Embedded newform 175.2.t.a.44.1

• Label: 175.2.t.a.44.1
• Level: $175$
• Weight: $2$
• Character: 175.44
• Analytic conductor: $1.397$
• Analytic rank: $0$
• Dimension: $144$
• 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			44.1
Character	\(\chi\)	\(=\)	175.44
Dual form			175.2.t.a.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.561630 + 2.64226i) q^{2} +(-0.981775 + 2.20510i) q^{3} +(-4.83903 - 2.15447i) q^{4} +(2.14180 + 0.642423i) q^{5} +(-5.27506 - 3.83256i) q^{6} +(-2.42948 + 1.04767i) q^{7} +(5.23487 - 7.20518i) q^{8} +(-1.89120 - 2.10039i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 5 q^{2} - 5 q^{3} - 19 q^{4} - 3 q^{5} - 12 q^{6} - 50 q^{8} - 17 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.561630	+	2.64226i	−0.397132	+	1.86836i	0.0913979	+	0.995814i	\(0.470866\pi\)
							−0.488530	+	0.872547i	\(0.662467\pi\)
\(3\)	−0.981775	+	2.20510i	−0.566828	+	1.27312i	0.371841	+	0.928296i	\(0.378727\pi\)
							−0.938669	+	0.344820i	\(0.887940\pi\)
\(5\)	2.14180	+	0.642423i	0.957841	+	0.287300i
							\(7\)	−2.42948	+	1.04767i	−0.918258	+	0.395982i
\(11\)	−0.529229	+	0.587768i	−0.159569	+	0.177219i								−0.817627	−	0.575748i	\(-0.804711\pi\)
							0.658059	+	0.752966i	\(0.271378\pi\)
\(13\)	3.83804	+	1.24706i	1.06448	+	0.345871i	0.788336	−	0.615245i	\(-0.210943\pi\)
							0.276145	+	0.961116i	\(0.410943\pi\)
\(17\)	1.05408	+	0.110788i	0.255652	+	0.0268701i	0.231487	−	0.972838i	\(-0.425641\pi\)
							0.0241643	+	0.999708i	\(0.492308\pi\)
\(19\)	−3.10717	+	1.38340i	−0.712834	+	0.317374i	−0.730921	−	0.682462i	\(-0.760909\pi\)
							0.0180870	+	0.999836i	\(0.494242\pi\)
\(23\)	0.333106	−	1.56714i	0.0694574	−	0.326771i	−0.929677	−	0.368376i	\(-0.879914\pi\)
							0.999134	+	0.0416051i	\(0.0132471\pi\)
\(29\)	−2.53946	+	1.84502i	−0.471565	+	0.342612i	−0.798051	−	0.602590i	\(-0.794135\pi\)
							0.326486	+	0.945202i	\(0.394135\pi\)
\(31\)	0.730149	−	6.94690i	0.131139	−	1.24770i	−0.708953	−	0.705256i	\(-0.750832\pi\)
							0.840091	−	0.542445i	\(-0.182501\pi\)
\(37\)	−5.99364	+	5.39669i	−0.985348	+	0.887211i	−0.993623	−	0.112755i	\(-0.964033\pi\)
							0.00827529	+	0.999966i	\(0.497366\pi\)
\(41\)	0.884702	−	2.72283i	0.138167	−	0.425235i	−0.857902	−	0.513813i	\(-0.828232\pi\)
							0.996069	+	0.0885784i	\(0.0282324\pi\)
\(43\)			5.98624i			0.912893i	0.889751	+	0.456446i	\(0.150878\pi\)
							−0.889751	+	0.456446i	\(0.849122\pi\)
\(47\)	1.29841	−	0.136468i	0.189392	−	0.0199059i	−0.00935757	−	0.999956i	\(-0.502979\pi\)
							0.198749	+	0.980050i	\(0.436312\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.2.t.a.44.1	yes	144
5.2	odd	4		875.2.q.b.401.1		288
5.3	odd	4		875.2.q.b.401.36		288
5.4	even	2		875.2.u.a.849.18		144
7.4	even	3	inner	175.2.t.a.144.1	yes	144
25.3	odd	20		875.2.q.b.226.1		288
25.4	even	10	inner	175.2.t.a.79.1	yes	144
25.21	even	5		875.2.u.a.149.18		144
25.22	odd	20		875.2.q.b.226.36		288
35.4	even	6		875.2.u.a.599.18		144
35.18	odd	12		875.2.q.b.151.1		288
35.32	odd	12		875.2.q.b.151.36		288
175.4	even	30	inner	175.2.t.a.4.1	✓	144
175.46	even	15		875.2.u.a.774.18		144
175.53	odd	60		875.2.q.b.851.36		288
175.172	odd	60		875.2.q.b.851.1		288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.t.a.4.1	✓	144	175.4	even	30	inner
175.2.t.a.44.1	yes	144	1.1	even	1	trivial
175.2.t.a.79.1	yes	144	25.4	even	10	inner
175.2.t.a.144.1	yes	144	7.4	even	3	inner
875.2.q.b.151.1		288	35.18	odd	12
875.2.q.b.151.36		288	35.32	odd	12
875.2.q.b.226.1		288	25.3	odd	20
875.2.q.b.226.36		288	25.22	odd	20
875.2.q.b.401.1		288	5.2	odd	4
875.2.q.b.401.36		288	5.3	odd	4
875.2.q.b.851.1		288	175.172	odd	60
875.2.q.b.851.36		288	175.53	odd	60
875.2.u.a.149.18		144	25.21	even	5
875.2.u.a.599.18		144	35.4	even	6
875.2.u.a.774.18		144	175.46	even	15
875.2.u.a.849.18		144	5.4	even	2