Embedded newform 175.2.t.a.9.4

• Label: 175.2.t.a.9.4
• Level: $175$
• Weight: $2$
• Character: 175.9
• 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.t (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.39738203537\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(18\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			9.4
Character	\(\chi\)	\(=\)	175.9
Dual form			175.2.t.a.39.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.715816 - 1.60775i) q^{2} +(0.245158 + 0.220741i) q^{3} +(-0.734203 + 0.815415i) q^{4} +(-0.794287 - 2.09024i) q^{5} +(0.179408 - 0.552162i) q^{6} +(2.55400 + 0.690709i) q^{7} +(-1.51099 - 0.490951i) q^{8} +(-0.302210 - 2.87533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 5 q^{2} - 5 q^{3} - 19 q^{4} - 3 q^{5} - 12 q^{6} - 50 q^{8} - 17 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.715816	−	1.60775i	−0.506158	−	1.13685i	−0.968242	−	0.250014i	\(-0.919565\pi\)
							0.462084	−	0.886836i	\(-0.347102\pi\)
\(3\)	0.245158	+	0.220741i	0.141542	+	0.127445i	0.736860	−	0.676045i	\(-0.236307\pi\)
							−0.595319	+	0.803490i	\(0.702974\pi\)
\(5\)	−0.794287	−	2.09024i	−0.355216	−	0.934784i
							\(7\)	2.55400	+	0.690709i	0.965322	+	0.261063i
\(11\)	−0.228756	+	2.17646i	−0.0689724	+	0.656229i								0.904351	+	0.426790i	\(0.140356\pi\)
							−0.973323	+	0.229439i	\(0.926311\pi\)
\(13\)	−2.54477	−	3.50257i	−0.705791	−	0.971438i	−0.999877	−	0.0156594i	\(-0.995015\pi\)
							0.294086	−	0.955779i	\(-0.404985\pi\)
\(17\)	−0.832775	−	3.91790i	−0.201978	−	0.950230i	−0.955997	−	0.293377i	\(-0.905221\pi\)
							0.754019	−	0.656852i	\(-0.228113\pi\)
\(19\)	3.45994	+	3.84266i	0.793765	+	0.881566i	0.995193	−	0.0979379i	\(-0.0312247\pi\)
							−0.201427	+	0.979503i	\(0.564558\pi\)
\(23\)	−0.560013	−	1.25781i	−0.116771	−	0.262271i	0.845699	−	0.533660i	\(-0.179184\pi\)
							−0.962470	+	0.271389i	\(0.912517\pi\)
\(29\)	2.10487	+	6.47814i	0.390865	+	1.20296i	0.932135	+	0.362111i	\(0.117944\pi\)
							−0.541270	+	0.840849i	\(0.682056\pi\)
\(31\)	6.89752	−	1.46611i	1.23883	−	0.263322i	0.458504	−	0.888693i	\(-0.348386\pi\)
							0.780328	+	0.625371i	\(0.215052\pi\)
\(37\)	4.38681	−	0.461073i	0.721188	−	0.0757999i	0.263178	−	0.964747i	\(-0.415229\pi\)
							0.458009	+	0.888947i	\(0.348563\pi\)
\(41\)	7.67501	−	5.57622i	1.19863	−	0.870859i	0.204485	−	0.978870i	\(-0.434448\pi\)
							0.994150	+	0.108011i	\(0.0344480\pi\)
\(43\)			0.649592i			0.0990619i	0.998773	+	0.0495309i	\(0.0157726\pi\)
							−0.998773	+	0.0495309i	\(0.984227\pi\)
\(47\)	−0.434269	+	2.04308i	−0.0633447	+	0.298013i	−0.998404	−	0.0564784i	\(-0.982013\pi\)
							0.935059	+	0.354492i	\(0.115346\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.2.t.a.9.4	✓	144
5.2	odd	4		875.2.q.b.576.29		288
5.3	odd	4		875.2.q.b.576.8		288
5.4	even	2		875.2.u.a.674.15		144
7.4	even	3	inner	175.2.t.a.109.15	yes	144
25.2	odd	20		875.2.q.b.51.8		288
25.11	even	5		875.2.u.a.324.4		144
25.14	even	10	inner	175.2.t.a.114.15	yes	144
25.23	odd	20		875.2.q.b.51.29		288
35.4	even	6		875.2.u.a.424.4		144
35.18	odd	12		875.2.q.b.326.29		288
35.32	odd	12		875.2.q.b.326.8		288
175.11	even	15		875.2.u.a.74.15		144
175.39	even	30	inner	175.2.t.a.39.4	yes	144
175.102	odd	60		875.2.q.b.676.29		288
175.123	odd	60		875.2.q.b.676.8		288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.t.a.9.4	✓	144	1.1	even	1	trivial
175.2.t.a.39.4	yes	144	175.39	even	30	inner
175.2.t.a.109.15	yes	144	7.4	even	3	inner
175.2.t.a.114.15	yes	144	25.14	even	10	inner
875.2.q.b.51.8		288	25.2	odd	20
875.2.q.b.51.29		288	25.23	odd	20
875.2.q.b.326.8		288	35.32	odd	12
875.2.q.b.326.29		288	35.18	odd	12
875.2.q.b.576.8		288	5.3	odd	4
875.2.q.b.576.29		288	5.2	odd	4
875.2.q.b.676.8		288	175.123	odd	60
875.2.q.b.676.29		288	175.102	odd	60
875.2.u.a.74.15		144	175.11	even	15
875.2.u.a.324.4		144	25.11	even	5
875.2.u.a.424.4		144	35.4	even	6
875.2.u.a.674.15		144	5.4	even	2