Embedded newform 430.2.t.a.9.3

• Label: 430.2.t.a.9.3
• 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			9.3
Character	\(\chi\)	\(=\)	430.9
Dual form			430.2.t.a.239.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 + 0.623490i) q^{2} +(-0.299194 + 1.98502i) q^{3} +(0.222521 - 0.974928i) q^{4} +(0.934065 - 2.03163i) q^{5} +(-1.00372 - 1.73850i) q^{6} +(1.47449 + 0.851296i) q^{7} +(0.433884 + 0.900969i) q^{8} +(-0.984076 - 0.303547i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

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	\(0.244371\pi\)
−0.892240	+	0.451561i	\(0.850867\pi\)
\(5\)	0.934065	−	2.03163i	0.417726	−	0.908573i
							\(7\)	1.47449	+	0.851296i	0.557304	+	0.321760i	0.752063	−	0.659091i	\(-0.229059\pi\)
−0.194759	+	0.980851i	\(0.562392\pi\)
\(11\)	−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	0.688010	−	0.725701i	\(-0.258484\pi\)
							0.572166	+	0.820138i	\(0.306103\pi\)
\(17\)	1.78056	+	2.61160i	0.431849	+	0.633407i	0.979056	−	0.203591i	\(-0.0652612\pi\)
							−0.547207	+	0.836998i	\(0.684309\pi\)
\(19\)	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	0.190529	−	0.981681i	\(-0.438980\pi\)
							−0.993175	+	0.116636i	\(0.962789\pi\)
\(29\)	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	−0.527235	−	0.849720i	\(-0.676771\pi\)
							0.472262	+	0.881458i	\(0.343438\pi\)
\(41\)	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	−0.119214	−	0.992869i	\(-0.538037\pi\)
−0.538197	+	0.842819i	\(0.680894\pi\)

(See \(a_n\) instead)

Twists

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

    

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