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Properties

Label 160.4.u.a
Level $160$
Weight $4$
Character orbit 160.u
Analytic conductor $9.440$
Analytic rank $0$
Dimension $280$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [160,4,Mod(43,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 5, 6]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.43");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.u (of order \(8\), degree \(4\), 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: \(9.44030560092\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(70\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 80 q^{8} + 8 q^{10} - 8 q^{11} + 44 q^{12} - 4 q^{13} + 64 q^{14} - 8 q^{15} - 8 q^{16} + 28 q^{18} - 48 q^{19} - 312 q^{20} - 8 q^{21} - 436 q^{22} + 736 q^{24}+ \cdots + 3480 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −2.82479 + 0.143339i −3.58549 + 8.65613i 7.95891 − 0.809808i 9.77656 + 5.42391i 8.88750 − 24.9657i − 20.1382i −22.3662 + 3.42837i −42.9811 − 42.9811i −28.3942 − 13.9201i
43.2 −2.82227 + 0.186555i −2.08687 + 5.03814i 7.93039 − 1.05302i −7.75655 + 8.05208i 4.94980 − 14.6083i 11.7199i −22.1853 + 4.45136i −1.93595 − 1.93595i 20.3889 − 24.1722i
43.3 −2.81921 + 0.228193i −1.83845 + 4.43841i 7.89586 − 1.28665i 0.493546 − 11.1694i 4.17015 − 12.9323i 25.5948i −21.9664 + 5.42911i 2.77233 + 2.77233i 1.15738 + 31.6016i
43.4 −2.81314 − 0.293650i 2.91738 − 7.04318i 7.82754 + 1.65216i −0.675996 + 11.1599i −10.2752 + 18.9568i − 16.4698i −21.5348 − 6.94631i −22.0034 − 22.0034i 5.17877 − 31.1958i
43.5 −2.74584 − 0.678514i 2.00978 − 4.85204i 7.07924 + 3.72618i −9.01416 − 6.61399i −8.81071 + 11.9592i 18.5009i −16.9102 − 15.0348i −0.411197 − 0.411197i 20.2637 + 24.2772i
43.6 −2.71454 + 0.794525i 1.40505 − 3.39208i 6.73746 − 4.31354i 10.5782 + 3.61974i −1.11896 + 10.3243i 12.0132i −14.8619 + 17.0624i 9.55980 + 9.55980i −31.5908 − 1.42133i
43.7 −2.69799 − 0.849028i 0.321918 − 0.777180i 6.55830 + 4.58134i 9.25335 − 6.27499i −1.52838 + 1.82351i − 14.9318i −13.8045 − 17.9286i 18.5915 + 18.5915i −30.2931 + 9.07351i
43.8 −2.69135 + 0.869846i 3.08842 − 7.45610i 6.48674 − 4.68212i −5.94465 − 9.46896i −1.82636 + 22.7534i − 23.1333i −13.3854 + 18.2437i −26.9632 − 26.9632i 24.2357 + 20.3134i
43.9 −2.58199 + 1.15470i −1.75736 + 4.24265i 5.33333 − 5.96285i 1.49092 − 11.0805i −0.361505 − 12.9837i − 25.1176i −6.88527 + 21.5544i 4.18013 + 4.18013i 8.94511 + 30.3313i
43.10 −2.56149 + 1.19949i 0.532873 − 1.28647i 5.12247 − 6.14494i −11.0603 + 1.63392i 0.178152 + 3.93445i 4.38872i −5.75039 + 21.8845i 17.7208 + 17.7208i 26.3710 − 17.4519i
43.11 −2.54222 − 1.23981i −0.377089 + 0.910374i 4.92575 + 6.30373i −9.59495 + 5.73907i 2.08733 − 1.84685i − 28.4398i −4.70694 − 22.1324i 18.4053 + 18.4053i 31.5078 − 2.69407i
43.12 −2.46943 − 1.37911i −3.62532 + 8.75229i 4.19613 + 6.81121i −9.83331 − 5.32035i 21.0228 − 16.6134i − 11.7495i −0.968637 − 22.6067i −44.3677 − 44.3677i 16.9453 + 26.6994i
43.13 −2.39368 − 1.50676i −1.47386 + 3.55822i 3.45936 + 7.21338i 6.48813 + 9.10518i 8.88932 − 6.29647i 18.1106i 2.58823 − 22.4789i 8.60323 + 8.60323i −1.81117 − 31.5709i
43.14 −2.34100 − 1.58737i 3.77736 − 9.11935i 2.96052 + 7.43205i 9.99136 − 5.01724i −23.3186 + 15.3523i 20.7744i 4.86682 − 22.0978i −49.8023 − 49.8023i −31.3539 − 4.11462i
43.15 −2.30320 + 1.64172i −0.643934 + 1.55459i 2.60950 − 7.56244i 2.79968 + 10.8241i −1.06910 − 4.63771i − 22.2482i 6.40519 + 21.7019i 17.0898 + 17.0898i −24.2184 − 20.3339i
43.16 −1.96007 + 2.03915i 3.68068 − 8.88596i −0.316262 − 7.99375i −6.53470 + 9.07181i 10.9054 + 24.9225i 30.1289i 16.9203 + 15.0234i −46.3209 − 46.3209i −5.69033 − 31.1066i
43.17 −1.93965 + 2.05859i −2.95022 + 7.12247i −0.475554 − 7.98585i 11.1803 − 0.0331836i −8.93983 − 19.8884i 27.8505i 17.3620 + 14.5108i −22.9339 − 22.9339i −21.6175 + 23.0800i
43.18 −1.88405 − 2.10958i −1.27026 + 3.06668i −0.900689 + 7.94914i −4.12150 − 10.3929i 8.86267 − 3.09807i 8.37858i 18.4663 − 13.0765i 11.3009 + 11.3009i −14.1596 + 28.2755i
43.19 −1.87785 + 2.11511i 1.73898 − 4.19826i −0.947364 − 7.94371i 4.36112 − 10.2947i 5.61424 + 11.5618i 11.4589i 18.5808 + 12.9133i 4.49050 + 4.49050i 13.5849 + 28.5561i
43.20 −1.79535 + 2.18557i −2.94201 + 7.10263i −1.55346 − 7.84772i −10.0432 − 4.91272i −10.2414 − 19.1817i − 2.95853i 19.9408 + 10.6942i −22.7001 − 22.7001i 28.7681 − 13.1300i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
160.u even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.4.u.a ✓ 280
5.c odd 4 1 160.4.ba.a yes 280
32.h odd 8 1 160.4.ba.a yes 280
160.u even 8 1 inner 160.4.u.a ✓ 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.u.a ✓ 280 1.a even 1 1 trivial
160.4.u.a ✓ 280 160.u even 8 1 inner
160.4.ba.a yes 280 5.c odd 4 1
160.4.ba.a yes 280 32.h odd 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(160, [\chi])\).