Embedded newform 49.4.g.a.11.10

• Label: 49.4.g.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	49.11
Dual form			49.4.g.a.9.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.83173 - 1.69960i) q^{2} +(-7.79144 + 1.17437i) q^{3} +(-0.131237 + 1.75124i) q^{4} +(-6.62603 + 16.8828i) q^{5} +(-12.2758 + 15.3934i) q^{6} +(-4.66166 - 17.9240i) q^{7} +(15.1997 + 19.0598i) q^{8} +(33.5269 - 10.3417i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.83173	−	1.69960i	0.647614	−	0.600898i	−0.286411	−	0.958107i	\(-0.592462\pi\)
							0.934024	+	0.357209i	\(0.116272\pi\)
\(3\)	−7.79144	+	1.17437i	−1.49946	+	0.226008i	−0.846910	−	0.531735i	\(-0.821540\pi\)
							−0.652553	+	0.757743i	\(0.726302\pi\)
\(5\)	−6.62603	+	16.8828i	−0.592650	+	1.51005i	0.248162	+	0.968719i	\(0.420174\pi\)
							−0.840811	+	0.541328i	\(0.817922\pi\)
\(7\)	−4.66166	−	17.9240i	−0.251706	−	0.967804i
							\(11\)	−42.9023	−	13.2336i	−1.17596	−	0.362734i	−0.355628	−	0.934628i	\(-0.615733\pi\)
−0.820328	+	0.571893i	\(0.806209\pi\)
\(13\)	8.81708	+	38.6301i	0.188109	+	0.824160i	0.977612	+	0.210413i	\(0.0674810\pi\)
							−0.789503	+	0.613746i	\(0.789662\pi\)
\(17\)	−37.1624	+	25.3369i	−0.530189	+	0.361477i	−0.798633	−	0.601818i	\(-0.794443\pi\)
							0.268444	+	0.963295i	\(0.413491\pi\)
\(19\)	8.67539	+	15.0262i	0.104751	+	0.181434i	0.913636	−	0.406532i	\(-0.133262\pi\)
							−0.808885	+	0.587966i	\(0.799929\pi\)
\(23\)	131.257	+	89.4894i	1.18995	+	0.811297i	0.985846	−	0.167651i	\(-0.0536183\pi\)
							0.204108	+	0.978948i	\(0.434571\pi\)
\(29\)	134.682	+	64.8596i	0.862409	+	0.415314i	0.812169	−	0.583423i	\(-0.198287\pi\)
							0.0502406	+	0.998737i	\(0.484001\pi\)
\(31\)	43.8072	−	75.8764i	0.253807	−	0.439606i	−0.710764	−	0.703431i	\(-0.751651\pi\)
							0.964571	+	0.263824i	\(0.0849839\pi\)
\(37\)	−4.16743	−	55.6105i	−0.0185168	−	0.247089i	−0.998780	−	0.0493903i	\(-0.984272\pi\)
							0.980263	−	0.197699i	\(-0.0633469\pi\)
\(41\)	−105.649	−	132.480i	−0.402430	−	0.504631i	0.538783	−	0.842444i	\(-0.318884\pi\)
							−0.941213	+	0.337813i	\(0.890313\pi\)
\(43\)	−274.626	+	344.371i	−0.973956	+	1.22130i	0.00124852	+	0.999999i	\(0.499603\pi\)
							−0.975205	+	0.221304i	\(0.928969\pi\)
\(47\)	−238.987	+	221.747i	−0.741698	+	0.688195i	−0.957562	−	0.288229i	\(-0.906934\pi\)
							0.215864	+	0.976423i	\(0.430743\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.11.10	yes	156
49.3	odd	42		2401.4.a.f.1.26		78
49.9	even	21	inner	49.4.g.a.9.10	✓	156
49.46	even	21		2401.4.a.g.1.26		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.9.10	✓	156	49.9	even	21	inner
49.4.g.a.11.10	yes	156	1.1	even	1	trivial
2401.4.a.f.1.26		78	49.3	odd	42
2401.4.a.g.1.26		78	49.46	even	21