Newform orbit 684.2.c.b

• Label: 684.2.c.b
• Level: $684$
• Weight: $2$
• Character orbit: 684.c
• Analytic conductor: $5.462$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.46176749826\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
647.1	−1.39577	−	0.227661i		0		1.89634	+	0.635524i			2.96037i		0			−	2.79268i	−2.50217	−	1.31877i		0		0.673961	−	4.13199i
647.2	−1.39577	+	0.227661i		0		1.89634	−	0.635524i		−	2.96037i		0				2.79268i	−2.50217	+	1.31877i		0		0.673961	+	4.13199i
647.3	−1.37167	−	0.344287i		0		1.76293	+	0.944493i			2.95253i		0				2.52587i	−2.09298	−	1.90248i		0		1.01652	−	4.04988i
647.4	−1.37167	+	0.344287i		0		1.76293	−	0.944493i		−	2.95253i		0			−	2.52587i	−2.09298	+	1.90248i		0		1.01652	+	4.04988i
647.5	−1.28221	−	0.596605i		0		1.28813	+	1.52995i		−	0.875520i		0				1.57625i	−0.738874	−	2.73021i		0		−0.522340	+	1.12260i
647.6	−1.28221	+	0.596605i		0		1.28813	−	1.52995i			0.875520i		0			−	1.57625i	−0.738874	+	2.73021i		0		−0.522340	−	1.12260i
647.7	−1.15329	−	0.818482i		0		0.660175	+	1.88790i		−	3.65888i		0			−	0.320351i	0.783836	−	2.71765i		0		−2.99473	+	4.21977i
647.8	−1.15329	+	0.818482i		0		0.660175	−	1.88790i			3.65888i		0				0.320351i	0.783836	+	2.71765i		0		−2.99473	−	4.21977i
647.9	−0.965364	−	1.03348i		0		−0.136143	+	1.99536i		−	0.0590006i		0				1.27229i	2.19358	−	1.78555i		0		−0.0609757	+	0.0569571i
647.10	−0.965364	+	1.03348i		0		−0.136143	−	1.99536i			0.0590006i		0			−	1.27229i	2.19358	+	1.78555i		0		−0.0609757	−	0.0569571i
647.11	−0.752610	−	1.19732i		0		−0.867158	+	1.80223i			2.94759i		0			−	2.73432i	2.81048	−	0.318110i		0		3.52922	−	2.21839i
647.12	−0.752610	+	1.19732i		0		−0.867158	−	1.80223i		−	2.94759i		0				2.73432i	2.81048	+	0.318110i		0		3.52922	+	2.21839i
647.13	−0.647279	−	1.25739i		0		−1.16206	+	1.62776i			1.07823i		0			−	3.32412i	2.79891	+	0.407546i		0		1.35575	−	0.697914i
647.14	−0.647279	+	1.25739i		0		−1.16206	−	1.62776i		−	1.07823i		0				3.32412i	2.79891	−	0.407546i		0		1.35575	+	0.697914i
647.15	−0.528104	−	1.31191i		0		−1.44221	+	1.38565i		−	3.80926i		0				3.88443i	2.57948	+	1.16029i		0		−4.99741	+	2.01168i
647.16	−0.528104	+	1.31191i		0		−1.44221	−	1.38565i			3.80926i		0			−	3.88443i	2.57948	−	1.16029i		0		−4.99741	−	2.01168i
647.17	0.528104	−	1.31191i		0		−1.44221	−	1.38565i		−	3.80926i		0			−	3.88443i	−2.57948	+	1.16029i		0		−4.99741	−	2.01168i
647.18	0.528104	+	1.31191i		0		−1.44221	+	1.38565i			3.80926i		0				3.88443i	−2.57948	−	1.16029i		0		−4.99741	+	2.01168i
647.19	0.647279	−	1.25739i		0		−1.16206	−	1.62776i			1.07823i		0				3.32412i	−2.79891	+	0.407546i		0		1.35575	+	0.697914i
647.20	0.647279	+	1.25739i		0		−1.16206	+	1.62776i		−	1.07823i		0			−	3.32412i	−2.79891	−	0.407546i		0		1.35575	−	0.697914i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.2.c.b	✓	32
3.b	odd	2	1	inner	684.2.c.b	✓	32
4.b	odd	2	1	inner	684.2.c.b	✓	32
12.b	even	2	1	inner	684.2.c.b	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.2.c.b	✓	32	1.a	even	1	1	trivial
684.2.c.b	✓	32	3.b	odd	2	1	inner
684.2.c.b	✓	32	4.b	odd	2	1	inner
684.2.c.b	✓	32	12.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^16 + 56*T5^14 + 1258*T5^12 + 14392*T5^10 + 87313*T5^8 + 261312*T5^6 + 305672*T5^4 + 115968*T5^2 + 400