Embedded newform 363.3.h.n.269.2

• Label: 363.3.h.n.269.2
• Level: $363$
• Weight: $3$
• Character: 363.269
• 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			269.2
Root			\(0.974642 + 1.34148i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.269
Dual form			363.3.h.n.251.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.57700 + 0.512399i) q^{2} +(-0.753215 - 2.90391i) q^{3} +(-1.01168 + 0.735029i) q^{4} +(-0.664316 - 0.215849i) q^{5} +(2.67578 + 4.19352i) q^{6} +(2.11081 - 1.53360i) q^{7} +(5.11736 - 7.04345i) q^{8} +(-7.86534 + 4.37453i) q^{9} +1.15823 q^{10} +(2.89647 + 2.38419i) q^{12} +(-5.28943 - 16.2792i) q^{13} +(-2.54295 + 3.50007i) q^{14} +(-0.126434 + 2.09169i) q^{15} +(-2.91533 + 8.97246i) q^{16} +(15.3239 + 4.97905i) q^{17} +(10.1622 - 10.9288i) q^{18} +(-12.8847 - 9.36127i) q^{19} +(0.830732 - 0.269921i) q^{20} +(-6.04332 - 4.97448i) q^{21} +23.1295i q^{23} +(-24.3080 - 9.55511i) q^{24} +(-19.8307 - 14.4078i) q^{25} +(16.6829 + 22.9620i) q^{26} +(18.6275 + 19.5452i) q^{27} +(-1.00823 + 3.10302i) q^{28} +(-3.15401 - 4.34112i) q^{29} +(-0.872396 - 3.36339i) q^{30} +(-1.25541 - 3.86376i) q^{31} +19.1813i q^{32} -26.7172 q^{34} +(-1.73327 + 0.563175i) q^{35} +(4.74180 - 10.2069i) q^{36} +(-51.3978 + 37.3427i) q^{37} +(25.1159 + 8.16065i) q^{38} +(-43.2891 + 27.6217i) q^{39} +(-4.91987 + 3.57450i) q^{40} +(-39.6389 + 54.5583i) q^{41} +(12.0793 + 4.74818i) q^{42} +22.6622 q^{43} +(6.16931 - 1.20834i) q^{45} +(-11.8515 - 36.4752i) q^{46} +(-43.3908 + 59.7223i) q^{47} +(28.2511 + 1.70765i) q^{48} +(-13.0382 + 40.1275i) q^{49} +(38.6557 + 12.5600i) q^{50} +(2.91647 - 48.2496i) q^{51} +(17.3169 + 12.5815i) q^{52} +(40.7539 - 13.2417i) q^{53} +(-39.3906 - 21.2782i) q^{54} -22.7154i q^{56} +(-17.4793 + 44.4670i) q^{57} +(7.19827 + 5.22985i) q^{58} +(-16.8918 - 23.2496i) q^{59} +(-1.40955 - 2.20906i) q^{60} +(14.3079 - 44.0351i) q^{61} +(3.95957 + 5.44989i) q^{62} +(-9.89350 + 21.2961i) q^{63} +(-21.4898 - 66.1388i) q^{64} +11.9562i q^{65} -77.2821 q^{67} +(-19.1627 + 6.22634i) q^{68} +(67.1657 - 17.4214i) q^{69} +(2.44481 - 1.77626i) q^{70} +(-39.2433 - 12.7509i) q^{71} +(-9.43802 + 77.7851i) q^{72} +(-45.9302 + 33.3703i) q^{73} +(61.9201 - 85.2257i) q^{74} +(-26.9023 + 68.4387i) q^{75} +19.9160 q^{76} +(54.1137 - 65.7409i) q^{78} +(15.7368 + 48.4329i) q^{79} +(3.87340 - 5.33128i) q^{80} +(42.7270 - 68.8143i) q^{81} +(34.5551 - 106.350i) q^{82} +(-56.3118 - 18.2968i) q^{83} +(9.77030 + 0.590571i) q^{84} +(-9.10522 - 6.61533i) q^{85} +(-35.7384 + 11.6121i) q^{86} +(-10.2306 + 12.4287i) q^{87} -38.1909i q^{89} +(-9.10987 + 5.06671i) q^{90} +(-36.1307 - 26.2505i) q^{91} +(-17.0008 - 23.3996i) q^{92} +(-10.2744 + 6.55583i) q^{93} +(37.8258 - 116.416i) q^{94} +(6.53888 + 9.00000i) q^{95} +(55.7007 - 14.4476i) 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 \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

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

Coefficient data

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

(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	\(-0.736057\pi\)
							−0.113036	+	0.993591i	\(0.536057\pi\)
\(3\)	−0.753215	−	2.90391i	−0.251072	−	0.967969i
							\(5\)	−0.664316	−	0.215849i	−0.132863	−	0.0431699i	0.241831	−	0.970318i	\(-0.422252\pi\)
−0.374694	+	0.927149i	\(0.622252\pi\)
\(7\)	2.11081	−	1.53360i	0.301545	−	0.219085i	−0.426715	−	0.904386i	\(-0.640329\pi\)
							0.728260	+	0.685301i	\(0.240329\pi\)
\(11\)		0			0
							\(13\)	−5.28943	−	16.2792i	−0.406879	−	1.25224i	−0.919316	−	0.393520i	\(-0.871257\pi\)
0.512437	−	0.858725i	\(-0.328743\pi\)
\(17\)	15.3239	+	4.97905i	0.901408	+	0.292885i	0.722818	−	0.691038i	\(-0.242846\pi\)
							0.178590	+	0.983924i	\(0.442846\pi\)
\(19\)	−12.8847	−	9.36127i	−0.678141	−	0.492698i	0.194599	−	0.980883i	\(-0.437659\pi\)
							−0.872741	+	0.488184i	\(0.837659\pi\)
\(23\)			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	\(-0.229491\pi\)
							−0.859927	+	0.510418i	\(0.829491\pi\)
\(31\)	−1.25541	−	3.86376i	−0.0404971	−	0.124637i	0.928764	−	0.370672i	\(-0.120872\pi\)
							−0.969261	+	0.246034i	\(0.920872\pi\)
\(37\)	−51.3978	+	37.3427i	−1.38913	+	1.00926i	−0.393169	+	0.919466i	\(0.628621\pi\)
							−0.995960	+	0.0897955i	\(0.971379\pi\)
\(41\)	−39.6389	+	54.5583i	−0.966803	+	1.33069i	−0.0231574	+	0.999732i	\(0.507372\pi\)
							−0.943645	+	0.330958i	\(0.892628\pi\)
\(43\)	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	0.0392417	+	0.999230i	\(0.487506\pi\)
							−0.962450	+	0.271458i	\(0.912494\pi\)

(See \(a_n\) instead)

Twists

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

    

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