Newform orbit 378.3.s.e

• Label: 378.3.s.e
• Level: $378$
• Weight: $3$
• Character orbit: 378.s
• Analytic conductor: $10.300$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.2997539928\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} + 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−1.22474	+	0.707107i		0		1.00000	−	1.73205i	−7.50889	+	4.33526i		0		−4.86680	+	5.03132i			2.82843i		0		6.13099	−	10.6192i
53.2	−1.22474	+	0.707107i		0		1.00000	−	1.73205i	−7.00128	+	4.04219i		0		6.42484	−	2.77874i			2.82843i		0		5.71652	−	9.90131i
53.3	−1.22474	+	0.707107i		0		1.00000	−	1.73205i	−0.470941	+	0.271898i		0		−4.45944	−	5.39568i			2.82843i		0		0.384522	−	0.666011i
53.4	−1.22474	+	0.707107i		0		1.00000	−	1.73205i	3.06283	−	1.76833i		0		−4.16770	+	5.62408i			2.82843i		0		−2.50079	+	4.33150i
53.5	−1.22474	+	0.707107i		0		1.00000	−	1.73205i	4.15701	−	2.40005i		0		3.96228	+	5.77064i			2.82843i		0		−3.39419	+	5.87891i
53.6	−1.22474	+	0.707107i		0		1.00000	−	1.73205i	5.31178	−	3.06676i		0		5.10682	−	4.78753i			2.82843i		0		−4.33705	+	7.51200i
53.7	1.22474	−	0.707107i		0		1.00000	−	1.73205i	−5.31178	+	3.06676i		0		5.10682	−	4.78753i		−	2.82843i		0		−4.33705	+	7.51200i
53.8	1.22474	−	0.707107i		0		1.00000	−	1.73205i	−4.15701	+	2.40005i		0		3.96228	+	5.77064i		−	2.82843i		0		−3.39419	+	5.87891i
53.9	1.22474	−	0.707107i		0		1.00000	−	1.73205i	−3.06283	+	1.76833i		0		−4.16770	+	5.62408i		−	2.82843i		0		−2.50079	+	4.33150i
53.10	1.22474	−	0.707107i		0		1.00000	−	1.73205i	0.470941	−	0.271898i		0		−4.45944	−	5.39568i		−	2.82843i		0		0.384522	−	0.666011i
53.11	1.22474	−	0.707107i		0		1.00000	−	1.73205i	7.00128	−	4.04219i		0		6.42484	−	2.77874i		−	2.82843i		0		5.71652	−	9.90131i
53.12	1.22474	−	0.707107i		0		1.00000	−	1.73205i	7.50889	−	4.33526i		0		−4.86680	+	5.03132i		−	2.82843i		0		6.13099	−	10.6192i
107.1	−1.22474	−	0.707107i		0		1.00000	+	1.73205i	−7.50889	−	4.33526i		0		−4.86680	−	5.03132i		−	2.82843i		0		6.13099	+	10.6192i
107.2	−1.22474	−	0.707107i		0		1.00000	+	1.73205i	−7.00128	−	4.04219i		0		6.42484	+	2.77874i		−	2.82843i		0		5.71652	+	9.90131i
107.3	−1.22474	−	0.707107i		0		1.00000	+	1.73205i	−0.470941	−	0.271898i		0		−4.45944	+	5.39568i		−	2.82843i		0		0.384522	+	0.666011i
107.4	−1.22474	−	0.707107i		0		1.00000	+	1.73205i	3.06283	+	1.76833i		0		−4.16770	−	5.62408i		−	2.82843i		0		−2.50079	−	4.33150i
107.5	−1.22474	−	0.707107i		0		1.00000	+	1.73205i	4.15701	+	2.40005i		0		3.96228	−	5.77064i		−	2.82843i		0		−3.39419	−	5.87891i
107.6	−1.22474	−	0.707107i		0		1.00000	+	1.73205i	5.31178	+	3.06676i		0		5.10682	+	4.78753i		−	2.82843i		0		−4.33705	−	7.51200i
107.7	1.22474	+	0.707107i		0		1.00000	+	1.73205i	−5.31178	−	3.06676i		0		5.10682	+	4.78753i			2.82843i		0		−4.33705	−	7.51200i
107.8	1.22474	+	0.707107i		0		1.00000	+	1.73205i	−4.15701	−	2.40005i		0		3.96228	−	5.77064i			2.82843i		0		−3.39419	−	5.87891i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
21.h	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.s.e	✓	24
3.b	odd	2	1	inner	378.3.s.e	✓	24
7.c	even	3	1	inner	378.3.s.e	✓	24
21.h	odd	6	1	inner	378.3.s.e	✓	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
378.3.s.e	✓	24	1.a	even	1	1	trivial
378.3.s.e	✓	24	3.b	odd	2	1	inner
378.3.s.e	✓	24	7.c	even	3	1	inner
378.3.s.e	✓	24	21.h	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^24 - 214*T5^22 + 28911*T5^20 - 2405842*T5^18 + 146207827*T5^16 - 6104445282*T5^14 + 188994834036*T5^12 - 3958652583882*T5^10 + 59725028164887*T5^8 - 524300446058508*T5^6 + 2992739959267902*T5^4 - 883480054030308*T5^2 + 248155780267521