Embedded newform 51.4.i.a.11.1

• Label: 51.4.i.a.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	51.11
Dual form			51.4.i.a.14.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87197 + 4.51934i) q^{2} +(5.11979 - 0.887546i) q^{3} +(-11.2633 - 11.2633i) q^{4} +(-7.31533 + 10.9482i) q^{5} +(-5.57298 + 24.7995i) q^{6} +(-22.3550 + 14.9371i) q^{7} +(35.8324 - 14.8423i) q^{8} +(25.4245 - 9.08810i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.87197	+	4.51934i	−0.661842	+	1.59783i	0.133073	+	0.991106i	\(0.457515\pi\)
							−0.794915	+	0.606721i	\(0.792485\pi\)
\(3\)	5.11979	−	0.887546i	0.985304	−	0.170808i
							\(5\)	−7.31533	+	10.9482i	−0.654303	+	0.979233i	0.344873	+	0.938649i	\(0.387922\pi\)
−0.999176	+	0.0405840i	\(0.987078\pi\)
\(7\)	−22.3550	+	14.9371i	−1.20706	+	0.806529i	−0.985675	−	0.168658i	\(-0.946057\pi\)
							−0.221382	+	0.975187i	\(0.571057\pi\)
\(11\)	−4.84408	+	24.3529i	−0.132777	+	0.667515i	0.855862	+	0.517204i	\(0.173027\pi\)
							−0.988639	+	0.150310i	\(0.951973\pi\)
\(13\)	33.8613	−	33.8613i	0.722419	−	0.722419i	−0.246679	−	0.969097i	\(-0.579339\pi\)
							0.969097	+	0.246679i	\(0.0793392\pi\)
\(17\)	7.31795	+	69.7097i	0.104404	+	0.994535i
							\(19\)	103.854	+	43.0179i	1.25399	+	0.519420i	0.908060	−	0.418840i	\(-0.137563\pi\)
0.345930	+	0.938260i	\(0.387563\pi\)
\(23\)	−8.78883	−	1.74821i	−0.0796782	−	0.0158490i	0.155090	−	0.987900i	\(-0.450433\pi\)
							−0.234768	+	0.972051i	\(0.575433\pi\)
\(29\)	−43.4732	−	29.0479i	−0.278371	−	0.186002i	0.408545	−	0.912738i	\(-0.366036\pi\)
							−0.686917	+	0.726736i	\(0.741036\pi\)
\(31\)	−93.3315	+	18.5648i	−0.540737	+	0.107559i	−0.457900	−	0.889004i	\(-0.651398\pi\)
							−0.0828371	+	0.996563i	\(0.526398\pi\)
\(37\)	55.7873	+	280.462i	0.247875	+	1.24615i	0.881378	+	0.472412i	\(0.156617\pi\)
							−0.633503	+	0.773741i	\(0.718383\pi\)
\(41\)	−138.353	−	207.060i	−0.527003	−	0.788716i	0.468498	−	0.883465i	\(-0.344795\pi\)
							−0.995501	+	0.0947483i	\(0.969795\pi\)
\(43\)	351.440	−	145.571i	1.24637	−	0.516265i	0.340674	−	0.940182i	\(-0.389345\pi\)
							0.905701	+	0.423916i	\(0.139345\pi\)
\(47\)	193.234	+	193.234i	0.599705	+	0.599705i	0.940234	−	0.340529i	\(-0.110606\pi\)
							−0.340529	+	0.940234i	\(0.610606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.4.i.a.11.1	✓	128
3.2	odd	2	inner	51.4.i.a.11.16	yes	128
17.14	odd	16	inner	51.4.i.a.14.16	yes	128
51.14	even	16	inner	51.4.i.a.14.1	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.i.a.11.1	✓	128	1.1	even	1	trivial
51.4.i.a.11.16	yes	128	3.2	odd	2	inner
51.4.i.a.14.1	yes	128	51.14	even	16	inner
51.4.i.a.14.16	yes	128	17.14	odd	16	inner