Embedded newform 336.3.bv.a.325.4

• Label: 336.3.bv.a.325.4
• 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			325.4
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94751 + 0.455206i) q^{2} +(1.67303 - 0.448288i) q^{3} +(3.58557 - 1.77304i) q^{4} +(-6.86280 - 1.83888i) q^{5} +(-3.05418 + 1.63462i) q^{6} +(6.65252 + 2.17808i) q^{7} +(-6.17584 + 5.08518i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94751	+	0.455206i	−0.973754	+	0.227603i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	−6.86280	−	1.83888i	−1.37256	−	0.367776i								−0.504147	−	0.863618i	\(-0.668193\pi\)
							−0.868413	+	0.495842i	\(0.834860\pi\)
\(7\)	6.65252	+	2.17808i	0.950359	+	0.311154i
							\(11\)	−9.67680	+	2.59289i	−0.879709	+	0.235717i	−0.670281	−	0.742107i	\(-0.733827\pi\)
−0.209427	+	0.977824i	\(0.567160\pi\)
\(13\)	8.11667	+	8.11667i	0.624359	+	0.624359i	0.946643	−	0.322284i	\(-0.104451\pi\)
							−0.322284	+	0.946643i	\(0.604451\pi\)
\(17\)	17.0873	+	9.86538i	1.00514	+	0.580317i	0.909764	−	0.415125i	\(-0.136262\pi\)
							0.0953736	+	0.995442i	\(0.469595\pi\)
\(19\)	1.49783	−	5.58999i	0.0788333	−	0.294210i	−0.915242	−	0.402905i	\(-0.868001\pi\)
							0.994075	+	0.108695i	\(0.0346672\pi\)
\(23\)	38.8030	−	22.4029i	1.68709	−	0.974041i	0.730358	−	0.683065i	\(-0.239353\pi\)
							0.956730	−	0.290976i	\(-0.0939799\pi\)
\(29\)	10.8563	−	10.8563i	0.374354	−	0.374354i	−0.494706	−	0.869060i	\(-0.664724\pi\)
							0.869060	+	0.494706i	\(0.164724\pi\)
\(31\)	−20.6649	−	11.9309i	−0.666610	−	0.384867i	0.128181	−	0.991751i	\(-0.459086\pi\)
							−0.794791	+	0.606884i	\(0.792419\pi\)
\(37\)	−0.325077	+	1.21320i	−0.00878587	+	0.0327893i	−0.970180	−	0.242387i	\(-0.922070\pi\)
							0.961394	+	0.275176i	\(0.0887363\pi\)
\(41\)	41.1304			1.00318			0.501590	−	0.865105i	\(-0.332748\pi\)
							0.501590	+	0.865105i	\(0.332748\pi\)
\(43\)	15.3514	+	15.3514i	0.357008	+	0.357008i	0.862709	−	0.505701i	\(-0.168766\pi\)
							−0.505701	+	0.862709i	\(0.668766\pi\)
\(47\)	73.8734	−	42.6508i	1.57177	−	0.907464i	0.575822	−	0.817575i	\(-0.304682\pi\)
							0.995952	−	0.0898886i	\(-0.0286511\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.4	yes	256
7.5	odd	6	inner	336.3.bv.a.229.47	yes	256
16.13	even	4	inner	336.3.bv.a.157.47	yes	256
112.61	odd	12	inner	336.3.bv.a.61.4	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.4	✓	256	112.61	odd	12	inner
336.3.bv.a.157.47	yes	256	16.13	even	4	inner
336.3.bv.a.229.47	yes	256	7.5	odd	6	inner
336.3.bv.a.325.4	yes	256	1.1	even	1	trivial