Properties

Label 177.14.a.b
Level $177$
Weight $14$
Character orbit 177.a
Self dual yes
Analytic conductor $189.799$
Analytic rank $1$
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: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31q - 52q^{2} - 22599q^{3} + 126886q^{4} + 33486q^{5} + 37908q^{6} - 1135539q^{7} - 1519749q^{8} + 16474671q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 31q - 52q^{2} - 22599q^{3} + 126886q^{4} + 33486q^{5} + 37908q^{6} - 1135539q^{7} - 1519749q^{8} + 16474671q^{9} - 3854663q^{10} + 3943968q^{11} - 92499894q^{12} - 48510022q^{13} - 51427459q^{14} - 24411294q^{15} + 370110498q^{16} + 83288419q^{17} - 27634932q^{18} - 180425297q^{19} + 753620445q^{20} + 827807931q^{21} + 2300196142q^{22} - 1305810279q^{23} + 1107897021q^{24} + 8070954867q^{25} + 464550322q^{26} - 12010035159q^{27} - 9887169562q^{28} + 6248352277q^{29} + 2810049327q^{30} - 26730150789q^{31} - 24001343230q^{32} - 2875152672q^{33} - 36571033348q^{34} + 10255900979q^{35} + 67432422726q^{36} - 43284776933q^{37} - 36293696947q^{38} + 35363806038q^{39} - 105980683856q^{40} - 9961079285q^{41} + 37490617611q^{42} - 51755851288q^{43} - 59623729442q^{44} + 17795833326q^{45} - 202287132683q^{46} - 82747063727q^{47} - 269810553042q^{48} + 535277836542q^{49} + 526974390461q^{50} - 60717257451q^{51} + 544982341446q^{52} + 561701818494q^{53} + 20145865428q^{54} - 521861534450q^{55} - 228056576664q^{56} + 131530041513q^{57} + 10555409160q^{58} - 1307596542871q^{59} - 549389304405q^{60} + 618193248201q^{61} - 1486611437386q^{62} - 603471981699q^{63} + 679062548045q^{64} - 1130583307122q^{65} - 1676842987518q^{66} - 4137387490592q^{67} - 3901389300295q^{68} + 951935693391q^{69} - 819291947844q^{70} - 3766439869810q^{71} - 807656928309q^{72} - 2386775553523q^{73} + 3060770694642q^{74} - 5883726098043q^{75} - 847741068784q^{76} + 1650423006137q^{77} - 338657184738q^{78} + 787155757766q^{79} + 13999832121779q^{80} + 8755315630911q^{81} + 10083281915577q^{82} + 8743877051639q^{83} + 7207746610698q^{84} + 15373177520565q^{85} + 18939443838984q^{86} - 4555048809933q^{87} + 39713314506713q^{88} + 11026795445259q^{89} - 2048525959383q^{90} + 23285721962531q^{91} + 40411079823254q^{92} + 19486279925181q^{93} + 35237377585624q^{94} + 13730236994039q^{95} + 17496979214670q^{96} + 10134565481560q^{97} + 70916776240976q^{98} + 2095986297888q^{99} + O(q^{100}) \)

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 −169.906 −729.000 20675.9 41498.5 123861. 222317. −2.12109e6 531441. −7.05083e6
1.2 −169.081 −729.000 20396.4 15031.4 123260. −257399. −2.06354e6 531441. −2.54153e6
1.3 −159.673 −729.000 17303.4 −10788.6 116402. 188818. −1.45485e6 531441. 1.72264e6
1.4 −151.259 −729.000 14687.4 64912.3 110268. −35238.1 −982489. 531441. −9.81859e6
1.5 −149.360 −729.000 14116.3 −48711.9 108883. −300158. −884852. 531441. 7.27559e6
1.6 −135.114 −729.000 10063.7 −36751.1 98497.9 −177254. −252896. 531441. 4.96558e6
1.7 −117.761 −729.000 5675.60 −28297.1 85847.6 297580. 296333. 531441. 3.33229e6
1.8 −98.0136 −729.000 1414.67 43954.8 71451.9 −396678. 664271. 531441. −4.30817e6
1.9 −80.4989 −729.000 −1711.93 −19074.6 58683.7 −213612. 797255. 531441. 1.53548e6
1.10 −73.1261 −729.000 −2844.57 −37898.1 53309.0 370148. 807062. 531441. 2.77134e6
1.11 −66.2972 −729.000 −3796.68 24188.3 48330.7 −395052. 794816. 531441. −1.60362e6
1.12 −63.8588 −729.000 −4114.05 18414.6 46553.1 120263. 785850. 531441. −1.17593e6
1.13 −57.5851 −729.000 −4875.95 31633.2 41979.6 267144. 752520. 531441. −1.82160e6
1.14 −42.0742 −729.000 −6421.76 −29646.8 30672.1 519485. 614862. 531441. 1.24737e6
1.15 −32.6149 −729.000 −7128.27 −49381.7 23776.2 −593230. 499669. 531441. 1.61058e6
1.16 13.8965 −729.000 −7998.89 24353.7 −10130.5 −38818.7 −224997. 531441. 338431.
1.17 20.8810 −729.000 −7755.98 27865.9 −15222.2 61118.3 −333010. 531441. 581867.
1.18 32.0352 −729.000 −7165.75 52348.2 −23353.7 485703. −491988. 531441. 1.67699e6
1.19 39.4833 −729.000 −6633.07 39747.6 −28783.4 −539057. −585343. 531441. 1.56937e6
1.20 52.9360 −729.000 −5389.78 −33150.0 −38590.4 −413060. −718965. 531441. −1.75483e6
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.b 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.14.a.b 31 1.a even 1 1 trivial