Properties

 Label 1305.2.c.d Level $1305$ Weight $2$ Character orbit 1305.c Analytic conductor $10.420$ 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] = [1305,2,Mod(784,1305)]

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, 0]))

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

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

chi := DirichletCharacter("1305.784");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.c (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: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 5$$ x^2 + 5 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 $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} - \beta q^{5} - 2 \beta q^{7} +O(q^{10})$$ q + 2 * q^4 - b * q^5 - 2*b * q^7 $$q + 2 q^{4} - \beta q^{5} - 2 \beta q^{7} + 5 q^{11} - 2 \beta q^{13} + 4 q^{16} + 2 \beta q^{17} - 4 q^{19} - 2 \beta q^{20} + 3 \beta q^{23} - 5 q^{25} - 4 \beta q^{28} + q^{29} - 2 q^{31} - 10 q^{35} - \beta q^{37} - 5 q^{41} + \beta q^{43} + 10 q^{44} - 4 \beta q^{47} - 13 q^{49} - 4 \beta q^{52} + \beta q^{53} - 5 \beta q^{55} - 10 q^{59} + 12 q^{61} + 8 q^{64} - 10 q^{65} + 4 \beta q^{67} + 4 \beta q^{68} + 10 q^{71} + 3 \beta q^{73} - 8 q^{76} - 10 \beta q^{77} + 14 q^{79} - 4 \beta q^{80} + \beta q^{83} + 10 q^{85} + 10 q^{89} - 20 q^{91} + 6 \beta q^{92} + 4 \beta q^{95} + 3 \beta q^{97} +O(q^{100})$$ q + 2 * q^4 - b * q^5 - 2*b * q^7 + 5 * q^11 - 2*b * q^13 + 4 * q^16 + 2*b * q^17 - 4 * q^19 - 2*b * q^20 + 3*b * q^23 - 5 * q^25 - 4*b * q^28 + q^29 - 2 * q^31 - 10 * q^35 - b * q^37 - 5 * q^41 + b * q^43 + 10 * q^44 - 4*b * q^47 - 13 * q^49 - 4*b * q^52 + b * q^53 - 5*b * q^55 - 10 * q^59 + 12 * q^61 + 8 * q^64 - 10 * q^65 + 4*b * q^67 + 4*b * q^68 + 10 * q^71 + 3*b * q^73 - 8 * q^76 - 10*b * q^77 + 14 * q^79 - 4*b * q^80 + b * q^83 + 10 * q^85 + 10 * q^89 - 20 * q^91 + 6*b * q^92 + 4*b * q^95 + 3*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4}+O(q^{10})$$ 2 * q + 4 * q^4 $$2 q + 4 q^{4} + 10 q^{11} + 8 q^{16} - 8 q^{19} - 10 q^{25} + 2 q^{29} - 4 q^{31} - 20 q^{35} - 10 q^{41} + 20 q^{44} - 26 q^{49} - 20 q^{59} + 24 q^{61} + 16 q^{64} - 20 q^{65} + 20 q^{71} - 16 q^{76} + 28 q^{79} + 20 q^{85} + 20 q^{89} - 40 q^{91}+O(q^{100})$$ 2 * q + 4 * q^4 + 10 * q^11 + 8 * q^16 - 8 * q^19 - 10 * q^25 + 2 * q^29 - 4 * q^31 - 20 * q^35 - 10 * q^41 + 20 * q^44 - 26 * q^49 - 20 * q^59 + 24 * q^61 + 16 * q^64 - 20 * q^65 + 20 * q^71 - 16 * q^76 + 28 * q^79 + 20 * q^85 + 20 * q^89 - 40 * q^91

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
784.1
 2.23607i − 2.23607i
0 0 2.00000 2.23607i 0 4.47214i 0 0 0
784.2 0 0 2.00000 2.23607i 0 4.47214i 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 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.2.c.d yes 2
3.b odd 2 1 1305.2.c.c 2
5.b even 2 1 inner 1305.2.c.d yes 2
5.c odd 4 2 6525.2.a.v 2
15.d odd 2 1 1305.2.c.c 2
15.e even 4 2 6525.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.c.c 2 3.b odd 2 1
1305.2.c.c 2 15.d odd 2 1
1305.2.c.d yes 2 1.a even 1 1 trivial
1305.2.c.d yes 2 5.b even 2 1 inner
6525.2.a.u 2 15.e even 4 2
6525.2.a.v 2 5.c odd 4 2

Hecke kernels

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

 $$T_{2}$$ T2 $$T_{7}^{2} + 20$$ T7^2 + 20 $$T_{11} - 5$$ T11 - 5

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5$$
$7$ $$T^{2} + 20$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2} + 20$$
$17$ $$T^{2} + 20$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 45$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 5$$
$41$ $$(T + 5)^{2}$$
$43$ $$T^{2} + 5$$
$47$ $$T^{2} + 80$$
$53$ $$T^{2} + 5$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2} + 80$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} + 45$$
$79$ $$(T - 14)^{2}$$
$83$ $$T^{2} + 5$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 45$$