Embedded newform 450.2.v.a.319.29

• Label: 450.2.v.a.319.29
• 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			319.29
Character	\(\chi\)	\(=\)	450.319
Dual form			450.2.v.a.79.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.743145 + 0.669131i) q^{2} +(1.71540 + 0.239586i) q^{3} +(0.104528 + 0.994522i) q^{4} +(-1.65115 + 1.50788i) q^{5} +(1.11448 + 1.32587i) q^{6} +(-2.80004 + 1.61661i) q^{7} +(-0.587785 + 0.809017i) q^{8} +(2.88520 + 0.821973i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	1.71540	+	0.239586i	0.990387	+	0.138325i
\(5\)	−1.65115	+	1.50788i	−0.738415	+	0.674346i
							\(7\)	−2.80004	+	1.61661i	−1.05832	+	0.611020i	−0.924966	−	0.380049i	\(-0.875907\pi\)
−0.133351	+	0.991069i	\(0.542574\pi\)
\(11\)	−1.71747	+	1.90744i	−0.517837	+	0.575116i	−0.944174	−	0.329448i	\(-0.893137\pi\)
							0.426337	+	0.904565i	\(0.359804\pi\)
\(13\)	−1.49301	+	1.34431i	−0.414086	+	0.372845i	−0.849680	−	0.527299i	\(-0.823205\pi\)
							0.435594	+	0.900143i	\(0.356538\pi\)
\(17\)	4.08061	−	5.61648i	0.989694	−	1.36220i	0.0582545	−	0.998302i	\(-0.481447\pi\)
							0.931440	−	0.363896i	\(-0.118553\pi\)
\(19\)	6.00863	+	4.36552i	1.37847	+	1.00152i	0.997021	+	0.0771314i	\(0.0245761\pi\)
							0.381453	+	0.924388i	\(0.375424\pi\)
\(23\)	0.496572	−	2.33619i	0.103542	−	0.487129i	−0.895567	−	0.444927i	\(-0.853230\pi\)
							0.999109	−	0.0422014i	\(-0.0134371\pi\)
\(29\)	1.79144	+	0.797600i	0.332662	+	0.148111i	0.566269	−	0.824221i	\(-0.308386\pi\)
							−0.233607	+	0.972331i	\(0.575053\pi\)
\(31\)	0.0214521	−	0.00955109i	0.00385291	−	0.00171543i	−0.404809	−	0.914401i	\(-0.632662\pi\)
							0.408662	+	0.912686i	\(0.365995\pi\)
\(37\)	6.63191	+	2.15484i	1.09028	+	0.354253i	0.798354	−	0.602188i	\(-0.205704\pi\)
							0.291925	+	0.956441i	\(0.405704\pi\)
\(41\)	−2.50176	−	2.77848i	−0.390709	−	0.433926i	0.515414	−	0.856941i	\(-0.327638\pi\)
							−0.906122	+	0.423015i	\(0.860972\pi\)
\(43\)	4.51112	−	2.60449i	0.687939	−	0.397182i	−0.114900	−	0.993377i	\(-0.536655\pi\)
							0.802839	+	0.596195i	\(0.203322\pi\)
\(47\)	1.96773	−	4.41958i	0.287022	−	0.644663i	−0.711280	−	0.702909i	\(-0.751884\pi\)
							0.998302	+	0.0582462i	\(0.0185508\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.319.29	yes	240
9.7	even	3	inner	450.2.v.a.169.23	yes	240
25.4	even	10	inner	450.2.v.a.229.23	yes	240
225.79	even	30	inner	450.2.v.a.79.29	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.29	✓	240	225.79	even	30	inner
450.2.v.a.169.23	yes	240	9.7	even	3	inner
450.2.v.a.229.23	yes	240	25.4	even	10	inner
450.2.v.a.319.29	yes	240	1.1	even	1	trivial