Embedded newform 637.2.a.f.1.2

• Label: 637.2.a.f.1.2
• Level: $637$
• Weight: $2$
• Character: 637.1
• Self dual: yes
• Analytic conductor: $5.086$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.08647060876\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{10})^+\)
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)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			\(-0.618034\) of defining polynomial
Character	\(\chi\)	\(=\)	637.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} +2.23607 q^{3} -1.85410 q^{4} -2.23607 q^{5} -0.854102 q^{6} +1.47214 q^{8} +2.00000 q^{9} +0.854102 q^{10} -3.00000 q^{11} -4.14590 q^{12} -1.00000 q^{13} -5.00000 q^{15} +3.14590 q^{16} -7.47214 q^{17} -0.763932 q^{18} +3.00000 q^{19} +4.14590 q^{20} +1.14590 q^{22} -3.76393 q^{23} +3.29180 q^{24} +0.381966 q^{26} -2.23607 q^{27} -4.47214 q^{29} +1.90983 q^{30} +5.00000 q^{31} -4.14590 q^{32} -6.70820 q^{33} +2.85410 q^{34} -3.70820 q^{36} -8.70820 q^{37} -1.14590 q^{38} -2.23607 q^{39} -3.29180 q^{40} +4.47214 q^{41} -8.00000 q^{43} +5.56231 q^{44} -4.47214 q^{45} +1.43769 q^{46} +1.47214 q^{47} +7.03444 q^{48} -16.7082 q^{51} +1.85410 q^{52} +1.47214 q^{53} +0.854102 q^{54} +6.70820 q^{55} +6.70820 q^{57} +1.70820 q^{58} +7.47214 q^{59} +9.27051 q^{60} +3.00000 q^{61} -1.90983 q^{62} -4.70820 q^{64} +2.23607 q^{65} +2.56231 q^{66} -3.00000 q^{67} +13.8541 q^{68} -8.41641 q^{69} +8.94427 q^{71} +2.94427 q^{72} +10.7082 q^{73} +3.32624 q^{74} -5.56231 q^{76} +0.854102 q^{78} +10.7082 q^{79} -7.03444 q^{80} -11.0000 q^{81} -1.70820 q^{82} +16.7082 q^{85} +3.05573 q^{86} -10.0000 q^{87} -4.41641 q^{88} -2.23607 q^{89} +1.70820 q^{90} +6.97871 q^{92} +11.1803 q^{93} -0.562306 q^{94} -6.70820 q^{95} -9.27051 q^{96} -17.4164 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 3 * q^2 + 3 * q^4 + 5 * q^6 - 6 * q^8 + 4 * q^9 - 5 * q^10 - 6 * q^11 - 15 * q^12 - 2 * q^13 - 10 * q^15 + 13 * q^16 - 6 * q^17 - 6 * q^18 + 6 * q^19 + 15 * q^20 + 9 * q^22 - 12 * q^23 + 20 * q^24 + 3 * q^26 + 15 * q^30 + 10 * q^31 - 15 * q^32 - q^34 + 6 * q^36 - 4 * q^37 - 9 * q^38 - 20 * q^40 - 16 * q^43 - 9 * q^44 + 23 * q^46 - 6 * q^47 - 15 * q^48 - 20 * q^51 - 3 * q^52 - 6 * q^53 - 5 * q^54 - 10 * q^58 + 6 * q^59 - 15 * q^60 + 6 * q^61 - 15 * q^62 + 4 * q^64 - 15 * q^66 - 6 * q^67 + 21 * q^68 + 10 * q^69 - 12 * q^72 + 8 * q^73 - 9 * q^74 + 9 * q^76 - 5 * q^78 + 8 * q^79 + 15 * q^80 - 22 * q^81 + 10 * q^82 + 20 * q^85 + 24 * q^86 - 20 * q^87 + 18 * q^88 - 10 * q^90 - 33 * q^92 + 19 * q^94 + 15 * q^96 - 8 * q^97 - 12 * q^99

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.381966	−0.270091	−0.135045	−	0.990839i	\(-0.543118\pi\)
			−0.135045	+	0.990839i	\(0.543118\pi\)
\(3\)	2.23607	1.29099	0.645497	−	0.763763i	\(-0.276650\pi\)
			0.645497	+	0.763763i	\(0.276650\pi\)
\(5\)	−2.23607	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(7\)	0	0
			\(11\)	−3.00000	−0.904534	−0.452267	−	0.891883i	\(-0.649385\pi\)
−0.452267	+	0.891883i				\(0.649385\pi\)
\(13\)	−1.00000	−0.277350
			\(17\)	−7.47214	−1.81226	−0.906130	−	0.423000i	\(-0.860977\pi\)
−0.906130	+	0.423000i				\(0.860977\pi\)
\(19\)	3.00000	0.688247	0.344124	−	0.938924i	\(-0.388176\pi\)
			0.344124	+	0.938924i	\(0.388176\pi\)
\(23\)	−3.76393	−0.784834	−0.392417	−	0.919787i	\(-0.628361\pi\)
			−0.392417	+	0.919787i	\(0.628361\pi\)
\(29\)	−4.47214	−0.830455	−0.415227	−	0.909718i	\(-0.636298\pi\)
			−0.415227	+	0.909718i	\(0.636298\pi\)
\(31\)	5.00000	0.898027	0.449013	−	0.893525i	\(-0.351776\pi\)
			0.449013	+	0.893525i	\(0.351776\pi\)
\(37\)	−8.70820	−1.43162	−0.715810	−	0.698295i	\(-0.753942\pi\)
			−0.715810	+	0.698295i	\(0.753942\pi\)
\(41\)	4.47214	0.698430	0.349215	−	0.937043i	\(-0.386448\pi\)
			0.349215	+	0.937043i	\(0.386448\pi\)
\(43\)	−8.00000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	1.47214	0.214733	0.107367	−	0.994220i	\(-0.465758\pi\)
			0.107367	+	0.994220i	\(0.465758\pi\)
\(53\)	1.47214	0.202213	0.101107	−	0.994876i	\(-0.467762\pi\)
			0.101107	+	0.994876i	\(0.467762\pi\)
\(59\)	7.47214	0.972789	0.486395	−	0.873739i	\(-0.338312\pi\)
			0.486395	+	0.873739i	\(0.338312\pi\)
\(61\)	3.00000	0.384111	0.192055	−	0.981384i	\(-0.438485\pi\)
			0.192055	+	0.981384i	\(0.438485\pi\)
\(67\)	−3.00000	−0.366508	−0.183254	−	0.983066i	\(-0.558663\pi\)
			−0.183254	+	0.983066i	\(0.558663\pi\)
\(71\)	8.94427	1.06149	0.530745	−	0.847532i	\(-0.321912\pi\)
			0.530745	+	0.847532i	\(0.321912\pi\)
\(73\)	10.7082	1.25330	0.626650	−	0.779301i	\(-0.284425\pi\)
			0.626650	+	0.779301i	\(0.284425\pi\)
\(79\)	10.7082	1.20477	0.602384	−	0.798207i	\(-0.294218\pi\)
			0.602384	+	0.798207i	\(0.294218\pi\)
\(83\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(89\)	−2.23607	−0.237023	−0.118511	−	0.992953i	\(-0.537812\pi\)
			−0.118511	+	0.992953i	\(0.537812\pi\)
\(97\)	−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	637.2.a.f.1.2		2
3.2	odd	2		5733.2.a.v.1.1		2
7.2	even	3		91.2.e.b.53.1	✓	4
7.3	odd	6		637.2.e.h.79.1		4
7.4	even	3		91.2.e.b.79.1	yes	4
7.5	odd	6		637.2.e.h.508.1		4
7.6	odd	2		637.2.a.e.1.2		2
13.12	even	2		8281.2.a.z.1.1		2
21.2	odd	6		819.2.j.c.235.2		4
21.11	odd	6		819.2.j.c.352.2		4
21.20	even	2		5733.2.a.w.1.1		2
28.11	odd	6		1456.2.r.j.625.2		4
28.23	odd	6		1456.2.r.j.417.2		4
91.25	even	6		1183.2.e.d.170.2		4
91.51	even	6		1183.2.e.d.508.2		4
91.90	odd	2		8281.2.a.ba.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.b.53.1	✓	4	7.2	even	3
91.2.e.b.79.1	yes	4	7.4	even	3
637.2.a.e.1.2		2	7.6	odd	2
637.2.a.f.1.2		2	1.1	even	1	trivial
637.2.e.h.79.1		4	7.3	odd	6
637.2.e.h.508.1		4	7.5	odd	6
819.2.j.c.235.2		4	21.2	odd	6
819.2.j.c.352.2		4	21.11	odd	6
1183.2.e.d.170.2		4	91.25	even	6
1183.2.e.d.508.2		4	91.51	even	6
1456.2.r.j.417.2		4	28.23	odd	6
1456.2.r.j.625.2		4	28.11	odd	6
5733.2.a.v.1.1		2	3.2	odd	2
5733.2.a.w.1.1		2	21.20	even	2
8281.2.a.z.1.1		2	13.12	even	2
8281.2.a.ba.1.1		2	91.90	odd	2