Embedded newform 75.10.e.c.68.1

• Label: 75.10.e.c.68.1
• Level: $75$
• Weight: $10$
• Character: 75.68
• Analytic conductor: $38.628$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -15
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(38.6276877123\)
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:	\( 3^{4} \)
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			75.10.e.c.32.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.0227 - 11.0227i) q^{2} +(-99.2043 + 99.2043i) q^{3} -269.000i q^{4} +2187.00 q^{6} +(-8608.73 + 8608.73i) q^{8} -19683.0i q^{9} +(26686.0 + 26686.0i) q^{12} +52055.0 q^{16} +(-210269. - 210269. i) q^{17} +(-216960. + 216960. i) q^{18} -1.03632e6i q^{19} +(347237. - 347237. i) q^{23} -1.70805e6i q^{24} +(1.95264e6 + 1.95264e6i) q^{27} -8.24737e6 q^{31} +(3.83388e6 + 3.83388e6i) q^{32} +4.63547e6i q^{34} -5.29473e6 q^{36} +(-1.14230e7 + 1.14230e7i) q^{38} -7.65499e6 q^{46} +(4.70102e7 + 4.70102e7i) q^{47} +(-5.16408e6 + 5.16408e6i) q^{48} +4.03536e7i q^{49} +4.17192e7 q^{51} +(-7.68054e7 + 7.68054e7i) q^{53} -4.30467e7i q^{54} +(1.02807e8 + 1.02807e8i) q^{57} +1.97895e8 q^{61} +(9.09083e7 + 9.09083e7i) q^{62} -1.11172e8i q^{64} +(-5.65624e7 + 5.65624e7i) q^{68} +6.88949e7i q^{69} +(1.69446e8 + 1.69446e8i) q^{72} -2.78769e8 q^{76} +4.21557e8i q^{79} -3.87420e8 q^{81} +(-6.04524e8 + 6.04524e8i) q^{83} +(-9.34068e7 - 9.34068e7i) q^{92} +(8.18175e8 - 8.18175e8i) q^{93} -1.03636e9i q^{94} -7.60676e8 q^{96} +(4.44806e8 - 4.44806e8i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8748 q^{6} + 208220 q^{16} - 32989472 q^{31} - 21178908 q^{36} - 30619944 q^{46} + 166876848 q^{51} + 791579528 q^{61} - 1115076016 q^{76} - 1549681956 q^{81} - 3042703116 q^{96}+O(q^{100}) \)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.0227	−	11.0227i	−0.487139	−	0.487139i	0.420263	−	0.907402i	\(-0.361938\pi\)
							−0.907402	+	0.420263i	\(0.861938\pi\)
\(3\)	−99.2043	+	99.2043i	−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	\(0.750000\pi\)
\(17\)	−210269.	−	210269.i	−0.610598	−	0.610598i	0.332504	−	0.943102i	\(-0.392107\pi\)
							−0.943102	+	0.332504i	\(0.892107\pi\)
\(19\)		−	1.03632e6i		−	1.82432i	−0.409834	−	0.912160i	\(-0.634414\pi\)
							0.409834	−	0.912160i	\(-0.365586\pi\)
\(23\)	347237.	−	347237.i	0.258733	−	0.258733i	−0.565806	−	0.824538i	\(-0.691435\pi\)
							0.824538	+	0.565806i	\(0.191435\pi\)
\(29\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(31\)	−8.24737e6			−1.60394			−0.801969	−	0.597365i	\(-0.796214\pi\)
							−0.801969	+	0.597365i	\(0.796214\pi\)
\(37\)		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	\(0.750000\pi\)
\(47\)	4.70102e7	+	4.70102e7i	1.40524	+	1.40524i	0.782132	+	0.623112i	\(0.214132\pi\)
							0.623112	+	0.782132i	\(0.285868\pi\)
\(53\)	−7.68054e7	+	7.68054e7i	−1.33706	+	1.33706i	−0.438163	+	0.898896i	\(0.644371\pi\)
							−0.898896	+	0.438163i	\(0.855629\pi\)
\(59\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(61\)	1.97895e8			1.83000			0.914998	−	0.403458i	\(-0.132192\pi\)
							0.914998	+	0.403458i	\(0.132192\pi\)
\(67\)		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	\(0.750000\pi\)
\(79\)			4.21557e8i			1.21768i	0.793292	+	0.608842i	\(0.208366\pi\)
							−0.793292	+	0.608842i	\(0.791634\pi\)
\(83\)	−6.04524e8	+	6.04524e8i	−1.39818	+	1.39818i	−0.592904	+	0.805273i	\(0.702019\pi\)
							−0.805273	+	0.592904i	\(0.797981\pi\)
\(89\)		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	75.10.e.c.68.1	yes	4
3.2	odd	2	inner	75.10.e.c.68.2	yes	4
5.2	odd	4	inner	75.10.e.c.32.2	yes	4
5.3	odd	4	inner	75.10.e.c.32.1	✓	4
5.4	even	2	inner	75.10.e.c.68.2	yes	4
15.2	even	4	inner	75.10.e.c.32.1	✓	4
15.8	even	4	inner	75.10.e.c.32.2	yes	4
15.14	odd	2	CM	75.10.e.c.68.1	yes	4

    

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