# Properties

 Label 798.2.k.i 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 $$q + ( - \zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + ( - 3 \zeta_{6} + 3) q^{5} - \zeta_{6} q^{6} - q^{7} - q^{8} - \zeta_{6} q^{9} - 3 \zeta_{6} q^{10} + 4 q^{11} - q^{12} - 3 \zeta_{6} q^{13} + (\zeta_{6} - 1) q^{14} - 3 \zeta_{6} q^{15} + (\zeta_{6} - 1) q^{16} + ( - 2 \zeta_{6} + 2) q^{17} - q^{18} + (5 \zeta_{6} - 3) q^{19} - 3 q^{20} + (\zeta_{6} - 1) q^{21} + ( - 4 \zeta_{6} + 4) q^{22} - \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{24} - 4 \zeta_{6} q^{25} - 3 q^{26} - q^{27} + \zeta_{6} q^{28} - 3 q^{30} - 6 q^{31} + \zeta_{6} q^{32} + ( - 4 \zeta_{6} + 4) q^{33} - 2 \zeta_{6} q^{34} + (3 \zeta_{6} - 3) q^{35} + (\zeta_{6} - 1) q^{36} + 4 q^{37} + (3 \zeta_{6} + 2) q^{38} - 3 q^{39} + (3 \zeta_{6} - 3) q^{40} + (4 \zeta_{6} - 4) q^{41} + \zeta_{6} q^{42} + ( - 6 \zeta_{6} + 6) q^{43} - 4 \zeta_{6} q^{44} - 3 q^{45} - q^{46} + 2 \zeta_{6} q^{47} + \zeta_{6} q^{48} + q^{49} - 4 q^{50} - 2 \zeta_{6} q^{51} + (3 \zeta_{6} - 3) q^{52} + 4 \zeta_{6} q^{53} + (\zeta_{6} - 1) q^{54} + ( - 12 \zeta_{6} + 12) q^{55} + q^{56} + (3 \zeta_{6} + 2) q^{57} + (13 \zeta_{6} - 13) q^{59} + (3 \zeta_{6} - 3) q^{60} - 5 \zeta_{6} q^{61} + (6 \zeta_{6} - 6) q^{62} + \zeta_{6} q^{63} + q^{64} - 9 q^{65} - 4 \zeta_{6} q^{66} + 12 \zeta_{6} q^{67} - 2 q^{68} - q^{69} + 3 \zeta_{6} q^{70} + ( - 7 \zeta_{6} + 7) q^{71} + \zeta_{6} q^{72} + ( - 2 \zeta_{6} + 2) q^{73} + ( - 4 \zeta_{6} + 4) q^{74} - 4 q^{75} + ( - 2 \zeta_{6} + 5) q^{76} - 4 q^{77} + (3 \zeta_{6} - 3) q^{78} + ( - 8 \zeta_{6} + 8) q^{79} + 3 \zeta_{6} q^{80} + (\zeta_{6} - 1) q^{81} + 4 \zeta_{6} q^{82} + 17 q^{83} + q^{84} - 6 \zeta_{6} q^{85} - 6 \zeta_{6} q^{86} - 4 q^{88} - 4 \zeta_{6} q^{89} + (3 \zeta_{6} - 3) q^{90} + 3 \zeta_{6} q^{91} + (\zeta_{6} - 1) q^{92} + (6 \zeta_{6} - 6) q^{93} + 2 q^{94} + (9 \zeta_{6} + 6) q^{95} + q^{96} + ( - 10 \zeta_{6} + 10) q^{97} + ( - \zeta_{6} + 1) q^{98} - 4 \zeta_{6} q^{99} +O(q^{100})$$ 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 + 4 * q^11 - q^12 - 3*z * q^13 + (z - 1) * q^14 - 3*z * q^15 + (z - 1) * q^16 + (-2*z + 2) * q^17 - q^18 + (5*z - 3) * q^19 - 3 * q^20 + (z - 1) * q^21 + (-4*z + 4) * q^22 - z * q^23 + (z - 1) * q^24 - 4*z * q^25 - 3 * q^26 - q^27 + z * q^28 - 3 * q^30 - 6 * q^31 + z * q^32 + (-4*z + 4) * q^33 - 2*z * q^34 + (3*z - 3) * q^35 + (z - 1) * q^36 + 4 * q^37 + (3*z + 2) * q^38 - 3 * q^39 + (3*z - 3) * q^40 + (4*z - 4) * q^41 + z * q^42 + (-6*z + 6) * q^43 - 4*z * q^44 - 3 * q^45 - q^46 + 2*z * q^47 + z * q^48 + q^49 - 4 * q^50 - 2*z * q^51 + (3*z - 3) * q^52 + 4*z * q^53 + (z - 1) * q^54 + (-12*z + 12) * q^55 + q^56 + (3*z + 2) * q^57 + (13*z - 13) * q^59 + (3*z - 3) * q^60 - 5*z * q^61 + (6*z - 6) * q^62 + z * q^63 + q^64 - 9 * q^65 - 4*z * q^66 + 12*z * q^67 - 2 * q^68 - q^69 + 3*z * q^70 + (-7*z + 7) * q^71 + z * q^72 + (-2*z + 2) * q^73 + (-4*z + 4) * q^74 - 4 * q^75 + (-2*z + 5) * q^76 - 4 * q^77 + (3*z - 3) * q^78 + (-8*z + 8) * q^79 + 3*z * q^80 + (z - 1) * q^81 + 4*z * q^82 + 17 * q^83 + q^84 - 6*z * q^85 - 6*z * q^86 - 4 * q^88 - 4*z * q^89 + (3*z - 3) * q^90 + 3*z * q^91 + (z - 1) * q^92 + (6*z - 6) * q^93 + 2 * q^94 + (9*z + 6) * q^95 + q^96 + (-10*z + 10) * q^97 + (-z + 1) * q^98 - 4*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 $$2 q + q^{2} + q^{3} - q^{4} + 3 q^{5} - q^{6} - 2 q^{7} - 2 q^{8} - q^{9} - 3 q^{10} + 8 q^{11} - 2 q^{12} - 3 q^{13} - q^{14} - 3 q^{15} - q^{16} + 2 q^{17} - 2 q^{18} - q^{19} - 6 q^{20} - q^{21} + 4 q^{22} - q^{23} - q^{24} - 4 q^{25} - 6 q^{26} - 2 q^{27} + q^{28} - 6 q^{30} - 12 q^{31} + q^{32} + 4 q^{33} - 2 q^{34} - 3 q^{35} - q^{36} + 8 q^{37} + 7 q^{38} - 6 q^{39} - 3 q^{40} - 4 q^{41} + q^{42} + 6 q^{43} - 4 q^{44} - 6 q^{45} - 2 q^{46} + 2 q^{47} + q^{48} + 2 q^{49} - 8 q^{50} - 2 q^{51} - 3 q^{52} + 4 q^{53} - q^{54} + 12 q^{55} + 2 q^{56} + 7 q^{57} - 13 q^{59} - 3 q^{60} - 5 q^{61} - 6 q^{62} + q^{63} + 2 q^{64} - 18 q^{65} - 4 q^{66} + 12 q^{67} - 4 q^{68} - 2 q^{69} + 3 q^{70} + 7 q^{71} + q^{72} + 2 q^{73} + 4 q^{74} - 8 q^{75} + 8 q^{76} - 8 q^{77} - 3 q^{78} + 8 q^{79} + 3 q^{80} - q^{81} + 4 q^{82} + 34 q^{83} + 2 q^{84} - 6 q^{85} - 6 q^{86} - 8 q^{88} - 4 q^{89} - 3 q^{90} + 3 q^{91} - q^{92} - 6 q^{93} + 4 q^{94} + 21 q^{95} + 2 q^{96} + 10 q^{97} + q^{98} - 4 q^{99}+O(q^{100})$$ 2 * q + q^2 + q^3 - q^4 + 3 * q^5 - q^6 - 2 * q^7 - 2 * q^8 - q^9 - 3 * q^10 + 8 * q^11 - 2 * q^12 - 3 * q^13 - q^14 - 3 * q^15 - q^16 + 2 * q^17 - 2 * q^18 - q^19 - 6 * q^20 - q^21 + 4 * q^22 - q^23 - q^24 - 4 * q^25 - 6 * q^26 - 2 * q^27 + q^28 - 6 * q^30 - 12 * q^31 + q^32 + 4 * q^33 - 2 * q^34 - 3 * q^35 - q^36 + 8 * q^37 + 7 * q^38 - 6 * q^39 - 3 * q^40 - 4 * q^41 + q^42 + 6 * q^43 - 4 * q^44 - 6 * q^45 - 2 * q^46 + 2 * q^47 + q^48 + 2 * q^49 - 8 * q^50 - 2 * q^51 - 3 * q^52 + 4 * q^53 - q^54 + 12 * q^55 + 2 * q^56 + 7 * q^57 - 13 * q^59 - 3 * q^60 - 5 * q^61 - 6 * q^62 + q^63 + 2 * q^64 - 18 * q^65 - 4 * q^66 + 12 * q^67 - 4 * q^68 - 2 * q^69 + 3 * q^70 + 7 * q^71 + q^72 + 2 * q^73 + 4 * q^74 - 8 * q^75 + 8 * q^76 - 8 * q^77 - 3 * q^78 + 8 * q^79 + 3 * q^80 - q^81 + 4 * q^82 + 34 * q^83 + 2 * q^84 - 6 * q^85 - 6 * q^86 - 8 * q^88 - 4 * q^89 - 3 * q^90 + 3 * q^91 - q^92 - 6 * q^93 + 4 * q^94 + 21 * q^95 + 2 * q^96 + 10 * q^97 + q^98 - 4 * 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.i 2
3.b odd 2 1 2394.2.o.a 2
19.c even 3 1 inner 798.2.k.i 2
57.h odd 6 1 2394.2.o.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.k.i 2 1.a even 1 1 trivial
798.2.k.i 2 19.c even 3 1 inner
2394.2.o.a 2 3.b odd 2 1
2394.2.o.a 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 $$T_{11} - 4$$ T11 - 4 $$T_{13}^{2} + 3T_{13} + 9$$ T13^2 + 3*T13 + 9 $$T_{17}^{2} - 2T_{17} + 4$$ T17^2 - 2*T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 3T + 9$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} + T + 19$$
$23$ $$T^{2} + T + 1$$
$29$ $$T^{2}$$
$31$ $$(T + 6)^{2}$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} + 4T + 16$$
$43$ $$T^{2} - 6T + 36$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$T^{2} - 4T + 16$$
$59$ $$T^{2} + 13T + 169$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} - 12T + 144$$
$71$ $$T^{2} - 7T + 49$$
$73$ $$T^{2} - 2T + 4$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T - 17)^{2}$$
$89$ $$T^{2} + 4T + 16$$
$97$ $$T^{2} - 10T + 100$$