Newform orbit 684.2.bp.a

• Label: 684.2.bp.a
• Level: $684$
• Weight: $2$
• Character orbit: 684.bp
• Analytic conductor: $5.462$
• Analytic rank: $0$
• Dimension: $120$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.bp (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.46176749826\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 120 * q + 6 * q^3 + 6 * q^9 + 3 * q^13 - 30 * q^15 - 9 * q^17 + 3 * q^19 + 12 * q^23 + 3 * q^27 + 18 * q^29 - 30 * q^33 + 18 * q^35 - 30 * q^39 - 6 * q^41 - 6 * q^43 + 36 * q^45 + 18 * q^47 - 60 * q^49 + 3 * q^51 + 24 * q^53 + 42 * q^57 - 33 * q^59 + 21 * q^61 - 6 * q^63 + 156 * q^65 - 60 * q^67 + 18 * q^69 + 42 * q^71 + 3 * q^73 - 6 * q^79 - 78 * q^81 - 12 * q^83 - 30 * q^87 + 18 * q^89 + 12 * q^91 - 30 * q^93 - 6 * q^95 - 18 * q^97 - 3 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
25.1		0		−1.73167	−	0.0365126i		0		−0.880504	+	0.738830i		0		−2.04100	−	3.53512i		0		2.99733	+	0.126455i		0
25.2		0		−1.72285	+	0.178326i		0		0.132874	−	0.111494i		0		0.0149516	+	0.0258970i		0		2.93640	−	0.614456i		0
25.3		0		−1.56681	+	0.738318i		0		2.91102	−	2.44264i		0		−0.122987	−	0.213019i		0		1.90977	−	2.31360i		0
25.4		0		−1.30541	−	1.13838i		0		2.74415	−	2.30262i		0		1.96091	+	3.39640i		0		0.408196	+	2.97210i		0
25.5		0		−1.15839	−	1.28768i		0		0.437491	−	0.367098i		0		0.365440	+	0.632960i		0		−0.316246	+	2.98328i		0
25.6		0		−1.06145	+	1.36870i		0		−2.64995	+	2.22357i		0		0.0672552	+	0.116489i		0		−0.746653	−	2.90560i		0
25.7		0		−0.973793	+	1.43238i		0		−0.607196	+	0.509498i		0		1.87368	+	3.24530i		0		−1.10345	−	2.78969i		0
25.8		0		−0.826604	−	1.52208i		0		−1.43329	+	1.20267i		0		0.340213	+	0.589266i		0		−1.63345	+	2.51631i		0
25.9		0		−0.150334	+	1.72551i		0		1.01376	−	0.850642i		0		−1.60643	−	2.78241i		0		−2.95480	−	0.518808i		0
25.10		0		−0.116346	−	1.72814i		0		−1.03386	+	0.867509i		0		−1.25184	−	2.16826i		0		−2.97293	+	0.402124i		0
25.11		0		0.457634	−	1.67050i		0		2.15968	−	1.81219i		0		−2.04635	−	3.54438i		0		−2.58114	−	1.52895i		0
25.12		0		0.716443	+	1.57693i		0		0.952957	−	0.799626i		0		2.20520	+	3.81951i		0		−1.97342	+	2.25956i		0
25.13		0		0.756282	+	1.55822i		0		−1.44074	+	1.20892i		0		−1.78700	−	3.09517i		0		−1.85607	+	2.35690i		0
25.14		0		0.935085	−	1.45795i		0		1.54593	−	1.29719i		0		1.14317	+	1.98002i		0		−1.25123	−	2.72661i		0
25.15		0		0.985048	−	1.42467i		0		−2.83316	+	2.37730i		0		2.45902	+	4.25915i		0		−1.05936	−	2.80673i		0
25.16		0		1.49486	−	0.874868i		0		−2.98951	+	2.50849i		0		−1.57275	−	2.72408i		0		1.46921	−	2.61561i		0
25.17		0		1.54019	+	0.792349i		0		3.08832	−	2.59141i		0		−1.26670	−	2.19399i		0		1.74437	+	2.44073i		0
25.18		0		1.54150	+	0.789789i		0		−0.943148	+	0.791395i		0		0.627040	+	1.08607i		0		1.75247	+	2.43493i		0
25.19		0		1.67956	−	0.423188i		0		1.40319	−	1.17742i		0		0.664031	+	1.15014i		0		2.64182	−	1.42154i		0
25.20		0		1.68070	+	0.418633i		0		−1.57803	+	1.32413i		0		0.147792	+	0.255983i		0		2.64949	+	1.40719i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.v	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.2.bp.a	✓	120
9.c	even	3	1		684.2.bq.a	yes	120
19.e	even	9	1		684.2.bq.a	yes	120
171.v	even	9	1	inner	684.2.bp.a	✓	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.2.bp.a	✓	120	1.a	even	1	1	trivial
684.2.bp.a	✓	120	171.v	even	9	1	inner
684.2.bq.a	yes	120	9.c	even	3	1
684.2.bq.a	yes	120	19.e	even	9	1

Hecke kernels

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