Embedded newform 75.9.j.a.11.6

• Label: 75.9.j.a.11.6
• 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			11.6
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.9218 - 23.2908i) q^{2} +(40.8105 - 69.9679i) q^{3} +(-177.008 + 544.774i) q^{4} +(356.232 + 513.541i) q^{5} +(-2320.20 + 233.469i) q^{6} +1515.19 q^{7} +(8674.23 - 2818.43i) q^{8} +(-3230.00 - 5710.85i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.9218	−	23.2908i	−1.05761	−	1.45568i	−0.882019	−	0.471214i	\(-0.843816\pi\)
							−0.175592	−	0.984463i	\(-0.556184\pi\)
\(3\)	40.8105	−	69.9679i	0.503834	−	0.863801i
							\(5\)	356.232	+	513.541i	0.569971	+	0.821665i
\(7\)	1515.19			0.631064										0.315532	−	0.948915i	\(-0.397817\pi\)
							0.315532	+	0.948915i	\(0.397817\pi\)
\(11\)	−11494.4	−	15820.7i	−0.785084	−	1.08058i	−0.994703	−	0.102793i	\(-0.967222\pi\)
							0.209619	−	0.977783i	\(-0.432778\pi\)
\(13\)	13099.7	+	9517.53i	0.458659	+	0.333235i	0.793005	−	0.609215i	\(-0.208515\pi\)
							−0.334346	+	0.942450i	\(0.608515\pi\)
\(17\)	−137644.	+	44723.4i	−1.64802	+	0.535475i	−0.978310	−	0.207146i	\(-0.933582\pi\)
							−0.669712	+	0.742621i	\(0.733582\pi\)
\(19\)	41701.2	+	128343.i	0.319989	+	0.984824i	0.973652	+	0.228039i	\(0.0732312\pi\)
							−0.653663	+	0.756785i	\(0.726769\pi\)
\(23\)	−109943.	−	151323.i	−0.392876	−	0.540748i	0.566062	−	0.824363i	\(-0.308466\pi\)
							−0.958938	+	0.283615i	\(0.908466\pi\)
\(29\)	−646262.	−	209983.i	−0.913727	−	0.296888i	−0.185836	−	0.982581i	\(-0.559499\pi\)
							−0.727891	+	0.685693i	\(0.759499\pi\)
\(31\)	−339318.	−	1.04431e6i	−0.367418	−	1.13080i	−0.948453	−	0.316917i	\(-0.897352\pi\)
							0.581035	−	0.813878i	\(-0.302648\pi\)
\(37\)	929222.	+	675119.i	0.495807	+	0.360225i	0.807413	−	0.589987i	\(-0.200867\pi\)
							−0.311606	+	0.950211i	\(0.600867\pi\)
\(41\)	−2.69298e6	+	3.70657e6i	−0.953010	+	1.31171i	−0.00283298	+	0.999996i	\(0.500902\pi\)
							−0.950177	+	0.311710i	\(0.899098\pi\)
\(43\)	−2.65859e6			−0.777639			−0.388820	−	0.921314i	\(-0.627117\pi\)
							−0.388820	+	0.921314i	\(0.627117\pi\)
\(47\)	−2.51996e6	−	818784.i	−0.516419	−	0.167795i	0.0392006	−	0.999231i	\(-0.487519\pi\)
							−0.555619	+	0.831437i	\(0.687519\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.6	✓	312
3.2	odd	2	inner	75.9.j.a.11.73	yes	312
25.16	even	5	inner	75.9.j.a.41.73	yes	312
75.41	odd	10	inner	75.9.j.a.41.6	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.6	✓	312	1.1	even	1	trivial
75.9.j.a.11.73	yes	312	3.2	odd	2	inner
75.9.j.a.41.6	yes	312	75.41	odd	10	inner
75.9.j.a.41.73	yes	312	25.16	even	5	inner