Embedded newform 1200.4.a.r.1.1

• Label: 1200.4.a.r.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

\(f(q)\)

\(=\)

\(q-3.00000 q^{3} +23.0000 q^{7} +9.00000 q^{9} +30.0000 q^{11} -29.0000 q^{13} -78.0000 q^{17} -149.000 q^{19} -69.0000 q^{21} +150.000 q^{23} -27.0000 q^{27} -234.000 q^{29} +217.000 q^{31} -90.0000 q^{33} -146.000 q^{37} +87.0000 q^{39} -156.000 q^{41} -433.000 q^{43} +30.0000 q^{47} +186.000 q^{49} +234.000 q^{51} +552.000 q^{53} +447.000 q^{57} +270.000 q^{59} +275.000 q^{61} +207.000 q^{63} +803.000 q^{67} -450.000 q^{69} -660.000 q^{71} +646.000 q^{73} +690.000 q^{77} -992.000 q^{79} +81.0000 q^{81} -846.000 q^{83} +702.000 q^{87} -1488.00 q^{89} -667.000 q^{91} -651.000 q^{93} +319.000 q^{97} +270.000 q^{99} +O(q^{100})\)

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
			\(7\)	23.0000	1.24188	0.620942	−	0.783857i	\(-0.286750\pi\)
0.620942	+	0.783857i				\(0.286750\pi\)
\(11\)	30.0000	0.822304	0.411152	−	0.911567i	\(-0.365127\pi\)
			0.411152	+	0.911567i	\(0.365127\pi\)
\(13\)	−29.0000	−0.618704	−0.309352	−	0.950948i	\(-0.600112\pi\)
			−0.309352	+	0.950948i	\(0.600112\pi\)
\(17\)	−78.0000	−1.11281	−0.556405	−	0.830911i	\(-0.687820\pi\)
			−0.556405	+	0.830911i	\(0.687820\pi\)
\(19\)	−149.000	−1.79910	−0.899551	−	0.436815i	\(-0.856106\pi\)
			−0.899551	+	0.436815i	\(0.856106\pi\)
\(23\)	150.000	1.35988	0.679938	−	0.733269i	\(-0.262007\pi\)
			0.679938	+	0.733269i	\(0.262007\pi\)
\(29\)	−234.000	−1.49837	−0.749185	−	0.662361i	\(-0.769554\pi\)
			−0.749185	+	0.662361i	\(0.769554\pi\)
\(31\)	217.000	1.25724	0.628619	−	0.777714i	\(-0.283621\pi\)
			0.628619	+	0.777714i	\(0.283621\pi\)
\(37\)	−146.000	−0.648710	−0.324355	−	0.945936i	\(-0.605147\pi\)
			−0.324355	+	0.945936i	\(0.605147\pi\)
\(41\)	−156.000	−0.594222	−0.297111	−	0.954843i	\(-0.596023\pi\)
			−0.297111	+	0.954843i	\(0.596023\pi\)
\(43\)	−433.000	−1.53563	−0.767813	−	0.640675i	\(-0.778655\pi\)
			−0.767813	+	0.640675i	\(0.778655\pi\)
\(47\)	30.0000	0.0931053	0.0465527	−	0.998916i	\(-0.485176\pi\)
			0.0465527	+	0.998916i	\(0.485176\pi\)

(See \(a_n\) instead)

Twists

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

    

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