Embedded newform 430.2.t.a.9.4

• Label: 430.2.t.a.9.4
• 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			9.4
Character	\(\chi\)	\(=\)	430.9
Dual form			430.2.t.a.239.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 + 0.623490i) q^{2} +(-0.192290 + 1.27576i) q^{3} +(0.222521 - 0.974928i) q^{4} +(2.15300 - 0.603819i) q^{5} +(-0.645086 - 1.11732i) q^{6} +(-4.41993 - 2.55185i) q^{7} +(0.433884 + 0.900969i) q^{8} +(1.27612 + 0.393632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.192290	+	1.27576i	−0.111019	+	0.736562i	0.862181	+	0.506600i	\(0.169098\pi\)
−0.973200	+	0.229961i	\(0.926140\pi\)
\(5\)	2.15300	−	0.603819i	0.962850	−	0.270036i
							\(7\)	−4.41993	−	2.55185i	−1.67058	−	0.964508i	−0.967315	−	0.253578i	\(-0.918392\pi\)
−0.703263	−	0.710930i	\(-0.748274\pi\)
\(11\)	0.949960	+	4.16205i	0.286424	+	1.25490i	0.889394	+	0.457141i	\(0.151127\pi\)
							−0.602970	+	0.797764i	\(0.706016\pi\)
\(13\)	4.44713	+	0.333266i	1.23341	+	0.0924314i	0.675431	−	0.737423i	\(-0.263958\pi\)
							0.557980	+	0.829854i	\(0.311577\pi\)
\(17\)	2.18240	+	3.20100i	0.529311	+	0.776357i	0.994031	−	0.109101i	\(-0.0347973\pi\)
							−0.464720	+	0.885458i	\(0.653845\pi\)
\(19\)	−0.758724	+	0.234035i	−0.174063	+	0.0536914i	−0.380562	−	0.924756i	\(-0.624269\pi\)
							0.206498	+	0.978447i	\(0.433793\pi\)
\(23\)	3.90443	+	4.20797i	0.814129	+	0.877422i	0.994344	−	0.106205i	\(-0.0338699\pi\)
							−0.180215	+	0.983627i	\(0.557679\pi\)
\(29\)	1.09445	−	0.164962i	0.203235	−	0.0306327i	−0.0466357	−	0.998912i	\(-0.514850\pi\)
							0.249871	+	0.968279i	\(0.419612\pi\)
\(31\)	1.73972	−	4.43274i	0.312463	−	0.796143i	−0.685234	−	0.728323i	\(-0.740300\pi\)
							0.997698	−	0.0678206i	\(-0.0216045\pi\)
\(37\)	−5.26327	+	3.03875i	−0.865277	+	0.499568i	−0.865776	−	0.500432i	\(-0.833174\pi\)
							0.000499191		1.00000i	\(0.499841\pi\)
\(41\)	−4.11570	−	5.16093i	−0.642765	−	0.806001i	0.348581	−	0.937279i	\(-0.386664\pi\)
							−0.991346	+	0.131277i	\(0.958092\pi\)
\(43\)	3.61442	+	5.47138i	0.551194	+	0.834377i
							\(47\)	8.13669	+	1.85715i	1.18686	+	0.270893i	0.769979	−	0.638069i	\(-0.220267\pi\)
0.416880	+	0.908962i	\(0.363124\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.t.a.9.4	✓	264
5.4	even	2	inner	430.2.t.a.9.19	yes	264
43.24	even	21	inner	430.2.t.a.239.19	yes	264
215.24	even	42	inner	430.2.t.a.239.4	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.9.4	✓	264	1.1	even	1	trivial
430.2.t.a.9.19	yes	264	5.4	even	2	inner
430.2.t.a.239.4	yes	264	215.24	even	42	inner
430.2.t.a.239.19	yes	264	43.24	even	21	inner