# Properties

 Label 1680.2.bg.o Level $1680$ Weight $2$ Character orbit 1680.bg Analytic conductor $13.415$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(961,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 2]))

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

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

chi := DirichletCharacter("1680.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.bg (of order $$3$$, degree $$2$$, not 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: $$13.4148675396$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{2} q^{3} + (\zeta_{12}^{2} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{7} + (\zeta_{12}^{2} - 1) q^{9} +O(q^{10})$$ q - z^2 * q^3 + (z^2 - 1) * q^5 + (-2*z^3 - z) * q^7 + (z^2 - 1) * q^9 $$q - \zeta_{12}^{2} q^{3} + (\zeta_{12}^{2} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{7} + (\zeta_{12}^{2} - 1) q^{9} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{11} + (\zeta_{12}^{3} - 2 \zeta_{12} + 4) q^{13} + q^{15} + (\zeta_{12}^{3} - 5 \zeta_{12}^{2} + \zeta_{12}) q^{17} + ( - 4 \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{19} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{21} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12} - 3) q^{23} - \zeta_{12}^{2} q^{25} + q^{27} + (3 \zeta_{12}^{3} - 6 \zeta_{12} + 1) q^{29} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12}) q^{31} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 1) q^{33} + ( - \zeta_{12}^{3} + 3 \zeta_{12}) q^{35} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{37} + (\zeta_{12}^{3} - 4 \zeta_{12}^{2} + \zeta_{12}) q^{39} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{41} + (3 \zeta_{12}^{3} - 6 \zeta_{12} + 2) q^{43} - \zeta_{12}^{2} q^{45} + ( - 2 \zeta_{12}^{2} + 2) q^{47} + (5 \zeta_{12}^{2} - 8) q^{49} + ( - 2 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + \zeta_{12} - 5) q^{51} + ( - 6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 6 \zeta_{12}) q^{53} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{55} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 1) q^{57} + ( - 3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 3 \zeta_{12}) q^{59} + (4 \zeta_{12}^{2} - 4) q^{61} + ( - \zeta_{12}^{3} + 3 \zeta_{12}) q^{63} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + \zeta_{12} - 4) q^{65} + ( - 5 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 5 \zeta_{12}) q^{67} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 3) q^{69} + (3 \zeta_{12}^{3} - 6 \zeta_{12} - 1) q^{71} + ( - 5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 5 \zeta_{12}) q^{73} + (\zeta_{12}^{2} - 1) q^{75} + (3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 2 \zeta_{12} + 5) q^{77} + (12 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 6 \zeta_{12} + 3) q^{79} - \zeta_{12}^{2} q^{81} + (7 \zeta_{12}^{3} - 14 \zeta_{12} - 3) q^{83} + (\zeta_{12}^{3} - 2 \zeta_{12} + 5) q^{85} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12}) q^{87} + (14 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 7 \zeta_{12} - 3) q^{89} + ( - 8 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} - 1) q^{91} + (4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 2 \zeta_{12} + 3) q^{93} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12}) q^{95} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 8) q^{97} + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{99} +O(q^{100})$$ q - z^2 * q^3 + (z^2 - 1) * q^5 + (-2*z^3 - z) * q^7 + (z^2 - 1) * q^9 + (z^3 - z^2 + z) * q^11 + (z^3 - 2*z + 4) * q^13 + q^15 + (z^3 - 5*z^2 + z) * q^17 + (-4*z^3 - z^2 + 2*z + 1) * q^19 + (3*z^3 - 2*z) * q^21 + (2*z^3 + 3*z^2 - z - 3) * q^23 - z^2 * q^25 + q^27 + (3*z^3 - 6*z + 1) * q^29 + (-2*z^3 + 3*z^2 - 2*z) * q^31 + (-2*z^3 + z^2 + z - 1) * q^33 + (-z^3 + 3*z) * q^35 + (-6*z^3 + 2*z^2 + 3*z - 2) * q^37 + (z^3 - 4*z^2 + z) * q^39 + (z^3 - 2*z + 1) * q^41 + (3*z^3 - 6*z + 2) * q^43 - z^2 * q^45 + (-2*z^2 + 2) * q^47 + (5*z^2 - 8) * q^49 + (-2*z^3 + 5*z^2 + z - 5) * q^51 + (-6*z^3 - 2*z^2 - 6*z) * q^53 + (z^3 - 2*z + 1) * q^55 + (2*z^3 - 4*z - 1) * q^57 + (-3*z^3 + 5*z^2 - 3*z) * q^59 + (4*z^2 - 4) * q^61 + (-z^3 + 3*z) * q^63 + (-2*z^3 + 4*z^2 + z - 4) * q^65 + (-5*z^3 - 6*z^2 - 5*z) * q^67 + (-z^3 + 2*z + 3) * q^69 + (3*z^3 - 6*z - 1) * q^71 + (-5*z^3 - 4*z^2 - 5*z) * q^73 + (z^2 - 1) * q^75 + (3*z^3 - 4*z^2 - 2*z + 5) * q^77 + (12*z^3 - 3*z^2 - 6*z + 3) * q^79 - z^2 * q^81 + (7*z^3 - 14*z - 3) * q^83 + (z^3 - 2*z + 5) * q^85 + (3*z^3 - z^2 + 3*z) * q^87 + (14*z^3 + 3*z^2 - 7*z - 3) * q^89 + (-8*z^3 + 5*z^2 - 4*z - 1) * q^91 + (4*z^3 - 3*z^2 - 2*z + 3) * q^93 + (2*z^3 + z^2 + 2*z) * q^95 + (-4*z^3 + 8*z + 8) * q^97 + (z^3 - 2*z + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 2 * q^5 - 2 * q^9 $$4 q - 2 q^{3} - 2 q^{5} - 2 q^{9} - 2 q^{11} + 16 q^{13} + 4 q^{15} - 10 q^{17} + 2 q^{19} - 6 q^{23} - 2 q^{25} + 4 q^{27} + 4 q^{29} + 6 q^{31} - 2 q^{33} - 4 q^{37} - 8 q^{39} + 4 q^{41} + 8 q^{43} - 2 q^{45} + 4 q^{47} - 22 q^{49} - 10 q^{51} - 4 q^{53} + 4 q^{55} - 4 q^{57} + 10 q^{59} - 8 q^{61} - 8 q^{65} - 12 q^{67} + 12 q^{69} - 4 q^{71} - 8 q^{73} - 2 q^{75} + 12 q^{77} + 6 q^{79} - 2 q^{81} - 12 q^{83} + 20 q^{85} - 2 q^{87} - 6 q^{89} + 6 q^{91} + 6 q^{93} + 2 q^{95} + 32 q^{97} + 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 2 * q^5 - 2 * q^9 - 2 * q^11 + 16 * q^13 + 4 * q^15 - 10 * q^17 + 2 * q^19 - 6 * q^23 - 2 * q^25 + 4 * q^27 + 4 * q^29 + 6 * q^31 - 2 * q^33 - 4 * q^37 - 8 * q^39 + 4 * q^41 + 8 * q^43 - 2 * q^45 + 4 * q^47 - 22 * q^49 - 10 * q^51 - 4 * q^53 + 4 * q^55 - 4 * q^57 + 10 * q^59 - 8 * q^61 - 8 * q^65 - 12 * q^67 + 12 * q^69 - 4 * q^71 - 8 * q^73 - 2 * q^75 + 12 * q^77 + 6 * q^79 - 2 * q^81 - 12 * q^83 + 20 * q^85 - 2 * q^87 - 6 * q^89 + 6 * q^91 + 6 * q^93 + 2 * q^95 + 32 * q^97 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −0.866025 2.50000i 0 −0.500000 + 0.866025i 0
961.2 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0.866025 + 2.50000i 0 −0.500000 + 0.866025i 0
1201.1 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −0.866025 + 2.50000i 0 −0.500000 0.866025i 0
1201.2 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0.866025 2.50000i 0 −0.500000 0.866025i 0
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.bg.o 4
4.b odd 2 1 105.2.i.d 4
7.c even 3 1 inner 1680.2.bg.o 4
12.b even 2 1 315.2.j.c 4
20.d odd 2 1 525.2.i.f 4
20.e even 4 1 525.2.r.a 4
20.e even 4 1 525.2.r.f 4
28.d even 2 1 735.2.i.l 4
28.f even 6 1 735.2.a.h 2
28.f even 6 1 735.2.i.l 4
28.g odd 6 1 105.2.i.d 4
28.g odd 6 1 735.2.a.g 2
84.j odd 6 1 2205.2.a.ba 2
84.n even 6 1 315.2.j.c 4
84.n even 6 1 2205.2.a.z 2
140.p odd 6 1 525.2.i.f 4
140.p odd 6 1 3675.2.a.bg 2
140.s even 6 1 3675.2.a.be 2
140.w even 12 1 525.2.r.a 4
140.w even 12 1 525.2.r.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 4.b odd 2 1
105.2.i.d 4 28.g odd 6 1
315.2.j.c 4 12.b even 2 1
315.2.j.c 4 84.n even 6 1
525.2.i.f 4 20.d odd 2 1
525.2.i.f 4 140.p odd 6 1
525.2.r.a 4 20.e even 4 1
525.2.r.a 4 140.w even 12 1
525.2.r.f 4 20.e even 4 1
525.2.r.f 4 140.w even 12 1
735.2.a.g 2 28.g odd 6 1
735.2.a.h 2 28.f even 6 1
735.2.i.l 4 28.d even 2 1
735.2.i.l 4 28.f even 6 1
1680.2.bg.o 4 1.a even 1 1 trivial
1680.2.bg.o 4 7.c even 3 1 inner
2205.2.a.z 2 84.n even 6 1
2205.2.a.ba 2 84.j odd 6 1
3675.2.a.be 2 140.s even 6 1
3675.2.a.bg 2 140.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1680, [\chi])$$:

 $$T_{11}^{4} + 2T_{11}^{3} + 6T_{11}^{2} - 4T_{11} + 4$$ T11^4 + 2*T11^3 + 6*T11^2 - 4*T11 + 4 $$T_{13}^{2} - 8T_{13} + 13$$ T13^2 - 8*T13 + 13 $$T_{17}^{4} + 10T_{17}^{3} + 78T_{17}^{2} + 220T_{17} + 484$$ T17^4 + 10*T17^3 + 78*T17^2 + 220*T17 + 484

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} + 11T^{2} + 49$$
$11$ $$T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4$$
$13$ $$(T^{2} - 8 T + 13)^{2}$$
$17$ $$T^{4} + 10 T^{3} + 78 T^{2} + \cdots + 484$$
$19$ $$T^{4} - 2 T^{3} + 15 T^{2} + 22 T + 121$$
$23$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$29$ $$(T^{2} - 2 T - 26)^{2}$$
$31$ $$T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9$$
$37$ $$T^{4} + 4 T^{3} + 39 T^{2} - 92 T + 529$$
$41$ $$(T^{2} - 2 T - 2)^{2}$$
$43$ $$(T^{2} - 4 T - 23)^{2}$$
$47$ $$(T^{2} - 2 T + 4)^{2}$$
$53$ $$T^{4} + 4 T^{3} + 120 T^{2} + \cdots + 10816$$
$59$ $$T^{4} - 10 T^{3} + 102 T^{2} + 20 T + 4$$
$61$ $$(T^{2} + 4 T + 16)^{2}$$
$67$ $$T^{4} + 12 T^{3} + 183 T^{2} + \cdots + 1521$$
$71$ $$(T^{2} + 2 T - 26)^{2}$$
$73$ $$T^{4} + 8 T^{3} + 123 T^{2} + \cdots + 3481$$
$79$ $$T^{4} - 6 T^{3} + 135 T^{2} + \cdots + 9801$$
$83$ $$(T^{2} + 6 T - 138)^{2}$$
$89$ $$T^{4} + 6 T^{3} + 174 T^{2} + \cdots + 19044$$
$97$ $$(T^{2} - 16 T + 16)^{2}$$