Embedded newform 1200.4.a.bb.1.1

• Label: 1200.4.a.bb.1.1
• Level: $1200$
• Weight: $4$
• Character: 1200.1
• Self dual: yes
• Analytic conductor: $70.802$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(70.8022920069\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1200.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+3.00000 q^{3} +1.00000 q^{7} +9.00000 q^{9} -42.0000 q^{11} -67.0000 q^{13} +54.0000 q^{17} +115.000 q^{19} +3.00000 q^{21} +162.000 q^{23} +27.0000 q^{27} -210.000 q^{29} +193.000 q^{31} -126.000 q^{33} -286.000 q^{37} -201.000 q^{39} +12.0000 q^{41} -263.000 q^{43} -414.000 q^{47} -342.000 q^{49} +162.000 q^{51} -192.000 q^{53} +345.000 q^{57} -690.000 q^{59} -733.000 q^{61} +9.00000 q^{63} -299.000 q^{67} +486.000 q^{69} +228.000 q^{71} +938.000 q^{73} -42.0000 q^{77} +160.000 q^{79} +81.0000 q^{81} +462.000 q^{83} -630.000 q^{87} -240.000 q^{89} -67.0000 q^{91} +579.000 q^{93} -511.000 q^{97} -378.000 q^{99} +O(q^{100})\)

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
\(5\)	0	0
			\(7\)	1.00000	0.0539949	0.0269975	−	0.999636i	\(-0.491405\pi\)
0.0269975	+	0.999636i				\(0.491405\pi\)
\(11\)	−42.0000	−1.15123	−0.575613	−	0.817723i	\(-0.695236\pi\)
			−0.575613	+	0.817723i	\(0.695236\pi\)
\(13\)	−67.0000	−1.42942	−0.714710	−	0.699421i	\(-0.753441\pi\)
			−0.714710	+	0.699421i	\(0.753441\pi\)
\(17\)	54.0000	0.770407	0.385204	−	0.922832i	\(-0.374131\pi\)
			0.385204	+	0.922832i	\(0.374131\pi\)
\(19\)	115.000	1.38857	0.694284	−	0.719701i	\(-0.255721\pi\)
			0.694284	+	0.719701i	\(0.255721\pi\)
\(23\)	162.000	1.46867	0.734333	−	0.678789i	\(-0.237495\pi\)
			0.734333	+	0.678789i	\(0.237495\pi\)
\(29\)	−210.000	−1.34469	−0.672345	−	0.740238i	\(-0.734713\pi\)
			−0.672345	+	0.740238i	\(0.734713\pi\)
\(31\)	193.000	1.11819	0.559094	−	0.829104i	\(-0.311149\pi\)
			0.559094	+	0.829104i	\(0.311149\pi\)
\(37\)	−286.000	−1.27076	−0.635380	−	0.772200i	\(-0.719156\pi\)
			−0.635380	+	0.772200i	\(0.719156\pi\)
\(41\)	12.0000	0.0457094	0.0228547	−	0.999739i	\(-0.492724\pi\)
			0.0228547	+	0.999739i	\(0.492724\pi\)
\(43\)	−263.000	−0.932724	−0.466362	−	0.884594i	\(-0.654436\pi\)
			−0.466362	+	0.884594i	\(0.654436\pi\)
\(47\)	−414.000	−1.28485	−0.642427	−	0.766347i	\(-0.722072\pi\)
			−0.642427	+	0.766347i	\(0.722072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.bb.1.1		1
4.3	odd	2		150.4.a.a.1.1	✓	1
5.2	odd	4		1200.4.f.c.49.1		2
5.3	odd	4		1200.4.f.c.49.2		2
5.4	even	2		1200.4.a.i.1.1		1
12.11	even	2		450.4.a.o.1.1		1
20.3	even	4		150.4.c.e.49.2		2
20.7	even	4		150.4.c.e.49.1		2
20.19	odd	2		150.4.a.h.1.1	yes	1
60.23	odd	4		450.4.c.a.199.1		2
60.47	odd	4		450.4.c.a.199.2		2
60.59	even	2		450.4.a.f.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.a.a.1.1	✓	1	4.3	odd	2
150.4.a.h.1.1	yes	1	20.19	odd	2
150.4.c.e.49.1		2	20.7	even	4
150.4.c.e.49.2		2	20.3	even	4
450.4.a.f.1.1		1	60.59	even	2
450.4.a.o.1.1		1	12.11	even	2
450.4.c.a.199.1		2	60.23	odd	4
450.4.c.a.199.2		2	60.47	odd	4
1200.4.a.i.1.1		1	5.4	even	2
1200.4.a.bb.1.1		1	1.1	even	1	trivial
1200.4.f.c.49.1		2	5.2	odd	4
1200.4.f.c.49.2		2	5.3	odd	4