Embedded newform 73.2.h.a.49.1

• Label: 73.2.h.a.49.1
• Level: $73$
• Weight: $2$
• Character: 73.49
• Analytic conductor: $0.583$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	73.h (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.582907934755\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	x^20 + 28*x^18 + 326*x^16 + 2044*x^14 + 7471*x^12 + 16090*x^10 + 19590*x^8 + 12030*x^6 + 2877*x^4 + 230*x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			49.1
Root			\(2.49160i\) of defining polynomial
Character	\(\chi\)	\(=\)	73.49
Dual form			73.2.h.a.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24580 - 2.15779i) q^{2} -2.90855i q^{3} +(-2.10403 + 3.64429i) q^{4} +(1.39079 - 0.372661i) q^{5} +(-6.27603 + 3.62347i) q^{6} +(2.87065 + 2.87065i) q^{7} +5.50161 q^{8} -5.45964 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} - 8 q^{4} - 4 q^{5} + 6 q^{6} - 2 q^{7} + 12 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/73\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24580	−	2.15779i	−0.880913	−	1.52579i	−0.850327	−	0.526254i	\(-0.823596\pi\)
							−0.0305858	−	0.999532i	\(-0.509737\pi\)
\(3\)		−	2.90855i		−	1.67925i	−0.543166	−	0.839625i	\(-0.682775\pi\)
							0.543166	−	0.839625i	\(-0.317225\pi\)
\(5\)	1.39079	−	0.372661i	0.621980	−	0.166659i	0.0659522	−	0.997823i	\(-0.478992\pi\)
							0.556028	+	0.831164i	\(0.312325\pi\)
\(7\)	2.87065	+	2.87065i	1.08500	+	1.08500i	0.996034	+	0.0889704i	\(0.0283576\pi\)
							0.0889704	+	0.996034i	\(0.471642\pi\)
\(11\)	−0.224728	−	0.838698i	−0.0677582	−	0.252877i	0.923736	−	0.383031i	\(-0.125120\pi\)
							−0.991494	+	0.130154i	\(0.958453\pi\)
\(13\)	−2.18693	−	0.585985i	−0.606544	−	0.162523i	−0.0575408	−	0.998343i	\(-0.518326\pi\)
							−0.549003	+	0.835820i	\(0.684993\pi\)
\(17\)	0.00830215	+	0.00830215i	0.00201357	+	0.00201357i	0.708113	−	0.706099i	\(-0.249547\pi\)
							−0.706099	+	0.708113i	\(0.749547\pi\)
\(19\)	−0.315382	+	0.182086i	−0.0723536	+	0.0417733i	−0.535740	−	0.844383i	\(-0.679967\pi\)
							0.463387	+	0.886156i	\(0.346634\pi\)
\(23\)	6.47077	+	3.73590i	1.34925	+	0.778989i	0.988143	−	0.153537i	\(-0.0490663\pi\)
							0.361105	+	0.932525i	\(0.382400\pi\)
\(29\)	−9.01865	−	2.41654i	−1.67472	−	0.448740i	−0.708344	−	0.705867i	\(-0.750558\pi\)
							−0.966378	+	0.257127i	\(0.917224\pi\)
\(31\)	4.51564	+	1.20996i	0.811033	+	0.217316i	0.640423	−	0.768023i	\(-0.278759\pi\)
							0.170611	+	0.985338i	\(0.445426\pi\)
\(37\)	2.95851	−	5.12430i	0.486377	−	0.842429i	−0.513501	−	0.858089i	\(-0.671652\pi\)
							0.999877	+	0.0156598i	\(0.00498489\pi\)
\(41\)	−1.95865	+	3.39248i	−0.305890	+	0.529816i	−0.977459	−	0.211125i	\(-0.932287\pi\)
							0.671569	+	0.740942i	\(0.265621\pi\)
\(43\)	−0.903686	+	0.903686i	−0.137811	+	0.137811i	−0.772647	−	0.634836i	\(-0.781068\pi\)
							0.634836	+	0.772647i	\(0.281068\pi\)
\(47\)	0.796053	+	2.97091i	0.116116	+	0.433352i	0.999368	−	0.0355472i	\(-0.0113174\pi\)
							−0.883252	+	0.468899i	\(0.844651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	73.2.h.a.49.1	yes	20
3.2	odd	2		657.2.be.c.487.5		20
73.3	even	12	inner	73.2.h.a.3.1	✓	20
73.21	odd	24		5329.2.a.m.1.19		20
73.52	odd	24		5329.2.a.m.1.20		20
219.149	odd	12		657.2.be.c.514.5		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
73.2.h.a.3.1	✓	20	73.3	even	12	inner
73.2.h.a.49.1	yes	20	1.1	even	1	trivial
657.2.be.c.487.5		20	3.2	odd	2
657.2.be.c.514.5		20	219.149	odd	12
5329.2.a.m.1.19		20	73.21	odd	24
5329.2.a.m.1.20		20	73.52	odd	24