Embedded newform 51.4.i.a.5.4

• Label: 51.4.i.a.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	51.5
Dual form			51.4.i.a.41.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.00477 + 1.24462i) q^{2} +(-4.72345 + 2.16542i) q^{3} +(1.82273 - 1.82273i) q^{4} +(-1.80227 - 0.358494i) q^{5} +(11.4978 - 12.3855i) q^{6} +(-0.238671 - 1.19988i) q^{7} +(6.74866 - 16.2927i) q^{8} +(17.6219 - 20.4565i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.00477	+	1.24462i	−1.06235	+	0.440039i	−0.844284	−	0.535896i	\(-0.819974\pi\)
							−0.218063	+	0.975935i	\(0.569974\pi\)
\(3\)	−4.72345	+	2.16542i	−0.909028	+	0.416735i
							\(5\)	−1.80227	−	0.358494i	−0.161200	−	0.0320647i	0.113830	−	0.993500i	\(-0.463688\pi\)
−0.275031	+	0.961436i	\(0.588688\pi\)
\(7\)	−0.238671	−	1.19988i	−0.0128870	−	0.0647875i	0.973807	−	0.227378i	\(-0.0730152\pi\)
							−0.986694	+	0.162590i	\(0.948015\pi\)
\(11\)	17.0307	−	25.4882i	0.466813	−	0.698634i	−0.521126	−	0.853480i	\(-0.674488\pi\)
							0.987939	+	0.154845i	\(0.0494879\pi\)
\(13\)	−13.5181	−	13.5181i	−0.288403	−	0.288403i	0.548046	−	0.836448i	\(-0.315372\pi\)
							−0.836448	+	0.548046i	\(0.815372\pi\)
\(17\)	−55.9390	−	42.2354i	−0.798071	−	0.602564i
							\(19\)	−3.30888	−	7.98835i	−0.0399532	−	0.0964555i	0.902643	−	0.430389i	\(-0.141624\pi\)
−0.942596	+	0.333934i	\(0.891624\pi\)
\(23\)	−4.45156	−	2.97444i	−0.0403572	−	0.0269658i	0.535228	−	0.844708i	\(-0.320226\pi\)
							−0.575585	+	0.817742i	\(0.695226\pi\)
\(29\)	31.5754	−	158.740i	0.202186	−	1.01646i	−0.737741	−	0.675084i	\(-0.764108\pi\)
							0.939927	−	0.341375i	\(-0.110892\pi\)
\(31\)	114.526	−	76.5236i	0.663530	−	0.443356i	−0.177663	−	0.984091i	\(-0.556854\pi\)
							0.841193	+	0.540735i	\(0.181854\pi\)
\(37\)	−188.835	−	282.611i	−0.839033	−	1.25570i	−0.964627	−	0.263620i	\(-0.915084\pi\)
							0.125594	−	0.992082i	\(-0.459916\pi\)
\(41\)	85.8414	−	17.0749i	0.326980	−	0.0650404i	−0.0288708	−	0.999583i	\(-0.509191\pi\)
							0.355851	+	0.934543i	\(0.384191\pi\)
\(43\)	130.932	−	316.097i	0.464347	−	1.12103i	−0.502248	−	0.864723i	\(-0.667494\pi\)
							0.966595	−	0.256309i	\(-0.0825063\pi\)
\(47\)	−200.519	+	200.519i	−0.622312	+	0.622312i	−0.946122	−	0.323810i	\(-0.895036\pi\)
							0.323810	+	0.946122i	\(0.395036\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.4.i.a.5.4	✓	128
3.2	odd	2	inner	51.4.i.a.5.13	yes	128
17.7	odd	16	inner	51.4.i.a.41.13	yes	128
51.41	even	16	inner	51.4.i.a.41.4	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.i.a.5.4	✓	128	1.1	even	1	trivial
51.4.i.a.5.13	yes	128	3.2	odd	2	inner
51.4.i.a.41.4	yes	128	51.41	even	16	inner
51.4.i.a.41.13	yes	128	17.7	odd	16	inner