↑

Properties

Label 285.2.be.a
Level $285$
Weight $2$
Character orbit 285.be
Analytic conductor $2.276$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [285,2,Mod(4,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.be (of order \(18\), degree \(6\), 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: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 6 q^{4} - 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 6 q^{4} - 6 q^{6} + 18 q^{10} + 12 q^{11} - 60 q^{14} - 12 q^{15} + 18 q^{16} - 24 q^{19} - 108 q^{20} + 24 q^{21} - 24 q^{24} - 12 q^{25} + 12 q^{26} - 24 q^{29} + 72 q^{31} + 6 q^{34} - 36 q^{35} - 6 q^{36} - 150 q^{40} - 36 q^{41} + 36 q^{44} + 6 q^{45} + 24 q^{46} + 48 q^{49} - 18 q^{50} - 12 q^{51} + 6 q^{54} + 18 q^{55} - 48 q^{59} - 24 q^{60} - 60 q^{61} + 54 q^{64} - 72 q^{69} + 108 q^{70} + 96 q^{71} - 60 q^{74} - 96 q^{76} - 24 q^{79} + 54 q^{80} - 108 q^{84} + 102 q^{85} + 168 q^{86} - 18 q^{90} + 12 q^{91} + 24 q^{94} + 90 q^{95} - 24 q^{96} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.928566 + 2.55121i 0.984808 − 0.173648i −4.11437 − 3.45237i −0.517404 − 2.17538i −0.471445 + 2.67370i 4.29857 + 2.48178i 7.92577 − 4.57594i 0.939693 − 0.342020i 6.03031 + 0.699978i
4.2 −0.892261 + 2.45147i −0.984808 + 0.173648i −3.68147 − 3.08912i 1.53740 − 1.62370i 0.453013 − 2.56916i −2.64336 − 1.52614i 6.33915 − 3.65991i 0.939693 − 0.342020i 2.60869 + 5.21765i
4.3 −0.743551 + 2.04289i −0.984808 + 0.173648i −2.08844 − 1.75241i 1.17871 + 1.90017i 0.377511 − 2.14097i 3.63754 + 2.10013i 1.36737 − 0.789454i 0.939693 − 0.342020i −4.75827 + 0.995106i
4.4 −0.642738 + 1.76591i −0.984808 + 0.173648i −1.17323 − 0.984456i −1.84371 − 1.26521i 0.326327 − 1.85069i 0.268978 + 0.155294i −0.762403 + 0.440173i 0.939693 − 0.342020i 3.41926 − 2.44262i
4.5 −0.517612 + 1.42213i 0.984808 − 0.173648i −0.222438 − 0.186648i 0.216290 + 2.22558i −0.262799 + 1.49041i 2.31097 + 1.33424i −2.24071 + 1.29367i 0.939693 − 0.342020i −3.27702 − 0.844398i
4.6 −0.513721 + 1.41144i 0.984808 − 0.173648i −0.196156 − 0.164594i 1.96066 − 1.07508i −0.260823 + 1.47920i −1.05300 − 0.607949i −2.26849 + 1.30971i 0.939693 − 0.342020i 0.510176 + 3.31964i
4.7 −0.442699 + 1.21631i −0.984808 + 0.173648i 0.248670 + 0.208659i −0.296783 + 2.21629i 0.224764 − 1.27470i −3.57134 − 2.06191i −2.60579 + 1.50445i 0.939693 − 0.342020i −2.56430 − 1.34213i
4.8 −0.419721 + 1.15317i 0.984808 − 0.173648i 0.378445 + 0.317553i −2.21620 + 0.297450i −0.213098 + 1.20854i 1.17230 + 0.676827i −2.65058 + 1.53031i 0.939693 − 0.342020i 0.587172 − 2.68050i
4.9 −0.141205 + 0.387956i −0.984808 + 0.173648i 1.40152 + 1.17601i 2.23352 + 0.106744i 0.0716914 − 0.406582i 0.754049 + 0.435350i −1.36923 + 0.790524i 0.939693 − 0.342020i −0.356795 + 0.851435i
4.10 −0.0411715 + 0.113118i −0.984808 + 0.173648i 1.52099 + 1.27626i −1.67801 − 1.47794i 0.0209033 − 0.118549i 1.56369 + 0.902797i −0.415489 + 0.239883i 0.939693 − 0.342020i 0.236267 − 0.128963i
4.11 0.0411715 − 0.113118i 0.984808 − 0.173648i 1.52099 + 1.27626i 1.16410 + 1.90915i 0.0209033 − 0.118549i −1.56369 − 0.902797i 0.415489 − 0.239883i 0.939693 − 0.342020i 0.263887 − 0.0530777i
4.12 0.141205 − 0.387956i 0.984808 − 0.173648i 1.40152 + 1.17601i 0.282724 − 2.21812i 0.0716914 − 0.406582i −0.754049 − 0.435350i 1.36923 − 0.790524i 0.939693 − 0.342020i −0.820613 − 0.422893i
4.13 0.419721 − 1.15317i −0.984808 + 0.173648i 0.378445 + 0.317553i −0.677769 + 2.13088i −0.213098 + 1.20854i −1.17230 − 0.676827i 2.65058 − 1.53031i 0.939693 − 0.342020i 2.17279 + 1.67596i
4.14 0.442699 − 1.21631i 0.984808 − 0.173648i 0.248670 + 0.208659i −2.23415 − 0.0925794i 0.224764 − 1.27470i 3.57134 + 2.06191i 2.60579 − 1.50445i 0.939693 − 0.342020i −1.10166 + 2.67643i
4.15 0.513721 − 1.41144i −0.984808 + 0.173648i −0.196156 − 0.164594i 1.39921 − 1.74419i −0.260823 + 1.47920i 1.05300 + 0.607949i 2.26849 − 1.30971i 0.939693 − 0.342020i −1.74301 − 2.87093i
4.16 0.517612 − 1.42213i −0.984808 + 0.173648i −0.222438 − 0.186648i −2.15421 − 0.599472i −0.262799 + 1.49041i −2.31097 − 1.33424i 2.24071 − 1.29367i 0.939693 − 0.342020i −1.96757 + 2.75327i
4.17 0.642738 − 1.76591i 0.984808 − 0.173648i −1.17323 − 0.984456i 0.925829 + 2.03540i 0.326327 − 1.85069i −0.268978 − 0.155294i 0.762403 − 0.440173i 0.939693 − 0.342020i 4.18939 − 0.326702i
4.18 0.743551 − 2.04289i 0.984808 − 0.173648i −2.08844 − 1.75241i −1.66662 − 1.49077i 0.377511 − 2.14097i −3.63754 − 2.10013i −1.36737 + 0.789454i 0.939693 − 0.342020i −4.28469 + 2.29626i
4.19 0.892261 − 2.45147i 0.984808 − 0.173648i −3.68147 − 3.08912i 1.86600 − 1.23209i 0.453013 − 2.56916i 2.64336 + 1.52614i −6.33915 + 3.65991i 0.939693 − 0.342020i −1.35546 − 5.67378i
4.20 0.928566 − 2.55121i −0.984808 + 0.173648i −4.11437 − 3.45237i 2.05249 + 0.887295i −0.471445 + 2.67370i −4.29857 − 2.48178i −7.92577 + 4.57594i 0.939693 − 0.342020i 4.16955 − 4.41242i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.e even 9 1 inner
95.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.be.a ✓ 120
3.b odd 2 1 855.2.da.d 120
5.b even 2 1 inner 285.2.be.a ✓ 120
15.d odd 2 1 855.2.da.d 120
19.e even 9 1 inner 285.2.be.a ✓ 120
57.l odd 18 1 855.2.da.d 120
95.p even 18 1 inner 285.2.be.a ✓ 120
285.bd odd 18 1 855.2.da.d 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.be.a ✓ 120 1.a even 1 1 trivial
285.2.be.a ✓ 120 5.b even 2 1 inner
285.2.be.a ✓ 120 19.e even 9 1 inner
285.2.be.a ✓ 120 95.p even 18 1 inner
855.2.da.d 120 3.b odd 2 1
855.2.da.d 120 15.d odd 2 1
855.2.da.d 120 57.l odd 18 1
855.2.da.d 120 285.bd odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(285, [\chi])\).