Embedded newform 430.2.r.a.323.10

• Label: 430.2.r.a.323.10
• 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			323.10
Character	\(\chi\)	\(=\)	430.323
Dual form			430.2.r.a.217.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.330279 - 0.943883i) q^{2} +(0.881492 - 2.51916i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(2.19614 - 0.420679i) q^{5} -2.66893 q^{6} +(-1.98369 - 1.98369i) q^{7} +(0.846724 + 0.532032i) q^{8} +(-3.22364 - 2.57077i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

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
							\(3\)	0.881492	−	2.51916i	0.508929	−	1.45444i	−0.347257	−	0.937770i	\(-0.612887\pi\)
0.856186	−	0.516667i	\(-0.172827\pi\)
\(5\)	2.19614	−	0.420679i	0.982144	−	0.188133i
							\(7\)	−1.98369	−	1.98369i	−0.749763	−	0.749763i	0.224671	−	0.974435i	\(-0.427869\pi\)
−0.974435	+	0.224671i	\(0.927869\pi\)
\(11\)	−2.09488	+	2.62689i	−0.631629	+	0.792038i	−0.989928	−	0.141569i	\(-0.954785\pi\)
							0.358300	+	0.933607i	\(0.383357\pi\)
\(13\)	1.09056	−	1.73561i	0.302466	−	0.481372i	−0.660562	−	0.750771i	\(-0.729682\pi\)
							0.963029	+	0.269399i	\(0.0868250\pi\)
\(17\)	−3.77235	−	6.00365i	−0.914928	−	1.45610i	−0.889766	−	0.456417i	\(-0.849133\pi\)
							−0.0251622	−	0.999683i	\(-0.508010\pi\)
\(19\)	3.05248	+	3.82769i	0.700288	+	0.878133i	0.997045	−	0.0768197i	\(-0.0244766\pi\)
							−0.296757	+	0.954953i	\(0.595905\pi\)
\(23\)	1.60488	−	0.180827i	0.334641	−	0.0377050i	0.0569545	−	0.998377i	\(-0.481861\pi\)
							0.277686	+	0.960672i	\(0.410432\pi\)
\(29\)	5.14194	+	2.47623i	0.954835	+	0.459824i	0.845379	−	0.534168i	\(-0.179375\pi\)
							0.109456	+	0.993992i	\(0.465089\pi\)
\(31\)	−5.99266	−	2.88591i	−1.07631	−	0.518325i	−0.190177	−	0.981750i	\(-0.560906\pi\)
							−0.886136	+	0.463425i	\(0.846620\pi\)
\(37\)	2.06191	+	2.06191i	0.338976	+	0.338976i	0.855982	−	0.517006i	\(-0.172953\pi\)
							−0.517006	+	0.855982i	\(0.672953\pi\)
\(41\)	7.12791	+	3.43262i	1.11319	+	0.536085i	0.897783	−	0.440439i	\(-0.145177\pi\)
							0.215409	+	0.976524i	\(0.430891\pi\)
\(43\)	−4.13360	+	5.09052i	−0.630368	+	0.776297i
							\(47\)	1.62565	+	0.183166i	0.237125	+	0.0267176i	0.229727	−	0.973255i	\(-0.426217\pi\)
0.00739774	+	0.999973i	\(0.497645\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.323.10	yes	264
5.2	odd	4	inner	430.2.r.a.237.13	yes	264
43.2	odd	14	inner	430.2.r.a.303.13	yes	264
215.2	even	28	inner	430.2.r.a.217.10	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.217.10	✓	264	215.2	even	28	inner
430.2.r.a.237.13	yes	264	5.2	odd	4	inner
430.2.r.a.303.13	yes	264	43.2	odd	14	inner
430.2.r.a.323.10	yes	264	1.1	even	1	trivial