Embedded newform 16.15.f.a.3.19

• Label: 16.15.f.a.3.19
• Level: $16$
• Weight: $15$
• Character: 16.3
• Analytic conductor: $19.893$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.8926349043\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.19
Character	\(\chi\)	\(=\)	16.3
Dual form			16.15.f.a.11.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(74.1318 + 104.348i) q^{2} +(-498.544 + 498.544i) q^{3} +(-5392.95 + 15471.0i) q^{4} +(40758.2 - 40758.2i) q^{5} +(-88979.9 - 15064.0i) q^{6} -589264. q^{7} +(-2.01415e6 + 584149. i) q^{8} +4.28588e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	74.1318	+	104.348i	0.579155	+	0.815218i
							\(3\)	−498.544	+	498.544i	−0.227958	+	0.227958i	−0.811839	−	0.583881i	\(-0.801533\pi\)
0.583881	+	0.811839i	\(0.301533\pi\)
\(5\)	40758.2	−	40758.2i	0.521705	−	0.521705i	−0.396381	−	0.918086i	\(-0.629734\pi\)
							0.918086	+	0.396381i	\(0.129734\pi\)
\(7\)	−589264.			−0.715523			−0.357762	−	0.933813i	\(-0.616460\pi\)
							−0.357762	+	0.933813i	\(0.616460\pi\)
\(11\)	−2.41357e7	−	2.41357e7i	−1.23854	−	1.23854i	−0.960597	−	0.277943i	\(-0.910347\pi\)
							−0.277943	−	0.960597i	\(-0.589653\pi\)
\(13\)	1.80328e6	+	1.80328e6i	0.0287382	+	0.0287382i	0.721330	−	0.692592i	\(-0.243531\pi\)
							−0.692592	+	0.721330i	\(0.743531\pi\)
\(17\)	−3.17335e8			−0.773349			−0.386675	−	0.922216i	\(-0.626376\pi\)
							−0.386675	+	0.922216i	\(0.626376\pi\)
\(19\)	1.06802e9	−	1.06802e9i	1.19482	−	1.19482i	0.219125	−	0.975697i	\(-0.429680\pi\)
							0.975697	−	0.219125i	\(-0.0703203\pi\)
\(23\)	−2.13381e8			−0.0626702			−0.0313351	−	0.999509i	\(-0.509976\pi\)
							−0.0313351	+	0.999509i	\(0.509976\pi\)
\(29\)	−1.97566e10	−	1.97566e10i	−1.14532	−	1.14532i	−0.987462	−	0.157857i	\(-0.949541\pi\)
							−0.157857	−	0.987462i	\(-0.550459\pi\)
\(31\)		−	1.06448e10i		−	0.386905i	−0.981110	−	0.193453i	\(-0.938031\pi\)
							0.981110	−	0.193453i	\(-0.0619686\pi\)
\(37\)	−6.70249e10	+	6.70249e10i	−0.706031	+	0.706031i	−0.965698	−	0.259667i	\(-0.916387\pi\)
							0.259667	+	0.965698i	\(0.416387\pi\)
\(41\)			1.36279e11i			0.699746i	0.936797	+	0.349873i	\(0.113775\pi\)
							−0.936797	+	0.349873i	\(0.886225\pi\)
\(43\)	7.96568e9	+	7.96568e9i	0.0293051	+	0.0293051i	0.721608	−	0.692302i	\(-0.243404\pi\)
							−0.692302	+	0.721608i	\(0.743404\pi\)
\(47\)			8.74309e11i			1.72576i	0.505411	+	0.862879i	\(0.331341\pi\)
							−0.505411	+	0.862879i	\(0.668659\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.19	✓	54
4.3	odd	2		64.15.f.a.47.16		54
16.5	even	4		64.15.f.a.15.16		54
16.11	odd	4	inner	16.15.f.a.11.19	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.19	✓	54	1.1	even	1	trivial
16.15.f.a.11.19	yes	54	16.11	odd	4	inner
64.15.f.a.15.16		54	16.5	even	4
64.15.f.a.47.16		54	4.3	odd	2