Embedded newform 99.2.a.a.1.1

• Label: 99.2.a.a.1.1
• Level: $99$
• Weight: $2$
• Character: 99.1
• Self dual: yes
• Analytic conductor: $0.791$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.790518980011\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	99.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} -1.00000 q^{4} -4.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} +4.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} -6.00000 q^{19} +4.00000 q^{20} +1.00000 q^{22} +4.00000 q^{23} +11.0000 q^{25} +2.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} -2.00000 q^{34} +8.00000 q^{35} -6.00000 q^{37} +6.00000 q^{38} -12.0000 q^{40} -10.0000 q^{41} +6.00000 q^{43} +1.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} -11.0000 q^{50} +2.00000 q^{52} +4.00000 q^{55} -6.00000 q^{56} +6.00000 q^{58} +4.00000 q^{59} -6.00000 q^{61} -4.00000 q^{62} +7.00000 q^{64} +8.00000 q^{65} +8.00000 q^{67} -2.00000 q^{68} -8.00000 q^{70} -2.00000 q^{73} +6.00000 q^{74} +6.00000 q^{76} +2.00000 q^{77} -10.0000 q^{79} +4.00000 q^{80} +10.0000 q^{82} +12.0000 q^{83} -8.00000 q^{85} -6.00000 q^{86} -3.00000 q^{88} +4.00000 q^{91} -4.00000 q^{92} +8.00000 q^{94} +24.0000 q^{95} +2.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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	−0.353553	−	0.935414i	\(-0.615027\pi\)
			−0.353553	+	0.935414i	\(0.615027\pi\)
\(3\)	0	0
			\(5\)	−4.00000	−1.78885	−0.894427	−	0.447214i	\(-0.852416\pi\)
−0.894427	+	0.447214i				\(0.852416\pi\)
\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
			−0.377964	+	0.925820i	\(0.623376\pi\)
\(11\)	−1.00000	−0.301511
			\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
−0.277350	+	0.960769i				\(0.589456\pi\)
\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	−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	\(-0.363071\pi\)
			0.417029	+	0.908893i	\(0.363071\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	−6.00000	−0.986394	−0.493197	−	0.869918i	\(-0.664172\pi\)
			−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	−10.0000	−1.56174	−0.780869	−	0.624695i	\(-0.785223\pi\)
			−0.780869	+	0.624695i	\(0.785223\pi\)
\(43\)	6.00000	0.914991	0.457496	−	0.889212i	\(-0.348747\pi\)
			0.457496	+	0.889212i	\(0.348747\pi\)
\(47\)	−8.00000	−1.16692	−0.583460	−	0.812142i	\(-0.698301\pi\)
			−0.583460	+	0.812142i	\(0.698301\pi\)
\(53\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(59\)	4.00000	0.520756	0.260378	−	0.965507i	\(-0.416153\pi\)
			0.260378	+	0.965507i	\(0.416153\pi\)
\(61\)	−6.00000	−0.768221	−0.384111	−	0.923287i	\(-0.625492\pi\)
			−0.384111	+	0.923287i	\(0.625492\pi\)
\(67\)	8.00000	0.977356	0.488678	−	0.872464i	\(-0.337479\pi\)
			0.488678	+	0.872464i	\(0.337479\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	−2.00000	−0.234082	−0.117041	−	0.993127i	\(-0.537341\pi\)
			−0.117041	+	0.993127i	\(0.537341\pi\)
\(79\)	−10.0000	−1.12509	−0.562544	−	0.826767i	\(-0.690177\pi\)
			−0.562544	+	0.826767i	\(0.690177\pi\)
\(83\)	12.0000	1.31717	0.658586	−	0.752506i	\(-0.271155\pi\)
			0.658586	+	0.752506i	\(0.271155\pi\)
\(89\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(97\)	2.00000	0.203069	0.101535	−	0.994832i	\(-0.467625\pi\)
			0.101535	+	0.994832i	\(0.467625\pi\)

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	99.2.a.a.1.1	✓	1
3.2	odd	2		99.2.a.c.1.1	yes	1
4.3	odd	2		1584.2.a.b.1.1		1
5.2	odd	4		2475.2.c.b.199.1		2
5.3	odd	4		2475.2.c.b.199.2		2
5.4	even	2		2475.2.a.j.1.1		1
7.6	odd	2		4851.2.a.g.1.1		1
8.3	odd	2		6336.2.a.cm.1.1		1
8.5	even	2		6336.2.a.cl.1.1		1
9.2	odd	6		891.2.e.c.595.1		2
9.4	even	3		891.2.e.j.298.1		2
9.5	odd	6		891.2.e.c.298.1		2
9.7	even	3		891.2.e.j.595.1		2
11.10	odd	2		1089.2.a.h.1.1		1
12.11	even	2		1584.2.a.r.1.1		1
15.2	even	4		2475.2.c.g.199.2		2
15.8	even	4		2475.2.c.g.199.1		2
15.14	odd	2		2475.2.a.c.1.1		1
21.20	even	2		4851.2.a.o.1.1		1
24.5	odd	2		6336.2.a.b.1.1		1
24.11	even	2		6336.2.a.f.1.1		1
33.32	even	2		1089.2.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.a.a.1.1	✓	1	1.1	even	1	trivial
99.2.a.c.1.1	yes	1	3.2	odd	2
891.2.e.c.298.1		2	9.5	odd	6
891.2.e.c.595.1		2	9.2	odd	6
891.2.e.j.298.1		2	9.4	even	3
891.2.e.j.595.1		2	9.7	even	3
1089.2.a.d.1.1		1	33.32	even	2
1089.2.a.h.1.1		1	11.10	odd	2
1584.2.a.b.1.1		1	4.3	odd	2
1584.2.a.r.1.1		1	12.11	even	2
2475.2.a.c.1.1		1	15.14	odd	2
2475.2.a.j.1.1		1	5.4	even	2
2475.2.c.b.199.1		2	5.2	odd	4
2475.2.c.b.199.2		2	5.3	odd	4
2475.2.c.g.199.1		2	15.8	even	4
2475.2.c.g.199.2		2	15.2	even	4
4851.2.a.g.1.1		1	7.6	odd	2
4851.2.a.o.1.1		1	21.20	even	2
6336.2.a.b.1.1		1	24.5	odd	2
6336.2.a.f.1.1		1	24.11	even	2
6336.2.a.cl.1.1		1	8.5	even	2
6336.2.a.cm.1.1		1	8.3	odd	2