Embedded newform 29.10.d.a.7.19

• Label: 29.10.d.a.7.19
• 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			7.19
Character	\(\chi\)	\(=\)	29.7
Dual form			29.10.d.a.25.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.92629 + 34.7274i) q^{2} +(-14.5980 - 7.03001i) q^{3} +(-681.868 + 328.370i) q^{4} +(-247.282 - 1083.41i) q^{5} +(128.426 - 562.671i) q^{6} +(-1483.34 - 714.340i) q^{7} +(-5437.11 - 6817.92i) q^{8} +(-12108.5 - 15183.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 23 q^{2} - 5 q^{3} - 4699 q^{4} - 1031 q^{5} + 12781 q^{6} - 4959 q^{7} - 43354 q^{8} - 76050 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.92629	+	34.7274i	0.350296	+	1.53475i	0.776498	+	0.630119i	\(0.216994\pi\)
							−0.426202	+	0.904628i	\(0.640149\pi\)
\(3\)	−14.5980	−	7.03001i	−0.104051	−	0.0501084i	0.381134	−	0.924520i	\(-0.375534\pi\)
							−0.485185	+	0.874411i	\(0.661248\pi\)
\(5\)	−247.282	−	1083.41i	−0.176941	−	0.775227i	−0.983032	−	0.183436i	\(-0.941278\pi\)
							0.806091	−	0.591791i	\(-0.201579\pi\)
\(7\)	−1483.34	−	714.340i	−0.233507	−	0.112451i	0.313472	−	0.949598i	\(-0.398508\pi\)
							−0.546979	+	0.837146i	\(0.684222\pi\)
\(11\)	33862.4	−	42462.1i	0.697350	−	0.874450i	−0.299472	−	0.954105i	\(-0.596811\pi\)
							0.996822	+	0.0796554i	\(0.0253820\pi\)
\(13\)	58356.8	−	73177.1i	0.566691	−	0.710608i	−0.413088	−	0.910691i	\(-0.635550\pi\)
							0.979779	+	0.200083i	\(0.0641211\pi\)
\(17\)	−259731.			−0.754230			−0.377115	−	0.926166i	\(-0.623084\pi\)
							−0.377115	+	0.926166i	\(0.623084\pi\)
\(19\)	43640.8	−	21016.3i	0.0768248	−	0.0369969i	−0.395077	−	0.918648i	\(-0.629282\pi\)
							0.471902	+	0.881651i	\(0.343568\pi\)
\(23\)	−175328.	+	768161.i	−0.130640	+	0.572370i	0.866658	+	0.498903i	\(0.166264\pi\)
							−0.997298	+	0.0734674i	\(0.976594\pi\)
\(29\)	236251.	−	3.80149e6i	0.0620273	−	0.998074i
							\(31\)	−1.08723e6	−	4.76347e6i	−0.211444	−	0.926395i	−0.963587	−	0.267395i	\(-0.913837\pi\)
0.752143	−	0.659000i	\(-0.229020\pi\)
\(37\)	−7.59550e6	−	9.52446e6i	−0.666268	−	0.835473i	0.327742	−	0.944767i	\(-0.393712\pi\)
							−0.994009	+	0.109294i	\(0.965141\pi\)
\(41\)	1.74551e6			0.0964709			0.0482354	−	0.998836i	\(-0.484640\pi\)
							0.0482354	+	0.998836i	\(0.484640\pi\)
\(43\)	1.43847e6	−	6.30235e6i	0.0641642	−	0.281122i	−0.932660	−	0.360757i	\(-0.882518\pi\)
							0.996824	+	0.0796352i	\(0.0253755\pi\)
\(47\)	−1.07072e7	+	1.34264e7i	−0.320064	+	0.401347i	−0.915671	−	0.401928i	\(-0.868340\pi\)
							0.595607	+	0.803276i	\(0.296911\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.d.a.7.19	✓	126
29.25	even	7	inner	29.10.d.a.25.19	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.d.a.7.19	✓	126	1.1	even	1	trivial
29.10.d.a.25.19	yes	126	29.25	even	7	inner