Embedded newform 45.4.e.b.16.2

• Label: 45.4.e.b.16.2
• Level: $45$
• Weight: $4$
• Character: 45.16
• Analytic conductor: $2.655$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.65508595026\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.15759792.1
Defining polynomial:	\( x^{6} - 3x^{5} + 16x^{4} - 27x^{3} + 52x^{2} - 39x + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			16.2
Root			\(0.500000 - 1.98116i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.16
Dual form			45.4.e.b.31.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0874923 - 0.151541i) q^{2} +(5.19394 + 0.151541i) q^{3} +(3.98469 + 6.90169i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(0.477395 - 0.773837i) q^{6} +(-4.23186 + 7.32979i) q^{7} +2.79440 q^{8} +(26.9541 + 1.57419i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 9 q^{3} - 11 q^{4} - 15 q^{5} + 84 q^{6} + 43 q^{7} - 54 q^{8} + 57 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\)	\(11\)	\(37\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0874923	−	0.151541i	0.0309332	−	0.0535779i	−0.850144	−	0.526550i	\(-0.823485\pi\)
							0.881078	+	0.472972i	\(0.156819\pi\)
\(3\)	5.19394	+	0.151541i	0.999575	+	0.0291641i
							\(5\)	−2.50000	−	4.33013i	−0.223607	−	0.387298i
\(7\)	−4.23186	+	7.32979i	−0.228499	+	0.395772i								−0.957363	−	0.288886i	\(-0.906715\pi\)
							0.728865	+	0.684658i	\(0.240048\pi\)
\(11\)	15.7541	−	27.2870i	0.431823	−	0.747939i	−0.565208	−	0.824949i	\(-0.691204\pi\)
							0.997030	+	0.0770098i	\(0.0245373\pi\)
\(13\)	−13.4348	−	23.2697i	−0.286626	−	0.496451i	0.686376	−	0.727247i	\(-0.259200\pi\)
							−0.973002	+	0.230796i	\(0.925867\pi\)
\(17\)	−44.3307			−0.632457			−0.316229	−	0.948683i	\(-0.602417\pi\)
							−0.316229	+	0.948683i	\(0.602417\pi\)
\(19\)	−90.2082			−1.08922			−0.544610	−	0.838689i	\(-0.683322\pi\)
							−0.544610	+	0.838689i	\(0.683322\pi\)
\(23\)	−97.1287	−	168.232i	−0.880553	−	1.52516i	−0.850727	−	0.525608i	\(-0.823838\pi\)
							−0.0298265	−	0.999555i	\(-0.509495\pi\)
\(29\)	−1.87186	+	3.24215i	−0.0119860	+	0.0207604i	−0.871956	−	0.489584i	\(-0.837149\pi\)
							0.859970	+	0.510344i	\(0.170482\pi\)
\(31\)	125.832	+	217.947i	0.729035	+	1.26273i	0.957292	+	0.289124i	\(0.0933641\pi\)
							−0.228257	+	0.973601i	\(0.573303\pi\)
\(37\)	−62.2293			−0.276498			−0.138249	−	0.990397i	\(-0.544147\pi\)
							−0.138249	+	0.990397i	\(0.544147\pi\)
\(41\)	102.173	+	176.969i	0.389188	+	0.674094i	0.992341	−	0.123532i	\(-0.0394223\pi\)
							−0.603152	+	0.797626i	\(0.706089\pi\)
\(43\)	263.831	−	456.968i	0.935669	−	1.62063i	0.162233	−	0.986752i	\(-0.448130\pi\)
							0.773436	−	0.633874i	\(-0.218536\pi\)
\(47\)	−77.8637	+	134.864i	−0.241651	+	0.418551i	−0.961185	−	0.275906i	\(-0.911022\pi\)
							0.719534	+	0.694457i	\(0.244355\pi\)