# Properties

 Label 1872.4.a.x Level $1872$ Weight $4$ Character orbit 1872.a Self dual yes Analytic conductor $110.452$ Analytic rank $0$ 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] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, 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("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1872 = 2^{4} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1872.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: $$110.451575531$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{113})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 28$$ x^2 - x - 28 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 312) 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 $$\beta = \sqrt{113}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 3) q^{5} + (\beta + 5) q^{7}+O(q^{10})$$ q + (-b - 3) * q^5 + (b + 5) * q^7 $$q + ( - \beta - 3) q^{5} + (\beta + 5) q^{7} + ( - 4 \beta - 8) q^{11} + 13 q^{13} - 2 q^{17} + ( - 5 \beta + 35) q^{19} - 64 q^{23} + (6 \beta - 3) q^{25} + (6 \beta + 40) q^{29} + ( - 15 \beta + 125) q^{31} + ( - 8 \beta - 128) q^{35} + ( - 18 \beta - 76) q^{37} + (7 \beta + 73) q^{41} + ( - 8 \beta + 252) q^{43} + ( - 6 \beta - 262) q^{47} + (10 \beta - 205) q^{49} + (8 \beta + 26) q^{53} + (20 \beta + 476) q^{55} + (10 \beta - 82) q^{59} + ( - 46 \beta - 152) q^{61} + ( - 13 \beta - 39) q^{65} + ( - 7 \beta + 457) q^{67} + 48 \beta q^{71} + ( - 6 \beta - 228) q^{73} + ( - 28 \beta - 492) q^{77} + (36 \beta + 412) q^{79} + (50 \beta + 414) q^{83} + (2 \beta + 6) q^{85} + (9 \beta - 413) q^{89} + (13 \beta + 65) q^{91} + ( - 20 \beta + 460) q^{95} + (122 \beta + 276) q^{97}+O(q^{100})$$ q + (-b - 3) * q^5 + (b + 5) * q^7 + (-4*b - 8) * q^11 + 13 * q^13 - 2 * q^17 + (-5*b + 35) * q^19 - 64 * q^23 + (6*b - 3) * q^25 + (6*b + 40) * q^29 + (-15*b + 125) * q^31 + (-8*b - 128) * q^35 + (-18*b - 76) * q^37 + (7*b + 73) * q^41 + (-8*b + 252) * q^43 + (-6*b - 262) * q^47 + (10*b - 205) * q^49 + (8*b + 26) * q^53 + (20*b + 476) * q^55 + (10*b - 82) * q^59 + (-46*b - 152) * q^61 + (-13*b - 39) * q^65 + (-7*b + 457) * q^67 + 48*b * q^71 + (-6*b - 228) * q^73 + (-28*b - 492) * q^77 + (36*b + 412) * q^79 + (50*b + 414) * q^83 + (2*b + 6) * q^85 + (9*b - 413) * q^89 + (13*b + 65) * q^91 + (-20*b + 460) * q^95 + (122*b + 276) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{5} + 10 q^{7}+O(q^{10})$$ 2 * q - 6 * q^5 + 10 * q^7 $$2 q - 6 q^{5} + 10 q^{7} - 16 q^{11} + 26 q^{13} - 4 q^{17} + 70 q^{19} - 128 q^{23} - 6 q^{25} + 80 q^{29} + 250 q^{31} - 256 q^{35} - 152 q^{37} + 146 q^{41} + 504 q^{43} - 524 q^{47} - 410 q^{49} + 52 q^{53} + 952 q^{55} - 164 q^{59} - 304 q^{61} - 78 q^{65} + 914 q^{67} - 456 q^{73} - 984 q^{77} + 824 q^{79} + 828 q^{83} + 12 q^{85} - 826 q^{89} + 130 q^{91} + 920 q^{95} + 552 q^{97}+O(q^{100})$$ 2 * q - 6 * q^5 + 10 * q^7 - 16 * q^11 + 26 * q^13 - 4 * q^17 + 70 * q^19 - 128 * q^23 - 6 * q^25 + 80 * q^29 + 250 * q^31 - 256 * q^35 - 152 * q^37 + 146 * q^41 + 504 * q^43 - 524 * q^47 - 410 * q^49 + 52 * q^53 + 952 * q^55 - 164 * q^59 - 304 * q^61 - 78 * q^65 + 914 * q^67 - 456 * q^73 - 984 * q^77 + 824 * q^79 + 828 * q^83 + 12 * q^85 - 826 * q^89 + 130 * q^91 + 920 * q^95 + 552 * q^97

## 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
 5.81507 −4.81507
0 0 0 −13.6301 0 15.6301 0 0 0
1.2 0 0 0 7.63015 0 −5.63015 0 0 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## 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 1872.4.a.x 2
3.b odd 2 1 624.4.a.q 2
4.b odd 2 1 936.4.a.d 2
12.b even 2 1 312.4.a.c 2
24.f even 2 1 2496.4.a.be 2
24.h odd 2 1 2496.4.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.a.c 2 12.b even 2 1
624.4.a.q 2 3.b odd 2 1
936.4.a.d 2 4.b odd 2 1
1872.4.a.x 2 1.a even 1 1 trivial
2496.4.a.v 2 24.h odd 2 1
2496.4.a.be 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1872))$$:

 $$T_{5}^{2} + 6T_{5} - 104$$ T5^2 + 6*T5 - 104 $$T_{7}^{2} - 10T_{7} - 88$$ T7^2 - 10*T7 - 88

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 6T - 104$$
$7$ $$T^{2} - 10T - 88$$
$11$ $$T^{2} + 16T - 1744$$
$13$ $$(T - 13)^{2}$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} - 70T - 1600$$
$23$ $$(T + 64)^{2}$$
$29$ $$T^{2} - 80T - 2468$$
$31$ $$T^{2} - 250T - 9800$$
$37$ $$T^{2} + 152T - 30836$$
$41$ $$T^{2} - 146T - 208$$
$43$ $$T^{2} - 504T + 56272$$
$47$ $$T^{2} + 524T + 64576$$
$53$ $$T^{2} - 52T - 6556$$
$59$ $$T^{2} + 164T - 4576$$
$61$ $$T^{2} + 304T - 216004$$
$67$ $$T^{2} - 914T + 203312$$
$71$ $$T^{2} - 260352$$
$73$ $$T^{2} + 456T + 47916$$
$79$ $$T^{2} - 824T + 23296$$
$83$ $$T^{2} - 828T - 111104$$
$89$ $$T^{2} + 826T + 161416$$
$97$ $$T^{2} - 552 T - 1605716$$