Embedded newform 243.2.e.c.28.1

• Label: 243.2.e.c.28.1
• Level: $243$
• Weight: $2$
• Character: 243.28
• 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			28.1
Root			\(0.500000 + 1.27297i\) of defining polynomial
Character	\(\chi\)	\(=\)	243.28
Dual form			243.2.e.c.217.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.139184 - 0.789350i) q^{2} +(1.27568 - 0.464311i) q^{4} +(-2.10650 - 1.76756i) q^{5} +(-2.23349 - 0.812925i) q^{7} +(-1.34559 - 2.33062i) q^{8} +(-1.10204 + 1.90878i) q^{10} +(-0.191633 + 0.160799i) q^{11} +(0.453566 - 2.57230i) q^{13} +(-0.330816 + 1.87615i) q^{14} +(0.427502 - 0.358716i) q^{16} +(0.146688 - 0.254072i) q^{17} +(1.39237 + 2.41166i) q^{19} +(-3.50793 - 1.27678i) q^{20} +(0.153599 + 0.128885i) q^{22} +(6.28639 - 2.28806i) q^{23} +(0.444822 + 2.52271i) q^{25} -2.09357 q^{26} -3.22668 q^{28} +(-0.0616550 - 0.349663i) q^{29} +(-2.59869 + 0.945845i) q^{31} +(-4.46577 - 3.74722i) q^{32} +(-0.220968 - 0.0804258i) q^{34} +(3.26796 + 5.66027i) q^{35} +(3.49619 - 6.05558i) q^{37} +(1.70985 - 1.43473i) q^{38} +(-1.28505 + 7.28786i) q^{40} +(-1.68744 + 9.56997i) q^{41} +(0.199713 - 0.167579i) q^{43} +(-0.169802 + 0.294106i) q^{44} +(-2.68104 - 4.64370i) q^{46} +(10.7365 + 3.90777i) q^{47} +(-1.03467 - 0.868188i) q^{49} +(1.92939 - 0.702240i) q^{50} +(-0.615741 - 3.49204i) q^{52} +5.43137 q^{53} +0.687897 q^{55} +(1.11073 + 6.29929i) q^{56} +(-0.267425 + 0.0973348i) q^{58} +(4.57859 + 3.84189i) q^{59} +(11.1323 + 4.05183i) q^{61} +(1.10830 + 1.91963i) q^{62} +(-1.77824 + 3.08001i) q^{64} +(-5.50214 + 4.61685i) q^{65} +(0.314356 - 1.78280i) q^{67} +(0.0691597 - 0.392224i) q^{68} +(4.01309 - 3.36738i) q^{70} +(0.185255 - 0.320871i) q^{71} +(-2.51339 - 4.35333i) q^{73} +(-5.26658 - 1.91688i) q^{74} +(2.89599 + 2.43002i) q^{76} +(0.558728 - 0.203360i) q^{77} +(0.139409 + 0.790625i) q^{79} -1.53459 q^{80} +7.78892 q^{82} +(-0.478514 - 2.71379i) q^{83} +(-0.758087 + 0.275921i) q^{85} +(-0.160075 - 0.134319i) q^{86} +(0.632620 + 0.230255i) q^{88} +(-5.22533 - 9.05054i) q^{89} +(-3.10412 + 5.37650i) q^{91} +(6.95708 - 5.83768i) q^{92} +(1.59025 - 9.01876i) q^{94} +(1.32973 - 7.54127i) q^{95} +(-11.3640 + 9.53550i) q^{97} +(-0.541296 + 0.937552i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.139184	−	0.789350i	−0.0984177	−	0.558155i	−0.993646	−	0.112548i	\(-0.964099\pi\)
							0.895229	−	0.445607i	\(-0.147012\pi\)
\(3\)		0			0
							\(5\)	−2.10650	−	1.76756i	−0.942056	−	0.790479i	0.0358862	−	0.999356i	\(-0.488575\pi\)
−0.977942	+	0.208877i	\(0.933019\pi\)
\(7\)	−2.23349	−	0.812925i	−0.844181	−	0.307257i	−0.116516	−	0.993189i	\(-0.537172\pi\)
							−0.727666	+	0.685932i	\(0.759395\pi\)
\(11\)	−0.191633	+	0.160799i	−0.0577795	+	0.0484827i	−0.671220	−	0.741258i	\(-0.734229\pi\)
							0.613441	+	0.789741i	\(0.289785\pi\)
\(13\)	0.453566	−	2.57230i	0.125797	−	0.713428i	−0.855035	−	0.518570i	\(-0.826464\pi\)
							0.980832	−	0.194858i	\(-0.0624245\pi\)
\(17\)	0.146688	−	0.254072i	0.0355772	−	0.0616215i	−0.847689	−	0.530494i	\(-0.822006\pi\)
							0.883266	+	0.468873i	\(0.155340\pi\)
\(19\)	1.39237	+	2.41166i	0.319432	+	0.553273i	0.980370	−	0.197168i	\(-0.0631745\pi\)
							−0.660937	+	0.750441i	\(0.729841\pi\)
\(23\)	6.28639	−	2.28806i	1.31080	−	0.477094i	0.410304	−	0.911949i	\(-0.365423\pi\)
							0.900500	+	0.434855i	\(0.143201\pi\)
\(29\)	−0.0616550	−	0.349663i	−0.0114490	−	0.0649308i	0.978548	−	0.206019i	\(-0.0660509\pi\)
							−0.989997	+	0.141088i	\(0.954940\pi\)
\(31\)	−2.59869	+	0.945845i	−0.466738	+	0.169879i	−0.564674	−	0.825314i	\(-0.690998\pi\)
							0.0979360	+	0.995193i	\(0.468776\pi\)
\(37\)	3.49619	−	6.05558i	0.574770	−	0.995531i	−0.421297	−	0.906923i	\(-0.638425\pi\)
							0.996067	−	0.0886080i	\(-0.0282418\pi\)
\(41\)	−1.68744	+	9.56997i	−0.263535	+	1.49458i	0.509641	+	0.860387i	\(0.329778\pi\)
							−0.773176	+	0.634192i	\(0.781333\pi\)
\(43\)	0.199713	−	0.167579i	0.0304559	−	0.0255555i	−0.627433	−	0.778671i	\(-0.715894\pi\)
							0.657889	+	0.753115i	\(0.271450\pi\)
\(47\)	10.7365	+	3.90777i	1.56608	+	0.570007i	0.972119	−	0.234486i	\(-0.0753407\pi\)
							0.593962	+	0.804493i	\(0.297563\pi\)
\(53\)	5.43137			0.746056			0.373028	−	0.927820i	\(-0.378320\pi\)
							0.373028	+	0.927820i	\(0.378320\pi\)
\(59\)	4.57859	+	3.84189i	0.596082	+	0.500172i	0.890184	−	0.455602i	\(-0.150576\pi\)
							−0.294102	+	0.955774i	\(0.595020\pi\)
\(61\)	11.1323	+	4.05183i	1.42535	+	0.518784i	0.935594	−	0.353078i	\(-0.114865\pi\)
							0.489753	+	0.871861i	\(0.337087\pi\)
\(67\)	0.314356	−	1.78280i	0.0384047	−	0.217804i	−0.959566	−	0.281485i	\(-0.909173\pi\)
							0.997970	+	0.0636814i	\(0.0202841\pi\)
\(71\)	0.185255	−	0.320871i	0.0219857	−	0.0380804i	−0.854823	−	0.518919i	\(-0.826334\pi\)
							0.876809	+	0.480839i	\(0.159668\pi\)
\(73\)	−2.51339	−	4.35333i	−0.294171	−	0.509518i	0.680621	−	0.732636i	\(-0.261710\pi\)
							−0.974792	+	0.223117i	\(0.928377\pi\)
\(79\)	0.139409	+	0.790625i	0.0156847	+	0.0889523i	0.991645	−	0.128995i	\(-0.0411751\pi\)
							−0.975961	+	0.217947i	\(0.930064\pi\)
\(83\)	−0.478514	−	2.71379i	−0.0525237	−	0.297877i	0.947218	−	0.320589i	\(-0.103881\pi\)
							−0.999742	+	0.0227124i	\(0.992770\pi\)
\(89\)	−5.22533	−	9.05054i	−0.553884	−	0.959356i	−0.997989	−	0.0633809i	\(-0.979812\pi\)
							0.444105	−	0.895975i	\(-0.353522\pi\)
\(97\)	−11.3640	+	9.53550i	−1.15384	+	0.968183i	−0.999802	−	0.0198821i	\(-0.993671\pi\)
							−0.154034	+	0.988066i	\(0.549226\pi\)

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.c.28.1		12
3.2	odd	2		243.2.e.b.28.2		12
9.2	odd	6		243.2.e.a.109.2		12
9.4	even	3		27.2.e.a.4.2	✓	12
9.5	odd	6		81.2.e.a.64.1		12
9.7	even	3		243.2.e.d.109.1		12
27.2	odd	18		243.2.e.b.217.2		12
27.4	even	9		729.2.c.e.487.4		12
27.5	odd	18		729.2.a.d.1.4		6
27.7	even	9		243.2.e.d.136.1		12
27.11	odd	18		81.2.e.a.19.1		12
27.13	even	9		729.2.c.e.244.4		12
27.14	odd	18		729.2.c.b.244.3		12
27.16	even	9		27.2.e.a.7.2	yes	12
27.20	odd	18		243.2.e.a.136.2		12
27.22	even	9		729.2.a.a.1.3		6
27.23	odd	18		729.2.c.b.487.3		12
27.25	even	9	inner	243.2.e.c.217.1		12
36.31	odd	6		432.2.u.c.193.2		12
45.4	even	6		675.2.l.c.301.1		12
45.13	odd	12		675.2.u.b.274.3		24
45.22	odd	12		675.2.u.b.274.2		24
108.43	odd	18		432.2.u.c.385.2		12
135.43	odd	36		675.2.u.b.574.2		24
135.97	odd	36		675.2.u.b.574.3		24
135.124	even	18		675.2.l.c.601.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.4.2	✓	12	9.4	even	3
27.2.e.a.7.2	yes	12	27.16	even	9
81.2.e.a.19.1		12	27.11	odd	18
81.2.e.a.64.1		12	9.5	odd	6
243.2.e.a.109.2		12	9.2	odd	6
243.2.e.a.136.2		12	27.20	odd	18
243.2.e.b.28.2		12	3.2	odd	2
243.2.e.b.217.2		12	27.2	odd	18
243.2.e.c.28.1		12	1.1	even	1	trivial
243.2.e.c.217.1		12	27.25	even	9	inner
243.2.e.d.109.1		12	9.7	even	3
243.2.e.d.136.1		12	27.7	even	9
432.2.u.c.193.2		12	36.31	odd	6
432.2.u.c.385.2		12	108.43	odd	18
675.2.l.c.301.1		12	45.4	even	6
675.2.l.c.601.1		12	135.124	even	18
675.2.u.b.274.2		24	45.22	odd	12
675.2.u.b.274.3		24	45.13	odd	12
675.2.u.b.574.2		24	135.43	odd	36
675.2.u.b.574.3		24	135.97	odd	36
729.2.a.a.1.3		6	27.22	even	9
729.2.a.d.1.4		6	27.5	odd	18
729.2.c.b.244.3		12	27.14	odd	18
729.2.c.b.487.3		12	27.23	odd	18
729.2.c.e.244.4		12	27.13	even	9
729.2.c.e.487.4		12	27.4	even	9