Embedded newform 17.3.e.b.12.1

• Label: 17.3.e.b.12.1
• Level: $17$
• Weight: $3$
• Character: 17.12
• Analytic conductor: $0.463$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	17.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.463216449413\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{16})\)
Defining polynomial:	\( x^{8} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			12.1
Root			\(0.382683 - 0.923880i\) of defining polynomial
Character	\(\chi\)	\(=\)	17.12
Dual form			17.3.e.b.10.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.324423 + 0.783227i) q^{2} +(-1.35595 + 0.906019i) q^{3} +(2.32023 - 2.32023i) q^{4} +(-6.70292 - 1.33329i) q^{5} +(-1.14952 - 0.768086i) q^{6} +(0.886687 - 0.176373i) q^{7} +(5.70292 + 2.36223i) q^{8} +(-2.42641 + 5.85788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 24 q^{5} - 16 q^{7} + 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.324423	+	0.783227i	0.162212	+	0.391614i	0.983997	−	0.178183i	\(-0.0570219\pi\)
							−0.821786	+	0.569797i	\(0.807022\pi\)
\(3\)	−1.35595	+	0.906019i	−0.451984	+	0.302006i	−0.760648	−	0.649165i	\(-0.775118\pi\)
							0.308663	+	0.951171i	\(0.400118\pi\)
\(5\)	−6.70292	−	1.33329i	−1.34058	−	0.266659i	−0.527871	−	0.849324i	\(-0.677010\pi\)
							−0.812712	+	0.582666i	\(0.802010\pi\)
\(7\)	0.886687	−	0.176373i	0.126670	−	0.0251962i	−0.131348	−	0.991336i	\(-0.541931\pi\)
							0.258018	+	0.966140i	\(0.416931\pi\)
\(11\)	3.73690	−	5.59267i	0.339718	−	0.508424i	−0.621796	−	0.783179i	\(-0.713597\pi\)
							0.961514	+	0.274755i	\(0.0885967\pi\)
\(13\)	10.5602	+	10.5602i	0.812320	+	0.812320i	0.984981	−	0.172662i	\(-0.0552367\pi\)
							−0.172662	+	0.984981i	\(0.555237\pi\)
\(17\)	−14.7921	+	8.37823i	−0.870122	+	0.492837i
							\(19\)	−12.9821	−	31.3415i	−0.683268	−	1.64955i	−0.757922	−	0.652345i	\(-0.773785\pi\)
0.0746542	−	0.997209i	\(-0.476215\pi\)
\(23\)	15.4758	+	10.3406i	0.672860	+	0.449590i	0.844490	−	0.535571i	\(-0.179904\pi\)
							−0.171631	+	0.985161i	\(0.554904\pi\)
\(29\)	4.13027	−	20.7643i	0.142423	−	0.716009i	−0.841900	−	0.539634i	\(-0.818563\pi\)
							0.984323	−	0.176376i	\(-0.0564373\pi\)
\(31\)	21.1305	+	31.6240i	0.681629	+	1.02013i	0.997454	+	0.0713127i	\(0.0227188\pi\)
							−0.315825	+	0.948818i	\(0.602281\pi\)
\(37\)	−33.3284	+	22.2693i	−0.900767	+	0.601873i	−0.917390	−	0.397988i	\(-0.869708\pi\)
							0.0166233	+	0.999862i	\(0.494708\pi\)
\(41\)	−3.70199	+	0.736372i	−0.0902925	+	0.0179603i	−0.240030	−	0.970766i	\(-0.577157\pi\)
							0.149737	+	0.988726i	\(0.452157\pi\)
\(43\)	−5.21542	+	12.5911i	−0.121289	+	0.292817i	−0.972849	−	0.231439i	\(-0.925657\pi\)
							0.851561	+	0.524256i	\(0.175657\pi\)
\(47\)	−9.20504	−	9.20504i	−0.195852	−	0.195852i	0.602367	−	0.798219i	\(-0.294224\pi\)
							−0.798219	+	0.602367i	\(0.794224\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.3.e.b.12.1	yes	8
3.2	odd	2		153.3.p.a.46.1		8
4.3	odd	2		272.3.bh.b.97.1		8
5.2	odd	4		425.3.t.b.199.1		8
5.3	odd	4		425.3.t.d.199.1		8
5.4	even	2		425.3.u.a.301.1		8
17.2	even	8		289.3.e.a.65.1		8
17.3	odd	16		289.3.e.j.40.1		8
17.4	even	4		289.3.e.f.158.1		8
17.5	odd	16		289.3.e.a.249.1		8
17.6	odd	16		289.3.e.h.75.1		8
17.7	odd	16		289.3.e.g.214.1		8
17.8	even	8		289.3.e.n.224.1		8
17.9	even	8		289.3.e.j.224.1		8
17.10	odd	16	inner	17.3.e.b.10.1	✓	8
17.11	odd	16		289.3.e.f.75.1		8
17.12	odd	16		289.3.e.e.249.1		8
17.13	even	4		289.3.e.h.158.1		8
17.14	odd	16		289.3.e.n.40.1		8
17.15	even	8		289.3.e.e.65.1		8
17.16	even	2		289.3.e.g.131.1		8
51.44	even	16		153.3.p.a.10.1		8
68.27	even	16		272.3.bh.b.129.1		8
85.27	even	16		425.3.t.d.299.1		8
85.44	odd	16		425.3.u.a.401.1		8
85.78	even	16		425.3.t.b.299.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.3.e.b.10.1	✓	8	17.10	odd	16	inner
17.3.e.b.12.1	yes	8	1.1	even	1	trivial
153.3.p.a.10.1		8	51.44	even	16
153.3.p.a.46.1		8	3.2	odd	2
272.3.bh.b.97.1		8	4.3	odd	2
272.3.bh.b.129.1		8	68.27	even	16
289.3.e.a.65.1		8	17.2	even	8
289.3.e.a.249.1		8	17.5	odd	16
289.3.e.e.65.1		8	17.15	even	8
289.3.e.e.249.1		8	17.12	odd	16
289.3.e.f.75.1		8	17.11	odd	16
289.3.e.f.158.1		8	17.4	even	4
289.3.e.g.131.1		8	17.16	even	2
289.3.e.g.214.1		8	17.7	odd	16
289.3.e.h.75.1		8	17.6	odd	16
289.3.e.h.158.1		8	17.13	even	4
289.3.e.j.40.1		8	17.3	odd	16
289.3.e.j.224.1		8	17.9	even	8
289.3.e.n.40.1		8	17.14	odd	16
289.3.e.n.224.1		8	17.8	even	8
425.3.t.b.199.1		8	5.2	odd	4
425.3.t.b.299.1		8	85.78	even	16
425.3.t.d.199.1		8	5.3	odd	4
425.3.t.d.299.1		8	85.27	even	16
425.3.u.a.301.1		8	5.4	even	2
425.3.u.a.401.1		8	85.44	odd	16