Embedded newform 429.2.bn.b.4.5

• Label: 429.2.bn.b.4.5
• 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			4.5
Character	\(\chi\)	\(=\)	429.4
Dual form			429.2.bn.b.322.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.425286 - 0.955208i) q^{2} +(0.669131 + 0.743145i) q^{3} +(0.606707 - 0.673817i) q^{4} +(2.35366 + 3.23953i) q^{5} +(0.425286 - 0.955208i) q^{6} +(-1.51333 - 1.36261i) q^{7} +(-2.89052 - 0.939186i) q^{8} +(-0.104528 + 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} - 8 q^{4} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.425286	−	0.955208i	−0.300723	−	0.675434i	0.698469	−	0.715640i	\(-0.253865\pi\)
							−0.999191	+	0.0402066i	\(0.987198\pi\)
\(3\)	0.669131	+	0.743145i	0.386323	+	0.429055i
							\(5\)	2.35366	+	3.23953i	1.05259	+	1.44876i	0.886544	+	0.462644i	\(0.153099\pi\)
0.166043	+	0.986118i	\(0.446901\pi\)
\(7\)	−1.51333	−	1.36261i	−0.571986	−	0.515019i	0.331603	−	0.943419i	\(-0.392411\pi\)
							−0.903589	+	0.428400i	\(0.859077\pi\)
\(11\)	3.06895	−	1.25758i	0.925325	−	0.379176i
							\(13\)	1.21609	−	3.39428i	0.337282	−	0.941404i
\(17\)	3.04954	+	1.35774i	0.739621	+	0.329301i								0.741731	−	0.670697i	\(-0.234005\pi\)
							−0.00210989	+	0.999998i	\(0.500672\pi\)
\(19\)	1.02583	+	4.82615i	0.235342	+	1.10720i	0.924084	+	0.382189i	\(0.124830\pi\)
							−0.688742	+	0.725006i	\(0.741837\pi\)
\(23\)	0.934625	−	1.61882i	0.194883	−	0.337547i	−0.751979	−	0.659187i	\(-0.770901\pi\)
							0.946862	+	0.321640i	\(0.104234\pi\)
\(29\)	4.84876	+	1.03064i	0.900392	+	0.191384i	0.634778	−	0.772695i	\(-0.281092\pi\)
							0.265615	+	0.964079i	\(0.414425\pi\)
\(31\)	−2.85262	+	3.92629i	−0.512345	+	0.705183i	−0.984313	−	0.176433i	\(-0.943544\pi\)
							0.471967	+	0.881616i	\(0.343544\pi\)
\(37\)	−0.00428725	+	0.0201699i	−0.000704820	+	0.00331592i	−0.978499	−	0.206253i	\(-0.933873\pi\)
							0.977794	+	0.209569i	\(0.0672062\pi\)
\(41\)	2.94868	−	2.65501i	0.460507	−	0.414643i	−0.405949	−	0.913896i	\(-0.633059\pi\)
							0.866456	+	0.499253i	\(0.166392\pi\)
\(43\)	−3.49161	−	6.04765i	−0.532466	−	0.922258i	−0.999281	−	0.0379031i	\(-0.987932\pi\)
							0.466816	−	0.884355i	\(-0.345401\pi\)
\(47\)	−5.14570	−	1.67194i	−0.750577	−	0.243877i	−0.0913474	−	0.995819i	\(-0.529117\pi\)
							−0.659229	+	0.751942i	\(0.729117\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.b.4.5	✓	112
11.3	even	5	inner	429.2.bn.b.355.5	yes	112
13.10	even	6	inner	429.2.bn.b.400.5	yes	112
143.36	even	30	inner	429.2.bn.b.322.5	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.b.4.5	✓	112	1.1	even	1	trivial
429.2.bn.b.322.5	yes	112	143.36	even	30	inner
429.2.bn.b.355.5	yes	112	11.3	even	5	inner
429.2.bn.b.400.5	yes	112	13.10	even	6	inner