# Properties

 Label 920.2.e.a Level $920$ Weight $2$ Character orbit 920.e Analytic conductor $7.346$ 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] = [920,2,Mod(369,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("920.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$920 = 2^{3} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 920.e (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: $$7.34623698596$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + ( - i + 2) q^{5} - 3 i q^{7} - q^{9} +O(q^{10})$$ q + 2*i * q^3 + (-i + 2) * q^5 - 3*i * q^7 - q^9 $$q + 2 i q^{3} + ( - i + 2) q^{5} - 3 i q^{7} - q^{9} + 6 i q^{13} + (4 i + 2) q^{15} - 7 i q^{17} + 4 q^{19} + 6 q^{21} + i q^{23} + ( - 4 i + 3) q^{25} + 4 i q^{27} + 9 q^{29} - 3 q^{31} + ( - 6 i - 3) q^{35} - 7 i q^{37} - 12 q^{39} + 9 q^{41} + 4 i q^{43} + (i - 2) q^{45} - 2 i q^{47} - 2 q^{49} + 14 q^{51} + 7 i q^{53} + 8 i q^{57} - 9 q^{59} - 2 q^{61} + 3 i q^{63} + (12 i + 6) q^{65} + 13 i q^{67} - 2 q^{69} - 13 q^{71} + 4 i q^{73} + (6 i + 8) q^{75} + 2 q^{79} - 11 q^{81} + 11 i q^{83} + ( - 14 i - 7) q^{85} + 18 i q^{87} + 10 q^{89} + 18 q^{91} - 6 i q^{93} + ( - 4 i + 8) q^{95} + 2 i q^{97} +O(q^{100})$$ q + 2*i * q^3 + (-i + 2) * q^5 - 3*i * q^7 - q^9 + 6*i * q^13 + (4*i + 2) * q^15 - 7*i * q^17 + 4 * q^19 + 6 * q^21 + i * q^23 + (-4*i + 3) * q^25 + 4*i * q^27 + 9 * q^29 - 3 * q^31 + (-6*i - 3) * q^35 - 7*i * q^37 - 12 * q^39 + 9 * q^41 + 4*i * q^43 + (i - 2) * q^45 - 2*i * q^47 - 2 * q^49 + 14 * q^51 + 7*i * q^53 + 8*i * q^57 - 9 * q^59 - 2 * q^61 + 3*i * q^63 + (12*i + 6) * q^65 + 13*i * q^67 - 2 * q^69 - 13 * q^71 + 4*i * q^73 + (6*i + 8) * q^75 + 2 * q^79 - 11 * q^81 + 11*i * q^83 + (-14*i - 7) * q^85 + 18*i * q^87 + 10 * q^89 + 18 * q^91 - 6*i * q^93 + (-4*i + 8) * q^95 + 2*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^5 - 2 * q^9 $$2 q + 4 q^{5} - 2 q^{9} + 4 q^{15} + 8 q^{19} + 12 q^{21} + 6 q^{25} + 18 q^{29} - 6 q^{31} - 6 q^{35} - 24 q^{39} + 18 q^{41} - 4 q^{45} - 4 q^{49} + 28 q^{51} - 18 q^{59} - 4 q^{61} + 12 q^{65} - 4 q^{69} - 26 q^{71} + 16 q^{75} + 4 q^{79} - 22 q^{81} - 14 q^{85} + 20 q^{89} + 36 q^{91} + 16 q^{95}+O(q^{100})$$ 2 * q + 4 * q^5 - 2 * q^9 + 4 * q^15 + 8 * q^19 + 12 * q^21 + 6 * q^25 + 18 * q^29 - 6 * q^31 - 6 * q^35 - 24 * q^39 + 18 * q^41 - 4 * q^45 - 4 * q^49 + 28 * q^51 - 18 * q^59 - 4 * q^61 + 12 * q^65 - 4 * q^69 - 26 * q^71 + 16 * q^75 + 4 * q^79 - 22 * q^81 - 14 * q^85 + 20 * q^89 + 36 * q^91 + 16 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/920\mathbb{Z}\right)^\times$$.

 $$n$$ $$231$$ $$281$$ $$461$$ $$737$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
369.1
 − 1.00000i 1.00000i
0 2.00000i 0 2.00000 + 1.00000i 0 3.00000i 0 −1.00000 0
369.2 0 2.00000i 0 2.00000 1.00000i 0 3.00000i 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 920.2.e.a 2
4.b odd 2 1 1840.2.e.a 2
5.b even 2 1 inner 920.2.e.a 2
5.c odd 4 1 4600.2.a.b 1
5.c odd 4 1 4600.2.a.o 1
20.d odd 2 1 1840.2.e.a 2
20.e even 4 1 9200.2.a.d 1
20.e even 4 1 9200.2.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.a 2 1.a even 1 1 trivial
920.2.e.a 2 5.b even 2 1 inner
1840.2.e.a 2 4.b odd 2 1
1840.2.e.a 2 20.d odd 2 1
4600.2.a.b 1 5.c odd 4 1
4600.2.a.o 1 5.c odd 4 1
9200.2.a.d 1 20.e even 4 1
9200.2.a.bi 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(920, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 9$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 49$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T - 9)^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 49$$
$41$ $$(T - 9)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 49$$
$59$ $$(T + 9)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 169$$
$71$ $$(T + 13)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 2)^{2}$$
$83$ $$T^{2} + 121$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 4$$