Embedded newform 98.2.c.a.67.1

• Label: 98.2.c.a.67.1
• Level: $98$
• Weight: $2$
• Character: 98.67
• Analytic conductor: $0.783$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.782533939809\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			67.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	98.67
Dual form			98.2.c.a.79.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-1.00000 + 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{4} -2.00000 q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} - q^{4} - 4 q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	+	0.866025i	0.353553	+	0.612372i
							\(3\)	−1.00000	+	1.73205i	−0.577350	+	1.00000i	0.418432	+	0.908248i	\(0.362580\pi\)
−0.995782	+	0.0917517i	\(0.970753\pi\)
\(5\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(7\)		0			0
							\(11\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
0.866025	+	0.500000i	\(0.166667\pi\)
\(13\)	4.00000			1.10940			0.554700	−	0.832050i	\(-0.312833\pi\)
							0.554700	+	0.832050i	\(0.312833\pi\)
\(17\)	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	\(-0.573966\pi\)
							0.957892	−	0.287129i	\(-0.0927008\pi\)
\(19\)	1.00000	+	1.73205i	0.229416	+	0.397360i	0.957635	−	0.287984i	\(-0.0929851\pi\)
							−0.728219	+	0.685344i	\(0.759652\pi\)
\(23\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(29\)	−6.00000			−1.11417			−0.557086	−	0.830455i	\(-0.688081\pi\)
							−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	\(-0.950287\pi\)
							0.628619	+	0.777714i	\(0.283621\pi\)
\(37\)	−1.00000	−	1.73205i	−0.164399	−	0.284747i	0.772043	−	0.635571i	\(-0.219235\pi\)
							−0.936442	+	0.350823i	\(0.885902\pi\)
\(41\)	−6.00000			−0.937043			−0.468521	−	0.883452i	\(-0.655213\pi\)
							−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	8.00000			1.21999			0.609994	−	0.792406i	\(-0.291172\pi\)
							0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.856560	−	0.516047i	\(-0.827403\pi\)
							−0.0186297	−	0.999826i	\(-0.505930\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.2.c.a.67.1		2
3.2	odd	2		882.2.g.d.361.1		2
4.3	odd	2		784.2.i.i.753.1		2
7.2	even	3	inner	98.2.c.a.79.1		2
7.3	odd	6		14.2.a.a.1.1	✓	1
7.4	even	3		98.2.a.a.1.1		1
7.5	odd	6		98.2.c.b.79.1		2
7.6	odd	2		98.2.c.b.67.1		2
21.2	odd	6		882.2.g.d.667.1		2
21.5	even	6		882.2.g.c.667.1		2
21.11	odd	6		882.2.a.i.1.1		1
21.17	even	6		126.2.a.b.1.1		1
21.20	even	2		882.2.g.c.361.1		2
28.3	even	6		112.2.a.c.1.1		1
28.11	odd	6		784.2.a.b.1.1		1
28.19	even	6		784.2.i.c.177.1		2
28.23	odd	6		784.2.i.i.177.1		2
28.27	even	2		784.2.i.c.753.1		2
35.3	even	12		350.2.c.d.99.2		2
35.4	even	6		2450.2.a.t.1.1		1
35.17	even	12		350.2.c.d.99.1		2
35.18	odd	12		2450.2.c.c.99.2		2
35.24	odd	6		350.2.a.f.1.1		1
35.32	odd	12		2450.2.c.c.99.1		2
56.3	even	6		448.2.a.a.1.1		1
56.11	odd	6		3136.2.a.z.1.1		1
56.45	odd	6		448.2.a.g.1.1		1
56.53	even	6		3136.2.a.e.1.1		1
63.31	odd	6		1134.2.f.l.379.1		2
63.38	even	6		1134.2.f.f.757.1		2
63.52	odd	6		1134.2.f.l.757.1		2
63.59	even	6		1134.2.f.f.379.1		2
77.10	even	6		1694.2.a.e.1.1		1
84.11	even	6		7056.2.a.bd.1.1		1
84.59	odd	6		1008.2.a.h.1.1		1
91.31	even	12		2366.2.d.b.337.2		2
91.38	odd	6		2366.2.a.j.1.1		1
91.73	even	12		2366.2.d.b.337.1		2
105.17	odd	12		3150.2.g.j.2899.2		2
105.38	odd	12		3150.2.g.j.2899.1		2
105.59	even	6		3150.2.a.i.1.1		1
112.3	even	12		1792.2.b.g.897.2		2
112.45	odd	12		1792.2.b.c.897.1		2
112.59	even	12		1792.2.b.g.897.1		2
112.101	odd	12		1792.2.b.c.897.2		2
119.101	odd	6		4046.2.a.f.1.1		1
133.94	even	6		5054.2.a.c.1.1		1
140.3	odd	12		2800.2.g.h.449.2		2
140.59	even	6		2800.2.a.g.1.1		1
140.87	odd	12		2800.2.g.h.449.1		2
161.45	even	6		7406.2.a.a.1.1		1
168.59	odd	6		4032.2.a.r.1.1		1
168.101	even	6		4032.2.a.w.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.2.a.a.1.1	✓	1	7.3	odd	6
98.2.a.a.1.1		1	7.4	even	3
98.2.c.a.67.1		2	1.1	even	1	trivial
98.2.c.a.79.1		2	7.2	even	3	inner
98.2.c.b.67.1		2	7.6	odd	2
98.2.c.b.79.1		2	7.5	odd	6
112.2.a.c.1.1		1	28.3	even	6
126.2.a.b.1.1		1	21.17	even	6
350.2.a.f.1.1		1	35.24	odd	6
350.2.c.d.99.1		2	35.17	even	12
350.2.c.d.99.2		2	35.3	even	12
448.2.a.a.1.1		1	56.3	even	6
448.2.a.g.1.1		1	56.45	odd	6
784.2.a.b.1.1		1	28.11	odd	6
784.2.i.c.177.1		2	28.19	even	6
784.2.i.c.753.1		2	28.27	even	2
784.2.i.i.177.1		2	28.23	odd	6
784.2.i.i.753.1		2	4.3	odd	2
882.2.a.i.1.1		1	21.11	odd	6
882.2.g.c.361.1		2	21.20	even	2
882.2.g.c.667.1		2	21.5	even	6
882.2.g.d.361.1		2	3.2	odd	2
882.2.g.d.667.1		2	21.2	odd	6
1008.2.a.h.1.1		1	84.59	odd	6
1134.2.f.f.379.1		2	63.59	even	6
1134.2.f.f.757.1		2	63.38	even	6
1134.2.f.l.379.1		2	63.31	odd	6
1134.2.f.l.757.1		2	63.52	odd	6
1694.2.a.e.1.1		1	77.10	even	6
1792.2.b.c.897.1		2	112.45	odd	12
1792.2.b.c.897.2		2	112.101	odd	12
1792.2.b.g.897.1		2	112.59	even	12
1792.2.b.g.897.2		2	112.3	even	12
2366.2.a.j.1.1		1	91.38	odd	6
2366.2.d.b.337.1		2	91.73	even	12
2366.2.d.b.337.2		2	91.31	even	12
2450.2.a.t.1.1		1	35.4	even	6
2450.2.c.c.99.1		2	35.32	odd	12
2450.2.c.c.99.2		2	35.18	odd	12
2800.2.a.g.1.1		1	140.59	even	6
2800.2.g.h.449.1		2	140.87	odd	12
2800.2.g.h.449.2		2	140.3	odd	12
3136.2.a.e.1.1		1	56.53	even	6
3136.2.a.z.1.1		1	56.11	odd	6
3150.2.a.i.1.1		1	105.59	even	6
3150.2.g.j.2899.1		2	105.38	odd	12
3150.2.g.j.2899.2		2	105.17	odd	12
4032.2.a.r.1.1		1	168.59	odd	6
4032.2.a.w.1.1		1	168.101	even	6
4046.2.a.f.1.1		1	119.101	odd	6
5054.2.a.c.1.1		1	133.94	even	6
7056.2.a.bd.1.1		1	84.11	even	6
7406.2.a.a.1.1		1	161.45	even	6