Embedded newform 22.12.c.a.9.4

• Label: 22.12.c.a.9.4
• Level: $22$
• Weight: $12$
• Character: 22.9
• Analytic conductor: $16.904$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.9035499723\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 4*x^19 + 339028*x^18 - 38195378*x^17 + 220926638311*x^16 - 35915431412664*x^15 + 115759899998314350*x^14 - 27226928185171403384*x^13 + 41831014975562739810959*x^12 - 3879797201571793549516048*x^11 + 3316840467475291520075063190*x^10 + 1067427514312498593848752050628*x^9 + 750967910898548932251904101148531*x^8 + 228128795234897755949671367783663114*x^7 + 90326787979451743483651549716111634592*x^6 + 21775725237124771428874092537573760059528*x^5 + 5115171268695044278298633768429412143699481*x^4 + 644893842489460735970879339730837145914702280*x^3 + 40307167403308519805773510178453296556136973760*x^2 - 118703222270772703791843925527006368174467110400*x + 1158220182319374358886221767374591631314546790400
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{24}\cdot 3^{4}\cdot 5\cdot 11^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			9.4
Root			\(-203.932 - 148.166i\) of defining polynomial
Character	\(\chi\)	\(=\)	22.9
Dual form			22.12.c.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.88854 - 30.4338i) q^{2} +(205.814 - 149.533i) q^{3} +(-828.433 - 601.892i) q^{4} +(-2975.41 - 9157.38i) q^{5} +(-2515.65 - 7742.38i) q^{6} +(94.4405 + 68.6150i) q^{7} +(-26509.9 + 19260.5i) q^{8} +(-34741.9 + 106925. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 160 q^{2} + 56 q^{3} - 5120 q^{4} - 362 q^{5} - 3488 q^{6} - 167178 q^{7} - 163840 q^{8} + 206411 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.88854	−	30.4338i	0.218508	−	0.672499i
							\(3\)	205.814	−	149.533i	0.489000	−	0.355279i	−0.315800	−	0.948826i	\(-0.602273\pi\)
0.804800	+	0.593547i	\(0.202273\pi\)
\(5\)	−2975.41	−	9157.38i	−0.425807	−	1.31050i	−0.902220	−	0.431277i	\(-0.858063\pi\)
							0.476413	−	0.879222i	\(-0.341937\pi\)
\(7\)	94.4405	+	68.6150i	0.00212383	+	0.00154305i	0.588847	−	0.808245i	\(-0.299582\pi\)
							−0.586723	+	0.809788i	\(0.699582\pi\)
\(11\)	−54030.4	−	531406.i	−0.101153	−	0.994871i
							\(13\)	−4721.05	+	14529.9i	−0.00352655	+	0.0108536i	−0.952804	−	0.303585i	\(-0.901816\pi\)
0.949278	+	0.314439i	\(0.101816\pi\)
\(17\)	160349.	+	493503.i	0.0273903	+	0.0842987i	0.963817	−	0.266564i	\(-0.0858884\pi\)
							−0.936427	+	0.350863i	\(0.885888\pi\)
\(19\)	−1.09783e7	+	7.97618e6i	−1.01716	+	0.739010i	−0.965699	−	0.259665i	\(-0.916388\pi\)
							−0.0514609	+	0.998675i	\(0.516388\pi\)
\(23\)	−1.75895e7			−0.569835			−0.284918	−	0.958552i	\(-0.591966\pi\)
							−0.284918	+	0.958552i	\(0.591966\pi\)
\(29\)	−6.38771e7	−	4.64094e7i	−0.578304	−	0.420162i	0.259808	−	0.965660i	\(-0.416341\pi\)
							−0.838112	+	0.545498i	\(0.816341\pi\)
\(31\)	6.32266e6	−	1.94591e7i	0.0396653	−	0.122077i	−0.929263	−	0.369419i	\(-0.879557\pi\)
							0.968928	+	0.247341i	\(0.0795569\pi\)
\(37\)	−5.49529e8	−	3.99256e8i	−1.30281	−	0.946547i	−0.302832	−	0.953044i	\(-0.597932\pi\)
							−0.999979	+	0.00649663i	\(0.997932\pi\)
\(41\)	7.53201e8	−	5.47232e8i	1.01531	−	0.737667i	0.0499952	−	0.998749i	\(-0.484079\pi\)
							0.965317	+	0.261082i	\(0.0840794\pi\)
\(43\)	9.71135e8			1.00740			0.503701	−	0.863878i	\(-0.331971\pi\)
							0.503701	+	0.863878i	\(0.331971\pi\)
\(47\)	1.69793e9	−	1.23362e9i	1.07989	−	0.784588i	0.102228	−	0.994761i	\(-0.467403\pi\)
							0.977664	+	0.210173i	\(0.0674028\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.12.c.a.9.4	yes	20
11.4	even	5		242.12.a.n.1.4		10
11.5	even	5	inner	22.12.c.a.5.4	✓	20
11.7	odd	10		242.12.a.m.1.4		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.12.c.a.5.4	✓	20	11.5	even	5	inner
22.12.c.a.9.4	yes	20	1.1	even	1	trivial
242.12.a.m.1.4		10	11.7	odd	10
242.12.a.n.1.4		10	11.4	even	5