Embedded newform 525.2.bo.a.319.40

• Label: 525.2.bo.a.319.40
• Level: $525$
• Weight: $2$
• Character: 525.319
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bo (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(40\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			319.40
Character	\(\chi\)	\(=\)	525.319
Dual form			525.2.bo.a.79.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00195 + 1.80257i) q^{2} +(0.994522 - 0.104528i) q^{3} +(0.549512 + 5.22826i) q^{4} +(-2.14439 + 0.633710i) q^{5} +(2.17940 + 1.58343i) q^{6} +(2.34235 - 1.23020i) q^{7} +(-5.15732 + 7.09845i) q^{8} +(0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{4} - 2 q^{5} + 8 q^{6} + 60 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00195	+	1.80257i	1.41559	+	1.27461i	0.911709	+	0.410836i	\(0.134763\pi\)
							0.503885	+	0.863771i	\(0.331904\pi\)
\(3\)	0.994522	−	0.104528i	0.574187	−	0.0603495i
							\(5\)	−2.14439	+	0.633710i	−0.959001	+	0.283404i
\(7\)	2.34235	−	1.23020i	0.885325	−	0.464973i
							\(11\)	3.71800	+	0.790286i	1.12102	+	0.238280i	0.730900	−	0.682484i	\(-0.239100\pi\)
0.390119	+	0.920764i	\(0.372434\pi\)
\(13\)	−5.67235	−	1.84306i	−1.57323	−	0.511172i	−0.612926	−	0.790140i	\(-0.710008\pi\)
							−0.960300	+	0.278968i	\(0.910008\pi\)
\(17\)	1.31020	+	2.94276i	0.317770	+	0.713724i	0.999846	−	0.0175579i	\(-0.00558914\pi\)
							−0.682076	+	0.731282i	\(0.738922\pi\)
\(19\)	0.249541	−	2.37423i	0.0572487	−	0.544685i	−0.927882	−	0.372875i	\(-0.878372\pi\)
							0.985130	−	0.171809i	\(-0.0549614\pi\)
\(23\)	−1.57767	−	1.42054i	−0.328966	−	0.296203i	0.488052	−	0.872815i	\(-0.337708\pi\)
							−0.817018	+	0.576612i	\(0.804374\pi\)
\(29\)	5.40259	−	3.92521i	1.00323	−	0.728893i	0.0404555	−	0.999181i	\(-0.487119\pi\)
							0.962779	+	0.270289i	\(0.0871191\pi\)
\(31\)	−4.83356	+	2.15204i	−0.868134	+	0.386518i	−0.791957	−	0.610576i	\(-0.790938\pi\)
							−0.0761763	+	0.997094i	\(0.524271\pi\)
\(37\)	−0.522636	−	2.45881i	−0.0859208	−	0.404226i	0.914078	−	0.405538i	\(-0.132916\pi\)
							−0.999999	+	0.00131202i	\(0.999582\pi\)
\(41\)	3.44611	−	10.6060i	0.538192	−	1.65638i	−0.198457	−	0.980110i	\(-0.563593\pi\)
							0.736649	−	0.676275i	\(-0.236407\pi\)
\(43\)			1.23974i			0.189059i	0.995522	+	0.0945293i	\(0.0301346\pi\)
							−0.995522	+	0.0945293i	\(0.969865\pi\)
\(47\)	−0.0912793	+	0.205017i	−0.0133145	+	0.0299048i	−0.920080	−	0.391730i	\(-0.871877\pi\)
							0.906766	+	0.421635i	\(0.138543\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bo.a.319.40	yes	320
7.2	even	3	inner	525.2.bo.a.394.40	yes	320
25.4	even	10	inner	525.2.bo.a.4.40	✓	320
175.79	even	30	inner	525.2.bo.a.79.40	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bo.a.4.40	✓	320	25.4	even	10	inner
525.2.bo.a.79.40	yes	320	175.79	even	30	inner
525.2.bo.a.319.40	yes	320	1.1	even	1	trivial
525.2.bo.a.394.40	yes	320	7.2	even	3	inner