Embedded newform 403.2.bn.a.60.13

• Label: 403.2.bn.a.60.13
• 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			60.13
Character	\(\chi\)	\(=\)	403.60
Dual form			403.2.bn.a.356.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.170304 - 1.07526i) q^{2} +(0.498908 + 0.686688i) q^{3} +(0.774943 - 0.251794i) q^{4} +(2.61824 + 2.61824i) q^{5} +(0.653399 - 0.653399i) q^{6} +(-0.113188 - 0.0576723i) q^{7} +(-1.39120 - 2.73038i) q^{8} +(0.704420 - 2.16798i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 272 q - 6 q^{2} - 20 q^{3} - 16 q^{5} - 18 q^{7} + 4 q^{8} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.170304	−	1.07526i	−0.120423	−	0.760320i	−0.971807	−	0.235778i	\(-0.924236\pi\)
							0.851384	−	0.524543i	\(-0.175764\pi\)
\(3\)	0.498908	+	0.686688i	0.288045	+	0.396460i	0.928378	−	0.371638i	\(-0.121204\pi\)
							−0.640333	+	0.768097i	\(0.721204\pi\)
\(5\)	2.61824	+	2.61824i	1.17091	+	1.17091i	0.981993	+	0.188918i	\(0.0604981\pi\)
							0.188918	+	0.981993i	\(0.439502\pi\)
\(7\)	−0.113188	−	0.0576723i	−0.0427811	−	0.0217981i	0.432469	−	0.901649i	\(-0.357642\pi\)
							−0.475250	+	0.879851i	\(0.657642\pi\)
\(11\)	−2.67995	+	5.25970i	−0.808036	+	1.58586i	0.00237469	+	0.999997i	\(0.499244\pi\)
							−0.810411	+	0.585862i	\(0.800756\pi\)
\(13\)	2.95523	−	2.06557i	0.819634	−	0.572887i
							\(17\)	−0.979046	+	3.01319i	−0.237454	+	0.730807i	0.759333	+	0.650702i	\(0.225525\pi\)
−0.996786	+	0.0801045i	\(0.974475\pi\)
\(19\)	−0.686027	−	4.33140i	−0.157385	−	0.993692i	−0.932316	−	0.361645i	\(-0.882215\pi\)
							0.774930	−	0.632047i	\(-0.217785\pi\)
\(23\)	−1.20734	+	3.71581i	−0.251748	+	0.774800i	0.742705	+	0.669619i	\(0.233542\pi\)
							−0.994453	+	0.105182i	\(0.966458\pi\)
\(29\)	3.24944	−	4.47247i	0.603406	−	0.830517i	−0.392609	−	0.919706i	\(-0.628427\pi\)
							0.996015	+	0.0891886i	\(0.0284274\pi\)
\(31\)	−3.05464	+	4.65502i	−0.548629	+	0.836066i
							\(37\)	−2.31305	−	2.31305i	−0.380263	−	0.380263i	0.490934	−	0.871197i	\(-0.336656\pi\)
−0.871197	+	0.490934i	\(0.836656\pi\)
\(41\)	−0.977562	−	6.17208i	−0.152670	−	0.963918i	−0.938451	−	0.345413i	\(-0.887739\pi\)
							0.785781	−	0.618505i	\(-0.212261\pi\)
\(43\)	−1.92838	−	1.40105i	−0.294075	−	0.213658i	0.430958	−	0.902372i	\(-0.358176\pi\)
							−0.725033	+	0.688714i	\(0.758176\pi\)
\(47\)	1.53317	−	9.68006i	0.223636	−	1.41198i	−0.578910	−	0.815391i	\(-0.696522\pi\)
							0.802546	−	0.596590i	\(-0.203478\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bn.a.60.13	✓	272
13.5	odd	4	inner	403.2.bn.a.122.13	yes	272
31.15	odd	10	inner	403.2.bn.a.294.13	yes	272
403.356	even	20	inner	403.2.bn.a.356.13	yes	272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bn.a.60.13	✓	272	1.1	even	1	trivial
403.2.bn.a.122.13	yes	272	13.5	odd	4	inner
403.2.bn.a.294.13	yes	272	31.15	odd	10	inner
403.2.bn.a.356.13	yes	272	403.356	even	20	inner