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Properties

Label 177.2.e.b
Level $177$
Weight $2$
Character orbit 177.e
Analytic conductor $1.413$
Analytic rank $0$
Dimension $140$
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] = [177,2,Mod(4,177)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(177, base_ring=CyclotomicField(58))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.e (of order \(29\), degree \(28\), 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: \(1.41335211578\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(5\) over \(\Q(\zeta_{29})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9} - 116 q^{10} + 2 q^{11} + q^{12} + 4 q^{13} - 43 q^{14} - 2 q^{15} + 7 q^{16} + q^{18} - 2 q^{19} + 4 q^{20} - 27 q^{21} + 6 q^{22} + 6 q^{23} + 3 q^{24} - 57 q^{25} + 12 q^{26} + 5 q^{27} - 10 q^{28} - 4 q^{29} - 12 q^{31} - 150 q^{32} - 2 q^{33} - 2 q^{34} + 6 q^{35} - q^{36} + 12 q^{37} - 12 q^{38} - 4 q^{39} - 66 q^{40} - 4 q^{41} + 14 q^{42} - 60 q^{43} + 20 q^{44} + 2 q^{45} + 76 q^{46} - 25 q^{47} - 7 q^{48} + 31 q^{49} + 137 q^{50} + 118 q^{52} + 48 q^{53} - q^{54} + 93 q^{55} + 228 q^{56} + 2 q^{57} - 120 q^{58} + 57 q^{59} - 4 q^{60} + 72 q^{61} - 179 q^{62} - 2 q^{63} + 249 q^{64} - 39 q^{65} - 6 q^{66} + 40 q^{67} + 94 q^{68} - 64 q^{69} + 94 q^{70} + 30 q^{71} - 3 q^{72} - 205 q^{73} + 66 q^{74} - q^{75} - 216 q^{76} - 46 q^{77} - 12 q^{78} + 4 q^{79} - 356 q^{80} - 5 q^{81} - 28 q^{82} + 4 q^{83} - 135 q^{84} + 50 q^{85} - 18 q^{86} - 54 q^{87} - 162 q^{88} + 26 q^{89} - 198 q^{91} + 10 q^{92} + 12 q^{93} - 4 q^{94} - 326 q^{95} + 5 q^{96} - 20 q^{97} - 143 q^{98} + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.53795 − 0.276019i −0.647386 + 0.762162i 4.41176 + 0.971102i −0.0920748 − 0.0699934i 1.85340 − 1.75564i 0.00524223 − 0.0131570i −6.09023 − 2.05204i −0.161782 − 0.986827i 0.214362 + 0.203054i
4.2 −0.756070 − 0.0822276i −0.647386 + 0.762162i −1.38836 − 0.305601i −0.128718 − 0.0978487i 0.552140 − 0.523015i 1.18304 − 2.96921i 2.46600 + 0.830893i −0.161782 − 0.986827i 0.0892739 + 0.0845647i
4.3 −0.233737 − 0.0254205i −0.647386 + 0.762162i −1.89925 − 0.418057i 1.40408 + 1.06736i 0.170693 − 0.161689i −0.853505 + 2.14214i 0.878915 + 0.296141i −0.161782 − 0.986827i −0.301054 − 0.285174i
4.4 1.81180 + 0.197046i −0.647386 + 0.762162i 1.29057 + 0.284075i 1.77642 + 1.35040i −1.32312 + 1.25332i −0.458287 + 1.15021i −1.17189 − 0.394857i −0.161782 − 0.986827i 2.95244 + 2.79670i
4.5 2.71009 + 0.294740i −0.647386 + 0.762162i 5.30448 + 1.16760i −2.37039 − 1.80192i −1.97912 + 1.87472i −0.223242 + 0.560295i 8.86474 + 2.98688i −0.161782 − 0.986827i −5.89286 − 5.58202i
7.1 −1.50851 − 2.22489i −0.0541389 + 0.998533i −1.93425 + 4.85460i 0.686980 − 0.317831i 2.30330 − 1.38585i −0.0286938 + 0.103346i 8.46834 − 1.86402i −0.994138 − 0.108119i −1.74346 − 1.04900i
7.2 −0.492250 − 0.726014i −0.0541389 + 0.998533i 0.455490 − 1.14319i −3.82749 + 1.77078i 0.751599 − 0.452222i −1.35341 + 4.87453i −2.76749 + 0.609171i −0.994138 − 0.108119i 3.16970 + 1.90714i
7.3 0.0649074 + 0.0957312i −0.0541389 + 0.998533i 0.735325 − 1.84553i 1.25197 − 0.579222i −0.0991048 + 0.0596294i 0.330724 − 1.19116i 0.450316 − 0.0991221i −0.994138 − 0.108119i 0.136712 + 0.0822566i
7.4 0.962498 + 1.41958i −0.0541389 + 0.998533i −0.348523 + 0.874727i 0.131897 − 0.0610220i −1.46960 + 0.884232i −0.721672 + 2.59923i 1.77283 − 0.390229i −0.994138 − 0.108119i 0.213576 + 0.128504i
7.5 1.53455 + 2.26329i −0.0541389 + 0.998533i −2.02735 + 5.08827i 1.27104 − 0.588045i −2.34304 + 1.40976i 1.26600 − 4.55972i −9.28622 + 2.04405i −0.994138 − 0.108119i 3.28138 + 1.97434i
16.1 −2.69139 − 0.592420i 0.161782 + 0.986827i 5.07747 + 2.34909i 0.982969 + 3.54033i 0.149197 − 2.75178i −0.252139 − 0.238838i −7.88603 − 5.99480i −0.947653 + 0.319302i −0.548189 − 10.1108i
16.2 −1.21818 − 0.268143i 0.161782 + 0.986827i −0.403079 − 0.186484i 0.0748466 + 0.269573i 0.0675300 − 1.24552i 0.579545 + 0.548975i 2.42703 + 1.84498i −0.947653 + 0.319302i −0.0188929 − 0.348459i
16.3 −0.0555609 − 0.0122299i 0.161782 + 0.986827i −1.81221 − 0.838419i −1.03314 − 3.72102i 0.00308001 − 0.0568075i −1.19345 − 1.13050i 0.181015 + 0.137604i −0.947653 + 0.319302i 0.0118943 + 0.219378i
16.4 0.821942 + 0.180923i 0.161782 + 0.986827i −1.17230 − 0.542362i 0.558556 + 2.01174i −0.0455643 + 0.840384i 2.86481 + 2.71369i −2.20545 − 1.67654i −0.947653 + 0.319302i 0.0951310 + 1.75459i
16.5 2.16657 + 0.476899i 0.161782 + 0.986827i 2.65145 + 1.22669i −0.194787 − 0.701560i −0.120104 + 2.21519i −1.18393 − 1.12147i 1.62739 + 1.23711i −0.947653 + 0.319302i −0.0874475 − 1.61287i
19.1 −1.07530 + 2.02823i 0.725995 − 0.687699i −1.83506 − 2.70652i 1.52678 − 0.336069i 0.614149 + 2.21197i 3.14104 − 2.38776i 2.89830 − 0.315209i 0.0541389 − 0.998533i −0.960116 + 3.45803i
19.2 −0.715981 + 1.35048i 0.725995 − 0.687699i −0.188804 − 0.278465i −3.13782 + 0.690687i 0.408928 + 1.47283i −2.11659 + 1.60899i −2.52792 + 0.274928i 0.0541389 − 0.998533i 1.31386 − 4.73210i
19.3 −0.166156 + 0.313404i 0.725995 − 0.687699i 1.05176 + 1.55123i 2.73415 − 0.601833i 0.0948990 + 0.341795i −2.80376 + 2.13136i −1.36621 + 0.148584i 0.0541389 − 0.998533i −0.265680 + 0.956892i
19.4 0.390110 − 0.735825i 0.725995 − 0.687699i 0.733121 + 1.08127i −0.577946 + 0.127216i −0.222808 − 0.802484i 0.804132 − 0.611285i 2.73754 − 0.297726i 0.0541389 − 0.998533i −0.131854 + 0.474895i
19.5 1.09892 − 2.07278i 0.725995 − 0.687699i −1.96641 − 2.90024i −2.10012 + 0.462272i −0.627639 − 2.26055i 0.717585 − 0.545494i −3.50786 + 0.381503i 0.0541389 − 0.998533i −1.34967 + 4.86109i
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.2.e.b ✓ 140
3.b odd 2 1 531.2.i.b 140
59.c even 29 1 inner 177.2.e.b ✓ 140
177.h odd 58 1 531.2.i.b 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.e.b ✓ 140 1.a even 1 1 trivial
177.2.e.b ✓ 140 59.c even 29 1 inner
531.2.i.b 140 3.b odd 2 1
531.2.i.b 140 177.h odd 58 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{140} - T_{2}^{139} + 6 T_{2}^{138} - 6 T_{2}^{137} + 28 T_{2}^{136} + 3 T_{2}^{135} + \cdots + 335952241 \) acting on \(S_{2}^{\mathrm{new}}(177, [\chi])\). Copy content Toggle raw display