Embedded newform 51.4.i.a.5.9

• Label: 51.4.i.a.5.9
• 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			5.9
Character	\(\chi\)	\(=\)	51.5
Dual form			51.4.i.a.41.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.358536 - 0.148510i) q^{2} +(-4.89946 - 1.73068i) q^{3} +(-5.55036 + 5.55036i) q^{4} +(18.8139 + 3.74231i) q^{5} +(-2.01366 + 0.107111i) q^{6} +(4.84695 + 24.3673i) q^{7} +(-2.35380 + 5.68258i) q^{8} +(21.0095 + 16.9588i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.358536	−	0.148510i	0.126762	−	0.0525064i	−0.318401	−	0.947956i	\(-0.603146\pi\)
							0.445163	+	0.895450i	\(0.353146\pi\)
\(3\)	−4.89946	−	1.73068i	−0.942902	−	0.333069i
							\(5\)	18.8139	+	3.74231i	1.68276	+	0.334722i	0.941633	−	0.336641i	\(-0.109291\pi\)
0.741129	+	0.671363i	\(0.234291\pi\)
\(7\)	4.84695	+	24.3673i	0.261711	+	1.31571i	0.858294	+	0.513158i	\(0.171524\pi\)
							−0.596583	+	0.802551i	\(0.703476\pi\)
\(11\)	−11.3987	+	17.0593i	−0.312439	+	0.467598i	−0.954142	−	0.299355i	\(-0.903228\pi\)
							0.641702	+	0.766954i	\(0.278228\pi\)
\(13\)	−42.2289	−	42.2289i	−0.900937	−	0.900937i	0.0945805	−	0.995517i	\(-0.469849\pi\)
							−0.995517	+	0.0945805i	\(0.969849\pi\)
\(17\)	51.8097	+	47.2097i	0.739158	+	0.673532i
							\(19\)	20.7824	+	50.1731i	0.250937	+	0.605815i	0.998280	−	0.0586228i	\(-0.0186709\pi\)
−0.747343	+	0.664438i	\(0.768671\pi\)
\(23\)	−1.16337	−	0.777336i	−0.0105469	−	0.00704721i	0.550286	−	0.834976i	\(-0.314519\pi\)
							−0.560833	+	0.827929i	\(0.689519\pi\)
\(29\)	32.9430	−	165.615i	0.210943	−	1.06048i	−0.719624	−	0.694363i	\(-0.755686\pi\)
							0.930568	−	0.366120i	\(-0.119314\pi\)
\(31\)	−27.1702	+	18.1546i	−0.157417	+	0.105182i	−0.631784	−	0.775145i	\(-0.717677\pi\)
							0.474367	+	0.880327i	\(0.342677\pi\)
\(37\)	−70.6414	−	105.722i	−0.313875	−	0.469747i	0.640669	−	0.767817i	\(-0.278657\pi\)
							−0.954544	+	0.298070i	\(0.903657\pi\)
\(41\)	338.265	−	67.2850i	1.28849	−	0.256297i	0.497145	−	0.867667i	\(-0.334382\pi\)
							0.791344	+	0.611371i	\(0.209382\pi\)
\(43\)	30.0560	−	72.5615i	0.106593	−	0.257338i	−0.861579	−	0.507623i	\(-0.830524\pi\)
							0.968172	+	0.250285i	\(0.0805243\pi\)
\(47\)	116.133	−	116.133i	0.360421	−	0.360421i	−0.503547	−	0.863968i	\(-0.667972\pi\)
							0.863968	+	0.503547i	\(0.167972\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.4.i.a.5.9	yes	128
3.2	odd	2	inner	51.4.i.a.5.8	✓	128
17.7	odd	16	inner	51.4.i.a.41.8	yes	128
51.41	even	16	inner	51.4.i.a.41.9	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.i.a.5.8	✓	128	3.2	odd	2	inner
51.4.i.a.5.9	yes	128	1.1	even	1	trivial
51.4.i.a.41.8	yes	128	17.7	odd	16	inner
51.4.i.a.41.9	yes	128	51.41	even	16	inner