Embedded newform 177.2.e.b.133.2

• Label: 177.2.e.b.133.2
• Level: $177$
• Weight: $2$
• Character: 177.133
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			133.2
Character	\(\chi\)	\(=\)	177.133
Dual form			177.2.e.b.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.756070 + 0.0822276i) q^{2} +(-0.647386 - 0.762162i) q^{3} +(-1.38836 + 0.305601i) q^{4} +(-0.128718 + 0.0978487i) q^{5} +(0.552140 + 0.523015i) q^{6} +(1.18304 + 2.96921i) q^{7} +(2.46600 - 0.830893i) q^{8} +(-0.161782 + 0.986827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{28}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.756070	+	0.0822276i	−0.534622	+	0.0581437i	−0.371448	−	0.928454i	\(-0.621139\pi\)
							−0.163174	+	0.986597i	\(0.552173\pi\)
\(3\)	−0.647386	−	0.762162i	−0.373769	−	0.440034i
							\(5\)	−0.128718	+	0.0978487i	−0.0575644	+	0.0437593i	−0.633561	−	0.773693i	\(-0.718407\pi\)
0.575996	+	0.817452i	\(0.304614\pi\)
\(7\)	1.18304	+	2.96921i	0.447147	+	1.12225i	0.963934	+	0.266141i	\(0.0857485\pi\)
							−0.516787	+	0.856114i	\(0.672872\pi\)
\(11\)	3.21368	+	1.48680i	0.968960	+	0.448288i	0.839538	−	0.543301i	\(-0.182826\pi\)
							0.129422	+	0.991590i	\(0.458688\pi\)
\(13\)	0.653255	+	3.98468i	0.181180	+	1.10515i	0.908201	+	0.418535i	\(0.137456\pi\)
							−0.727020	+	0.686616i	\(0.759095\pi\)
\(17\)	0.702422	−	1.76295i	0.170362	−	0.427577i	−0.818861	−	0.573992i	\(-0.805394\pi\)
							0.989223	+	0.146415i	\(0.0467734\pi\)
\(19\)	−2.07230	+	3.05641i	−0.475418	+	0.701189i	−0.987477	−	0.157761i	\(-0.949572\pi\)
							0.512059	+	0.858950i	\(0.328883\pi\)
\(23\)	−0.0779331	−	1.43739i	−0.0162502	−	0.299717i	−0.995349	−	0.0963327i	\(-0.969289\pi\)
							0.979099	−	0.203384i	\(-0.0651940\pi\)
\(29\)	−0.693292	−	0.0754000i	−0.128741	−	0.0140014i	0.0435216	−	0.999052i	\(-0.486142\pi\)
							−0.172263	+	0.985051i	\(0.555108\pi\)
\(31\)	0.294127	+	0.433805i	0.0528267	+	0.0779136i	0.853185	−	0.521608i	\(-0.174668\pi\)
							−0.800358	+	0.599522i	\(0.795357\pi\)
\(37\)	−1.54938	−	0.522048i	−0.254717	−	0.0858241i	0.189046	−	0.981968i	\(-0.439460\pi\)
							−0.443763	+	0.896144i	\(0.646357\pi\)
\(41\)	0.240369	−	4.43335i	0.0375394	−	0.692373i	−0.917833	−	0.396968i	\(-0.870063\pi\)
							0.955372	−	0.295405i	\(-0.0954547\pi\)
\(43\)	6.48573	−	3.00062i	0.989064	−	0.457590i	0.142453	−	0.989802i	\(-0.454501\pi\)
							0.846611	+	0.532212i	\(0.178639\pi\)
\(47\)	4.38426	+	3.33283i	0.639511	+	0.486143i	0.873968	−	0.485983i	\(-0.161538\pi\)
							−0.234458	+	0.972126i	\(0.575331\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.133.2	yes	140
3.2	odd	2		531.2.i.b.487.4		140
59.4	even	29	inner	177.2.e.b.4.2	✓	140
177.122	odd	58		531.2.i.b.181.4		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.4.2	✓	140	59.4	even	29	inner
177.2.e.b.133.2	yes	140	1.1	even	1	trivial
531.2.i.b.181.4		140	177.122	odd	58
531.2.i.b.487.4		140	3.2	odd	2