# Properties

 Label 40.3.l.b Level $40$ Weight $3$ Character orbit 40.l Analytic conductor $1.090$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [40,3,Mod(17,40)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(40, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("40.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$40 = 2^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 40.l (of order $$4$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$1.08992105744$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 21x^{2} + 100$$ x^4 + 21*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{3} + (\beta_{3} - 2 \beta_1 - 1) q^{5} + ( - \beta_{3} - 3 \beta_1 + 3) q^{7} + ( - \beta_{3} - \beta_{2} + 12 \beta_1) q^{9}+O(q^{10})$$ q + (b2 - 1) * q^3 + (b3 - 2*b1 - 1) * q^5 + (-b3 - 3*b1 + 3) * q^7 + (-b3 - b2 + 12*b1) * q^9 $$q + (\beta_{2} - 1) q^{3} + (\beta_{3} - 2 \beta_1 - 1) q^{5} + ( - \beta_{3} - 3 \beta_1 + 3) q^{7} + ( - \beta_{3} - \beta_{2} + 12 \beta_1) q^{9} + (\beta_{3} - \beta_{2} - \beta_1 + 6) q^{11} + ( - 2 \beta_{2} - 9 \beta_1 - 7) q^{13} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 21) q^{15} + (2 \beta_{3} - 5 \beta_1 + 5) q^{17} + (2 \beta_{3} + 2 \beta_{2} + 10 \beta_1) q^{19} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 14) q^{21} + ( - 3 \beta_{2} + 8 \beta_1 + 11) q^{23} + ( - 3 \beta_{3} + 3 \beta_{2} - 13 \beta_1 - 3) q^{25} + (4 \beta_{3} - 24 \beta_1 + 24) q^{27} + (4 \beta_{3} + 4 \beta_{2} + 20 \beta_1) q^{29} + ( - \beta_{3} + \beta_{2} + \beta_1 - 10) q^{31} + (6 \beta_{2} - 20 \beta_1 - 26) q^{33} + (5 \beta_{3} + 2 \beta_{2} + 19 \beta_1 - 9) q^{35} + ( - 8 \beta_{3} + 19 \beta_1 - 19) q^{37} + ( - 7 \beta_{3} - 7 \beta_{2} - 33 \beta_1) q^{39} + (9 \beta_{3} - 9 \beta_{2} - 9 \beta_1 + 22) q^{41} + (\beta_{2} + 24 \beta_1 + 23) q^{43} + (3 \beta_{3} - 12 \beta_{2} - 5 \beta_1 + 46) q^{45} + ( - 5 \beta_{3} + 29 \beta_1 - 29) q^{47} + ( - 7 \beta_{3} - 7 \beta_{2} + 4 \beta_1) q^{49} + ( - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 50) q^{51} + ( - 3 \beta_1 - 3) q^{53} + (5 \beta_{3} + 3 \beta_{2} - 29 \beta_1 + 14) q^{55} + (8 \beta_{3} + 32 \beta_1 - 32) q^{57} + (2 \beta_{3} + 2 \beta_{2} - 54 \beta_1) q^{59} + (\beta_{3} - \beta_{2} - \beta_1 - 50) q^{61} + (5 \beta_{2} + 24 \beta_1 + 19) q^{63} + ( - 5 \beta_{3} + 11 \beta_{2} + 32 \beta_1 + 33) q^{65} + ( - \beta_{3} - 31 \beta_1 + 31) q^{67} + (11 \beta_{3} + 11 \beta_{2} - 71 \beta_1) q^{69} + ( - \beta_{3} + \beta_{2} + \beta_1 + 38) q^{71} + ( - 8 \beta_{2} + 33 \beta_1 + 41) q^{73} + ( - 16 \beta_{3} - 3 \beta_{2} + 76 \beta_1 + 47) q^{75} + (2 \beta_{3} + 2 \beta_1 - 2) q^{77} + (16 \beta_{3} + 16 \beta_{2} - 16 \beta_1) q^{79} + ( - 15 \beta_{3} + 15 \beta_{2} + 15 \beta_1 - 11) q^{81} + ( - 11 \beta_{2} - 52 \beta_1 - 41) q^{83} + (\beta_{3} + 7 \beta_{2} - 38 \beta_1 - 15) q^{85} + (16 \beta_{3} + 64 \beta_1 - 64) q^{87} + ( - 8 \beta_{3} - 8 \beta_{2} - 40 \beta_1) q^{89} + (15 \beta_{3} - 15 \beta_{2} - 15 \beta_1 - 82) q^{91} + ( - 10 \beta_{2} + 20 \beta_1 + 30) q^{93} + ( - 6 \beta_{3} - 10 \beta_{2} - 58 \beta_1 - 24) q^{95} + ( - 49 \beta_1 + 49) q^{97} + ( - 17 \beta_{3} - 17 \beta_{2} + 101 \beta_1) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^3 + (b3 - 2*b1 - 1) * q^5 + (-b3 - 3*b1 + 3) * q^7 + (-b3 - b2 + 12*b1) * q^9 + (b3 - b2 - b1 + 6) * q^11 + (-2*b2 - 9*b1 - 7) * q^13 + (-2*b3 - b2 + 2*b1 - 21) * q^15 + (2*b3 - 5*b1 + 5) * q^17 + (2*b3 + 2*b2 + 10*b1) * q^19 + (-3*b3 + 3*b2 + 3*b1 + 14) * q^21 + (-3*b2 + 8*b1 + 11) * q^23 + (-3*b3 + 3*b2 - 13*b1 - 3) * q^25 + (4*b3 - 24*b1 + 24) * q^27 + (4*b3 + 4*b2 + 20*b1) * q^29 + (-b3 + b2 + b1 - 10) * q^31 + (6*b2 - 20*b1 - 26) * q^33 + (5*b3 + 2*b2 + 19*b1 - 9) * q^35 + (-8*b3 + 19*b1 - 19) * q^37 + (-7*b3 - 7*b2 - 33*b1) * q^39 + (9*b3 - 9*b2 - 9*b1 + 22) * q^41 + (b2 + 24*b1 + 23) * q^43 + (3*b3 - 12*b2 - 5*b1 + 46) * q^45 + (-5*b3 + 29*b1 - 29) * q^47 + (-7*b3 - 7*b2 + 4*b1) * q^49 + (-5*b3 + 5*b2 + 5*b1 - 50) * q^51 + (-3*b1 - 3) * q^53 + (5*b3 + 3*b2 - 29*b1 + 14) * q^55 + (8*b3 + 32*b1 - 32) * q^57 + (2*b3 + 2*b2 - 54*b1) * q^59 + (b3 - b2 - b1 - 50) * q^61 + (5*b2 + 24*b1 + 19) * q^63 + (-5*b3 + 11*b2 + 32*b1 + 33) * q^65 + (-b3 - 31*b1 + 31) * q^67 + (11*b3 + 11*b2 - 71*b1) * q^69 + (-b3 + b2 + b1 + 38) * q^71 + (-8*b2 + 33*b1 + 41) * q^73 + (-16*b3 - 3*b2 + 76*b1 + 47) * q^75 + (2*b3 + 2*b1 - 2) * q^77 + (16*b3 + 16*b2 - 16*b1) * q^79 + (-15*b3 + 15*b2 + 15*b1 - 11) * q^81 + (-11*b2 - 52*b1 - 41) * q^83 + (b3 + 7*b2 - 38*b1 - 15) * q^85 + (16*b3 + 64*b1 - 64) * q^87 + (-8*b3 - 8*b2 - 40*b1) * q^89 + (15*b3 - 15*b2 - 15*b1 - 82) * q^91 + (-10*b2 + 20*b1 + 30) * q^93 + (-6*b3 - 10*b2 - 58*b1 - 24) * q^95 + (-49*b1 + 49) * q^97 + (-17*b3 - 17*b2 + 101*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 6 q^{5} + 14 q^{7}+O(q^{10})$$ 4 * q - 2 * q^3 - 6 * q^5 + 14 * q^7 $$4 q - 2 q^{3} - 6 q^{5} + 14 q^{7} + 20 q^{11} - 32 q^{13} - 82 q^{15} + 16 q^{17} + 68 q^{21} + 38 q^{23} + 88 q^{27} - 36 q^{31} - 92 q^{33} - 42 q^{35} - 60 q^{37} + 52 q^{41} + 94 q^{43} + 154 q^{45} - 106 q^{47} - 180 q^{51} - 12 q^{53} + 52 q^{55} - 144 q^{57} - 204 q^{61} + 86 q^{63} + 164 q^{65} + 126 q^{67} + 156 q^{71} + 148 q^{73} + 214 q^{75} - 12 q^{77} + 16 q^{81} - 186 q^{83} - 48 q^{85} - 288 q^{87} - 388 q^{91} + 100 q^{93} - 104 q^{95} + 196 q^{97}+O(q^{100})$$ 4 * q - 2 * q^3 - 6 * q^5 + 14 * q^7 + 20 * q^11 - 32 * q^13 - 82 * q^15 + 16 * q^17 + 68 * q^21 + 38 * q^23 + 88 * q^27 - 36 * q^31 - 92 * q^33 - 42 * q^35 - 60 * q^37 + 52 * q^41 + 94 * q^43 + 154 * q^45 - 106 * q^47 - 180 * q^51 - 12 * q^53 + 52 * q^55 - 144 * q^57 - 204 * q^61 + 86 * q^63 + 164 * q^65 + 126 * q^67 + 156 * q^71 + 148 * q^73 + 214 * q^75 - 12 * q^77 + 16 * q^81 - 186 * q^83 - 48 * q^85 - 288 * q^87 - 388 * q^91 + 100 * q^93 - 104 * q^95 + 196 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 11\nu ) / 10$$ (v^3 + 11*v) / 10 $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu + 11$$ v^2 + v + 11 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 10\nu^{2} + 21\nu - 110 ) / 10$$ (v^3 - 10*v^2 + 21*v - 110) / 10
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (b3 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + \beta_{2} + \beta _1 - 22 ) / 2$$ (-b3 + b2 + b1 - 22) / 2 $$\nu^{3}$$ $$=$$ $$( -11\beta_{3} - 11\beta_{2} + 31\beta_1 ) / 2$$ (-11*b3 - 11*b2 + 31*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/40\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$21$$ $$31$$ $$\chi(n)$$ $$-\beta_{1}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 3.70156i − 2.70156i − 3.70156i 2.70156i
0 −3.70156 + 3.70156i 0 1.70156 + 4.70156i 0 0.298438 + 0.298438i 0 18.4031i 0
17.2 0 2.70156 2.70156i 0 −4.70156 1.70156i 0 6.70156 + 6.70156i 0 5.59688i 0
33.1 0 −3.70156 3.70156i 0 1.70156 4.70156i 0 0.298438 0.298438i 0 18.4031i 0
33.2 0 2.70156 + 2.70156i 0 −4.70156 + 1.70156i 0 6.70156 6.70156i 0 5.59688i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.3.l.b 4
3.b odd 2 1 360.3.v.c 4
4.b odd 2 1 80.3.p.d 4
5.b even 2 1 200.3.l.e 4
5.c odd 4 1 inner 40.3.l.b 4
5.c odd 4 1 200.3.l.e 4
8.b even 2 1 320.3.p.l 4
8.d odd 2 1 320.3.p.i 4
12.b even 2 1 720.3.bh.l 4
15.d odd 2 1 1800.3.v.k 4
15.e even 4 1 360.3.v.c 4
15.e even 4 1 1800.3.v.k 4
20.d odd 2 1 400.3.p.i 4
20.e even 4 1 80.3.p.d 4
20.e even 4 1 400.3.p.i 4
40.i odd 4 1 320.3.p.l 4
40.k even 4 1 320.3.p.i 4
60.l odd 4 1 720.3.bh.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.l.b 4 1.a even 1 1 trivial
40.3.l.b 4 5.c odd 4 1 inner
80.3.p.d 4 4.b odd 2 1
80.3.p.d 4 20.e even 4 1
200.3.l.e 4 5.b even 2 1
200.3.l.e 4 5.c odd 4 1
320.3.p.i 4 8.d odd 2 1
320.3.p.i 4 40.k even 4 1
320.3.p.l 4 8.b even 2 1
320.3.p.l 4 40.i odd 4 1
360.3.v.c 4 3.b odd 2 1
360.3.v.c 4 15.e even 4 1
400.3.p.i 4 20.d odd 2 1
400.3.p.i 4 20.e even 4 1
720.3.bh.l 4 12.b even 2 1
720.3.bh.l 4 60.l odd 4 1
1800.3.v.k 4 15.d odd 2 1
1800.3.v.k 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 40T_{3} + 400$$ acting on $$S_{3}^{\mathrm{new}}(40, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + 2 T^{2} - 40 T + 400$$
$5$ $$T^{4} + 6 T^{3} + 18 T^{2} + 150 T + 625$$
$7$ $$T^{4} - 14 T^{3} + 98 T^{2} - 56 T + 16$$
$11$ $$(T^{2} - 10 T - 16)^{2}$$
$13$ $$T^{4} + 32 T^{3} + 512 T^{2} + \cdots + 2116$$
$17$ $$T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 2500$$
$19$ $$T^{4} + 528T^{2} + 4096$$
$23$ $$T^{4} - 38 T^{3} + 722 T^{2} + \cdots + 16$$
$29$ $$T^{4} + 2112 T^{2} + 65536$$
$31$ $$(T^{2} + 18 T + 40)^{2}$$
$37$ $$T^{4} + 60 T^{3} + 1800 T^{2} + \cdots + 743044$$
$41$ $$(T^{2} - 26 T - 3152)^{2}$$
$43$ $$T^{4} - 94 T^{3} + 4418 T^{2} + \cdots + 1175056$$
$47$ $$T^{4} + 106 T^{3} + 5618 T^{2} + \cdots + 795664$$
$53$ $$(T^{2} + 6 T + 18)^{2}$$
$59$ $$T^{4} + 6160 T^{2} + \cdots + 7573504$$
$61$ $$(T^{2} + 102 T + 2560)^{2}$$
$67$ $$T^{4} - 126 T^{3} + 7938 T^{2} + \cdots + 3857296$$
$71$ $$(T^{2} - 78 T + 1480)^{2}$$
$73$ $$T^{4} - 148 T^{3} + 10952 T^{2} + \cdots + 2033476$$
$79$ $$T^{4} + 21504 T^{2} + \cdots + 104857600$$
$83$ $$T^{4} + 186 T^{3} + 17298 T^{2} + \cdots + 3400336$$
$89$ $$T^{4} + 8448 T^{2} + \cdots + 1048576$$
$97$ $$(T^{2} - 98 T + 4802)^{2}$$