Embedded newform 784.2.i.c.177.1

• Label: 784.2.i.c.177.1
• Level: $784$
• Weight: $2$
• Character: 784.177
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$784 = 2^{4} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	784.i (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			177.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	784.177
Dual form			784.2.i.c.753.1

$q$-expansion

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{9} -4.00000 q^{13} +(-3.00000 - 5.19615i) q^{17} +(1.00000 - 1.73205i) q^{19} +(2.50000 + 4.33013i) q^{25} -4.00000 q^{27} -6.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(-1.00000 + 1.73205i) q^{37} +(4.00000 + 6.92820i) q^{39} +6.00000 q^{41} -8.00000 q^{43} +(-6.00000 + 10.3923i) q^{47} +(-6.00000 + 10.3923i) q^{51} +(-3.00000 - 5.19615i) q^{53} -4.00000 q^{57} +(-3.00000 - 5.19615i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(-2.00000 - 3.46410i) q^{67} +(-1.00000 - 1.73205i) q^{73} +(5.00000 - 8.66025i) q^{75} +(4.00000 - 6.92820i) q^{79} +(5.50000 + 9.52628i) q^{81} +6.00000 q^{83} +(6.00000 + 10.3923i) q^{87} +(3.00000 - 5.19615i) q^{89} +(-4.00000 + 6.92820i) q^{93} -10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 2 * q^3 - q^9 - 8 * q^13 - 6 * q^17 + 2 * q^19 + 5 * q^25 - 8 * q^27 - 12 * q^29 - 4 * q^31 - 2 * q^37 + 8 * q^39 + 12 * q^41 - 16 * q^43 - 12 * q^47 - 12 * q^51 - 6 * q^53 - 8 * q^57 - 6 * q^59 - 8 * q^61 - 4 * q^67 - 2 * q^73 + 10 * q^75 + 8 * q^79 + 11 * q^81 + 12 * q^83 + 12 * q^87 + 6 * q^89 - 8 * q^93 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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			0
							$3$	−1.00000	−	1.73205i	−0.577350	−	1.00000i	−0.995782	−	0.0917517i	$-0.970753\pi$
0.418432	−	0.908248i	$-0.362580\pi$
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
							0.866025	+	0.500000i	$0.166667\pi$
$7$		0			0
							$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
−0.866025	+	0.500000i	$0.833333\pi$
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
							−0.554700	+	0.832050i	$0.687167\pi$
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	−0.957892	−	0.287129i	$-0.907299\pi$
							0.230285	−	0.973123i	$-0.426034\pi$
$19$	1.00000	−	1.73205i	0.229416	−	0.397360i	−0.728219	−	0.685344i	$-0.759652\pi$
							0.957635	+	0.287984i	$0.0929851\pi$
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
							0.866025	+	0.500000i	$0.166667\pi$
$29$	−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	0.628619	−	0.777714i	$-0.283621\pi$
							−0.987829	+	0.155543i	$0.950287\pi$
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	−0.936442	−	0.350823i	$-0.885902\pi$
							0.772043	+	0.635571i	$0.219235\pi$
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
							0.468521	+	0.883452i	$0.344787\pi$
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
							−0.609994	+	0.792406i	$0.708828\pi$
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.0186297	+	0.999826i	$0.505930\pi$
							−0.856560	+	0.516047i	$0.827403\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.i.c.177.1		2
4.3	odd	2		98.2.c.b.79.1		2
7.2	even	3		112.2.a.c.1.1		1
7.3	odd	6		784.2.i.i.753.1		2
7.4	even	3	inner	784.2.i.c.753.1		2
7.5	odd	6		784.2.a.b.1.1		1
7.6	odd	2		784.2.i.i.177.1		2
12.11	even	2		882.2.g.c.667.1		2
21.2	odd	6		1008.2.a.h.1.1		1
21.5	even	6		7056.2.a.bd.1.1		1
28.3	even	6		98.2.c.a.67.1		2
28.11	odd	6		98.2.c.b.67.1		2
28.19	even	6		98.2.a.a.1.1		1
28.23	odd	6		14.2.a.a.1.1	✓	1
28.27	even	2		98.2.c.a.79.1		2
35.2	odd	12		2800.2.g.h.449.1		2
35.9	even	6		2800.2.a.g.1.1		1
35.23	odd	12		2800.2.g.h.449.2		2
56.5	odd	6		3136.2.a.z.1.1		1
56.19	even	6		3136.2.a.e.1.1		1
56.37	even	6		448.2.a.a.1.1		1
56.51	odd	6		448.2.a.g.1.1		1
84.11	even	6		882.2.g.c.361.1		2
84.23	even	6		126.2.a.b.1.1		1
84.47	odd	6		882.2.a.i.1.1		1
84.59	odd	6		882.2.g.d.361.1		2
84.83	odd	2		882.2.g.d.667.1		2
112.37	even	12		1792.2.b.g.897.1		2
112.51	odd	12		1792.2.b.c.897.1		2
112.93	even	12		1792.2.b.g.897.2		2
112.107	odd	12		1792.2.b.c.897.2		2
140.19	even	6		2450.2.a.t.1.1		1
140.23	even	12		350.2.c.d.99.2		2
140.47	odd	12		2450.2.c.c.99.1		2
140.79	odd	6		350.2.a.f.1.1		1
140.103	odd	12		2450.2.c.c.99.2		2
140.107	even	12		350.2.c.d.99.1		2
168.107	even	6		4032.2.a.w.1.1		1
168.149	odd	6		4032.2.a.r.1.1		1
252.23	even	6		1134.2.f.f.379.1		2
252.79	odd	6		1134.2.f.l.757.1		2
252.191	even	6		1134.2.f.f.757.1		2
252.247	odd	6		1134.2.f.l.379.1		2
308.219	even	6		1694.2.a.e.1.1		1
364.51	odd	6		2366.2.a.j.1.1		1
364.135	even	12		2366.2.d.b.337.2		2
364.359	even	12		2366.2.d.b.337.1		2
420.23	odd	12		3150.2.g.j.2899.1		2
420.107	odd	12		3150.2.g.j.2899.2		2
420.359	even	6		3150.2.a.i.1.1		1
476.135	odd	6		4046.2.a.f.1.1		1
532.303	even	6		5054.2.a.c.1.1		1
644.275	even	6		7406.2.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.2.a.a.1.1	✓	1	28.23	odd	6
98.2.a.a.1.1		1	28.19	even	6
98.2.c.a.67.1		2	28.3	even	6
98.2.c.a.79.1		2	28.27	even	2
98.2.c.b.67.1		2	28.11	odd	6
98.2.c.b.79.1		2	4.3	odd	2
112.2.a.c.1.1		1	7.2	even	3
126.2.a.b.1.1		1	84.23	even	6
350.2.a.f.1.1		1	140.79	odd	6
350.2.c.d.99.1		2	140.107	even	12
350.2.c.d.99.2		2	140.23	even	12
448.2.a.a.1.1		1	56.37	even	6
448.2.a.g.1.1		1	56.51	odd	6
784.2.a.b.1.1		1	7.5	odd	6
784.2.i.c.177.1		2	1.1	even	1	trivial
784.2.i.c.753.1		2	7.4	even	3	inner
784.2.i.i.177.1		2	7.6	odd	2
784.2.i.i.753.1		2	7.3	odd	6
882.2.a.i.1.1		1	84.47	odd	6
882.2.g.c.361.1		2	84.11	even	6
882.2.g.c.667.1		2	12.11	even	2
882.2.g.d.361.1		2	84.59	odd	6
882.2.g.d.667.1		2	84.83	odd	2
1008.2.a.h.1.1		1	21.2	odd	6
1134.2.f.f.379.1		2	252.23	even	6
1134.2.f.f.757.1		2	252.191	even	6
1134.2.f.l.379.1		2	252.247	odd	6
1134.2.f.l.757.1		2	252.79	odd	6
1694.2.a.e.1.1		1	308.219	even	6
1792.2.b.c.897.1		2	112.51	odd	12
1792.2.b.c.897.2		2	112.107	odd	12
1792.2.b.g.897.1		2	112.37	even	12
1792.2.b.g.897.2		2	112.93	even	12
2366.2.a.j.1.1		1	364.51	odd	6
2366.2.d.b.337.1		2	364.359	even	12
2366.2.d.b.337.2		2	364.135	even	12
2450.2.a.t.1.1		1	140.19	even	6
2450.2.c.c.99.1		2	140.47	odd	12
2450.2.c.c.99.2		2	140.103	odd	12
2800.2.a.g.1.1		1	35.9	even	6
2800.2.g.h.449.1		2	35.2	odd	12
2800.2.g.h.449.2		2	35.23	odd	12
3136.2.a.e.1.1		1	56.19	even	6
3136.2.a.z.1.1		1	56.5	odd	6
3150.2.a.i.1.1		1	420.359	even	6
3150.2.g.j.2899.1		2	420.23	odd	12
3150.2.g.j.2899.2		2	420.107	odd	12
4032.2.a.r.1.1		1	168.149	odd	6
4032.2.a.w.1.1		1	168.107	even	6
4046.2.a.f.1.1		1	476.135	odd	6
5054.2.a.c.1.1		1	532.303	even	6
7056.2.a.bd.1.1		1	21.5	even	6
7406.2.a.a.1.1		1	644.275	even	6