# Properties

 Label 798.2.j.h Level $798$ Weight $2$ Character orbit 798.j Analytic conductor $6.372$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$798 = 2 \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 798.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.37206208130$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{5} - q^{6} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q + b2 * q^2 + (b2 + 1) * q^3 + (-b2 - 1) * q^4 + (b3 - 3*b2 + b1) * q^5 - q^6 + (2*b3 - b2 + b1 - 1) * q^7 + q^8 + b2 * q^9 $$q + \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{5} - q^{6} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + q^{8} + \beta_{2} q^{9} + (3 \beta_{2} - \beta_1 + 3) q^{10} + (\beta_{2} + 1) q^{11} - \beta_{2} q^{12} - 2 \beta_{3} q^{13} + ( - \beta_{3} - 2 \beta_1 + 1) q^{14} + (\beta_{3} + 3) q^{15} + \beta_{2} q^{16} + 3 \beta_1 q^{17} + ( - \beta_{2} - 1) q^{18} + \beta_{2} q^{19} + ( - \beta_{3} - 3) q^{20} + (\beta_{3} - \beta_{2} - \beta_1) q^{21} - q^{22} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{23} + (\beta_{2} + 1) q^{24} + ( - 6 \beta_{2} + 6 \beta_1 - 6) q^{25} + (2 \beta_{3} + 2 \beta_1) q^{26} - q^{27} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{28} + (2 \beta_{3} + 5) q^{29} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{30} + (5 \beta_{2} + 4 \beta_1 + 5) q^{31} + ( - \beta_{2} - 1) q^{32} + \beta_{2} q^{33} + 3 \beta_{3} q^{34} + (2 \beta_{3} - 4 \beta_{2} + 6 \beta_1 - 5) q^{35} + q^{36} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{37} + ( - \beta_{2} - 1) q^{38} + 2 \beta_1 q^{39} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{40} + ( - \beta_{3} - 2) q^{41} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{42} + ( - 3 \beta_{3} + 2) q^{43} - \beta_{2} q^{44} + (3 \beta_{2} - \beta_1 + 3) q^{45} + ( - 2 \beta_{2} - \beta_1 - 2) q^{46} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{47} - q^{48} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{49} + (6 \beta_{3} + 6) q^{50} + (3 \beta_{3} + 3 \beta_1) q^{51} - 2 \beta_1 q^{52} + (\beta_{2} - 2 \beta_1 + 1) q^{53} - \beta_{2} q^{54} + (\beta_{3} + 3) q^{55} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{56} - q^{57} + ( - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1) q^{58} + (\beta_{2} - 5 \beta_1 + 1) q^{59} + ( - 3 \beta_{2} + \beta_1 - 3) q^{60} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{61} + (4 \beta_{3} - 5) q^{62} + ( - \beta_{3} - 2 \beta_1 + 1) q^{63} + q^{64} + ( - 6 \beta_{3} + 4 \beta_{2} - 6 \beta_1) q^{65} + ( - \beta_{2} - 1) q^{66} + (2 \beta_{2} + 2) q^{67} + ( - 3 \beta_{3} - 3 \beta_1) q^{68} + (\beta_{3} - 2) q^{69} + (4 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{70} + (7 \beta_{3} + 2) q^{71} + \beta_{2} q^{72} + (12 \beta_{2} + 2 \beta_1 + 12) q^{73} + (4 \beta_{2} - 3 \beta_1 + 4) q^{74} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{75} + q^{76} + (\beta_{3} - \beta_{2} - \beta_1) q^{77} + 2 \beta_{3} q^{78} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{79} + (3 \beta_{2} - \beta_1 + 3) q^{80} + ( - \beta_{2} - 1) q^{81} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{82} + (10 \beta_{3} - 3) q^{83} + (\beta_{3} + 2 \beta_1 - 1) q^{84} + ( - 9 \beta_{3} - 6) q^{85} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{86} + (5 \beta_{2} - 2 \beta_1 + 5) q^{87} + (\beta_{2} + 1) q^{88} + ( - 3 \beta_{3} + 8 \beta_{2} - 3 \beta_1) q^{89} + ( - \beta_{3} - 3) q^{90} + (4 \beta_{2} - 2 \beta_1 - 4) q^{91} + ( - \beta_{3} + 2) q^{92} + (4 \beta_{3} + 5 \beta_{2} + 4 \beta_1) q^{93} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{94} + (3 \beta_{2} - \beta_1 + 3) q^{95} - \beta_{2} q^{96} + ( - \beta_{3} - 7) q^{97} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 5) q^{98} - q^{99}+O(q^{100})$$ q + b2 * q^2 + (b2 + 1) * q^3 + (-b2 - 1) * q^4 + (b3 - 3*b2 + b1) * q^5 - q^6 + (2*b3 - b2 + b1 - 1) * q^7 + q^8 + b2 * q^9 + (3*b2 - b1 + 3) * q^10 + (b2 + 1) * q^11 - b2 * q^12 - 2*b3 * q^13 + (-b3 - 2*b1 + 1) * q^14 + (b3 + 3) * q^15 + b2 * q^16 + 3*b1 * q^17 + (-b2 - 1) * q^18 + b2 * q^19 + (-b3 - 3) * q^20 + (b3 - b2 - b1) * q^21 - q^22 + (b3 + 2*b2 + b1) * q^23 + (b2 + 1) * q^24 + (-6*b2 + 6*b1 - 6) * q^25 + (2*b3 + 2*b1) * q^26 - q^27 + (-b3 + b2 + b1) * q^28 + (2*b3 + 5) * q^29 + (-b3 + 3*b2 - b1) * q^30 + (5*b2 + 4*b1 + 5) * q^31 + (-b2 - 1) * q^32 + b2 * q^33 + 3*b3 * q^34 + (2*b3 - 4*b2 + 6*b1 - 5) * q^35 + q^36 + (3*b3 - 4*b2 + 3*b1) * q^37 + (-b2 - 1) * q^38 + 2*b1 * q^39 + (b3 - 3*b2 + b1) * q^40 + (-b3 - 2) * q^41 + (-2*b3 + b2 - b1 + 1) * q^42 + (-3*b3 + 2) * q^43 - b2 * q^44 + (3*b2 - b1 + 3) * q^45 + (-2*b2 - b1 - 2) * q^46 + (-2*b3 + 6*b2 - 2*b1) * q^47 - q^48 + (-2*b3 - 5*b2 + 2*b1) * q^49 + (6*b3 + 6) * q^50 + (3*b3 + 3*b1) * q^51 - 2*b1 * q^52 + (b2 - 2*b1 + 1) * q^53 - b2 * q^54 + (b3 + 3) * q^55 + (2*b3 - b2 + b1 - 1) * q^56 - q^57 + (-2*b3 + 5*b2 - 2*b1) * q^58 + (b2 - 5*b1 + 1) * q^59 + (-3*b2 + b1 - 3) * q^60 + (-2*b3 - 2*b2 - 2*b1) * q^61 + (4*b3 - 5) * q^62 + (-b3 - 2*b1 + 1) * q^63 + q^64 + (-6*b3 + 4*b2 - 6*b1) * q^65 + (-b2 - 1) * q^66 + (2*b2 + 2) * q^67 + (-3*b3 - 3*b1) * q^68 + (b3 - 2) * q^69 + (4*b3 - b2 - 2*b1 + 4) * q^70 + (7*b3 + 2) * q^71 + b2 * q^72 + (12*b2 + 2*b1 + 12) * q^73 + (4*b2 - 3*b1 + 4) * q^74 + (6*b3 - 6*b2 + 6*b1) * q^75 + q^76 + (b3 - b2 - b1) * q^77 + 2*b3 * q^78 + (2*b3 - 3*b2 + 2*b1) * q^79 + (3*b2 - b1 + 3) * q^80 + (-b2 - 1) * q^81 + (b3 - 2*b2 + b1) * q^82 + (10*b3 - 3) * q^83 + (b3 + 2*b1 - 1) * q^84 + (-9*b3 - 6) * q^85 + (3*b3 + 2*b2 + 3*b1) * q^86 + (5*b2 - 2*b1 + 5) * q^87 + (b2 + 1) * q^88 + (-3*b3 + 8*b2 - 3*b1) * q^89 + (-b3 - 3) * q^90 + (4*b2 - 2*b1 - 4) * q^91 + (-b3 + 2) * q^92 + (4*b3 + 5*b2 + 4*b1) * q^93 + (-6*b2 + 2*b1 - 6) * q^94 + (3*b2 - b1 + 3) * q^95 - b2 * q^96 + (-b3 - 7) * q^97 + (4*b3 + 5*b2 + 2*b1 + 5) * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 6 q^{5} - 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 6 * q^5 - 4 * q^6 - 2 * q^7 + 4 * q^8 - 2 * q^9 $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 6 q^{5} - 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9} + 6 q^{10} + 2 q^{11} + 2 q^{12} + 4 q^{14} + 12 q^{15} - 2 q^{16} - 2 q^{18} - 2 q^{19} - 12 q^{20} + 2 q^{21} - 4 q^{22} - 4 q^{23} + 2 q^{24} - 12 q^{25} - 4 q^{27} - 2 q^{28} + 20 q^{29} - 6 q^{30} + 10 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{35} + 4 q^{36} + 8 q^{37} - 2 q^{38} + 6 q^{40} - 8 q^{41} + 2 q^{42} + 8 q^{43} + 2 q^{44} + 6 q^{45} - 4 q^{46} - 12 q^{47} - 4 q^{48} + 10 q^{49} + 24 q^{50} + 2 q^{53} + 2 q^{54} + 12 q^{55} - 2 q^{56} - 4 q^{57} - 10 q^{58} + 2 q^{59} - 6 q^{60} + 4 q^{61} - 20 q^{62} + 4 q^{63} + 4 q^{64} - 8 q^{65} - 2 q^{66} + 4 q^{67} - 8 q^{69} + 18 q^{70} + 8 q^{71} - 2 q^{72} + 24 q^{73} + 8 q^{74} + 12 q^{75} + 4 q^{76} + 2 q^{77} + 6 q^{79} + 6 q^{80} - 2 q^{81} + 4 q^{82} - 12 q^{83} - 4 q^{84} - 24 q^{85} - 4 q^{86} + 10 q^{87} + 2 q^{88} - 16 q^{89} - 12 q^{90} - 24 q^{91} + 8 q^{92} - 10 q^{93} - 12 q^{94} + 6 q^{95} + 2 q^{96} - 28 q^{97} + 10 q^{98} - 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 6 * q^5 - 4 * q^6 - 2 * q^7 + 4 * q^8 - 2 * q^9 + 6 * q^10 + 2 * q^11 + 2 * q^12 + 4 * q^14 + 12 * q^15 - 2 * q^16 - 2 * q^18 - 2 * q^19 - 12 * q^20 + 2 * q^21 - 4 * q^22 - 4 * q^23 + 2 * q^24 - 12 * q^25 - 4 * q^27 - 2 * q^28 + 20 * q^29 - 6 * q^30 + 10 * q^31 - 2 * q^32 - 2 * q^33 - 12 * q^35 + 4 * q^36 + 8 * q^37 - 2 * q^38 + 6 * q^40 - 8 * q^41 + 2 * q^42 + 8 * q^43 + 2 * q^44 + 6 * q^45 - 4 * q^46 - 12 * q^47 - 4 * q^48 + 10 * q^49 + 24 * q^50 + 2 * q^53 + 2 * q^54 + 12 * q^55 - 2 * q^56 - 4 * q^57 - 10 * q^58 + 2 * q^59 - 6 * q^60 + 4 * q^61 - 20 * q^62 + 4 * q^63 + 4 * q^64 - 8 * q^65 - 2 * q^66 + 4 * q^67 - 8 * q^69 + 18 * q^70 + 8 * q^71 - 2 * q^72 + 24 * q^73 + 8 * q^74 + 12 * q^75 + 4 * q^76 + 2 * q^77 + 6 * q^79 + 6 * q^80 - 2 * q^81 + 4 * q^82 - 12 * q^83 - 4 * q^84 - 24 * q^85 - 4 * q^86 + 10 * q^87 + 2 * q^88 - 16 * q^89 - 12 * q^90 - 24 * q^91 + 8 * q^92 - 10 * q^93 - 12 * q^94 + 6 * q^95 + 2 * q^96 - 28 * q^97 + 10 * q^98 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/798\mathbb{Z}\right)^\times$$.

 $$n$$ $$115$$ $$211$$ $$533$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
457.1
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.792893 1.37333i −1.00000 −2.62132 + 0.358719i 1.00000 −0.500000 + 0.866025i 0.792893 + 1.37333i
457.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 2.20711 3.82282i −1.00000 1.62132 2.09077i 1.00000 −0.500000 + 0.866025i 2.20711 + 3.82282i
571.1 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.792893 + 1.37333i −1.00000 −2.62132 0.358719i 1.00000 −0.500000 0.866025i 0.792893 1.37333i
571.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 2.20711 + 3.82282i −1.00000 1.62132 + 2.09077i 1.00000 −0.500000 0.866025i 2.20711 3.82282i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.j.h 4
7.c even 3 1 inner 798.2.j.h 4
7.c even 3 1 5586.2.a.bg 2
7.d odd 6 1 5586.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.h 4 1.a even 1 1 trivial
798.2.j.h 4 7.c even 3 1 inner
5586.2.a.bg 2 7.c even 3 1
5586.2.a.br 2 7.d odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 6T_{5}^{3} + 29T_{5}^{2} - 42T_{5} + 49$$ T5^4 - 6*T5^3 + 29*T5^2 - 42*T5 + 49 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1 $$T_{13}^{2} - 8$$ T13^2 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$T^{4} - 6 T^{3} + 29 T^{2} - 42 T + 49$$
$7$ $$T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49$$
$11$ $$(T^{2} - T + 1)^{2}$$
$13$ $$(T^{2} - 8)^{2}$$
$17$ $$T^{4} + 18T^{2} + 324$$
$19$ $$(T^{2} + T + 1)^{2}$$
$23$ $$T^{4} + 4 T^{3} + 14 T^{2} + 8 T + 4$$
$29$ $$(T^{2} - 10 T + 17)^{2}$$
$31$ $$T^{4} - 10 T^{3} + 107 T^{2} + \cdots + 49$$
$37$ $$T^{4} - 8 T^{3} + 66 T^{2} + 16 T + 4$$
$41$ $$(T^{2} + 4 T + 2)^{2}$$
$43$ $$(T^{2} - 4 T - 14)^{2}$$
$47$ $$T^{4} + 12 T^{3} + 116 T^{2} + \cdots + 784$$
$53$ $$T^{4} - 2 T^{3} + 11 T^{2} + 14 T + 49$$
$59$ $$T^{4} - 2 T^{3} + 53 T^{2} + \cdots + 2401$$
$61$ $$T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16$$
$67$ $$(T^{2} - 2 T + 4)^{2}$$
$71$ $$(T^{2} - 4 T - 94)^{2}$$
$73$ $$T^{4} - 24 T^{3} + 440 T^{2} + \cdots + 18496$$
$79$ $$T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1$$
$83$ $$(T^{2} + 6 T - 191)^{2}$$
$89$ $$T^{4} + 16 T^{3} + 210 T^{2} + \cdots + 2116$$
$97$ $$(T^{2} + 14 T + 47)^{2}$$