# Properties

 Label 1305.1.l.a Level $1305$ Weight $1$ Character orbit 1305.l Analytic conductor $0.651$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ RM discriminant 5 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,1,Mod(244,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1305.244");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1305.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: $$0.651279841486$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.121945.1 Artin image: $C_4^2:C_2^2$ Artin field: Galois closure of 16.0.1524467112922265625.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - i q^{4} - i q^{5} +O(q^{10})$$ q - z * q^4 - z * q^5 $$q - i q^{4} - i q^{5} + ( - i + 1) q^{11} - q^{16} + (i - 1) q^{19} - q^{20} - q^{25} + i q^{29} + ( - i + 1) q^{31} + ( - i - 1) q^{41} + ( - i - 1) q^{44} + q^{49} + ( - i - 1) q^{55} + ( - i + 1) q^{61} + i q^{64} + 2 i q^{71} + (i + 1) q^{76} + (i - 1) q^{79} + i q^{80} + ( - i + 1) q^{89} + (i + 1) q^{95} +O(q^{100})$$ q - z * q^4 - z * q^5 + (-z + 1) * q^11 - q^16 + (z - 1) * q^19 - q^20 - q^25 + z * q^29 + (-z + 1) * q^31 + (-z - 1) * q^41 + (-z - 1) * q^44 + q^49 + (-z - 1) * q^55 + (-z + 1) * q^61 + z * q^64 + 2*z * q^71 + (z + 1) * q^76 + (z - 1) * q^79 + z * q^80 + (-z + 1) * q^89 + (z + 1) * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 2 q^{11} - 2 q^{16} - 2 q^{19} - 2 q^{20} - 2 q^{25} + 2 q^{31} - 2 q^{41} - 2 q^{44} + 2 q^{49} - 2 q^{55} + 2 q^{61} + 2 q^{76} - 2 q^{79} + 2 q^{89} + 2 q^{95}+O(q^{100})$$ 2 * q + 2 * q^11 - 2 * q^16 - 2 * q^19 - 2 * q^20 - 2 * q^25 + 2 * q^31 - 2 * q^41 - 2 * q^44 + 2 * q^49 - 2 * q^55 + 2 * q^61 + 2 * q^76 - 2 * q^79 + 2 * q^89 + 2 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$$.

 $$n$$ $$146$$ $$262$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
244.1
 − 1.00000i 1.00000i
0 0 1.00000i 1.00000i 0 0 0 0 0
829.1 0 0 1.00000i 1.00000i 0 0 0 0 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 RM by $$\Q(\sqrt{5})$$
29.c odd 4 1 inner
145.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.1.l.a 2
3.b odd 2 1 145.1.f.a 2
5.b even 2 1 RM 1305.1.l.a 2
12.b even 2 1 2320.1.bj.a 2
15.d odd 2 1 145.1.f.a 2
15.e even 4 2 725.1.g.a 2
29.c odd 4 1 inner 1305.1.l.a 2
60.h even 2 1 2320.1.bj.a 2
87.f even 4 1 145.1.f.a 2
145.f odd 4 1 inner 1305.1.l.a 2
348.k odd 4 1 2320.1.bj.a 2
435.i odd 4 1 725.1.g.a 2
435.l even 4 1 145.1.f.a 2
435.t odd 4 1 725.1.g.a 2
1740.bm odd 4 1 2320.1.bj.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.1.f.a 2 3.b odd 2 1
145.1.f.a 2 15.d odd 2 1
145.1.f.a 2 87.f even 4 1
145.1.f.a 2 435.l even 4 1
725.1.g.a 2 15.e even 4 2
725.1.g.a 2 435.i odd 4 1
725.1.g.a 2 435.t odd 4 1
1305.1.l.a 2 1.a even 1 1 trivial
1305.1.l.a 2 5.b even 2 1 RM
1305.1.l.a 2 29.c odd 4 1 inner
1305.1.l.a 2 145.f odd 4 1 inner
2320.1.bj.a 2 12.b even 2 1
2320.1.bj.a 2 60.h even 2 1
2320.1.bj.a 2 348.k odd 4 1
2320.1.bj.a 2 1740.bm odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1305, [\chi])$$.

## Hecke characteristic polynomials

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