Embedded newform 430.2.r.a.323.8

• Label: 430.2.r.a.323.8
• Level: $430$
• Weight: $2$
• Character: 430.323
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $264$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 430 = 2 \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	430.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			323.8
Character	\(\chi\)	\(=\)	430.323
Dual form			430.2.r.a.217.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.330279 - 0.943883i) q^{2} +(0.239530 - 0.684537i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(-2.22909 - 0.176456i) q^{5} -0.725235 q^{6} +(2.27292 + 2.27292i) q^{7} +(0.846724 + 0.532032i) q^{8} +(1.93428 + 1.54254i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.330279	−	0.943883i	−0.233543	−	0.667426i
							\(3\)	0.239530	−	0.684537i	0.138293	−	0.395218i	−0.853820	−	0.520569i	\(-0.825720\pi\)
0.992112	+	0.125351i	\(0.0400057\pi\)
\(5\)	−2.22909	−	0.176456i	−0.996881	−	0.0789133i
							\(7\)	2.27292	+	2.27292i	0.859081	+	0.859081i	0.991230	−	0.132149i	\(-0.0421876\pi\)
−0.132149	+	0.991230i	\(0.542188\pi\)
\(11\)	−3.95951	+	4.96507i	−1.19384	+	1.49702i	−0.371107	+	0.928590i	\(0.621022\pi\)
							−0.822729	+	0.568434i	\(0.807550\pi\)
\(13\)	−0.739124	+	1.17631i	−0.204996	+	0.326249i	−0.933360	−	0.358941i	\(-0.883138\pi\)
							0.728364	+	0.685190i	\(0.240281\pi\)
\(17\)	−0.0746745	−	0.118844i	−0.0181112	−	0.0288239i	0.837546	−	0.546367i	\(-0.183990\pi\)
							−0.855657	+	0.517543i	\(0.826847\pi\)
\(19\)	−0.365716	−	0.458593i	−0.0839010	−	0.105209i	0.738109	−	0.674681i	\(-0.235719\pi\)
							−0.822010	+	0.569473i	\(0.807147\pi\)
\(23\)	1.59696	−	0.179934i	0.332989	−	0.0375189i	0.0561130	−	0.998424i	\(-0.482129\pi\)
							0.276876	+	0.960906i	\(0.410701\pi\)
\(29\)	3.14195	+	1.51308i	0.583445	+	0.280972i	0.702233	−	0.711947i	\(-0.252187\pi\)
							−0.118788	+	0.992920i	\(0.537901\pi\)
\(31\)	1.11828	+	0.538533i	0.200848	+	0.0967234i	0.531604	−	0.846993i	\(-0.321589\pi\)
							−0.330756	+	0.943716i	\(0.607304\pi\)
\(37\)	5.32912	+	5.32912i	0.876102	+	0.876102i	0.993129	−	0.117027i	\(-0.0373364\pi\)
							−0.117027	+	0.993129i	\(0.537336\pi\)
\(41\)	−6.98589	−	3.36423i	−1.09101	−	0.525404i	−0.200191	−	0.979757i	\(-0.564156\pi\)
							−0.890822	+	0.454352i	\(0.849871\pi\)
\(43\)	−1.23954	+	6.43922i	−0.189027	+	0.981972i
							\(47\)	2.44716	+	0.275729i	0.356955	+	0.0402192i	0.288621	−	0.957443i	\(-0.406803\pi\)
0.0683337	+	0.997663i	\(0.478232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.323.8	yes	264
5.2	odd	4	inner	430.2.r.a.237.15	yes	264
43.2	odd	14	inner	430.2.r.a.303.15	yes	264
215.2	even	28	inner	430.2.r.a.217.8	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.217.8	✓	264	215.2	even	28	inner
430.2.r.a.237.15	yes	264	5.2	odd	4	inner
430.2.r.a.303.15	yes	264	43.2	odd	14	inner
430.2.r.a.323.8	yes	264	1.1	even	1	trivial