Embedded newform 429.2.bs.b.145.11

• Label: 429.2.bs.b.145.11
• Level: $429$
• Weight: $2$
• Character: 429.145
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bs (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(14\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			145.11
Character	\(\chi\)	\(=\)	429.145
Dual form			429.2.bs.b.358.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.17030 + 0.759999i) q^{2} +(0.913545 + 0.406737i) q^{3} +(-0.0214796 - 0.0482440i) q^{4} +(0.436783 + 0.222552i) q^{5} +(0.759999 + 1.17030i) q^{6} +(1.37613 - 3.58494i) q^{7} +(0.448111 - 2.82926i) q^{8} +(0.669131 + 0.743145i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q + 28 q^{3} + 6 q^{5} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.17030	+	0.759999i	0.827524	+	0.537400i	0.887504	−	0.460799i	\(-0.152437\pi\)
							−0.0599804	+	0.998200i	\(0.519104\pi\)
\(3\)	0.913545	+	0.406737i	0.527436	+	0.234830i
							\(5\)	0.436783	+	0.222552i	0.195335	+	0.0995283i	0.548922	−	0.835873i	\(-0.315038\pi\)
−0.353587	+	0.935402i	\(0.615038\pi\)
\(7\)	1.37613	−	3.58494i	0.520128	−	1.35498i	−0.382259	−	0.924055i	\(-0.624854\pi\)
							0.902388	−	0.430925i	\(-0.141813\pi\)
\(11\)	−2.08840	+	2.57654i	−0.629678	+	0.776857i
							\(13\)	3.31820	+	1.41051i	0.920303	+	0.391206i
\(17\)	4.63612	+	0.985438i	1.12442	+	0.239004i								0.732349	−	0.680929i	\(-0.238424\pi\)
							0.392075	+	0.919933i	\(0.371757\pi\)
\(19\)	−0.479057	+	0.591586i	−0.109903	+	0.135719i	−0.829129	−	0.559057i	\(-0.811163\pi\)
							0.719226	+	0.694776i	\(0.244497\pi\)
\(23\)	−7.92658	+	4.57641i	−1.65281	+	0.954248i	−0.676894	+	0.736081i	\(0.736674\pi\)
							−0.975912	+	0.218167i	\(0.929992\pi\)
\(29\)	−6.92007	−	0.727328i	−1.28502	−	0.135061i	−0.562726	−	0.826643i	\(-0.690247\pi\)
							−0.722298	+	0.691582i	\(0.756914\pi\)
\(31\)	2.59260	+	5.08826i	0.465645	+	0.913879i	0.997740	+	0.0671944i	\(0.0214048\pi\)
							−0.532095	+	0.846684i	\(0.678595\pi\)
\(37\)	2.19010	−	1.77351i	0.360050	−	0.291563i	−0.432178	−	0.901788i	\(-0.642255\pi\)
							0.792227	+	0.610226i	\(0.208921\pi\)
\(41\)	1.61992	+	4.22004i	0.252989	+	0.659059i	0.999983	−	0.00577141i	\(-0.00183711\pi\)
							−0.746994	+	0.664830i	\(0.768504\pi\)
\(43\)	−2.23571	+	3.87236i	−0.340943	+	0.590530i	−0.984608	−	0.174777i	\(-0.944080\pi\)
							0.643665	+	0.765307i	\(0.277413\pi\)
\(47\)	−2.44138	−	0.386677i	−0.356112	−	0.0564026i	−0.0241846	−	0.999708i	\(-0.507699\pi\)
							−0.331928	+	0.943305i	\(0.607699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bs.b.145.11	yes	224
11.6	odd	10	inner	429.2.bs.b.28.11	✓	224
13.7	odd	12	inner	429.2.bs.b.46.11	yes	224
143.72	even	60	inner	429.2.bs.b.358.11	yes	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bs.b.28.11	✓	224	11.6	odd	10	inner
429.2.bs.b.46.11	yes	224	13.7	odd	12	inner
429.2.bs.b.145.11	yes	224	1.1	even	1	trivial
429.2.bs.b.358.11	yes	224	143.72	even	60	inner