Embedded newform 430.2.t.a.9.17

• Label: 430.2.t.a.9.17
• Level: $430$
• Weight: $2$
• Character: 430.9
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 430 = 2 \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	430.t (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			9.17
Character	\(\chi\)	\(=\)	430.9
Dual form			430.2.t.a.239.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.781831 - 0.623490i) q^{2} +(-0.0612820 + 0.406579i) q^{3} +(0.222521 - 0.974928i) q^{4} +(1.30031 + 1.81912i) q^{5} +(0.205586 + 0.356085i) q^{6} +(-1.79116 - 1.03412i) q^{7} +(-0.433884 - 0.900969i) q^{8} +(2.70517 + 0.834434i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.781831	−	0.623490i	0.552838	−	0.440874i
							\(3\)	−0.0612820	+	0.406579i	−0.0353812	+	0.234739i	−0.999531	−	0.0306193i	\(-0.990252\pi\)
0.964150	+	0.265358i	\(0.0854901\pi\)
\(5\)	1.30031	+	1.81912i	0.581518	+	0.813534i
							\(7\)	−1.79116	−	1.03412i	−0.676993	−	0.390862i	0.121728	−	0.992564i	\(-0.461156\pi\)
−0.798721	+	0.601701i	\(0.794490\pi\)
\(11\)	−0.164406	−	0.720308i	−0.0495701	−	0.217181i	0.944076	−	0.329727i	\(-0.106957\pi\)
							−0.993647	+	0.112546i	\(0.964099\pi\)
\(13\)	4.81095	+	0.360531i	1.33432	+	0.0999933i	0.722701	−	0.691161i	\(-0.242900\pi\)
							0.611617	+	0.791154i	\(0.290519\pi\)
\(17\)	2.01262	+	2.95196i	0.488131	+	0.715957i	0.988728	−	0.149724i	\(-0.0478385\pi\)
							−0.500597	+	0.865680i	\(0.666886\pi\)
\(19\)	3.40026	−	1.04884i	0.780073	−	0.240621i	0.120953	−	0.992658i	\(-0.461405\pi\)
							0.659120	+	0.752038i	\(0.270929\pi\)
\(23\)	−4.68581	−	5.05010i	−0.977059	−	1.05302i	−0.998581	−	0.0532508i	\(-0.983042\pi\)
							0.0215222	−	0.999768i	\(-0.493149\pi\)
\(29\)	3.53084	−	0.532189i	0.655661	−	0.0988250i	0.187217	−	0.982318i	\(-0.440053\pi\)
							0.468443	+	0.883494i	\(0.344815\pi\)
\(31\)	−2.81187	+	7.16452i	−0.505026	+	1.28679i	0.420222	+	0.907421i	\(0.361952\pi\)
							−0.925248	+	0.379364i	\(0.876143\pi\)
\(37\)	−2.68547	+	1.55046i	−0.441489	+	0.254894i	−0.704229	−	0.709973i	\(-0.748707\pi\)
							0.262740	+	0.964867i	\(0.415374\pi\)
\(41\)	−6.88900	−	8.63853i	−1.07588	−	1.34911i	−0.933206	−	0.359343i	\(-0.883001\pi\)
							−0.142675	−	0.989770i	\(-0.545571\pi\)
\(43\)	−2.14015	−	6.19837i	−0.326370	−	0.945242i
							\(47\)	4.78384	+	1.09188i	0.697795	+	0.159267i	0.556681	−	0.830726i	\(-0.312074\pi\)
0.141114	+	0.989993i	\(0.454932\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.t.a.9.17	yes	264
5.4	even	2	inner	430.2.t.a.9.6	✓	264
43.24	even	21	inner	430.2.t.a.239.6	yes	264
215.24	even	42	inner	430.2.t.a.239.17	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.9.6	✓	264	5.4	even	2	inner
430.2.t.a.9.17	yes	264	1.1	even	1	trivial
430.2.t.a.239.6	yes	264	43.24	even	21	inner
430.2.t.a.239.17	yes	264	215.24	even	42	inner