# Properties

 Label 1840.4.a.l Level $1840$ Weight $4$ Character orbit 1840.a Self dual yes Analytic conductor $108.564$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$108.563514411$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 84x^{2} - 11x + 1242$$ x^4 - 84*x^2 - 11*x + 1242 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{3} - 5 q^{5} + (2 \beta_{2} + \beta_1 - 6) q^{7} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 16) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^3 - 5 * q^5 + (2*b2 + b1 - 6) * q^7 + (b3 + b2 - 2*b1 + 16) * q^9 $$q + (\beta_1 - 1) q^{3} - 5 q^{5} + (2 \beta_{2} + \beta_1 - 6) q^{7} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 16) q^{9} + ( - 3 \beta_{2} - \beta_1 - 24) q^{11} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 9) q^{13} + ( - 5 \beta_1 + 5) q^{15} + ( - \beta_{3} - 5 \beta_{2} + \cdots + 26) q^{17}+ \cdots + ( - 11 \beta_{3} - 38 \beta_{2} + \cdots - 510) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^3 - 5 * q^5 + (2*b2 + b1 - 6) * q^7 + (b3 + b2 - 2*b1 + 16) * q^9 + (-3*b2 - b1 - 24) * q^11 + (-3*b3 + b2 + 2*b1 + 9) * q^13 + (-5*b1 + 5) * q^15 + (-b3 - 5*b2 + 4*b1 + 26) * q^17 + (b3 - 6*b2 - 48) * q^19 + (5*b3 + 7*b2 - 5*b1 + 76) * q^21 - 23 * q^23 + 25 * q^25 + (-5*b3 + 10*b2 - b1 - 61) * q^27 + (7*b3 - 11*b2 + 23*b1 + 69) * q^29 + (2*b3 + 11*b2 - 13*b1 + 55) * q^31 + (-7*b3 - 10*b2 - 26*b1 - 60) * q^33 + (-10*b2 - 5*b1 + 30) * q^35 + (-7*b3 - 4*b2 + 17*b1 + 2) * q^37 + (19*b3 - 22*b2 - 13*b1 + 95) * q^39 + (3*b3 + 8*b2 - 16*b1 - 91) * q^41 + (-16*b3 + 12*b2 + 4*b1 + 32) * q^43 + (-5*b3 - 5*b2 + 10*b1 - 80) * q^45 + (-6*b3 - 13*b2 - 50*b1 + 89) * q^47 + (9*b3 - 19*b2 + 32*b1 + 143) * q^49 + (-b3 - 20*b2 + 10*b1 + 74) * q^51 + (3*b3 + 38*b2 - 41*b1 + 14) * q^53 + (15*b2 + 5*b1 + 120) * q^55 + (-17*b3 - 9*b2 - 47*b1 - 38) * q^57 + (9*b3 - 24*b2 - 53*b1) * q^59 + (-10*b3 - 23*b2 - 51*b1 - 80) * q^61 + (-16*b3 + 7*b2 + 96*b1 - 36) * q^63 + (15*b3 - 5*b2 - 10*b1 - 45) * q^65 + (-b3 - 52*b2 + 35*b1 - 194) * q^67 + (-23*b1 + 23) * q^69 + (-b3 + 18*b2 - 56*b1 + 319) * q^71 + (-10*b3 - 3*b2 - 102*b1 + 129) * q^73 + (25*b1 - 25) * q^75 + (-11*b3 - 33*b2 - 83*b1 - 496) * q^77 + (-28*b3 + 62*b2 - 80*b1 - 278) * q^79 + (17*b3 - 43*b2 - 31*b1 - 263) * q^81 + (37*b3 - 60*b2 + 29*b1 + 320) * q^83 + (5*b3 + 25*b2 - 20*b1 - 130) * q^85 + (-34*b3 + 53*b2 + 84*b1 + 729) * q^87 + (-12*b3 + 60*b2 - 228) * q^89 + (36*b3 + 25*b2 - 131*b1 + 246) * q^91 + (-b3 + 38*b2 + 93*b1 - 451) * q^93 + (-5*b3 + 30*b2 + 240) * q^95 + (44*b3 + 59*b2 + 63*b1 + 38) * q^97 + (-11*b3 - 38*b2 - 66*b1 - 510) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 20 * q^5 - 26 * q^7 + 64 * q^9 $$4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9} - 93 q^{11} + 32 q^{13} + 20 q^{15} + 108 q^{17} - 185 q^{19} + 302 q^{21} - 92 q^{23} + 100 q^{25} - 259 q^{27} + 294 q^{29} + 211 q^{31} - 237 q^{33} + 130 q^{35} + 5 q^{37} + 421 q^{39} - 369 q^{41} + 100 q^{43} - 320 q^{45} + 363 q^{47} + 600 q^{49} + 315 q^{51} + 21 q^{53} + 465 q^{55} - 160 q^{57} + 33 q^{59} - 307 q^{61} - 167 q^{63} - 160 q^{65} - 725 q^{67} + 92 q^{69} + 1257 q^{71} + 509 q^{73} - 100 q^{75} - 1962 q^{77} - 1202 q^{79} - 992 q^{81} + 1377 q^{83} - 540 q^{85} + 2829 q^{87} - 984 q^{89} + 995 q^{91} - 1843 q^{93} + 925 q^{95} + 137 q^{97} - 2013 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 - 20 * q^5 - 26 * q^7 + 64 * q^9 - 93 * q^11 + 32 * q^13 + 20 * q^15 + 108 * q^17 - 185 * q^19 + 302 * q^21 - 92 * q^23 + 100 * q^25 - 259 * q^27 + 294 * q^29 + 211 * q^31 - 237 * q^33 + 130 * q^35 + 5 * q^37 + 421 * q^39 - 369 * q^41 + 100 * q^43 - 320 * q^45 + 363 * q^47 + 600 * q^49 + 315 * q^51 + 21 * q^53 + 465 * q^55 - 160 * q^57 + 33 * q^59 - 307 * q^61 - 167 * q^63 - 160 * q^65 - 725 * q^67 + 92 * q^69 + 1257 * q^71 + 509 * q^73 - 100 * q^75 - 1962 * q^77 - 1202 * q^79 - 992 * q^81 + 1377 * q^83 - 540 * q^85 + 2829 * q^87 - 984 * q^89 + 995 * q^91 - 1843 * q^93 + 925 * q^95 + 137 * q^97 - 2013 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 84x^{2} - 11x + 1242$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 50\nu - 96 ) / 15$$ (v^3 + 2*v^2 - 50*v - 96) / 15 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 13\nu^{2} + 50\nu - 534 ) / 15$$ (-v^3 + 13*v^2 + 50*v - 534) / 15
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 42$$ b3 + b2 + 42 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + 13\beta_{2} + 50\beta _1 + 12$$ -2*b3 + 13*b2 + 50*b1 + 12

## 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
 −7.92791 −4.50148 4.26018 8.16920
0 −8.92791 0 −5.00000 0 −23.5524 0 52.7075 0
1.2 0 −5.50148 0 −5.00000 0 −0.0500526 0 3.26627 0
1.3 0 3.26018 0 −5.00000 0 −27.7921 0 −16.3712 0
1.4 0 7.16920 0 −5.00000 0 25.3945 0 24.3974 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
$$2$$ $$-1$$
$$5$$ $$+1$$
$$23$$ $$+1$$

## 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 1840.4.a.l 4
4.b odd 2 1 230.4.a.i 4
12.b even 2 1 2070.4.a.bi 4
20.d odd 2 1 1150.4.a.o 4
20.e even 4 2 1150.4.b.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.i 4 4.b odd 2 1
1150.4.a.o 4 20.d odd 2 1
1150.4.b.m 8 20.e even 4 2
1840.4.a.l 4 1.a even 1 1 trivial
2070.4.a.bi 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{4} + 4T_{3}^{3} - 78T_{3}^{2} - 175T_{3} + 1148$$ T3^4 + 4*T3^3 - 78*T3^2 - 175*T3 + 1148 $$T_{7}^{4} + 26T_{7}^{3} - 648T_{7}^{2} - 16655T_{7} - 832$$ T7^4 + 26*T7^3 - 648*T7^2 - 16655*T7 - 832

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 4 T^{3} + \cdots + 1148$$
$5$ $$(T + 5)^{4}$$
$7$ $$T^{4} + 26 T^{3} + \cdots - 832$$
$11$ $$T^{4} + 93 T^{3} + \cdots - 41700$$
$13$ $$T^{4} - 32 T^{3} + \cdots + 682562$$
$17$ $$T^{4} - 108 T^{3} + \cdots + 96120$$
$19$ $$T^{4} + 185 T^{3} + \cdots + 1404820$$
$23$ $$(T + 23)^{4}$$
$29$ $$T^{4} + \cdots - 1735042500$$
$31$ $$T^{4} - 211 T^{3} + \cdots - 311259365$$
$37$ $$T^{4} - 5 T^{3} + \cdots - 14187104$$
$41$ $$T^{4} + 369 T^{3} + \cdots - 114199911$$
$43$ $$T^{4} - 100 T^{3} + \cdots + 93158400$$
$47$ $$T^{4} + \cdots - 9732540576$$
$53$ $$T^{4} + \cdots - 8104397736$$
$59$ $$T^{4} - 33 T^{3} + \cdots - 623752800$$
$61$ $$T^{4} + \cdots - 5277943032$$
$67$ $$T^{4} + \cdots - 49731103520$$
$71$ $$T^{4} + \cdots - 3703975749$$
$73$ $$T^{4} + \cdots + 80640087688$$
$79$ $$T^{4} + \cdots - 498954065920$$
$83$ $$T^{4} + \cdots - 424764340752$$
$89$ $$T^{4} + \cdots + 92383027200$$
$97$ $$T^{4} + \cdots + 734679597128$$