Properties

Label 177.14.a.c
Level $177$
Weight $14$
Character orbit 177.a
Self dual yes
Analytic conductor $189.799$
Analytic rank $0$
Dimension $31$
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: \(0\)
Dimension: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31q + 310q^{2} + 22599q^{3} + 126886q^{4} + 81008q^{5} + 225990q^{6} + 1002941q^{7} + 4632723q^{8} + 16474671q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 31q + 310q^{2} + 22599q^{3} + 126886q^{4} + 81008q^{5} + 225990q^{6} + 1002941q^{7} + 4632723q^{8} + 16474671q^{9} + 4647481q^{10} + 17937316q^{11} + 92499894q^{12} + 40664720q^{13} + 139193613q^{14} + 59054832q^{15} + 370110498q^{16} + 213442823q^{17} + 164746710q^{18} - 62592329q^{19} + 1637085153q^{20} + 731143989q^{21} + 4142028314q^{22} + 1873486387q^{23} + 3377255067q^{24} + 8307272395q^{25} - 534777728q^{26} + 12010035159q^{27} + 766416778q^{28} + 13765513563q^{29} + 3388013649q^{30} + 14274077235q^{31} + 30574460156q^{32} + 13076303364q^{33} - 677551028q^{34} + 36023610185q^{35} + 67432422726q^{36} - 18278838391q^{37} - 23650502933q^{38} + 29644580880q^{39} + 10045447572q^{40} + 34748006725q^{41} + 101472143877q^{42} + 40350158146q^{43} + 163101196592q^{44} + 43050972528q^{45} + 296118466353q^{46} + 233954631099q^{47} + 269810553042q^{48} + 324065402790q^{49} - 102960745787q^{50} + 155599817967q^{51} + 668297695096q^{52} + 500927963876q^{53} + 120100351590q^{54} + 884972340924q^{55} + 1392234478810q^{56} - 45629807841q^{57} + 689262776200q^{58} - 1307596542871q^{59} + 1193435076537q^{60} + 1716832157925q^{61} + 1816094290366q^{62} + 533003967981q^{63} + 4381780009133q^{64} + 1457007885906q^{65} + 3019538640906q^{66} + 1212131702006q^{67} + 6552992665503q^{68} + 1365771576123q^{69} + 8806714081634q^{70} + 6074000239936q^{71} + 2462018943843q^{72} + 3756145185973q^{73} + 8066450143602q^{74} + 6056001575955q^{75} + 7913230001992q^{76} + 6031241575915q^{77} - 389852963712q^{78} + 11377744190862q^{79} + 16473302366969q^{80} + 8755315630911q^{81} + 10413363680159q^{82} + 19915461517429q^{83} + 558717831162q^{84} + 15280981141573q^{85} + 7573325358452q^{86} + 10035059387427q^{87} + 19271409121081q^{88} + 14115863121241q^{89} + 2469861950121q^{90} + 18296287784699q^{91} + 15158951168774q^{92} + 10405802304315q^{93} - 18637923572412q^{94} - 2294034679397q^{95} + 22288781453724q^{96} + 38558536599054q^{97} - 1998410212380q^{98} + 9532625152356q^{99} + O(q^{100}) \)

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 −170.157 729.000 20761.4 7033.25 −124044. −166219. −2.13877e6 531441. −1.19676e6
1.2 −166.716 729.000 19602.3 43558.7 −121536. −395203. −1.90228e6 531441. −7.26193e6
1.3 −138.949 729.000 11114.9 −41433.2 −101294. −70948.7 −406128. 531441. 5.75710e6
1.4 −126.798 729.000 7885.84 57088.2 −92436.0 543807. 38821.0 531441. −7.23869e6
1.5 −121.100 729.000 6473.28 −61050.1 −88282.1 274195. 208137. 531441. 7.39319e6
1.6 −118.157 729.000 5769.07 218.169 −86136.4 217680. 286287. 531441. −25778.2
1.7 −107.990 729.000 3469.94 5614.85 −78725.0 −441395. 509938. 531441. −606350.
1.8 −102.312 729.000 2275.81 −9860.18 −74585.7 439885. 605299. 531441. 1.00882e6
1.9 −83.5640 729.000 −1209.07 55860.4 −60918.1 −515369. 785590. 531441. −4.66791e6
1.10 −60.0941 729.000 −4580.70 −50851.6 −43808.6 −259334. 767564. 531441. 3.05588e6
1.11 −59.5570 729.000 −4644.96 −45189.8 −43417.1 −134821. 764531. 531441. 2.69137e6
1.12 −55.7627 729.000 −5082.53 35389.9 −40651.0 112882. 740223. 531441. −1.97344e6
1.13 −50.5493 729.000 −5636.77 37474.1 −36850.4 131476. 699034. 531441. −1.89429e6
1.14 −7.03559 729.000 −8142.50 12466.0 −5128.95 −179621. 114923. 531441. −87705.8
1.15 −1.12446 729.000 −8190.74 −53521.6 −819.732 445275. 18421.7 531441. 60183.0
1.16 15.7636 729.000 −7943.51 −18232.0 11491.7 −139545. −254354. 531441. −287402.
1.17 28.6118 729.000 −7373.36 60045.3 20858.0 428045. −445354. 531441. 1.71801e6
1.18 28.6646 729.000 −7370.34 −9309.96 20896.5 216532. −446089. 531441. −266867.
1.19 66.7973 729.000 −3730.11 36075.7 48695.3 −269744. −796366. 531441. 2.40976e6
1.20 68.2207 729.000 −3537.94 −14307.6 49732.9 −16077.5 −800224. 531441. −976072.
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
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.c 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.14.a.c 31 1.a even 1 1 trivial