Embedded newform 12.13.d.a.7.4

• Label: 12.13.d.a.7.4
• Level: $12$
• Weight: $13$
• Character: 12.7
• Analytic conductor: $10.968$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	12.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9679258073\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 3*x^11 + 1570*x^10 - 4077*x^9 + 1884069*x^8 - 3551868*x^7 + 881574992*x^6 + 2167220880*x^5 + 304613421264*x^4 + 27792635520*x^3 + 5806366144000*x^2 + 1120796928000*x + 104882177440000
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{57}\cdot 3^{25} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			7.4
Root			\(2.19768 + 3.80649i\) of defining polynomial
Character	\(\chi\)	\(=\)	12.7
Dual form			12.13.d.a.7.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-46.3006 + 44.1844i) q^{2} -420.888i q^{3} +(191.485 - 4091.52i) q^{4} +3884.37 q^{5} +(18596.7 + 19487.4i) q^{6} +57100.7i q^{7} +(171915. + 197900. i) q^{8} -177147. q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 90 q^{2} - 4692 q^{4} - 10296 q^{5} - 48114 q^{6} - 648000 q^{8} - 2125764 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−46.3006	+	44.1844i	−0.723446	+	0.690381i
							\(3\)		−	420.888i		−	0.577350i
\(5\)	3884.37			0.248599										0.124300	−	0.992245i	\(-0.460332\pi\)
							0.124300	+	0.992245i	\(0.460332\pi\)
\(7\)			57100.7i			0.485348i	0.970108	+	0.242674i	\(0.0780245\pi\)
							−0.970108	+	0.242674i	\(0.921976\pi\)
\(11\)		−	682381.i		−	0.385187i	−0.981279	−	0.192593i	\(-0.938310\pi\)
							0.981279	−	0.192593i	\(-0.0616898\pi\)
\(13\)	−8.14207e6			−1.68684			−0.843422	−	0.537252i	\(-0.819463\pi\)
							−0.843422	+	0.537252i	\(0.819463\pi\)
\(17\)	−2.04604e7			−0.847658			−0.423829	−	0.905742i	\(-0.639314\pi\)
							−0.423829	+	0.905742i	\(0.639314\pi\)
\(19\)			6.40410e7i			1.36125i	0.732634	+	0.680623i	\(0.238291\pi\)
							−0.732634	+	0.680623i	\(0.761709\pi\)
\(23\)			6.19286e6i			0.0418335i	0.999781	+	0.0209168i	\(0.00665850\pi\)
							−0.999781	+	0.0209168i	\(0.993342\pi\)
\(29\)	−6.00461e7			−0.100948			−0.0504739	−	0.998725i	\(-0.516073\pi\)
							−0.0504739	+	0.998725i	\(0.516073\pi\)
\(31\)			1.27401e9i			1.43550i	0.696301	+	0.717750i	\(0.254828\pi\)
							−0.696301	+	0.717750i	\(0.745172\pi\)
\(37\)	−1.65103e9			−0.643493			−0.321746	−	0.946826i	\(-0.604270\pi\)
							−0.321746	+	0.946826i	\(0.604270\pi\)
\(41\)	−5.89164e9			−1.24032			−0.620159	−	0.784476i	\(-0.712932\pi\)
							−0.620159	+	0.784476i	\(0.712932\pi\)
\(43\)		−	1.66763e9i		−	0.263809i	−0.991262	−	0.131904i	\(-0.957891\pi\)
							0.991262	−	0.131904i	\(-0.0421092\pi\)
\(47\)		−	1.58435e10i		−	1.46982i	−0.678163	−	0.734912i	\(-0.737224\pi\)
							0.678163	−	0.734912i	\(-0.262776\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.13.d.a.7.4	yes	12
3.2	odd	2		36.13.d.d.19.9		12
4.3	odd	2	inner	12.13.d.a.7.3	✓	12
8.3	odd	2		192.13.g.e.127.3		12
8.5	even	2		192.13.g.e.127.9		12
12.11	even	2		36.13.d.d.19.10		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.13.d.a.7.3	✓	12	4.3	odd	2	inner
12.13.d.a.7.4	yes	12	1.1	even	1	trivial
36.13.d.d.19.9		12	3.2	odd	2
36.13.d.d.19.10		12	12.11	even	2
192.13.g.e.127.3		12	8.3	odd	2
192.13.g.e.127.9		12	8.5	even	2