Embedded newform 51.4.i.a.5.6

• Label: 51.4.i.a.5.6
• 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			5.6
Character	\(\chi\)	\(=\)	51.5
Dual form			51.4.i.a.41.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.54951 + 0.641828i) q^{2} +(0.682978 - 5.15107i) q^{3} +(-3.66781 + 3.66781i) q^{4} +(11.5824 + 2.30388i) q^{5} +(2.24782 + 8.42000i) q^{6} +(-7.03184 - 35.3514i) q^{7} +(8.46384 - 20.4335i) q^{8} +(-26.0671 - 7.03614i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.54951	+	0.641828i	−0.547835	+	0.226921i	−0.639394	−	0.768879i	\(-0.720815\pi\)
							0.0915595	+	0.995800i	\(0.470815\pi\)
\(3\)	0.682978	−	5.15107i	0.131439	−	0.991324i
							\(5\)	11.5824	+	2.30388i	1.03596	+	0.206065i	0.683643	−	0.729816i	\(-0.260394\pi\)
0.352316	+	0.935881i	\(0.385394\pi\)
\(7\)	−7.03184	−	35.3514i	−0.379684	−	1.90880i	−0.415811	−	0.909451i	\(-0.636502\pi\)
							0.0361269	−	0.999347i	\(-0.488498\pi\)
\(11\)	21.8822	−	32.7490i	0.599793	−	0.897653i	−0.400029	−	0.916502i	\(-0.631000\pi\)
							0.999822	+	0.0188488i	\(0.00600012\pi\)
\(13\)	15.5736	+	15.5736i	0.332256	+	0.332256i	0.853443	−	0.521187i	\(-0.174510\pi\)
							−0.521187	+	0.853443i	\(0.674510\pi\)
\(17\)	−16.2279	+	68.1884i	−0.231520	+	0.972830i
							\(19\)	−8.01190	−	19.3424i	−0.0967398	−	0.233550i	0.868100	−	0.496389i	\(-0.165341\pi\)
−0.964840	+	0.262839i	\(0.915341\pi\)
\(23\)	109.889	+	73.4256i	0.996239	+	0.665665i	0.942957	−	0.332914i	\(-0.108032\pi\)
							0.0532816	+	0.998580i	\(0.483032\pi\)
\(29\)	14.7040	−	73.9219i	0.0941538	−	0.473343i	−0.904725	−	0.425997i	\(-0.859924\pi\)
							0.998879	−	0.0473464i	\(-0.0150765\pi\)
\(31\)	−48.7720	+	32.5884i	−0.282571	+	0.188808i	−0.688776	−	0.724974i	\(-0.741852\pi\)
							0.406205	+	0.913782i	\(0.366852\pi\)
\(37\)	88.0107	+	131.717i	0.391051	+	0.585248i	0.973801	−	0.227403i	\(-0.0730234\pi\)
							−0.582750	+	0.812651i	\(0.698023\pi\)
\(41\)	172.474	−	34.3072i	0.656973	−	0.130680i	0.144663	−	0.989481i	\(-0.453790\pi\)
							0.512310	+	0.858801i	\(0.328790\pi\)
\(43\)	72.0268	−	173.888i	0.255441	−	0.616690i	−0.743185	−	0.669086i	\(-0.766686\pi\)
							0.998626	+	0.0523958i	\(0.0166857\pi\)
\(47\)	−189.372	+	189.372i	−0.587718	+	0.587718i	−0.937013	−	0.349295i	\(-0.886421\pi\)
							0.349295	+	0.937013i	\(0.386421\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.4.i.a.5.6	✓	128
3.2	odd	2	inner	51.4.i.a.5.11	yes	128
17.7	odd	16	inner	51.4.i.a.41.11	yes	128
51.41	even	16	inner	51.4.i.a.41.6	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.i.a.5.6	✓	128	1.1	even	1	trivial
51.4.i.a.5.11	yes	128	3.2	odd	2	inner
51.4.i.a.41.6	yes	128	51.41	even	16	inner
51.4.i.a.41.11	yes	128	17.7	odd	16	inner