Embedded newform 45.4.e.c.16.1

• Label: 45.4.e.c.16.1
• Level: $45$
• Weight: $4$
• Character: 45.16
• Analytic conductor: $2.655$
• Analytic rank: $0$
• Dimension: $14$
• 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:	\(14\)
Relative dimension:	\(7\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial:	x^14 - 2*x^13 + 48*x^12 - 60*x^11 + 1605*x^10 - 1800*x^9 + 23232*x^8 - 2346*x^7 + 209529*x^6 - 55412*x^5 + 765088*x^4 + 276096*x^3 + 1572480*x^2 - 345600*x + 82944
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			16.1
Root			\(-2.69252 - 4.66357i\) of defining polynomial
Character	\(\chi\)	\(=\)	45.16
Dual form			45.4.e.c.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.69252 + 4.66357i) q^{2} +(-4.09553 + 3.19791i) q^{3} +(-10.4993 - 18.1853i) q^{4} +(2.50000 + 4.33013i) q^{5} +(-3.88642 - 27.7102i) q^{6} +(-6.28510 + 10.8861i) q^{7} +69.9976 q^{8} +(6.54673 - 26.1943i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} - 5 q^{3} - 36 q^{4} + 35 q^{5} - 31 q^{6} - 22 q^{7} - 36 q^{8} + 17 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.69252	+	4.66357i	−0.951948	+	1.64882i	−0.210745	+	0.977541i	\(0.567589\pi\)
							−0.741203	+	0.671281i	\(0.765744\pi\)
\(3\)	−4.09553	+	3.19791i	−0.788185	+	0.615438i
							\(5\)	2.50000	+	4.33013i	0.223607	+	0.387298i
\(7\)	−6.28510	+	10.8861i	−0.339363	+	0.587795i								−0.984313	−	0.176430i	\(-0.943545\pi\)
							0.644950	+	0.764225i	\(0.276878\pi\)
\(11\)	−6.41534	+	11.1117i	−0.175845	+	0.304573i	−0.940454	−	0.339922i	\(-0.889599\pi\)
							0.764608	+	0.644495i	\(0.222933\pi\)
\(13\)	−29.9322	−	51.8442i	−0.638593	−	1.10608i	−0.985742	−	0.168266i	\(-0.946183\pi\)
							0.347148	−	0.937810i	\(-0.387150\pi\)
\(17\)	−110.011			−1.56950			−0.784752	−	0.619810i	\(-0.787210\pi\)
							−0.784752	+	0.619810i	\(0.787210\pi\)
\(19\)	−12.0872			−0.145947			−0.0729733	−	0.997334i	\(-0.523249\pi\)
							−0.0729733	+	0.997334i	\(0.523249\pi\)
\(23\)	33.8608	+	58.6485i	0.306976	+	0.531699i	0.977699	−	0.210009i	\(-0.0673494\pi\)
							−0.670723	+	0.741708i	\(0.734016\pi\)
\(29\)	−99.9790	+	173.169i	−0.640194	+	1.10885i	0.345195	+	0.938531i	\(0.387813\pi\)
							−0.985389	+	0.170318i	\(0.945521\pi\)
\(31\)	−38.3143	−	66.3624i	−0.221982	−	0.384485i	0.733427	−	0.679768i	\(-0.237919\pi\)
							−0.955410	+	0.295283i	\(0.904586\pi\)
\(37\)	−22.4815			−0.0998900			−0.0499450	−	0.998752i	\(-0.515905\pi\)
							−0.0499450	+	0.998752i	\(0.515905\pi\)
\(41\)	−43.8807	−	76.0037i	−0.167147	−	0.289507i	0.770269	−	0.637719i	\(-0.220122\pi\)
							−0.937416	+	0.348213i	\(0.886789\pi\)
\(43\)	−59.7050	+	103.412i	−0.211743	+	0.366749i	−0.952260	−	0.305288i	\(-0.901247\pi\)
							0.740517	+	0.672037i	\(0.234581\pi\)
\(47\)	−121.578	+	210.579i	−0.377317	+	0.653533i	−0.990671	−	0.136276i	\(-0.956487\pi\)
							0.613354	+	0.789808i	\(0.289820\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.e.c.16.1	✓	14
3.2	odd	2		135.4.e.c.46.7		14
5.2	odd	4		225.4.k.d.124.1		28
5.3	odd	4		225.4.k.d.124.14		28
5.4	even	2		225.4.e.d.151.7		14
9.2	odd	6		405.4.a.n.1.1		7
9.4	even	3	inner	45.4.e.c.31.1	yes	14
9.5	odd	6		135.4.e.c.91.7		14
9.7	even	3		405.4.a.m.1.7		7
45.4	even	6		225.4.e.d.76.7		14
45.13	odd	12		225.4.k.d.49.1		28
45.22	odd	12		225.4.k.d.49.14		28
45.29	odd	6		2025.4.a.ba.1.7		7
45.34	even	6		2025.4.a.bb.1.1		7

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.1	✓	14	1.1	even	1	trivial
45.4.e.c.31.1	yes	14	9.4	even	3	inner
135.4.e.c.46.7		14	3.2	odd	2
135.4.e.c.91.7		14	9.5	odd	6
225.4.e.d.76.7		14	45.4	even	6
225.4.e.d.151.7		14	5.4	even	2
225.4.k.d.49.1		28	45.13	odd	12
225.4.k.d.49.14		28	45.22	odd	12
225.4.k.d.124.1		28	5.2	odd	4
225.4.k.d.124.14		28	5.3	odd	4
405.4.a.m.1.7		7	9.7	even	3
405.4.a.n.1.1		7	9.2	odd	6
2025.4.a.ba.1.7		7	45.29	odd	6
2025.4.a.bb.1.1		7	45.34	even	6