Embedded newform 17.10.d.a.15.11

• Label: 17.10.d.a.15.11
• Level: $17$
• Weight: $10$
• Character: 17.15
• 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			15.11
Character	\(\chi\)	\(=\)	17.15
Dual form			17.10.d.a.8.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(17.9968 + 17.9968i) q^{2} +(-30.9190 - 74.6450i) q^{3} +135.770i q^{4} +(723.808 - 299.811i) q^{5} +(786.929 - 1899.81i) q^{6} +(5306.77 + 2198.13i) q^{7} +(6770.93 - 6770.93i) q^{8} +(9302.09 - 9302.09i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	17.9968	+	17.9968i	0.795354	+	0.795354i	0.982359	−	0.187005i	\(-0.0598780\pi\)
							−0.187005	+	0.982359i	\(0.559878\pi\)
\(3\)	−30.9190	−	74.6450i	−0.220384	−	0.532053i	0.774559	−	0.632502i	\(-0.217972\pi\)
							−0.994942	+	0.100449i	\(0.967972\pi\)
\(5\)	723.808	−	299.811i	0.517915	−	0.214527i	−0.108386	−	0.994109i	\(-0.534568\pi\)
							0.626301	+	0.779582i	\(0.284568\pi\)
\(7\)	5306.77	+	2198.13i	0.835389	+	0.346029i	0.759034	−	0.651051i	\(-0.225672\pi\)
							0.0763552	+	0.997081i	\(0.475672\pi\)
\(11\)	992.353	−	2395.75i	0.0204362	−	0.0493372i	−0.913332	−	0.407216i	\(-0.866500\pi\)
							0.933768	+	0.357878i	\(0.116500\pi\)
\(13\)			108940.i			1.05790i	0.848654	+	0.528949i	\(0.177414\pi\)
							−0.848654	+	0.528949i	\(0.822586\pi\)
\(17\)	−49428.2	−	340800.i	−0.143534	−	0.989645i
							\(19\)	−67635.2	−	67635.2i	−0.119064	−	0.119064i	0.645064	−	0.764128i	\(-0.276831\pi\)
−0.764128	+	0.645064i	\(0.776831\pi\)
\(23\)	35562.3	−	85855.1i	0.0264981	−	0.0639721i	−0.910077	−	0.414440i	\(-0.863977\pi\)
							0.936575	+	0.350468i	\(0.113977\pi\)
\(29\)	−2.69221e6	+	1.11515e6i	−0.706834	+	0.292780i	−0.706994	−	0.707220i	\(-0.749949\pi\)
							0.000160047		1.00000i	\(0.499949\pi\)
\(31\)	−834349.	−	2.01430e6i	−0.162263	−	0.391738i	0.821746	−	0.569853i	\(-0.193000\pi\)
							−0.984010	+	0.178115i	\(0.943000\pi\)
\(37\)	4.20614e6	+	1.01545e7i	0.368957	+	0.890741i	0.993922	+	0.110088i	\(0.0351134\pi\)
							−0.624965	+	0.780653i	\(0.714887\pi\)
\(41\)	1.02771e7	+	4.25693e6i	0.567996	+	0.235271i	0.648152	−	0.761511i	\(-0.275542\pi\)
							−0.0801567	+	0.996782i	\(0.525542\pi\)
\(43\)	−8.53406e6	+	8.53406e6i	−0.380669	+	0.380669i	−0.871343	−	0.490674i	\(-0.836751\pi\)
							0.490674	+	0.871343i	\(0.336751\pi\)
\(47\)			5.88161e7i			1.75815i	0.476684	+	0.879075i	\(0.341839\pi\)
							−0.476684	+	0.879075i	\(0.658161\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.15.11	yes	52
17.5	odd	16		289.10.a.i.1.14		52
17.8	even	8	inner	17.10.d.a.8.11	✓	52
17.12	odd	16		289.10.a.i.1.13		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.8.11	✓	52	17.8	even	8	inner
17.10.d.a.15.11	yes	52	1.1	even	1	trivial
289.10.a.i.1.13		52	17.12	odd	16
289.10.a.i.1.14		52	17.5	odd	16