Embedded newform 13.7.f.a.6.1

• Label: 13.7.f.a.6.1
• Level: $13$
• Weight: $7$
• Character: 13.6
• Analytic conductor: $2.991$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	13.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.99070308706\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			6.1
Character	\(\chi\)	\(=\)	13.6
Dual form			13.7.f.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.92596 - 10.9198i) q^{2} +(14.1081 - 24.4359i) q^{3} +(-55.2556 + 31.9018i) q^{4} +(-47.2751 + 47.2751i) q^{5} +(-308.115 - 82.5592i) q^{6} +(27.7639 - 103.616i) q^{7} +(-1.56966 - 1.56966i) q^{8} +(-33.5751 - 58.1539i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{2} - 2 q^{3} - 6 q^{4} - 114 q^{5} + 634 q^{6} + 316 q^{7} - 1398 q^{8} - 1946 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.92596	−	10.9198i	−0.365745	−	1.36498i	−0.866409	−	0.499336i	\(-0.833577\pi\)
							0.500664	−	0.865642i	\(-0.333089\pi\)
\(3\)	14.1081	−	24.4359i	0.522521	−	0.905033i	−0.477136	−	0.878830i	\(-0.658325\pi\)
							0.999657	−	0.0262033i	\(-0.00834172\pi\)
\(5\)	−47.2751	+	47.2751i	−0.378201	+	0.378201i	−0.870453	−	0.492252i	\(-0.836174\pi\)
							0.492252	+	0.870453i	\(0.336174\pi\)
\(7\)	27.7639	−	103.616i	0.0809443	−	0.302088i	−0.913571	−	0.406679i	\(-0.866687\pi\)
							0.994515	+	0.104591i	\(0.0333534\pi\)
\(11\)	2246.66	−	601.991i	1.68795	−	0.452285i	0.718091	−	0.695949i	\(-0.245016\pi\)
							0.969860	+	0.243664i	\(0.0783494\pi\)
\(13\)	−980.172	−	1966.23i	−0.446141	−	0.894963i
							\(17\)	2976.88	−	1718.70i	0.605919	−	0.349827i	−0.165448	−	0.986219i	\(-0.552907\pi\)
0.771367	+	0.636391i	\(0.219574\pi\)
\(19\)	−4738.10	−	1269.57i	−0.690786	−	0.185096i	−0.103686	−	0.994610i	\(-0.533064\pi\)
							−0.587100	+	0.809514i	\(0.699730\pi\)
\(23\)	16806.1	+	9702.98i	1.38128	+	0.797483i	0.992311	−	0.123767i	\(-0.0394975\pi\)
							0.388970	+	0.921250i	\(0.372831\pi\)
\(29\)	−14053.0	+	24340.6i	−0.576204	+	0.998015i	0.419705	+	0.907660i	\(0.362133\pi\)
							−0.995910	+	0.0903546i	\(0.971200\pi\)
\(31\)	−954.688	+	954.688i	−0.0320462	+	0.0320462i	−0.722948	−	0.690902i	\(-0.757214\pi\)
							0.690902	+	0.722948i	\(0.257214\pi\)
\(37\)	−5811.31	+	1557.13i	−0.114728	+	0.0307412i	−0.315726	−	0.948850i	\(-0.602248\pi\)
							0.200998	+	0.979592i	\(0.435581\pi\)
\(41\)	−16034.2	−	59840.5i	−0.232646	−	0.868248i	−0.979196	−	0.202918i	\(-0.934957\pi\)
							0.746549	−	0.665330i	\(-0.231709\pi\)
\(43\)	−45726.2	+	26400.0i	−0.575121	+	0.332046i	−0.759192	−	0.650867i	\(-0.774406\pi\)
							0.184071	+	0.982913i	\(0.441072\pi\)
\(47\)	−28376.8	−	28376.8i	−0.273319	−	0.273319i	0.557116	−	0.830435i	\(-0.311908\pi\)
							−0.830435	+	0.557116i	\(0.811908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.7.f.a.6.1	✓	24
13.11	odd	12	inner	13.7.f.a.11.1	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.7.f.a.6.1	✓	24	1.1	even	1	trivial
13.7.f.a.11.1	yes	24	13.11	odd	12	inner