Properties

Label 675.2.bd.a
Level $675$
Weight $2$
Character orbit 675.bd
Analytic conductor $5.390$
Analytic rank $0$
Dimension $448$
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] = [675,2,Mod(8,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.bd (of order \(60\), degree \(16\), not 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(448\)
Relative dimension: \(28\) over \(\Q(\zeta_{60})\)
Twist minimal: no (minimal twist has level 225)
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.

\(\operatorname{Tr}(f)(q) = \) \( 448 q + 24 q^{2} - 10 q^{4} + 24 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 448 q + 24 q^{2} - 10 q^{4} + 24 q^{5} - 8 q^{7} - 32 q^{10} + 18 q^{11} - 8 q^{13} + 30 q^{14} - 50 q^{16} - 40 q^{19} + 48 q^{20} + 48 q^{23} + 16 q^{25} - 24 q^{28} + 30 q^{29} - 6 q^{31} + 60 q^{32} - 10 q^{34} - 44 q^{37} - 16 q^{40} + 18 q^{41} - 8 q^{43} - 24 q^{46} + 18 q^{47} - 24 q^{50} + 24 q^{52} - 24 q^{55} + 18 q^{56} - 4 q^{58} - 6 q^{61} - 40 q^{64} + 96 q^{65} - 14 q^{67} - 288 q^{68} - 28 q^{70} - 32 q^{73} - 32 q^{76} - 216 q^{77} - 10 q^{79} - 72 q^{82} - 36 q^{83} - 32 q^{85} + 18 q^{86} - 28 q^{88} - 24 q^{91} - 30 q^{92} - 130 q^{94} - 6 q^{95} - 38 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −1.42019 2.18691i 0 −1.95214 + 4.38458i −0.865670 2.06170i 0 0.0498024 + 0.185865i 7.21013 1.14197i 0 −3.27933 + 4.82116i
8.2 −1.35622 2.08840i 0 −1.70861 + 3.83760i −2.16393 + 0.563377i 0 0.669023 + 2.49683i 5.41275 0.857295i 0 4.11134 + 3.75510i
8.3 −1.23818 1.90664i 0 −1.28869 + 2.89444i 2.20972 0.342224i 0 0.640500 + 2.39038i 2.62344 0.415512i 0 −3.38854 3.78940i
8.4 −1.18213 1.82032i 0 −1.10267 + 2.47663i 2.17921 0.501050i 0 −0.367194 1.37039i 1.52424 0.241416i 0 −3.48818 3.37456i
8.5 −1.03790 1.59823i 0 −0.663624 + 1.49052i −0.300573 2.21577i 0 −0.911535 3.40190i −0.693439 + 0.109830i 0 −3.22935 + 2.78014i
8.6 −1.01926 1.56953i 0 −0.611049 + 1.37244i −0.303082 + 2.21543i 0 −0.0934709 0.348838i −0.919915 + 0.145700i 0 3.78611 1.78242i
8.7 −0.754539 1.16189i 0 0.0328191 0.0737129i −0.331994 + 2.21128i 0 0.541248 + 2.01996i −2.84708 + 0.450933i 0 2.81977 1.28276i
8.8 −0.752499 1.15875i 0 0.0370346 0.0831810i 1.32432 1.80172i 0 1.05097 + 3.92229i −2.85353 + 0.451954i 0 −3.08428 0.178760i
8.9 −0.745007 1.14721i 0 0.0524168 0.117730i −2.22341 0.237578i 0 −1.22522 4.57257i −2.87621 + 0.455548i 0 1.38391 + 2.72772i
8.10 −0.431818 0.664942i 0 0.557793 1.25282i 1.15892 + 1.91230i 0 0.246375 + 0.919483i −2.64010 + 0.418151i 0 0.771129 1.59638i
8.11 −0.326894 0.503372i 0 0.666949 1.49799i −0.293865 2.21667i 0 −0.105719 0.394547i −2.15769 + 0.341745i 0 −1.01975 + 0.872540i
8.12 −0.270032 0.415813i 0 0.713490 1.60253i 2.01262 + 0.974349i 0 −0.995692 3.71597i −1.83841 + 0.291175i 0 −0.138325 1.09998i
8.13 −0.0328244 0.0505452i 0 0.811996 1.82377i −2.19134 + 0.444987i 0 0.835175 + 3.11691i −0.237889 + 0.0376779i 0 0.0944215 + 0.0961555i
8.14 0.154007 + 0.237149i 0 0.780951 1.75405i 1.48145 1.67490i 0 1.05334 + 3.93112i 1.09482 0.173402i 0 0.625356 + 0.0933787i
8.15 0.191087 + 0.294249i 0 0.763405 1.71464i 1.24308 + 1.85869i 0 −0.223000 0.832248i 1.34347 0.212785i 0 −0.309381 + 0.720948i
8.16 0.299974 + 0.461920i 0 0.690088 1.54996i −1.56435 + 1.59775i 0 −0.838741 3.13022i 2.01096 0.318504i 0 −1.20730 0.243322i
8.17 0.336419 + 0.518040i 0 0.658286 1.47853i 1.21725 1.87571i 0 −0.707664 2.64104i 2.20757 0.349645i 0 1.38120 0.000442522i
8.18 0.386810 + 0.595635i 0 0.608314 1.36630i −2.18497 0.475279i 0 0.140157 + 0.523074i 2.45205 0.388367i 0 −0.562077 1.48529i
8.19 0.617042 + 0.950162i 0 0.291407 0.654510i 2.22710 0.200038i 0 0.126426 + 0.471827i 3.03968 0.481438i 0 1.56428 + 1.99268i
8.20 0.822576 + 1.26666i 0 −0.114312 + 0.256749i −1.20491 + 1.88366i 0 0.0589271 + 0.219919i 2.56419 0.406129i 0 −3.37708 + 0.0232479i
See next 80 embeddings (of 448 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
25.f odd 20 1 inner
225.w even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.bd.a 448
3.b odd 2 1 225.2.w.a 448
9.c even 3 1 225.2.w.a 448
9.d odd 6 1 inner 675.2.bd.a 448
25.f odd 20 1 inner 675.2.bd.a 448
75.l even 20 1 225.2.w.a 448
225.w even 60 1 inner 675.2.bd.a 448
225.x odd 60 1 225.2.w.a 448
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.w.a 448 3.b odd 2 1
225.2.w.a 448 9.c even 3 1
225.2.w.a 448 75.l even 20 1
225.2.w.a 448 225.x odd 60 1
675.2.bd.a 448 1.a even 1 1 trivial
675.2.bd.a 448 9.d odd 6 1 inner
675.2.bd.a 448 25.f odd 20 1 inner
675.2.bd.a 448 225.w even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(675, [\chi])\).