Embedded newform 572.2.bm.a.35.11

• Label: 572.2.bm.a.35.11
• Level: $572$
• Weight: $2$
• Character: 572.35
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.bm (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(80\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			35.11
Character	\(\chi\)	\(=\)	572.35
Dual form			572.2.bm.a.523.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30497 + 0.545018i) q^{2} +(3.13196 - 0.329183i) q^{3} +(1.40591 - 1.42247i) q^{4} +(-0.288212 - 0.887025i) q^{5} +(-3.90772 + 2.13655i) q^{6} +(0.347803 - 3.30913i) q^{7} +(-1.05941 + 2.62253i) q^{8} +(6.76639 - 1.43824i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 5 q^{2} - 3 q^{4} - 24 q^{5} - 5 q^{6} - 20 q^{8} - 78 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30497	+	0.545018i	−0.922755	+	0.385386i
							\(3\)	3.13196	−	0.329183i	1.80824	−	0.190054i	0.860576	−	0.509322i	\(-0.170104\pi\)
0.947664	+	0.319269i	\(0.103437\pi\)
\(5\)	−0.288212	−	0.887025i	−0.128892	−	0.396690i	0.865698	−	0.500567i	\(-0.166875\pi\)
							−0.994590	+	0.103877i	\(0.966875\pi\)
\(7\)	0.347803	−	3.30913i	0.131457	−	1.25073i	−0.707570	−	0.706643i	\(-0.750209\pi\)
							0.839027	−	0.544089i	\(-0.183125\pi\)
\(11\)	−2.66395	−	1.97570i	−0.803210	−	0.595696i
							\(13\)	−0.501647	+	3.57048i	−0.139132	+	0.990274i
\(17\)	−0.890495	−	0.801805i	−0.215977	−	0.194466i								0.554040	−	0.832490i	\(-0.313085\pi\)
							−0.770017	+	0.638024i	\(0.779752\pi\)
\(19\)	2.71233	+	1.20761i	0.622250	+	0.277044i	0.693547	−	0.720412i	\(-0.256047\pi\)
							−0.0712967	+	0.997455i	\(0.522714\pi\)
\(23\)	−0.953128	−	0.550289i	−0.198741	−	0.114743i	0.397327	−	0.917677i	\(-0.369938\pi\)
							−0.596068	+	0.802934i	\(0.703271\pi\)
\(29\)	0.221744	+	0.498045i	0.0411768	+	0.0924846i	0.932963	−	0.359973i	\(-0.117214\pi\)
							−0.891786	+	0.452458i	\(0.850547\pi\)
\(31\)	2.58599	+	0.840239i	0.464457	+	0.150911i	0.531892	−	0.846812i	\(-0.321481\pi\)
							−0.0674345	+	0.997724i	\(0.521481\pi\)
\(37\)	−4.64949	+	2.07009i	−0.764372	+	0.340320i	−0.751618	−	0.659598i	\(-0.770726\pi\)
							−0.0127537	+	0.999919i	\(0.504060\pi\)
\(41\)	−5.55689	+	0.584052i	−0.867840	+	0.0912136i	−0.527974	−	0.849260i	\(-0.677048\pi\)
							−0.339866	+	0.940474i	\(0.610382\pi\)
\(43\)	1.78152	+	3.08569i	0.271680	+	0.470563i	0.969292	−	0.245912i	\(-0.0790876\pi\)
							−0.697612	+	0.716475i	\(0.745754\pi\)
\(47\)	−7.25433	−	9.98473i	−1.05815	−	1.45642i	−0.881520	−	0.472146i	\(-0.843479\pi\)
							−0.176633	−	0.984277i	\(-0.556521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bm.a.35.11	✓	640
4.3	odd	2	inner	572.2.bm.a.35.28	yes	640
11.6	odd	10	inner	572.2.bm.a.347.25	yes	640
13.3	even	3	inner	572.2.bm.a.211.44	yes	640
44.39	even	10	inner	572.2.bm.a.347.44	yes	640
52.3	odd	6	inner	572.2.bm.a.211.25	yes	640
143.94	odd	30	inner	572.2.bm.a.523.28	yes	640
572.523	even	30	inner	572.2.bm.a.523.11	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bm.a.35.11	✓	640	1.1	even	1	trivial
572.2.bm.a.35.28	yes	640	4.3	odd	2	inner
572.2.bm.a.211.25	yes	640	52.3	odd	6	inner
572.2.bm.a.211.44	yes	640	13.3	even	3	inner
572.2.bm.a.347.25	yes	640	11.6	odd	10	inner
572.2.bm.a.347.44	yes	640	44.39	even	10	inner
572.2.bm.a.523.11	yes	640	572.523	even	30	inner
572.2.bm.a.523.28	yes	640	143.94	odd	30	inner