Embedded newform 75.4.g.b.46.4

• Label: 75.4.g.b.46.4
• Level: $75$
• Weight: $4$
• Character: 75.46
• 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			46.4
Character	\(\chi\)	\(=\)	75.46
Dual form			75.4.g.b.31.4

$q$-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 \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{5}\right)\)

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	−0.977526	−	0.210813i	\(-0.932389\pi\)
							0.914748	+	0.404024i	\(0.132389\pi\)
\(3\)	−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	0.211924	+	0.977286i	\(0.432027\pi\)
							−0.745885	+	0.666075i	\(0.767973\pi\)
\(13\)	4.38017	+	13.4808i	0.0934493	+	0.287607i	0.986846	−	0.161660i	\(-0.0516849\pi\)
							−0.893397	+	0.449268i	\(0.851685\pi\)
\(17\)	−24.5906	+	17.8661i	−0.350829	+	0.254892i	−0.749217	−	0.662325i	\(-0.769570\pi\)
							0.398388	+	0.917217i	\(0.369570\pi\)
\(19\)	−22.2647	+	16.1762i	−0.268835	+	0.195320i	−0.714033	−	0.700112i	\(-0.753133\pi\)
							0.445198	+	0.895432i	\(0.353133\pi\)
\(23\)	35.9246	−	110.565i	0.325687	−	1.00236i	−0.645443	−	0.763809i	\(-0.723327\pi\)
							0.971130	−	0.238553i	\(-0.0766729\pi\)
\(29\)	32.5312	+	23.6353i	0.208307	+	0.151344i	0.687047	−	0.726613i	\(-0.258907\pi\)
							−0.478740	+	0.877956i	\(0.658907\pi\)
\(31\)	180.304	−	130.998i	1.04463	−	0.758967i	0.0734444	−	0.997299i	\(-0.476601\pi\)
							0.971184	+	0.238333i	\(0.0766008\pi\)
\(37\)	25.6160	+	78.8378i	0.113817	+	0.350293i	0.991698	−	0.128585i	\(-0.0410435\pi\)
							−0.877881	+	0.478878i	\(0.841044\pi\)
\(41\)	44.5269	+	137.040i	0.169608	+	0.522000i	0.999346	−	0.0361527i	\(-0.0115103\pi\)
							−0.829738	+	0.558153i	\(0.811510\pi\)
\(43\)	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	−0.164218	−	0.986424i	\(-0.552510\pi\)
							−0.988891	+	0.148641i	\(0.952510\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.31.4	✓	28	25.6	even	5	inner
75.4.g.b.46.4	yes	28	1.1	even	1	trivial
225.4.h.a.46.4		28	3.2	odd	2
225.4.h.a.181.4		28	75.56	odd	10
1875.4.a.f.1.8		14	25.9	even	10
1875.4.a.g.1.7		14	25.16	even	5