Embedded newform 354.3.h.a.71.20

• Label: 354.3.h.a.71.20
• 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			71.20
Character	\(\chi\)	\(=\)	354.71
Dual form			354.3.h.a.5.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 - 1.34018i) q^{2} +(2.95183 - 0.535437i) q^{3} +(-1.59219 + 1.21035i) q^{4} +(-1.84046 + 0.733305i) q^{5} +(-2.05051 - 3.71421i) q^{6} +(-10.2065 - 4.72201i) q^{7} +(2.34106 + 1.58728i) q^{8} +(8.42661 - 3.16104i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

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
							\(3\)	2.95183	−	0.535437i	0.983944	−	0.178479i
\(5\)	−1.84046	+	0.733305i	−0.368091	+	0.146661i								−0.546847	−	0.837232i	\(-0.684172\pi\)
							0.178756	+	0.983893i	\(0.442793\pi\)
\(7\)	−10.2065	−	4.72201i	−1.45806	−	0.674572i	−0.478943	−	0.877846i	\(-0.658980\pi\)
							−0.979122	+	0.203274i	\(0.934842\pi\)
\(11\)	13.7546	−	3.81894i	1.25042	−	0.347176i	0.421643	−	0.906762i	\(-0.361454\pi\)
							0.828772	+	0.559586i	\(0.189040\pi\)
\(13\)	−5.71863	−	10.7865i	−0.439895	−	0.829730i	−0.999999	−	0.00130757i	\(-0.999584\pi\)
							0.560104	−	0.828422i	\(-0.310761\pi\)
\(17\)	−3.27733	−	7.08382i	−0.192784	−	0.416696i	0.786945	−	0.617023i	\(-0.211661\pi\)
							−0.979729	+	0.200327i	\(0.935799\pi\)
\(19\)	−23.9818	−	5.27880i	−1.26220	−	0.277832i	−0.467044	−	0.884234i	\(-0.654681\pi\)
							−0.795158	+	0.606402i	\(0.792612\pi\)
\(23\)	−39.7305	+	6.51348i	−1.72741	+	0.283195i	−0.941959	−	0.335727i	\(-0.891018\pi\)
							−0.785453	+	0.618922i	\(0.787570\pi\)
\(29\)	7.02588	−	20.8521i	0.242272	−	0.719037i	−0.755673	−	0.654950i	\(-0.772690\pi\)
							0.997944	−	0.0640873i	\(-0.0204136\pi\)
\(31\)	38.5985	−	8.49616i	1.24511	−	0.274070i	0.456925	−	0.889505i	\(-0.348951\pi\)
							0.788187	+	0.615436i	\(0.211020\pi\)
\(37\)	−26.7124	−	39.3978i	−0.721956	−	1.06481i	−0.994710	−	0.102722i	\(-0.967245\pi\)
							0.272754	−	0.962084i	\(-0.412065\pi\)
\(41\)	−42.1646	−	6.91253i	−1.02840	−	0.168598i	−0.376118	−	0.926572i	\(-0.622741\pi\)
							−0.652285	+	0.757973i	\(0.726190\pi\)
\(43\)	12.4232	−	44.7444i	0.288912	−	1.04057i	−0.666551	−	0.745459i	\(-0.732230\pi\)
							0.955464	−	0.295109i	\(-0.0953560\pi\)
\(47\)	38.5151	+	15.3458i	0.819470	+	0.326507i	0.741925	−	0.670482i	\(-0.233913\pi\)
							0.0775441	+	0.996989i	\(0.475292\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.71.20	yes	1120
3.2	odd	2	inner	354.3.h.a.71.34	yes	1120
59.5	even	29	inner	354.3.h.a.5.34	yes	1120
177.5	odd	58	inner	354.3.h.a.5.20	✓	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.20	✓	1120	177.5	odd	58	inner
354.3.h.a.5.34	yes	1120	59.5	even	29	inner
354.3.h.a.71.20	yes	1120	1.1	even	1	trivial
354.3.h.a.71.34	yes	1120	3.2	odd	2	inner