Embedded newform 17.10.d.a.2.11

• Label: 17.10.d.a.2.11
• Level: $17$
• Weight: $10$
• Character: 17.2
• Analytic conductor: $8.756$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.75560921479\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(13\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			2.11
Character	\(\chi\)	\(=\)	17.2
Dual form			17.10.d.a.9.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(21.3276 + 21.3276i) q^{2} +(-21.6241 + 8.95698i) q^{3} +397.734i q^{4} +(288.795 + 697.212i) q^{5} +(-652.221 - 270.159i) q^{6} +(-4324.98 + 10441.4i) q^{7} +(2437.03 - 2437.03i) q^{8} +(-13530.6 + 13530.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	21.3276	+	21.3276i	0.942556	+	0.942556i	0.998437	−	0.0558814i	\(-0.0177969\pi\)
							−0.0558814	+	0.998437i	\(0.517797\pi\)
\(3\)	−21.6241	+	8.95698i	−0.154132	+	0.0638434i	−0.458416	−	0.888738i	\(-0.651583\pi\)
							0.304284	+	0.952581i	\(0.401583\pi\)
\(5\)	288.795	+	697.212i	0.206645	+	0.498884i	0.992891	−	0.119029i	\(-0.0379782\pi\)
							−0.786246	+	0.617913i	\(0.787978\pi\)
\(7\)	−4324.98	+	10441.4i	−0.680836	+	1.64368i	0.0816355	+	0.996662i	\(0.473986\pi\)
							−0.762472	+	0.647022i	\(0.776014\pi\)
\(11\)	5738.85	+	2377.11i	0.118184	+	0.0489533i	0.440992	−	0.897511i	\(-0.354627\pi\)
							−0.322808	+	0.946464i	\(0.604627\pi\)
\(13\)		−	150348.i		−	1.46000i	−0.683448	−	0.729999i	\(-0.739520\pi\)
							0.683448	−	0.729999i	\(-0.260480\pi\)
\(17\)	36644.8	+	342411.i	0.106412	+	0.994322i
							\(19\)	389452.	+	389452.i	0.685587	+	0.685587i	0.961253	−	0.275666i	\(-0.0888985\pi\)
−0.275666	+	0.961253i	\(0.588899\pi\)
\(23\)	602669.	+	249634.i	0.449059	+	0.186006i	0.595740	−	0.803177i	\(-0.296859\pi\)
							−0.146681	+	0.989184i	\(0.546859\pi\)
\(29\)	−592209.	−	1.42972e6i	−0.155483	−	0.375370i	0.826873	−	0.562389i	\(-0.190117\pi\)
							−0.982356	+	0.187019i	\(0.940117\pi\)
\(31\)	6.78653e6	−	2.81107e6i	1.31984	−	0.546694i	0.392097	−	0.919924i	\(-0.371750\pi\)
							0.927739	+	0.373230i	\(0.121750\pi\)
\(37\)	−2.84768e6	+	1.17955e6i	−0.249794	+	0.103468i	−0.504068	−	0.863664i	\(-0.668164\pi\)
							0.254273	+	0.967132i	\(0.418164\pi\)
\(41\)	−6.19031e6	+	1.49447e7i	−0.342125	+	0.825963i	0.655375	+	0.755303i	\(0.272510\pi\)
							−0.997500	+	0.0706598i	\(0.977490\pi\)
\(43\)	−2.79657e6	+	2.79657e6i	−0.124743	+	0.124743i	−0.766722	−	0.641979i	\(-0.778114\pi\)
							0.641979	+	0.766722i	\(0.278114\pi\)
\(47\)		−	4.93322e7i		−	1.47465i	−0.675536	−	0.737327i	\(-0.736088\pi\)
							0.675536	−	0.737327i	\(-0.263912\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.2.11	✓	52
17.3	odd	16		289.10.a.i.1.41		52
17.9	even	8	inner	17.10.d.a.9.11	yes	52
17.14	odd	16		289.10.a.i.1.42		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.11	✓	52	1.1	even	1	trivial
17.10.d.a.9.11	yes	52	17.9	even	8	inner
289.10.a.i.1.41		52	17.3	odd	16
289.10.a.i.1.42		52	17.14	odd	16