Embedded newform 309.2.o.b.5.1

• Label: 309.2.o.b.5.1
• Level: $309$
• Weight: $2$
• Character: 309.5
• Analytic conductor: $2.467$
• Analytic rank: $0$
• Dimension: $1024$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 309 = 3 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	309.o (of order \(102\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.46737742246\)
Analytic rank:	\(0\)
Dimension:	\(1024\)
Relative dimension:	\(32\) over \(\Q(\zeta_{102})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{102}]$

Embedding invariants

Embedding label			5.1
Character	\(\chi\)	\(=\)	309.5
Dual form			309.2.o.b.62.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.72076 + 0.595855i) q^{2} +(1.53394 + 0.804387i) q^{3} +(5.23053 - 2.40643i) q^{4} +(0.0101383 - 0.329063i) q^{5} +(-4.65276 - 1.27454i) q^{6} +(-0.926125 - 0.114679i) q^{7} +(-8.35179 + 6.30698i) q^{8} +(1.70592 + 2.46776i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 34 q^{3} - 100 q^{4} - 31 q^{6} - 66 q^{7} - 26 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/309\mathbb{Z}\right)^\times\).

\(n\)	\(104\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{102}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.72076	+	0.595855i	−1.92386	+	0.421333i	−0.924689	+	0.380723i	\(0.875675\pi\)
							−0.999175	+	0.0406102i	\(0.987070\pi\)
\(3\)	1.53394	+	0.804387i	0.885619	+	0.464413i
							\(5\)	0.0101383	−	0.329063i	0.00453400	−	0.147162i	−0.994547	−	0.104292i	\(-0.966742\pi\)
0.999081	−	0.0428693i	\(-0.0136499\pi\)
\(7\)	−0.926125	−	0.114679i	−0.350042	−	0.0433446i	−0.0540590	−	0.998538i	\(-0.517216\pi\)
							−0.295983	+	0.955193i	\(0.595647\pi\)
\(11\)	2.07481	+	0.660045i	0.625579	+	0.199011i	0.599146	−	0.800640i	\(-0.295507\pi\)
							0.0264327	+	0.999651i	\(0.491585\pi\)
\(13\)	−3.80894	+	5.04385i	−1.05641	+	1.39891i	−0.144028	+	0.989574i	\(0.546006\pi\)
							−0.912382	+	0.409340i	\(0.865759\pi\)
\(17\)	1.78306	+	1.18150i	0.432456	+	0.286556i	0.749687	−	0.661793i	\(-0.230204\pi\)
							−0.317231	+	0.948348i	\(0.602753\pi\)
\(19\)	2.88740	−	2.32348i	0.662414	−	0.533042i	−0.236243	−	0.971694i	\(-0.575916\pi\)
							0.898657	+	0.438651i	\(0.144544\pi\)
\(23\)	1.83298	−	2.01069i	0.382204	−	0.419257i	−0.517552	−	0.855652i	\(-0.673157\pi\)
							0.899755	+	0.436395i	\(0.143745\pi\)
\(29\)	3.36288	+	6.26247i	0.624471	+	1.16291i	0.974120	+	0.226031i	\(0.0725752\pi\)
							−0.349649	+	0.936881i	\(0.613699\pi\)
\(31\)	−0.970240	+	5.19033i	−0.174260	+	0.932210i	0.778307	+	0.627883i	\(0.216079\pi\)
							−0.952568	+	0.304327i	\(0.901569\pi\)
\(37\)	−4.60192	+	1.30936i	−0.756552	+	0.215258i	−0.629886	−	0.776688i	\(-0.716898\pi\)
							−0.126666	+	0.991945i	\(0.540428\pi\)
\(41\)	1.54375	−	0.0475623i	0.241093	−	0.00742799i	0.0899761	−	0.995944i	\(-0.471321\pi\)
							0.151117	+	0.988516i	\(0.451713\pi\)
\(43\)	2.64592	−	2.56565i	0.403498	−	0.391258i	−0.466331	−	0.884610i	\(-0.654425\pi\)
							0.869830	+	0.493352i	\(0.164229\pi\)
\(47\)	−0.263017	−	0.455559i	−0.0383650	−	0.0664502i	0.846205	−	0.532857i	\(-0.178882\pi\)
							−0.884570	+	0.466407i	\(0.845548\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	309.2.o.b.5.1	✓	1024
3.2	odd	2	inner	309.2.o.b.5.32	yes	1024
103.62	odd	102	inner	309.2.o.b.62.32	yes	1024
309.62	even	102	inner	309.2.o.b.62.1	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.2.o.b.5.1	✓	1024	1.1	even	1	trivial
309.2.o.b.5.32	yes	1024	3.2	odd	2	inner
309.2.o.b.62.1	yes	1024	309.62	even	102	inner
309.2.o.b.62.32	yes	1024	103.62	odd	102	inner