Embedded newform 85.2.m.a.49.4

• Label: 85.2.m.a.49.4
• Level: $85$
• Weight: $2$
• Character: 85.49
• Analytic conductor: $0.679$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.m (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.678728417181\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			49.4
Character	\(\chi\)	\(=\)	85.49
Dual form			85.2.m.a.59.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.176012 - 0.176012i) q^{2} +(-1.51856 + 0.629010i) q^{3} +1.93804i q^{4} +(1.62749 + 1.53338i) q^{5} +(-0.156572 + 0.377999i) q^{6} +(0.547653 - 1.32215i) q^{7} +(0.693142 + 0.693142i) q^{8} +(-0.210935 + 0.210935i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(71\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.176012	−	0.176012i	0.124459	−	0.124459i	−0.642134	−	0.766593i	\(-0.721951\pi\)
							0.766593	+	0.642134i	\(0.221951\pi\)
\(3\)	−1.51856	+	0.629010i	−0.876744	+	0.363159i	−0.775233	−	0.631675i	\(-0.782368\pi\)
							−0.101511	+	0.994834i	\(0.532368\pi\)
\(5\)	1.62749	+	1.53338i	0.727837	+	0.685750i
							\(7\)	0.547653	−	1.32215i	0.206993	−	0.499726i	−0.785954	−	0.618286i	\(-0.787828\pi\)
0.992947	+	0.118559i	\(0.0378275\pi\)
\(11\)	1.03080	−	2.48857i	0.310798	−	0.750333i	−0.688878	−	0.724878i	\(-0.741896\pi\)
							0.999676	−	0.0254558i	\(-0.00810370\pi\)
\(13\)	0.174664			0.0484432			0.0242216	−	0.999707i	\(-0.492289\pi\)
							0.0242216	+	0.999707i	\(0.492289\pi\)
\(17\)	2.27165	−	3.44088i	0.550955	−	0.834535i
							\(19\)	−2.69404	−	2.69404i	−0.618055	−	0.618055i	0.326978	−	0.945032i	\(-0.393970\pi\)
−0.945032	+	0.326978i	\(0.893970\pi\)
\(23\)	2.54807	+	1.05544i	0.531309	+	0.220075i	0.632176	−	0.774825i	\(-0.282162\pi\)
							−0.100867	+	0.994900i	\(0.532162\pi\)
\(29\)	5.94408	−	2.46212i	1.10379	−	0.457204i	0.244994	−	0.969525i	\(-0.421214\pi\)
							0.858794	+	0.512321i	\(0.171214\pi\)
\(31\)	−0.188417	−	0.454879i	−0.0338407	−	0.0816987i	0.906056	−	0.423159i	\(-0.139079\pi\)
							−0.939896	+	0.341460i	\(0.889079\pi\)
\(37\)	−5.30576	+	2.19772i	−0.872261	+	0.361302i	−0.773490	−	0.633808i	\(-0.781491\pi\)
							−0.0987704	+	0.995110i	\(0.531491\pi\)
\(41\)	5.54000	+	2.29474i	0.865203	+	0.358379i	0.770740	−	0.637150i	\(-0.219887\pi\)
							0.0944627	+	0.995528i	\(0.469887\pi\)
\(43\)	−8.00004	−	8.00004i	−1.22000	−	1.22000i	−0.967633	−	0.252363i	\(-0.918792\pi\)
							−0.252363	−	0.967633i	\(-0.581208\pi\)
\(47\)	2.65291			0.386967			0.193483	−	0.981104i	\(-0.438021\pi\)
							0.193483	+	0.981104i	\(0.438021\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.2.m.a.49.4	yes	24
3.2	odd	2		765.2.bh.b.559.3		24
5.2	odd	4		425.2.m.e.151.4		24
5.3	odd	4		425.2.m.e.151.3		24
5.4	even	2	inner	85.2.m.a.49.3	✓	24
15.14	odd	2		765.2.bh.b.559.4		24
17.5	odd	16		1445.2.b.i.579.13		24
17.8	even	8	inner	85.2.m.a.59.3	yes	24
17.12	odd	16		1445.2.b.i.579.14		24
51.8	odd	8		765.2.bh.b.739.4		24
85.8	odd	8		425.2.m.e.76.3		24
85.12	even	16		7225.2.a.by.1.12		24
85.22	even	16		7225.2.a.by.1.11		24
85.29	odd	16		1445.2.b.i.579.11		24
85.39	odd	16		1445.2.b.i.579.12		24
85.42	odd	8		425.2.m.e.76.4		24
85.59	even	8	inner	85.2.m.a.59.4	yes	24
85.63	even	16		7225.2.a.by.1.13		24
85.73	even	16		7225.2.a.by.1.14		24
255.59	odd	8		765.2.bh.b.739.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.m.a.49.3	✓	24	5.4	even	2	inner
85.2.m.a.49.4	yes	24	1.1	even	1	trivial
85.2.m.a.59.3	yes	24	17.8	even	8	inner
85.2.m.a.59.4	yes	24	85.59	even	8	inner
425.2.m.e.76.3		24	85.8	odd	8
425.2.m.e.76.4		24	85.42	odd	8
425.2.m.e.151.3		24	5.3	odd	4
425.2.m.e.151.4		24	5.2	odd	4
765.2.bh.b.559.3		24	3.2	odd	2
765.2.bh.b.559.4		24	15.14	odd	2
765.2.bh.b.739.3		24	255.59	odd	8
765.2.bh.b.739.4		24	51.8	odd	8
1445.2.b.i.579.11		24	85.29	odd	16
1445.2.b.i.579.12		24	85.39	odd	16
1445.2.b.i.579.13		24	17.5	odd	16
1445.2.b.i.579.14		24	17.12	odd	16
7225.2.a.by.1.11		24	85.22	even	16
7225.2.a.by.1.12		24	85.12	even	16
7225.2.a.by.1.13		24	85.63	even	16
7225.2.a.by.1.14		24	85.73	even	16