Embedded newform 49.4.e.a.36.7

• Label: 49.4.e.a.36.7
• Level: $49$
• Weight: $4$
• Character: 49.36
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $78$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(78\)
Relative dimension:	\(13\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			36.7
Character	\(\chi\)	\(=\)	49.36
Dual form			49.4.e.a.15.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0613960 + 0.268993i) q^{2} +(2.25069 + 2.82228i) q^{3} +(7.13916 - 3.43804i) q^{4} +(-2.36861 - 2.97014i) q^{5} +(-0.620990 + 0.778697i) q^{6} +(9.47466 + 15.9132i) q^{7} +(2.73935 + 3.43503i) q^{8} +(3.10843 - 13.6189i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0613960	+	0.268993i	0.0217068	+	0.0951035i	0.984621	−	0.174704i	\(-0.0558970\pi\)
							−0.962914	+	0.269808i	\(0.913040\pi\)
\(3\)	2.25069	+	2.82228i	0.433146	+	0.543147i	0.949722	−	0.313094i	\(-0.101365\pi\)
							−0.516577	+	0.856241i	\(0.672794\pi\)
\(5\)	−2.36861	−	2.97014i	−0.211855	−	0.265657i	0.664538	−	0.747254i	\(-0.268628\pi\)
							−0.876393	+	0.481597i	\(0.840057\pi\)
\(7\)	9.47466	+	15.9132i	0.511583	+	0.859234i
							\(11\)	12.3443	+	54.0839i	0.338358	+	1.48244i	0.802483	+	0.596675i	\(0.203512\pi\)
−0.464125	+	0.885770i	\(0.653631\pi\)
\(13\)	−10.7609	−	47.1465i	−0.229580	−	1.00585i	−0.949984	−	0.312299i	\(-0.898901\pi\)
							0.720404	−	0.693554i	\(-0.243956\pi\)
\(17\)	−51.6949	−	24.8950i	−0.737521	−	0.355172i	0.0271161	−	0.999632i	\(-0.491368\pi\)
							−0.764637	+	0.644461i	\(0.777082\pi\)
\(19\)	−112.013			−1.35250			−0.676251	−	0.736671i	\(-0.736396\pi\)
							−0.676251	+	0.736671i	\(0.736396\pi\)
\(23\)	−108.715	+	52.3546i	−0.985596	+	0.474638i	−0.856026	−	0.516932i	\(-0.827074\pi\)
							−0.129570	+	0.991570i	\(0.541360\pi\)
\(29\)	−154.212	−	74.2644i	−0.987462	−	0.475536i	−0.130796	−	0.991409i	\(-0.541753\pi\)
							−0.856665	+	0.515873i	\(0.827468\pi\)
\(31\)	−75.8179			−0.439268			−0.219634	−	0.975582i	\(-0.570486\pi\)
							−0.219634	+	0.975582i	\(0.570486\pi\)
\(37\)	358.390	+	172.591i	1.59240	+	0.766861i	0.999268	−	0.0382431i	\(-0.0121761\pi\)
							0.593134	+	0.805104i	\(0.297890\pi\)
\(41\)	−53.9698	−	67.6759i	−0.205577	−	0.257786i	0.668345	−	0.743851i	\(-0.267003\pi\)
							−0.873922	+	0.486066i	\(0.838432\pi\)
\(43\)	29.5287	−	37.0279i	0.104723	−	0.131319i	−0.726708	−	0.686946i	\(-0.758951\pi\)
							0.831431	+	0.555628i	\(0.187522\pi\)
\(47\)	92.7165	+	406.217i	0.287747	+	1.26070i	0.887609	+	0.460597i	\(0.152365\pi\)
							−0.599863	+	0.800103i	\(0.704778\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.36.7	yes	78
49.8	even	7		2401.4.a.d.1.19		39
49.15	even	7	inner	49.4.e.a.15.7	✓	78
49.41	odd	14		2401.4.a.c.1.19		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.7	✓	78	49.15	even	7	inner
49.4.e.a.36.7	yes	78	1.1	even	1	trivial
2401.4.a.c.1.19		39	49.41	odd	14
2401.4.a.d.1.19		39	49.8	even	7