Embedded newform 363.2.e.n.202.3

• Label: 363.2.e.n.202.3
• Level: $363$
• Weight: $2$
• Character: 363.202
• Analytic conductor: $2.899$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.e (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89856959337\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{5})\)
Coefficient field:	16.0.22502537891856000000000000.3
Defining polynomial:	\( x^{16} + 7x^{14} + 45x^{12} + 287x^{10} + 1829x^{8} + 1148x^{6} + 720x^{4} + 448x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			202.3
Root			\(0.640974 - 0.465695i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.202
Dual form			363.2.e.n.124.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.640974 + 0.465695i) q^{2} +(0.309017 - 0.951057i) q^{3} +(-0.424058 - 1.30512i) q^{4} +(-2.72823 + 1.98218i) q^{5} +(0.640974 - 0.465695i) q^{6} +(-0.780063 - 2.40079i) q^{7} +(0.825636 - 2.54105i) q^{8} +(-0.809017 - 0.587785i) q^{9} -2.67181 q^{10} -1.37228 q^{12} +(-4.72544 - 3.43323i) q^{13} +(0.618034 - 1.90211i) q^{14} +(1.04209 + 3.20723i) q^{15} +(-0.507835 + 0.368964i) q^{16} +(2.16154 - 1.57045i) q^{17} +(-0.244830 - 0.753510i) q^{18} +(0.290403 - 0.893769i) q^{19} +(3.74390 + 2.72010i) q^{20} -2.52434 q^{21} +2.00000 q^{23} +(-2.16154 - 1.57045i) q^{24} +(1.96914 - 6.06040i) q^{25} +(-1.43004 - 4.40122i) q^{26} +(-0.809017 + 0.587785i) q^{27} +(-2.80252 + 2.03615i) q^{28} +(0.244830 + 0.753510i) q^{29} +(-0.825636 + 2.54105i) q^{30} +(-1.31685 - 0.956749i) q^{31} -5.84096 q^{32} +2.11684 q^{34} +(6.88698 + 5.00368i) q^{35} +(-0.424058 + 1.30512i) q^{36} +(1.54508 + 4.75528i) q^{37} +(0.602364 - 0.437643i) q^{38} +(-4.72544 + 3.43323i) q^{39} +(2.78428 + 8.56912i) q^{40} +(3.36508 - 10.3567i) q^{41} +(-1.61803 - 1.17557i) q^{42} +6.63325 q^{43} +3.37228 q^{45} +(1.28195 + 0.931389i) q^{46} +(-3.93829 + 12.1208i) q^{47} +(0.193976 + 0.596996i) q^{48} +(0.507835 - 0.368964i) q^{49} +(4.08446 - 2.96754i) q^{50} +(-0.825636 - 2.54105i) q^{51} +(-2.47691 + 7.62314i) q^{52} +(3.33060 + 2.41982i) q^{53} -0.792287 q^{54} -6.74456 q^{56} +(-0.760285 - 0.552379i) q^{57} +(-0.193976 + 0.596996i) q^{58} +(-1.85410 - 5.70634i) q^{59} +(3.74390 - 2.72010i) q^{60} +(4.84475 - 3.51992i) q^{61} +(-0.398515 - 1.22650i) q^{62} +(-0.780063 + 2.40079i) q^{63} +(-2.72823 - 1.98218i) q^{64} +19.6974 q^{65} -1.11684 q^{67} +(-2.96625 - 2.15510i) q^{68} +(0.618034 - 1.90211i) q^{69} +(2.08418 + 6.41446i) q^{70} +(8.69253 - 6.31550i) q^{71} +(-2.16154 + 1.57045i) q^{72} +(-2.82985 - 8.70938i) q^{73} +(-1.22415 + 3.76755i) q^{74} +(-5.15528 - 3.74553i) q^{75} -1.28962 q^{76} -4.62772 q^{78} +(3.32418 + 2.41516i) q^{79} +(0.654141 - 2.01324i) q^{80} +(0.309017 + 0.951057i) q^{81} +(6.97997 - 5.07125i) q^{82} +(-1.52057 + 1.10476i) q^{83} +(1.07047 + 3.29456i) q^{84} +(-2.78428 + 8.56912i) q^{85} +(4.25174 + 3.08907i) q^{86} +0.792287 q^{87} -0.627719 q^{89} +(2.16154 + 1.57045i) q^{90} +(-4.55632 + 14.0229i) q^{91} +(-0.848116 - 2.61023i) q^{92} +(-1.31685 + 0.956749i) q^{93} +(-8.16893 + 5.93507i) q^{94} +(0.979321 + 3.01404i) q^{95} +(-1.80496 + 5.55509i) q^{96} +(-8.48588 - 6.16535i) q^{97} +0.497333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	16 * q - 4 * q^3 - 6 * q^4 - 2 * q^5 - 4 * q^9 + 24 * q^12 - 8 * q^14 - 2 * q^15 - 14 * q^16 + 30 * q^20 + 32 * q^23 - 14 * q^25 + 30 * q^26 - 4 * q^27 - 18 * q^31 - 104 * q^34 - 6 * q^36 - 20 * q^37 - 20 * q^38 - 8 * q^42 + 8 * q^45 + 28 * q^47 - 14 * q^48 + 14 * q^49 - 18 * q^53 - 16 * q^56 + 14 * q^58 + 24 * q^59 + 30 * q^60 - 2 * q^64 + 120 * q^67 - 8 * q^69 - 4 * q^70 + 20 * q^71 - 14 * q^75 - 120 * q^78 + 26 * q^80 - 4 * q^81 + 46 * q^82 + 44 * q^86 - 56 * q^89 + 36 * q^91 - 12 * q^92 - 18 * q^93 + 4 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)

Coefficient data

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

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

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.640974	+	0.465695i	0.453237	+	0.329296i	0.790872	−	0.611981i	\(-0.209627\pi\)
							−0.337636	+	0.941277i	\(0.609627\pi\)
\(3\)	0.309017	−	0.951057i	0.178411	−	0.549093i
							\(5\)	−2.72823	+	1.98218i	−1.22010	+	0.886457i	−0.996109	−	0.0881339i	\(-0.971910\pi\)
−0.223994	+	0.974590i	\(0.571910\pi\)
\(7\)	−0.780063	−	2.40079i	−0.294836	−	0.907413i	−0.983276	−	0.182119i	\(-0.941704\pi\)
							0.688440	−	0.725293i	\(-0.258296\pi\)
\(11\)		0			0
							\(13\)	−4.72544	−	3.43323i	−1.31060	−	0.952207i	−0.999999	−	0.00166711i	\(-0.999469\pi\)
−0.310602	−	0.950540i	\(-0.600531\pi\)
\(17\)	2.16154	−	1.57045i	0.524251	−	0.380891i	−0.293952	−	0.955820i	\(-0.594970\pi\)
							0.818203	+	0.574929i	\(0.194970\pi\)
\(19\)	0.290403	−	0.893769i	0.0666230	−	0.205045i	−0.912203	−	0.409739i	\(-0.865620\pi\)
							0.978826	+	0.204694i	\(0.0656199\pi\)
\(23\)	2.00000			0.417029			0.208514	−	0.978019i	\(-0.433137\pi\)
							0.208514	+	0.978019i	\(0.433137\pi\)
\(29\)	0.244830	+	0.753510i	0.0454638	+	0.139923i	0.971212	−	0.238218i	\(-0.0765633\pi\)
							−0.925748	+	0.378141i	\(0.876563\pi\)
\(31\)	−1.31685	−	0.956749i	−0.236514	−	0.171837i	0.463215	−	0.886246i	\(-0.346696\pi\)
							−0.699729	+	0.714409i	\(0.746696\pi\)
\(37\)	1.54508	+	4.75528i	0.254010	+	0.781764i	0.994023	+	0.109171i	\(0.0348195\pi\)
							−0.740013	+	0.672593i	\(0.765181\pi\)
\(41\)	3.36508	−	10.3567i	0.525538	−	1.61744i	−0.237712	−	0.971336i	\(-0.576398\pi\)
							0.763250	−	0.646103i	\(-0.223602\pi\)
\(43\)	6.63325			1.01156			0.505781	−	0.862662i	\(-0.331205\pi\)
							0.505781	+	0.862662i	\(0.331205\pi\)
\(47\)	−3.93829	+	12.1208i	−0.574458	+	1.76800i	0.0635590	+	0.997978i	\(0.479755\pi\)
							−0.638017	+	0.770022i	\(0.720245\pi\)
\(53\)	3.33060	+	2.41982i	0.457493	+	0.332388i	0.792547	−	0.609811i	\(-0.208755\pi\)
							−0.335054	+	0.942199i	\(0.608755\pi\)
\(59\)	−1.85410	−	5.70634i	−0.241384	−	0.742902i	−0.996210	−	0.0869778i	\(-0.972279\pi\)
							0.754827	−	0.655924i	\(-0.227721\pi\)
\(61\)	4.84475	−	3.51992i	0.620307	−	0.450679i	−0.232722	−	0.972543i	\(-0.574763\pi\)
							0.853029	+	0.521864i	\(0.174763\pi\)
\(67\)	−1.11684			−0.136444			−0.0682221	−	0.997670i	\(-0.521733\pi\)
							−0.0682221	+	0.997670i	\(0.521733\pi\)
\(71\)	8.69253	−	6.31550i	1.03161	−	0.749511i	0.0629829	−	0.998015i	\(-0.479939\pi\)
							0.968631	+	0.248503i	\(0.0799386\pi\)
\(73\)	−2.82985	−	8.70938i	−0.331209	−	1.01936i	−0.968559	−	0.248782i	\(-0.919970\pi\)
							0.637351	−	0.770574i	\(-0.280030\pi\)
\(79\)	3.32418	+	2.41516i	0.373999	+	0.271726i	0.758867	−	0.651245i	\(-0.225753\pi\)
							−0.384868	+	0.922972i	\(0.625753\pi\)
\(83\)	−1.52057	+	1.10476i	−0.166904	+	0.121263i	−0.668102	−	0.744070i	\(-0.732893\pi\)
							0.501198	+	0.865333i	\(0.332893\pi\)
\(89\)	−0.627719			−0.0665380			−0.0332690	−	0.999446i	\(-0.510592\pi\)
							−0.0332690	+	0.999446i	\(0.510592\pi\)
\(97\)	−8.48588	−	6.16535i	−0.861611	−	0.625997i	0.0667120	−	0.997772i	\(-0.478749\pi\)
							−0.928323	+	0.371776i	\(0.878749\pi\)

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	363.2.e.n.202.3		16
11.2	odd	10	inner	363.2.e.n.148.3		16
11.3	even	5	inner	363.2.e.n.124.3		16
11.4	even	5	inner	363.2.e.n.130.2		16
11.5	even	5		363.2.a.j.1.2	✓	4
11.6	odd	10		363.2.a.j.1.3	yes	4
11.7	odd	10	inner	363.2.e.n.130.3		16
11.8	odd	10	inner	363.2.e.n.124.2		16
11.9	even	5	inner	363.2.e.n.148.2		16
11.10	odd	2	inner	363.2.e.n.202.2		16
33.5	odd	10		1089.2.a.u.1.3		4
33.17	even	10		1089.2.a.u.1.2		4
44.27	odd	10		5808.2.a.ck.1.4		4
44.39	even	10		5808.2.a.ck.1.3		4
55.39	odd	10		9075.2.a.cv.1.2		4
55.49	even	10		9075.2.a.cv.1.3		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.j.1.2	✓	4	11.5	even	5
363.2.a.j.1.3	yes	4	11.6	odd	10
363.2.e.n.124.2		16	11.8	odd	10	inner
363.2.e.n.124.3		16	11.3	even	5	inner
363.2.e.n.130.2		16	11.4	even	5	inner
363.2.e.n.130.3		16	11.7	odd	10	inner
363.2.e.n.148.2		16	11.9	even	5	inner
363.2.e.n.148.3		16	11.2	odd	10	inner
363.2.e.n.202.2		16	11.10	odd	2	inner
363.2.e.n.202.3		16	1.1	even	1	trivial
1089.2.a.u.1.2		4	33.17	even	10
1089.2.a.u.1.3		4	33.5	odd	10
5808.2.a.ck.1.3		4	44.39	even	10
5808.2.a.ck.1.4		4	44.27	odd	10
9075.2.a.cv.1.2		4	55.39	odd	10
9075.2.a.cv.1.3		4	55.49	even	10