Embedded newform 203.3.i.a.115.8

• Label: 203.3.i.a.115.8
• Level: $203$
• Weight: $3$
• Character: 203.115
• Analytic conductor: $5.531$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 203 = 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	203.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53134936651\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			115.8
Character	\(\chi\)	\(=\)	203.115
Dual form			203.3.i.a.173.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.13772 + 1.23421i) q^{2} +(0.485316 - 0.840593i) q^{3} +(1.04657 - 1.81272i) q^{4} +(-8.09956 + 4.67628i) q^{5} +2.39594i q^{6} +(-6.15229 - 3.33906i) q^{7} -4.70694i q^{8} +(4.02894 + 6.97832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 64 q^{4} - 6 q^{5} - 12 q^{7} - 116 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/203\mathbb{Z}\right)^\times\).

\(n\)	\(59\)	\(176\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.13772	+	1.23421i	−1.06886	+	0.617107i	−0.927870	−	0.372903i	\(-0.878362\pi\)
							−0.140991	+	0.990011i	\(0.545029\pi\)
\(3\)	0.485316	−	0.840593i	0.161772	−	0.280198i	−0.773732	−	0.633513i	\(-0.781612\pi\)
							0.935504	+	0.353315i	\(0.114946\pi\)
\(5\)	−8.09956	+	4.67628i	−1.61991	+	0.935256i	−0.632970	+	0.774177i	\(0.718164\pi\)
							−0.986941	+	0.161080i	\(0.948502\pi\)
\(7\)	−6.15229	−	3.33906i	−0.878898	−	0.477009i
							\(11\)	7.80871	+	4.50836i	0.709883	+	0.409851i	0.811018	−	0.585022i	\(-0.198914\pi\)
−0.101135	+	0.994873i	\(0.532247\pi\)
\(13\)		−	9.17342i		−	0.705648i	−0.935690	−	0.352824i	\(-0.885221\pi\)
							0.935690	−	0.352824i	\(-0.114779\pi\)
\(17\)	6.71081	−	11.6235i	0.394754	−	0.683733i	−0.598316	−	0.801260i	\(-0.704163\pi\)
							0.993070	+	0.117527i	\(0.0374966\pi\)
\(19\)	−10.0541	−	17.4142i	−0.529163	−	0.916538i	−0.999422	−	0.0340091i	\(-0.989172\pi\)
							0.470258	−	0.882529i	\(-0.344161\pi\)
\(23\)	−1.76131	−	3.05067i	−0.0765785	−	0.132638i	0.825193	−	0.564851i	\(-0.191066\pi\)
							−0.901772	+	0.432213i	\(0.857733\pi\)
\(29\)	6.80277	+	28.1908i	0.234578	+	0.972097i
							\(31\)	17.4903	−	30.2940i	0.564203	−	0.977227i	−0.432921	−	0.901432i	\(-0.642517\pi\)
0.997123	−	0.0757955i	\(-0.0241496\pi\)
\(37\)	−43.5717	+	25.1562i	−1.17761	+	0.679896i	−0.955462	−	0.295116i	\(-0.904642\pi\)
							−0.222153	+	0.975012i	\(0.571308\pi\)
\(41\)	61.8472			1.50847			0.754234	−	0.656605i	\(-0.228008\pi\)
							0.754234	+	0.656605i	\(0.228008\pi\)
\(43\)		−	85.0683i		−	1.97833i	−0.146800	−	0.989166i	\(-0.546897\pi\)
							0.146800	−	0.989166i	\(-0.453103\pi\)
\(47\)	−11.7032	−	20.2706i	−0.249005	−	0.431289i	0.714245	−	0.699896i	\(-0.246770\pi\)
							−0.963250	+	0.268607i	\(0.913437\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	203.3.i.a.115.8	✓	76
7.5	odd	6	inner	203.3.i.a.173.31	yes	76
29.28	even	2	inner	203.3.i.a.115.31	yes	76
203.173	odd	6	inner	203.3.i.a.173.8	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
203.3.i.a.115.8	✓	76	1.1	even	1	trivial
203.3.i.a.115.31	yes	76	29.28	even	2	inner
203.3.i.a.173.8	yes	76	203.173	odd	6	inner
203.3.i.a.173.31	yes	76	7.5	odd	6	inner