Embedded newform 429.2.bq.a.29.4

• Label: 429.2.bq.a.29.4
• Level: $429$
• Weight: $2$
• Character: 429.29
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $416$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bq (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(416\)
Relative dimension:	\(52\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			29.4
Character	\(\chi\)	\(=\)	429.29
Dual form			429.2.bq.a.74.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.260719 + 2.48057i) q^{2} +(1.72227 + 0.183844i) q^{3} +(-4.12897 - 0.877640i) q^{4} +(-2.22258 - 3.05912i) q^{5} +(-0.905066 + 4.22428i) q^{6} +(0.668489 - 3.14500i) q^{7} +(1.71203 - 5.26908i) q^{8} +(2.93240 + 0.633258i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 3 q^{3} + 42 q^{4} - 5 q^{6} - 10 q^{7} + 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.260719	+	2.48057i	−0.184356	+	1.75403i	0.376777	+	0.926304i	\(0.377032\pi\)
							−0.561133	+	0.827726i	\(0.689634\pi\)
\(3\)	1.72227	+	0.183844i	0.994351	+	0.106143i
							\(5\)	−2.22258	−	3.05912i	−0.993968	−	1.36808i	−0.928954	−	0.370195i	\(-0.879291\pi\)
−0.0650142	−	0.997884i	\(-0.520709\pi\)
\(7\)	0.668489	−	3.14500i	0.252665	−	1.18870i	−0.650535	−	0.759476i	\(-0.725455\pi\)
							0.903200	−	0.429220i	\(-0.141211\pi\)
\(11\)	−2.72327	−	1.89309i	−0.821097	−	0.570789i
							\(13\)	−1.04613	−	3.45045i	−0.290144	−	0.956983i
\(17\)	−0.114794	−	1.09219i	−0.0278415	−	0.264895i								−0.999584	−	0.0288499i	\(-0.990816\pi\)
							0.971742	−	0.236045i	\(-0.0758512\pi\)
\(19\)	−4.07213	+	3.66656i	−0.934211	+	0.841167i	−0.987533	−	0.157410i	\(-0.949686\pi\)
							0.0533226	+	0.998577i	\(0.483019\pi\)
\(23\)	6.52755	−	3.76868i	1.36109	−	0.785824i	0.371319	−	0.928505i	\(-0.378906\pi\)
							0.989769	+	0.142681i	\(0.0455724\pi\)
\(29\)	−3.46572	+	3.84908i	−0.643569	+	0.714755i	−0.973356	−	0.229300i	\(-0.926356\pi\)
							0.329787	+	0.944055i	\(0.393023\pi\)
\(31\)	3.17762	+	2.30868i	0.570718	+	0.414651i	0.835366	−	0.549694i	\(-0.185256\pi\)
							−0.264648	+	0.964345i	\(0.585256\pi\)
\(37\)	−0.767530	+	0.852428i	−0.126181	+	0.140138i	−0.802925	−	0.596080i	\(-0.796724\pi\)
							0.676744	+	0.736219i	\(0.263391\pi\)
\(41\)	1.13659	−	0.241589i	0.177505	−	0.0377299i	−0.118302	−	0.992978i	\(-0.537745\pi\)
							0.295807	+	0.955248i	\(0.404412\pi\)
\(43\)	−3.12015	−	1.80142i	−0.475818	−	0.274714i	0.242854	−	0.970063i	\(-0.421916\pi\)
							−0.718672	+	0.695349i	\(0.755250\pi\)
\(47\)	−4.39359	−	1.42756i	−0.640870	−	0.208231i	−0.0294861	−	0.999565i	\(-0.509387\pi\)
							−0.611384	+	0.791334i	\(0.709387\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bq.a.29.4	✓	416
3.2	odd	2	inner	429.2.bq.a.29.49	yes	416
11.8	odd	10	inner	429.2.bq.a.107.4	yes	416
13.9	even	3	inner	429.2.bq.a.425.49	yes	416
33.8	even	10	inner	429.2.bq.a.107.49	yes	416
39.35	odd	6	inner	429.2.bq.a.425.4	yes	416
143.74	odd	30	inner	429.2.bq.a.74.49	yes	416
429.74	even	30	inner	429.2.bq.a.74.4	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bq.a.29.4	✓	416	1.1	even	1	trivial
429.2.bq.a.29.49	yes	416	3.2	odd	2	inner
429.2.bq.a.74.4	yes	416	429.74	even	30	inner
429.2.bq.a.74.49	yes	416	143.74	odd	30	inner
429.2.bq.a.107.4	yes	416	11.8	odd	10	inner
429.2.bq.a.107.49	yes	416	33.8	even	10	inner
429.2.bq.a.425.4	yes	416	39.35	odd	6	inner
429.2.bq.a.425.49	yes	416	13.9	even	3	inner