Embedded newform 75.4.g.b.61.6

• Label: 75.4.g.b.61.6
• 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			61.6
Character	\(\chi\)	\(=\)	75.61
Dual form			75.4.g.b.16.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.25026 - 2.36146i) q^{2} +(0.927051 - 2.85317i) q^{3} +(2.51561 - 7.74226i) q^{4} +(-6.00716 - 9.42943i) q^{5} +(-3.72447 - 11.4627i) q^{6} +1.75849 q^{7} +(-0.174670 - 0.537580i) q^{8} +(-7.28115 - 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.25026	−	2.36146i	1.14914	−	0.834901i	0.160776	−	0.986991i	\(-0.448600\pi\)
							0.988367	+	0.152090i	\(0.0486005\pi\)
\(3\)	0.927051	−	2.85317i	0.178411	−	0.549093i
							\(5\)	−6.00716	−	9.42943i	−0.537296	−	0.843393i
\(7\)	1.75849			0.0949496										0.0474748	−	0.998872i	\(-0.484883\pi\)
							0.0474748	+	0.998872i	\(0.484883\pi\)
\(11\)	18.9070	−	13.7367i	0.518243	−	0.376526i	−0.297699	−	0.954660i	\(-0.596219\pi\)
							0.815942	+	0.578134i	\(0.196219\pi\)
\(13\)	43.1036	+	31.3166i	0.919598	+	0.668127i	0.943424	−	0.331589i	\(-0.107585\pi\)
							−0.0238256	+	0.999716i	\(0.507585\pi\)
\(17\)	−6.45676	−	19.8719i	−0.0921173	−	0.283508i	0.894374	−	0.447319i	\(-0.147621\pi\)
							−0.986492	+	0.163811i	\(0.947621\pi\)
\(19\)	5.67829	+	17.4760i	0.0685626	+	0.211014i	0.979467	−	0.201603i	\(-0.0646149\pi\)
							−0.910905	+	0.412617i	\(0.864615\pi\)
\(23\)	−34.8681	+	25.3331i	−0.316108	+	0.229666i	−0.734513	−	0.678595i	\(-0.762589\pi\)
							0.418405	+	0.908261i	\(0.362589\pi\)
\(29\)	41.2605	−	126.987i	0.264203	−	0.813133i	−0.727673	−	0.685924i	\(-0.759398\pi\)
							0.991876	−	0.127209i	\(-0.0406018\pi\)
\(31\)	74.8546	+	230.379i	0.433686	+	1.33475i	0.894427	+	0.447214i	\(0.147584\pi\)
							−0.460740	+	0.887535i	\(0.652416\pi\)
\(37\)	−288.826	−	209.844i	−1.28332	−	0.932384i	−0.283669	−	0.958922i	\(-0.591552\pi\)
							−0.999648	+	0.0265384i	\(0.991552\pi\)
\(41\)	343.417	+	249.507i	1.30812	+	0.950401i	1.00000		0.000947731i	\(-0.000301672\pi\)
							0.308116	+	0.951349i	\(0.400302\pi\)
\(43\)	−93.5407			−0.331740			−0.165870	−	0.986148i	\(-0.553043\pi\)
							−0.165870	+	0.986148i	\(0.553043\pi\)
\(47\)	−72.1996	+	222.207i	−0.224072	+	0.689623i	0.774312	+	0.632804i	\(0.218096\pi\)
							−0.998384	+	0.0568194i	\(0.981904\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.61.6	yes	28
3.2	odd	2		225.4.h.a.136.2		28
25.4	even	10		1875.4.a.f.1.12		14
25.16	even	5	inner	75.4.g.b.16.6	✓	28
25.21	even	5		1875.4.a.g.1.3		14
75.41	odd	10		225.4.h.a.91.2		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.16.6	✓	28	25.16	even	5	inner
75.4.g.b.61.6	yes	28	1.1	even	1	trivial
225.4.h.a.91.2		28	75.41	odd	10
225.4.h.a.136.2		28	3.2	odd	2
1875.4.a.f.1.12		14	25.4	even	10
1875.4.a.g.1.3		14	25.21	even	5