# Properties

 Label 1305.2.f.e Level $1305$ Weight $2$ Character orbit 1305.f Analytic conductor $10.420$ Analytic rank $0$ Dimension $4$ CM no 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 145) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + (\beta_1 - 1) q^{5} - \beta_{3} q^{7} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 + (b1 - 1) * q^5 - b3 * q^7 + 3 * q^8 $$q - q^{2} - q^{4} + (\beta_1 - 1) q^{5} - \beta_{3} q^{7} + 3 q^{8} + ( - \beta_1 + 1) q^{10} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{11} + (\beta_{2} + 2 \beta_1 - 1) q^{13} + \beta_{3} q^{14} - q^{16} + 2 q^{17} - \beta_{3} q^{19} + ( - \beta_1 + 1) q^{20} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{22} + \beta_{3} q^{23} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{25} + ( - \beta_{2} - 2 \beta_1 + 1) q^{26} + \beta_{3} q^{28} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{29} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{31} - 5 q^{32} - 2 q^{34} + (\beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{35} + ( - 2 \beta_{2} + 4) q^{37} + \beta_{3} q^{38} + (3 \beta_1 - 3) q^{40} + (2 \beta_{2} + 4 \beta_1 - 2) q^{41} + (\beta_{2} + 7) q^{43} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{44} - \beta_{3} q^{46} + ( - 3 \beta_{2} + 7) q^{47} - 5 q^{49} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{50} + ( - \beta_{2} - 2 \beta_1 + 1) q^{52} + (\beta_{2} + 2 \beta_1 - 1) q^{53} + (2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{55} - 3 \beta_{3} q^{56} + (\beta_{3} - 2 \beta_{2} + 1) q^{58} + ( - 2 \beta_{2} - 2) q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{61}+ \cdots + 5 q^{98}+O(q^{100})$$ q - q^2 - q^4 + (b1 - 1) * q^5 - b3 * q^7 + 3 * q^8 + (-b1 + 1) * q^10 + (-b3 + b2 + 2*b1 - 1) * q^11 + (b2 + 2*b1 - 1) * q^13 + b3 * q^14 - q^16 + 2 * q^17 - b3 * q^19 + (-b1 + 1) * q^20 + (b3 - b2 - 2*b1 + 1) * q^22 + b3 * q^23 + (b3 + b2 - b1 - 2) * q^25 + (-b2 - 2*b1 + 1) * q^26 + b3 * q^28 + (-b3 + 2*b2 - 1) * q^29 + (-b3 - b2 - 2*b1 + 1) * q^31 - 5 * q^32 - 2 * q^34 + (b3 - 4*b2 - 2*b1 + 4) * q^35 + (-2*b2 + 4) * q^37 + b3 * q^38 + (3*b1 - 3) * q^40 + (2*b2 + 4*b1 - 2) * q^41 + (b2 + 7) * q^43 + (b3 - b2 - 2*b1 + 1) * q^44 - b3 * q^46 + (-3*b2 + 7) * q^47 - 5 * q^49 + (-b3 - b2 + b1 + 2) * q^50 + (-b2 - 2*b1 + 1) * q^52 + (b2 + 2*b1 - 1) * q^53 + (2*b3 - 3*b2 - 3*b1 - 3) * q^55 - 3*b3 * q^56 + (b3 - 2*b2 + 1) * q^58 + (-2*b2 - 2) * q^59 + (-2*b3 + 2*b2 + 4*b1 - 2) * q^61 + (b3 + b2 + 2*b1 - 1) * q^62 + 7 * q^64 + (b3 + b2 - b1 - 7) * q^65 + (b3 + 2*b2 + 4*b1 - 2) * q^67 - 2 * q^68 + (-b3 + 4*b2 + 2*b1 - 4) * q^70 + (-4*b2 - 4) * q^71 + (2*b2 - 4) * q^73 + (2*b2 - 4) * q^74 + b3 * q^76 + (-6*b2 - 6) * q^77 + (3*b3 + b2 + 2*b1 - 1) * q^79 + (-b1 + 1) * q^80 + (-2*b2 - 4*b1 + 2) * q^82 + 3*b3 * q^83 + (2*b1 - 2) * q^85 + (-b2 - 7) * q^86 + (-3*b3 + 3*b2 + 6*b1 - 3) * q^88 + (2*b2 + 4*b1 - 2) * q^89 + (-6*b2 + 6) * q^91 - b3 * q^92 + (3*b2 - 7) * q^94 + (b3 - 4*b2 - 2*b1 + 4) * q^95 + (-4*b2 + 2) * q^97 + 5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 4 q^{4} - 3 q^{5} + 12 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 - 4 * q^4 - 3 * q^5 + 12 * q^8 $$4 q - 4 q^{2} - 4 q^{4} - 3 q^{5} + 12 q^{8} + 3 q^{10} - 4 q^{16} + 8 q^{17} + 3 q^{20} - 7 q^{25} - 20 q^{32} - 8 q^{34} + 6 q^{35} + 12 q^{37} - 9 q^{40} + 30 q^{43} + 22 q^{47} - 20 q^{49} + 7 q^{50} - 21 q^{55} - 12 q^{59} + 28 q^{64} - 27 q^{65} - 8 q^{68} - 6 q^{70} - 24 q^{71} - 12 q^{73} - 12 q^{74} - 36 q^{77} + 3 q^{80} - 6 q^{85} - 30 q^{86} + 12 q^{91} - 22 q^{94} + 6 q^{95} + 20 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 - 4 * q^4 - 3 * q^5 + 12 * q^8 + 3 * q^10 - 4 * q^16 + 8 * q^17 + 3 * q^20 - 7 * q^25 - 20 * q^32 - 8 * q^34 + 6 * q^35 + 12 * q^37 - 9 * q^40 + 30 * q^43 + 22 * q^47 - 20 * q^49 + 7 * q^50 - 21 * q^55 - 12 * q^59 + 28 * q^64 - 27 * q^65 - 8 * q^68 - 6 * q^70 - 24 * q^71 - 12 * q^73 - 12 * q^74 - 36 * q^77 + 3 * q^80 - 6 * q^85 - 30 * q^86 + 12 * q^91 - 22 * q^94 + 6 * q^95 + 20 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 25\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 25*v + 16) / 20 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} - 4 ) / 5$$ (-v^3 - 4) / 5 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 6 ) / 5$$ (-v^3 + 5*v^2 - 5*v + 6) / 5
 $$\nu$$ $$=$$ $$( \beta_{3} - 4\beta _1 + 2 ) / 4$$ (b3 - 4*b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( 5\beta_{3} - 4\beta_{2} - 4\beta _1 - 6 ) / 4$$ (5*b3 - 4*b2 - 4*b1 - 6) / 4 $$\nu^{3}$$ $$=$$ $$-5\beta_{2} - 4$$ -5*b2 - 4

## 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
 1.28078 + 2.21837i 1.28078 − 2.21837i −0.780776 + 1.35234i −0.780776 − 1.35234i
−1.00000 0 −1.00000 −1.78078 1.35234i 0 3.46410i 3.00000 0 1.78078 + 1.35234i
289.2 −1.00000 0 −1.00000 −1.78078 + 1.35234i 0 3.46410i 3.00000 0 1.78078 1.35234i
289.3 −1.00000 0 −1.00000 0.280776 2.21837i 0 3.46410i 3.00000 0 −0.280776 + 2.21837i
289.4 −1.00000 0 −1.00000 0.280776 + 2.21837i 0 3.46410i 3.00000 0 −0.280776 2.21837i
 $$n$$: 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.e 4
3.b odd 2 1 145.2.d.c yes 4
5.b even 2 1 1305.2.f.i 4
12.b even 2 1 2320.2.j.c 4
15.d odd 2 1 145.2.d.a 4
15.e even 4 2 725.2.c.f 8
29.b even 2 1 1305.2.f.i 4
60.h even 2 1 2320.2.j.a 4
87.d odd 2 1 145.2.d.a 4
145.d even 2 1 inner 1305.2.f.e 4
348.b even 2 1 2320.2.j.a 4
435.b odd 2 1 145.2.d.c yes 4
435.p even 4 2 725.2.c.f 8
1740.k even 2 1 2320.2.j.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.d.a 4 15.d odd 2 1
145.2.d.a 4 87.d odd 2 1
145.2.d.c yes 4 3.b odd 2 1
145.2.d.c yes 4 435.b odd 2 1
725.2.c.f 8 15.e even 4 2
725.2.c.f 8 435.p even 4 2
1305.2.f.e 4 1.a even 1 1 trivial
1305.2.f.e 4 145.d even 2 1 inner
1305.2.f.i 4 5.b even 2 1
1305.2.f.i 4 29.b even 2 1
2320.2.j.a 4 60.h even 2 1
2320.2.j.a 4 348.b even 2 1
2320.2.j.c 4 12.b even 2 1
2320.2.j.c 4 1740.k even 2 1

## Hecke kernels

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

 $$T_{2} + 1$$ T2 + 1 $$T_{7}^{2} + 12$$ T7^2 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 3 T^{3} + \cdots + 25$$
$7$ $$(T^{2} + 12)^{2}$$
$11$ $$T^{4} + 39T^{2} + 36$$
$13$ $$T^{4} + 27T^{2} + 144$$
$17$ $$(T - 2)^{4}$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$(T^{2} + 12)^{2}$$
$29$ $$T^{4} - 10T^{2} + 841$$
$31$ $$T^{4} + 63T^{2} + 36$$
$37$ $$(T^{2} - 6 T - 8)^{2}$$
$41$ $$T^{4} + 108T^{2} + 2304$$
$43$ $$(T^{2} - 15 T + 52)^{2}$$
$47$ $$(T^{2} - 11 T - 8)^{2}$$
$53$ $$T^{4} + 27T^{2} + 144$$
$59$ $$(T^{2} + 6 T - 8)^{2}$$
$61$ $$T^{4} + 156T^{2} + 576$$
$67$ $$T^{4} + 156T^{2} + 576$$
$71$ $$(T^{2} + 12 T - 32)^{2}$$
$73$ $$(T^{2} + 6 T - 8)^{2}$$
$79$ $$T^{4} + 279 T^{2} + 12996$$
$83$ $$(T^{2} + 108)^{2}$$
$89$ $$T^{4} + 108T^{2} + 2304$$
$97$ $$(T^{2} - 68)^{2}$$