Embedded newform 243.2.e.b.217.1

• Label: 243.2.e.b.217.1
• Level: $243$
• Weight: $2$
• Character: 243.217
• Analytic conductor: $1.940$
• Analytic rank: $0$
• Dimension: $12$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94036476912\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	12.0.1952986685049.1
Defining polynomial:	x^12 - 6*x^11 + 27*x^10 - 80*x^9 + 186*x^8 - 330*x^7 + 463*x^6 - 504*x^5 + 420*x^4 - 258*x^3 + 108*x^2 - 27*x + 3
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			217.1
Root			\(0.500000 + 1.27297i\) of defining polynomial
Character	\(\chi\)	\(=\)	243.217
Dual form			243.2.e.b.28.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.291887 + 1.65538i) q^{2} +(-0.775684 - 0.282326i) q^{4} +(0.865281 - 0.726057i) q^{5} +(3.67319 - 1.33693i) q^{7} +(-0.987144 + 1.70978i) q^{8} +(0.949332 + 1.64429i) q^{10} +(1.43285 + 1.20231i) q^{11} +(-0.127214 - 0.721468i) q^{13} +(1.14097 + 6.47073i) q^{14} +(-3.80689 - 3.19436i) q^{16} +(0.944822 + 1.63648i) q^{17} +(-1.37143 + 2.37538i) q^{19} +(-0.876169 + 0.318900i) q^{20} +(-2.40850 + 2.02097i) q^{22} +(-5.47625 - 1.99319i) q^{23} +(-0.646688 + 3.66755i) q^{25} +1.23143 q^{26} -3.22668 q^{28} +(0.923797 - 5.23911i) q^{29} +(-1.25975 - 0.458512i) q^{31} +(3.37426 - 2.83134i) q^{32} +(-2.98477 + 1.08637i) q^{34} +(2.20765 - 3.82376i) q^{35} +(-1.69806 - 2.94112i) q^{37} +(-3.53185 - 2.96357i) q^{38} +(0.387244 + 2.19617i) q^{40} +(-0.311930 - 1.76904i) q^{41} +(3.85332 + 3.23332i) q^{43} +(-0.771999 - 1.33714i) q^{44} +(4.89793 - 8.48346i) q^{46} +(1.60563 - 0.584402i) q^{47} +(6.34260 - 5.32207i) q^{49} +(-5.88242 - 2.14102i) q^{50} +(-0.105011 + 0.595547i) q^{52} -2.84494 q^{53} +2.11276 q^{55} +(-1.34010 + 7.60010i) q^{56} +(8.40305 + 3.05846i) q^{58} +(8.62570 - 7.23782i) q^{59} +(-4.91543 + 1.78907i) q^{61} +(1.12672 - 1.95153i) q^{62} +(-1.26751 - 2.19540i) q^{64} +(-0.633903 - 0.531908i) q^{65} +(-0.328026 - 1.86033i) q^{67} +(-0.270863 - 1.53614i) q^{68} +(5.68538 + 4.77060i) q^{70} +(-6.09193 - 10.5515i) q^{71} +(-4.94384 + 8.56298i) q^{73} +(5.36430 - 1.95245i) q^{74} +(1.73443 - 1.45536i) q^{76} +(6.87053 + 2.50067i) q^{77} +(2.14505 - 12.1652i) q^{79} -5.61331 q^{80} +3.01948 q^{82} +(2.02876 - 11.5057i) q^{83} +(2.00571 + 0.730020i) q^{85} +(-6.47709 + 5.43493i) q^{86} +(-3.47012 + 1.26302i) q^{88} +(-2.86437 + 4.96123i) q^{89} +(-1.43183 - 2.48001i) q^{91} +(3.68511 + 3.09217i) q^{92} +(0.498741 + 2.82850i) q^{94} +(0.537993 + 3.05111i) q^{95} +(-0.263043 - 0.220719i) q^{97} +(6.95870 + 12.0528i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	12 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 6 * q^8 - 3 * q^10 - 3 * q^11 + 3 * q^13 - 6 * q^14 - 9 * q^16 - 9 * q^17 - 3 * q^19 + 21 * q^20 - 15 * q^22 - 24 * q^23 - 15 * q^25 + 30 * q^26 - 12 * q^28 - 30 * q^29 - 15 * q^31 + 27 * q^32 - 9 * q^34 - 12 * q^35 - 3 * q^37 + 12 * q^38 - 6 * q^40 + 21 * q^41 + 12 * q^43 - 3 * q^44 - 3 * q^46 - 3 * q^47 + 21 * q^49 - 12 * q^50 + 36 * q^52 + 18 * q^53 - 12 * q^55 - 3 * q^56 + 30 * q^58 - 15 * q^59 + 21 * q^61 + 12 * q^62 + 12 * q^64 + 24 * q^65 + 21 * q^67 + 18 * q^68 + 30 * q^70 - 27 * q^71 + 6 * q^73 + 12 * q^74 + 42 * q^76 + 3 * q^77 + 21 * q^79 - 42 * q^80 - 12 * q^82 + 33 * q^83 - 9 * q^85 + 30 * q^86 - 12 * q^88 - 9 * q^89 + 6 * q^91 - 42 * q^92 - 33 * q^94 - 30 * q^95 - 42 * q^97 + 45 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.291887	+	1.65538i	−0.206396	+	1.17053i	0.688833	+	0.724920i	\(0.258123\pi\)
							−0.895229	+	0.445607i	\(0.852988\pi\)
\(3\)		0			0
							\(5\)	0.865281	−	0.726057i	0.386965	−	0.324703i	−0.428464	−	0.903559i	\(-0.640945\pi\)
0.815430	+	0.578856i	\(0.196501\pi\)
\(7\)	3.67319	−	1.33693i	1.38833	−	0.505312i	0.463640	−	0.886024i	\(-0.346543\pi\)
							0.924694	+	0.380712i	\(0.124321\pi\)
\(11\)	1.43285	+	1.20231i	0.432021	+	0.362509i	0.832714	−	0.553704i	\(-0.186786\pi\)
							−0.400693	+	0.916213i	\(0.631230\pi\)
\(13\)	−0.127214	−	0.721468i	−0.0352829	−	0.200099i	0.962071	−	0.272799i	\(-0.0879495\pi\)
							−0.997354	+	0.0727001i	\(0.976838\pi\)
\(17\)	0.944822	+	1.63648i	0.229153	+	0.396905i	0.957557	−	0.288243i	\(-0.0930711\pi\)
							−0.728404	+	0.685147i	\(0.759738\pi\)
\(19\)	−1.37143	+	2.37538i	−0.314627	+	0.544950i	−0.979358	−	0.202133i	\(-0.935213\pi\)
							0.664731	+	0.747083i	\(0.268546\pi\)
\(23\)	−5.47625	−	1.99319i	−1.14188	−	0.415609i	−0.299286	−	0.954164i	\(-0.596748\pi\)
							−0.842591	+	0.538555i	\(0.818971\pi\)
\(29\)	0.923797	−	5.23911i	0.171545	−	0.972879i	−0.770512	−	0.637426i	\(-0.779999\pi\)
							0.942057	−	0.335453i	\(-0.108890\pi\)
\(31\)	−1.25975	−	0.458512i	−0.226258	−	0.0823513i	0.226403	−	0.974034i	\(-0.427303\pi\)
							−0.452662	+	0.891682i	\(0.649525\pi\)
\(37\)	−1.69806	−	2.94112i	−0.279159	−	0.483517i	0.692017	−	0.721881i	\(-0.256722\pi\)
							−0.971176	+	0.238364i	\(0.923389\pi\)
\(41\)	−0.311930	−	1.76904i	−0.0487153	−	0.276278i	0.950714	−	0.310070i	\(-0.100353\pi\)
							−0.999429	+	0.0337924i	\(0.989241\pi\)
\(43\)	3.85332	+	3.23332i	0.587626	+	0.493077i	0.887441	−	0.460921i	\(-0.152481\pi\)
							−0.299816	+	0.953997i	\(0.596925\pi\)
\(47\)	1.60563	−	0.584402i	0.234205	−	0.0852438i	−0.222251	−	0.974989i	\(-0.571341\pi\)
							0.456457	+	0.889746i	\(0.349118\pi\)
\(53\)	−2.84494			−0.390783			−0.195391	−	0.980725i	\(-0.562598\pi\)
							−0.195391	+	0.980725i	\(0.562598\pi\)
\(59\)	8.62570	−	7.23782i	1.12297	−	0.942284i	0.124219	−	0.992255i	\(-0.460357\pi\)
							0.998751	+	0.0499712i	\(0.0159130\pi\)
\(61\)	−4.91543	+	1.78907i	−0.629357	+	0.229067i	−0.636951	−	0.770904i	\(-0.719805\pi\)
							0.00759462	+	0.999971i	\(0.497583\pi\)
\(67\)	−0.328026	−	1.86033i	−0.0400748	−	0.227275i	0.958192	−	0.286126i	\(-0.0923676\pi\)
							−0.998267	+	0.0588505i	\(0.981256\pi\)
\(71\)	−6.09193	−	10.5515i	−0.722980	−	1.25224i	−0.959800	−	0.280684i	\(-0.909439\pi\)
							0.236821	−	0.971553i	\(-0.423895\pi\)
\(73\)	−4.94384	+	8.56298i	−0.578633	+	1.00222i	0.417004	+	0.908905i	\(0.363080\pi\)
							−0.995637	+	0.0933164i	\(0.970253\pi\)
\(79\)	2.14505	−	12.1652i	0.241337	−	1.36869i	−0.587509	−	0.809217i	\(-0.699892\pi\)
							0.828847	−	0.559475i	\(-0.188997\pi\)
\(83\)	2.02876	−	11.5057i	0.222685	−	1.26291i	−0.644376	−	0.764709i	\(-0.722883\pi\)
							0.867061	−	0.498202i	\(-0.166006\pi\)
\(89\)	−2.86437	+	4.96123i	−0.303622	+	0.525889i	−0.976954	−	0.213452i	\(-0.931529\pi\)
							0.673331	+	0.739341i	\(0.264863\pi\)
\(97\)	−0.263043	−	0.220719i	−0.0267080	−	0.0224107i	0.629336	−	0.777133i	\(-0.283327\pi\)
							−0.656044	+	0.754723i	\(0.727771\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.e.b.217.1		12
3.2	odd	2		243.2.e.c.217.2		12
9.2	odd	6		27.2.e.a.7.1	yes	12
9.4	even	3		243.2.e.a.136.1		12
9.5	odd	6		243.2.e.d.136.2		12
9.7	even	3		81.2.e.a.19.2		12
27.2	odd	18		729.2.c.e.487.1		12
27.4	even	9		243.2.e.a.109.1		12
27.5	odd	18		27.2.e.a.4.1	✓	12
27.7	even	9		729.2.c.b.244.6		12
27.11	odd	18		729.2.a.a.1.6		6
27.13	even	9	inner	243.2.e.b.28.1		12
27.14	odd	18		243.2.e.c.28.2		12
27.16	even	9		729.2.a.d.1.1		6
27.20	odd	18		729.2.c.e.244.1		12
27.22	even	9		81.2.e.a.64.2		12
27.23	odd	18		243.2.e.d.109.2		12
27.25	even	9		729.2.c.b.487.6		12
36.11	even	6		432.2.u.c.385.1		12
45.2	even	12		675.2.u.b.574.1		24
45.29	odd	6		675.2.l.c.601.2		12
45.38	even	12		675.2.u.b.574.4		24
108.59	even	18		432.2.u.c.193.1		12
135.32	even	36		675.2.u.b.274.4		24
135.59	odd	18		675.2.l.c.301.2		12
135.113	even	36		675.2.u.b.274.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.4.1	✓	12	27.5	odd	18
27.2.e.a.7.1	yes	12	9.2	odd	6
81.2.e.a.19.2		12	9.7	even	3
81.2.e.a.64.2		12	27.22	even	9
243.2.e.a.109.1		12	27.4	even	9
243.2.e.a.136.1		12	9.4	even	3
243.2.e.b.28.1		12	27.13	even	9	inner
243.2.e.b.217.1		12	1.1	even	1	trivial
243.2.e.c.28.2		12	27.14	odd	18
243.2.e.c.217.2		12	3.2	odd	2
243.2.e.d.109.2		12	27.23	odd	18
243.2.e.d.136.2		12	9.5	odd	6
432.2.u.c.193.1		12	108.59	even	18
432.2.u.c.385.1		12	36.11	even	6
675.2.l.c.301.2		12	135.59	odd	18
675.2.l.c.601.2		12	45.29	odd	6
675.2.u.b.274.1		24	135.113	even	36
675.2.u.b.274.4		24	135.32	even	36
675.2.u.b.574.1		24	45.2	even	12
675.2.u.b.574.4		24	45.38	even	12
729.2.a.a.1.6		6	27.11	odd	18
729.2.a.d.1.1		6	27.16	even	9
729.2.c.b.244.6		12	27.7	even	9
729.2.c.b.487.6		12	27.25	even	9
729.2.c.e.244.1		12	27.20	odd	18
729.2.c.e.487.1		12	27.2	odd	18