Embedded newform 637.2.f.i.393.2

• Label: 637.2.f.i.393.2
• Level: $637$
• Weight: $2$
• Character: 637.393
• Analytic conductor: $5.086$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.08647060876\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.59066497296.1
Defining polynomial:	\( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \)
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			393.2
Root			\(-0.115680 - 0.200364i\) of defining polynomial
Character	\(\chi\)	\(=\)	637.393
Dual form			637.2.f.i.295.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.115680 - 0.200364i) q^{2} +(-1.66113 - 2.87716i) q^{3} +(0.973236 - 1.68569i) q^{4} +2.23136 q^{5} +(-0.384320 + 0.665661i) q^{6} -0.913059 q^{8} +(-4.01868 + 6.96056i) q^{9} +(-0.258125 - 0.447085i) q^{10} +(-1.66113 - 2.87716i) q^{11} -6.46667 q^{12} +(-3.40300 + 1.19146i) q^{13} +(-3.70657 - 6.41997i) q^{15} +(-1.84085 - 3.18844i) q^{16} +(-0.687890 + 1.19146i) q^{17} +1.85953 q^{18} +(1.61766 - 2.80186i) q^{19} +(2.17164 - 3.76139i) q^{20} +(-0.384320 + 0.665661i) q^{22} +(-0.419251 - 0.726164i) q^{23} +(1.51671 + 2.62701i) q^{24} -0.0210289 q^{25} +(0.632387 + 0.544012i) q^{26} +16.7354 q^{27} +(0.303571 + 0.525800i) q^{29} +(-0.857556 + 1.48533i) q^{30} -1.71511 q^{31} +(-1.33896 + 2.31915i) q^{32} +(-5.51868 + 9.55864i) q^{33} +0.318302 q^{34} +(7.82225 + 13.5485i) q^{36} +(-0.776807 - 1.34547i) q^{37} -0.748524 q^{38} +(9.08083 + 7.81180i) q^{39} -2.03736 q^{40} +(-4.58892 - 7.94824i) q^{41} +(-0.615680 + 1.06639i) q^{43} -6.46667 q^{44} +(-8.96713 + 15.5315i) q^{45} +(-0.0969983 + 0.168006i) q^{46} +1.62817 q^{47} +(-6.11577 + 10.5928i) q^{48} +(0.00243263 + 0.00421343i) q^{50} +4.57069 q^{51} +(-1.30348 + 6.89599i) q^{52} +8.39607 q^{53} +(-1.93596 - 3.35318i) q^{54} +(-3.70657 - 6.41997i) q^{55} -10.7485 q^{57} +(0.0702344 - 0.121650i) q^{58} +(4.41117 - 7.64037i) q^{59} -14.4295 q^{60} +(2.73334 - 4.73428i) q^{61} +(0.198405 + 0.343647i) q^{62} -6.74383 q^{64} +(-7.59332 + 2.65858i) q^{65} +2.55361 q^{66} +(5.09287 + 8.82111i) q^{67} +(1.33896 + 2.31915i) q^{68} +(-1.39286 + 2.41250i) q^{69} +(2.60714 - 4.51570i) q^{71} +(3.66929 - 6.35540i) q^{72} +3.96355 q^{73} +(-0.179723 + 0.311289i) q^{74} +(0.0349316 + 0.0605033i) q^{75} +(-3.14872 - 5.45375i) q^{76} +(0.514731 - 2.72315i) q^{78} +6.45051 q^{79} +(-4.10760 - 7.11457i) q^{80} +(-15.7436 - 27.2687i) q^{81} +(-1.06170 + 1.83891i) q^{82} +4.64055 q^{83} +(-1.53493 + 2.65858i) q^{85} +0.284889 q^{86} +(1.00854 - 1.74684i) q^{87} +(1.51671 + 2.62701i) q^{88} +(4.56413 + 7.90530i) q^{89} +4.14929 q^{90} -1.63212 q^{92} +(2.84902 + 4.93464i) q^{93} +(-0.188347 - 0.326227i) q^{94} +(3.60957 - 6.25197i) q^{95} +8.89672 q^{96} +(-7.67944 + 13.3012i) q^{97} +26.7022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q + q^2 + q^3 - 5 * q^4 + 14 * q^5 - 5 * q^6 - 12 * q^8 - 7 * q^9 - 11 * q^10 + q^11 - 24 * q^12 - 4 * q^13 - 3 * q^15 - 19 * q^16 - 4 * q^17 - 6 * q^18 + q^19 - 2 * q^20 - 5 * q^22 + 2 * q^23 - 3 * q^24 + 10 * q^25 - 12 * q^26 + 52 * q^27 - q^29 + 4 * q^30 + 8 * q^31 + 33 * q^32 - 19 * q^33 - 6 * q^34 + 34 * q^36 + 10 * q^37 + 46 * q^38 + 20 * q^39 + 34 * q^40 - 22 * q^41 - 3 * q^43 - 24 * q^44 - 11 * q^45 - 24 * q^46 - 4 * q^47 + 11 * q^48 - 43 * q^50 + 14 * q^51 - 65 * q^52 + 4 * q^53 + 5 * q^54 - 3 * q^55 - 34 * q^57 + 11 * q^58 - 8 * q^59 - 22 * q^60 + 8 * q^61 - 5 * q^62 + 28 * q^64 + 7 * q^65 - 12 * q^66 + 6 * q^67 - 33 * q^68 - 18 * q^69 + 14 * q^71 - 5 * q^72 + 16 * q^73 - 20 * q^74 - 7 * q^75 + 32 * q^76 - q^78 - 52 * q^79 + 7 * q^80 - 24 * q^81 - 14 * q^82 - 5 * q^85 + 24 * q^86 + 13 * q^87 - 3 * q^88 - q^89 - 52 * q^90 + 24 * q^92 + 7 * q^93 + 33 * q^94 - 21 * q^95 + 116 * q^96 + 3 * q^97 + 46 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(248\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)

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 \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.115680	−	0.200364i	−0.0817984	−	0.141679i	0.822224	−	0.569164i	\(-0.192733\pi\)
							−0.904022	+	0.427485i	\(0.859400\pi\)
\(3\)	−1.66113	−	2.87716i	−0.959052	−	1.66113i	−0.724811	−	0.688948i	\(-0.758073\pi\)
							−0.234241	−	0.972179i	\(-0.575260\pi\)
\(5\)	2.23136			0.997895			0.498947	−	0.866632i	\(-0.333720\pi\)
							0.498947	+	0.866632i	\(0.333720\pi\)
\(7\)		0			0
							\(11\)	−1.66113	−	2.87716i	−0.500848	−	0.867495i	−1.00000		0.000980003i	\(-0.999688\pi\)
0.499151	−	0.866515i	\(-0.333645\pi\)
\(13\)	−3.40300	+	1.19146i	−0.943823	+	0.330452i
							\(17\)	−0.687890	+	1.19146i	−0.166838	+	0.288972i	−0.937306	−	0.348506i	\(-0.886689\pi\)
0.770469	+	0.637478i	\(0.220022\pi\)
\(19\)	1.61766	−	2.80186i	0.371116	−	0.642791i	−0.618622	−	0.785689i	\(-0.712309\pi\)
							0.989737	+	0.142898i	\(0.0456420\pi\)
\(23\)	−0.419251	−	0.726164i	−0.0874199	−	0.151416i	0.819000	−	0.573794i	\(-0.194529\pi\)
							−0.906420	+	0.422378i	\(0.861196\pi\)
\(29\)	0.303571	+	0.525800i	0.0563717	+	0.0976386i	0.892834	−	0.450386i	\(-0.148713\pi\)
							−0.836462	+	0.548024i	\(0.815380\pi\)
\(31\)	−1.71511			−0.308043			−0.154022	−	0.988067i	\(-0.549223\pi\)
							−0.154022	+	0.988067i	\(0.549223\pi\)
\(37\)	−0.776807	−	1.34547i	−0.127706	−	0.221194i	0.795081	−	0.606503i	\(-0.207428\pi\)
							−0.922788	+	0.385309i	\(0.874095\pi\)
\(41\)	−4.58892	−	7.94824i	−0.716668	−	1.24131i	−0.962313	−	0.271946i	\(-0.912333\pi\)
							0.245644	−	0.969360i	\(-0.421001\pi\)
\(43\)	−0.615680	+	1.06639i	−0.0938904	+	0.162623i	−0.909145	−	0.416480i	\(-0.863264\pi\)
							0.815255	+	0.579103i	\(0.196597\pi\)
\(47\)	1.62817			0.237493			0.118747	−	0.992925i	\(-0.462112\pi\)
							0.118747	+	0.992925i	\(0.462112\pi\)
\(53\)	8.39607			1.15329			0.576644	−	0.816995i	\(-0.304362\pi\)
							0.576644	+	0.816995i	\(0.304362\pi\)
\(59\)	4.41117	−	7.64037i	0.574285	−	0.994691i	−0.421834	−	0.906673i	\(-0.638613\pi\)
							0.996119	−	0.0880181i	\(-0.0280533\pi\)
\(61\)	2.73334	−	4.73428i	0.349968	−	0.606162i	−0.636276	−	0.771462i	\(-0.719526\pi\)
							0.986243	+	0.165300i	\(0.0528592\pi\)
\(67\)	5.09287	+	8.82111i	0.622193	+	1.07767i	0.989077	+	0.147403i	\(0.0470913\pi\)
							−0.366884	+	0.930267i	\(0.619575\pi\)
\(71\)	2.60714	−	4.51570i	0.309411	−	0.535915i	−0.668823	−	0.743422i	\(-0.733202\pi\)
							0.978234	+	0.207507i	\(0.0665349\pi\)
\(73\)	3.96355			0.463898			0.231949	−	0.972728i	\(-0.425490\pi\)
							0.231949	+	0.972728i	\(0.425490\pi\)
\(79\)	6.45051			0.725739			0.362869	−	0.931840i	\(-0.381797\pi\)
							0.362869	+	0.931840i	\(0.381797\pi\)
\(83\)	4.64055			0.509367			0.254684	−	0.967024i	\(-0.418029\pi\)
							0.254684	+	0.967024i	\(0.418029\pi\)
\(89\)	4.56413	+	7.90530i	0.483797	+	0.837960i	0.999827	−	0.0186101i	\(-0.00592411\pi\)
							−0.516030	+	0.856570i	\(0.672591\pi\)
\(97\)	−7.67944	+	13.3012i	−0.779729	+	1.35053i	0.152369	+	0.988324i	\(0.451310\pi\)
							−0.932098	+	0.362206i	\(0.882024\pi\)

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	637.2.f.i.393.2		8
7.2	even	3		637.2.g.j.263.2		8
7.3	odd	6		637.2.h.h.471.3		8
7.4	even	3		637.2.h.i.471.3		8
7.5	odd	6		637.2.g.k.263.2		8
7.6	odd	2		91.2.f.c.29.2	yes	8
13.3	even	3		8281.2.a.bp.1.3		4
13.9	even	3	inner	637.2.f.i.295.2		8
13.10	even	6		8281.2.a.bt.1.2		4
21.20	even	2		819.2.o.h.757.3		8
28.27	even	2		1456.2.s.q.1121.1		8
91.9	even	3		637.2.h.i.165.3		8
91.41	even	12		1183.2.c.g.337.4		8
91.48	odd	6		91.2.f.c.22.2	✓	8
91.55	odd	6		1183.2.a.k.1.3		4
91.61	odd	6		637.2.h.h.165.3		8
91.62	odd	6		1183.2.a.l.1.2		4
91.74	even	3		637.2.g.j.373.2		8
91.76	even	12		1183.2.c.g.337.5		8
91.87	odd	6		637.2.g.k.373.2		8
273.230	even	6		819.2.o.h.568.3		8
364.139	even	6		1456.2.s.q.113.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.f.c.22.2	✓	8	91.48	odd	6
91.2.f.c.29.2	yes	8	7.6	odd	2
637.2.f.i.295.2		8	13.9	even	3	inner
637.2.f.i.393.2		8	1.1	even	1	trivial
637.2.g.j.263.2		8	7.2	even	3
637.2.g.j.373.2		8	91.74	even	3
637.2.g.k.263.2		8	7.5	odd	6
637.2.g.k.373.2		8	91.87	odd	6
637.2.h.h.165.3		8	91.61	odd	6
637.2.h.h.471.3		8	7.3	odd	6
637.2.h.i.165.3		8	91.9	even	3
637.2.h.i.471.3		8	7.4	even	3
819.2.o.h.568.3		8	273.230	even	6
819.2.o.h.757.3		8	21.20	even	2
1183.2.a.k.1.3		4	91.55	odd	6
1183.2.a.l.1.2		4	91.62	odd	6
1183.2.c.g.337.4		8	91.41	even	12
1183.2.c.g.337.5		8	91.76	even	12
1456.2.s.q.113.1		8	364.139	even	6
1456.2.s.q.1121.1		8	28.27	even	2
8281.2.a.bp.1.3		4	13.3	even	3
8281.2.a.bt.1.2		4	13.10	even	6