Embedded newform 819.2.j.c.352.2

• Label: 819.2.j.c.352.2
• Level: $819$
• Weight: $2$
• Character: 819.352
• Analytic conductor: $6.540$
• Analytic rank: $1$
• Dimension: $4$
• 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:	\(1\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial:	\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			352.2
Root			\(-0.309017 + 0.535233i\) of defining polynomial
Character	\(\chi\)	\(=\)	819.352
Dual form			819.2.j.c.235.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.190983 + 0.330792i) q^{2} +(0.927051 + 1.60570i) q^{4} +(-1.11803 + 1.93649i) q^{5} +(-2.00000 - 1.73205i) q^{7} -1.47214 q^{8} +(-0.427051 - 0.739674i) q^{10} +(-1.50000 - 2.59808i) q^{11} -1.00000 q^{13} +(0.954915 - 0.330792i) q^{14} +(-1.57295 + 2.72443i) q^{16} +(-3.73607 - 6.47106i) q^{17} +(-1.50000 + 2.59808i) q^{19} -4.14590 q^{20} +1.14590 q^{22} +(-1.88197 + 3.25966i) q^{23} +(0.190983 - 0.330792i) q^{26} +(0.927051 - 4.81710i) q^{28} +4.47214 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-2.07295 - 3.59045i) q^{32} +2.85410 q^{34} +(5.59017 - 1.93649i) q^{35} +(4.35410 - 7.54153i) q^{37} +(-0.572949 - 0.992377i) q^{38} +(1.64590 - 2.85078i) q^{40} -4.47214 q^{41} -8.00000 q^{43} +(2.78115 - 4.81710i) q^{44} +(-0.718847 - 1.24508i) q^{46} +(0.736068 - 1.27491i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-0.927051 - 1.60570i) q^{52} +(0.736068 + 1.27491i) q^{53} +6.70820 q^{55} +(2.94427 + 2.54981i) q^{56} +(-0.854102 + 1.47935i) q^{58} +(3.73607 + 6.47106i) q^{59} +(-1.50000 + 2.59808i) q^{61} +1.90983 q^{62} -4.70820 q^{64} +(1.11803 - 1.93649i) q^{65} +(1.50000 + 2.59808i) q^{67} +(6.92705 - 11.9980i) q^{68} +(-0.427051 + 2.21902i) q^{70} -8.94427 q^{71} +(-5.35410 - 9.27358i) q^{73} +(1.66312 + 2.88061i) q^{74} -5.56231 q^{76} +(-1.50000 + 7.79423i) q^{77} +(-5.35410 + 9.27358i) q^{79} +(-3.51722 - 6.09201i) q^{80} +(0.854102 - 1.47935i) q^{82} +16.7082 q^{85} +(1.52786 - 2.64634i) q^{86} +(2.20820 + 3.82472i) q^{88} +(-1.11803 + 1.93649i) q^{89} +(2.00000 + 1.73205i) q^{91} -6.97871 q^{92} +(0.281153 + 0.486971i) q^{94} +(-3.35410 - 5.80948i) q^{95} -17.4164 q^{97} +(-2.48278 - 0.992377i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q - 3 * q^2 - 3 * q^4 - 8 * q^7 + 12 * q^8 + 5 * q^10 - 6 * q^11 - 4 * q^13 + 15 * q^14 - 13 * q^16 - 6 * q^17 - 6 * q^19 - 30 * q^20 + 18 * q^22 - 12 * q^23 + 3 * q^26 - 3 * q^28 - 10 * q^31 - 15 * q^32 - 2 * q^34 + 4 * q^37 - 9 * q^38 + 20 * q^40 - 32 * q^43 - 9 * q^44 - 23 * q^46 - 6 * q^47 + 4 * q^49 + 3 * q^52 - 6 * q^53 - 24 * q^56 + 10 * q^58 + 6 * q^59 - 6 * q^61 + 30 * q^62 + 8 * q^64 + 6 * q^67 + 21 * q^68 + 5 * q^70 - 8 * q^73 - 9 * q^74 + 18 * q^76 - 6 * q^77 - 8 * q^79 + 15 * q^80 - 10 * q^82 + 40 * q^85 + 24 * q^86 - 18 * q^88 + 8 * q^91 + 66 * q^92 - 19 * q^94 - 16 * q^97 - 39 * q^98

Character values

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

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

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.190983	+	0.330792i	−0.135045	+	0.233905i	−0.925615	−	0.378467i	\(-0.876451\pi\)
							0.790569	+	0.612372i	\(0.209785\pi\)
\(3\)		0			0
							\(5\)	−1.11803	+	1.93649i	−0.500000	+	0.866025i	0.500000	+	0.866025i	\(0.333333\pi\)
−1.00000			\(\pi\)
\(7\)	−2.00000	−	1.73205i	−0.755929	−	0.654654i
							\(11\)	−1.50000	−	2.59808i	−0.452267	−	0.783349i	0.546259	−	0.837616i	\(-0.316051\pi\)
−0.998526	+	0.0542666i	\(0.982718\pi\)
\(13\)	−1.00000			−0.277350
							\(17\)	−3.73607	−	6.47106i	−0.906130	−	1.56946i	−0.819394	−	0.573231i	\(-0.805690\pi\)
−0.0867359	−	0.996231i	\(-0.527644\pi\)
\(19\)	−1.50000	+	2.59808i	−0.344124	+	0.596040i	−0.985194	−	0.171442i	\(-0.945157\pi\)
							0.641071	+	0.767482i	\(0.278491\pi\)
\(23\)	−1.88197	+	3.25966i	−0.392417	+	0.679686i	−0.992768	−	0.120051i	\(-0.961694\pi\)
							0.600351	+	0.799737i	\(0.295028\pi\)
\(29\)	4.47214			0.830455			0.415227	−	0.909718i	\(-0.363702\pi\)
							0.415227	+	0.909718i	\(0.363702\pi\)
\(31\)	−2.50000	−	4.33013i	−0.449013	−	0.777714i	0.549309	−	0.835619i	\(-0.314891\pi\)
							−0.998322	+	0.0579057i	\(0.981558\pi\)
\(37\)	4.35410	−	7.54153i	0.715810	−	1.23982i	−0.246836	−	0.969057i	\(-0.579391\pi\)
							0.962646	−	0.270762i	\(-0.0872757\pi\)
\(41\)	−4.47214			−0.698430			−0.349215	−	0.937043i	\(-0.613552\pi\)
							−0.349215	+	0.937043i	\(0.613552\pi\)
\(43\)	−8.00000			−1.21999			−0.609994	−	0.792406i	\(-0.708828\pi\)
							−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	0.736068	−	1.27491i	0.107367	−	0.185964i	−0.807336	−	0.590092i	\(-0.799091\pi\)
							0.914703	+	0.404128i	\(0.132425\pi\)
\(53\)	0.736068	+	1.27491i	0.101107	+	0.175122i	0.912141	−	0.409877i	\(-0.134428\pi\)
							−0.811034	+	0.584999i	\(0.801095\pi\)
\(59\)	3.73607	+	6.47106i	0.486395	+	0.842460i	0.999878	−	0.0156395i	\(-0.00497842\pi\)
							−0.513483	+	0.858100i	\(0.671645\pi\)
\(61\)	−1.50000	+	2.59808i	−0.192055	+	0.332650i	−0.945931	−	0.324367i	\(-0.894849\pi\)
							0.753876	+	0.657017i	\(0.228182\pi\)
\(67\)	1.50000	+	2.59808i	0.183254	+	0.317406i	0.942987	−	0.332830i	\(-0.108004\pi\)
							−0.759733	+	0.650236i	\(0.774670\pi\)
\(71\)	−8.94427			−1.06149			−0.530745	−	0.847532i	\(-0.678088\pi\)
							−0.530745	+	0.847532i	\(0.678088\pi\)
\(73\)	−5.35410	−	9.27358i	−0.626650	−	1.08539i	−0.988219	−	0.153045i	\(-0.951092\pi\)
							0.361569	−	0.932345i	\(-0.382241\pi\)
\(79\)	−5.35410	+	9.27358i	−0.602384	+	1.04336i	0.390076	+	0.920783i	\(0.372449\pi\)
							−0.992459	+	0.122576i	\(0.960884\pi\)
\(83\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(89\)	−1.11803	+	1.93649i	−0.118511	+	0.205268i	−0.919178	−	0.393842i	\(-0.871146\pi\)
							0.800667	+	0.599110i	\(0.204479\pi\)
\(97\)	−17.4164			−1.76837			−0.884184	−	0.467139i	\(-0.845285\pi\)
							−0.884184	+	0.467139i	\(0.845285\pi\)

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.c.352.2		4
3.2	odd	2		91.2.e.b.79.1	yes	4
7.2	even	3		5733.2.a.v.1.1		2
7.4	even	3	inner	819.2.j.c.235.2		4
7.5	odd	6		5733.2.a.w.1.1		2
12.11	even	2		1456.2.r.j.625.2		4
21.2	odd	6		637.2.a.f.1.2		2
21.5	even	6		637.2.a.e.1.2		2
21.11	odd	6		91.2.e.b.53.1	✓	4
21.17	even	6		637.2.e.h.508.1		4
21.20	even	2		637.2.e.h.79.1		4
39.38	odd	2		1183.2.e.d.170.2		4
84.11	even	6		1456.2.r.j.417.2		4
273.116	odd	6		1183.2.e.d.508.2		4
273.194	even	6		8281.2.a.ba.1.1		2
273.233	odd	6		8281.2.a.z.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.b.53.1	✓	4	21.11	odd	6
91.2.e.b.79.1	yes	4	3.2	odd	2
637.2.a.e.1.2		2	21.5	even	6
637.2.a.f.1.2		2	21.2	odd	6
637.2.e.h.79.1		4	21.20	even	2
637.2.e.h.508.1		4	21.17	even	6
819.2.j.c.235.2		4	7.4	even	3	inner
819.2.j.c.352.2		4	1.1	even	1	trivial
1183.2.e.d.170.2		4	39.38	odd	2
1183.2.e.d.508.2		4	273.116	odd	6
1456.2.r.j.417.2		4	84.11	even	6
1456.2.r.j.625.2		4	12.11	even	2
5733.2.a.v.1.1		2	7.2	even	3
5733.2.a.w.1.1		2	7.5	odd	6
8281.2.a.z.1.1		2	273.233	odd	6
8281.2.a.ba.1.1		2	273.194	even	6