Embedded newform 450.2.q.c.121.8

• Label: 450.2.q.c.121.8
• Level: $450$
• Weight: $2$
• Character: 450.121
• Analytic conductor: $3.593$
• Analytic rank: $0$
• Dimension: $120$
• 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:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			121.8
Character	\(\chi\)	\(=\)	450.121
Dual form			450.2.q.c.331.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.978148 - 0.207912i) q^{2} +(0.200855 + 1.72037i) q^{3} +(0.913545 - 0.406737i) q^{4} +(1.75274 + 1.38849i) q^{5} +(0.554150 + 1.64101i) q^{6} +(0.746133 + 1.29234i) q^{7} +(0.809017 - 0.587785i) q^{8} +(-2.91931 + 0.691088i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{5}\right)\)

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
							\(3\)	0.200855	+	1.72037i	0.115964	+	0.993253i
\(5\)	1.75274	+	1.38849i	0.783850	+	0.620951i
							\(7\)	0.746133	+	1.29234i	0.282012	+	0.488458i	0.971880	−	0.235476i	\(-0.0756650\pi\)
−0.689869	+	0.723935i	\(0.742332\pi\)
\(11\)	0.796977	−	0.169403i	0.240298	−	0.0510768i	−0.0861884	−	0.996279i	\(-0.527469\pi\)
							0.326486	+	0.945202i	\(0.394135\pi\)
\(13\)	−4.01220	−	0.852820i	−1.11278	−	0.236530i	−0.385393	−	0.922752i	\(-0.625934\pi\)
							−0.727392	+	0.686223i	\(0.759268\pi\)
\(17\)	0.388751	−	0.282444i	0.0942860	−	0.0685028i	−0.539643	−	0.841894i	\(-0.681441\pi\)
							0.633929	+	0.773391i	\(0.281441\pi\)
\(19\)	3.79739	−	2.75897i	0.871181	−	0.632950i	−0.0597223	−	0.998215i	\(-0.519022\pi\)
							0.930904	+	0.365265i	\(0.119022\pi\)
\(23\)	−3.41579	−	3.79362i	−0.712242	−	0.791024i	0.273033	−	0.962005i	\(-0.411973\pi\)
							−0.985274	+	0.170980i	\(0.945307\pi\)
\(29\)	0.142961	−	1.36018i	0.0265472	−	0.252580i	−0.973197	−	0.229972i	\(-0.926136\pi\)
							0.999744	−	0.0226076i	\(-0.00719685\pi\)
\(31\)	0.328041	+	3.12110i	0.0589179	+	0.560566i	0.983668	+	0.179992i	\(0.0576071\pi\)
							−0.924750	+	0.380575i	\(0.875726\pi\)
\(37\)	1.63213	+	5.02317i	0.268320	+	0.825804i	0.990910	+	0.134527i	\(0.0429515\pi\)
							−0.722590	+	0.691277i	\(0.757049\pi\)
\(41\)	8.51755	+	1.81046i	1.33022	+	0.282746i	0.817558	−	0.575846i	\(-0.195327\pi\)
							0.512659	+	0.858592i	\(0.328660\pi\)
\(43\)	−0.371715	−	0.643829i	−0.0566860	−	0.0981830i	0.836290	−	0.548288i	\(-0.184720\pi\)
							−0.892976	+	0.450105i	\(0.851387\pi\)
\(47\)	0.405033	−	3.85363i	0.0590801	−	0.562110i	−0.924442	−	0.381322i	\(-0.875469\pi\)
							0.983522	−	0.180787i	\(-0.0578646\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.q.c.121.8	yes	120
9.7	even	3	inner	450.2.q.c.421.14	yes	120
25.6	even	5	inner	450.2.q.c.31.14	✓	120
225.106	even	15	inner	450.2.q.c.331.8	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.q.c.31.14	✓	120	25.6	even	5	inner
450.2.q.c.121.8	yes	120	1.1	even	1	trivial
450.2.q.c.331.8	yes	120	225.106	even	15	inner
450.2.q.c.421.14	yes	120	9.7	even	3	inner