# Properties

 Label 1305.2.a.l Level $1305$ Weight $2$ Character orbit 1305.a Self dual yes Analytic conductor $10.420$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta - 1) q^{4} + q^{5} + ( - 2 \beta - 1) q^{7} + ( - 2 \beta + 1) q^{8}+O(q^{10})$$ q + b * q^2 + (b - 1) * q^4 + q^5 + (-2*b - 1) * q^7 + (-2*b + 1) * q^8 $$q + \beta q^{2} + (\beta - 1) q^{4} + q^{5} + ( - 2 \beta - 1) q^{7} + ( - 2 \beta + 1) q^{8} + \beta q^{10} + (2 \beta - 3) q^{11} + q^{13} + ( - 3 \beta - 2) q^{14} - 3 \beta q^{16} + ( - 4 \beta - 1) q^{17} + (4 \beta - 4) q^{19} + (\beta - 1) q^{20} + ( - \beta + 2) q^{22} + ( - 4 \beta + 2) q^{23} + q^{25} + \beta q^{26} + ( - \beta - 1) q^{28} + q^{29} - 8 q^{31} + (\beta - 5) q^{32} + ( - 5 \beta - 4) q^{34} + ( - 2 \beta - 1) q^{35} + 4 q^{38} + ( - 2 \beta + 1) q^{40} + 6 q^{41} - 6 q^{43} + ( - 3 \beta + 5) q^{44} + ( - 2 \beta - 4) q^{46} + (2 \beta - 7) q^{47} + (8 \beta - 2) q^{49} + \beta q^{50} + (\beta - 1) q^{52} + ( - 4 \beta + 4) q^{53} + (2 \beta - 3) q^{55} + (4 \beta + 3) q^{56} + \beta q^{58} + 6 q^{59} + ( - 4 \beta - 2) q^{61} - 8 \beta q^{62} + (2 \beta + 1) q^{64} + q^{65} + (6 \beta - 11) q^{67} + ( - \beta - 3) q^{68} + ( - 3 \beta - 2) q^{70} + 4 \beta q^{71} - 6 q^{73} + ( - 4 \beta + 8) q^{76} - q^{77} - 6 q^{79} - 3 \beta q^{80} + 6 \beta q^{82} + (4 \beta - 4) q^{83} + ( - 4 \beta - 1) q^{85} - 6 \beta q^{86} + (4 \beta - 7) q^{88} + (8 \beta - 3) q^{89} + ( - 2 \beta - 1) q^{91} + (2 \beta - 6) q^{92} + ( - 5 \beta + 2) q^{94} + (4 \beta - 4) q^{95} + (12 \beta - 4) q^{97} + (6 \beta + 8) q^{98} +O(q^{100})$$ q + b * q^2 + (b - 1) * q^4 + q^5 + (-2*b - 1) * q^7 + (-2*b + 1) * q^8 + b * q^10 + (2*b - 3) * q^11 + q^13 + (-3*b - 2) * q^14 - 3*b * q^16 + (-4*b - 1) * q^17 + (4*b - 4) * q^19 + (b - 1) * q^20 + (-b + 2) * q^22 + (-4*b + 2) * q^23 + q^25 + b * q^26 + (-b - 1) * q^28 + q^29 - 8 * q^31 + (b - 5) * q^32 + (-5*b - 4) * q^34 + (-2*b - 1) * q^35 + 4 * q^38 + (-2*b + 1) * q^40 + 6 * q^41 - 6 * q^43 + (-3*b + 5) * q^44 + (-2*b - 4) * q^46 + (2*b - 7) * q^47 + (8*b - 2) * q^49 + b * q^50 + (b - 1) * q^52 + (-4*b + 4) * q^53 + (2*b - 3) * q^55 + (4*b + 3) * q^56 + b * q^58 + 6 * q^59 + (-4*b - 2) * q^61 - 8*b * q^62 + (2*b + 1) * q^64 + q^65 + (6*b - 11) * q^67 + (-b - 3) * q^68 + (-3*b - 2) * q^70 + 4*b * q^71 - 6 * q^73 + (-4*b + 8) * q^76 - q^77 - 6 * q^79 - 3*b * q^80 + 6*b * q^82 + (4*b - 4) * q^83 + (-4*b - 1) * q^85 - 6*b * q^86 + (4*b - 7) * q^88 + (8*b - 3) * q^89 + (-2*b - 1) * q^91 + (2*b - 6) * q^92 + (-5*b + 2) * q^94 + (4*b - 4) * q^95 + (12*b - 4) * q^97 + (6*b + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10})$$ 2 * q + q^2 - q^4 + 2 * q^5 - 4 * q^7 $$2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7} + q^{10} - 4 q^{11} + 2 q^{13} - 7 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{19} - q^{20} + 3 q^{22} + 2 q^{25} + q^{26} - 3 q^{28} + 2 q^{29} - 16 q^{31} - 9 q^{32} - 13 q^{34} - 4 q^{35} + 8 q^{38} + 12 q^{41} - 12 q^{43} + 7 q^{44} - 10 q^{46} - 12 q^{47} + 4 q^{49} + q^{50} - q^{52} + 4 q^{53} - 4 q^{55} + 10 q^{56} + q^{58} + 12 q^{59} - 8 q^{61} - 8 q^{62} + 4 q^{64} + 2 q^{65} - 16 q^{67} - 7 q^{68} - 7 q^{70} + 4 q^{71} - 12 q^{73} + 12 q^{76} - 2 q^{77} - 12 q^{79} - 3 q^{80} + 6 q^{82} - 4 q^{83} - 6 q^{85} - 6 q^{86} - 10 q^{88} + 2 q^{89} - 4 q^{91} - 10 q^{92} - q^{94} - 4 q^{95} + 4 q^{97} + 22 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 + 2 * q^5 - 4 * q^7 + q^10 - 4 * q^11 + 2 * q^13 - 7 * q^14 - 3 * q^16 - 6 * q^17 - 4 * q^19 - q^20 + 3 * q^22 + 2 * q^25 + q^26 - 3 * q^28 + 2 * q^29 - 16 * q^31 - 9 * q^32 - 13 * q^34 - 4 * q^35 + 8 * q^38 + 12 * q^41 - 12 * q^43 + 7 * q^44 - 10 * q^46 - 12 * q^47 + 4 * q^49 + q^50 - q^52 + 4 * q^53 - 4 * q^55 + 10 * q^56 + q^58 + 12 * q^59 - 8 * q^61 - 8 * q^62 + 4 * q^64 + 2 * q^65 - 16 * q^67 - 7 * q^68 - 7 * q^70 + 4 * q^71 - 12 * q^73 + 12 * q^76 - 2 * q^77 - 12 * q^79 - 3 * q^80 + 6 * q^82 - 4 * q^83 - 6 * q^85 - 6 * q^86 - 10 * q^88 + 2 * q^89 - 4 * q^91 - 10 * q^92 - q^94 - 4 * q^95 + 4 * q^97 + 22 * 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
 −0.618034 1.61803
−0.618034 0 −1.61803 1.00000 0 0.236068 2.23607 0 −0.618034
1.2 1.61803 0 0.618034 1.00000 0 −4.23607 −2.23607 0 1.61803
 $$n$$: 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.l yes 2
3.b odd 2 1 1305.2.a.h 2
5.b even 2 1 6525.2.a.r 2
15.d odd 2 1 6525.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.a.h 2 3.b odd 2 1
1305.2.a.l yes 2 1.a even 1 1 trivial
6525.2.a.r 2 5.b even 2 1
6525.2.a.bb 2 15.d odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 1$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 4T - 1$$
$11$ $$T^{2} + 4T - 1$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 6T - 11$$
$19$ $$T^{2} + 4T - 16$$
$23$ $$T^{2} - 20$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2}$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 12T + 31$$
$53$ $$T^{2} - 4T - 16$$
$59$ $$(T - 6)^{2}$$
$61$ $$T^{2} + 8T - 4$$
$67$ $$T^{2} + 16T + 19$$
$71$ $$T^{2} - 4T - 16$$
$73$ $$(T + 6)^{2}$$
$79$ $$(T + 6)^{2}$$
$83$ $$T^{2} + 4T - 16$$
$89$ $$T^{2} - 2T - 79$$
$97$ $$T^{2} - 4T - 176$$