Newform orbit 177.9.c.a

• Label: 177.9.c.a
• Level: $177$
• Weight: $9$
• Character orbit: 177.c
• Analytic conductor: $72.106$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(72.1060139808\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 10240 q^{4} + 160 q^{7} + 174960 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
58.1		−	31.6572i	46.7654			−746.176			733.576				−	1480.46i	−1197.79					15517.6i	2187.00				−	23222.9i
58.2		−	31.4468i	−46.7654			−732.900			37.7655					1470.62i	1285.94					14997.0i	2187.00				−	1187.60i
58.3		−	29.8918i	46.7654			−637.520			−247.630				−	1397.90i	4057.23					11404.3i	2187.00					7402.10i
58.4		−	29.4383i	−46.7654			−610.615			−616.312					1376.69i	−3233.41					10439.3i	2187.00					18143.2i
58.5		−	29.1592i	46.7654			−594.257			−890.286				−	1363.64i	−1275.54					9863.29i	2187.00					25960.0i
58.6		−	28.5404i	−46.7654			−558.553			1019.37					1334.70i	1132.75					8634.97i	2187.00				−	29093.1i
58.7		−	28.1432i	−46.7654			−536.039			−406.960					1316.13i	2744.16					7881.18i	2187.00					11453.1i
58.8		−	27.0356i	46.7654			−474.926			21.6634				−	1264.33i	−141.547					5918.81i	2187.00				−	585.684i
58.9		−	25.6810i	46.7654			−403.515			951.178				−	1200.98i	4182.78					3788.34i	2187.00				−	24427.2i
58.10		−	25.6289i	46.7654			−400.842			−239.734				−	1198.55i	−1755.35					3712.14i	2187.00					6144.13i
58.11		−	25.1007i	−46.7654			−374.043			458.555					1173.84i	−1157.18					2962.95i	2187.00				−	11510.0i
58.12		−	24.0665i	−46.7654			−323.199			−984.641					1125.48i	−1144.76					1617.24i	2187.00					23696.9i
58.13		−	23.8187i	46.7654			−311.333			925.940				−	1113.89i	−3176.58					1317.95i	2187.00				−	22054.7i
58.14		−	23.2103i	−46.7654			−282.716			684.951					1085.44i	−2372.31					620.092i	2187.00				−	15897.9i
58.15		−	20.7900i	46.7654			−176.226			−602.288				−	972.254i	1280.77				−	1658.51i	2187.00					12521.6i
58.16		−	20.7097i	46.7654			−172.891			−840.056				−	968.496i	−4543.72				−	1721.16i	2187.00					17397.3i
58.17		−	20.4347i	−46.7654			−161.576			−84.1426					955.636i	4222.23				−	1929.52i	2187.00					1719.43i
58.18		−	20.4085i	−46.7654			−160.507			−745.364					954.411i	−9.58116				−	1948.88i	2187.00					15211.8i
58.19		−	19.3849i	46.7654			−119.774			778.298				−	906.541i	1420.14				−	2640.73i	2187.00				−	15087.2i
58.20		−	19.2811i	−46.7654			−115.759			525.267					901.686i	658.593				−	2703.99i	2187.00				−	10127.7i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
59.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	177.9.c.a	✓	80
59.b	odd	2	1	inner	177.9.c.a	✓	80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
177.9.c.a	✓	80	1.a	even	1	1	trivial
177.9.c.a	✓	80	59.b	odd	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(177, [\chi])\).