# Properties

 Label 1872.4.a.bb 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: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{5} + (11 \beta - 1) q^{7}+O(q^{10})$$ q + (b + 1) * q^5 + (11*b - 1) * q^7 $$q + (\beta + 1) q^{5} + (11 \beta - 1) q^{7} + ( - 12 \beta + 46) q^{11} - 13 q^{13} + ( - 17 \beta - 1) q^{17} + ( - 32 \beta + 58) q^{19} + (12 \beta + 92) q^{23} + (3 \beta - 120) q^{25} + (96 \beta - 26) q^{29} + ( - 34 \beta + 60) q^{31} + (21 \beta + 43) q^{35} + ( - 5 \beta + 107) q^{37} + (22 \beta + 104) q^{41} + (143 \beta - 215) q^{43} + (121 \beta + 157) q^{47} + (99 \beta + 142) q^{49} + (30 \beta + 44) q^{53} + (22 \beta - 2) q^{55} + ( - 124 \beta - 122) q^{59} + (190 \beta - 624) q^{61} + ( - 13 \beta - 13) q^{65} + ( - 232 \beta + 82) q^{67} + (231 \beta - 181) q^{71} + ( - 260 \beta + 358) q^{73} + (386 \beta - 574) q^{77} + (40 \beta + 484) q^{79} + (182 \beta + 888) q^{83} + ( - 35 \beta - 69) q^{85} + ( - 388 \beta + 554) q^{89} + ( - 143 \beta + 13) q^{91} + ( - 6 \beta - 70) q^{95} + ( - 508 \beta - 210) q^{97}+O(q^{100})$$ q + (b + 1) * q^5 + (11*b - 1) * q^7 + (-12*b + 46) * q^11 - 13 * q^13 + (-17*b - 1) * q^17 + (-32*b + 58) * q^19 + (12*b + 92) * q^23 + (3*b - 120) * q^25 + (96*b - 26) * q^29 + (-34*b + 60) * q^31 + (21*b + 43) * q^35 + (-5*b + 107) * q^37 + (22*b + 104) * q^41 + (143*b - 215) * q^43 + (121*b + 157) * q^47 + (99*b + 142) * q^49 + (30*b + 44) * q^53 + (22*b - 2) * q^55 + (-124*b - 122) * q^59 + (190*b - 624) * q^61 + (-13*b - 13) * q^65 + (-232*b + 82) * q^67 + (231*b - 181) * q^71 + (-260*b + 358) * q^73 + (386*b - 574) * q^77 + (40*b + 484) * q^79 + (182*b + 888) * q^83 + (-35*b - 69) * q^85 + (-388*b + 554) * q^89 + (-143*b + 13) * q^91 + (-6*b - 70) * q^95 + (-508*b - 210) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{5} + 9 q^{7}+O(q^{10})$$ 2 * q + 3 * q^5 + 9 * q^7 $$2 q + 3 q^{5} + 9 q^{7} + 80 q^{11} - 26 q^{13} - 19 q^{17} + 84 q^{19} + 196 q^{23} - 237 q^{25} + 44 q^{29} + 86 q^{31} + 107 q^{35} + 209 q^{37} + 230 q^{41} - 287 q^{43} + 435 q^{47} + 383 q^{49} + 118 q^{53} + 18 q^{55} - 368 q^{59} - 1058 q^{61} - 39 q^{65} - 68 q^{67} - 131 q^{71} + 456 q^{73} - 762 q^{77} + 1008 q^{79} + 1958 q^{83} - 173 q^{85} + 720 q^{89} - 117 q^{91} - 146 q^{95} - 928 q^{97}+O(q^{100})$$ 2 * q + 3 * q^5 + 9 * q^7 + 80 * q^11 - 26 * q^13 - 19 * q^17 + 84 * q^19 + 196 * q^23 - 237 * q^25 + 44 * q^29 + 86 * q^31 + 107 * q^35 + 209 * q^37 + 230 * q^41 - 287 * q^43 + 435 * q^47 + 383 * q^49 + 118 * q^53 + 18 * q^55 - 368 * q^59 - 1058 * q^61 - 39 * q^65 - 68 * q^67 - 131 * q^71 + 456 * q^73 - 762 * q^77 + 1008 * q^79 + 1958 * q^83 - 173 * q^85 + 720 * q^89 - 117 * q^91 - 146 * q^95 - 928 * 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
 −1.56155 2.56155
0 0 0 −0.561553 0 −18.1771 0 0 0
1.2 0 0 0 3.56155 0 27.1771 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.bb 2
3.b odd 2 1 208.4.a.h 2
4.b odd 2 1 117.4.a.d 2
12.b even 2 1 13.4.a.b 2
24.f even 2 1 832.4.a.s 2
24.h odd 2 1 832.4.a.z 2
52.b odd 2 1 1521.4.a.r 2
60.h even 2 1 325.4.a.f 2
60.l odd 4 2 325.4.b.e 4
84.h odd 2 1 637.4.a.b 2
132.d odd 2 1 1573.4.a.b 2
156.h even 2 1 169.4.a.g 2
156.l odd 4 2 169.4.b.f 4
156.p even 6 2 169.4.c.g 4
156.r even 6 2 169.4.c.j 4
156.v odd 12 4 169.4.e.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 12.b even 2 1
117.4.a.d 2 4.b odd 2 1
169.4.a.g 2 156.h even 2 1
169.4.b.f 4 156.l odd 4 2
169.4.c.g 4 156.p even 6 2
169.4.c.j 4 156.r even 6 2
169.4.e.f 8 156.v odd 12 4
208.4.a.h 2 3.b odd 2 1
325.4.a.f 2 60.h even 2 1
325.4.b.e 4 60.l odd 4 2
637.4.a.b 2 84.h odd 2 1
832.4.a.s 2 24.f even 2 1
832.4.a.z 2 24.h odd 2 1
1521.4.a.r 2 52.b odd 2 1
1573.4.a.b 2 132.d odd 2 1
1872.4.a.bb 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{5}^{2} - 3T_{5} - 2$$ T5^2 - 3*T5 - 2 $$T_{7}^{2} - 9T_{7} - 494$$ T7^2 - 9*T7 - 494

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T - 2$$
$7$ $$T^{2} - 9T - 494$$
$11$ $$T^{2} - 80T + 988$$
$13$ $$(T + 13)^{2}$$
$17$ $$T^{2} + 19T - 1138$$
$19$ $$T^{2} - 84T - 2588$$
$23$ $$T^{2} - 196T + 8992$$
$29$ $$T^{2} - 44T - 38684$$
$31$ $$T^{2} - 86T - 3064$$
$37$ $$T^{2} - 209T + 10814$$
$41$ $$T^{2} - 230T + 11168$$
$43$ $$T^{2} + 287T - 66316$$
$47$ $$T^{2} - 435T - 14918$$
$53$ $$T^{2} - 118T - 344$$
$59$ $$T^{2} + 368T - 31492$$
$61$ $$T^{2} + 1058 T + 126416$$
$67$ $$T^{2} + 68T - 227596$$
$71$ $$T^{2} + 131T - 222494$$
$73$ $$T^{2} - 456T - 235316$$
$79$ $$T^{2} - 1008 T + 247216$$
$83$ $$T^{2} - 1958 T + 817664$$
$89$ $$T^{2} - 720T - 510212$$
$97$ $$T^{2} + 928T - 881476$$