Embedded newform 29.10.e.a.4.18

• Label: 29.10.e.a.4.18
• Level: $29$
• Weight: $10$
• Character: 29.4
• Analytic conductor: $14.936$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.9360392488\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			4.18
Character	\(\chi\)	\(=\)	29.4
Dual form			29.10.e.a.22.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(28.1428 + 6.42341i) q^{2} +(-39.5780 - 82.1846i) q^{3} +(289.462 + 139.397i) q^{4} +(-3.26559 + 14.3075i) q^{5} +(-585.931 - 2567.13i) q^{6} +(2325.76 - 1120.03i) q^{7} +(-4304.36 - 3432.61i) q^{8} +(7084.26 - 8883.38i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 7 q^{2} - 7 q^{3} + 5797 q^{4} + 1367 q^{5} - 7607 q^{6} + 4949 q^{7} + 31766 q^{8} + 112237 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	28.1428	+	6.42341i	1.24375	+	0.283877i	0.793248	−	0.608899i	\(-0.208388\pi\)
							0.450500	+	0.892776i	\(0.351246\pi\)
\(3\)	−39.5780	−	82.1846i	−0.282104	−	0.585794i	0.710979	−	0.703213i	\(-0.248252\pi\)
							−0.993082	+	0.117419i	\(0.962538\pi\)
\(5\)	−3.26559	+	14.3075i	−0.00233666	+	0.0102376i	−0.976083	−	0.217399i	\(-0.930243\pi\)
							0.973746	+	0.227637i	\(0.0730998\pi\)
\(7\)	2325.76	−	1120.03i	0.366121	−	0.176314i	−0.241773	−	0.970333i	\(-0.577729\pi\)
							0.607894	+	0.794018i	\(0.292015\pi\)
\(11\)	35850.4	−	28589.7i	0.738290	−	0.588767i	−0.180471	−	0.983580i	\(-0.557762\pi\)
							0.918761	+	0.394814i	\(0.129191\pi\)
\(13\)	−104495.	−	131033.i	−1.01473	−	1.27243i	−0.961777	−	0.273835i	\(-0.911708\pi\)
							−0.0529539	−	0.998597i	\(-0.516864\pi\)
\(17\)		−	211092.i		−	0.612987i	−0.951873	−	0.306494i	\(-0.900844\pi\)
							0.951873	−	0.306494i	\(-0.0991558\pi\)
\(19\)	−56274.5	+	116855.i	−0.0990650	+	0.205711i	−0.944611	−	0.328192i	\(-0.893561\pi\)
							0.845546	+	0.533902i	\(0.179275\pi\)
\(23\)	494162.	+	2.16506e6i	0.368208	+	1.61323i	0.731699	+	0.681628i	\(0.238728\pi\)
							−0.363490	+	0.931598i	\(0.618415\pi\)
\(29\)	2.77973e6	−	2.60389e6i	0.729813	−	0.683647i
							\(31\)	780115.	+	178056.i	0.151716	+	0.0346281i	0.297704	−	0.954658i	\(-0.403779\pi\)
−0.145988	+	0.989286i	\(0.546636\pi\)
\(37\)	2.42321e6	+	1.93245e6i	0.212561	+	0.169512i	0.723983	−	0.689818i	\(-0.242310\pi\)
							−0.511422	+	0.859330i	\(0.670881\pi\)
\(41\)			2.45838e7i			1.35870i	0.733816	+	0.679348i	\(0.237737\pi\)
							−0.733816	+	0.679348i	\(0.762263\pi\)
\(43\)	7.04162e6	−	1.60720e6i	0.314098	−	0.0716908i	−0.0625660	−	0.998041i	\(-0.519928\pi\)
							0.376664	+	0.926350i	\(0.377071\pi\)
\(47\)	−50569.2	+	40327.6i	−0.00151163	+	0.00120549i	−0.624245	−	0.781228i	\(-0.714593\pi\)
							0.622734	+	0.782434i	\(0.286022\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.e.a.4.18	✓	132
29.22	even	14	inner	29.10.e.a.22.18	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.e.a.4.18	✓	132	1.1	even	1	trivial
29.10.e.a.22.18	yes	132	29.22	even	14	inner