Embedded newform 75.9.j.a.11.7

• Label: 75.9.j.a.11.7
• Level: $75$
• Weight: $9$
• Character: 75.11
• Analytic conductor: $30.553$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(30.5533957546\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(78\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			11.7
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.6151 - 22.8688i) q^{2} +(76.4502 - 26.7649i) q^{3} +(-167.810 + 516.465i) q^{4} +(-553.809 + 289.691i) q^{5} +(-1882.31 - 1303.62i) q^{6} -219.465 q^{7} +(7716.82 - 2507.35i) q^{8} +(5128.28 - 4092.37i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.6151	−	22.8688i	−1.03845	−	1.42930i	−0.898413	−	0.439151i	\(-0.855279\pi\)
							−0.140032	−	0.990147i	\(-0.544721\pi\)
\(3\)	76.4502	−	26.7649i	0.943830	−	0.330431i
							\(5\)	−553.809	+	289.691i	−0.886094	+	0.463505i
\(7\)	−219.465			−0.0914056										−0.0457028	−	0.998955i	\(-0.514553\pi\)
							−0.0457028	+	0.998955i	\(0.514553\pi\)
\(11\)	12227.9	+	16830.2i	0.835181	+	1.14953i	0.986936	+	0.161111i	\(0.0515076\pi\)
							−0.151755	+	0.988418i	\(0.548492\pi\)
\(13\)	−25472.2	−	18506.7i	−0.891853	−	0.647969i	0.0445073	−	0.999009i	\(-0.485828\pi\)
							−0.936361	+	0.351040i	\(0.885828\pi\)
\(17\)	80022.9	−	26001.0i	0.958117	−	0.311311i	0.212108	−	0.977246i	\(-0.431967\pi\)
							0.746010	+	0.665935i	\(0.231967\pi\)
\(19\)	−13774.3	−	42392.9i	−0.105695	−	0.325296i	0.884198	−	0.467113i	\(-0.154706\pi\)
							−0.989893	+	0.141816i	\(0.954706\pi\)
\(23\)	123988.	+	170654.i	0.443065	+	0.609826i	0.970890	−	0.239527i	\(-0.0769923\pi\)
							−0.527825	+	0.849353i	\(0.676992\pi\)
\(29\)	−837962.	−	272270.i	−1.18477	−	0.384954i	−0.350631	−	0.936514i	\(-0.614033\pi\)
							−0.834135	+	0.551560i	\(0.814033\pi\)
\(31\)	−519347.	−	1.59839e6i	−0.562356	−	1.73075i	−0.675680	−	0.737195i	\(-0.736150\pi\)
							0.113325	−	0.993558i	\(-0.463850\pi\)
\(37\)	−812312.	−	590180.i	−0.433427	−	0.314903i	0.349591	−	0.936903i	\(-0.386321\pi\)
							−0.783018	+	0.621999i	\(0.786321\pi\)
\(41\)	−660570.	+	909197.i	−0.233767	+	0.321753i	−0.909744	−	0.415170i	\(-0.863722\pi\)
							0.675977	+	0.736923i	\(0.263722\pi\)
\(43\)	2.88699e6			0.844446			0.422223	−	0.906492i	\(-0.361250\pi\)
							0.422223	+	0.906492i	\(0.361250\pi\)
\(47\)	6.26587e6	+	2.03591e6i	1.28407	+	0.417221i	0.870013	−	0.493028i	\(-0.164110\pi\)
							0.414061	+	0.910249i	\(0.364110\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.7	✓	312
3.2	odd	2	inner	75.9.j.a.11.72	yes	312
25.16	even	5	inner	75.9.j.a.41.72	yes	312
75.41	odd	10	inner	75.9.j.a.41.7	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.7	✓	312	1.1	even	1	trivial
75.9.j.a.11.72	yes	312	3.2	odd	2	inner
75.9.j.a.41.7	yes	312	75.41	odd	10	inner
75.9.j.a.41.72	yes	312	25.16	even	5	inner