Embedded newform 13.7.f.a.6.5

• Label: 13.7.f.a.6.5
• 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			6.5
Character	\(\chi\)	\(=\)	13.6
Dual form			13.7.f.a.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.99444 + 7.44337i) q^{2} +(25.6304 - 44.3931i) q^{3} +(3.99974 - 2.30925i) q^{4} +(-23.8066 + 23.8066i) q^{5} +(381.553 + 102.237i) q^{6} +(-75.6362 + 282.278i) q^{7} +(373.897 + 373.897i) q^{8} +(-949.333 - 1644.29i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{2} - 2 q^{3} - 6 q^{4} - 114 q^{5} + 634 q^{6} + 316 q^{7} - 1398 q^{8} - 1946 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.99444	+	7.44337i	0.249305	+	0.930421i	0.971170	+	0.238387i	\(0.0766186\pi\)
							−0.721865	+	0.692034i	\(0.756715\pi\)
\(3\)	25.6304	−	44.3931i	0.949273	−	1.64419i	0.202313	−	0.979321i	\(-0.435154\pi\)
							0.746960	−	0.664869i	\(-0.231513\pi\)
\(5\)	−23.8066	+	23.8066i	−0.190453	+	0.190453i	−0.795892	−	0.605439i	\(-0.792998\pi\)
							0.605439	+	0.795892i	\(0.292998\pi\)
\(7\)	−75.6362	+	282.278i	−0.220514	+	0.822969i	0.763639	+	0.645644i	\(0.223411\pi\)
							−0.984152	+	0.177325i	\(0.943256\pi\)
\(11\)	−427.378	+	114.516i	−0.321095	+	0.0860373i	−0.415767	−	0.909471i	\(-0.636487\pi\)
							0.0946712	+	0.995509i	\(0.469820\pi\)
\(13\)	40.6056	+	2196.62i	0.0184823	+	0.999829i
							\(17\)	2798.48	−	1615.70i	0.569607	−	0.328863i	−0.187385	−	0.982287i	\(-0.560001\pi\)
0.756992	+	0.653424i	\(0.226668\pi\)
\(19\)	−10133.1	−	2715.15i	−1.47734	−	0.395853i	−0.571900	−	0.820323i	\(-0.693794\pi\)
							−0.905442	+	0.424470i	\(0.860460\pi\)
\(23\)	2637.16	+	1522.57i	0.216747	+	0.125139i	0.604443	−	0.796648i	\(-0.293396\pi\)
							−0.387696	+	0.921787i	\(0.626729\pi\)
\(29\)	17327.5	−	30012.0i	0.710462	−	1.23056i	−0.254222	−	0.967146i	\(-0.581819\pi\)
							0.964684	−	0.263411i	\(-0.0848474\pi\)
\(31\)	5703.31	−	5703.31i	0.191444	−	0.191444i	−0.604876	−	0.796320i	\(-0.706777\pi\)
							0.796320	+	0.604876i	\(0.206777\pi\)
\(37\)	3123.71	−	836.996i	0.0616689	−	0.0165241i	−0.227853	−	0.973696i	\(-0.573170\pi\)
							0.289522	+	0.957171i	\(0.406504\pi\)
\(41\)	−14330.9	−	53483.8i	−0.207933	−	0.776016i	−0.988536	−	0.150988i	\(-0.951754\pi\)
							0.780603	−	0.625028i	\(-0.214912\pi\)
\(43\)	58195.9	−	33599.4i	0.731960	−	0.422597i	−0.0871790	−	0.996193i	\(-0.527785\pi\)
							0.819139	+	0.573596i	\(0.194452\pi\)
\(47\)	−70573.1	−	70573.1i	−0.679744	−	0.679744i	0.280198	−	0.959942i	\(-0.409600\pi\)
							−0.959942	+	0.280198i	\(0.909600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.7.f.a.6.5	✓	24
13.11	odd	12	inner	13.7.f.a.11.5	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.7.f.a.6.5	✓	24	1.1	even	1	trivial
13.7.f.a.11.5	yes	24	13.11	odd	12	inner