Embedded newform 429.2.x.a.248.2

• Label: 429.2.x.a.248.2
• Level: $429$
• Weight: $2$
• Character: 429.248
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			248.2
Character	\(\chi\)	\(=\)	429.248
Dual form			429.2.x.a.365.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.833208 - 2.56435i) q^{2} +(1.66893 + 0.463321i) q^{3} +(-4.26363 + 3.09771i) q^{4} +(-3.14288 - 1.02118i) q^{5} +(-0.202449 - 4.66577i) q^{6} +(-1.19340 - 1.64257i) q^{7} +(7.13336 + 5.18269i) q^{8} +(2.57067 + 1.54650i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 6 q^{3} - 48 q^{4} - 10 q^{6} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.833208	−	2.56435i	−0.589167	−	1.81327i	−0.581849	−	0.813297i	\(-0.697671\pi\)
							−0.00731792	−	0.999973i	\(-0.502329\pi\)
\(3\)	1.66893	+	0.463321i	0.963558	+	0.267499i
							\(5\)	−3.14288	−	1.02118i	−1.40554	−	0.456687i	−0.494560	−	0.869143i	\(-0.664671\pi\)
−0.910977	+	0.412457i	\(0.864671\pi\)
\(7\)	−1.19340	−	1.64257i	−0.451062	−	0.620833i	0.521564	−	0.853212i	\(-0.325349\pi\)
							−0.972625	+	0.232379i	\(0.925349\pi\)
\(11\)	−2.77140	+	1.82190i	−0.835609	+	0.549325i
							\(13\)	−0.951057	+	0.309017i	−0.263776	+	0.0857059i
\(17\)	−2.08794	+	6.42601i	−0.506399	+	1.55854i								0.292006	+	0.956416i	\(0.405677\pi\)
							−0.798406	+	0.602120i	\(0.794323\pi\)
\(19\)	−1.52286	+	2.09603i	−0.349367	+	0.480862i	−0.947148	−	0.320797i	\(-0.896049\pi\)
							0.597781	+	0.801659i	\(0.296049\pi\)
\(23\)		−	5.76411i		−	1.20190i	−0.799286	−	0.600950i	\(-0.794789\pi\)
							0.799286	−	0.600950i	\(-0.205211\pi\)
\(29\)	−5.83818	+	4.24168i	−1.08412	+	0.787661i	−0.978397	−	0.206735i	\(-0.933716\pi\)
							−0.105725	+	0.994395i	\(0.533716\pi\)
\(31\)	−0.868993	−	2.67449i	−0.156076	−	0.480352i	0.842193	−	0.539177i	\(-0.181265\pi\)
							−0.998268	+	0.0588251i	\(0.981265\pi\)
\(37\)	−1.72134	+	1.25063i	−0.282987	+	0.205602i	−0.720219	−	0.693747i	\(-0.755959\pi\)
							0.437232	+	0.899349i	\(0.355959\pi\)
\(41\)	1.31152	+	0.952878i	0.204826	+	0.148814i	0.685469	−	0.728101i	\(-0.259597\pi\)
							−0.480644	+	0.876916i	\(0.659597\pi\)
\(43\)			0.343293i			0.0523517i	0.999657	+	0.0261759i	\(0.00833298\pi\)
							−0.999657	+	0.0261759i	\(0.991667\pi\)
\(47\)	−0.933118	+	1.28433i	−0.136109	+	0.187338i	−0.871631	−	0.490163i	\(-0.836937\pi\)
							0.735521	+	0.677501i	\(0.236937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.x.a.248.2	✓	192
3.2	odd	2	inner	429.2.x.a.248.47	yes	192
11.2	odd	10	inner	429.2.x.a.365.47	yes	192
33.2	even	10	inner	429.2.x.a.365.2	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.x.a.248.2	✓	192	1.1	even	1	trivial
429.2.x.a.248.47	yes	192	3.2	odd	2	inner
429.2.x.a.365.2	yes	192	33.2	even	10	inner
429.2.x.a.365.47	yes	192	11.2	odd	10	inner