Embedded newform 450.2.q.c.61.7

• Label: 450.2.q.c.61.7
• Level: $450$
• Weight: $2$
• Character: 450.61
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	450.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59326809096\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			61.7
Character	\(\chi\)	\(=\)	450.61
Dual form			450.2.q.c.391.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.994522i) q^{2} +(0.0496062 + 1.73134i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(2.00551 - 0.988911i) q^{5} +(-1.71667 + 0.230309i) q^{6} +(0.476155 - 0.824725i) q^{7} +(-0.309017 - 0.951057i) q^{8} +(-2.99508 + 0.171770i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.104528	+	0.994522i	0.0739128	+	0.703233i
							\(3\)	0.0496062	+	1.73134i	0.0286401	+	0.999590i
\(5\)	2.00551	−	0.988911i	0.896890	−	0.442255i
							\(7\)	0.476155	−	0.824725i	0.179970	−	0.311717i	−0.761900	−	0.647694i	\(-0.775733\pi\)
0.941870	+	0.335978i	\(0.109067\pi\)
\(11\)	0.274950	+	2.61597i	0.0829005	+	0.788745i	0.954437	+	0.298414i	\(0.0964576\pi\)
							−0.871536	+	0.490331i	\(0.836876\pi\)
\(13\)	−0.702834	+	6.68702i	−0.194931	+	1.85465i	0.261903	+	0.965094i	\(0.415650\pi\)
							−0.456834	+	0.889552i	\(0.651017\pi\)
\(17\)	1.79610	+	5.52784i	0.435619	+	1.34070i	0.892451	+	0.451145i	\(0.148984\pi\)
							−0.456831	+	0.889553i	\(0.651016\pi\)
\(19\)	−0.0188855	−	0.0581237i	−0.00433264	−	0.0133345i	0.948867	−	0.315677i	\(-0.102231\pi\)
							−0.953199	+	0.302342i	\(0.902231\pi\)
\(23\)	−2.02271	−	0.900568i	−0.421764	−	0.187781i	0.184870	−	0.982763i	\(-0.440814\pi\)
							−0.606633	+	0.794982i	\(0.707480\pi\)
\(29\)	−5.25885	−	5.84055i	−0.976544	−	1.08456i	−0.996401	−	0.0847592i	\(-0.972988\pi\)
							0.0198574	−	0.999803i	\(-0.493679\pi\)
\(31\)	3.19590	−	3.54941i	0.574001	−	0.637493i	−0.384315	−	0.923202i	\(-0.625562\pi\)
							0.958316	+	0.285709i	\(0.0922291\pi\)
\(37\)	6.28202	+	4.56415i	1.03276	+	0.750342i	0.968859	−	0.247615i	\(-0.0796467\pi\)
							0.0638984	+	0.997956i	\(0.479647\pi\)
\(41\)	−0.479577	+	4.56287i	−0.0748973	+	0.712600i	0.891058	+	0.453890i	\(0.149964\pi\)
							−0.965955	+	0.258710i	\(0.916703\pi\)
\(43\)	4.60441	−	7.97507i	0.702166	−	1.21619i	−0.265539	−	0.964100i	\(-0.585550\pi\)
							0.967705	−	0.252086i	\(-0.0811167\pi\)
\(47\)	−1.80893	−	2.00902i	−0.263860	−	0.293046i	0.596627	−	0.802519i	\(-0.296507\pi\)
							−0.860487	+	0.509473i	\(0.829840\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.c.61.7	✓	120
9.4	even	3	inner	450.2.q.c.211.5	yes	120
25.16	even	5	inner	450.2.q.c.241.5	yes	120
225.166	even	15	inner	450.2.q.c.391.7	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.c.61.7	✓	120	1.1	even	1	trivial
450.2.q.c.211.5	yes	120	9.4	even	3	inner
450.2.q.c.241.5	yes	120	25.16	even	5	inner
450.2.q.c.391.7	yes	120	225.166	even	15	inner