Embedded newform 430.2.r.a.27.8

• Label: 430.2.r.a.27.8
• Level: $430$
• Weight: $2$
• Character: 430.27
• 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.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			27.8
Character	\(\chi\)	\(=\)	430.27
Dual form			430.2.r.a.223.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.846724 - 0.532032i) q^{2} +(1.28668 - 0.808477i) q^{3} +(0.433884 + 0.900969i) q^{4} +(2.21966 + 0.270414i) q^{5} -1.51960 q^{6} +(2.16897 - 2.16897i) q^{7} +(0.111964 - 0.993712i) q^{8} +(-0.299731 + 0.622397i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.846724	−	0.532032i	−0.598724	−	0.376203i
							\(3\)	1.28668	−	0.808477i	0.742867	−	0.466775i	−0.106676	−	0.994294i	\(-0.534021\pi\)
0.849543	+	0.527519i	\(0.176878\pi\)
\(5\)	2.21966	+	0.270414i	0.992661	+	0.120933i
							\(7\)	2.16897	−	2.16897i	0.819794	−	0.819794i	−0.166284	−	0.986078i	\(-0.553177\pi\)
0.986078	+	0.166284i	\(0.0531770\pi\)
\(11\)	0.682167	+	0.328515i	0.205681	+	0.0990509i	0.533889	−	0.845555i	\(-0.320730\pi\)
							−0.328207	+	0.944606i	\(0.606444\pi\)
\(13\)	−0.801352	−	0.0902907i	−0.222255	−	0.0250421i	0.000134607	−	1.00000i	\(-0.499957\pi\)
							−0.222390	+	0.974958i	\(0.571386\pi\)
\(17\)	4.75875	−	0.536183i	1.15417	−	0.130043i	0.485934	−	0.873995i	\(-0.338479\pi\)
							0.668233	+	0.743952i	\(0.267051\pi\)
\(19\)	−4.37331	+	2.10608i	−1.00331	+	0.483167i	−0.862059	−	0.506807i	\(-0.830826\pi\)
							−0.141247	+	0.989974i	\(0.545111\pi\)
\(23\)	−4.92817	+	1.72444i	−1.02759	+	0.359571i	−0.790856	−	0.612002i	\(-0.790364\pi\)
							−0.236739	+	0.971573i	\(0.576079\pi\)
\(29\)	−1.89002	−	8.28073i	−0.350968	−	1.53769i	−0.774951	−	0.632022i	\(-0.782225\pi\)
							0.423982	−	0.905670i	\(-0.360632\pi\)
\(31\)	1.34020	+	5.87178i	0.240706	+	1.05460i	0.940376	+	0.340136i	\(0.110473\pi\)
							−0.699670	+	0.714466i	\(0.746670\pi\)
\(37\)	3.18696	−	3.18696i	0.523933	−	0.523933i	−0.394824	−	0.918757i	\(-0.629194\pi\)
							0.918757	+	0.394824i	\(0.129194\pi\)
\(41\)	−1.00503	−	4.40333i	−0.156959	−	0.687684i	−0.990761	−	0.135620i	\(-0.956697\pi\)
							0.833802	−	0.552064i	\(-0.186160\pi\)
\(43\)	−1.25809	−	6.43562i	−0.191857	−	0.981423i
							\(47\)	−10.4140	−	3.64403i	−1.51904	−	0.531536i	−0.563431	−	0.826163i	\(-0.690519\pi\)
−0.955612	+	0.294627i	\(0.904805\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.27.8	✓	264
5.3	odd	4	inner	430.2.r.a.113.8	yes	264
43.8	odd	14	inner	430.2.r.a.137.8	yes	264
215.8	even	28	inner	430.2.r.a.223.8	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.27.8	✓	264	1.1	even	1	trivial
430.2.r.a.113.8	yes	264	5.3	odd	4	inner
430.2.r.a.137.8	yes	264	43.8	odd	14	inner
430.2.r.a.223.8	yes	264	215.8	even	28	inner