Properties

Label 177.10.a.c
Level $177$
Weight $10$
Character orbit 177.a
Self dual yes
Analytic conductor $91.161$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.1613430010\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22q + 36q^{2} - 1782q^{3} + 5718q^{4} + 808q^{5} - 2916q^{6} + 21249q^{7} + 9435q^{8} + 144342q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 22q + 36q^{2} - 1782q^{3} + 5718q^{4} + 808q^{5} - 2916q^{6} + 21249q^{7} + 9435q^{8} + 144342q^{9} + 68441q^{10} - 68033q^{11} - 463158q^{12} + 283817q^{13} + 80285q^{14} - 65448q^{15} + 1067674q^{16} + 436893q^{17} + 236196q^{18} + 1207580q^{19} + 4209677q^{20} - 1721169q^{21} + 5460442q^{22} + 2421966q^{23} - 764235q^{24} + 7441842q^{25} - 2736526q^{26} - 11691702q^{27} + 4095246q^{28} - 2320594q^{29} - 5543721q^{30} - 3178024q^{31} - 20786874q^{32} + 5510673q^{33} - 13809336q^{34} - 2630800q^{35} + 37515798q^{36} + 3981807q^{37} - 24156377q^{38} - 22989177q^{39} - 29544450q^{40} - 885225q^{41} - 6503085q^{42} + 12360835q^{43} - 117711882q^{44} + 5301288q^{45} + 161066949q^{46} + 75901252q^{47} - 86481594q^{48} + 170907951q^{49} - 61318927q^{50} - 35388333q^{51} - 100762q^{52} - 34790192q^{53} - 19131876q^{54} + 151773316q^{55} - 417630344q^{56} - 97813980q^{57} - 432929294q^{58} + 266581942q^{59} - 340983837q^{60} - 290555332q^{61} + 158267098q^{62} + 139414689q^{63} - 131794443q^{64} - 650690086q^{65} - 442295802q^{66} + 86645184q^{67} + 62738541q^{68} - 196179246q^{69} + 429714610q^{70} - 36567631q^{71} + 61903035q^{72} + 907807228q^{73} - 171827242q^{74} - 602789202q^{75} + 1744504396q^{76} - 310688725q^{77} + 221658606q^{78} + 2508604687q^{79} + 3509441927q^{80} + 947027862q^{81} + 1759214793q^{82} + 2185672083q^{83} - 331714926q^{84} + 2868860198q^{85} + 2397001564q^{86} + 187968114q^{87} + 7683735877q^{88} + 1320145942q^{89} + 449041401q^{90} + 3894639897q^{91} + 3505964640q^{92} + 257419944q^{93} + 5406355552q^{94} + 3093659122q^{95} + 1683736794q^{96} + 3904552980q^{97} + 6137683116q^{98} - 446364513q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −43.0478 −81.0000 1341.12 1406.99 3486.87 9392.26 −35691.7 6561.00 −60567.9
1.2 −38.2133 −81.0000 948.259 −2363.53 3095.28 4562.96 −16670.9 6561.00 90318.2
1.3 −36.7798 −81.0000 840.750 2305.05 2979.16 −1509.72 −12091.4 6561.00 −84779.2
1.4 −36.1747 −81.0000 796.607 898.892 2930.15 −4674.69 −10295.5 6561.00 −32517.1
1.5 −27.1769 −81.0000 226.586 −1812.37 2201.33 −9882.92 7756.68 6561.00 49254.6
1.6 −20.5236 −81.0000 −90.7810 −622.721 1662.41 1394.83 12371.2 6561.00 12780.5
1.7 −17.2166 −81.0000 −215.588 470.596 1394.55 −5413.27 12526.6 6561.00 −8102.07
1.8 −13.7066 −81.0000 −324.128 −1287.82 1110.24 3218.96 11460.5 6561.00 17651.6
1.9 −9.19311 −81.0000 −427.487 −1961.85 744.642 11918.6 8636.80 6561.00 18035.5
1.10 −7.89550 −81.0000 −449.661 876.449 639.536 −1202.42 7592.80 6561.00 −6920.01
1.11 −3.04761 −81.0000 −502.712 −901.977 246.857 1315.27 3092.45 6561.00 2748.88
1.12 1.33644 −81.0000 −510.214 2090.38 −108.251 107.167 −1366.13 6561.00 2793.66
1.13 13.0004 −81.0000 −342.991 −5.38377 −1053.03 −6261.56 −11115.2 6561.00 −69.9909
1.14 17.2254 −81.0000 −215.286 27.4764 −1395.26 8666.20 −12527.8 6561.00 473.292
1.15 18.6113 −81.0000 −165.619 1492.75 −1507.52 11801.9 −12611.4 6561.00 27782.1
1.16 25.6716 −81.0000 147.031 −2019.49 −2079.40 7281.90 −9369.35 6561.00 −51843.5
1.17 26.7529 −81.0000 203.719 −302.298 −2166.99 −5739.31 −8247.43 6561.00 −8087.35
1.18 31.5568 −81.0000 483.832 1956.26 −2556.10 −4902.75 −888.901 6561.00 61733.3
1.19 34.5007 −81.0000 678.300 −2125.49 −2794.56 −11083.2 5737.48 6561.00 −73331.0
1.20 38.7622 −81.0000 990.507 −782.368 −3139.74 8972.92 18548.0 6561.00 −30326.3
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(59\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.10.a.c 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.10.a.c 22 1.a even 1 1 trivial