# Properties

 Label 2790.2.a.bh Level $2790$ Weight $2$ Character orbit 2790.a Self dual yes Analytic conductor $22.278$ 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] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2790.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: $$22.2782621639$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 310) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} + 2 \beta q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^5 + 2*b * q^7 + q^8 $$q + q^{2} + q^{4} + q^{5} + 2 \beta q^{7} + q^{8} + q^{10} + ( - \beta + 1) q^{11} + ( - \beta - 3) q^{13} + 2 \beta q^{14} + q^{16} + 4 q^{17} + ( - 2 \beta + 2) q^{19} + q^{20} + ( - \beta + 1) q^{22} + (2 \beta + 4) q^{23} + q^{25} + ( - \beta - 3) q^{26} + 2 \beta q^{28} + (\beta + 1) q^{29} - q^{31} + q^{32} + 4 q^{34} + 2 \beta q^{35} + (\beta - 5) q^{37} + ( - 2 \beta + 2) q^{38} + q^{40} + (2 \beta + 6) q^{41} + (\beta + 5) q^{43} + ( - \beta + 1) q^{44} + (2 \beta + 4) q^{46} - 2 \beta q^{47} + 5 q^{49} + q^{50} + ( - \beta - 3) q^{52} + (\beta + 7) q^{53} + ( - \beta + 1) q^{55} + 2 \beta q^{56} + (\beta + 1) q^{58} + (2 \beta - 2) q^{59} + ( - 7 \beta - 3) q^{61} - q^{62} + q^{64} + ( - \beta - 3) q^{65} + ( - 6 \beta - 2) q^{67} + 4 q^{68} + 2 \beta q^{70} - 8 \beta q^{71} + (6 \beta + 6) q^{73} + (\beta - 5) q^{74} + ( - 2 \beta + 2) q^{76} + (2 \beta - 6) q^{77} + ( - 2 \beta + 14) q^{79} + q^{80} + (2 \beta + 6) q^{82} + ( - \beta + 3) q^{83} + 4 q^{85} + (\beta + 5) q^{86} + ( - \beta + 1) q^{88} + (4 \beta + 6) q^{89} + ( - 6 \beta - 6) q^{91} + (2 \beta + 4) q^{92} - 2 \beta q^{94} + ( - 2 \beta + 2) q^{95} + 4 q^{97} + 5 q^{98} +O(q^{100})$$ q + q^2 + q^4 + q^5 + 2*b * q^7 + q^8 + q^10 + (-b + 1) * q^11 + (-b - 3) * q^13 + 2*b * q^14 + q^16 + 4 * q^17 + (-2*b + 2) * q^19 + q^20 + (-b + 1) * q^22 + (2*b + 4) * q^23 + q^25 + (-b - 3) * q^26 + 2*b * q^28 + (b + 1) * q^29 - q^31 + q^32 + 4 * q^34 + 2*b * q^35 + (b - 5) * q^37 + (-2*b + 2) * q^38 + q^40 + (2*b + 6) * q^41 + (b + 5) * q^43 + (-b + 1) * q^44 + (2*b + 4) * q^46 - 2*b * q^47 + 5 * q^49 + q^50 + (-b - 3) * q^52 + (b + 7) * q^53 + (-b + 1) * q^55 + 2*b * q^56 + (b + 1) * q^58 + (2*b - 2) * q^59 + (-7*b - 3) * q^61 - q^62 + q^64 + (-b - 3) * q^65 + (-6*b - 2) * q^67 + 4 * q^68 + 2*b * q^70 - 8*b * q^71 + (6*b + 6) * q^73 + (b - 5) * q^74 + (-2*b + 2) * q^76 + (2*b - 6) * q^77 + (-2*b + 14) * q^79 + q^80 + (2*b + 6) * q^82 + (-b + 3) * q^83 + 4 * q^85 + (b + 5) * q^86 + (-b + 1) * q^88 + (4*b + 6) * q^89 + (-6*b - 6) * q^91 + (2*b + 4) * q^92 - 2*b * q^94 + (-2*b + 2) * q^95 + 4 * q^97 + 5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 2 q^{11} - 6 q^{13} + 2 q^{16} + 8 q^{17} + 4 q^{19} + 2 q^{20} + 2 q^{22} + 8 q^{23} + 2 q^{25} - 6 q^{26} + 2 q^{29} - 2 q^{31} + 2 q^{32} + 8 q^{34} - 10 q^{37} + 4 q^{38} + 2 q^{40} + 12 q^{41} + 10 q^{43} + 2 q^{44} + 8 q^{46} + 10 q^{49} + 2 q^{50} - 6 q^{52} + 14 q^{53} + 2 q^{55} + 2 q^{58} - 4 q^{59} - 6 q^{61} - 2 q^{62} + 2 q^{64} - 6 q^{65} - 4 q^{67} + 8 q^{68} + 12 q^{73} - 10 q^{74} + 4 q^{76} - 12 q^{77} + 28 q^{79} + 2 q^{80} + 12 q^{82} + 6 q^{83} + 8 q^{85} + 10 q^{86} + 2 q^{88} + 12 q^{89} - 12 q^{91} + 8 q^{92} + 4 q^{95} + 8 q^{97} + 10 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 2 * q^11 - 6 * q^13 + 2 * q^16 + 8 * q^17 + 4 * q^19 + 2 * q^20 + 2 * q^22 + 8 * q^23 + 2 * q^25 - 6 * q^26 + 2 * q^29 - 2 * q^31 + 2 * q^32 + 8 * q^34 - 10 * q^37 + 4 * q^38 + 2 * q^40 + 12 * q^41 + 10 * q^43 + 2 * q^44 + 8 * q^46 + 10 * q^49 + 2 * q^50 - 6 * q^52 + 14 * q^53 + 2 * q^55 + 2 * q^58 - 4 * q^59 - 6 * q^61 - 2 * q^62 + 2 * q^64 - 6 * q^65 - 4 * q^67 + 8 * q^68 + 12 * q^73 - 10 * q^74 + 4 * q^76 - 12 * q^77 + 28 * q^79 + 2 * q^80 + 12 * q^82 + 6 * q^83 + 8 * q^85 + 10 * q^86 + 2 * q^88 + 12 * q^89 - 12 * q^91 + 8 * q^92 + 4 * q^95 + 8 * q^97 + 10 * q^98

## 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
 −1.73205 1.73205
1.00000 0 1.00000 1.00000 0 −3.46410 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 3.46410 1.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$31$$ $$1$$

## 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 2790.2.a.bh 2
3.b odd 2 1 310.2.a.c 2
12.b even 2 1 2480.2.a.s 2
15.d odd 2 1 1550.2.a.j 2
15.e even 4 2 1550.2.b.h 4
24.f even 2 1 9920.2.a.bl 2
24.h odd 2 1 9920.2.a.bt 2
93.c even 2 1 9610.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.c 2 3.b odd 2 1
1550.2.a.j 2 15.d odd 2 1
1550.2.b.h 4 15.e even 4 2
2480.2.a.s 2 12.b even 2 1
2790.2.a.bh 2 1.a even 1 1 trivial
9610.2.a.j 2 93.c even 2 1
9920.2.a.bl 2 24.f even 2 1
9920.2.a.bt 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2790))$$:

 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{11}^{2} - 2T_{11} - 2$$ T11^2 - 2*T11 - 2 $$T_{13}^{2} + 6T_{13} + 6$$ T13^2 + 6*T13 + 6 $$T_{17} - 4$$ T17 - 4 $$T_{19}^{2} - 4T_{19} - 8$$ T19^2 - 4*T19 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2} - 2T - 2$$
$13$ $$T^{2} + 6T + 6$$
$17$ $$(T - 4)^{2}$$
$19$ $$T^{2} - 4T - 8$$
$23$ $$T^{2} - 8T + 4$$
$29$ $$T^{2} - 2T - 2$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 10T + 22$$
$41$ $$T^{2} - 12T + 24$$
$43$ $$T^{2} - 10T + 22$$
$47$ $$T^{2} - 12$$
$53$ $$T^{2} - 14T + 46$$
$59$ $$T^{2} + 4T - 8$$
$61$ $$T^{2} + 6T - 138$$
$67$ $$T^{2} + 4T - 104$$
$71$ $$T^{2} - 192$$
$73$ $$T^{2} - 12T - 72$$
$79$ $$T^{2} - 28T + 184$$
$83$ $$T^{2} - 6T + 6$$
$89$ $$T^{2} - 12T - 12$$
$97$ $$(T - 4)^{2}$$