Embedded newform 5070.2.a.b.1.1

• Label: 5070.2.a.b.1.1
• Level: $5070$
• Weight: $2$
• Character: 5070.1
• Self dual: yes
• Analytic conductor: $40.484$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(40.4841538248\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	5070.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +3.00000 q^{19} -1.00000 q^{20} -3.00000 q^{21} +3.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.00000 q^{28} -4.00000 q^{29} -1.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -3.00000 q^{35} +1.00000 q^{36} +9.00000 q^{37} -3.00000 q^{38} +1.00000 q^{40} -10.0000 q^{41} +3.00000 q^{42} -10.0000 q^{43} -3.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +9.00000 q^{53} +1.00000 q^{54} +3.00000 q^{55} -3.00000 q^{56} -3.00000 q^{57} +4.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} -6.00000 q^{61} -6.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} -8.00000 q^{67} +4.00000 q^{69} +3.00000 q^{70} -14.0000 q^{71} -1.00000 q^{72} -8.00000 q^{73} -9.00000 q^{74} -1.00000 q^{75} +3.00000 q^{76} -9.00000 q^{77} +6.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +16.0000 q^{83} -3.00000 q^{84} +10.0000 q^{86} +4.00000 q^{87} +3.00000 q^{88} -3.00000 q^{89} +1.00000 q^{90} -4.00000 q^{92} -6.00000 q^{93} +3.00000 q^{94} -3.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} -2.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107
			\(3\)	−1.00000	−0.577350
\(5\)	−1.00000	−0.447214
			\(7\)	3.00000	1.13389	0.566947	−	0.823754i	\(-0.308125\pi\)
0.566947	+	0.823754i				\(0.308125\pi\)
\(11\)	−3.00000	−0.904534	−0.452267	−	0.891883i	\(-0.649385\pi\)
			−0.452267	+	0.891883i	\(0.649385\pi\)
\(13\)	0	0
			\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(19\)	3.00000	0.688247	0.344124	−	0.938924i	\(-0.388176\pi\)
			0.344124	+	0.938924i	\(0.388176\pi\)
\(23\)	−4.00000	−0.834058	−0.417029	−	0.908893i	\(-0.636929\pi\)
			−0.417029	+	0.908893i	\(0.636929\pi\)
\(29\)	−4.00000	−0.742781	−0.371391	−	0.928477i	\(-0.621119\pi\)
			−0.371391	+	0.928477i	\(0.621119\pi\)
\(31\)	6.00000	1.07763	0.538816	−	0.842424i	\(-0.318872\pi\)
			0.538816	+	0.842424i	\(0.318872\pi\)
\(37\)	9.00000	1.47959	0.739795	−	0.672832i	\(-0.234922\pi\)
			0.739795	+	0.672832i	\(0.234922\pi\)
\(41\)	−10.0000	−1.56174	−0.780869	−	0.624695i	\(-0.785223\pi\)
			−0.780869	+	0.624695i	\(0.785223\pi\)
\(43\)	−10.0000	−1.52499	−0.762493	−	0.646997i	\(-0.776025\pi\)
			−0.762493	+	0.646997i	\(0.776025\pi\)
\(47\)	−3.00000	−0.437595	−0.218797	−	0.975770i	\(-0.570213\pi\)
			−0.218797	+	0.975770i	\(0.570213\pi\)
\(53\)	9.00000	1.23625	0.618123	−	0.786082i	\(-0.287894\pi\)
			0.618123	+	0.786082i	\(0.287894\pi\)
\(59\)	12.0000	1.56227	0.781133	−	0.624364i	\(-0.214642\pi\)
			0.781133	+	0.624364i	\(0.214642\pi\)
\(61\)	−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	\(-0.662521\pi\)
			−0.488678	+	0.872464i	\(0.662521\pi\)
\(71\)	−14.0000	−1.66149	−0.830747	−	0.556650i	\(-0.812086\pi\)
			−0.830747	+	0.556650i	\(0.812086\pi\)
\(73\)	−8.00000	−0.936329	−0.468165	−	0.883641i	\(-0.655085\pi\)
			−0.468165	+	0.883641i	\(0.655085\pi\)
\(79\)	6.00000	0.675053	0.337526	−	0.941316i	\(-0.390410\pi\)
			0.337526	+	0.941316i	\(0.390410\pi\)
\(83\)	16.0000	1.75623	0.878114	−	0.478451i	\(-0.158802\pi\)
			0.878114	+	0.478451i	\(0.158802\pi\)
\(89\)	−3.00000	−0.317999	−0.159000	−	0.987279i	\(-0.550827\pi\)
			−0.159000	+	0.987279i	\(0.550827\pi\)
\(97\)	8.00000	0.812277	0.406138	−	0.913812i	\(-0.366875\pi\)
			0.406138	+	0.913812i	\(0.366875\pi\)

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	5070.2.a.b.1.1		1
13.3	even	3		390.2.i.e.61.1	✓	2
13.5	odd	4		5070.2.b.b.1351.2		2
13.8	odd	4		5070.2.b.b.1351.1		2
13.9	even	3		390.2.i.e.211.1	yes	2
13.12	even	2		5070.2.a.r.1.1		1
39.29	odd	6		1170.2.i.c.451.1		2
39.35	odd	6		1170.2.i.c.991.1		2
65.3	odd	12		1950.2.z.d.1699.1		4
65.9	even	6		1950.2.i.f.601.1		2
65.22	odd	12		1950.2.z.d.1849.1		4
65.29	even	6		1950.2.i.f.451.1		2
65.42	odd	12		1950.2.z.d.1699.2		4
65.48	odd	12		1950.2.z.d.1849.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.e.61.1	✓	2	13.3	even	3
390.2.i.e.211.1	yes	2	13.9	even	3
1170.2.i.c.451.1		2	39.29	odd	6
1170.2.i.c.991.1		2	39.35	odd	6
1950.2.i.f.451.1		2	65.29	even	6
1950.2.i.f.601.1		2	65.9	even	6
1950.2.z.d.1699.1		4	65.3	odd	12
1950.2.z.d.1699.2		4	65.42	odd	12
1950.2.z.d.1849.1		4	65.22	odd	12
1950.2.z.d.1849.2		4	65.48	odd	12
5070.2.a.b.1.1		1	1.1	even	1	trivial
5070.2.a.r.1.1		1	13.12	even	2
5070.2.b.b.1351.1		2	13.8	odd	4
5070.2.b.b.1351.2		2	13.5	odd	4