Embedded newform 731.2.n.a.608.49

• Label: 731.2.n.a.608.49
• Level: $731$
• Weight: $2$
• Character: 731.608
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
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			608.49
Character	\(\chi\)	\(=\)	731.608
Dual form			731.2.n.a.208.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.73334i q^{2} +(2.03902 - 0.546353i) q^{3} -1.00446 q^{4} +(3.75820 - 1.00701i) q^{5} +(0.947015 + 3.53431i) q^{6} +(0.573555 + 0.153684i) q^{7} +1.72560i q^{8} +(1.26102 - 0.728048i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			1.73334i			1.22566i	0.790216	+	0.612828i	\(0.209968\pi\)
							−0.790216	+	0.612828i	\(0.790032\pi\)
\(3\)	2.03902	−	0.546353i	1.17723	−	0.315437i	0.383401	−	0.923582i	\(-0.374753\pi\)
							0.793826	+	0.608145i	\(0.208086\pi\)
\(5\)	3.75820	−	1.00701i	1.68072	−	0.450347i	0.712748	−	0.701420i	\(-0.247450\pi\)
							0.967968	+	0.251074i	\(0.0807836\pi\)
\(7\)	0.573555	+	0.153684i	0.216783	+	0.0580870i	0.365576	−	0.930781i	\(-0.380872\pi\)
							−0.148793	+	0.988868i	\(0.547539\pi\)
\(11\)	−1.44312	+	1.44312i	−0.435116	+	0.435116i	−0.890365	−	0.455248i	\(-0.849551\pi\)
							0.455248	+	0.890365i	\(0.349551\pi\)
\(13\)	1.48111	+	2.56535i	0.410785	+	0.711501i	0.994976	−	0.100115i	\(-0.0319212\pi\)
							−0.584190	+	0.811617i	\(0.698588\pi\)
\(17\)	−3.35715	−	2.39365i	−0.814228	−	0.580545i
							\(19\)	−6.58291	−	3.80064i	−1.51022	−	0.871928i	−0.999929	−	0.0119274i	\(-0.996203\pi\)
−0.510294	−	0.860000i	\(-0.670463\pi\)
\(23\)	−1.41278	−	5.27257i	−0.294585	−	1.09941i	−0.941546	−	0.336883i	\(-0.890627\pi\)
							0.646962	−	0.762523i	\(-0.276039\pi\)
\(29\)	−1.46507	+	5.46773i	−0.272057	+	1.01533i	0.685730	+	0.727856i	\(0.259483\pi\)
							−0.957788	+	0.287476i	\(0.907184\pi\)
\(31\)	7.66923	−	2.05496i	1.37743	−	0.369083i	0.507246	−	0.861801i	\(-0.330663\pi\)
							0.870189	+	0.492719i	\(0.163997\pi\)
\(37\)	−1.95198	+	0.523030i	−0.320903	+	0.0859857i	−0.415675	−	0.909513i	\(-0.636455\pi\)
							0.0947718	+	0.995499i	\(0.469788\pi\)
\(41\)	5.44379	−	5.44379i	0.850177	−	0.850177i	−0.139978	−	0.990155i	\(-0.544703\pi\)
							0.990155	+	0.139978i	\(0.0447031\pi\)
\(43\)	−6.20944	−	2.10782i	−0.946930	−	0.321440i
							\(47\)	−2.34592			−0.342188			−0.171094	−	0.985255i	\(-0.554730\pi\)
−0.171094	+	0.985255i	\(0.554730\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.608.49	yes	256
17.4	even	4	inner	731.2.n.a.565.16	yes	256
43.36	even	3	inner	731.2.n.a.251.49	yes	256
731.208	even	12	inner	731.2.n.a.208.16	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.16	✓	256	731.208	even	12	inner
731.2.n.a.251.49	yes	256	43.36	even	3	inner
731.2.n.a.565.16	yes	256	17.4	even	4	inner
731.2.n.a.608.49	yes	256	1.1	even	1	trivial