Newform orbit 620.3.s.a

• Label: 620.3.s.a
• Level: $620$
• Weight: $3$
• Character orbit: 620.s
• Analytic conductor: $16.894$
• Analytic rank: $0$
• Dimension: $44$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 620 = 2^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	620.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.8937763903\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 6 q^{3} - 10 q^{7} + 72 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
161.1		0		−5.11404	+	2.95259i		0		1.11803	−	1.93649i		0		−5.21198	−	9.02741i		0		12.9356	−	22.4051i		0
161.2		0		−4.42933	+	2.55728i		0		1.11803	−	1.93649i		0		4.65784	+	8.06762i		0		8.57932	−	14.8598i		0
161.3		0		−4.20282	+	2.42650i		0		−1.11803	+	1.93649i		0		2.40662	+	4.16839i		0		7.27581	−	12.6021i		0
161.4		0		−3.87893	+	2.23950i		0		−1.11803	+	1.93649i		0		−6.50423	−	11.2657i		0		5.53073	−	9.57950i		0
161.5		0		−3.84656	+	2.22081i		0		−1.11803	+	1.93649i		0		1.41631	+	2.45313i		0		5.36402	−	9.29076i		0
161.6		0		−2.60077	+	1.50155i		0		1.11803	−	1.93649i		0		−1.60430	−	2.77873i		0		0.00932825	−	0.0161570i		0
161.7		0		−2.59455	+	1.49797i		0		1.11803	−	1.93649i		0		−1.38320	−	2.39578i		0		−0.0121976	+	0.0211269i		0
161.8		0		−1.48316	+	0.856302i		0		−1.11803	+	1.93649i		0		−5.04001	−	8.72955i		0		−3.03349	+	5.25416i		0
161.9		0		−0.938793	+	0.542012i		0		−1.11803	+	1.93649i		0		5.83715	+	10.1102i		0		−3.91245	+	6.77655i		0
161.10		0		−0.790506	+	0.456399i		0		−1.11803	+	1.93649i		0		−1.01357	−	1.75555i		0		−4.08340	+	7.07266i		0
161.11		0		−0.348582	+	0.201254i		0		1.11803	−	1.93649i		0		−0.876221	−	1.51766i		0		−4.41899	+	7.65392i		0
161.12		0		0.0183569	−	0.0105984i		0		1.11803	−	1.93649i		0		6.29590	+	10.9048i		0		−4.49978	+	7.79384i		0
161.13		0		0.516880	−	0.298421i		0		−1.11803	+	1.93649i		0		0.276387	+	0.478717i		0		−4.32189	+	7.48573i		0
161.14		0		1.36942	−	0.790636i		0		1.11803	−	1.93649i		0		−1.50268	−	2.60272i		0		−3.24979	+	5.62880i		0
161.15		0		1.72056	−	0.993368i		0		1.11803	−	1.93649i		0		2.79068	+	4.83359i		0		−2.52644	+	4.37592i		0
161.16		0		1.98376	−	1.14532i		0		−1.11803	+	1.93649i		0		−3.99521	−	6.91990i		0		−1.87647	+	3.25015i		0
161.17		0		2.10633	−	1.21609i		0		1.11803	−	1.93649i		0		−4.30936	−	7.46404i		0		−1.54224	+	2.67123i		0
161.18		0		2.88764	−	1.66718i		0		−1.11803	+	1.93649i		0		4.54290	+	7.86853i		0		1.05896	−	1.83417i		0
161.19		0		3.51857	−	2.03145i		0		1.11803	−	1.93649i		0		4.03660	+	6.99159i		0		3.75354	−	6.50133i		0
161.20		0		3.91258	−	2.25893i		0		−1.11803	+	1.93649i		0		−1.18344	−	2.04977i		0		5.70553	−	9.88226i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.e	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	620.3.s.a	✓	44
31.e	odd	6	1	inner	620.3.s.a	✓	44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
620.3.s.a	✓	44	1.a	even	1	1	trivial
620.3.s.a	✓	44	31.e	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(620, [\chi])\).