Embedded newform 1850.2.a.e.1.1

• Label: 1850.2.a.e.1.1
• Level: $1850$
• Weight: $2$
• Character: 1850.1
• Self dual: yes
• Analytic conductor: $14.772$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.7723243739\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1850.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} -2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +3.00000 q^{18} -6.00000 q^{19} -4.00000 q^{23} +2.00000 q^{26} +2.00000 q^{28} -4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} -3.00000 q^{36} +1.00000 q^{37} +6.00000 q^{38} -10.0000 q^{41} +4.00000 q^{43} +4.00000 q^{46} +2.00000 q^{47} -3.00000 q^{49} -2.00000 q^{52} -2.00000 q^{53} -2.00000 q^{56} -6.00000 q^{59} +4.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} -8.00000 q^{67} +6.00000 q^{68} +3.00000 q^{72} -8.00000 q^{73} -1.00000 q^{74} -6.00000 q^{76} +4.00000 q^{79} +9.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} -4.00000 q^{86} +6.00000 q^{89} -4.00000 q^{91} -4.00000 q^{92} -2.00000 q^{94} +10.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107
			\(3\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(5\)	0	0
			\(7\)	2.00000	0.755929	0.377964	−	0.925820i	\(-0.376624\pi\)
0.377964	+	0.925820i				\(0.376624\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
			0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	−6.00000	−1.37649	−0.688247	−	0.725476i	\(-0.741620\pi\)
			−0.688247	+	0.725476i	\(0.741620\pi\)
\(23\)	−4.00000	−0.834058	−0.417029	−	0.908893i	\(-0.636929\pi\)
			−0.417029	+	0.908893i	\(0.636929\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	1.00000	0.164399
			\(41\)	−10.0000	−1.56174	−0.780869	−	0.624695i	\(-0.785223\pi\)
−0.780869	+	0.624695i				\(0.785223\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	2.00000	0.291730	0.145865	−	0.989305i	\(-0.453403\pi\)
			0.145865	+	0.989305i	\(0.453403\pi\)
\(53\)	−2.00000	−0.274721	−0.137361	−	0.990521i	\(-0.543862\pi\)
			−0.137361	+	0.990521i	\(0.543862\pi\)
\(59\)	−6.00000	−0.781133	−0.390567	−	0.920575i	\(-0.627721\pi\)
			−0.390567	+	0.920575i	\(0.627721\pi\)
\(61\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(67\)	−8.00000	−0.977356	−0.488678	−	0.872464i	\(-0.662521\pi\)
			−0.488678	+	0.872464i	\(0.662521\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	−8.00000	−0.936329	−0.468165	−	0.883641i	\(-0.655085\pi\)
			−0.468165	+	0.883641i	\(0.655085\pi\)
\(79\)	4.00000	0.450035	0.225018	−	0.974355i	\(-0.427756\pi\)
			0.225018	+	0.974355i	\(0.427756\pi\)
\(83\)	12.0000	1.31717	0.658586	−	0.752506i	\(-0.271155\pi\)
			0.658586	+	0.752506i	\(0.271155\pi\)
\(89\)	6.00000	0.635999	0.317999	−	0.948091i	\(-0.396989\pi\)
			0.317999	+	0.948091i	\(0.396989\pi\)
\(97\)	10.0000	1.01535	0.507673	−	0.861550i	\(-0.330506\pi\)
			0.507673	+	0.861550i	\(0.330506\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.e.1.1		1
5.2	odd	4		370.2.b.a.149.1	✓	2
5.3	odd	4		370.2.b.a.149.2	yes	2
5.4	even	2		1850.2.a.j.1.1		1
15.2	even	4		3330.2.d.f.1999.2		2
15.8	even	4		3330.2.d.f.1999.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.b.a.149.1	✓	2	5.2	odd	4
370.2.b.a.149.2	yes	2	5.3	odd	4
1850.2.a.e.1.1		1	1.1	even	1	trivial
1850.2.a.j.1.1		1	5.4	even	2
3330.2.d.f.1999.1		2	15.8	even	4
3330.2.d.f.1999.2		2	15.2	even	4