Properties

Label 177.5.b.a
Level $177$
Weight $5$
Character orbit 177.b
Analytic conductor $18.296$
Analytic rank $0$
Dimension $78$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 177.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.2964834658\)
Analytic rank: \(0\)
Dimension: \(78\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 78q - 612q^{4} + 64q^{6} + 76q^{7} - 100q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 78q - 612q^{4} + 64q^{6} + 76q^{7} - 100q^{9} - 156q^{10} - 4q^{13} + 13q^{15} + 4948q^{16} + 22q^{18} + 812q^{19} - 173q^{21} - 1644q^{22} - 678q^{24} - 8238q^{25} + 777q^{27} - 3764q^{28} + 1374q^{30} + 4664q^{31} - 1042q^{33} + 3244q^{34} + 3648q^{36} - 3960q^{37} - 7078q^{39} - 1576q^{40} + 4934q^{42} - 1492q^{43} - 2063q^{45} - 2036q^{46} - 2620q^{48} + 24274q^{49} + 7300q^{51} + 8408q^{52} - 14766q^{54} + 9780q^{55} + 6939q^{57} - 3856q^{58} + 4712q^{60} - 212q^{61} - 7438q^{63} - 45760q^{64} + 3048q^{66} - 12972q^{67} + 21672q^{69} + 5828q^{70} - 866q^{72} - 5240q^{73} - 20922q^{75} + 12368q^{76} - 16508q^{78} - 14976q^{79} + 25524q^{81} - 14484q^{82} + 9540q^{84} + 11572q^{85} + 5695q^{87} + 62160q^{88} - 31672q^{90} + 8284q^{91} - 9590q^{93} - 10992q^{94} + 34102q^{96} - 55000q^{97} - 14254q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
119.1 7.85862i 1.36439 + 8.89598i −45.7580 33.8616i 69.9102 10.7222i 13.8137 233.857i −77.2769 + 24.2751i 266.105
119.2 7.73459i −8.52815 2.87588i −43.8239 0.620129i −22.2437 + 65.9618i 71.8012 215.207i 64.4587 + 49.0518i −4.79645
119.3 7.49534i −6.13251 + 6.58728i −40.1801 24.9668i 49.3739 + 45.9653i −52.3496 181.238i −5.78454 80.7932i −187.134
119.4 7.42214i 2.94091 8.50594i −39.0882 22.5647i −63.1323 21.8278i 11.4871 171.364i −63.7021 50.0304i 167.478
119.5 7.37175i 8.47141 3.03895i −38.3427 40.5786i −22.4024 62.4491i 39.1148 164.705i 62.5296 51.4883i −299.136
119.6 7.08630i −1.58963 8.85850i −34.2156 8.65783i −62.7740 + 11.2646i −70.0243 129.081i −75.9462 + 28.1634i −61.3520
119.7 6.87418i 8.98827 + 0.459381i −31.2543 23.0718i 3.15787 61.7870i 23.0938 104.861i 80.5779 + 8.25808i 158.600
119.8 6.80672i 7.51401 + 4.95376i −30.3314 9.67205i 33.7188 51.1457i −60.1761 97.5501i 31.9206 + 74.4451i −65.8349
119.9 6.47467i −8.82514 1.76549i −25.9214 26.7513i −11.4310 + 57.1399i −48.5383 64.2378i 74.7661 + 31.1614i 173.206
119.10 6.42743i 2.05646 + 8.76190i −25.3118 39.0573i 56.3165 13.2178i 65.2885 59.8512i −72.5419 + 36.0371i −251.038
119.11 6.25305i −4.60229 7.73427i −23.1007 36.4421i −48.3628 + 28.7783i 14.4138 44.4009i −38.6379 + 71.1907i −227.875
119.12 5.88235i −8.00524 + 4.11293i −18.6020 15.2405i 24.1937 + 47.0896i 49.3247 15.3061i 47.1676 65.8500i −89.6502
119.13 5.74797i −1.10979 + 8.93131i −17.0392 1.93242i 51.3369 + 6.37903i −31.0983 5.97306i −78.5367 19.8237i 11.1075
119.14 5.57893i −6.24762 + 6.47821i −15.1244 38.3444i 36.1415 + 34.8550i 12.9947 4.88479i −2.93443 80.9468i 213.920
119.15 5.52059i 4.22607 + 7.94609i −14.4769 14.8774i 43.8671 23.3304i 83.7732 8.40835i −45.2807 + 67.1614i 82.1318
119.16 5.41049i 7.81854 4.45762i −13.2734 41.9044i −24.1179 42.3021i −64.0047 14.7525i 41.2592 69.7042i 226.723
119.17 5.25412i 4.74448 7.64787i −11.6058 7.04330i −40.1829 24.9281i 35.8552 23.0875i −35.9798 72.5703i −37.0064
119.18 5.04906i −3.85410 8.13301i −9.49305 33.9624i −41.0641 + 19.4596i 88.4932 32.8540i −51.2919 + 62.6909i 171.478
119.19 4.62880i 7.14239 5.47598i −5.42582 33.7032i −25.3472 33.0607i −90.1786 48.9458i 21.0273 78.2231i −156.006
119.20 4.27932i −6.98370 5.67697i −2.31259 6.15778i −24.2936 + 29.8855i −22.1503 58.5728i 16.5440 + 79.2925i −26.3511
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 119.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.5.b.a 78
3.b odd 2 1 inner 177.5.b.a 78
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.5.b.a 78 1.a even 1 1 trivial
177.5.b.a 78 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(177, [\chi])\).