Embedded newform 429.2.bn.a.4.2

• Label: 429.2.bn.a.4.2
• 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			4.2
Character	\(\chi\)	\(=\)	429.4
Dual form			429.2.bn.a.322.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.866606 - 1.94643i) q^{2} +(-0.669131 - 0.743145i) q^{3} +(-1.69932 + 1.88729i) q^{4} +(1.80442 + 2.48357i) q^{5} +(-0.866606 + 1.94643i) q^{6} +(-3.61054 - 3.25094i) q^{7} +(1.09341 + 0.355271i) q^{8} +(-0.104528 + 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} - 20 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.866606	−	1.94643i	−0.612783	−	1.37633i	−0.907217	−	0.420662i	\(-0.861798\pi\)
							0.294434	−	0.955672i	\(-0.404869\pi\)
\(3\)	−0.669131	−	0.743145i	−0.386323	−	0.429055i
							\(5\)	1.80442	+	2.48357i	0.806960	+	1.11069i	0.991785	+	0.127914i	\(0.0408281\pi\)
−0.184825	+	0.982771i	\(0.559172\pi\)
\(7\)	−3.61054	−	3.25094i	−1.36465	−	1.22874i	−0.947441	−	0.319932i	\(-0.896340\pi\)
							−0.417214	−	0.908808i	\(-0.636993\pi\)
\(11\)	−0.927339	−	3.18434i	−0.279603	−	0.960116i
							\(13\)	−0.373885	+	3.58611i	−0.103697	+	0.994609i
\(17\)	−5.84955	−	2.60439i	−1.41872	−	0.631657i								−0.453067	−	0.891477i	\(-0.649670\pi\)
							−0.965657	+	0.259820i	\(0.916337\pi\)
\(19\)	1.27837	+	6.01424i	0.293277	+	1.37976i	0.840063	+	0.542489i	\(0.182518\pi\)
							−0.546786	+	0.837273i	\(0.684149\pi\)
\(23\)	−2.33482	+	4.04404i	−0.486845	+	0.843240i	−0.999886	−	0.0151245i	\(-0.995186\pi\)
							0.513041	+	0.858364i	\(0.328519\pi\)
\(29\)	−6.11647	−	1.30010i	−1.13580	−	0.241422i	−0.398619	−	0.917117i	\(-0.630510\pi\)
							−0.737182	+	0.675695i	\(0.763844\pi\)
\(31\)	0.455055	−	0.626329i	0.0817302	−	0.112492i	−0.766194	−	0.642609i	\(-0.777852\pi\)
							0.847925	+	0.530117i	\(0.177852\pi\)
\(37\)	−0.139581	+	0.656679i	−0.0229470	+	0.107957i	−0.988131	−	0.153616i	\(-0.950908\pi\)
							0.965184	+	0.261574i	\(0.0842414\pi\)
\(41\)	−0.321489	+	0.289470i	−0.0502081	+	0.0452076i	−0.693853	−	0.720117i	\(-0.744088\pi\)
							0.643645	+	0.765324i	\(0.277421\pi\)
\(43\)	−0.613294	−	1.06226i	−0.0935265	−	0.161993i	0.815466	−	0.578805i	\(-0.196481\pi\)
							−0.908993	+	0.416812i	\(0.863147\pi\)
\(47\)	5.86219	+	1.90474i	0.855089	+	0.277835i	0.703576	−	0.710620i	\(-0.251586\pi\)
							0.151513	+	0.988455i	\(0.451586\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.a.4.2	✓	112
11.3	even	5	inner	429.2.bn.a.355.2	yes	112
13.10	even	6	inner	429.2.bn.a.400.2	yes	112
143.36	even	30	inner	429.2.bn.a.322.2	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.a.4.2	✓	112	1.1	even	1	trivial
429.2.bn.a.322.2	yes	112	143.36	even	30	inner
429.2.bn.a.355.2	yes	112	11.3	even	5	inner
429.2.bn.a.400.2	yes	112	13.10	even	6	inner