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Properties

Label 465.2.bs.a
Level $465$
Weight $2$
Character orbit 465.bs
Analytic conductor $3.713$
Analytic rank $0$
Dimension $512$
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] = [465,2,Mod(13,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([0, 45, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.bs (of order \(60\), degree \(16\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(512\)
Relative dimension: \(32\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 512 q - 12 q^{6} - 24 q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 512 q - 12 q^{6} - 24 q^{7} + 24 q^{8} - 8 q^{10} + 128 q^{16} - 60 q^{20} + 16 q^{21} - 12 q^{22} + 8 q^{25} + 24 q^{28} + 16 q^{31} - 232 q^{32} + 24 q^{33} + 72 q^{35} + 252 q^{36} + 36 q^{37} - 48 q^{38} - 44 q^{40} - 48 q^{41} - 28 q^{42} - 72 q^{43} + 20 q^{46} + 24 q^{47} - 312 q^{50} - 8 q^{51} - 72 q^{52} + 40 q^{53} - 100 q^{55} - 80 q^{56} - 24 q^{57} - 40 q^{62} - 8 q^{63} - 132 q^{65} - 24 q^{66} - 80 q^{67} + 132 q^{68} - 328 q^{70} + 144 q^{71} - 12 q^{72} + 16 q^{73} - 192 q^{76} - 40 q^{77} - 60 q^{80} - 64 q^{81} + 4 q^{82} + 132 q^{83} - 120 q^{85} - 28 q^{87} - 204 q^{88} - 12 q^{90} - 40 q^{91} - 88 q^{93} + 144 q^{95} - 44 q^{96} + 80 q^{97} - 124 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.433529 + 2.73720i −0.358368 + 0.933580i −5.40219 − 1.75528i 1.96269 − 1.07138i −2.40003 − 1.38566i −3.72756 − 0.195353i 4.63025 − 9.08737i −0.743145 − 0.669131i 2.08169 + 5.83674i
13.2 −0.426078 + 2.69015i 0.358368 − 0.933580i −5.15325 − 1.67439i −1.65615 − 1.50238i 2.35878 + 1.36184i 1.41184 + 0.0739914i 4.22699 − 8.29594i −0.743145 − 0.669131i 4.74728 − 3.81517i
13.3 −0.382075 + 2.41233i 0.358368 − 0.933580i −3.77124 − 1.22535i 0.298852 + 2.21601i 2.11518 + 1.22120i −4.05355 − 0.212437i 2.17919 − 4.27691i −0.743145 − 0.669131i −5.45992 − 0.125753i
13.4 −0.355765 + 2.24621i −0.358368 + 0.933580i −3.01677 − 0.980209i −2.00132 − 0.997358i −1.96952 − 1.13710i 0.538218 + 0.0282068i 1.21008 − 2.37491i −0.743145 − 0.669131i 2.95227 − 4.14056i
13.5 −0.338674 + 2.13831i 0.358368 − 0.933580i −2.55554 − 0.830344i 0.609287 − 2.15146i 1.87491 + 1.08248i 0.166773 + 0.00874023i 0.675282 − 1.32531i −0.743145 − 0.669131i 4.39412 + 2.03149i
13.6 −0.304997 + 1.92567i −0.358368 + 0.933580i −1.71308 − 0.556614i −1.74140 + 1.40269i −1.68847 − 0.974839i 0.217239 + 0.0113850i −0.175929 + 0.345280i −0.743145 − 0.669131i −2.17000 − 3.78118i
13.7 −0.304926 + 1.92523i −0.358368 + 0.933580i −1.71141 − 0.556072i 1.91129 + 1.16060i −1.68808 − 0.974613i 0.858087 + 0.0449704i −0.177442 + 0.348249i −0.743145 − 0.669131i −2.81721 + 3.32577i
13.8 −0.286130 + 1.80656i 0.358368 − 0.933580i −1.27966 − 0.415788i 0.397012 + 2.20054i 1.58403 + 0.914538i 4.46526 + 0.234014i −0.543472 + 1.06662i −0.743145 − 0.669131i −4.08900 + 0.0875831i
13.9 −0.225735 + 1.42523i 0.358368 − 0.933580i −0.0782209 − 0.0254155i 2.10884 − 0.743497i 1.24967 + 0.721500i −1.03173 − 0.0540706i −1.25634 + 2.46570i −0.743145 − 0.669131i 0.583618 + 3.17342i
13.10 −0.209306 + 1.32151i 0.358368 − 0.933580i 0.199544 + 0.0648357i −2.16726 + 0.550423i 1.15872 + 0.668989i 0.463549 + 0.0242936i −1.34231 + 2.63442i −0.743145 − 0.669131i −0.273766 − 2.97926i
13.11 −0.190621 + 1.20353i −0.358368 + 0.933580i 0.489958 + 0.159197i −0.599862 + 2.15410i −1.05528 − 0.609268i −4.09367 − 0.214540i −1.39140 + 2.73078i −0.743145 − 0.669131i −2.47819 − 1.13257i
13.12 −0.166776 + 1.05299i 0.358368 − 0.933580i 0.821150 + 0.266808i −1.75047 − 1.39135i 0.923279 + 0.533055i −4.91394 − 0.257529i −1.38590 + 2.71998i −0.743145 − 0.669131i 1.75701 − 1.61118i
13.13 −0.123437 + 0.779348i −0.358368 + 0.933580i 1.30997 + 0.425634i 0.487229 − 2.18234i −0.683348 − 0.394531i 0.341042 + 0.0178733i −1.20987 + 2.37450i −0.743145 − 0.669131i 1.64066 + 0.649101i
13.14 −0.0611943 + 0.386366i −0.358368 + 0.933580i 1.75658 + 0.570747i −2.09338 − 0.785982i −0.338773 − 0.195591i 3.22656 + 0.169097i −0.683196 + 1.34085i −0.743145 − 0.669131i 0.431779 − 0.760712i
13.15 −0.0181530 + 0.114614i −0.358368 + 0.933580i 1.88931 + 0.613873i 2.10455 + 0.755568i −0.100496 − 0.0580211i −4.98017 − 0.261000i −0.210019 + 0.412185i −0.743145 − 0.669131i −0.124802 + 0.227494i
13.16 −0.00559847 + 0.0353474i 0.358368 − 0.933580i 1.90089 + 0.617638i 0.593971 − 2.15574i 0.0309933 + 0.0178940i 1.74650 + 0.0915300i −0.0649688 + 0.127508i −0.743145 − 0.669131i 0.0728743 + 0.0330642i
13.17 0.0323886 − 0.204494i −0.358368 + 0.933580i 1.86134 + 0.604787i 0.285479 + 2.21777i 0.179304 + 0.103521i 3.35389 + 0.175770i 0.371953 − 0.729998i −0.743145 − 0.669131i 0.462767 + 0.0134518i
13.18 0.0380813 − 0.240436i 0.358368 − 0.933580i 1.84575 + 0.599722i −2.21164 − 0.329627i −0.210819 − 0.121716i 2.75330 + 0.144295i 0.435516 − 0.854748i −0.743145 − 0.669131i −0.163476 + 0.519205i
13.19 0.0578387 − 0.365179i 0.358368 − 0.933580i 1.77210 + 0.575791i 1.35928 + 1.77549i −0.320196 − 0.184865i −0.380516 − 0.0199420i 0.648472 − 1.27270i −0.743145 − 0.669131i 0.726989 − 0.393689i
13.20 0.150206 − 0.948362i −0.358368 + 0.933580i 1.02528 + 0.333135i −0.378150 − 2.20386i 0.831543 + 0.480092i −2.42313 − 0.126991i 1.34176 − 2.63336i −0.743145 − 0.669131i −2.14686 + 0.0275903i
See next 80 embeddings (of 512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.h odd 30 1 inner
155.x even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.bs.a ✓ 512
5.c odd 4 1 inner 465.2.bs.a ✓ 512
31.h odd 30 1 inner 465.2.bs.a ✓ 512
155.x even 60 1 inner 465.2.bs.a ✓ 512
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.bs.a ✓ 512 1.a even 1 1 trivial
465.2.bs.a ✓ 512 5.c odd 4 1 inner
465.2.bs.a ✓ 512 31.h odd 30 1 inner
465.2.bs.a ✓ 512 155.x even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(465, [\chi])\).