Newform orbit 684.3.by.a

• Label: 684.3.by.a
• Level: $684$
• Weight: $3$
• Character orbit: 684.by
• Analytic conductor: $18.638$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.by (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.6376500822\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 84 q+O(q^{10}) \)

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			0			0		−3.11345	−	8.55412i		0		−5.95307	−	10.3110i		0			0			0
17.2		0			0			0		−3.00681	−	8.26115i		0		4.42220	+	7.65948i		0			0			0
17.3		0			0			0		−1.65531	−	4.54792i		0		2.47208	+	4.28177i		0			0			0
17.4		0			0			0		−1.65383	−	4.54386i		0		−5.81218	−	10.0670i		0			0			0
17.5		0			0			0		−1.41546	−	3.88895i		0		2.99209	+	5.18245i		0			0			0
17.6		0			0			0		−1.16814	−	3.20943i		0		0.346892	+	0.600835i		0			0			0
17.7		0			0			0		−0.128255	−	0.352378i		0		−1.11345	−	1.92856i		0			0			0
17.8		0			0			0		0.128255	+	0.352378i		0		−1.11345	−	1.92856i		0			0			0
17.9		0			0			0		1.16814	+	3.20943i		0		0.346892	+	0.600835i		0			0			0
17.10		0			0			0		1.41546	+	3.88895i		0		2.99209	+	5.18245i		0			0			0
17.11		0			0			0		1.65383	+	4.54386i		0		−5.81218	−	10.0670i		0			0			0
17.12		0			0			0		1.65531	+	4.54792i		0		2.47208	+	4.28177i		0			0			0
17.13		0			0			0		3.00681	+	8.26115i		0		4.42220	+	7.65948i		0			0			0
17.14		0			0			0		3.11345	+	8.55412i		0		−5.95307	−	10.3110i		0			0			0
161.1		0			0			0		−3.11345	+	8.55412i		0		−5.95307	+	10.3110i		0			0			0
161.2		0			0			0		−3.00681	+	8.26115i		0		4.42220	−	7.65948i		0			0			0
161.3		0			0			0		−1.65531	+	4.54792i		0		2.47208	−	4.28177i		0			0			0
161.4		0			0			0		−1.65383	+	4.54386i		0		−5.81218	+	10.0670i		0			0			0
161.5		0			0			0		−1.41546	+	3.88895i		0		2.99209	−	5.18245i		0			0			0
161.6		0			0			0		−1.16814	+	3.20943i		0		0.346892	−	0.600835i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
19.e	even	9	1	inner
57.l	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.3.by.a	✓	84
3.b	odd	2	1	inner	684.3.by.a	✓	84
19.e	even	9	1	inner	684.3.by.a	✓	84
57.l	odd	18	1	inner	684.3.by.a	✓	84

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.3.by.a	✓	84	1.a	even	1	1	trivial
684.3.by.a	✓	84	3.b	odd	2	1	inner
684.3.by.a	✓	84	19.e	even	9	1	inner
684.3.by.a	✓	84	57.l	odd	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(684, [\chi])\).