Embedded newform 75.9.j.a.11.4

• Label: 75.9.j.a.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-17.3034 - 23.8161i) q^{2} +(79.9396 + 13.0637i) q^{3} +(-188.690 + 580.728i) q^{4} +(518.937 - 348.324i) q^{5} +(-1072.10 - 2129.89i) q^{6} -3638.55 q^{7} +(9928.28 - 3225.89i) q^{8} +(6219.68 + 2088.62i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−17.3034	−	23.8161i	−1.08146	−	1.48851i	−0.857901	−	0.513815i	\(-0.828232\pi\)
							−0.223561	−	0.974690i	\(-0.571768\pi\)
\(3\)	79.9396	+	13.0637i	0.986909	+	0.161280i
							\(5\)	518.937	−	348.324i	0.830299	−	0.557319i
\(7\)	−3638.55			−1.51543										−0.757715	−	0.652586i	\(-0.773684\pi\)
							−0.757715	+	0.652586i	\(0.773684\pi\)
\(11\)	−1965.68	−	2705.53i	−0.134259	−	0.184791i	0.736594	−	0.676335i	\(-0.236433\pi\)
							−0.870853	+	0.491544i	\(0.836433\pi\)
\(13\)	32195.2	+	23391.2i	1.12725	+	0.818992i	0.985292	−	0.170882i	\(-0.0546616\pi\)
							0.141954	+	0.989873i	\(0.454662\pi\)
\(17\)	1651.08	−	536.468i	0.0197684	−	0.00642315i	−0.299116	−	0.954217i	\(-0.596692\pi\)
							0.318885	+	0.947794i	\(0.396692\pi\)
\(19\)	−64395.0	−	198187.i	−0.494126	−	1.52076i	−0.818314	−	0.574771i	\(-0.805091\pi\)
							0.324188	−	0.945993i	\(-0.394909\pi\)
\(23\)	−219581.	−	302228.i	−0.784664	−	1.08000i	−0.994752	−	0.102315i	\(-0.967375\pi\)
							0.210088	−	0.977682i	\(-0.432625\pi\)
\(29\)	292243.	+	94955.6i	0.413193	+	0.134254i	0.508234	−	0.861219i	\(-0.330298\pi\)
							−0.0950416	+	0.995473i	\(0.530298\pi\)
\(31\)	−38226.0	−	117648.i	−0.0413916	−	0.127390i	0.928225	−	0.372018i	\(-0.121334\pi\)
							−0.969617	+	0.244628i	\(0.921334\pi\)
\(37\)	−2.05471e6	−	1.49283e6i	−1.09634	−	0.796535i	−0.115878	−	0.993263i	\(-0.536968\pi\)
							−0.980458	+	0.196729i	\(0.936968\pi\)
\(41\)	1.81131e6	−	2.49305e6i	0.640998	−	0.882258i	−0.357670	−	0.933848i	\(-0.616429\pi\)
							0.998668	+	0.0515896i	\(0.0164288\pi\)
\(43\)	−338774.			−0.0990915			−0.0495457	−	0.998772i	\(-0.515777\pi\)
							−0.0495457	+	0.998772i	\(0.515777\pi\)
\(47\)	−8.00262e6	−	2.60021e6i	−1.63999	−	0.532864i	−0.663452	−	0.748219i	\(-0.730909\pi\)
							−0.976536	+	0.215355i	\(0.930909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.4	✓	312
3.2	odd	2	inner	75.9.j.a.11.75	yes	312
25.16	even	5	inner	75.9.j.a.41.75	yes	312
75.41	odd	10	inner	75.9.j.a.41.4	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.4	✓	312	1.1	even	1	trivial
75.9.j.a.11.75	yes	312	3.2	odd	2	inner
75.9.j.a.41.4	yes	312	75.41	odd	10	inner
75.9.j.a.41.75	yes	312	25.16	even	5	inner