Embedded newform 7105.2.a.e.1.2

• Label: 7105.2.a.e.1.2
• Level: $7105$
• Weight: $2$
• Character: 7105.1
• Self dual: yes
• Analytic conductor: $56.734$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7105.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(56.7337106361\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{8})^+\)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			\(1.41421\) of defining polynomial
Character	\(\chi\)	\(=\)	7105.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.828427 q^{6} -1.58579 q^{8} +1.00000 q^{9} -0.414214 q^{10} +0.828427 q^{11} -3.65685 q^{12} +2.00000 q^{13} -2.00000 q^{15} +3.00000 q^{16} -2.82843 q^{17} +0.414214 q^{18} +4.82843 q^{19} +1.82843 q^{20} +0.343146 q^{22} -3.17157 q^{23} -3.17157 q^{24} +1.00000 q^{25} +0.828427 q^{26} -4.00000 q^{27} +1.00000 q^{29} -0.828427 q^{30} -6.48528 q^{31} +4.41421 q^{32} +1.65685 q^{33} -1.17157 q^{34} -1.82843 q^{36} -8.48528 q^{37} +2.00000 q^{38} +4.00000 q^{39} +1.58579 q^{40} +6.00000 q^{41} -6.00000 q^{43} -1.51472 q^{44} -1.00000 q^{45} -1.31371 q^{46} +11.6569 q^{47} +6.00000 q^{48} +0.414214 q^{50} -5.65685 q^{51} -3.65685 q^{52} -3.65685 q^{53} -1.65685 q^{54} -0.828427 q^{55} +9.65685 q^{57} +0.414214 q^{58} +3.65685 q^{60} +3.65685 q^{61} -2.68629 q^{62} -4.17157 q^{64} -2.00000 q^{65} +0.686292 q^{66} +6.48528 q^{67} +5.17157 q^{68} -6.34315 q^{69} -15.3137 q^{71} -1.58579 q^{72} -8.48528 q^{73} -3.51472 q^{74} +2.00000 q^{75} -8.82843 q^{76} +1.65685 q^{78} -2.48528 q^{79} -3.00000 q^{80} -11.0000 q^{81} +2.48528 q^{82} -7.17157 q^{83} +2.82843 q^{85} -2.48528 q^{86} +2.00000 q^{87} -1.31371 q^{88} +7.65685 q^{89} -0.414214 q^{90} +5.79899 q^{92} -12.9706 q^{93} +4.82843 q^{94} -4.82843 q^{95} +8.82843 q^{96} +12.4853 q^{97} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 2 * q^2 + 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^6 - 6 * q^8 + 2 * q^9 + 2 * q^10 - 4 * q^11 + 4 * q^12 + 4 * q^13 - 4 * q^15 + 6 * q^16 - 2 * q^18 + 4 * q^19 - 2 * q^20 + 12 * q^22 - 12 * q^23 - 12 * q^24 + 2 * q^25 - 4 * q^26 - 8 * q^27 + 2 * q^29 + 4 * q^30 + 4 * q^31 + 6 * q^32 - 8 * q^33 - 8 * q^34 + 2 * q^36 + 4 * q^38 + 8 * q^39 + 6 * q^40 + 12 * q^41 - 12 * q^43 - 20 * q^44 - 2 * q^45 + 20 * q^46 + 12 * q^47 + 12 * q^48 - 2 * q^50 + 4 * q^52 + 4 * q^53 + 8 * q^54 + 4 * q^55 + 8 * q^57 - 2 * q^58 - 4 * q^60 - 4 * q^61 - 28 * q^62 - 14 * q^64 - 4 * q^65 + 24 * q^66 - 4 * q^67 + 16 * q^68 - 24 * q^69 - 8 * q^71 - 6 * q^72 - 24 * q^74 + 4 * q^75 - 12 * q^76 - 8 * q^78 + 12 * q^79 - 6 * q^80 - 22 * q^81 - 12 * q^82 - 20 * q^83 + 12 * q^86 + 4 * q^87 + 20 * q^88 + 4 * q^89 + 2 * q^90 - 28 * q^92 + 8 * q^93 + 4 * q^94 - 4 * q^95 + 12 * q^96 + 8 * q^97 - 4 * 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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0.414214	0.292893	0.146447	−	0.989219i	\(-0.453216\pi\)
			0.146447	+	0.989219i	\(0.453216\pi\)
\(3\)	2.00000	1.15470	0.577350	−	0.816497i	\(-0.304087\pi\)
			0.577350	+	0.816497i	\(0.304087\pi\)
\(5\)	−1.00000	−0.447214
			\(7\)	0	0
\(11\)	0.828427	0.249780				0.124890	−	0.992171i	\(-0.460142\pi\)
			0.124890	+	0.992171i	\(0.460142\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	−2.82843	−0.685994	−0.342997	−	0.939336i	\(-0.611442\pi\)
			−0.342997	+	0.939336i	\(0.611442\pi\)
\(19\)	4.82843	1.10772	0.553859	−	0.832611i	\(-0.313155\pi\)
			0.553859	+	0.832611i	\(0.313155\pi\)
\(23\)	−3.17157	−0.661319	−0.330659	−	0.943750i	\(-0.607271\pi\)
			−0.330659	+	0.943750i	\(0.607271\pi\)
\(29\)	1.00000	0.185695
			\(31\)	−6.48528	−1.16479	−0.582395	−	0.812906i	\(-0.697884\pi\)
−0.582395	+	0.812906i				\(0.697884\pi\)
\(37\)	−8.48528	−1.39497	−0.697486	−	0.716599i	\(-0.745698\pi\)
			−0.697486	+	0.716599i	\(0.745698\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	−6.00000	−0.914991	−0.457496	−	0.889212i	\(-0.651253\pi\)
			−0.457496	+	0.889212i	\(0.651253\pi\)
\(47\)	11.6569	1.70033	0.850163	−	0.526519i	\(-0.176503\pi\)
			0.850163	+	0.526519i	\(0.176503\pi\)
\(53\)	−3.65685	−0.502308	−0.251154	−	0.967947i	\(-0.580810\pi\)
			−0.251154	+	0.967947i	\(0.580810\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	3.65685	0.468212	0.234106	−	0.972211i	\(-0.424784\pi\)
			0.234106	+	0.972211i	\(0.424784\pi\)
\(67\)	6.48528	0.792303	0.396152	−	0.918185i	\(-0.370345\pi\)
			0.396152	+	0.918185i	\(0.370345\pi\)
\(71\)	−15.3137	−1.81740	−0.908701	−	0.417447i	\(-0.862925\pi\)
			−0.908701	+	0.417447i	\(0.862925\pi\)
\(73\)	−8.48528	−0.993127	−0.496564	−	0.868000i	\(-0.665405\pi\)
			−0.496564	+	0.868000i	\(0.665405\pi\)
\(79\)	−2.48528	−0.279616	−0.139808	−	0.990179i	\(-0.544649\pi\)
			−0.139808	+	0.990179i	\(0.544649\pi\)
\(83\)	−7.17157	−0.787182	−0.393591	−	0.919286i	\(-0.628767\pi\)
			−0.393591	+	0.919286i	\(0.628767\pi\)
\(89\)	7.65685	0.811625	0.405812	−	0.913956i	\(-0.366989\pi\)
			0.405812	+	0.913956i	\(0.366989\pi\)
\(97\)	12.4853	1.26769	0.633844	−	0.773461i	\(-0.281476\pi\)
			0.633844	+	0.773461i	\(0.281476\pi\)

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	7105.2.a.e.1.2		2
7.6	odd	2		145.2.a.b.1.2	✓	2
21.20	even	2		1305.2.a.n.1.1		2
28.27	even	2		2320.2.a.k.1.2		2
35.13	even	4		725.2.b.c.349.2		4
35.27	even	4		725.2.b.c.349.3		4
35.34	odd	2		725.2.a.c.1.1		2
56.13	odd	2		9280.2.a.be.1.1		2
56.27	even	2		9280.2.a.w.1.2		2
105.104	even	2		6525.2.a.p.1.2		2
203.202	odd	2		4205.2.a.d.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.b.1.2	✓	2	7.6	odd	2
725.2.a.c.1.1		2	35.34	odd	2
725.2.b.c.349.2		4	35.13	even	4
725.2.b.c.349.3		4	35.27	even	4
1305.2.a.n.1.1		2	21.20	even	2
2320.2.a.k.1.2		2	28.27	even	2
4205.2.a.d.1.1		2	203.202	odd	2
6525.2.a.p.1.2		2	105.104	even	2
7105.2.a.e.1.2		2	1.1	even	1	trivial
9280.2.a.w.1.2		2	56.27	even	2
9280.2.a.be.1.1		2	56.13	odd	2