Embedded newform 75.6.e.b.32.1

• Label: 75.6.e.b.32.1
• Level: $75$
• Weight: $6$
• Character: 75.32
• 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			32.1
Root			\(-1.22474 + 1.22474i\) of defining polynomial
Character	\(\chi\)	\(=\)	75.32
Dual form			75.6.e.b.68.1

$q$-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 \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

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

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	\(-0.798923\pi\)
							0.590518	+	0.807024i	\(0.298923\pi\)
\(3\)	−11.0227	−	11.0227i	−0.707107	−	0.707107i
							\(5\)		0			0
\(7\)		0			0									0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(11\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(13\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(17\)	1621.56	−	1621.56i	1.36085	−	1.36085i	0.488022	−	0.872831i	\(-0.337719\pi\)
							0.872831	−	0.488022i	\(-0.162281\pi\)
\(19\)		−	2164.00i		−	1.37522i	−0.726079	−	0.687612i	\(-0.758659\pi\)
							0.726079	−	0.687612i	\(-0.241341\pi\)
\(23\)	−3451.33	−	3451.33i	−1.36040	−	1.36040i	−0.873403	−	0.486998i	\(-0.838092\pi\)
							−0.486998	−	0.873403i	\(-0.661908\pi\)
\(29\)		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	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(41\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(43\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(47\)	−19627.8	+	19627.8i	−1.29606	+	1.29606i	−0.365091	+	0.930972i	\(0.618962\pi\)
							−0.930972	+	0.365091i	\(0.881038\pi\)
\(53\)	284.141	+	284.141i	0.0138945	+	0.0138945i	0.714020	−	0.700125i	\(-0.246873\pi\)
							−0.700125	+	0.714020i	\(0.746873\pi\)
\(59\)		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	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(71\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(73\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(79\)		−	70064.0i		−	1.26307i	−0.775348	−	0.631535i	\(-0.782425\pi\)
							0.775348	−	0.631535i	\(-0.217575\pi\)
\(83\)	−72526.9	−	72526.9i	−1.15559	−	1.15559i	−0.985413	−	0.170178i	\(-0.945566\pi\)
							−0.170178	−	0.985413i	\(-0.554434\pi\)
\(89\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(97\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)

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.32.1	✓	4
3.2	odd	2	inner	75.6.e.b.32.2	yes	4
5.2	odd	4	inner	75.6.e.b.68.1	yes	4
5.3	odd	4	inner	75.6.e.b.68.2	yes	4
5.4	even	2	inner	75.6.e.b.32.2	yes	4
15.2	even	4	inner	75.6.e.b.68.2	yes	4
15.8	even	4	inner	75.6.e.b.68.1	yes	4
15.14	odd	2	CM	75.6.e.b.32.1	✓	4

    

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