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Properties

Label 286.2.r.a
Level $286$
Weight $2$
Character orbit 286.r
Analytic conductor $2.284$
Analytic rank $0$
Dimension $112$
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(57,286)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(286, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([2, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("286.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 286 = 2 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 286.r (of order \(20\), degree \(8\), 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: \(112\)
Relative dimension: \(14\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 28 q^{9} - 28 q^{11} + 20 q^{13} - 8 q^{14} - 8 q^{15} + 28 q^{16} + 4 q^{22} - 4 q^{26} + 48 q^{27} - 20 q^{29} - 28 q^{31} + 96 q^{33} - 16 q^{34} - 200 q^{35} + 24 q^{37} + 40 q^{39} - 20 q^{41} + 8 q^{44} - 64 q^{45} - 40 q^{46} - 28 q^{47} + 32 q^{53} - 8 q^{55} - 4 q^{58} - 36 q^{59} - 8 q^{60} - 80 q^{61} + 80 q^{63} + 96 q^{66} + 56 q^{67} + 20 q^{68} - 52 q^{70} - 100 q^{71} - 40 q^{72} + 60 q^{73} - 64 q^{78} - 80 q^{79} + 84 q^{81} - 80 q^{84} - 80 q^{85} + 48 q^{86} - 88 q^{89} + 72 q^{91} - 16 q^{92} + 24 q^{93} - 180 q^{94} + 12 q^{97} - 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1 −0.891007 + 0.453990i −2.17229 + 1.57826i 0.587785 − 0.809017i 3.10905 + 1.58414i 1.21901 − 2.39244i 2.30980 − 0.365837i −0.156434 + 0.987688i 1.30089 − 4.00373i −3.48936
57.2 −0.891007 + 0.453990i −1.35717 + 0.986043i 0.587785 − 0.809017i −0.848174 − 0.432166i 0.761595 − 1.49471i −0.662815 + 0.104980i −0.156434 + 0.987688i −0.0574167 + 0.176710i 0.951928
57.3 −0.891007 + 0.453990i −1.03548 + 0.752322i 0.587785 − 0.809017i −1.31397 − 0.669501i 0.581074 − 1.14042i 2.38819 − 0.378253i −0.156434 + 0.987688i −0.420816 + 1.29514i 1.47470
57.4 −0.891007 + 0.453990i 0.318541 − 0.231433i 0.587785 − 0.809017i −0.147555 − 0.0751832i −0.178753 + 0.350823i −4.72206 + 0.747901i −0.156434 + 0.987688i −0.879144 + 2.70573i 0.165605
57.5 −0.891007 + 0.453990i 0.673001 − 0.488964i 0.587785 − 0.809017i 2.56224 + 1.30553i −0.377663 + 0.741206i −0.864666 + 0.136950i −0.156434 + 0.987688i −0.713206 + 2.19502i −2.87567
57.6 −0.891007 + 0.453990i 1.44540 − 1.05014i 0.587785 − 0.809017i −3.80146 − 1.93694i −0.811103 + 1.59188i 1.81275 − 0.287111i −0.156434 + 0.987688i 0.0593206 − 0.182570i 4.26647
57.7 −0.891007 + 0.453990i 2.12801 − 1.54609i 0.587785 − 0.809017i 0.439866 + 0.224123i −1.19416 + 2.34367i 0.349221 − 0.0553112i −0.156434 + 0.987688i 1.21098 − 3.72702i −0.493673
57.8 0.891007 − 0.453990i −2.65028 + 1.92554i 0.587785 − 0.809017i 0.195996 + 0.0998650i −1.48724 + 2.91887i −4.33747 + 0.686987i 0.156434 − 0.987688i 2.38922 − 7.35326i 0.219972
57.9 0.891007 − 0.453990i −2.02840 + 1.47372i 0.587785 − 0.809017i −3.16054 − 1.61038i −1.13826 + 2.23397i 5.02157 − 0.795339i 0.156434 − 0.987688i 1.01551 − 3.12542i −3.54716
57.10 0.891007 − 0.453990i −0.910439 + 0.661473i 0.587785 − 0.809017i 2.19509 + 1.11846i −0.510905 + 1.00271i 0.298470 − 0.0472730i 0.156434 − 0.987688i −0.535698 + 1.64871i 2.46361
57.11 0.891007 − 0.453990i 0.206543 − 0.150062i 0.587785 − 0.809017i 2.16866 + 1.10499i 0.115904 − 0.227475i 0.742989 − 0.117678i 0.156434 − 0.987688i −0.906910 + 2.79118i 2.43395
57.12 0.891007 − 0.453990i 1.48748 − 1.08072i 0.587785 − 0.809017i −1.68948 − 0.860833i 0.834719 − 1.63823i 3.72156 − 0.589437i 0.156434 − 0.987688i 0.117594 − 0.361919i −1.89615
57.13 0.891007 − 0.453990i 1.80779 − 1.31344i 0.587785 − 0.809017i −2.18918 − 1.11544i 1.01447 − 1.99100i −2.15733 + 0.341687i 0.156434 − 0.987688i 0.615940 − 1.89567i −2.45697
57.14 0.891007 − 0.453990i 2.08731 − 1.51652i 0.587785 − 0.809017i 2.47945 + 1.26334i 1.17132 − 2.29884i −3.90022 + 0.617734i 0.156434 − 0.987688i 1.12997 − 3.47769i 2.78275
73.1 −0.987688 + 0.156434i −0.850175 + 2.61657i 0.951057 − 0.309017i 0.0558876 + 0.00885173i 0.430386 − 2.71735i −1.50487 + 2.95348i −0.891007 + 0.453990i −3.69659 − 2.68573i −0.0565843
73.2 −0.987688 + 0.156434i −0.718371 + 2.21092i 0.951057 − 0.309017i −3.48754 − 0.552372i 0.363663 − 2.29608i 1.37505 − 2.69869i −0.891007 + 0.453990i −1.94506 − 1.41317i 3.53101
73.3 −0.987688 + 0.156434i −0.207718 + 0.639292i 0.951057 − 0.309017i 3.47491 + 0.550372i 0.105154 − 0.663915i 0.612653 − 1.20240i −0.891007 + 0.453990i 2.06150 + 1.49777i −3.51823
73.4 −0.987688 + 0.156434i 0.0834123 − 0.256717i 0.951057 − 0.309017i −2.48299 − 0.393268i −0.0422260 + 0.266604i −0.527791 + 1.03585i −0.891007 + 0.453990i 2.36811 + 1.72053i 2.51394
73.5 −0.987688 + 0.156434i 0.357868 − 1.10140i 0.951057 − 0.309017i 2.83914 + 0.449676i −0.181164 + 1.14383i −1.94552 + 3.81829i −0.891007 + 0.453990i 1.34203 + 0.975041i −2.87453
73.6 −0.987688 + 0.156434i 0.512362 − 1.57689i 0.951057 − 0.309017i −1.30092 − 0.206045i −0.259374 + 1.63763i 0.503600 − 0.988370i −0.891007 + 0.453990i 0.202987 + 0.147479i 1.31714
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
13.d odd 4 1 inner
143.s even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 286.2.r.a ✓ 112
11.d odd 10 1 inner 286.2.r.a ✓ 112
13.d odd 4 1 inner 286.2.r.a ✓ 112
143.s even 20 1 inner 286.2.r.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
286.2.r.a ✓ 112 1.a even 1 1 trivial
286.2.r.a ✓ 112 11.d odd 10 1 inner
286.2.r.a ✓ 112 13.d odd 4 1 inner
286.2.r.a ✓ 112 143.s even 20 1 inner

Hecke kernels

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