# Properties

 Label 1305.2.d.b Level $1305$ Weight $2$ Character orbit 1305.d Analytic conductor $10.420$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(811,1305)]

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, 0, 1]))

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

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

chi := DirichletCharacter("1305.811");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.d (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: $$6$$ Coefficient field: 6.0.16516096.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 9x^{4} + 13x^{2} + 1$$ x^6 + 9*x^4 + 13*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 145) 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} - \beta_{2} q^{7} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 - 1) * q^4 - q^5 - b2 * q^7 + (b5 + b4 - 2*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} - \beta_{2} q^{7} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{8} - \beta_1 q^{10} + (\beta_{5} - \beta_1) q^{11} + (\beta_{3} - 1) q^{13} + ( - \beta_{5} - \beta_{4} + 3 \beta_1) q^{14} + (\beta_{3} - 2 \beta_{2} + 4) q^{16} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{17} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{19} + ( - \beta_{2} + 1) q^{20} + ( - \beta_{2} + 2) q^{22} + (\beta_{3} - \beta_{2} + 1) q^{23} + q^{25} + ( - \beta_{4} - 2 \beta_1) q^{26} + ( - \beta_{3} + 3 \beta_{2} - 9) q^{28} + (\beta_{5} - \beta_{4} - \beta_{2} + \cdots + 1) q^{29}+ \cdots + ( - 2 \beta_{5} - 3 \beta_{4} + 7 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 - 1) * q^4 - q^5 - b2 * q^7 + (b5 + b4 - 2*b1) * q^8 - b1 * q^10 + (b5 - b1) * q^11 + (b3 - 1) * q^13 + (-b5 - b4 + 3*b1) * q^14 + (b3 - 2*b2 + 4) * q^16 + (-b5 - b4 - b1) * q^17 + (-b5 - b4 + b1) * q^19 + (-b2 + 1) * q^20 + (-b2 + 2) * q^22 + (b3 - b2 + 1) * q^23 + q^25 + (-b4 - 2*b1) * q^26 + (-b3 + 3*b2 - 9) * q^28 + (b5 - b4 - b2 - b1 + 1) * q^29 + (3*b5 + b4 + b1) * q^31 + (-b4 + 5*b1) * q^32 + (-b3 + b2 + 3) * q^34 + b2 * q^35 + (-b5 + b4 + 3*b1) * q^37 + (-b3 + 3*b2 - 3) * q^38 + (-b5 - b4 + 2*b1) * q^40 + (2*b5 - b4 + 2*b1) * q^41 + (b5 - 2*b4 - b1) * q^43 + (b5 - b4 + 3*b1) * q^44 + (-b5 - 2*b4 + 3*b1) * q^46 + (b5 - b1) * q^47 + (b3 - 2*b2 + 2) * q^49 + b1 * q^50 + (b3 + 3) * q^52 + (b3 + 2*b2 - 5) * q^53 + (-b5 + b1) * q^55 + (b5 + 2*b4 - 11*b1) * q^56 + (-b5 - b4 - b3 + b2 + 4*b1 + 1) * q^58 + (-b3 + 3) * q^59 + (2*b5 - b4 - 2*b1) * q^61 + (b3 - b2 - 5) * q^62 + (b3 + 3*b2 - 8) * q^64 + (-b3 + 1) * q^65 + (b3 - b2 - 3) * q^67 + (-b5 - b1) * q^68 + (b5 + b4 - 3*b1) * q^70 + (b3 + 9) * q^71 + (-3*b5 - 3*b1) * q^73 + (b3 + b2 - 7) * q^74 + (b5 + 2*b4 - 9*b1) * q^76 + (-2*b5 + b4 - 2*b1) * q^77 + (b5 - 2*b4 - b1) * q^79 + (-b3 + 2*b2 - 4) * q^80 + (-b3 + 4*b2 - 9) * q^82 + (-b2 + 4) * q^83 + (b5 + b4 + b1) * q^85 + (-2*b3 + 3*b2) * q^86 + (-b3 + 3*b2 - 7) * q^88 + 2*b4 * q^89 + (-2*b3 - 2) * q^91 + (5*b2 - 8) * q^92 + (-b2 + 2) * q^94 + (b5 + b4 - b1) * q^95 + (b5 + b1) * q^97 + (-2*b5 - 3*b4 + 7*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 6 q^{5}+O(q^{10})$$ 6 * q - 6 * q^4 - 6 * q^5 $$6 q - 6 q^{4} - 6 q^{5} - 4 q^{13} + 26 q^{16} + 6 q^{20} + 12 q^{22} + 8 q^{23} + 6 q^{25} - 56 q^{28} + 6 q^{29} + 16 q^{34} - 20 q^{38} + 14 q^{49} + 20 q^{52} - 28 q^{53} + 4 q^{58} + 16 q^{59} - 28 q^{62} - 46 q^{64} + 4 q^{65} - 16 q^{67} + 56 q^{71} - 40 q^{74} - 26 q^{80} - 56 q^{82} + 24 q^{83} - 4 q^{86} - 44 q^{88} - 16 q^{91} - 48 q^{92} + 12 q^{94}+O(q^{100})$$ 6 * q - 6 * q^4 - 6 * q^5 - 4 * q^13 + 26 * q^16 + 6 * q^20 + 12 * q^22 + 8 * q^23 + 6 * q^25 - 56 * q^28 + 6 * q^29 + 16 * q^34 - 20 * q^38 + 14 * q^49 + 20 * q^52 - 28 * q^53 + 4 * q^58 + 16 * q^59 - 28 * q^62 - 46 * q^64 + 4 * q^65 - 16 * q^67 + 56 * q^71 - 40 * q^74 - 26 * q^80 - 56 * q^82 + 24 * q^83 - 4 * q^86 - 44 * q^88 - 16 * q^91 - 48 * q^92 + 12 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9x^{4} + 13x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ v^2 + 3 $$\beta_{3}$$ $$=$$ $$\nu^{4} + 8\nu^{2} + 6$$ v^4 + 8*v^2 + 6 $$\beta_{4}$$ $$=$$ $$-\nu^{5} - 8\nu^{3} - 7\nu$$ -v^5 - 8*v^3 - 7*v $$\beta_{5}$$ $$=$$ $$\nu^{5} + 9\nu^{3} + 13\nu$$ v^5 + 9*v^3 + 13*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ b2 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} - 6\beta_1$$ b5 + b4 - 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{3} - 8\beta_{2} + 18$$ b3 - 8*b2 + 18 $$\nu^{5}$$ $$=$$ $$-8\beta_{5} - 9\beta_{4} + 41\beta_1$$ -8*b5 - 9*b4 + 41*b1

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
811.1
 − 2.68667i − 1.30397i − 0.285442i 0.285442i 1.30397i 2.68667i
2.68667i 0 −5.21819 −1.00000 0 4.21819 8.64620i 0 2.68667i
811.2 1.30397i 0 0.299664 −1.00000 0 −1.29966 2.99869i 0 1.30397i
811.3 0.285442i 0 1.91852 −1.00000 0 −2.91852 1.11851i 0 0.285442i
811.4 0.285442i 0 1.91852 −1.00000 0 −2.91852 1.11851i 0 0.285442i
811.5 1.30397i 0 0.299664 −1.00000 0 −1.29966 2.99869i 0 1.30397i
811.6 2.68667i 0 −5.21819 −1.00000 0 4.21819 8.64620i 0 2.68667i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 811.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.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.d.b 6
3.b odd 2 1 145.2.c.b 6
12.b even 2 1 2320.2.g.i 6
15.d odd 2 1 725.2.c.e 6
15.e even 4 2 725.2.d.c 12
29.b even 2 1 inner 1305.2.d.b 6
87.d odd 2 1 145.2.c.b 6
87.f even 4 2 4205.2.a.m 6
348.b even 2 1 2320.2.g.i 6
435.b odd 2 1 725.2.c.e 6
435.p even 4 2 725.2.d.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.c.b 6 3.b odd 2 1
145.2.c.b 6 87.d odd 2 1
725.2.c.e 6 15.d odd 2 1
725.2.c.e 6 435.b odd 2 1
725.2.d.c 12 15.e even 4 2
725.2.d.c 12 435.p even 4 2
1305.2.d.b 6 1.a even 1 1 trivial
1305.2.d.b 6 29.b even 2 1 inner
2320.2.g.i 6 12.b even 2 1
2320.2.g.i 6 348.b even 2 1
4205.2.a.m 6 87.f even 4 2

## Hecke kernels

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

 $$T_{2}^{6} + 9T_{2}^{4} + 13T_{2}^{2} + 1$$ T2^6 + 9*T2^4 + 13*T2^2 + 1 $$T_{23}^{3} - 4T_{23}^{2} - 26T_{23} + 96$$ T23^3 - 4*T23^2 - 26*T23 + 96

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 9 T^{4} + \cdots + 1$$
$3$ $$T^{6}$$
$5$ $$(T + 1)^{6}$$
$7$ $$(T^{3} - 14 T - 16)^{2}$$
$11$ $$T^{6} + 16 T^{4} + \cdots + 16$$
$13$ $$(T^{3} + 2 T^{2} - 24 T - 16)^{2}$$
$17$ $$T^{6} + 52 T^{4} + \cdots + 64$$
$19$ $$T^{6} + 56 T^{4} + \cdots + 1296$$
$23$ $$(T^{3} - 4 T^{2} - 26 T + 96)^{2}$$
$29$ $$T^{6} - 6 T^{5} + \cdots + 24389$$
$31$ $$T^{6} + 152 T^{4} + \cdots + 144$$
$37$ $$T^{6} + 100 T^{4} + \cdots + 20736$$
$41$ $$T^{6} + 184 T^{4} + \cdots + 4096$$
$43$ $$T^{6} + 176 T^{4} + \cdots + 121104$$
$47$ $$T^{6} + 16 T^{4} + \cdots + 16$$
$53$ $$(T^{3} + 14 T^{2} + \cdots - 576)^{2}$$
$59$ $$(T^{3} - 8 T^{2} - 4 T + 48)^{2}$$
$61$ $$T^{6} + 104 T^{4} + \cdots + 2304$$
$67$ $$(T^{3} + 8 T^{2} - 10 T - 8)^{2}$$
$71$ $$(T^{3} - 28 T^{2} + \cdots - 576)^{2}$$
$73$ $$T^{6} + 252 T^{4} + \cdots + 419904$$
$79$ $$T^{6} + 176 T^{4} + \cdots + 121104$$
$83$ $$(T^{3} - 12 T^{2} + \cdots - 24)^{2}$$
$89$ $$T^{6} + 160 T^{4} + \cdots + 65536$$
$97$ $$T^{6} + 28 T^{4} + \cdots + 576$$