Newform orbit 684.2.cl.a

• Label: 684.2.cl.a
• Level: $684$
• Weight: $2$
• Character orbit: 684.cl
• Analytic conductor: $5.462$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.46176749826\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 3 q^{3} - 3 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
173.1		0		−1.73138	−	0.0482696i		0		1.17608	−	1.40160i		0		−2.53374				0		2.99534	+	0.167146i		0
173.2		0		−1.71798	−	0.220334i		0		−1.73689	+	2.06995i		0		1.02060				0		2.90291	+	0.757059i		0
173.3		0		−1.48936	+	0.884193i		0		−0.523365	+	0.623723i		0		−0.627048				0		1.43641	−	2.63377i		0
173.4		0		−1.33577	−	1.10259i		0		0.938512	−	1.11847i		0		3.72893				0		0.568575	+	2.94563i		0
173.5		0		−1.33193	+	1.10723i		0		2.73396	−	3.25821i		0		4.48977				0		0.548082	−	2.94951i		0
173.6		0		−1.06122	−	1.36887i		0		1.04221	−	1.24206i		0		−4.01697				0		−0.747632	+	2.90535i		0
173.7		0		−1.04255	+	1.38314i		0		−1.03490	+	1.23335i		0		−1.49222				0		−0.826167	−	2.88400i		0
173.8		0		−0.296047	+	1.70656i		0		0.160156	−	0.190867i		0		2.22789				0		−2.82471	−	1.01045i		0
173.9		0		−0.217417	−	1.71835i		0		−1.57261	+	1.87416i		0		2.65103				0		−2.90546	+	0.747197i		0
173.10		0		−0.206887	−	1.71965i		0		−1.75355	+	2.08980i		0		0.622286				0		−2.91440	+	0.711548i		0
173.11		0		0.0101717	+	1.73202i		0		2.40941	−	2.87142i		0		−4.42209				0		−2.99979	+	0.0352353i		0
173.12		0		0.653303	+	1.60412i		0		−2.67958	+	3.19339i		0		−2.96691				0		−2.14639	+	2.09595i		0
173.13		0		0.677405	−	1.59409i		0		1.55971	−	1.85879i		0		0.376193				0		−2.08224	−	2.15969i		0
173.14		0		0.908317	+	1.47477i		0		−0.151615	+	0.180688i		0		4.23406				0		−1.34992	+	2.67913i		0
173.15		0		1.04562	−	1.38082i		0		−0.827799	+	0.986533i		0		−4.63457				0		−0.813352	−	2.88764i		0
173.16		0		1.23831	+	1.21102i		0		0.0615564	−	0.0733601i		0		−3.02198				0		0.0668417	+	2.99926i		0
173.17		0		1.54696	−	0.779042i		0		−0.880894	+	1.04981i		0		−0.932536				0		1.78619	−	2.41030i		0
173.18		0		1.56169	+	0.749084i		0		0.835144	−	0.995286i		0		1.05537				0		1.87775	+	2.33967i		0
173.19		0		1.73037	−	0.0762044i		0		−2.24289	+	2.67297i		0		2.59492				0		2.98839	−	0.263724i		0
173.20		0		1.73204	+	0.00691491i		0		2.48734	−	2.96430i		0		−0.232357				0		2.99990	+	0.0239538i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.bd	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.2.cl.a	yes	120
9.d	odd	6	1		684.2.bv.a	✓	120
19.f	odd	18	1		684.2.bv.a	✓	120
171.bd	even	18	1	inner	684.2.cl.a	yes	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.2.bv.a	✓	120	9.d	odd	6	1
684.2.bv.a	✓	120	19.f	odd	18	1
684.2.cl.a	yes	120	1.a	even	1	1	trivial
684.2.cl.a	yes	120	171.bd	even	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(684, [\chi])\).