Embedded newform 363.3.h.n.251.3

• Label: 363.3.h.n.251.3
• Level: $363$
• Weight: $3$
• Character: 363.251
• 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			251.3
Root			\(-0.974642 + 1.34148i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.251
Dual form			363.3.h.n.269.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.57700 + 0.512399i) q^{2} +(-2.99453 - 0.181006i) q^{3} +(-1.01168 - 0.735029i) q^{4} +(0.664316 - 0.215849i) q^{5} +(-4.62964 - 1.81985i) q^{6} +(2.11081 + 1.53360i) q^{7} +(-5.11736 - 7.04345i) q^{8} +(8.93447 + 1.08406i) q^{9} +1.15823 q^{10} +(2.89647 + 2.38419i) q^{12} +(-5.28943 + 16.2792i) q^{13} +(2.54295 + 3.50007i) q^{14} +(-2.02839 + 0.526123i) q^{15} +(-2.91533 - 8.97246i) q^{16} +(-15.3239 + 4.97905i) q^{17} +(13.5342 + 6.28759i) q^{18} +(-12.8847 + 9.36127i) q^{19} +(-0.830732 - 0.269921i) q^{20} +(-6.04332 - 4.97448i) q^{21} +23.1295i q^{23} +(14.0492 + 22.0181i) q^{24} +(-19.8307 + 14.4078i) q^{25} +(-16.6829 + 22.9620i) q^{26} +(-26.5584 - 4.86345i) q^{27} +(-1.00823 - 3.10302i) q^{28} +(3.15401 - 4.34112i) q^{29} +(-3.46836 - 0.209647i) q^{30} +(-1.25541 + 3.86376i) q^{31} +19.1813i q^{32} -26.7172 q^{34} +(1.73327 + 0.563175i) q^{35} +(-8.24202 - 7.66382i) q^{36} +(-51.3978 - 37.3427i) q^{37} +(-25.1159 + 8.16065i) q^{38} +(18.7860 - 47.7911i) q^{39} +(-4.91987 - 3.57450i) q^{40} +(39.6389 + 54.5583i) q^{41} +(-6.98141 - 10.9414i) q^{42} +22.6622 q^{43} +(6.16931 - 1.20834i) q^{45} +(-11.8515 + 36.4752i) q^{46} +(43.3908 + 59.7223i) q^{47} +(7.10598 + 27.3960i) q^{48} +(-13.0382 - 40.1275i) q^{49} +(-38.6557 + 12.5600i) q^{50} +(46.7893 - 12.1362i) q^{51} +(17.3169 - 12.5815i) q^{52} +(-40.7539 - 13.2417i) q^{53} +(-39.3906 - 21.2782i) q^{54} -22.7154i q^{56} +(40.2781 - 25.7004i) q^{57} +(7.19827 - 5.22985i) q^{58} +(16.8918 - 23.2496i) q^{59} +(2.43880 + 0.958656i) q^{60} +(14.3079 + 44.0351i) q^{61} +(-3.95957 + 5.44989i) q^{62} +(17.1965 + 15.9901i) q^{63} +(-21.4898 + 66.1388i) q^{64} +11.9562i q^{65} -77.2821 q^{67} +(19.1627 + 6.22634i) q^{68} +(4.18658 - 69.2619i) q^{69} +(2.44481 + 1.77626i) q^{70} +(39.2433 - 12.7509i) q^{71} +(-38.0854 - 68.4770i) q^{72} +(-45.9302 - 33.3703i) q^{73} +(-61.9201 - 85.2257i) q^{74} +(61.9916 - 39.5553i) q^{75} +19.9160 q^{76} +(54.1137 - 65.7409i) q^{78} +(15.7368 - 48.4329i) q^{79} +(-3.87340 - 5.33128i) q^{80} +(78.6496 + 19.3710i) q^{81} +(34.5551 + 106.350i) q^{82} +(56.3118 - 18.2968i) q^{83} +(2.45752 + 9.47460i) q^{84} +(-9.10522 + 6.61533i) q^{85} +(35.7384 + 11.6121i) q^{86} +(-10.2306 + 12.4287i) q^{87} -38.1909i q^{89} +(10.3482 + 1.25559i) q^{90} +(-36.1307 + 26.2505i) q^{91} +(17.0008 - 23.3996i) q^{92} +(4.45874 - 11.3429i) q^{93} +(37.8258 + 116.416i) q^{94} +(-6.53888 + 9.00000i) q^{95} +(3.47194 - 57.4390i) q^{96} +(5.03463 - 15.4950i) q^{97} -69.9620i 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 \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.57700	+	0.512399i	0.788502	+	0.256200i	0.675466	−	0.737391i	\(-0.263943\pi\)
							0.113036	+	0.993591i	\(0.463943\pi\)
\(3\)	−2.99453	−	0.181006i	−0.998178	−	0.0603355i
							\(5\)	0.664316	−	0.215849i	0.132863	−	0.0431699i	−0.241831	−	0.970318i	\(-0.577748\pi\)
0.374694	+	0.927149i	\(0.377748\pi\)
\(7\)	2.11081	+	1.53360i	0.301545	+	0.219085i	0.728260	−	0.685301i	\(-0.240329\pi\)
							−0.426715	+	0.904386i	\(0.640329\pi\)
\(11\)		0			0
							\(13\)	−5.28943	+	16.2792i	−0.406879	+	1.25224i	0.512437	+	0.858725i	\(0.328743\pi\)
−0.919316	+	0.393520i	\(0.871257\pi\)
\(17\)	−15.3239	+	4.97905i	−0.901408	+	0.292885i	−0.722818	−	0.691038i	\(-0.757154\pi\)
							−0.178590	+	0.983924i	\(0.557154\pi\)
\(19\)	−12.8847	+	9.36127i	−0.678141	+	0.492698i	−0.872741	−	0.488184i	\(-0.837659\pi\)
							0.194599	+	0.980883i	\(0.437659\pi\)
\(23\)			23.1295i			1.00563i	0.864395	+	0.502814i	\(0.167702\pi\)
							−0.864395	+	0.502814i	\(0.832298\pi\)
\(29\)	3.15401	−	4.34112i	0.108759	−	0.149694i	−0.751168	−	0.660111i	\(-0.770509\pi\)
							0.859927	+	0.510418i	\(0.170509\pi\)
\(31\)	−1.25541	+	3.86376i	−0.0404971	+	0.124637i	−0.969261	−	0.246034i	\(-0.920872\pi\)
							0.928764	+	0.370672i	\(0.120872\pi\)
\(37\)	−51.3978	−	37.3427i	−1.38913	−	1.00926i	−0.995960	−	0.0897955i	\(-0.971379\pi\)
							−0.393169	−	0.919466i	\(-0.628621\pi\)
\(41\)	39.6389	+	54.5583i	0.966803	+	1.33069i	0.943645	+	0.330958i	\(0.107372\pi\)
							0.0231574	+	0.999732i	\(0.492628\pi\)
\(43\)	22.6622			0.527028			0.263514	−	0.964656i	\(-0.415118\pi\)
							0.263514	+	0.964656i	\(0.415118\pi\)
\(47\)	43.3908	+	59.7223i	0.923209	+	1.27069i	0.962450	+	0.271458i	\(0.0875058\pi\)
							−0.0392417	+	0.999230i	\(0.512494\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.h.n.251.3		16
3.2	odd	2	inner	363.3.h.n.251.2		16
11.2	odd	10		33.3.h.b.14.3	yes	16
11.3	even	5		363.3.h.j.323.3		16
11.4	even	5		363.3.b.l.122.5		8
11.5	even	5	inner	363.3.h.n.269.2		16
11.6	odd	10		363.3.h.o.269.3		16
11.7	odd	10		363.3.b.m.122.4		8
11.8	odd	10		33.3.h.b.26.2	yes	16
11.9	even	5		363.3.h.j.245.2		16
11.10	odd	2		363.3.h.o.251.2		16
33.2	even	10		33.3.h.b.14.2	✓	16
33.5	odd	10	inner	363.3.h.n.269.3		16
33.8	even	10		33.3.h.b.26.3	yes	16
33.14	odd	10		363.3.h.j.323.2		16
33.17	even	10		363.3.h.o.269.2		16
33.20	odd	10		363.3.h.j.245.3		16
33.26	odd	10		363.3.b.l.122.4		8
33.29	even	10		363.3.b.m.122.5		8
33.32	even	2		363.3.h.o.251.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.h.b.14.2	✓	16	33.2	even	10
33.3.h.b.14.3	yes	16	11.2	odd	10
33.3.h.b.26.2	yes	16	11.8	odd	10
33.3.h.b.26.3	yes	16	33.8	even	10
363.3.b.l.122.4		8	33.26	odd	10
363.3.b.l.122.5		8	11.4	even	5
363.3.b.m.122.4		8	11.7	odd	10
363.3.b.m.122.5		8	33.29	even	10
363.3.h.j.245.2		16	11.9	even	5
363.3.h.j.245.3		16	33.20	odd	10
363.3.h.j.323.2		16	33.14	odd	10
363.3.h.j.323.3		16	11.3	even	5
363.3.h.n.251.2		16	3.2	odd	2	inner
363.3.h.n.251.3		16	1.1	even	1	trivial
363.3.h.n.269.2		16	11.5	even	5	inner
363.3.h.n.269.3		16	33.5	odd	10	inner
363.3.h.o.251.2		16	11.10	odd	2
363.3.h.o.251.3		16	33.32	even	2
363.3.h.o.269.2		16	33.17	even	10
363.3.h.o.269.3		16	11.6	odd	10