Embedded newform 429.2.y.b.233.2

• Label: 429.2.y.b.233.2
• Level: $429$
• Weight: $2$
• Character: 429.233
• 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			233.2
Character	\(\chi\)	\(=\)	429.233
Dual form			429.2.y.b.116.2

$q$-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 \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.y.b.116.1	✓	176	11.6	odd	10	inner
429.2.y.b.116.2	yes	176	429.116	even	10	inner
429.2.y.b.116.43	yes	176	143.116	odd	10	inner
429.2.y.b.116.44	yes	176	33.17	even	10	inner
429.2.y.b.233.1	yes	176	39.38	odd	2	inner
429.2.y.b.233.2	yes	176	1.1	even	1	trivial
429.2.y.b.233.43	yes	176	3.2	odd	2	inner
429.2.y.b.233.44	yes	176	13.12	even	2	inner