Properties

 Label 798.2.a.k Level $798$ Weight $2$ Character orbit 798.a Self dual yes Analytic conductor $6.372$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$798 = 2 \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 798.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$6.37206208130$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + b * q^5 + q^6 - q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - \beta q^{10} + (\beta - 2) q^{11} - q^{12} + \beta q^{13} + q^{14} - \beta q^{15} + q^{16} + ( - \beta + 4) q^{17} - q^{18} + q^{19} + \beta q^{20} + q^{21} + ( - \beta + 2) q^{22} + (\beta - 2) q^{23} + q^{24} + 7 q^{25} - \beta q^{26} - q^{27} - q^{28} - 6 q^{29} + \beta q^{30} - 2 \beta q^{31} - q^{32} + ( - \beta + 2) q^{33} + (\beta - 4) q^{34} - \beta q^{35} + q^{36} + 10 q^{37} - q^{38} - \beta q^{39} - \beta q^{40} - 2 q^{41} - q^{42} + 2 \beta q^{43} + (\beta - 2) q^{44} + \beta q^{45} + ( - \beta + 2) q^{46} + ( - 2 \beta + 4) q^{47} - q^{48} + q^{49} - 7 q^{50} + (\beta - 4) q^{51} + \beta q^{52} + 2 q^{53} + q^{54} + ( - 2 \beta + 12) q^{55} + q^{56} - q^{57} + 6 q^{58} + 4 q^{59} - \beta q^{60} + (2 \beta + 2) q^{61} + 2 \beta q^{62} - q^{63} + q^{64} + 12 q^{65} + (\beta - 2) q^{66} + ( - \beta + 6) q^{67} + ( - \beta + 4) q^{68} + ( - \beta + 2) q^{69} + \beta q^{70} + ( - 2 \beta - 4) q^{71} - q^{72} + 10 q^{73} - 10 q^{74} - 7 q^{75} + q^{76} + ( - \beta + 2) q^{77} + \beta q^{78} + (3 \beta - 2) q^{79} + \beta q^{80} + q^{81} + 2 q^{82} + 8 q^{83} + q^{84} + (4 \beta - 12) q^{85} - 2 \beta q^{86} + 6 q^{87} + ( - \beta + 2) q^{88} - 2 q^{89} - \beta q^{90} - \beta q^{91} + (\beta - 2) q^{92} + 2 \beta q^{93} + (2 \beta - 4) q^{94} + \beta q^{95} + q^{96} + (\beta + 8) q^{97} - q^{98} + (\beta - 2) q^{99} +O(q^{100})$$ q - q^2 - q^3 + q^4 + b * q^5 + q^6 - q^7 - q^8 + q^9 - b * q^10 + (b - 2) * q^11 - q^12 + b * q^13 + q^14 - b * q^15 + q^16 + (-b + 4) * q^17 - q^18 + q^19 + b * q^20 + q^21 + (-b + 2) * q^22 + (b - 2) * q^23 + q^24 + 7 * q^25 - b * q^26 - q^27 - q^28 - 6 * q^29 + b * q^30 - 2*b * q^31 - q^32 + (-b + 2) * q^33 + (b - 4) * q^34 - b * q^35 + q^36 + 10 * q^37 - q^38 - b * q^39 - b * q^40 - 2 * q^41 - q^42 + 2*b * q^43 + (b - 2) * q^44 + b * q^45 + (-b + 2) * q^46 + (-2*b + 4) * q^47 - q^48 + q^49 - 7 * q^50 + (b - 4) * q^51 + b * q^52 + 2 * q^53 + q^54 + (-2*b + 12) * q^55 + q^56 - q^57 + 6 * q^58 + 4 * q^59 - b * q^60 + (2*b + 2) * q^61 + 2*b * q^62 - q^63 + q^64 + 12 * q^65 + (b - 2) * q^66 + (-b + 6) * q^67 + (-b + 4) * q^68 + (-b + 2) * q^69 + b * q^70 + (-2*b - 4) * q^71 - q^72 + 10 * q^73 - 10 * q^74 - 7 * q^75 + q^76 + (-b + 2) * q^77 + b * q^78 + (3*b - 2) * q^79 + b * q^80 + q^81 + 2 * q^82 + 8 * q^83 + q^84 + (4*b - 12) * q^85 - 2*b * q^86 + 6 * q^87 + (-b + 2) * q^88 - 2 * q^89 - b * q^90 - b * q^91 + (b - 2) * q^92 + 2*b * q^93 + (2*b - 4) * q^94 + b * q^95 + q^96 + (b + 8) * q^97 - q^98 + (b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 4 q^{11} - 2 q^{12} + 2 q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 2 q^{19} + 2 q^{21} + 4 q^{22} - 4 q^{23} + 2 q^{24} + 14 q^{25} - 2 q^{27} - 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{33} - 8 q^{34} + 2 q^{36} + 20 q^{37} - 2 q^{38} - 4 q^{41} - 2 q^{42} - 4 q^{44} + 4 q^{46} + 8 q^{47} - 2 q^{48} + 2 q^{49} - 14 q^{50} - 8 q^{51} + 4 q^{53} + 2 q^{54} + 24 q^{55} + 2 q^{56} - 2 q^{57} + 12 q^{58} + 8 q^{59} + 4 q^{61} - 2 q^{63} + 2 q^{64} + 24 q^{65} - 4 q^{66} + 12 q^{67} + 8 q^{68} + 4 q^{69} - 8 q^{71} - 2 q^{72} + 20 q^{73} - 20 q^{74} - 14 q^{75} + 2 q^{76} + 4 q^{77} - 4 q^{79} + 2 q^{81} + 4 q^{82} + 16 q^{83} + 2 q^{84} - 24 q^{85} + 12 q^{87} + 4 q^{88} - 4 q^{89} - 4 q^{92} - 8 q^{94} + 2 q^{96} + 16 q^{97} - 2 q^{98} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 - 4 * q^11 - 2 * q^12 + 2 * q^14 + 2 * q^16 + 8 * q^17 - 2 * q^18 + 2 * q^19 + 2 * q^21 + 4 * q^22 - 4 * q^23 + 2 * q^24 + 14 * q^25 - 2 * q^27 - 2 * q^28 - 12 * q^29 - 2 * q^32 + 4 * q^33 - 8 * q^34 + 2 * q^36 + 20 * q^37 - 2 * q^38 - 4 * q^41 - 2 * q^42 - 4 * q^44 + 4 * q^46 + 8 * q^47 - 2 * q^48 + 2 * q^49 - 14 * q^50 - 8 * q^51 + 4 * q^53 + 2 * q^54 + 24 * q^55 + 2 * q^56 - 2 * q^57 + 12 * q^58 + 8 * q^59 + 4 * q^61 - 2 * q^63 + 2 * q^64 + 24 * q^65 - 4 * q^66 + 12 * q^67 + 8 * q^68 + 4 * q^69 - 8 * q^71 - 2 * q^72 + 20 * q^73 - 20 * q^74 - 14 * q^75 + 2 * q^76 + 4 * q^77 - 4 * q^79 + 2 * q^81 + 4 * q^82 + 16 * q^83 + 2 * q^84 - 24 * q^85 + 12 * q^87 + 4 * q^88 - 4 * q^89 - 4 * q^92 - 8 * q^94 + 2 * q^96 + 16 * q^97 - 2 * q^98 - 4 * q^99

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−1.00000 −1.00000 1.00000 −3.46410 1.00000 −1.00000 −1.00000 1.00000 3.46410
1.2 −1.00000 −1.00000 1.00000 3.46410 1.00000 −1.00000 −1.00000 1.00000 −3.46410
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$1$$
$$19$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.a.k 2
3.b odd 2 1 2394.2.a.x 2
4.b odd 2 1 6384.2.a.br 2
7.b odd 2 1 5586.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.a.k 2 1.a even 1 1 trivial
2394.2.a.x 2 3.b odd 2 1
5586.2.a.bd 2 7.b odd 2 1
6384.2.a.br 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(798))$$:

 $$T_{5}^{2} - 12$$ T5^2 - 12 $$T_{11}^{2} + 4T_{11} - 8$$ T11^2 + 4*T11 - 8 $$T_{17}^{2} - 8T_{17} + 4$$ T17^2 - 8*T17 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - 12$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 4T - 8$$
$13$ $$T^{2} - 12$$
$17$ $$T^{2} - 8T + 4$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 4T - 8$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} - 48$$
$37$ $$(T - 10)^{2}$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} - 48$$
$47$ $$T^{2} - 8T - 32$$
$53$ $$(T - 2)^{2}$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} - 4T - 44$$
$67$ $$T^{2} - 12T + 24$$
$71$ $$T^{2} + 8T - 32$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2} + 4T - 104$$
$83$ $$(T - 8)^{2}$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} - 16T + 52$$