Embedded newform 429.2.y.b.116.7

• Label: 429.2.y.b.116.7
• 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			116.7
Character	\(\chi\)	\(=\)	429.116
Dual form			429.2.y.b.233.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.03433 + 0.660995i) q^{2} +(-0.778983 + 1.54699i) q^{3} +(2.08356 - 1.51379i) q^{4} +(-0.766534 + 2.35915i) q^{5} +(0.562157 - 3.66200i) q^{6} +(-2.59152 + 1.88285i) q^{7} +(-0.723470 + 0.995771i) q^{8} +(-1.78637 - 2.41016i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 6 q^{3} + 20 q^{4} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.03433	+	0.660995i	−1.43849	+	0.467394i	−0.921426	−	0.388553i	\(-0.872975\pi\)
							−0.517064	+	0.855947i	\(0.672975\pi\)
\(3\)	−0.778983	+	1.54699i	−0.449746	+	0.893156i
							\(5\)	−0.766534	+	2.35915i	−0.342804	+	1.05504i	0.619944	+	0.784646i	\(0.287155\pi\)
−0.962749	+	0.270398i	\(0.912845\pi\)
\(7\)	−2.59152	+	1.88285i	−0.979504	+	0.711651i	−0.957598	−	0.288109i	\(-0.906973\pi\)
							−0.0219060	+	0.999760i	\(0.506973\pi\)
\(11\)	−1.84787	−	2.75416i	−0.557153	−	0.830410i
							\(13\)	0.207315	+	3.59959i	0.0574989	+	0.998346i
\(17\)	0.140131	−	0.431280i	0.0339868	−	0.104601i								−0.932624	−	0.360850i	\(-0.882487\pi\)
							0.966611	+	0.256249i	\(0.0824867\pi\)
\(19\)	−3.20947	−	2.33181i	−0.736302	−	0.534955i	0.155249	−	0.987875i	\(-0.450382\pi\)
							−0.891551	+	0.452921i	\(0.850382\pi\)
\(23\)			1.90436i			0.397087i	0.980092	+	0.198543i	\(0.0636210\pi\)
							−0.980092	+	0.198543i	\(0.936379\pi\)
\(29\)	−4.42690	+	3.21633i	−0.822055	+	0.597258i	−0.917301	−	0.398196i	\(-0.869637\pi\)
							0.0952451	+	0.995454i	\(0.469637\pi\)
\(31\)	2.59964	−	0.844676i	0.466910	−	0.151708i	−0.0661074	−	0.997813i	\(-0.521058\pi\)
							0.533017	+	0.846104i	\(0.321058\pi\)
\(37\)	2.63407	+	3.62549i	0.433039	+	0.596027i	0.968647	−	0.248439i	\(-0.0799176\pi\)
							−0.535608	+	0.844467i	\(0.679918\pi\)
\(41\)	6.03999	−	8.31333i	0.943288	−	1.29832i	−0.0111568	−	0.999938i	\(-0.503551\pi\)
							0.954445	−	0.298387i	\(-0.0964486\pi\)
\(43\)		−	10.5693i		−	1.61181i	−0.592048	−	0.805903i	\(-0.701681\pi\)
							0.592048	−	0.805903i	\(-0.298319\pi\)
\(47\)	5.62334	+	4.08559i	0.820248	+	0.595945i	0.916784	−	0.399384i	\(-0.130776\pi\)
							−0.0965355	+	0.995330i	\(0.530776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.y.b.116.7	✓	176
3.2	odd	2	inner	429.2.y.b.116.38	yes	176
11.2	odd	10	inner	429.2.y.b.233.8	yes	176
13.12	even	2	inner	429.2.y.b.116.37	yes	176
33.2	even	10	inner	429.2.y.b.233.37	yes	176
39.38	odd	2	inner	429.2.y.b.116.8	yes	176
143.90	odd	10	inner	429.2.y.b.233.38	yes	176
429.233	even	10	inner	429.2.y.b.233.7	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.y.b.116.7	✓	176	1.1	even	1	trivial
429.2.y.b.116.8	yes	176	39.38	odd	2	inner
429.2.y.b.116.37	yes	176	13.12	even	2	inner
429.2.y.b.116.38	yes	176	3.2	odd	2	inner
429.2.y.b.233.7	yes	176	429.233	even	10	inner
429.2.y.b.233.8	yes	176	11.2	odd	10	inner
429.2.y.b.233.37	yes	176	33.2	even	10	inner
429.2.y.b.233.38	yes	176	143.90	odd	10	inner