Embedded newform 336.3.bv.a.325.26

• Label: 336.3.bv.a.325.26
• 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			325.26
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.615355 + 1.90298i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(-3.24268 - 2.34202i) q^{4} +(-5.34622 - 1.43252i) q^{5} +(0.176425 - 3.45961i) q^{6} +(5.50340 - 4.32581i) q^{7} +(6.45221 - 4.72958i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.615355	+	1.90298i	−0.307677	+	0.951491i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	−5.34622	−	1.43252i	−1.06924	−	0.286503i								−0.319061	−	0.947734i	\(-0.603368\pi\)
							−0.750182	+	0.661231i	\(0.770034\pi\)
\(7\)	5.50340	−	4.32581i	0.786200	−	0.617972i
							\(11\)	−5.05324	+	1.35401i	−0.459385	+	0.123092i	−0.481087	−	0.876673i	\(-0.659758\pi\)
0.0217024	+	0.999764i	\(0.493091\pi\)
\(13\)	9.27346	+	9.27346i	0.713343	+	0.713343i	0.967233	−	0.253890i	\(-0.0817101\pi\)
							−0.253890	+	0.967233i	\(0.581710\pi\)
\(17\)	−1.68910	−	0.975200i	−0.0993586	−	0.0573647i	0.449497	−	0.893282i	\(-0.351603\pi\)
							−0.548856	+	0.835917i	\(0.684936\pi\)
\(19\)	−1.00314	+	3.74377i	−0.0527968	+	0.197040i	−0.987287	−	0.158949i	\(-0.949189\pi\)
							0.934490	+	0.355989i	\(0.115856\pi\)
\(23\)	−17.2587	+	9.96434i	−0.750380	+	0.433232i	−0.825831	−	0.563917i	\(-0.809294\pi\)
							0.0754513	+	0.997149i	\(0.475960\pi\)
\(29\)	−9.03154	+	9.03154i	−0.311432	+	0.311432i	−0.845464	−	0.534032i	\(-0.820676\pi\)
							0.534032	+	0.845464i	\(0.320676\pi\)
\(31\)	38.5034	+	22.2299i	1.24204	+	0.717094i	0.969510	−	0.245052i	\(-0.0788049\pi\)
							0.272534	+	0.962146i	\(0.412138\pi\)
\(37\)	−6.93699	+	25.8892i	−0.187486	+	0.699708i	0.806598	+	0.591100i	\(0.201306\pi\)
							−0.994085	+	0.108608i	\(0.965361\pi\)
\(41\)	63.7106			1.55392			0.776959	−	0.629552i	\(-0.216761\pi\)
							0.776959	+	0.629552i	\(0.216761\pi\)
\(43\)	42.1358	+	42.1358i	0.979902	+	0.979902i	0.999802	−	0.0199002i	\(-0.00633486\pi\)
							−0.0199002	+	0.999802i	\(0.506335\pi\)
\(47\)	4.79821	−	2.77025i	0.102089	−	0.0589414i	−0.448086	−	0.893990i	\(-0.647894\pi\)
							0.550175	+	0.835049i	\(0.314561\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.26	yes	256
7.5	odd	6	inner	336.3.bv.a.229.61	yes	256
16.13	even	4	inner	336.3.bv.a.157.61	yes	256
112.61	odd	12	inner	336.3.bv.a.61.26	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.26	✓	256	112.61	odd	12	inner
336.3.bv.a.157.61	yes	256	16.13	even	4	inner
336.3.bv.a.229.61	yes	256	7.5	odd	6	inner
336.3.bv.a.325.26	yes	256	1.1	even	1	trivial