# Properties

 Label 1305.2.d.a Level $1305$ Weight $2$ Character orbit 1305.d Analytic conductor $10.420$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + (\beta_{2} - 1) q^{7} + \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + b2 * q^4 + q^5 + (b2 - 1) * q^7 + b3 * q^8 $$q + \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + (\beta_{2} - 1) q^{7} + \beta_{3} q^{8} + \beta_1 q^{10} + ( - 3 \beta_{3} + \beta_1) q^{11} + ( - 2 \beta_{2} + 2) q^{13} + (\beta_{3} - 3 \beta_1) q^{14} + (2 \beta_{2} - 1) q^{16} + (3 \beta_{3} + \beta_1) q^{17} + ( - 3 \beta_{3} + 3 \beta_1) q^{19} + \beta_{2} q^{20} + (\beta_{2} + 1) q^{22} + (3 \beta_{2} + 3) q^{23} + q^{25} + ( - 2 \beta_{3} + 6 \beta_1) q^{26} + ( - \beta_{2} + 3) q^{28} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{29} + ( - 3 \beta_{3} + 3 \beta_1) q^{31} + (4 \beta_{3} - 5 \beta_1) q^{32} + (\beta_{2} - 5) q^{34} + (\beta_{2} - 1) q^{35} + (3 \beta_{3} - 3 \beta_1) q^{37} + (3 \beta_{2} - 3) q^{38} + \beta_{3} q^{40} + (4 \beta_{3} + 2 \beta_1) q^{41} + ( - 3 \beta_{3} + 3 \beta_1) q^{43} + ( - 5 \beta_{3} + \beta_1) q^{44} + (3 \beta_{3} - 3 \beta_1) q^{46} + ( - 3 \beta_{3} - 5 \beta_1) q^{47} + ( - 2 \beta_{2} - 3) q^{49} + \beta_1 q^{50} + (2 \beta_{2} - 6) q^{52} + ( - 3 \beta_{3} + \beta_1) q^{55} + (\beta_{3} - \beta_1) q^{56} + (3 \beta_{3} + \beta_{2} - 6 \beta_1 - 1) q^{58} - 6 q^{59} + 6 \beta_1 q^{61} + (3 \beta_{2} - 3) q^{62} + ( - \beta_{2} + 4) q^{64} + ( - 2 \beta_{2} + 2) q^{65} + (7 \beta_{2} - 1) q^{67} + (7 \beta_{3} - 5 \beta_1) q^{68} + (\beta_{3} - 3 \beta_1) q^{70} - 6 q^{71} + ( - 3 \beta_{3} + 9 \beta_1) q^{73} + ( - 3 \beta_{2} + 3) q^{74} + ( - 3 \beta_{3} - 3 \beta_1) q^{76} - 2 \beta_{3} q^{77} + (9 \beta_{3} - 3 \beta_1) q^{79} + (2 \beta_{2} - 1) q^{80} + (2 \beta_{2} - 8) q^{82} + ( - 3 \beta_{2} + 3) q^{83} + (3 \beta_{3} + \beta_1) q^{85} + (3 \beta_{2} - 3) q^{86} + (3 \beta_{2} + 5) q^{88} + 4 \beta_1 q^{89} + (4 \beta_{2} - 8) q^{91} + (3 \beta_{2} + 9) q^{92} + ( - 5 \beta_{2} + 13) q^{94} + ( - 3 \beta_{3} + 3 \beta_1) q^{95} + ( - 3 \beta_{3} - 3 \beta_1) q^{97} + ( - 2 \beta_{3} + \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + b2 * q^4 + q^5 + (b2 - 1) * q^7 + b3 * q^8 + b1 * q^10 + (-3*b3 + b1) * q^11 + (-2*b2 + 2) * q^13 + (b3 - 3*b1) * q^14 + (2*b2 - 1) * q^16 + (3*b3 + b1) * q^17 + (-3*b3 + 3*b1) * q^19 + b2 * q^20 + (b2 + 1) * q^22 + (3*b2 + 3) * q^23 + q^25 + (-2*b3 + 6*b1) * q^26 + (-b2 + 3) * q^28 + (-b3 + 3*b2 + b1) * q^29 + (-3*b3 + 3*b1) * q^31 + (4*b3 - 5*b1) * q^32 + (b2 - 5) * q^34 + (b2 - 1) * q^35 + (3*b3 - 3*b1) * q^37 + (3*b2 - 3) * q^38 + b3 * q^40 + (4*b3 + 2*b1) * q^41 + (-3*b3 + 3*b1) * q^43 + (-5*b3 + b1) * q^44 + (3*b3 - 3*b1) * q^46 + (-3*b3 - 5*b1) * q^47 + (-2*b2 - 3) * q^49 + b1 * q^50 + (2*b2 - 6) * q^52 + (-3*b3 + b1) * q^55 + (b3 - b1) * q^56 + (3*b3 + b2 - 6*b1 - 1) * q^58 - 6 * q^59 + 6*b1 * q^61 + (3*b2 - 3) * q^62 + (-b2 + 4) * q^64 + (-2*b2 + 2) * q^65 + (7*b2 - 1) * q^67 + (7*b3 - 5*b1) * q^68 + (b3 - 3*b1) * q^70 - 6 * q^71 + (-3*b3 + 9*b1) * q^73 + (-3*b2 + 3) * q^74 + (-3*b3 - 3*b1) * q^76 - 2*b3 * q^77 + (9*b3 - 3*b1) * q^79 + (2*b2 - 1) * q^80 + (2*b2 - 8) * q^82 + (-3*b2 + 3) * q^83 + (3*b3 + b1) * q^85 + (3*b2 - 3) * q^86 + (3*b2 + 5) * q^88 + 4*b1 * q^89 + (4*b2 - 8) * q^91 + (3*b2 + 9) * q^92 + (-5*b2 + 13) * q^94 + (-3*b3 + 3*b1) * q^95 + (-3*b3 - 3*b1) * q^97 + (-2*b3 + b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} - 4 q^{7}+O(q^{10})$$ 4 * q + 4 * q^5 - 4 * q^7 $$4 q + 4 q^{5} - 4 q^{7} + 8 q^{13} - 4 q^{16} + 4 q^{22} + 12 q^{23} + 4 q^{25} + 12 q^{28} - 20 q^{34} - 4 q^{35} - 12 q^{38} - 12 q^{49} - 24 q^{52} - 4 q^{58} - 24 q^{59} - 12 q^{62} + 16 q^{64} + 8 q^{65} - 4 q^{67} - 24 q^{71} + 12 q^{74} - 4 q^{80} - 32 q^{82} + 12 q^{83} - 12 q^{86} + 20 q^{88} - 32 q^{91} + 36 q^{92} + 52 q^{94}+O(q^{100})$$ 4 * q + 4 * q^5 - 4 * q^7 + 8 * q^13 - 4 * q^16 + 4 * q^22 + 12 * q^23 + 4 * q^25 + 12 * q^28 - 20 * q^34 - 4 * q^35 - 12 * q^38 - 12 * q^49 - 24 * q^52 - 4 * q^58 - 24 * q^59 - 12 * q^62 + 16 * q^64 + 8 * q^65 - 4 * q^67 - 24 * q^71 + 12 * q^74 - 4 * q^80 - 32 * q^82 + 12 * q^83 - 12 * q^86 + 20 * q^88 - 32 * q^91 + 36 * q^92 + 52 * q^94

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4\beta_1$$ b3 - 4*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
 − 1.93185i − 0.517638i 0.517638i 1.93185i
1.93185i 0 −1.73205 1.00000 0 −2.73205 0.517638i 0 1.93185i
811.2 0.517638i 0 1.73205 1.00000 0 0.732051 1.93185i 0 0.517638i
811.3 0.517638i 0 1.73205 1.00000 0 0.732051 1.93185i 0 0.517638i
811.4 1.93185i 0 −1.73205 1.00000 0 −2.73205 0.517638i 0 1.93185i
 $$n$$: 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
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.a 4
3.b odd 2 1 145.2.c.a 4
12.b even 2 1 2320.2.g.e 4
15.d odd 2 1 725.2.c.d 4
15.e even 4 2 725.2.d.b 8
29.b even 2 1 inner 1305.2.d.a 4
87.d odd 2 1 145.2.c.a 4
87.f even 4 2 4205.2.a.g 4
348.b even 2 1 2320.2.g.e 4
435.b odd 2 1 725.2.c.d 4
435.p even 4 2 725.2.d.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.c.a 4 3.b odd 2 1
145.2.c.a 4 87.d odd 2 1
725.2.c.d 4 15.d odd 2 1
725.2.c.d 4 435.b odd 2 1
725.2.d.b 8 15.e even 4 2
725.2.d.b 8 435.p even 4 2
1305.2.d.a 4 1.a even 1 1 trivial
1305.2.d.a 4 29.b even 2 1 inner
2320.2.g.e 4 12.b even 2 1
2320.2.g.e 4 348.b even 2 1
4205.2.a.g 4 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 $$S_{2}^{\mathrm{new}}(1305, [\chi])$$:

 $$T_{2}^{4} + 4T_{2}^{2} + 1$$ T2^4 + 4*T2^2 + 1 $$T_{23}^{2} - 6T_{23} - 18$$ T23^2 - 6*T23 - 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$(T - 1)^{4}$$
$7$ $$(T^{2} + 2 T - 2)^{2}$$
$11$ $$T^{4} + 28T^{2} + 4$$
$13$ $$(T^{2} - 4 T - 8)^{2}$$
$17$ $$T^{4} + 52T^{2} + 484$$
$19$ $$(T^{2} + 18)^{2}$$
$23$ $$(T^{2} - 6 T - 18)^{2}$$
$29$ $$T^{4} - 50T^{2} + 841$$
$31$ $$(T^{2} + 18)^{2}$$
$37$ $$(T^{2} + 18)^{2}$$
$41$ $$T^{4} + 112T^{2} + 2704$$
$43$ $$(T^{2} + 18)^{2}$$
$47$ $$T^{4} + 196T^{2} + 8836$$
$53$ $$T^{4}$$
$59$ $$(T + 6)^{4}$$
$61$ $$T^{4} + 144T^{2} + 1296$$
$67$ $$(T^{2} + 2 T - 146)^{2}$$
$71$ $$(T + 6)^{4}$$
$73$ $$T^{4} + 252T^{2} + 324$$
$79$ $$T^{4} + 252T^{2} + 324$$
$83$ $$(T^{2} - 6 T - 18)^{2}$$
$89$ $$T^{4} + 64T^{2} + 256$$
$97$ $$(T^{2} + 54)^{2}$$