Embedded newform 525.2.bs.a.2.3

• Label: 525.2.bs.a.2.3
• Level: $525$
• Weight: $2$
• Character: 525.2
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $1216$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bs (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(1216\)
Relative dimension:	\(76\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			2.3
Character	\(\chi\)	\(=\)	525.2
Dual form			525.2.bs.a.263.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.02843 - 1.64259i) q^{2} +(-1.68808 - 0.387799i) q^{3} +(1.00060 + 4.70747i) q^{4} +(2.22126 - 0.256881i) q^{5} +(2.78716 + 3.55945i) q^{6} +(2.15806 - 1.53061i) q^{7} +(3.33287 - 6.54113i) q^{8} +(2.69922 + 1.30927i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1216 q - 8 q^{3} - 20 q^{4} - 24 q^{6} - 28 q^{7} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.02843	−	1.64259i	−1.43432	−	1.16149i	−0.959607	−	0.281346i	\(-0.909219\pi\)
							−0.474711	−	0.880142i	\(-0.657447\pi\)
\(3\)	−1.68808	−	0.387799i	−0.974613	−	0.223896i
							\(5\)	2.22126	−	0.256881i	0.993379	−	0.114881i
\(7\)	2.15806	−	1.53061i	0.815671	−	0.578516i
							\(11\)	−1.60977	+	3.61559i	−0.485363	+	1.09014i	0.490437	+	0.871477i	\(0.336837\pi\)
−0.975800	+	0.218666i	\(0.929830\pi\)
\(13\)	−0.929070	+	0.147150i	−0.257678	+	0.0408121i	−0.283937	−	0.958843i	\(-0.591641\pi\)
							0.0262590	+	0.999655i	\(0.491641\pi\)
\(17\)	0.277086	+	5.28711i	0.0672032	+	1.28231i	0.799225	+	0.601032i	\(0.205243\pi\)
							−0.732022	+	0.681281i	\(0.761423\pi\)
\(19\)	−0.884917	+	4.16321i	−0.203014	+	0.955105i	0.752142	+	0.659001i	\(0.229021\pi\)
							−0.955156	+	0.296104i	\(0.904313\pi\)
\(23\)	−5.01083	+	6.18786i	−1.04483	+	1.29026i	−0.0890963	+	0.996023i	\(0.528398\pi\)
							−0.955733	+	0.294234i	\(0.904935\pi\)
\(29\)	1.37710	−	4.23829i	0.255722	−	0.787031i	−0.737965	−	0.674839i	\(-0.764213\pi\)
							0.993687	−	0.112192i	\(-0.0357870\pi\)
\(31\)	−4.25643	+	4.72724i	−0.764477	+	0.849038i	−0.992196	−	0.124689i	\(-0.960207\pi\)
							0.227719	+	0.973727i	\(0.426873\pi\)
\(37\)	0.319026	+	0.122463i	0.0524476	+	0.0201327i	0.384450	−	0.923146i	\(-0.374391\pi\)
							−0.332003	+	0.943278i	\(0.607724\pi\)
\(41\)	−3.61589	+	4.97685i	−0.564707	+	0.777253i	−0.991915	−	0.126900i	\(-0.959497\pi\)
							0.427208	+	0.904153i	\(0.359497\pi\)
\(43\)	−5.96034	+	5.96034i	−0.908944	+	0.908944i	−0.996187	−	0.0872430i	\(-0.972194\pi\)
							0.0872430	+	0.996187i	\(0.472194\pi\)
\(47\)	5.76838	+	0.302308i	0.841405	+	0.0440962i	0.468168	−	0.883640i	\(-0.344914\pi\)
							0.373238	+	0.927736i	\(0.378248\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bs.a.2.3	✓	1216
3.2	odd	2	inner	525.2.bs.a.2.74	yes	1216
7.4	even	3	inner	525.2.bs.a.452.74	yes	1216
21.11	odd	6	inner	525.2.bs.a.452.3	yes	1216
25.13	odd	20	inner	525.2.bs.a.338.3	yes	1216
75.38	even	20	inner	525.2.bs.a.338.74	yes	1216
175.88	odd	60	inner	525.2.bs.a.263.74	yes	1216
525.263	even	60	inner	525.2.bs.a.263.3	yes	1216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bs.a.2.3	✓	1216	1.1	even	1	trivial
525.2.bs.a.2.74	yes	1216	3.2	odd	2	inner
525.2.bs.a.263.3	yes	1216	525.263	even	60	inner
525.2.bs.a.263.74	yes	1216	175.88	odd	60	inner
525.2.bs.a.338.3	yes	1216	25.13	odd	20	inner
525.2.bs.a.338.74	yes	1216	75.38	even	20	inner
525.2.bs.a.452.3	yes	1216	21.11	odd	6	inner
525.2.bs.a.452.74	yes	1216	7.4	even	3	inner