# Properties

 Label 740.2.bc.a Level $740$ Weight $2$ Character orbit 740.bc Analytic conductor $5.909$ Analytic rank $0$ Dimension $6$ Inner twists $2$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(81,740)]

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([0, 0, 16]))

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

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

chi := DirichletCharacter("740.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.bc (of order $$9$$, 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: $$5.90892974957$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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 $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{3} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{5} + ( - 2 \zeta_{18}^{5} + \cdots + 2 \zeta_{18}^{2}) q^{7}+ \cdots + (2 \zeta_{18}^{5} + 3 \zeta_{18}^{3} + \cdots - 1) q^{9}+O(q^{10})$$ q + (-z^3 - z^2 - z) * q^3 + (z^5 - z^2) * q^5 + (-2*z^5 - z^4 + z^3 + 2*z^2) * q^7 + (2*z^5 + 3*z^3 + z^2 - 1) * q^9 $$q + ( - \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{3} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{5} + ( - 2 \zeta_{18}^{5} + \cdots + 2 \zeta_{18}^{2}) q^{7}+ \cdots + (9 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + \cdots - 9) q^{99}+O(q^{100})$$ q + (-z^3 - z^2 - z) * q^3 + (z^5 - z^2) * q^5 + (-2*z^5 - z^4 + z^3 + 2*z^2) * q^7 + (2*z^5 + 3*z^3 + z^2 - 1) * q^9 + (-z^4 + 4*z^3 - z^2) * q^11 + (z^5 + z^4 + z^3 + z - 2) * q^13 + (z^2 + z + 1) * q^15 + (-3*z^5 - z^4 - z^3 + 2*z^2 - 2) * q^17 + (3*z^3 + 2*z^2 + 3*z) * q^19 + (-2*z^2 - 3*z - 2) * q^21 + (z^3 + z^2 - z - 1) * q^23 + (-z^4 + z) * q^25 + (-3*z^5 - 3*z^4 - 2*z^3 + 3*z^2 + 2) * q^27 + (-2*z^4 - z^3 - 2*z^2) * q^29 + (-2*z^4 + 2*z^2 + 2*z + 7) * q^31 + (-2*z^5 - 2*z^4 - 2*z^3 - z + 3) * q^33 + (2*z^4 + z^3 - z^2 - 2*z) * q^35 + (-2*z^5 - 4*z^4 + 2*z^3 - 2*z^2 - z) * q^37 + (-3*z^5 - 4*z^4 - 2*z^3 + 2*z^2 + 4*z + 3) * q^39 + (-2*z^5 + z - 1) * q^41 + (5*z^5 - z^4 - 4*z^2 - 4*z + 5) * q^43 + (-z^5 - 2*z^4 - 2*z^2 - z) * q^45 + (-2*z^5 - 2*z^4 - 9*z^3 - 2*z^2 + 4*z + 9) * q^47 + (z^5 + z^4 - 3*z^3 + 3*z^2 - z - 1) * q^49 + (3*z^5 + 3*z^4 + 5*z^3 - z^2 - 2*z - 5) * q^51 + (-4*z^5 + 4*z^3 + 4*z^2 + 7*z) * q^53 + (z^3 - 4*z^2 + z) * q^55 + (-5*z^5 - 8*z^4 - 8*z^3 - 3*z^2 + 3) * q^57 + (5*z^2 - 5*z + 5) * q^59 + (-5*z^5 + 5*z^4 + 5*z^3 - 4*z - 1) * q^61 + (5*z^5 + 2*z^4 + z^3 + 2*z^2 + 5*z) * q^63 + (-2*z^5 - z^4 - z^3 + z^2 - 1) * q^65 + (-z^5 + 3*z^4 - 6*z^3 + z^2 + 3*z + 3) * q^67 + (-2*z^5 - z^4 + 2*z^2 + z + 1) * q^69 + (-6*z^4 + 3*z^3 - 3*z^2 + 3*z - 6) * q^71 + (-2*z^5 - 3*z^4 + 5*z^2 + 5*z) * q^73 + (z^5 - z^2 - z - 1) * q^75 + (-5*z^4 + 3*z^3 + 7*z^2 + 3*z - 5) * q^77 + (3*z^5 + 6*z^4 - 5*z^3 - 3*z^2 - z - 1) * q^79 + (5*z^5 - z^4 - 5*z^2 + z + 1) * q^81 + (z^5 - 3*z^4 + 5*z^3 + 4*z^2 - 4) * q^83 + (-2*z^5 + 3*z^4 + z^3 + 3*z^2 - 2*z) * q^85 + (5*z^5 + 5*z^4 + 5*z^3 - 2*z - 3) * q^87 + (-4*z^5 + 4*z^3 + 3*z^2 - 4*z - 1) * q^89 + (3*z^5 + 5*z^4 + z^3 - 2*z^2 + 2) * q^91 + (-2*z^4 - 9*z^3 - 9*z^2 - 9*z - 2) * q^93 + (-3*z^2 - 2*z - 3) * q^95 + (4*z^5 + 4*z^4 - 2*z^3 + 3*z^2 - 7*z + 2) * q^97 + (9*z^5 - 5*z^4 + 7*z^3 - 7*z^2 + 5*z - 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10})$$ 6 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 $$6 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 12 q^{11} - 9 q^{13} + 6 q^{15} - 15 q^{17} + 9 q^{19} - 12 q^{21} - 3 q^{23} + 6 q^{27} - 3 q^{29} + 42 q^{31} + 12 q^{33} + 3 q^{35} + 6 q^{37} + 12 q^{39} - 6 q^{41} + 30 q^{43} + 27 q^{47} - 15 q^{49} - 15 q^{51} + 12 q^{53} + 3 q^{55} - 6 q^{57} + 30 q^{59} + 9 q^{61} + 3 q^{63} - 9 q^{65} + 6 q^{69} - 27 q^{71} - 6 q^{75} - 21 q^{77} - 21 q^{79} + 6 q^{81} - 9 q^{83} + 3 q^{85} - 3 q^{87} + 6 q^{89} + 15 q^{91} - 39 q^{93} - 18 q^{95} + 6 q^{97} - 33 q^{99}+O(q^{100})$$ 6 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 12 * q^11 - 9 * q^13 + 6 * q^15 - 15 * q^17 + 9 * q^19 - 12 * q^21 - 3 * q^23 + 6 * q^27 - 3 * q^29 + 42 * q^31 + 12 * q^33 + 3 * q^35 + 6 * q^37 + 12 * q^39 - 6 * q^41 + 30 * q^43 + 27 * q^47 - 15 * q^49 - 15 * q^51 + 12 * q^53 + 3 * q^55 - 6 * q^57 + 30 * q^59 + 9 * q^61 + 3 * q^63 - 9 * q^65 + 6 * q^69 - 27 * q^71 - 6 * q^75 - 21 * q^77 - 21 * q^79 + 6 * q^81 - 9 * q^83 + 3 * q^85 - 3 * q^87 + 6 * q^89 + 15 * q^91 - 39 * q^93 - 18 * q^95 + 6 * q^97 - 33 * q^99

## Character values

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

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 −0.173648 + 0.984808i −0.766044 − 0.642788i −0.173648 − 0.984808i 0.939693 − 0.342020i −0.766044 + 0.642788i 0.939693 + 0.342020i
0 0.613341 + 0.223238i 0 0.173648 + 0.984808i 0 −0.613341 3.47843i 0 −1.97178 1.65452i 0
181.1 0 0.0923963 + 0.524005i 0 0.766044 0.642788i 0 −0.0923963 + 0.0775297i 0 2.55303 0.929228i 0
201.1 0 0.613341 0.223238i 0 0.173648 0.984808i 0 −0.613341 + 3.47843i 0 −1.97178 + 1.65452i 0
441.1 0 −2.20574 + 1.85083i 0 −0.939693 0.342020i 0 2.20574 + 0.802823i 0 0.918748 5.21048i 0
601.1 0 0.0923963 0.524005i 0 0.766044 + 0.642788i 0 −0.0923963 0.0775297i 0 2.55303 + 0.929228i 0
641.1 0 −2.20574 1.85083i 0 −0.939693 + 0.342020i 0 2.20574 0.802823i 0 0.918748 + 5.21048i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 81.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.bc.a 6
37.f even 9 1 inner 740.2.bc.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bc.a 6 1.a even 1 1 trivial
740.2.bc.a 6 37.f even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 3T_{3}^{5} + 3T_{3}^{4} - 8T_{3}^{3} + 6T_{3}^{2} - 3T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(740, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$5$ $$T^{6} + T^{3} + 1$$
$7$ $$T^{6} - 3 T^{5} + \cdots + 1$$
$11$ $$T^{6} - 12 T^{5} + \cdots + 2601$$
$13$ $$T^{6} + 9 T^{5} + \cdots + 289$$
$17$ $$T^{6} + 15 T^{5} + \cdots + 3249$$
$19$ $$T^{6} - 9 T^{5} + \cdots + 361$$
$23$ $$T^{6} + 3 T^{5} + \cdots + 9$$
$29$ $$T^{6} + 3 T^{5} + \cdots + 9$$
$31$ $$(T^{3} - 21 T^{2} + \cdots - 251)^{2}$$
$37$ $$T^{6} - 6 T^{5} + \cdots + 50653$$
$41$ $$T^{6} + 6 T^{5} + \cdots + 9$$
$43$ $$(T^{3} - 15 T^{2} + \cdots + 181)^{2}$$
$47$ $$T^{6} - 27 T^{5} + \cdots + 110889$$
$53$ $$T^{6} - 12 T^{5} + \cdots + 3249$$
$59$ $$T^{6} - 30 T^{5} + \cdots + 140625$$
$61$ $$T^{6} - 9 T^{5} + \cdots + 39601$$
$67$ $$T^{6} + 54 T^{4} + \cdots + 104329$$
$71$ $$T^{6} + 27 T^{5} + \cdots + 210681$$
$73$ $$(T^{3} - 57 T - 107)^{2}$$
$79$ $$T^{6} + 21 T^{5} + \cdots + 292681$$
$83$ $$T^{6} + 9 T^{5} + \cdots + 29241$$
$89$ $$T^{6} - 6 T^{5} + \cdots + 9$$
$97$ $$T^{6} - 6 T^{5} + \cdots + 11881$$
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