# Properties

 Label 1305.2.f.g Level $1305$ Weight $2$ Character orbit 1305.f Analytic conductor $10.420$ Analytic rank $0$ Dimension $4$ CM discriminant -435 Inner twists $8$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(289,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1305.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.f (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: $$10.4204774638$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-5}, \sqrt{-29})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 17x^{2} + 36$$ x^4 + 17*x^2 + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{4} - \beta_1 q^{5}+O(q^{10})$$ q - 2 * q^4 - b1 * q^5 $$q - 2 q^{4} - \beta_1 q^{5} - \beta_{2} q^{11} + 4 q^{16} + 2 \beta_1 q^{20} + \beta_1 q^{23} - 5 q^{25} + \beta_{2} q^{29} - \beta_{3} q^{37} - \beta_{2} q^{41} - \beta_{3} q^{43} + 2 \beta_{2} q^{44} + 7 q^{49} - 5 \beta_1 q^{53} - \beta_{3} q^{55} - 8 q^{64} - \beta_{3} q^{73} - 4 \beta_1 q^{80} + 7 \beta_1 q^{83} + 2 \beta_{2} q^{89} - 2 \beta_1 q^{92} - \beta_{3} q^{97}+O(q^{100})$$ q - 2 * q^4 - b1 * q^5 - b2 * q^11 + 4 * q^16 + 2*b1 * q^20 + b1 * q^23 - 5 * q^25 + b2 * q^29 - b3 * q^37 - b2 * q^41 - b3 * q^43 + 2*b2 * q^44 + 7 * q^49 - 5*b1 * q^53 - b3 * q^55 - 8 * q^64 - b3 * q^73 - 4*b1 * q^80 + 7*b1 * q^83 + 2*b2 * q^89 - 2*b1 * q^92 - b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4}+O(q^{10})$$ 4 * q - 8 * q^4 $$4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} - 32 q^{64}+O(q^{100})$$ 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 28 * q^49 - 32 * q^64

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 17x^{2} + 36$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 11\nu ) / 6$$ (v^3 + 11*v) / 6 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 23\nu ) / 6$$ (v^3 + 23*v) / 6 $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 17$$ 2*v^2 + 17
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 17 ) / 2$$ (b3 - 17) / 2 $$\nu^{3}$$ $$=$$ $$( -11\beta_{2} + 23\beta_1 ) / 2$$ (-11*b2 + 23*b1) / 2

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 1.57455i − 3.81062i 3.81062i − 1.57455i
0 0 −2.00000 2.23607i 0 0 0 0 0
289.2 0 0 −2.00000 2.23607i 0 0 0 0 0
289.3 0 0 −2.00000 2.23607i 0 0 0 0 0
289.4 0 0 −2.00000 2.23607i 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
435.b odd 2 1 CM by $$\Q(\sqrt{-435})$$
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
29.b even 2 1 inner
87.d odd 2 1 inner
145.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.2.f.g 4
3.b odd 2 1 inner 1305.2.f.g 4
5.b even 2 1 inner 1305.2.f.g 4
15.d odd 2 1 inner 1305.2.f.g 4
29.b even 2 1 inner 1305.2.f.g 4
87.d odd 2 1 inner 1305.2.f.g 4
145.d even 2 1 inner 1305.2.f.g 4
435.b odd 2 1 CM 1305.2.f.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.f.g 4 1.a even 1 1 trivial
1305.2.f.g 4 3.b odd 2 1 inner
1305.2.f.g 4 5.b even 2 1 inner
1305.2.f.g 4 15.d odd 2 1 inner
1305.2.f.g 4 29.b even 2 1 inner
1305.2.f.g 4 87.d odd 2 1 inner
1305.2.f.g 4 145.d even 2 1 inner
1305.2.f.g 4 435.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1305, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 29)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 5)^{2}$$
$29$ $$(T^{2} + 29)^{2}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} - 145)^{2}$$
$41$ $$(T^{2} + 29)^{2}$$
$43$ $$(T^{2} - 145)^{2}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 125)^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 145)^{2}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 245)^{2}$$
$89$ $$(T^{2} + 116)^{2}$$
$97$ $$(T^{2} - 145)^{2}$$
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