Embedded newform 75.9.j.a.11.10

• Label: 75.9.j.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.9990 - 20.6444i) q^{2} +(74.9382 + 30.7451i) q^{3} +(-122.112 + 375.821i) q^{4} +(-233.811 - 579.619i) q^{5} +(-489.288 - 2008.20i) q^{6} +4241.10 q^{7} +(3377.31 - 1097.36i) q^{8} +(4670.48 + 4607.96i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.9990	−	20.6444i	−0.937439	−	1.29027i	−0.956886	−	0.290464i	\(-0.906190\pi\)
							0.0194466	−	0.999811i	\(-0.493810\pi\)
\(3\)	74.9382	+	30.7451i	0.925164	+	0.379569i
							\(5\)	−233.811	−	579.619i	−0.374097	−	0.927390i
\(7\)	4241.10			1.76639										0.883195	−	0.469006i	\(-0.155388\pi\)
							0.883195	+	0.469006i	\(0.155388\pi\)
\(11\)	1932.57	+	2659.96i	0.131997	+	0.181679i	0.869900	−	0.493229i	\(-0.164183\pi\)
							−0.737902	+	0.674908i	\(0.764183\pi\)
\(13\)	11580.5	+	8413.75i	0.405466	+	0.294589i	0.771764	−	0.635909i	\(-0.219375\pi\)
							−0.366297	+	0.930498i	\(0.619375\pi\)
\(17\)	−125868.	+	40897.0i	−1.50702	+	0.489662i	−0.942057	−	0.335452i	\(-0.891111\pi\)
							−0.564966	+	0.825114i	\(0.691111\pi\)
\(19\)	57572.5	+	177190.i	0.441774	+	1.35964i	0.885983	+	0.463718i	\(0.153485\pi\)
							−0.444209	+	0.895923i	\(0.646515\pi\)
\(23\)	161897.	+	222832.i	0.578533	+	0.796282i	0.993534	−	0.113539i	\(-0.0362188\pi\)
							−0.415001	+	0.909821i	\(0.636219\pi\)
\(29\)	1.08358e6	+	352077.i	1.53204	+	0.497789i	0.949166	−	0.314775i	\(-0.101929\pi\)
							0.582872	+	0.812564i	\(0.301929\pi\)
\(31\)	−14349.6	−	44163.6i	−0.0155380	−	0.0478209i	0.942987	−	0.332830i	\(-0.108003\pi\)
							−0.958525	+	0.285009i	\(0.908003\pi\)
\(37\)	−962618.	−	699383.i	−0.513626	−	0.373171i	0.300571	−	0.953759i	\(-0.402823\pi\)
							−0.814197	+	0.580588i	\(0.802823\pi\)
\(41\)	−130268.	+	179298.i	−0.0461001	+	0.0634513i	−0.831445	−	0.555607i	\(-0.812486\pi\)
							0.785345	+	0.619059i	\(0.212486\pi\)
\(43\)	1.02237e6			0.299043			0.149522	−	0.988758i	\(-0.452227\pi\)
							0.149522	+	0.988758i	\(0.452227\pi\)
\(47\)	1.45794e6	+	473712.i	0.298777	+	0.0970785i	0.454570	−	0.890711i	\(-0.349793\pi\)
							−0.155792	+	0.987790i	\(0.549793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.10	✓	312
3.2	odd	2	inner	75.9.j.a.11.69	yes	312
25.16	even	5	inner	75.9.j.a.41.69	yes	312
75.41	odd	10	inner	75.9.j.a.41.10	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.10	✓	312	1.1	even	1	trivial
75.9.j.a.11.69	yes	312	3.2	odd	2	inner
75.9.j.a.41.10	yes	312	75.41	odd	10	inner
75.9.j.a.41.69	yes	312	25.16	even	5	inner