# Properties

 Label 1350.2.a.x Level $1350$ Weight $2$ Character orbit 1350.a Self dual yes Analytic conductor $10.780$ 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] = [1350,2,Mod(1,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1350.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: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 19$$ x^2 - 19 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) 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 $$\beta = \sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + b * q^7 + q^8 $$q + q^{2} + q^{4} + \beta q^{7} + q^{8} - \beta q^{11} + \beta q^{14} + q^{16} + 4 q^{17} + 6 q^{19} - \beta q^{22} - 2 q^{23} + \beta q^{28} + 7 q^{31} + q^{32} + 4 q^{34} - 2 \beta q^{37} + 6 q^{38} + 2 \beta q^{41} - 2 \beta q^{43} - \beta q^{44} - 2 q^{46} + 2 q^{47} + 12 q^{49} - 3 q^{53} + \beta q^{56} - 2 \beta q^{59} - 4 q^{61} + 7 q^{62} + q^{64} + 2 \beta q^{67} + 4 q^{68} + \beta q^{73} - 2 \beta q^{74} + 6 q^{76} - 19 q^{77} + 2 \beta q^{82} - 5 q^{83} - 2 \beta q^{86} - \beta q^{88} + 2 \beta q^{89} - 2 q^{92} + 2 q^{94} - \beta q^{97} + 12 q^{98} +O(q^{100})$$ q + q^2 + q^4 + b * q^7 + q^8 - b * q^11 + b * q^14 + q^16 + 4 * q^17 + 6 * q^19 - b * q^22 - 2 * q^23 + b * q^28 + 7 * q^31 + q^32 + 4 * q^34 - 2*b * q^37 + 6 * q^38 + 2*b * q^41 - 2*b * q^43 - b * q^44 - 2 * q^46 + 2 * q^47 + 12 * q^49 - 3 * q^53 + b * q^56 - 2*b * q^59 - 4 * q^61 + 7 * q^62 + q^64 + 2*b * q^67 + 4 * q^68 + b * q^73 - 2*b * q^74 + 6 * q^76 - 19 * q^77 + 2*b * q^82 - 5 * q^83 - 2*b * q^86 - b * q^88 + 2*b * q^89 - 2 * q^92 + 2 * q^94 - b * q^97 + 12 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{16} + 8 q^{17} + 12 q^{19} - 4 q^{23} + 14 q^{31} + 2 q^{32} + 8 q^{34} + 12 q^{38} - 4 q^{46} + 4 q^{47} + 24 q^{49} - 6 q^{53} - 8 q^{61} + 14 q^{62} + 2 q^{64} + 8 q^{68} + 12 q^{76} - 38 q^{77} - 10 q^{83} - 4 q^{92} + 4 q^{94} + 24 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 2 * q^16 + 8 * q^17 + 12 * q^19 - 4 * q^23 + 14 * q^31 + 2 * q^32 + 8 * q^34 + 12 * q^38 - 4 * q^46 + 4 * q^47 + 24 * q^49 - 6 * q^53 - 8 * q^61 + 14 * q^62 + 2 * q^64 + 8 * q^68 + 12 * q^76 - 38 * q^77 - 10 * q^83 - 4 * q^92 + 4 * q^94 + 24 * 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
 −4.35890 4.35890
1.00000 0 1.00000 0 0 −4.35890 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 4.35890 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.a.x 2
3.b odd 2 1 1350.2.a.w 2
5.b even 2 1 1350.2.a.w 2
5.c odd 4 2 270.2.c.c 4
15.d odd 2 1 inner 1350.2.a.x 2
15.e even 4 2 270.2.c.c 4
20.e even 4 2 2160.2.f.m 4
45.k odd 12 4 810.2.i.h 8
45.l even 12 4 810.2.i.h 8
60.l odd 4 2 2160.2.f.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.c.c 4 5.c odd 4 2
270.2.c.c 4 15.e even 4 2
810.2.i.h 8 45.k odd 12 4
810.2.i.h 8 45.l even 12 4
1350.2.a.w 2 3.b odd 2 1
1350.2.a.w 2 5.b even 2 1
1350.2.a.x 2 1.a even 1 1 trivial
1350.2.a.x 2 15.d odd 2 1 inner
2160.2.f.m 4 20.e even 4 2
2160.2.f.m 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1350))$$:

 $$T_{7}^{2} - 19$$ T7^2 - 19 $$T_{11}^{2} - 19$$ T11^2 - 19 $$T_{13}$$ T13 $$T_{17} - 4$$ T17 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 19$$
$11$ $$T^{2} - 19$$
$13$ $$T^{2}$$
$17$ $$(T - 4)^{2}$$
$19$ $$(T - 6)^{2}$$
$23$ $$(T + 2)^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 7)^{2}$$
$37$ $$T^{2} - 76$$
$41$ $$T^{2} - 76$$
$43$ $$T^{2} - 76$$
$47$ $$(T - 2)^{2}$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2} - 76$$
$61$ $$(T + 4)^{2}$$
$67$ $$T^{2} - 76$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 19$$
$79$ $$T^{2}$$
$83$ $$(T + 5)^{2}$$
$89$ $$T^{2} - 76$$
$97$ $$T^{2} - 19$$