Embedded newform 29.11.f.a.2.12

• Label: 29.11.f.a.2.12
• Level: $29$
• Weight: $11$
• Character: 29.2
• Analytic conductor: $18.425$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.4253603275\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(24\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			2.12
Character	\(\chi\)	\(=\)	29.2
Dual form			29.11.f.a.15.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.757424 + 0.0853412i) q^{2} +(157.893 + 99.2110i) q^{3} +(-997.760 - 227.732i) q^{4} +(-3734.49 + 2978.16i) q^{5} +(111.125 + 88.6196i) q^{6} +(-313.352 - 1372.88i) q^{7} +(-1473.00 - 515.426i) q^{8} +(-10532.9 - 21871.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 22 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} - 76784 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.757424	+	0.0853412i	0.0236695	+	0.00266691i	0.123791	−	0.992308i	\(-0.460495\pi\)
							−0.100122	+	0.994975i	\(0.531923\pi\)
\(3\)	157.893	+	99.2110i	0.649767	+	0.408276i	0.816194	−	0.577778i	\(-0.196080\pi\)
							−0.166427	+	0.986054i	\(0.553223\pi\)
\(5\)	−3734.49	+	2978.16i	−1.19504	+	0.953010i	−0.999616	−	0.0277096i	\(-0.991179\pi\)
							−0.195421	+	0.980720i	\(0.562607\pi\)
\(7\)	−313.352	−	1372.88i	−0.0186441	−	0.0816852i	0.964750	−	0.263168i	\(-0.0847674\pi\)
							−0.983394	+	0.181483i	\(0.941910\pi\)
\(11\)	268904.	−	94093.7i	1.66968	−	0.584248i	0.680738	−	0.732527i	\(-0.261659\pi\)
							0.988946	+	0.148279i	\(0.0473734\pi\)
\(13\)	116601.	−	242125.i	0.314042	−	0.652114i	−0.682879	−	0.730532i	\(-0.739272\pi\)
							0.996921	+	0.0784172i	\(0.0249866\pi\)
\(17\)	−723016.	+	723016.i	−0.509218	+	0.509218i	−0.914286	−	0.405069i	\(-0.867248\pi\)
							0.405069	+	0.914286i	\(0.367248\pi\)
\(19\)	−233849.	−	372169.i	−0.0944425	−	0.150304i	0.796161	−	0.605085i	\(-0.206861\pi\)
							−0.890604	+	0.454780i	\(0.849718\pi\)
\(23\)	3.08200e6	−	3.86471e6i	0.478844	−	0.600451i	−0.482468	−	0.875914i	\(-0.660259\pi\)
							0.961312	+	0.275462i	\(0.0888309\pi\)
\(29\)	4.81167e6	−	1.99388e7i	0.234588	−	0.972095i
							\(31\)	−1.30812e7	−	1.47389e6i	−0.456918	−	0.0514823i	−0.119494	−	0.992835i	\(-0.538127\pi\)
−0.337424	+	0.941353i	\(0.609556\pi\)
\(37\)	−4.72019e6	−	1.65167e6i	−0.0680692	−	0.0238185i	0.296030	−	0.955179i	\(-0.404337\pi\)
							−0.364099	+	0.931360i	\(0.618623\pi\)
\(41\)	−9.20473e7	−	9.20473e7i	−0.794496	−	0.794496i	0.187725	−	0.982222i	\(-0.439888\pi\)
							−0.982222	+	0.187725i	\(0.939888\pi\)
\(43\)	1.89302e7	+	1.68010e8i	0.128769	+	1.14286i	0.877362	+	0.479829i	\(0.159301\pi\)
							−0.748593	+	0.663030i	\(0.769270\pi\)
\(47\)	−9.51103e7	−	2.71810e8i	−0.414704	−	1.18516i	−0.942040	−	0.335500i	\(-0.891095\pi\)
							0.527336	−	0.849657i	\(-0.323191\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.11.f.a.2.12	✓	288
29.15	odd	28	inner	29.11.f.a.15.12	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.11.f.a.2.12	✓	288	1.1	even	1	trivial
29.11.f.a.15.12	yes	288	29.15	odd	28	inner