Embedded newform 4400.2.a.i.1.1

• Label: 4400.2.a.i.1.1
• Level: $4400$
• Weight: $2$
• Character: 4400.1
• Self dual: yes
• Analytic conductor: $35.134$
• Analytic rank: $0$
• 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:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	4400.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} -2.00000 q^{7} -2.00000 q^{9} -1.00000 q^{11} -4.00000 q^{13} +2.00000 q^{17} +2.00000 q^{21} -1.00000 q^{23} +5.00000 q^{27} -7.00000 q^{31} +1.00000 q^{33} -3.00000 q^{37} +4.00000 q^{39} -8.00000 q^{41} -6.00000 q^{43} +8.00000 q^{47} -3.00000 q^{49} -2.00000 q^{51} +6.00000 q^{53} -5.00000 q^{59} +12.0000 q^{61} +4.00000 q^{63} -7.00000 q^{67} +1.00000 q^{69} +3.00000 q^{71} -4.00000 q^{73} +2.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +15.0000 q^{89} +8.00000 q^{91} +7.00000 q^{93} +7.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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	\(-0.593215\pi\)
−0.288675	+	0.957427i				\(0.593215\pi\)
\(5\)	0	0
			\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
−0.377964	+	0.925820i				\(0.623376\pi\)
\(11\)	−1.00000	−0.301511
			\(13\)	−4.00000	−1.10940	−0.554700	−	0.832050i	\(-0.687167\pi\)
−0.554700	+	0.832050i				\(0.687167\pi\)
\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	−1.00000	−0.208514	−0.104257	−	0.994550i	\(-0.533247\pi\)
			−0.104257	+	0.994550i	\(0.533247\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	−7.00000	−1.25724	−0.628619	−	0.777714i	\(-0.716379\pi\)
			−0.628619	+	0.777714i	\(0.716379\pi\)
\(37\)	−3.00000	−0.493197	−0.246598	−	0.969118i	\(-0.579313\pi\)
			−0.246598	+	0.969118i	\(0.579313\pi\)
\(41\)	−8.00000	−1.24939	−0.624695	−	0.780869i	\(-0.714777\pi\)
			−0.624695	+	0.780869i	\(0.714777\pi\)
\(43\)	−6.00000	−0.914991	−0.457496	−	0.889212i	\(-0.651253\pi\)
			−0.457496	+	0.889212i	\(0.651253\pi\)
\(47\)	8.00000	1.16692	0.583460	−	0.812142i	\(-0.301699\pi\)
			0.583460	+	0.812142i	\(0.301699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.i.1.1		1
4.3	odd	2		275.2.a.b.1.1		1
5.2	odd	4		4400.2.b.h.4049.2		2
5.3	odd	4		4400.2.b.h.4049.1		2
5.4	even	2		176.2.a.b.1.1		1
12.11	even	2		2475.2.a.a.1.1		1
15.14	odd	2		1584.2.a.g.1.1		1
20.3	even	4		275.2.b.a.199.1		2
20.7	even	4		275.2.b.a.199.2		2
20.19	odd	2		11.2.a.a.1.1	✓	1
35.34	odd	2		8624.2.a.j.1.1		1
40.19	odd	2		704.2.a.h.1.1		1
40.29	even	2		704.2.a.c.1.1		1
44.43	even	2		3025.2.a.a.1.1		1
55.54	odd	2		1936.2.a.i.1.1		1
60.23	odd	4		2475.2.c.a.199.2		2
60.47	odd	4		2475.2.c.a.199.1		2
60.59	even	2		99.2.a.d.1.1		1
80.19	odd	4		2816.2.c.j.1409.1		2
80.29	even	4		2816.2.c.f.1409.2		2
80.59	odd	4		2816.2.c.j.1409.2		2
80.69	even	4		2816.2.c.f.1409.1		2
120.29	odd	2		6336.2.a.bu.1.1		1
120.59	even	2		6336.2.a.br.1.1		1
140.19	even	6		539.2.e.g.67.1		2
140.39	odd	6		539.2.e.h.177.1		2
140.59	even	6		539.2.e.g.177.1		2
140.79	odd	6		539.2.e.h.67.1		2
140.139	even	2		539.2.a.a.1.1		1
180.59	even	6		891.2.e.b.298.1		2
180.79	odd	6		891.2.e.k.595.1		2
180.119	even	6		891.2.e.b.595.1		2
180.139	odd	6		891.2.e.k.298.1		2
220.19	even	10		121.2.c.a.9.1		4
220.39	even	10		121.2.c.a.3.1		4
220.59	odd	10		121.2.c.e.27.1		4
220.79	even	10		121.2.c.a.81.1		4
220.119	odd	10		121.2.c.e.81.1		4
220.139	even	10		121.2.c.a.27.1		4
220.159	odd	10		121.2.c.e.3.1		4
220.179	odd	10		121.2.c.e.9.1		4
220.219	even	2		121.2.a.d.1.1		1
260.259	odd	2		1859.2.a.b.1.1		1
340.339	odd	2		3179.2.a.a.1.1		1
380.379	even	2		3971.2.a.b.1.1		1
420.419	odd	2		4851.2.a.t.1.1		1
440.109	odd	2		7744.2.a.k.1.1		1
440.219	even	2		7744.2.a.x.1.1		1
460.459	even	2		5819.2.a.a.1.1		1
580.579	odd	2		9251.2.a.d.1.1		1
660.659	odd	2		1089.2.a.b.1.1		1
1540.1539	odd	2		5929.2.a.h.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.2.a.a.1.1	✓	1	20.19	odd	2
99.2.a.d.1.1		1	60.59	even	2
121.2.a.d.1.1		1	220.219	even	2
121.2.c.a.3.1		4	220.39	even	10
121.2.c.a.9.1		4	220.19	even	10
121.2.c.a.27.1		4	220.139	even	10
121.2.c.a.81.1		4	220.79	even	10
121.2.c.e.3.1		4	220.159	odd	10
121.2.c.e.9.1		4	220.179	odd	10
121.2.c.e.27.1		4	220.59	odd	10
121.2.c.e.81.1		4	220.119	odd	10
176.2.a.b.1.1		1	5.4	even	2
275.2.a.b.1.1		1	4.3	odd	2
275.2.b.a.199.1		2	20.3	even	4
275.2.b.a.199.2		2	20.7	even	4
539.2.a.a.1.1		1	140.139	even	2
539.2.e.g.67.1		2	140.19	even	6
539.2.e.g.177.1		2	140.59	even	6
539.2.e.h.67.1		2	140.79	odd	6
539.2.e.h.177.1		2	140.39	odd	6
704.2.a.c.1.1		1	40.29	even	2
704.2.a.h.1.1		1	40.19	odd	2
891.2.e.b.298.1		2	180.59	even	6
891.2.e.b.595.1		2	180.119	even	6
891.2.e.k.298.1		2	180.139	odd	6
891.2.e.k.595.1		2	180.79	odd	6
1089.2.a.b.1.1		1	660.659	odd	2
1584.2.a.g.1.1		1	15.14	odd	2
1859.2.a.b.1.1		1	260.259	odd	2
1936.2.a.i.1.1		1	55.54	odd	2
2475.2.a.a.1.1		1	12.11	even	2
2475.2.c.a.199.1		2	60.47	odd	4
2475.2.c.a.199.2		2	60.23	odd	4
2816.2.c.f.1409.1		2	80.69	even	4
2816.2.c.f.1409.2		2	80.29	even	4
2816.2.c.j.1409.1		2	80.19	odd	4
2816.2.c.j.1409.2		2	80.59	odd	4
3025.2.a.a.1.1		1	44.43	even	2
3179.2.a.a.1.1		1	340.339	odd	2
3971.2.a.b.1.1		1	380.379	even	2
4400.2.a.i.1.1		1	1.1	even	1	trivial
4400.2.b.h.4049.1		2	5.3	odd	4
4400.2.b.h.4049.2		2	5.2	odd	4
4851.2.a.t.1.1		1	420.419	odd	2
5819.2.a.a.1.1		1	460.459	even	2
5929.2.a.h.1.1		1	1540.1539	odd	2
6336.2.a.br.1.1		1	120.59	even	2
6336.2.a.bu.1.1		1	120.29	odd	2
7744.2.a.k.1.1		1	440.109	odd	2
7744.2.a.x.1.1		1	440.219	even	2
8624.2.a.j.1.1		1	35.34	odd	2
9251.2.a.d.1.1		1	580.579	odd	2