Embedded newform 354.3.h.a.5.4

• Label: 354.3.h.a.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	354.5
Dual form			354.3.h.a.71.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 + 1.34018i) q^{2} +(-2.24157 - 1.99383i) q^{3} +(-1.59219 - 1.21035i) q^{4} +(-1.72634 - 0.687837i) q^{5} +(3.68430 - 2.10378i) q^{6} +(6.44007 - 2.97949i) q^{7} +(2.34106 - 1.58728i) q^{8} +(1.04928 + 8.93862i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.24157	−	1.99383i	−0.747190	−	0.664610i
\(5\)	−1.72634	−	0.687837i	−0.345268	−	0.137567i								0.191057	−	0.981579i	\(-0.438808\pi\)
							−0.536325	+	0.844012i	\(0.680188\pi\)
\(7\)	6.44007	−	2.97949i	0.920009	−	0.425642i	0.0980460	−	0.995182i	\(-0.468741\pi\)
							0.821963	+	0.569540i	\(0.192879\pi\)
\(11\)	−4.39328	−	1.21979i	−0.399389	−	0.110890i	0.0620255	−	0.998075i	\(-0.480244\pi\)
							−0.461414	+	0.887185i	\(0.652658\pi\)
\(13\)	5.60427	−	10.5708i	0.431098	−	0.813137i	−0.568859	−	0.822435i	\(-0.692615\pi\)
							0.999957	+	0.00929844i	\(0.00295983\pi\)
\(17\)	−5.70044	+	12.3213i	−0.335320	+	0.724782i	−0.999745	−	0.0225724i	\(-0.992814\pi\)
							0.664425	+	0.747355i	\(0.268676\pi\)
\(19\)	−4.78993	+	1.05434i	−0.252101	+	0.0554917i	−0.339223	−	0.940706i	\(-0.610164\pi\)
							0.0871217	+	0.996198i	\(0.472233\pi\)
\(23\)	−14.7216	−	2.41348i	−0.640068	−	0.104934i	−0.166996	−	0.985958i	\(-0.553407\pi\)
							−0.473072	+	0.881024i	\(0.656855\pi\)
\(29\)	−0.344489	−	1.02241i	−0.0118789	−	0.0352555i	0.941550	−	0.336874i	\(-0.109370\pi\)
							−0.953429	+	0.301619i	\(0.902473\pi\)
\(31\)	−49.8053	−	10.9630i	−1.60662	−	0.353645i	−0.681057	−	0.732230i	\(-0.738480\pi\)
							−0.925566	+	0.378585i	\(0.876411\pi\)
\(37\)	0.156427	−	0.230712i	0.00422774	−	0.00623545i	−0.825569	−	0.564301i	\(-0.809146\pi\)
							0.829797	+	0.558065i	\(0.188456\pi\)
\(41\)	−64.0236	+	10.4961i	−1.56155	+	0.256003i	−0.879715	−	0.475502i	\(-0.842266\pi\)
							−0.681836	+	0.731505i	\(0.738818\pi\)
\(43\)	−19.9320	−	71.7886i	−0.463535	−	1.66950i	−0.713036	−	0.701128i	\(-0.752680\pi\)
							0.249500	−	0.968375i	\(-0.419734\pi\)
\(47\)	24.7503	−	9.86143i	0.526603	−	0.209818i	−0.0916584	−	0.995791i	\(-0.529217\pi\)
							0.618261	+	0.785973i	\(0.287837\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.5.4	✓	1120
3.2	odd	2	inner	354.3.h.a.5.23	yes	1120
59.12	even	29	inner	354.3.h.a.71.23	yes	1120
177.71	odd	58	inner	354.3.h.a.71.4	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.4	✓	1120	1.1	even	1	trivial
354.3.h.a.5.23	yes	1120	3.2	odd	2	inner
354.3.h.a.71.4	yes	1120	177.71	odd	58	inner
354.3.h.a.71.23	yes	1120	59.12	even	29	inner