Embedded newform 450.2.v.a.319.12

• Label: 450.2.v.a.319.12
• 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			319.12
Character	\(\chi\)	\(=\)	450.319
Dual form			450.2.v.a.79.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743145 - 0.669131i) q^{2} +(1.33209 - 1.10704i) q^{3} +(0.104528 + 0.994522i) q^{4} +(-0.575676 + 2.16069i) q^{5} +(-1.73069 - 0.0686527i) q^{6} +(-4.13897 + 2.38963i) q^{7} +(0.587785 - 0.809017i) q^{8} +(0.548931 - 2.94935i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 30 q^{4} - 8 q^{5} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	1.33209	−	1.10704i	0.769083	−	0.639149i
\(5\)	−0.575676	+	2.16069i	−0.257450	+	0.966292i
							\(7\)	−4.13897	+	2.38963i	−1.56438	+	0.903196i	−0.567577	+	0.823320i	\(0.692119\pi\)
−0.996805	+	0.0798764i	\(0.974547\pi\)
\(11\)	−2.32912	+	2.58675i	−0.702255	+	0.779933i	−0.983733	−	0.179634i	\(-0.942509\pi\)
							0.281479	+	0.959568i	\(0.409175\pi\)
\(13\)	0.661707	−	0.595804i	0.183525	−	0.165246i	−0.572248	−	0.820081i	\(-0.693928\pi\)
							0.755772	+	0.654835i	\(0.227262\pi\)
\(17\)	−4.69516	+	6.46234i	−1.13874	+	1.56735i	−0.368464	+	0.929642i	\(0.620116\pi\)
							−0.770281	+	0.637705i	\(0.779884\pi\)
\(19\)	−2.13435	−	1.55070i	−0.489654	−	0.355755i	0.315397	−	0.948960i	\(-0.397862\pi\)
							−0.805051	+	0.593205i	\(0.797862\pi\)
\(23\)	−0.480141	+	2.25889i	−0.100116	+	0.471011i	0.899315	+	0.437301i	\(0.144066\pi\)
							−0.999432	+	0.0337098i	\(0.989268\pi\)
\(29\)	0.923075	+	0.410979i	0.171411	+	0.0763170i	0.490648	−	0.871358i	\(-0.336760\pi\)
							−0.319237	+	0.947675i	\(0.603427\pi\)
\(31\)	6.37531	−	2.83847i	1.14504	−	0.509804i	0.255566	−	0.966792i	\(-0.417738\pi\)
							0.889473	+	0.456988i	\(0.151072\pi\)
\(37\)	9.43641	+	3.06608i	1.55134	+	0.504060i	0.954477	−	0.298285i	\(-0.0964147\pi\)
							0.596860	+	0.802345i	\(0.296415\pi\)
\(41\)	−0.719955	−	0.799592i	−0.112438	−	0.124875i	0.684298	−	0.729202i	\(-0.260109\pi\)
							−0.796736	+	0.604327i	\(0.793442\pi\)
\(43\)	−7.00177	+	4.04248i	−1.06776	+	0.616472i	−0.927569	−	0.373653i	\(-0.878105\pi\)
							−0.140192	+	0.990124i	\(0.544772\pi\)
\(47\)	−1.51487	+	3.40246i	−0.220967	+	0.496300i	−0.989680	−	0.143294i	\(-0.954231\pi\)
							0.768713	+	0.639594i	\(0.220897\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.v.a.319.12	yes	240
9.7	even	3	inner	450.2.v.a.169.3	yes	240
25.4	even	10	inner	450.2.v.a.229.3	yes	240
225.79	even	30	inner	450.2.v.a.79.12	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.v.a.79.12	✓	240	225.79	even	30	inner
450.2.v.a.169.3	yes	240	9.7	even	3	inner
450.2.v.a.229.3	yes	240	25.4	even	10	inner
450.2.v.a.319.12	yes	240	1.1	even	1	trivial