Newform orbit 693.4.g.a

• Label: 693.4.g.a
• Level: $693$
• Weight: $4$
• Character orbit: 693.g
• Analytic conductor: $40.888$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(40.8883236340\)
Analytic rank:	\(0\)
Dimension:	\(72\)
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.

\(\operatorname{Tr}(f)(q) = \) 72 * q + 288 * q^4 + 1320 * q^16 - 588 * q^22 - 2328 * q^25 - 384 * q^31 + 576 * q^34 - 240 * q^37 - 3528 * q^49 - 3456 * q^55 + 4560 * q^58 + 5016 * q^64 + 48 * q^67 + 1512 * q^70 - 11760 * q^82 - 6900 * q^88 - 768 * q^97

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
197.1	−5.54443				0		22.7407					20.7224i		0				7.00000i	−81.7287				0			−	114.894i
197.2	−5.54443				0		22.7407				−	20.7224i		0			−	7.00000i	−81.7287				0				114.894i
197.3	−5.33722				0		20.4859					11.1564i		0			−	7.00000i	−66.6400				0			−	59.5440i
197.4	−5.33722				0		20.4859				−	11.1564i		0				7.00000i	−66.6400				0				59.5440i
197.5	−5.01604				0		17.1607					3.75726i		0				7.00000i	−45.9502				0			−	18.8466i
197.6	−5.01604				0		17.1607				−	3.75726i		0			−	7.00000i	−45.9502				0				18.8466i
197.7	−4.88592				0		15.8722					11.9509i		0			−	7.00000i	−38.4628				0			−	58.3910i
197.8	−4.88592				0		15.8722				−	11.9509i		0				7.00000i	−38.4628				0				58.3910i
197.9	−4.18662				0		9.52782					6.47352i		0				7.00000i	−6.39641				0			−	27.1022i
197.10	−4.18662				0		9.52782				−	6.47352i		0			−	7.00000i	−6.39641				0				27.1022i
197.11	−3.95244				0		7.62177					18.3807i		0			−	7.00000i	1.49495				0			−	72.6485i
197.12	−3.95244				0		7.62177				−	18.3807i		0				7.00000i	1.49495				0				72.6485i
197.13	−3.84503				0		6.78427					20.1727i		0				7.00000i	4.67454				0			−	77.5645i
197.14	−3.84503				0		6.78427				−	20.1727i		0			−	7.00000i	4.67454				0				77.5645i
197.15	−3.82250				0		6.61151					8.82392i		0			−	7.00000i	5.30751				0			−	33.7294i
197.16	−3.82250				0		6.61151				−	8.82392i		0				7.00000i	5.30751				0				33.7294i
197.17	−3.62176				0		5.11711				−	1.96682i		0				7.00000i	10.4411				0				7.12333i
197.18	−3.62176				0		5.11711					1.96682i		0			−	7.00000i	10.4411				0			−	7.12333i
197.19	−3.57555				0		4.78456					13.0328i		0				7.00000i	11.4970				0			−	46.5995i
197.20	−3.57555				0		4.78456				−	13.0328i		0			−	7.00000i	11.4970				0				46.5995i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.b	odd	2	1	inner
33.d	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.g.a	✓	72
3.b	odd	2	1	inner	693.4.g.a	✓	72
11.b	odd	2	1	inner	693.4.g.a	✓	72
33.d	even	2	1	inner	693.4.g.a	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.4.g.a	✓	72	1.a	even	1	1	trivial
693.4.g.a	✓	72	3.b	odd	2	1	inner
693.4.g.a	✓	72	11.b	odd	2	1	inner
693.4.g.a	✓	72	33.d	even	2	1	inner

Hecke kernels

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