Embedded newform 49.4.g.a.11.4

• Label: 49.4.g.a.11.4
• Level: $49$
• Weight: $4$
• Character: 49.11
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(13\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			11.4
Character	\(\chi\)	\(=\)	49.11
Dual form			49.4.g.a.9.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.83456 + 1.70222i) q^{2} +(-1.89225 + 0.285211i) q^{3} +(-0.129795 + 1.73200i) q^{4} +(-0.225715 + 0.575114i) q^{5} +(2.98595 - 3.74426i) q^{6} +(-16.4056 - 8.59392i) q^{7} +(-15.1930 - 19.0515i) q^{8} +(-22.3012 + 6.87901i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.83456	+	1.70222i	−0.648614	+	0.601825i	−0.934296	−	0.356499i	\(-0.883970\pi\)
							0.285682	+	0.958324i	\(0.407780\pi\)
\(3\)	−1.89225	+	0.285211i	−0.364164	+	0.0548889i	−0.328576	−	0.944478i	\(-0.606569\pi\)
							−0.0355883	+	0.999367i	\(0.511331\pi\)
\(5\)	−0.225715	+	0.575114i	−0.0201886	+	0.0514397i	−0.940613	−	0.339482i	\(-0.889748\pi\)
							0.920424	+	0.390921i	\(0.127844\pi\)
\(7\)	−16.4056	−	8.59392i	−0.885821	−	0.464028i
							\(11\)	−0.936861	−	0.288983i	−0.0256795	−	0.00792107i	0.281889	−	0.959447i	\(-0.409039\pi\)
−0.307568	+	0.951526i	\(0.599515\pi\)
\(13\)	−3.83183	−	16.7884i	−0.0817507	−	0.358173i	0.917463	−	0.397821i	\(-0.130234\pi\)
							−0.999214	+	0.0396479i	\(0.987376\pi\)
\(17\)	−75.8345	+	51.7031i	−1.08192	+	0.737638i	−0.966754	−	0.255709i	\(-0.917691\pi\)
							−0.115162	+	0.993347i	\(0.536739\pi\)
\(19\)	64.5895	+	111.872i	0.779886	+	1.35080i	0.932007	+	0.362441i	\(0.118056\pi\)
							−0.152121	+	0.988362i	\(0.548610\pi\)
\(23\)	6.96190	+	4.74655i	0.0631155	+	0.0430314i	0.594467	−	0.804120i	\(-0.297363\pi\)
							−0.531351	+	0.847152i	\(0.678316\pi\)
\(29\)	−244.935	−	117.954i	−1.56839	−	0.755295i	−0.570564	−	0.821253i	\(-0.693276\pi\)
							−0.997822	+	0.0659575i	\(0.978990\pi\)
\(31\)	−86.4282	+	149.698i	−0.500741	+	0.867309i	0.499259	+	0.866453i	\(0.333606\pi\)
							−1.00000		0.000855805i	\(0.999728\pi\)
\(37\)	2.36428	+	31.5492i	0.0105050	+	0.140180i	0.999990	−	0.00445540i	\(-0.00141820\pi\)
							−0.989485	+	0.144635i	\(0.953799\pi\)
\(41\)	−116.192	−	145.700i	−0.442590	−	0.554990i	0.509634	−	0.860391i	\(-0.329781\pi\)
							−0.952224	+	0.305401i	\(0.901209\pi\)
\(43\)	55.9341	−	70.1392i	0.198369	−	0.248747i	−0.672691	−	0.739924i	\(-0.734862\pi\)
							0.871060	+	0.491177i	\(0.163433\pi\)
\(47\)	−238.417	+	221.218i	−0.739929	+	0.686554i	−0.957158	−	0.289567i	\(-0.906489\pi\)
							0.217229	+	0.976121i	\(0.430298\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.11.4	yes	156
49.3	odd	42		2401.4.a.f.1.55		78
49.9	even	21	inner	49.4.g.a.9.4	✓	156
49.46	even	21		2401.4.a.g.1.55		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.9.4	✓	156	49.9	even	21	inner
49.4.g.a.11.4	yes	156	1.1	even	1	trivial
2401.4.a.f.1.55		78	49.3	odd	42
2401.4.a.g.1.55		78	49.46	even	21