Embedded newform 624.4.a.d.1.1

• Label: 624.4.a.d.1.1
• Level: $624$
• Weight: $4$
• Character: 624.1
• Self dual: yes
• Analytic conductor: $36.817$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(36.8171918436\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	624.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{3} +10.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} -40.0000 q^{11} +13.0000 q^{13} -30.0000 q^{15} +130.000 q^{17} +20.0000 q^{19} -24.0000 q^{21} -25.0000 q^{25} -27.0000 q^{27} -18.0000 q^{29} +184.000 q^{31} +120.000 q^{33} +80.0000 q^{35} -74.0000 q^{37} -39.0000 q^{39} -362.000 q^{41} -76.0000 q^{43} +90.0000 q^{45} +452.000 q^{47} -279.000 q^{49} -390.000 q^{51} +382.000 q^{53} -400.000 q^{55} -60.0000 q^{57} -464.000 q^{59} +358.000 q^{61} +72.0000 q^{63} +130.000 q^{65} +700.000 q^{67} +748.000 q^{71} +1058.00 q^{73} +75.0000 q^{75} -320.000 q^{77} +976.000 q^{79} +81.0000 q^{81} +1008.00 q^{83} +1300.00 q^{85} +54.0000 q^{87} -386.000 q^{89} +104.000 q^{91} -552.000 q^{93} +200.000 q^{95} -614.000 q^{97} -360.000 q^{99} +O(q^{100})\)

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
			\(3\)	−3.00000	−0.577350
\(5\)	10.0000	0.894427				0.447214	−	0.894427i	\(-0.352416\pi\)
			0.447214	+	0.894427i	\(0.352416\pi\)
\(7\)	8.00000	0.431959	0.215980	−	0.976398i	\(-0.430705\pi\)
			0.215980	+	0.976398i	\(0.430705\pi\)
\(11\)	−40.0000	−1.09640	−0.548202	−	0.836346i	\(-0.684688\pi\)
			−0.548202	+	0.836346i	\(0.684688\pi\)
\(13\)	13.0000	0.277350
			\(17\)	130.000	1.85468	0.927342	−	0.374215i	\(-0.122088\pi\)
0.927342	+	0.374215i				\(0.122088\pi\)
\(19\)	20.0000	0.241490	0.120745	−	0.992684i	\(-0.461472\pi\)
			0.120745	+	0.992684i	\(0.461472\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−18.0000	−0.115259	−0.0576296	−	0.998338i	\(-0.518354\pi\)
			−0.0576296	+	0.998338i	\(0.518354\pi\)
\(31\)	184.000	1.06604	0.533022	−	0.846101i	\(-0.321056\pi\)
			0.533022	+	0.846101i	\(0.321056\pi\)
\(37\)	−74.0000	−0.328798	−0.164399	−	0.986394i	\(-0.552568\pi\)
			−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	−362.000	−1.37890	−0.689450	−	0.724333i	\(-0.742148\pi\)
			−0.689450	+	0.724333i	\(0.742148\pi\)
\(43\)	−76.0000	−0.269532	−0.134766	−	0.990877i	\(-0.543028\pi\)
			−0.134766	+	0.990877i	\(0.543028\pi\)
\(47\)	452.000	1.40279	0.701393	−	0.712774i	\(-0.252562\pi\)
			0.701393	+	0.712774i	\(0.252562\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.d.1.1		1
3.2	odd	2		1872.4.a.d.1.1		1
4.3	odd	2		78.4.a.c.1.1	✓	1
8.3	odd	2		2496.4.a.a.1.1		1
8.5	even	2		2496.4.a.j.1.1		1
12.11	even	2		234.4.a.h.1.1		1
20.19	odd	2		1950.4.a.l.1.1		1
52.31	even	4		1014.4.b.h.337.2		2
52.47	even	4		1014.4.b.h.337.1		2
52.51	odd	2		1014.4.a.j.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.c.1.1	✓	1	4.3	odd	2
234.4.a.h.1.1		1	12.11	even	2
624.4.a.d.1.1		1	1.1	even	1	trivial
1014.4.a.j.1.1		1	52.51	odd	2
1014.4.b.h.337.1		2	52.47	even	4
1014.4.b.h.337.2		2	52.31	even	4
1872.4.a.d.1.1		1	3.2	odd	2
1950.4.a.l.1.1		1	20.19	odd	2
2496.4.a.a.1.1		1	8.3	odd	2
2496.4.a.j.1.1		1	8.5	even	2