Embedded newform 75.4.g.b.16.3

• Label: 75.4.g.b.16.3
• 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			16.3
Character	\(\chi\)	\(=\)	75.16
Dual form			75.4.g.b.61.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08389 - 0.787491i) q^{2} +(0.927051 + 2.85317i) q^{3} +(-1.91746 - 5.90135i) q^{4} +(-3.83217 + 10.5031i) q^{5} +(1.24203 - 3.82256i) q^{6} -12.2101 q^{7} +(-5.88101 + 18.0999i) q^{8} +(-7.28115 + 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08389	−	0.787491i	−0.383212	−	0.278420i	0.379456	−	0.925210i	\(-0.376111\pi\)
							−0.762668	+	0.646790i	\(0.776111\pi\)
\(3\)	0.927051	+	2.85317i	0.178411	+	0.549093i
							\(5\)	−3.83217	+	10.5031i	−0.342760	+	0.939423i
\(7\)	−12.2101			−0.659284										−0.329642	−	0.944106i	\(-0.606928\pi\)
							−0.329642	+	0.944106i	\(0.606928\pi\)
\(11\)	2.00821	+	1.45905i	0.0550454	+	0.0399928i	0.614968	−	0.788552i	\(-0.289169\pi\)
							−0.559922	+	0.828545i	\(0.689169\pi\)
\(13\)	−69.2682	+	50.3263i	−1.47781	+	1.07369i	−0.499557	+	0.866281i	\(0.666504\pi\)
							−0.978254	+	0.207411i	\(0.933496\pi\)
\(17\)	−1.71458	+	5.27694i	−0.0244616	+	0.0752850i	−0.962542	−	0.271132i	\(-0.912602\pi\)
							0.938080	+	0.346417i	\(0.112602\pi\)
\(19\)	−7.45887	+	22.9561i	−0.0900623	+	0.277183i	−0.985935	−	0.167127i	\(-0.946551\pi\)
							0.895873	+	0.444310i	\(0.146551\pi\)
\(23\)	83.6143	+	60.7493i	0.758034	+	0.550744i	0.898307	−	0.439369i	\(-0.144798\pi\)
							−0.140273	+	0.990113i	\(0.544798\pi\)
\(29\)	−35.3035	−	108.653i	−0.226059	−	0.695737i	−0.998182	−	0.0602646i	\(-0.980806\pi\)
							0.772124	−	0.635472i	\(-0.219194\pi\)
\(31\)	47.5487	−	146.340i	0.275484	−	0.847852i	−0.713607	−	0.700546i	\(-0.752940\pi\)
							0.989091	−	0.147306i	\(-0.0470602\pi\)
\(37\)	277.229	−	201.419i	1.23179	−	0.894948i	0.234767	−	0.972052i	\(-0.424567\pi\)
							0.997023	+	0.0771043i	\(0.0245674\pi\)
\(41\)	−28.0530	+	20.3817i	−0.106857	+	0.0776363i	−0.639931	−	0.768433i	\(-0.721037\pi\)
							0.533073	+	0.846069i	\(0.321037\pi\)
\(43\)	−154.740			−0.548783			−0.274391	−	0.961618i	\(-0.588476\pi\)
							−0.274391	+	0.961618i	\(0.588476\pi\)
\(47\)	104.613	+	321.965i	0.324666	+	0.999221i	0.971591	+	0.236667i	\(0.0760549\pi\)
							−0.646924	+	0.762554i	\(0.723945\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.16.3	✓	28
3.2	odd	2		225.4.h.a.91.5		28
25.6	even	5		1875.4.a.g.1.9		14
25.11	even	5	inner	75.4.g.b.61.3	yes	28
25.19	even	10		1875.4.a.f.1.6		14
75.11	odd	10		225.4.h.a.136.5		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.16.3	✓	28	1.1	even	1	trivial
75.4.g.b.61.3	yes	28	25.11	even	5	inner
225.4.h.a.91.5		28	3.2	odd	2
225.4.h.a.136.5		28	75.11	odd	10
1875.4.a.f.1.6		14	25.19	even	10
1875.4.a.g.1.9		14	25.6	even	5