# Properties

 Label 740.1.cb.a Level $740$ Weight $1$ Character orbit 740.cb Analytic conductor $0.369$ Analytic rank $0$ Dimension $12$ Projective image $D_{36}$ 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(87,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([18, 9, 11]))

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

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

chi := DirichletCharacter("740.87");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 740.cb (of order $$36$$, degree $$12$$, 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: $$12$$ Coefficient field: $$\Q(\zeta_{36})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{6} + 1$$ x^12 - x^6 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{36}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{36} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{36}^{8} q^{2} + \zeta_{36}^{16} q^{4} + \zeta_{36}^{14} q^{5} + \zeta_{36}^{6} q^{8} + \zeta_{36}^{11} q^{9} +O(q^{10})$$ q - z^8 * q^2 + z^16 * q^4 + z^14 * q^5 + z^6 * q^8 + z^11 * q^9 $$q - \zeta_{36}^{8} q^{2} + \zeta_{36}^{16} q^{4} + \zeta_{36}^{14} q^{5} + \zeta_{36}^{6} q^{8} + \zeta_{36}^{11} q^{9} + \zeta_{36}^{4} q^{10} + (\zeta_{36}^{13} - \zeta_{36}) q^{13} - \zeta_{36}^{14} q^{16} + (\zeta_{36}^{15} + \zeta_{36}^{7}) q^{17} + \zeta_{36} q^{18} - \zeta_{36}^{12} q^{20} - \zeta_{36}^{10} q^{25} + (\zeta_{36}^{9} + \zeta_{36}^{3}) q^{26} + ( - \zeta_{36}^{13} + \zeta_{36}^{8}) q^{29} - \zeta_{36}^{4} q^{32} + ( - \zeta_{36}^{15} + \zeta_{36}^{5}) q^{34} - \zeta_{36}^{9} q^{36} - \zeta_{36}^{16} q^{37} - \zeta_{36}^{2} q^{40} + (\zeta_{36}^{12} + \zeta_{36}^{2}) q^{41} - \zeta_{36}^{7} q^{45} - \zeta_{36}^{17} q^{49} - q^{50} + ( - \zeta_{36}^{17} - \zeta_{36}^{11}) q^{52} + (\zeta_{36}^{17} - \zeta_{36}^{2}) q^{53} + ( - \zeta_{36}^{16} - \zeta_{36}^{3}) q^{58} + ( - \zeta_{36}^{5} - 1) q^{61} + \zeta_{36}^{12} q^{64} + ( - \zeta_{36}^{15} - \zeta_{36}^{9}) q^{65} + ( - \zeta_{36}^{13} - \zeta_{36}^{5}) q^{68} + \zeta_{36}^{17} q^{72} + ( - \zeta_{36}^{6} + \zeta_{36}^{3}) q^{73} - \zeta_{36}^{6} q^{74} + \zeta_{36}^{10} q^{80} - \zeta_{36}^{4} q^{81} + ( - \zeta_{36}^{10} + \zeta_{36}^{2}) q^{82} + ( - \zeta_{36}^{11} - \zeta_{36}^{3}) q^{85} + (\zeta_{36}^{10} + \zeta_{36}^{9}) q^{89} + \zeta_{36}^{15} q^{90} + ( - \zeta_{36}^{14} + \zeta_{36}^{10}) q^{97} - \zeta_{36}^{7} q^{98} +O(q^{100})$$ q - z^8 * q^2 + z^16 * q^4 + z^14 * q^5 + z^6 * q^8 + z^11 * q^9 + z^4 * q^10 + (z^13 - z) * q^13 - z^14 * q^16 + (z^15 + z^7) * q^17 + z * q^18 - z^12 * q^20 - z^10 * q^25 + (z^9 + z^3) * q^26 + (-z^13 + z^8) * q^29 - z^4 * q^32 + (-z^15 + z^5) * q^34 - z^9 * q^36 - z^16 * q^37 - z^2 * q^40 + (z^12 + z^2) * q^41 - z^7 * q^45 - z^17 * q^49 - q^50 + (-z^17 - z^11) * q^52 + (z^17 - z^2) * q^53 + (-z^16 - z^3) * q^58 + (-z^5 - 1) * q^61 + z^12 * q^64 + (-z^15 - z^9) * q^65 + (-z^13 - z^5) * q^68 + z^17 * q^72 + (-z^6 + z^3) * q^73 - z^6 * q^74 + z^10 * q^80 - z^4 * q^81 + (-z^10 + z^2) * q^82 + (-z^11 - z^3) * q^85 + (z^10 + z^9) * q^89 + z^15 * q^90 + (-z^14 + z^10) * q^97 - z^7 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{8}+O(q^{10})$$ 12 * q + 6 * q^8 $$12 q + 6 q^{8} + 6 q^{20} - 6 q^{41} - 12 q^{50} - 12 q^{61} - 6 q^{64} - 6 q^{73} - 6 q^{74}+O(q^{100})$$ 12 * q + 6 * q^8 + 6 * q^20 - 6 * q^41 - 12 * q^50 - 12 * q^61 - 6 * q^64 - 6 * q^73 - 6 * q^74

## Character values

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

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$-\zeta_{36}^{7}$$ $$-\zeta_{36}^{9}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
87.1
 0.984808 − 0.173648i 0.342020 − 0.939693i −0.342020 − 0.939693i −0.642788 − 0.766044i 0.642788 + 0.766044i 0.342020 + 0.939693i −0.342020 + 0.939693i −0.984808 + 0.173648i −0.984808 − 0.173648i −0.642788 + 0.766044i 0.642788 − 0.766044i 0.984808 + 0.173648i
−0.173648 + 0.984808i 0 −0.939693 0.342020i −0.766044 0.642788i 0 0 0.500000 0.866025i −0.342020 0.939693i 0.766044 0.642788i
143.1 0.939693 0.342020i 0 0.766044 0.642788i −0.173648 + 0.984808i 0 0 0.500000 0.866025i 0.642788 0.766044i 0.173648 + 0.984808i
163.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i −0.173648 0.984808i 0 0 0.500000 + 0.866025i −0.642788 0.766044i 0.173648 0.984808i
167.1 −0.766044 0.642788i 0 0.173648 + 0.984808i 0.939693 0.342020i 0 0 0.500000 0.866025i 0.984808 + 0.173648i −0.939693 0.342020i
203.1 −0.766044 0.642788i 0 0.173648 + 0.984808i 0.939693 0.342020i 0 0 0.500000 0.866025i −0.984808 0.173648i −0.939693 0.342020i
207.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i −0.173648 0.984808i 0 0 0.500000 + 0.866025i 0.642788 + 0.766044i 0.173648 0.984808i
227.1 0.939693 0.342020i 0 0.766044 0.642788i −0.173648 + 0.984808i 0 0 0.500000 0.866025i −0.642788 + 0.766044i 0.173648 + 0.984808i
283.1 −0.173648 + 0.984808i 0 −0.939693 0.342020i −0.766044 0.642788i 0 0 0.500000 0.866025i 0.342020 + 0.939693i 0.766044 0.642788i
387.1 −0.173648 0.984808i 0 −0.939693 + 0.342020i −0.766044 + 0.642788i 0 0 0.500000 + 0.866025i 0.342020 0.939693i 0.766044 + 0.642788i
483.1 −0.766044 + 0.642788i 0 0.173648 0.984808i 0.939693 + 0.342020i 0 0 0.500000 + 0.866025i 0.984808 0.173648i −0.939693 + 0.342020i
627.1 −0.766044 + 0.642788i 0 0.173648 0.984808i 0.939693 + 0.342020i 0 0 0.500000 + 0.866025i −0.984808 + 0.173648i −0.939693 + 0.342020i
723.1 −0.173648 0.984808i 0 −0.939693 + 0.342020i −0.766044 + 0.642788i 0 0 0.500000 + 0.866025i −0.342020 + 0.939693i 0.766044 + 0.642788i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 87.1 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.z even 36 1 inner
740.cb odd 36 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.cb.a yes 12
4.b odd 2 1 CM 740.1.cb.a yes 12
5.b even 2 1 3700.1.dd.a 12
5.c odd 4 1 740.1.bw.a 12
5.c odd 4 1 3700.1.cy.a 12
20.d odd 2 1 3700.1.dd.a 12
20.e even 4 1 740.1.bw.a 12
20.e even 4 1 3700.1.cy.a 12
37.i odd 36 1 740.1.bw.a 12
148.q even 36 1 740.1.bw.a 12
185.z even 36 1 inner 740.1.cb.a yes 12
185.ba odd 36 1 3700.1.cy.a 12
185.bc even 36 1 3700.1.dd.a 12
740.bw odd 36 1 3700.1.dd.a 12
740.ca even 36 1 3700.1.cy.a 12
740.cb odd 36 1 inner 740.1.cb.a yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.bw.a 12 5.c odd 4 1
740.1.bw.a 12 20.e even 4 1
740.1.bw.a 12 37.i odd 36 1
740.1.bw.a 12 148.q even 36 1
740.1.cb.a yes 12 1.a even 1 1 trivial
740.1.cb.a yes 12 4.b odd 2 1 CM
740.1.cb.a yes 12 185.z even 36 1 inner
740.1.cb.a yes 12 740.cb odd 36 1 inner
3700.1.cy.a 12 5.c odd 4 1
3700.1.cy.a 12 20.e even 4 1
3700.1.cy.a 12 185.ba odd 36 1
3700.1.cy.a 12 740.ca even 36 1
3700.1.dd.a 12 5.b even 2 1
3700.1.dd.a 12 20.d odd 2 1
3700.1.dd.a 12 185.bc even 36 1
3700.1.dd.a 12 740.bw odd 36 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(740, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} - T^{3} + 1)^{2}$$
$3$ $$T^{12}$$
$5$ $$(T^{6} - T^{3} + 1)^{2}$$
$7$ $$T^{12}$$
$11$ $$T^{12}$$
$13$ $$T^{12} + 27T^{6} + 729$$
$17$ $$T^{12} - 3 T^{10} + \cdots + 1$$
$19$ $$T^{12}$$
$23$ $$T^{12}$$
$29$ $$T^{12} + 2 T^{9} + \cdots + 1$$
$31$ $$T^{12}$$
$37$ $$(T^{6} - T^{3} + 1)^{2}$$
$41$ $$(T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 3)^{2}$$
$43$ $$T^{12}$$
$47$ $$T^{12}$$
$53$ $$T^{12} + 14 T^{9} + \cdots + 1$$
$59$ $$T^{12}$$
$61$ $$T^{12} + 12 T^{11} + \cdots + 1$$
$67$ $$T^{12}$$
$71$ $$T^{12}$$
$73$ $$(T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{3}$$
$79$ $$T^{12}$$
$83$ $$T^{12}$$
$89$ $$T^{12} + 6 T^{10} + \cdots + 1$$
$97$ $$(T^{6} - 3 T^{4} + 9 T^{2} + \cdots + 3)^{2}$$