Embedded newform 73.2.h.a.3.3

• Label: 73.2.h.a.3.3
• Level: $73$
• Weight: $2$
• Character: 73.3
• 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			3.3
Root			\(-0.467725i\) of defining polynomial
Character	\(\chi\)	\(=\)	73.3
Dual form			73.2.h.a.49.3

$q$-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 \(\left(\mathbb{Z}/73\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
73.2.h.a.3.3	✓	20	1.1	even	1	trivial
73.2.h.a.49.3	yes	20	73.49	even	12	inner
657.2.be.c.487.3		20	219.122	odd	12
657.2.be.c.514.3		20	3.2	odd	2
5329.2.a.m.1.11		20	73.66	odd	24
5329.2.a.m.1.12		20	73.7	odd	24