Embedded newform 403.2.bs.a.4.12

• Label: 403.2.bs.a.4.12
• 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			4.12
Character	\(\chi\)	\(=\)	403.4
Dual form			403.2.bs.a.101.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.254380 + 1.19677i) q^{2} +(-1.60706 - 1.78482i) q^{3} +(0.459552 + 0.204606i) q^{4} +0.244018i q^{5} +(2.54482 - 1.46925i) q^{6} +(-0.0246816 + 0.0554357i) q^{7} +(-1.80008 + 2.47760i) q^{8} +(-0.289361 + 2.75309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.254380	+	1.19677i	−0.179874	+	0.846241i	0.791960	+	0.610573i	\(0.209061\pi\)
							−0.971834	+	0.235668i	\(0.924272\pi\)
\(3\)	−1.60706	−	1.78482i	−0.927839	−	1.03047i	−0.999454	−	0.0330368i	\(-0.989482\pi\)
							0.0716156	−	0.997432i	\(-0.477185\pi\)
\(5\)			0.244018i			0.109128i	0.998510	+	0.0545642i	\(0.0173769\pi\)
							−0.998510	+	0.0545642i	\(0.982623\pi\)
\(7\)	−0.0246816	+	0.0554357i	−0.00932876	+	0.0209527i	−0.918150	−	0.396234i	\(-0.870317\pi\)
							0.908821	+	0.417186i	\(0.136984\pi\)
\(11\)	0.706812	−	0.0742889i	0.213112	−	0.0223990i	0.00262918	−	0.999997i	\(-0.499163\pi\)
							0.210483	+	0.977598i	\(0.432496\pi\)
\(13\)	3.31073	+	1.42797i	0.918230	+	0.396048i
							\(17\)	0.244382	−	2.32514i	0.0592714	−	0.563930i	−0.924078	−	0.382204i	\(-0.875165\pi\)
0.983349	−	0.181726i	\(-0.0581683\pi\)
\(19\)	5.00196	+	4.50379i	1.14753	+	1.03324i	0.999017	+	0.0443288i	\(0.0141149\pi\)
							0.148512	+	0.988911i	\(0.452552\pi\)
\(23\)	5.12835	−	2.28329i	1.06934	−	0.476099i	0.204869	−	0.978789i	\(-0.434323\pi\)
							0.864466	+	0.502691i	\(0.167657\pi\)
\(29\)	4.00706	+	0.851726i	0.744092	+	0.158162i	0.564331	−	0.825549i	\(-0.309134\pi\)
							0.179761	+	0.983710i	\(0.442468\pi\)
\(31\)	−5.02847	+	2.39049i	−0.903141	+	0.429345i
							\(37\)	5.20056	+	3.00255i	0.854967	+	0.493615i	0.862324	−	0.506358i	\(-0.169008\pi\)
−0.00735671	+	0.999973i	\(0.502342\pi\)
\(41\)	−1.88013	+	8.84531i	−0.293627	+	1.38141i	0.545792	+	0.837920i	\(0.316229\pi\)
							−0.839419	+	0.543485i	\(0.817105\pi\)
\(43\)	4.79029	−	5.32015i	0.730512	−	0.811316i	−0.257405	−	0.966304i	\(-0.582867\pi\)
							0.987916	+	0.154988i	\(0.0495340\pi\)
\(47\)	−2.18159	−	0.708843i	−0.318218	−	0.103395i	0.145553	−	0.989350i	\(-0.453504\pi\)
							−0.463771	+	0.885955i	\(0.653504\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.4.12	✓	288
13.10	even	6	inner	403.2.bs.a.283.25	yes	288
31.8	even	5	inner	403.2.bs.a.225.25	yes	288
403.101	even	30	inner	403.2.bs.a.101.12	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.12	✓	288	1.1	even	1	trivial
403.2.bs.a.101.12	yes	288	403.101	even	30	inner
403.2.bs.a.225.25	yes	288	31.8	even	5	inner
403.2.bs.a.283.25	yes	288	13.10	even	6	inner