Embedded newform 450.2.v.a.319.27

• Label: 450.2.v.a.319.27
• 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			319.27
Character	\(\chi\)	\(=\)	450.319
Dual form			450.2.v.a.79.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.743145 + 0.669131i) q^{2} +(1.29047 + 1.15528i) q^{3} +(0.104528 + 0.994522i) q^{4} +(1.39024 - 1.75135i) q^{5} +(0.185973 + 1.72204i) q^{6} +(0.451321 - 0.260570i) q^{7} +(-0.587785 + 0.809017i) q^{8} +(0.330640 + 2.98172i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	1.29047	+	1.15528i	0.745055	+	0.667003i
\(5\)	1.39024	−	1.75135i	0.621734	−	0.783229i
							\(7\)	0.451321	−	0.260570i	0.170583	−	0.0984863i	−0.412278	−	0.911058i	\(-0.635267\pi\)
0.582861	+	0.812572i	\(0.301933\pi\)
\(11\)	−0.961757	+	1.06814i	−0.289981	+	0.322056i	−0.870479	−	0.492205i	\(-0.836191\pi\)
							0.580499	+	0.814261i	\(0.302858\pi\)
\(13\)	1.22089	−	1.09929i	0.338613	−	0.304889i	−0.482227	−	0.876046i	\(-0.660172\pi\)
							0.820840	+	0.571158i	\(0.193505\pi\)
\(17\)	−0.973546	+	1.33997i	−0.236120	+	0.324991i	−0.910590	−	0.413311i	\(-0.864372\pi\)
							0.674470	+	0.738302i	\(0.264372\pi\)
\(19\)	1.05798	+	0.768670i	0.242718	+	0.176345i	0.702493	−	0.711690i	\(-0.252070\pi\)
							−0.459775	+	0.888035i	\(0.652070\pi\)
\(23\)	0.705240	−	3.31789i	0.147053	−	0.691829i	−0.841414	−	0.540391i	\(-0.818276\pi\)
							0.988467	−	0.151438i	\(-0.0483903\pi\)
\(29\)	−0.514922	−	0.229258i	−0.0956186	−	0.0425721i	0.358370	−	0.933580i	\(-0.383333\pi\)
							−0.453989	+	0.891008i	\(0.649999\pi\)
\(31\)	0.186285	−	0.0829396i	0.0334578	−	0.0148964i	−0.389940	−	0.920840i	\(-0.627504\pi\)
							0.423397	+	0.905944i	\(0.360837\pi\)
\(37\)	−1.59731	−	0.518999i	−0.262597	−	0.0853228i	0.174760	−	0.984611i	\(-0.444085\pi\)
							−0.437356	+	0.899288i	\(0.644085\pi\)
\(41\)	−7.48172	−	8.30929i	−1.16845	−	1.29769i	−0.946525	−	0.322629i	\(-0.895433\pi\)
							−0.221923	−	0.975064i	\(-0.571233\pi\)
\(43\)	0.0631205	−	0.0364426i	0.00962578	−	0.00555745i	−0.495179	−	0.868791i	\(-0.664898\pi\)
							0.504805	+	0.863233i	\(0.331564\pi\)
\(47\)	−3.96138	+	8.89740i	−0.577827	+	1.29782i	0.354112	+	0.935203i	\(0.384783\pi\)
							−0.931938	+	0.362617i	\(0.881883\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.319.27	yes	240
9.7	even	3	inner	450.2.v.a.169.25	yes	240
25.4	even	10	inner	450.2.v.a.229.25	yes	240
225.79	even	30	inner	450.2.v.a.79.27	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.27	✓	240	225.79	even	30	inner
450.2.v.a.169.25	yes	240	9.7	even	3	inner
450.2.v.a.229.25	yes	240	25.4	even	10	inner
450.2.v.a.319.27	yes	240	1.1	even	1	trivial