Embedded newform 73.2.h.a.49.3

• Label: 73.2.h.a.49.3
• 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			49.3
Root			\(0.467725i\) of defining polynomial
Character	\(\chi\)	\(=\)	73.49
Dual form			73.2.h.a.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.233863 - 0.405062i) q^{2} +2.48355i q^{3} +(0.890617 - 1.54259i) q^{4} +(-0.733428 + 0.196522i) q^{5} +(1.00599 - 0.580811i) q^{6} +(3.06980 + 3.06980i) q^{7} -1.76858 q^{8} -3.16804 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} - 8 q^{4} - 4 q^{5} + 6 q^{6} - 2 q^{7} + 12 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.233863	−	0.405062i	−0.165366	−	0.286422i	0.771419	−	0.636327i	\(-0.219547\pi\)
							−0.936785	+	0.349905i	\(0.886214\pi\)
\(3\)			2.48355i			1.43388i	0.697134	+	0.716940i	\(0.254458\pi\)
							−0.697134	+	0.716940i	\(0.745542\pi\)
\(5\)	−0.733428	+	0.196522i	−0.327999	+	0.0878871i	−0.419061	−	0.907958i	\(-0.637641\pi\)
							0.0910617	+	0.995845i	\(0.470974\pi\)
\(7\)	3.06980	+	3.06980i	1.16028	+	1.16028i	0.984415	+	0.175861i	\(0.0562710\pi\)
							0.175861	+	0.984415i	\(0.443729\pi\)
\(11\)	−1.54223	−	5.75567i	−0.464999	−	1.73540i	−0.656893	−	0.753984i	\(-0.728130\pi\)
							0.191894	−	0.981416i	\(-0.438537\pi\)
\(13\)	−4.06719	−	1.08980i	−1.12804	−	0.302256i	−0.353905	−	0.935281i	\(-0.615146\pi\)
							−0.774131	+	0.633025i	\(0.781813\pi\)
\(17\)	0.444813	+	0.444813i	0.107883	+	0.107883i	0.758988	−	0.651105i	\(-0.225694\pi\)
							−0.651105	+	0.758988i	\(0.725694\pi\)
\(19\)	2.34924	−	1.35633i	0.538952	−	0.311164i	−0.205702	−	0.978615i	\(-0.565948\pi\)
							0.744654	+	0.667450i	\(0.232614\pi\)
\(23\)	−2.49269	−	1.43915i	−0.519761	−	0.300084i	0.217076	−	0.976155i	\(-0.430348\pi\)
							−0.736837	+	0.676071i	\(0.763681\pi\)
\(29\)	5.16250	+	1.38329i	0.958652	+	0.256870i	0.704030	−	0.710170i	\(-0.251382\pi\)
							0.254622	+	0.967041i	\(0.418049\pi\)
\(31\)	3.60251	+	0.965290i	0.647030	+	0.173371i	0.567386	−	0.823452i	\(-0.307955\pi\)
							0.0796444	+	0.996823i	\(0.474622\pi\)
\(37\)	2.97841	−	5.15877i	0.489648	−	0.848096i	−0.510281	−	0.860008i	\(-0.670458\pi\)
							0.999929	+	0.0119121i	\(0.00379182\pi\)
\(41\)	−3.90784	+	6.76858i	−0.610303	+	1.05708i	0.380887	+	0.924622i	\(0.375619\pi\)
							−0.991189	+	0.132453i	\(0.957715\pi\)
\(43\)	0.273614	−	0.273614i	0.0417257	−	0.0417257i	−0.685936	−	0.727662i	\(-0.740607\pi\)
							0.727662	+	0.685936i	\(0.240607\pi\)
\(47\)	−0.269417	−	1.00548i	−0.0392986	−	0.146664i	0.943489	−	0.331404i	\(-0.107522\pi\)
							−0.982787	+	0.184740i	\(0.940856\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	73.2.h.a.49.3	yes	20
3.2	odd	2		657.2.be.c.487.3		20
73.3	even	12	inner	73.2.h.a.3.3	✓	20
73.21	odd	24		5329.2.a.m.1.12		20
73.52	odd	24		5329.2.a.m.1.11		20
219.149	odd	12		657.2.be.c.514.3		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
73.2.h.a.3.3	✓	20	73.3	even	12	inner
73.2.h.a.49.3	yes	20	1.1	even	1	trivial
657.2.be.c.487.3		20	3.2	odd	2
657.2.be.c.514.3		20	219.149	odd	12
5329.2.a.m.1.11		20	73.52	odd	24
5329.2.a.m.1.12		20	73.21	odd	24