Embedded newform 450.2.v.a.319.30

• Label: 450.2.v.a.319.30
• 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			319.30
Character	\(\chi\)	\(=\)	450.319
Dual form			450.2.v.a.79.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.743145 + 0.669131i) q^{2} +(1.73129 - 0.0512006i) q^{3} +(0.104528 + 0.994522i) q^{4} +(-2.18131 - 0.491815i) q^{5} +(1.32086 + 1.12041i) q^{6} +(4.14021 - 2.39035i) q^{7} +(-0.587785 + 0.809017i) q^{8} +(2.99476 - 0.177287i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	1.73129	−	0.0512006i	0.999563	−	0.0295607i
\(5\)	−2.18131	−	0.491815i	−0.975512	−	0.219947i
							\(7\)	4.14021	−	2.39035i	1.56485	−	0.903468i	0.568098	−	0.822961i	\(-0.307680\pi\)
0.996754	−	0.0805064i	\(-0.0256537\pi\)
\(11\)	0.281842	−	0.313017i	0.0849784	−	0.0943781i	−0.699156	−	0.714969i	\(-0.746441\pi\)
							0.784134	+	0.620591i	\(0.213107\pi\)
\(13\)	1.35380	−	1.21897i	0.375477	−	0.338081i	−0.459688	−	0.888080i	\(-0.652039\pi\)
							0.835165	+	0.550000i	\(0.185372\pi\)
\(17\)	−2.58717	+	3.56094i	−0.627482	+	0.863655i	−0.997871	−	0.0652221i	\(-0.979224\pi\)
							0.370389	+	0.928877i	\(0.379224\pi\)
\(19\)	−3.94318	−	2.86489i	−0.904627	−	0.657250i	0.0350233	−	0.999386i	\(-0.488849\pi\)
							−0.939650	+	0.342137i	\(0.888849\pi\)
\(23\)	−1.60033	+	7.52897i	−0.333692	+	1.56990i	0.416787	+	0.909004i	\(0.363156\pi\)
							−0.750479	+	0.660894i	\(0.770177\pi\)
\(29\)	3.82881	+	1.70470i	0.710993	+	0.316554i	0.730174	−	0.683262i	\(-0.239439\pi\)
							−0.0191808	+	0.999816i	\(0.506106\pi\)
\(31\)	−4.15921	+	1.85180i	−0.747017	+	0.332593i	−0.744696	−	0.667404i	\(-0.767405\pi\)
							−0.00232095	+	0.999997i	\(0.500739\pi\)
\(37\)	−0.0686089	−	0.0222924i	−0.0112792	−	0.00366484i	0.303372	−	0.952872i	\(-0.401888\pi\)
							−0.314651	+	0.949207i	\(0.601888\pi\)
\(41\)	−4.45649	−	4.94943i	−0.695986	−	0.772971i	0.286745	−	0.958007i	\(-0.407427\pi\)
							−0.982732	+	0.185036i	\(0.940760\pi\)
\(43\)	−8.78943	+	5.07458i	−1.34038	+	0.773867i	−0.986862	−	0.161563i	\(-0.948347\pi\)
							−0.353514	+	0.935429i	\(0.615013\pi\)
\(47\)	5.02904	−	11.2954i	0.733561	−	1.64760i	−0.0280801	−	0.999606i	\(-0.508939\pi\)
							0.761641	−	0.647999i	\(-0.224394\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.319.30	yes	240
9.7	even	3	inner	450.2.v.a.169.22	yes	240
25.4	even	10	inner	450.2.v.a.229.22	yes	240
225.79	even	30	inner	450.2.v.a.79.30	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.30	✓	240	225.79	even	30	inner
450.2.v.a.169.22	yes	240	9.7	even	3	inner
450.2.v.a.229.22	yes	240	25.4	even	10	inner
450.2.v.a.319.30	yes	240	1.1	even	1	trivial