Embedded newform 29.9.f.a.15.14

• Label: 29.9.f.a.15.14
• Level: $29$
• Weight: $9$
• Character: 29.15
• 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			15.14
Character	\(\chi\)	\(=\)	29.15
Dual form			29.9.f.a.2.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.6914 - 1.65532i) q^{2} +(1.69586 - 1.06558i) q^{3} +(-36.4848 + 8.32741i) q^{4} +(784.225 + 625.399i) q^{5} +(23.1506 - 18.4620i) q^{6} +(313.531 - 1373.67i) q^{7} +(-4094.63 + 1432.77i) q^{8} +(-2844.97 + 5907.64i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 3664 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{27}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.6914	−	1.65532i	0.918212	−	0.103458i	0.359807	−	0.933027i	\(-0.382842\pi\)
							0.558404	+	0.829569i	\(0.311414\pi\)
\(3\)	1.69586	−	1.06558i	0.0209365	−	0.0131553i	−0.521523	−	0.853237i	\(-0.674636\pi\)
							0.542460	+	0.840082i	\(0.317493\pi\)
\(5\)	784.225	+	625.399i	1.25476	+	1.00064i	0.999429	+	0.0337846i	\(0.0107560\pi\)
							0.255331	+	0.966854i	\(0.417815\pi\)
\(7\)	313.531	−	1373.67i	0.130583	−	0.572123i	−0.866724	−	0.498788i	\(-0.833779\pi\)
							0.997307	−	0.0733352i	\(-0.0233643\pi\)
\(11\)	22594.0	+	7905.97i	1.54320	+	0.539988i	0.961829	−	0.273652i	\(-0.0882315\pi\)
							0.581369	+	0.813640i	\(0.302517\pi\)
\(13\)	7336.91	+	15235.3i	0.256886	+	0.533428i	0.989029	−	0.147722i	\(-0.0471941\pi\)
							−0.732143	+	0.681151i	\(0.761480\pi\)
\(17\)	50076.2	+	50076.2i	0.599564	+	0.599564i	0.940196	−	0.340633i	\(-0.110641\pi\)
							−0.340633	+	0.940196i	\(0.610641\pi\)
\(19\)	62833.2	−	99998.4i	0.482142	−	0.767324i	−0.513744	−	0.857944i	\(-0.671742\pi\)
							0.995886	+	0.0906196i	\(0.0288848\pi\)
\(23\)	−116662.	−	146289.i	−0.416886	−	0.522758i	0.528403	−	0.848994i	\(-0.322791\pi\)
							−0.945288	+	0.326236i	\(0.894220\pi\)
\(29\)	−81017.8	−	702625.i	−0.114548	−	0.993418i
							\(31\)	−1.40656e6	+	158481.i	−1.52304	+	0.171605i	−0.833449	−	0.552597i	\(-0.813637\pi\)
−0.689588	+	0.724202i	\(0.742208\pi\)
\(37\)	−1.81821e6	+	636219.i	−0.970146	+	0.339469i	−0.768400	−	0.639969i	\(-0.778947\pi\)
							−0.201745	+	0.979438i	\(0.564661\pi\)
\(41\)	423253.	−	423253.i	0.149784	−	0.149784i	−0.628238	−	0.778021i	\(-0.716223\pi\)
							0.778021	+	0.628238i	\(0.216223\pi\)
\(43\)	596583.	−	5.29482e6i	0.174501	−	1.54874i	−0.535500	−	0.844535i	\(-0.679877\pi\)
							0.710001	−	0.704201i	\(-0.248695\pi\)
\(47\)	1.80425e6	−	5.15626e6i	0.369748	−	1.05668i	−0.597285	−	0.802029i	\(-0.703754\pi\)
							0.967033	−	0.254650i	\(-0.0819604\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.9.f.a.15.14	yes	228
29.2	odd	28	inner	29.9.f.a.2.14	✓	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.9.f.a.2.14	✓	228	29.2	odd	28	inner
29.9.f.a.15.14	yes	228	1.1	even	1	trivial