Embedded newform 49.4.e.a.15.5

• Label: 49.4.e.a.15.5
• Level: $49$
• Weight: $4$
• Character: 49.15
• 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			15.5
Character	\(\chi\)	\(=\)	49.15
Dual form			49.4.e.a.36.5

$q$-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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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	0.842699	+	0.538385i	\(0.180965\pi\)
							−0.992842	+	0.119435i	\(0.961892\pi\)
\(3\)	0.113640	−	0.142500i	0.0218701	−	0.0274242i	−0.770776	−	0.637106i	\(-0.780131\pi\)
							0.792646	+	0.609682i	\(0.208703\pi\)
\(5\)	11.1399	−	13.9690i	0.996381	−	1.24942i	0.0280878	−	0.999605i	\(-0.491058\pi\)
							0.968293	−	0.249817i	\(-0.0803704\pi\)
\(7\)	1.18252	+	18.4825i	0.0638498	+	0.997960i
							\(11\)	4.46612	−	19.5674i	0.122417	−	0.536344i	−0.876111	−	0.482109i	\(-0.839871\pi\)
0.998528	−	0.0542349i	\(-0.0172720\pi\)
\(13\)	9.93444	−	43.5256i	0.211948	−	0.928603i	−0.751294	−	0.659967i	\(-0.770570\pi\)
							0.963242	−	0.268636i	\(-0.0865728\pi\)
\(17\)	9.17124	−	4.41664i	0.130844	−	0.0630113i	−0.367316	−	0.930096i	\(-0.619723\pi\)
							0.498161	+	0.867085i	\(0.334009\pi\)
\(19\)	−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	−0.327392	−	0.944889i	\(-0.606170\pi\)
							−0.942869	+	0.333163i	\(0.891884\pi\)
\(29\)	235.771	−	113.541i	1.50971	−	0.727037i	0.517980	−	0.855393i	\(-0.326684\pi\)
							0.991728	+	0.128356i	\(0.0409700\pi\)
\(31\)	−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	−0.953348	−	0.301874i	\(-0.902388\pi\)
							−0.358388	+	0.933573i	\(0.616674\pi\)
\(41\)	157.381	−	197.349i	0.599481	−	0.751726i	−0.385816	−	0.922576i	\(-0.626080\pi\)
							0.985297	+	0.170850i	\(0.0546514\pi\)
\(43\)	−12.1868	−	15.2817i	−0.0432201	−	0.0541963i	0.759752	−	0.650213i	\(-0.225320\pi\)
							−0.802972	+	0.596017i	\(0.796749\pi\)
\(47\)	−1.76856	+	7.74855i	−0.00548874	+	0.0240477i	−0.977598	−	0.210480i	\(-0.932497\pi\)
							0.972109	+	0.234528i	\(0.0753544\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.5	✓	78	1.1	even	1	trivial
49.4.e.a.36.5	yes	78	49.36	even	7	inner
2401.4.a.c.1.26		39	49.6	odd	14
2401.4.a.d.1.26		39	49.43	even	7