Embedded newform 450.2.v.a.319.1

• Label: 450.2.v.a.319.1
• 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			319.1
Character	\(\chi\)	\(=\)	450.319
Dual form			450.2.v.a.79.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743145 - 0.669131i) q^{2} +(-1.69642 + 0.349507i) q^{3} +(0.104528 + 0.994522i) q^{4} +(0.950392 + 2.02404i) q^{5} +(1.49455 + 0.875393i) q^{6} +(-0.590039 + 0.340659i) q^{7} +(0.587785 - 0.809017i) q^{8} +(2.75569 - 1.18582i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.69642	+	0.349507i	−0.979429	+	0.201788i
\(5\)	0.950392	+	2.02404i	0.425028	+	0.905180i
							\(7\)	−0.590039	+	0.340659i	−0.223014	+	0.128757i	−0.607345	−	0.794438i	\(-0.707765\pi\)
0.384331	+	0.923195i	\(0.374432\pi\)
\(11\)	1.29308	−	1.43611i	0.389877	−	0.433003i	−0.515970	−	0.856607i	\(-0.672568\pi\)
							0.905848	+	0.423604i	\(0.139235\pi\)
\(13\)	−3.42751	+	3.08614i	−0.950621	+	0.855943i	−0.989680	−	0.143294i	\(-0.954230\pi\)
							0.0390596	+	0.999237i	\(0.487564\pi\)
\(17\)	−0.520172	+	0.715956i	−0.126160	+	0.173645i	−0.867425	−	0.497568i	\(-0.834226\pi\)
							0.741265	+	0.671213i	\(0.234226\pi\)
\(19\)	−2.48455	−	1.80513i	−0.569994	−	0.414125i	0.265109	−	0.964218i	\(-0.414592\pi\)
							−0.835103	+	0.550094i	\(0.814592\pi\)
\(23\)	−1.03795	+	4.88317i	−0.216428	+	1.01821i	0.727001	+	0.686636i	\(0.240914\pi\)
							−0.943429	+	0.331576i	\(0.892420\pi\)
\(29\)	0.725967	+	0.323221i	0.134809	+	0.0600207i	0.473032	−	0.881045i	\(-0.343159\pi\)
							−0.338224	+	0.941066i	\(0.609826\pi\)
\(31\)	−5.66755	+	2.52335i	−1.01792	+	0.453208i	−0.846726	−	0.532029i	\(-0.821430\pi\)
							−0.171195	+	0.985237i	\(0.554763\pi\)
\(37\)	0.232625	+	0.0755845i	0.0382433	+	0.0124260i	0.328076	−	0.944651i	\(-0.393600\pi\)
							−0.289833	+	0.957077i	\(0.593600\pi\)
\(41\)	4.34301	+	4.82341i	0.678265	+	0.753290i	0.979759	−	0.200178i	\(-0.0641522\pi\)
							−0.301494	+	0.953468i	\(0.597485\pi\)
\(43\)	−7.14949	+	4.12776i	−1.09029	+	0.629477i	−0.933653	−	0.358179i	\(-0.883398\pi\)
							−0.156634	+	0.987657i	\(0.550064\pi\)
\(47\)	1.95384	−	4.38840i	0.284997	−	0.640114i	−0.713147	−	0.701015i	\(-0.752731\pi\)
							0.998144	+	0.0609010i	\(0.0193974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.319.1	yes	240
9.7	even	3	inner	450.2.v.a.169.11	yes	240
25.4	even	10	inner	450.2.v.a.229.11	yes	240
225.79	even	30	inner	450.2.v.a.79.1	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.1	✓	240	225.79	even	30	inner
450.2.v.a.169.11	yes	240	9.7	even	3	inner
450.2.v.a.229.11	yes	240	25.4	even	10	inner
450.2.v.a.319.1	yes	240	1.1	even	1	trivial