Newform orbit 378.3.o.a

• Label: 378.3.o.a
• Level: $378$
• Weight: $3$
• Character orbit: 378.o
• Analytic conductor: $10.300$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.2997539928\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} - 2 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
181.1	−0.707107	+	1.22474i		0		−1.00000	−	1.73205i	−6.78854	+	3.91937i		0		3.39779	+	6.12005i	2.82843				0			−	11.0856i
181.2	−0.707107	+	1.22474i		0		−1.00000	−	1.73205i	−4.98058	+	2.87554i		0		−6.71501	−	1.97703i	2.82843				0			−	8.13325i
181.3	−0.707107	+	1.22474i		0		−1.00000	−	1.73205i	−2.71575	+	1.56794i		0		2.86183	−	6.38827i	2.82843				0			−	4.43480i
181.4	−0.707107	+	1.22474i		0		−1.00000	−	1.73205i	−2.31738	+	1.33794i		0		6.01390	+	3.58231i	2.82843				0			−	3.78427i
181.5	−0.707107	+	1.22474i		0		−1.00000	−	1.73205i	2.31738	−	1.33794i		0		−6.10933	−	3.41704i	2.82843				0				3.78427i
181.6	−0.707107	+	1.22474i		0		−1.00000	−	1.73205i	2.71575	−	1.56794i		0		4.10149	−	5.67255i	2.82843				0				4.43480i
181.7	−0.707107	+	1.22474i		0		−1.00000	−	1.73205i	4.98058	−	2.87554i		0		5.06966	+	4.82685i	2.82843				0				8.13325i
181.8	−0.707107	+	1.22474i		0		−1.00000	−	1.73205i	6.78854	−	3.91937i		0		−6.99901	+	0.117453i	2.82843				0				11.0856i
181.9	0.707107	−	1.22474i		0		−1.00000	−	1.73205i	−8.58770	+	4.95811i		0		6.88643	+	1.25583i	−2.82843				0				14.0237i
181.10	0.707107	−	1.22474i		0		−1.00000	−	1.73205i	−4.10393	+	2.36941i		0		1.32093	−	6.87424i	−2.82843				0				6.70169i
181.11	0.707107	−	1.22474i		0		−1.00000	−	1.73205i	−2.08306	+	1.20266i		0		−3.20945	+	6.22089i	−2.82843				0				3.40163i
181.12	0.707107	−	1.22474i		0		−1.00000	−	1.73205i	−1.19747	+	0.691358i		0		2.27133	+	6.62126i	−2.82843				0				1.95546i
181.13	0.707107	−	1.22474i		0		−1.00000	−	1.73205i	1.19747	−	0.691358i		0		−6.86984	+	1.34360i	−2.82843				0			−	1.95546i
181.14	0.707107	−	1.22474i		0		−1.00000	−	1.73205i	2.08306	−	1.20266i		0		−3.78272	+	5.88991i	−2.82843				0			−	3.40163i
181.15	0.707107	−	1.22474i		0		−1.00000	−	1.73205i	4.10393	−	2.36941i		0		5.29280	−	4.58108i	−2.82843				0			−	6.70169i
181.16	0.707107	−	1.22474i		0		−1.00000	−	1.73205i	8.58770	−	4.95811i		0		−4.53080	−	5.33591i	−2.82843				0			−	14.0237i
307.1	−0.707107	−	1.22474i		0		−1.00000	+	1.73205i	−6.78854	−	3.91937i		0		3.39779	−	6.12005i	2.82843				0				11.0856i
307.2	−0.707107	−	1.22474i		0		−1.00000	+	1.73205i	−4.98058	−	2.87554i		0		−6.71501	+	1.97703i	2.82843				0				8.13325i
307.3	−0.707107	−	1.22474i		0		−1.00000	+	1.73205i	−2.71575	−	1.56794i		0		2.86183	+	6.38827i	2.82843				0				4.43480i
307.4	−0.707107	−	1.22474i		0		−1.00000	+	1.73205i	−2.31738	−	1.33794i		0		6.01390	−	3.58231i	2.82843				0				3.78427i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
9.c	even	3	1	inner
63.l	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	378.3.o.a		32
3.b	odd	2	1		126.3.o.a	✓	32
7.b	odd	2	1	inner	378.3.o.a		32
9.c	even	3	1	inner	378.3.o.a		32
9.c	even	3	1		1134.3.c.d		16
9.d	odd	6	1		126.3.o.a	✓	32
9.d	odd	6	1		1134.3.c.e		16
21.c	even	2	1		126.3.o.a	✓	32
63.l	odd	6	1	inner	378.3.o.a		32
63.l	odd	6	1		1134.3.c.d		16
63.o	even	6	1		126.3.o.a	✓	32
63.o	even	6	1		1134.3.c.e		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
126.3.o.a	✓	32	3.b	odd	2	1
126.3.o.a	✓	32	9.d	odd	6	1
126.3.o.a	✓	32	21.c	even	2	1
126.3.o.a	✓	32	63.o	even	6	1
378.3.o.a		32	1.a	even	1	1	trivial
378.3.o.a		32	7.b	odd	2	1	inner
378.3.o.a		32	9.c	even	3	1	inner
378.3.o.a		32	63.l	odd	6	1	inner
1134.3.c.d		16	9.c	even	3	1
1134.3.c.d		16	63.l	odd	6	1
1134.3.c.e		16	9.d	odd	6	1
1134.3.c.e		16	63.o	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(378, [\chi])\).