Embedded newform 363.3.h.n.323.2

• Label: 363.3.h.n.323.2
• Level: $363$
• Weight: $3$
• Character: 363.323
• Analytic conductor: $9.891$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.h (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89103359628\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 12x^{14} + 180x^{12} - 2562x^{10} + 25179x^{8} - 96398x^{6} + 239275x^{4} - 536393x^{2} + 1771561 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			323.2
Root			\(-1.90610 + 0.619331i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.323
Dual form			363.3.h.n.245.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17804 + 1.62143i) q^{2} +(-0.157363 - 2.99587i) q^{3} +(-0.00519352 - 0.0159840i) q^{4} +(3.22364 + 4.43696i) q^{5} +(5.04297 + 3.27409i) q^{6} +(-1.72766 - 5.31721i) q^{7} +(-7.59238 - 2.46692i) q^{8} +(-8.95047 + 0.942880i) q^{9} -10.9918 q^{10} +(-0.0470687 + 0.0180744i) q^{12} +(-7.86105 - 5.71139i) q^{13} +(10.6567 + 3.46258i) q^{14} +(12.7853 - 10.3558i) q^{15} +(12.9984 - 9.44391i) q^{16} +(-10.5047 - 14.4584i) q^{17} +(9.01518 - 15.6233i) q^{18} +(-5.76592 + 17.7457i) q^{19} +(0.0541783 - 0.0745701i) q^{20} +(-15.6578 + 6.01259i) q^{21} +12.3649i q^{23} +(-6.19579 + 23.1340i) q^{24} +(-1.56932 + 4.82989i) q^{25} +(18.5212 - 6.01791i) q^{26} +(4.23322 + 26.6661i) q^{27} +(-0.0760176 + 0.0552300i) q^{28} +(-2.35608 + 0.765535i) q^{29} +(1.72970 + 32.9300i) q^{30} +(-39.8286 - 28.9372i) q^{31} +0.268903i q^{32} +35.8182 q^{34} +(18.0228 - 24.8063i) q^{35} +(0.0615554 + 0.138168i) q^{36} +(-12.1539 - 37.4057i) q^{37} +(-21.9809 - 30.2541i) q^{38} +(-15.8735 + 24.4494i) q^{39} +(-13.5295 - 41.6395i) q^{40} +(-53.7571 - 17.4667i) q^{41} +(8.69646 - 32.4711i) q^{42} +43.9060 q^{43} +(-33.0366 - 36.6734i) q^{45} +(-20.0488 - 14.5663i) q^{46} +(-54.9558 - 17.8562i) q^{47} +(-30.3382 - 37.4555i) q^{48} +(14.3540 - 10.4288i) q^{49} +(-5.98260 - 8.23434i) q^{50} +(-41.6625 + 33.7458i) q^{51} +(-0.0504643 + 0.155313i) q^{52} +(25.3428 - 34.8814i) q^{53} +(-48.2241 - 24.5498i) q^{54} +44.6323i q^{56} +(54.0710 + 14.4814i) q^{57} +(1.53428 - 4.72204i) q^{58} +(-85.6247 + 27.8212i) q^{59} +(-0.231928 - 0.150577i) q^{60} +(-24.9740 + 18.1446i) q^{61} +(93.8392 - 30.4902i) q^{62} +(20.4769 + 45.9625i) q^{63} +(51.5577 + 37.4589i) q^{64} -53.2906i q^{65} +34.0775 q^{67} +(-0.176547 + 0.242996i) q^{68} +(37.0435 - 1.94578i) q^{69} +(18.9901 + 58.4456i) q^{70} +(-22.0985 - 30.4160i) q^{71} +(70.2814 + 14.9214i) q^{72} +(3.74717 + 11.5326i) q^{73} +(74.9685 + 24.3587i) q^{74} +(14.7167 + 3.94145i) q^{75} +0.313592 q^{76} +(-20.9434 - 54.5402i) q^{78} +(-51.0852 - 37.1155i) q^{79} +(83.8045 + 27.2297i) q^{80} +(79.2220 - 16.8784i) q^{81} +(91.6490 - 66.5869i) q^{82} +(5.70238 + 7.84865i) q^{83} +(0.177424 + 0.219048i) q^{84} +(30.2882 - 93.2174i) q^{85} +(-51.7229 + 71.1904i) q^{86} +(2.66420 + 6.93803i) q^{87} +34.1289i q^{89} +(98.3817 - 10.3639i) q^{90} +(-16.7874 + 51.6662i) q^{91} +(0.197640 - 0.0642171i) q^{92} +(-80.4244 + 123.875i) q^{93} +(93.6927 - 68.0717i) q^{94} +(-97.3240 + 31.6225i) q^{95} +(0.805600 - 0.0423155i) q^{96} +(30.5992 + 22.2316i) q^{97} +35.5595i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	16 * q + 5 * q^3 + 18 * q^4 - 32 * q^6 + 34 * q^7 + 17 * q^9 + 12 * q^10 + 106 * q^12 + 2 * q^13 - 28 * q^15 + 102 * q^16 - 42 * q^18 - 66 * q^19 + 12 * q^21 + 74 * q^24 - 176 * q^25 - 55 * q^27 + 146 * q^28 - 110 * q^30 - 126 * q^31 - 132 * q^34 + 226 * q^36 - 230 * q^37 - 136 * q^39 - 226 * q^40 - 72 * q^42 + 156 * q^43 - 72 * q^45 - 308 * q^46 + 255 * q^48 + 170 * q^49 + 169 * q^51 + 224 * q^52 - 1046 * q^54 + 259 * q^57 + 184 * q^58 - 316 * q^60 - 104 * q^61 + 108 * q^63 - 184 * q^64 + 368 * q^67 - 22 * q^69 - 52 * q^70 - 73 * q^72 - 354 * q^73 + 54 * q^75 + 900 * q^76 - 492 * q^78 - 566 * q^79 + 377 * q^81 + 200 * q^82 + 720 * q^84 + 162 * q^85 - 132 * q^87 + 774 * q^90 - 226 * q^91 - 370 * q^93 + 530 * q^94 + 67 * q^96 + 252 * 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 \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17804	+	1.62143i	−0.589019	+	0.810715i	−0.994648	−	0.103323i	\(-0.967052\pi\)
							0.405629	+	0.914038i	\(0.367052\pi\)
\(3\)	−0.157363	−	2.99587i	−0.0524544	−	0.998623i
							\(5\)	3.22364	+	4.43696i	0.644728	+	0.887392i	0.998857	−	0.0478050i	\(-0.0152226\pi\)
−0.354129	+	0.935197i	\(0.615223\pi\)
\(7\)	−1.72766	−	5.31721i	−0.246809	−	0.759601i	−0.995334	−	0.0964935i	\(-0.969237\pi\)
							0.748524	−	0.663107i	\(-0.230763\pi\)
\(11\)		0			0
							\(13\)	−7.86105	−	5.71139i	−0.604696	−	0.439338i	0.242846	−	0.970065i	\(-0.421919\pi\)
−0.847543	+	0.530727i	\(0.821919\pi\)
\(17\)	−10.5047	−	14.4584i	−0.617921	−	0.850495i	0.379279	−	0.925282i	\(-0.376172\pi\)
							−0.997199	+	0.0747875i	\(0.976172\pi\)
\(19\)	−5.76592	+	17.7457i	−0.303469	+	0.933982i	0.676775	+	0.736190i	\(0.263377\pi\)
							−0.980244	+	0.197792i	\(0.936623\pi\)
\(23\)			12.3649i			0.537603i	0.963196	+	0.268801i	\(0.0866275\pi\)
							−0.963196	+	0.268801i	\(0.913372\pi\)
\(29\)	−2.35608	+	0.765535i	−0.0812440	+	0.0263978i	−0.349357	−	0.936990i	\(-0.613600\pi\)
							0.268113	+	0.963387i	\(0.413600\pi\)
\(31\)	−39.8286	−	28.9372i	−1.28479	−	0.933457i	−0.285107	−	0.958496i	\(-0.592029\pi\)
							−0.999686	+	0.0250387i	\(0.992029\pi\)
\(37\)	−12.1539	−	37.4057i	−0.328483	−	1.01097i	−0.969844	−	0.243727i	\(-0.921630\pi\)
							0.641361	−	0.767239i	\(-0.278370\pi\)
\(41\)	−53.7571	−	17.4667i	−1.31115	−	0.426018i	−0.431703	−	0.902016i	\(-0.642087\pi\)
							−0.879446	+	0.475998i	\(0.842087\pi\)
\(43\)	43.9060			1.02107			0.510534	−	0.859857i	\(-0.329448\pi\)
							0.510534	+	0.859857i	\(0.329448\pi\)
\(47\)	−54.9558	−	17.8562i	−1.16927	−	0.379920i	−0.340902	−	0.940099i	\(-0.610732\pi\)
							−0.828372	+	0.560179i	\(0.810732\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.h.n.323.2		16
3.2	odd	2	inner	363.3.h.n.323.3		16
11.2	odd	10		33.3.h.b.5.2	✓	16
11.3	even	5	inner	363.3.h.n.245.3		16
11.4	even	5		363.3.h.j.251.2		16
11.5	even	5		363.3.b.l.122.3		8
11.6	odd	10		363.3.b.m.122.6		8
11.7	odd	10		33.3.h.b.20.3	yes	16
11.8	odd	10		363.3.h.o.245.2		16
11.9	even	5		363.3.h.j.269.3		16
11.10	odd	2		363.3.h.o.323.3		16
33.2	even	10		33.3.h.b.5.3	yes	16
33.5	odd	10		363.3.b.l.122.6		8
33.8	even	10		363.3.h.o.245.3		16
33.14	odd	10	inner	363.3.h.n.245.2		16
33.17	even	10		363.3.b.m.122.3		8
33.20	odd	10		363.3.h.j.269.2		16
33.26	odd	10		363.3.h.j.251.3		16
33.29	even	10		33.3.h.b.20.2	yes	16
33.32	even	2		363.3.h.o.323.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.h.b.5.2	✓	16	11.2	odd	10
33.3.h.b.5.3	yes	16	33.2	even	10
33.3.h.b.20.2	yes	16	33.29	even	10
33.3.h.b.20.3	yes	16	11.7	odd	10
363.3.b.l.122.3		8	11.5	even	5
363.3.b.l.122.6		8	33.5	odd	10
363.3.b.m.122.3		8	33.17	even	10
363.3.b.m.122.6		8	11.6	odd	10
363.3.h.j.251.2		16	11.4	even	5
363.3.h.j.251.3		16	33.26	odd	10
363.3.h.j.269.2		16	33.20	odd	10
363.3.h.j.269.3		16	11.9	even	5
363.3.h.n.245.2		16	33.14	odd	10	inner
363.3.h.n.245.3		16	11.3	even	5	inner
363.3.h.n.323.2		16	1.1	even	1	trivial
363.3.h.n.323.3		16	3.2	odd	2	inner
363.3.h.o.245.2		16	11.8	odd	10
363.3.h.o.245.3		16	33.8	even	10
363.3.h.o.323.2		16	33.32	even	2
363.3.h.o.323.3		16	11.10	odd	2