Embedded newform 731.2.bl.a.9.4

• Label: 731.2.bl.a.9.4
• Level: $731$
• Weight: $2$
• Character: 731.9
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $3072$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bl (of order \(168\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(3072\)
Relative dimension:	\(64\) over \(\Q(\zeta_{168})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{168}]$

Embedding invariants

Embedding label			9.4
Character	\(\chi\)	\(=\)	731.9
Dual form			731.2.bl.a.325.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.54211 - 0.286426i) q^{2} +(2.35858 + 1.42118i) q^{3} +(4.43040 + 1.01121i) q^{4} +(1.38702 - 0.908135i) q^{5} +(-5.58870 - 4.28836i) q^{6} +(3.61781 + 0.476294i) q^{7} +(-6.14365 - 2.14976i) q^{8} +(2.14134 + 4.05160i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3072 q - 40 q^{2} - 52 q^{3} - 44 q^{5} - 24 q^{6} - 24 q^{7} - 72 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.54211	−	0.286426i	−1.79754	−	0.202534i	−0.851010	−	0.525150i	\(-0.824009\pi\)
							−0.946530	+	0.322615i	\(0.895438\pi\)
\(3\)	2.35858	+	1.42118i	1.36173	+	0.820520i	0.993997	−	0.109412i	\(-0.0348967\pi\)
							0.367731	+	0.929932i	\(0.380135\pi\)
\(5\)	1.38702	−	0.908135i	0.620294	−	0.406130i	−0.198595	−	0.980082i	\(-0.563638\pi\)
							0.818889	+	0.573951i	\(0.194590\pi\)
\(7\)	3.61781	+	0.476294i	1.36740	+	0.180022i	0.778228	−	0.627982i	\(-0.216119\pi\)
							0.589176	+	0.808005i	\(0.299452\pi\)
\(11\)	−1.08142	−	1.52412i	−0.326060	−	0.459538i	0.619495	−	0.785000i	\(-0.287337\pi\)
							−0.945555	+	0.325462i	\(0.894480\pi\)
\(13\)	0.343246	+	0.0257227i	0.0951993	+	0.00713420i	0.122245	−	0.992500i	\(-0.460991\pi\)
							−0.0270453	+	0.999634i	\(0.508610\pi\)
\(17\)	3.53697	+	2.11892i	0.857842	+	0.513914i
							\(19\)	−3.32043	+	6.28256i	−0.761758	+	1.44132i	0.131284	+	0.991345i	\(0.458090\pi\)
−0.893042	+	0.449972i	\(0.851434\pi\)
\(23\)	1.95863	−	4.26958i	0.408403	−	0.890270i	−0.588262	−	0.808670i	\(-0.700188\pi\)
							0.996665	−	0.0815993i	\(-0.0260028\pi\)
\(29\)	1.79143	−	7.22338i	0.332660	−	1.34135i	−0.535278	−	0.844676i	\(-0.679793\pi\)
							0.867938	−	0.496672i	\(-0.165445\pi\)
\(31\)	0.708026	−	0.735013i	0.127165	−	0.132012i	−0.653592	−	0.756847i	\(-0.726739\pi\)
							0.780757	+	0.624835i	\(0.214834\pi\)
\(37\)	−0.201581	−	1.53116i	−0.0331397	−	0.251721i	0.966854	−	0.255331i	\(-0.0821843\pi\)
							−0.999993	+	0.00360972i	\(0.998851\pi\)
\(41\)	−3.29558	+	5.96290i	−0.514683	+	0.931248i	0.483746	+	0.875208i	\(0.339276\pi\)
							−0.998429	+	0.0560396i	\(0.982153\pi\)
\(43\)	−6.54339	−	0.428996i	−0.997858	−	0.0654212i
							\(47\)	−7.93874	−	1.81196i	−1.15798	−	0.264302i	−0.399962	−	0.916532i	\(-0.630977\pi\)
−0.758022	+	0.652229i	\(0.773834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bl.a.9.4	✓	3072
17.2	even	8	inner	731.2.bl.a.138.61	yes	3072
43.24	even	21	inner	731.2.bl.a.196.61	yes	3072
731.325	even	168	inner	731.2.bl.a.325.4	yes	3072

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bl.a.9.4	✓	3072	1.1	even	1	trivial
731.2.bl.a.138.61	yes	3072	17.2	even	8	inner
731.2.bl.a.196.61	yes	3072	43.24	even	21	inner
731.2.bl.a.325.4	yes	3072	731.325	even	168	inner