Embedded newform 429.2.x.a.248.7

• Label: 429.2.x.a.248.7
• 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			248.7
Character	\(\chi\)	\(=\)	429.248
Dual form			429.2.x.a.365.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.653756 - 2.01206i) q^{2} +(1.33811 + 1.09976i) q^{3} +(-2.00293 + 1.45522i) q^{4} +(0.460929 + 0.149765i) q^{5} +(1.33797 - 3.41132i) q^{6} +(1.17111 + 1.61190i) q^{7} +(0.814296 + 0.591620i) q^{8} +(0.581071 + 2.94319i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 6 q^{3} - 48 q^{4} - 10 q^{6} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.653756	−	2.01206i	−0.462276	−	1.42274i	−0.862377	−	0.506267i	\(-0.831025\pi\)
							0.400101	−	0.916471i	\(-0.368975\pi\)
\(3\)	1.33811	+	1.09976i	0.772558	+	0.634945i
							\(5\)	0.460929	+	0.149765i	0.206134	+	0.0669769i	0.410264	−	0.911967i	\(-0.365437\pi\)
−0.204130	+	0.978944i	\(0.565437\pi\)
\(7\)	1.17111	+	1.61190i	0.442639	+	0.609240i	0.970796	−	0.239907i	\(-0.0771168\pi\)
							−0.528157	+	0.849147i	\(0.677117\pi\)
\(11\)	2.69669	+	1.93077i	0.813082	+	0.582149i
							\(13\)	0.951057	−	0.309017i	0.263776	−	0.0857059i
\(17\)	−1.02016	+	3.13974i	−0.247426	+	0.761499i								0.747802	+	0.663922i	\(0.231109\pi\)
							−0.995228	+	0.0975770i	\(0.968891\pi\)
\(19\)	2.72777	−	3.75445i	0.625792	−	0.861329i	−0.371966	−	0.928246i	\(-0.621316\pi\)
							0.997759	+	0.0669170i	\(0.0213163\pi\)
\(23\)			1.80265i			0.375879i	0.982181	+	0.187939i	\(0.0601808\pi\)
							−0.982181	+	0.187939i	\(0.939819\pi\)
\(29\)	7.22187	−	5.24700i	1.34107	−	0.974343i	0.341664	−	0.939822i	\(-0.389009\pi\)
							0.999404	−	0.0345209i	\(-0.0109905\pi\)
\(31\)	1.18879	+	3.65872i	0.213513	+	0.657125i	0.999256	+	0.0385719i	\(0.0122809\pi\)
							−0.785743	+	0.618553i	\(0.787719\pi\)
\(37\)	5.92036	−	4.30139i	0.973301	−	0.707145i	0.0170998	−	0.999854i	\(-0.494557\pi\)
							0.956202	+	0.292709i	\(0.0945567\pi\)
\(41\)	−4.99303	−	3.62765i	−0.779781	−	0.566544i	0.125132	−	0.992140i	\(-0.460064\pi\)
							−0.904913	+	0.425596i	\(0.860064\pi\)
\(43\)		−	5.28834i		−	0.806464i	−0.915098	−	0.403232i	\(-0.867887\pi\)
							0.915098	−	0.403232i	\(-0.132113\pi\)
\(47\)	−2.56074	+	3.52455i	−0.373522	+	0.514109i	−0.953854	−	0.300271i	\(-0.902923\pi\)
							0.580332	+	0.814380i	\(0.302923\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.x.a.248.7	✓	192
3.2	odd	2	inner	429.2.x.a.248.42	yes	192
11.2	odd	10	inner	429.2.x.a.365.42	yes	192
33.2	even	10	inner	429.2.x.a.365.7	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.x.a.248.7	✓	192	1.1	even	1	trivial
429.2.x.a.248.42	yes	192	3.2	odd	2	inner
429.2.x.a.365.7	yes	192	33.2	even	10	inner
429.2.x.a.365.42	yes	192	11.2	odd	10	inner