# Properties

 Label 1305.2.a.b Level $1305$ Weight $2$ Character orbit 1305.a Self dual yes Analytic conductor $10.420$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} - q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 - q^5 + 4 * q^7 + 3 * q^8 $$q - q^{2} - q^{4} - q^{5} + 4 q^{7} + 3 q^{8} + q^{10} + 4 q^{11} + 6 q^{13} - 4 q^{14} - q^{16} - 6 q^{17} - 4 q^{19} + q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 6 q^{26} - 4 q^{28} - q^{29} - 8 q^{31} - 5 q^{32} + 6 q^{34} - 4 q^{35} + 2 q^{37} + 4 q^{38} - 3 q^{40} + 6 q^{41} + 4 q^{43} - 4 q^{44} - 4 q^{46} + 9 q^{49} - q^{50} - 6 q^{52} + 10 q^{53} - 4 q^{55} + 12 q^{56} + q^{58} + 12 q^{59} - 10 q^{61} + 8 q^{62} + 7 q^{64} - 6 q^{65} + 8 q^{67} + 6 q^{68} + 4 q^{70} + 8 q^{71} - 2 q^{73} - 2 q^{74} + 4 q^{76} + 16 q^{77} + q^{80} - 6 q^{82} - 8 q^{83} + 6 q^{85} - 4 q^{86} + 12 q^{88} + 6 q^{89} + 24 q^{91} - 4 q^{92} + 4 q^{95} - 2 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 - q^4 - q^5 + 4 * q^7 + 3 * q^8 + q^10 + 4 * q^11 + 6 * q^13 - 4 * q^14 - q^16 - 6 * q^17 - 4 * q^19 + q^20 - 4 * q^22 + 4 * q^23 + q^25 - 6 * q^26 - 4 * q^28 - q^29 - 8 * q^31 - 5 * q^32 + 6 * q^34 - 4 * q^35 + 2 * q^37 + 4 * q^38 - 3 * q^40 + 6 * q^41 + 4 * q^43 - 4 * q^44 - 4 * q^46 + 9 * q^49 - q^50 - 6 * q^52 + 10 * q^53 - 4 * q^55 + 12 * q^56 + q^58 + 12 * q^59 - 10 * q^61 + 8 * q^62 + 7 * q^64 - 6 * q^65 + 8 * q^67 + 6 * q^68 + 4 * q^70 + 8 * q^71 - 2 * q^73 - 2 * q^74 + 4 * q^76 + 16 * q^77 + q^80 - 6 * q^82 - 8 * q^83 + 6 * q^85 - 4 * q^86 + 12 * q^88 + 6 * q^89 + 24 * q^91 - 4 * q^92 + 4 * q^95 - 2 * q^97 - 9 * q^98

## 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
−1.00000 0 −1.00000 −1.00000 0 4.00000 3.00000 0 1.00000
 $$n$$: 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.b 1
3.b odd 2 1 435.2.a.d 1
5.b even 2 1 6525.2.a.j 1
12.b even 2 1 6960.2.a.l 1
15.d odd 2 1 2175.2.a.b 1
15.e even 4 2 2175.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.d 1 3.b odd 2 1
1305.2.a.b 1 1.a even 1 1 trivial
2175.2.a.b 1 15.d odd 2 1
2175.2.c.b 2 15.e even 4 2
6525.2.a.j 1 5.b even 2 1
6960.2.a.l 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1305))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 4$$
$11$ $$T - 4$$
$13$ $$T - 6$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T - 4$$
$29$ $$T + 1$$
$31$ $$T + 8$$
$37$ $$T - 2$$
$41$ $$T - 6$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T - 10$$
$59$ $$T - 12$$
$61$ $$T + 10$$
$67$ $$T - 8$$
$71$ $$T - 8$$
$73$ $$T + 2$$
$79$ $$T$$
$83$ $$T + 8$$
$89$ $$T - 6$$
$97$ $$T + 2$$