Embedded newform 338.3.f.h.249.2

• Label: 338.3.f.h.249.2
• Level: $338$
• Weight: $3$
• Character: 338.249
• Analytic conductor: $9.210$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	338.f (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.20983293538\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	8.0.612074651904.1
Defining polynomial:	\( x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			249.2
Root			\(3.90972 + 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	338.249
Dual form			338.3.f.h.319.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(2.38787 + 4.13592i) q^{3} +(1.73205 + 1.00000i) q^{4} +(-5.88981 + 5.88981i) q^{5} +(-1.74804 - 6.52379i) q^{6} +(-0.344179 + 0.0922225i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-6.90386 + 11.9578i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 6 q^{5} - 6 q^{6} - 8 q^{7} - 16 q^{8} - 42 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36603	−	0.366025i	−0.683013	−	0.183013i
							\(3\)	2.38787	+	4.13592i	0.795957	+	1.37864i	0.922230	+	0.386643i	\(0.126365\pi\)
−0.126272	+	0.991996i	\(0.540301\pi\)
\(5\)	−5.88981	+	5.88981i	−1.17796	+	1.17796i	−0.197700	+	0.980263i	\(0.563347\pi\)
							−0.980263	+	0.197700i	\(0.936653\pi\)
\(7\)	−0.344179	+	0.0922225i	−0.0491684	+	0.0131746i	−0.283319	−	0.959026i	\(-0.591436\pi\)
							0.234151	+	0.972200i	\(0.424769\pi\)
\(11\)	−0.191720	+	0.715507i	−0.0174290	+	0.0650461i	−0.974093	−	0.226149i	\(-0.927386\pi\)
							0.956664	+	0.291195i	\(0.0940530\pi\)
\(13\)		0			0
							\(17\)	2.49470	+	1.44031i	0.146747	+	0.0847244i	0.571576	−	0.820549i	\(-0.306332\pi\)
−0.424829	+	0.905274i	\(0.639666\pi\)
\(19\)	2.81216	+	10.4951i	0.148009	+	0.552375i	0.999603	+	0.0281743i	\(0.00896933\pi\)
							−0.851594	+	0.524201i	\(0.824364\pi\)
\(23\)	−19.8278	+	11.4476i	−0.862076	+	0.497720i	−0.864707	−	0.502277i	\(-0.832496\pi\)
							0.00263083	+	0.999997i	\(0.499163\pi\)
\(29\)	−15.2092	−	26.3432i	−0.524457	−	0.908386i	−0.999595	−	0.0284745i	\(-0.990935\pi\)
							0.475138	−	0.879911i	\(-0.342398\pi\)
\(31\)	14.8631	−	14.8631i	0.479456	−	0.479456i	−0.425502	−	0.904958i	\(-0.639902\pi\)
							0.904958	+	0.425502i	\(0.139902\pi\)
\(37\)	10.7928	−	40.2792i	0.291697	−	1.08863i	−0.652108	−	0.758126i	\(-0.726115\pi\)
							0.943805	−	0.330502i	\(-0.107218\pi\)
\(41\)	24.8893	+	6.66907i	0.607056	+	0.162660i	0.549237	−	0.835667i	\(-0.314919\pi\)
							0.0578194	+	0.998327i	\(0.481585\pi\)
\(43\)	25.3907	+	14.6593i	0.590480	+	0.340914i	0.765287	−	0.643689i	\(-0.222597\pi\)
							−0.174807	+	0.984603i	\(0.555930\pi\)
\(47\)	21.2000	+	21.2000i	0.451065	+	0.451065i	0.895708	−	0.444643i	\(-0.146670\pi\)
							−0.444643	+	0.895708i	\(0.646670\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.3.f.h.249.2		8
13.2	odd	12		338.3.d.f.99.1		8
13.3	even	3		338.3.d.g.239.1		8
13.4	even	6		338.3.f.i.19.2		8
13.5	odd	4		338.3.f.i.89.2		8
13.6	odd	12		338.3.f.j.319.2		8
13.7	odd	12	inner	338.3.f.h.319.2		8
13.8	odd	4		26.3.f.b.11.2	✓	8
13.9	even	3		26.3.f.b.19.2	yes	8
13.10	even	6		338.3.d.f.239.1		8
13.11	odd	12		338.3.d.g.99.1		8
13.12	even	2		338.3.f.j.249.2		8
39.8	even	4		234.3.bb.f.37.2		8
39.35	odd	6		234.3.bb.f.19.2		8
52.35	odd	6		208.3.bd.f.97.1		8
52.47	even	4		208.3.bd.f.193.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.3.f.b.11.2	✓	8	13.8	odd	4
26.3.f.b.19.2	yes	8	13.9	even	3
208.3.bd.f.97.1		8	52.35	odd	6
208.3.bd.f.193.1		8	52.47	even	4
234.3.bb.f.19.2		8	39.35	odd	6
234.3.bb.f.37.2		8	39.8	even	4
338.3.d.f.99.1		8	13.2	odd	12
338.3.d.f.239.1		8	13.10	even	6
338.3.d.g.99.1		8	13.11	odd	12
338.3.d.g.239.1		8	13.3	even	3
338.3.f.h.249.2		8	1.1	even	1	trivial
338.3.f.h.319.2		8	13.7	odd	12	inner
338.3.f.i.19.2		8	13.4	even	6
338.3.f.i.89.2		8	13.5	odd	4
338.3.f.j.249.2		8	13.12	even	2
338.3.f.j.319.2		8	13.6	odd	12