# Properties

 Label 1680.4.a.bg Level $1680$ Weight $4$ Character orbit 1680.a Self dual yes Analytic conductor $99.123$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,4,Mod(1,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1680.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1680.a (trivial)

## 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: yes Analytic conductor: $$99.1232088096$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{65}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 5 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 5 * q^5 + 7 * q^7 + 9 * q^9 $$q - 3 q^{3} + 5 q^{5} + 7 q^{7} + 9 q^{9} + ( - \beta + 11) q^{11} + ( - \beta - 11) q^{13} - 15 q^{15} + (8 \beta + 58) q^{17} + ( - 7 \beta - 51) q^{19} - 21 q^{21} + ( - 10 \beta - 130) q^{23} + 25 q^{25} - 27 q^{27} + (24 \beta - 98) q^{29} + (21 \beta - 75) q^{31} + (3 \beta - 33) q^{33} + 35 q^{35} + (18 \beta - 48) q^{37} + (3 \beta + 33) q^{39} + ( - 50 \beta - 88) q^{41} + ( - 16 \beta + 172) q^{43} + 45 q^{45} + ( - 24 \beta - 280) q^{47} + 49 q^{49} + ( - 24 \beta - 174) q^{51} + ( - 43 \beta + 163) q^{53} + ( - 5 \beta + 55) q^{55} + (21 \beta + 153) q^{57} + (42 \beta + 422) q^{59} + ( - 12 \beta - 102) q^{61} + 63 q^{63} + ( - 5 \beta - 55) q^{65} + (32 \beta + 52) q^{67} + (30 \beta + 390) q^{69} + (21 \beta - 835) q^{71} + (101 \beta - 193) q^{73} - 75 q^{75} + ( - 7 \beta + 77) q^{77} + ( - 52 \beta + 444) q^{79} + 81 q^{81} + (108 \beta - 464) q^{83} + (40 \beta + 290) q^{85} + ( - 72 \beta + 294) q^{87} + ( - 4 \beta + 294) q^{89} + ( - 7 \beta - 77) q^{91} + ( - 63 \beta + 225) q^{93} + ( - 35 \beta - 255) q^{95} + ( - 157 \beta + 261) q^{97} + ( - 9 \beta + 99) q^{99}+O(q^{100})$$ q - 3 * q^3 + 5 * q^5 + 7 * q^7 + 9 * q^9 + (-b + 11) * q^11 + (-b - 11) * q^13 - 15 * q^15 + (8*b + 58) * q^17 + (-7*b - 51) * q^19 - 21 * q^21 + (-10*b - 130) * q^23 + 25 * q^25 - 27 * q^27 + (24*b - 98) * q^29 + (21*b - 75) * q^31 + (3*b - 33) * q^33 + 35 * q^35 + (18*b - 48) * q^37 + (3*b + 33) * q^39 + (-50*b - 88) * q^41 + (-16*b + 172) * q^43 + 45 * q^45 + (-24*b - 280) * q^47 + 49 * q^49 + (-24*b - 174) * q^51 + (-43*b + 163) * q^53 + (-5*b + 55) * q^55 + (21*b + 153) * q^57 + (42*b + 422) * q^59 + (-12*b - 102) * q^61 + 63 * q^63 + (-5*b - 55) * q^65 + (32*b + 52) * q^67 + (30*b + 390) * q^69 + (21*b - 835) * q^71 + (101*b - 193) * q^73 - 75 * q^75 + (-7*b + 77) * q^77 + (-52*b + 444) * q^79 + 81 * q^81 + (108*b - 464) * q^83 + (40*b + 290) * q^85 + (-72*b + 294) * q^87 + (-4*b + 294) * q^89 + (-7*b - 77) * q^91 + (-63*b + 225) * q^93 + (-35*b - 255) * q^95 + (-157*b + 261) * q^97 + (-9*b + 99) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 $$2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} - 22 q^{13} - 30 q^{15} + 116 q^{17} - 102 q^{19} - 42 q^{21} - 260 q^{23} + 50 q^{25} - 54 q^{27} - 196 q^{29} - 150 q^{31} - 66 q^{33} + 70 q^{35} - 96 q^{37} + 66 q^{39} - 176 q^{41} + 344 q^{43} + 90 q^{45} - 560 q^{47} + 98 q^{49} - 348 q^{51} + 326 q^{53} + 110 q^{55} + 306 q^{57} + 844 q^{59} - 204 q^{61} + 126 q^{63} - 110 q^{65} + 104 q^{67} + 780 q^{69} - 1670 q^{71} - 386 q^{73} - 150 q^{75} + 154 q^{77} + 888 q^{79} + 162 q^{81} - 928 q^{83} + 580 q^{85} + 588 q^{87} + 588 q^{89} - 154 q^{91} + 450 q^{93} - 510 q^{95} + 522 q^{97} + 198 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 22 * q^11 - 22 * q^13 - 30 * q^15 + 116 * q^17 - 102 * q^19 - 42 * q^21 - 260 * q^23 + 50 * q^25 - 54 * q^27 - 196 * q^29 - 150 * q^31 - 66 * q^33 + 70 * q^35 - 96 * q^37 + 66 * q^39 - 176 * q^41 + 344 * q^43 + 90 * q^45 - 560 * q^47 + 98 * q^49 - 348 * q^51 + 326 * q^53 + 110 * q^55 + 306 * q^57 + 844 * q^59 - 204 * q^61 + 126 * q^63 - 110 * q^65 + 104 * q^67 + 780 * q^69 - 1670 * q^71 - 386 * q^73 - 150 * q^75 + 154 * q^77 + 888 * q^79 + 162 * q^81 - 928 * q^83 + 580 * q^85 + 588 * q^87 + 588 * q^89 - 154 * q^91 + 450 * q^93 - 510 * q^95 + 522 * q^97 + 198 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 4.53113 −3.53113
0 −3.00000 0 5.00000 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 5.00000 0 7.00000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$+1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.4.a.bg 2
4.b odd 2 1 105.4.a.f 2
12.b even 2 1 315.4.a.i 2
20.d odd 2 1 525.4.a.k 2
20.e even 4 2 525.4.d.h 4
28.d even 2 1 735.4.a.p 2
60.h even 2 1 1575.4.a.w 2
84.h odd 2 1 2205.4.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.f 2 4.b odd 2 1
315.4.a.i 2 12.b even 2 1
525.4.a.k 2 20.d odd 2 1
525.4.d.h 4 20.e even 4 2
735.4.a.p 2 28.d even 2 1
1575.4.a.w 2 60.h even 2 1
1680.4.a.bg 2 1.a even 1 1 trivial
2205.4.a.z 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1680))$$:

 $$T_{11}^{2} - 22T_{11} + 56$$ T11^2 - 22*T11 + 56 $$T_{13}^{2} + 22T_{13} + 56$$ T13^2 + 22*T13 + 56 $$T_{17}^{2} - 116T_{17} - 796$$ T17^2 - 116*T17 - 796

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$(T - 7)^{2}$$
$11$ $$T^{2} - 22T + 56$$
$13$ $$T^{2} + 22T + 56$$
$17$ $$T^{2} - 116T - 796$$
$19$ $$T^{2} + 102T - 584$$
$23$ $$T^{2} + 260T + 10400$$
$29$ $$T^{2} + 196T - 27836$$
$31$ $$T^{2} + 150T - 23040$$
$37$ $$T^{2} + 96T - 18756$$
$41$ $$T^{2} + 176T - 154756$$
$43$ $$T^{2} - 344T + 12944$$
$47$ $$T^{2} + 560T + 40960$$
$53$ $$T^{2} - 326T - 93616$$
$59$ $$T^{2} - 844T + 63424$$
$61$ $$T^{2} + 204T + 1044$$
$67$ $$T^{2} - 104T - 63856$$
$71$ $$T^{2} + 1670 T + 668560$$
$73$ $$T^{2} + 386T - 625816$$
$79$ $$T^{2} - 888T + 21376$$
$83$ $$T^{2} + 928T - 542864$$
$89$ $$T^{2} - 588T + 85396$$
$97$ $$T^{2} - 522 T - 1534064$$