Embedded newform 450.2.v.a.319.2

• Label: 450.2.v.a.319.2
• Level: $450$
• Weight: $2$
• Character: 450.319
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.v (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59326809096\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			319.2
Character	\(\chi\)	\(=\)	450.319
Dual form			450.2.v.a.79.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743145 - 0.669131i) q^{2} +(-1.53471 + 0.802914i) q^{3} +(0.104528 + 0.994522i) q^{4} +(1.99429 - 1.01134i) q^{5} +(1.67776 + 0.430239i) q^{6} +(1.01614 - 0.586669i) q^{7} +(0.587785 - 0.809017i) q^{8} +(1.71066 - 2.46448i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.743145	−	0.669131i	−0.525483	−	0.473147i
							\(3\)	−1.53471	+	0.802914i	−0.886064	+	0.463562i
\(5\)	1.99429	−	1.01134i	0.891874	−	0.452284i
							\(7\)	1.01614	−	0.586669i	0.384065	−	0.221740i	−0.295520	−	0.955336i	\(-0.595493\pi\)
0.679586	+	0.733596i	\(0.262160\pi\)
\(11\)	−4.11667	+	4.57203i	−1.24122	+	1.37852i	−0.342788	+	0.939413i	\(0.611371\pi\)
							−0.898435	+	0.439106i	\(0.855295\pi\)
\(13\)	2.61200	−	2.35186i	0.724439	−	0.652288i	−0.222046	−	0.975036i	\(-0.571273\pi\)
							0.946485	+	0.322748i	\(0.104607\pi\)
\(17\)	0.704605	−	0.969806i	0.170892	−	0.235213i	−0.714977	−	0.699148i	\(-0.753563\pi\)
							0.885869	+	0.463935i	\(0.153563\pi\)
\(19\)	1.98402	+	1.44148i	0.455166	+	0.330697i	0.791632	−	0.610998i	\(-0.209232\pi\)
							−0.336466	+	0.941696i	\(0.609232\pi\)
\(23\)	0.953457	−	4.48566i	0.198809	−	0.935325i	−0.759702	−	0.650271i	\(-0.774655\pi\)
							0.958512	−	0.285054i	\(-0.0920115\pi\)
\(29\)	5.63103	+	2.50710i	1.04566	+	0.465556i	0.856368	−	0.516365i	\(-0.172715\pi\)
							0.189288	+	0.981922i	\(0.439382\pi\)
\(31\)	5.04485	−	2.24611i	0.906082	−	0.403414i	0.0998430	−	0.995003i	\(-0.468166\pi\)
							0.806239	+	0.591589i	\(0.201499\pi\)
\(37\)	9.29901	+	3.02143i	1.52875	+	0.496720i	0.948245	−	0.317538i	\(-0.102856\pi\)
							0.580502	+	0.814259i	\(0.302856\pi\)
\(41\)	−2.09892	−	2.33109i	−0.327796	−	0.364054i	0.556609	−	0.830775i	\(-0.312102\pi\)
							−0.884405	+	0.466720i	\(0.845436\pi\)
\(43\)	8.90899	−	5.14361i	1.35861	−	0.784393i	0.369172	−	0.929361i	\(-0.379641\pi\)
							0.989436	+	0.144968i	\(0.0463079\pi\)
\(47\)	1.54947	−	3.48017i	0.226014	−	0.507635i	−0.764571	−	0.644540i	\(-0.777049\pi\)
							0.990584	+	0.136905i	\(0.0437156\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.319.2	yes	240
9.7	even	3	inner	450.2.v.a.169.12	yes	240
25.4	even	10	inner	450.2.v.a.229.12	yes	240
225.79	even	30	inner	450.2.v.a.79.2	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.2	✓	240	225.79	even	30	inner
450.2.v.a.169.12	yes	240	9.7	even	3	inner
450.2.v.a.229.12	yes	240	25.4	even	10	inner
450.2.v.a.319.2	yes	240	1.1	even	1	trivial