Embedded newform 1800.4.a.c.1.1

• Label: 1800.4.a.c.1.1
• Level: $1800$
• Weight: $4$
• Character: 1800.1
• Self dual: yes
• Analytic conductor: $106.203$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(106.203438010\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1800.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-26.0000 q^{7} +59.0000 q^{11} -28.0000 q^{13} +5.00000 q^{17} +109.000 q^{19} -194.000 q^{23} +32.0000 q^{29} +10.0000 q^{31} +198.000 q^{37} -117.000 q^{41} -388.000 q^{43} -68.0000 q^{47} +333.000 q^{49} -18.0000 q^{53} -392.000 q^{59} -710.000 q^{61} +253.000 q^{67} +612.000 q^{71} +549.000 q^{73} -1534.00 q^{77} +414.000 q^{79} -121.000 q^{83} +81.0000 q^{89} +728.000 q^{91} +1502.00 q^{97} +O(q^{100})\)

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
			\(3\)	0	0
\(5\)	0	0
			\(7\)	−26.0000	−1.40387	−0.701934	−	0.712242i	\(-0.747680\pi\)
−0.701934	+	0.712242i				\(0.747680\pi\)
\(11\)	59.0000	1.61720	0.808599	−	0.588361i	\(-0.200226\pi\)
			0.808599	+	0.588361i	\(0.200226\pi\)
\(13\)	−28.0000	−0.597369	−0.298685	−	0.954352i	\(-0.596548\pi\)
			−0.298685	+	0.954352i	\(0.596548\pi\)
\(17\)	5.00000	0.0713340	0.0356670	−	0.999364i	\(-0.488644\pi\)
			0.0356670	+	0.999364i	\(0.488644\pi\)
\(19\)	109.000	1.31612	0.658061	−	0.752965i	\(-0.271377\pi\)
			0.658061	+	0.752965i	\(0.271377\pi\)
\(23\)	−194.000	−1.75877	−0.879387	−	0.476108i	\(-0.842047\pi\)
			−0.879387	+	0.476108i	\(0.842047\pi\)
\(29\)	32.0000	0.204905	0.102453	−	0.994738i	\(-0.467331\pi\)
			0.102453	+	0.994738i	\(0.467331\pi\)
\(31\)	10.0000	0.0579372	0.0289686	−	0.999580i	\(-0.490778\pi\)
			0.0289686	+	0.999580i	\(0.490778\pi\)
\(37\)	198.000	0.879757	0.439878	−	0.898057i	\(-0.355022\pi\)
			0.439878	+	0.898057i	\(0.355022\pi\)
\(41\)	−117.000	−0.445667	−0.222833	−	0.974857i	\(-0.571531\pi\)
			−0.222833	+	0.974857i	\(0.571531\pi\)
\(43\)	−388.000	−1.37603	−0.688017	−	0.725695i	\(-0.741518\pi\)
			−0.688017	+	0.725695i	\(0.741518\pi\)
\(47\)	−68.0000	−0.211039	−0.105519	−	0.994417i	\(-0.533650\pi\)
			−0.105519	+	0.994417i	\(0.533650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.a.c.1.1		1
3.2	odd	2		200.4.a.b.1.1	✓	1
5.2	odd	4		1800.4.f.w.649.1		2
5.3	odd	4		1800.4.f.w.649.2		2
5.4	even	2		1800.4.a.bh.1.1		1
12.11	even	2		400.4.a.t.1.1		1
15.2	even	4		200.4.c.b.49.2		2
15.8	even	4		200.4.c.b.49.1		2
15.14	odd	2		200.4.a.j.1.1	yes	1
24.5	odd	2		1600.4.a.bz.1.1		1
24.11	even	2		1600.4.a.b.1.1		1
60.23	odd	4		400.4.c.b.49.2		2
60.47	odd	4		400.4.c.b.49.1		2
60.59	even	2		400.4.a.a.1.1		1
120.29	odd	2		1600.4.a.c.1.1		1
120.59	even	2		1600.4.a.by.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.4.a.b.1.1	✓	1	3.2	odd	2
200.4.a.j.1.1	yes	1	15.14	odd	2
200.4.c.b.49.1		2	15.8	even	4
200.4.c.b.49.2		2	15.2	even	4
400.4.a.a.1.1		1	60.59	even	2
400.4.a.t.1.1		1	12.11	even	2
400.4.c.b.49.1		2	60.47	odd	4
400.4.c.b.49.2		2	60.23	odd	4
1600.4.a.b.1.1		1	24.11	even	2
1600.4.a.c.1.1		1	120.29	odd	2
1600.4.a.by.1.1		1	120.59	even	2
1600.4.a.bz.1.1		1	24.5	odd	2
1800.4.a.c.1.1		1	1.1	even	1	trivial
1800.4.a.bh.1.1		1	5.4	even	2
1800.4.f.w.649.1		2	5.2	odd	4
1800.4.f.w.649.2		2	5.3	odd	4