Embedded newform 7104.2.a.u.1.1

• Label: 7104.2.a.u.1.1
• Level: $7104$
• Weight: $2$
• Character: 7104.1
• Self dual: yes
• Analytic conductor: $56.726$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7104 = 2^{6} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7104.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(56.7257255959\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 222)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	7104.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{13} -3.00000 q^{17} -7.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} -5.00000 q^{25} +1.00000 q^{27} -2.00000 q^{31} +3.00000 q^{33} -1.00000 q^{37} +1.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} -6.00000 q^{47} -6.00000 q^{49} -3.00000 q^{51} -9.00000 q^{53} -7.00000 q^{57} +10.0000 q^{61} +1.00000 q^{63} +2.00000 q^{67} -3.00000 q^{69} -12.0000 q^{71} +5.00000 q^{73} -5.00000 q^{75} +3.00000 q^{77} -2.00000 q^{79} +1.00000 q^{81} +3.00000 q^{83} -3.00000 q^{89} +1.00000 q^{91} -2.00000 q^{93} +2.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	1.00000	0.577350
\(5\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	1.00000	0.377964	0.188982	−	0.981981i	\(-0.439481\pi\)
			0.188982	+	0.981981i	\(0.439481\pi\)
\(11\)	3.00000	0.904534	0.452267	−	0.891883i	\(-0.350615\pi\)
			0.452267	+	0.891883i	\(0.350615\pi\)
\(13\)	1.00000	0.277350	0.138675	−	0.990338i	\(-0.455716\pi\)
			0.138675	+	0.990338i	\(0.455716\pi\)
\(17\)	−3.00000	−0.727607	−0.363803	−	0.931476i	\(-0.618522\pi\)
			−0.363803	+	0.931476i	\(0.618522\pi\)
\(19\)	−7.00000	−1.60591	−0.802955	−	0.596040i	\(-0.796740\pi\)
			−0.802955	+	0.596040i	\(0.796740\pi\)
\(23\)	−3.00000	−0.625543	−0.312772	−	0.949828i	\(-0.601257\pi\)
			−0.312772	+	0.949828i	\(0.601257\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	−2.00000	−0.359211	−0.179605	−	0.983739i	\(-0.557482\pi\)
			−0.179605	+	0.983739i	\(0.557482\pi\)
\(37\)	−1.00000	−0.164399
			\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
−0.468521	+	0.883452i				\(0.655213\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	−6.00000	−0.875190	−0.437595	−	0.899172i	\(-0.644170\pi\)
			−0.437595	+	0.899172i	\(0.644170\pi\)
\(53\)	−9.00000	−1.23625	−0.618123	−	0.786082i	\(-0.712106\pi\)
			−0.618123	+	0.786082i	\(0.712106\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	10.0000	1.28037	0.640184	−	0.768221i	\(-0.278858\pi\)
			0.640184	+	0.768221i	\(0.278858\pi\)
\(67\)	2.00000	0.244339	0.122169	−	0.992509i	\(-0.461015\pi\)
			0.122169	+	0.992509i	\(0.461015\pi\)
\(71\)	−12.0000	−1.42414	−0.712069	−	0.702109i	\(-0.752242\pi\)
			−0.712069	+	0.702109i	\(0.752242\pi\)
\(73\)	5.00000	0.585206	0.292603	−	0.956234i	\(-0.405479\pi\)
			0.292603	+	0.956234i	\(0.405479\pi\)
\(79\)	−2.00000	−0.225018	−0.112509	−	0.993651i	\(-0.535889\pi\)
			−0.112509	+	0.993651i	\(0.535889\pi\)
\(83\)	3.00000	0.329293	0.164646	−	0.986353i	\(-0.447352\pi\)
			0.164646	+	0.986353i	\(0.447352\pi\)
\(89\)	−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	7104.2.a.u.1.1		1
4.3	odd	2		7104.2.a.g.1.1		1
8.3	odd	2		222.2.a.e.1.1	✓	1
8.5	even	2		1776.2.a.c.1.1		1
24.5	odd	2		5328.2.a.l.1.1		1
24.11	even	2		666.2.a.a.1.1		1
40.19	odd	2		5550.2.a.h.1.1		1
296.147	odd	2		8214.2.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.a.e.1.1	✓	1	8.3	odd	2
666.2.a.a.1.1		1	24.11	even	2
1776.2.a.c.1.1		1	8.5	even	2
5328.2.a.l.1.1		1	24.5	odd	2
5550.2.a.h.1.1		1	40.19	odd	2
7104.2.a.g.1.1		1	4.3	odd	2
7104.2.a.u.1.1		1	1.1	even	1	trivial
8214.2.a.d.1.1		1	296.147	odd	2