Embedded newform 429.2.bn.a.49.1

• Label: 429.2.bn.a.49.1
• Level: $429$
• Weight: $2$
• Character: 429.49
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			49.1
Character	\(\chi\)	\(=\)	429.49
Dual form			429.2.bn.a.394.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.97287 - 1.77638i) q^{2} +(0.104528 - 0.994522i) q^{3} +(0.527630 + 5.02007i) q^{4} +(1.64057 + 0.533054i) q^{5} +(-1.97287 + 1.77638i) q^{6} +(-2.55155 + 0.268179i) q^{7} +(4.75574 - 6.54571i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} - 20 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.97287	−	1.77638i	−1.39503	−	1.25609i	−0.928374	−	0.371648i	\(-0.878793\pi\)
							−0.466654	−	0.884440i	\(-0.654541\pi\)
\(3\)	0.104528	−	0.994522i	0.0603495	−	0.574187i
							\(5\)	1.64057	+	0.533054i	0.733686	+	0.238389i	0.651947	−	0.758265i	\(-0.273952\pi\)
0.0817395	+	0.996654i	\(0.473952\pi\)
\(7\)	−2.55155	+	0.268179i	−0.964397	+	0.101362i	−0.573618	−	0.819123i	\(-0.694461\pi\)
							−0.390778	+	0.920485i	\(0.627794\pi\)
\(11\)	2.71672	+	1.90249i	0.819121	+	0.573621i
							\(13\)	−3.54211	−	0.673406i	−0.982404	−	0.186769i
\(17\)	−2.72144	−	3.02247i	−0.660047	−	0.733057i								0.316445	−	0.948611i	\(-0.397511\pi\)
							−0.976492	+	0.215554i	\(0.930844\pi\)
\(19\)	−3.12329	−	7.01503i	−0.716532	−	1.60936i	−0.790767	−	0.612117i	\(-0.790318\pi\)
							0.0742353	−	0.997241i	\(-0.476348\pi\)
\(23\)	−4.54932	−	7.87965i	−0.948599	−	1.64302i	−0.748380	−	0.663270i	\(-0.769168\pi\)
							−0.200218	−	0.979751i	\(-0.564165\pi\)
\(29\)	−2.22896	−	0.992398i	−0.413908	−	0.184284i	0.189208	−	0.981937i	\(-0.439408\pi\)
							−0.603116	+	0.797653i	\(0.706074\pi\)
\(31\)	3.00859	−	0.977552i	0.540360	−	0.175573i	−0.0261055	−	0.999659i	\(-0.508311\pi\)
							0.566465	+	0.824086i	\(0.308311\pi\)
\(37\)	−1.20068	+	2.69676i	−0.197390	+	0.443345i	−0.984938	−	0.172908i	\(-0.944684\pi\)
							0.787548	+	0.616253i	\(0.211350\pi\)
\(41\)	9.07409	+	0.953726i	1.41714	+	0.148947i	0.781916	−	0.623384i	\(-0.214243\pi\)
							0.635220	+	0.772331i	\(0.280909\pi\)
\(43\)	−3.68996	+	6.39119i	−0.562713	+	0.974648i	0.434545	+	0.900650i	\(0.356909\pi\)
							−0.997258	+	0.0739977i	\(0.976424\pi\)
\(47\)	4.70169	−	6.47132i	0.685811	−	0.943938i	−0.314174	−	0.949365i	\(-0.601728\pi\)
							0.999985	+	0.00542711i	\(0.00172751\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.a.49.1	✓	112
11.9	even	5	inner	429.2.bn.a.361.14	yes	112
13.4	even	6	inner	429.2.bn.a.82.14	yes	112
143.108	even	30	inner	429.2.bn.a.394.1	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.a.49.1	✓	112	1.1	even	1	trivial
429.2.bn.a.82.14	yes	112	13.4	even	6	inner
429.2.bn.a.361.14	yes	112	11.9	even	5	inner
429.2.bn.a.394.1	yes	112	143.108	even	30	inner