4 1 inner 405.2.x.a ✓ 1872
81.h odd 54 1 inner 405.2.x.a ✓ 1872
405.x even 108 1 inner 405.2.x.a ✓ 1872
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.x.a ✓ 1872 1.a even 1 1 trivial
405.2.x.a ✓ 1872 5.c odd 4 1 inner
405.2.x.a ✓ 1872 81.h odd 54 1 inner
405.2.x.a ✓ 1872 405.x even 108 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(405, [\chi])\).