Embedded newform 67.2.e.c.62.2

• Label: 67.2.e.c.62.2
• Level: $67$
• Weight: $2$
• Character: 67.62
• Analytic conductor: $0.535$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	67.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.534997693543\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(2\) over \(\Q(\zeta_{11})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 7*x^19 + 39*x^18 - 148*x^17 + 492*x^16 - 1282*x^15 + 2921*x^14 - 4316*x^13 + 2696*x^12 + 9361*x^11 - 20998*x^10 + 10813*x^9 + 44155*x^8 - 46933*x^7 - 36976*x^6 - 49131*x^5 + 105828*x^4 + 108234*x^3 + 143215*x^2 + 41004*x + 4489
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			62.2
Root			\(0.441163 + 3.06836i\) of defining polynomial
Character	\(\chi\)	\(=\)	67.62
Dual form			67.2.e.c.40.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10181 + 1.27155i) q^{2} +(0.372334 + 0.815299i) q^{3} +(-0.118239 - 0.822373i) q^{4} +(1.36958 - 0.402144i) q^{5} +(-1.44694 - 0.424859i) q^{6} +(-3.35003 + 3.86614i) q^{7} +(-1.65486 - 1.06351i) q^{8} +(1.43850 - 1.66012i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} - 4 q^{3} - 4 q^{4} + q^{5} + 3 q^{6} - 8 q^{7} - 22 q^{8} + 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/67\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.10181	+	1.27155i	−0.779096	+	0.899124i	−0.997044	−	0.0768359i	\(-0.975518\pi\)
							0.217948	+	0.975960i	\(0.430064\pi\)
\(3\)	0.372334	+	0.815299i	0.214967	+	0.470713i	0.986140	−	0.165913i	\(-0.0530570\pi\)
							−0.771173	+	0.636626i	\(0.780330\pi\)
\(5\)	1.36958	−	0.402144i	0.612493	−	0.179844i	0.0392496	−	0.999229i	\(-0.487503\pi\)
							0.573243	+	0.819385i	\(0.305685\pi\)
\(7\)	−3.35003	+	3.86614i	−1.26619	+	1.46126i	−0.439878	+	0.898058i	\(0.644978\pi\)
							−0.826316	+	0.563207i	\(0.809567\pi\)
\(11\)	3.02375	−	0.887854i	0.911696	−	0.267698i	0.207941	−	0.978141i	\(-0.433324\pi\)
							0.703755	+	0.710443i	\(0.251506\pi\)
\(13\)	2.17878	−	1.40022i	0.604286	−	0.388351i	−0.202425	−	0.979298i	\(-0.564882\pi\)
							0.806710	+	0.590947i	\(0.201246\pi\)
\(17\)	0.327698	−	2.27919i	0.0794783	−	0.552784i	−0.910710	−	0.413046i	\(-0.864465\pi\)
							0.990189	−	0.139738i	\(-0.0446260\pi\)
\(19\)	1.98974	+	2.29629i	0.456478	+	0.526804i	0.936601	−	0.350397i	\(-0.113953\pi\)
							−0.480123	+	0.877201i	\(0.659408\pi\)
\(23\)	−3.12440	−	6.84148i	−0.651482	−	1.42655i	−0.890251	−	0.455470i	\(-0.849471\pi\)
							0.238769	−	0.971076i	\(-0.423256\pi\)
\(29\)	0.448242			0.0832364			0.0416182	−	0.999134i	\(-0.486749\pi\)
							0.0416182	+	0.999134i	\(0.486749\pi\)
\(31\)	−2.78825	−	1.79190i	−0.500784	−	0.321835i	0.265746	−	0.964043i	\(-0.414382\pi\)
							−0.766530	+	0.642209i	\(0.778018\pi\)
\(37\)	−3.87210			−0.636569			−0.318284	−	0.947995i	\(-0.603107\pi\)
							−0.318284	+	0.947995i	\(0.603107\pi\)
\(41\)	−1.15914	+	8.06201i	−0.181028	+	1.25907i	0.673312	+	0.739359i	\(0.264871\pi\)
							−0.854339	+	0.519716i	\(0.826038\pi\)
\(43\)	−0.217535	+	1.51299i	−0.0331738	+	0.230729i	−0.999662	−	0.0259806i	\(-0.991729\pi\)
							0.966489	+	0.256709i	\(0.0826383\pi\)
\(47\)	−0.776469	−	1.70023i	−0.113260	−	0.248004i	0.844511	−	0.535539i	\(-0.179891\pi\)
							−0.957770	+	0.287535i	\(0.907164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.2.e.c.62.2	yes	20
3.2	odd	2		603.2.u.c.397.2		20
67.24	even	11		4489.2.a.l.1.10		10
67.40	even	11	inner	67.2.e.c.40.2	✓	20
67.43	odd	22		4489.2.a.m.1.1		10
201.107	odd	22		603.2.u.c.442.2		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.e.c.40.2	✓	20	67.40	even	11	inner
67.2.e.c.62.2	yes	20	1.1	even	1	trivial
603.2.u.c.397.2		20	3.2	odd	2
603.2.u.c.442.2		20	201.107	odd	22
4489.2.a.l.1.10		10	67.24	even	11
4489.2.a.m.1.1		10	67.43	odd	22