Embedded newform 29.9.f.a.2.4

• Label: 29.9.f.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	29.2
Dual form			29.9.f.a.15.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-21.5736 - 2.43076i) q^{2} +(-17.4548 - 10.9676i) q^{3} +(209.928 + 47.9147i) q^{4} +(-119.794 + 95.5326i) q^{5} +(349.902 + 279.037i) q^{6} +(-857.743 - 3758.02i) q^{7} +(833.466 + 291.642i) q^{8} +(-2662.33 - 5528.38i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 3664 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−21.5736	−	2.43076i	−1.34835	−	0.151922i	−0.591918	−	0.805998i	\(-0.701629\pi\)
							−0.756429	+	0.654076i	\(0.773058\pi\)
\(3\)	−17.4548	−	10.9676i	−0.215491	−	0.135402i	0.419962	−	0.907542i	\(-0.362044\pi\)
							−0.635453	+	0.772140i	\(0.719186\pi\)
\(5\)	−119.794	+	95.5326i	−0.191671	+	0.152852i	−0.714623	−	0.699509i	\(-0.753402\pi\)
							0.522953	+	0.852362i	\(0.324830\pi\)
\(7\)	−857.743	−	3758.02i	−0.357244	−	1.56519i	−0.760026	−	0.649892i	\(-0.774814\pi\)
							0.402783	−	0.915296i	\(-0.368043\pi\)
\(11\)	4900.31	−	1714.69i	0.334698	−	0.117116i	−0.157698	−	0.987487i	\(-0.550407\pi\)
							0.492395	+	0.870372i	\(0.336121\pi\)
\(13\)	1512.62	−	3141.00i	0.0529612	−	0.109975i	−0.872809	−	0.488062i	\(-0.837704\pi\)
							0.925770	+	0.378087i	\(0.123418\pi\)
\(17\)	−97651.2	+	97651.2i	−1.16918	+	1.16918i	−0.186779	+	0.982402i	\(0.559805\pi\)
							−0.982402	+	0.186779i	\(0.940195\pi\)
\(19\)	77454.2	+	123268.i	0.594334	+	0.945877i	0.999500	+	0.0316336i	\(0.0100710\pi\)
							−0.405166	+	0.914243i	\(0.632786\pi\)
\(23\)	72766.2	−	91246.0i	0.260027	−	0.326064i	−0.634631	−	0.772815i	\(-0.718848\pi\)
							0.894658	+	0.446752i	\(0.147419\pi\)
\(29\)	330661.	−	625228.i	0.467509	−	0.883988i
							\(31\)	−1.02261e6	−	115221.i	−1.10730	−	0.124762i	−0.460672	−	0.887571i	\(-0.652391\pi\)
−0.646625	+	0.762808i	\(0.723820\pi\)
\(37\)	−2.60573e6	−	911783.i	−1.39034	−	0.486502i	−0.471759	−	0.881728i	\(-0.656381\pi\)
							−0.918584	+	0.395226i	\(0.870666\pi\)
\(41\)	2.18287e6	+	2.18287e6i	0.772490	+	0.772490i	0.978541	−	0.206051i	\(-0.0660612\pi\)
							−0.206051	+	0.978541i	\(0.566061\pi\)
\(43\)	646585.	+	5.73860e6i	0.189126	+	1.67854i	0.626624	+	0.779321i	\(0.284436\pi\)
							−0.437498	+	0.899219i	\(0.644135\pi\)
\(47\)	535026.	+	1.52902e6i	0.109644	+	0.313343i	0.985612	−	0.169026i	\(-0.0540621\pi\)
							−0.875968	+	0.482369i	\(0.839776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.9.f.a.2.4	✓	228
29.15	odd	28	inner	29.9.f.a.15.4	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.9.f.a.2.4	✓	228	1.1	even	1	trivial
29.9.f.a.15.4	yes	228	29.15	odd	28	inner