Embedded newform 403.2.bs.a.4.16

• Label: 403.2.bs.a.4.16
• Level: $403$
• Weight: $2$
• Character: 403.4
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bs (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.16
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0758408 + 0.356803i) q^{2} +(1.47267 + 1.63557i) q^{3} +(1.70553 + 0.759353i) q^{4} -1.67783i q^{5} +(-0.695264 + 0.401411i) q^{6} +(0.425932 - 0.956658i) q^{7} +(-0.829106 + 1.14117i) q^{8} +(-0.192734 + 1.83374i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0758408	+	0.356803i	−0.0536275	+	0.252298i	−0.996793	−	0.0800266i	\(-0.974499\pi\)
							0.943165	+	0.332324i	\(0.107833\pi\)
\(3\)	1.47267	+	1.63557i	0.850247	+	0.944295i	0.999006	−	0.0445782i	\(-0.0141944\pi\)
							−0.148759	+	0.988874i	\(0.547528\pi\)
\(5\)		−	1.67783i		−	0.750348i	−0.926954	−	0.375174i	\(-0.877583\pi\)
							0.926954	−	0.375174i	\(-0.122417\pi\)
\(7\)	0.425932	−	0.956658i	0.160987	−	0.361583i	−0.814984	−	0.579484i	\(-0.803254\pi\)
							0.975971	+	0.217901i	\(0.0699209\pi\)
\(11\)	0.312518	−	0.0328469i	0.0942276	−	0.00990372i	−0.0572973	−	0.998357i	\(-0.518248\pi\)
							0.151525	+	0.988453i	\(0.451582\pi\)
\(13\)	3.55997	−	0.571497i	0.987358	−	0.158505i
							\(17\)	−0.488474	+	4.64752i	−0.118472	+	1.12719i	0.760175	+	0.649719i	\(0.225113\pi\)
−0.878647	+	0.477471i	\(0.841553\pi\)
\(19\)	−4.83893	−	4.35699i	−1.11013	−	0.999562i	−0.999984	−	0.00564665i	\(-0.998203\pi\)
							−0.110143	−	0.993916i	\(-0.535131\pi\)
\(23\)	−4.53929	+	2.02102i	−0.946507	+	0.421412i	−0.821158	−	0.570701i	\(-0.806672\pi\)
							−0.125349	+	0.992113i	\(0.540005\pi\)
\(29\)	−6.18119	−	1.31385i	−1.14782	−	0.243976i	−0.405546	−	0.914075i	\(-0.632919\pi\)
							−0.742272	+	0.670098i	\(0.766252\pi\)
\(31\)	−5.56765	+	0.0358235i	−0.999979	+	0.00643409i
							\(37\)	−8.51834	−	4.91807i	−1.40041	−	0.808525i	−0.405973	−	0.913885i	\(-0.633067\pi\)
−0.994434	+	0.105360i	\(0.966400\pi\)
\(41\)	−1.56139	+	7.34576i	−0.243848	+	1.14721i	0.670381	+	0.742017i	\(0.266131\pi\)
							−0.914229	+	0.405198i	\(0.867203\pi\)
\(43\)	5.13351	−	5.70135i	0.782854	−	0.869447i	−0.211300	−	0.977421i	\(-0.567770\pi\)
							0.994154	+	0.107974i	\(0.0344364\pi\)
\(47\)	5.48737	+	1.78295i	0.800415	+	0.260071i	0.680533	−	0.732718i	\(-0.261748\pi\)
							0.119882	+	0.992788i	\(0.461748\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.16	✓	288
13.10	even	6	inner	403.2.bs.a.283.21	yes	288
31.8	even	5	inner	403.2.bs.a.225.21	yes	288
403.101	even	30	inner	403.2.bs.a.101.16	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.16	✓	288	1.1	even	1	trivial
403.2.bs.a.101.16	yes	288	403.101	even	30	inner
403.2.bs.a.225.21	yes	288	31.8	even	5	inner
403.2.bs.a.283.21	yes	288	13.10	even	6	inner