Embedded newform 429.2.bs.a.184.14

• Label: 429.2.bs.a.184.14
• Level: $429$
• Weight: $2$
• Character: 429.184
• 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			184.14
Character	\(\chi\)	\(=\)	429.184
Dual form			429.2.bs.a.7.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.957096 + 2.49332i) q^{2} +(-0.669131 + 0.743145i) q^{3} +(-3.81432 + 3.43443i) q^{4} +(-2.91905 + 0.462332i) q^{5} +(-2.49332 - 0.957096i) q^{6} +(1.75832 + 0.0921498i) q^{7} +(-7.45456 - 3.79829i) q^{8} +(-0.104528 - 0.994522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 224 q - 28 q^{3} - 6 q^{5} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.957096	+	2.49332i	0.676769	+	1.76304i	0.645780	+	0.763523i	\(0.276532\pi\)
							0.0309885	+	0.999520i	\(0.490134\pi\)
\(3\)	−0.669131	+	0.743145i	−0.386323	+	0.429055i
							\(5\)	−2.91905	+	0.462332i	−1.30544	+	0.206761i	−0.770126	−	0.637892i	\(-0.779807\pi\)
−0.535314	+	0.844653i	\(0.679807\pi\)
\(7\)	1.75832	+	0.0921498i	0.664584	+	0.0348294i	0.381645	−	0.924309i	\(-0.375358\pi\)
							0.282938	+	0.959138i	\(0.408691\pi\)
\(11\)	3.31643	−	0.0362395i	0.999940	−	0.0109266i
							\(13\)	−3.56093	−	0.565478i	−0.987625	−	0.156835i
\(17\)	−3.35811	+	1.49513i	−0.814461	+	0.362621i								−0.771320	−	0.636448i	\(-0.780403\pi\)
							−0.0431411	+	0.999069i	\(0.513737\pi\)
\(19\)	3.01602	+	4.64427i	0.691923	+	1.06547i	0.993698	+	0.112089i	\(0.0357543\pi\)
							−0.301775	+	0.953379i	\(0.597579\pi\)
\(23\)	−3.42126	+	1.97527i	−0.713383	+	0.411872i	−0.812312	−	0.583223i	\(-0.801791\pi\)
							0.0989297	+	0.995094i	\(0.468458\pi\)
\(29\)	0.264998	+	1.24672i	0.0492089	+	0.231510i	0.995879	−	0.0906958i	\(-0.0289091\pi\)
							−0.946670	+	0.322206i	\(0.895576\pi\)
\(31\)	0.313797	−	1.98124i	0.0563597	−	0.355841i	−0.943351	−	0.331796i	\(-0.892345\pi\)
							0.999711	−	0.0240451i	\(-0.00765453\pi\)
\(37\)	−1.23838	−	0.804213i	−0.203588	−	0.132212i	0.438818	−	0.898576i	\(-0.355397\pi\)
							−0.642406	+	0.766364i	\(0.722064\pi\)
\(41\)	−4.45161	+	0.233299i	−0.695224	+	0.0364352i	−0.396675	−	0.917959i	\(-0.629836\pi\)
							−0.298549	+	0.954394i	\(0.596503\pi\)
\(43\)	6.20323	−	10.7443i	0.945984	−	1.63849i	0.192214	−	0.981353i	\(-0.438433\pi\)
							0.753770	−	0.657138i	\(-0.228233\pi\)
\(47\)	−4.30469	+	8.44843i	−0.627904	+	1.23233i	0.329658	+	0.944100i	\(0.393067\pi\)
							−0.957561	+	0.288230i	\(0.906933\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bs.a.184.14	yes	224
11.7	odd	10	inner	429.2.bs.a.106.14	yes	224
13.7	odd	12	inner	429.2.bs.a.85.14	yes	224
143.7	even	60	inner	429.2.bs.a.7.14	✓	224

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bs.a.7.14	✓	224	143.7	even	60	inner
429.2.bs.a.85.14	yes	224	13.7	odd	12	inner
429.2.bs.a.106.14	yes	224	11.7	odd	10	inner
429.2.bs.a.184.14	yes	224	1.1	even	1	trivial