Embedded newform 336.2.bs.a.19.8

• Label: 336.2.bs.a.19.8
• Level: $336$
• Weight: $2$
• Character: 336.19
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.bs (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			19.8
Character	\(\chi\)	\(=\)	336.19
Dual form			336.2.bs.a.283.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09426 + 0.895880i) q^{2} +(0.258819 - 0.965926i) q^{3} +(0.394798 - 1.96065i) q^{4} +(-2.90593 + 0.778643i) q^{5} +(0.582139 + 1.28884i) q^{6} +(2.58569 + 0.560554i) q^{7} +(1.32449 + 2.49914i) q^{8} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 4 q^{4} - 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09426	+	0.895880i	−0.773757	+	0.633483i
							\(3\)	0.258819	−	0.965926i	0.149429	−	0.557678i
\(5\)	−2.90593	+	0.778643i	−1.29957	+	0.348220i								−0.841289	−	0.540586i	\(-0.818203\pi\)
							−0.458285	+	0.888805i	\(0.651536\pi\)
\(7\)	2.58569	+	0.560554i	0.977298	+	0.211869i
							\(11\)	−0.213523	+	0.796878i	−0.0643795	+	0.240268i	−0.990616	−	0.136676i	\(-0.956358\pi\)
0.926236	+	0.376943i	\(0.123025\pi\)
\(13\)	1.43628	−	1.43628i	0.398353	−	0.398353i	−0.479299	−	0.877652i	\(-0.659109\pi\)
							0.877652	+	0.479299i	\(0.159109\pi\)
\(17\)	6.92432	−	3.99776i	1.67940	−	0.969599i	0.717348	−	0.696715i	\(-0.245356\pi\)
							0.962047	−	0.272884i	\(-0.0879776\pi\)
\(19\)	5.11403	−	1.37030i	1.17324	−	0.314368i	0.380998	−	0.924576i	\(-0.375581\pi\)
							0.792242	+	0.610208i	\(0.208914\pi\)
\(23\)	2.40531	−	4.16612i	0.501542	−	0.868697i	−0.498456	−	0.866915i	\(-0.666099\pi\)
							0.999998	−	0.00178199i	\(-0.000567224\pi\)
\(29\)	−3.35248	−	3.35248i	−0.622539	−	0.622539i	0.323641	−	0.946180i	\(-0.395093\pi\)
							−0.946180	+	0.323641i	\(0.895093\pi\)
\(31\)	0.102170	+	0.176964i	0.0183503	+	0.0317837i	0.875055	−	0.484024i	\(-0.160825\pi\)
							−0.856704	+	0.515808i	\(0.827492\pi\)
\(37\)	1.86150	+	6.94722i	0.306029	+	1.14212i	0.932056	+	0.362315i	\(0.118013\pi\)
							−0.626027	+	0.779802i	\(0.715320\pi\)
\(41\)	10.6633			1.66532			0.832661	−	0.553783i	\(-0.186816\pi\)
							0.832661	+	0.553783i	\(0.186816\pi\)
\(43\)	−5.29413	−	5.29413i	−0.807348	−	0.807348i	0.176884	−	0.984232i	\(-0.443398\pi\)
							−0.984232	+	0.176884i	\(0.943398\pi\)
\(47\)	−5.93316	+	10.2765i	−0.865440	+	1.49899i	0.00116919	+	0.999999i	\(0.499628\pi\)
							−0.866609	+	0.498987i	\(0.833705\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bs.a.19.8	✓	128
7.3	odd	6	inner	336.2.bs.a.115.14	yes	128
16.11	odd	4	inner	336.2.bs.a.187.14	yes	128
112.59	even	12	inner	336.2.bs.a.283.8	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bs.a.19.8	✓	128	1.1	even	1	trivial
336.2.bs.a.115.14	yes	128	7.3	odd	6	inner
336.2.bs.a.187.14	yes	128	16.11	odd	4	inner
336.2.bs.a.283.8	yes	128	112.59	even	12	inner