Embedded newform 450.2.q.c.331.13

• Label: 450.2.q.c.331.13
• 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			331.13
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.c.121.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.978148 + 0.207912i) q^{2} +(1.55198 - 0.768997i) q^{3} +(0.913545 + 0.406737i) q^{4} +(-0.618563 - 2.14881i) q^{5} +(1.67795 - 0.429517i) q^{6} +(-0.0993321 + 0.172048i) q^{7} +(0.809017 + 0.587785i) q^{8} +(1.81729 - 2.38694i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9}+O(q^{10}) \)

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
							\(3\)	1.55198	−	0.768997i	0.896036	−	0.443980i
\(5\)	−0.618563	−	2.14881i	−0.276630	−	0.960977i
							\(7\)	−0.0993321	+	0.172048i	−0.0375440	+	0.0650281i	−0.884187	−	0.467133i	\(-0.845287\pi\)
0.846643	+	0.532162i	\(0.178620\pi\)
\(11\)	−6.13530	−	1.30410i	−1.84986	−	0.393200i	−0.857287	−	0.514839i	\(-0.827852\pi\)
							−0.992574	+	0.121639i	\(0.961185\pi\)
\(13\)	5.94749	−	1.26418i	1.64954	−	0.350620i	0.712994	−	0.701170i	\(-0.247339\pi\)
							0.936544	+	0.350550i	\(0.114005\pi\)
\(17\)	4.66220	+	3.38729i	1.13075	+	0.821538i	0.985804	−	0.167902i	\(-0.0536991\pi\)
							0.144946	+	0.989440i	\(0.453699\pi\)
\(19\)	−0.00256372	−	0.00186265i	−0.000588158	−	0.000427322i	0.587491	−	0.809231i	\(-0.300116\pi\)
							−0.588079	+	0.808803i	\(0.700116\pi\)
\(23\)	−4.01539	+	4.45955i	−0.837268	+	0.929880i	−0.998371	−	0.0570576i	\(-0.981828\pi\)
							0.161103	+	0.986938i	\(0.448495\pi\)
\(29\)	0.309061	+	2.94051i	0.0573911	+	0.546040i	0.985008	+	0.172507i	\(0.0551869\pi\)
							−0.927617	+	0.373532i	\(0.878146\pi\)
\(31\)	−0.379852	+	3.61405i	−0.0682234	+	0.649102i	0.905963	+	0.423357i	\(0.139148\pi\)
							−0.974186	+	0.225745i	\(0.927518\pi\)
\(37\)	0.959704	−	2.95366i	0.157774	−	0.485579i	−0.840657	−	0.541568i	\(-0.817831\pi\)
							0.998431	+	0.0559883i	\(0.0178310\pi\)
\(41\)	0.197876	−	0.0420599i	0.0309031	−	0.00656865i	−0.192434	−	0.981310i	\(-0.561638\pi\)
							0.223337	+	0.974741i	\(0.428305\pi\)
\(43\)	−0.889010	+	1.53981i	−0.135573	+	0.234819i	−0.925816	−	0.377974i	\(-0.876621\pi\)
							0.790243	+	0.612793i	\(0.209954\pi\)
\(47\)	−0.345071	−	3.28313i	−0.0503338	−	0.478894i	−0.990433	−	0.137994i	\(-0.955935\pi\)
							0.940099	−	0.340901i	\(-0.110732\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.c.331.13	yes	120
9.4	even	3	inner	450.2.q.c.31.12	✓	120
25.21	even	5	inner	450.2.q.c.421.12	yes	120
225.121	even	15	inner	450.2.q.c.121.13	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.c.31.12	✓	120	9.4	even	3	inner
450.2.q.c.121.13	yes	120	225.121	even	15	inner
450.2.q.c.331.13	yes	120	1.1	even	1	trivial
450.2.q.c.421.12	yes	120	25.21	even	5	inner