Embedded newform 203.3.i.a.115.9

• Label: 203.3.i.a.115.9
• 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			115.9
Character	\(\chi\)	\(=\)	203.115
Dual form			203.3.i.a.173.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.10154 + 1.21333i) q^{2} +(0.807411 - 1.39848i) q^{3} +(0.944318 - 1.63561i) q^{4} +(-3.24085 + 1.87110i) q^{5} +3.91861i q^{6} +(6.99488 - 0.267582i) q^{7} -5.12354i q^{8} +(3.19618 + 5.53594i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 64 q^{4} - 6 q^{5} - 12 q^{7} - 116 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.10154	+	1.21333i	−1.05077	+	0.606663i	−0.922865	−	0.385123i	\(-0.874159\pi\)
							−0.127906	+	0.991786i	\(0.540826\pi\)
\(3\)	0.807411	−	1.39848i	0.269137	−	0.466159i	−0.699502	−	0.714630i	\(-0.746595\pi\)
							0.968639	+	0.248472i	\(0.0799282\pi\)
\(5\)	−3.24085	+	1.87110i	−0.648169	+	0.374221i	−0.787754	−	0.615989i	\(-0.788756\pi\)
							0.139585	+	0.990210i	\(0.455423\pi\)
\(7\)	6.99488	−	0.267582i	0.999269	−	0.0382260i
							\(11\)	−7.00409	−	4.04382i	−0.636736	−	0.367620i	0.146620	−	0.989193i	\(-0.453160\pi\)
−0.783356	+	0.621573i	\(0.786494\pi\)
\(13\)			1.59985i			0.123065i	0.998105	+	0.0615327i	\(0.0195988\pi\)
							−0.998105	+	0.0615327i	\(0.980401\pi\)
\(17\)	−4.24327	+	7.34956i	−0.249604	+	0.432327i	−0.963416	−	0.268010i	\(-0.913634\pi\)
							0.713812	+	0.700338i	\(0.246967\pi\)
\(19\)	10.0415	+	17.3925i	0.528502	+	0.915392i	0.999448	+	0.0332302i	\(0.0105794\pi\)
							−0.470946	+	0.882162i	\(0.656087\pi\)
\(23\)	10.8255	+	18.7502i	0.470672	+	0.815228i	0.999437	−	0.0335401i	\(-0.0106781\pi\)
							−0.528765	+	0.848768i	\(0.677345\pi\)
\(29\)	−23.1958	+	17.4056i	−0.799854	+	0.600194i
							\(31\)	−18.7777	+	32.5240i	−0.605733	+	1.04916i	0.386202	+	0.922414i	\(0.373787\pi\)
−0.991935	+	0.126747i	\(0.959547\pi\)
\(37\)	4.83152	−	2.78948i	0.130581	−	0.0753913i	−0.433286	−	0.901256i	\(-0.642646\pi\)
							0.563868	+	0.825865i	\(0.309313\pi\)
\(41\)	0.383694			0.00935839			0.00467920	−	0.999989i	\(-0.498511\pi\)
							0.00467920	+	0.999989i	\(0.498511\pi\)
\(43\)			49.9631i			1.16193i	0.813928	+	0.580966i	\(0.197325\pi\)
							−0.813928	+	0.580966i	\(0.802675\pi\)
\(47\)	−3.77492	−	6.53835i	−0.0803174	−	0.139114i	0.823069	−	0.567942i	\(-0.192260\pi\)
							−0.903386	+	0.428828i	\(0.858927\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	203.3.i.a.115.9	✓	76
7.5	odd	6	inner	203.3.i.a.173.30	yes	76
29.28	even	2	inner	203.3.i.a.115.30	yes	76
203.173	odd	6	inner	203.3.i.a.173.9	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
203.3.i.a.115.9	✓	76	1.1	even	1	trivial
203.3.i.a.115.30	yes	76	29.28	even	2	inner
203.3.i.a.173.9	yes	76	203.173	odd	6	inner
203.3.i.a.173.30	yes	76	7.5	odd	6	inner