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Properties

Label 177.2.f.a
Level $177$
Weight $2$
Character orbit 177.f
Analytic conductor $1.413$
Analytic rank $0$
Dimension $504$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [177,2,Mod(2,177)]
 
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([29, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.f (of order \(58\), 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: \(504\)
Relative dimension: \(18\) over \(\Q(\zeta_{58})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{58}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9} - 58 q^{10} - 15 q^{12} - 58 q^{13} - 38 q^{15} - 66 q^{16} - 29 q^{18} - 66 q^{19} - 24 q^{21} - 62 q^{22} - 29 q^{24} - 20 q^{25} - 54 q^{27} - 26 q^{28} - 29 q^{30} - 58 q^{31} - 29 q^{33} - 58 q^{34} + 13 q^{36} - 58 q^{37} - 29 q^{39} - 58 q^{40} - 29 q^{42} - 58 q^{43} - q^{45} - 46 q^{46} + 147 q^{48} - 48 q^{49} + 59 q^{51} - 58 q^{52} + 174 q^{54} - 58 q^{55} + 83 q^{57} + 250 q^{60} - 58 q^{61} + 82 q^{63} + 10 q^{64} + 226 q^{66} - 58 q^{67} + 87 q^{69} - 58 q^{70} + 145 q^{72} - 58 q^{73} - 28 q^{75} - 150 q^{76} - 13 q^{78} - 30 q^{79} + 13 q^{81} - 58 q^{82} - 69 q^{84} - 86 q^{85} - 36 q^{87} + 22 q^{88} - 29 q^{90} - 58 q^{91} - 29 q^{93} - 162 q^{94} - 29 q^{96} - 58 q^{97} - 29 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.139199 + 2.56737i 1.16564 + 1.28112i −4.58374 − 0.498512i −0.00781348 + 0.0231896i −3.45137 + 2.81431i 0.224941 + 0.331763i 1.08599 − 6.62424i −0.282552 + 2.98666i −0.0584487 − 0.0232881i
2.2 −0.132697 + 2.44745i −1.61630 − 0.622564i −3.98412 − 0.433299i −0.468889 + 1.39161i 1.73817 − 3.87319i −2.78535 − 4.10808i 0.796087 − 4.85592i 2.22483 + 2.01250i −3.34368 − 1.33224i
2.3 −0.127399 + 2.34974i 0.849015 − 1.50969i −3.51679 − 0.382473i −0.844245 + 2.50563i 3.43923 + 2.18730i 1.82720 + 2.69492i 0.585342 − 3.57043i −1.55835 − 2.56350i −5.78003 − 2.30297i
2.4 −0.106315 + 1.96087i −1.24892 − 1.20008i −1.84543 − 0.200702i 1.09139 − 3.23912i 2.48598 − 2.32139i 2.29285 + 3.38170i −0.0456507 + 0.278457i 0.119624 + 2.99761i 6.23546 + 2.48444i
2.5 −0.0852373 + 1.57211i 1.69601 − 0.351511i −0.475985 − 0.0517665i 0.628919 − 1.86657i 0.408051 + 2.69627i −0.690899 − 1.01900i −0.387471 + 2.36347i 2.75288 − 1.19233i 2.88084 + 1.14783i
2.6 −0.0767664 + 1.41587i −0.135861 + 1.72671i −0.0105295 − 0.00114515i −0.435891 + 1.29368i −2.43438 − 0.324915i −0.863119 − 1.27300i −0.456369 + 2.78373i −2.96308 − 0.469186i −1.79822 − 0.716478i
2.7 −0.0596516 + 1.10021i −1.61700 + 0.620740i 0.781375 + 0.0849796i −0.182962 + 0.543013i −0.586487 − 1.81606i 1.24114 + 1.83055i −0.496616 + 3.02923i 2.22936 − 2.00747i −0.586514 − 0.233688i
2.8 −0.0266055 + 0.490710i 0.0777289 − 1.73031i 1.74819 + 0.190127i 0.431333 − 1.28015i 0.847010 + 0.0841780i −1.31835 − 1.94442i −0.298818 + 1.82271i −2.98792 − 0.268989i 0.616706 + 0.245718i
2.9 −0.00955626 + 0.176255i −1.25381 − 1.19497i 1.95730 + 0.212869i −1.32719 + 3.93896i 0.222601 − 0.209571i 0.403437 + 0.595025i −0.113337 + 0.691326i 0.144093 + 2.99654i −0.681577 − 0.271565i
2.10 0.00955626 − 0.176255i 0.0990596 + 1.72922i 1.95730 + 0.212869i 1.32719 − 3.93896i 0.305729 0.000934904i 0.403437 + 0.595025i 0.113337 − 0.691326i −2.98037 + 0.342591i −0.681577 − 0.271565i
2.11 0.0266055 − 0.490710i 1.36909 + 1.06093i 1.74819 + 0.190127i −0.431333 + 1.28015i 0.557036 − 0.643601i −1.31835 − 1.94442i 0.298818 − 1.82271i 0.748837 + 2.90504i 0.616706 + 0.245718i
2.12 0.0596516 − 1.10021i −1.51993 + 0.830556i 0.781375 + 0.0849796i 0.182962 − 0.543013i 0.823119 + 1.72178i 1.24114 + 1.83055i 0.496616 − 3.02923i 1.62035 − 2.52477i −0.586514 − 0.233688i
2.13 0.0767664 − 1.41587i −1.40399 − 1.01430i −0.0105295 − 0.00114515i 0.435891 − 1.29368i −1.54390 + 1.91001i −0.863119 − 1.27300i 0.456369 − 2.78373i 0.942378 + 2.84814i −1.79822 − 0.716478i
2.14 0.0852373 − 1.57211i 1.36588 − 1.06507i −0.475985 − 0.0517665i −0.628919 + 1.86657i −1.55798 − 2.23810i −0.690899 − 1.01900i 0.387471 − 2.36347i 0.731257 − 2.90951i 2.88084 + 1.14783i
2.15 0.106315 − 1.96087i 0.106118 + 1.72880i −1.84543 − 0.200702i −1.09139 + 3.23912i 3.40123 − 0.0242854i 2.29285 + 3.38170i 0.0456507 − 0.278457i −2.97748 + 0.366912i 6.23546 + 2.48444i
2.16 0.127399 − 2.34974i 1.70027 + 0.330268i −3.51679 − 0.382473i 0.844245 − 2.50563i 0.992658 − 3.95313i 1.82720 + 2.69492i −0.585342 + 3.57043i 2.78185 + 1.12309i −5.78003 − 2.30297i
2.17 0.132697 − 2.44745i −0.571873 + 1.63492i −3.98412 − 0.433299i 0.468889 − 1.39161i 3.92549 + 1.61658i −2.78535 − 4.10808i −0.796087 + 4.85592i −2.34592 − 1.86993i −3.34368 − 1.33224i
2.18 0.139199 − 2.56737i −0.221802 − 1.71779i −4.58374 − 0.498512i 0.00781348 − 0.0231896i −4.44108 + 0.330333i 0.224941 + 0.331763i −1.08599 + 6.62424i −2.90161 + 0.762018i −0.0584487 − 0.0232881i
8.1 −0.438895 + 2.67714i −0.255734 − 1.71307i −5.07914 − 1.71136i 2.43047 − 1.64790i 4.69836 + 0.0672212i 3.44243 − 0.757736i 4.26929 − 8.05274i −2.86920 + 0.876179i 3.34494 + 7.22997i
8.2 −0.426187 + 2.59963i 1.72906 − 0.101732i −4.68111 − 1.57725i −3.43308 + 2.32768i −0.472438 + 4.53827i −0.862947 + 0.189949i 3.62740 − 6.84201i 2.97930 − 0.351802i −4.58797 − 9.91674i
See next 80 embeddings (of 504 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
59.d odd 58 1 inner
177.f even 58 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.2.f.a ✓ 504
3.b odd 2 1 inner 177.2.f.a ✓ 504
59.d odd 58 1 inner 177.2.f.a ✓ 504
177.f even 58 1 inner 177.2.f.a ✓ 504
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.f.a ✓ 504 1.a even 1 1 trivial
177.2.f.a ✓ 504 3.b odd 2 1 inner
177.2.f.a ✓ 504 59.d odd 58 1 inner
177.2.f.a ✓ 504 177.f even 58 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(177, [\chi])\).