Embedded newform 450.2.v.a.319.28

• Label: 450.2.v.a.319.28
• 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			319.28
Character	\(\chi\)	\(=\)	450.319
Dual form			450.2.v.a.79.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.743145 + 0.669131i) q^{2} +(1.67230 - 0.451013i) q^{3} +(0.104528 + 0.994522i) q^{4} +(2.14390 + 0.635362i) q^{5} +(1.54455 + 0.783819i) q^{6} +(-2.49058 + 1.43794i) q^{7} +(-0.587785 + 0.809017i) q^{8} +(2.59317 - 1.50846i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	1.67230	−	0.451013i	0.965503	−	0.260393i
\(5\)	2.14390	+	0.635362i	0.958782	+	0.284143i
							\(7\)	−2.49058	+	1.43794i	−0.941350	+	0.543489i	−0.890383	−	0.455211i	\(-0.849564\pi\)
−0.0509670	+	0.998700i	\(0.516230\pi\)
\(11\)	3.62627	−	4.02738i	1.09336	−	1.21430i	0.118158	−	0.992995i	\(-0.462301\pi\)
							0.975204	−	0.221307i	\(-0.0710324\pi\)
\(13\)	−1.92594	+	1.73413i	−0.534160	+	0.480960i	−0.891504	−	0.453013i	\(-0.850349\pi\)
							0.357344	+	0.933973i	\(0.383682\pi\)
\(17\)	−1.80748	+	2.48778i	−0.438378	+	0.603376i	−0.969851	−	0.243699i	\(-0.921639\pi\)
							0.531472	+	0.847076i	\(0.321639\pi\)
\(19\)	−4.37513	−	3.17872i	−1.00372	−	0.729247i	−0.0408396	−	0.999166i	\(-0.513003\pi\)
							−0.962883	+	0.269918i	\(0.913003\pi\)
\(23\)	−1.05825	+	4.97867i	−0.220660	+	1.03813i	0.718734	+	0.695286i	\(0.244722\pi\)
							−0.939394	+	0.342840i	\(0.888611\pi\)
\(29\)	−7.34106	−	3.26845i	−1.36320	−	0.606936i	−0.410783	−	0.911733i	\(-0.634745\pi\)
							−0.952417	+	0.304797i	\(0.901411\pi\)
\(31\)	9.17749	−	4.08608i	1.64833	−	0.733882i	0.648695	−	0.761049i	\(-0.275315\pi\)
							0.999631	+	0.0271667i	\(0.00864849\pi\)
\(37\)	−5.78953	−	1.88113i	−0.951792	−	0.309256i	−0.208349	−	0.978055i	\(-0.566809\pi\)
							−0.743444	+	0.668799i	\(0.766809\pi\)
\(41\)	−1.46891	−	1.63139i	−0.229405	−	0.254780i	0.617442	−	0.786616i	\(-0.288169\pi\)
							−0.846847	+	0.531836i	\(0.821502\pi\)
\(43\)	5.97220	−	3.44805i	0.910752	−	0.525823i	0.0300792	−	0.999548i	\(-0.490424\pi\)
							0.880673	+	0.473724i	\(0.157091\pi\)
\(47\)	−0.616643	+	1.38500i	−0.0899467	+	0.202024i	−0.952917	−	0.303231i	\(-0.901935\pi\)
							0.862970	+	0.505254i	\(0.168601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.319.28	yes	240
9.7	even	3	inner	450.2.v.a.169.20	yes	240
25.4	even	10	inner	450.2.v.a.229.20	yes	240
225.79	even	30	inner	450.2.v.a.79.28	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.28	✓	240	225.79	even	30	inner
450.2.v.a.169.20	yes	240	9.7	even	3	inner
450.2.v.a.229.20	yes	240	25.4	even	10	inner
450.2.v.a.319.28	yes	240	1.1	even	1	trivial