Embedded newform 203.3.i.a.115.33

• Label: 203.3.i.a.115.33
• Level: $203$
• Weight: $3$
• Character: 203.115
• Analytic conductor: $5.531$
• Analytic rank: $0$
• Dimension: $76$
• 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			115.33
Character	\(\chi\)	\(=\)	203.115
Dual form			203.3.i.a.173.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.75567 - 1.59098i) q^{2} +(1.54266 - 2.67196i) q^{3} +(3.06246 - 5.30434i) q^{4} +(0.876658 - 0.506139i) q^{5} -9.81739i q^{6} +(5.64064 + 4.14526i) q^{7} -6.76145i q^{8} +(-0.259596 - 0.449633i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 64 q^{4} - 6 q^{5} - 12 q^{7} - 116 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.75567	−	1.59098i	1.37783	−	0.795492i	0.385935	−	0.922526i	\(-0.373879\pi\)
							0.991898	+	0.127034i	\(0.0405457\pi\)
\(3\)	1.54266	−	2.67196i	0.514220	−	0.890655i	−0.485644	−	0.874157i	\(-0.661415\pi\)
							0.999864	−	0.0164982i	\(-0.00525176\pi\)
\(5\)	0.876658	−	0.506139i	0.175332	−	0.101228i	−0.409766	−	0.912191i	\(-0.634390\pi\)
							0.585097	+	0.810963i	\(0.301056\pi\)
\(7\)	5.64064	+	4.14526i	0.805806	+	0.592180i
							\(11\)	−14.1898	−	8.19249i	−1.28998	−	0.744772i	−0.311332	−	0.950301i	\(-0.600775\pi\)
−0.978651	+	0.205529i	\(0.934109\pi\)
\(13\)			10.0751i			0.775010i	0.921868	+	0.387505i	\(0.126663\pi\)
							−0.921868	+	0.387505i	\(0.873337\pi\)
\(17\)	−11.6680	+	20.2096i	−0.686353	+	1.18880i	0.286656	+	0.958034i	\(0.407456\pi\)
							−0.973009	+	0.230765i	\(0.925877\pi\)
\(19\)	−8.40984	−	14.5663i	−0.442623	−	0.766646i	0.555260	−	0.831677i	\(-0.312619\pi\)
							−0.997883	+	0.0650309i	\(0.979285\pi\)
\(23\)	−16.2963	−	28.2260i	−0.708534	−	1.22722i	−0.965401	−	0.260770i	\(-0.916023\pi\)
							0.256867	−	0.966447i	\(-0.417310\pi\)
\(29\)	16.6705	−	23.7296i	0.574846	−	0.818262i
							\(31\)	−25.1602	+	43.5787i	−0.811619	+	1.40577i	0.100111	+	0.994976i	\(0.468080\pi\)
−0.911730	+	0.410789i	\(0.865253\pi\)
\(37\)	55.8064	−	32.2198i	1.50828	−	0.870807i	0.508328	−	0.861164i	\(-0.330264\pi\)
							0.999954	−	0.00964318i	\(-0.00306957\pi\)
\(41\)	15.6936			0.382770			0.191385	−	0.981515i	\(-0.438702\pi\)
							0.191385	+	0.981515i	\(0.438702\pi\)
\(43\)		−	18.3878i		−	0.427623i	−0.976875	−	0.213811i	\(-0.931412\pi\)
							0.976875	−	0.213811i	\(-0.0685878\pi\)
\(47\)	−16.9313	−	29.3259i	−0.360241	−	0.623956i	0.627759	−	0.778408i	\(-0.283972\pi\)
							−0.988000	+	0.154451i	\(0.950639\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	203.3.i.a.115.33	yes	76
7.5	odd	6	inner	203.3.i.a.173.6	yes	76
29.28	even	2	inner	203.3.i.a.115.6	✓	76
203.173	odd	6	inner	203.3.i.a.173.33	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
203.3.i.a.115.6	✓	76	29.28	even	2	inner
203.3.i.a.115.33	yes	76	1.1	even	1	trivial
203.3.i.a.173.6	yes	76	7.5	odd	6	inner
203.3.i.a.173.33	yes	76	203.173	odd	6	inner