Embedded newform 16.10.e.a.13.3

• Label: 16.10.e.a.13.3
• Level: $16$
• Weight: $10$
• Character: 16.13
• Analytic conductor: $8.241$
• Analytic rank: $0$
• Dimension: $34$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.24057337862\)
Analytic rank:	\(0\)
Dimension:	\(34\)
Relative dimension:	\(17\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	16.13
Dual form			16.10.e.a.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-20.7667 - 8.98584i) q^{2} +(155.975 + 155.975i) q^{3} +(350.509 + 373.212i) q^{4} +(-404.055 + 404.055i) q^{5} +(-1837.51 - 4640.65i) q^{6} -637.759i q^{7} +(-3925.29 - 10900.0i) q^{8} +28973.4i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} - 2 q^{3} + 168 q^{4} - 2 q^{5} - 2192 q^{6} - 716 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−20.7667	−	8.98584i	−0.917766	−	0.397122i
							\(3\)	155.975	+	155.975i	1.11176	+	1.11176i	0.992913	+	0.118842i	\(0.0379182\pi\)
0.118842	+	0.992913i	\(0.462082\pi\)
\(5\)	−404.055	+	404.055i	−0.289119	+	0.289119i	−0.836732	−	0.547613i	\(-0.815537\pi\)
							0.547613	+	0.836732i	\(0.315537\pi\)
\(7\)		−	637.759i		−	0.100396i	−0.998739	−	0.0501978i	\(-0.984015\pi\)
							0.998739	−	0.0501978i	\(-0.0159852\pi\)
\(11\)	−51449.9	+	51449.9i	−1.05954	+	1.05954i	−0.0614283	+	0.998111i	\(0.519566\pi\)
							−0.998111	+	0.0614283i	\(0.980434\pi\)
\(13\)	75573.0	+	75573.0i	0.733874	+	0.733874i	0.971385	−	0.237511i	\(-0.0763316\pi\)
							−0.237511	+	0.971385i	\(0.576332\pi\)
\(17\)	−6136.34			−0.0178193			−0.00890963	−	0.999960i	\(-0.502836\pi\)
							−0.00890963	+	0.999960i	\(0.502836\pi\)
\(19\)	−618458.	−	618458.i	−1.08873	−	1.08873i	−0.995660	−	0.0930664i	\(-0.970333\pi\)
							−0.0930664	−	0.995660i	\(-0.529667\pi\)
\(23\)			2.24096e6i			1.66978i	0.550416	+	0.834891i	\(0.314469\pi\)
							−0.550416	+	0.834891i	\(0.685531\pi\)
\(29\)	−2.13077e6	−	2.13077e6i	−0.559429	−	0.559429i	0.369716	−	0.929145i	\(-0.379455\pi\)
							−0.929145	+	0.369716i	\(0.879455\pi\)
\(31\)	7.58926e6			1.47595			0.737976	−	0.674827i	\(-0.235782\pi\)
							0.737976	+	0.674827i	\(0.235782\pi\)
\(37\)	1.13688e7	−	1.13688e7i	0.997253	−	0.997253i	−0.00274356	−	0.999996i	\(-0.500873\pi\)
							0.999996	+	0.00274356i	\(0.000873304\pi\)
\(41\)		−	5.78849e6i		−	0.319917i	−0.987124	−	0.159959i	\(-0.948864\pi\)
							0.987124	−	0.159959i	\(-0.0511361\pi\)
\(43\)	−2.45770e6	+	2.45770e6i	−0.109628	+	0.109628i	−0.759793	−	0.650165i	\(-0.774700\pi\)
							0.650165	+	0.759793i	\(0.274700\pi\)
\(47\)	2.88187e7			0.861458			0.430729	−	0.902481i	\(-0.358256\pi\)
							0.430729	+	0.902481i	\(0.358256\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.10.e.a.13.3	yes	34
4.3	odd	2		64.10.e.a.17.2		34
8.3	odd	2		128.10.e.a.33.16		34
8.5	even	2		128.10.e.b.33.2		34
16.3	odd	4		128.10.e.a.97.16		34
16.5	even	4	inner	16.10.e.a.5.3	✓	34
16.11	odd	4		64.10.e.a.49.2		34
16.13	even	4		128.10.e.b.97.2		34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.10.e.a.5.3	✓	34	16.5	even	4	inner
16.10.e.a.13.3	yes	34	1.1	even	1	trivial
64.10.e.a.17.2		34	4.3	odd	2
64.10.e.a.49.2		34	16.11	odd	4
128.10.e.a.33.16		34	8.3	odd	2
128.10.e.a.97.16		34	16.3	odd	4
128.10.e.b.33.2		34	8.5	even	2
128.10.e.b.97.2		34	16.13	even	4