Embedded newform 429.2.be.a.89.15

• Label: 429.2.be.a.89.15
• Level: $429$
• Weight: $2$
• Character: 429.89
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(46\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			89.15
Character	\(\chi\)	\(=\)	429.89
Dual form			429.2.be.a.188.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22945 - 0.329429i) q^{2} +(-1.72326 - 0.174256i) q^{3} +(-0.329037 - 0.189970i) q^{4} +(0.992009 - 0.992009i) q^{5} +(2.06125 + 0.781932i) q^{6} +(-0.778711 - 2.90619i) q^{7} +(2.14199 + 2.14199i) q^{8} +(2.93927 + 0.600579i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 12 q^{6} + 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22945	−	0.329429i	−0.869349	−	0.232941i	−0.203543	−	0.979066i	\(-0.565246\pi\)
							−0.665806	+	0.746125i	\(0.731912\pi\)
\(3\)	−1.72326	−	0.174256i	−0.994926	−	0.100607i
							\(5\)	0.992009	−	0.992009i	0.443640	−	0.443640i	−0.449593	−	0.893233i	\(-0.648431\pi\)
0.893233	+	0.449593i	\(0.148431\pi\)
\(7\)	−0.778711	−	2.90619i	−0.294325	−	1.09844i	−0.941752	−	0.336309i	\(-0.890821\pi\)
							0.647427	−	0.762128i	\(-0.275845\pi\)
\(11\)	0.258819	−	0.965926i	0.0780369	−	0.291238i
							\(13\)	1.86065	−	3.08836i	0.516051	−	0.856558i
\(17\)	−1.14960	+	1.99117i	−0.278819	+	0.482929i								−0.971092	−	0.238707i	\(-0.923276\pi\)
							0.692272	+	0.721636i	\(0.256610\pi\)
\(19\)	2.42924	−	0.650914i	0.557307	−	0.149330i	0.0308381	−	0.999524i	\(-0.490182\pi\)
							0.526469	+	0.850194i	\(0.323516\pi\)
\(23\)	−3.73353	−	6.46667i	−0.778496	−	1.34839i	−0.932809	−	0.360372i	\(-0.882650\pi\)
							0.154313	−	0.988022i	\(-0.450684\pi\)
\(29\)	−5.27539	+	3.04575i	−0.979616	+	0.565581i	−0.902154	−	0.431414i	\(-0.858015\pi\)
							−0.0774615	+	0.996995i	\(0.524681\pi\)
\(31\)	−2.54164	−	2.54164i	−0.456491	−	0.456491i	0.441011	−	0.897502i	\(-0.354620\pi\)
							−0.897502	+	0.441011i	\(0.854620\pi\)
\(37\)	−0.695182	−	0.186274i	−0.114287	−	0.0306232i	0.201222	−	0.979546i	\(-0.435509\pi\)
							−0.315509	+	0.948922i	\(0.602175\pi\)
\(41\)	−6.08659	−	1.63090i	−0.950565	−	0.254703i	−0.249963	−	0.968255i	\(-0.580418\pi\)
							−0.700602	+	0.713552i	\(0.747085\pi\)
\(43\)	−4.42243	−	2.55329i	−0.674414	−	0.389373i	0.123333	−	0.992365i	\(-0.460642\pi\)
							−0.797747	+	0.602992i	\(0.793975\pi\)
\(47\)	1.89645	+	1.89645i	0.276625	+	0.276625i	0.831760	−	0.555135i	\(-0.187333\pi\)
							−0.555135	+	0.831760i	\(0.687333\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.be.a.89.15	✓	184
3.2	odd	2	inner	429.2.be.a.89.32	yes	184
13.6	odd	12	inner	429.2.be.a.188.32	yes	184
39.32	even	12	inner	429.2.be.a.188.15	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.be.a.89.15	✓	184	1.1	even	1	trivial
429.2.be.a.89.32	yes	184	3.2	odd	2	inner
429.2.be.a.188.15	yes	184	39.32	even	12	inner
429.2.be.a.188.32	yes	184	13.6	odd	12	inner