# Properties

 Label 40.2.c.a Level $40$ Weight $2$ Character orbit 40.c Analytic conductor $0.319$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [40,2,Mod(9,40)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(40, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("40.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$40 = 2^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 40.c (of order $$2$$, degree $$1$$, 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: $$0.319401608085$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( - \beta - 1) q^{5} - \beta q^{7} - q^{9} +O(q^{10})$$ q + b * q^3 + (-b - 1) * q^5 - b * q^7 - q^9 $$q + \beta q^{3} + ( - \beta - 1) q^{5} - \beta q^{7} - q^{9} - 4 q^{11} + 2 \beta q^{13} + ( - \beta + 4) q^{15} + 4 q^{19} + 4 q^{21} - \beta q^{23} + (2 \beta - 3) q^{25} + 2 \beta q^{27} - 2 q^{29} - 4 \beta q^{33} + (\beta - 4) q^{35} - 2 \beta q^{37} - 8 q^{39} + 2 q^{41} - 3 \beta q^{43} + (\beta + 1) q^{45} + 3 \beta q^{47} + 3 q^{49} - 2 \beta q^{53} + (4 \beta + 4) q^{55} + 4 \beta q^{57} + 12 q^{59} - 10 q^{61} + \beta q^{63} + ( - 2 \beta + 8) q^{65} - 7 \beta q^{67} + 4 q^{69} + 8 q^{71} + 4 \beta q^{73} + ( - 3 \beta - 8) q^{75} + 4 \beta q^{77} - 16 q^{79} - 11 q^{81} + \beta q^{83} - 2 \beta q^{87} - 6 q^{89} + 8 q^{91} + ( - 4 \beta - 4) q^{95} - 8 \beta q^{97} + 4 q^{99} +O(q^{100})$$ q + b * q^3 + (-b - 1) * q^5 - b * q^7 - q^9 - 4 * q^11 + 2*b * q^13 + (-b + 4) * q^15 + 4 * q^19 + 4 * q^21 - b * q^23 + (2*b - 3) * q^25 + 2*b * q^27 - 2 * q^29 - 4*b * q^33 + (b - 4) * q^35 - 2*b * q^37 - 8 * q^39 + 2 * q^41 - 3*b * q^43 + (b + 1) * q^45 + 3*b * q^47 + 3 * q^49 - 2*b * q^53 + (4*b + 4) * q^55 + 4*b * q^57 + 12 * q^59 - 10 * q^61 + b * q^63 + (-2*b + 8) * q^65 - 7*b * q^67 + 4 * q^69 + 8 * q^71 + 4*b * q^73 + (-3*b - 8) * q^75 + 4*b * q^77 - 16 * q^79 - 11 * q^81 + b * q^83 - 2*b * q^87 - 6 * q^89 + 8 * q^91 + (-4*b - 4) * q^95 - 8*b * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^9 $$2 q - 2 q^{5} - 2 q^{9} - 8 q^{11} + 8 q^{15} + 8 q^{19} + 8 q^{21} - 6 q^{25} - 4 q^{29} - 8 q^{35} - 16 q^{39} + 4 q^{41} + 2 q^{45} + 6 q^{49} + 8 q^{55} + 24 q^{59} - 20 q^{61} + 16 q^{65} + 8 q^{69} + 16 q^{71} - 16 q^{75} - 32 q^{79} - 22 q^{81} - 12 q^{89} + 16 q^{91} - 8 q^{95} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^9 - 8 * q^11 + 8 * q^15 + 8 * q^19 + 8 * q^21 - 6 * q^25 - 4 * q^29 - 8 * q^35 - 16 * q^39 + 4 * q^41 + 2 * q^45 + 6 * q^49 + 8 * q^55 + 24 * q^59 - 20 * q^61 + 16 * q^65 + 8 * q^69 + 16 * q^71 - 16 * q^75 - 32 * q^79 - 22 * q^81 - 12 * q^89 + 16 * q^91 - 8 * q^95 + 8 * q^99

## Character values

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

 $$n$$ $$17$$ $$21$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 − 1.00000i 1.00000i
0 2.00000i 0 −1.00000 + 2.00000i 0 2.00000i 0 −1.00000 0
9.2 0 2.00000i 0 −1.00000 2.00000i 0 2.00000i 0 −1.00000 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.2.c.a 2
3.b odd 2 1 360.2.f.c 2
4.b odd 2 1 80.2.c.a 2
5.b even 2 1 inner 40.2.c.a 2
5.c odd 4 1 200.2.a.b 1
5.c odd 4 1 200.2.a.d 1
7.b odd 2 1 1960.2.g.b 2
8.b even 2 1 320.2.c.c 2
8.d odd 2 1 320.2.c.b 2
12.b even 2 1 720.2.f.e 2
15.d odd 2 1 360.2.f.c 2
15.e even 4 1 1800.2.a.j 1
15.e even 4 1 1800.2.a.s 1
16.e even 4 1 1280.2.f.a 2
16.e even 4 1 1280.2.f.f 2
16.f odd 4 1 1280.2.f.b 2
16.f odd 4 1 1280.2.f.e 2
20.d odd 2 1 80.2.c.a 2
20.e even 4 1 400.2.a.b 1
20.e even 4 1 400.2.a.g 1
24.f even 2 1 2880.2.f.i 2
24.h odd 2 1 2880.2.f.h 2
35.c odd 2 1 1960.2.g.b 2
35.f even 4 1 9800.2.a.d 1
35.f even 4 1 9800.2.a.bf 1
40.e odd 2 1 320.2.c.b 2
40.f even 2 1 320.2.c.c 2
40.i odd 4 1 1600.2.a.f 1
40.i odd 4 1 1600.2.a.v 1
40.k even 4 1 1600.2.a.d 1
40.k even 4 1 1600.2.a.u 1
60.h even 2 1 720.2.f.e 2
60.l odd 4 1 3600.2.a.k 1
60.l odd 4 1 3600.2.a.bb 1
80.k odd 4 1 1280.2.f.b 2
80.k odd 4 1 1280.2.f.e 2
80.q even 4 1 1280.2.f.a 2
80.q even 4 1 1280.2.f.f 2
120.i odd 2 1 2880.2.f.h 2
120.m even 2 1 2880.2.f.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a 2 1.a even 1 1 trivial
40.2.c.a 2 5.b even 2 1 inner
80.2.c.a 2 4.b odd 2 1
80.2.c.a 2 20.d odd 2 1
200.2.a.b 1 5.c odd 4 1
200.2.a.d 1 5.c odd 4 1
320.2.c.b 2 8.d odd 2 1
320.2.c.b 2 40.e odd 2 1
320.2.c.c 2 8.b even 2 1
320.2.c.c 2 40.f even 2 1
360.2.f.c 2 3.b odd 2 1
360.2.f.c 2 15.d odd 2 1
400.2.a.b 1 20.e even 4 1
400.2.a.g 1 20.e even 4 1
720.2.f.e 2 12.b even 2 1
720.2.f.e 2 60.h even 2 1
1280.2.f.a 2 16.e even 4 1
1280.2.f.a 2 80.q even 4 1
1280.2.f.b 2 16.f odd 4 1
1280.2.f.b 2 80.k odd 4 1
1280.2.f.e 2 16.f odd 4 1
1280.2.f.e 2 80.k odd 4 1
1280.2.f.f 2 16.e even 4 1
1280.2.f.f 2 80.q even 4 1
1600.2.a.d 1 40.k even 4 1
1600.2.a.f 1 40.i odd 4 1
1600.2.a.u 1 40.k even 4 1
1600.2.a.v 1 40.i odd 4 1
1800.2.a.j 1 15.e even 4 1
1800.2.a.s 1 15.e even 4 1
1960.2.g.b 2 7.b odd 2 1
1960.2.g.b 2 35.c odd 2 1
2880.2.f.h 2 24.h odd 2 1
2880.2.f.h 2 120.i odd 2 1
2880.2.f.i 2 24.f even 2 1
2880.2.f.i 2 120.m even 2 1
3600.2.a.k 1 60.l odd 4 1
3600.2.a.bb 1 60.l odd 4 1
9800.2.a.d 1 35.f even 4 1
9800.2.a.bf 1 35.f even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(40, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 16$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 64$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} + 4$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 256$$