Embedded newform 450.2.q.c.331.15

• Label: 450.2.q.c.331.15
• Level: $450$
• Weight: $2$
• Character: 450.331
• 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			331.15
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.c.121.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.978148 + 0.207912i) q^{2} +(1.73040 - 0.0756829i) q^{3} +(0.913545 + 0.406737i) q^{4} +(-1.82709 - 1.28908i) q^{5} +(1.70832 + 0.285741i) q^{6} +(0.989724 - 1.71425i) q^{7} +(0.809017 + 0.587785i) q^{8} +(2.98854 - 0.261923i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	1.73040	−	0.0756829i	0.999045	−	0.0436955i
\(5\)	−1.82709	−	1.28908i	−0.817102	−	0.576494i
							\(7\)	0.989724	−	1.71425i	0.374081	−	0.647927i	−0.616108	−	0.787661i	\(-0.711292\pi\)
0.990189	+	0.139735i	\(0.0446250\pi\)
\(11\)	5.25090	+	1.11611i	1.58320	+	0.336521i	0.913732	−	0.406316i	\(-0.133187\pi\)
							0.669472	+	0.742837i	\(0.266520\pi\)
\(13\)	−5.39049	+	1.14578i	−1.49505	+	0.317783i	−0.881618	−	0.471963i	\(-0.843545\pi\)
							−0.613434	+	0.789746i	\(0.710212\pi\)
\(17\)	−3.72407	−	2.70569i	−0.903219	−	0.656227i	0.0360721	−	0.999349i	\(-0.488515\pi\)
							−0.939291	+	0.343123i	\(0.888515\pi\)
\(19\)	3.17629	+	2.30771i	0.728691	+	0.529425i	0.889149	−	0.457617i	\(-0.151297\pi\)
							−0.160458	+	0.987043i	\(0.551297\pi\)
\(23\)	0.0477847	−	0.0530703i	0.00996380	−	0.0110659i	−0.738142	−	0.674645i	\(-0.764297\pi\)
							0.748106	+	0.663579i	\(0.230963\pi\)
\(29\)	−0.863692	−	8.21748i	−0.160384	−	1.52595i	−0.718114	−	0.695925i	\(-0.754994\pi\)
							0.557731	−	0.830022i	\(-0.311672\pi\)
\(31\)	−0.542342	+	5.16004i	−0.0974075	+	0.926770i	0.831267	+	0.555874i	\(0.187616\pi\)
							−0.928674	+	0.370897i	\(0.879050\pi\)
\(37\)	−1.38137	+	4.25143i	−0.227097	+	0.698931i	0.770975	+	0.636865i	\(0.219769\pi\)
							−0.998072	+	0.0620665i	\(0.980231\pi\)
\(41\)	−10.8293	+	2.30184i	−1.69126	+	0.359488i	−0.950124	−	0.311872i	\(-0.899044\pi\)
							−0.741132	+	0.671360i	\(0.765711\pi\)
\(43\)	−3.17600	+	5.50100i	−0.484336	+	0.838894i	−0.999838	−	0.0179938i	\(-0.994272\pi\)
							0.515502	+	0.856888i	\(0.327605\pi\)
\(47\)	−0.103779	−	0.987395i	−0.0151378	−	0.144026i	0.984342	−	0.176267i	\(-0.0564023\pi\)
							−0.999480	+	0.0322409i	\(0.989736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.c.331.15	yes	120
9.4	even	3	inner	450.2.q.c.31.9	✓	120
25.21	even	5	inner	450.2.q.c.421.9	yes	120
225.121	even	15	inner	450.2.q.c.121.15	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.c.31.9	✓	120	9.4	even	3	inner
450.2.q.c.121.15	yes	120	225.121	even	15	inner
450.2.q.c.331.15	yes	120	1.1	even	1	trivial
450.2.q.c.421.9	yes	120	25.21	even	5	inner