Embedded newform 175.2.q.a.81.12

• Label: 175.2.q.a.81.12
• Level: $175$
• Weight: $2$
• Character: 175.81
• 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.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.39738203537\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(18\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			81.12
Character	\(\chi\)	\(=\)	175.81
Dual form			175.2.q.a.121.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.599774 - 0.666116i) q^{2} +(0.0597744 + 0.568715i) q^{3} +(0.125075 + 1.19001i) q^{4} +(-1.98154 + 1.03610i) q^{5} +(0.414682 + 0.301284i) q^{6} +(0.154468 + 2.64124i) q^{7} +(2.31802 + 1.68414i) q^{8} +(2.61458 - 0.555746i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 3 q^{2} - 3 q^{3} + 13 q^{4} - 3 q^{5} - 12 q^{6} - 22 q^{7} - 2 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.599774	−	0.666116i	0.424104	−	0.471015i	−0.492789	−	0.870149i	\(-0.664022\pi\)
							0.916893	+	0.399134i	\(0.130689\pi\)
\(3\)	0.0597744	+	0.568715i	0.0345108	+	0.328348i	0.998133	+	0.0610796i	\(0.0194543\pi\)
							−0.963622	+	0.267268i	\(0.913879\pi\)
\(5\)	−1.98154	+	1.03610i	−0.886171	+	0.463357i
							\(7\)	0.154468	+	2.64124i	0.0583834	+	0.998294i
\(11\)	−0.258419	−	0.0549287i	−0.0779164	−	0.0165616i								0.168789	−	0.985652i	\(-0.446015\pi\)
							−0.246705	+	0.969091i	\(0.579348\pi\)
\(13\)	1.90812	−	5.87259i	0.529217	−	1.62876i	−0.226606	−	0.973987i	\(-0.572763\pi\)
							0.755823	−	0.654776i	\(-0.227237\pi\)
\(17\)	−4.81335	+	2.14304i	−1.16741	+	0.519764i	−0.896588	−	0.442866i	\(-0.853962\pi\)
							−0.270821	+	0.962630i	\(0.587295\pi\)
\(19\)	0.284381	−	2.70570i	0.0652415	−	0.620731i	−0.912232	−	0.409674i	\(-0.865643\pi\)
							0.977474	−	0.211058i	\(-0.0676908\pi\)
\(23\)	−1.44645	+	1.60645i	−0.301607	+	0.334968i	−0.874829	−	0.484432i	\(-0.839026\pi\)
							0.573223	+	0.819400i	\(0.305693\pi\)
\(29\)	3.07350	−	2.23303i	0.570735	−	0.414663i	−0.264637	−	0.964348i	\(-0.585252\pi\)
							0.835372	+	0.549685i	\(0.185252\pi\)
\(31\)	7.89124	−	3.51341i	1.41731	−	0.631026i	0.451971	−	0.892032i	\(-0.350721\pi\)
							0.965337	+	0.261006i	\(0.0840542\pi\)
\(37\)	1.24144	−	0.263875i	0.204091	−	0.0433808i	−0.104732	−	0.994501i	\(-0.533398\pi\)
							0.308823	+	0.951120i	\(0.400065\pi\)
\(41\)	−0.210759	+	0.648651i	−0.0329151	+	0.101302i	−0.966164	−	0.257927i	\(-0.916960\pi\)
							0.933249	+	0.359230i	\(0.116960\pi\)
\(43\)	−0.914477			−0.139456			−0.0697282	−	0.997566i	\(-0.522213\pi\)
							−0.0697282	+	0.997566i	\(0.522213\pi\)
\(47\)	−7.35545	−	3.27486i	−1.07290	−	0.477687i	−0.207227	−	0.978293i	\(-0.566444\pi\)
							−0.865675	+	0.500606i	\(0.833111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.2.q.a.81.12	yes	144
5.2	odd	4		875.2.u.b.599.24		288
5.3	odd	4		875.2.u.b.599.13		288
5.4	even	2		875.2.q.a.151.7		144
7.2	even	3	inner	175.2.q.a.156.7	yes	144
25.3	odd	20		875.2.u.b.774.24		288
25.4	even	10		875.2.q.a.851.12		144
25.21	even	5	inner	175.2.q.a.46.7	✓	144
25.22	odd	20		875.2.u.b.774.13		288
35.2	odd	12		875.2.u.b.849.24		288
35.9	even	6		875.2.q.a.401.12		144
35.23	odd	12		875.2.u.b.849.13		288
175.72	odd	60		875.2.u.b.149.13		288
175.79	even	30		875.2.q.a.226.7		144
175.121	even	15	inner	175.2.q.a.121.12	yes	144
175.128	odd	60		875.2.u.b.149.24		288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.q.a.46.7	✓	144	25.21	even	5	inner
175.2.q.a.81.12	yes	144	1.1	even	1	trivial
175.2.q.a.121.12	yes	144	175.121	even	15	inner
175.2.q.a.156.7	yes	144	7.2	even	3	inner
875.2.q.a.151.7		144	5.4	even	2
875.2.q.a.226.7		144	175.79	even	30
875.2.q.a.401.12		144	35.9	even	6
875.2.q.a.851.12		144	25.4	even	10
875.2.u.b.149.13		288	175.72	odd	60
875.2.u.b.149.24		288	175.128	odd	60
875.2.u.b.599.13		288	5.3	odd	4
875.2.u.b.599.24		288	5.2	odd	4
875.2.u.b.774.13		288	25.22	odd	20
875.2.u.b.774.24		288	25.3	odd	20
875.2.u.b.849.13		288	35.23	odd	12
875.2.u.b.849.24		288	35.2	odd	12