Embedded newform 403.2.e.a.191.18

• Label: 403.2.e.a.191.18
• Level: $403$
• Weight: $2$
• Character: 403.191
• 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.e (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			191.18
Character	\(\chi\)	\(=\)	403.191
Dual form			403.2.e.a.211.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0861040 + 0.149137i) q^{2} +(1.23907 + 2.14613i) q^{3} +(0.985172 + 1.70637i) q^{4} +(-0.342759 - 0.593675i) q^{5} -0.426756 q^{6} -3.89977 q^{7} -0.683725 q^{8} +(-1.57059 + 2.72035i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0861040	+	0.149137i	−0.0608847	+	0.105455i	−0.894861	−	0.446345i	\(-0.852726\pi\)
							0.833976	+	0.551800i	\(0.186059\pi\)
\(3\)	1.23907	+	2.14613i	0.715378	+	1.23907i	0.962814	+	0.270167i	\(0.0870788\pi\)
							−0.247436	+	0.968904i	\(0.579588\pi\)
\(5\)	−0.342759	−	0.593675i	−0.153286	−	0.265500i	0.779147	−	0.626841i	\(-0.215652\pi\)
							−0.932434	+	0.361341i	\(0.882319\pi\)
\(7\)	−3.89977			−1.47398			−0.736988	−	0.675906i	\(-0.763752\pi\)
							−0.736988	+	0.675906i	\(0.763752\pi\)
\(11\)	−5.31337			−1.60204			−0.801020	−	0.598638i	\(-0.795709\pi\)
							−0.801020	+	0.598638i	\(0.795709\pi\)
\(13\)	2.23395	+	2.83010i	0.619587	+	0.784928i
							\(17\)	3.82240			0.927067			0.463534	−	0.886079i	\(-0.346581\pi\)
0.463534	+	0.886079i	\(0.346581\pi\)
\(19\)	5.71511			1.31114			0.655568	−	0.755136i	\(-0.272429\pi\)
							0.655568	+	0.755136i	\(0.272429\pi\)
\(23\)	−1.17980	+	2.04347i	−0.246005	+	0.426094i	−0.962414	−	0.271587i	\(-0.912451\pi\)
							0.716409	+	0.697681i	\(0.245785\pi\)
\(29\)	−1.54270	+	2.67204i	−0.286473	+	0.496186i	−0.972965	−	0.230951i	\(-0.925816\pi\)
							0.686492	+	0.727137i	\(0.259150\pi\)
\(31\)	5.48758	+	0.941521i	0.985599	+	0.169102i
							\(37\)	4.12683	+	7.14789i	0.678447	+	1.17511i	0.975448	+	0.220228i	\(0.0706802\pi\)
−0.297001	+	0.954877i	\(0.595987\pi\)
\(41\)	−2.46650			−0.385203			−0.192602	−	0.981277i	\(-0.561693\pi\)
							−0.192602	+	0.981277i	\(0.561693\pi\)
\(43\)	−1.01490			−0.154770			−0.0773852	−	0.997001i	\(-0.524657\pi\)
							−0.0773852	+	0.997001i	\(0.524657\pi\)
\(47\)	13.5708			1.97951			0.989755	−	0.142774i	\(-0.0456021\pi\)
							0.989755	+	0.142774i	\(0.0456021\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.e.a.191.18	✓	70
13.3	even	3		403.2.g.a.315.18	yes	70
31.25	even	3		403.2.g.a.87.18	yes	70
403.211	even	3	inner	403.2.e.a.211.18	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.18	✓	70	1.1	even	1	trivial
403.2.e.a.211.18	yes	70	403.211	even	3	inner
403.2.g.a.87.18	yes	70	31.25	even	3
403.2.g.a.315.18	yes	70	13.3	even	3