Embedded newform 361.3.f.b.299.1

• Label: 361.3.f.b.299.1
• Level: $361$
• Weight: $3$
• Character: 361.299
• Analytic conductor: $9.837$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	361.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.83653754341\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 24x^{10} + 216x^{8} + 905x^{6} + 1770x^{4} + 1395x^{2} + 361 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			299.1
Root			\(0.728740i\) of defining polynomial
Character	\(\chi\)	\(=\)	361.299
Dual form			361.3.f.b.262.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.249244 - 0.684791i) q^{2} +(-3.28847 - 0.579846i) q^{3} +(2.65736 - 2.22979i) q^{4} +(-4.98985 - 4.18698i) q^{5} +(0.422557 + 2.39644i) q^{6} +(-5.47943 - 9.49065i) q^{7} +(-4.71370 - 2.72146i) q^{8} +(2.02057 + 0.735427i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} - 9 q^{3} + 9 q^{4} + 3 q^{5} + 27 q^{6} + 6 q^{7} + 9 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.249244	−	0.684791i	−0.124622	−	0.342396i	0.861655	−	0.507494i	\(-0.169428\pi\)
							−0.986277	+	0.165098i	\(0.947206\pi\)
\(3\)	−3.28847	−	0.579846i	−1.09616	−	0.193282i	−0.403807	−	0.914844i	\(-0.632313\pi\)
							−0.692349	+	0.721562i	\(0.743424\pi\)
\(5\)	−4.98985	−	4.18698i	−0.997970	−	0.837396i	−0.0112680	−	0.999937i	\(-0.503587\pi\)
							−0.986702	+	0.162540i	\(0.948031\pi\)
\(7\)	−5.47943	−	9.49065i	−0.782776	−	1.35581i	−0.930319	−	0.366752i	\(-0.880470\pi\)
							0.147543	−	0.989056i	\(-0.452864\pi\)
\(11\)	−2.46116	+	4.26286i	−0.223742	+	0.387533i	−0.955941	−	0.293558i	\(-0.905161\pi\)
							0.732199	+	0.681090i	\(0.238494\pi\)
\(13\)	6.95678	−	1.22667i	0.535137	−	0.0943591i	0.100453	−	0.994942i	\(-0.467971\pi\)
							0.434685	+	0.900583i	\(0.356860\pi\)
\(17\)	5.14864	−	1.87395i	0.302861	−	0.110233i	−0.186119	−	0.982527i	\(-0.559591\pi\)
							0.488981	+	0.872295i	\(0.337369\pi\)
\(19\)		0			0
							\(23\)	15.8715	−	13.3177i	0.690064	−	0.579032i	−0.228864	−	0.973458i	\(-0.573501\pi\)
0.918928	+	0.394426i	\(0.129057\pi\)
\(29\)	−8.71139	+	23.9343i	−0.300393	+	0.825322i	0.694039	+	0.719937i	\(0.255830\pi\)
							−0.994432	+	0.105385i	\(0.966393\pi\)
\(31\)	4.26339	−	2.46147i	0.137529	−	0.0794022i	−0.429657	−	0.902992i	\(-0.641366\pi\)
							0.567186	+	0.823590i	\(0.308032\pi\)
\(37\)		−	25.2454i		−	0.682309i	−0.940007	−	0.341155i	\(-0.889182\pi\)
							0.940007	−	0.341155i	\(-0.110818\pi\)
\(41\)	−70.5832	−	12.4457i	−1.72154	−	0.303554i	−0.776406	−	0.630233i	\(-0.782959\pi\)
							−0.945135	+	0.326679i	\(0.894071\pi\)
\(43\)	20.2735	+	17.0115i	0.471476	+	0.395615i	0.847333	−	0.531062i	\(-0.178207\pi\)
							−0.375857	+	0.926678i	\(0.622651\pi\)
\(47\)	43.0055	+	15.6527i	0.915010	+	0.333036i	0.756251	−	0.654281i	\(-0.227029\pi\)
							0.158758	+	0.987317i	\(0.449251\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	361.3.f.b.299.1		12
19.2	odd	18		361.3.b.c.360.6		12
19.3	odd	18		361.3.d.d.69.3		12
19.4	even	9		361.3.f.f.262.2		12
19.5	even	9		361.3.d.d.293.3		12
19.6	even	9		361.3.f.e.333.1		12
19.7	even	3		361.3.f.c.116.2		12
19.8	odd	6		19.3.f.a.3.2	✓	12
19.9	even	9		19.3.f.a.13.2	yes	12
19.10	odd	18		361.3.f.g.127.1		12
19.11	even	3		361.3.f.g.307.1		12
19.12	odd	6		361.3.f.e.116.1		12
19.13	odd	18		361.3.f.c.333.2		12
19.14	odd	18		361.3.d.f.293.4		12
19.15	odd	18	inner	361.3.f.b.262.1		12
19.16	even	9		361.3.d.f.69.4		12
19.17	even	9		361.3.b.c.360.7		12
19.18	odd	2		361.3.f.f.299.2		12
57.8	even	6		171.3.ba.b.136.1		12
57.47	odd	18		171.3.ba.b.127.1		12
76.27	even	6		304.3.z.a.193.2		12
76.47	odd	18		304.3.z.a.241.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.f.a.3.2	✓	12	19.8	odd	6
19.3.f.a.13.2	yes	12	19.9	even	9
171.3.ba.b.127.1		12	57.47	odd	18
171.3.ba.b.136.1		12	57.8	even	6
304.3.z.a.193.2		12	76.27	even	6
304.3.z.a.241.2		12	76.47	odd	18
361.3.b.c.360.6		12	19.2	odd	18
361.3.b.c.360.7		12	19.17	even	9
361.3.d.d.69.3		12	19.3	odd	18
361.3.d.d.293.3		12	19.5	even	9
361.3.d.f.69.4		12	19.16	even	9
361.3.d.f.293.4		12	19.14	odd	18
361.3.f.b.262.1		12	19.15	odd	18	inner
361.3.f.b.299.1		12	1.1	even	1	trivial
361.3.f.c.116.2		12	19.7	even	3
361.3.f.c.333.2		12	19.13	odd	18
361.3.f.e.116.1		12	19.12	odd	6
361.3.f.e.333.1		12	19.6	even	9
361.3.f.f.262.2		12	19.4	even	9
361.3.f.f.299.2		12	19.18	odd	2
361.3.f.g.127.1		12	19.10	odd	18
361.3.f.g.307.1		12	19.11	even	3