Embedded newform 49.4.g.a.2.4

• Label: 49.4.g.a.2.4
• Level: $49$
• Weight: $4$
• Character: 49.2
• 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			2.4
Character	\(\chi\)	\(=\)	49.2
Dual form			49.4.g.a.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.15708 - 0.973830i) q^{2} +(-0.0268992 + 0.0685381i) q^{3} +(2.40889 + 1.64236i) q^{4} +(-0.551282 + 0.0830924i) q^{5} +(0.151667 - 0.190185i) q^{6} +(-15.0526 + 10.7898i) q^{7} +(10.4737 + 13.1336i) q^{8} +(19.7884 + 18.3610i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.15708	−	0.973830i	−1.11620	−	0.344301i	−0.318839	−	0.947809i	\(-0.603293\pi\)
							−0.797357	+	0.603508i	\(0.793769\pi\)
\(3\)	−0.0268992	+	0.0685381i	−0.00517676	+	0.0131902i	−0.933439	−	0.358737i	\(-0.883208\pi\)
							0.928262	+	0.371927i	\(0.121303\pi\)
\(5\)	−0.551282	+	0.0830924i	−0.0493082	+	0.00743201i	−0.173650	−	0.984807i	\(-0.555556\pi\)
							0.124342	+	0.992239i	\(0.460318\pi\)
\(7\)	−15.0526	+	10.7898i	−0.812765	+	0.582592i
							\(11\)	−32.2163	+	29.8923i	−0.883052	+	0.819353i	−0.984481	−	0.175490i	\(-0.943849\pi\)
0.101429	+	0.994843i	\(0.467659\pi\)
\(13\)	14.1253	+	61.8869i	0.301358	+	1.32033i	0.868079	+	0.496425i	\(0.165354\pi\)
							−0.566722	+	0.823909i	\(0.691789\pi\)
\(17\)	−8.02927	−	107.143i	−0.114552	−	1.52859i	−0.697798	−	0.716295i	\(-0.745837\pi\)
							0.583246	−	0.812296i	\(-0.301782\pi\)
\(19\)	−54.2138	+	93.9010i	−0.654605	+	1.13381i	0.327388	+	0.944890i	\(0.393832\pi\)
							−0.981993	+	0.188919i	\(0.939502\pi\)
\(23\)	8.12082	−	108.365i	0.0736221	−	0.982419i	−0.831019	−	0.556244i	\(-0.812242\pi\)
							0.904641	−	0.426175i	\(-0.140139\pi\)
\(29\)	132.759	+	63.9334i	0.850094	+	0.409384i	0.807613	−	0.589713i	\(-0.200759\pi\)
							0.0424816	+	0.999097i	\(0.486474\pi\)
\(31\)	24.8195	+	42.9887i	0.143797	+	0.249064i	0.928924	−	0.370271i	\(-0.120735\pi\)
							−0.785126	+	0.619336i	\(0.787402\pi\)
\(37\)	−203.657	+	138.851i	−0.904893	+	0.616946i	−0.923800	−	0.382876i	\(-0.874934\pi\)
							0.0189069	+	0.999821i	\(0.493981\pi\)
\(41\)	171.095	+	214.546i	0.651720	+	0.817232i	0.992414	−	0.122943i	\(-0.0392332\pi\)
							−0.340693	+	0.940174i	\(0.610662\pi\)
\(43\)	−39.3286	+	49.3165i	−0.139478	+	0.174900i	−0.846664	−	0.532127i	\(-0.821393\pi\)
							0.707186	+	0.707027i	\(0.249964\pi\)
\(47\)	−177.107	−	54.6302i	−0.549652	−	0.169545i	0.00747546	−	0.999972i	\(-0.497620\pi\)
							−0.557128	+	0.830427i	\(0.688097\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.2.4	✓	156
49.5	odd	42		2401.4.a.f.1.20		78
49.25	even	21	inner	49.4.g.a.25.4	yes	156
49.44	even	21		2401.4.a.g.1.20		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.2.4	✓	156	1.1	even	1	trivial
49.4.g.a.25.4	yes	156	49.25	even	21	inner
2401.4.a.f.1.20		78	49.5	odd	42
2401.4.a.g.1.20		78	49.44	even	21