Properties

 Label 1305.2.f.b Level $1305$ Weight $2$ Character orbit 1305.f 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(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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 435) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

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

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
 1.00000i − 1.00000i
−1.00000 0 −1.00000 1.00000 2.00000i 0 2.00000i 3.00000 0 −1.00000 + 2.00000i
289.2 −1.00000 0 −1.00000 1.00000 + 2.00000i 0 2.00000i 3.00000 0 −1.00000 2.00000i
 $$n$$: 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
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.b 2
3.b odd 2 1 435.2.f.c yes 2
5.b even 2 1 1305.2.f.c 2
15.d odd 2 1 435.2.f.b 2
15.e even 4 1 2175.2.d.c 2
15.e even 4 1 2175.2.d.d 2
29.b even 2 1 1305.2.f.c 2
87.d odd 2 1 435.2.f.b 2
145.d even 2 1 inner 1305.2.f.b 2
435.b odd 2 1 435.2.f.c yes 2
435.p even 4 1 2175.2.d.c 2
435.p even 4 1 2175.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.f.b 2 15.d odd 2 1
435.2.f.b 2 87.d odd 2 1
435.2.f.c yes 2 3.b odd 2 1
435.2.f.c yes 2 435.b odd 2 1
1305.2.f.b 2 1.a even 1 1 trivial
1305.2.f.b 2 145.d even 2 1 inner
1305.2.f.c 2 5.b even 2 1
1305.2.f.c 2 29.b even 2 1
2175.2.d.c 2 15.e even 4 1
2175.2.d.c 2 435.p even 4 1
2175.2.d.d 2 15.e even 4 1
2175.2.d.d 2 435.p even 4 1

Hecke kernels

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

 $$T_{2} + 1$$ T2 + 1 $$T_{7}^{2} + 4$$ T7^2 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2} + 16$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$T^{2} + 36$$
$29$ $$T^{2} - 10T + 29$$
$31$ $$T^{2} + 4$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 144$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} + 144$$
$67$ $$T^{2} + 36$$
$71$ $$(T + 8)^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} + 4$$
$83$ $$T^{2} + 4$$
$89$ $$T^{2} + 256$$
$97$ $$(T - 14)^{2}$$