# Properties

 Label 1305.2.a.c Level $1305$ Weight $2$ Character orbit 1305.a Self dual yes Analytic conductor $10.420$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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, 0]))

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

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

chi := DirichletCharacter("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.a (trivial)

## 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: yes Analytic conductor: $$10.4204774638$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{4} + q^{5} - 2 q^{7}+O(q^{10})$$ q - 2 * q^4 + q^5 - 2 * q^7 $$q - 2 q^{4} + q^{5} - 2 q^{7} - q^{11} + 6 q^{13} + 4 q^{16} - 4 q^{17} - 2 q^{19} - 2 q^{20} - 3 q^{23} + q^{25} + 4 q^{28} - q^{29} - 4 q^{31} - 2 q^{35} - 3 q^{37} - 7 q^{41} + 5 q^{43} + 2 q^{44} - 6 q^{47} - 3 q^{49} - 12 q^{52} - 13 q^{53} - q^{55} - 8 q^{64} + 6 q^{65} - 10 q^{67} + 8 q^{68} - 6 q^{71} + 3 q^{73} + 4 q^{76} + 2 q^{77} + 4 q^{80} - 9 q^{83} - 4 q^{85} + 10 q^{89} - 12 q^{91} + 6 q^{92} - 2 q^{95} + 17 q^{97}+O(q^{100})$$ q - 2 * q^4 + q^5 - 2 * q^7 - q^11 + 6 * q^13 + 4 * q^16 - 4 * q^17 - 2 * q^19 - 2 * q^20 - 3 * q^23 + q^25 + 4 * q^28 - q^29 - 4 * q^31 - 2 * q^35 - 3 * q^37 - 7 * q^41 + 5 * q^43 + 2 * q^44 - 6 * q^47 - 3 * q^49 - 12 * q^52 - 13 * q^53 - q^55 - 8 * q^64 + 6 * q^65 - 10 * q^67 + 8 * q^68 - 6 * q^71 + 3 * q^73 + 4 * q^76 + 2 * q^77 + 4 * q^80 - 9 * q^83 - 4 * q^85 + 10 * q^89 - 12 * q^91 + 6 * q^92 - 2 * q^95 + 17 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 −2.00000 1.00000 0 −2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$29$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.2.a.c 1
3.b odd 2 1 435.2.a.b 1
5.b even 2 1 6525.2.a.i 1
12.b even 2 1 6960.2.a.bc 1
15.d odd 2 1 2175.2.a.g 1
15.e even 4 2 2175.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.b 1 3.b odd 2 1
1305.2.a.c 1 1.a even 1 1 trivial
2175.2.a.g 1 15.d odd 2 1
2175.2.c.d 2 15.e even 4 2
6525.2.a.i 1 5.b even 2 1
6960.2.a.bc 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1305))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T + 1$$
$13$ $$T - 6$$
$17$ $$T + 4$$
$19$ $$T + 2$$
$23$ $$T + 3$$
$29$ $$T + 1$$
$31$ $$T + 4$$
$37$ $$T + 3$$
$41$ $$T + 7$$
$43$ $$T - 5$$
$47$ $$T + 6$$
$53$ $$T + 13$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T + 10$$
$71$ $$T + 6$$
$73$ $$T - 3$$
$79$ $$T$$
$83$ $$T + 9$$
$89$ $$T - 10$$
$97$ $$T - 17$$