# Properties

 Label 1680.4.a.y 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: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 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 $$\beta = \sqrt{41}$$. 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 $$q - 3 q^{3} - 5 q^{5} - 7 q^{7} + 9 q^{9} + ( - \beta - 31) q^{11} + (5 \beta - 3) q^{13} + 15 q^{15} + (6 \beta + 20) q^{17} + ( - \beta + 61) q^{19} + 21 q^{21} + ( - 24 \beta - 8) q^{23} + 25 q^{25} - 27 q^{27} + ( - 6 \beta + 176) q^{29} + ( - 19 \beta - 33) q^{31} + (3 \beta + 93) q^{33} + 35 q^{35} + ( - 40 \beta - 94) q^{37} + ( - 15 \beta + 9) q^{39} + (54 \beta + 8) q^{41} + (50 \beta + 198) q^{43} - 45 q^{45} + (70 \beta + 94) q^{47} + 49 q^{49} + ( - 18 \beta - 60) q^{51} + ( - 29 \beta + 491) q^{53} + (5 \beta + 155) q^{55} + (3 \beta - 183) q^{57} + (38 \beta - 258) q^{59} + (42 \beta - 440) q^{61} - 63 q^{63} + ( - 25 \beta + 15) q^{65} + ( - 114 \beta + 178) q^{67} + (72 \beta + 24) q^{69} + (43 \beta - 155) q^{71} + (19 \beta + 163) q^{73} - 75 q^{75} + (7 \beta + 217) q^{77} + ( - 4 \beta - 916) q^{79} + 81 q^{81} + ( - 112 \beta + 340) q^{83} + ( - 30 \beta - 100) q^{85} + (18 \beta - 528) q^{87} + ( - 168 \beta + 398) q^{89} + ( - 35 \beta + 21) q^{91} + (57 \beta + 99) q^{93} + (5 \beta - 305) q^{95} + ( - 139 \beta - 335) q^{97} + ( - 9 \beta - 279) q^{99}+O(q^{100})$$ q - 3 * q^3 - 5 * q^5 - 7 * q^7 + 9 * q^9 + (-b - 31) * q^11 + (5*b - 3) * q^13 + 15 * q^15 + (6*b + 20) * q^17 + (-b + 61) * q^19 + 21 * q^21 + (-24*b - 8) * q^23 + 25 * q^25 - 27 * q^27 + (-6*b + 176) * q^29 + (-19*b - 33) * q^31 + (3*b + 93) * q^33 + 35 * q^35 + (-40*b - 94) * q^37 + (-15*b + 9) * q^39 + (54*b + 8) * q^41 + (50*b + 198) * q^43 - 45 * q^45 + (70*b + 94) * q^47 + 49 * q^49 + (-18*b - 60) * q^51 + (-29*b + 491) * q^53 + (5*b + 155) * q^55 + (3*b - 183) * q^57 + (38*b - 258) * q^59 + (42*b - 440) * q^61 - 63 * q^63 + (-25*b + 15) * q^65 + (-114*b + 178) * q^67 + (72*b + 24) * q^69 + (43*b - 155) * q^71 + (19*b + 163) * q^73 - 75 * q^75 + (7*b + 217) * q^77 + (-4*b - 916) * q^79 + 81 * q^81 + (-112*b + 340) * q^83 + (-30*b - 100) * q^85 + (18*b - 528) * q^87 + (-168*b + 398) * q^89 + (-35*b + 21) * q^91 + (57*b + 99) * q^93 + (5*b - 305) * q^95 + (-139*b - 335) * q^97 + (-9*b - 279) * 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 $$2 q - 6 q^{3} - 10 q^{5} - 14 q^{7} + 18 q^{9} - 62 q^{11} - 6 q^{13} + 30 q^{15} + 40 q^{17} + 122 q^{19} + 42 q^{21} - 16 q^{23} + 50 q^{25} - 54 q^{27} + 352 q^{29} - 66 q^{31} + 186 q^{33} + 70 q^{35} - 188 q^{37} + 18 q^{39} + 16 q^{41} + 396 q^{43} - 90 q^{45} + 188 q^{47} + 98 q^{49} - 120 q^{51} + 982 q^{53} + 310 q^{55} - 366 q^{57} - 516 q^{59} - 880 q^{61} - 126 q^{63} + 30 q^{65} + 356 q^{67} + 48 q^{69} - 310 q^{71} + 326 q^{73} - 150 q^{75} + 434 q^{77} - 1832 q^{79} + 162 q^{81} + 680 q^{83} - 200 q^{85} - 1056 q^{87} + 796 q^{89} + 42 q^{91} + 198 q^{93} - 610 q^{95} - 670 q^{97} - 558 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 10 * q^5 - 14 * q^7 + 18 * q^9 - 62 * q^11 - 6 * q^13 + 30 * q^15 + 40 * q^17 + 122 * q^19 + 42 * q^21 - 16 * q^23 + 50 * q^25 - 54 * q^27 + 352 * q^29 - 66 * q^31 + 186 * q^33 + 70 * q^35 - 188 * q^37 + 18 * q^39 + 16 * q^41 + 396 * q^43 - 90 * q^45 + 188 * q^47 + 98 * q^49 - 120 * q^51 + 982 * q^53 + 310 * q^55 - 366 * q^57 - 516 * q^59 - 880 * q^61 - 126 * q^63 + 30 * q^65 + 356 * q^67 + 48 * q^69 - 310 * q^71 + 326 * q^73 - 150 * q^75 + 434 * q^77 - 1832 * q^79 + 162 * q^81 + 680 * q^83 - 200 * q^85 - 1056 * q^87 + 796 * q^89 + 42 * q^91 + 198 * q^93 - 610 * q^95 - 670 * q^97 - 558 * 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
 3.70156 −2.70156
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.y 2
4.b odd 2 1 105.4.a.g 2
12.b even 2 1 315.4.a.g 2
20.d odd 2 1 525.4.a.i 2
20.e even 4 2 525.4.d.j 4
28.d even 2 1 735.4.a.q 2
60.h even 2 1 1575.4.a.y 2
84.h odd 2 1 2205.4.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 4.b odd 2 1
315.4.a.g 2 12.b even 2 1
525.4.a.i 2 20.d odd 2 1
525.4.d.j 4 20.e even 4 2
735.4.a.q 2 28.d even 2 1
1575.4.a.y 2 60.h even 2 1
1680.4.a.y 2 1.a even 1 1 trivial
2205.4.a.v 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1680))$$:

 $$T_{11}^{2} + 62T_{11} + 920$$ T11^2 + 62*T11 + 920 $$T_{13}^{2} + 6T_{13} - 1016$$ T13^2 + 6*T13 - 1016 $$T_{17}^{2} - 40T_{17} - 1076$$ T17^2 - 40*T17 - 1076

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$(T + 5)^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} + 62T + 920$$
$13$ $$T^{2} + 6T - 1016$$
$17$ $$T^{2} - 40T - 1076$$
$19$ $$T^{2} - 122T + 3680$$
$23$ $$T^{2} + 16T - 23552$$
$29$ $$T^{2} - 352T + 29500$$
$31$ $$T^{2} + 66T - 13712$$
$37$ $$T^{2} + 188T - 56764$$
$41$ $$T^{2} - 16T - 119492$$
$43$ $$T^{2} - 396T - 63296$$
$47$ $$T^{2} - 188T - 192064$$
$53$ $$T^{2} - 982T + 206600$$
$59$ $$T^{2} + 516T + 7360$$
$61$ $$T^{2} + 880T + 121276$$
$67$ $$T^{2} - 356T - 501152$$
$71$ $$T^{2} + 310T - 51784$$
$73$ $$T^{2} - 326T + 11768$$
$79$ $$T^{2} + 1832 T + 838400$$
$83$ $$T^{2} - 680T - 398704$$
$89$ $$T^{2} - 796T - 998780$$
$97$ $$T^{2} + 670T - 679936$$