Embedded newform 4400.2.a.v.1.1

• Label: 4400.2.a.v.1.1
• Level: $4400$
• Weight: $2$
• Character: 4400.1
• Self dual: yes
• Analytic conductor: $35.134$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(35.1341768894\)
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	\(\chi\)	\(=\)	4400.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{3} +2.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} +4.00000 q^{13} -6.00000 q^{17} -8.00000 q^{19} +2.00000 q^{21} -3.00000 q^{23} -5.00000 q^{27} -5.00000 q^{31} +1.00000 q^{33} +1.00000 q^{37} +4.00000 q^{39} -10.0000 q^{43} -3.00000 q^{49} -6.00000 q^{51} +6.00000 q^{53} -8.00000 q^{57} -3.00000 q^{59} -4.00000 q^{61} -4.00000 q^{63} -1.00000 q^{67} -3.00000 q^{69} -15.0000 q^{71} +4.00000 q^{73} +2.00000 q^{77} -2.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -9.00000 q^{89} +8.00000 q^{91} -5.00000 q^{93} +7.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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
			\(3\)	1.00000	0.577350	0.288675	−	0.957427i	\(-0.406785\pi\)
0.288675	+	0.957427i				\(0.406785\pi\)
\(5\)	0	0
			\(7\)	2.00000	0.755929	0.377964	−	0.925820i	\(-0.376624\pi\)
0.377964	+	0.925820i				\(0.376624\pi\)
\(11\)	1.00000	0.301511
			\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
0.554700	+	0.832050i				\(0.312833\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	−8.00000	−1.83533	−0.917663	−	0.397360i	\(-0.869927\pi\)
			−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	−3.00000	−0.625543	−0.312772	−	0.949828i	\(-0.601257\pi\)
			−0.312772	+	0.949828i	\(0.601257\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	−5.00000	−0.898027	−0.449013	−	0.893525i	\(-0.648224\pi\)
			−0.449013	+	0.893525i	\(0.648224\pi\)
\(37\)	1.00000	0.164399	0.0821995	−	0.996616i	\(-0.473806\pi\)
			0.0821995	+	0.996616i	\(0.473806\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	−10.0000	−1.52499	−0.762493	−	0.646997i	\(-0.776025\pi\)
			−0.762493	+	0.646997i	\(0.776025\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.v.1.1		1
4.3	odd	2		1100.2.a.b.1.1		1
5.2	odd	4		4400.2.b.k.4049.1		2
5.3	odd	4		4400.2.b.k.4049.2		2
5.4	even	2		176.2.a.a.1.1		1
12.11	even	2		9900.2.a.h.1.1		1
15.14	odd	2		1584.2.a.p.1.1		1
20.3	even	4		1100.2.b.c.749.1		2
20.7	even	4		1100.2.b.c.749.2		2
20.19	odd	2		44.2.a.a.1.1	✓	1
35.34	odd	2		8624.2.a.w.1.1		1
40.19	odd	2		704.2.a.f.1.1		1
40.29	even	2		704.2.a.i.1.1		1
55.54	odd	2		1936.2.a.c.1.1		1
60.23	odd	4		9900.2.c.g.5149.2		2
60.47	odd	4		9900.2.c.g.5149.1		2
60.59	even	2		396.2.a.c.1.1		1
80.19	odd	4		2816.2.c.e.1409.2		2
80.29	even	4		2816.2.c.k.1409.1		2
80.59	odd	4		2816.2.c.e.1409.1		2
80.69	even	4		2816.2.c.k.1409.2		2
120.29	odd	2		6336.2.a.i.1.1		1
120.59	even	2		6336.2.a.j.1.1		1
140.19	even	6		2156.2.i.c.1145.1		2
140.39	odd	6		2156.2.i.b.177.1		2
140.59	even	6		2156.2.i.c.177.1		2
140.79	odd	6		2156.2.i.b.1145.1		2
140.139	even	2		2156.2.a.a.1.1		1
180.59	even	6		3564.2.i.a.1189.1		2
180.79	odd	6		3564.2.i.j.2377.1		2
180.119	even	6		3564.2.i.a.2377.1		2
180.139	odd	6		3564.2.i.j.1189.1		2
220.19	even	10		484.2.e.b.9.1		4
220.39	even	10		484.2.e.b.245.1		4
220.59	odd	10		484.2.e.a.269.1		4
220.79	even	10		484.2.e.b.81.1		4
220.119	odd	10		484.2.e.a.81.1		4
220.139	even	10		484.2.e.b.269.1		4
220.159	odd	10		484.2.e.a.245.1		4
220.179	odd	10		484.2.e.a.9.1		4
220.219	even	2		484.2.a.a.1.1		1
260.259	odd	2		7436.2.a.d.1.1		1
440.109	odd	2		7744.2.a.bc.1.1		1
440.219	even	2		7744.2.a.m.1.1		1
660.659	odd	2		4356.2.a.j.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.2.a.a.1.1	✓	1	20.19	odd	2
176.2.a.a.1.1		1	5.4	even	2
396.2.a.c.1.1		1	60.59	even	2
484.2.a.a.1.1		1	220.219	even	2
484.2.e.a.9.1		4	220.179	odd	10
484.2.e.a.81.1		4	220.119	odd	10
484.2.e.a.245.1		4	220.159	odd	10
484.2.e.a.269.1		4	220.59	odd	10
484.2.e.b.9.1		4	220.19	even	10
484.2.e.b.81.1		4	220.79	even	10
484.2.e.b.245.1		4	220.39	even	10
484.2.e.b.269.1		4	220.139	even	10
704.2.a.f.1.1		1	40.19	odd	2
704.2.a.i.1.1		1	40.29	even	2
1100.2.a.b.1.1		1	4.3	odd	2
1100.2.b.c.749.1		2	20.3	even	4
1100.2.b.c.749.2		2	20.7	even	4
1584.2.a.p.1.1		1	15.14	odd	2
1936.2.a.c.1.1		1	55.54	odd	2
2156.2.a.a.1.1		1	140.139	even	2
2156.2.i.b.177.1		2	140.39	odd	6
2156.2.i.b.1145.1		2	140.79	odd	6
2156.2.i.c.177.1		2	140.59	even	6
2156.2.i.c.1145.1		2	140.19	even	6
2816.2.c.e.1409.1		2	80.59	odd	4
2816.2.c.e.1409.2		2	80.19	odd	4
2816.2.c.k.1409.1		2	80.29	even	4
2816.2.c.k.1409.2		2	80.69	even	4
3564.2.i.a.1189.1		2	180.59	even	6
3564.2.i.a.2377.1		2	180.119	even	6
3564.2.i.j.1189.1		2	180.139	odd	6
3564.2.i.j.2377.1		2	180.79	odd	6
4356.2.a.j.1.1		1	660.659	odd	2
4400.2.a.v.1.1		1	1.1	even	1	trivial
4400.2.b.k.4049.1		2	5.2	odd	4
4400.2.b.k.4049.2		2	5.3	odd	4
6336.2.a.i.1.1		1	120.29	odd	2
6336.2.a.j.1.1		1	120.59	even	2
7436.2.a.d.1.1		1	260.259	odd	2
7744.2.a.m.1.1		1	440.219	even	2
7744.2.a.bc.1.1		1	440.109	odd	2
8624.2.a.w.1.1		1	35.34	odd	2
9900.2.a.h.1.1		1	12.11	even	2
9900.2.c.g.5149.1		2	60.47	odd	4
9900.2.c.g.5149.2		2	60.23	odd	4