Embedded newform 403.2.g.a.87.9

• Label: 403.2.g.a.87.9
• Level: $403$
• Weight: $2$
• Character: 403.87
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			87.9
Character	\(\chi\)	\(=\)	403.87
Dual form			403.2.g.a.315.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.925450 + 1.60293i) q^{2} +3.11839 q^{3} +(-0.712914 - 1.23480i) q^{4} +(-0.0505045 + 0.0874763i) q^{5} +(-2.88591 + 4.99854i) q^{6} +(-2.21347 + 3.83384i) q^{7} -1.06274 q^{8} +6.72433 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} - 8 q^{3} - 34 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 58 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.925450	+	1.60293i	−0.654392	+	1.13344i	0.327654	+	0.944798i	\(0.393742\pi\)
							−0.982046	+	0.188642i	\(0.939591\pi\)
\(3\)	3.11839			1.80040			0.900201	−	0.435475i	\(-0.143420\pi\)
							0.900201	+	0.435475i	\(0.143420\pi\)
\(5\)	−0.0505045	+	0.0874763i	−0.0225863	+	0.0391206i	−0.877098	−	0.480312i	\(-0.840523\pi\)
							0.854511	+	0.519433i	\(0.173857\pi\)
\(7\)	−2.21347	+	3.83384i	−0.836612	+	1.44906i	0.0560985	+	0.998425i	\(0.482134\pi\)
							−0.892711	+	0.450630i	\(0.851199\pi\)
\(11\)	0.850075	+	1.47237i	0.256307	+	0.443937i	0.965250	−	0.261329i	\(-0.0841608\pi\)
							−0.708943	+	0.705266i	\(0.750827\pi\)
\(13\)	−3.59657	−	0.254401i	−0.997508	−	0.0705581i
							\(17\)	−1.54384	+	2.67401i	−0.374436	+	0.648542i	−0.990242	−	0.139355i	\(-0.955497\pi\)
0.615807	+	0.787897i	\(0.288830\pi\)
\(19\)	3.50166	−	6.06505i	0.803335	−	1.39142i	−0.114074	−	0.993472i	\(-0.536390\pi\)
							0.917409	−	0.397945i	\(-0.130276\pi\)
\(23\)	−0.0352514	+	0.0610573i	−0.00735043	+	0.0127313i	−0.869677	−	0.493621i	\(-0.835673\pi\)
							0.862327	+	0.506352i	\(0.169006\pi\)
\(29\)	3.87316	−	6.70851i	0.719227	−	1.24574i	−0.242079	−	0.970257i	\(-0.577829\pi\)
							0.961306	−	0.275482i	\(-0.0888374\pi\)
\(31\)	3.46018	+	4.36201i	0.621467	+	0.783440i
							\(37\)	10.1717			1.67222			0.836109	−	0.548563i	\(-0.184825\pi\)
0.836109	+	0.548563i	\(0.184825\pi\)
\(41\)	−1.44794	−	2.50790i	−0.226130	−	0.391669i	0.730528	−	0.682883i	\(-0.239274\pi\)
							−0.956658	+	0.291214i	\(0.905941\pi\)
\(43\)	−0.699937	+	1.21233i	−0.106739	+	0.184878i	−0.914448	−	0.404705i	\(-0.867374\pi\)
							0.807708	+	0.589583i	\(0.200708\pi\)
\(47\)	−10.7041			−1.56135			−0.780675	−	0.624938i	\(-0.785124\pi\)
							−0.780675	+	0.624938i	\(0.785124\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.g.a.87.9	yes	70
13.3	even	3		403.2.e.a.211.9	yes	70
31.5	even	3		403.2.e.a.191.9	✓	70
403.315	even	3	inner	403.2.g.a.315.9	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.9	✓	70	31.5	even	3
403.2.e.a.211.9	yes	70	13.3	even	3
403.2.g.a.87.9	yes	70	1.1	even	1	trivial
403.2.g.a.315.9	yes	70	403.315	even	3	inner