Embedded newform 196.6.a.c.1.1

• Label: 196.6.a.c.1.1
• Level: $196$
• Weight: $6$
• Character: 196.1
• Self dual: yes
• Analytic conductor: $31.435$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	196.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(31.4352286833\)
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	\(\chi\)	\(=\)	196.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-16.0000 q^{3} +16.0000 q^{5} +13.0000 q^{9} -76.0000 q^{11} +880.000 q^{13} -256.000 q^{15} -1056.00 q^{17} +1936.00 q^{19} +936.000 q^{23} -2869.00 q^{25} +3680.00 q^{27} -3982.00 q^{29} +1568.00 q^{31} +1216.00 q^{33} +4938.00 q^{37} -14080.0 q^{39} -15840.0 q^{41} -16412.0 q^{43} +208.000 q^{45} -20768.0 q^{47} +16896.0 q^{51} -37402.0 q^{53} -1216.00 q^{55} -30976.0 q^{57} +21136.0 q^{59} -2992.00 q^{61} +14080.0 q^{65} -45836.0 q^{67} -14976.0 q^{69} -49840.0 q^{71} -56320.0 q^{73} +45904.0 q^{75} +40744.0 q^{79} -62039.0 q^{81} +112464. q^{83} -16896.0 q^{85} +63712.0 q^{87} +64256.0 q^{89} -25088.0 q^{93} +30976.0 q^{95} -2272.00 q^{97} -988.000 q^{99} +O(q^{100})\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	−16.0000	−1.02640	−0.513200	−	0.858269i	\(-0.671540\pi\)
−0.513200	+	0.858269i				\(0.671540\pi\)
\(5\)	16.0000	0.286217	0.143108	−	0.989707i	\(-0.454290\pi\)
			0.143108	+	0.989707i	\(0.454290\pi\)
\(7\)	0	0
			\(11\)	−76.0000	−0.189379	−0.0946895	−	0.995507i	\(-0.530186\pi\)
−0.0946895	+	0.995507i				\(0.530186\pi\)
\(13\)	880.000	1.44419	0.722095	−	0.691794i	\(-0.243179\pi\)
			0.722095	+	0.691794i	\(0.243179\pi\)
\(17\)	−1056.00	−0.886220	−0.443110	−	0.896467i	\(-0.646125\pi\)
			−0.443110	+	0.896467i	\(0.646125\pi\)
\(19\)	1936.00	1.23033	0.615165	−	0.788399i	\(-0.289090\pi\)
			0.615165	+	0.788399i	\(0.289090\pi\)
\(23\)	936.000	0.368940	0.184470	−	0.982838i	\(-0.440943\pi\)
			0.184470	+	0.982838i	\(0.440943\pi\)
\(29\)	−3982.00	−0.879238	−0.439619	−	0.898184i	\(-0.644886\pi\)
			−0.439619	+	0.898184i	\(0.644886\pi\)
\(31\)	1568.00	0.293050	0.146525	−	0.989207i	\(-0.453191\pi\)
			0.146525	+	0.989207i	\(0.453191\pi\)
\(37\)	4938.00	0.592989	0.296495	−	0.955035i	\(-0.404182\pi\)
			0.296495	+	0.955035i	\(0.404182\pi\)
\(41\)	−15840.0	−1.47162	−0.735810	−	0.677188i	\(-0.763198\pi\)
			−0.735810	+	0.677188i	\(0.763198\pi\)
\(43\)	−16412.0	−1.35360	−0.676800	−	0.736167i	\(-0.736634\pi\)
			−0.676800	+	0.736167i	\(0.736634\pi\)
\(47\)	−20768.0	−1.37136	−0.685678	−	0.727905i	\(-0.740494\pi\)
			−0.685678	+	0.727905i	\(0.740494\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.6.a.c.1.1	✓	1
4.3	odd	2		784.6.a.j.1.1		1
7.2	even	3		196.6.e.h.165.1		2
7.3	odd	6		196.6.e.c.177.1		2
7.4	even	3		196.6.e.h.177.1		2
7.5	odd	6		196.6.e.c.165.1		2
7.6	odd	2		196.6.a.f.1.1	yes	1
28.27	even	2		784.6.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.6.a.c.1.1	✓	1	1.1	even	1	trivial
196.6.a.f.1.1	yes	1	7.6	odd	2
196.6.e.c.165.1		2	7.5	odd	6
196.6.e.c.177.1		2	7.3	odd	6
196.6.e.h.165.1		2	7.2	even	3
196.6.e.h.177.1		2	7.4	even	3
784.6.a.b.1.1		1	28.27	even	2
784.6.a.j.1.1		1	4.3	odd	2