Embedded newform 403.2.r.a.218.16

• Label: 403.2.r.a.218.16
• Level: $403$
• Weight: $2$
• Character: 403.218
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			218.16
Character	\(\chi\)	\(=\)	403.218
Dual form			403.2.r.a.342.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.441172 + 0.254711i) q^{2} +(-1.38443 - 2.39789i) q^{3} +(-0.870245 + 1.50731i) q^{4} +1.59074i q^{5} +(1.22154 + 0.705256i) q^{6} +(0.253842 + 0.146556i) q^{7} -1.90549i q^{8} +(-2.33327 + 4.04134i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 32 q^{4} - 34 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.441172	+	0.254711i	−0.311956	+	0.180108i	−0.647801	−	0.761809i	\(-0.724311\pi\)
							0.335846	+	0.941917i	\(0.390978\pi\)
\(3\)	−1.38443	−	2.39789i	−0.799298	−	1.38443i	−0.920074	−	0.391745i	\(-0.871871\pi\)
							0.120775	−	0.992680i	\(-0.461462\pi\)
\(5\)			1.59074i			0.711402i	0.934600	+	0.355701i	\(0.115758\pi\)
							−0.934600	+	0.355701i	\(0.884242\pi\)
\(7\)	0.253842	+	0.146556i	0.0959432	+	0.0553928i	0.547204	−	0.836999i	\(-0.315692\pi\)
							−0.451261	+	0.892392i	\(0.649026\pi\)
\(11\)	1.95980	−	1.13149i	0.590902	−	0.341158i	−0.174552	−	0.984648i	\(-0.555848\pi\)
							0.765454	+	0.643490i	\(0.222514\pi\)
\(13\)	3.01316	−	1.98013i	0.835699	−	0.549188i
							\(17\)	1.26912	−	2.19819i	0.307808	−	0.533139i	−0.670075	−	0.742294i	\(-0.733738\pi\)
0.977883	+	0.209155i	\(0.0670712\pi\)
\(19\)	−6.32788	−	3.65340i	−1.45172	−	0.838148i	−0.453136	−	0.891441i	\(-0.649695\pi\)
							−0.998579	+	0.0532930i	\(0.983028\pi\)
\(23\)	−2.81628	−	4.87794i	−0.587234	−	1.01712i	−0.994593	−	0.103852i	\(-0.966883\pi\)
							0.407358	−	0.913268i	\(-0.366450\pi\)
\(29\)	0.708957	+	1.22795i	0.131650	+	0.228024i	0.924313	−	0.381636i	\(-0.124639\pi\)
							−0.792663	+	0.609660i	\(0.791306\pi\)
\(31\)		−	1.00000i		−	0.179605i
							\(37\)	−2.75450	+	1.59031i	−0.452837	+	0.261446i	−0.709028	−	0.705181i	\(-0.750866\pi\)
0.256191	+	0.966626i	\(0.417533\pi\)
\(41\)	3.80791	−	2.19850i	0.594696	−	0.343348i	−0.172256	−	0.985052i	\(-0.555106\pi\)
							0.766952	+	0.641704i	\(0.221772\pi\)
\(43\)	5.79062	−	10.0296i	0.883061	−	1.52951i	0.0351416	−	0.999382i	\(-0.488812\pi\)
							0.847920	−	0.530125i	\(-0.177855\pi\)
\(47\)		−	7.44680i		−	1.08623i	−0.839659	−	0.543113i	\(-0.817245\pi\)
							0.839659	−	0.543113i	\(-0.182755\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.r.a.218.16	✓	68
13.2	odd	12		5239.2.a.r.1.14		34
13.4	even	6	inner	403.2.r.a.342.16	yes	68
13.11	odd	12		5239.2.a.q.1.21		34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.r.a.218.16	✓	68	1.1	even	1	trivial
403.2.r.a.342.16	yes	68	13.4	even	6	inner
5239.2.a.q.1.21		34	13.11	odd	12
5239.2.a.r.1.14		34	13.2	odd	12