Embedded newform 75.9.k.a.13.3

• Label: 75.9.k.a.13.3
• Level: $75$
• Weight: $9$
• Character: 75.13
• Analytic conductor: $30.553$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.k (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(30.5533957546\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(40\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	75.13
Dual form			75.9.k.a.52.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-28.6989 + 4.54547i) q^{2} +(-21.2310 + 41.6683i) q^{3} +(559.498 - 181.792i) q^{4} +(-53.0876 - 622.741i) q^{5} +(419.907 - 1292.34i) q^{6} +(-1163.90 + 1163.90i) q^{7} +(-8602.90 + 4383.40i) q^{8} +(-1285.49 - 1769.32i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 444 q^{5} - 4540 q^{7} - 17460 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−28.6989	+	4.54547i	−1.79368	+	0.284092i	−0.962376	−	0.271721i	\(-0.912407\pi\)
							−0.831308	+	0.555812i	\(0.812407\pi\)
\(3\)	−21.2310	+	41.6683i	−0.262112	+	0.514423i
							\(5\)	−53.0876	−	622.741i	−0.0849401	−	0.996386i
\(7\)	−1163.90	+	1163.90i	−0.484758	+	0.484758i								−0.906647	−	0.421890i	\(-0.861367\pi\)
							0.421890	+	0.906647i	\(0.361367\pi\)
\(11\)	20561.8	+	14939.0i	1.40440	+	1.02035i	0.994108	+	0.108399i	\(0.0345723\pi\)
							0.410289	+	0.911955i	\(0.365428\pi\)
\(13\)	−42315.6	−	6702.13i	−1.48159	−	0.234660i	−0.637326	−	0.770595i	\(-0.719959\pi\)
							−0.844260	+	0.535935i	\(0.819959\pi\)
\(17\)	−18129.4	−	35580.9i	−0.217064	−	0.426012i	0.756639	−	0.653833i	\(-0.226840\pi\)
							−0.973703	+	0.227821i	\(0.926840\pi\)
\(19\)	−50710.2	−	16476.8i	−0.389118	−	0.126432i	0.107923	−	0.994159i	\(-0.465580\pi\)
							−0.497041	+	0.867727i	\(0.665580\pi\)
\(23\)	−36448.0	−	230123.i	−0.130245	−	0.822336i	−0.963159	−	0.268933i	\(-0.913329\pi\)
							0.832914	−	0.553403i	\(-0.186671\pi\)
\(29\)	432227.	−	140439.i	0.611111	−	0.198562i	0.0129214	−	0.999917i	\(-0.495887\pi\)
							0.598190	+	0.801354i	\(0.295887\pi\)
\(31\)	18205.6	−	56031.1i	0.0197133	−	0.0606712i	−0.940716	−	0.339195i	\(-0.889845\pi\)
							0.960429	+	0.278524i	\(0.0898452\pi\)
\(37\)	306451.	−	1.93486e6i	0.163514	−	1.03239i	−0.760308	−	0.649563i	\(-0.774952\pi\)
							0.923822	−	0.382823i	\(-0.125048\pi\)
\(41\)	−1.71487e6	+	1.24593e6i	−0.606871	+	0.440918i	−0.848311	−	0.529498i	\(-0.822380\pi\)
							0.241440	+	0.970416i	\(0.422380\pi\)
\(43\)	−926141.	−	926141.i	−0.270896	−	0.270896i	0.558565	−	0.829461i	\(-0.311352\pi\)
							−0.829461	+	0.558565i	\(0.811352\pi\)
\(47\)	2.84465e6	+	1.44942e6i	0.582958	+	0.297032i	0.720490	−	0.693465i	\(-0.243917\pi\)
							−0.137532	+	0.990497i	\(0.543917\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.k.a.13.3	✓	320
25.2	odd	20	inner	75.9.k.a.52.3	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.k.a.13.3	✓	320	1.1	even	1	trivial
75.9.k.a.52.3	yes	320	25.2	odd	20	inner