Embedded newform 29.9.f.a.2.7

• Label: 29.9.f.a.2.7
• Level: $29$
• Weight: $9$
• Character: 29.2
• Analytic conductor: $11.814$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8139796918\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(19\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			2.7
Character	\(\chi\)	\(=\)	29.2
Dual form			29.9.f.a.15.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.14136 - 0.804638i) q^{2} +(-31.1173 - 19.5523i) q^{3} +(-199.230 - 45.4729i) q^{4} +(-619.444 + 493.990i) q^{5} +(206.487 + 164.668i) q^{6} +(232.502 + 1018.66i) q^{7} +(3122.70 + 1092.68i) q^{8} +(-2260.72 - 4694.43i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 3664 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.14136	−	0.804638i	−0.446335	−	0.0502899i	−0.114063	−	0.993474i	\(-0.536387\pi\)
							−0.332272	+	0.943184i	\(0.607815\pi\)
\(3\)	−31.1173	−	19.5523i	−0.384164	−	0.241386i	0.326079	−	0.945342i	\(-0.394272\pi\)
							−0.710243	+	0.703956i	\(0.751415\pi\)
\(5\)	−619.444	+	493.990i	−0.991110	+	0.790384i	−0.977803	−	0.209527i	\(-0.932807\pi\)
							−0.0133076	+	0.999911i	\(0.504236\pi\)
\(7\)	232.502	+	1018.66i	0.0968355	+	0.424264i	0.999987	−	0.00505756i	\(-0.00160988\pi\)
							−0.903152	+	0.429321i	\(0.858753\pi\)
\(11\)	−5468.60	+	1913.55i	−0.373513	+	0.130698i	−0.510506	−	0.859874i	\(-0.670542\pi\)
							0.136993	+	0.990572i	\(0.456256\pi\)
\(13\)	5187.02	−	10770.9i	0.181612	−	0.377121i	−0.790210	−	0.612836i	\(-0.790029\pi\)
							0.971822	+	0.235715i	\(0.0757431\pi\)
\(17\)	83423.7	−	83423.7i	0.998836	−	0.998836i	−0.00116372	−	0.999999i	\(-0.500370\pi\)
							0.999999	+	0.00116372i	\(0.000370423\pi\)
\(19\)	51464.2	+	81904.9i	0.394904	+	0.628486i	0.983705	−	0.179793i	\(-0.0575426\pi\)
							−0.588801	+	0.808278i	\(0.700400\pi\)
\(23\)	95782.2	−	120107.i	0.342274	−	0.429198i	−0.580666	−	0.814142i	\(-0.697208\pi\)
							0.922940	+	0.384944i	\(0.125779\pi\)
\(29\)	303364.	+	638918.i	0.428916	+	0.903344i
							\(31\)	−31594.1	−	3559.80i	−0.0342105	−	0.00385459i	0.0948427	−	0.995492i	\(-0.469765\pi\)
−0.129053	+	0.991638i	\(0.541194\pi\)
\(37\)	−783247.	−	274070.i	−0.417919	−	0.146236i	0.113124	−	0.993581i	\(-0.463914\pi\)
							−0.531043	+	0.847345i	\(0.678200\pi\)
\(41\)	−391202.	−	391202.i	−0.138441	−	0.138441i	0.634490	−	0.772931i	\(-0.281210\pi\)
							−0.772931	+	0.634490i	\(0.781210\pi\)
\(43\)	227352.	+	2.01781e6i	0.0665007	+	0.590210i	0.982132	+	0.188191i	\(0.0602626\pi\)
							−0.915632	+	0.402018i	\(0.868309\pi\)
\(47\)	−1.78860e6	−	5.11152e6i	−0.366540	−	1.04751i	−0.968506	−	0.248989i	\(-0.919902\pi\)
							0.601966	−	0.798521i	\(-0.294384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.9.f.a.2.7	✓	228
29.15	odd	28	inner	29.9.f.a.15.7	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.9.f.a.2.7	✓	228	1.1	even	1	trivial
29.9.f.a.15.7	yes	228	29.15	odd	28	inner