Embedded newform 731.2.n.a.208.3

• Label: 731.2.n.a.208.3
• Level: $731$
• Weight: $2$
• Character: 731.208
• 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			208.3
Character	\(\chi\)	\(=\)	731.208
Dual form			731.2.n.a.608.62

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.67476i q^{2} +(-2.11059 - 0.565531i) q^{3} -5.15436 q^{4} +(2.13934 + 0.573235i) q^{5} +(-1.51266 + 5.64533i) q^{6} +(0.737688 - 0.197663i) q^{7} +8.43716i q^{8} +(1.53669 + 0.887206i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.67476i		−	1.89134i	−0.325123	−	0.945672i	\(-0.605406\pi\)
							0.325123	−	0.945672i	\(-0.394594\pi\)
\(3\)	−2.11059	−	0.565531i	−1.21855	−	0.326509i	−0.408438	−	0.912786i	\(-0.633927\pi\)
							−0.810111	+	0.586277i	\(0.800593\pi\)
\(5\)	2.13934	+	0.573235i	0.956744	+	0.256359i	0.703221	−	0.710971i	\(-0.251744\pi\)
							0.253522	+	0.967330i	\(0.418411\pi\)
\(7\)	0.737688	−	0.197663i	0.278820	−	0.0747095i	−0.116699	−	0.993167i	\(-0.537231\pi\)
							0.395519	+	0.918458i	\(0.370565\pi\)
\(11\)	4.20765	+	4.20765i	1.26865	+	1.26865i	0.946784	+	0.321870i	\(0.104311\pi\)
							0.321870	+	0.946784i	\(0.395689\pi\)
\(13\)	2.31304	−	4.00631i	0.641522	−	1.11115i	−0.343571	−	0.939127i	\(-0.611636\pi\)
							0.985093	−	0.172023i	\(-0.0550302\pi\)
\(17\)	1.61094	+	3.79538i	0.390709	+	0.920514i
							\(19\)	3.64127	−	2.10229i	0.835365	−	0.482298i	−0.0203209	−	0.999794i	\(-0.506469\pi\)
0.855686	+	0.517495i	\(0.173135\pi\)
\(23\)	−1.79603	+	6.70287i	−0.374498	+	1.39764i	0.479580	+	0.877498i	\(0.340789\pi\)
							−0.854077	+	0.520146i	\(0.825878\pi\)
\(29\)	−0.860862	−	3.21278i	−0.159858	−	0.596598i	−0.998640	−	0.0521311i	\(-0.983399\pi\)
							0.838782	−	0.544467i	\(-0.183268\pi\)
\(31\)	3.46139	+	0.927476i	0.621683	+	0.166580i	0.555893	−	0.831254i	\(-0.312376\pi\)
							0.0657904	+	0.997833i	\(0.479043\pi\)
\(37\)	−5.62546	−	1.50734i	−0.924819	−	0.247805i	−0.235175	−	0.971953i	\(-0.575566\pi\)
							−0.689644	+	0.724148i	\(0.742233\pi\)
\(41\)	7.28041	+	7.28041i	1.13701	+	1.13701i	0.988984	+	0.148025i	\(0.0472915\pi\)
							0.148025	+	0.988984i	\(0.452708\pi\)
\(43\)	2.18587	−	6.18239i	0.333342	−	0.942806i
							\(47\)	3.34796			0.488350			0.244175	−	0.969731i	\(-0.421483\pi\)
0.244175	+	0.969731i	\(0.421483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.208.3	✓	256
17.13	even	4	inner	731.2.n.a.251.62	yes	256
43.6	even	3	inner	731.2.n.a.565.3	yes	256
731.608	even	12	inner	731.2.n.a.608.62	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.3	✓	256	1.1	even	1	trivial
731.2.n.a.251.62	yes	256	17.13	even	4	inner
731.2.n.a.565.3	yes	256	43.6	even	3	inner
731.2.n.a.608.62	yes	256	731.608	even	12	inner