Embedded newform 1600.4.a.bq.1.1

• Label: 1600.4.a.bq.1.1
• Level: $1600$
• Weight: $4$
• Character: 1600.1
• Self dual: yes
• Analytic conductor: $94.403$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(94.4030560092\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 800)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1600.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{3} +10.0000 q^{7} -2.00000 q^{9} +15.0000 q^{11} -8.00000 q^{13} -21.0000 q^{17} -105.000 q^{19} +50.0000 q^{21} +10.0000 q^{23} -145.000 q^{27} +20.0000 q^{29} -230.000 q^{31} +75.0000 q^{33} +54.0000 q^{37} -40.0000 q^{39} -195.000 q^{41} -300.000 q^{43} +480.000 q^{47} -243.000 q^{49} -105.000 q^{51} -322.000 q^{53} -525.000 q^{57} -560.000 q^{59} +730.000 q^{61} -20.0000 q^{63} +255.000 q^{67} +50.0000 q^{69} -40.0000 q^{71} +317.000 q^{73} +150.000 q^{77} -830.000 q^{79} -671.000 q^{81} +75.0000 q^{83} +100.000 q^{87} -705.000 q^{89} -80.0000 q^{91} -1150.00 q^{93} -1434.00 q^{97} -30.0000 q^{99} +O(q^{100})\)

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
			\(7\)	10.0000	0.539949	0.269975	−	0.962867i	\(-0.412985\pi\)
0.269975	+	0.962867i				\(0.412985\pi\)
\(11\)	15.0000	0.411152	0.205576	−	0.978641i	\(-0.434093\pi\)
			0.205576	+	0.978641i	\(0.434093\pi\)
\(13\)	−8.00000	−0.170677	−0.0853385	−	0.996352i	\(-0.527197\pi\)
			−0.0853385	+	0.996352i	\(0.527197\pi\)
\(17\)	−21.0000	−0.299603	−0.149801	−	0.988716i	\(-0.547863\pi\)
			−0.149801	+	0.988716i	\(0.547863\pi\)
\(19\)	−105.000	−1.26782	−0.633912	−	0.773405i	\(-0.718552\pi\)
			−0.633912	+	0.773405i	\(0.718552\pi\)
\(23\)	10.0000	0.0906584	0.0453292	−	0.998972i	\(-0.485566\pi\)
			0.0453292	+	0.998972i	\(0.485566\pi\)
\(29\)	20.0000	0.128066	0.0640329	−	0.997948i	\(-0.479604\pi\)
			0.0640329	+	0.997948i	\(0.479604\pi\)
\(31\)	−230.000	−1.33256	−0.666278	−	0.745704i	\(-0.732113\pi\)
			−0.666278	+	0.745704i	\(0.732113\pi\)
\(37\)	54.0000	0.239934	0.119967	−	0.992778i	\(-0.461721\pi\)
			0.119967	+	0.992778i	\(0.461721\pi\)
\(41\)	−195.000	−0.742778	−0.371389	−	0.928477i	\(-0.621118\pi\)
			−0.371389	+	0.928477i	\(0.621118\pi\)
\(43\)	−300.000	−1.06394	−0.531972	−	0.846762i	\(-0.678549\pi\)
			−0.531972	+	0.846762i	\(0.678549\pi\)
\(47\)	480.000	1.48969	0.744843	−	0.667240i	\(-0.232525\pi\)
			0.744843	+	0.667240i	\(0.232525\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.4.a.bq.1.1		1
4.3	odd	2		1600.4.a.k.1.1		1
5.4	even	2		1600.4.a.l.1.1		1
8.3	odd	2		800.4.a.j.1.1	yes	1
8.5	even	2		800.4.a.b.1.1	✓	1
20.19	odd	2		1600.4.a.bp.1.1		1
40.3	even	4		800.4.c.d.449.2		2
40.13	odd	4		800.4.c.c.449.1		2
40.19	odd	2		800.4.a.c.1.1	yes	1
40.27	even	4		800.4.c.d.449.1		2
40.29	even	2		800.4.a.i.1.1	yes	1
40.37	odd	4		800.4.c.c.449.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.b.1.1	✓	1	8.5	even	2
800.4.a.c.1.1	yes	1	40.19	odd	2
800.4.a.i.1.1	yes	1	40.29	even	2
800.4.a.j.1.1	yes	1	8.3	odd	2
800.4.c.c.449.1		2	40.13	odd	4
800.4.c.c.449.2		2	40.37	odd	4
800.4.c.d.449.1		2	40.27	even	4
800.4.c.d.449.2		2	40.3	even	4
1600.4.a.k.1.1		1	4.3	odd	2
1600.4.a.l.1.1		1	5.4	even	2
1600.4.a.bp.1.1		1	20.19	odd	2
1600.4.a.bq.1.1		1	1.1	even	1	trivial