Embedded newform 403.2.cd.a.21.3

• Label: 403.2.cd.a.21.3
• Level: $403$
• Weight: $2$
• Character: 403.21
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $576$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.cd (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(576\)
Relative dimension:	\(36\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			21.3
Character	\(\chi\)	\(=\)	403.21
Dual form			403.2.cd.a.96.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.375607 - 2.37149i) q^{2} +(2.23445 - 0.234850i) q^{3} +(-3.58075 + 1.16346i) q^{4} +(0.997640 - 3.72324i) q^{5} +(-1.39622 - 5.21075i) q^{6} +(0.525344 - 0.341162i) q^{7} +(1.92397 + 3.77601i) q^{8} +(2.00316 - 0.425785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 576 q - 12 q^{2} - 28 q^{3} - 8 q^{5} + 6 q^{6} - 16 q^{7} - 46 q^{8} - 104 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{29}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.375607	−	2.37149i	−0.265594	−	1.67689i	−0.654846	−	0.755763i	\(-0.727266\pi\)
							0.389252	−	0.921131i	\(-0.372734\pi\)
\(3\)	2.23445	−	0.234850i	1.29006	−	0.135591i	0.565468	−	0.824770i	\(-0.308695\pi\)
							0.724591	+	0.689179i	\(0.242029\pi\)
\(5\)	0.997640	−	3.72324i	0.446158	−	1.66508i	−0.266703	−	0.963779i	\(-0.585934\pi\)
							0.712861	−	0.701306i	\(-0.247399\pi\)
\(7\)	0.525344	−	0.341162i	0.198561	−	0.128947i	−0.441529	−	0.897247i	\(-0.645564\pi\)
							0.640090	+	0.768300i	\(0.278897\pi\)
\(11\)	−0.201501	+	0.0105602i	−0.0607547	+	0.00318402i	−0.0826891	−	0.996575i	\(-0.526351\pi\)
							0.0219344	+	0.999759i	\(0.493018\pi\)
\(13\)	1.93585	+	3.04179i	0.536908	+	0.843641i
							\(17\)	4.36373	+	4.84642i	1.05836	+	1.17543i	0.983995	+	0.178197i	\(0.0570266\pi\)
0.0743658	+	0.997231i	\(0.476307\pi\)
\(19\)	0.207715	+	0.0797345i	0.0476532	+	0.0182923i	0.382078	−	0.924130i	\(-0.375209\pi\)
							−0.334425	+	0.942423i	\(0.608542\pi\)
\(23\)	−0.0209280	+	0.0644097i	−0.00436379	+	0.0134303i	−0.953215	−	0.302294i	\(-0.902248\pi\)
							0.948851	+	0.315724i	\(0.102248\pi\)
\(29\)	−3.34860	+	4.60895i	−0.621819	+	0.855860i	−0.997484	−	0.0708955i	\(-0.977414\pi\)
							0.375665	+	0.926755i	\(0.377414\pi\)
\(31\)	−2.72658	−	4.85446i	−0.489708	−	0.871887i
							\(37\)	−4.43978	+	1.18964i	−0.729896	+	0.195575i	−0.604583	−	0.796542i	\(-0.706660\pi\)
−0.125313	+	0.992117i	\(0.539993\pi\)
\(41\)	4.83635	−	3.91640i	0.755311	−	0.611639i	−0.172361	−	0.985034i	\(-0.555140\pi\)
							0.927672	+	0.373395i	\(0.121806\pi\)
\(43\)	5.34561	−	2.38002i	0.815197	−	0.362949i	0.0435909	−	0.999049i	\(-0.486120\pi\)
							0.771606	+	0.636100i	\(0.219454\pi\)
\(47\)	−0.0470708	+	0.297193i	−0.00686598	+	0.0433501i	−0.990883	−	0.134722i	\(-0.956986\pi\)
							0.984017	+	0.178072i	\(0.0569859\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.cd.a.21.3	✓	576
13.5	odd	4	inner	403.2.cd.a.83.3	yes	576
31.3	odd	30	inner	403.2.cd.a.34.3	yes	576
403.96	even	60	inner	403.2.cd.a.96.3	yes	576

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cd.a.21.3	✓	576	1.1	even	1	trivial
403.2.cd.a.34.3	yes	576	31.3	odd	30	inner
403.2.cd.a.83.3	yes	576	13.5	odd	4	inner
403.2.cd.a.96.3	yes	576	403.96	even	60	inner