Embedded newform 430.2.r.a.323.21

• Label: 430.2.r.a.323.21
• Level: $430$
• Weight: $2$
• Character: 430.323
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $264$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 430 = 2 \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	430.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			323.21
Character	\(\chi\)	\(=\)	430.323
Dual form			430.2.r.a.217.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.330279 + 0.943883i) q^{2} +(0.751711 - 2.14827i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(2.11770 - 0.717883i) q^{5} +2.27599 q^{6} +(1.96966 + 1.96966i) q^{7} +(-0.846724 - 0.532032i) q^{8} +(-1.70448 - 1.35928i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.330279	+	0.943883i	0.233543	+	0.667426i
							\(3\)	0.751711	−	2.14827i	0.434001	−	1.24030i	−0.494567	−	0.869139i	\(-0.664673\pi\)
0.928568	−	0.371163i	\(-0.121041\pi\)
\(5\)	2.11770	−	0.717883i	0.947063	−	0.321047i
							\(7\)	1.96966	+	1.96966i	0.744462	+	0.744462i	0.973433	−	0.228971i	\(-0.0735362\pi\)
−0.228971	+	0.973433i	\(0.573536\pi\)
\(11\)	−0.0258033	+	0.0323563i	−0.00777998	+	0.00975579i	−0.785706	−	0.618600i	\(-0.787700\pi\)
							0.777926	+	0.628356i	\(0.216272\pi\)
\(13\)	−0.0324261	+	0.0516058i	−0.00899337	+	0.0143129i	−0.851191	−	0.524857i	\(-0.824119\pi\)
							0.842197	+	0.539170i	\(0.181262\pi\)
\(17\)	−0.00982881	−	0.0156425i	−0.00238384	−	0.00379385i	0.845530	−	0.533928i	\(-0.179285\pi\)
							−0.847914	+	0.530134i	\(0.822142\pi\)
\(19\)	−1.07389	−	1.34661i	−0.246367	−	0.308934i	0.643237	−	0.765667i	\(-0.277591\pi\)
							−0.889604	+	0.456733i	\(0.849019\pi\)
\(23\)	−3.86438	+	0.435411i	−0.805778	+	0.0907894i	−0.505238	−	0.862980i	\(-0.668595\pi\)
							−0.300540	+	0.953769i	\(0.597167\pi\)
\(29\)	−4.83712	−	2.32944i	−0.898231	−	0.432565i	−0.0729815	−	0.997333i	\(-0.523251\pi\)
							−0.825250	+	0.564768i	\(0.808966\pi\)
\(31\)	3.46918	+	1.67067i	0.623083	+	0.300061i	0.718661	−	0.695361i	\(-0.244755\pi\)
							−0.0955774	+	0.995422i	\(0.530470\pi\)
\(37\)	−0.756125	−	0.756125i	−0.124306	−	0.124306i	0.642217	−	0.766523i	\(-0.278015\pi\)
							−0.766523	+	0.642217i	\(0.778015\pi\)
\(41\)	6.53050	+	3.14492i	1.01989	+	0.491154i	0.867643	−	0.497188i	\(-0.165634\pi\)
							0.152249	+	0.988342i	\(0.451348\pi\)
\(43\)	−6.13151	+	2.32477i	−0.935047	+	0.354524i
							\(47\)	−0.870885	−	0.0981252i	−0.127032	−	0.0143130i	0.0482197	−	0.998837i	\(-0.484645\pi\)
−0.175251	+	0.984524i	\(0.556074\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.323.21	yes	264
5.2	odd	4	inner	430.2.r.a.237.2	yes	264
43.2	odd	14	inner	430.2.r.a.303.2	yes	264
215.2	even	28	inner	430.2.r.a.217.21	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.217.21	✓	264	215.2	even	28	inner
430.2.r.a.237.2	yes	264	5.2	odd	4	inner
430.2.r.a.303.2	yes	264	43.2	odd	14	inner
430.2.r.a.323.21	yes	264	1.1	even	1	trivial