Embedded newform 75.3.j.a.11.6

• Label: 75.3.j.a.11.6
• Level: $75$
• Weight: $3$
• Character: 75.11
• Analytic conductor: $2.044$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	75.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.04360198270\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			11.6
Character	\(\chi\)	\(=\)	75.11
Dual form			75.3.j.a.41.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06164 - 1.46122i) q^{2} +(2.88836 - 0.810770i) q^{3} +(0.227982 - 0.701657i) q^{4} +(-4.39665 - 2.38107i) q^{5} +(-4.25111 - 3.35979i) q^{6} +6.57297 q^{7} +(-8.13837 + 2.64432i) q^{8} +(7.68530 - 4.68360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - q^{3} + 26 q^{4} - 11 q^{6} - 8 q^{7} - 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06164	−	1.46122i	−0.530819	−	0.730609i	0.456436	−	0.889756i	\(-0.349126\pi\)
							−0.987255	+	0.159147i	\(0.949126\pi\)
\(3\)	2.88836	−	0.810770i	0.962788	−	0.270257i
							\(5\)	−4.39665	−	2.38107i	−0.879329	−	0.476214i
\(7\)	6.57297			0.938996										0.469498	−	0.882934i	\(-0.344435\pi\)
							0.469498	+	0.882934i	\(0.344435\pi\)
\(11\)	2.41938	+	3.32999i	0.219943	+	0.302726i	0.904703	−	0.426043i	\(-0.140093\pi\)
							−0.684760	+	0.728769i	\(0.740093\pi\)
\(13\)	−16.8626	−	12.2514i	−1.29712	−	0.942415i	−0.297200	−	0.954815i	\(-0.596053\pi\)
							−0.999923	+	0.0124000i	\(0.996053\pi\)
\(17\)	21.5162	−	6.99103i	1.26566	−	0.411237i	0.402151	−	0.915573i	\(-0.368263\pi\)
							0.863507	+	0.504336i	\(0.168263\pi\)
\(19\)	9.46504	+	29.1304i	0.498160	+	1.53318i	0.811973	+	0.583694i	\(0.198393\pi\)
							−0.313813	+	0.949485i	\(0.601607\pi\)
\(23\)	6.13998	+	8.45096i	0.266956	+	0.367433i	0.921359	−	0.388713i	\(-0.127080\pi\)
							−0.654403	+	0.756146i	\(0.727080\pi\)
\(29\)	19.3497	+	6.28711i	0.667232	+	0.216797i	0.622997	−	0.782224i	\(-0.285915\pi\)
							0.0442353	+	0.999021i	\(0.485915\pi\)
\(31\)	9.70116	+	29.8571i	0.312941	+	0.963132i	0.976594	+	0.215092i	\(0.0690051\pi\)
							−0.663653	+	0.748040i	\(0.730995\pi\)
\(37\)	−30.2167	−	21.9537i	−0.816667	−	0.593343i	0.0990889	−	0.995079i	\(-0.468407\pi\)
							−0.915756	+	0.401735i	\(0.868407\pi\)
\(41\)	3.35250	−	4.61432i	0.0817682	−	0.112544i	−0.766173	−	0.642634i	\(-0.777842\pi\)
							0.847942	+	0.530089i	\(0.177842\pi\)
\(43\)	−4.60482			−0.107089			−0.0535444	−	0.998565i	\(-0.517052\pi\)
							−0.0535444	+	0.998565i	\(0.517052\pi\)
\(47\)	17.7926	+	5.78116i	0.378566	+	0.123003i	0.492118	−	0.870529i	\(-0.336223\pi\)
							−0.113552	+	0.993532i	\(0.536223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.3.j.a.11.6	✓	72
3.2	odd	2	inner	75.3.j.a.11.13	yes	72
5.2	odd	4		375.3.h.b.74.25		144
5.3	odd	4		375.3.h.b.74.12		144
5.4	even	2		375.3.j.a.176.13		72
15.2	even	4		375.3.h.b.74.11		144
15.8	even	4		375.3.h.b.74.26		144
15.14	odd	2		375.3.j.a.176.6		72
25.9	even	10		375.3.j.a.326.6		72
25.12	odd	20		375.3.h.b.299.26		144
25.13	odd	20		375.3.h.b.299.11		144
25.16	even	5	inner	75.3.j.a.41.13	yes	72
75.38	even	20		375.3.h.b.299.25		144
75.41	odd	10	inner	75.3.j.a.41.6	yes	72
75.59	odd	10		375.3.j.a.326.13		72
75.62	even	20		375.3.h.b.299.12		144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.j.a.11.6	✓	72	1.1	even	1	trivial
75.3.j.a.11.13	yes	72	3.2	odd	2	inner
75.3.j.a.41.6	yes	72	75.41	odd	10	inner
75.3.j.a.41.13	yes	72	25.16	even	5	inner
375.3.h.b.74.11		144	15.2	even	4
375.3.h.b.74.12		144	5.3	odd	4
375.3.h.b.74.25		144	5.2	odd	4
375.3.h.b.74.26		144	15.8	even	4
375.3.h.b.299.11		144	25.13	odd	20
375.3.h.b.299.12		144	75.62	even	20
375.3.h.b.299.25		144	75.38	even	20
375.3.h.b.299.26		144	25.12	odd	20
375.3.j.a.176.6		72	15.14	odd	2
375.3.j.a.176.13		72	5.4	even	2
375.3.j.a.326.6		72	25.9	even	10
375.3.j.a.326.13		72	75.59	odd	10