# Properties

 Label 1368.1.eh.a Level $1368$ Weight $1$ Character orbit 1368.eh Analytic conductor $0.683$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,1,Mod(595,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 0, 14]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1368.595");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1368.eh (of order $$18$$, degree $$6$$, 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: $$0.682720937282$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.69564674215936.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{5} q^{2} - \zeta_{18} q^{4} - \zeta_{18}^{6} q^{8} +O(q^{10})$$ q + z^5 * q^2 - z * q^4 - z^6 * q^8 $$q + \zeta_{18}^{5} q^{2} - \zeta_{18} q^{4} - \zeta_{18}^{6} q^{8} + ( - \zeta_{18}^{8} - \zeta_{18}^{4}) q^{11} + \zeta_{18}^{2} q^{16} - \zeta_{18}^{5} q^{17} - \zeta_{18} q^{19} + (\zeta_{18}^{4} + 1) q^{22} - \zeta_{18}^{7} q^{25} + \zeta_{18}^{7} q^{32} + \zeta_{18} q^{34} - \zeta_{18}^{6} q^{38} + (\zeta_{18}^{3} + \zeta_{18}) q^{41} - \zeta_{18}^{8} q^{43} + (\zeta_{18}^{5} - 1) q^{44} + \zeta_{18}^{6} q^{49} + \zeta_{18}^{3} q^{50} + (\zeta_{18}^{7} + \zeta_{18}^{3}) q^{59} - \zeta_{18}^{3} q^{64} + ( - \zeta_{18}^{5} - \zeta_{18}^{3}) q^{67} + \zeta_{18}^{6} q^{68} + (\zeta_{18}^{4} + 1) q^{73} + \zeta_{18}^{2} q^{76} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{82} + ( - \zeta_{18}^{4} - \zeta_{18}^{2}) q^{83} + \zeta_{18}^{4} q^{86} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{88} - \zeta_{18}^{7} q^{89} + (\zeta_{18}^{6} + \zeta_{18}^{4}) q^{97} - \zeta_{18}^{2} q^{98} +O(q^{100})$$ q + z^5 * q^2 - z * q^4 - z^6 * q^8 + (-z^8 - z^4) * q^11 + z^2 * q^16 - z^5 * q^17 - z * q^19 + (z^4 + 1) * q^22 - z^7 * q^25 + z^7 * q^32 + z * q^34 - z^6 * q^38 + (z^3 + z) * q^41 - z^8 * q^43 + (z^5 - 1) * q^44 + z^6 * q^49 + z^3 * q^50 + (z^7 + z^3) * q^59 - z^3 * q^64 + (-z^5 - z^3) * q^67 + z^6 * q^68 + (z^4 + 1) * q^73 + z^2 * q^76 + (z^8 + z^6) * q^82 + (-z^4 - z^2) * q^83 + z^4 * q^86 + (-z^5 - z) * q^88 - z^7 * q^89 + (z^6 + z^4) * q^97 - z^2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{8}+O(q^{10})$$ 6 * q + 3 * q^8 $$6 q + 3 q^{8} + 6 q^{22} + 3 q^{38} + 3 q^{41} - 6 q^{44} - 3 q^{49} + 3 q^{50} + 3 q^{59} - 3 q^{64} - 3 q^{67} - 3 q^{68} + 6 q^{73} - 3 q^{82} - 3 q^{97}+O(q^{100})$$ 6 * q + 3 * q^8 + 6 * q^22 + 3 * q^38 + 3 * q^41 - 6 * q^44 - 3 * q^49 + 3 * q^50 + 3 * q^59 - 3 * q^64 - 3 * q^67 - 3 * q^68 + 6 * q^73 - 3 * q^82 - 3 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$$.

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{18}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
595.1
 −0.173648 + 0.984808i 0.939693 + 0.342020i 0.939693 − 0.342020i −0.766044 + 0.642788i −0.173648 − 0.984808i −0.766044 − 0.642788i
−0.766044 + 0.642788i 0 0.173648 0.984808i 0 0 0 0.500000 + 0.866025i 0 0
739.1 −0.173648 + 0.984808i 0 −0.939693 0.342020i 0 0 0 0.500000 0.866025i 0 0
883.1 −0.173648 0.984808i 0 −0.939693 + 0.342020i 0 0 0 0.500000 + 0.866025i 0 0
955.1 0.939693 0.342020i 0 0.766044 0.642788i 0 0 0 0.500000 0.866025i 0 0
1099.1 −0.766044 0.642788i 0 0.173648 + 0.984808i 0 0 0 0.500000 0.866025i 0 0
1315.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i 0 0 0 0.500000 + 0.866025i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 595.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
19.e even 9 1 inner
152.u odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.1.eh.a 6
3.b odd 2 1 152.1.u.a 6
8.d odd 2 1 CM 1368.1.eh.a 6
12.b even 2 1 608.1.bg.a 6
15.d odd 2 1 3800.1.cv.c 6
15.e even 4 2 3800.1.cq.b 12
19.e even 9 1 inner 1368.1.eh.a 6
24.f even 2 1 152.1.u.a 6
24.h odd 2 1 608.1.bg.a 6
57.d even 2 1 2888.1.u.e 6
57.f even 6 1 2888.1.u.a 6
57.f even 6 1 2888.1.u.f 6
57.h odd 6 1 2888.1.u.b 6
57.h odd 6 1 2888.1.u.g 6
57.j even 18 1 2888.1.f.c 3
57.j even 18 2 2888.1.k.c 6
57.j even 18 1 2888.1.u.a 6
57.j even 18 1 2888.1.u.e 6
57.j even 18 1 2888.1.u.f 6
57.l odd 18 1 152.1.u.a 6
57.l odd 18 1 2888.1.f.d 3
57.l odd 18 2 2888.1.k.b 6
57.l odd 18 1 2888.1.u.b 6
57.l odd 18 1 2888.1.u.g 6
120.m even 2 1 3800.1.cv.c 6
120.q odd 4 2 3800.1.cq.b 12
152.u odd 18 1 inner 1368.1.eh.a 6
228.v even 18 1 608.1.bg.a 6
285.bd odd 18 1 3800.1.cv.c 6
285.bi even 36 2 3800.1.cq.b 12
456.l odd 2 1 2888.1.u.e 6
456.s odd 6 1 2888.1.u.a 6
456.s odd 6 1 2888.1.u.f 6
456.u even 6 1 2888.1.u.b 6
456.u even 6 1 2888.1.u.g 6
456.bh odd 18 1 608.1.bg.a 6
456.bt odd 18 1 2888.1.f.c 3
456.bt odd 18 2 2888.1.k.c 6
456.bt odd 18 1 2888.1.u.a 6
456.bt odd 18 1 2888.1.u.e 6
456.bt odd 18 1 2888.1.u.f 6
456.bu even 18 1 152.1.u.a 6
456.bu even 18 1 2888.1.f.d 3
456.bu even 18 2 2888.1.k.b 6
456.bu even 18 1 2888.1.u.b 6
456.bu even 18 1 2888.1.u.g 6
2280.eg even 18 1 3800.1.cv.c 6
2280.fl odd 36 2 3800.1.cq.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 3.b odd 2 1
152.1.u.a 6 24.f even 2 1
152.1.u.a 6 57.l odd 18 1
152.1.u.a 6 456.bu even 18 1
608.1.bg.a 6 12.b even 2 1
608.1.bg.a 6 24.h odd 2 1
608.1.bg.a 6 228.v even 18 1
608.1.bg.a 6 456.bh odd 18 1
1368.1.eh.a 6 1.a even 1 1 trivial
1368.1.eh.a 6 8.d odd 2 1 CM
1368.1.eh.a 6 19.e even 9 1 inner
1368.1.eh.a 6 152.u odd 18 1 inner
2888.1.f.c 3 57.j even 18 1
2888.1.f.c 3 456.bt odd 18 1
2888.1.f.d 3 57.l odd 18 1
2888.1.f.d 3 456.bu even 18 1
2888.1.k.b 6 57.l odd 18 2
2888.1.k.b 6 456.bu even 18 2
2888.1.k.c 6 57.j even 18 2
2888.1.k.c 6 456.bt odd 18 2
2888.1.u.a 6 57.f even 6 1
2888.1.u.a 6 57.j even 18 1
2888.1.u.a 6 456.s odd 6 1
2888.1.u.a 6 456.bt odd 18 1
2888.1.u.b 6 57.h odd 6 1
2888.1.u.b 6 57.l odd 18 1
2888.1.u.b 6 456.u even 6 1
2888.1.u.b 6 456.bu even 18 1
2888.1.u.e 6 57.d even 2 1
2888.1.u.e 6 57.j even 18 1
2888.1.u.e 6 456.l odd 2 1
2888.1.u.e 6 456.bt odd 18 1
2888.1.u.f 6 57.f even 6 1
2888.1.u.f 6 57.j even 18 1
2888.1.u.f 6 456.s odd 6 1
2888.1.u.f 6 456.bt odd 18 1
2888.1.u.g 6 57.h odd 6 1
2888.1.u.g 6 57.l odd 18 1
2888.1.u.g 6 456.u even 6 1
2888.1.u.g 6 456.bu even 18 1
3800.1.cq.b 12 15.e even 4 2
3800.1.cq.b 12 120.q odd 4 2
3800.1.cq.b 12 285.bi even 36 2
3800.1.cq.b 12 2280.fl odd 36 2
3800.1.cv.c 6 15.d odd 2 1
3800.1.cv.c 6 120.m even 2 1
3800.1.cv.c 6 285.bd odd 18 1
3800.1.cv.c 6 2280.eg even 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1368, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{3} + 1$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6} + 3 T^{4} + \cdots + 1$$
$13$ $$T^{6}$$
$17$ $$T^{6} + T^{3} + 1$$
$19$ $$T^{6} + T^{3} + 1$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6}$$
$37$ $$T^{6}$$
$41$ $$T^{6} - 3 T^{5} + \cdots + 1$$
$43$ $$T^{6} - T^{3} + 1$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6} - 3 T^{5} + \cdots + 1$$
$61$ $$T^{6}$$
$67$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$71$ $$T^{6}$$
$73$ $$T^{6} - 6 T^{5} + \cdots + 1$$
$79$ $$T^{6}$$
$83$ $$T^{6} + 3 T^{4} + \cdots + 1$$
$89$ $$T^{6} + T^{3} + 1$$
$97$ $$T^{6} + 3 T^{5} + \cdots + 1$$