# Properties

 Label 760.1.bm.b Level $760$ Weight $1$ Character orbit 760.bm Analytic conductor $0.379$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -40 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [760,1,Mod(539,760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(760, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 3, 2]))

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

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

chi := DirichletCharacter("760.539");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$760 = 2^{3} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 760.bm (of order $$6$$, degree $$2$$, 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: $$0.379289409601$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.14440.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} + \zeta_{6} q^{5} + q^{7} - q^{8} + \zeta_{6}^{2} q^{9}+O(q^{10})$$ q + z * q^2 + z^2 * q^4 + z * q^5 + q^7 - q^8 + z^2 * q^9 $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} + \zeta_{6} q^{5} + q^{7} - q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6}^{2} q^{10} - q^{11} - 2 \zeta_{6}^{2} q^{13} + \zeta_{6} q^{14} - \zeta_{6} q^{16} - q^{18} - \zeta_{6} q^{19} - q^{20} - \zeta_{6} q^{22} + \zeta_{6}^{2} q^{23} + \zeta_{6}^{2} q^{25} + 2 q^{26} + \zeta_{6}^{2} q^{28} - \zeta_{6}^{2} q^{32} + \zeta_{6} q^{35} - \zeta_{6} q^{36} + q^{37} - \zeta_{6}^{2} q^{38} - \zeta_{6} q^{40} + \zeta_{6} q^{41} - \zeta_{6}^{2} q^{44} - q^{45} - q^{46} - 2 \zeta_{6}^{2} q^{47} - q^{50} + 2 \zeta_{6} q^{52} + \zeta_{6}^{2} q^{53} - \zeta_{6} q^{55} - q^{56} - 2 \zeta_{6} q^{59} + \zeta_{6}^{2} q^{63} + q^{64} + 2 q^{65} + \zeta_{6}^{2} q^{70} - \zeta_{6}^{2} q^{72} + \zeta_{6} q^{74} + q^{76} - q^{77} - \zeta_{6}^{2} q^{80} - \zeta_{6} q^{81} + \zeta_{6}^{2} q^{82} + q^{88} - \zeta_{6}^{2} q^{89} - \zeta_{6} q^{90} - 2 \zeta_{6}^{2} q^{91} - \zeta_{6} q^{92} + 2 q^{94} - \zeta_{6}^{2} q^{95} - \zeta_{6}^{2} q^{99} +O(q^{100})$$ q + z * q^2 + z^2 * q^4 + z * q^5 + q^7 - q^8 + z^2 * q^9 + z^2 * q^10 - q^11 - 2*z^2 * q^13 + z * q^14 - z * q^16 - q^18 - z * q^19 - q^20 - z * q^22 + z^2 * q^23 + z^2 * q^25 + 2 * q^26 + z^2 * q^28 - z^2 * q^32 + z * q^35 - z * q^36 + q^37 - z^2 * q^38 - z * q^40 + z * q^41 - z^2 * q^44 - q^45 - q^46 - 2*z^2 * q^47 - q^50 + 2*z * q^52 + z^2 * q^53 - z * q^55 - q^56 - 2*z * q^59 + z^2 * q^63 + q^64 + 2 * q^65 + z^2 * q^70 - z^2 * q^72 + z * q^74 + q^76 - q^77 - z^2 * q^80 - z * q^81 + z^2 * q^82 + q^88 - z^2 * q^89 - z * q^90 - 2*z^2 * q^91 - z * q^92 + 2 * q^94 - z^2 * q^95 - z^2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} + q^{5} + 2 q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 - q^4 + q^5 + 2 * q^7 - 2 * q^8 - q^9 $$2 q + q^{2} - q^{4} + q^{5} + 2 q^{7} - 2 q^{8} - q^{9} - q^{10} - 2 q^{11} + 2 q^{13} + q^{14} - q^{16} - 2 q^{18} - q^{19} - 2 q^{20} - q^{22} - q^{23} - q^{25} + 4 q^{26} - q^{28} + q^{32} + q^{35} - q^{36} + 2 q^{37} + q^{38} - q^{40} + q^{41} + q^{44} - 2 q^{45} - 2 q^{46} + 2 q^{47} - 2 q^{50} + 2 q^{52} - q^{53} - q^{55} - 2 q^{56} - 2 q^{59} - q^{63} + 2 q^{64} + 4 q^{65} - q^{70} + q^{72} + q^{74} + 2 q^{76} - 2 q^{77} + q^{80} - q^{81} - q^{82} + 2 q^{88} + q^{89} - q^{90} + 2 q^{91} - q^{92} + 4 q^{94} + q^{95} + q^{99}+O(q^{100})$$ 2 * q + q^2 - q^4 + q^5 + 2 * q^7 - 2 * q^8 - q^9 - q^10 - 2 * q^11 + 2 * q^13 + q^14 - q^16 - 2 * q^18 - q^19 - 2 * q^20 - q^22 - q^23 - q^25 + 4 * q^26 - q^28 + q^32 + q^35 - q^36 + 2 * q^37 + q^38 - q^40 + q^41 + q^44 - 2 * q^45 - 2 * q^46 + 2 * q^47 - 2 * q^50 + 2 * q^52 - q^53 - q^55 - 2 * q^56 - 2 * q^59 - q^63 + 2 * q^64 + 4 * q^65 - q^70 + q^72 + q^74 + 2 * q^76 - 2 * q^77 + q^80 - q^81 - q^82 + 2 * q^88 + q^89 - q^90 + 2 * q^91 - q^92 + 4 * q^94 + q^95 + q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$381$$ $$401$$ $$457$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{6}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
539.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 1.00000 −1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
619.1 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 1.00000 −1.00000 −0.500000 0.866025i −0.500000 0.866025i
 $$n$$: 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
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
19.c even 3 1 inner
760.bm odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.1.bm.b yes 2
4.b odd 2 1 3040.1.cc.b 2
5.b even 2 1 760.1.bm.a 2
5.c odd 4 2 3800.1.bd.f 4
8.b even 2 1 3040.1.cc.a 2
8.d odd 2 1 760.1.bm.a 2
19.c even 3 1 inner 760.1.bm.b yes 2
20.d odd 2 1 3040.1.cc.a 2
40.e odd 2 1 CM 760.1.bm.b yes 2
40.f even 2 1 3040.1.cc.b 2
40.k even 4 2 3800.1.bd.f 4
76.g odd 6 1 3040.1.cc.b 2
95.i even 6 1 760.1.bm.a 2
95.m odd 12 2 3800.1.bd.f 4
152.k odd 6 1 760.1.bm.a 2
152.p even 6 1 3040.1.cc.a 2
380.p odd 6 1 3040.1.cc.a 2
760.z even 6 1 3040.1.cc.b 2
760.bm odd 6 1 inner 760.1.bm.b yes 2
760.bw even 12 2 3800.1.bd.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.bm.a 2 5.b even 2 1
760.1.bm.a 2 8.d odd 2 1
760.1.bm.a 2 95.i even 6 1
760.1.bm.a 2 152.k odd 6 1
760.1.bm.b yes 2 1.a even 1 1 trivial
760.1.bm.b yes 2 19.c even 3 1 inner
760.1.bm.b yes 2 40.e odd 2 1 CM
760.1.bm.b yes 2 760.bm odd 6 1 inner
3040.1.cc.a 2 8.b even 2 1
3040.1.cc.a 2 20.d odd 2 1
3040.1.cc.a 2 152.p even 6 1
3040.1.cc.a 2 380.p odd 6 1
3040.1.cc.b 2 4.b odd 2 1
3040.1.cc.b 2 40.f even 2 1
3040.1.cc.b 2 76.g odd 6 1
3040.1.cc.b 2 760.z even 6 1
3800.1.bd.f 4 5.c odd 4 2
3800.1.bd.f 4 40.k even 4 2
3800.1.bd.f 4 95.m odd 12 2
3800.1.bd.f 4 760.bw even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 1$$ acting on $$S_{1}^{\mathrm{new}}(760, [\chi])$$.

## Hecke characteristic polynomials

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