Embedded newform 354.3.h.a.5.16

• Label: 354.3.h.a.5.16
• Level: $354$
• Weight: $3$
• Character: 354.5
• 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			5.16
Character	\(\chi\)	\(=\)	354.5
Dual form			354.3.h.a.71.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 + 1.34018i) q^{2} +(2.35308 - 1.86092i) q^{3} +(-1.59219 - 1.21035i) q^{4} +(-1.37348 - 0.547244i) q^{5} +(1.43141 + 3.99388i) q^{6} +(-4.55846 + 2.10897i) q^{7} +(2.34106 - 1.58728i) q^{8} +(2.07397 - 8.75778i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{29}\right)\)	\(-1\)

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
							\(3\)	2.35308	−	1.86092i	0.784360	−	0.620306i
\(5\)	−1.37348	−	0.547244i	−0.274696	−	0.109449i								0.228723	−	0.973492i	\(-0.426545\pi\)
							−0.503418	+	0.864043i	\(0.667924\pi\)
\(7\)	−4.55846	+	2.10897i	−0.651208	+	0.301281i	−0.717541	−	0.696516i	\(-0.754732\pi\)
							0.0663324	+	0.997798i	\(0.478870\pi\)
\(11\)	−4.49940	−	1.24925i	−0.409037	−	0.113568i	0.0569143	−	0.998379i	\(-0.481874\pi\)
							−0.465951	+	0.884811i	\(0.654288\pi\)
\(13\)	3.32957	−	6.28024i	0.256121	−	0.483095i	−0.721791	−	0.692111i	\(-0.756681\pi\)
							0.977912	+	0.209015i	\(0.0670259\pi\)
\(17\)	13.6316	−	29.4641i	0.801857	−	1.73318i	0.131279	−	0.991345i	\(-0.458092\pi\)
							0.670578	−	0.741839i	\(-0.266046\pi\)
\(19\)	−16.1828	+	3.56210i	−0.851725	+	0.187479i	−0.619319	−	0.785140i	\(-0.712591\pi\)
							−0.232407	+	0.972619i	\(0.574660\pi\)
\(23\)	28.7112	+	4.70696i	1.24831	+	0.204650i	0.749467	−	0.662041i	\(-0.230310\pi\)
							0.498844	+	0.866692i	\(0.333758\pi\)
\(29\)	−10.9858	−	32.6046i	−0.378820	−	1.12430i	−0.952323	−	0.305093i	\(-0.901312\pi\)
							0.573503	−	0.819203i	\(-0.305584\pi\)
\(31\)	−57.4191	−	12.6389i	−1.85223	−	0.407706i	−0.857877	−	0.513855i	\(-0.828217\pi\)
							−0.994350	+	0.106149i	\(0.966148\pi\)
\(37\)	17.3623	−	25.6075i	0.469251	−	0.692094i	−0.517252	−	0.855833i	\(-0.673045\pi\)
							0.986504	+	0.163739i	\(0.0523555\pi\)
\(41\)	−21.2638	+	3.48603i	−0.518630	+	0.0850251i	−0.415412	−	0.909634i	\(-0.636362\pi\)
							−0.103219	+	0.994659i	\(0.532914\pi\)
\(43\)	12.0388	+	43.3599i	0.279972	+	1.00837i	0.961099	+	0.276203i	\(0.0890763\pi\)
							−0.681127	+	0.732165i	\(0.738510\pi\)
\(47\)	58.4602	−	23.2927i	1.24383	−	0.495589i	0.346963	−	0.937879i	\(-0.387213\pi\)
							0.896872	+	0.442290i	\(0.145834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.5.16	✓	1120
3.2	odd	2	inner	354.3.h.a.5.30	yes	1120
59.12	even	29	inner	354.3.h.a.71.30	yes	1120
177.71	odd	58	inner	354.3.h.a.71.16	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.16	✓	1120	1.1	even	1	trivial
354.3.h.a.5.30	yes	1120	3.2	odd	2	inner
354.3.h.a.71.16	yes	1120	177.71	odd	58	inner
354.3.h.a.71.30	yes	1120	59.12	even	29	inner