Newform orbit 120.4.r.a

• Label: 120.4.r.a
• Level: $120$
• Weight: $4$
• Character orbit: 120.r
• Analytic conductor: $7.080$
• Analytic rank: $0$
• Dimension: $36$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	120.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.08022920069\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 12 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1		0		−5.18112	−	0.394983i		0		−11.1795	+	0.136761i		0		10.5113	+	10.5113i		0		26.6880	+	4.09291i		0
17.2		0		−4.96635	+	1.52818i		0		−1.11114	−	11.1250i		0		9.09927	+	9.09927i		0		22.3293	−	15.1790i		0
17.3		0		−4.83816	−	1.89532i		0		8.68358	+	7.04240i		0		−20.0016	−	20.0016i		0		19.8155	+	18.3397i		0
17.4		0		−4.59064	+	2.43434i		0		−4.14870	+	10.3821i		0		−10.3236	−	10.3236i		0		15.1480	−	22.3504i		0
17.5		0		−3.82735	−	3.51445i		0		11.0949	+	1.37965i		0		7.07013	+	7.07013i		0		2.29721	+	26.9021i		0
17.6		0		−2.43434	+	4.59064i		0		4.14870	−	10.3821i		0		−10.3236	−	10.3236i		0		−15.1480	−	22.3504i		0
17.7		0		−1.52818	+	4.96635i		0		1.11114	+	11.1250i		0		9.09927	+	9.09927i		0		−22.3293	−	15.1790i		0
17.8		0		−1.43528	−	4.99399i		0		−0.901387	−	11.1439i		0		0.0607868	+	0.0607868i		0		−22.8800	+	14.3355i		0
17.9		0		−1.07808	−	5.08308i		0		−10.2050	+	4.56698i		0		1.68165	+	1.68165i		0		−24.6755	+	10.9599i		0
17.10		0		0.394983	+	5.18112i		0		11.1795	−	0.136761i		0		10.5113	+	10.5113i		0		−26.6880	+	4.09291i		0
17.11		0		1.89532	+	4.83816i		0		−8.68358	−	7.04240i		0		−20.0016	−	20.0016i		0		−19.8155	+	18.3397i		0
17.12		0		1.90672	−	4.83368i		0		6.19347	+	9.30811i		0		20.8011	+	20.8011i		0		−19.7289	−	18.4329i		0
17.13		0		3.05156	−	4.20571i		0		9.28746	−	6.22439i		0		−21.8991	−	21.8991i		0		−8.37595	−	25.6679i		0
17.14		0		3.51445	+	3.82735i		0		−11.0949	−	1.37965i		0		7.07013	+	7.07013i		0		−2.29721	+	26.9021i		0
17.15		0		4.20571	−	3.05156i		0		−9.28746	+	6.22439i		0		−21.8991	−	21.8991i		0		8.37595	−	25.6679i		0
17.16		0		4.83368	−	1.90672i		0		−6.19347	−	9.30811i		0		20.8011	+	20.8011i		0		19.7289	−	18.4329i		0
17.17		0		4.99399	+	1.43528i		0		0.901387	+	11.1439i		0		0.0607868	+	0.0607868i		0		22.8800	+	14.3355i		0
17.18		0		5.08308	+	1.07808i		0		10.2050	−	4.56698i		0		1.68165	+	1.68165i		0		24.6755	+	10.9599i		0
113.1		0		−5.18112	+	0.394983i		0		−11.1795	−	0.136761i		0		10.5113	−	10.5113i		0		26.6880	−	4.09291i		0
113.2		0		−4.96635	−	1.52818i		0		−1.11114	+	11.1250i		0		9.09927	−	9.09927i		0		22.3293	+	15.1790i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	120.4.r.a	✓	36
3.b	odd	2	1	inner	120.4.r.a	✓	36
4.b	odd	2	1		240.4.v.e		36
5.c	odd	4	1	inner	120.4.r.a	✓	36
12.b	even	2	1		240.4.v.e		36
15.e	even	4	1	inner	120.4.r.a	✓	36
20.e	even	4	1		240.4.v.e		36
60.l	odd	4	1		240.4.v.e		36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.4.r.a	✓	36	1.a	even	1	1	trivial
120.4.r.a	✓	36	3.b	odd	2	1	inner
120.4.r.a	✓	36	5.c	odd	4	1	inner
120.4.r.a	✓	36	15.e	even	4	1	inner
240.4.v.e		36	4.b	odd	2	1
240.4.v.e		36	12.b	even	2	1
240.4.v.e		36	20.e	even	4	1
240.4.v.e		36	60.l	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(120, [\chi])\).