Embedded newform 196.2.j.a.27.5

• Label: 196.2.j.a.27.5
• Level: $196$
• Weight: $2$
• Character: 196.27
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $156$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	196.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.56506787962\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(26\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			27.5
Character	\(\chi\)	\(=\)	196.27
Dual form			196.2.j.a.167.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23433 - 0.690243i) q^{2} +(-0.0211920 + 0.0928483i) q^{3} +(1.04713 + 1.70397i) q^{4} +(-3.74506 - 0.854785i) q^{5} +(0.0902457 - 0.0999775i) q^{6} +(1.51239 + 2.17087i) q^{7} +(-0.116345 - 2.82603i) q^{8} +(2.69473 + 1.29772i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23433	−	0.690243i	−0.872801	−	0.488076i
							\(3\)	−0.0211920	+	0.0928483i	−0.0122352	+	0.0536060i	−0.980677	−	0.195634i	\(-0.937324\pi\)
0.968442	+	0.249240i	\(0.0801807\pi\)
\(5\)	−3.74506	−	0.854785i	−1.67484	−	0.382271i	−0.723478	−	0.690347i	\(-0.757458\pi\)
							−0.951362	+	0.308076i	\(0.900315\pi\)
\(7\)	1.51239	+	2.17087i	0.571629	+	0.820512i
							\(11\)	1.95278	+	4.05499i	0.588786	+	1.22263i	0.956240	+	0.292584i	\(0.0945153\pi\)
−0.367454	+	0.930042i	\(0.619770\pi\)
\(13\)	0.0239750	+	0.0497845i	0.00664946	+	0.0138077i	0.904267	−	0.426967i	\(-0.140418\pi\)
							−0.897618	+	0.440775i	\(0.854704\pi\)
\(17\)	−4.87147	+	3.88487i	−1.18151	+	0.942219i	−0.999159	−	0.0410147i	\(-0.986941\pi\)
							−0.182347	+	0.983234i	\(0.558370\pi\)
\(19\)	4.34022			0.995715			0.497857	−	0.867259i	\(-0.334120\pi\)
							0.497857	+	0.867259i	\(0.334120\pi\)
\(23\)	3.53943	+	2.82260i	0.738022	+	0.588553i	0.918685	−	0.394992i	\(-0.129253\pi\)
							−0.180662	+	0.983545i	\(0.557824\pi\)
\(29\)	−1.63571	−	2.05111i	−0.303743	−	0.380882i	0.606411	−	0.795151i	\(-0.292609\pi\)
							−0.910154	+	0.414269i	\(0.864037\pi\)
\(31\)	−3.07426			−0.552154			−0.276077	−	0.961136i	\(-0.589034\pi\)
							−0.276077	+	0.961136i	\(0.589034\pi\)
\(37\)	−4.15924	−	5.21552i	−0.683775	−	0.857427i	0.311921	−	0.950108i	\(-0.399028\pi\)
							−0.995696	+	0.0926812i	\(0.970456\pi\)
\(41\)	3.50699	+	0.800447i	0.547699	+	0.125009i	0.487412	−	0.873172i	\(-0.337941\pi\)
							0.0602876	+	0.998181i	\(0.480798\pi\)
\(43\)	−10.2424	+	2.33777i	−1.56196	+	0.356506i	−0.914175	−	0.405321i	\(-0.867160\pi\)
							−0.647781	+	0.761827i	\(0.724303\pi\)
\(47\)	3.67011	−	1.76743i	0.535340	−	0.257806i	−0.146624	−	0.989192i	\(-0.546841\pi\)
							0.681963	+	0.731386i	\(0.261126\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.j.a.27.5	✓	156
4.3	odd	2	inner	196.2.j.a.27.13	yes	156
49.20	odd	14	inner	196.2.j.a.167.13	yes	156
196.167	even	14	inner	196.2.j.a.167.5	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.j.a.27.5	✓	156	1.1	even	1	trivial
196.2.j.a.27.13	yes	156	4.3	odd	2	inner
196.2.j.a.167.5	yes	156	196.167	even	14	inner
196.2.j.a.167.13	yes	156	49.20	odd	14	inner