# Properties

 Label 1305.2.a.o Level $1305$ Weight $2$ Character orbit 1305.a Self dual yes Analytic conductor $10.420$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + (3 \beta_1 - 4) q^{8}+O(q^{10})$$ q + (-b2 - 1) * q^2 + (b2 - b1 + 2) * q^4 + q^5 + (-b2 + b1 - 1) * q^7 + (3*b1 - 4) * q^8 $$q + ( - \beta_{2} - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + (3 \beta_1 - 4) q^{8} + ( - \beta_{2} - 1) q^{10} + (\beta_{2} + \beta_1 - 3) q^{11} + (2 \beta_{2} - 2) q^{13} + (\beta_{2} - 3 \beta_1 + 5) q^{14} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + (\beta_{2} - 3 \beta_1 + 1) q^{17} + ( - \beta_{2} - 3 \beta_1 + 1) q^{19} + (\beta_{2} - \beta_1 + 2) q^{20} + (3 \beta_{2} - \beta_1 + 1) q^{22} + (\beta_{2} + \beta_1 - 5) q^{23} + q^{25} + (2 \beta_{2} + 2 \beta_1 - 4) q^{26} + ( - 3 \beta_{2} + 5 \beta_1 - 9) q^{28} - q^{29} + (\beta_{2} + 3 \beta_1 + 3) q^{31} + ( - 3 \beta_{2} + 4 \beta_1 - 5) q^{32} + ( - \beta_{2} + 7 \beta_1 - 7) q^{34} + ( - \beta_{2} + \beta_1 - 1) q^{35} + ( - 3 \beta_{2} + \beta_1 + 1) q^{37} + ( - \beta_{2} + 5 \beta_1 - 1) q^{38} + (3 \beta_1 - 4) q^{40} + ( - 2 \beta_1 + 4) q^{41} + ( - \beta_{2} - \beta_1 - 3) q^{43} + ( - 3 \beta_{2} + 3 \beta_1 - 5) q^{44} + (5 \beta_{2} - \beta_1 + 3) q^{46} + (3 \beta_{2} + 3 \beta_1 - 7) q^{47} + (2 \beta_{2} - 4 \beta_1 + 1) q^{49} + ( - \beta_{2} - 1) q^{50} + ( - 2 \beta_1 + 4) q^{52} + (2 \beta_1 - 4) q^{53} + (\beta_{2} + \beta_1 - 3) q^{55} + (7 \beta_{2} - 7 \beta_1 + 13) q^{56} + (\beta_{2} + 1) q^{58} + ( - 2 \beta_{2} - 4 \beta_1) q^{59} + (2 \beta_{2} + 2) q^{61} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{62} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + (2 \beta_{2} - 2) q^{65} + ( - \beta_{2} - \beta_1 - 3) q^{67} + (5 \beta_{2} - 9 \beta_1 + 15) q^{68} + (\beta_{2} - 3 \beta_1 + 5) q^{70} + ( - 2 \beta_{2} - 8) q^{71} + ( - 7 \beta_{2} - \beta_1 - 1) q^{73} + ( - \beta_{2} - 5 \beta_1 + 9) q^{74} + (3 \beta_{2} - 5 \beta_1 + 7) q^{76} + (4 \beta_{2} - 2 \beta_1 + 2) q^{77} + (\beta_{2} + 5 \beta_1 + 1) q^{79} + (2 \beta_{2} - 4 \beta_1 + 3) q^{80} + ( - 4 \beta_{2} + 4 \beta_1 - 6) q^{82} + ( - 3 \beta_{2} - \beta_1 + 1) q^{83} + (\beta_{2} - 3 \beta_1 + 1) q^{85} + (3 \beta_{2} + \beta_1 + 5) q^{86} + ( - \beta_{2} - 7 \beta_1 + 15) q^{88} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{89} + (2 \beta_{2} + 2 \beta_1 - 6) q^{91} + ( - 5 \beta_{2} + 5 \beta_1 - 9) q^{92} + (7 \beta_{2} - 3 \beta_1 + 1) q^{94} + ( - \beta_{2} - 3 \beta_1 + 1) q^{95} + (3 \beta_{2} - 3 \beta_1 - 11) q^{97} + ( - \beta_{2} + 10 \beta_1 - 11) q^{98}+O(q^{100})$$ q + (-b2 - 1) * q^2 + (b2 - b1 + 2) * q^4 + q^5 + (-b2 + b1 - 1) * q^7 + (3*b1 - 4) * q^8 + (-b2 - 1) * q^10 + (b2 + b1 - 3) * q^11 + (2*b2 - 2) * q^13 + (b2 - 3*b1 + 5) * q^14 + (2*b2 - 4*b1 + 3) * q^16 + (b2 - 3*b1 + 1) * q^17 + (-b2 - 3*b1 + 1) * q^19 + (b2 - b1 + 2) * q^20 + (3*b2 - b1 + 1) * q^22 + (b2 + b1 - 5) * q^23 + q^25 + (2*b2 + 2*b1 - 4) * q^26 + (-3*b2 + 5*b1 - 9) * q^28 - q^29 + (b2 + 3*b1 + 3) * q^31 + (-3*b2 + 4*b1 - 5) * q^32 + (-b2 + 7*b1 - 7) * q^34 + (-b2 + b1 - 1) * q^35 + (-3*b2 + b1 + 1) * q^37 + (-b2 + 5*b1 - 1) * q^38 + (3*b1 - 4) * q^40 + (-2*b1 + 4) * q^41 + (-b2 - b1 - 3) * q^43 + (-3*b2 + 3*b1 - 5) * q^44 + (5*b2 - b1 + 3) * q^46 + (3*b2 + 3*b1 - 7) * q^47 + (2*b2 - 4*b1 + 1) * q^49 + (-b2 - 1) * q^50 + (-2*b1 + 4) * q^52 + (2*b1 - 4) * q^53 + (b2 + b1 - 3) * q^55 + (7*b2 - 7*b1 + 13) * q^56 + (b2 + 1) * q^58 + (-2*b2 - 4*b1) * q^59 + (2*b2 + 2) * q^61 + (-3*b2 - 5*b1 - 3) * q^62 + (b2 - 3*b1 + 12) * q^64 + (2*b2 - 2) * q^65 + (-b2 - b1 - 3) * q^67 + (5*b2 - 9*b1 + 15) * q^68 + (b2 - 3*b1 + 5) * q^70 + (-2*b2 - 8) * q^71 + (-7*b2 - b1 - 1) * q^73 + (-b2 - 5*b1 + 9) * q^74 + (3*b2 - 5*b1 + 7) * q^76 + (4*b2 - 2*b1 + 2) * q^77 + (b2 + 5*b1 + 1) * q^79 + (2*b2 - 4*b1 + 3) * q^80 + (-4*b2 + 4*b1 - 6) * q^82 + (-3*b2 - b1 + 1) * q^83 + (b2 - 3*b1 + 1) * q^85 + (3*b2 + b1 + 5) * q^86 + (-b2 - 7*b1 + 15) * q^88 + (-2*b2 + 2*b1 - 8) * q^89 + (2*b2 + 2*b1 - 6) * q^91 + (-5*b2 + 5*b1 - 9) * q^92 + (7*b2 - 3*b1 + 1) * q^94 + (-b2 - 3*b1 + 1) * q^95 + (3*b2 - 3*b1 - 11) * q^97 + (-b2 + 10*b1 - 11) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 5 q^{4} + 3 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 5 * q^4 + 3 * q^5 - 2 * q^7 - 9 * q^8 $$3 q - 3 q^{2} + 5 q^{4} + 3 q^{5} - 2 q^{7} - 9 q^{8} - 3 q^{10} - 8 q^{11} - 6 q^{13} + 12 q^{14} + 5 q^{16} + 5 q^{20} + 2 q^{22} - 14 q^{23} + 3 q^{25} - 10 q^{26} - 22 q^{28} - 3 q^{29} + 12 q^{31} - 11 q^{32} - 14 q^{34} - 2 q^{35} + 4 q^{37} + 2 q^{38} - 9 q^{40} + 10 q^{41} - 10 q^{43} - 12 q^{44} + 8 q^{46} - 18 q^{47} - q^{49} - 3 q^{50} + 10 q^{52} - 10 q^{53} - 8 q^{55} + 32 q^{56} + 3 q^{58} - 4 q^{59} + 6 q^{61} - 14 q^{62} + 33 q^{64} - 6 q^{65} - 10 q^{67} + 36 q^{68} + 12 q^{70} - 24 q^{71} - 4 q^{73} + 22 q^{74} + 16 q^{76} + 4 q^{77} + 8 q^{79} + 5 q^{80} - 14 q^{82} + 2 q^{83} + 16 q^{86} + 38 q^{88} - 22 q^{89} - 16 q^{91} - 22 q^{92} - 36 q^{97} - 23 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 5 * q^4 + 3 * q^5 - 2 * q^7 - 9 * q^8 - 3 * q^10 - 8 * q^11 - 6 * q^13 + 12 * q^14 + 5 * q^16 + 5 * q^20 + 2 * q^22 - 14 * q^23 + 3 * q^25 - 10 * q^26 - 22 * q^28 - 3 * q^29 + 12 * q^31 - 11 * q^32 - 14 * q^34 - 2 * q^35 + 4 * q^37 + 2 * q^38 - 9 * q^40 + 10 * q^41 - 10 * q^43 - 12 * q^44 + 8 * q^46 - 18 * q^47 - q^49 - 3 * q^50 + 10 * q^52 - 10 * q^53 - 8 * q^55 + 32 * q^56 + 3 * q^58 - 4 * q^59 + 6 * q^61 - 14 * q^62 + 33 * q^64 - 6 * q^65 - 10 * q^67 + 36 * q^68 + 12 * q^70 - 24 * q^71 - 4 * q^73 + 22 * q^74 + 16 * q^76 + 4 * q^77 + 8 * q^79 + 5 * q^80 - 14 * q^82 + 2 * q^83 + 16 * q^86 + 38 * q^88 - 22 * q^89 - 16 * q^91 - 22 * q^92 - 36 * q^97 - 23 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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
 −1.48119 2.17009 0.311108
−2.67513 0 5.15633 1.00000 0 −4.15633 −8.44358 0 −2.67513
1.2 −1.53919 0 0.369102 1.00000 0 0.630898 2.51026 0 −1.53919
1.3 1.21432 0 −0.525428 1.00000 0 1.52543 −3.06668 0 1.21432
 $$n$$: 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.o 3
3.b odd 2 1 145.2.a.d 3
5.b even 2 1 6525.2.a.bh 3
12.b even 2 1 2320.2.a.s 3
15.d odd 2 1 725.2.a.d 3
15.e even 4 2 725.2.b.d 6
21.c even 2 1 7105.2.a.p 3
24.f even 2 1 9280.2.a.bm 3
24.h odd 2 1 9280.2.a.bu 3
87.d odd 2 1 4205.2.a.e 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.d 3 3.b odd 2 1
725.2.a.d 3 15.d odd 2 1
725.2.b.d 6 15.e even 4 2
1305.2.a.o 3 1.a even 1 1 trivial
2320.2.a.s 3 12.b even 2 1
4205.2.a.e 3 87.d odd 2 1
6525.2.a.bh 3 5.b even 2 1
7105.2.a.p 3 21.c even 2 1
9280.2.a.bm 3 24.f even 2 1
9280.2.a.bu 3 24.h odd 2 1

## Hecke kernels

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

 $$T_{2}^{3} + 3T_{2}^{2} - T_{2} - 5$$ T2^3 + 3*T2^2 - T2 - 5 $$T_{7}^{3} + 2T_{7}^{2} - 8T_{7} + 4$$ T7^3 + 2*T7^2 - 8*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 3T^{2} - T - 5$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$11$ $$T^{3} + 8 T^{2} + \cdots + 4$$
$13$ $$T^{3} + 6 T^{2} + \cdots - 8$$
$17$ $$T^{3} - 40T - 76$$
$19$ $$T^{3} - 28T + 52$$
$23$ $$T^{3} + 14 T^{2} + \cdots + 76$$
$29$ $$(T + 1)^{3}$$
$31$ $$T^{3} - 12 T^{2} + \cdots - 4$$
$37$ $$T^{3} - 4 T^{2} + \cdots + 68$$
$41$ $$T^{3} - 10 T^{2} + \cdots + 8$$
$43$ $$T^{3} + 10 T^{2} + \cdots + 20$$
$47$ $$T^{3} + 18 T^{2} + \cdots - 92$$
$53$ $$T^{3} + 10 T^{2} + \cdots - 8$$
$59$ $$T^{3} + 4 T^{2} + \cdots + 80$$
$61$ $$T^{3} - 6 T^{2} + \cdots + 40$$
$67$ $$T^{3} + 10 T^{2} + \cdots + 20$$
$71$ $$T^{3} + 24 T^{2} + \cdots + 368$$
$73$ $$T^{3} + 4 T^{2} + \cdots - 1108$$
$79$ $$T^{3} - 8 T^{2} + \cdots + 20$$
$83$ $$T^{3} - 2 T^{2} + \cdots - 52$$
$89$ $$T^{3} + 22 T^{2} + \cdots + 200$$
$97$ $$T^{3} + 36 T^{2} + \cdots + 452$$