Embedded newform 196.2.p.a.103.6

• Label: 196.2.p.a.103.6
• Level: $196$
• Weight: $2$
• Character: 196.103
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	196.p (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.56506787962\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(26\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			103.6
Character	\(\chi\)	\(=\)	196.103
Dual form			196.2.p.a.59.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19983 - 0.748605i) q^{2} +(0.318723 + 0.217301i) q^{3} +(0.879181 + 1.79640i) q^{4} +(-2.23221 - 0.167281i) q^{5} +(-0.219740 - 0.499322i) q^{6} +(-2.62649 + 0.318706i) q^{7} +(0.289926 - 2.81353i) q^{8} +(-1.04166 - 2.65410i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 13 q^{2} - 13 q^{4} - 22 q^{5} - 14 q^{6} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19983	−	0.748605i	−0.848407	−	0.529344i
							\(3\)	0.318723	+	0.217301i	0.184015	+	0.125459i	0.651824	−	0.758370i	\(-0.274004\pi\)
−0.467809	+	0.883829i	\(0.654957\pi\)
\(5\)	−2.23221	−	0.167281i	−0.998276	−	0.0748105i	−0.434443	−	0.900699i	\(-0.643055\pi\)
							−0.563833	+	0.825889i	\(0.690674\pi\)
\(7\)	−2.62649	+	0.318706i	−0.992718	+	0.120459i
							\(11\)	−4.62345	−	1.81457i	−1.39402	−	0.547113i	−0.454852	−	0.890567i	\(-0.650308\pi\)
−0.939170	+	0.343454i	\(0.888403\pi\)
\(13\)	4.05935	−	3.23722i	1.12586	−	0.897845i	0.130255	−	0.991481i	\(-0.458420\pi\)
							0.995606	+	0.0936360i	\(0.0298490\pi\)
\(17\)	−1.83646	−	1.97924i	−0.445408	−	0.480035i	0.469941	−	0.882698i	\(-0.344275\pi\)
							−0.915348	+	0.402663i	\(0.868085\pi\)
\(19\)	−1.20408	+	2.08553i	−0.276235	+	0.478453i	−0.970446	−	0.241318i	\(-0.922420\pi\)
							0.694211	+	0.719772i	\(0.255754\pi\)
\(23\)	−2.13014	+	2.29575i	−0.444165	+	0.478696i	−0.914959	−	0.403547i	\(-0.867777\pi\)
							0.470794	+	0.882243i	\(0.343968\pi\)
\(29\)	1.93620	+	8.48305i	0.359544	+	1.57526i	0.754333	+	0.656492i	\(0.227960\pi\)
							−0.394790	+	0.918771i	\(0.629183\pi\)
\(31\)	−0.0585377	−	0.101390i	−0.0105137	−	0.0182102i	0.860721	−	0.509077i	\(-0.170013\pi\)
							−0.871234	+	0.490867i	\(0.836680\pi\)
\(37\)	−0.160245	−	0.0494289i	−0.0263441	−	0.00812606i	0.281555	−	0.959545i	\(-0.409150\pi\)
							−0.307899	+	0.951419i	\(0.599626\pi\)
\(41\)	−4.98272	−	10.3467i	−0.778171	−	1.61589i	−0.787797	−	0.615935i	\(-0.788778\pi\)
							0.00962633	−	0.999954i	\(-0.496936\pi\)
\(43\)	−1.97048	+	4.09173i	−0.300495	+	0.623984i	−0.995473	−	0.0950473i	\(-0.969700\pi\)
							0.694978	+	0.719031i	\(0.255414\pi\)
\(47\)	10.2787	−	1.54926i	1.49930	−	0.225983i	0.652457	−	0.757826i	\(-0.273738\pi\)
							0.846843	+	0.531843i	\(0.178500\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.p.a.103.6	yes	312
4.3	odd	2	inner	196.2.p.a.103.9	yes	312
49.10	odd	42	inner	196.2.p.a.59.9	yes	312
196.59	even	42	inner	196.2.p.a.59.6	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.p.a.59.6	✓	312	196.59	even	42	inner
196.2.p.a.59.9	yes	312	49.10	odd	42	inner
196.2.p.a.103.6	yes	312	1.1	even	1	trivial
196.2.p.a.103.9	yes	312	4.3	odd	2	inner