Embedded newform 2400.4.a.h.1.1

• Label: 2400.4.a.h.1.1
• Level: $2400$
• Weight: $4$
• Character: 2400.1
• Self dual: yes
• Analytic conductor: $141.605$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(141.604584014\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2400.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{3} +16.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} +14.0000 q^{13} +18.0000 q^{17} -36.0000 q^{19} -48.0000 q^{21} +104.000 q^{23} -27.0000 q^{27} -250.000 q^{29} +28.0000 q^{31} +72.0000 q^{33} +54.0000 q^{37} -42.0000 q^{39} +354.000 q^{41} +228.000 q^{43} +408.000 q^{47} -87.0000 q^{49} -54.0000 q^{51} -262.000 q^{53} +108.000 q^{57} +64.0000 q^{59} +374.000 q^{61} +144.000 q^{63} +300.000 q^{67} -312.000 q^{69} -1016.00 q^{71} -274.000 q^{73} -384.000 q^{77} -788.000 q^{79} +81.0000 q^{81} -396.000 q^{83} +750.000 q^{87} +786.000 q^{89} +224.000 q^{91} -84.0000 q^{93} +1086.00 q^{97} -216.000 q^{99} +O(q^{100})\)

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
			\(7\)	16.0000	0.863919	0.431959	−	0.901893i	\(-0.357822\pi\)
0.431959	+	0.901893i				\(0.357822\pi\)
\(11\)	−24.0000	−0.657843	−0.328921	−	0.944357i	\(-0.606685\pi\)
			−0.328921	+	0.944357i	\(0.606685\pi\)
\(13\)	14.0000	0.298685	0.149342	−	0.988786i	\(-0.452284\pi\)
			0.149342	+	0.988786i	\(0.452284\pi\)
\(17\)	18.0000	0.256802	0.128401	−	0.991722i	\(-0.459015\pi\)
			0.128401	+	0.991722i	\(0.459015\pi\)
\(19\)	−36.0000	−0.434682	−0.217341	−	0.976096i	\(-0.569738\pi\)
			−0.217341	+	0.976096i	\(0.569738\pi\)
\(23\)	104.000	0.942848	0.471424	−	0.881907i	\(-0.343740\pi\)
			0.471424	+	0.881907i	\(0.343740\pi\)
\(29\)	−250.000	−1.60082	−0.800411	−	0.599452i	\(-0.795385\pi\)
			−0.800411	+	0.599452i	\(0.795385\pi\)
\(31\)	28.0000	0.162224	0.0811121	−	0.996705i	\(-0.474153\pi\)
			0.0811121	+	0.996705i	\(0.474153\pi\)
\(37\)	54.0000	0.239934	0.119967	−	0.992778i	\(-0.461721\pi\)
			0.119967	+	0.992778i	\(0.461721\pi\)
\(41\)	354.000	1.34843	0.674214	−	0.738536i	\(-0.264483\pi\)
			0.674214	+	0.738536i	\(0.264483\pi\)
\(43\)	228.000	0.808597	0.404299	−	0.914627i	\(-0.367516\pi\)
			0.404299	+	0.914627i	\(0.367516\pi\)
\(47\)	408.000	1.26623	0.633116	−	0.774057i	\(-0.281776\pi\)
			0.633116	+	0.774057i	\(0.281776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.h.1.1		1
4.3	odd	2		2400.4.a.o.1.1		1
5.4	even	2		480.4.a.i.1.1	yes	1
15.14	odd	2		1440.4.a.c.1.1		1
20.19	odd	2		480.4.a.f.1.1	✓	1
40.19	odd	2		960.4.a.z.1.1		1
40.29	even	2		960.4.a.c.1.1		1
60.59	even	2		1440.4.a.h.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.f.1.1	✓	1	20.19	odd	2
480.4.a.i.1.1	yes	1	5.4	even	2
960.4.a.c.1.1		1	40.29	even	2
960.4.a.z.1.1		1	40.19	odd	2
1440.4.a.c.1.1		1	15.14	odd	2
1440.4.a.h.1.1		1	60.59	even	2
2400.4.a.h.1.1		1	1.1	even	1	trivial
2400.4.a.o.1.1		1	4.3	odd	2