Embedded newform 75.4.g.b.31.4

• Label: 75.4.g.b.31.4
• 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			31.4
Character	\(\chi\)	\(=\)	75.31
Dual form			75.4.g.b.46.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.177563 - 0.546483i) q^{2} +(-2.42705 + 1.76336i) q^{3} +(6.20502 - 4.50821i) q^{4} +(-9.45304 - 5.96993i) q^{5} +(1.39460 + 1.01324i) q^{6} -2.67744 q^{7} +(-7.28438 - 5.29241i) q^{8} +(2.78115 - 8.55951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.177563	−	0.546483i	−0.0627781	−	0.193211i	0.914748	−	0.404024i	\(-0.132389\pi\)
							−0.977526	+	0.210813i	\(0.932389\pi\)
\(3\)	−2.42705	+	1.76336i	−0.467086	+	0.339358i
							\(5\)	−9.45304	−	5.96993i	−0.845506	−	0.533967i
\(7\)	−2.67744			−0.144568										−0.0722840	−	0.997384i	\(-0.523029\pi\)
							−0.0722840	+	0.997384i	\(0.523029\pi\)
\(11\)	−19.4804	−	59.9545i	−0.533960	−	1.64336i	−0.745885	−	0.666075i	\(-0.767973\pi\)
							0.211924	−	0.977286i	\(-0.432027\pi\)
\(13\)	4.38017	−	13.4808i	0.0934493	−	0.287607i	−0.893397	−	0.449268i	\(-0.851685\pi\)
							0.986846	+	0.161660i	\(0.0516849\pi\)
\(17\)	−24.5906	−	17.8661i	−0.350829	−	0.254892i	0.398388	−	0.917217i	\(-0.369570\pi\)
							−0.749217	+	0.662325i	\(0.769570\pi\)
\(19\)	−22.2647	−	16.1762i	−0.268835	−	0.195320i	0.445198	−	0.895432i	\(-0.353133\pi\)
							−0.714033	+	0.700112i	\(0.753133\pi\)
\(23\)	35.9246	+	110.565i	0.325687	+	1.00236i	0.971130	+	0.238553i	\(0.0766729\pi\)
							−0.645443	+	0.763809i	\(0.723327\pi\)
\(29\)	32.5312	−	23.6353i	0.208307	−	0.151344i	−0.478740	−	0.877956i	\(-0.658907\pi\)
							0.687047	+	0.726613i	\(0.258907\pi\)
\(31\)	180.304	+	130.998i	1.04463	+	0.758967i	0.971184	−	0.238333i	\(-0.0766008\pi\)
							0.0734444	+	0.997299i	\(0.476601\pi\)
\(37\)	25.6160	−	78.8378i	0.113817	−	0.350293i	−0.877881	−	0.478878i	\(-0.841044\pi\)
							0.991698	+	0.128585i	\(0.0410435\pi\)
\(41\)	44.5269	−	137.040i	0.169608	−	0.522000i	−0.829738	−	0.558153i	\(-0.811510\pi\)
							0.999346	+	0.0361527i	\(0.0115103\pi\)
\(43\)	433.956			1.53901			0.769507	−	0.638638i	\(-0.220502\pi\)
							0.769507	+	0.638638i	\(0.220502\pi\)
\(47\)	−371.550	+	269.947i	−1.15311	+	0.837783i	−0.988891	−	0.148641i	\(-0.952510\pi\)
							−0.164218	+	0.986424i	\(0.552510\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.31.4	✓	28
3.2	odd	2		225.4.h.a.181.4		28
25.11	even	5		1875.4.a.g.1.7		14
25.14	even	10		1875.4.a.f.1.8		14
25.21	even	5	inner	75.4.g.b.46.4	yes	28
75.71	odd	10		225.4.h.a.46.4		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.31.4	✓	28	1.1	even	1	trivial
75.4.g.b.46.4	yes	28	25.21	even	5	inner
225.4.h.a.46.4		28	75.71	odd	10
225.4.h.a.181.4		28	3.2	odd	2
1875.4.a.f.1.8		14	25.14	even	10
1875.4.a.g.1.7		14	25.11	even	5