Embedded newform 430.2.t.a.9.20

• Label: 430.2.t.a.9.20
• 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			9.20
Character	\(\chi\)	\(=\)	430.9
Dual form			430.2.t.a.239.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.781831 - 0.623490i) q^{2} +(0.299194 - 1.98502i) q^{3} +(0.222521 - 0.974928i) q^{4} +(-1.54994 + 1.61173i) q^{5} +(-1.00372 - 1.73850i) q^{6} +(-1.47449 - 0.851296i) q^{7} +(-0.433884 - 0.900969i) q^{8} +(-0.984076 - 0.303547i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

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
							\(3\)	0.299194	−	1.98502i	0.172740	−	1.14605i	−0.719501	−	0.694492i	\(-0.755629\pi\)
0.892240	−	0.451561i	\(-0.149133\pi\)
\(5\)	−1.54994	+	1.61173i	−0.693154	+	0.720790i
							\(7\)	−1.47449	−	0.851296i	−0.557304	−	0.321760i	0.194759	−	0.980851i	\(-0.437608\pi\)
−0.752063	+	0.659091i	\(0.770941\pi\)
\(11\)	−1.34524	−	5.89389i	−0.405606	−	1.77708i	−0.604032	−	0.796960i	\(-0.706440\pi\)
							0.198426	−	0.980116i	\(-0.436417\pi\)
\(13\)	−4.54363	−	0.340498i	−1.26018	−	0.0944371i	−0.572166	−	0.820138i	\(-0.693897\pi\)
							−0.688010	+	0.725701i	\(0.741516\pi\)
\(17\)	−1.78056	−	2.61160i	−0.431849	−	0.633407i	0.547207	−	0.836998i	\(-0.315691\pi\)
							−0.979056	+	0.203591i	\(0.934739\pi\)
\(19\)	5.14335	−	1.58651i	1.17997	−	0.363971i	0.358117	−	0.933677i	\(-0.383419\pi\)
							0.821849	+	0.569705i	\(0.192943\pi\)
\(23\)	3.84935	+	4.14861i	0.802645	+	0.865046i	0.993175	−	0.116636i	\(-0.0372109\pi\)
							−0.190529	+	0.981681i	\(0.561020\pi\)
\(29\)	5.49258	−	0.827873i	1.01995	−	0.153732i	0.382281	−	0.924046i	\(-0.375139\pi\)
							0.637665	+	0.770314i	\(0.279901\pi\)
\(31\)	−3.38921	+	8.63556i	−0.608720	+	1.55099i	0.210549	+	0.977583i	\(0.432475\pi\)
							−0.819269	+	0.573410i	\(0.805620\pi\)
\(37\)	0.334388	−	0.193059i	0.0549730	−	0.0317387i	−0.472262	−	0.881458i	\(-0.656562\pi\)
							0.527235	+	0.849720i	\(0.323229\pi\)
\(41\)	6.43226	+	8.06579i	1.00455	+	1.25967i	0.965493	+	0.260428i	\(0.0838638\pi\)
							0.0390564	+	0.999237i	\(0.487565\pi\)
\(43\)	4.81520	−	4.45128i	0.734311	−	0.678813i
							\(47\)	4.50698	+	1.02869i	0.657411	+	0.150050i	0.538197	−	0.842819i	\(-0.319106\pi\)
0.119214	+	0.992869i	\(0.461963\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.t.a.9.20	yes	264
5.4	even	2	inner	430.2.t.a.9.3	✓	264
43.24	even	21	inner	430.2.t.a.239.3	yes	264
215.24	even	42	inner	430.2.t.a.239.20	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.9.3	✓	264	5.4	even	2	inner
430.2.t.a.9.20	yes	264	1.1	even	1	trivial
430.2.t.a.239.3	yes	264	43.24	even	21	inner
430.2.t.a.239.20	yes	264	215.24	even	42	inner