Embedded newform 75.4.g.a.61.3

• Label: 75.4.g.a.61.3
• 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			61.3
Character	\(\chi\)	\(=\)	75.61
Dual form			75.4.g.a.16.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.01897 + 1.46687i) q^{2} +(-0.927051 + 2.85317i) q^{3} +(-0.547592 + 1.68532i) q^{4} +(-10.1178 + 4.75718i) q^{5} +(-2.31354 - 7.12033i) q^{6} +10.8656 q^{7} +(-7.53599 - 23.1934i) q^{8} +(-7.28115 - 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} + 21 q^{3} - 18 q^{4} + 15 q^{5} + 12 q^{6} + 58 q^{7} - 111 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.01897	+	1.46687i	−0.713814	+	0.518617i	−0.884402	−	0.466726i	\(-0.845433\pi\)
							0.170587	+	0.985343i	\(0.445433\pi\)
\(3\)	−0.927051	+	2.85317i	−0.178411	+	0.549093i
							\(5\)	−10.1178	+	4.75718i	−0.904961	+	0.425495i
\(7\)	10.8656			0.586688										0.293344	−	0.956007i	\(-0.405232\pi\)
							0.293344	+	0.956007i	\(0.405232\pi\)
\(11\)	−7.74816	+	5.62936i	−0.212378	+	0.154302i	−0.688889	−	0.724867i	\(-0.741901\pi\)
							0.476511	+	0.879168i	\(0.341901\pi\)
\(13\)	−16.6627	−	12.1062i	−0.355493	−	0.258281i	0.395677	−	0.918390i	\(-0.370510\pi\)
							−0.751170	+	0.660109i	\(0.770510\pi\)
\(17\)	−38.9351	−	119.830i	−0.555479	−	1.70959i	−0.694675	−	0.719323i	\(-0.744452\pi\)
							0.139196	−	0.990265i	\(-0.455548\pi\)
\(19\)	2.83997	+	8.74054i	0.0342913	+	0.105538i	0.966737	−	0.255772i	\(-0.0823297\pi\)
							−0.932446	+	0.361310i	\(0.882330\pi\)
\(23\)	−154.737	+	112.423i	−1.40282	+	1.01921i	−0.408502	+	0.912757i	\(0.633949\pi\)
							−0.994318	+	0.106451i	\(0.966051\pi\)
\(29\)	−91.1005	+	280.379i	−0.583343	+	1.79534i	0.0224833	+	0.999747i	\(0.492843\pi\)
							−0.605826	+	0.795597i	\(0.707157\pi\)
\(31\)	48.3027	+	148.660i	0.279852	+	0.861296i	0.987895	+	0.155126i	\(0.0495784\pi\)
							−0.708043	+	0.706170i	\(0.750422\pi\)
\(37\)	−26.0772	−	18.9462i	−0.115867	−	0.0841821i	0.528343	−	0.849031i	\(-0.322814\pi\)
							−0.644210	+	0.764849i	\(0.722814\pi\)
\(41\)	−84.1243	−	61.1199i	−0.320439	−	0.232813i	0.415924	−	0.909399i	\(-0.363458\pi\)
							−0.736363	+	0.676587i	\(0.763458\pi\)
\(43\)	−357.900			−1.26928			−0.634642	−	0.772807i	\(-0.718852\pi\)
							−0.634642	+	0.772807i	\(0.718852\pi\)
\(47\)	70.2011	−	216.057i	0.217870	−	0.670535i	−0.781067	−	0.624447i	\(-0.785325\pi\)
							0.998937	−	0.0460880i	\(-0.0146755\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.a.61.3	yes	28
3.2	odd	2		225.4.h.c.136.5		28
25.4	even	10		1875.4.a.e.1.6		14
25.16	even	5	inner	75.4.g.a.16.3	✓	28
25.21	even	5		1875.4.a.h.1.9		14
75.41	odd	10		225.4.h.c.91.5		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.a.16.3	✓	28	25.16	even	5	inner
75.4.g.a.61.3	yes	28	1.1	even	1	trivial
225.4.h.c.91.5		28	75.41	odd	10
225.4.h.c.136.5		28	3.2	odd	2
1875.4.a.e.1.6		14	25.4	even	10
1875.4.a.h.1.9		14	25.21	even	5