Embedded newform 363.2.e.n.148.2

• Label: 363.2.e.n.148.2
• Level: $363$
• Weight: $2$
• Character: 363.148
• 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			148.2
Root			\(-0.244830 + 0.753510i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.148
Dual form			363.2.e.n.130.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.244830 - 0.753510i) q^{2} +(-0.809017 - 0.587785i) q^{3} +(1.11020 - 0.806607i) q^{4} +(1.04209 - 3.20723i) q^{5} +(-0.244830 + 0.753510i) q^{6} +(2.04223 - 1.48377i) q^{7} +(-2.16154 - 1.57045i) q^{8} +(0.309017 + 0.951057i) q^{9} -2.67181 q^{10} -1.37228 q^{12} +(1.80496 + 5.55509i) q^{13} +(-1.61803 - 1.17557i) q^{14} +(-2.72823 + 1.98218i) q^{15} +(0.193976 - 0.596996i) q^{16} +(-0.825636 + 2.54105i) q^{17} +(0.640974 - 0.465695i) q^{18} +(-0.760285 - 0.552379i) q^{19} +(-1.43004 - 4.40122i) q^{20} -2.52434 q^{21} +2.00000 q^{23} +(0.825636 + 2.54105i) q^{24} +(-5.15528 - 3.74553i) q^{25} +(3.74390 - 2.72010i) q^{26} +(0.309017 - 0.951057i) q^{27} +(1.07047 - 3.29456i) q^{28} +(-0.640974 + 0.465695i) q^{29} +(2.16154 + 1.57045i) q^{30} +(0.502993 + 1.54805i) q^{31} -5.84096 q^{32} +2.11684 q^{34} +(-2.63059 - 8.09613i) q^{35} +(1.11020 + 0.806607i) q^{36} +(-4.04508 + 2.93893i) q^{37} +(-0.230083 + 0.708121i) q^{38} +(1.80496 - 5.55509i) q^{39} +(-7.28933 + 5.29601i) q^{40} +(-8.80990 - 6.40077i) q^{41} +(0.618034 + 1.90211i) q^{42} +6.63325 q^{43} +3.37228 q^{45} +(-0.489660 - 1.50702i) q^{46} +(10.3106 + 7.49107i) q^{47} +(-0.507835 + 0.368964i) q^{48} +(-0.193976 + 0.596996i) q^{49} +(-1.56013 + 4.80158i) q^{50} +(2.16154 - 1.57045i) q^{51} +(6.48463 + 4.71136i) q^{52} +(-1.27217 - 3.91535i) q^{53} -0.792287 q^{54} -6.74456 q^{56} +(0.290403 + 0.893769i) q^{57} +(0.507835 + 0.368964i) q^{58} +(4.85410 - 3.52671i) q^{59} +(-1.43004 + 4.40122i) q^{60} +(-1.85053 + 5.69534i) q^{61} +(1.04332 - 0.758020i) q^{62} +(2.04223 + 1.48377i) q^{63} +(1.04209 + 3.20723i) q^{64} +19.6974 q^{65} -1.11684 q^{67} +(1.13301 + 3.48703i) q^{68} +(-1.61803 - 1.17557i) q^{69} +(-5.45647 + 3.96435i) q^{70} +(-3.32025 + 10.2187i) q^{71} +(0.825636 - 2.54105i) q^{72} +(7.40864 - 5.38269i) q^{73} +(3.20487 + 2.32847i) q^{74} +(1.96914 + 6.06040i) q^{75} -1.28962 q^{76} -4.62772 q^{78} +(-1.26972 - 3.90781i) q^{79} +(-1.71256 - 1.24425i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(-2.66611 + 8.20545i) q^{82} +(0.580806 - 1.78754i) q^{83} +(-2.80252 + 2.03615i) q^{84} +(7.28933 + 5.29601i) q^{85} +(-1.62402 - 4.99822i) q^{86} +0.792287 q^{87} -0.627719 q^{89} +(-0.825636 - 2.54105i) q^{90} +(11.9286 + 8.66664i) q^{91} +(2.22040 - 1.61321i) q^{92} +(0.502993 - 1.54805i) q^{93} +(3.12025 - 9.60315i) q^{94} +(-2.56389 + 1.86278i) q^{95} +(4.72544 + 3.43323i) q^{96} +(3.24132 + 9.97575i) 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 \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)

Coefficient data

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

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

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.244830	−	0.753510i	−0.173121	−	0.532812i	0.826422	−	0.563052i	\(-0.190373\pi\)
							−0.999543	+	0.0302400i	\(0.990373\pi\)
\(3\)	−0.809017	−	0.587785i	−0.467086	−	0.339358i
							\(5\)	1.04209	−	3.20723i	0.466038	−	1.43432i	−0.391635	−	0.920121i	\(-0.628090\pi\)
0.857673	−	0.514196i	\(-0.171910\pi\)
\(7\)	2.04223	−	1.48377i	0.771891	−	0.560812i	−0.130643	−	0.991429i	\(-0.541704\pi\)
							0.902534	+	0.430618i	\(0.141704\pi\)
\(11\)		0			0
							\(13\)	1.80496	+	5.55509i	0.500605	+	1.54070i	0.808036	+	0.589133i	\(0.200531\pi\)
−0.307431	+	0.951570i	\(0.599469\pi\)
\(17\)	−0.825636	+	2.54105i	−0.200246	+	0.616294i	0.799629	+	0.600494i	\(0.205030\pi\)
							−0.999875	+	0.0158000i	\(0.994970\pi\)
\(19\)	−0.760285	−	0.552379i	−0.174421	−	0.126725i	0.497149	−	0.867665i	\(-0.334380\pi\)
							−0.671571	+	0.740940i	\(0.734380\pi\)
\(23\)	2.00000			0.417029			0.208514	−	0.978019i	\(-0.433137\pi\)
							0.208514	+	0.978019i	\(0.433137\pi\)
\(29\)	−0.640974	+	0.465695i	−0.119026	+	0.0864773i	−0.645706	−	0.763586i	\(-0.723437\pi\)
							0.526680	+	0.850064i	\(0.323437\pi\)
\(31\)	0.502993	+	1.54805i	0.0903402	+	0.278038i	0.986011	−	0.166679i	\(-0.0533042\pi\)
							−0.895671	+	0.444717i	\(0.853304\pi\)
\(37\)	−4.04508	+	2.93893i	−0.665008	+	0.483157i	−0.868350	−	0.495951i	\(-0.834819\pi\)
							0.203343	+	0.979108i	\(0.434819\pi\)
\(41\)	−8.80990	−	6.40077i	−1.37588	−	0.999632i	−0.997252	−	0.0740813i	\(-0.976398\pi\)
							−0.378623	−	0.925551i	\(-0.623602\pi\)
\(43\)	6.63325			1.01156			0.505781	−	0.862662i	\(-0.331205\pi\)
							0.505781	+	0.862662i	\(0.331205\pi\)
\(47\)	10.3106	+	7.49107i	1.50395	+	1.09268i	0.968774	+	0.247944i	\(0.0797549\pi\)
							0.535177	+	0.844740i	\(0.320245\pi\)
\(53\)	−1.27217	−	3.91535i	−0.174747	−	0.537815i	0.824875	−	0.565315i	\(-0.191245\pi\)
							−0.999622	+	0.0275001i	\(0.991245\pi\)
\(59\)	4.85410	−	3.52671i	0.631950	−	0.459139i	−0.225125	−	0.974330i	\(-0.572279\pi\)
							0.857075	+	0.515191i	\(0.172279\pi\)
\(61\)	−1.85053	+	5.69534i	−0.236936	+	0.729214i	0.759923	+	0.650013i	\(0.225237\pi\)
							−0.996859	+	0.0792006i	\(0.974763\pi\)
\(67\)	−1.11684			−0.136444			−0.0682221	−	0.997670i	\(-0.521733\pi\)
							−0.0682221	+	0.997670i	\(0.521733\pi\)
\(71\)	−3.32025	+	10.2187i	−0.394041	+	1.21273i	0.535664	+	0.844431i	\(0.320061\pi\)
							−0.929706	+	0.368304i	\(0.879939\pi\)
\(73\)	7.40864	−	5.38269i	0.867116	−	0.629997i	−0.0626954	−	0.998033i	\(-0.519970\pi\)
							0.929812	+	0.368036i	\(0.119970\pi\)
\(79\)	−1.26972	−	3.90781i	−0.142855	−	0.439663i	0.853874	−	0.520480i	\(-0.174247\pi\)
							−0.996729	+	0.0808173i	\(0.974247\pi\)
\(83\)	0.580806	−	1.78754i	0.0637517	−	0.196208i	−0.914107	−	0.405472i	\(-0.867107\pi\)
							0.977859	+	0.209265i	\(0.0671070\pi\)
\(89\)	−0.627719			−0.0665380			−0.0332690	−	0.999446i	\(-0.510592\pi\)
							−0.0332690	+	0.999446i	\(0.510592\pi\)
\(97\)	3.24132	+	9.97575i	0.329106	+	1.01288i	0.969553	+	0.244882i	\(0.0787491\pi\)
							−0.640447	+	0.768002i	\(0.721251\pi\)

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.148.2		16
11.2	odd	10	inner	363.2.e.n.130.3		16
11.3	even	5		363.2.a.j.1.2	✓	4
11.4	even	5	inner	363.2.e.n.124.3		16
11.5	even	5	inner	363.2.e.n.202.3		16
11.6	odd	10	inner	363.2.e.n.202.2		16
11.7	odd	10	inner	363.2.e.n.124.2		16
11.8	odd	10		363.2.a.j.1.3	yes	4
11.9	even	5	inner	363.2.e.n.130.2		16
11.10	odd	2	inner	363.2.e.n.148.3		16
33.8	even	10		1089.2.a.u.1.2		4
33.14	odd	10		1089.2.a.u.1.3		4
44.3	odd	10		5808.2.a.ck.1.4		4
44.19	even	10		5808.2.a.ck.1.3		4
55.14	even	10		9075.2.a.cv.1.3		4
55.19	odd	10		9075.2.a.cv.1.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.j.1.2	✓	4	11.3	even	5
363.2.a.j.1.3	yes	4	11.8	odd	10
363.2.e.n.124.2		16	11.7	odd	10	inner
363.2.e.n.124.3		16	11.4	even	5	inner
363.2.e.n.130.2		16	11.9	even	5	inner
363.2.e.n.130.3		16	11.2	odd	10	inner
363.2.e.n.148.2		16	1.1	even	1	trivial
363.2.e.n.148.3		16	11.10	odd	2	inner
363.2.e.n.202.2		16	11.6	odd	10	inner
363.2.e.n.202.3		16	11.5	even	5	inner
1089.2.a.u.1.2		4	33.8	even	10
1089.2.a.u.1.3		4	33.14	odd	10
5808.2.a.ck.1.3		4	44.19	even	10
5808.2.a.ck.1.4		4	44.3	odd	10
9075.2.a.cv.1.2		4	55.19	odd	10
9075.2.a.cv.1.3		4	55.14	even	10