Embedded newform 49.4.e.a.43.13

• Label: 49.4.e.a.43.13
• Level: $49$
• Weight: $4$
• Character: 49.43
• 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			43.13
Character	\(\chi\)	\(=\)	49.43
Dual form			49.4.e.a.8.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.18020 - 3.98784i) q^{2} +(-4.68696 - 2.25712i) q^{3} +(-4.00905 - 17.5648i) q^{4} +(4.76744 + 2.29588i) q^{5} +(-23.9065 + 11.5128i) q^{6} +(7.38357 + 16.9848i) q^{7} +(-46.0311 - 22.1674i) q^{8} +(0.0387732 + 0.0486201i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.18020	−	3.98784i	1.12437	−	1.40991i	0.224108	−	0.974564i	\(-0.428053\pi\)
							0.900261	−	0.435350i	\(-0.143375\pi\)
\(3\)	−4.68696	−	2.25712i	−0.902006	−	0.434383i	−0.0753929	−	0.997154i	\(-0.524021\pi\)
							−0.826613	+	0.562771i	\(0.809735\pi\)
\(5\)	4.76744	+	2.29588i	0.426413	+	0.205350i	0.634767	−	0.772704i	\(-0.281096\pi\)
							−0.208354	+	0.978054i	\(0.566811\pi\)
\(7\)	7.38357	+	16.9848i	0.398675	+	0.917092i
							\(11\)	33.6056	−	42.1400i	0.921133	−	1.15506i	−0.0664227	−	0.997792i	\(-0.521159\pi\)
0.987555	−	0.157272i	\(-0.0502700\pi\)
\(13\)	17.2120	−	21.5832i	0.367212	−	0.460470i	−0.563557	−	0.826077i	\(-0.690567\pi\)
							0.930769	+	0.365608i	\(0.119139\pi\)
\(17\)	−22.3063	+	97.7302i	−0.318239	+	1.39430i	0.522399	+	0.852701i	\(0.325037\pi\)
							−0.840638	+	0.541597i	\(0.817820\pi\)
\(19\)	70.0510			0.845831			0.422916	−	0.906169i	\(-0.361007\pi\)
							0.422916	+	0.906169i	\(0.361007\pi\)
\(23\)	32.8243	+	143.813i	0.297580	+	1.30378i	0.873719	+	0.486431i	\(0.161702\pi\)
							−0.576139	+	0.817352i	\(0.695441\pi\)
\(29\)	−17.4325	+	76.3767i	−0.111625	+	0.489062i	0.887951	+	0.459939i	\(0.152129\pi\)
							−0.999576	+	0.0291231i	\(0.990729\pi\)
\(31\)	−79.8101			−0.462397			−0.231199	−	0.972907i	\(-0.574265\pi\)
							−0.231199	+	0.972907i	\(0.574265\pi\)
\(37\)	−54.6848	+	239.590i	−0.242976	+	1.06455i	0.695315	+	0.718705i	\(0.255265\pi\)
							−0.938292	+	0.345844i	\(0.887592\pi\)
\(41\)	−277.770	−	133.767i	−1.05806	−	0.509535i	−0.177819	−	0.984063i	\(-0.556904\pi\)
							−0.880240	+	0.474529i	\(0.842619\pi\)
\(43\)	327.199	−	157.571i	1.16040	−	0.558821i	0.248261	−	0.968693i	\(-0.420141\pi\)
							0.912143	+	0.409872i	\(0.134427\pi\)
\(47\)	165.122	−	207.056i	0.512457	−	0.642600i	−0.456532	−	0.889707i	\(-0.650908\pi\)
							0.968988	+	0.247107i	\(0.0794799\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.43.13	yes	78
49.8	even	7	inner	49.4.e.a.8.13	✓	78
49.20	odd	14		2401.4.a.c.1.38		39
49.29	even	7		2401.4.a.d.1.38		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.13	✓	78	49.8	even	7	inner
49.4.e.a.43.13	yes	78	1.1	even	1	trivial
2401.4.a.c.1.38		39	49.20	odd	14
2401.4.a.d.1.38		39	49.29	even	7