Embedded newform 429.2.bi.a.5.6

• Label: 429.2.bi.a.5.6
• Level: $429$
• Weight: $2$
• Character: 429.5
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $416$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bi (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(416\)
Relative dimension:	\(52\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	429.5
Dual form			429.2.bi.a.86.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04506 + 2.05104i) q^{2} +(1.56544 + 0.741218i) q^{3} +(-1.93906 - 2.66889i) q^{4} +(0.172910 - 0.0881019i) q^{5} +(-3.15625 + 2.43617i) q^{6} +(3.48662 + 0.552227i) q^{7} +(2.95325 - 0.467749i) q^{8} +(1.90119 + 2.32066i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 12 q^{3} - 14 q^{6} - 12 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04506	+	2.05104i	−0.738968	+	1.45031i	0.148246	+	0.988950i	\(0.452637\pi\)
							−0.887215	+	0.461357i	\(0.847363\pi\)
\(3\)	1.56544	+	0.741218i	0.903806	+	0.427942i
							\(5\)	0.172910	−	0.0881019i	0.0773275	−	0.0394003i	−0.414899	−	0.909867i	\(-0.636183\pi\)
0.492227	+	0.870467i	\(0.336183\pi\)
\(7\)	3.48662	+	0.552227i	1.31782	+	0.208722i	0.775445	−	0.631415i	\(-0.217526\pi\)
							0.542374	+	0.840137i	\(0.317526\pi\)
\(11\)	3.25167	+	0.653155i	0.980417	+	0.196934i
							\(13\)	1.91195	−	3.05687i	0.530279	−	0.847823i
\(17\)	0.171964	−	0.529251i	0.0417074	−	0.128362i								−0.928035	−	0.372494i	\(-0.878503\pi\)
							0.969742	+	0.244131i	\(0.0785028\pi\)
\(19\)	−0.0860666	+	0.0136316i	−0.0197450	+	0.00312731i	−0.166299	−	0.986075i	\(-0.553182\pi\)
							0.146554	+	0.989203i	\(0.453182\pi\)
\(23\)	−4.88370			−1.01832			−0.509161	−	0.860671i	\(-0.670044\pi\)
							−0.509161	+	0.860671i	\(0.670044\pi\)
\(29\)	0.732422	+	1.00809i	0.136007	+	0.187198i	0.871588	−	0.490239i	\(-0.163091\pi\)
							−0.735581	+	0.677437i	\(0.763091\pi\)
\(31\)	−5.24944	−	2.67472i	−0.942828	−	0.480395i	−0.0861702	−	0.996280i	\(-0.527463\pi\)
							−0.856657	+	0.515886i	\(0.827463\pi\)
\(37\)	−9.45943	−	1.49823i	−1.55512	−	0.246307i	−0.681098	−	0.732192i	\(-0.738497\pi\)
							−0.874023	+	0.485885i	\(0.838497\pi\)
\(41\)	−1.56359	−	9.87212i	−0.244192	−	1.54177i	−0.739565	−	0.673085i	\(-0.764969\pi\)
							0.495373	−	0.868680i	\(-0.335031\pi\)
\(43\)			5.62598i			0.857953i	0.903316	+	0.428977i	\(0.141126\pi\)
							−0.903316	+	0.428977i	\(0.858874\pi\)
\(47\)	10.5174	−	1.66579i	1.53412	−	0.242981i	0.668513	−	0.743700i	\(-0.266931\pi\)
							0.865610	+	0.500719i	\(0.166931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bi.a.5.6	✓	416
3.2	odd	2	inner	429.2.bi.a.5.47	yes	416
11.9	even	5	inner	429.2.bi.a.317.47	yes	416
13.8	odd	4	inner	429.2.bi.a.203.6	yes	416
33.20	odd	10	inner	429.2.bi.a.317.6	yes	416
39.8	even	4	inner	429.2.bi.a.203.47	yes	416
143.86	odd	20	inner	429.2.bi.a.86.47	yes	416
429.86	even	20	inner	429.2.bi.a.86.6	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bi.a.5.6	✓	416	1.1	even	1	trivial
429.2.bi.a.5.47	yes	416	3.2	odd	2	inner
429.2.bi.a.86.6	yes	416	429.86	even	20	inner
429.2.bi.a.86.47	yes	416	143.86	odd	20	inner
429.2.bi.a.203.6	yes	416	13.8	odd	4	inner
429.2.bi.a.203.47	yes	416	39.8	even	4	inner
429.2.bi.a.317.6	yes	416	33.20	odd	10	inner
429.2.bi.a.317.47	yes	416	11.9	even	5	inner