Embedded newform 75.2.l.a.23.2

• Label: 75.2.l.a.23.2
• Level: $75$
• Weight: $2$
• Character: 75.23
• Analytic conductor: $0.599$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.598878015160\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(8\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			23.2
Character	\(\chi\)	\(=\)	75.23
Dual form			75.2.l.a.62.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78150 - 0.282162i) q^{2} +(0.0717082 + 1.73057i) q^{3} +(1.19203 + 0.387313i) q^{4} +(-2.23260 + 0.124491i) q^{5} +(0.360552 - 3.10324i) q^{6} +(-2.65364 + 2.65364i) q^{7} +(1.19992 + 0.611391i) q^{8} +(-2.98972 + 0.248191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} - 20 q^{4} - 6 q^{6} - 20 q^{7} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78150	−	0.282162i	−1.25971	−	0.199519i	−0.509358	−	0.860554i	\(-0.670117\pi\)
							−0.750355	+	0.661036i	\(0.770117\pi\)
\(3\)	0.0717082	+	1.73057i	0.0414007	+	0.999143i
							\(5\)	−2.23260	+	0.124491i	−0.998449	+	0.0556739i
\(7\)	−2.65364	+	2.65364i	−1.00298	+	1.00298i								−0.00298642	+	0.999996i	\(0.500951\pi\)
							−0.999996	+	0.00298642i	\(0.999049\pi\)
\(11\)	1.51304	+	2.08252i	0.456198	+	0.627903i	0.973715	−	0.227770i	\(-0.0731435\pi\)
							−0.517517	+	0.855673i	\(0.673143\pi\)
\(13\)	0.0649985	+	0.410385i	0.0180273	+	0.113820i	0.995061	−	0.0992645i	\(-0.0316490\pi\)
							−0.977034	+	0.213085i	\(0.931649\pi\)
\(17\)	0.931829	−	1.82882i	0.226002	−	0.443554i	−0.749964	−	0.661479i	\(-0.769929\pi\)
							0.975965	+	0.217926i	\(0.0699290\pi\)
\(19\)	4.65508	−	1.51253i	1.06795	−	0.346997i	0.278260	−	0.960506i	\(-0.410242\pi\)
							0.789688	+	0.613509i	\(0.210242\pi\)
\(23\)	−0.345599	+	2.18202i	−0.0720623	+	0.454983i	0.925101	+	0.379720i	\(0.123980\pi\)
							−0.997164	+	0.0752633i	\(0.976020\pi\)
\(29\)	−1.43698	+	4.42258i	−0.266841	+	0.821252i	0.724423	+	0.689356i	\(0.242107\pi\)
							−0.991264	+	0.131896i	\(0.957893\pi\)
\(31\)	2.77301	+	8.53446i	0.498048	+	1.53283i	0.812152	+	0.583446i	\(0.198296\pi\)
							−0.314104	+	0.949389i	\(0.601704\pi\)
\(37\)	−4.34815	+	0.688680i	−0.714832	+	0.113218i	−0.503249	−	0.864142i	\(-0.667862\pi\)
							−0.211583	+	0.977360i	\(0.567862\pi\)
\(41\)	−2.81270	+	3.87134i	−0.439269	+	0.604602i	−0.970050	−	0.242907i	\(-0.921899\pi\)
							0.530780	+	0.847510i	\(0.321899\pi\)
\(43\)	−2.89094	−	2.89094i	−0.440865	−	0.440865i	0.451438	−	0.892303i	\(-0.350911\pi\)
							−0.892303	+	0.451438i	\(0.850911\pi\)
\(47\)	4.60775	−	2.34777i	0.672110	−	0.342457i	−0.0844068	−	0.996431i	\(-0.526900\pi\)
							0.756517	+	0.653974i	\(0.226900\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.2.l.a.23.2	✓	64
3.2	odd	2	inner	75.2.l.a.23.7	yes	64
5.2	odd	4		375.2.l.a.32.7		64
5.3	odd	4		375.2.l.b.32.2		64
5.4	even	2		375.2.l.c.218.7		64
15.2	even	4		375.2.l.a.32.2		64
15.8	even	4		375.2.l.b.32.7		64
15.14	odd	2		375.2.l.c.218.2		64
25.9	even	10		375.2.l.b.293.7		64
25.12	odd	20	inner	75.2.l.a.62.7	yes	64
25.13	odd	20		375.2.l.c.332.2		64
25.16	even	5		375.2.l.a.293.2		64
75.38	even	20		375.2.l.c.332.7		64
75.41	odd	10		375.2.l.a.293.7		64
75.59	odd	10		375.2.l.b.293.2		64
75.62	even	20	inner	75.2.l.a.62.2	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.l.a.23.2	✓	64	1.1	even	1	trivial
75.2.l.a.23.7	yes	64	3.2	odd	2	inner
75.2.l.a.62.2	yes	64	75.62	even	20	inner
75.2.l.a.62.7	yes	64	25.12	odd	20	inner
375.2.l.a.32.2		64	15.2	even	4
375.2.l.a.32.7		64	5.2	odd	4
375.2.l.a.293.2		64	25.16	even	5
375.2.l.a.293.7		64	75.41	odd	10
375.2.l.b.32.2		64	5.3	odd	4
375.2.l.b.32.7		64	15.8	even	4
375.2.l.b.293.2		64	75.59	odd	10
375.2.l.b.293.7		64	25.9	even	10
375.2.l.c.218.2		64	15.14	odd	2
375.2.l.c.218.7		64	5.4	even	2
375.2.l.c.332.2		64	25.13	odd	20
375.2.l.c.332.7		64	75.38	even	20