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Properties

Label 285.2.q.d
Level $285$
Weight $2$
Character orbit 285.q
Analytic conductor $2.276$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [285,2,Mod(164,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.164");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.q (of order \(6\), degree \(2\), 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: \(2.27573645761\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 24 q^{4} - 4 q^{6} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 24 q^{4} - 4 q^{6} - 22 q^{9} - 36 q^{10} - 24 q^{15} + 32 q^{16} + 4 q^{19} + 24 q^{21} - 2 q^{24} - 8 q^{25} - 36 q^{30} - 60 q^{34} + 24 q^{36} - 4 q^{39} - 48 q^{40} + 16 q^{45} - 80 q^{49} - 12 q^{51} + 12 q^{54} - 12 q^{55} + 36 q^{60} + 36 q^{61} + 72 q^{64} + 18 q^{66} - 36 q^{70} + 72 q^{76} + 24 q^{79} - 22 q^{81} - 8 q^{85} - 30 q^{90} + 108 q^{91} + 4 q^{96} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
164.1 −2.12728 + 1.22819i −0.400301 − 1.68516i 2.01688 − 3.49334i 1.92782 + 1.13293i 2.92124 + 3.09316i 0.866312i 4.99566i −2.67952 + 1.34914i −5.49245 − 0.0423318i
164.2 −2.12728 + 1.22819i 1.25924 + 1.18925i 2.01688 − 3.49334i 0.0172335 + 2.23600i −4.13937 − 0.983287i − 0.866312i 4.99566i 0.171370 + 2.99510i −2.78288 − 4.73543i
164.3 −1.92143 + 1.10934i −1.22352 − 1.22597i 1.46126 − 2.53097i −2.23592 + 0.0252969i 3.71092 + 0.998308i − 0.860043i 2.04675i −0.00598857 + 2.99999i 4.26810 − 2.52900i
164.4 −1.92143 + 1.10934i 0.449957 + 1.67258i 1.46126 − 2.53097i 1.13987 − 1.92372i −2.72002 − 2.71460i 0.860043i 2.04675i −2.59508 + 1.50518i −0.0561255 + 4.96079i
164.5 −1.58878 + 0.917285i −1.72839 + 0.112615i 0.682824 − 1.18269i −0.277856 + 2.21874i 2.64273 − 1.76434i 4.07456i − 1.16376i 2.97464 − 0.389283i −1.59376 − 3.77997i
164.6 −1.58878 + 0.917285i −0.961720 + 1.44052i 0.682824 − 1.18269i 2.06041 + 0.868738i 0.206599 − 3.17085i − 4.07456i − 1.16376i −1.15019 − 2.77075i −4.07043 + 0.509747i
164.7 −1.46819 + 0.847662i 1.23638 − 1.21300i 0.437063 − 0.757015i −1.32408 + 1.80189i −0.787033 + 2.82895i − 3.64009i − 1.90872i 0.0572710 − 2.99945i 0.416619 − 3.76790i
164.8 −1.46819 + 0.847662i 1.66868 − 0.464237i 0.437063 − 0.757015i 2.22252 − 0.245746i −2.05643 + 2.09607i 3.64009i − 1.90872i 2.56897 − 1.54932i −3.05479 + 2.24475i
164.9 −1.16557 + 0.672945i 0.370002 − 1.69207i −0.0942902 + 0.163315i −1.81800 − 1.30187i 0.707405 + 2.22122i 3.25990i − 2.94559i −2.72620 − 1.25214i 2.99510 + 0.294015i
164.10 −1.16557 + 0.672945i 1.65038 + 0.525604i −0.0942902 + 0.163315i −0.218454 − 2.22537i −2.27734 + 0.497981i − 3.25990i − 2.94559i 2.44748 + 1.73489i 1.75218 + 2.44683i
164.11 −0.621247 + 0.358677i −1.44719 − 0.951645i −0.742701 + 1.28640i 1.73750 + 1.40751i 1.24040 + 0.0721314i − 3.62465i − 2.50027i 1.18874 + 2.75443i −1.58426 − 0.251213i
164.12 −0.621247 + 0.358677i 0.100552 + 1.72913i −0.742701 + 1.28640i 0.350194 + 2.20848i −0.682667 − 1.03815i 3.62465i − 2.50027i −2.97978 + 0.347735i −1.00969 − 1.24640i
164.13 −0.598712 + 0.345666i −1.14198 − 1.30226i −0.761030 + 1.31814i 1.22036 − 1.87369i 1.13386 + 0.384938i 1.97295i − 2.43491i −0.391784 + 2.97431i −0.0829680 + 1.54364i
164.14 −0.598712 + 0.345666i 0.556806 + 1.64011i −0.761030 + 1.31814i −2.23285 + 0.120012i −0.900298 − 0.789485i − 1.97295i − 2.43491i −2.37993 + 1.82645i 1.29535 − 0.843672i
164.15 0.598712 − 0.345666i −0.556806 − 1.64011i −0.761030 + 1.31814i −1.22036 + 1.87369i −0.900298 − 0.789485i 1.97295i 2.43491i −2.37993 + 1.82645i −0.0829680 + 1.54364i
164.16 0.598712 − 0.345666i 1.14198 + 1.30226i −0.761030 + 1.31814i 2.23285 − 0.120012i 1.13386 + 0.384938i − 1.97295i 2.43491i −0.391784 + 2.97431i 1.29535 − 0.843672i
164.17 0.621247 − 0.358677i −0.100552 − 1.72913i −0.742701 + 1.28640i −1.73750 − 1.40751i −0.682667 − 1.03815i − 3.62465i 2.50027i −2.97978 + 0.347735i −1.58426 − 0.251213i
164.18 0.621247 − 0.358677i 1.44719 + 0.951645i −0.742701 + 1.28640i −0.350194 − 2.20848i 1.24040 + 0.0721314i 3.62465i 2.50027i 1.18874 + 2.75443i −1.00969 − 1.24640i
164.19 1.16557 − 0.672945i −1.65038 − 0.525604i −0.0942902 + 0.163315i 1.81800 + 1.30187i −2.27734 + 0.497981i 3.25990i 2.94559i 2.44748 + 1.73489i 2.99510 + 0.294015i
164.20 1.16557 − 0.672945i −0.370002 + 1.69207i −0.0942902 + 0.163315i 0.218454 + 2.22537i 0.707405 + 2.22122i − 3.25990i 2.94559i −2.72620 − 1.25214i 1.75218 + 2.44683i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 164.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner
95.h odd 6 1 inner
285.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.q.d ✓ 56
3.b odd 2 1 inner 285.2.q.d ✓ 56
5.b even 2 1 inner 285.2.q.d ✓ 56
15.d odd 2 1 inner 285.2.q.d ✓ 56
19.d odd 6 1 inner 285.2.q.d ✓ 56
57.f even 6 1 inner 285.2.q.d ✓ 56
95.h odd 6 1 inner 285.2.q.d ✓ 56
285.q even 6 1 inner 285.2.q.d ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.q.d ✓ 56 1.a even 1 1 trivial
285.2.q.d ✓ 56 3.b odd 2 1 inner
285.2.q.d ✓ 56 5.b even 2 1 inner
285.2.q.d ✓ 56 15.d odd 2 1 inner
285.2.q.d ✓ 56 19.d odd 6 1 inner
285.2.q.d ✓ 56 57.f even 6 1 inner
285.2.q.d ✓ 56 95.h odd 6 1 inner
285.2.q.d ✓ 56 285.q even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 20 T_{2}^{26} + 242 T_{2}^{24} - 1902 T_{2}^{22} + 11052 T_{2}^{20} - 47551 T_{2}^{18} + \cdots + 16384 \) acting on \(S_{2}^{\mathrm{new}}(285, [\chi])\). Copy content Toggle raw display