Embedded newform 51.4.i.a.5.1

• Label: 51.4.i.a.5.1
• Level: $51$
• Weight: $4$
• Character: 51.5
• Analytic conductor: $3.009$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	51.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.00909741029\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			5.1
Character	\(\chi\)	\(=\)	51.5
Dual form			51.4.i.a.41.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.62770 + 1.91686i) q^{2} +(1.47526 + 4.98233i) q^{3} +(12.0844 - 12.0844i) q^{4} +(20.2465 + 4.02728i) q^{5} +(-16.3775 - 20.2289i) q^{6} +(1.72626 + 8.67851i) q^{7} +(-17.4241 + 42.0654i) q^{8} +(-22.6472 + 14.7004i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/51\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(37\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.62770	+	1.91686i	−1.63614	+	0.677711i	−0.995900	−	0.0904654i	\(-0.971165\pi\)
							−0.640239	+	0.768176i	\(0.721165\pi\)
\(3\)	1.47526	+	4.98233i	0.283913	+	0.958850i
							\(5\)	20.2465	+	4.02728i	1.81090	+	0.360211i	0.980433	−	0.196855i	\(-0.0630728\pi\)
0.830468	+	0.557066i	\(0.188073\pi\)
\(7\)	1.72626	+	8.67851i	0.0932094	+	0.468595i	0.998993	+	0.0448557i	\(0.0142828\pi\)
							−0.905784	+	0.423740i	\(0.860717\pi\)
\(11\)	9.14398	−	13.6849i	0.250638	−	0.375106i	−0.684722	−	0.728805i	\(-0.740076\pi\)
							0.935359	+	0.353699i	\(0.115076\pi\)
\(13\)	9.79673	+	9.79673i	0.209010	+	0.209010i	0.803846	−	0.594837i	\(-0.202783\pi\)
							−0.594837	+	0.803846i	\(0.702783\pi\)
\(17\)	−31.6480	+	62.5412i	−0.451515	+	0.892263i
							\(19\)	−45.4069	−	109.622i	−0.548266	−	1.32363i	−0.918767	−	0.394799i	\(-0.870814\pi\)
0.370502	−	0.928832i	\(-0.379186\pi\)
\(23\)	3.03218	+	2.02604i	0.0274893	+	0.0183678i	0.569239	−	0.822172i	\(-0.307238\pi\)
							−0.541750	+	0.840540i	\(0.682238\pi\)
\(29\)	−4.62574	+	23.2552i	−0.0296200	+	0.148910i	−0.992765	−	0.120073i	\(-0.961687\pi\)
							0.963145	+	0.268982i	\(0.0866873\pi\)
\(31\)	−78.6407	+	52.5460i	−0.455622	+	0.304437i	−0.762122	−	0.647434i	\(-0.775842\pi\)
							0.306500	+	0.951871i	\(0.400842\pi\)
\(37\)	72.2201	+	108.085i	0.320890	+	0.480245i	0.956487	−	0.291776i	\(-0.0942461\pi\)
							−0.635597	+	0.772021i	\(0.719246\pi\)
\(41\)	−81.8537	+	16.2817i	−0.311790	+	0.0620190i	−0.348505	−	0.937307i	\(-0.613311\pi\)
							0.0367149	+	0.999326i	\(0.488311\pi\)
\(43\)	180.873	−	436.667i	0.641463	−	1.54863i	−0.183243	−	0.983068i	\(-0.558659\pi\)
							0.824706	−	0.565561i	\(-0.191341\pi\)
\(47\)	155.027	−	155.027i	0.481128	−	0.481128i	−0.424364	−	0.905492i	\(-0.639502\pi\)
							0.905492	+	0.424364i	\(0.139502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.4.i.a.5.1	✓	128
3.2	odd	2	inner	51.4.i.a.5.16	yes	128
17.7	odd	16	inner	51.4.i.a.41.16	yes	128
51.41	even	16	inner	51.4.i.a.41.1	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.i.a.5.1	✓	128	1.1	even	1	trivial
51.4.i.a.5.16	yes	128	3.2	odd	2	inner
51.4.i.a.41.1	yes	128	51.41	even	16	inner
51.4.i.a.41.16	yes	128	17.7	odd	16	inner