Embedded newform 354.3.h.a.5.11

• Label: 354.3.h.a.5.11
• 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			5.11
Character	\(\chi\)	\(=\)	354.5
Dual form			354.3.h.a.71.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.451561 + 1.34018i) q^{2} +(0.908343 - 2.85918i) q^{3} +(-1.59219 - 1.21035i) q^{4} +(4.52215 + 1.80179i) q^{5} +(3.42166 + 2.50844i) q^{6} +(-8.14687 + 3.76914i) q^{7} +(2.34106 - 1.58728i) q^{8} +(-7.34983 - 5.19423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1120 q + 80 q^{4} - 8 q^{6} - 8 q^{7} + 24 q^{9}+O(q^{10}) \)

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
							\(3\)	0.908343	−	2.85918i	0.302781	−	0.953060i
\(5\)	4.52215	+	1.80179i	0.904430	+	0.360358i								0.775538	−	0.631301i	\(-0.217479\pi\)
							0.128892	+	0.991659i	\(0.458858\pi\)
\(7\)	−8.14687	+	3.76914i	−1.16384	+	0.538449i	−0.904138	−	0.427241i	\(-0.859486\pi\)
							−0.259700	+	0.965689i	\(0.583624\pi\)
\(11\)	−19.7070	−	5.47161i	−1.79154	−	0.497419i	−0.797207	−	0.603706i	\(-0.793690\pi\)
							−0.994336	+	0.106286i	\(0.966104\pi\)
\(13\)	−2.77123	+	5.22710i	−0.213172	+	0.402085i	−0.966920	−	0.255078i	\(-0.917899\pi\)
							0.753749	+	0.657163i	\(0.228244\pi\)
\(17\)	−11.5893	+	25.0499i	−0.681725	+	1.47353i	0.189113	+	0.981955i	\(0.439439\pi\)
							−0.870838	+	0.491570i	\(0.836423\pi\)
\(19\)	10.6810	−	2.35106i	0.562157	−	0.123740i	0.0752038	−	0.997168i	\(-0.476039\pi\)
							0.486953	+	0.873428i	\(0.338108\pi\)
\(23\)	−1.42616	−	0.233807i	−0.0620070	−	0.0101655i	0.130699	−	0.991422i	\(-0.458278\pi\)
							−0.192706	+	0.981257i	\(0.561726\pi\)
\(29\)	−5.48489	−	16.2786i	−0.189134	−	0.561330i	0.810570	−	0.585641i	\(-0.199157\pi\)
							−0.999704	+	0.0243114i	\(0.992261\pi\)
\(31\)	40.6107	+	8.93909i	1.31002	+	0.288358i	0.814462	−	0.580216i	\(-0.197032\pi\)
							0.495560	+	0.868574i	\(0.334963\pi\)
\(37\)	−20.4760	+	30.1999i	−0.553407	+	0.816214i	−0.996785	−	0.0801207i	\(-0.974469\pi\)
							0.443379	+	0.896334i	\(0.353780\pi\)
\(41\)	−73.7793	+	12.0955i	−1.79949	+	0.295012i	−0.966196	−	0.257810i	\(-0.916999\pi\)
							−0.833299	+	0.552822i	\(0.813551\pi\)
\(43\)	−12.8858	−	46.4106i	−0.299671	−	1.07932i	−0.948082	−	0.318026i	\(-0.896980\pi\)
							0.648411	−	0.761290i	\(-0.275434\pi\)
\(47\)	−27.7140	+	11.0423i	−0.589661	+	0.234942i	−0.645833	−	0.763479i	\(-0.723490\pi\)
							0.0561722	+	0.998421i	\(0.482110\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.h.a.5.11	✓	1120
3.2	odd	2	inner	354.3.h.a.5.25	yes	1120
59.12	even	29	inner	354.3.h.a.71.25	yes	1120
177.71	odd	58	inner	354.3.h.a.71.11	yes	1120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.h.a.5.11	✓	1120	1.1	even	1	trivial
354.3.h.a.5.25	yes	1120	3.2	odd	2	inner
354.3.h.a.71.11	yes	1120	177.71	odd	58	inner
354.3.h.a.71.25	yes	1120	59.12	even	29	inner