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Properties

Label 286.2.x.a
Level $286$
Weight $2$
Character orbit 286.x
Analytic conductor $2.284$
Analytic rank $0$
Dimension $224$
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] = [286,2,Mod(7,286)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(286, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([42, 55]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("286.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 286 = 2 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 286.x (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: \(2.28372149781\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 28 q^{9} - 20 q^{11} - 20 q^{13} - 16 q^{14} + 56 q^{15} - 28 q^{16} - 4 q^{22} - 48 q^{23} + 4 q^{26} - 48 q^{27} + 20 q^{29} + 16 q^{31} - 36 q^{33} - 32 q^{34} - 100 q^{35} + 48 q^{36} - 24 q^{37} - 40 q^{39} + 20 q^{41} + 16 q^{44} - 56 q^{45} - 80 q^{46} + 64 q^{47} - 8 q^{53} + 20 q^{55} - 24 q^{56} + 4 q^{58} - 96 q^{59} + 8 q^{60} + 80 q^{61} - 80 q^{63} - 48 q^{66} - 152 q^{67} - 20 q^{68} - 56 q^{70} - 44 q^{71} + 40 q^{72} - 60 q^{73} - 120 q^{75} - 8 q^{78} + 80 q^{79} - 108 q^{81} + 48 q^{82} - 180 q^{83} + 80 q^{84} + 80 q^{85} - 48 q^{86} + 112 q^{89} + 24 q^{91} + 16 q^{92} - 84 q^{93} - 120 q^{94} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.358368 + 0.933580i −2.17218 − 2.41245i −0.743145 − 0.669131i −3.77635 − 0.598115i 3.03065 − 1.16336i 1.60589 − 0.0841610i 0.891007 − 0.453990i −0.787961 + 7.49695i 1.91171 − 3.31118i
7.2 −0.358368 + 0.933580i −1.78486 − 1.98229i −0.743145 − 0.669131i 3.31385 + 0.524862i 2.49026 − 0.955922i −1.57906 + 0.0827550i 0.891007 − 0.453990i −0.430155 + 4.09265i −1.67758 + 2.90565i
7.3 −0.358368 + 0.933580i −0.204079 − 0.226653i −0.743145 − 0.669131i −1.92750 − 0.305285i 0.284734 − 0.109299i 4.46519 − 0.234010i 0.891007 − 0.453990i 0.303862 − 2.89106i 0.975762 − 1.69007i
7.4 −0.358368 + 0.933580i −0.127174 − 0.141241i −0.743145 − 0.669131i −1.99895 − 0.316603i 0.177435 − 0.0681108i −1.47577 + 0.0773420i 0.891007 − 0.453990i 0.309810 − 2.94764i 1.01194 − 1.75272i
7.5 −0.358368 + 0.933580i 0.432264 + 0.480078i −0.743145 − 0.669131i 1.94540 + 0.308121i −0.603101 + 0.231509i 0.613388 − 0.0321463i 0.891007 − 0.453990i 0.269963 − 2.56852i −0.984825 + 1.70577i
7.6 −0.358368 + 0.933580i 1.86528 + 2.07160i −0.743145 − 0.669131i 2.51352 + 0.398102i −2.60246 + 0.998992i −4.62009 + 0.242129i 0.891007 − 0.453990i −0.498684 + 4.74466i −1.27243 + 2.20391i
7.7 −0.358368 + 0.933580i 1.99075 + 2.21095i −0.743145 − 0.669131i −0.0699649 − 0.0110813i −2.77752 + 1.06619i 3.67528 − 0.192613i 0.891007 − 0.453990i −0.611635 + 5.81932i 0.0354185 − 0.0613466i
7.8 0.358368 − 0.933580i −2.04702 − 2.27345i −0.743145 − 0.669131i 0.860358 + 0.136267i −2.85604 + 1.09633i −4.51620 + 0.236684i −0.891007 + 0.453990i −0.664684 + 6.32404i 0.435541 − 0.754380i
7.9 0.358368 − 0.933580i −1.44400 − 1.60372i −0.743145 − 0.669131i −2.34923 − 0.372082i −2.01469 + 0.773367i 0.426714 − 0.0223631i −0.891007 + 0.453990i −0.173211 + 1.64799i −1.18926 + 2.05986i
7.10 0.358368 − 0.933580i −0.831479 − 0.923451i −0.743145 − 0.669131i 1.38143 + 0.218797i −1.16009 + 0.445317i 2.87390 − 0.150615i −0.891007 + 0.453990i 0.152181 − 1.44790i 0.699324 − 1.21127i
7.11 0.358368 − 0.933580i 0.0586282 + 0.0651133i −0.743145 − 0.669131i 1.99408 + 0.315832i 0.0817989 − 0.0313997i 0.892946 − 0.0467973i −0.891007 + 0.453990i 0.312783 − 2.97593i 1.00947 − 1.74845i
7.12 0.358368 − 0.933580i 0.536728 + 0.596096i −0.743145 − 0.669131i −4.15552 − 0.658170i 0.748850 − 0.287457i −1.04310 + 0.0546667i −0.891007 + 0.453990i 0.246331 − 2.34368i −2.10366 + 3.64365i
7.13 0.358368 − 0.933580i 1.47813 + 1.64163i −0.743145 − 0.669131i 3.48759 + 0.552380i 2.06230 − 0.791644i −2.97879 + 0.156112i −0.891007 + 0.453990i −0.196492 + 1.86950i 1.76553 − 3.05799i
7.14 0.358368 − 0.933580i 2.24902 + 2.49779i −0.743145 − 0.669131i −1.21870 − 0.193024i 3.13786 − 1.20451i 1.65972 − 0.0869823i −0.891007 + 0.453990i −0.867278 + 8.25160i −0.616948 + 1.06859i
19.1 −0.629320 + 0.777146i −2.12892 − 0.452517i −0.207912 − 0.978148i 1.14087 − 0.180696i 1.69145 − 1.36971i −0.257998 + 0.397282i 0.891007 + 0.453990i 1.58691 + 0.706538i −0.577546 + 1.00034i
19.2 −0.629320 + 0.777146i −2.05751 − 0.437337i −0.207912 − 0.978148i −2.17647 + 0.344719i 1.63471 − 1.32376i 0.687525 − 1.05870i 0.891007 + 0.453990i 1.30143 + 0.579436i 1.10180 − 1.90837i
19.3 −0.629320 + 0.777146i −0.500227 − 0.106327i −0.207912 − 0.978148i −1.34681 + 0.213314i 0.397434 − 0.321836i −0.641515 + 0.987846i 0.891007 + 0.453990i −2.50171 − 1.11384i 0.681801 − 1.18091i
19.4 −0.629320 + 0.777146i 0.0425990 + 0.00905470i −0.207912 − 0.978148i 2.85784 − 0.452637i −0.0338453 + 0.0274074i 2.08053 − 3.20374i 0.891007 + 0.453990i −2.73890 − 1.21944i −1.44673 + 2.50581i
19.5 −0.629320 + 0.777146i 0.835990 + 0.177695i −0.207912 − 0.978148i 3.64026 − 0.576560i −0.664201 + 0.537859i −2.40956 + 3.71040i 0.891007 + 0.453990i −2.07333 − 0.923107i −1.84282 + 3.19185i
19.6 −0.629320 + 0.777146i 1.66538 + 0.353988i −0.207912 − 0.978148i −2.81382 + 0.445665i −1.32316 + 1.07147i −2.13858 + 3.29313i 0.891007 + 0.453990i −0.0924512 − 0.0411619i 1.42445 − 2.46721i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
13.f odd 12 1 inner
143.w even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 286.2.x.a ✓ 224
11.d odd 10 1 inner 286.2.x.a ✓ 224
13.f odd 12 1 inner 286.2.x.a ✓ 224
143.w even 60 1 inner 286.2.x.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
286.2.x.a ✓ 224 1.a even 1 1 trivial
286.2.x.a ✓ 224 11.d odd 10 1 inner
286.2.x.a ✓ 224 13.f odd 12 1 inner
286.2.x.a ✓ 224 143.w even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(286, [\chi])\).