Embedded newform 403.2.l.c.25.2

• Label: 403.2.l.c.25.2
• Level: $403$
• Weight: $2$
• Character: 403.25
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.l (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			25.2
Character	\(\chi\)	\(=\)	403.25
Dual form			403.2.l.c.129.33

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.59015i q^{2} +(0.881022 + 1.52597i) q^{3} -4.70886 q^{4} +(-1.89705 - 1.09526i) q^{5} +(3.95250 - 2.28198i) q^{6} +(3.93715 - 2.27311i) q^{7} +7.01635i q^{8} +(-0.0523990 + 0.0907577i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 6 q^{3} - 76 q^{4} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.59015i		−	1.83151i	−0.401737	−	0.915755i	\(-0.631593\pi\)
							0.401737	−	0.915755i	\(-0.368407\pi\)
\(3\)	0.881022	+	1.52597i	0.508658	+	0.881022i	0.999950	+	0.0100267i	\(0.00319164\pi\)
							−0.491292	+	0.870995i	\(0.663475\pi\)
\(5\)	−1.89705	−	1.09526i	−0.848385	−	0.489815i	0.0117207	−	0.999931i	\(-0.496269\pi\)
							−0.860106	+	0.510116i	\(0.829602\pi\)
\(7\)	3.93715	−	2.27311i	1.48810	−	0.859157i	0.488195	−	0.872735i	\(-0.337655\pi\)
							0.999908	+	0.0135779i	\(0.00432212\pi\)
\(11\)	−1.50994	−	0.871764i	−0.455264	−	0.262847i	0.254787	−	0.966997i	\(-0.417995\pi\)
							−0.710051	+	0.704150i	\(0.751328\pi\)
\(13\)	−2.16097	−	2.88621i	−0.599346	−	0.800490i
							\(17\)	−0.732809	−	1.26926i	−0.177732	−	0.307841i	0.763371	−	0.645960i	\(-0.223543\pi\)
−0.941103	+	0.338119i	\(0.890209\pi\)
\(19\)	−1.45805	+	0.841805i	−0.334499	+	0.193123i	−0.657837	−	0.753160i	\(-0.728528\pi\)
							0.323338	+	0.946284i	\(0.395195\pi\)
\(23\)	−4.43950			−0.925699			−0.462850	−	0.886437i	\(-0.653173\pi\)
							−0.462850	+	0.886437i	\(0.653173\pi\)
\(29\)	2.97762			0.552931			0.276466	−	0.961024i	\(-0.410837\pi\)
							0.276466	+	0.961024i	\(0.410837\pi\)
\(31\)	−5.06790	−	2.30574i	−0.910221	−	0.414122i
							\(37\)	6.89567	−	3.98122i	1.13364	−	0.654508i	0.188793	−	0.982017i	\(-0.439542\pi\)
0.944848	+	0.327509i	\(0.106209\pi\)
\(41\)	5.50738	+	3.17969i	0.860108	+	0.496584i	0.864049	−	0.503408i	\(-0.167921\pi\)
							−0.00394025	+	0.999992i	\(0.501254\pi\)
\(43\)	4.48900	+	7.77517i	0.684566	+	1.18570i	0.973573	+	0.228375i	\(0.0733414\pi\)
							−0.289008	+	0.957327i	\(0.593325\pi\)
\(47\)		−	7.02370i		−	1.02451i	−0.858833	−	0.512256i	\(-0.828810\pi\)
							0.858833	−	0.512256i	\(-0.171190\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.l.c.25.2	✓	68
13.12	even	2	inner	403.2.l.c.25.33	yes	68
31.5	even	3	inner	403.2.l.c.129.2	yes	68
403.129	even	6	inner	403.2.l.c.129.33	yes	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.l.c.25.2	✓	68	1.1	even	1	trivial
403.2.l.c.25.33	yes	68	13.12	even	2	inner
403.2.l.c.129.2	yes	68	31.5	even	3	inner
403.2.l.c.129.33	yes	68	403.129	even	6	inner