Embedded newform 429.2.y.b.116.2

• Label: 429.2.y.b.116.2
• 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			116.2
Character	\(\chi\)	\(=\)	429.116
Dual form			429.2.y.b.233.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.50188 + 0.812912i) q^{2} +(1.55965 - 0.753314i) q^{3} +(3.98057 - 2.89205i) q^{4} +(0.944274 - 2.90618i) q^{5} +(-3.28969 + 3.15256i) q^{6} +(2.69270 - 1.95636i) q^{7} +(-4.51544 + 6.21497i) q^{8} +(1.86504 - 2.34982i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 6 q^{3} + 20 q^{4} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.50188	+	0.812912i	−1.76910	+	0.574815i	−0.998077	−	0.0619920i	\(-0.980255\pi\)
							−0.771023	+	0.636807i	\(0.780255\pi\)
\(3\)	1.55965	−	0.753314i	0.900466	−	0.434926i
							\(5\)	0.944274	−	2.90618i	0.422292	−	1.29968i	−0.483271	−	0.875471i	\(-0.660551\pi\)
0.905563	−	0.424211i	\(-0.139449\pi\)
\(7\)	2.69270	−	1.95636i	1.01774	−	0.739435i	0.0519255	−	0.998651i	\(-0.483464\pi\)
							0.965819	+	0.259216i	\(0.0834642\pi\)
\(11\)	3.16176	+	1.00163i	0.953307	+	0.302004i
							\(13\)	1.09244	+	3.43607i	0.302987	+	0.952995i
\(17\)	0.313445	−	0.964684i	0.0760216	−	0.233970i								−0.905823	−	0.423655i	\(-0.860747\pi\)
							0.981845	+	0.189685i	\(0.0607467\pi\)
\(19\)	−1.38863	−	1.00890i	−0.318573	−	0.231457i	0.416993	−	0.908910i	\(-0.363084\pi\)
							−0.735566	+	0.677453i	\(0.763084\pi\)
\(23\)			5.67180i			1.18265i	0.806432	+	0.591326i	\(0.201395\pi\)
							−0.806432	+	0.591326i	\(0.798605\pi\)
\(29\)	−7.29720	+	5.30173i	−1.35506	+	0.984506i	−0.356314	+	0.934366i	\(0.615967\pi\)
							−0.998742	+	0.0501398i	\(0.984033\pi\)
\(31\)	−7.38677	+	2.40011i	−1.32670	+	0.431072i	−0.884792	−	0.465987i	\(-0.845699\pi\)
							−0.441911	+	0.897059i	\(0.645699\pi\)
\(37\)	−0.978230	−	1.34642i	−0.160820	−	0.221350i	0.721001	−	0.692934i	\(-0.243682\pi\)
							−0.881821	+	0.471584i	\(0.843682\pi\)
\(41\)	1.77989	−	2.44981i	0.277973	−	0.382597i	−0.647088	−	0.762415i	\(-0.724014\pi\)
							0.925061	+	0.379818i	\(0.124014\pi\)
\(43\)			0.0954413i			0.0145547i	0.999974	+	0.00727733i	\(0.00231647\pi\)
							−0.999974	+	0.00727733i	\(0.997684\pi\)
\(47\)	9.10002	+	6.61155i	1.32737	+	0.964394i	0.999809	+	0.0195659i	\(0.00622842\pi\)
							0.327566	+	0.944828i	\(0.393772\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.y.b.116.2	yes	176
3.2	odd	2	inner	429.2.y.b.116.43	yes	176
11.2	odd	10	inner	429.2.y.b.233.1	yes	176
13.12	even	2	inner	429.2.y.b.116.44	yes	176
33.2	even	10	inner	429.2.y.b.233.44	yes	176
39.38	odd	2	inner	429.2.y.b.116.1	✓	176
143.90	odd	10	inner	429.2.y.b.233.43	yes	176
429.233	even	10	inner	429.2.y.b.233.2	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.y.b.116.1	✓	176	39.38	odd	2	inner
429.2.y.b.116.2	yes	176	1.1	even	1	trivial
429.2.y.b.116.43	yes	176	3.2	odd	2	inner
429.2.y.b.116.44	yes	176	13.12	even	2	inner
429.2.y.b.233.1	yes	176	11.2	odd	10	inner
429.2.y.b.233.2	yes	176	429.233	even	10	inner
429.2.y.b.233.43	yes	176	143.90	odd	10	inner
429.2.y.b.233.44	yes	176	33.2	even	10	inner