Embedded newform 49.4.e.a.36.5

• Label: 49.4.e.a.36.5
• 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			36.5
Character	\(\chi\)	\(=\)	49.36
Dual form			49.4.e.a.15.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.424669 - 1.86059i) q^{2} +(0.113640 + 0.142500i) q^{3} +(3.92628 - 1.89080i) q^{4} +(11.1399 + 13.9690i) q^{5} +(0.216876 - 0.271954i) q^{6} +(1.18252 - 18.4825i) q^{7} +(-14.7045 - 18.4389i) q^{8} +(6.00067 - 26.2907i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.424669	−	1.86059i	−0.150143	−	0.657820i	−0.992842	−	0.119435i	\(-0.961892\pi\)
							0.842699	−	0.538385i	\(-0.180965\pi\)
\(3\)	0.113640	+	0.142500i	0.0218701	+	0.0274242i	0.792646	−	0.609682i	\(-0.208703\pi\)
							−0.770776	+	0.637106i	\(0.780131\pi\)
\(5\)	11.1399	+	13.9690i	0.996381	+	1.24942i	0.968293	+	0.249817i	\(0.0803704\pi\)
							0.0280878	+	0.999605i	\(0.491058\pi\)
\(7\)	1.18252	−	18.4825i	0.0638498	−	0.997960i
							\(11\)	4.46612	+	19.5674i	0.122417	+	0.536344i	0.998528	+	0.0542349i	\(0.0172720\pi\)
−0.876111	+	0.482109i	\(0.839871\pi\)
\(13\)	9.93444	+	43.5256i	0.211948	+	0.928603i	0.963242	+	0.268636i	\(0.0865728\pi\)
							−0.751294	+	0.659967i	\(0.770570\pi\)
\(17\)	9.17124	+	4.41664i	0.130844	+	0.0630113i	0.498161	−	0.867085i	\(-0.334009\pi\)
							−0.367316	+	0.930096i	\(0.619723\pi\)
\(19\)	−51.5845			−0.622858			−0.311429	−	0.950269i	\(-0.600808\pi\)
							−0.311429	+	0.950269i	\(0.600808\pi\)
\(23\)	−140.115	+	67.4759i	−1.27026	+	0.611726i	−0.942869	−	0.333163i	\(-0.891884\pi\)
							−0.327392	+	0.944889i	\(0.606170\pi\)
\(29\)	235.771	+	113.541i	1.50971	+	0.727037i	0.991728	−	0.128356i	\(-0.0409700\pi\)
							0.517980	+	0.855393i	\(0.326684\pi\)
\(31\)	−211.476			−1.22523			−0.612616	−	0.790381i	\(-0.709883\pi\)
							−0.612616	+	0.790381i	\(0.709883\pi\)
\(37\)	−295.222	−	142.172i	−1.31174	−	0.631699i	−0.358388	−	0.933573i	\(-0.616674\pi\)
							−0.953348	+	0.301874i	\(0.902388\pi\)
\(41\)	157.381	+	197.349i	0.599481	+	0.751726i	0.985297	−	0.170850i	\(-0.0546514\pi\)
							−0.385816	+	0.922576i	\(0.626080\pi\)
\(43\)	−12.1868	+	15.2817i	−0.0432201	+	0.0541963i	−0.802972	−	0.596017i	\(-0.796749\pi\)
							0.759752	+	0.650213i	\(0.225320\pi\)
\(47\)	−1.76856	−	7.74855i	−0.00548874	−	0.0240477i	0.972109	−	0.234528i	\(-0.0753544\pi\)
							−0.977598	+	0.210480i	\(0.932497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.36.5	yes	78
49.8	even	7		2401.4.a.d.1.26		39
49.15	even	7	inner	49.4.e.a.15.5	✓	78
49.41	odd	14		2401.4.a.c.1.26		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.5	✓	78	49.15	even	7	inner
49.4.e.a.36.5	yes	78	1.1	even	1	trivial
2401.4.a.c.1.26		39	49.41	odd	14
2401.4.a.d.1.26		39	49.8	even	7