Properties

Label 177.3.g.a
Level $177$
Weight $3$
Character orbit 177.g
Analytic conductor $4.823$
Analytic rank $0$
Dimension $560$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.82290067918\)
Analytic rank: \(0\)
Dimension: \(560\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 560q + 40q^{4} + 8q^{7} - 60q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 560q + 40q^{4} + 8q^{7} - 60q^{9} + 24q^{12} - 24q^{15} + 8q^{16} - 16q^{17} + 60q^{19} + 164q^{20} - 40q^{22} - 100q^{25} + 156q^{26} - 200q^{28} + 60q^{29} + 32q^{35} + 120q^{36} - 28q^{41} - 1572q^{46} - 638q^{47} + 96q^{48} - 1328q^{49} - 1856q^{50} + 24q^{51} - 1392q^{52} - 572q^{53} - 522q^{55} - 928q^{56} - 24q^{57} + 268q^{59} + 72q^{60} + 348q^{61} + 472q^{62} + 24q^{63} + 2580q^{64} + 1218q^{65} + 120q^{66} + 1044q^{67} + 1936q^{68} + 2784q^{70} + 1416q^{71} + 870q^{73} + 1752q^{74} - 240q^{75} - 120q^{76} + 468q^{78} + 420q^{79} - 376q^{80} - 180q^{81} - 168q^{84} + 348q^{85} - 232q^{86} - 144q^{87} + 212q^{88} - 152q^{94} - 788q^{95} - 3306q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −3.55013 1.41450i 1.72190 + 0.187268i 7.69866 + 7.29256i 2.93856 3.45954i −5.84808 3.10045i −1.56353 0.940746i −10.5974 22.9059i 2.92986 + 0.644911i −15.3258 + 8.12523i
10.2 −3.48729 1.38946i −1.72190 0.187268i 7.32661 + 6.94014i 0.951158 1.11979i 5.74456 + 3.04557i 1.10368 + 0.664063i −9.60207 20.7545i 2.92986 + 0.644911i −4.87287 + 2.58343i
10.3 −2.66127 1.06035i −1.72190 0.187268i 3.05402 + 2.89292i −5.63025 + 6.62844i 4.38386 + 2.32417i −9.66711 5.81651i −0.248591 0.537320i 2.92986 + 0.644911i 22.0120 11.6700i
10.4 −2.31503 0.922394i 1.72190 + 0.187268i 1.60459 + 1.51995i −0.698140 + 0.821913i −3.81352 2.02180i −4.78757 2.88058i 1.87281 + 4.04801i 2.92986 + 0.644911i 2.37434 1.25880i
10.5 −2.17964 0.868447i −1.72190 0.187268i 1.09264 + 1.03500i 5.68569 6.69371i 3.59048 + 1.90355i 0.838143 + 0.504294i 2.45800 + 5.31287i 2.92986 + 0.644911i −18.2059 + 9.65215i
10.6 −2.06048 0.820970i 1.72190 + 0.187268i 0.667604 + 0.632388i 4.71743 5.55378i −3.39419 1.79949i 8.57272 + 5.15803i 2.86886 + 6.20094i 2.92986 + 0.644911i −14.2797 + 7.57059i
10.7 −1.49154 0.594282i −1.72190 0.187268i −1.03247 0.978012i −2.12045 + 2.49638i 2.45698 + 1.30261i 6.56905 + 3.95247i 3.65540 + 7.90102i 2.92986 + 0.644911i 4.64627 2.46330i
10.8 −1.45950 0.581520i 1.72190 + 0.187268i −1.11199 1.05334i −2.61612 + 3.07994i −2.40422 1.27464i −1.72503 1.03791i 3.64916 + 7.88752i 2.92986 + 0.644911i 5.60928 2.97385i
10.9 −0.911464 0.363160i −1.72190 0.187268i −2.20510 2.08878i 0.969991 1.14196i 1.50144 + 0.796012i −4.53786 2.73034i 2.89920 + 6.26652i 2.92986 + 0.644911i −1.29883 + 0.688594i
10.10 0.0935508 + 0.0372741i 1.72190 + 0.187268i −2.89662 2.74382i −5.55524 + 6.54013i 0.154105 + 0.0817011i 5.52684 + 3.32539i −0.337844 0.730238i 2.92986 + 0.644911i −0.763474 + 0.404768i
10.11 0.140866 + 0.0561262i 1.72190 + 0.187268i −2.88729 2.73499i 1.82294 2.14613i 0.232047 + 0.123023i −8.41764 5.06473i −0.507898 1.09780i 2.92986 + 0.644911i 0.377244 0.200002i
10.12 0.782131 + 0.311630i −1.72190 0.187268i −2.38937 2.26333i −2.21295 + 2.60528i −1.28839 0.683062i 9.58665 + 5.76810i −2.57754 5.57127i 2.92986 + 0.644911i −2.54270 + 1.34805i
10.13 0.819309 + 0.326442i 1.72190 + 0.187268i −2.33928 2.21588i 1.32036 1.55445i 1.34963 + 0.715530i 7.54931 + 4.54227i −2.67451 5.78087i 2.92986 + 0.644911i 1.58922 0.842551i
10.14 1.61637 + 0.644021i −1.72190 0.187268i −0.706089 0.668843i −2.42613 + 2.85626i −2.66262 1.41163i −4.88728 2.94058i −3.63289 7.85237i 2.92986 + 0.644911i −5.76103 + 3.05430i
10.15 1.89981 + 0.756953i −1.72190 0.187268i 0.132311 + 0.125332i 3.62948 4.27295i −3.12952 1.65917i −2.21876 1.33498i −3.27829 7.08590i 2.92986 + 0.644911i 10.1297 5.37045i
10.16 2.13762 + 0.851704i 1.72190 + 0.187268i 0.940023 + 0.890437i 4.37246 5.14766i 3.52126 + 1.86685i −0.447582 0.269301i −2.61371 5.64945i 2.92986 + 0.644911i 13.7309 7.27968i
10.17 2.78888 + 1.11119i 1.72190 + 0.187268i 3.63913 + 3.44717i −3.64648 + 4.29297i 4.59408 + 2.43563i 3.88004 + 2.33454i 1.27644 + 2.75897i 2.92986 + 0.644911i −14.9399 + 7.92064i
10.18 3.16149 + 1.25965i −1.72190 0.187268i 5.50432 + 5.21397i 3.15169 3.71045i −5.20787 2.76104i 6.22205 + 3.74368i 5.11821 + 11.0628i 2.92986 + 0.644911i 14.6379 7.76053i
10.19 3.27139 + 1.30344i −1.72190 0.187268i 6.09906 + 5.77734i −5.84142 + 6.87706i −5.38891 2.85702i −0.280020 0.168482i 6.50743 + 14.0656i 2.92986 + 0.644911i −28.0734 + 14.8836i
10.20 3.40493 + 1.35665i 1.72190 + 0.187268i 6.84907 + 6.48778i 1.18743 1.39795i 5.60888 + 2.97364i −5.85901 3.52525i 8.36297 + 18.0763i 2.92986 + 0.644911i 5.93964 3.14900i
See next 80 embeddings (of 560 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 172.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.d odd 58 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.3.g.a 560
59.d odd 58 1 inner 177.3.g.a 560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.3.g.a 560 1.a even 1 1 trivial
177.3.g.a 560 59.d odd 58 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(177, [\chi])\).