Properties

Label 160.3.bb.a
Level $160$
Weight $3$
Character orbit 160.bb
Analytic conductor $4.360$
Analytic rank $0$
Dimension $184$
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] = [160,3,Mod(53,160)]
 
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([0, 5, 6]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 160.bb (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: \(4.35968422976\)
Analytic rank: \(0\)
Dimension: \(184\)
Relative dimension: \(46\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} - 8 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} - 8 q^{7} - 16 q^{8} - 40 q^{10} - 8 q^{11} + 44 q^{12} - 4 q^{13} - 32 q^{14} - 8 q^{16} + 12 q^{18} + 64 q^{19} - 88 q^{20} - 8 q^{21} - 116 q^{22} - 8 q^{23} - 32 q^{24} - 4 q^{25} - 8 q^{26} - 40 q^{27} - 68 q^{28} - 240 q^{30} - 16 q^{31} + 136 q^{32} - 8 q^{33} + 88 q^{34} - 200 q^{35} - 8 q^{36} - 4 q^{37} + 276 q^{38} + 176 q^{40} - 8 q^{41} - 28 q^{42} + 124 q^{43} - 176 q^{44} - 4 q^{45} - 8 q^{46} + 232 q^{48} + 952 q^{49} - 20 q^{50} - 200 q^{51} - 104 q^{52} - 4 q^{53} + 56 q^{54} - 260 q^{55} - 344 q^{56} - 8 q^{57} - 356 q^{58} - 68 q^{60} + 56 q^{61} + 232 q^{62} - 8 q^{63} + 432 q^{64} - 8 q^{65} - 280 q^{66} - 292 q^{67} - 376 q^{68} - 72 q^{69} - 524 q^{70} + 248 q^{71} - 680 q^{72} - 8 q^{73} + 32 q^{75} + 312 q^{76} - 200 q^{77} + 336 q^{78} + 344 q^{80} - 388 q^{82} - 164 q^{83} - 392 q^{84} - 4 q^{85} + 216 q^{86} + 456 q^{88} + 316 q^{90} - 8 q^{91} + 172 q^{92} - 40 q^{93} + 288 q^{94} - 8 q^{95} + 536 q^{96} - 8 q^{97} - 472 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −1.99895 0.0647219i 2.83311 + 1.17351i 3.99162 + 0.258752i −3.94886 3.06700i −5.58730 2.52916i −10.8175 −7.96232 0.775578i 0.285413 + 0.285413i 7.69509 + 6.38637i
53.2 −1.99869 + 0.0723062i −0.937700 0.388408i 3.98954 0.289036i 3.72867 + 3.33122i 1.90226 + 0.708507i 2.36359 −7.95297 + 0.866163i −5.63554 5.63554i −7.69333 6.38848i
53.3 −1.97081 + 0.340438i −2.80621 1.16237i 3.76820 1.34188i 1.18740 4.85696i 5.92623 + 1.33547i 1.22791 −6.96959 + 3.92744i 0.159755 + 0.159755i −0.686648 + 9.97640i
53.4 −1.95783 + 0.408524i −4.62398 1.91532i 3.66622 1.59964i −3.33317 + 3.72692i 9.83544 + 1.86086i −10.0343 −6.52435 + 4.62957i 11.3488 + 11.3488i 5.00326 8.65837i
53.5 −1.94738 + 0.455756i 5.04734 + 2.09068i 3.58457 1.77506i 1.86459 + 4.63932i −10.7819 1.77098i 5.36005 −6.17153 + 5.09041i 14.7407 + 14.7407i −5.74546 8.18472i
53.6 −1.87059 0.707744i −1.23386 0.511082i 2.99820 + 2.64779i −4.18639 + 2.73389i 1.94633 + 1.82928i 11.1763 −3.73443 7.07489i −5.10275 5.10275i 9.76590 2.15109i
53.7 −1.81433 0.841556i 2.98308 + 1.23563i 2.58357 + 3.05372i 4.00144 2.99808i −4.37243 4.75227i 3.22829 −2.11756 7.71466i 1.00802 + 1.00802i −9.78297 + 2.07205i
53.8 −1.75049 + 0.967352i 1.84758 + 0.765291i 2.12846 3.38669i −4.83879 1.25939i −3.97448 + 0.447618i 8.29433 −0.449741 + 7.98735i −3.53610 3.53610i 9.68856 2.47625i
53.9 −1.60685 1.19081i −5.49113 2.27450i 1.16395 + 3.82691i 4.99245 0.274630i 6.11495 + 10.1937i 6.10452 2.68682 7.53532i 18.6152 + 18.6152i −8.34917 5.50376i
53.10 −1.54645 1.26826i −0.233667 0.0967881i 0.783021 + 3.92261i 3.06987 + 3.94663i 0.238602 + 0.446029i −11.9595 3.76400 7.05920i −6.31873 6.31873i 0.257954 9.99667i
53.11 −1.52127 + 1.29836i −0.105501 0.0436998i 0.628542 3.95031i 4.49805 2.18348i 0.217233 0.0704980i −3.47322 4.17272 + 6.82557i −6.35474 6.35474i −4.00782 + 9.16174i
53.12 −1.46084 1.36599i −2.98434 1.23615i 0.268122 + 3.99100i −4.69474 1.72030i 2.67107 + 5.88241i −5.77525 5.06000 6.19648i 1.01424 + 1.01424i 4.50836 + 8.92607i
53.13 −1.39655 + 1.43166i 1.12933 + 0.467783i −0.0993084 3.99877i −1.30374 + 4.82703i −2.24687 + 0.963533i −5.91992 5.86357 + 5.44229i −5.30740 5.30740i −5.08994 8.60770i
53.14 −1.28127 1.53569i 4.44428 + 1.84088i −0.716715 + 3.93527i −3.55622 + 3.51473i −2.86727 9.18371i −1.60224 6.96167 3.94147i 9.99881 + 9.99881i 9.95401 + 0.957970i
53.15 −1.22944 + 1.57749i −3.88606 1.60966i −0.976973 3.87886i −3.87919 3.15466i 7.31689 4.15126i 6.72772 7.32000 + 3.22764i 6.14650 + 6.14650i 9.74567 2.24095i
53.16 −1.11009 + 1.66364i 4.97343 + 2.06006i −1.53540 3.69358i 1.42035 4.79402i −8.94815 + 5.98714i −2.45065 7.84922 + 1.54586i 14.1272 + 14.1272i 6.39881 + 7.68474i
53.17 −0.963660 1.75253i 1.59304 + 0.659857i −2.14272 + 3.37768i −2.41012 4.38079i −0.378725 3.42772i 7.82512 7.98434 + 0.500244i −4.26161 4.26161i −5.35493 + 8.44540i
53.18 −0.628589 1.89865i −1.59431 0.660386i −3.20975 + 2.38694i 3.34875 3.71293i −0.251675 + 3.44216i −6.24134 6.54958 + 4.59379i −4.25824 4.25824i −9.15455 4.02421i
53.19 −0.575434 + 1.91543i −4.38228 1.81520i −3.33775 2.20441i 4.90866 0.951362i 5.99861 7.34943i −8.87308 6.14305 5.12474i 9.54550 + 9.54550i −1.00234 + 9.94964i
53.20 −0.483550 1.94066i −2.23592 0.926148i −3.53236 + 1.87682i 0.525403 + 4.97232i −0.716165 + 4.78701i 7.42339 5.35034 + 5.94759i −2.22238 2.22238i 9.39554 3.42399i
See next 80 embeddings (of 184 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.46
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.3.bb.a yes 184
5.c odd 4 1 160.3.v.a 184
32.g even 8 1 160.3.v.a 184
160.bb odd 8 1 inner 160.3.bb.a yes 184
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.v.a 184 5.c odd 4 1
160.3.v.a 184 32.g even 8 1
160.3.bb.a yes 184 1.a even 1 1 trivial
160.3.bb.a yes 184 160.bb odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(160, [\chi])\).