# Properties

 Label 1305.2.f.a Level $1305$ Weight $2$ Character orbit 1305.f Analytic conductor $10.420$ Analytic rank $1$ 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: $$1$$ 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: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} + ( - i - 2) q^{5} + 4 i q^{7}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 + (-i - 2) * q^5 + 4*i * q^7 $$q - 2 q^{2} + 2 q^{4} + ( - i - 2) q^{5} + 4 i q^{7} + (2 i + 4) q^{10} + i q^{11} + 2 i q^{13} - 8 i q^{14} - 4 q^{16} - 6 q^{17} - 4 i q^{19} + ( - 2 i - 4) q^{20} - 2 i q^{22} + 9 i q^{23} + (4 i + 3) q^{25} - 4 i q^{26} + 8 i q^{28} + ( - 5 i + 2) q^{29} - 2 i q^{31} + 8 q^{32} + 12 q^{34} + ( - 8 i + 4) q^{35} + q^{37} + 8 i q^{38} + 9 i q^{41} + q^{43} + 2 i q^{44} - 18 i q^{46} + 8 q^{47} - 9 q^{49} + ( - 8 i - 6) q^{50} + 4 i q^{52} - 9 i q^{53} + ( - 2 i + 1) q^{55} + (10 i - 4) q^{58} - 8 q^{59} - 6 i q^{61} + 4 i q^{62} - 8 q^{64} + ( - 4 i + 2) q^{65} - 12 i q^{67} - 12 q^{68} + (16 i - 8) q^{70} - 2 q^{71} - 15 q^{73} - 2 q^{74} - 8 i q^{76} - 4 q^{77} + 4 i q^{79} + (4 i + 8) q^{80} - 18 i q^{82} + 7 i q^{83} + (6 i + 12) q^{85} - 2 q^{86} - 2 i q^{89} - 8 q^{91} + 18 i q^{92} - 16 q^{94} + (8 i - 4) q^{95} - 11 q^{97} + 18 q^{98} +O(q^{100})$$ q - 2 * q^2 + 2 * q^4 + (-i - 2) * q^5 + 4*i * q^7 + (2*i + 4) * q^10 + i * q^11 + 2*i * q^13 - 8*i * q^14 - 4 * q^16 - 6 * q^17 - 4*i * q^19 + (-2*i - 4) * q^20 - 2*i * q^22 + 9*i * q^23 + (4*i + 3) * q^25 - 4*i * q^26 + 8*i * q^28 + (-5*i + 2) * q^29 - 2*i * q^31 + 8 * q^32 + 12 * q^34 + (-8*i + 4) * q^35 + q^37 + 8*i * q^38 + 9*i * q^41 + q^43 + 2*i * q^44 - 18*i * q^46 + 8 * q^47 - 9 * q^49 + (-8*i - 6) * q^50 + 4*i * q^52 - 9*i * q^53 + (-2*i + 1) * q^55 + (10*i - 4) * q^58 - 8 * q^59 - 6*i * q^61 + 4*i * q^62 - 8 * q^64 + (-4*i + 2) * q^65 - 12*i * q^67 - 12 * q^68 + (16*i - 8) * q^70 - 2 * q^71 - 15 * q^73 - 2 * q^74 - 8*i * q^76 - 4 * q^77 + 4*i * q^79 + (4*i + 8) * q^80 - 18*i * q^82 + 7*i * q^83 + (6*i + 12) * q^85 - 2 * q^86 - 2*i * q^89 - 8 * q^91 + 18*i * q^92 - 16 * q^94 + (8*i - 4) * q^95 - 11 * q^97 + 18 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 4 q^{4} - 4 q^{5}+O(q^{10})$$ 2 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 $$2 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 8 q^{10} - 8 q^{16} - 12 q^{17} - 8 q^{20} + 6 q^{25} + 4 q^{29} + 16 q^{32} + 24 q^{34} + 8 q^{35} + 2 q^{37} + 2 q^{43} + 16 q^{47} - 18 q^{49} - 12 q^{50} + 2 q^{55} - 8 q^{58} - 16 q^{59} - 16 q^{64} + 4 q^{65} - 24 q^{68} - 16 q^{70} - 4 q^{71} - 30 q^{73} - 4 q^{74} - 8 q^{77} + 16 q^{80} + 24 q^{85} - 4 q^{86} - 16 q^{91} - 32 q^{94} - 8 q^{95} - 22 q^{97} + 36 q^{98}+O(q^{100})$$ 2 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 8 * q^10 - 8 * q^16 - 12 * q^17 - 8 * q^20 + 6 * q^25 + 4 * q^29 + 16 * q^32 + 24 * q^34 + 8 * q^35 + 2 * q^37 + 2 * q^43 + 16 * q^47 - 18 * q^49 - 12 * q^50 + 2 * q^55 - 8 * q^58 - 16 * q^59 - 16 * q^64 + 4 * q^65 - 24 * q^68 - 16 * q^70 - 4 * q^71 - 30 * q^73 - 4 * q^74 - 8 * q^77 + 16 * q^80 + 24 * q^85 - 4 * q^86 - 16 * q^91 - 32 * q^94 - 8 * q^95 - 22 * q^97 + 36 * 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
−2.00000 0 2.00000 −2.00000 1.00000i 0 4.00000i 0 0 4.00000 + 2.00000i
289.2 −2.00000 0 2.00000 −2.00000 + 1.00000i 0 4.00000i 0 0 4.00000 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.a 2
3.b odd 2 1 435.2.f.d yes 2
5.b even 2 1 1305.2.f.d 2
15.d odd 2 1 435.2.f.a 2
15.e even 4 1 2175.2.d.a 2
15.e even 4 1 2175.2.d.b 2
29.b even 2 1 1305.2.f.d 2
87.d odd 2 1 435.2.f.a 2
145.d even 2 1 inner 1305.2.f.a 2
435.b odd 2 1 435.2.f.d yes 2
435.p even 4 1 2175.2.d.a 2
435.p even 4 1 2175.2.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.f.a 2 15.d odd 2 1
435.2.f.a 2 87.d odd 2 1
435.2.f.d yes 2 3.b odd 2 1
435.2.f.d yes 2 435.b odd 2 1
1305.2.f.a 2 1.a even 1 1 trivial
1305.2.f.a 2 145.d even 2 1 inner
1305.2.f.d 2 5.b even 2 1
1305.2.f.d 2 29.b even 2 1
2175.2.d.a 2 15.e even 4 1
2175.2.d.a 2 435.p even 4 1
2175.2.d.b 2 15.e even 4 1
2175.2.d.b 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 $$S_{2}^{\mathrm{new}}(1305, [\chi])$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{7}^{2} + 16$$ T7^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} + 1$$
$13$ $$T^{2} + 4$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$T^{2} + 81$$
$29$ $$T^{2} - 4T + 29$$
$31$ $$T^{2} + 4$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} + 81$$
$43$ $$(T - 1)^{2}$$
$47$ $$(T - 8)^{2}$$
$53$ $$T^{2} + 81$$
$59$ $$(T + 8)^{2}$$
$61$ $$T^{2} + 36$$
$67$ $$T^{2} + 144$$
$71$ $$(T + 2)^{2}$$
$73$ $$(T + 15)^{2}$$
$79$ $$T^{2} + 16$$
$83$ $$T^{2} + 49$$
$89$ $$T^{2} + 4$$
$97$ $$(T + 11)^{2}$$