Embedded newform 75.4.g.b.61.7

• Label: 75.4.g.b.61.7
• 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			61.7
Character	\(\chi\)	\(=\)	75.61
Dual form			75.4.g.b.16.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.87763 - 2.81726i) q^{2} +(0.927051 - 2.85317i) q^{3} +(4.62691 - 14.2402i) q^{4} +(7.51887 + 8.27445i) q^{5} +(-4.44337 - 13.6753i) q^{6} +0.140520 q^{7} +(-10.3279 - 31.7859i) q^{8} +(-7.28115 - 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.87763	−	2.81726i	1.37095	−	0.996053i	0.373287	−	0.927716i	\(-0.378231\pi\)
							0.997662	−	0.0683371i	\(-0.0217693\pi\)
\(3\)	0.927051	−	2.85317i	0.178411	−	0.549093i
							\(5\)	7.51887	+	8.27445i	0.672509	+	0.740089i
\(7\)	0.140520			0.00758737										0.00379368	−	0.999993i	\(-0.498792\pi\)
							0.00379368	+	0.999993i	\(0.498792\pi\)
\(11\)	−38.3073	+	27.8319i	−1.05001	+	0.762875i	−0.972214	−	0.234094i	\(-0.924788\pi\)
							−0.0777938	+	0.996969i	\(0.524788\pi\)
\(13\)	−27.7836	−	20.1859i	−0.592752	−	0.430660i	0.250547	−	0.968104i	\(-0.419390\pi\)
							−0.843299	+	0.537445i	\(0.819390\pi\)
\(17\)	5.10385	+	15.7080i	0.0728156	+	0.224103i	0.980840	−	0.194813i	\(-0.0624100\pi\)
							−0.908025	+	0.418916i	\(0.862410\pi\)
\(19\)	28.1212	+	86.5481i	0.339549	+	1.04503i	0.964437	+	0.264311i	\(0.0851446\pi\)
							−0.624888	+	0.780714i	\(0.714855\pi\)
\(23\)	130.722	−	94.9752i	1.18511	−	0.861031i	0.192368	−	0.981323i	\(-0.438383\pi\)
							0.992739	+	0.120292i	\(0.0383831\pi\)
\(29\)	−3.81101	+	11.7291i	−0.0244030	+	0.0751048i	−0.962516	−	0.271224i	\(-0.912572\pi\)
							0.938113	+	0.346328i	\(0.112572\pi\)
\(31\)	−102.914	−	316.738i	−0.596257	−	1.83509i	−0.548366	−	0.836239i	\(-0.684750\pi\)
							−0.0478915	−	0.998853i	\(-0.515250\pi\)
\(37\)	227.659	+	165.404i	1.01154	+	0.734926i	0.964531	−	0.263968i	\(-0.0850315\pi\)
							0.0470077	+	0.998895i	\(0.485031\pi\)
\(41\)	−101.620	−	73.8315i	−0.387084	−	0.281233i	0.377176	−	0.926142i	\(-0.376895\pi\)
							−0.764259	+	0.644909i	\(0.776895\pi\)
\(43\)	−529.821			−1.87900			−0.939500	−	0.342549i	\(-0.888710\pi\)
							−0.939500	+	0.342549i	\(0.888710\pi\)
\(47\)	23.1983	−	71.3970i	0.0719962	−	0.221581i	−0.908583	−	0.417704i	\(-0.862835\pi\)
							0.980579	+	0.196122i	\(0.0628350\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.g.b.61.7	yes	28
3.2	odd	2		225.4.h.a.136.1		28
25.4	even	10		1875.4.a.f.1.13		14
25.16	even	5	inner	75.4.g.b.16.7	✓	28
25.21	even	5		1875.4.a.g.1.2		14
75.41	odd	10		225.4.h.a.91.1		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.g.b.16.7	✓	28	25.16	even	5	inner
75.4.g.b.61.7	yes	28	1.1	even	1	trivial
225.4.h.a.91.1		28	75.41	odd	10
225.4.h.a.136.1		28	3.2	odd	2
1875.4.a.f.1.13		14	25.4	even	10
1875.4.a.g.1.2		14	25.21	even	5