Embedded newform 75.9.j.a.11.15

• Label: 75.9.j.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.0487 - 17.9600i) q^{2} +(76.2368 + 27.3669i) q^{3} +(-73.1849 + 225.240i) q^{4} +(326.265 + 533.081i) q^{5} +(-503.283 - 1726.32i) q^{6} +791.787 q^{7} +(-404.710 + 131.498i) q^{8} +(5063.11 + 4172.73i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.0487	−	17.9600i	−0.815545	−	1.12250i	−0.990444	−	0.137915i	\(-0.955960\pi\)
							0.174899	−	0.984586i	\(-0.444040\pi\)
\(3\)	76.2368	+	27.3669i	0.941195	+	0.337863i
							\(5\)	326.265	+	533.081i	0.522025	+	0.852930i
\(7\)	791.787			0.329774										0.164887	−	0.986312i	\(-0.447274\pi\)
							0.164887	+	0.986312i	\(0.447274\pi\)
\(11\)	−9972.89	−	13726.5i	−0.681162	−	0.937539i	0.318785	−	0.947827i	\(-0.396725\pi\)
							−0.999947	+	0.0102878i	\(0.996725\pi\)
\(13\)	−22016.6	−	15996.0i	−0.770861	−	0.560063i	0.131362	−	0.991335i	\(-0.458065\pi\)
							−0.902222	+	0.431271i	\(0.858065\pi\)
\(17\)	121456.	−	39463.3i	1.45419	−	0.472495i	0.527901	−	0.849306i	\(-0.322979\pi\)
							0.926290	+	0.376810i	\(0.122979\pi\)
\(19\)	4783.19	+	14721.2i	0.0367032	+	0.112961i	0.967730	−	0.251991i	\(-0.0810854\pi\)
							−0.931026	+	0.364952i	\(0.881085\pi\)
\(23\)	140643.	+	193579.i	0.502583	+	0.691746i	0.982647	−	0.185488i	\(-0.0593864\pi\)
							−0.480064	+	0.877234i	\(0.659386\pi\)
\(29\)	351094.	+	114077.i	0.496399	+	0.161290i	0.546508	−	0.837454i	\(-0.315957\pi\)
							−0.0501087	+	0.998744i	\(0.515957\pi\)
\(31\)	561539.	+	1.72824e6i	0.608042	+	1.87136i	0.474347	+	0.880338i	\(0.342684\pi\)
							0.133695	+	0.991022i	\(0.457316\pi\)
\(37\)	2.51605e6	+	1.82802e6i	1.34249	+	0.975379i	0.999348	+	0.0360966i	\(0.0114924\pi\)
							0.343146	+	0.939282i	\(0.388508\pi\)
\(41\)	2.99500e6	−	4.12227e6i	1.05989	−	1.45882i	0.179988	−	0.983669i	\(-0.442394\pi\)
							0.879905	−	0.475149i	\(-0.157606\pi\)
\(43\)	1.39237e6			0.407269			0.203635	−	0.979047i	\(-0.434725\pi\)
							0.203635	+	0.979047i	\(0.434725\pi\)
\(47\)	−146170.	−	47493.4i	−0.0299548	−	0.00973289i	0.294001	−	0.955805i	\(-0.405013\pi\)
							−0.323956	+	0.946072i	\(0.605013\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.15	✓	312
3.2	odd	2	inner	75.9.j.a.11.64	yes	312
25.16	even	5	inner	75.9.j.a.41.64	yes	312
75.41	odd	10	inner	75.9.j.a.41.15	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.15	✓	312	1.1	even	1	trivial
75.9.j.a.11.64	yes	312	3.2	odd	2	inner
75.9.j.a.41.15	yes	312	75.41	odd	10	inner
75.9.j.a.41.64	yes	312	25.16	even	5	inner