Embedded newform 429.2.bn.a.4.8

• Label: 429.2.bn.a.4.8
• 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			4.8
Character	\(\chi\)	\(=\)	429.4
Dual form			429.2.bn.a.322.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.250968 + 0.563684i) q^{2} +(-0.669131 - 0.743145i) q^{3} +(1.08351 - 1.20336i) q^{4} +(-1.79971 - 2.47709i) q^{5} +(0.250968 - 0.563684i) q^{6} +(-2.23596 - 2.01327i) q^{7} +(2.12390 + 0.690096i) q^{8} +(-0.104528 + 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} - 20 q^{4} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.250968	+	0.563684i	0.177461	+	0.398585i	0.980273	−	0.197648i	\(-0.0633302\pi\)
							−0.802812	+	0.596233i	\(0.796664\pi\)
\(3\)	−0.669131	−	0.743145i	−0.386323	−	0.429055i
							\(5\)	−1.79971	−	2.47709i	−0.804855	−	1.10779i	−0.992097	−	0.125474i	\(-0.959955\pi\)
0.187242	−	0.982314i	\(-0.440045\pi\)
\(7\)	−2.23596	−	2.01327i	−0.845113	−	0.760944i	0.127877	−	0.991790i	\(-0.459184\pi\)
							−0.972990	+	0.230847i	\(0.925850\pi\)
\(11\)	−0.468330	+	3.28339i	−0.141207	+	0.989980i
							\(13\)	−1.14466	+	3.41903i	−0.317471	+	0.948268i
\(17\)	−6.64520	−	2.95863i	−1.61170	−	0.717574i								−0.614259	−	0.789104i	\(-0.710545\pi\)
							−0.997438	+	0.0715300i	\(0.977212\pi\)
\(19\)	−1.17514	−	5.52860i	−0.269596	−	1.26835i	−0.879510	−	0.475881i	\(-0.842129\pi\)
							0.609914	−	0.792468i	\(-0.291204\pi\)
\(23\)	−2.48611	+	4.30606i	−0.518389	+	0.897876i	0.481383	+	0.876511i	\(0.340135\pi\)
							−0.999772	+	0.0213657i	\(0.993199\pi\)
\(29\)	7.03971	+	1.49634i	1.30724	+	0.277863i	0.808299	−	0.588773i	\(-0.200389\pi\)
							0.498942	+	0.866635i	\(0.333722\pi\)
\(31\)	4.27727	−	5.88716i	0.768220	−	1.05736i	−0.228265	−	0.973599i	\(-0.573305\pi\)
							0.996485	−	0.0837655i	\(-0.0266947\pi\)
\(37\)	2.19115	−	10.3085i	0.360222	−	1.69471i	−0.308520	−	0.951218i	\(-0.599834\pi\)
							0.668742	−	0.743494i	\(-0.266833\pi\)
\(41\)	1.22814	−	1.10582i	0.191804	−	0.172701i	−0.567634	−	0.823281i	\(-0.692141\pi\)
							0.759437	+	0.650580i	\(0.225474\pi\)
\(43\)	2.59745	+	4.49892i	0.396108	+	0.686079i	0.993242	−	0.116063i	\(-0.0370273\pi\)
							−0.597134	+	0.802141i	\(0.703694\pi\)
\(47\)	−1.12484	−	0.365482i	−0.164075	−	0.0533111i	0.225828	−	0.974167i	\(-0.427491\pi\)
							−0.389902	+	0.920856i	\(0.627491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.a.4.8	✓	112
11.3	even	5	inner	429.2.bn.a.355.8	yes	112
13.10	even	6	inner	429.2.bn.a.400.8	yes	112
143.36	even	30	inner	429.2.bn.a.322.8	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.a.4.8	✓	112	1.1	even	1	trivial
429.2.bn.a.322.8	yes	112	143.36	even	30	inner
429.2.bn.a.355.8	yes	112	11.3	even	5	inner
429.2.bn.a.400.8	yes	112	13.10	even	6	inner