Embedded newform 75.4.g.b.61.2

• Label: 75.4.g.b.61.2
• 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			61.2
Character	\(\chi\)	\(=\)	75.61
Dual form			75.4.g.b.16.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.16070 + 2.29638i) q^{2} +(0.927051 - 2.85317i) q^{3} +(2.24451 - 6.90789i) q^{4} +(-11.0852 + 1.45535i) q^{5} +(3.62184 + 11.1469i) q^{6} +22.0918 q^{7} +(-0.889297 - 2.73697i) q^{8} +(-7.28115 - 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.16070	+	2.29638i	−1.11748	+	0.811894i	−0.983825	−	0.179135i	\(-0.942670\pi\)
							−0.133651	+	0.991028i	\(0.542670\pi\)
\(3\)	0.927051	−	2.85317i	0.178411	−	0.549093i
							\(5\)	−11.0852	+	1.45535i	−0.991492	+	0.130171i
\(7\)	22.0918			1.19284										0.596422	−	0.802671i	\(-0.296589\pi\)
							0.596422	+	0.802671i	\(0.296589\pi\)
\(11\)	31.5999	−	22.9587i	0.866158	−	0.629301i	−0.0633953	−	0.997988i	\(-0.520193\pi\)
							0.929553	+	0.368688i	\(0.120193\pi\)
\(13\)	55.3321	+	40.2012i	1.18049	+	0.857676i	0.992227	−	0.124442i	\(-0.0397141\pi\)
							0.188264	+	0.982119i	\(0.439714\pi\)
\(17\)	31.1744	+	95.9450i	0.444759	+	1.36883i	0.882747	+	0.469848i	\(0.155691\pi\)
							−0.437988	+	0.898981i	\(0.644309\pi\)
\(19\)	−29.0621	−	89.4438i	−0.350910	−	1.07999i	−0.958343	−	0.285619i	\(-0.907801\pi\)
							0.607433	−	0.794371i	\(-0.292199\pi\)
\(23\)	130.565	−	94.8612i	1.18368	−	0.859997i	0.191102	−	0.981570i	\(-0.438794\pi\)
							0.992582	+	0.121573i	\(0.0387939\pi\)
\(29\)	−15.8462	+	48.7695i	−0.101468	+	0.312285i	−0.988885	−	0.148681i	\(-0.952497\pi\)
							0.887418	+	0.460966i	\(0.152497\pi\)
\(31\)	−80.4233	−	247.517i	−0.465950	−	1.43405i	−0.857784	−	0.514011i	\(-0.828159\pi\)
							0.391834	−	0.920036i	\(-0.371841\pi\)
\(37\)	70.1129	+	50.9400i	0.311527	+	0.226337i	0.732551	−	0.680712i	\(-0.238329\pi\)
							−0.421025	+	0.907049i	\(0.638329\pi\)
\(41\)	42.4765	+	30.8610i	0.161798	+	0.117553i	0.665738	−	0.746185i	\(-0.268117\pi\)
							−0.503940	+	0.863739i	\(0.668117\pi\)
\(43\)	−53.3748			−0.189293			−0.0946463	−	0.995511i	\(-0.530172\pi\)
							−0.0946463	+	0.995511i	\(0.530172\pi\)
\(47\)	74.7809	−	230.152i	0.232083	−	0.714279i	−0.765412	−	0.643541i	\(-0.777465\pi\)
							0.997495	−	0.0707380i	\(-0.0225354\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.61.2	yes	28
3.2	odd	2		225.4.h.a.136.6		28
25.4	even	10		1875.4.a.f.1.4		14
25.16	even	5	inner	75.4.g.b.16.2	✓	28
25.21	even	5		1875.4.a.g.1.11		14
75.41	odd	10		225.4.h.a.91.6		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.16.2	✓	28	25.16	even	5	inner
75.4.g.b.61.2	yes	28	1.1	even	1	trivial
225.4.h.a.91.6		28	75.41	odd	10
225.4.h.a.136.6		28	3.2	odd	2
1875.4.a.f.1.4		14	25.4	even	10
1875.4.a.g.1.11		14	25.21	even	5