Properties

Label 177.14.a.a
Level $177$
Weight $14$
Character orbit 177.a
Self dual yes
Analytic conductor $189.799$
Analytic rank $1$
Dimension $30$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(189.798744245\)
Analytic rank: \(1\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30q - 138q^{2} + 21870q^{3} + 114598q^{4} - 137742q^{5} - 100602q^{6} - 879443q^{7} - 872301q^{8} + 15943230q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q - 138q^{2} + 21870q^{3} + 114598q^{4} - 137742q^{5} - 100602q^{6} - 879443q^{7} - 872301q^{8} + 15943230q^{9} - 5352519q^{10} - 13950782q^{11} + 83541942q^{12} - 17256988q^{13} + 33780109q^{14} - 100413918q^{15} + 499996762q^{16} - 317583695q^{17} - 73338858q^{18} - 863401469q^{19} - 1841280623q^{20} - 641113947q^{21} - 2723764842q^{22} - 3142075981q^{23} - 635907429q^{24} + 5435751692q^{25} - 6441414040q^{26} + 11622614670q^{27} - 7538400046q^{28} - 4604589283q^{29} - 3901986351q^{30} + 4308675373q^{31} + 6094556360q^{32} - 10170120078q^{33} + 38097713432q^{34} - 15447827315q^{35} + 60902075718q^{36} - 19633376949q^{37} - 18152222923q^{38} - 12580344252q^{39} + 14680384170q^{40} - 103644439493q^{41} + 24625699461q^{42} - 64494894924q^{43} - 199714496208q^{44} - 73201746222q^{45} - 265425792847q^{46} - 293365585139q^{47} + 364497639498q^{48} + 414396765797q^{49} - 126058522207q^{50} - 231518513655q^{51} + 156029960316q^{52} - 76747013118q^{53} - 53464027482q^{54} - 433465885754q^{55} - 502955241518q^{56} - 629419670901q^{57} - 1755031845830q^{58} + 1265416009230q^{59} - 1342293574167q^{60} - 2022612531219q^{61} - 3816005187046q^{62} - 467372067363q^{63} - 3570205594131q^{64} - 3889749040576q^{65} - 1985624569818q^{66} - 502618987776q^{67} - 8953998390517q^{68} - 2290573390149q^{69} - 6805178272420q^{70} - 1599540605456q^{71} - 463576515741q^{72} - 3826795087235q^{73} - 7573387813210q^{74} + 3962662983468q^{75} - 19498723328388q^{76} - 9088623115219q^{77} - 4695790835160q^{78} - 8595482172338q^{79} - 17452527463963q^{80} + 8472886094430q^{81} - 11181116792901q^{82} - 13548556984389q^{83} - 5495493633534q^{84} - 12851795888367q^{85} + 8539949468848q^{86} - 3356745587307q^{87} - 25134826741387q^{88} - 21826401667403q^{89} - 2844548049879q^{90} - 26577050621355q^{91} - 34908210763168q^{92} + 3141024346917q^{93} - 26426808959500q^{94} - 29105233533993q^{95} + 4442931586440q^{96} + 417815797414q^{97} + 29159956938360q^{98} - 7414017536862q^{99} + O(q^{100}) \)

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 −169.531 729.000 20548.9 −21413.6 −123588. 361154. −2.09488e6 531441. 3.63027e6
1.2 −158.048 729.000 16787.0 −61907.4 −115217. 140047. −1.35842e6 531441. 9.78431e6
1.3 −153.212 729.000 15282.0 58431.3 −111692. 57012.1 −1.08627e6 531441. −8.95239e6
1.4 −152.210 729.000 14976.0 −42581.4 −110961. −571520. −1.03259e6 531441. 6.48132e6
1.5 −150.711 729.000 14521.7 40403.2 −109868. −111720. −953949. 531441. −6.08919e6
1.6 −141.008 729.000 11691.3 −32545.4 −102795. −339369. −493432. 531441. 4.58917e6
1.7 −112.878 729.000 4549.48 37606.3 −82288.2 −60488.5 411161. 531441. −4.24493e6
1.8 −108.701 729.000 3623.83 −15679.8 −79242.8 −222995. 496563. 531441. 1.70441e6
1.9 −102.341 729.000 2281.63 1529.46 −74606.4 −106408. 604872. 531441. −156526.
1.10 −73.9227 729.000 −2727.43 −22721.6 −53889.7 405591. 807194. 531441. 1.67964e6
1.11 −49.4802 729.000 −5743.71 45167.9 −36071.0 116012. 689542. 531441. −2.23492e6
1.12 −44.2174 729.000 −6236.82 14897.9 −32234.5 473501. 638005. 531441. −658745.
1.13 −41.9317 729.000 −6433.74 −10428.5 −30568.2 −484911. 613281. 531441. 437283.
1.14 −28.5091 729.000 −7379.23 −13887.1 −20783.1 351614. 443921. 531441. 395908.
1.15 −17.8149 729.000 −7874.63 5823.96 −12987.1 −495710. 286225. 531441. −103753.
1.16 9.64157 729.000 −8099.04 −58650.0 7028.71 −68208.9 −157071. 531441. −565478.
1.17 20.2681 729.000 −7781.20 37235.2 14775.5 −260203. −323747. 531441. 754688.
1.18 23.4632 729.000 −7641.48 −42940.2 17104.7 465285. −371505. 531441. −1.00752e6
1.19 31.8985 729.000 −7174.48 61104.8 23254.0 −236717. −490168. 531441. 1.94915e6
1.20 43.5631 729.000 −6294.26 −53819.8 31757.5 −600256. −631066. 531441. −2.34456e6
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
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 177.14.a.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.14.a.a 30 1.a even 1 1 trivial