Embedded newform 67.2.e.c.15.1

• Label: 67.2.e.c.15.1
• Level: $67$
• Weight: $2$
• Character: 67.15
• 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			15.1
Root			\(-1.27213 + 2.78557i\) of defining polynomial
Character	\(\chi\)	\(=\)	67.15
Dual form			67.2.e.c.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.239446 + 0.153882i) q^{2} +(-2.47877 + 0.727832i) q^{3} +(-0.797176 + 1.74557i) q^{4} +(1.21919 + 1.40702i) q^{5} +(0.481530 - 0.555715i) q^{6} +(-1.14774 + 0.737606i) q^{7} +(-0.158746 - 1.10411i) q^{8} +(3.09080 - 1.98634i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} - 4 q^{3} - 4 q^{4} + q^{5} + 3 q^{6} - 8 q^{7} - 22 q^{8} + 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.239446	+	0.153882i	−0.169314	+	0.108811i	−0.622553	−	0.782577i	\(-0.713905\pi\)
							0.453240	+	0.891389i	\(0.350268\pi\)
\(3\)	−2.47877	+	0.727832i	−1.43112	+	0.420214i	−0.903251	−	0.429113i	\(-0.858826\pi\)
							−0.527867	+	0.849327i	\(0.677008\pi\)
\(5\)	1.21919	+	1.40702i	0.545237	+	0.629237i	0.959767	−	0.280799i	\(-0.0905994\pi\)
							−0.414530	+	0.910036i	\(0.636054\pi\)
\(7\)	−1.14774	+	0.737606i	−0.433804	+	0.278789i	−0.739264	−	0.673416i	\(-0.764826\pi\)
							0.305460	+	0.952205i	\(0.401190\pi\)
\(11\)	1.59592	+	1.84179i	0.481188	+	0.555320i	0.943490	−	0.331402i	\(-0.107522\pi\)
							−0.462302	+	0.886723i	\(0.652976\pi\)
\(13\)	0.666114	−	4.63292i	0.184747	−	1.28494i	−0.660606	−	0.750733i	\(-0.729701\pi\)
							0.845353	−	0.534209i	\(-0.179390\pi\)
\(17\)	2.99601	+	6.56035i	0.726640	+	1.59112i	0.804359	+	0.594144i	\(0.202509\pi\)
							−0.0777190	+	0.996975i	\(0.524764\pi\)
\(19\)	4.79541	+	3.08183i	1.10014	+	0.707019i	0.959124	−	0.282988i	\(-0.0913255\pi\)
							0.141020	+	0.990007i	\(0.454962\pi\)
\(23\)	−5.17917	+	1.52074i	−1.07993	+	0.317097i	−0.772852	−	0.634587i	\(-0.781170\pi\)
							−0.307081	+	0.951683i	\(0.599352\pi\)
\(29\)	4.41509			0.819862			0.409931	−	0.912117i	\(-0.365553\pi\)
							0.409931	+	0.912117i	\(0.365553\pi\)
\(31\)	0.185095	+	1.28737i	0.0332441	+	0.231218i	0.999669	−	0.0257319i	\(-0.00819162\pi\)
							−0.966425	+	0.256950i	\(0.917283\pi\)
\(37\)	−4.48772			−0.737776			−0.368888	−	0.929474i	\(-0.620261\pi\)
							−0.368888	+	0.929474i	\(0.620261\pi\)
\(41\)	1.16726	+	2.55595i	0.182296	+	0.399172i	0.978614	−	0.205706i	\(-0.0659491\pi\)
							−0.796318	+	0.604878i	\(0.793222\pi\)
\(43\)	−4.94442	−	10.8268i	−0.754016	−	1.65107i	−0.759011	−	0.651078i	\(-0.774317\pi\)
							0.00499410	−	0.999988i	\(-0.498410\pi\)
\(47\)	6.10467	−	1.79249i	0.890457	−	0.261462i	0.195664	−	0.980671i	\(-0.437314\pi\)
							0.694793	+	0.719209i	\(0.255496\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.2.e.c.15.1	yes	20
3.2	odd	2		603.2.u.c.82.1		20
67.3	odd	22		4489.2.a.m.1.5		10
67.9	even	11	inner	67.2.e.c.9.1	✓	20
67.64	even	11		4489.2.a.l.1.6		10
201.143	odd	22		603.2.u.c.478.1		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.e.c.9.1	✓	20	67.9	even	11	inner
67.2.e.c.15.1	yes	20	1.1	even	1	trivial
603.2.u.c.82.1		20	3.2	odd	2
603.2.u.c.478.1		20	201.143	odd	22
4489.2.a.l.1.6		10	67.64	even	11
4489.2.a.m.1.5		10	67.3	odd	22