Embedded newform 731.2.u.a.52.5

• Label: 731.2.u.a.52.5
• Level: $731$
• Weight: $2$
• Character: 731.52
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.u (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(29\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			52.5
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.a.239.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27132 - 1.59418i) q^{2} +(0.488205 + 0.0735850i) q^{3} +(-0.480123 + 2.10356i) q^{4} +(-1.73427 - 1.18240i) q^{5} +(-0.503355 - 0.871836i) q^{6} +(1.29743 - 2.24721i) q^{7} +(0.289629 - 0.139478i) q^{8} +(-2.63379 - 0.812416i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 2 q^{2} + 5 q^{3} - 54 q^{4} + 3 q^{5} + 8 q^{6} - 71 q^{7} - 26 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27132	−	1.59418i	−0.898956	−	1.12726i	−0.991312	−	0.131529i	\(-0.958011\pi\)
							0.0923560	−	0.995726i	\(-0.470560\pi\)
\(3\)	0.488205	+	0.0735850i	0.281865	+	0.0424843i	0.288453	−	0.957494i	\(-0.406859\pi\)
							−0.00658821	+	0.999978i	\(0.502097\pi\)
\(5\)	−1.73427	−	1.18240i	−0.775588	−	0.528787i	0.109625	−	0.993973i	\(-0.465035\pi\)
							−0.885213	+	0.465186i	\(0.845987\pi\)
\(7\)	1.29743	−	2.24721i	0.490382	−	0.849366i	−0.509557	−	0.860437i	\(-0.670191\pi\)
							0.999939	+	0.0110706i	\(0.00352395\pi\)
\(11\)	1.23200	+	5.39775i	0.371462	+	1.62748i	0.722676	+	0.691187i	\(0.242912\pi\)
							−0.351213	+	0.936295i	\(0.614231\pi\)
\(13\)	−0.299875	+	4.00156i	−0.0831705	+	1.10983i	0.786958	+	0.617007i	\(0.211655\pi\)
							−0.870128	+	0.492826i	\(0.835964\pi\)
\(17\)	0.826239	−	0.563320i	0.200392	−	0.136625i
							\(19\)	−2.71506	+	0.837485i	−0.622877	+	0.192132i	−0.590100	−	0.807330i	\(-0.700912\pi\)
−0.0327778	+	0.999463i	\(0.510435\pi\)
\(23\)	−5.67596	+	5.26652i	−1.18352	+	1.09814i	−0.190319	+	0.981722i	\(0.560952\pi\)
							−0.993200	+	0.116423i	\(0.962857\pi\)
\(29\)	−0.566783	+	0.0854288i	−0.105249	+	0.0158637i	−0.201456	−	0.979498i	\(-0.564567\pi\)
							0.0962066	+	0.995361i	\(0.469329\pi\)
\(31\)	2.33133	−	5.94012i	0.418719	−	1.06688i	−0.553510	−	0.832843i	\(-0.686712\pi\)
							0.972228	−	0.234035i	\(-0.0751930\pi\)
\(37\)	1.75641	+	3.04220i	0.288753	+	0.500135i	0.973512	−	0.228635i	\(-0.0734261\pi\)
							−0.684759	+	0.728769i	\(0.740093\pi\)
\(41\)	2.46042	+	3.08527i	0.384253	+	0.481838i	0.935913	−	0.352231i	\(-0.114577\pi\)
							−0.551660	+	0.834069i	\(0.686005\pi\)
\(43\)	5.92213	+	2.81574i	0.903117	+	0.429396i
							\(47\)	−1.65375	+	7.24554i	−0.241224	+	1.05687i	0.698681	+	0.715433i	\(0.253771\pi\)
−0.939905	+	0.341437i	\(0.889086\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.a.52.5	✓	348
43.24	even	21	inner	731.2.u.a.239.5	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.a.52.5	✓	348	1.1	even	1	trivial
731.2.u.a.239.5	yes	348	43.24	even	21	inner