Embedded newform 175.2.x.a.152.9

• Label: 175.2.x.a.152.9
• Level: $175$
• Weight: $2$
• Character: 175.152
• 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			152.9
Character	\(\chi\)	\(=\)	175.152
Dual form			175.2.x.a.38.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.169876 + 0.209779i) q^{2} +(1.64499 + 1.06827i) q^{3} +(0.400674 + 1.88502i) q^{4} +(2.20759 - 0.355734i) q^{5} +(-0.503546 + 0.163612i) q^{6} +(-2.64466 - 0.0759508i) q^{7} +(-0.944532 - 0.481263i) q^{8} +(0.344591 + 0.773964i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 30 q^{5} - 10 q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.169876	+	0.209779i	−0.120120	+	0.148336i	−0.833659	−	0.552280i	\(-0.813758\pi\)
							0.713538	+	0.700616i	\(0.247091\pi\)
\(3\)	1.64499	+	1.06827i	0.949738	+	0.616767i	0.923791	−	0.382897i	\(-0.125073\pi\)
							0.0259464	+	0.999663i	\(0.491740\pi\)
\(5\)	2.20759	−	0.355734i	0.987264	−	0.159089i
							\(7\)	−2.64466	−	0.0759508i	−0.999588	−	0.0287067i
\(11\)	−1.18476	−	0.527490i	−0.357219	−	0.159044i								0.220277	−	0.975437i	\(-0.429304\pi\)
							−0.577497	+	0.816393i	\(0.695970\pi\)
\(13\)	−0.951469	−	6.00734i	−0.263890	−	1.66614i	−0.662535	−	0.749031i	\(-0.730520\pi\)
							0.398645	−	0.917105i	\(-0.369480\pi\)
\(17\)	0.0784852	+	1.49759i	0.0190354	+	0.363218i	0.991495	+	0.130146i	\(0.0415445\pi\)
							−0.972459	+	0.233072i	\(0.925122\pi\)
\(19\)	2.33888	+	0.497143i	0.536575	+	0.114052i	0.468226	−	0.883609i	\(-0.344893\pi\)
							0.0683492	+	0.997661i	\(0.478227\pi\)
\(23\)	4.05937	+	3.28721i	0.846438	+	0.685432i	0.951035	−	0.309083i	\(-0.100022\pi\)
							−0.104598	+	0.994515i	\(0.533355\pi\)
\(29\)	−2.77357	−	0.901187i	−0.515039	−	0.167346i	0.0399538	−	0.999202i	\(-0.487279\pi\)
							−0.554993	+	0.831855i	\(0.687279\pi\)
\(31\)	5.67348	+	5.10843i	1.01899	+	0.917500i	0.996612	−	0.0822493i	\(-0.0262104\pi\)
							0.0223756	+	0.999750i	\(0.492877\pi\)
\(37\)	−9.12185	−	3.50155i	−1.49962	−	0.575651i	−0.536307	−	0.844023i	\(-0.680181\pi\)
							−0.963315	+	0.268372i	\(0.913514\pi\)
\(41\)	−4.30696	+	5.92802i	−0.672634	+	0.925801i	−0.999816	−	0.0191600i	\(-0.993901\pi\)
							0.327183	+	0.944961i	\(0.393901\pi\)
\(43\)	2.10433	−	2.10433i	0.320908	−	0.320908i	−0.528208	−	0.849115i	\(-0.677136\pi\)
							0.849115	+	0.528208i	\(0.177136\pi\)
\(47\)	−12.4176	−	0.650780i	−1.81130	−	0.0949259i	−0.883595	−	0.468252i	\(-0.844884\pi\)
							−0.927700	+	0.373326i	\(0.878217\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.2.x.a.152.9	yes	288
5.2	odd	4		875.2.bb.a.593.9		288
5.3	odd	4		875.2.bb.b.593.10		288
5.4	even	2		875.2.bb.c.782.10		288
7.3	odd	6	inner	175.2.x.a.52.9	yes	288
25.9	even	10		875.2.bb.a.82.10		288
25.12	odd	20		875.2.bb.c.418.10		288
25.13	odd	20	inner	175.2.x.a.138.9	yes	288
25.16	even	5		875.2.bb.b.82.9		288
35.3	even	12		875.2.bb.b.843.9		288
35.17	even	12		875.2.bb.a.843.10		288
35.24	odd	6		875.2.bb.c.157.10		288
175.38	even	60	inner	175.2.x.a.38.9	✓	288
175.59	odd	30		875.2.bb.a.332.9		288
175.66	odd	30		875.2.bb.b.332.10		288
175.87	even	60		875.2.bb.c.668.10		288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.x.a.38.9	✓	288	175.38	even	60	inner
175.2.x.a.52.9	yes	288	7.3	odd	6	inner
175.2.x.a.138.9	yes	288	25.13	odd	20	inner
175.2.x.a.152.9	yes	288	1.1	even	1	trivial
875.2.bb.a.82.10		288	25.9	even	10
875.2.bb.a.332.9		288	175.59	odd	30
875.2.bb.a.593.9		288	5.2	odd	4
875.2.bb.a.843.10		288	35.17	even	12
875.2.bb.b.82.9		288	25.16	even	5
875.2.bb.b.332.10		288	175.66	odd	30
875.2.bb.b.593.10		288	5.3	odd	4
875.2.bb.b.843.9		288	35.3	even	12
875.2.bb.c.157.10		288	35.24	odd	6
875.2.bb.c.418.10		288	25.12	odd	20
875.2.bb.c.668.10		288	175.87	even	60
875.2.bb.c.782.10		288	5.4	even	2