# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [798,2,Mod(463,798)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(798, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("798.463");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$6.37206208130$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$115$$ $$211$$ $$533$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
463.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.50000 + 2.59808i 0.500000 0.866025i 1.00000 −1.00000 −0.500000 + 0.866025i −1.50000 + 2.59808i
505.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.50000 2.59808i 0.500000 + 0.866025i 1.00000 −1.00000 −0.500000 0.866025i −1.50000 2.59808i
 $$n$$: 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
19.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.k.h 2
3.b odd 2 1 2394.2.o.b 2
19.c even 3 1 inner 798.2.k.h 2
57.h odd 6 1 2394.2.o.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.k.h 2 1.a even 1 1 trivial
798.2.k.h 2 19.c even 3 1 inner
2394.2.o.b 2 3.b odd 2 1
2394.2.o.b 2 57.h odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{11} + 6$$ T11 + 6 $$T_{13}^{2} + 5T_{13} + 25$$ T13^2 + 5*T13 + 25 $$T_{17}^{2} - 6T_{17} + 36$$ T17^2 - 6*T17 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T + 6)^{2}$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + T + 19$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2} - 9T + 81$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$T^{2} - 3T + 9$$
$73$ $$T^{2} - 4T + 16$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$(T - 15)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$T^{2} + 8T + 64$$