Embedded newform 354.3.h.a.5.7

• Label: 354.3.h.a.5.7
• 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			5.7
Character	\(\chi\)	\(=\)	354.5
Dual form			354.3.h.a.71.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 + 1.34018i) q^{2} +(-1.59946 + 2.53806i) q^{3} +(-1.59219 - 1.21035i) q^{4} +(2.85503 + 1.13755i) q^{5} +(-2.67921 - 3.28965i) q^{6} +(-8.37649 + 3.87538i) q^{7} +(2.34106 - 1.58728i) q^{8} +(-3.88348 - 8.11903i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.59946	+	2.53806i	−0.533152	+	0.846019i
\(5\)	2.85503	+	1.13755i	0.571007	+	0.227510i								0.637728	−	0.770262i	\(-0.279874\pi\)
							−0.0667211	+	0.997772i	\(0.521254\pi\)
\(7\)	−8.37649	+	3.87538i	−1.19664	+	0.553625i	−0.914045	−	0.405613i	\(-0.867058\pi\)
							−0.282597	+	0.959239i	\(0.591196\pi\)
\(11\)	−3.85190	−	1.06947i	−0.350173	−	0.0972250i	0.0879883	−	0.996122i	\(-0.471956\pi\)
							−0.438161	+	0.898897i	\(0.644370\pi\)
\(13\)	1.39600	−	2.63313i	0.107384	−	0.202548i	−0.823996	−	0.566595i	\(-0.808260\pi\)
							0.931381	+	0.364047i	\(0.118605\pi\)
\(17\)	6.50248	−	14.0549i	0.382499	−	0.826758i	−0.616718	−	0.787185i	\(-0.711538\pi\)
							0.999217	−	0.0395737i	\(-0.0126000\pi\)
\(19\)	−20.2663	+	4.46094i	−1.06665	+	0.234787i	−0.713409	−	0.700747i	\(-0.752850\pi\)
							−0.353237	+	0.935534i	\(0.614919\pi\)
\(23\)	−4.75736	−	0.779929i	−0.206842	−	0.0339100i	0.0574702	−	0.998347i	\(-0.481697\pi\)
							−0.264312	+	0.964437i	\(0.585145\pi\)
\(29\)	1.16642	+	3.46181i	0.0402214	+	0.119373i	0.965882	−	0.258981i	\(-0.0833867\pi\)
							−0.925661	+	0.378354i	\(0.876490\pi\)
\(31\)	6.06320	+	1.33461i	0.195587	+	0.0430520i	0.311684	−	0.950186i	\(-0.399107\pi\)
							−0.116096	+	0.993238i	\(0.537038\pi\)
\(37\)	22.1383	−	32.6516i	0.598333	−	0.882475i	−0.401104	−	0.916033i	\(-0.631373\pi\)
							0.999437	+	0.0335575i	\(0.0106837\pi\)
\(41\)	11.9841	−	1.96469i	0.292295	−	0.0479193i	−0.0138504	−	0.999904i	\(-0.504409\pi\)
							0.306145	+	0.951985i	\(0.400961\pi\)
\(43\)	−11.4259	−	41.1525i	−0.265719	−	0.957034i	−0.969191	−	0.246311i	\(-0.920782\pi\)
							0.703472	−	0.710723i	\(-0.251632\pi\)
\(47\)	36.5861	−	14.5772i	0.778428	−	0.310154i	0.0531288	−	0.998588i	\(-0.483081\pi\)
							0.725299	+	0.688433i	\(0.241701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.5.7	✓	1120
3.2	odd	2	inner	354.3.h.a.5.33	yes	1120
59.12	even	29	inner	354.3.h.a.71.33	yes	1120
177.71	odd	58	inner	354.3.h.a.71.7	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.7	✓	1120	1.1	even	1	trivial
354.3.h.a.5.33	yes	1120	3.2	odd	2	inner
354.3.h.a.71.7	yes	1120	177.71	odd	58	inner
354.3.h.a.71.33	yes	1120	59.12	even	29	inner