Embedded newform 336.3.bv.a.325.2

• Label: 336.3.bv.a.325.2
• Level: $336$
• Weight: $3$
• Character: 336.325
• Analytic conductor: $9.155$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.15533688251\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			325.2
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99949 + 0.0450665i) q^{2} +(1.67303 - 0.448288i) q^{3} +(3.99594 - 0.180220i) q^{4} +(5.27205 + 1.41264i) q^{5} +(-3.32501 + 0.971746i) q^{6} +(6.64245 - 2.20860i) q^{7} +(-7.98172 + 0.540432i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99949	+	0.0450665i	−0.999746	+	0.0225333i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	5.27205	+	1.41264i	1.05441	+	0.282528i								0.744073	−	0.668099i	\(-0.232892\pi\)
							0.310336	+	0.950627i	\(0.399558\pi\)
\(7\)	6.64245	−	2.20860i	0.948921	−	0.315514i
							\(11\)	11.6751	−	3.12834i	1.06138	−	0.284395i	0.314430	−	0.949281i	\(-0.398187\pi\)
0.746945	+	0.664886i	\(0.231520\pi\)
\(13\)	7.75719	+	7.75719i	0.596707	+	0.596707i	0.939435	−	0.342728i	\(-0.111351\pi\)
							−0.342728	+	0.939435i	\(0.611351\pi\)
\(17\)	−16.8017	−	9.70049i	−0.988338	−	0.570617i	−0.0835607	−	0.996503i	\(-0.526629\pi\)
							−0.904777	+	0.425886i	\(0.859963\pi\)
\(19\)	0.915502	−	3.41670i	0.0481843	−	0.179826i	−0.937640	−	0.347608i	\(-0.886994\pi\)
							0.985824	+	0.167782i	\(0.0536605\pi\)
\(23\)	−26.0144	+	15.0194i	−1.13106	+	0.653017i	−0.944201	−	0.329370i	\(-0.893164\pi\)
							−0.186858	+	0.982387i	\(0.559830\pi\)
\(29\)	−24.5609	+	24.5609i	−0.846929	+	0.846929i	−0.989749	−	0.142820i	\(-0.954383\pi\)
							0.142820	+	0.989749i	\(0.454383\pi\)
\(31\)	16.1467	+	9.32232i	0.520862	+	0.300720i	0.737287	−	0.675579i	\(-0.236106\pi\)
							−0.216425	+	0.976299i	\(0.569440\pi\)
\(37\)	15.9985	−	59.7074i	0.432393	−	1.61371i	−0.314836	−	0.949146i	\(-0.601950\pi\)
							0.747229	−	0.664567i	\(-0.231384\pi\)
\(41\)	1.84886			0.0450943			0.0225471	−	0.999746i	\(-0.492822\pi\)
							0.0225471	+	0.999746i	\(0.492822\pi\)
\(43\)	13.4144	+	13.4144i	0.311964	+	0.311964i	0.845670	−	0.533706i	\(-0.179201\pi\)
							−0.533706	+	0.845670i	\(0.679201\pi\)
\(47\)	−3.95656	+	2.28432i	−0.0841821	+	0.0486025i	−0.541500	−	0.840701i	\(-0.682143\pi\)
							0.457318	+	0.889303i	\(0.348810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.2	yes	256
7.5	odd	6	inner	336.3.bv.a.229.45	yes	256
16.13	even	4	inner	336.3.bv.a.157.45	yes	256
112.61	odd	12	inner	336.3.bv.a.61.2	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.2	✓	256	112.61	odd	12	inner
336.3.bv.a.157.45	yes	256	16.13	even	4	inner
336.3.bv.a.229.45	yes	256	7.5	odd	6	inner
336.3.bv.a.325.2	yes	256	1.1	even	1	trivial