Embedded newform 403.2.bz.a.38.8

• Label: 403.2.bz.a.38.8
• Level: $403$
• Weight: $2$
• Character: 403.38
• 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.bz (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			38.8
Character	\(\chi\)	\(=\)	403.38
Dual form			403.2.bz.a.350.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65648 + 0.538222i) q^{2} +(-2.26380 + 0.481186i) q^{3} +(0.836200 - 0.607535i) q^{4} +(-0.549117 - 0.317033i) q^{5} +(3.49095 - 2.01550i) q^{6} +(-1.65725 + 3.72223i) q^{7} +(0.989361 - 1.36174i) q^{8} +(2.15263 - 0.958411i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 16 q^{3} + 60 q^{4} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65648	+	0.538222i	−1.17131	+	0.380580i	−0.829130	−	0.559055i	\(-0.811164\pi\)
							−0.342176	+	0.939636i	\(0.611164\pi\)
\(3\)	−2.26380	+	0.481186i	−1.30701	+	0.277813i	−0.808204	−	0.588902i	\(-0.799560\pi\)
							−0.498803	+	0.866715i	\(0.666227\pi\)
\(5\)	−0.549117	−	0.317033i	−0.245572	−	0.141781i	0.372163	−	0.928167i	\(-0.378616\pi\)
							−0.617735	+	0.786386i	\(0.711950\pi\)
\(7\)	−1.65725	+	3.72223i	−0.626380	+	1.40687i	0.269685	+	0.962949i	\(0.413080\pi\)
							−0.896065	+	0.443923i	\(0.853586\pi\)
\(11\)	−6.08508	+	0.639568i	−1.83472	+	0.192837i	−0.957730	−	0.287670i	\(-0.907119\pi\)
							−0.876991	+	0.480507i	\(0.840453\pi\)
\(13\)	−2.81013	+	2.25902i	−0.779390	+	0.626539i
							\(17\)	0.0694106	−	0.660398i	0.0168345	−	0.160170i	−0.982875	−	0.184274i	\(-0.941007\pi\)
0.999709	+	0.0241040i	\(0.00767329\pi\)
\(19\)	−0.445412	−	0.401050i	−0.102184	−	0.0920073i	0.616455	−	0.787390i	\(-0.288568\pi\)
							−0.718640	+	0.695383i	\(0.755235\pi\)
\(23\)	5.95097	+	4.32363i	1.24086	+	0.901539i	0.997656	−	0.0684352i	\(-0.0218006\pi\)
							0.243207	+	0.969974i	\(0.421801\pi\)
\(29\)	−1.38339	−	4.25764i	−0.256889	−	0.790624i	−0.993452	−	0.114254i	\(-0.963552\pi\)
							0.736562	−	0.676370i	\(-0.236448\pi\)
\(31\)	5.56239	−	0.244587i	0.999035	−	0.0439291i
							\(37\)	1.23354	−	0.712184i	0.202793	−	0.117082i	−0.395165	−	0.918610i	\(-0.629313\pi\)
0.597957	+	0.801528i	\(0.295979\pi\)
\(41\)	0.451756	−	2.12535i	0.0705525	−	0.331923i	−0.928689	−	0.370859i	\(-0.879063\pi\)
							0.999242	+	0.0389356i	\(0.0123967\pi\)
\(43\)	−6.56991	+	7.29662i	−1.00190	+	1.11272i	−0.00827948	+	0.999966i	\(0.502635\pi\)
							−0.993622	+	0.112759i	\(0.964031\pi\)
\(47\)	8.32503	+	2.70497i	1.21433	+	0.394560i	0.845014	−	0.534744i	\(-0.179592\pi\)
							0.369316	+	0.929304i	\(0.379592\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bz.a.38.8	✓	288
13.12	even	2	inner	403.2.bz.a.38.29	yes	288
31.9	even	15	inner	403.2.bz.a.350.29	yes	288
403.350	even	30	inner	403.2.bz.a.350.8	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bz.a.38.8	✓	288	1.1	even	1	trivial
403.2.bz.a.38.29	yes	288	13.12	even	2	inner
403.2.bz.a.350.8	yes	288	403.350	even	30	inner
403.2.bz.a.350.29	yes	288	31.9	even	15	inner