Embedded newform 525.2.bo.a.319.30

• Label: 525.2.bo.a.319.30
• 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			319.30
Character	\(\chi\)	\(=\)	525.319
Dual form			525.2.bo.a.79.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.07234 + 0.965540i) q^{2} +(0.994522 - 0.104528i) q^{3} +(0.00859042 + 0.0817324i) q^{4} +(1.18152 + 1.89842i) q^{5} +(1.16739 + 0.848161i) q^{6} +(0.513464 - 2.59545i) q^{7} +(1.62662 - 2.23885i) q^{8} +(0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{4} - 2 q^{5} + 8 q^{6} + 60 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.07234	+	0.965540i	0.758260	+	0.682740i	0.954632	−	0.297790i	\(-0.0962494\pi\)
							−0.196372	+	0.980529i	\(0.562916\pi\)
\(3\)	0.994522	−	0.104528i	0.574187	−	0.0603495i
							\(5\)	1.18152	+	1.89842i	0.528392	+	0.849000i
\(7\)	0.513464	−	2.59545i	0.194071	−	0.980987i
							\(11\)	3.57015	+	0.758859i	1.07644	+	0.228805i	0.711828	−	0.702354i	\(-0.247868\pi\)
0.364614	+	0.931159i	\(0.381201\pi\)
\(13\)	−5.06979	−	1.64727i	−1.40611	−	0.456872i	−0.494946	−	0.868924i	\(-0.664812\pi\)
							−0.911160	+	0.412052i	\(0.864812\pi\)
\(17\)	−1.82786	−	4.10544i	−0.443321	−	0.995715i	−0.987626	−	0.156828i	\(-0.949873\pi\)
							0.544305	−	0.838887i	\(-0.316793\pi\)
\(19\)	−0.833854	+	7.93359i	−0.191299	+	1.82009i	0.305463	+	0.952204i	\(0.401189\pi\)
							−0.496763	+	0.867886i	\(0.665478\pi\)
\(23\)	1.04733	+	0.943016i	0.218382	+	0.196632i	0.771060	−	0.636763i	\(-0.219727\pi\)
							−0.552677	+	0.833395i	\(0.686394\pi\)
\(29\)	−6.30305	+	4.57943i	−1.17045	+	0.850379i	−0.991062	−	0.133400i	\(-0.957411\pi\)
							−0.179384	+	0.983779i	\(0.557411\pi\)
\(31\)	1.29026	−	0.574459i	0.231737	−	0.103176i	−0.287583	−	0.957756i	\(-0.592852\pi\)
							0.519320	+	0.854580i	\(0.326185\pi\)
\(37\)	1.14185	+	5.37200i	0.187720	+	0.883152i	0.966662	+	0.256056i	\(0.0824232\pi\)
							−0.778942	+	0.627096i	\(0.784243\pi\)
\(41\)	−0.420217	+	1.29329i	−0.0656268	+	0.201979i	−0.978493	−	0.206281i	\(-0.933864\pi\)
							0.912866	+	0.408259i	\(0.133864\pi\)
\(43\)		−	4.76617i		−	0.726835i	−0.931626	−	0.363417i	\(-0.881610\pi\)
							0.931626	−	0.363417i	\(-0.118390\pi\)
\(47\)	−2.15446	+	4.83899i	−0.314260	+	0.705840i	−0.999753	−	0.0222400i	\(-0.992920\pi\)
							0.685493	+	0.728080i	\(0.259587\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bo.a.319.30	yes	320
7.2	even	3	inner	525.2.bo.a.394.30	yes	320
25.4	even	10	inner	525.2.bo.a.4.30	✓	320
175.79	even	30	inner	525.2.bo.a.79.30	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bo.a.4.30	✓	320	25.4	even	10	inner
525.2.bo.a.79.30	yes	320	175.79	even	30	inner
525.2.bo.a.319.30	yes	320	1.1	even	1	trivial
525.2.bo.a.394.30	yes	320	7.2	even	3	inner