Properties

 Label 740.1.bu.a Level $740$ Weight $1$ Character orbit 740.bu Analytic conductor $0.369$ Analytic rank $0$ Dimension $6$ Projective image $D_{18}$ CM discriminant -4 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,1,Mod(99,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 5]))

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

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

chi := DirichletCharacter("740.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 740.bu (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: $$0.369308109348$$ 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: yes Projective image: $$D_{18}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{18} - \cdots)$$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{8} q^{2} - \zeta_{18}^{7} q^{4} - \zeta_{18}^{5} q^{5} + \zeta_{18}^{6} q^{8} - \zeta_{18}^{2} q^{9} +O(q^{10})$$ q + z^8 * q^2 - z^7 * q^4 - z^5 * q^5 + z^6 * q^8 - z^2 * q^9 $$q + \zeta_{18}^{8} q^{2} - \zeta_{18}^{7} q^{4} - \zeta_{18}^{5} q^{5} + \zeta_{18}^{6} q^{8} - \zeta_{18}^{2} q^{9} + \zeta_{18}^{4} q^{10} - \zeta_{18}^{7} q^{13} - \zeta_{18}^{5} q^{16} + (\zeta_{18}^{7} - \zeta_{18}^{6}) q^{17} + \zeta_{18} q^{18} - \zeta_{18}^{3} q^{20} - \zeta_{18} q^{25} + \zeta_{18}^{6} q^{26} + (\zeta_{18}^{8} - \zeta_{18}^{4}) q^{29} + \zeta_{18}^{4} q^{32} + ( - \zeta_{18}^{6} + \zeta_{18}^{5}) q^{34} - q^{36} - \zeta_{18}^{7} q^{37} + \zeta_{18}^{2} q^{40} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{41} + \zeta_{18}^{7} q^{45} - \zeta_{18}^{8} q^{49} + q^{50} - \zeta_{18}^{5} q^{52} + ( - \zeta_{18}^{8} + \zeta_{18}^{2}) q^{53} + ( - \zeta_{18}^{7} + \zeta_{18}^{3}) q^{58} + (\zeta_{18}^{5} + 1) q^{61} - \zeta_{18}^{3} q^{64} - \zeta_{18}^{3} q^{65} + (\zeta_{18}^{5} - \zeta_{18}^{4}) q^{68} - \zeta_{18}^{8} q^{72} + (\zeta_{18}^{6} + \zeta_{18}^{3}) q^{73} + \zeta_{18}^{6} q^{74} - \zeta_{18} q^{80} + \zeta_{18}^{4} q^{81} + (\zeta_{18}^{2} - \zeta_{18}) q^{82} + (\zeta_{18}^{3} - \zeta_{18}^{2}) q^{85} + ( - \zeta_{18} - 1) q^{89} - \zeta_{18}^{6} q^{90} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{97} + \zeta_{18}^{7} q^{98} +O(q^{100})$$ q + z^8 * q^2 - z^7 * q^4 - z^5 * q^5 + z^6 * q^8 - z^2 * q^9 + z^4 * q^10 - z^7 * q^13 - z^5 * q^16 + (z^7 - z^6) * q^17 + z * q^18 - z^3 * q^20 - z * q^25 + z^6 * q^26 + (z^8 - z^4) * q^29 + z^4 * q^32 + (-z^6 + z^5) * q^34 - q^36 - z^7 * q^37 + z^2 * q^40 + (-z^3 + z^2) * q^41 + z^7 * q^45 - z^8 * q^49 + q^50 - z^5 * q^52 + (-z^8 + z^2) * q^53 + (-z^7 + z^3) * q^58 + (z^5 + 1) * q^61 - z^3 * q^64 - z^3 * q^65 + (z^5 - z^4) * q^68 - z^8 * q^72 + (z^6 + z^3) * q^73 + z^6 * q^74 - z * q^80 + z^4 * q^81 + (z^2 - z) * q^82 + (z^3 - z^2) * q^85 + (-z - 1) * q^89 - z^6 * q^90 + (-z^5 - z) * q^97 + z^7 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{8}+O(q^{10})$$ 6 * q - 3 * q^8 $$6 q - 3 q^{8} + 3 q^{17} - 3 q^{20} - 3 q^{26} + 3 q^{34} - 6 q^{36} - 3 q^{41} + 6 q^{50} + 3 q^{58} + 6 q^{61} - 3 q^{64} - 3 q^{65} - 3 q^{74} + 3 q^{85} - 6 q^{89} + 3 q^{90}+O(q^{100})$$ 6 * q - 3 * q^8 + 3 * q^17 - 3 * q^20 - 3 * q^26 + 3 * q^34 - 6 * q^36 - 3 * q^41 + 6 * q^50 + 3 * q^58 + 6 * q^61 - 3 * q^64 - 3 * q^65 - 3 * q^74 + 3 * q^85 - 6 * q^89 + 3 * q^90

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$\zeta_{18}^{7}$$ $$-1$$ $$-1$$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 −0.766044 − 0.642788i −0.173648 + 0.984808i −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 − 0.984808i 0.939693 + 0.342020i
0.766044 0.642788i 0 0.173648 0.984808i −0.939693 0.342020i 0 0 −0.500000 0.866025i −0.173648 0.984808i −0.939693 + 0.342020i
139.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 0.766044 0.642788i 0 0 −0.500000 0.866025i 0.939693 + 0.342020i 0.766044 + 0.642788i
299.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.939693 + 0.342020i 0 0 −0.500000 + 0.866025i −0.173648 + 0.984808i −0.939693 0.342020i
539.1 −0.939693 0.342020i 0 0.766044 + 0.642788i 0.173648 + 0.984808i 0 0 −0.500000 0.866025i −0.766044 + 0.642788i 0.173648 0.984808i
559.1 0.173648 0.984808i 0 −0.939693 0.342020i 0.766044 + 0.642788i 0 0 −0.500000 + 0.866025i 0.939693 0.342020i 0.766044 0.642788i
659.1 −0.939693 + 0.342020i 0 0.766044 0.642788i 0.173648 0.984808i 0 0 −0.500000 + 0.866025i −0.766044 0.642788i 0.173648 + 0.984808i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 99.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
185.v even 18 1 inner
740.bu odd 18 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.bu.a 6
4.b odd 2 1 CM 740.1.bu.a 6
5.b even 2 1 740.1.bu.b yes 6
5.c odd 4 2 3700.1.cc.a 12
20.d odd 2 1 740.1.bu.b yes 6
20.e even 4 2 3700.1.cc.a 12
37.h even 18 1 740.1.bu.b yes 6
148.o odd 18 1 740.1.bu.b yes 6
185.v even 18 1 inner 740.1.bu.a 6
185.y odd 36 2 3700.1.cc.a 12
740.bu odd 18 1 inner 740.1.bu.a 6
740.ce even 36 2 3700.1.cc.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.bu.a 6 1.a even 1 1 trivial
740.1.bu.a 6 4.b odd 2 1 CM
740.1.bu.a 6 185.v even 18 1 inner
740.1.bu.a 6 740.bu odd 18 1 inner
740.1.bu.b yes 6 5.b even 2 1
740.1.bu.b yes 6 20.d odd 2 1
740.1.bu.b yes 6 37.h even 18 1
740.1.bu.b yes 6 148.o odd 18 1
3700.1.cc.a 12 5.c odd 4 2
3700.1.cc.a 12 20.e even 4 2
3700.1.cc.a 12 185.y odd 36 2
3700.1.cc.a 12 740.ce even 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{6} + T_{13}^{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(740, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{3} + 1$$
$3$ $$T^{6}$$
$5$ $$T^{6} + T^{3} + 1$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$T^{6} + T^{3} + 1$$
$17$ $$T^{6} - 3 T^{5} + \cdots + 1$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$T^{6} - 3 T^{4} + \cdots + 3$$
$31$ $$T^{6}$$
$37$ $$T^{6} + T^{3} + 1$$
$41$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$43$ $$T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6} - 9T^{3} + 27$$
$59$ $$T^{6}$$
$61$ $$T^{6} - 6 T^{5} + \cdots + 3$$
$67$ $$T^{6}$$
$71$ $$T^{6}$$
$73$ $$(T^{2} + 3)^{3}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6} + 6 T^{5} + \cdots + 3$$
$97$ $$T^{6} + 3 T^{4} + \cdots + 1$$