# Properties

 Label 1782.2.e.f Level $1782$ Weight $2$ Character orbit 1782.e Analytic conductor $14.229$ 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] = [1782,2,Mod(595,1782)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1782, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1782.595");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1782 = 2 \cdot 3^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1782.e (of order $$3$$, degree $$2$$, not 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: $$14.2293416402$$ 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: no (minimal twist has level 66) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (2 \zeta_{6} - 2) q^{7} + q^{8} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 + (2*z - 2) * q^7 + q^8 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (2 \zeta_{6} - 2) q^{7} + q^{8} + (\zeta_{6} - 1) q^{11} + 4 \zeta_{6} q^{13} - 2 \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} + 6 q^{17} - 4 q^{19} - \zeta_{6} q^{22} + 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} - 4 q^{26} + 2 q^{28} + ( - 6 \zeta_{6} + 6) q^{29} - 8 \zeta_{6} q^{31} - \zeta_{6} q^{32} + (6 \zeta_{6} - 6) q^{34} - 10 q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + 6 \zeta_{6} q^{41} + (8 \zeta_{6} - 8) q^{43} + q^{44} - 6 q^{46} + (6 \zeta_{6} - 6) q^{47} + 3 \zeta_{6} q^{49} + 5 \zeta_{6} q^{50} + ( - 4 \zeta_{6} + 4) q^{52} + (2 \zeta_{6} - 2) q^{56} + 6 \zeta_{6} q^{58} + (8 \zeta_{6} - 8) q^{61} + 8 q^{62} + q^{64} + 4 \zeta_{6} q^{67} - 6 \zeta_{6} q^{68} - 6 q^{71} + 2 q^{73} + ( - 10 \zeta_{6} + 10) q^{74} + 4 \zeta_{6} q^{76} - 2 \zeta_{6} q^{77} + (14 \zeta_{6} - 14) q^{79} - 6 q^{82} + (12 \zeta_{6} - 12) q^{83} - 8 \zeta_{6} q^{86} + (\zeta_{6} - 1) q^{88} + 6 q^{89} - 8 q^{91} + ( - 6 \zeta_{6} + 6) q^{92} - 6 \zeta_{6} q^{94} + (14 \zeta_{6} - 14) q^{97} - 3 q^{98} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 + (2*z - 2) * q^7 + q^8 + (z - 1) * q^11 + 4*z * q^13 - 2*z * q^14 + (z - 1) * q^16 + 6 * q^17 - 4 * q^19 - z * q^22 + 6*z * q^23 + (-5*z + 5) * q^25 - 4 * q^26 + 2 * q^28 + (-6*z + 6) * q^29 - 8*z * q^31 - z * q^32 + (6*z - 6) * q^34 - 10 * q^37 + (-4*z + 4) * q^38 + 6*z * q^41 + (8*z - 8) * q^43 + q^44 - 6 * q^46 + (6*z - 6) * q^47 + 3*z * q^49 + 5*z * q^50 + (-4*z + 4) * q^52 + (2*z - 2) * q^56 + 6*z * q^58 + (8*z - 8) * q^61 + 8 * q^62 + q^64 + 4*z * q^67 - 6*z * q^68 - 6 * q^71 + 2 * q^73 + (-10*z + 10) * q^74 + 4*z * q^76 - 2*z * q^77 + (14*z - 14) * q^79 - 6 * q^82 + (12*z - 12) * q^83 - 8*z * q^86 + (z - 1) * q^88 + 6 * q^89 - 8 * q^91 + (-6*z + 6) * q^92 - 6*z * q^94 + (14*z - 14) * q^97 - 3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 - 2 * q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} - 2 q^{7} + 2 q^{8} - q^{11} + 4 q^{13} - 2 q^{14} - q^{16} + 12 q^{17} - 8 q^{19} - q^{22} + 6 q^{23} + 5 q^{25} - 8 q^{26} + 4 q^{28} + 6 q^{29} - 8 q^{31} - q^{32} - 6 q^{34} - 20 q^{37} + 4 q^{38} + 6 q^{41} - 8 q^{43} + 2 q^{44} - 12 q^{46} - 6 q^{47} + 3 q^{49} + 5 q^{50} + 4 q^{52} - 2 q^{56} + 6 q^{58} - 8 q^{61} + 16 q^{62} + 2 q^{64} + 4 q^{67} - 6 q^{68} - 12 q^{71} + 4 q^{73} + 10 q^{74} + 4 q^{76} - 2 q^{77} - 14 q^{79} - 12 q^{82} - 12 q^{83} - 8 q^{86} - q^{88} + 12 q^{89} - 16 q^{91} + 6 q^{92} - 6 q^{94} - 14 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - 2 * q^7 + 2 * q^8 - q^11 + 4 * q^13 - 2 * q^14 - q^16 + 12 * q^17 - 8 * q^19 - q^22 + 6 * q^23 + 5 * q^25 - 8 * q^26 + 4 * q^28 + 6 * q^29 - 8 * q^31 - q^32 - 6 * q^34 - 20 * q^37 + 4 * q^38 + 6 * q^41 - 8 * q^43 + 2 * q^44 - 12 * q^46 - 6 * q^47 + 3 * q^49 + 5 * q^50 + 4 * q^52 - 2 * q^56 + 6 * q^58 - 8 * q^61 + 16 * q^62 + 2 * q^64 + 4 * q^67 - 6 * q^68 - 12 * q^71 + 4 * q^73 + 10 * q^74 + 4 * q^76 - 2 * q^77 - 14 * q^79 - 12 * q^82 - 12 * q^83 - 8 * q^86 - q^88 + 12 * q^89 - 16 * q^91 + 6 * q^92 - 6 * q^94 - 14 * q^97 - 6 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1782\mathbb{Z}\right)^\times$$.

 $$n$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
595.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −1.00000 1.73205i 1.00000 0 0
1189.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −1.00000 + 1.73205i 1.00000 0 0
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1782.2.e.f 2
3.b odd 2 1 1782.2.e.s 2
9.c even 3 1 198.2.a.e 1
9.c even 3 1 inner 1782.2.e.f 2
9.d odd 6 1 66.2.a.a 1
9.d odd 6 1 1782.2.e.s 2
36.f odd 6 1 1584.2.a.h 1
36.h even 6 1 528.2.a.d 1
45.h odd 6 1 1650.2.a.m 1
45.j even 6 1 4950.2.a.g 1
45.k odd 12 2 4950.2.c.r 2
45.l even 12 2 1650.2.c.d 2
63.l odd 6 1 9702.2.a.bu 1
63.o even 6 1 3234.2.a.d 1
72.j odd 6 1 2112.2.a.i 1
72.l even 6 1 2112.2.a.v 1
72.n even 6 1 6336.2.a.bj 1
72.p odd 6 1 6336.2.a.bf 1
99.g even 6 1 726.2.a.i 1
99.h odd 6 1 2178.2.a.b 1
99.n odd 30 4 726.2.e.k 4
99.p even 30 4 726.2.e.b 4
396.o odd 6 1 5808.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 9.d odd 6 1
198.2.a.e 1 9.c even 3 1
528.2.a.d 1 36.h even 6 1
726.2.a.i 1 99.g even 6 1
726.2.e.b 4 99.p even 30 4
726.2.e.k 4 99.n odd 30 4
1584.2.a.h 1 36.f odd 6 1
1650.2.a.m 1 45.h odd 6 1
1650.2.c.d 2 45.l even 12 2
1782.2.e.f 2 1.a even 1 1 trivial
1782.2.e.f 2 9.c even 3 1 inner
1782.2.e.s 2 3.b odd 2 1
1782.2.e.s 2 9.d odd 6 1
2112.2.a.i 1 72.j odd 6 1
2112.2.a.v 1 72.l even 6 1
2178.2.a.b 1 99.h odd 6 1
3234.2.a.d 1 63.o even 6 1
4950.2.a.g 1 45.j even 6 1
4950.2.c.r 2 45.k odd 12 2
5808.2.a.l 1 396.o odd 6 1
6336.2.a.bf 1 72.p odd 6 1
6336.2.a.bj 1 72.n even 6 1
9702.2.a.bu 1 63.l odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1782, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4 $$T_{13}^{2} - 4T_{13} + 16$$ T13^2 - 4*T13 + 16 $$T_{17} - 6$$ T17 - 6 $$T_{23}^{2} - 6T_{23} + 36$$ T23^2 - 6*T23 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} + T + 1$$
$13$ $$T^{2} - 4T + 16$$
$17$ $$(T - 6)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} + 8T + 64$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T + 6)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$T^{2} + 12T + 144$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 14T + 196$$