Embedded newform 525.2.bf.g.32.15

• Label: 525.2.bf.g.32.15
• Level: $525$
• Weight: $2$
• Character: 525.32
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(20\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			32.15
Character	\(\chi\)	\(=\)	525.32
Dual form			525.2.bf.g.443.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.78672 + 0.478751i) q^{2} +(-1.10735 - 1.33183i) q^{3} +(1.23113 + 0.710791i) q^{4} +(-1.34091 - 2.90976i) q^{6} +(0.585067 - 2.58025i) q^{7} +(-0.756555 - 0.756555i) q^{8} +(-0.547566 + 2.94961i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 16 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.78672	+	0.478751i	1.26340	+	0.338528i	0.827500	−	0.561465i	\(-0.189762\pi\)
							0.435904	+	0.899993i	\(0.356429\pi\)
\(3\)	−1.10735	−	1.33183i	−0.639327	−	0.768935i
							\(5\)		0			0
\(7\)	0.585067	−	2.58025i	0.221134	−	0.975243i
							\(11\)	−4.61054	−	2.66189i	−1.39013	−	0.802591i	−0.396800	−	0.917905i	\(-0.629879\pi\)
−0.993329	+	0.115314i	\(0.963213\pi\)
\(13\)	3.44938	−	3.44938i	0.956686	−	0.956686i	−0.0424138	−	0.999100i	\(-0.513505\pi\)
							0.999100	+	0.0424138i	\(0.0135048\pi\)
\(17\)	1.13091	+	4.22062i	0.274286	+	1.02365i	0.956318	+	0.292328i	\(0.0944297\pi\)
							−0.682032	+	0.731322i	\(0.738904\pi\)
\(19\)	2.14466	−	1.23822i	0.492018	−	0.284067i	−0.233393	−	0.972382i	\(-0.574983\pi\)
							0.725411	+	0.688316i	\(0.241650\pi\)
\(23\)	0.533436	−	1.99081i	0.111229	−	0.415113i	−0.887748	−	0.460330i	\(-0.847731\pi\)
							0.998977	+	0.0452168i	\(0.0143979\pi\)
\(29\)	−2.05014			−0.380701			−0.190350	−	0.981716i	\(-0.560962\pi\)
							−0.190350	+	0.981716i	\(0.560962\pi\)
\(31\)	1.18652	−	2.05512i	0.213106	−	0.369110i	−0.739579	−	0.673069i	\(-0.764975\pi\)
							0.952685	+	0.303960i	\(0.0983088\pi\)
\(37\)	−1.26256	+	4.71194i	−0.207564	+	0.774639i	0.781089	+	0.624420i	\(0.214665\pi\)
							−0.988653	+	0.150219i	\(0.952002\pi\)
\(41\)			4.06556i			0.634934i	0.948269	+	0.317467i	\(0.102832\pi\)
							−0.948269	+	0.317467i	\(0.897168\pi\)
\(43\)	0.254109	−	0.254109i	0.0387513	−	0.0387513i	−0.687466	−	0.726217i	\(-0.741277\pi\)
							0.726217	+	0.687466i	\(0.241277\pi\)
\(47\)	8.31648	+	2.22839i	1.21308	+	0.325045i	0.807971	−	0.589222i	\(-0.200566\pi\)
							0.405112	+	0.914267i	\(0.367232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bf.g.32.15	yes	80
3.2	odd	2	inner	525.2.bf.g.32.5	✓	80
5.2	odd	4	inner	525.2.bf.g.368.5	yes	80
5.3	odd	4	inner	525.2.bf.g.368.16	yes	80
5.4	even	2	inner	525.2.bf.g.32.6	yes	80
7.2	even	3	inner	525.2.bf.g.107.6	yes	80
15.2	even	4	inner	525.2.bf.g.368.15	yes	80
15.8	even	4	inner	525.2.bf.g.368.6	yes	80
15.14	odd	2	inner	525.2.bf.g.32.16	yes	80
21.2	odd	6	inner	525.2.bf.g.107.16	yes	80
35.2	odd	12	inner	525.2.bf.g.443.16	yes	80
35.9	even	6	inner	525.2.bf.g.107.15	yes	80
35.23	odd	12	inner	525.2.bf.g.443.5	yes	80
105.2	even	12	inner	525.2.bf.g.443.6	yes	80
105.23	even	12	inner	525.2.bf.g.443.15	yes	80
105.44	odd	6	inner	525.2.bf.g.107.5	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bf.g.32.5	✓	80	3.2	odd	2	inner
525.2.bf.g.32.6	yes	80	5.4	even	2	inner
525.2.bf.g.32.15	yes	80	1.1	even	1	trivial
525.2.bf.g.32.16	yes	80	15.14	odd	2	inner
525.2.bf.g.107.5	yes	80	105.44	odd	6	inner
525.2.bf.g.107.6	yes	80	7.2	even	3	inner
525.2.bf.g.107.15	yes	80	35.9	even	6	inner
525.2.bf.g.107.16	yes	80	21.2	odd	6	inner
525.2.bf.g.368.5	yes	80	5.2	odd	4	inner
525.2.bf.g.368.6	yes	80	15.8	even	4	inner
525.2.bf.g.368.15	yes	80	15.2	even	4	inner
525.2.bf.g.368.16	yes	80	5.3	odd	4	inner
525.2.bf.g.443.5	yes	80	35.23	odd	12	inner
525.2.bf.g.443.6	yes	80	105.2	even	12	inner
525.2.bf.g.443.15	yes	80	105.23	even	12	inner
525.2.bf.g.443.16	yes	80	35.2	odd	12	inner