Embedded newform 403.2.ch.a.11.14

• Label: 403.2.ch.a.11.14
• Level: $403$
• Weight: $2$
• Character: 403.11
• 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.ch (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			11.14
Character	\(\chi\)	\(=\)	403.11
Dual form			403.2.ch.a.110.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.636159 + 0.0333397i) q^{2} +(3.08296 + 1.00171i) q^{3} +(-1.58546 + 0.166638i) q^{4} +(0.0795633 - 0.296934i) q^{5} +(-1.99465 - 0.534465i) q^{6} +(2.99900 - 1.15121i) q^{7} +(2.26143 - 0.358175i) q^{8} +(6.07415 + 4.41312i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q - 12 q^{2} - 10 q^{3} - 18 q^{4} - 8 q^{5} - 36 q^{6} - 32 q^{7} + 50 q^{8} + 114 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{23}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.636159	+	0.0333397i	−0.449833	+	0.0235747i	−0.275908	−	0.961184i	\(-0.588978\pi\)
							−0.173925	+	0.984759i	\(0.555645\pi\)
\(3\)	3.08296	+	1.00171i	1.77995	+	0.578340i	0.998933	−	0.0461752i	\(-0.0147033\pi\)
							0.781013	+	0.624515i	\(0.214703\pi\)
\(5\)	0.0795633	−	0.296934i	0.0355818	−	0.132793i	−0.945850	−	0.324603i	\(-0.894769\pi\)
							0.981432	+	0.191810i	\(0.0614358\pi\)
\(7\)	2.99900	−	1.15121i	1.13352	−	0.435116i	0.281984	−	0.959419i	\(-0.409007\pi\)
							0.851532	+	0.524303i	\(0.175674\pi\)
\(11\)	−2.92104	+	3.60719i	−0.880728	+	1.08761i	0.114823	+	0.993386i	\(0.463370\pi\)
							−0.995551	+	0.0942227i	\(0.969963\pi\)
\(13\)	−0.589240	−	3.55708i	−0.163426	−	0.986556i
							\(17\)	−0.0797097	+	0.758388i	−0.0193325	+	0.183936i	−0.999927	−	0.0121126i	\(-0.996144\pi\)
0.980594	+	0.196049i	\(0.0628110\pi\)
\(19\)	−2.44187	+	0.127973i	−0.560204	+	0.0293590i	−0.330337	−	0.943863i	\(-0.607162\pi\)
							−0.229867	+	0.973222i	\(0.573829\pi\)
\(23\)	0.412738	−	3.92694i	0.0860619	−	0.818824i	−0.863311	−	0.504672i	\(-0.831613\pi\)
							0.949373	−	0.314152i	\(-0.101720\pi\)
\(29\)	4.26499	+	3.84021i	0.791988	+	0.713109i	0.962214	−	0.272296i	\(-0.0877829\pi\)
							−0.170225	+	0.985405i	\(0.554450\pi\)
\(31\)	−3.64495	−	4.20884i	−0.654653	−	0.755929i
							\(37\)	−3.00137	−	3.00137i	−0.493422	−	0.493422i	0.415960	−	0.909383i	\(-0.363446\pi\)
−0.909383	+	0.415960i	\(0.863446\pi\)
\(41\)	1.90586	+	2.93476i	0.297645	+	0.458333i	0.955576	−	0.294745i	\(-0.0952348\pi\)
							−0.657931	+	0.753078i	\(0.728568\pi\)
\(43\)	−5.08624	+	5.64885i	−0.775645	+	0.861441i	−0.993416	−	0.114559i	\(-0.963455\pi\)
							0.217772	+	0.976000i	\(0.430121\pi\)
\(47\)	0.355909	+	0.698511i	0.0519146	+	0.101888i	0.915506	−	0.402304i	\(-0.131790\pi\)
							−0.863592	+	0.504192i	\(0.831790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ch.a.11.14	yes	560
13.6	odd	12		403.2.cc.a.383.22	yes	560
31.17	odd	30		403.2.cc.a.141.22	✓	560
403.110	even	60	inner	403.2.ch.a.110.14	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.cc.a.141.22	✓	560	31.17	odd	30
403.2.cc.a.383.22	yes	560	13.6	odd	12
403.2.ch.a.11.14	yes	560	1.1	even	1	trivial
403.2.ch.a.110.14	yes	560	403.110	even	60	inner