Embedded newform 361.3.f.e.307.2

• Label: 361.3.f.e.307.2
• Level: $361$
• Weight: $3$
• Character: 361.307
• Analytic conductor: $9.837$
• Analytic rank: $0$
• Dimension: $12$
• 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			307.2
Root			\(2.57727i\) of defining polynomial
Character	\(\chi\)	\(=\)	361.307
Dual form			361.3.f.e.127.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.21695 - 1.45030i) q^{2} +(0.664575 + 1.82591i) q^{3} +(0.0721792 + 0.409349i) q^{4} +(-0.485005 + 2.75060i) q^{5} +(3.45687 + 1.25820i) q^{6} +(-1.20796 - 2.09224i) q^{7} +(7.23987 + 4.17994i) q^{8} +(4.00213 - 3.35818i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} - 9 q^{3} - 9 q^{4} + 3 q^{5} + 9 q^{6} + 6 q^{7} - 9 q^{8} + 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.21695	−	1.45030i	0.608474	−	0.725151i	−0.370569	−	0.928805i	\(-0.620837\pi\)
							0.979043	+	0.203654i	\(0.0652818\pi\)
\(3\)	0.664575	+	1.82591i	0.221525	+	0.608635i	0.999814	−	0.0192707i	\(-0.00613443\pi\)
							−0.778289	+	0.627906i	\(0.783912\pi\)
\(5\)	−0.485005	+	2.75060i	−0.0970011	+	0.550121i	0.897115	+	0.441798i	\(0.145659\pi\)
							−0.994116	+	0.108323i	\(0.965452\pi\)
\(7\)	−1.20796	−	2.09224i	−0.172565	−	0.298892i	0.766751	−	0.641945i	\(-0.221872\pi\)
							−0.939316	+	0.343053i	\(0.888539\pi\)
\(11\)	−9.60360	+	16.6339i	−0.873055	+	1.51218i	−0.0142343	+	0.999899i	\(0.504531\pi\)
							−0.858821	+	0.512277i	\(0.828802\pi\)
\(13\)	−4.96498	+	13.6412i	−0.381921	+	1.04932i	0.588625	+	0.808406i	\(0.299670\pi\)
							−0.970546	+	0.240914i	\(0.922553\pi\)
\(17\)	−13.1432	−	11.0285i	−0.773132	−	0.648735i	0.168377	−	0.985723i	\(-0.446147\pi\)
							−0.941509	+	0.336988i	\(0.890592\pi\)
\(19\)		0			0
							\(23\)	−1.32476	−	7.51307i	−0.0575981	−	0.326655i	0.942371	−	0.334571i	\(-0.108591\pi\)
−0.999969	+	0.00791590i	\(0.997480\pi\)
\(29\)	5.29504	+	6.31038i	0.182587	+	0.217599i	0.849573	−	0.527472i	\(-0.176860\pi\)
							−0.666985	+	0.745071i	\(0.732416\pi\)
\(31\)	5.19837	−	3.00128i	0.167689	−	0.0968155i	−0.413806	−	0.910365i	\(-0.635801\pi\)
							0.581496	+	0.813549i	\(0.302468\pi\)
\(37\)			59.5153i			1.60852i	0.594277	+	0.804260i	\(0.297438\pi\)
							−0.594277	+	0.804260i	\(0.702562\pi\)
\(41\)	10.5649	+	29.0269i	0.257681	+	0.707974i	0.999309	+	0.0371621i	\(0.0118318\pi\)
							−0.741628	+	0.670811i	\(0.765946\pi\)
\(43\)	4.26712	−	24.2001i	0.0992354	−	0.562792i	−0.894131	−	0.447805i	\(-0.852206\pi\)
							0.993367	−	0.114988i	\(-0.0366828\pi\)
\(47\)	56.0539	−	47.0348i	1.19264	−	1.00074i	0.192826	−	0.981233i	\(-0.438235\pi\)
							0.999810	−	0.0195069i	\(-0.00620964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	361.3.f.e.307.2		12
19.2	odd	18		361.3.d.f.69.5		12
19.3	odd	18		361.3.d.d.293.2		12
19.4	even	9		361.3.f.b.333.2		12
19.5	even	9		361.3.b.c.360.4		12
19.6	even	9		361.3.f.c.127.1		12
19.7	even	3		19.3.f.a.14.2	✓	12
19.8	odd	6		361.3.f.b.116.2		12
19.9	even	9		361.3.f.g.262.1		12
19.10	odd	18		19.3.f.a.15.2	yes	12
19.11	even	3		361.3.f.f.116.1		12
19.12	odd	6		361.3.f.g.299.1		12
19.13	odd	18	inner	361.3.f.e.127.2		12
19.14	odd	18		361.3.b.c.360.9		12
19.15	odd	18		361.3.f.f.333.1		12
19.16	even	9		361.3.d.f.293.5		12
19.17	even	9		361.3.d.d.69.2		12
19.18	odd	2		361.3.f.c.307.1		12
57.26	odd	6		171.3.ba.b.109.1		12
57.29	even	18		171.3.ba.b.91.1		12
76.7	odd	6		304.3.z.a.33.2		12
76.67	even	18		304.3.z.a.129.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.f.a.14.2	✓	12	19.7	even	3
19.3.f.a.15.2	yes	12	19.10	odd	18
171.3.ba.b.91.1		12	57.29	even	18
171.3.ba.b.109.1		12	57.26	odd	6
304.3.z.a.33.2		12	76.7	odd	6
304.3.z.a.129.2		12	76.67	even	18
361.3.b.c.360.4		12	19.5	even	9
361.3.b.c.360.9		12	19.14	odd	18
361.3.d.d.69.2		12	19.17	even	9
361.3.d.d.293.2		12	19.3	odd	18
361.3.d.f.69.5		12	19.2	odd	18
361.3.d.f.293.5		12	19.16	even	9
361.3.f.b.116.2		12	19.8	odd	6
361.3.f.b.333.2		12	19.4	even	9
361.3.f.c.127.1		12	19.6	even	9
361.3.f.c.307.1		12	19.18	odd	2
361.3.f.e.127.2		12	19.13	odd	18	inner
361.3.f.e.307.2		12	1.1	even	1	trivial
361.3.f.f.116.1		12	19.11	even	3
361.3.f.f.333.1		12	19.15	odd	18
361.3.f.g.262.1		12	19.9	even	9
361.3.f.g.299.1		12	19.12	odd	6