Embedded newform 450.2.q.b.331.14

• Label: 450.2.q.b.331.14
• Level: $450$
• Weight: $2$
• Character: 450.331
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $112$
• 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:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			331.14
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.b.121.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.978148 - 0.207912i) q^{2} +(1.70739 + 0.291246i) q^{3} +(0.913545 + 0.406737i) q^{4} +(0.742996 + 2.10902i) q^{5} +(-1.60952 - 0.639868i) q^{6} +(-1.24661 + 2.15919i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(2.83035 + 0.994541i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{2} + 2 q^{3} + 14 q^{4} + 9 q^{5} - 7 q^{6} - 8 q^{7} - 28 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.978148	−	0.207912i	−0.691655	−	0.147016i
							\(3\)	1.70739	+	0.291246i	0.985761	+	0.168151i
\(5\)	0.742996	+	2.10902i	0.332278	+	0.943182i
							\(7\)	−1.24661	+	2.15919i	−0.471174	+	0.816097i	−0.999456	−	0.0329716i	\(-0.989503\pi\)
0.528282	+	0.849069i	\(0.322836\pi\)
\(11\)	1.89191	+	0.402138i	0.570433	+	0.121249i	0.484093	−	0.875016i	\(-0.339150\pi\)
							0.0863398	+	0.996266i	\(0.472483\pi\)
\(13\)	−6.05172	+	1.28633i	−1.67844	+	0.356764i	−0.946026	−	0.324091i	\(-0.894942\pi\)
							−0.732418	+	0.680855i	\(0.761608\pi\)
\(17\)	2.42955	+	1.76517i	0.589252	+	0.428117i	0.842048	−	0.539403i	\(-0.181350\pi\)
							−0.252796	+	0.967520i	\(0.581350\pi\)
\(19\)	−2.08855	−	1.51742i	−0.479147	−	0.348121i	0.321848	−	0.946791i	\(-0.395696\pi\)
							−0.800995	+	0.598671i	\(0.795696\pi\)
\(23\)	0.874034	−	0.970714i	0.182249	−	0.202408i	−0.645097	−	0.764101i	\(-0.723183\pi\)
							0.827346	+	0.561693i	\(0.189850\pi\)
\(29\)	−0.809198	−	7.69901i	−0.150264	−	1.42967i	−0.766569	−	0.642162i	\(-0.778038\pi\)
							0.616305	−	0.787508i	\(-0.288629\pi\)
\(31\)	−0.638995	+	6.07963i	−0.114767	+	1.09193i	0.773877	+	0.633336i	\(0.218315\pi\)
							−0.888644	+	0.458598i	\(0.848352\pi\)
\(37\)	1.73908	−	5.35234i	0.285903	−	0.879920i	−0.700223	−	0.713924i	\(-0.746916\pi\)
							0.986126	−	0.165996i	\(-0.0530838\pi\)
\(41\)	9.61764	−	2.04429i	1.50202	−	0.319265i	0.617802	−	0.786334i	\(-0.288023\pi\)
							0.884221	+	0.467069i	\(0.154690\pi\)
\(43\)	5.32586	−	9.22466i	0.812186	−	1.40675i	−0.0991444	−	0.995073i	\(-0.531611\pi\)
							0.911331	−	0.411675i	\(-0.135056\pi\)
\(47\)	1.16046	+	11.0411i	0.169271	+	1.61051i	0.668281	+	0.743909i	\(0.267031\pi\)
							−0.499010	+	0.866597i	\(0.666303\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.b.331.14	yes	112
9.4	even	3	inner	450.2.q.b.31.9	✓	112
25.21	even	5	inner	450.2.q.b.421.9	yes	112
225.121	even	15	inner	450.2.q.b.121.14	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.b.31.9	✓	112	9.4	even	3	inner
450.2.q.b.121.14	yes	112	225.121	even	15	inner
450.2.q.b.331.14	yes	112	1.1	even	1	trivial
450.2.q.b.421.9	yes	112	25.21	even	5	inner