Properties

Label 177.10.a.a
Level $177$
Weight $10$
Character orbit 177.a
Self dual yes
Analytic conductor $91.161$
Analytic rank $1$
Dimension $21$
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: \(1\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} - 54663q^{10} - 151769q^{11} + 421686q^{12} - 153611q^{13} - 286771q^{14} - 240084q^{15} + 805530q^{16} - 723621q^{17} - 433026q^{18} - 549388q^{19} - 527311q^{20} - 2492775q^{21} + 2973158q^{22} + 169962q^{23} - 1994301q^{24} + 8035779q^{25} - 2337392q^{26} + 11160261q^{27} - 22659054q^{28} - 16845442q^{29} - 4427703q^{30} - 19307976q^{31} - 44923568q^{32} - 12293289q^{33} - 35547496q^{34} - 34882596q^{35} + 34156566q^{36} - 41561129q^{37} - 52335371q^{38} - 12442491q^{39} - 125735038q^{40} - 68169291q^{41} - 23228451q^{42} - 25719587q^{43} - 126277032q^{44} - 19446804q^{45} - 292814271q^{46} - 174095332q^{47} + 65247930q^{48} + 7479350q^{49} - 227877439q^{50} - 58613301q^{51} - 232397708q^{52} - 228390500q^{53} - 35075106q^{54} - 29426208q^{55} + 326778474q^{56} - 44500428q^{57} + 480343762q^{58} + 254464581q^{59} - 42712191q^{60} - 183928964q^{61} - 21753862q^{62} - 201914775q^{63} + 310571245q^{64} + 5308466q^{65} + 240825798q^{66} - 82724114q^{67} - 138336205q^{68} + 13766922q^{69} + 1030274876q^{70} - 404721965q^{71} - 161538381q^{72} + 154162574q^{73} + 36352054q^{74} + 650898099q^{75} + 1068940636q^{76} - 448535481q^{77} - 189328752q^{78} + 272529635q^{79} - 345587859q^{80} + 903981141q^{81} - 38412637q^{82} + 432518643q^{83} - 1835383374q^{84} - 126211490q^{85} - 3699273072q^{86} - 1364480802q^{87} + 170111045q^{88} - 1255621070q^{89} - 358643943q^{90} + 1448885849q^{91} + 1568933320q^{92} - 1563946056q^{93} - 1908445164q^{94} - 2896546490q^{95} - 3638809008q^{96} + 1007235486q^{97} - 9506868248q^{98} - 995756409q^{99} + O(q^{100}) \)

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 −43.9134 81.0000 1416.39 −993.635 −3556.99 3425.90 −39714.8 6561.00 43633.9
1.2 −43.5628 81.0000 1385.72 1597.21 −3528.58 −10966.0 −38061.5 6561.00 −69578.8
1.3 −35.8479 81.0000 773.074 1881.24 −2903.68 −9737.47 −9358.98 6561.00 −67438.4
1.4 −31.8892 81.0000 504.921 −1889.12 −2583.03 12272.3 225.735 6561.00 60242.4
1.5 −29.8118 81.0000 376.741 1198.15 −2414.75 3753.28 4032.31 6561.00 −35719.0
1.6 −26.8579 81.0000 209.348 −2742.53 −2175.49 −8878.91 8128.61 6561.00 73658.6
1.7 −24.8760 81.0000 106.816 −222.031 −2014.96 −2494.49 10079.4 6561.00 5523.25
1.8 −18.1702 81.0000 −181.842 1096.53 −1471.79 −2885.78 12607.3 6561.00 −19924.2
1.9 −16.6893 81.0000 −233.469 −1238.34 −1351.83 5162.18 12441.3 6561.00 20667.0
1.10 −12.7580 81.0000 −349.234 1739.29 −1033.40 7509.10 10987.6 6561.00 −22189.8
1.11 −3.19317 81.0000 −501.804 −2710.01 −258.647 −1841.65 3237.25 6561.00 8653.54
1.12 1.56630 81.0000 −509.547 −1262.61 126.871 −7655.10 −1600.05 6561.00 −1977.63
1.13 2.31191 81.0000 −506.655 −66.4563 187.265 −1904.69 −2355.04 6561.00 −153.641
1.14 6.97380 81.0000 −463.366 1656.32 564.878 2037.84 −6802.01 6561.00 11550.8
1.15 15.9818 81.0000 −256.580 61.0966 1294.53 4757.03 −12283.3 6561.00 976.437
1.16 24.3661 81.0000 81.7051 1969.67 1973.65 −11418.4 −10484.6 6561.00 47993.2
1.17 27.0996 81.0000 222.388 −1517.97 2195.07 2210.22 −7848.37 6561.00 −41136.3
1.18 30.4984 81.0000 418.150 1050.93 2470.37 −5890.68 −2862.26 6561.00 32051.7
1.19 35.2875 81.0000 733.210 388.913 2858.29 −2376.48 7805.95 6561.00 13723.8
1.20 37.7088 81.0000 909.951 −1229.57 3054.41 −1968.46 15006.2 6561.00 −46365.5
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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.a 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.10.a.a 21 1.a even 1 1 trivial