Embedded newform 403.2.bn.a.60.3

• Label: 403.2.bn.a.60.3
• Level: $403$
• Weight: $2$
• Character: 403.60
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $272$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bn (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(272\)
Relative dimension:	\(34\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			60.3
Character	\(\chi\)	\(=\)	403.60
Dual form			403.2.bn.a.356.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.390481 - 2.46540i) q^{2} +(-1.53915 - 2.11846i) q^{3} +(-4.02361 + 1.30735i) q^{4} +(-1.32495 - 1.32495i) q^{5} +(-4.62184 + 4.62184i) q^{6} +(-2.80514 - 1.42929i) q^{7} +(2.52785 + 4.96118i) q^{8} +(-1.19183 + 3.66808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 6 q^{2} - 20 q^{3} - 16 q^{5} - 18 q^{7} + 4 q^{8} + 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.390481	−	2.46540i	−0.276112	−	1.74330i	−0.602548	−	0.798082i	\(-0.705848\pi\)
							0.326436	−	0.945219i	\(-0.394152\pi\)
\(3\)	−1.53915	−	2.11846i	−0.888628	−	1.22309i	−0.973956	−	0.226739i	\(-0.927194\pi\)
							0.0853271	−	0.996353i	\(-0.472806\pi\)
\(5\)	−1.32495	−	1.32495i	−0.592535	−	0.592535i	0.345780	−	0.938316i	\(-0.387614\pi\)
							−0.938316	+	0.345780i	\(0.887614\pi\)
\(7\)	−2.80514	−	1.42929i	−1.06024	−	0.540221i	−0.165227	−	0.986256i	\(-0.552836\pi\)
							−0.895016	+	0.446035i	\(0.852836\pi\)
\(11\)	2.22333	−	4.36353i	0.670359	−	1.31565i	−0.265784	−	0.964033i	\(-0.585631\pi\)
							0.936143	−	0.351620i	\(-0.114369\pi\)
\(13\)	2.78470	−	2.29030i	0.772337	−	0.635214i
							\(17\)	2.02833	−	6.24257i	0.491943	−	1.51404i	−0.329724	−	0.944077i	\(-0.606956\pi\)
0.821667	−	0.569968i	\(-0.193044\pi\)
\(19\)	0.657057	+	4.14849i	0.150739	+	0.951730i	0.940864	+	0.338784i	\(0.110016\pi\)
							−0.790125	+	0.612946i	\(0.789984\pi\)
\(23\)	−0.585955	+	1.80338i	−0.122180	+	0.376032i	−0.993377	−	0.114903i	\(-0.963344\pi\)
							0.871197	+	0.490934i	\(0.163344\pi\)
\(29\)	3.92427	−	5.40130i	0.728719	−	1.00300i	−0.270470	−	0.962729i	\(-0.587179\pi\)
							0.999189	−	0.0402676i	\(-0.0128210\pi\)
\(31\)	1.12009	+	5.45393i	0.201175	+	0.979555i
							\(37\)	−3.07402	−	3.07402i	−0.505365	−	0.505365i	0.407735	−	0.913100i	\(-0.366319\pi\)
−0.913100	+	0.407735i	\(0.866319\pi\)
\(41\)	0.364017	+	2.29831i	0.0568499	+	0.358936i	0.999671	+	0.0256377i	\(0.00816164\pi\)
							−0.942821	+	0.333298i	\(0.891838\pi\)
\(43\)	0.923152	+	0.670709i	0.140779	+	0.102282i	0.655946	−	0.754808i	\(-0.272270\pi\)
							−0.515167	+	0.857090i	\(0.672270\pi\)
\(47\)	0.328232	−	2.07237i	0.0478775	−	0.302287i	−0.952117	−	0.305735i	\(-0.901098\pi\)
							0.999994	+	0.00344863i	\(0.00109774\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bn.a.60.3	✓	272
13.5	odd	4	inner	403.2.bn.a.122.3	yes	272
31.15	odd	10	inner	403.2.bn.a.294.3	yes	272
403.356	even	20	inner	403.2.bn.a.356.3	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bn.a.60.3	✓	272	1.1	even	1	trivial
403.2.bn.a.122.3	yes	272	13.5	odd	4	inner
403.2.bn.a.294.3	yes	272	31.15	odd	10	inner
403.2.bn.a.356.3	yes	272	403.356	even	20	inner