Embedded newform 547.2.o.a.4.5

• Label: 547.2.o.a.4.5
• Level: $547$
• Weight: $2$
• Character: 547.4
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $6480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.o (of order \(273\), degree \(144\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(6480\)
Relative dimension:	\(45\) over \(\Q(\zeta_{273})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{273}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.46368 - 0.0283525i) q^{2} +(-0.550970 + 2.41396i) q^{3} +(4.06944 + 0.0936762i) q^{4} +(-1.20693 + 2.26781i) q^{5} +(1.42585 - 5.93159i) q^{6} +(0.105582 + 0.260175i) q^{7} +(-5.09840 - 0.176082i) q^{8} +(-2.82071 - 1.35838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/547\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{273}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.46368	−	0.0283525i	−1.74208	−	0.0200482i	−0.865391	−	0.501097i	\(-0.832930\pi\)
							−0.876694	+	0.481049i	\(0.840256\pi\)
\(3\)	−0.550970	+	2.41396i	−0.318103	+	1.39370i	0.522774	+	0.852471i	\(0.324897\pi\)
							−0.840877	+	0.541227i	\(0.817960\pi\)
\(5\)	−1.20693	+	2.26781i	−0.539755	+	1.01419i	0.452709	+	0.891658i	\(0.350458\pi\)
							−0.992464	+	0.122535i	\(0.960898\pi\)
\(7\)	0.105582	+	0.260175i	0.0399063	+	0.0983369i	0.945256	−	0.326331i	\(-0.105812\pi\)
							−0.905349	+	0.424667i	\(0.860391\pi\)
\(11\)	2.90178	+	3.02107i	0.874918	+	0.910886i	0.996659	−	0.0816755i	\(-0.0260271\pi\)
							−0.121741	+	0.992562i	\(0.538848\pi\)
\(13\)	−3.88004	−	0.584822i	−1.07613	−	0.162200i	−0.413019	−	0.910722i	\(-0.635526\pi\)
							−0.663110	+	0.748522i	\(0.730764\pi\)
\(17\)	−0.818051	+	4.85700i	−0.198407	+	1.17800i	0.691680	+	0.722204i	\(0.256871\pi\)
							−0.890086	+	0.455792i	\(0.849356\pi\)
\(19\)	−2.44198	+	1.31771i	−0.560229	+	0.302304i	−0.730298	−	0.683128i	\(-0.760619\pi\)
							0.170070	+	0.985432i	\(0.445601\pi\)
\(23\)	−3.18309	−	0.745814i	−0.663720	−	0.155513i	−0.117394	−	0.993085i	\(-0.537454\pi\)
							−0.546326	+	0.837572i	\(0.683974\pi\)
\(29\)	5.21072	+	2.96857i	0.967606	+	0.551249i	0.894978	−	0.446110i	\(-0.147191\pi\)
							0.0726279	+	0.997359i	\(0.476861\pi\)
\(31\)	1.61737	+	0.581660i	0.290489	+	0.104469i	0.479602	−	0.877486i	\(-0.340781\pi\)
							−0.189113	+	0.981955i	\(0.560561\pi\)
\(37\)	4.30012	−	3.46990i	0.706936	−	0.570448i	−0.205999	−	0.978552i	\(-0.566044\pi\)
							0.912935	+	0.408104i	\(0.133810\pi\)
\(41\)	0.317043	+	0.549135i	0.0495138	+	0.0857605i	0.889720	−	0.456506i	\(-0.150899\pi\)
							−0.840206	+	0.542267i	\(0.817566\pi\)
\(43\)	0.700166	−	7.13521i	0.106774	−	1.08811i	−0.779960	−	0.625829i	\(-0.784761\pi\)
							0.886734	−	0.462280i	\(-0.152968\pi\)
\(47\)	−13.3807	+	1.08020i	−1.95178	+	0.157564i	−0.992268	−	0.124117i	\(-0.960390\pi\)
							−0.959508	+	0.281680i	\(0.909108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.5	✓	6480
547.137	even	273	inner	547.2.o.a.137.5	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.5	✓	6480	1.1	even	1	trivial
547.2.o.a.137.5	yes	6480	547.137	even	273	inner