Newform orbit 693.4.e.a

• Label: 693.4.e.a
• Level: $693$
• Weight: $4$
• Character orbit: 693.e
• Analytic conductor: $40.888$
• Analytic rank: $0$
• Dimension: $80$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(40.8883236340\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 80 * q - 320 * q^4 + 64 * q^7 + 1280 * q^16 + 2000 * q^25 - 1600 * q^28 + 1568 * q^37 + 848 * q^43 + 1200 * q^46 + 368 * q^49 + 3240 * q^58 - 6056 * q^64 - 5456 * q^67 - 5616 * q^70 + 2464 * q^79 + 2688 * q^85 - 792 * q^88 + 1488 * q^91

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
188.1		−	5.46769i		0		−21.8956			21.9870				0		15.1433	−	10.6621i			75.9770i		0			−	120.218i
188.2			5.46769i		0		−21.8956			21.9870				0		15.1433	+	10.6621i		−	75.9770i		0				120.218i
188.3		−	3.94483i		0		−7.56165			19.8810				0		−12.9500	−	13.2400i		−	1.72920i		0			−	78.4272i
188.4			3.94483i		0		−7.56165			19.8810				0		−12.9500	+	13.2400i			1.72920i		0				78.4272i
188.5		−	0.695650i		0		7.51607			19.4181				0		18.2291	−	3.27124i		−	10.7937i		0			−	13.5082i
188.6			0.695650i		0		7.51607			19.4181				0		18.2291	+	3.27124i			10.7937i		0				13.5082i
188.7		−	2.07073i		0		3.71209			17.4770				0		2.95852	+	18.2824i		−	24.2525i		0			−	36.1900i
188.8			2.07073i		0		3.71209			17.4770				0		2.95852	−	18.2824i			24.2525i		0				36.1900i
188.9		−	3.89407i		0		−7.16376			−13.1480				0		15.5143	+	10.1146i		−	3.25637i		0				51.1990i
188.10			3.89407i		0		−7.16376			−13.1480				0		15.5143	−	10.1146i			3.25637i		0			−	51.1990i
188.11		−	0.533031i		0		7.71588			14.1882				0		−15.3659	−	10.3388i		−	8.37705i		0			−	7.56272i
188.12			0.533031i		0		7.71588			14.1882				0		−15.3659	+	10.3388i			8.37705i		0				7.56272i
188.13		−	3.06392i		0		−1.38760			12.9790				0		18.4701	−	1.36272i		−	20.2599i		0			−	39.7667i
188.14			3.06392i		0		−1.38760			12.9790				0		18.4701	+	1.36272i			20.2599i		0				39.7667i
188.15		−	4.96430i		0		−16.6442			12.0169				0		−8.15265	−	16.6293i			42.9125i		0			−	59.6553i
188.16			4.96430i		0		−16.6442			12.0169				0		−8.15265	+	16.6293i		−	42.9125i		0				59.6553i
188.17		−	5.54811i		0		−22.7815			11.4124				0		5.93379	+	17.5439i			82.0097i		0			−	63.3171i
188.18			5.54811i		0		−22.7815			11.4124				0		5.93379	−	17.5439i		−	82.0097i		0				63.3171i
188.19		−	3.08092i		0		−1.49208			−11.0749				0		−18.3326	+	2.62950i		−	20.0504i		0				34.1210i
188.20			3.08092i		0		−1.49208			−11.0749				0		−18.3326	−	2.62950i			20.0504i		0			−	34.1210i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.4.e.a	✓	80
3.b	odd	2	1	inner	693.4.e.a	✓	80
7.b	odd	2	1	inner	693.4.e.a	✓	80
21.c	even	2	1	inner	693.4.e.a	✓	80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.4.e.a	✓	80	1.a	even	1	1	trivial
693.4.e.a	✓	80	3.b	odd	2	1	inner
693.4.e.a	✓	80	7.b	odd	2	1	inner
693.4.e.a	✓	80	21.c	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(693, [\chi])\).