Embedded newform 75.6.e.b.68.2

• Label: 75.6.e.b.68.2
• Level: $75$
• Weight: $6$
• Character: 75.68
• Analytic conductor: $12.029$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -15
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.0287864860\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(i)\)
Coefficient field:	\(\Q(i, \sqrt{6})\)
Defining polynomial:	\( x^{4} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			68.2
Root			\(1.22474 + 1.22474i\) of defining polynomial
Character	\(\chi\)	\(=\)	75.68
Dual form			75.6.e.b.32.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{2} +(11.0227 - 11.0227i) q^{3} -29.0000i q^{4} +27.0000 q^{6} +(74.7094 - 74.7094i) q^{8} -243.000i q^{9} +(-319.658 - 319.658i) q^{12} -745.000 q^{16} +(-1621.56 - 1621.56i) q^{17} +(297.613 - 297.613i) q^{18} +2164.00i q^{19} +(3451.33 - 3451.33i) q^{23} -1647.00i q^{24} +(-2678.52 - 2678.52i) q^{27} +8152.00 q^{31} +(-3303.14 - 3303.14i) q^{32} -3972.00i q^{34} -7047.00 q^{36} +(-2650.35 + 2650.35i) q^{38} +8454.00 q^{46} +(19627.8 + 19627.8i) q^{47} +(-8211.91 + 8211.91i) q^{48} +16807.0i q^{49} -35748.0 q^{51} +(-284.141 + 284.141i) q^{53} -6561.00i q^{54} +(23853.1 + 23853.1i) q^{57} +34802.0 q^{61} +(9984.12 + 9984.12i) q^{62} +15749.0i q^{64} +(-47025.3 + 47025.3i) q^{68} -76086.0i q^{69} +(-18154.4 - 18154.4i) q^{72} +62756.0 q^{76} +70064.0i q^{79} -59049.0 q^{81} +(72526.9 - 72526.9i) q^{83} +(-100089. - 100089. i) q^{92} +(89857.1 - 89857.1i) q^{93} +48078.0i q^{94} -72819.0 q^{96} +(-20584.3 + 20584.3i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{6} - 2980 q^{16} + 32608 q^{31} - 28188 q^{36} + 33816 q^{46} - 142992 q^{51} + 139208 q^{61} + 251024 q^{76} - 236196 q^{81} - 291276 q^{96}+O(q^{100}) \)

Character values

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

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.22474	+	1.22474i	0.216506	+	0.216506i	0.807024	−	0.590518i	\(-0.201077\pi\)
							−0.590518	+	0.807024i	\(0.701077\pi\)
\(3\)	11.0227	−	11.0227i	0.707107	−	0.707107i
							\(5\)		0			0
\(7\)		0			0									−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(11\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(13\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(17\)	−1621.56	−	1621.56i	−1.36085	−	1.36085i	−0.872831	−	0.488022i	\(-0.837719\pi\)
							−0.488022	−	0.872831i	\(-0.662281\pi\)
\(19\)			2164.00i			1.37522i	0.726079	+	0.687612i	\(0.241341\pi\)
							−0.726079	+	0.687612i	\(0.758659\pi\)
\(23\)	3451.33	−	3451.33i	1.36040	−	1.36040i	0.486998	−	0.873403i	\(-0.338092\pi\)
							0.873403	−	0.486998i	\(-0.161908\pi\)
\(29\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(31\)	8152.00			1.52356			0.761781	−	0.647835i	\(-0.224325\pi\)
							0.761781	+	0.647835i	\(0.224325\pi\)
\(37\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(41\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(43\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(47\)	19627.8	+	19627.8i	1.29606	+	1.29606i	0.930972	+	0.365091i	\(0.118962\pi\)
							0.365091	+	0.930972i	\(0.381038\pi\)
\(53\)	−284.141	+	284.141i	−0.0138945	+	0.0138945i	−0.714020	−	0.700125i	\(-0.753127\pi\)
							0.700125	+	0.714020i	\(0.253127\pi\)
\(59\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(61\)	34802.0			1.19751			0.598756	−	0.800932i	\(-0.295662\pi\)
							0.598756	+	0.800932i	\(0.295662\pi\)
\(67\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(71\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(73\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(79\)			70064.0i			1.26307i	0.775348	+	0.631535i	\(0.217575\pi\)
							−0.775348	+	0.631535i	\(0.782425\pi\)
\(83\)	72526.9	−	72526.9i	1.15559	−	1.15559i	0.170178	−	0.985413i	\(-0.445566\pi\)
							0.985413	−	0.170178i	\(-0.0544341\pi\)
\(89\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(97\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.e.b.68.2	yes	4
3.2	odd	2	inner	75.6.e.b.68.1	yes	4
5.2	odd	4	inner	75.6.e.b.32.1	✓	4
5.3	odd	4	inner	75.6.e.b.32.2	yes	4
5.4	even	2	inner	75.6.e.b.68.1	yes	4
15.2	even	4	inner	75.6.e.b.32.2	yes	4
15.8	even	4	inner	75.6.e.b.32.1	✓	4
15.14	odd	2	CM	75.6.e.b.68.2	yes	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.e.b.32.1	✓	4	5.2	odd	4	inner
75.6.e.b.32.1	✓	4	15.8	even	4	inner
75.6.e.b.32.2	yes	4	5.3	odd	4	inner
75.6.e.b.32.2	yes	4	15.2	even	4	inner
75.6.e.b.68.1	yes	4	3.2	odd	2	inner
75.6.e.b.68.1	yes	4	5.4	even	2	inner
75.6.e.b.68.2	yes	4	1.1	even	1	trivial
75.6.e.b.68.2	yes	4	15.14	odd	2	CM