Embedded newform 16.15.f.a.3.11

• Label: 16.15.f.a.3.11
• 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			3.11
Character	\(\chi\)	\(=\)	16.3
Dual form			16.15.f.a.11.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-40.7689 + 121.334i) q^{2} +(-2416.37 + 2416.37i) q^{3} +(-13059.8 - 9893.30i) q^{4} +(-20189.0 + 20189.0i) q^{5} +(-194674. - 391700. i) q^{6} -480275. q^{7} +(1.73283e6 - 1.18125e6i) q^{8} -6.89468e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−40.7689	+	121.334i	−0.318507	+	0.947920i
							\(3\)	−2416.37	+	2416.37i	−1.10488	+	1.10488i	−0.111064	+	0.993813i	\(0.535426\pi\)
−0.993813	+	0.111064i	\(0.964574\pi\)
\(5\)	−20189.0	+	20189.0i	−0.258420	+	0.258420i	−0.824411	−	0.565991i	\(-0.808494\pi\)
							0.565991	+	0.824411i	\(0.308494\pi\)
\(7\)	−480275.			−0.583181			−0.291590	−	0.956543i	\(-0.594184\pi\)
							−0.291590	+	0.956543i	\(0.594184\pi\)
\(11\)	−1.80847e7	−	1.80847e7i	−0.928033	−	0.928033i	0.0695456	−	0.997579i	\(-0.477845\pi\)
							−0.997579	+	0.0695456i	\(0.977845\pi\)
\(13\)	3.53141e7	+	3.53141e7i	0.562787	+	0.562787i	0.930098	−	0.367311i	\(-0.119721\pi\)
							−0.367311	+	0.930098i	\(0.619721\pi\)
\(17\)	−4.80313e8			−1.17053			−0.585264	−	0.810843i	\(-0.699009\pi\)
							−0.585264	+	0.810843i	\(0.699009\pi\)
\(19\)	−6.99872e8	+	6.99872e8i	−0.782967	+	0.782967i	−0.980330	−	0.197364i	\(-0.936762\pi\)
							0.197364	+	0.980330i	\(0.436762\pi\)
\(23\)	6.80752e8			0.199938			0.0999688	−	0.994991i	\(-0.468126\pi\)
							0.0999688	+	0.994991i	\(0.468126\pi\)
\(29\)	2.07702e10	+	2.07702e10i	1.20408	+	1.20408i	0.972914	+	0.231166i	\(0.0742542\pi\)
							0.231166	+	0.972914i	\(0.425746\pi\)
\(31\)		−	3.27392e10i		−	1.18997i	−0.803736	−	0.594986i	\(-0.797158\pi\)
							0.803736	−	0.594986i	\(-0.202842\pi\)
\(37\)	3.47364e10	−	3.47364e10i	0.365909	−	0.365909i	−0.500074	−	0.865983i	\(-0.666694\pi\)
							0.865983	+	0.500074i	\(0.166694\pi\)
\(41\)		−	2.51758e10i		−	0.129269i	−0.997909	−	0.0646347i	\(-0.979412\pi\)
							0.997909	−	0.0646347i	\(-0.0205882\pi\)
\(43\)	−2.39402e11	−	2.39402e11i	−0.880742	−	0.880742i	0.112868	−	0.993610i	\(-0.463996\pi\)
							−0.993610	+	0.112868i	\(0.963996\pi\)
\(47\)		−	7.85983e8i		−	0.00155142i	−1.00000		0.000775708i	\(-0.999753\pi\)
							1.00000		0.000775708i	\(-0.000246916\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.11	✓	54
4.3	odd	2		64.15.f.a.47.25		54
16.5	even	4		64.15.f.a.15.25		54
16.11	odd	4	inner	16.15.f.a.11.11	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.11	✓	54	1.1	even	1	trivial
16.15.f.a.11.11	yes	54	16.11	odd	4	inner
64.15.f.a.15.25		54	16.5	even	4
64.15.f.a.47.25		54	4.3	odd	2