Embedded newform 75.8.e.a.68.1

• Label: 75.8.e.a.68.1
• Level: $75$
• Weight: $8$
• Character: 75.68
• Analytic conductor: $23.429$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -15
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.4288769113\)
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			68.1
Root			\(-1.22474 - 1.22474i\) of defining polynomial
Character	\(\chi\)	\(=\)	75.68
Dual form			75.8.e.a.32.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.9217 - 15.9217i) q^{2} +(33.0681 - 33.0681i) q^{3} +379.000i q^{4} -1053.00 q^{6} +(3996.34 - 3996.34i) q^{8} -2187.00i q^{9} +(12532.8 + 12532.8i) q^{12} -78745.0 q^{16} +(-18454.5 - 18454.5i) q^{17} +(-34820.7 + 34820.7i) q^{18} +11596.0i q^{19} +(-43485.8 + 43485.8i) q^{23} -264303. i q^{24} +(-72320.0 - 72320.0i) q^{27} -206648. q^{31} +(742221. + 742221. i) q^{32} +587652. i q^{34} +828873. q^{36} +(184628. - 184628. i) q^{38} +1.38473e6 q^{46} +(-260893. - 260893. i) q^{47} +(-2.60395e6 + 2.60395e6i) q^{48} +823543. i q^{49} -1.22051e6 q^{51} +(883805. - 883805. i) q^{53} +2.30291e6i q^{54} +(383458. + 383458. i) q^{57} -2.77452e6 q^{61} +(3.29018e6 + 3.29018e6i) q^{62} -1.35555e7i q^{64} +(6.99424e6 - 6.99424e6i) q^{68} +2.87599e6i q^{69} +(-8.74000e6 - 8.74000e6i) q^{72} -4.39488e6 q^{76} +8.76354e6i q^{79} -4.78297e6 q^{81} +(-7.16988e6 + 7.16988e6i) q^{83} +(-1.64811e7 - 1.64811e7i) q^{92} +(-6.83346e6 + 6.83346e6i) q^{93} +8.30770e6i q^{94} +4.90877e7 q^{96} +(1.31122e7 - 1.31122e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4212 q^{6} - 314980 q^{16} - 826592 q^{31} + 3315492 q^{36} + 5538936 q^{46} - 4882032 q^{51} - 11098072 q^{61} - 17579536 q^{76} - 19131876 q^{81} + 196350804 q^{96}+O(q^{100}) \)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.9217	−	15.9217i	−1.40729	−	1.40729i	−0.773523	−	0.633769i	\(-0.781507\pi\)
							−0.633769	−	0.773523i	\(-0.718493\pi\)
\(3\)	33.0681	−	33.0681i	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	\(0.750000\pi\)
\(17\)	−18454.5	−	18454.5i	−0.911024	−	0.911024i	0.0853285	−	0.996353i	\(-0.472806\pi\)
							−0.996353	+	0.0853285i	\(0.972806\pi\)
\(19\)			11596.0i			0.387856i	0.981016	+	0.193928i	\(0.0621228\pi\)
							−0.981016	+	0.193928i	\(0.937877\pi\)
\(23\)	−43485.8	+	43485.8i	−0.745247	+	0.745247i	−0.973582	−	0.228336i	\(-0.926672\pi\)
							0.228336	+	0.973582i	\(0.426672\pi\)
\(29\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(31\)	−206648.			−1.24585			−0.622924	−	0.782282i	\(-0.714056\pi\)
							−0.622924	+	0.782282i	\(0.714056\pi\)
\(37\)		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	\(0.750000\pi\)
\(47\)	−260893.	−	260893.i	−0.366538	−	0.366538i	0.499675	−	0.866213i	\(-0.333453\pi\)
							−0.866213	+	0.499675i	\(0.833453\pi\)
\(53\)	883805.	−	883805.i	0.815438	−	0.815438i	−0.170005	−	0.985443i	\(-0.554378\pi\)
							0.985443	+	0.170005i	\(0.0543784\pi\)
\(59\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(61\)	−2.77452e6			−1.56507			−0.782534	−	0.622608i	\(-0.786073\pi\)
							−0.782534	+	0.622608i	\(0.786073\pi\)
\(67\)		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	\(0.750000\pi\)
\(79\)			8.76354e6i			1.99979i	0.0144650	+	0.999895i	\(0.495395\pi\)
							−0.0144650	+	0.999895i	\(0.504605\pi\)
\(83\)	−7.16988e6	+	7.16988e6i	−1.37638	+	1.37638i	−0.525730	+	0.850652i	\(0.676208\pi\)
							−0.850652	+	0.525730i	\(0.823792\pi\)
\(89\)		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	75.8.e.a.68.1	yes	4
3.2	odd	2	inner	75.8.e.a.68.2	yes	4
5.2	odd	4	inner	75.8.e.a.32.2	yes	4
5.3	odd	4	inner	75.8.e.a.32.1	✓	4
5.4	even	2	inner	75.8.e.a.68.2	yes	4
15.2	even	4	inner	75.8.e.a.32.1	✓	4
15.8	even	4	inner	75.8.e.a.32.2	yes	4
15.14	odd	2	CM	75.8.e.a.68.1	yes	4

    

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