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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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 504q - 27q^{3} - 70q^{4} - 29q^{6} - 58q^{7} - 19q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 504q - 27q^{3} - 70q^{4} - 29q^{6} - 58q^{7} - 19q^{9} - 58q^{10} - 15q^{12} - 58q^{13} - 38q^{15} - 66q^{16} - 29q^{18} - 66q^{19} - 24q^{21} - 62q^{22} - 29q^{24} - 20q^{25} - 54q^{27} - 26q^{28} - 29q^{30} - 58q^{31} - 29q^{33} - 58q^{34} + 13q^{36} - 58q^{37} - 29q^{39} - 58q^{40} - 29q^{42} - 58q^{43} - q^{45} - 46q^{46} + 147q^{48} - 48q^{49} + 59q^{51} - 58q^{52} + 174q^{54} - 58q^{55} + 83q^{57} + 250q^{60} - 58q^{61} + 82q^{63} + 10q^{64} + 226q^{66} - 58q^{67} + 87q^{69} - 58q^{70} + 145q^{72} - 58q^{73} - 28q^{75} - 150q^{76} - 13q^{78} - 30q^{79} + 13q^{81} - 58q^{82} - 69q^{84} - 86q^{85} - 36q^{87} + 22q^{88} - 29q^{90} - 58q^{91} - 29q^{93} - 162q^{94} - 29q^{96} - 58q^{97} - 29q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 173.18
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 \(S_{2}^{\mathrm{new}}(177, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database