Embedded newform 22.12.c.a.5.3

• Label: 22.12.c.a.5.3
• Level: $22$
• Weight: $12$
• Character: 22.5
• 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			5.3
Root			\(-104.347 + 75.8125i\) of defining polynomial
Character	\(\chi\)	\(=\)	22.5
Dual form			22.12.c.a.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.88854 + 30.4338i) q^{2} +(106.229 + 77.1798i) q^{3} +(-828.433 + 601.892i) q^{4} +(1487.63 - 4578.47i) q^{5} +(-1298.43 + 3996.15i) q^{6} +(9589.00 - 6966.82i) q^{7} +(-26509.9 - 19260.5i) q^{8} +(-49413.6 - 152079. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 160 q^{2} + 56 q^{3} - 5120 q^{4} - 362 q^{5} - 3488 q^{6} - 167178 q^{7} - 163840 q^{8} + 206411 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)

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
							\(3\)	106.229	+	77.1798i	0.252392	+	0.183374i	0.706786	−	0.707427i	\(-0.250144\pi\)
−0.454394	+	0.890801i	\(0.650144\pi\)
\(5\)	1487.63	−	4578.47i	0.212893	−	0.655217i	−0.786404	−	0.617713i	\(-0.788059\pi\)
							0.999296	−	0.0375038i	\(-0.0119406\pi\)
\(7\)	9589.00	−	6966.82i	0.215642	−	0.156673i	−0.474720	−	0.880137i	\(-0.657451\pi\)
							0.690363	+	0.723463i	\(0.257451\pi\)
\(11\)	399017.	−	355101.i	0.747020	−	0.664802i
							\(13\)	−21634.1	−	66583.1i	−0.0161604	−	0.0497365i	0.942651	−	0.333780i	\(-0.108324\pi\)
−0.958812	+	0.284043i	\(0.908324\pi\)
\(17\)	702031.	−	2.16063e6i	0.119919	−	0.369072i	−0.873022	−	0.487680i	\(-0.837843\pi\)
							0.992941	+	0.118608i	\(0.0378432\pi\)
\(19\)	1.67576e6	+	1.21751e6i	0.155263	+	0.112805i	0.662704	−	0.748881i	\(-0.269409\pi\)
							−0.507441	+	0.861686i	\(0.669409\pi\)
\(23\)	5.38810e7			1.74555			0.872775	−	0.488123i	\(-0.162318\pi\)
							0.872775	+	0.488123i	\(0.162318\pi\)
\(29\)	3.05311e7	−	2.21821e7i	0.276410	−	0.200823i	−0.440940	−	0.897537i	\(-0.645355\pi\)
							0.717350	+	0.696713i	\(0.245355\pi\)
\(31\)	−5.00396e7	−	1.54006e8i	−0.313924	−	0.966159i	−0.976195	−	0.216895i	\(-0.930407\pi\)
							0.662271	−	0.749265i	\(-0.269593\pi\)
\(37\)	−1.37000e8	+	9.95362e7i	−0.324796	+	0.235978i	−0.738219	−	0.674561i	\(-0.764333\pi\)
							0.413423	+	0.910539i	\(0.364333\pi\)
\(41\)	−6.10278e8	−	4.43393e8i	−0.822653	−	0.597692i	0.0948181	−	0.995495i	\(-0.469773\pi\)
							−0.917471	+	0.397802i	\(0.869773\pi\)
\(43\)	−1.16483e9			−1.20834			−0.604168	−	0.796857i	\(-0.706494\pi\)
							−0.604168	+	0.796857i	\(0.706494\pi\)
\(47\)	−5.46220e8	−	3.96852e8i	−0.347400	−	0.252401i	0.400378	−	0.916350i	\(-0.368879\pi\)
							−0.747777	+	0.663950i	\(0.768879\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.12.c.a.5.3	✓	20
11.3	even	5		242.12.a.n.1.5		10
11.8	odd	10		242.12.a.m.1.5		10
11.9	even	5	inner	22.12.c.a.9.3	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.12.c.a.5.3	✓	20	1.1	even	1	trivial
22.12.c.a.9.3	yes	20	11.9	even	5	inner
242.12.a.m.1.5		10	11.8	odd	10
242.12.a.n.1.5		10	11.3	even	5