Embedded newform 429.2.bn.b.4.7

• Label: 429.2.bn.b.4.7
• Level: $429$
• Weight: $2$
• Character: 429.4
• 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			4.7
Character	\(\chi\)	\(=\)	429.4
Dual form			429.2.bn.b.322.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0203023 + 0.0455997i) q^{2} +(0.669131 + 0.743145i) q^{3} +(1.33659 - 1.48444i) q^{4} +(1.77101 + 2.43759i) q^{5} +(-0.0203023 + 0.0455997i) q^{6} +(0.890300 + 0.801630i) q^{7} +(0.189770 + 0.0616600i) q^{8} +(-0.104528 + 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} - 8 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0203023	+	0.0455997i	0.0143559	+	0.0322439i	0.920581	−	0.390551i	\(-0.127716\pi\)
							−0.906225	+	0.422795i	\(0.861049\pi\)
\(3\)	0.669131	+	0.743145i	0.386323	+	0.429055i
							\(5\)	1.77101	+	2.43759i	0.792022	+	1.09012i	0.993853	+	0.110704i	\(0.0353105\pi\)
−0.201832	+	0.979420i	\(0.564689\pi\)
\(7\)	0.890300	+	0.801630i	0.336502	+	0.302988i	0.820007	−	0.572354i	\(-0.193970\pi\)
							−0.483505	+	0.875342i	\(0.660636\pi\)
\(11\)	−3.12854	+	1.10102i	−0.943289	+	0.331971i
							\(13\)	2.40887	+	2.68279i	0.668101	+	0.744071i
\(17\)	−4.05319	−	1.80459i	−0.983042	−	0.437678i								−0.148674	−	0.988886i	\(-0.547500\pi\)
							−0.834368	+	0.551208i	\(0.814167\pi\)
\(19\)	−0.828760	−	3.89901i	−0.190131	−	0.894494i	−0.964975	−	0.262342i	\(-0.915505\pi\)
							0.774844	−	0.632152i	\(-0.217828\pi\)
\(23\)	−0.393084	+	0.680842i	−0.0819638	+	0.141965i	−0.904093	−	0.427335i	\(-0.859453\pi\)
							0.822130	+	0.569300i	\(0.192786\pi\)
\(29\)	−2.03665	−	0.432902i	−0.378196	−	0.0803880i	0.0148917	−	0.999889i	\(-0.495260\pi\)
							−0.393087	+	0.919501i	\(0.628593\pi\)
\(31\)	5.95827	−	8.20086i	1.07014	−	1.47292i	0.200200	−	0.979755i	\(-0.435841\pi\)
							0.869937	−	0.493162i	\(-0.164159\pi\)
\(37\)	−2.05838	+	9.68392i	−0.338396	+	1.59203i	0.399252	+	0.916841i	\(0.369270\pi\)
							−0.737648	+	0.675186i	\(0.764063\pi\)
\(41\)	2.95939	−	2.66464i	0.462179	−	0.416148i	−0.404867	−	0.914375i	\(-0.632682\pi\)
							0.867046	+	0.498228i	\(0.166016\pi\)
\(43\)	−3.37657	−	5.84838i	−0.514922	−	0.891870i	−0.999850	−	0.0173164i	\(-0.994488\pi\)
							0.484929	−	0.874554i	\(-0.338846\pi\)
\(47\)	11.5582	+	3.75550i	1.68594	+	0.547796i	0.986050	−	0.166451i	\(-0.0532308\pi\)
							0.699893	+	0.714247i	\(0.253231\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.b.4.7	✓	112
11.3	even	5	inner	429.2.bn.b.355.7	yes	112
13.10	even	6	inner	429.2.bn.b.400.7	yes	112
143.36	even	30	inner	429.2.bn.b.322.7	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.b.4.7	✓	112	1.1	even	1	trivial
429.2.bn.b.322.7	yes	112	143.36	even	30	inner
429.2.bn.b.355.7	yes	112	11.3	even	5	inner
429.2.bn.b.400.7	yes	112	13.10	even	6	inner