Embedded newform 296.2.a.a.1.1

• Label: 296.2.a.a.1.1
• Level: $296$
• Weight: $2$
• Character: 296.1
• Self dual: yes
• Analytic conductor: $2.364$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 296 = 2^{3} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	296.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(2.36357189983\)
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	\(\chi\)	\(=\)	296.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} -2.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} -6.00000 q^{13} +2.00000 q^{15} -4.00000 q^{17} -8.00000 q^{19} -1.00000 q^{21} +6.00000 q^{23} -1.00000 q^{25} +5.00000 q^{27} +2.00000 q^{29} -4.00000 q^{31} -1.00000 q^{33} -2.00000 q^{35} -1.00000 q^{37} +6.00000 q^{39} +7.00000 q^{41} +2.00000 q^{43} +4.00000 q^{45} +9.00000 q^{47} -6.00000 q^{49} +4.00000 q^{51} -3.00000 q^{53} -2.00000 q^{55} +8.00000 q^{57} -12.0000 q^{59} +4.00000 q^{61} -2.00000 q^{63} +12.0000 q^{65} -6.00000 q^{69} +7.00000 q^{71} +7.00000 q^{73} +1.00000 q^{75} +1.00000 q^{77} +1.00000 q^{81} +3.00000 q^{83} +8.00000 q^{85} -2.00000 q^{87} -12.0000 q^{89} -6.00000 q^{91} +4.00000 q^{93} +16.0000 q^{95} -8.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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
			\(3\)	−1.00000	−0.577350	−0.288675	−	0.957427i	\(-0.593215\pi\)
−0.288675	+	0.957427i				\(0.593215\pi\)
\(5\)	−2.00000	−0.894427	−0.447214	−	0.894427i	\(-0.647584\pi\)
			−0.447214	+	0.894427i	\(0.647584\pi\)
\(7\)	1.00000	0.377964	0.188982	−	0.981981i	\(-0.439481\pi\)
			0.188982	+	0.981981i	\(0.439481\pi\)
\(11\)	1.00000	0.301511	0.150756	−	0.988571i	\(-0.451829\pi\)
			0.150756	+	0.988571i	\(0.451829\pi\)
\(13\)	−6.00000	−1.66410	−0.832050	−	0.554700i	\(-0.812833\pi\)
			−0.832050	+	0.554700i	\(0.812833\pi\)
\(17\)	−4.00000	−0.970143	−0.485071	−	0.874475i	\(-0.661206\pi\)
			−0.485071	+	0.874475i	\(0.661206\pi\)
\(19\)	−8.00000	−1.83533	−0.917663	−	0.397360i	\(-0.869927\pi\)
			−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	2.00000	0.371391	0.185695	−	0.982607i	\(-0.440546\pi\)
			0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	−1.00000	−0.164399
			\(41\)	7.00000	1.09322	0.546608	−	0.837389i	\(-0.315919\pi\)
0.546608	+	0.837389i				\(0.315919\pi\)
\(43\)	2.00000	0.304997	0.152499	−	0.988304i	\(-0.451268\pi\)
			0.152499	+	0.988304i	\(0.451268\pi\)
\(47\)	9.00000	1.31278	0.656392	−	0.754420i	\(-0.272082\pi\)
			0.656392	+	0.754420i	\(0.272082\pi\)
\(53\)	−3.00000	−0.412082	−0.206041	−	0.978543i	\(-0.566058\pi\)
			−0.206041	+	0.978543i	\(0.566058\pi\)
\(59\)	−12.0000	−1.56227	−0.781133	−	0.624364i	\(-0.785358\pi\)
			−0.781133	+	0.624364i	\(0.785358\pi\)
\(61\)	4.00000	0.512148	0.256074	−	0.966657i	\(-0.417571\pi\)
			0.256074	+	0.966657i	\(0.417571\pi\)
\(67\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(71\)	7.00000	0.830747	0.415374	−	0.909651i	\(-0.363651\pi\)
			0.415374	+	0.909651i	\(0.363651\pi\)
\(73\)	7.00000	0.819288	0.409644	−	0.912245i	\(-0.365653\pi\)
			0.409644	+	0.912245i	\(0.365653\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	3.00000	0.329293	0.164646	−	0.986353i	\(-0.447352\pi\)
			0.164646	+	0.986353i	\(0.447352\pi\)
\(89\)	−12.0000	−1.27200	−0.635999	−	0.771690i	\(-0.719412\pi\)
			−0.635999	+	0.771690i	\(0.719412\pi\)
\(97\)	−8.00000	−0.812277	−0.406138	−	0.913812i	\(-0.633125\pi\)
			−0.406138	+	0.913812i	\(0.633125\pi\)

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	296.2.a.a.1.1	✓	1
3.2	odd	2		2664.2.a.f.1.1		1
4.3	odd	2		592.2.a.c.1.1		1
5.4	even	2		7400.2.a.f.1.1		1
8.3	odd	2		2368.2.a.g.1.1		1
8.5	even	2		2368.2.a.n.1.1		1
12.11	even	2		5328.2.a.o.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.a.a.1.1	✓	1	1.1	even	1	trivial
592.2.a.c.1.1		1	4.3	odd	2
2368.2.a.g.1.1		1	8.3	odd	2
2368.2.a.n.1.1		1	8.5	even	2
2664.2.a.f.1.1		1	3.2	odd	2
5328.2.a.o.1.1		1	12.11	even	2
7400.2.a.f.1.1		1	5.4	even	2