Embedded newform 75.4.g.b.61.4

• Label: 75.4.g.b.61.4
• Level: $75$
• Weight: $4$
• Character: 75.61
• 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			61.4
Character	\(\chi\)	\(=\)	75.61
Dual form			75.4.g.b.16.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.109191 - 0.0793317i) q^{2} +(0.927051 - 2.85317i) q^{3} +(-2.46651 + 7.59113i) q^{4} +(-6.22327 + 9.28822i) q^{5} +(-0.125121 - 0.385084i) q^{6} -17.3099 q^{7} +(0.666555 + 2.05145i) q^{8} +(-7.28115 - 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.109191	−	0.0793317i	0.0386048	−	0.0280480i	−0.568316	−	0.822811i	\(-0.692405\pi\)
							0.606920	+	0.794763i	\(0.292405\pi\)
\(3\)	0.927051	−	2.85317i	0.178411	−	0.549093i
							\(5\)	−6.22327	+	9.28822i	−0.556626	+	0.830763i
\(7\)	−17.3099			−0.934647										−0.467323	−	0.884087i	\(-0.654782\pi\)
							−0.467323	+	0.884087i	\(0.654782\pi\)
\(11\)	−34.1037	+	24.7778i	−0.934785	+	0.679161i	−0.947160	−	0.320762i	\(-0.896061\pi\)
							0.0123744	+	0.999923i	\(0.496061\pi\)
\(13\)	68.3813	+	49.6819i	1.45889	+	1.05995i	0.983650	+	0.180089i	\(0.0576385\pi\)
							0.475239	+	0.879857i	\(0.342361\pi\)
\(17\)	−13.9374	−	42.8949i	−0.198842	−	0.611974i	−0.999910	−	0.0134002i	\(-0.995734\pi\)
							0.801068	−	0.598573i	\(-0.204266\pi\)
\(19\)	25.8534	+	79.5686i	0.312167	+	0.960753i	0.976905	+	0.213675i	\(0.0685435\pi\)
							−0.664737	+	0.747077i	\(0.731457\pi\)
\(23\)	−16.9333	+	12.3027i	−0.153514	+	0.111535i	−0.661891	−	0.749600i	\(-0.730246\pi\)
							0.508377	+	0.861135i	\(0.330246\pi\)
\(29\)	68.0039	−	209.295i	0.435449	−	1.34017i	−0.457177	−	0.889376i	\(-0.651139\pi\)
							0.892626	−	0.450798i	\(-0.148861\pi\)
\(31\)	49.5106	+	152.378i	0.286850	+	0.882834i	0.985838	+	0.167701i	\(0.0536342\pi\)
							−0.698988	+	0.715134i	\(0.746366\pi\)
\(37\)	212.955	+	154.721i	0.946206	+	0.687459i	0.949907	−	0.312534i	\(-0.101178\pi\)
							−0.00370013	+	0.999993i	\(0.501178\pi\)
\(41\)	−331.812	−	241.075i	−1.26391	−	0.918284i	−0.264967	−	0.964257i	\(-0.585361\pi\)
							−0.998943	+	0.0459734i	\(0.985361\pi\)
\(43\)	290.699			1.03096			0.515479	−	0.856902i	\(-0.327614\pi\)
							0.515479	+	0.856902i	\(0.327614\pi\)
\(47\)	−142.456	+	438.434i	−0.442114	+	1.36069i	0.443505	+	0.896272i	\(0.353735\pi\)
							−0.885618	+	0.464414i	\(0.846265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.61.4	yes	28
3.2	odd	2		225.4.h.a.136.4		28
25.4	even	10		1875.4.a.f.1.7		14
25.16	even	5	inner	75.4.g.b.16.4	✓	28
25.21	even	5		1875.4.a.g.1.8		14
75.41	odd	10		225.4.h.a.91.4		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.16.4	✓	28	25.16	even	5	inner
75.4.g.b.61.4	yes	28	1.1	even	1	trivial
225.4.h.a.91.4		28	75.41	odd	10
225.4.h.a.136.4		28	3.2	odd	2
1875.4.a.f.1.7		14	25.4	even	10
1875.4.a.g.1.8		14	25.21	even	5