Properties

 Label 160.2.u.a Level $160$ Weight $2$ Character orbit 160.u Analytic conductor $1.278$ Analytic rank $0$ Dimension $88$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 160.u (of order $$8$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$1.27760643234$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$22$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$88q - 4q^{2} - 4q^{3} - 4q^{5} - 8q^{6} - 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q - 4q^{2} - 4q^{3} - 4q^{5} - 8q^{6} - 16q^{8} - 16q^{10} - 8q^{11} - 20q^{12} - 4q^{13} + 16q^{14} - 8q^{15} - 8q^{16} + 4q^{18} + 16q^{19} + 24q^{20} - 8q^{21} - 20q^{22} - 32q^{24} - 4q^{25} - 8q^{26} - 16q^{27} + 12q^{28} + 8q^{30} - 24q^{32} - 8q^{33} + 8q^{34} - 8q^{35} - 8q^{36} - 4q^{37} - 60q^{38} + 16q^{40} - 8q^{41} + 60q^{42} - 36q^{43} + 16q^{44} - 4q^{45} - 8q^{46} - 8q^{47} - 24q^{48} - 40q^{49} - 12q^{50} + 8q^{51} - 56q^{52} - 4q^{53} - 56q^{54} - 36q^{55} + 40q^{56} - 8q^{57} - 36q^{58} + 12q^{60} - 40q^{61} + 40q^{62} + 56q^{63} + 48q^{64} - 8q^{65} + 72q^{66} + 20q^{67} + 8q^{68} + 24q^{69} + 60q^{70} + 24q^{71} + 24q^{72} - 8q^{73} + 8q^{75} + 56q^{76} + 24q^{77} + 32q^{78} - 8q^{80} + 84q^{82} - 44q^{83} + 56q^{84} - 4q^{85} + 24q^{86} - 120q^{87} + 104q^{88} + 148q^{90} - 8q^{91} + 124q^{92} + 8q^{93} - 32q^{94} + 24q^{96} - 8q^{97} - 16q^{98} + O(q^{100})$$

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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −1.41312 0.0556644i −0.220591 + 0.532554i 1.99380 + 0.157321i −1.81739 1.30273i 0.341365 0.740282i 3.48272i −2.80872 0.333297i 1.88637 + 1.88637i 2.49567 + 1.94207i
43.2 −1.40216 0.184269i 0.458849 1.10776i 1.93209 + 0.516749i −0.222064 + 2.22501i −0.847505 + 1.46870i 4.27741i −2.61387 1.08059i 1.10473 + 1.10473i 0.721371 3.07890i
43.3 −1.37186 + 0.343520i 1.11476 2.69126i 1.76399 0.942521i 1.50867 1.65043i −0.604785 + 4.07496i 0.518179i −2.09617 + 1.89897i −3.87887 3.87887i −1.50273 + 2.78241i
43.4 −1.32278 0.500260i −1.03647 + 2.50226i 1.49948 + 1.32347i 1.53173 1.62905i 2.62280 2.79143i 2.65674i −1.32140 2.50078i −3.06572 3.06572i −2.84109 + 1.38861i
43.5 −1.19101 + 0.762560i −1.11089 + 2.68192i 0.837005 1.81643i −1.09565 + 1.94924i −0.722048 4.04131i 0.874514i 0.388258 + 2.80165i −3.83732 3.83732i −0.181489 3.15707i
43.6 −1.04802 + 0.949552i −0.255424 + 0.616647i 0.196703 1.99030i 2.08980 0.795458i −0.317849 0.888798i 2.27809i 1.68375 + 2.27266i 1.80631 + 1.80631i −1.43482 + 2.81803i
43.7 −0.998231 1.00177i 0.509063 1.22899i −0.00706901 + 1.99999i 2.15631 + 0.591867i −1.73932 + 0.716851i 2.73471i 2.01058 1.98937i 0.870059 + 0.870059i −1.55959 2.75094i
43.8 −0.759267 1.19311i −0.252131 + 0.608697i −0.847028 + 1.81178i −2.19888 + 0.406097i 0.917677 0.161344i 1.49067i 2.80477 0.365026i 1.81438 + 1.81438i 2.15406 + 2.31517i
43.9 −0.491080 + 1.32621i 0.896482 2.16430i −1.51768 1.30255i −2.04305 0.908815i 2.43008 + 2.25177i 0.225996i 2.47277 1.37311i −1.75919 1.75919i 2.20858 2.26322i
43.10 −0.458371 + 1.33787i 0.218753 0.528116i −1.57979 1.22648i 1.03318 + 1.98306i 0.606280 + 0.534736i 0.814088i 2.36500 1.55137i 1.89027 + 1.89027i −3.12666 + 0.473287i
43.11 −0.201889 1.39973i 0.867966 2.09546i −1.91848 + 0.565181i −0.324298 2.21243i −3.10830 0.791867i 1.82364i 1.17842 + 2.57125i −1.51625 1.51625i −3.03132 + 0.900595i
43.12 0.0754956 1.41220i −0.698571 + 1.68650i −1.98860 0.213229i 1.86202 + 1.23810i 2.32893 + 1.11384i 2.70081i −0.451252 + 2.79220i −0.234961 0.234961i 1.88901 2.53606i
43.13 0.375898 + 1.36334i −0.508197 + 1.22690i −1.71740 + 1.02496i −2.07129 + 0.842463i −1.86371 0.231658i 0.810621i −2.04293 1.95613i 0.874309 + 0.874309i −1.92716 2.50720i
43.14 0.400521 1.35631i 0.627770 1.51557i −1.67917 1.08646i −1.04216 + 1.97836i −1.80415 1.45847i 4.80429i −2.14613 + 1.84232i 0.218458 + 0.218458i 2.26587 + 2.20587i
43.15 0.640905 + 1.26065i 0.0983610 0.237464i −1.17848 + 1.61591i 1.44168 1.70926i 0.362400 0.0281932i 4.12414i −2.79240 0.450008i 2.07461 + 2.07461i 3.07876 + 0.721985i
43.16 0.816290 + 1.15485i 1.28377 3.09930i −0.667342 + 1.88538i 1.06380 + 1.96680i 4.62714 1.04737i 0.906290i −2.72207 + 0.768338i −5.83625 5.83625i −1.40298 + 2.83401i
43.17 0.992527 1.00742i −0.278775 + 0.673021i −0.0297801 1.99978i 0.336737 2.21057i 0.401322 + 0.948834i 0.467309i −2.04417 1.95483i 1.74608 + 1.74608i −1.89274 2.53328i
43.18 1.02775 + 0.971453i −1.05354 + 2.54348i 0.112559 + 1.99683i 2.23055 + 0.156938i −3.55365 + 1.59060i 4.43630i −1.82414 + 2.16160i −3.23800 3.23800i 2.14000 + 2.32817i
43.19 1.30208 0.551886i −0.577765 + 1.39485i 1.39084 1.43720i 0.315793 + 2.21366i 0.0174990 + 2.13507i 1.62907i 1.01782 2.63895i 0.509534 + 0.509534i 1.63288 + 2.70808i
43.20 1.30876 + 0.535863i 0.461737 1.11473i 1.42570 + 1.40263i −1.20640 1.88271i 1.20165 1.21149i 2.85280i 1.11428 + 2.59969i 1.09189 + 1.09189i −0.570021 3.11048i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 147.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 160.2.u.a 88
4.b odd 2 1 640.2.u.a 88
5.b even 2 1 800.2.v.b 88
5.c odd 4 1 160.2.ba.a yes 88
5.c odd 4 1 800.2.bb.b 88
20.e even 4 1 640.2.ba.a 88
32.g even 8 1 640.2.ba.a 88
32.h odd 8 1 160.2.ba.a yes 88
160.u even 8 1 inner 160.2.u.a 88
160.y odd 8 1 800.2.bb.b 88
160.ba even 8 1 800.2.v.b 88
160.bb odd 8 1 640.2.u.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.u.a 88 1.a even 1 1 trivial
160.2.u.a 88 160.u even 8 1 inner
160.2.ba.a yes 88 5.c odd 4 1
160.2.ba.a yes 88 32.h odd 8 1
640.2.u.a 88 4.b odd 2 1
640.2.u.a 88 160.bb odd 8 1
640.2.ba.a 88 20.e even 4 1
640.2.ba.a 88 32.g even 8 1
800.2.v.b 88 5.b even 2 1
800.2.v.b 88 160.ba even 8 1
800.2.bb.b 88 5.c odd 4 1
800.2.bb.b 88 160.y odd 8 1

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(160, [\chi])$$.