Embedded newform 430.2.r.a.323.18

• Label: 430.2.r.a.323.18
• 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			323.18
Character	\(\chi\)	\(=\)	430.323
Dual form			430.2.r.a.217.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.330279 + 0.943883i) q^{2} +(0.166855 - 0.476845i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(-1.39381 - 1.74851i) q^{5} +0.505194 q^{6} +(2.20495 + 2.20495i) q^{7} +(-0.846724 - 0.532032i) q^{8} +(2.14595 + 1.71134i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

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
							\(3\)	0.166855	−	0.476845i	0.0963339	−	0.275306i	−0.885625	−	0.464401i	\(-0.846270\pi\)
0.981959	+	0.189095i	\(0.0605553\pi\)
\(5\)	−1.39381	−	1.74851i	−0.623331	−	0.781958i
							\(7\)	2.20495	+	2.20495i	0.833394	+	0.833394i	0.987979	−	0.154585i	\(-0.0494041\pi\)
−0.154585	+	0.987979i	\(0.549404\pi\)
\(11\)	1.86146	−	2.33420i	0.561252	−	0.703788i	−0.417537	−	0.908660i	\(-0.637106\pi\)
							0.978789	+	0.204872i	\(0.0656779\pi\)
\(13\)	0.817471	−	1.30100i	0.226726	−	0.360832i	−0.713917	−	0.700230i	\(-0.753081\pi\)
							0.940643	+	0.339399i	\(0.110224\pi\)
\(17\)	−0.294593	−	0.468843i	−0.0714494	−	0.113711i	0.809089	−	0.587687i	\(-0.199961\pi\)
							−0.880538	+	0.473976i	\(0.842818\pi\)
\(19\)	3.02215	+	3.78966i	0.693329	+	0.869407i	0.996505	−	0.0835275i	\(-0.0266186\pi\)
							−0.303177	+	0.952934i	\(0.598047\pi\)
\(23\)	3.39715	−	0.382767i	0.708354	−	0.0798124i	0.249566	−	0.968358i	\(-0.419712\pi\)
							0.458788	+	0.888546i	\(0.348284\pi\)
\(29\)	3.68891	+	1.77648i	0.685013	+	0.329885i	0.743816	−	0.668385i	\(-0.233014\pi\)
							−0.0588028	+	0.998270i	\(0.518728\pi\)
\(31\)	2.60671	+	1.25533i	0.468179	+	0.225463i	0.653073	−	0.757295i	\(-0.273480\pi\)
							−0.184893	+	0.982759i	\(0.559194\pi\)
\(37\)	3.71772	+	3.71772i	0.611190	+	0.611190i	0.943256	−	0.332066i	\(-0.107746\pi\)
							−0.332066	+	0.943256i	\(0.607746\pi\)
\(41\)	−9.63950	−	4.64214i	−1.50544	−	0.724980i	−0.514273	−	0.857626i	\(-0.671938\pi\)
							−0.991163	+	0.132646i	\(0.957653\pi\)
\(43\)	−4.57855	−	4.69435i	−0.698222	−	0.715881i
							\(47\)	−10.7256	−	1.20848i	−1.56449	−	0.176275i	−0.713194	−	0.700967i	\(-0.752752\pi\)
−0.851293	+	0.524691i	\(0.824181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.323.18	yes	264
5.2	odd	4	inner	430.2.r.a.237.5	yes	264
43.2	odd	14	inner	430.2.r.a.303.5	yes	264
215.2	even	28	inner	430.2.r.a.217.18	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.217.18	✓	264	215.2	even	28	inner
430.2.r.a.237.5	yes	264	5.2	odd	4	inner
430.2.r.a.303.5	yes	264	43.2	odd	14	inner
430.2.r.a.323.18	yes	264	1.1	even	1	trivial