Embedded newform 75.9.j.a.11.11

• Label: 75.9.j.a.11.11
• 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			11.11
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.0772 - 19.3756i) q^{2} +(-55.8311 + 58.6847i) q^{3} +(-98.1381 + 302.038i) q^{4} +(-216.170 - 586.426i) q^{5} +(1923.00 + 255.644i) q^{6} -1564.18 q^{7} +(1402.67 - 455.754i) q^{8} +(-326.786 - 6552.86i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.0772	−	19.3756i	−0.879825	−	1.21098i	−0.976469	−	0.215656i	\(-0.930811\pi\)
							0.0966444	−	0.995319i	\(-0.469189\pi\)
\(3\)	−55.8311	+	58.6847i	−0.689272	+	0.724502i
							\(5\)	−216.170	−	586.426i	−0.345872	−	0.938282i
\(7\)	−1564.18			−0.651470										−0.325735	−	0.945461i	\(-0.605612\pi\)
							−0.325735	+	0.945461i	\(0.605612\pi\)
\(11\)	12801.4	+	17619.7i	0.874356	+	1.20345i	0.977952	+	0.208828i	\(0.0669649\pi\)
							−0.103597	+	0.994619i	\(0.533035\pi\)
\(13\)	13186.3	+	9580.39i	0.461688	+	0.335436i	0.794193	−	0.607665i	\(-0.207894\pi\)
							−0.332505	+	0.943102i	\(0.607894\pi\)
\(17\)	61836.0	−	20091.7i	0.740365	−	0.240559i	0.0855349	−	0.996335i	\(-0.472740\pi\)
							0.654830	+	0.755776i	\(0.272740\pi\)
\(19\)	9471.38	+	29149.9i	0.0726773	+	0.223678i	0.980796	−	0.195034i	\(-0.0624817\pi\)
							−0.908119	+	0.418712i	\(0.862482\pi\)
\(23\)	−185337.	−	255094.i	−0.662293	−	0.911568i	0.337261	−	0.941411i	\(-0.390499\pi\)
							−0.999555	+	0.0298427i	\(0.990499\pi\)
\(29\)	568686.	+	184777.i	0.804046	+	0.261250i	0.682074	−	0.731283i	\(-0.261078\pi\)
							0.121972	+	0.992534i	\(0.461078\pi\)
\(31\)	−80313.2	−	247179.i	−0.0869642	−	0.267648i	0.898112	−	0.439767i	\(-0.144939\pi\)
							−0.985076	+	0.172118i	\(0.944939\pi\)
\(37\)	1.50405e6	+	1.09276e6i	0.802521	+	0.583065i	0.911653	−	0.410962i	\(-0.134807\pi\)
							−0.109132	+	0.994027i	\(0.534807\pi\)
\(41\)	−899753.	+	1.23840e6i	−0.318411	+	0.438255i	−0.937981	−	0.346686i	\(-0.887307\pi\)
							0.619570	+	0.784941i	\(0.287307\pi\)
\(43\)	−80725.5			−0.0236122			−0.0118061	−	0.999930i	\(-0.503758\pi\)
							−0.0118061	+	0.999930i	\(0.503758\pi\)
\(47\)	−4.60205e6	−	1.49530e6i	−0.943105	−	0.306433i	−0.203194	−	0.979138i	\(-0.565132\pi\)
							−0.739911	+	0.672705i	\(0.765132\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.11	✓	312
3.2	odd	2	inner	75.9.j.a.11.68	yes	312
25.16	even	5	inner	75.9.j.a.41.68	yes	312
75.41	odd	10	inner	75.9.j.a.41.11	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.11	✓	312	1.1	even	1	trivial
75.9.j.a.11.68	yes	312	3.2	odd	2	inner
75.9.j.a.41.11	yes	312	75.41	odd	10	inner
75.9.j.a.41.68	yes	312	25.16	even	5	inner