Embedded newform 252.6.a.d.1.1

• Label: 252.6.a.d.1.1
• Level: $252$
• Weight: $6$
• Character: 252.1
• Self dual: yes
• Analytic conductor: $40.417$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(40.4167225929\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	252.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+96.0000 q^{5} +49.0000 q^{7} +720.000 q^{11} +572.000 q^{13} -1254.00 q^{17} -94.0000 q^{19} -96.0000 q^{23} +6091.00 q^{25} +4374.00 q^{29} -6244.00 q^{31} +4704.00 q^{35} -10798.0 q^{37} -12006.0 q^{41} -9160.00 q^{43} +25836.0 q^{47} +2401.00 q^{49} -1014.00 q^{53} +69120.0 q^{55} -1242.00 q^{59} +7592.00 q^{61} +54912.0 q^{65} +41132.0 q^{67} +37632.0 q^{71} -13438.0 q^{73} +35280.0 q^{77} +6248.00 q^{79} +25254.0 q^{83} -120384. q^{85} +45126.0 q^{89} +28028.0 q^{91} -9024.00 q^{95} +107222. q^{97} +O(q^{100})\)

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
			\(3\)	0	0
\(5\)	96.0000	1.71730				0.858650	−	0.512562i	\(-0.171304\pi\)
			0.858650	+	0.512562i	\(0.171304\pi\)
\(7\)	49.0000	0.377964
			\(11\)	720.000	1.79412	0.897059	−	0.441912i	\(-0.145700\pi\)
0.897059	+	0.441912i				\(0.145700\pi\)
\(13\)	572.000	0.938723	0.469362	−	0.883006i	\(-0.344484\pi\)
			0.469362	+	0.883006i	\(0.344484\pi\)
\(17\)	−1254.00	−1.05239	−0.526193	−	0.850365i	\(-0.676381\pi\)
			−0.526193	+	0.850365i	\(0.676381\pi\)
\(19\)	−94.0000	−0.0597371	−0.0298685	−	0.999554i	\(-0.509509\pi\)
			−0.0298685	+	0.999554i	\(0.509509\pi\)
\(23\)	−96.0000	−0.0378400	−0.0189200	−	0.999821i	\(-0.506023\pi\)
			−0.0189200	+	0.999821i	\(0.506023\pi\)
\(29\)	4374.00	0.965792	0.482896	−	0.875678i	\(-0.339585\pi\)
			0.482896	+	0.875678i	\(0.339585\pi\)
\(31\)	−6244.00	−1.16697	−0.583484	−	0.812125i	\(-0.698311\pi\)
			−0.583484	+	0.812125i	\(0.698311\pi\)
\(37\)	−10798.0	−1.29670	−0.648349	−	0.761343i	\(-0.724540\pi\)
			−0.648349	+	0.761343i	\(0.724540\pi\)
\(41\)	−12006.0	−1.11542	−0.557710	−	0.830036i	\(-0.688320\pi\)
			−0.557710	+	0.830036i	\(0.688320\pi\)
\(43\)	−9160.00	−0.755482	−0.377741	−	0.925911i	\(-0.623299\pi\)
			−0.377741	+	0.925911i	\(0.623299\pi\)
\(47\)	25836.0	1.70601	0.853003	−	0.521906i	\(-0.174779\pi\)
			0.853003	+	0.521906i	\(0.174779\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.6.a.d.1.1		1
3.2	odd	2		28.6.a.a.1.1	✓	1
4.3	odd	2		1008.6.a.bb.1.1		1
12.11	even	2		112.6.a.e.1.1		1
15.2	even	4		700.6.e.d.449.2		2
15.8	even	4		700.6.e.d.449.1		2
15.14	odd	2		700.6.a.d.1.1		1
21.2	odd	6		196.6.e.f.165.1		2
21.5	even	6		196.6.e.e.165.1		2
21.11	odd	6		196.6.e.f.177.1		2
21.17	even	6		196.6.e.e.177.1		2
21.20	even	2		196.6.a.d.1.1		1
24.5	odd	2		448.6.a.i.1.1		1
24.11	even	2		448.6.a.h.1.1		1
84.83	odd	2		784.6.a.f.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.a.a.1.1	✓	1	3.2	odd	2
112.6.a.e.1.1		1	12.11	even	2
196.6.a.d.1.1		1	21.20	even	2
196.6.e.e.165.1		2	21.5	even	6
196.6.e.e.177.1		2	21.17	even	6
196.6.e.f.165.1		2	21.2	odd	6
196.6.e.f.177.1		2	21.11	odd	6
252.6.a.d.1.1		1	1.1	even	1	trivial
448.6.a.h.1.1		1	24.11	even	2
448.6.a.i.1.1		1	24.5	odd	2
700.6.a.d.1.1		1	15.14	odd	2
700.6.e.d.449.1		2	15.8	even	4
700.6.e.d.449.2		2	15.2	even	4
784.6.a.f.1.1		1	84.83	odd	2
1008.6.a.bb.1.1		1	4.3	odd	2