Embedded newform 49.6.c.a.18.1

• Label: 49.6.c.a.18.1
• Level: $49$
• Weight: $6$
• Character: 49.18
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -7
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			18.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	49.18
Dual form			49.6.c.a.30.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.50000 - 9.52628i) q^{2} +(-44.5000 + 77.0763i) q^{4} +627.000 q^{8} +(121.500 + 210.444i) q^{9} +(38.0000 - 65.8179i) q^{11} +(-2024.50 - 3506.54i) q^{16} +(1336.50 - 2314.89i) q^{18} -836.000 q^{22} +(2476.00 + 4288.56i) q^{23} +(1562.50 - 2706.33i) q^{25} +7282.00 q^{29} +(-12237.5 + 21196.0i) q^{32} -21627.0 q^{36} +(4443.00 + 7695.50i) q^{37} +11748.0 q^{43} +(3382.00 + 5857.80i) q^{44} +(27236.0 - 47174.1i) q^{46} -34375.0 q^{50} +(-12275.0 + 21260.9i) q^{53} +(-40051.0 - 69370.4i) q^{58} +139657. q^{64} +(-34682.0 + 60071.0i) q^{67} -2224.00 q^{71} +(76180.5 + 131948. i) q^{72} +(48873.0 - 84650.5i) q^{74} +(-40084.0 - 69427.5i) q^{79} +(-29524.5 + 51137.9i) q^{81} +(-64614.0 - 111915. i) q^{86} +(23826.0 - 41267.8i) q^{88} -440728. q^{92} +18468.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 11 * q^2 - 89 * q^4 + 1254 * q^8 + 243 * q^9 + 76 * q^11 - 4049 * q^16 + 2673 * q^18 - 1672 * q^22 + 4952 * q^23 + 3125 * q^25 + 14564 * q^29 - 24475 * q^32 - 43254 * q^36 + 8886 * q^37 + 23496 * q^43 + 6764 * q^44 + 54472 * q^46 - 68750 * q^50 - 24550 * q^53 - 80102 * q^58 + 279314 * q^64 - 69364 * q^67 - 4448 * q^71 + 152361 * q^72 + 97746 * q^74 - 80168 * q^79 - 59049 * q^81 - 129228 * q^86 + 47652 * q^88 - 881456 * q^92 + 36936 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.50000	−	9.52628i	−0.972272	−	1.68402i	−0.688659	−	0.725085i	\(-0.741800\pi\)
							−0.283613	−	0.958939i	\(-0.591533\pi\)
\(3\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(5\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(7\)		0			0
							\(11\)	38.0000	−	65.8179i	0.0946895	−	0.164007i	−0.814789	−	0.579757i	\(-0.803148\pi\)
0.909479	+	0.415750i	\(0.136481\pi\)
\(13\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(17\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(19\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(23\)	2476.00	+	4288.56i	0.975958	+	1.69041i	0.676737	+	0.736225i	\(0.263393\pi\)
							0.299220	+	0.954184i	\(0.403273\pi\)
\(29\)	7282.00			1.60789			0.803944	−	0.594705i	\(-0.202731\pi\)
							0.803944	+	0.594705i	\(0.202731\pi\)
\(31\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(37\)	4443.00	+	7695.50i	0.533546	+	0.924129i	0.999232	+	0.0391791i	\(0.0124743\pi\)
							−0.465686	+	0.884950i	\(0.654192\pi\)
\(41\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(43\)	11748.0			0.968931			0.484465	−	0.874810i	\(-0.339014\pi\)
							0.484465	+	0.874810i	\(0.339014\pi\)
\(47\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(53\)	−12275.0	+	21260.9i	−0.600250	+	1.03966i	0.392533	+	0.919738i	\(0.371599\pi\)
							−0.992783	+	0.119925i	\(0.961735\pi\)
\(59\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(61\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(67\)	−34682.0	+	60071.0i	−0.943881	+	1.63485i	−0.185904	+	0.982568i	\(0.559521\pi\)
							−0.757977	+	0.652281i	\(0.773812\pi\)
\(71\)	−2224.00			−0.0523587			−0.0261794	−	0.999657i	\(-0.508334\pi\)
							−0.0261794	+	0.999657i	\(0.508334\pi\)
\(73\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(79\)	−40084.0	−	69427.5i	−0.722609	−	1.25160i	−0.959951	−	0.280169i	\(-0.909609\pi\)
							0.237342	−	0.971426i	\(-0.423724\pi\)
\(83\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(89\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(97\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.c.a.18.1		2
7.2	even	3	inner	49.6.c.a.30.1		2
7.3	odd	6		49.6.a.b.1.1	✓	1
7.4	even	3		49.6.a.b.1.1	✓	1
7.5	odd	6	inner	49.6.c.a.30.1		2
7.6	odd	2	CM	49.6.c.a.18.1		2
21.11	odd	6		441.6.a.a.1.1		1
21.17	even	6		441.6.a.a.1.1		1
28.3	even	6		784.6.a.g.1.1		1
28.11	odd	6		784.6.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.a.b.1.1	✓	1	7.3	odd	6
49.6.a.b.1.1	✓	1	7.4	even	3
49.6.c.a.18.1		2	1.1	even	1	trivial
49.6.c.a.18.1		2	7.6	odd	2	CM
49.6.c.a.30.1		2	7.2	even	3	inner
49.6.c.a.30.1		2	7.5	odd	6	inner
441.6.a.a.1.1		1	21.11	odd	6
441.6.a.a.1.1		1	21.17	even	6
784.6.a.g.1.1		1	28.3	even	6
784.6.a.g.1.1		1	28.11	odd	6