Embedded newform 354.3.h.a.71.12

• Label: 354.3.h.a.71.12
• Level: $354$
• Weight: $3$
• Character: 354.71
• Analytic conductor: $9.646$
• Analytic rank: $0$
• Dimension: $1120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	354.h (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.64580135835\)
Analytic rank:	\(0\)
Dimension:	\(1120\)
Relative dimension:	\(40\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			71.12
Character	\(\chi\)	\(=\)	354.71
Dual form			354.3.h.a.5.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 - 1.34018i) q^{2} +(0.924357 - 2.85404i) q^{3} +(-1.59219 + 1.21035i) q^{4} +(-0.606021 + 0.241461i) q^{5} +(-4.24235 + 0.0499646i) q^{6} +(6.70675 + 3.10287i) q^{7} +(2.34106 + 1.58728i) q^{8} +(-7.29113 - 5.27631i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{26}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.451561	−	1.34018i	−0.225780	−	0.670092i
							\(3\)	0.924357	−	2.85404i	0.308119	−	0.951348i
\(5\)	−0.606021	+	0.241461i	−0.121204	+	0.0482922i								−0.429952	−	0.902852i	\(-0.641469\pi\)
							0.308748	+	0.951144i	\(0.400090\pi\)
\(7\)	6.70675	+	3.10287i	0.958108	+	0.443268i	0.835684	−	0.549210i	\(-0.185071\pi\)
							0.122423	+	0.992478i	\(0.460933\pi\)
\(11\)	−16.9218	+	4.69831i	−1.53834	+	0.427119i	−0.930305	−	0.366788i	\(-0.880458\pi\)
							−0.608040	+	0.793907i	\(0.708044\pi\)
\(13\)	−8.35854	−	15.7659i	−0.642965	−	1.21276i	−0.964053	−	0.265711i	\(-0.914393\pi\)
							0.321088	−	0.947049i	\(-0.395951\pi\)
\(17\)	−7.23501	−	15.6382i	−0.425589	−	0.919896i	−0.995189	−	0.0979736i	\(-0.968764\pi\)
							0.569600	−	0.821922i	\(-0.307098\pi\)
\(19\)	−23.1206	−	5.08922i	−1.21687	−	0.267854i	−0.440282	−	0.897860i	\(-0.645121\pi\)
							−0.776590	+	0.630006i	\(0.783052\pi\)
\(23\)	2.68360	−	0.439954i	0.116678	−	0.0191284i	−0.103160	−	0.994665i	\(-0.532895\pi\)
							0.219838	+	0.975536i	\(0.429447\pi\)
\(29\)	9.66368	−	28.6808i	0.333230	−	0.988992i	−0.641791	−	0.766880i	\(-0.721808\pi\)
							0.975021	−	0.222113i	\(-0.0712952\pi\)
\(31\)	11.7046	−	2.57639i	0.377569	−	0.0831092i	−0.0221313	−	0.999755i	\(-0.507045\pi\)
							0.399700	+	0.916646i	\(0.369114\pi\)
\(37\)	13.2262	+	19.5072i	0.357466	+	0.527222i	0.963321	−	0.268353i	\(-0.0864792\pi\)
							−0.605855	+	0.795575i	\(0.707169\pi\)
\(41\)	−26.4294	−	4.33287i	−0.644618	−	0.105680i	−0.169400	−	0.985547i	\(-0.554183\pi\)
							−0.475219	+	0.879868i	\(0.657631\pi\)
\(43\)	5.94003	−	21.3941i	0.138140	−	0.497536i	−0.861813	−	0.507226i	\(-0.830671\pi\)
							0.999953	+	0.00969024i	\(0.00308455\pi\)
\(47\)	64.0593	+	25.5235i	1.36296	+	0.543054i	0.933027	−	0.359805i	\(-0.117157\pi\)
							0.429936	+	0.902860i	\(0.358536\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.71.12	yes	1120
3.2	odd	2	inner	354.3.h.a.71.38	yes	1120
59.5	even	29	inner	354.3.h.a.5.38	yes	1120
177.5	odd	58	inner	354.3.h.a.5.12	✓	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.12	✓	1120	177.5	odd	58	inner
354.3.h.a.5.38	yes	1120	59.5	even	29	inner
354.3.h.a.71.12	yes	1120	1.1	even	1	trivial
354.3.h.a.71.38	yes	1120	3.2	odd	2	inner