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Properties

Label 287.2.u.a
Level $287$
Weight $2$
Character orbit 287.u
Analytic conductor $2.292$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [287,2,Mod(8,287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(287, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 19]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("287.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 287 = 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 287.u (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: \(2.29170653801\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 4 q^{3} + 36 q^{4} - 28 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 4 q^{3} + 36 q^{4} - 28 q^{6} - 48 q^{10} + 8 q^{11} - 16 q^{12} - 16 q^{13} - 32 q^{15} - 28 q^{16} - 20 q^{17} + 12 q^{18} - 40 q^{19} - 4 q^{22} - 36 q^{23} - 108 q^{24} + 40 q^{25} + 20 q^{26} + 20 q^{27} + 12 q^{29} - 4 q^{30} + 20 q^{31} - 32 q^{34} - 4 q^{35} + 140 q^{36} - 44 q^{38} + 80 q^{39} - 64 q^{40} + 4 q^{41} - 72 q^{42} - 8 q^{44} + 8 q^{45} + 40 q^{46} - 56 q^{47} + 120 q^{48} + 24 q^{51} + 16 q^{52} + 12 q^{53} - 52 q^{54} - 72 q^{55} + 48 q^{57} + 36 q^{58} - 48 q^{59} - 52 q^{60} + 8 q^{63} - 96 q^{64} - 96 q^{65} + 24 q^{66} + 20 q^{67} - 80 q^{68} - 68 q^{69} + 8 q^{70} + 60 q^{71} + 100 q^{72} - 40 q^{74} - 120 q^{75} + 4 q^{76} - 12 q^{78} + 12 q^{79} + 200 q^{80} - 56 q^{81} - 68 q^{82} - 40 q^{83} + 132 q^{85} + 80 q^{86} - 16 q^{88} + 8 q^{89} + 140 q^{92} + 64 q^{93} + 128 q^{94} - 68 q^{96} + 4 q^{98} + 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −1.40231 + 1.93011i −1.46692 − 1.46692i −1.14082 − 3.51109i −2.49680 + 0.811260i 4.88839 − 0.774244i −0.987688 − 0.156434i 3.83861 + 1.24724i 1.30371i 1.93546 − 5.95673i
8.2 −1.35859 + 1.86993i −1.46234 − 1.46234i −1.03286 − 3.17882i 0.803450 − 0.261057i 4.72118 − 0.747761i 0.987688 + 0.156434i 2.95092 + 0.958814i 1.27685i −0.603397 + 1.85706i
8.3 −1.21144 + 1.66740i −0.0854640 − 0.0854640i −0.694607 − 2.13778i 1.91874 − 0.623438i 0.246037 − 0.0389684i 0.987688 + 0.156434i 0.485715 + 0.157818i − 2.98539i −1.28492 + 3.95457i
8.4 −1.17054 + 1.61111i 1.85139 + 1.85139i −0.607480 − 1.86963i 3.26052 − 1.05941i −5.14992 + 0.815666i −0.987688 − 0.156434i −0.0646837 − 0.0210170i 3.85527i −2.10975 + 6.49315i
8.5 −0.950420 + 1.30814i 1.68439 + 1.68439i −0.189901 − 0.584454i −2.48593 + 0.807729i −3.80430 + 0.602541i 0.987688 + 0.156434i −2.13059 − 0.692271i 2.67433i 1.30606 − 4.01963i
8.6 −0.835728 + 1.15028i −2.24559 − 2.24559i −0.00667062 − 0.0205301i 3.16801 − 1.02935i 4.45975 − 0.706356i −0.987688 − 0.156434i −2.67528 − 0.869252i 7.08532i −1.46355 + 4.50436i
8.7 −0.657012 + 0.904299i −0.174416 − 0.174416i 0.231942 + 0.713843i −2.00897 + 0.652754i 0.272317 − 0.0431308i −0.987688 − 0.156434i −2.92405 − 0.950082i − 2.93916i 0.729633 − 2.24558i
8.8 −0.540570 + 0.744030i −1.20050 − 1.20050i 0.356668 + 1.09771i −2.96817 + 0.964417i 1.54217 − 0.244255i 0.987688 + 0.156434i −2.75886 − 0.896407i − 0.117584i 0.886947 − 2.72974i
8.9 −0.158161 + 0.217690i 1.91716 + 1.91716i 0.595660 + 1.83325i −0.605064 + 0.196597i −0.720570 + 0.114127i −0.987688 − 0.156434i −1.00511 − 0.326581i 4.35104i 0.0529004 − 0.162811i
8.10 −0.00319615 + 0.00439912i 1.15829 + 1.15829i 0.618025 + 1.90208i 2.71750 − 0.882970i −0.00879749 + 0.00139339i 0.987688 + 0.156434i −0.0206857 − 0.00672121i − 0.316748i −0.00480125 + 0.0147767i
8.11 0.200055 − 0.275352i 0.707260 + 0.707260i 0.582237 + 1.79194i −1.14034 + 0.370520i 0.336236 − 0.0532545i 0.987688 + 0.156434i 1.25728 + 0.408516i − 1.99957i −0.126108 + 0.388120i
8.12 0.335454 − 0.461712i −2.30133 − 2.30133i 0.517385 + 1.59235i −2.26795 + 0.736900i −1.83454 + 0.290563i −0.987688 − 0.156434i 1.99432 + 0.647992i 7.59224i −0.420555 + 1.29433i
8.13 0.531572 − 0.731646i −1.44322 − 1.44322i 0.365297 + 1.12427i 2.38170 − 0.773861i −1.82310 + 0.288751i 0.987688 + 0.156434i 2.73695 + 0.889289i 1.16578i 0.699852 − 2.15392i
8.14 0.754715 − 1.03878i −0.345510 − 0.345510i 0.108573 + 0.334152i −0.00591493 + 0.00192188i −0.619669 + 0.0981459i −0.987688 − 0.156434i 2.87136 + 0.932962i − 2.76125i −0.00246769 + 0.00759477i
8.15 0.880705 − 1.21219i 1.40144 + 1.40144i −0.0757208 − 0.233045i 0.589683 − 0.191600i 2.93306 − 0.464551i −0.987688 − 0.156434i 2.50084 + 0.812572i 0.928067i 0.287082 − 0.883548i
8.16 1.09913 − 1.51282i 2.20211 + 2.20211i −0.462515 − 1.42347i −3.74866 + 1.21801i 5.75183 − 0.911000i 0.987688 + 0.156434i 0.895034 + 0.290814i 6.69862i −2.27763 + 7.00981i
8.17 1.28791 − 1.77266i −1.22984 − 1.22984i −0.865568 − 2.66394i 3.10142 − 1.00771i −3.76400 + 0.596159i −0.987688 − 0.156434i −1.66927 − 0.542379i 0.0249934i 2.20802 − 6.79560i
8.18 1.40653 − 1.93593i −2.19462 − 2.19462i −1.15144 − 3.54378i −0.741595 + 0.240959i −7.33543 + 1.16182i 0.987688 + 0.156434i −3.92839 − 1.27641i 6.63271i −0.576599 + 1.77459i
8.19 1.41470 − 1.94717i 0.651505 + 0.651505i −1.17206 − 3.60722i 0.379466 − 0.123296i 2.19028 − 0.346906i 0.987688 + 0.156434i −4.10391 − 1.33344i − 2.15108i 0.296753 − 0.913312i
8.20 1.55275 − 2.13717i −0.217409 − 0.217409i −1.53845 − 4.73486i −2.92878 + 0.951619i −0.802220 + 0.127059i −0.987688 − 0.156434i −7.48325 − 2.43145i − 2.90547i −2.51388 + 7.73693i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.2.u.a ✓ 160
41.g even 20 1 inner 287.2.u.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.u.a ✓ 160 1.a even 1 1 trivial
287.2.u.a ✓ 160 41.g even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(287, [\chi])\).