Embedded newform 1100.4.a.d.1.1

• Label: 1100.4.a.d.1.1
• Level: $1100$
• Weight: $4$
• Character: 1100.1
• Self dual: yes
• Analytic conductor: $64.902$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1100.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(64.9021010063\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1100.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{3} +26.0000 q^{7} -2.00000 q^{9} -11.0000 q^{11} -52.0000 q^{13} -46.0000 q^{17} -96.0000 q^{19} +130.000 q^{21} -27.0000 q^{23} -145.000 q^{27} +16.0000 q^{29} -293.000 q^{31} -55.0000 q^{33} +29.0000 q^{37} -260.000 q^{39} -472.000 q^{41} +110.000 q^{43} +224.000 q^{47} +333.000 q^{49} -230.000 q^{51} -754.000 q^{53} -480.000 q^{57} +825.000 q^{59} -548.000 q^{61} -52.0000 q^{63} +123.000 q^{67} -135.000 q^{69} +1001.00 q^{71} +1020.00 q^{73} -286.000 q^{77} +526.000 q^{79} -671.000 q^{81} +158.000 q^{83} +80.0000 q^{87} -1217.00 q^{89} -1352.00 q^{91} -1465.00 q^{93} +263.000 q^{97} +22.0000 q^{99} +O(q^{100})\)

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
			\(3\)	5.00000	0.962250	0.481125	−	0.876652i	\(-0.340228\pi\)
0.481125	+	0.876652i				\(0.340228\pi\)
\(5\)	0	0
			\(7\)	26.0000	1.40387	0.701934	−	0.712242i	\(-0.252320\pi\)
0.701934	+	0.712242i				\(0.252320\pi\)
\(11\)	−11.0000	−0.301511
			\(13\)	−52.0000	−1.10940	−0.554700	−	0.832050i	\(-0.687167\pi\)
−0.554700	+	0.832050i				\(0.687167\pi\)
\(17\)	−46.0000	−0.656273	−0.328136	−	0.944630i	\(-0.606421\pi\)
			−0.328136	+	0.944630i	\(0.606421\pi\)
\(19\)	−96.0000	−1.15915	−0.579577	−	0.814918i	\(-0.696782\pi\)
			−0.579577	+	0.814918i	\(0.696782\pi\)
\(23\)	−27.0000	−0.244778	−0.122389	−	0.992482i	\(-0.539056\pi\)
			−0.122389	+	0.992482i	\(0.539056\pi\)
\(29\)	16.0000	0.102453	0.0512263	−	0.998687i	\(-0.483687\pi\)
			0.0512263	+	0.998687i	\(0.483687\pi\)
\(31\)	−293.000	−1.69756	−0.848780	−	0.528746i	\(-0.822662\pi\)
			−0.848780	+	0.528746i	\(0.822662\pi\)
\(37\)	29.0000	0.128853	0.0644266	−	0.997922i	\(-0.479478\pi\)
			0.0644266	+	0.997922i	\(0.479478\pi\)
\(41\)	−472.000	−1.79790	−0.898951	−	0.438048i	\(-0.855670\pi\)
			−0.898951	+	0.438048i	\(0.855670\pi\)
\(43\)	110.000	0.390113	0.195056	−	0.980792i	\(-0.437511\pi\)
			0.195056	+	0.980792i	\(0.437511\pi\)
\(47\)	224.000	0.695186	0.347593	−	0.937645i	\(-0.386999\pi\)
			0.347593	+	0.937645i	\(0.386999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.4.a.d.1.1		1
5.2	odd	4		1100.4.b.c.749.1		2
5.3	odd	4		1100.4.b.c.749.2		2
5.4	even	2		44.4.a.a.1.1	✓	1
15.14	odd	2		396.4.a.e.1.1		1
20.19	odd	2		176.4.a.e.1.1		1
35.34	odd	2		2156.4.a.b.1.1		1
40.19	odd	2		704.4.a.c.1.1		1
40.29	even	2		704.4.a.j.1.1		1
55.54	odd	2		484.4.a.a.1.1		1
60.59	even	2		1584.4.a.p.1.1		1
220.219	even	2		1936.4.a.m.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.4.a.a.1.1	✓	1	5.4	even	2
176.4.a.e.1.1		1	20.19	odd	2
396.4.a.e.1.1		1	15.14	odd	2
484.4.a.a.1.1		1	55.54	odd	2
704.4.a.c.1.1		1	40.19	odd	2
704.4.a.j.1.1		1	40.29	even	2
1100.4.a.d.1.1		1	1.1	even	1	trivial
1100.4.b.c.749.1		2	5.2	odd	4
1100.4.b.c.749.2		2	5.3	odd	4
1584.4.a.p.1.1		1	60.59	even	2
1936.4.a.m.1.1		1	220.219	even	2
2156.4.a.b.1.1		1	35.34	odd	2