Embedded newform 75.9.j.a.11.9

• Label: 75.9.j.a.11.9
• 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			11.9
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.8525 - 21.8191i) q^{2} +(-5.79551 - 80.7924i) q^{3} +(-145.663 + 448.304i) q^{4} +(426.206 - 457.136i) q^{5} +(-1670.94 + 1407.21i) q^{6} -1651.43 q^{7} +(5524.34 - 1794.97i) q^{8} +(-6493.82 + 936.466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.8525	−	21.8191i	−0.990781	−	1.36369i	−0.930814	−	0.365493i	\(-0.880900\pi\)
							−0.0599674	−	0.998200i	\(-0.519100\pi\)
\(3\)	−5.79551	−	80.7924i	−0.0715495	−	0.997437i
							\(5\)	426.206	−	457.136i	0.681930	−	0.731418i
\(7\)	−1651.43			−0.687811										−0.343906	−	0.939004i	\(-0.611750\pi\)
							−0.343906	+	0.939004i	\(0.611750\pi\)
\(11\)	8975.80	+	12354.1i	0.613059	+	0.843804i	0.996825	−	0.0796270i	\(-0.0253729\pi\)
							−0.383766	+	0.923431i	\(0.625373\pi\)
\(13\)	−30161.4	−	21913.5i	−1.05603	−	0.767253i	−0.0826830	−	0.996576i	\(-0.526349\pi\)
							−0.973350	+	0.229323i	\(0.926349\pi\)
\(17\)	25130.8	−	8165.49i	0.300892	−	0.0977657i	−0.154680	−	0.987965i	\(-0.549435\pi\)
							0.455572	+	0.890199i	\(0.349435\pi\)
\(19\)	74262.1	+	228555.i	0.569840	+	1.75379i	0.653113	+	0.757260i	\(0.273462\pi\)
							−0.0832736	+	0.996527i	\(0.526538\pi\)
\(23\)	223428.	+	307523.i	0.798412	+	1.09892i	0.993009	+	0.118037i	\(0.0376601\pi\)
							−0.194597	+	0.980883i	\(0.562340\pi\)
\(29\)	161584.	+	52501.7i	0.228457	+	0.0742303i	0.421009	−	0.907057i	\(-0.361676\pi\)
							−0.192551	+	0.981287i	\(0.561676\pi\)
\(31\)	305210.	+	939338.i	0.330485	+	1.01713i	0.968904	+	0.247438i	\(0.0795887\pi\)
							−0.638419	+	0.769689i	\(0.720411\pi\)
\(37\)	−2.21824e6	−	1.61165e6i	−1.18359	−	0.859930i	−0.191020	−	0.981586i	\(-0.561180\pi\)
							−0.992572	+	0.121656i	\(0.961180\pi\)
\(41\)	−391805.	+	539273.i	−0.138655	+	0.190842i	−0.872697	−	0.488262i	\(-0.837631\pi\)
							0.734043	+	0.679103i	\(0.237631\pi\)
\(43\)	−87882.8			−0.0257057			−0.0128529	−	0.999917i	\(-0.504091\pi\)
							−0.0128529	+	0.999917i	\(0.504091\pi\)
\(47\)	−5.68035e6	−	1.84566e6i	−1.16408	−	0.378233i	−0.337651	−	0.941271i	\(-0.609632\pi\)
							−0.826431	+	0.563038i	\(0.809632\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.9	✓	312
3.2	odd	2	inner	75.9.j.a.11.70	yes	312
25.16	even	5	inner	75.9.j.a.41.70	yes	312
75.41	odd	10	inner	75.9.j.a.41.9	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.9	✓	312	1.1	even	1	trivial
75.9.j.a.11.70	yes	312	3.2	odd	2	inner
75.9.j.a.41.9	yes	312	75.41	odd	10	inner
75.9.j.a.41.70	yes	312	25.16	even	5	inner