Embedded newform 75.4.g.b.61.5

• Label: 75.4.g.b.61.5
• 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			61.5
Character	\(\chi\)	\(=\)	75.61
Dual form			75.4.g.b.16.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.772797 - 0.561470i) q^{2} +(0.927051 - 2.85317i) q^{3} +(-2.19017 + 6.74065i) q^{4} +(11.0970 + 1.36218i) q^{5} +(-0.885547 - 2.72543i) q^{6} +12.4836 q^{7} +(4.45357 + 13.7067i) q^{8} +(-7.28115 - 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.772797	−	0.561470i	0.273225	−	0.198510i	−0.442732	−	0.896654i	\(-0.645991\pi\)
							0.715957	+	0.698144i	\(0.245991\pi\)
\(3\)	0.927051	−	2.85317i	0.178411	−	0.549093i
							\(5\)	11.0970	+	1.36218i	0.992550	+	0.121837i
\(7\)	12.4836			0.674049										0.337024	−	0.941496i	\(-0.390580\pi\)
							0.337024	+	0.941496i	\(0.390580\pi\)
\(11\)	36.1845	−	26.2896i	0.991823	−	0.720601i	0.0315032	−	0.999504i	\(-0.489971\pi\)
							0.960319	+	0.278902i	\(0.0899705\pi\)
\(13\)	5.77338	+	4.19460i	0.123173	+	0.0894903i	0.647666	−	0.761924i	\(-0.275745\pi\)
							−0.524493	+	0.851415i	\(0.675745\pi\)
\(17\)	8.19254	+	25.2140i	0.116881	+	0.359724i	0.992335	−	0.123579i	\(-0.0394371\pi\)
							−0.875453	+	0.483302i	\(0.839437\pi\)
\(19\)	−14.3353	−	44.1196i	−0.173092	−	0.532723i	0.826449	−	0.563012i	\(-0.190357\pi\)
							−0.999541	+	0.0302886i	\(0.990357\pi\)
\(23\)	−117.674	+	85.4955i	−1.06682	+	0.775089i	−0.975338	−	0.220718i	\(-0.929160\pi\)
							−0.0914808	+	0.995807i	\(0.529160\pi\)
\(29\)	−65.9299	+	202.911i	−0.422168	+	1.29930i	0.483512	+	0.875338i	\(0.339361\pi\)
							−0.905680	+	0.423962i	\(0.860639\pi\)
\(31\)	−45.7598	−	140.834i	−0.265120	−	0.815954i	−0.991666	−	0.128837i	\(-0.958876\pi\)
							0.726546	−	0.687118i	\(-0.241124\pi\)
\(37\)	−325.478	−	236.473i	−1.44617	−	1.05070i	−0.986708	−	0.162503i	\(-0.948043\pi\)
							−0.459459	−	0.888199i	\(-0.651957\pi\)
\(41\)	−189.031	−	137.339i	−0.720043	−	0.523142i	0.166355	−	0.986066i	\(-0.446800\pi\)
							−0.886398	+	0.462924i	\(0.846800\pi\)
\(43\)	87.5080			0.310345			0.155173	−	0.987887i	\(-0.450407\pi\)
							0.155173	+	0.987887i	\(0.450407\pi\)
\(47\)	−23.4804	+	72.2653i	−0.0728717	+	0.224276i	−0.980858	−	0.194723i	\(-0.937619\pi\)
							0.907986	+	0.418999i	\(0.137619\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.61.5	yes	28
3.2	odd	2		225.4.h.a.136.3		28
25.4	even	10		1875.4.a.f.1.9		14
25.16	even	5	inner	75.4.g.b.16.5	✓	28
25.21	even	5		1875.4.a.g.1.6		14
75.41	odd	10		225.4.h.a.91.3		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.16.5	✓	28	25.16	even	5	inner
75.4.g.b.61.5	yes	28	1.1	even	1	trivial
225.4.h.a.91.3		28	75.41	odd	10
225.4.h.a.136.3		28	3.2	odd	2
1875.4.a.f.1.9		14	25.4	even	10
1875.4.a.g.1.6		14	25.21	even	5