Embedded newform 75.4.g.b.16.1

• Label: 75.4.g.b.16.1
• Level: $75$
• Weight: $4$
• Character: 75.16
• Analytic conductor: $4.425$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.42514325043\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			16.1
Character	\(\chi\)	\(=\)	75.16
Dual form			75.4.g.b.61.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.76530 - 2.73565i) q^{2} +(0.927051 + 2.85317i) q^{3} +(4.22155 + 12.9926i) q^{4} +(5.34090 - 9.82216i) q^{5} +(4.31465 - 13.2791i) q^{6} -26.0445 q^{7} +(8.14206 - 25.0587i) q^{8} +(-7.28115 + 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.76530	−	2.73565i	−1.33123	−	0.967198i	−0.999718	−	0.0237477i	\(-0.992440\pi\)
							−0.331515	−	0.943450i	\(-0.607560\pi\)
\(3\)	0.927051	+	2.85317i	0.178411	+	0.549093i
							\(5\)	5.34090	−	9.82216i	0.477705	−	0.878520i
\(7\)	−26.0445			−1.40627										−0.703136	−	0.711056i	\(-0.748217\pi\)
							−0.703136	+	0.711056i	\(0.748217\pi\)
\(11\)	−2.03543	−	1.47883i	−0.0557915	−	0.0405349i	0.559540	−	0.828803i	\(-0.310978\pi\)
							−0.615331	+	0.788268i	\(0.710978\pi\)
\(13\)	−32.0534	+	23.2881i	−0.683846	+	0.496843i	−0.874632	−	0.484788i	\(-0.838897\pi\)
							0.190785	+	0.981632i	\(0.438897\pi\)
\(17\)	−26.2638	+	80.8316i	−0.374700	+	1.15321i	0.568981	+	0.822351i	\(0.307338\pi\)
							−0.943681	+	0.330857i	\(0.892662\pi\)
\(19\)	−44.1360	+	135.837i	−0.532921	+	1.64016i	0.215179	+	0.976575i	\(0.430967\pi\)
							−0.748099	+	0.663587i	\(0.769033\pi\)
\(23\)	−127.177	−	92.3993i	−1.15296	−	0.837678i	−0.164092	−	0.986445i	\(-0.552469\pi\)
							−0.988872	+	0.148768i	\(0.952469\pi\)
\(29\)	−30.9737	−	95.3273i	−0.198334	−	0.610408i	−0.999921	−	0.0125312i	\(-0.996011\pi\)
							0.801588	−	0.597877i	\(-0.203989\pi\)
\(31\)	−53.5715	+	164.876i	−0.310378	+	0.955246i	0.667237	+	0.744846i	\(0.267477\pi\)
							−0.977615	+	0.210401i	\(0.932523\pi\)
\(37\)	48.0520	−	34.9118i	0.213506	−	0.155121i	−0.475893	−	0.879503i	\(-0.657875\pi\)
							0.689398	+	0.724382i	\(0.257875\pi\)
\(41\)	287.546	−	208.914i	1.09530	−	0.795779i	0.115011	−	0.993364i	\(-0.463310\pi\)
							0.980286	+	0.197585i	\(0.0633098\pi\)
\(43\)	109.742			0.389198			0.194599	−	0.980883i	\(-0.437659\pi\)
							0.194599	+	0.980883i	\(0.437659\pi\)
\(47\)	−17.9575	−	55.2674i	−0.0557312	−	0.171523i	0.919316	−	0.393520i	\(-0.128743\pi\)
							−0.975047	+	0.221997i	\(0.928743\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.16.1	✓	28
3.2	odd	2		225.4.h.a.91.7		28
25.6	even	5		1875.4.a.g.1.13		14
25.11	even	5	inner	75.4.g.b.61.1	yes	28
25.19	even	10		1875.4.a.f.1.2		14
75.11	odd	10		225.4.h.a.136.7		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.16.1	✓	28	1.1	even	1	trivial
75.4.g.b.61.1	yes	28	25.11	even	5	inner
225.4.h.a.91.7		28	3.2	odd	2
225.4.h.a.136.7		28	75.11	odd	10
1875.4.a.f.1.2		14	25.19	even	10
1875.4.a.g.1.13		14	25.6	even	5