Embedded newform 16.15.f.a.3.5

• Label: 16.15.f.a.3.5
• 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			3.5
Character	\(\chi\)	\(=\)	16.3
Dual form			16.15.f.a.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-108.596 + 67.7560i) q^{2} +(-1164.54 + 1164.54i) q^{3} +(7202.24 - 14716.1i) q^{4} +(98063.2 - 98063.2i) q^{5} +(47560.0 - 205370. i) q^{6} -431458. q^{7} +(214968. + 2.08611e6i) q^{8} +2.07065e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−108.596	+	67.7560i	−0.848407	+	0.529344i
							\(3\)	−1164.54	+	1164.54i	−0.532484	+	0.532484i	−0.921311	−	0.388827i	\(-0.872880\pi\)
0.388827	+	0.921311i	\(0.372880\pi\)
\(5\)	98063.2	−	98063.2i	1.25521	−	1.25521i	0.301855	−	0.953354i	\(-0.402394\pi\)
							0.953354	−	0.301855i	\(-0.0976059\pi\)
\(7\)	−431458.			−0.523904			−0.261952	−	0.965081i	\(-0.584366\pi\)
							−0.261952	+	0.965081i	\(0.584366\pi\)
\(11\)	−1.02345e6	−	1.02345e6i	−0.0525193	−	0.0525193i	0.680359	−	0.732879i	\(-0.261824\pi\)
							−0.732879	+	0.680359i	\(0.761824\pi\)
\(13\)	−4.31640e7	−	4.31640e7i	−0.687888	−	0.687888i	0.273877	−	0.961765i	\(-0.411694\pi\)
							−0.961765	+	0.273877i	\(0.911694\pi\)
\(17\)	5.03362e8			1.22670			0.613349	−	0.789812i	\(-0.289822\pi\)
							0.613349	+	0.789812i	\(0.289822\pi\)
\(19\)	−7.43156e8	+	7.43156e8i	−0.831390	+	0.831390i	−0.987707	−	0.156317i	\(-0.950038\pi\)
							0.156317	+	0.987707i	\(0.450038\pi\)
\(23\)	−2.29456e9			−0.673914			−0.336957	−	0.941520i	\(-0.609398\pi\)
							−0.336957	+	0.941520i	\(0.609398\pi\)
\(29\)	−3.72132e9	−	3.72132e9i	−0.215730	−	0.215730i	0.590966	−	0.806696i	\(-0.298747\pi\)
							−0.806696	+	0.590966i	\(0.798747\pi\)
\(31\)		−	1.04382e10i		−	0.379396i	−0.981842	−	0.189698i	\(-0.939249\pi\)
							0.981842	−	0.189698i	\(-0.0607510\pi\)
\(37\)	−7.60088e10	+	7.60088e10i	−0.800667	+	0.800667i	−0.983200	−	0.182533i	\(-0.941570\pi\)
							0.182533	+	0.983200i	\(0.441570\pi\)
\(41\)		−	3.02287e11i		−	1.55215i	−0.630644	−	0.776073i	\(-0.717209\pi\)
							0.630644	−	0.776073i	\(-0.282791\pi\)
\(43\)	−2.71074e11	−	2.71074e11i	−0.997262	−	0.997262i	0.00273401	−	0.999996i	\(-0.499130\pi\)
							−0.999996	+	0.00273401i	\(0.999130\pi\)
\(47\)			4.94211e11i			0.975500i	0.872983	+	0.487750i	\(0.162182\pi\)
							−0.872983	+	0.487750i	\(0.837818\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.5	✓	54
4.3	odd	2		64.15.f.a.47.20		54
16.5	even	4		64.15.f.a.15.20		54
16.11	odd	4	inner	16.15.f.a.11.5	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.5	✓	54	1.1	even	1	trivial
16.15.f.a.11.5	yes	54	16.11	odd	4	inner
64.15.f.a.15.20		54	16.5	even	4
64.15.f.a.47.20		54	4.3	odd	2