Embedded newform 429.2.y.b.116.20

• Label: 429.2.y.b.116.20
• Level: $429$
• Weight: $2$
• Character: 429.116
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.y (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			116.20
Character	\(\chi\)	\(=\)	429.116
Dual form			429.2.y.b.233.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.473784 + 0.153942i) q^{2} +(1.72665 + 0.136713i) q^{3} +(-1.41726 + 1.02970i) q^{4} +(0.269770 - 0.830268i) q^{5} +(-0.839104 + 0.201031i) q^{6} +(3.14053 - 2.28173i) q^{7} +(1.09859 - 1.51208i) q^{8} +(2.96262 + 0.472112i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 6 q^{3} + 20 q^{4} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.473784	+	0.153942i	−0.335016	+	0.108853i	−0.471694	−	0.881762i	\(-0.656357\pi\)
							0.136678	+	0.990616i	\(0.456357\pi\)
\(3\)	1.72665	+	0.136713i	0.996880	+	0.0789316i
							\(5\)	0.269770	−	0.830268i	0.120645	−	0.371307i	−0.872437	−	0.488726i	\(-0.837462\pi\)
0.993083	+	0.117418i	\(0.0374619\pi\)
\(7\)	3.14053	−	2.28173i	1.18701	−	0.862413i	0.194065	−	0.980989i	\(-0.437833\pi\)
							0.992945	+	0.118575i	\(0.0378327\pi\)
\(11\)	−0.601650	−	3.26160i	−0.181404	−	0.983409i
							\(13\)	−2.02735	+	2.98159i	−0.562286	+	0.826943i
\(17\)	0.569262	−	1.75201i	0.138066	−	0.424924i								−0.857988	−	0.513670i	\(-0.828286\pi\)
							0.996054	+	0.0887453i	\(0.0282857\pi\)
\(19\)	−1.56886	−	1.13984i	−0.359920	−	0.261497i	0.393099	−	0.919496i	\(-0.371403\pi\)
							−0.753019	+	0.657999i	\(0.771403\pi\)
\(23\)			4.93276i			1.02855i	0.857625	+	0.514275i	\(0.171939\pi\)
							−0.857625	+	0.514275i	\(0.828061\pi\)
\(29\)	6.90338	−	5.01560i	1.28193	−	0.931373i	0.282316	−	0.959322i	\(-0.408897\pi\)
							0.999609	+	0.0279485i	\(0.00889744\pi\)
\(31\)	1.93495	−	0.628704i	0.347527	−	0.112919i	−0.130052	−	0.991507i	\(-0.541514\pi\)
							0.477579	+	0.878589i	\(0.341514\pi\)
\(37\)	2.13779	+	2.94242i	0.351451	+	0.483731i	0.947742	−	0.319037i	\(-0.103360\pi\)
							−0.596291	+	0.802768i	\(0.703360\pi\)
\(41\)	−3.08517	+	4.24637i	−0.481823	+	0.663172i	−0.978854	−	0.204561i	\(-0.934423\pi\)
							0.497031	+	0.867733i	\(0.334423\pi\)
\(43\)			2.50389i			0.381839i	0.981606	+	0.190920i	\(0.0611469\pi\)
							−0.981606	+	0.190920i	\(0.938853\pi\)
\(47\)	−8.32016	−	6.04495i	−1.21362	−	0.881747i	−0.218066	−	0.975934i	\(-0.569975\pi\)
							−0.995555	+	0.0941870i	\(0.969975\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.y.b.116.20	yes	176
3.2	odd	2	inner	429.2.y.b.116.25	yes	176
11.2	odd	10	inner	429.2.y.b.233.19	yes	176
13.12	even	2	inner	429.2.y.b.116.26	yes	176
33.2	even	10	inner	429.2.y.b.233.26	yes	176
39.38	odd	2	inner	429.2.y.b.116.19	✓	176
143.90	odd	10	inner	429.2.y.b.233.25	yes	176
429.233	even	10	inner	429.2.y.b.233.20	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.y.b.116.19	✓	176	39.38	odd	2	inner
429.2.y.b.116.20	yes	176	1.1	even	1	trivial
429.2.y.b.116.25	yes	176	3.2	odd	2	inner
429.2.y.b.116.26	yes	176	13.12	even	2	inner
429.2.y.b.233.19	yes	176	11.2	odd	10	inner
429.2.y.b.233.20	yes	176	429.233	even	10	inner
429.2.y.b.233.25	yes	176	143.90	odd	10	inner
429.2.y.b.233.26	yes	176	33.2	even	10	inner