Embedded newform 2156.4.a.c.1.1

• Label: 2156.4.a.c.1.1
• Level: $2156$
• Weight: $4$
• Character: 2156.1
• Self dual: yes
• Analytic conductor: $127.208$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2156.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(127.208117972\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 308)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2156.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+7.00000 q^{3} +1.00000 q^{5} +22.0000 q^{9} +11.0000 q^{11} -12.0000 q^{13} +7.00000 q^{15} -2.00000 q^{17} -82.0000 q^{19} +7.00000 q^{23} -124.000 q^{25} -35.0000 q^{27} -102.000 q^{29} +171.000 q^{31} +77.0000 q^{33} -357.000 q^{37} -84.0000 q^{39} +114.000 q^{41} -344.000 q^{43} +22.0000 q^{45} -96.0000 q^{47} -14.0000 q^{51} -430.000 q^{53} +11.0000 q^{55} -574.000 q^{57} +201.000 q^{59} +2.00000 q^{61} -12.0000 q^{65} +313.000 q^{67} +49.0000 q^{69} -579.000 q^{71} +438.000 q^{73} -868.000 q^{75} +494.000 q^{79} -839.000 q^{81} -748.000 q^{83} -2.00000 q^{85} -714.000 q^{87} -457.000 q^{89} +1197.00 q^{93} -82.0000 q^{95} +1037.00 q^{97} +242.000 q^{99} +O(q^{100})\)

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
			\(3\)	7.00000	1.34715	0.673575	−	0.739119i	\(-0.264758\pi\)
0.673575	+	0.739119i				\(0.264758\pi\)
\(5\)	1.00000	0.0894427	0.0447214	−	0.998999i	\(-0.485760\pi\)
			0.0447214	+	0.998999i	\(0.485760\pi\)
\(7\)	0	0
			\(11\)	11.0000	0.301511
\(13\)	−12.0000	−0.256015				−0.128008	−	0.991773i	\(-0.540858\pi\)
			−0.128008	+	0.991773i	\(0.540858\pi\)
\(17\)	−2.00000	−0.0285336	−0.0142668	−	0.999898i	\(-0.504541\pi\)
			−0.0142668	+	0.999898i	\(0.504541\pi\)
\(19\)	−82.0000	−0.990110	−0.495055	−	0.868862i	\(-0.664852\pi\)
			−0.495055	+	0.868862i	\(0.664852\pi\)
\(23\)	7.00000	0.0634609	0.0317305	−	0.999496i	\(-0.489898\pi\)
			0.0317305	+	0.999496i	\(0.489898\pi\)
\(29\)	−102.000	−0.653135	−0.326568	−	0.945174i	\(-0.605892\pi\)
			−0.326568	+	0.945174i	\(0.605892\pi\)
\(31\)	171.000	0.990726	0.495363	−	0.868686i	\(-0.335035\pi\)
			0.495363	+	0.868686i	\(0.335035\pi\)
\(37\)	−357.000	−1.58623	−0.793114	−	0.609073i	\(-0.791542\pi\)
			−0.793114	+	0.609073i	\(0.791542\pi\)
\(41\)	114.000	0.434239	0.217120	−	0.976145i	\(-0.430334\pi\)
			0.217120	+	0.976145i	\(0.430334\pi\)
\(43\)	−344.000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	−96.0000	−0.297937	−0.148969	−	0.988842i	\(-0.547595\pi\)
			−0.148969	+	0.988842i	\(0.547595\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.4.a.c.1.1		1
7.6	odd	2		308.4.a.a.1.1	✓	1
28.27	even	2		1232.4.a.h.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.4.a.a.1.1	✓	1	7.6	odd	2
1232.4.a.h.1.1		1	28.27	even	2
2156.4.a.c.1.1		1	1.1	even	1	trivial