Embedded newform 9801.2.a.g.1.1

• Label: 9801.2.a.g.1.1
• Level: $9801$
• Weight: $2$
• Character: 9801.1
• Self dual: yes
• Analytic conductor: $78.261$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(78.2613790211\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	9801.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{4} +3.00000 q^{5} +4.00000 q^{7} -2.00000 q^{13} +4.00000 q^{16} -6.00000 q^{17} -2.00000 q^{19} -6.00000 q^{20} -3.00000 q^{23} +4.00000 q^{25} -8.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} +12.0000 q^{35} +2.00000 q^{37} -8.00000 q^{43} -3.00000 q^{47} +9.00000 q^{49} +4.00000 q^{52} -3.00000 q^{53} -8.00000 q^{61} -8.00000 q^{64} -6.00000 q^{65} -13.0000 q^{67} +12.0000 q^{68} -2.00000 q^{73} +4.00000 q^{76} -2.00000 q^{79} +12.0000 q^{80} -18.0000 q^{83} -18.0000 q^{85} -3.00000 q^{89} -8.00000 q^{91} +6.00000 q^{92} -6.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)

Coefficient data

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

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

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(3\)	0	0
			\(5\)	3.00000	1.34164	0.670820	−	0.741620i	\(-0.265942\pi\)
0.670820	+	0.741620i				\(0.265942\pi\)
\(7\)	4.00000	1.51186	0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	0	0
			\(13\)	−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	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	−2.00000	−0.458831	−0.229416	−	0.973329i	\(-0.573682\pi\)
			−0.229416	+	0.973329i	\(0.573682\pi\)
\(23\)	−3.00000	−0.625543	−0.312772	−	0.949828i	\(-0.601257\pi\)
			−0.312772	+	0.949828i	\(0.601257\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	8.00000	1.43684	0.718421	−	0.695608i	\(-0.244865\pi\)
			0.718421	+	0.695608i	\(0.244865\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	−8.00000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	−3.00000	−0.437595	−0.218797	−	0.975770i	\(-0.570213\pi\)
			−0.218797	+	0.975770i	\(0.570213\pi\)
\(53\)	−3.00000	−0.412082	−0.206041	−	0.978543i	\(-0.566058\pi\)
			−0.206041	+	0.978543i	\(0.566058\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	−8.00000	−1.02430	−0.512148	−	0.858898i	\(-0.671150\pi\)
			−0.512148	+	0.858898i	\(0.671150\pi\)
\(67\)	−13.0000	−1.58820	−0.794101	−	0.607785i	\(-0.792058\pi\)
			−0.794101	+	0.607785i	\(0.792058\pi\)
\(71\)	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	\(0.537341\pi\)
\(79\)	−2.00000	−0.225018	−0.112509	−	0.993651i	\(-0.535889\pi\)
			−0.112509	+	0.993651i	\(0.535889\pi\)
\(83\)	−18.0000	−1.97576	−0.987878	−	0.155230i	\(-0.950388\pi\)
			−0.987878	+	0.155230i	\(0.950388\pi\)
\(89\)	−3.00000	−0.317999	−0.159000	−	0.987279i	\(-0.550827\pi\)
			−0.159000	+	0.987279i	\(0.550827\pi\)
\(97\)	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	9801.2.a.g.1.1		1
3.2	odd	2		9801.2.a.f.1.1		1
9.2	odd	6		1089.2.e.b.364.1		2
9.5	odd	6		1089.2.e.b.727.1		2
11.10	odd	2		891.2.a.e.1.1		1
33.32	even	2		891.2.a.d.1.1		1
99.32	even	6		99.2.e.b.34.1	✓	2
99.43	odd	6		297.2.e.b.199.1		2
99.65	even	6		99.2.e.b.67.1	yes	2
99.76	odd	6		297.2.e.b.100.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.b.34.1	✓	2	99.32	even	6
99.2.e.b.67.1	yes	2	99.65	even	6
297.2.e.b.100.1		2	99.76	odd	6
297.2.e.b.199.1		2	99.43	odd	6
891.2.a.d.1.1		1	33.32	even	2
891.2.a.e.1.1		1	11.10	odd	2
1089.2.e.b.364.1		2	9.2	odd	6
1089.2.e.b.727.1		2	9.5	odd	6
9801.2.a.f.1.1		1	3.2	odd	2
9801.2.a.g.1.1		1	1.1	even	1	trivial