Embedded newform 429.2.y.b.116.9

• Label: 429.2.y.b.116.9
• Level: $429$
• Weight: $2$
• Character: 429.116
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.y (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			116.9
Character	\(\chi\)	\(=\)	429.116
Dual form			429.2.y.b.233.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39922 + 0.454633i) q^{2} +(0.606125 + 1.62253i) q^{3} +(0.133079 - 0.0966876i) q^{4} +(0.724806 - 2.23072i) q^{5} +(-1.58576 - 1.99471i) q^{6} +(-0.197187 + 0.143265i) q^{7} +(1.58728 - 2.18470i) q^{8} +(-2.26523 + 1.96691i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 6 q^{3} + 20 q^{4} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39922	+	0.454633i	−0.989395	+	0.321474i	−0.758620	−	0.651533i	\(-0.774126\pi\)
							−0.230775	+	0.973007i	\(0.574126\pi\)
\(3\)	0.606125	+	1.62253i	0.349946	+	0.936770i
							\(5\)	0.724806	−	2.23072i	0.324143	−	0.997611i	−0.647683	−	0.761910i	\(-0.724262\pi\)
0.971826	−	0.235700i	\(-0.0757383\pi\)
\(7\)	−0.197187	+	0.143265i	−0.0745297	+	0.0541490i	−0.624426	−	0.781084i	\(-0.714667\pi\)
							0.549897	+	0.835233i	\(0.314667\pi\)
\(11\)	−1.96181	+	2.67419i	−0.591507	+	0.806300i
							\(13\)	2.40195	+	2.68899i	0.666180	+	0.745791i
\(17\)	−1.44366	+	4.44314i	−0.350140	+	1.07762i								0.608635	+	0.793451i	\(0.291718\pi\)
							−0.958774	+	0.284169i	\(0.908282\pi\)
\(19\)	−0.342154	−	0.248590i	−0.0784956	−	0.0570304i	0.547845	−	0.836580i	\(-0.315448\pi\)
							−0.626341	+	0.779549i	\(0.715448\pi\)
\(23\)			7.63341i			1.59168i	0.605510	+	0.795838i	\(0.292969\pi\)
							−0.605510	+	0.795838i	\(0.707031\pi\)
\(29\)	7.03467	−	5.11099i	1.30631	−	0.949087i	0.306310	−	0.951932i	\(-0.400905\pi\)
							0.999996	+	0.00284467i	\(0.000905487\pi\)
\(31\)	−9.96917	+	3.23918i	−1.79052	+	0.581774i	−0.999547	−	0.0300824i	\(-0.990423\pi\)
							−0.790969	+	0.611856i	\(0.790423\pi\)
\(37\)	−2.17673	−	2.99602i	−0.357853	−	0.492542i	0.591696	−	0.806161i	\(-0.298459\pi\)
							−0.949549	+	0.313619i	\(0.898459\pi\)
\(41\)	−3.71574	+	5.11428i	−0.580302	+	0.798717i	−0.993728	−	0.111821i	\(-0.964332\pi\)
							0.413427	+	0.910537i	\(0.364332\pi\)
\(43\)		−	10.1773i		−	1.55203i	−0.630717	−	0.776013i	\(-0.717239\pi\)
							0.630717	−	0.776013i	\(-0.282761\pi\)
\(47\)	6.94011	+	5.04228i	1.01232	+	0.735493i	0.964694	−	0.263373i	\(-0.0848348\pi\)
							0.0476248	+	0.998865i	\(0.484835\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.y.b.116.9	✓	176
3.2	odd	2	inner	429.2.y.b.116.36	yes	176
11.2	odd	10	inner	429.2.y.b.233.10	yes	176
13.12	even	2	inner	429.2.y.b.116.35	yes	176
33.2	even	10	inner	429.2.y.b.233.35	yes	176
39.38	odd	2	inner	429.2.y.b.116.10	yes	176
143.90	odd	10	inner	429.2.y.b.233.36	yes	176
429.233	even	10	inner	429.2.y.b.233.9	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.y.b.116.9	✓	176	1.1	even	1	trivial
429.2.y.b.116.10	yes	176	39.38	odd	2	inner
429.2.y.b.116.35	yes	176	13.12	even	2	inner
429.2.y.b.116.36	yes	176	3.2	odd	2	inner
429.2.y.b.233.9	yes	176	429.233	even	10	inner
429.2.y.b.233.10	yes	176	11.2	odd	10	inner
429.2.y.b.233.35	yes	176	33.2	even	10	inner
429.2.y.b.233.36	yes	176	143.90	odd	10	inner