Embedded newform 75.2.l.a.38.8

• Label: 75.2.l.a.38.8
• Level: $75$
• Weight: $2$
• Character: 75.38
• 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			38.8
Character	\(\chi\)	\(=\)	75.38
Dual form			75.2.l.a.2.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.400841 + 2.53081i) q^{2} +(1.67651 - 0.435117i) q^{3} +(-4.34220 + 1.41087i) q^{4} +(-1.84517 - 1.26307i) q^{5} +(1.77321 + 4.06850i) q^{6} +(1.29796 - 1.29796i) q^{7} +(-2.98460 - 5.85760i) q^{8} +(2.62135 - 1.45895i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 10 q^{3} - 20 q^{4} - 6 q^{6} - 20 q^{7} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{19}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.400841	+	2.53081i	0.283437	+	1.78955i	0.559935	+	0.828537i	\(0.310826\pi\)
							−0.276497	+	0.961015i	\(0.589174\pi\)
\(3\)	1.67651	−	0.435117i	0.967931	−	0.251215i
							\(5\)	−1.84517	−	1.26307i	−0.825185	−	0.564862i
\(7\)	1.29796	−	1.29796i	0.490582	−	0.490582i								−0.417907	−	0.908490i	\(-0.637236\pi\)
							0.908490	+	0.417907i	\(0.137236\pi\)
\(11\)	−1.30491	+	1.79605i	−0.393444	+	0.541529i	−0.959084	−	0.283123i	\(-0.908629\pi\)
							0.565639	+	0.824653i	\(0.308629\pi\)
\(13\)	−2.84658	−	0.450854i	−0.789500	−	0.125044i	−0.251353	−	0.967896i	\(-0.580875\pi\)
							−0.538147	+	0.842851i	\(0.680875\pi\)
\(17\)	−0.546966	+	0.278693i	−0.132659	+	0.0675930i	−0.519060	−	0.854738i	\(-0.673718\pi\)
							0.386401	+	0.922331i	\(0.373718\pi\)
\(19\)	3.64082	+	1.18297i	0.835262	+	0.271393i	0.695260	−	0.718758i	\(-0.255289\pi\)
							0.140002	+	0.990151i	\(0.455289\pi\)
\(23\)	−2.80121	+	0.443669i	−0.584093	+	0.0925113i	−0.441486	−	0.897268i	\(-0.645549\pi\)
							−0.142607	+	0.989779i	\(0.545549\pi\)
\(29\)	−1.23999	−	3.81629i	−0.230260	−	0.708668i	−0.997715	−	0.0675649i	\(-0.978477\pi\)
							0.767455	−	0.641103i	\(-0.221523\pi\)
\(31\)	−1.33062	+	4.09523i	−0.238987	+	0.735525i	0.757581	+	0.652741i	\(0.226381\pi\)
							−0.996568	+	0.0827840i	\(0.973619\pi\)
\(37\)	−0.289237	+	1.82617i	−0.0475502	+	0.300220i	−0.999990	−	0.00450714i	\(-0.998565\pi\)
							0.952440	+	0.304727i	\(0.0985653\pi\)
\(41\)	3.99821	+	5.50306i	0.624415	+	0.859434i	0.997665	−	0.0682969i	\(-0.0217565\pi\)
							−0.373250	+	0.927731i	\(0.621757\pi\)
\(43\)	−9.16523	−	9.16523i	−1.39768	−	1.39768i	−0.806637	−	0.591047i	\(-0.798715\pi\)
							−0.591047	−	0.806637i	\(-0.701285\pi\)
\(47\)	3.43326	−	6.73816i	0.500793	−	0.982861i	−0.492832	−	0.870124i	\(-0.664038\pi\)
							0.993625	−	0.112737i	\(-0.0359617\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.2.l.a.38.8	yes	64
3.2	odd	2	inner	75.2.l.a.38.1	yes	64
5.2	odd	4		375.2.l.a.332.1		64
5.3	odd	4		375.2.l.b.332.8		64
5.4	even	2		375.2.l.c.293.1		64
15.2	even	4		375.2.l.a.332.8		64
15.8	even	4		375.2.l.b.332.1		64
15.14	odd	2		375.2.l.c.293.8		64
25.2	odd	20	inner	75.2.l.a.2.1	✓	64
25.11	even	5		375.2.l.a.218.8		64
25.14	even	10		375.2.l.b.218.1		64
25.23	odd	20		375.2.l.c.32.8		64
75.2	even	20	inner	75.2.l.a.2.8	yes	64
75.11	odd	10		375.2.l.a.218.1		64
75.14	odd	10		375.2.l.b.218.8		64
75.23	even	20		375.2.l.c.32.1		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.l.a.2.1	✓	64	25.2	odd	20	inner
75.2.l.a.2.8	yes	64	75.2	even	20	inner
75.2.l.a.38.1	yes	64	3.2	odd	2	inner
75.2.l.a.38.8	yes	64	1.1	even	1	trivial
375.2.l.a.218.1		64	75.11	odd	10
375.2.l.a.218.8		64	25.11	even	5
375.2.l.a.332.1		64	5.2	odd	4
375.2.l.a.332.8		64	15.2	even	4
375.2.l.b.218.1		64	25.14	even	10
375.2.l.b.218.8		64	75.14	odd	10
375.2.l.b.332.1		64	15.8	even	4
375.2.l.b.332.8		64	5.3	odd	4
375.2.l.c.32.1		64	75.23	even	20
375.2.l.c.32.8		64	25.23	odd	20
375.2.l.c.293.1		64	5.4	even	2
375.2.l.c.293.8		64	15.14	odd	2