Embedded newform 450.2.v.a.79.6

• Label: 450.2.v.a.79.6
• Level: $450$
• Weight: $2$
• Character: 450.79
• 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			79.6
Character	\(\chi\)	\(=\)	450.79
Dual form			450.2.v.a.319.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743145 + 0.669131i) q^{2} +(-0.429668 + 1.67791i) q^{3} +(0.104528 - 0.994522i) q^{4} +(0.680748 - 2.12993i) q^{5} +(-0.803436 - 1.53444i) q^{6} +(-0.636266 - 0.367348i) q^{7} +(0.587785 + 0.809017i) q^{8} +(-2.63077 - 1.44189i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{10}\right)\)

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
							\(3\)	−0.429668	+	1.67791i	−0.248069	+	0.968742i
\(5\)	0.680748	−	2.12993i	0.304440	−	0.952532i
							\(7\)	−0.636266	−	0.367348i	−0.240486	−	0.138845i	0.374914	−	0.927060i	\(-0.377672\pi\)
−0.615400	+	0.788215i	\(0.711006\pi\)
\(11\)	1.82129	+	2.02275i	0.549139	+	0.609881i	0.952268	−	0.305262i	\(-0.0987441\pi\)
							−0.403129	+	0.915143i	\(0.632077\pi\)
\(13\)	1.59870	+	1.43948i	0.443400	+	0.399239i	0.860361	−	0.509685i	\(-0.170238\pi\)
							−0.416961	+	0.908924i	\(0.636905\pi\)
\(17\)	4.42112	+	6.08515i	1.07228	+	1.47587i	0.867747	+	0.497007i	\(0.165568\pi\)
							0.204533	+	0.978860i	\(0.434432\pi\)
\(19\)	5.79150	−	4.20777i	1.32866	−	0.965328i	0.328880	−	0.944372i	\(-0.393329\pi\)
							0.999780	−	0.0209565i	\(-0.00667116\pi\)
\(23\)	0.786599	+	3.70066i	0.164017	+	0.771640i	0.980850	+	0.194766i	\(0.0623946\pi\)
							−0.816833	+	0.576875i	\(0.804272\pi\)
\(29\)	−6.14239	+	2.73477i	−1.14061	+	0.507834i	−0.888050	−	0.459746i	\(-0.847940\pi\)
							−0.252564	+	0.967580i	\(0.581274\pi\)
\(31\)	7.30960	+	3.25444i	1.31284	+	0.584515i	0.939300	−	0.343097i	\(-0.111476\pi\)
							0.373543	+	0.927613i	\(0.378143\pi\)
\(37\)	3.70690	−	1.20445i	0.609411	−	0.198010i	0.0119777	−	0.999928i	\(-0.496187\pi\)
							0.597433	+	0.801919i	\(0.296187\pi\)
\(41\)	−4.07276	+	4.52326i	−0.636058	+	0.706414i	−0.971871	−	0.235515i	\(-0.924322\pi\)
							0.335813	+	0.941929i	\(0.390989\pi\)
\(43\)	5.20702	+	3.00628i	0.794064	+	0.458453i	0.841391	−	0.540427i	\(-0.181737\pi\)
							−0.0473275	+	0.998879i	\(0.515070\pi\)
\(47\)	−0.0735445	−	0.165184i	−0.0107276	−	0.0240945i	0.908102	−	0.418749i	\(-0.137531\pi\)
							−0.918830	+	0.394654i	\(0.870864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.79.6	✓	240
9.4	even	3	inner	450.2.v.a.229.4	yes	240
25.19	even	10	inner	450.2.v.a.169.4	yes	240
225.94	even	30	inner	450.2.v.a.319.6	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.6	✓	240	1.1	even	1	trivial
450.2.v.a.169.4	yes	240	25.19	even	10	inner
450.2.v.a.229.4	yes	240	9.4	even	3	inner
450.2.v.a.319.6	yes	240	225.94	even	30	inner