Embedded newform 403.2.bg.a.123.15

• Label: 403.2.bg.a.123.15
• Level: $403$
• Weight: $2$
• Character: 403.123
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bg (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			123.15
Character	\(\chi\)	\(=\)	403.123
Dual form			403.2.bg.a.154.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.137428 - 0.512890i) q^{2} +(-2.33567 - 1.34850i) q^{3} +(1.48788 - 0.859029i) q^{4} +(0.158725 - 0.158725i) q^{5} +(-0.370644 + 1.38326i) q^{6} +(0.491600 - 1.83468i) q^{7} +(-1.39599 - 1.39599i) q^{8} +(2.13689 + 3.70121i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 8 q^{2} - 12 q^{4} - 8 q^{5} - 16 q^{7} - 28 q^{8} + 64 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.137428	−	0.512890i	−0.0971765	−	0.362668i	0.900164	−	0.435551i	\(-0.143447\pi\)
							−0.997340	+	0.0728836i	\(0.976780\pi\)
\(3\)	−2.33567	−	1.34850i	−1.34850	−	0.778556i	−0.360461	−	0.932774i	\(-0.617381\pi\)
							−0.988037	+	0.154219i	\(0.950714\pi\)
\(5\)	0.158725	−	0.158725i	0.0709839	−	0.0709839i	−0.670724	−	0.741707i	\(-0.734016\pi\)
							0.741707	+	0.670724i	\(0.234016\pi\)
\(7\)	0.491600	−	1.83468i	0.185807	−	0.693442i	−0.808649	−	0.588291i	\(-0.799801\pi\)
							0.994456	−	0.105151i	\(-0.0335325\pi\)
\(11\)	−1.38074	−	5.15299i	−0.416309	−	1.55369i	−0.782199	−	0.623028i	\(-0.785902\pi\)
							0.365891	−	0.930658i	\(-0.380764\pi\)
\(13\)	2.54829	+	2.55073i	0.706768	+	0.707445i
							\(17\)	1.71871	+	2.97689i	0.416848	+	0.722003i	0.995621	−	0.0934868i	\(-0.0298013\pi\)
−0.578772	+	0.815489i	\(0.696468\pi\)
\(19\)	−6.07853	−	1.62874i	−1.39451	−	0.373658i	−0.518140	−	0.855296i	\(-0.673376\pi\)
							−0.876371	+	0.481637i	\(0.840042\pi\)
\(23\)	1.87701	−	3.25108i	0.391384	−	0.677898i	−0.601248	−	0.799063i	\(-0.705330\pi\)
							0.992632	+	0.121165i	\(0.0386629\pi\)
\(29\)	−8.37177	−	4.83344i	−1.55460	−	0.897548i	−0.997758	−	0.0669291i	\(-0.978680\pi\)
							−0.556841	−	0.830619i	\(-0.687987\pi\)
\(31\)	−3.11549	+	4.61451i	−0.559558	+	0.828791i
							\(37\)	−0.247001	−	0.921820i	−0.0406067	−	0.151546i	0.942646	−	0.333795i	\(-0.108329\pi\)
−0.983253	+	0.182248i	\(0.941663\pi\)
\(41\)	−1.38174	−	5.15672i	−0.215791	−	0.805344i	−0.985887	−	0.167415i	\(-0.946458\pi\)
							0.770095	−	0.637929i	\(-0.220209\pi\)
\(43\)	2.80096	+	4.85141i	0.427143	+	0.739833i	0.996618	−	0.0821757i	\(-0.0261869\pi\)
							−0.569475	+	0.822009i	\(0.692854\pi\)
\(47\)	3.95545	+	3.95545i	0.576961	+	0.576961i	0.934065	−	0.357104i	\(-0.116236\pi\)
							−0.357104	+	0.934065i	\(0.616236\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bg.a.123.15	✓	144
13.11	odd	12	inner	403.2.bg.a.154.16	yes	144
31.30	odd	2	inner	403.2.bg.a.123.16	yes	144
403.154	even	12	inner	403.2.bg.a.154.15	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bg.a.123.15	✓	144	1.1	even	1	trivial
403.2.bg.a.123.16	yes	144	31.30	odd	2	inner
403.2.bg.a.154.15	yes	144	403.154	even	12	inner
403.2.bg.a.154.16	yes	144	13.11	odd	12	inner