Embedded newform 75.4.g.b.31.2

• Label: 75.4.g.b.31.2
• Level: $75$
• Weight: $4$
• Character: 75.31
• 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			31.2
Character	\(\chi\)	\(=\)	75.31
Dual form			75.4.g.b.46.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.907834 - 2.79403i) q^{2} +(-2.42705 + 1.76336i) q^{3} +(-0.510282 + 0.370741i) q^{4} +(10.7234 + 3.16365i) q^{5} +(7.13022 + 5.18041i) q^{6} +18.9115 q^{7} +(-17.5148 - 12.7253i) q^{8} +(2.78115 - 8.55951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.907834	−	2.79403i	−0.320968	−	0.987837i	−0.973228	−	0.229842i	\(-0.926179\pi\)
							0.652260	−	0.757995i	\(-0.273821\pi\)
\(3\)	−2.42705	+	1.76336i	−0.467086	+	0.339358i
							\(5\)	10.7234	+	3.16365i	0.959130	+	0.282966i
\(7\)	18.9115			1.02113										0.510563	−	0.859840i	\(-0.329437\pi\)
							0.510563	+	0.859840i	\(0.329437\pi\)
\(11\)	−1.87505	−	5.77082i	−0.0513955	−	0.158179i	0.922065	−	0.387036i	\(-0.126501\pi\)
							−0.973460	+	0.228857i	\(0.926501\pi\)
\(13\)	24.1805	−	74.4201i	0.515883	−	1.58772i	−0.265787	−	0.964032i	\(-0.585632\pi\)
							0.781670	−	0.623692i	\(-0.214368\pi\)
\(17\)	−31.0670	−	22.5715i	−0.443227	−	0.322023i	0.343689	−	0.939084i	\(-0.388324\pi\)
							−0.786916	+	0.617060i	\(0.788324\pi\)
\(19\)	75.2030	+	54.6382i	0.908040	+	0.659730i	0.940518	−	0.339743i	\(-0.110340\pi\)
							−0.0324785	+	0.999472i	\(0.510340\pi\)
\(23\)	−25.0398	−	77.0645i	−0.227007	−	0.698655i	−0.998082	−	0.0619104i	\(-0.980281\pi\)
							0.771075	−	0.636744i	\(-0.219719\pi\)
\(29\)	9.02201	−	6.55487i	0.0577705	−	0.0419727i	−0.558525	−	0.829488i	\(-0.688633\pi\)
							0.616296	+	0.787515i	\(0.288633\pi\)
\(31\)	181.125	+	131.595i	1.04939	+	0.762425i	0.972096	−	0.234583i	\(-0.0753725\pi\)
							0.0772923	+	0.997008i	\(0.475373\pi\)
\(37\)	−33.2987	+	102.483i	−0.147954	+	0.455354i	−0.997379	−	0.0723545i	\(-0.976949\pi\)
							0.849425	+	0.527709i	\(0.176949\pi\)
\(41\)	−108.379	+	333.556i	−0.412828	+	1.27056i	0.501350	+	0.865244i	\(0.332837\pi\)
							−0.914179	+	0.405311i	\(0.867163\pi\)
\(43\)	−356.550			−1.26450			−0.632249	−	0.774765i	\(-0.717868\pi\)
							−0.632249	+	0.774765i	\(0.717868\pi\)
\(47\)	236.019	−	171.478i	0.732488	−	0.532184i	−0.157861	−	0.987461i	\(-0.550460\pi\)
							0.890350	+	0.455277i	\(0.150460\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.31.2	✓	28
3.2	odd	2		225.4.h.a.181.6		28
25.11	even	5		1875.4.a.g.1.4		14
25.14	even	10		1875.4.a.f.1.11		14
25.21	even	5	inner	75.4.g.b.46.2	yes	28
75.71	odd	10		225.4.h.a.46.6		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.31.2	✓	28	1.1	even	1	trivial
75.4.g.b.46.2	yes	28	25.21	even	5	inner
225.4.h.a.46.6		28	75.71	odd	10
225.4.h.a.181.6		28	3.2	odd	2
1875.4.a.f.1.11		14	25.14	even	10
1875.4.a.g.1.4		14	25.11	even	5