Embedded newform 819.2.j.h.235.5

• Label: 819.2.j.h.235.5
• Level: $819$
• Weight: $2$
• Character: 819.235
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $10$
• 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:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	\( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			235.5
Root			\(-0.862625 - 1.49411i\) of defining polynomial
Character	\(\chi\)	\(=\)	819.235
Dual form			819.2.j.h.352.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36263 + 2.36014i) q^{2} +(-2.71349 + 4.69991i) q^{4} +(1.09358 + 1.89414i) q^{5} +(-2.19729 - 1.47375i) q^{7} -9.33940 q^{8} +(-2.98028 + 5.16200i) q^{10} +(-0.524077 + 0.907729i) q^{11} +1.00000 q^{13} +(0.484172 - 7.19406i) q^{14} +(-7.29912 - 12.6424i) q^{16} +(-2.64562 + 4.58236i) q^{17} +(-0.378453 - 0.655500i) q^{19} -11.8697 q^{20} -2.85648 q^{22} +(0.326792 + 0.566020i) q^{23} +(0.108157 - 0.187333i) q^{25} +(1.36263 + 2.36014i) q^{26} +(12.8888 - 6.32803i) q^{28} +3.10408 q^{29} +(-0.513956 + 0.890198i) q^{31} +(10.5525 - 18.2775i) q^{32} -14.4200 q^{34} +(0.388575 - 5.77363i) q^{35} +(5.44661 + 9.43381i) q^{37} +(1.03138 - 1.78640i) q^{38} +(-10.2134 - 17.6901i) q^{40} -7.32040 q^{41} +0.887771 q^{43} +(-2.84416 - 4.92623i) q^{44} +(-0.890590 + 1.54255i) q^{46} +(1.16875 + 2.02434i) q^{47} +(2.65613 + 6.47650i) q^{49} +0.589510 q^{50} +(-2.71349 + 4.69991i) q^{52} +(2.44407 - 4.23325i) q^{53} -2.29249 q^{55} +(20.5213 + 13.7639i) q^{56} +(4.22970 + 7.32606i) q^{58} +(-0.524077 + 0.907729i) q^{59} +(6.24989 + 10.8251i) q^{61} -2.80132 q^{62} +28.3200 q^{64} +(1.09358 + 1.89414i) q^{65} +(-2.23944 + 3.87883i) q^{67} +(-14.3578 - 24.8684i) q^{68} +(14.1560 - 6.95021i) q^{70} +6.60274 q^{71} +(4.14174 - 7.17370i) q^{73} +(-14.8434 + 25.7095i) q^{74} +4.10772 q^{76} +(2.48931 - 1.22218i) q^{77} +(-1.07007 - 1.85342i) q^{79} +(15.9644 - 27.6511i) q^{80} +(-9.97496 - 17.2771i) q^{82} +6.66558 q^{83} -11.5728 q^{85} +(1.20970 + 2.09526i) q^{86} +(4.89457 - 8.47765i) q^{88} +(-2.88388 - 4.99503i) q^{89} +(-2.19729 - 1.47375i) q^{91} -3.54699 q^{92} +(-3.18515 + 5.51684i) q^{94} +(0.827739 - 1.43369i) q^{95} -2.88777 q^{97} +(-11.6661 + 15.0939i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	10 * q + 4 * q^2 - 8 * q^4 + 2 * q^5 + q^7 - 18 * q^8 + 5 * q^10 + 11 * q^11 + 10 * q^13 - 10 * q^14 - 10 * q^16 - 5 * q^17 - 9 * q^19 - 2 * q^20 + 16 * q^22 + 10 * q^23 - 9 * q^25 + 4 * q^26 + 37 * q^28 + 6 * q^29 + 6 * q^31 + 22 * q^32 - 44 * q^34 + 4 * q^35 - 4 * q^37 - 10 * q^38 - 28 * q^40 - 28 * q^41 + 4 * q^43 - 3 * q^46 + q^47 - 11 * q^49 - 18 * q^50 - 8 * q^52 + 17 * q^53 + 21 * q^56 + 27 * q^58 + 11 * q^59 + 11 * q^61 + 46 * q^62 + 18 * q^64 + 2 * q^65 - 13 * q^67 - 32 * q^68 + 49 * q^70 - 30 * q^71 - 33 * q^74 + 16 * q^76 + 46 * q^77 - 2 * q^79 + 55 * q^80 - 34 * q^82 - 12 * q^83 - 44 * q^85 + 28 * q^86 + 3 * q^88 - 4 * q^89 + q^91 - 42 * q^92 - 20 * q^94 - 12 * q^95 - 24 * q^97 + 7 * 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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36263	+	2.36014i	0.963521	+	1.66887i	0.713536	+	0.700619i	\(0.247093\pi\)
							0.249986	+	0.968250i	\(0.419574\pi\)
\(3\)		0			0
							\(5\)	1.09358	+	1.89414i	0.489065	+	0.847085i	0.999921	−	0.0125813i	\(-0.00400485\pi\)
−0.510856	+	0.859666i	\(0.670672\pi\)
\(7\)	−2.19729	−	1.47375i	−0.830496	−	0.557025i
							\(11\)	−0.524077	+	0.907729i	−0.158015	+	0.273691i	−0.934153	−	0.356873i	\(-0.883843\pi\)
0.776138	+	0.630564i	\(0.217176\pi\)
\(13\)	1.00000			0.277350
							\(17\)	−2.64562	+	4.58236i	−0.641658	+	1.11138i	0.343404	+	0.939188i	\(0.388420\pi\)
−0.985063	+	0.172197i	\(0.944913\pi\)
\(19\)	−0.378453	−	0.655500i	−0.0868231	−	0.150382i	0.819344	−	0.573303i	\(-0.194338\pi\)
							−0.906167	+	0.422921i	\(0.861005\pi\)
\(23\)	0.326792	+	0.566020i	0.0681408	+	0.118023i	0.898083	−	0.439826i	\(-0.144960\pi\)
							−0.829942	+	0.557850i	\(0.811627\pi\)
\(29\)	3.10408			0.576414			0.288207	−	0.957568i	\(-0.406941\pi\)
							0.288207	+	0.957568i	\(0.406941\pi\)
\(31\)	−0.513956	+	0.890198i	−0.0923092	+	0.159884i	−0.908482	−	0.417923i	\(-0.862758\pi\)
							0.816173	+	0.577807i	\(0.196091\pi\)
\(37\)	5.44661	+	9.43381i	0.895418	+	1.55091i	0.833287	+	0.552841i	\(0.186456\pi\)
							0.0621309	+	0.998068i	\(0.480210\pi\)
\(41\)	−7.32040			−1.14325			−0.571627	−	0.820514i	\(-0.693688\pi\)
							−0.571627	+	0.820514i	\(0.693688\pi\)
\(43\)	0.887771			0.135384			0.0676919	−	0.997706i	\(-0.478437\pi\)
							0.0676919	+	0.997706i	\(0.478437\pi\)
\(47\)	1.16875	+	2.02434i	0.170480	+	0.295281i	0.938588	−	0.345040i	\(-0.112135\pi\)
							−0.768108	+	0.640321i	\(0.778801\pi\)
\(53\)	2.44407	−	4.23325i	0.335719	−	0.581482i	−0.647904	−	0.761722i	\(-0.724354\pi\)
							0.983623	+	0.180240i	\(0.0576875\pi\)
\(59\)	−0.524077	+	0.907729i	−0.0682291	+	0.118176i	−0.898122	−	0.439747i	\(-0.855068\pi\)
							0.829893	+	0.557923i	\(0.188402\pi\)
\(61\)	6.24989	+	10.8251i	0.800217	+	1.38602i	0.919473	+	0.393153i	\(0.128616\pi\)
							−0.119256	+	0.992864i	\(0.538051\pi\)
\(67\)	−2.23944	+	3.87883i	−0.273592	+	0.473875i	−0.969779	−	0.243986i	\(-0.921545\pi\)
							0.696187	+	0.717860i	\(0.254878\pi\)
\(71\)	6.60274			0.783601			0.391801	−	0.920050i	\(-0.371852\pi\)
							0.391801	+	0.920050i	\(0.371852\pi\)
\(73\)	4.14174	−	7.17370i	0.484754	−	0.839618i	−0.515093	−	0.857134i	\(-0.672243\pi\)
							0.999847	+	0.0175164i	\(0.00557593\pi\)
\(79\)	−1.07007	−	1.85342i	−0.120392	−	0.208526i	0.799530	−	0.600626i	\(-0.205082\pi\)
							−0.919922	+	0.392100i	\(0.871749\pi\)
\(83\)	6.66558			0.731642			0.365821	−	0.930685i	\(-0.380788\pi\)
							0.365821	+	0.930685i	\(0.380788\pi\)
\(89\)	−2.88388	−	4.99503i	−0.305691	−	0.529472i	0.671724	−	0.740802i	\(-0.265554\pi\)
							−0.977415	+	0.211329i	\(0.932221\pi\)
\(97\)	−2.88777			−0.293209			−0.146604	−	0.989195i	\(-0.546834\pi\)
							−0.146604	+	0.989195i	\(0.546834\pi\)

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.h.235.5		10
3.2	odd	2		91.2.e.c.53.1	✓	10
7.2	even	3	inner	819.2.j.h.352.5		10
7.3	odd	6		5733.2.a.bm.1.1		5
7.4	even	3		5733.2.a.bl.1.1		5
12.11	even	2		1456.2.r.p.417.2		10
21.2	odd	6		91.2.e.c.79.1	yes	10
21.5	even	6		637.2.e.m.79.1		10
21.11	odd	6		637.2.a.l.1.5		5
21.17	even	6		637.2.a.k.1.5		5
21.20	even	2		637.2.e.m.508.1		10
39.38	odd	2		1183.2.e.f.508.5		10
84.23	even	6		1456.2.r.p.625.2		10
273.38	even	6		8281.2.a.bx.1.1		5
273.116	odd	6		8281.2.a.bw.1.1		5
273.233	odd	6		1183.2.e.f.170.5		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.c.53.1	✓	10	3.2	odd	2
91.2.e.c.79.1	yes	10	21.2	odd	6
637.2.a.k.1.5		5	21.17	even	6
637.2.a.l.1.5		5	21.11	odd	6
637.2.e.m.79.1		10	21.5	even	6
637.2.e.m.508.1		10	21.20	even	2
819.2.j.h.235.5		10	1.1	even	1	trivial
819.2.j.h.352.5		10	7.2	even	3	inner
1183.2.e.f.170.5		10	273.233	odd	6
1183.2.e.f.508.5		10	39.38	odd	2
1456.2.r.p.417.2		10	12.11	even	2
1456.2.r.p.625.2		10	84.23	even	6
5733.2.a.bl.1.1		5	7.4	even	3
5733.2.a.bm.1.1		5	7.3	odd	6
8281.2.a.bw.1.1		5	273.116	odd	6
8281.2.a.bx.1.1		5	273.38	even	6