Properties

Label 177.12.a.d
Level $177$
Weight $12$
Character orbit 177.a
Self dual yes
Analytic conductor $135.997$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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 −88.1033 243.000 5714.20 1093.28 −21409.1 −20521.5 −323004. 59049.0 −96322.0
1.2 −79.8853 243.000 4333.66 −5377.20 −19412.1 −58607.4 −182591. 59049.0 429560.
1.3 −76.9358 243.000 3871.11 1565.71 −18695.4 51126.9 −140263. 59049.0 −120459.
1.4 −67.4662 243.000 2503.69 −169.543 −16394.3 65772.9 −30743.5 59049.0 11438.4
1.5 −67.3482 243.000 2487.78 9961.02 −16365.6 −37608.5 −29618.6 59049.0 −670857.
1.6 −47.6933 243.000 226.652 10067.2 −11589.5 19897.8 86866.1 59049.0 −480139.
1.7 −44.5190 243.000 −66.0554 −2490.42 −10818.1 −10217.0 94115.7 59049.0 110871.
1.8 −40.3996 243.000 −415.876 13699.0 −9817.09 81088.1 99539.5 59049.0 −553432.
1.9 −39.1439 243.000 −515.752 158.720 −9511.98 −70159.1 100355. 59049.0 −6212.92
1.10 −33.4729 243.000 −927.563 −11827.5 −8133.92 55304.7 99600.8 59049.0 395902.
1.11 −33.1334 243.000 −950.178 −7615.06 −8051.42 31854.1 99339.8 59049.0 252313.
1.12 −12.4112 243.000 −1893.96 −2629.42 −3015.92 23296.0 48924.4 59049.0 32634.3
1.13 −7.28530 243.000 −1994.92 6136.55 −1770.33 −33718.8 29453.9 59049.0 −44706.7
1.14 −5.21206 243.000 −2020.83 −8551.43 −1266.53 −16524.4 21207.0 59049.0 44570.6
1.15 10.5323 243.000 −1937.07 8300.31 2559.34 46494.5 −41971.8 59049.0 87421.0
1.16 23.0914 243.000 −1514.79 32.6758 5611.21 −80813.9 −82269.7 59049.0 754.528
1.17 23.7768 243.000 −1482.66 6921.85 5777.76 24486.4 −83947.8 59049.0 164579.
1.18 31.6204 243.000 −1048.15 −7920.33 7683.75 55386.7 −97901.5 59049.0 −250444.
1.19 31.7299 243.000 −1041.21 1823.69 7710.37 −52038.7 −98020.5 59049.0 57865.4
1.20 45.0643 243.000 −17.2112 −12444.4 10950.6 −11322.2 −93067.2 59049.0 −560797.
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
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 177.12.a.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.12.a.d 28 1.a even 1 1 trivial