Newform orbit 177.7.c.a

• Label: 177.7.c.a
• Level: $177$
• Weight: $7$
• Character orbit: 177.c
• Analytic conductor: $40.720$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(40.7195728007\)
Analytic rank:	\(0\)
Dimension:	\(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 1920 q^{4} - 408 q^{7} + 14580 q^{9}+O(q^{10}) \)

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		−	15.3575i	15.5885			−171.853			−142.367				−	239.400i	584.638					1656.35i	243.000					2186.40i
58.2		−	15.1313i	15.5885			−164.955			−92.6753				−	235.873i	−595.836					1527.58i	243.000					1402.29i
58.3		−	14.9223i	−15.5885			−158.675			237.797					232.615i	−568.491					1412.76i	243.000				−	3548.48i
58.4		−	14.6202i	−15.5885			−149.749			−98.4329					227.906i	−154.279					1253.67i	243.000					1439.11i
58.5		−	13.8068i	15.5885			−126.629			76.4303				−	215.227i	−5.71721					864.707i	243.000				−	1055.26i
58.6		−	13.7128i	−15.5885			−124.040			78.2520					213.761i	452.461					823.310i	243.000				−	1073.05i
58.7		−	13.2670i	15.5885			−112.012			130.160				−	206.811i	304.309					636.976i	243.000				−	1726.83i
58.8		−	13.2493i	−15.5885			−111.544			−224.166					206.536i	460.347					629.930i	243.000					2970.05i
58.9		−	12.5406i	−15.5885			−93.2673			48.4743					195.489i	−222.016					367.030i	243.000				−	607.898i
58.10		−	11.8923i	15.5885			−77.4273			54.6144				−	185.383i	−559.219					159.682i	243.000				−	649.492i
58.11		−	11.3092i	15.5885			−63.8971			−168.463				−	176.292i	92.9299				−	1.16321i	243.000					1905.18i
58.12		−	10.4738i	−15.5885			−45.7013			65.8108					163.271i	49.8494				−	191.658i	243.000				−	689.291i
58.13		−	10.1526i	−15.5885			−39.0745			−104.301					158.263i	−654.718				−	253.057i	243.000					1058.93i
58.14		−	9.91595i	15.5885			−34.3260			−70.2821				−	154.574i	315.675				−	294.246i	243.000					696.914i
58.15		−	8.95127i	15.5885			−16.1253			−71.7905				−	139.537i	97.3010				−	428.540i	243.000					642.617i
58.16		−	8.91504i	15.5885			−15.4779			175.637				−	138.972i	−256.162				−	432.577i	243.000				−	1565.81i
58.17		−	8.09269i	−15.5885			−1.49168			−181.057					126.153i	502.973				−	505.861i	243.000					1465.24i
58.18		−	7.10973i	−15.5885			13.4517			−5.28151					110.830i	−92.4639				−	550.661i	243.000					37.5501i
58.19		−	7.02990i	−15.5885			14.5805			242.375					109.585i	243.138				−	552.413i	243.000				−	1703.87i
58.20		−	6.65433i	15.5885			19.7199			168.519				−	103.731i	568.245				−	557.100i	243.000				−	1121.38i

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	177.7.c.a	✓	60
59.b	odd	2	1	inner	177.7.c.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
177.7.c.a	✓	60	1.a	even	1	1	trivial
177.7.c.a	✓	60	59.b	odd	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(177, [\chi])\).