Embedded newform 196.10.e.a.177.1

• Label: 196.10.e.a.177.1
• Level: $196$
• Weight: $10$
• Character: 196.177
• Analytic conductor: $100.947$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	196.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(100.947023888\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			177.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	196.177
Dual form			196.10.e.a.165.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-114.000 - 197.454i) q^{3} +(333.000 - 576.773i) q^{5} +(-16150.5 + 27973.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 228 q^{3} + 666 q^{5} - 32301 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(99\)	\(101\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)	−114.000	−	197.454i	−0.812567	−	1.40741i	−0.911062	−	0.412270i	\(-0.864736\pi\)
0.0984947	−	0.995138i	\(-0.468597\pi\)
\(5\)	333.000	−	576.773i	0.238275	−	0.412705i	−0.721944	−	0.691951i	\(-0.756751\pi\)
							0.960220	+	0.279246i	\(0.0900846\pi\)
\(7\)		0			0
							\(11\)	15210.0	+	26344.5i	0.313229	+	0.542529i	0.979059	−	0.203574i	\(-0.0652559\pi\)
−0.665830	+	0.746103i	\(0.731923\pi\)
\(13\)	−32338.0			−0.314028			−0.157014	−	0.987596i	\(-0.550187\pi\)
							−0.157014	+	0.987596i	\(0.550187\pi\)
\(17\)	−295497.	−	511816.i	−0.858090	−	1.48626i	−0.873748	−	0.486378i	\(-0.838318\pi\)
							0.0156581	−	0.999877i	\(-0.495016\pi\)
\(19\)	−17338.0	+	30030.3i	−0.0305216	+	0.0528650i	−0.880883	−	0.473335i	\(-0.843050\pi\)
							0.850361	+	0.526200i	\(0.176384\pi\)
\(23\)	−524268.	+	908059.i	−0.390641	+	0.676610i	−0.992534	−	0.121966i	\(-0.961080\pi\)
							0.601893	+	0.798577i	\(0.294413\pi\)
\(29\)	4.40941e6			1.15768			0.578841	−	0.815441i	\(-0.303505\pi\)
							0.578841	+	0.815441i	\(0.303505\pi\)
\(31\)	3.70059e6	+	6.40961e6i	0.719687	+	1.24653i	0.961124	+	0.276118i	\(0.0890480\pi\)
							−0.241437	+	0.970417i	\(0.577619\pi\)
\(37\)	−5.11725e6	+	8.86334e6i	−0.448879	+	0.777481i	−0.998313	−	0.0580560i	\(-0.981510\pi\)
							0.549435	+	0.835537i	\(0.314843\pi\)
\(41\)	1.83527e7			1.01432			0.507158	−	0.861853i	\(-0.330696\pi\)
							0.507158	+	0.861853i	\(0.330696\pi\)
\(43\)	−252340.			−0.0112558			−0.00562792	−	0.999984i	\(-0.501791\pi\)
							−0.00562792	+	0.999984i	\(0.501791\pi\)
\(47\)	2.47586e7	−	4.28831e7i	0.740091	−	1.28188i	−0.212362	−	0.977191i	\(-0.568116\pi\)
							0.952453	−	0.304684i	\(-0.0985510\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.10.e.a.177.1		2
7.2	even	3		4.10.a.a.1.1	✓	1
7.3	odd	6		196.10.e.b.165.1		2
7.4	even	3	inner	196.10.e.a.165.1		2
7.5	odd	6		196.10.a.a.1.1		1
7.6	odd	2		196.10.e.b.177.1		2
21.2	odd	6		36.10.a.b.1.1		1
28.23	odd	6		16.10.a.a.1.1		1
35.2	odd	12		100.10.c.a.49.1		2
35.9	even	6		100.10.a.a.1.1		1
35.23	odd	12		100.10.c.a.49.2		2
56.37	even	6		64.10.a.a.1.1		1
56.51	odd	6		64.10.a.i.1.1		1
63.2	odd	6		324.10.e.b.109.1		2
63.16	even	3		324.10.e.e.109.1		2
63.23	odd	6		324.10.e.b.217.1		2
63.58	even	3		324.10.e.e.217.1		2
84.23	even	6		144.10.a.j.1.1		1
112.37	even	12		256.10.b.j.129.1		2
112.51	odd	12		256.10.b.b.129.1		2
112.93	even	12		256.10.b.j.129.2		2
112.107	odd	12		256.10.b.b.129.2		2
140.23	even	12		400.10.c.a.49.1		2
140.79	odd	6		400.10.a.k.1.1		1
140.107	even	12		400.10.c.a.49.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.10.a.a.1.1	✓	1	7.2	even	3
16.10.a.a.1.1		1	28.23	odd	6
36.10.a.b.1.1		1	21.2	odd	6
64.10.a.a.1.1		1	56.37	even	6
64.10.a.i.1.1		1	56.51	odd	6
100.10.a.a.1.1		1	35.9	even	6
100.10.c.a.49.1		2	35.2	odd	12
100.10.c.a.49.2		2	35.23	odd	12
144.10.a.j.1.1		1	84.23	even	6
196.10.a.a.1.1		1	7.5	odd	6
196.10.e.a.165.1		2	7.4	even	3	inner
196.10.e.a.177.1		2	1.1	even	1	trivial
196.10.e.b.165.1		2	7.3	odd	6
196.10.e.b.177.1		2	7.6	odd	2
256.10.b.b.129.1		2	112.51	odd	12
256.10.b.b.129.2		2	112.107	odd	12
256.10.b.j.129.1		2	112.37	even	12
256.10.b.j.129.2		2	112.93	even	12
324.10.e.b.109.1		2	63.2	odd	6
324.10.e.b.217.1		2	63.23	odd	6
324.10.e.e.109.1		2	63.16	even	3
324.10.e.e.217.1		2	63.58	even	3
400.10.a.k.1.1		1	140.79	odd	6
400.10.c.a.49.1		2	140.23	even	12
400.10.c.a.49.2		2	140.107	even	12