Embedded newform 29.10.d.a.7.4

• Label: 29.10.d.a.7.4
• Level: $29$
• Weight: $10$
• Character: 29.7
• Analytic conductor: $14.936$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.9360392488\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(21\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	29.7
Dual form			29.10.d.a.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.87345 - 25.7333i) q^{2} +(26.9577 + 12.9821i) q^{3} +(-166.408 + 80.1377i) q^{4} +(-430.743 - 1887.21i) q^{5} +(175.738 - 769.959i) q^{6} +(11130.5 + 5360.17i) q^{7} +(-5386.41 - 6754.35i) q^{8} +(-11714.0 - 14688.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 23 q^{2} - 5 q^{3} - 4699 q^{4} - 1031 q^{5} + 12781 q^{6} - 4959 q^{7} - 43354 q^{8} - 76050 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.87345	−	25.7333i	−0.259572	−	1.13726i	−0.921710	−	0.387879i	\(-0.873208\pi\)
							0.662138	−	0.749382i	\(-0.269649\pi\)
\(3\)	26.9577	+	12.9821i	0.192148	+	0.0925338i	0.527484	−	0.849565i	\(-0.323135\pi\)
							−0.335336	+	0.942099i	\(0.608850\pi\)
\(5\)	−430.743	−	1887.21i	−0.308214	−	1.35038i	−0.857389	−	0.514668i	\(-0.827915\pi\)
							0.549175	−	0.835707i	\(-0.314942\pi\)
\(7\)	11130.5	+	5360.17i	1.75216	+	0.843796i	0.977358	+	0.211592i	\(0.0678649\pi\)
							0.774802	+	0.632204i	\(0.217849\pi\)
\(11\)	27293.3	−	34224.7i	0.562068	−	0.704811i	−0.416871	−	0.908966i	\(-0.636873\pi\)
							0.978939	+	0.204155i	\(0.0654446\pi\)
\(13\)	35444.6	−	44446.2i	0.344196	−	0.431608i	−0.579360	−	0.815072i	\(-0.696698\pi\)
							0.923556	+	0.383464i	\(0.125269\pi\)
\(17\)	−78500.1			−0.227955			−0.113978	−	0.993483i	\(-0.536359\pi\)
							−0.113978	+	0.993483i	\(0.536359\pi\)
\(19\)	−861655.	+	414951.i	−1.51685	+	0.730476i	−0.992639	−	0.121113i	\(-0.961354\pi\)
							−0.524210	+	0.851589i	\(0.675639\pi\)
\(23\)	−59886.9	+	262382.i	−0.0446228	+	0.195505i	−0.992326	−	0.123647i	\(-0.960541\pi\)
							0.947704	+	0.319152i	\(0.103398\pi\)
\(29\)	796300.	+	3.72465e6i	0.209067	+	0.977901i
							\(31\)	−400200.	−	1.75339e6i	−0.0778304	−	0.340997i	0.920988	−	0.389590i	\(-0.127383\pi\)
−0.998819	+	0.0485931i	\(0.984526\pi\)
\(37\)	−1.11824e7	−	1.40223e7i	−0.980905	−	1.23002i	−0.973180	−	0.230045i	\(-0.926113\pi\)
							−0.00772479	−	0.999970i	\(-0.502459\pi\)
\(41\)	1.17381e7			0.648740			0.324370	−	0.945930i	\(-0.394848\pi\)
							0.324370	+	0.945930i	\(0.394848\pi\)
\(43\)	3.86916e6	−	1.69519e7i	0.172587	−	0.756154i	−0.812340	−	0.583184i	\(-0.801807\pi\)
							0.984927	−	0.172970i	\(-0.0553362\pi\)
\(47\)	3.19997e7	−	4.01263e7i	0.956544	−	1.19947i	−0.0233050	−	0.999728i	\(-0.507419\pi\)
							0.979849	−	0.199740i	\(-0.0640097\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.d.a.7.4	✓	126
29.25	even	7	inner	29.10.d.a.25.4	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.d.a.7.4	✓	126	1.1	even	1	trivial
29.10.d.a.25.4	yes	126	29.25	even	7	inner