Embedded newform 8100.2.a.f.1.1

• Label: 8100.2.a.f.1.1
• Level: $8100$
• Weight: $2$
• Character: 8100.1
• Self dual: yes
• Analytic conductor: $64.679$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(64.6788256372\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 324)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	8100.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{7} +6.00000 q^{11} -5.00000 q^{13} -3.00000 q^{17} +2.00000 q^{19} +6.00000 q^{23} -3.00000 q^{29} -4.00000 q^{31} -5.00000 q^{37} +6.00000 q^{41} +10.0000 q^{43} -3.00000 q^{49} -6.00000 q^{53} +12.0000 q^{59} +5.00000 q^{61} -2.00000 q^{67} -6.00000 q^{71} +1.00000 q^{73} -12.0000 q^{77} -10.0000 q^{79} +3.00000 q^{89} +10.0000 q^{91} +10.0000 q^{97} +O(q^{100})\)

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
			\(3\)	0	0
\(5\)	0	0
			\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
−0.377964	+	0.925820i				\(0.623376\pi\)
\(11\)	6.00000	1.80907	0.904534	−	0.426401i	\(-0.140219\pi\)
			0.904534	+	0.426401i	\(0.140219\pi\)
\(13\)	−5.00000	−1.38675	−0.693375	−	0.720577i	\(-0.743877\pi\)
			−0.693375	+	0.720577i	\(0.743877\pi\)
\(17\)	−3.00000	−0.727607	−0.363803	−	0.931476i	\(-0.618522\pi\)
			−0.363803	+	0.931476i	\(0.618522\pi\)
\(19\)	2.00000	0.458831	0.229416	−	0.973329i	\(-0.426318\pi\)
			0.229416	+	0.973329i	\(0.426318\pi\)
\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	−3.00000	−0.557086	−0.278543	−	0.960424i	\(-0.589851\pi\)
			−0.278543	+	0.960424i	\(0.589851\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	−5.00000	−0.821995	−0.410997	−	0.911636i	\(-0.634819\pi\)
			−0.410997	+	0.911636i	\(0.634819\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	10.0000	1.52499	0.762493	−	0.646997i	\(-0.223975\pi\)
			0.762493	+	0.646997i	\(0.223975\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	−6.00000	−0.824163	−0.412082	−	0.911147i	\(-0.635198\pi\)
			−0.412082	+	0.911147i	\(0.635198\pi\)
\(59\)	12.0000	1.56227	0.781133	−	0.624364i	\(-0.214642\pi\)
			0.781133	+	0.624364i	\(0.214642\pi\)
\(61\)	5.00000	0.640184	0.320092	−	0.947386i	\(-0.396286\pi\)
			0.320092	+	0.947386i	\(0.396286\pi\)
\(67\)	−2.00000	−0.244339	−0.122169	−	0.992509i	\(-0.538985\pi\)
			−0.122169	+	0.992509i	\(0.538985\pi\)
\(71\)	−6.00000	−0.712069	−0.356034	−	0.934473i	\(-0.615871\pi\)
			−0.356034	+	0.934473i	\(0.615871\pi\)
\(73\)	1.00000	0.117041	0.0585206	−	0.998286i	\(-0.481362\pi\)
			0.0585206	+	0.998286i	\(0.481362\pi\)
\(79\)	−10.0000	−1.12509	−0.562544	−	0.826767i	\(-0.690177\pi\)
			−0.562544	+	0.826767i	\(0.690177\pi\)
\(83\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(89\)	3.00000	0.317999	0.159000	−	0.987279i	\(-0.449173\pi\)
			0.159000	+	0.987279i	\(0.449173\pi\)
\(97\)	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	8100.2.a.f.1.1		1
3.2	odd	2		8100.2.a.a.1.1		1
5.2	odd	4		8100.2.d.j.649.1		2
5.3	odd	4		8100.2.d.j.649.2		2
5.4	even	2		324.2.a.b.1.1	✓	1
15.2	even	4		8100.2.d.a.649.1		2
15.8	even	4		8100.2.d.a.649.2		2
15.14	odd	2		324.2.a.d.1.1	yes	1
20.19	odd	2		1296.2.a.a.1.1		1
40.19	odd	2		5184.2.a.z.1.1		1
40.29	even	2		5184.2.a.bc.1.1		1
45.4	even	6		324.2.e.d.217.1		2
45.14	odd	6		324.2.e.a.217.1		2
45.29	odd	6		324.2.e.a.109.1		2
45.34	even	6		324.2.e.d.109.1		2
60.59	even	2		1296.2.a.j.1.1		1
120.29	odd	2		5184.2.a.g.1.1		1
120.59	even	2		5184.2.a.d.1.1		1
180.59	even	6		1296.2.i.d.865.1		2
180.79	odd	6		1296.2.i.p.433.1		2
180.119	even	6		1296.2.i.d.433.1		2
180.139	odd	6		1296.2.i.p.865.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.2.a.b.1.1	✓	1	5.4	even	2
324.2.a.d.1.1	yes	1	15.14	odd	2
324.2.e.a.109.1		2	45.29	odd	6
324.2.e.a.217.1		2	45.14	odd	6
324.2.e.d.109.1		2	45.34	even	6
324.2.e.d.217.1		2	45.4	even	6
1296.2.a.a.1.1		1	20.19	odd	2
1296.2.a.j.1.1		1	60.59	even	2
1296.2.i.d.433.1		2	180.119	even	6
1296.2.i.d.865.1		2	180.59	even	6
1296.2.i.p.433.1		2	180.79	odd	6
1296.2.i.p.865.1		2	180.139	odd	6
5184.2.a.d.1.1		1	120.59	even	2
5184.2.a.g.1.1		1	120.29	odd	2
5184.2.a.z.1.1		1	40.19	odd	2
5184.2.a.bc.1.1		1	40.29	even	2
8100.2.a.a.1.1		1	3.2	odd	2
8100.2.a.f.1.1		1	1.1	even	1	trivial
8100.2.d.a.649.1		2	15.2	even	4
8100.2.d.a.649.2		2	15.8	even	4
8100.2.d.j.649.1		2	5.2	odd	4
8100.2.d.j.649.2		2	5.3	odd	4