Embedded newform 21.11.f.a.19.2

• Label: 21.11.f.a.19.2
• Level: $21$
• Weight: $11$
• Character: 21.19
• Analytic conductor: $13.343$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	21.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.3425023061\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - x^11 + 3773*x^10 + 44516*x^9 + 11068388*x^8 + 100480832*x^7 + 11177140432*x^6 + 67553728512*x^5 + 8140577253696*x^4 + 32158758551040*x^3 + 2022248850888960*x^2 - 9516227338368000*x + 300517801880601600
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{10}\cdot 3^{7}\cdot 7^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			19.2
Root			\(14.5243 - 25.1569i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.19
Dual form			21.11.f.a.10.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.5243 + 23.4248i) q^{2} +(-121.500 + 70.1481i) q^{3} +(146.185 + 253.199i) q^{4} +(-1211.43 - 699.418i) q^{5} -3794.82i q^{6} +(-12945.1 + 10719.1i) q^{7} -35606.0 q^{8} +(9841.50 - 17046.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 11 q^{2} - 1458 q^{3} - 1421 q^{4} - 1287 q^{5} + 20090 q^{7} + 99310 q^{8} + 118098 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.5243	+	23.4248i	−0.422635	+	0.732026i	−0.996196	−	0.0871368i	\(-0.972228\pi\)
							0.573561	+	0.819163i	\(0.305562\pi\)
\(3\)	−121.500	+	70.1481i	−0.500000	+	0.288675i
							\(5\)	−1211.43	−	699.418i	−0.387657	−	0.223814i	0.293488	−	0.955963i	\(-0.405184\pi\)
−0.681144	+	0.732149i	\(0.738517\pi\)
\(7\)	−12945.1	+	10719.1i	−0.770222	+	0.637776i
							\(11\)	−3967.94	−	6872.68i	−0.0246378	−	0.0426739i	0.853444	−	0.521185i	\(-0.174510\pi\)
−0.878081	+	0.478511i	\(0.841177\pi\)
\(13\)		−	80828.8i		−	0.217696i	−0.994058	−	0.108848i	\(-0.965284\pi\)
							0.994058	−	0.108848i	\(-0.0347161\pi\)
\(17\)	1.93863e6	−	1.11927e6i	1.36537	−	0.788298i	0.375039	−	0.927009i	\(-0.377629\pi\)
							0.990333	+	0.138712i	\(0.0442961\pi\)
\(19\)	541175.	+	312447.i	0.218559	+	0.126185i	0.605283	−	0.796010i	\(-0.293060\pi\)
							−0.386724	+	0.922196i	\(0.626393\pi\)
\(23\)	2.73171e6	−	4.73146e6i	0.424420	−	0.735116i	−0.571946	−	0.820291i	\(-0.693811\pi\)
							0.996366	+	0.0851746i	\(0.0271448\pi\)
\(29\)	2.70806e6			0.132029			0.0660145	−	0.997819i	\(-0.478972\pi\)
							0.0660145	+	0.997819i	\(0.478972\pi\)
\(31\)	−3.76608e7	+	2.17435e7i	−1.31547	+	0.759487i	−0.982996	−	0.183625i	\(-0.941217\pi\)
							−0.332474	+	0.943112i	\(0.607884\pi\)
\(37\)	−1.77646e7	+	3.07691e7i	−0.256180	+	0.443718i	−0.965215	−	0.261456i	\(-0.915797\pi\)
							0.709035	+	0.705173i	\(0.249131\pi\)
\(41\)		−	4.22840e7i		−	0.364969i	−0.983209	−	0.182485i	\(-0.941586\pi\)
							0.983209	−	0.182485i	\(-0.0584140\pi\)
\(43\)	−7.33629e7			−0.499039			−0.249519	−	0.968370i	\(-0.580273\pi\)
							−0.249519	+	0.968370i	\(0.580273\pi\)
\(47\)	−1.77607e8	−	1.02542e8i	−0.774412	−	0.447107i	0.0600344	−	0.998196i	\(-0.480879\pi\)
							−0.834446	+	0.551089i	\(0.814212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.11.f.a.19.2	yes	12
3.2	odd	2		63.11.m.b.19.5		12
7.2	even	3		147.11.d.a.97.9		12
7.3	odd	6	inner	21.11.f.a.10.2	✓	12
7.5	odd	6		147.11.d.a.97.10		12
21.17	even	6		63.11.m.b.10.5		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.11.f.a.10.2	✓	12	7.3	odd	6	inner
21.11.f.a.19.2	yes	12	1.1	even	1	trivial
63.11.m.b.10.5		12	21.17	even	6
63.11.m.b.19.5		12	3.2	odd	2
147.11.d.a.97.9		12	7.2	even	3
147.11.d.a.97.10		12	7.5	odd	6