Embedded newform 403.2.cc.a.24.3

• Label: 403.2.cc.a.24.3
• Level: $403$
• Weight: $2$
• Character: 403.24
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(35\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			24.3
Character	\(\chi\)	\(=\)	403.24
Dual form			403.2.cc.a.84.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.48692 + 0.130334i) q^{2} +(-0.0686740 + 0.0618344i) q^{3} +(4.17875 - 0.439204i) q^{4} +(-3.44926 + 0.924226i) q^{5} +(0.162728 - 0.162728i) q^{6} +(-0.267734 + 1.69041i) q^{7} +(-5.41563 + 0.857752i) q^{8} +(-0.312693 + 2.97507i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 12 q^{2} - 4 q^{3} - 18 q^{4} - 8 q^{5} - 42 q^{6} - 8 q^{7} - 40 q^{8} - 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{13}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.48692	+	0.130334i	−1.75852	+	0.0921601i	−0.904073	−	0.427379i	\(-0.859437\pi\)
							−0.854447	+	0.519539i	\(0.826104\pi\)
\(3\)	−0.0686740	+	0.0618344i	−0.0396490	+	0.0357001i	−0.688717	−	0.725030i	\(-0.741826\pi\)
							0.649068	+	0.760730i	\(0.275159\pi\)
\(5\)	−3.44926	+	0.924226i	−1.54256	+	0.413326i	−0.927090	−	0.374838i	\(-0.877698\pi\)
							−0.615465	+	0.788164i	\(0.711032\pi\)
\(7\)	−0.267734	+	1.69041i	−0.101194	+	0.638913i	0.884002	+	0.467483i	\(0.154839\pi\)
							−0.985196	+	0.171431i	\(0.945161\pi\)
\(11\)	−0.458765	−	0.0726612i	−0.138323	−	0.0219082i	0.0868891	−	0.996218i	\(-0.472307\pi\)
							−0.225212	+	0.974310i	\(0.572307\pi\)
\(13\)	1.82749	−	3.10810i	0.506854	−	0.862032i
							\(17\)	−1.00903	−	0.733103i	−0.244726	−	0.177804i	0.458660	−	0.888612i	\(-0.348329\pi\)
−0.703386	+	0.710808i	\(0.748329\pi\)
\(19\)	−1.00133	+	1.96523i	−0.229722	+	0.450854i	−0.976879	−	0.213793i	\(-0.931418\pi\)
							0.747157	+	0.664647i	\(0.231418\pi\)
\(23\)	−0.0480232	+	0.456910i	−0.0100135	+	0.0952723i	−0.998387	−	0.0567723i	\(-0.981919\pi\)
							0.988374	+	0.152045i	\(0.0485857\pi\)
\(29\)	−0.731660	−	0.658789i	−0.135866	−	0.122334i	0.598396	−	0.801200i	\(-0.295805\pi\)
							−0.734262	+	0.678866i	\(0.762472\pi\)
\(31\)	−1.03357	−	5.47099i	−0.185635	−	0.982619i
							\(37\)	−10.4810	+	2.80837i	−1.72306	+	0.461693i	−0.978566	−	0.205936i	\(-0.933976\pi\)
−0.744495	+	0.667628i	\(0.767310\pi\)
\(41\)	2.49804	−	4.90268i	0.390129	−	0.765671i	−0.609504	−	0.792783i	\(-0.708631\pi\)
							0.999632	+	0.0271126i	\(0.00863127\pi\)
\(43\)	−3.59625	−	11.0681i	−0.548423	−	1.68787i	−0.712708	−	0.701461i	\(-0.752532\pi\)
							0.164285	−	0.986413i	\(-0.447468\pi\)
\(47\)	−4.07872	−	8.00493i	−0.594942	−	1.16764i	−0.970559	−	0.240864i	\(-0.922569\pi\)
							0.375617	−	0.926775i	\(-0.377431\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.cc.a.24.3	✓	560
13.6	odd	12		403.2.ch.a.396.33	yes	560
31.22	odd	30		403.2.ch.a.115.33	yes	560
403.84	even	60	inner	403.2.cc.a.84.3	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cc.a.24.3	✓	560	1.1	even	1	trivial
403.2.cc.a.84.3	yes	560	403.84	even	60	inner
403.2.ch.a.115.33	yes	560	31.22	odd	30
403.2.ch.a.396.33	yes	560	13.6	odd	12