Embedded newform 354.3.h.a.5.3

• Label: 354.3.h.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	354.5
Dual form			354.3.h.a.71.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 + 1.34018i) q^{2} +(-2.91198 - 0.721372i) q^{3} +(-1.59219 - 1.21035i) q^{4} +(8.94495 + 3.56399i) q^{5} +(2.28171 - 3.57684i) q^{6} +(-3.17617 + 1.46945i) q^{7} +(2.34106 - 1.58728i) q^{8} +(7.95924 + 4.20124i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.91198	−	0.721372i	−0.970660	−	0.240457i
\(5\)	8.94495	+	3.56399i	1.78899	+	0.712799i								0.994404	+	0.105648i	\(0.0336918\pi\)
							0.794587	+	0.607151i	\(0.207688\pi\)
\(7\)	−3.17617	+	1.46945i	−0.453739	+	0.209922i	−0.633428	−	0.773801i	\(-0.718353\pi\)
							0.179689	+	0.983723i	\(0.442491\pi\)
\(11\)	−2.20232	−	0.611471i	−0.200211	−	0.0555882i	0.165975	−	0.986130i	\(-0.446923\pi\)
							−0.366186	+	0.930542i	\(0.619337\pi\)
\(13\)	7.92324	−	14.9448i	0.609480	−	1.14960i	−0.366228	−	0.930525i	\(-0.619351\pi\)
							0.975708	−	0.219076i	\(-0.0703043\pi\)
\(17\)	0.0763348	−	0.164995i	0.00449028	−	0.00970559i	−0.905317	−	0.424736i	\(-0.860367\pi\)
							0.909808	+	0.415030i	\(0.136229\pi\)
\(19\)	7.06586	−	1.55531i	0.371888	−	0.0818587i	−0.0250943	−	0.999685i	\(-0.507989\pi\)
							0.396982	+	0.917826i	\(0.370058\pi\)
\(23\)	16.6052	+	2.72229i	0.721967	+	0.118360i	0.511553	−	0.859252i	\(-0.329071\pi\)
							0.210414	+	0.977612i	\(0.432519\pi\)
\(29\)	11.9052	+	35.3332i	0.410523	+	1.21839i	0.931160	+	0.364610i	\(0.118798\pi\)
							−0.520638	+	0.853778i	\(0.674306\pi\)
\(31\)	−2.36671	−	0.520953i	−0.0763456	−	0.0168049i	0.176633	−	0.984277i	\(-0.443479\pi\)
							−0.252979	+	0.967472i	\(0.581410\pi\)
\(37\)	−27.1036	+	39.9748i	−0.732529	+	1.08040i	0.260834	+	0.965384i	\(0.416002\pi\)
							−0.993364	+	0.115016i	\(0.963308\pi\)
\(41\)	−13.5648	+	2.22383i	−0.330848	+	0.0542397i	−0.324917	−	0.945743i	\(-0.605336\pi\)
							−0.00593119	+	0.999982i	\(0.501888\pi\)
\(43\)	17.3748	+	62.5782i	0.404064	+	1.45531i	0.833656	+	0.552284i	\(0.186244\pi\)
							−0.429592	+	0.903023i	\(0.641343\pi\)
\(47\)	75.9976	−	30.2802i	1.61697	−	0.644260i	0.626182	−	0.779677i	\(-0.284617\pi\)
							0.990789	+	0.135417i	\(0.0432375\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.5.3	✓	1120
3.2	odd	2	inner	354.3.h.a.5.26	yes	1120
59.12	even	29	inner	354.3.h.a.71.26	yes	1120
177.71	odd	58	inner	354.3.h.a.71.3	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.3	✓	1120	1.1	even	1	trivial
354.3.h.a.5.26	yes	1120	3.2	odd	2	inner
354.3.h.a.71.3	yes	1120	177.71	odd	58	inner
354.3.h.a.71.26	yes	1120	59.12	even	29	inner