Embedded newform 450.2.v.a.319.11

• Label: 450.2.v.a.319.11
• 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			319.11
Character	\(\chi\)	\(=\)	450.319
Dual form			450.2.v.a.79.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743145 - 0.669131i) q^{2} +(1.31955 - 1.12196i) q^{3} +(0.104528 + 0.994522i) q^{4} +(2.22159 + 0.254058i) q^{5} +(-1.73135 - 0.0491696i) q^{6} +(1.56138 - 0.901461i) q^{7} +(0.587785 - 0.809017i) q^{8} +(0.482407 - 2.96096i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	1.31955	−	1.12196i	0.761841	−	0.647764i
\(5\)	2.22159	+	0.254058i	0.993524	+	0.113618i
							\(7\)	1.56138	−	0.901461i	0.590144	−	0.340720i	−0.175010	−	0.984567i	\(-0.555996\pi\)
0.765155	+	0.643847i	\(0.222663\pi\)
\(11\)	0.991781	−	1.10148i	0.299033	−	0.332110i	−0.574838	−	0.818267i	\(-0.694935\pi\)
							0.873871	+	0.486157i	\(0.161602\pi\)
\(13\)	−3.29373	+	2.96569i	−0.913516	+	0.822533i	−0.984579	−	0.174939i	\(-0.944027\pi\)
							0.0710637	+	0.997472i	\(0.477361\pi\)
\(17\)	−1.34830	+	1.85578i	−0.327012	+	0.450093i	−0.940592	−	0.339539i	\(-0.889729\pi\)
							0.613580	+	0.789632i	\(0.289729\pi\)
\(19\)	3.61199	+	2.62427i	0.828648	+	0.602048i	0.919177	−	0.393846i	\(-0.128856\pi\)
							−0.0905285	+	0.995894i	\(0.528856\pi\)
\(23\)	1.18799	−	5.58906i	0.247714	−	1.16540i	−0.661785	−	0.749694i	\(-0.730201\pi\)
							0.909499	−	0.415707i	\(-0.136466\pi\)
\(29\)	8.63152	+	3.84300i	1.60283	+	0.713627i	0.996657	−	0.0817033i	\(-0.0260360\pi\)
							0.606176	+	0.795330i	\(0.292703\pi\)
\(31\)	−7.30427	+	3.25207i	−1.31189	+	0.584089i	−0.939041	−	0.343804i	\(-0.888284\pi\)
							−0.372845	+	0.927894i	\(0.621618\pi\)
\(37\)	−11.2399	−	3.65208i	−1.84783	−	0.600398i	−0.997212	−	0.0746203i	\(-0.976226\pi\)
							−0.850622	−	0.525777i	\(-0.823774\pi\)
\(41\)	−7.70404	−	8.55620i	−1.20317	−	1.33625i	−0.926962	−	0.375156i	\(-0.877589\pi\)
							−0.276207	−	0.961098i	\(-0.589077\pi\)
\(43\)	−2.62855	+	1.51760i	−0.400851	+	0.231431i	−0.686851	−	0.726798i	\(-0.741008\pi\)
							0.286000	+	0.958230i	\(0.407674\pi\)
\(47\)	2.42653	−	5.45008i	0.353946	−	0.794976i	−0.645565	−	0.763705i	\(-0.723378\pi\)
							0.999511	−	0.0312703i	\(-0.00995528\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.319.11	yes	240
9.7	even	3	inner	450.2.v.a.169.2	yes	240
25.4	even	10	inner	450.2.v.a.229.2	yes	240
225.79	even	30	inner	450.2.v.a.79.11	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.11	✓	240	225.79	even	30	inner
450.2.v.a.169.2	yes	240	9.7	even	3	inner
450.2.v.a.229.2	yes	240	25.4	even	10	inner
450.2.v.a.319.11	yes	240	1.1	even	1	trivial