Embedded newform 175.2.s.a.48.16

• Label: 175.2.s.a.48.16
• Level: $175$
• Weight: $2$
• Character: 175.48
• Analytic conductor: $1.397$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	175.s (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.39738203537\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(18\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			48.16
Character	\(\chi\)	\(=\)	175.48
Dual form			175.2.s.a.62.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.321665 - 2.03092i) q^{2} +(1.39246 + 2.73286i) q^{3} +(-2.11904 - 0.688517i) q^{4} +(-0.390914 - 2.20163i) q^{5} +(5.99812 - 1.94891i) q^{6} +(1.56983 + 2.12970i) q^{7} +(-0.212921 + 0.417881i) q^{8} +(-3.76622 + 5.18376i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 16 q^{2} - 20 q^{4} - 14 q^{7} - 12 q^{8} - 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.321665	−	2.03092i	0.227452	−	1.43607i	−0.564471	−	0.825453i	\(-0.690920\pi\)
							0.791923	−	0.610621i	\(-0.209080\pi\)
\(3\)	1.39246	+	2.73286i	0.803939	+	1.57782i	0.816113	+	0.577892i	\(0.196125\pi\)
							−0.0121746	+	0.999926i	\(0.503875\pi\)
\(5\)	−0.390914	−	2.20163i	−0.174822	−	0.984600i
							\(7\)	1.56983	+	2.12970i	0.593341	+	0.804951i
\(11\)	2.92802	−	2.12733i	0.882833	−	0.641415i								−0.0511668	−	0.998690i	\(-0.516294\pi\)
							0.933999	+	0.357275i	\(0.116294\pi\)
\(13\)	−2.07973	+	0.329397i	−0.576814	+	0.0913583i	−0.438024	−	0.898963i	\(-0.644321\pi\)
							−0.138790	+	0.990322i	\(0.544321\pi\)
\(17\)	−0.704948	+	1.38354i	−0.170975	+	0.335557i	−0.960555	−	0.278090i	\(-0.910299\pi\)
							0.789580	+	0.613648i	\(0.210299\pi\)
\(19\)	−0.298017	−	0.917202i	−0.0683698	−	0.210420i	0.911034	−	0.412331i	\(-0.135285\pi\)
							−0.979404	+	0.201910i	\(0.935285\pi\)
\(23\)	−2.57698	−	0.408154i	−0.537338	−	0.0851061i	−0.118134	−	0.992998i	\(-0.537691\pi\)
							−0.419205	+	0.907892i	\(0.637691\pi\)
\(29\)	−4.65189	−	1.51149i	−0.863834	−	0.280677i	−0.156605	−	0.987661i	\(-0.550055\pi\)
							−0.707229	+	0.706985i	\(0.750055\pi\)
\(31\)	−9.20646	+	2.99136i	−1.65353	+	0.537264i	−0.979501	−	0.201440i	\(-0.935438\pi\)
							−0.674029	+	0.738705i	\(0.735438\pi\)
\(37\)	9.27516	−	1.46904i	1.52483	−	0.241509i	0.662964	−	0.748651i	\(-0.269298\pi\)
							0.861862	+	0.507142i	\(0.169298\pi\)
\(41\)	−3.01756	+	4.15332i	−0.471264	+	0.648640i	−0.976797	−	0.214168i	\(-0.931296\pi\)
							0.505533	+	0.862807i	\(0.331296\pi\)
\(43\)	0.604853	+	0.604853i	0.0922393	+	0.0922393i	0.751721	−	0.659482i	\(-0.229224\pi\)
							−0.659482	+	0.751721i	\(0.729224\pi\)
\(47\)	7.74129	−	3.94438i	1.12918	−	0.575347i	0.213377	−	0.976970i	\(-0.431554\pi\)
							0.915805	+	0.401623i	\(0.131554\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.2.s.a.48.16	yes	144
5.2	odd	4		875.2.s.b.657.16		144
5.3	odd	4		875.2.s.a.657.3		144
5.4	even	2		875.2.s.c.468.3		144
7.6	odd	2	inner	175.2.s.a.48.15	✓	144
25.9	even	10		875.2.s.a.293.4		144
25.12	odd	20	inner	175.2.s.a.62.15	yes	144
25.13	odd	20		875.2.s.c.832.4		144
25.16	even	5		875.2.s.b.293.15		144
35.13	even	4		875.2.s.a.657.4		144
35.27	even	4		875.2.s.b.657.15		144
35.34	odd	2		875.2.s.c.468.4		144
175.13	even	20		875.2.s.c.832.3		144
175.34	odd	10		875.2.s.a.293.3		144
175.41	odd	10		875.2.s.b.293.16		144
175.62	even	20	inner	175.2.s.a.62.16	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.s.a.48.15	✓	144	7.6	odd	2	inner
175.2.s.a.48.16	yes	144	1.1	even	1	trivial
175.2.s.a.62.15	yes	144	25.12	odd	20	inner
175.2.s.a.62.16	yes	144	175.62	even	20	inner
875.2.s.a.293.3		144	175.34	odd	10
875.2.s.a.293.4		144	25.9	even	10
875.2.s.a.657.3		144	5.3	odd	4
875.2.s.a.657.4		144	35.13	even	4
875.2.s.b.293.15		144	25.16	even	5
875.2.s.b.293.16		144	175.41	odd	10
875.2.s.b.657.15		144	35.27	even	4
875.2.s.b.657.16		144	5.2	odd	4
875.2.s.c.468.3		144	5.4	even	2
875.2.s.c.468.4		144	35.34	odd	2
875.2.s.c.832.3		144	175.13	even	20
875.2.s.c.832.4		144	25.13	odd	20