Embedded newform 51.4.i.a.11.2

• Label: 51.4.i.a.11.2
• Level: $51$
• Weight: $4$
• Character: 51.11
• 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			11.2
Character	\(\chi\)	\(=\)	51.11
Dual form			51.4.i.a.14.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77125 + 4.27617i) q^{2} +(-4.82349 + 1.93234i) q^{3} +(-9.49143 - 9.49143i) q^{4} +(0.436777 - 0.653684i) q^{5} +(0.280586 - 24.0487i) q^{6} +(-5.84895 + 3.90814i) q^{7} +(23.1892 - 9.60530i) q^{8} +(19.5321 - 18.6412i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(37\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77125	+	4.27617i	−0.626230	+	1.51185i	0.218043	+	0.975939i	\(0.430033\pi\)
							−0.844273	+	0.535914i	\(0.819967\pi\)
\(3\)	−4.82349	+	1.93234i	−0.928281	+	0.371879i
							\(5\)	0.436777	−	0.653684i	0.0390666	−	0.0584672i	−0.811422	−	0.584460i	\(-0.801306\pi\)
0.850489	+	0.525993i	\(0.176306\pi\)
\(7\)	−5.84895	+	3.90814i	−0.315813	+	0.211020i	−0.703366	−	0.710828i	\(-0.748321\pi\)
							0.387553	+	0.921848i	\(0.373321\pi\)
\(11\)	5.31641	−	26.7274i	0.145723	−	0.732601i	−0.836954	−	0.547274i	\(-0.815666\pi\)
							0.982677	−	0.185327i	\(-0.0593344\pi\)
\(13\)	−31.6492	+	31.6492i	−0.675224	+	0.675224i	−0.958916	−	0.283692i	\(-0.908441\pi\)
							0.283692	+	0.958916i	\(0.408441\pi\)
\(17\)	−70.0040	−	3.52616i	−0.998734	−	0.0503070i
							\(19\)	−93.8813	−	38.8869i	−1.13357	−	0.469540i	−0.264578	−	0.964364i	\(-0.585233\pi\)
−0.868993	+	0.494824i	\(0.835233\pi\)
\(23\)	−168.442	−	33.5052i	−1.52707	−	0.303752i	−0.641084	−	0.767471i	\(-0.721515\pi\)
							−0.885983	+	0.463718i	\(0.846515\pi\)
\(29\)	203.236	+	135.798i	1.30138	+	0.869552i	0.996561	−	0.0828598i	\(-0.0264054\pi\)
							0.304815	+	0.952412i	\(0.401405\pi\)
\(31\)	−15.8826	+	3.15924i	−0.0920191	+	0.0183037i	−0.240885	−	0.970554i	\(-0.577438\pi\)
							0.148866	+	0.988857i	\(0.452438\pi\)
\(37\)	−38.1084	−	191.584i	−0.169324	−	0.851248i	−0.968282	−	0.249861i	\(-0.919615\pi\)
							0.798958	−	0.601387i	\(-0.205385\pi\)
\(41\)	6.05508	+	9.06206i	0.0230645	+	0.0345184i	0.842823	−	0.538191i	\(-0.180892\pi\)
							−0.819758	+	0.572710i	\(0.805892\pi\)
\(43\)	−373.218	+	154.592i	−1.32361	+	0.548258i	−0.928826	−	0.370516i	\(-0.879181\pi\)
							−0.394785	+	0.918774i	\(0.629181\pi\)
\(47\)	205.749	+	205.749i	0.638544	+	0.638544i	0.950196	−	0.311652i	\(-0.100882\pi\)
							−0.311652	+	0.950196i	\(0.600882\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.4.i.a.11.2	✓	128
3.2	odd	2	inner	51.4.i.a.11.15	yes	128
17.14	odd	16	inner	51.4.i.a.14.15	yes	128
51.14	even	16	inner	51.4.i.a.14.2	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.i.a.11.2	✓	128	1.1	even	1	trivial
51.4.i.a.11.15	yes	128	3.2	odd	2	inner
51.4.i.a.14.2	yes	128	51.14	even	16	inner
51.4.i.a.14.15	yes	128	17.14	odd	16	inner