Embedded newform 819.2.j.b.235.1

• Label: 819.2.j.b.235.1
• Level: $819$
• Weight: $2$
• Character: 819.235
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			235.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	819.235
Dual form			819.2.j.b.352.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.500000 - 2.59808i) q^{7} +3.00000 q^{8} +(-1.50000 + 2.59808i) q^{11} -1.00000 q^{13} +(2.50000 - 0.866025i) q^{14} +(0.500000 + 0.866025i) q^{16} +(3.50000 - 6.06218i) q^{17} +(3.50000 + 6.06218i) q^{19} -3.00000 q^{22} +(-3.00000 - 5.19615i) q^{23} +(2.50000 - 4.33013i) q^{25} +(-0.500000 - 0.866025i) q^{26} +(-2.00000 - 1.73205i) q^{28} +5.00000 q^{29} +(2.50000 - 4.33013i) q^{32} +7.00000 q^{34} +(-4.00000 - 6.92820i) q^{37} +(-3.50000 + 6.06218i) q^{38} +2.00000 q^{43} +(1.50000 + 2.59808i) q^{44} +(3.00000 - 5.19615i) q^{46} +(3.50000 + 6.06218i) q^{47} +(-6.50000 - 2.59808i) q^{49} +5.00000 q^{50} +(-0.500000 + 0.866025i) q^{52} +(-1.50000 + 2.59808i) q^{53} +(1.50000 - 7.79423i) q^{56} +(2.50000 + 4.33013i) q^{58} +(-3.50000 + 6.06218i) q^{59} +(3.50000 + 6.06218i) q^{61} +7.00000 q^{64} +(1.50000 - 2.59808i) q^{67} +(-3.50000 - 6.06218i) q^{68} +5.00000 q^{71} +(-7.00000 + 12.1244i) q^{73} +(4.00000 - 6.92820i) q^{74} +7.00000 q^{76} +(6.00000 + 5.19615i) q^{77} +(3.00000 + 5.19615i) q^{79} +(1.00000 + 1.73205i) q^{86} +(-4.50000 + 7.79423i) q^{88} +(-0.500000 + 2.59808i) q^{91} -6.00000 q^{92} +(-3.50000 + 6.06218i) q^{94} -14.0000 q^{97} +(-1.00000 - 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 + q^4 + q^7 + 6 * q^8 - 3 * q^11 - 2 * q^13 + 5 * q^14 + q^16 + 7 * q^17 + 7 * q^19 - 6 * q^22 - 6 * q^23 + 5 * q^25 - q^26 - 4 * q^28 + 10 * q^29 + 5 * q^32 + 14 * q^34 - 8 * q^37 - 7 * q^38 + 4 * q^43 + 3 * q^44 + 6 * q^46 + 7 * q^47 - 13 * q^49 + 10 * q^50 - q^52 - 3 * q^53 + 3 * q^56 + 5 * q^58 - 7 * q^59 + 7 * q^61 + 14 * q^64 + 3 * q^67 - 7 * q^68 + 10 * q^71 - 14 * q^73 + 8 * q^74 + 14 * q^76 + 12 * q^77 + 6 * q^79 + 2 * q^86 - 9 * q^88 - q^91 - 12 * q^92 - 7 * q^94 - 28 * q^97 - 2 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	+	0.866025i	0.353553	+	0.612372i	0.986869	−	0.161521i	\(-0.0516399\pi\)
							−0.633316	+	0.773893i	\(0.718307\pi\)
\(3\)		0			0
							\(5\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
−0.866025	+	0.500000i	\(0.833333\pi\)
\(7\)	0.500000	−	2.59808i	0.188982	−	0.981981i
							\(11\)	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	\(-0.982718\pi\)
0.546259	+	0.837616i	\(0.316051\pi\)
\(13\)	−1.00000			−0.277350
							\(17\)	3.50000	−	6.06218i	0.848875	−	1.47029i	−0.0333386	−	0.999444i	\(-0.510614\pi\)
0.882213	−	0.470850i	\(-0.156053\pi\)
\(19\)	3.50000	+	6.06218i	0.802955	+	1.39076i	0.917663	+	0.397360i	\(0.130073\pi\)
							−0.114708	+	0.993399i	\(0.536593\pi\)
\(23\)	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	\(-0.951544\pi\)
							0.362892	−	0.931831i	\(-0.381789\pi\)
\(29\)	5.00000			0.928477			0.464238	−	0.885710i	\(-0.346328\pi\)
							0.464238	+	0.885710i	\(0.346328\pi\)
\(31\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(37\)	−4.00000	−	6.92820i	−0.657596	−	1.13899i	−0.981236	−	0.192809i	\(-0.938240\pi\)
							0.323640	−	0.946180i	\(-0.395093\pi\)
\(41\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(43\)	2.00000			0.304997			0.152499	−	0.988304i	\(-0.451268\pi\)
							0.152499	+	0.988304i	\(0.451268\pi\)
\(47\)	3.50000	+	6.06218i	0.510527	+	0.884260i	0.999926	+	0.0121990i	\(0.00388317\pi\)
							−0.489398	+	0.872060i	\(0.662783\pi\)
\(53\)	−1.50000	+	2.59808i	−0.206041	+	0.356873i	−0.950464	−	0.310835i	\(-0.899391\pi\)
							0.744423	+	0.667708i	\(0.232725\pi\)
\(59\)	−3.50000	+	6.06218i	−0.455661	+	0.789228i	−0.998726	−	0.0504625i	\(-0.983930\pi\)
							0.543065	+	0.839691i	\(0.317264\pi\)
\(61\)	3.50000	+	6.06218i	0.448129	+	0.776182i	0.998264	−	0.0588933i	\(-0.0187572\pi\)
							−0.550135	+	0.835076i	\(0.685424\pi\)
\(67\)	1.50000	−	2.59808i	0.183254	−	0.317406i	−0.759733	−	0.650236i	\(-0.774670\pi\)
							0.942987	+	0.332830i	\(0.108004\pi\)
\(71\)	5.00000			0.593391			0.296695	−	0.954972i	\(-0.404115\pi\)
							0.296695	+	0.954972i	\(0.404115\pi\)
\(73\)	−7.00000	+	12.1244i	−0.819288	+	1.41905i	0.0869195	+	0.996215i	\(0.472298\pi\)
							−0.906208	+	0.422833i	\(0.861036\pi\)
\(79\)	3.00000	+	5.19615i	0.337526	+	0.584613i	0.983967	−	0.178352i	\(-0.0570765\pi\)
							−0.646440	+	0.762964i	\(0.723743\pi\)
\(83\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(89\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(97\)	−14.0000			−1.42148			−0.710742	−	0.703452i	\(-0.751641\pi\)
							−0.710742	+	0.703452i	\(0.751641\pi\)

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	819.2.j.b.235.1		2
3.2	odd	2		91.2.e.a.53.1	✓	2
7.2	even	3	inner	819.2.j.b.352.1		2
7.3	odd	6		5733.2.a.d.1.1		1
7.4	even	3		5733.2.a.c.1.1		1
12.11	even	2		1456.2.r.g.417.1		2
21.2	odd	6		91.2.e.a.79.1	yes	2
21.5	even	6		637.2.e.a.79.1		2
21.11	odd	6		637.2.a.c.1.1		1
21.17	even	6		637.2.a.d.1.1		1
21.20	even	2		637.2.e.a.508.1		2
39.38	odd	2		1183.2.e.b.508.1		2
84.23	even	6		1456.2.r.g.625.1		2
273.38	even	6		8281.2.a.e.1.1		1
273.116	odd	6		8281.2.a.f.1.1		1
273.233	odd	6		1183.2.e.b.170.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.a.53.1	✓	2	3.2	odd	2
91.2.e.a.79.1	yes	2	21.2	odd	6
637.2.a.c.1.1		1	21.11	odd	6
637.2.a.d.1.1		1	21.17	even	6
637.2.e.a.79.1		2	21.5	even	6
637.2.e.a.508.1		2	21.20	even	2
819.2.j.b.235.1		2	1.1	even	1	trivial
819.2.j.b.352.1		2	7.2	even	3	inner
1183.2.e.b.170.1		2	273.233	odd	6
1183.2.e.b.508.1		2	39.38	odd	2
1456.2.r.g.417.1		2	12.11	even	2
1456.2.r.g.625.1		2	84.23	even	6
5733.2.a.c.1.1		1	7.4	even	3
5733.2.a.d.1.1		1	7.3	odd	6
8281.2.a.e.1.1		1	273.38	even	6
8281.2.a.f.1.1		1	273.116	odd	6