Embedded newform 403.2.bs.a.4.18

• Label: 403.2.bs.a.4.18
• 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			4.18
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.00361138 + 0.0169902i) q^{2} +(-2.04909 - 2.27575i) q^{3} +(1.82682 + 0.813351i) q^{4} +1.58318i q^{5} +(0.0460655 - 0.0265959i) q^{6} +(-0.684412 + 1.53721i) q^{7} +(-0.0408357 + 0.0562056i) q^{8} +(-0.666663 + 6.34287i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.00361138	+	0.0169902i	−0.00255363	+	0.0120139i	−0.979406	−	0.201901i	\(-0.935288\pi\)
							0.976852	+	0.213915i	\(0.0686215\pi\)
\(3\)	−2.04909	−	2.27575i	−1.18304	−	1.31390i	−0.938910	−	0.344162i	\(-0.888163\pi\)
							−0.244134	−	0.969742i	\(-0.578504\pi\)
\(5\)			1.58318i			0.708018i	0.935242	+	0.354009i	\(0.115182\pi\)
							−0.935242	+	0.354009i	\(0.884818\pi\)
\(7\)	−0.684412	+	1.53721i	−0.258683	+	0.581012i	−0.995466	−	0.0951174i	\(-0.969677\pi\)
							0.736783	+	0.676129i	\(0.236344\pi\)
\(11\)	−3.90292	+	0.410213i	−1.17677	+	0.123684i	−0.672659	−	0.739953i	\(-0.734848\pi\)
							−0.504115	+	0.863637i	\(0.668181\pi\)
\(13\)	−3.21245	+	1.63712i	−0.890973	+	0.454056i
							\(17\)	−0.290680	+	2.76563i	−0.0705002	+	0.670764i	0.901015	+	0.433788i	\(0.142823\pi\)
−0.971515	+	0.236977i	\(0.923843\pi\)
\(19\)	−0.583515	−	0.525399i	−0.133868	−	0.120535i	0.599477	−	0.800392i	\(-0.295375\pi\)
							−0.733345	+	0.679857i	\(0.762042\pi\)
\(23\)	−6.33244	+	2.81938i	−1.32041	+	0.587882i	−0.941331	−	0.337484i	\(-0.890424\pi\)
							−0.379074	+	0.925366i	\(0.623757\pi\)
\(29\)	7.80463	+	1.65892i	1.44928	+	0.308055i	0.864294	−	0.502987i	\(-0.167766\pi\)
							0.584989	+	0.811041i	\(0.301099\pi\)
\(31\)	5.48285	−	0.968681i	0.984749	−	0.173980i
							\(37\)	−2.59348	−	1.49734i	−0.426365	−	0.246162i	0.271432	−	0.962458i	\(-0.412503\pi\)
−0.697797	+	0.716296i	\(0.745836\pi\)
\(41\)	−1.09741	+	5.16290i	−0.171387	+	0.806310i	0.805507	+	0.592586i	\(0.201893\pi\)
							−0.976894	+	0.213725i	\(0.931440\pi\)
\(43\)	−1.21969	+	1.35461i	−0.186001	+	0.206575i	−0.828933	−	0.559348i	\(-0.811051\pi\)
							0.642931	+	0.765924i	\(0.277718\pi\)
\(47\)	−7.70331	−	2.50296i	−1.12364	−	0.365094i	−0.312486	−	0.949922i	\(-0.601162\pi\)
							−0.811157	+	0.584828i	\(0.801162\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.18	✓	288
13.10	even	6	inner	403.2.bs.a.283.19	yes	288
31.8	even	5	inner	403.2.bs.a.225.19	yes	288
403.101	even	30	inner	403.2.bs.a.101.18	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.18	✓	288	1.1	even	1	trivial
403.2.bs.a.101.18	yes	288	403.101	even	30	inner
403.2.bs.a.225.19	yes	288	31.8	even	5	inner
403.2.bs.a.283.19	yes	288	13.10	even	6	inner