Embedded newform 430.2.r.a.27.17

• Label: 430.2.r.a.27.17
• 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			27.17
Character	\(\chi\)	\(=\)	430.27
Dual form			430.2.r.a.223.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.846724 + 0.532032i) q^{2} +(-0.126989 + 0.0797926i) q^{3} +(0.433884 + 0.900969i) q^{4} +(1.11185 - 1.94005i) q^{5} -0.149977 q^{6} +(-0.830650 + 0.830650i) q^{7} +(-0.111964 + 0.993712i) q^{8} +(-1.29189 + 2.68264i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

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
							\(3\)	−0.126989	+	0.0797926i	−0.0733172	+	0.0460683i	−0.568192	−	0.822896i	\(-0.692357\pi\)
0.494875	+	0.868964i	\(0.335214\pi\)
\(5\)	1.11185	−	1.94005i	0.497235	−	0.867616i
							\(7\)	−0.830650	+	0.830650i	−0.313956	+	0.313956i	−0.846440	−	0.532484i	\(-0.821259\pi\)
0.532484	+	0.846440i	\(0.321259\pi\)
\(11\)	2.12380	+	1.02277i	0.640351	+	0.308377i	0.725738	−	0.687971i	\(-0.241498\pi\)
							−0.0853873	+	0.996348i	\(0.527213\pi\)
\(13\)	5.12103	+	0.577001i	1.42032	+	0.160031i	0.788476	−	0.615066i	\(-0.210871\pi\)
							0.631842	+	0.775097i	\(0.282299\pi\)
\(17\)	6.28685	−	0.708357i	1.52478	−	0.171802i	0.690579	−	0.723257i	\(-0.257356\pi\)
							0.834205	+	0.551455i	\(0.185927\pi\)
\(19\)	2.03918	−	0.982016i	0.467819	−	0.225290i	−0.185097	−	0.982720i	\(-0.559260\pi\)
							0.652916	+	0.757430i	\(0.273546\pi\)
\(23\)	−5.36094	+	1.87587i	−1.11783	+	0.391147i	−0.825066	−	0.565037i	\(-0.808862\pi\)
							−0.292768	+	0.956184i	\(0.594576\pi\)
\(29\)	−1.26818	−	5.55624i	−0.235494	−	1.03177i	−0.945001	−	0.327069i	\(-0.893939\pi\)
							0.709506	−	0.704699i	\(-0.248918\pi\)
\(31\)	−0.765409	−	3.35347i	−0.137471	−	0.602302i	−0.995986	−	0.0895122i	\(-0.971469\pi\)
							0.858514	−	0.512790i	\(-0.171388\pi\)
\(37\)	−6.66575	+	6.66575i	−1.09584	+	1.09584i	−0.100952	+	0.994891i	\(0.532189\pi\)
							−0.994891	+	0.100952i	\(0.967811\pi\)
\(41\)	−1.98989	−	8.71828i	−0.310769	−	1.36157i	−0.853251	−	0.521500i	\(-0.825373\pi\)
							0.542482	−	0.840067i	\(-0.317485\pi\)
\(43\)	3.34651	+	5.63923i	0.510338	+	0.859974i
							\(47\)	−12.3071	−	4.30644i	−1.79518	−	0.628159i	−0.999756	−	0.0221021i	\(-0.992964\pi\)
−0.795421	−	0.606057i	\(-0.792750\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.27.17	✓	264
5.3	odd	4	inner	430.2.r.a.113.17	yes	264
43.8	odd	14	inner	430.2.r.a.137.17	yes	264
215.8	even	28	inner	430.2.r.a.223.17	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.27.17	✓	264	1.1	even	1	trivial
430.2.r.a.113.17	yes	264	5.3	odd	4	inner
430.2.r.a.137.17	yes	264	43.8	odd	14	inner
430.2.r.a.223.17	yes	264	215.8	even	28	inner