Embedded newform 175.2.x.a.12.12

• Label: 175.2.x.a.12.12
• Level: $175$
• Weight: $2$
• Character: 175.12
• Analytic conductor: $1.397$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	175.x (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.39738203537\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(18\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			12.12
Character	\(\chi\)	\(=\)	175.12
Dual form			175.2.x.a.73.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.947352 + 0.363654i) q^{2} +(-3.05010 + 0.159849i) q^{3} +(-0.721058 - 0.649244i) q^{4} +(2.14062 - 0.646323i) q^{5} +(-2.94765 - 0.957749i) q^{6} +(2.27212 - 1.35553i) q^{7} +(-1.36837 - 2.68558i) q^{8} +(6.29399 - 0.661525i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 30 q^{5} - 10 q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{9}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.947352	+	0.363654i	0.669879	+	0.257142i	0.669450	−	0.742857i	\(-0.266530\pi\)
							0.000429364		1.00000i	\(0.499863\pi\)
\(3\)	−3.05010	+	0.159849i	−1.76098	+	0.0922888i	−0.905181	−	0.425026i	\(-0.860265\pi\)
							−0.855795	+	0.517315i	\(0.826932\pi\)
\(5\)	2.14062	−	0.646323i	0.957316	−	0.289045i
							\(7\)	2.27212	−	1.35553i	0.858782	−	0.512341i
\(11\)	0.471446	−	4.48551i	0.142146	−	1.35243i								−0.658177	−	0.752863i	\(-0.728672\pi\)
							0.800323	−	0.599569i	\(-0.204661\pi\)
\(13\)	−0.803090	−	0.127197i	−0.222737	−	0.0352781i	0.0440683	−	0.999029i	\(-0.485968\pi\)
							−0.266805	+	0.963750i	\(0.585968\pi\)
\(17\)	−2.71146	+	4.17529i	−0.657627	+	1.01266i	0.339767	+	0.940510i	\(0.389652\pi\)
							−0.997394	+	0.0721469i	\(0.977015\pi\)
\(19\)	−1.55511	−	1.72712i	−0.356766	−	0.396229i	0.537868	−	0.843029i	\(-0.319230\pi\)
							−0.894634	+	0.446801i	\(0.852563\pi\)
\(23\)	−0.483910	+	1.26063i	−0.100902	+	0.262859i	−0.974734	−	0.223369i	\(-0.928294\pi\)
							0.873832	+	0.486229i	\(0.161628\pi\)
\(29\)	7.43700	−	2.41643i	1.38102	−	0.448720i	0.478013	−	0.878353i	\(-0.341357\pi\)
							0.903004	+	0.429633i	\(0.141357\pi\)
\(31\)	0.0572779	+	0.269471i	0.0102874	+	0.0483985i	0.982992	−	0.183649i	\(-0.0587909\pi\)
							−0.972705	+	0.232047i	\(0.925458\pi\)
\(37\)	−0.645491	+	0.797115i	−0.106118	+	0.131045i	−0.827438	−	0.561557i	\(-0.810202\pi\)
							0.721320	+	0.692602i	\(0.243536\pi\)
\(41\)	3.95694	+	5.44626i	0.617970	+	0.850563i	0.997203	−	0.0747384i	\(-0.0238122\pi\)
							−0.379233	+	0.925301i	\(0.623812\pi\)
\(43\)	−6.40923	+	6.40923i	−0.977399	+	0.977399i	−0.999750	−	0.0223513i	\(-0.992885\pi\)
							0.0223513	+	0.999750i	\(0.492885\pi\)
\(47\)	5.61967	−	3.64946i	0.819714	−	0.532328i	−0.0653151	−	0.997865i	\(-0.520805\pi\)
							0.885029	+	0.465536i	\(0.154139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.2.x.a.12.12	✓	288
5.2	odd	4		875.2.bb.a.418.7		288
5.3	odd	4		875.2.bb.b.418.12		288
5.4	even	2		875.2.bb.c.82.7		288
7.3	odd	6	inner	175.2.x.a.87.7	yes	288
25.2	odd	20		875.2.bb.c.593.12		288
25.11	even	5		875.2.bb.b.782.12		288
25.14	even	10		875.2.bb.a.782.7		288
25.23	odd	20	inner	175.2.x.a.173.7	yes	288
35.3	even	12		875.2.bb.b.668.12		288
35.17	even	12		875.2.bb.a.668.7		288
35.24	odd	6		875.2.bb.c.332.12		288
175.52	even	60		875.2.bb.c.843.7		288
175.73	even	60	inner	175.2.x.a.73.12	yes	288
175.136	odd	30		875.2.bb.b.157.12		288
175.164	odd	30		875.2.bb.a.157.7		288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.x.a.12.12	✓	288	1.1	even	1	trivial
175.2.x.a.73.12	yes	288	175.73	even	60	inner
175.2.x.a.87.7	yes	288	7.3	odd	6	inner
175.2.x.a.173.7	yes	288	25.23	odd	20	inner
875.2.bb.a.157.7		288	175.164	odd	30
875.2.bb.a.418.7		288	5.2	odd	4
875.2.bb.a.668.7		288	35.17	even	12
875.2.bb.a.782.7		288	25.14	even	10
875.2.bb.b.157.12		288	175.136	odd	30
875.2.bb.b.418.12		288	5.3	odd	4
875.2.bb.b.668.12		288	35.3	even	12
875.2.bb.b.782.12		288	25.11	even	5
875.2.bb.c.82.7		288	5.4	even	2
875.2.bb.c.332.12		288	35.24	odd	6
875.2.bb.c.593.12		288	25.2	odd	20
875.2.bb.c.843.7		288	175.52	even	60