Embedded newform 450.2.q.b.331.2

• Label: 450.2.q.b.331.2
• Level: $450$
• Weight: $2$
• Character: 450.331
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $112$
• 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:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			331.2
Character	\(\chi\)	\(=\)	450.331
Dual form			450.2.q.b.121.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.978148 - 0.207912i) q^{2} +(-1.68741 - 0.390691i) q^{3} +(0.913545 + 0.406737i) q^{4} +(1.82525 - 1.29169i) q^{5} +(1.56931 + 0.732986i) q^{6} +(1.61076 - 2.78992i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(2.69472 + 1.31851i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{2} + 2 q^{3} + 14 q^{4} + 9 q^{5} - 7 q^{6} - 8 q^{7} - 28 q^{8} + 8 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.68741	−	0.390691i	−0.974228	−	0.225566i
\(5\)	1.82525	−	1.29169i	0.816278	−	0.577659i
							\(7\)	1.61076	−	2.78992i	0.608810	−	1.05449i	−0.382627	−	0.923903i	\(-0.624981\pi\)
0.991437	−	0.130587i	\(-0.0416861\pi\)
\(11\)	3.49230	+	0.742310i	1.05297	+	0.223815i	0.701709	−	0.712464i	\(-0.252421\pi\)
							0.351258	+	0.936279i	\(0.385754\pi\)
\(13\)	−6.84091	+	1.45408i	−1.89733	+	0.403289i	−0.999326	−	0.0367033i	\(-0.988314\pi\)
							−0.898001	+	0.439993i	\(0.854981\pi\)
\(17\)	3.96133	+	2.87807i	0.960763	+	0.698035i	0.953328	−	0.301937i	\(-0.0976334\pi\)
							0.00743493	+	0.999972i	\(0.497633\pi\)
\(19\)	−0.471271	−	0.342398i	−0.108117	−	0.0785515i	0.532413	−	0.846485i	\(-0.321285\pi\)
							−0.640530	+	0.767933i	\(0.721285\pi\)
\(23\)	5.46055	−	6.06455i	1.13860	−	1.26455i	0.178795	−	0.983886i	\(-0.442780\pi\)
							0.959807	−	0.280660i	\(-0.0905533\pi\)
\(29\)	−0.317733	−	3.02303i	−0.0590015	−	0.561362i	−0.983593	−	0.180402i	\(-0.942260\pi\)
							0.924591	−	0.380960i	\(-0.124406\pi\)
\(31\)	0.442438	−	4.20951i	0.0794642	−	0.756051i	−0.880143	−	0.474708i	\(-0.842554\pi\)
							0.959607	−	0.281343i	\(-0.0907798\pi\)
\(37\)	1.37019	−	4.21702i	0.225258	−	0.693274i	−0.773007	−	0.634398i	\(-0.781248\pi\)
							0.998265	−	0.0588765i	\(-0.0187518\pi\)
\(41\)	−7.11698	+	1.51276i	−1.11149	+	0.236254i	−0.726837	−	0.686810i	\(-0.759011\pi\)
							−0.384648	+	0.923063i	\(0.625677\pi\)
\(43\)	−2.62623	+	4.54876i	−0.400496	+	0.693680i	−0.993786	−	0.111309i	\(-0.964496\pi\)
							0.593290	+	0.804989i	\(0.297829\pi\)
\(47\)	−0.320959	−	3.05372i	−0.0468166	−	0.445430i	−0.992672	−	0.120841i	\(-0.961441\pi\)
							0.945855	−	0.324589i	\(-0.105226\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.b.331.2	yes	112
9.4	even	3	inner	450.2.q.b.31.7	✓	112
25.21	even	5	inner	450.2.q.b.421.7	yes	112
225.121	even	15	inner	450.2.q.b.121.2	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.b.31.7	✓	112	9.4	even	3	inner
450.2.q.b.121.2	yes	112	225.121	even	15	inner
450.2.q.b.331.2	yes	112	1.1	even	1	trivial
450.2.q.b.421.7	yes	112	25.21	even	5	inner