Embedded newform 430.2.t.a.239.20

• Label: 430.2.t.a.239.20
• Level: $430$
• Weight: $2$
• Character: 430.239
• 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			239.20
Character	\(\chi\)	\(=\)	430.239
Dual form			430.2.t.a.9.20

$q$-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 \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{20}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.9.3	✓	264	43.9	even	21	inner
430.2.t.a.9.20	yes	264	215.9	even	42	inner
430.2.t.a.239.3	yes	264	5.4	even	2	inner
430.2.t.a.239.20	yes	264	1.1	even	1	trivial