Embedded newform 203.3.i.a.115.3

• Label: 203.3.i.a.115.3
• 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			115.3
Character	\(\chi\)	\(=\)	203.115
Dual form			203.3.i.a.173.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.81256 + 1.62383i) q^{2} +(-0.770122 + 1.33389i) q^{3} +(3.27367 - 5.67017i) q^{4} +(3.06569 - 1.76998i) q^{5} -5.00220i q^{6} +(-5.85179 - 3.84142i) q^{7} +8.27293i q^{8} +(3.31382 + 5.73971i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 64 q^{4} - 6 q^{5} - 12 q^{7} - 116 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.81256	+	1.62383i	−1.40628	+	0.811917i	−0.995027	−	0.0996038i	\(-0.968242\pi\)
							−0.411254	+	0.911521i	\(0.634909\pi\)
\(3\)	−0.770122	+	1.33389i	−0.256707	+	0.444630i	−0.965358	−	0.260930i	\(-0.915971\pi\)
							0.708651	+	0.705560i	\(0.249304\pi\)
\(5\)	3.06569	−	1.76998i	0.613138	−	0.353996i	−0.161054	−	0.986946i	\(-0.551489\pi\)
							0.774193	+	0.632950i	\(0.218156\pi\)
\(7\)	−5.85179	−	3.84142i	−0.835970	−	0.548775i
							\(11\)	3.51053	+	2.02681i	0.319139	+	0.184255i	0.651009	−	0.759070i	\(-0.274346\pi\)
−0.331870	+	0.943325i	\(0.607679\pi\)
\(13\)			11.5263i			0.886640i	0.896363	+	0.443320i	\(0.146199\pi\)
							−0.896363	+	0.443320i	\(0.853801\pi\)
\(17\)	−5.92333	+	10.2595i	−0.348431	+	0.603501i	−0.985971	−	0.166917i	\(-0.946619\pi\)
							0.637540	+	0.770418i	\(0.279952\pi\)
\(19\)	−0.749147	−	1.29756i	−0.0394288	−	0.0682927i	0.845638	−	0.533758i	\(-0.179220\pi\)
							−0.885066	+	0.465465i	\(0.845887\pi\)
\(23\)	0.607982	+	1.05306i	0.0264340	+	0.0457850i	0.878940	−	0.476933i	\(-0.158251\pi\)
							−0.852506	+	0.522718i	\(0.824918\pi\)
\(29\)	−18.8895	−	22.0042i	−0.651363	−	0.758766i
							\(31\)	−25.5458	+	44.2466i	−0.824057	+	1.42731i	0.0785804	+	0.996908i	\(0.474961\pi\)
−0.902638	+	0.430401i	\(0.858372\pi\)
\(37\)	2.79632	−	1.61446i	0.0755762	−	0.0436339i	−0.461736	−	0.887018i	\(-0.652773\pi\)
							0.537312	+	0.843384i	\(0.319440\pi\)
\(41\)	38.6483			0.942641			0.471321	−	0.881962i	\(-0.343777\pi\)
							0.471321	+	0.881962i	\(0.343777\pi\)
\(43\)			20.2996i			0.472084i	0.971743	+	0.236042i	\(0.0758502\pi\)
							−0.971743	+	0.236042i	\(0.924150\pi\)
\(47\)	−19.3190	−	33.4615i	−0.411043	−	0.711948i	0.583961	−	0.811782i	\(-0.301502\pi\)
							−0.995004	+	0.0998341i	\(0.968169\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	203.3.i.a.115.3	✓	76
7.5	odd	6	inner	203.3.i.a.173.36	yes	76
29.28	even	2	inner	203.3.i.a.115.36	yes	76
203.173	odd	6	inner	203.3.i.a.173.3	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
203.3.i.a.115.3	✓	76	1.1	even	1	trivial
203.3.i.a.115.36	yes	76	29.28	even	2	inner
203.3.i.a.173.3	yes	76	203.173	odd	6	inner
203.3.i.a.173.36	yes	76	7.5	odd	6	inner