Embedded newform 17.10.d.a.8.9

• Label: 17.10.d.a.8.9
• Level: $17$
• Weight: $10$
• Character: 17.8
• 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			8.9
Character	\(\chi\)	\(=\)	17.8
Dual form			17.10.d.a.15.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.30554 - 6.30554i) q^{2} +(12.9872 - 31.3540i) q^{3} +432.480i q^{4} +(-32.2201 - 13.3460i) q^{5} +(-115.812 - 279.595i) q^{6} +(-7889.84 + 3268.08i) q^{7} +(5955.46 + 5955.46i) q^{8} +(13103.6 + 13103.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.30554	−	6.30554i	0.278668	−	0.278668i	−0.553909	−	0.832577i	\(-0.686864\pi\)
							0.832577	+	0.553909i	\(0.186864\pi\)
\(3\)	12.9872	−	31.3540i	0.0925702	−	0.223484i	−0.870812	−	0.491616i	\(-0.836406\pi\)
							0.963382	+	0.268132i	\(0.0864063\pi\)
\(5\)	−32.2201	−	13.3460i	−0.0230548	−	0.00954962i	0.371126	−	0.928582i	\(-0.378972\pi\)
							−0.394181	+	0.919033i	\(0.628972\pi\)
\(7\)	−7889.84	+	3268.08i	−1.24202	+	0.514460i	−0.904344	−	0.426804i	\(-0.859639\pi\)
							−0.337672	+	0.941264i	\(0.609639\pi\)
\(11\)	317.169	+	765.715i	0.00653167	+	0.0157688i	0.927112	−	0.374785i	\(-0.122283\pi\)
							−0.920580	+	0.390554i	\(0.872283\pi\)
\(13\)			147504.i			1.43238i	0.697905	+	0.716190i	\(0.254116\pi\)
							−0.697905	+	0.716190i	\(0.745884\pi\)
\(17\)	338719.	+	62107.4i	0.983602	+	0.180353i
							\(19\)	−272849.	+	272849.i	−0.480321	+	0.480321i	−0.905234	−	0.424913i	\(-0.860305\pi\)
0.424913	+	0.905234i	\(0.360305\pi\)
\(23\)	−757938.	−	1.82982e6i	−0.564752	−	1.36343i	−0.905927	−	0.423433i	\(-0.860825\pi\)
							0.341175	−	0.940000i	\(-0.389175\pi\)
\(29\)	2.43674e6	+	1.00933e6i	0.639762	+	0.264998i	0.678894	−	0.734236i	\(-0.262459\pi\)
							−0.0391324	+	0.999234i	\(0.512459\pi\)
\(31\)	2.45568e6	−	5.92853e6i	0.477577	−	1.15297i	−0.483165	−	0.875529i	\(-0.660513\pi\)
							0.960742	−	0.277444i	\(-0.0894872\pi\)
\(37\)	−7.24517e6	+	1.74914e7i	−0.635537	+	1.53432i	0.197031	+	0.980397i	\(0.436870\pi\)
							−0.832567	+	0.553924i	\(0.813130\pi\)
\(41\)	8.61660e6	−	3.56911e6i	0.476221	−	0.197257i	−0.131645	−	0.991297i	\(-0.542026\pi\)
							0.607866	+	0.794040i	\(0.292026\pi\)
\(43\)	1.21629e7	+	1.21629e7i	0.542535	+	0.542535i	0.924271	−	0.381736i	\(-0.124674\pi\)
							−0.381736	+	0.924271i	\(0.624674\pi\)
\(47\)			1.46297e7i			0.437317i	0.975801	+	0.218658i	\(0.0701680\pi\)
							−0.975801	+	0.218658i	\(0.929832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.8.9	✓	52
17.7	odd	16		289.10.a.i.1.19		52
17.10	odd	16		289.10.a.i.1.20		52
17.15	even	8	inner	17.10.d.a.15.9	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.8.9	✓	52	1.1	even	1	trivial
17.10.d.a.15.9	yes	52	17.15	even	8	inner
289.10.a.i.1.19		52	17.7	odd	16
289.10.a.i.1.20		52	17.10	odd	16