# Properties

 Label 1305.2.f.h Level $1305$ Weight $2$ Character orbit 1305.f Analytic conductor $10.420$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

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: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu^{3} + 4\nu$$ 2*v^3 + 4*v $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 8\nu$$ 2*v^3 + 8*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 4$$ (b2 - b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 2\beta_1 ) / 2$$ (-b2 + 2*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
 1.61803i − 1.61803i − 0.618034i 0.618034i
−2.23607 0 3.00000 1.00000 2.00000i 0 2.00000i −2.23607 0 −2.23607 + 4.47214i
289.2 −2.23607 0 3.00000 1.00000 + 2.00000i 0 2.00000i −2.23607 0 −2.23607 4.47214i
289.3 2.23607 0 3.00000 1.00000 2.00000i 0 2.00000i 2.23607 0 2.23607 4.47214i
289.4 2.23607 0 3.00000 1.00000 + 2.00000i 0 2.00000i 2.23607 0 2.23607 + 4.47214i
 $$n$$: 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
29.b even 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.h 4
3.b odd 2 1 145.2.d.b 4
5.b even 2 1 inner 1305.2.f.h 4
12.b even 2 1 2320.2.j.b 4
15.d odd 2 1 145.2.d.b 4
15.e even 4 1 725.2.c.a 2
15.e even 4 1 725.2.c.b 2
29.b even 2 1 inner 1305.2.f.h 4
60.h even 2 1 2320.2.j.b 4
87.d odd 2 1 145.2.d.b 4
145.d even 2 1 inner 1305.2.f.h 4
348.b even 2 1 2320.2.j.b 4
435.b odd 2 1 145.2.d.b 4
435.p even 4 1 725.2.c.a 2
435.p even 4 1 725.2.c.b 2
1740.k even 2 1 2320.2.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.d.b 4 3.b odd 2 1
145.2.d.b 4 15.d odd 2 1
145.2.d.b 4 87.d odd 2 1
145.2.d.b 4 435.b odd 2 1
725.2.c.a 2 15.e even 4 1
725.2.c.a 2 435.p even 4 1
725.2.c.b 2 15.e even 4 1
725.2.c.b 2 435.p even 4 1
1305.2.f.h 4 1.a even 1 1 trivial
1305.2.f.h 4 5.b even 2 1 inner
1305.2.f.h 4 29.b even 2 1 inner
1305.2.f.h 4 145.d even 2 1 inner
2320.2.j.b 4 12.b even 2 1
2320.2.j.b 4 60.h even 2 1
2320.2.j.b 4 348.b even 2 1
2320.2.j.b 4 1740.k even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{7}^{2} + 4$$ T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 5)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 2 T + 5)^{2}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} + 20)^{2}$$
$13$ $$(T^{2} + 16)^{2}$$
$17$ $$(T^{2} - 20)^{2}$$
$19$ $$(T^{2} + 20)^{2}$$
$23$ $$(T^{2} + 36)^{2}$$
$29$ $$(T^{2} + 6 T + 29)^{2}$$
$31$ $$(T^{2} + 20)^{2}$$
$37$ $$(T^{2} - 20)^{2}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} - 80)^{2}$$
$53$ $$(T^{2} + 16)^{2}$$
$59$ $$(T + 4)^{4}$$
$61$ $$(T^{2} + 80)^{2}$$
$67$ $$(T^{2} + 4)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 180)^{2}$$
$79$ $$(T^{2} + 180)^{2}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} + 320)^{2}$$
$97$ $$(T^{2} - 20)^{2}$$